Dislocations in Solids, Volume 15

Dislocations in Solids Volume 15 Edited by J. P. HIRTH Hereford, AZ, USA and L. KUBIN ´ tude des Microstructures, La...

Author: John P. Hirth | Ladislas Kubin

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Dislocations in Solids Volume 15

Edited by

J. P. HIRTH Hereford, AZ, USA and

L. KUBIN ´ tude des Microstructures, Laboratoire d’E CNRS-ONERA, Chatillon Cedex, France

Amsterdam Boston Heidelberg London New York Oxford Paris San Diego San Francisco Singapore Sydney 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 Linacre House, Jordan Hill, Oxford OX2 8DP, UK First edition 2009 Copyright r 2009 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 (þ44) (0) 1865 843830; fax (þ44) (0) 1865 853333; email: [email protected]. Alternatively you can submit your request online by visiting the Elsevier web site at http://www.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: 978-0-444-53285-5 ISSN: 1572-4859 For information on all North-Holland publications visit our website at elsevierdirect.com

Printed and bound in Great Britain 09 10 11 12 13

10 9 8 7 6 5 4 3 2 1

Preface This volume continues the tradition of presenting new position papers in key areas of dislocation behavior. Bacon et al. present an atomistic model of dislocation– particle interactions in metal systems, including irradiated materials. This work is important in simulating actual behavior, removing earlier reliance on assumed mechanisms for dislocation motion. Meyers et al. present the new mechanisms for dislocation generation under shock loading. These models provide a basis for understanding the constitutive behavior of shocked material. Saada and Dirras provide a new perspective on the Hall–Petch relation, with particular emphasis on nanocrystals. Of particular significance, deviations from the traditional stress proportional to the square root of grain size relation are explained. Robertson et al. consider a number of effects of hydrogen on plastic flow and provide a model that further provides an explanation of the broad range of properties. Thus, as with other recent volumes in the series, this volume includes chapters on a wide range of topics, all at the leading edge of new research in the dislocation area. J.P. Hirth L. Kubin

v

CHAPTER 88

Dislocation–Obstacle Interactions at the Atomic Level D.J. BACON Department of Engineering, The University of Liverpool, Brownlow Hill, Liverpool L69 3GH, UK

Y.N. OSETSKY Materials Sciences and Technology, ORNL, Oak Ridge, TN 37831, USA and

D. RODNEY Science et Inge´nierie des Mate´riaux et Proce´de´s, INP Grenoble, CNRS/UJF, Domaine Universitaire, Boıˆte Postale 46, 38402 Saint Martin d’He`res, France

r 2009 Elsevier B.V. All rights reserved 1572-4859, DOI: 10.1016/S1572-4859(09)01501-0

Dislocations in Solids Edited by J. P. Hirth and L. Kubin

Contents 1. Introduction 4 2. Structure of models used to simulate dislocations at the atomic level 8 2.1. Rigid boundary model 8 2.2. Flexible boundary model 9 2.3. Periodic array model 10 2.3.1. Formal description of creating periodicity 10 2.3.2. Model for an edge dislocation 12 2.3.3. Model for a screw dislocation 13 2.4. Boundary conditions in the z-direction and loading techniques 15 2.5. Restrictions on model parameters 17 2.6. Other practical issues 19 3. Dislocation glide in pure metals and solid solutions 21 3.1. Glide in pure crystals 21 3.1.1. Glide at 0 K: the Peierls stress 21 3.1.2. Glide at ﬁnite temperature 23 3.1.3. Comparison with experiment 26 3.2. Glide in solid solutions 26 3.2.1. Background 26 3.2.2. Substitutional solute atoms 27 3.2.3. Interstitial solute atoms 32 3.2.4. Extension to microscopic models 35 4. Voids and precipitates 35 4.1. Introduction 35 4.2. Edge dislocation–obstacle interaction at T ¼ 0 K 37 4.2.1. Voids 37 4.2.2. Precipitates 43 4.2.3. Comparison of atomistic and continuum results at 0 K 45 4.3. Temperature effects for voids and precipitates 48 4.4. Bubbles and loose clusters of vacancies 55 4.5. Conclusions 56 5. Obstacles having dislocation character 57 5.1. Dislocation loops and SFTs 57 5.2. Classiﬁcation of main reactions 59 5.2.1. Reaction R1: the obstacle is crossed by the dislocation and both are unchanged 59 5.2.2. Reaction R2: the obstacle is crossed and modiﬁed and the dislocation is unchanged 59 5.2.3. Reaction R3: partial or full absorption of the obstacle by an edge dislocation that acquires a double superjog 59 5.2.4. Reaction R4: temporary absorption of part or the entire obstacle into a helical turn on a screw dislocation 60 5.2.5. Other reactions 60 5.3. Loops in FCC metals 61 5.3.1. Perfect interstitial loops 61 5.3.2. Interstitial Frank loops 62

5.4. Interstitial loops in BCC metals 66 5.4.1. ½/1 1 1S loops 66 5.4.2. /1 0 0S loops 70 5.4.3. Comparison of obstacle strength for voids and loops in iron 75 5.5. Stacking fault tetrahedra 76 5.5.1. Particular reactions 76 5.5.2. Other cases 80 5.6. Conclusions 82 6. Concluding remarks 83 Acknowledgements 85 References 85

1. Introduction The techniques of atomic-scale simulation by computer offer ways of investi gating properties of crystal defects that are not usually open to direct study by experiment. Some have been in use for over 40 years and have provided important information on the atomic structure and energy of crystalline defects. This chapter is concerned with dislocation–obstacle interactions that resist the glide of dislocations in metals and hence increase the applied stress necessary to cause plastic deformation. Computer simulation of the atomic mechanisms involved in this area of materials science is fairly recent and the aim of this chapter is to highlight the understanding that has been achieved over the past decade. A recent handbook [1] provides a comprehensive introduction to many aspects of modelling at both the atomic and continuum scales. It includes descriptions of methods and examples of codes for a variety of techniques. We have tried to avoid unnecessary overlap except inasmuch that a clear description of the methods that have led to the results we present is required in order that their strength and weakness can be appreciated. We assume a prior knowledge of basic properties of dislocations, such as the Burgers vector, b; their edge, screw or mixed character; their glide (or slip) plane; the process of cross-slip; consequences of dissociation into partials; and the form of b of perfect dislocations in body-centred cubic (BCC) and face-centred cubic (FCC) metals, i.e. ½/1 1 1S and ½/1 1 0S, respectively. (See Ref. [2] for an introduction and Refs [3,4] for more advanced and detailed presentations.) We do not go into detail about the atomic structure of dislocation cores: recent reviews of progress in modelling core structure in different metals and the inﬂuence of core structure on dislocation motion are to be found in Refs [5,6]. Nor do we study effects of grain or interphase boundaries on dislocation properties and behaviour, for the materials considered here are single crystals: simulation of boundary effects in nanocrystals and other materials are discussed in Ref. [7]. In the multiscale framework for simulating the properties of dislocations in metals, there are three distinct spatial scales, each involving distinct methods. The ﬁnest scale uses ab initio (ﬁrst principles) calculations in which Schro¨dinger’s equation for interacting electrons is used to compute the position of atoms [8]. It is restricted to a few hundred atoms at most and is therefore limited to the core region around a dislocation. The coarsest scale is the continuum, in which dislocations are treated as though in an elastic, rather than atomic, medium. The distortion ﬁeld produced by a long dislocation varies inversely with distance and is thus longranged, and many of the important properties arising from this, such as stress, strain and strain energy, can be modelled using linear elasticity. Indeed, most of the applications of dislocation theory over the past 70 years have been based on this [2–4].

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Modern approaches to continuum-scale computer simulation of dislocations are not dealt with here. (For a broad introduction see Ref. [1].) Atomic-scale simulation treated in this chapter sits between the ab initio and continuum scales. It uses model sizes that are large enough to allow for the main effects of the distortion ﬁeld, but also provides full resolution of the atomic structure of the dislocation core and the obstacles a dislocation may encounter as it moves under external loading. Although the strain energy of a dislocation is affected by the ﬁnite size of the atomic model, its core properties (structure, energy), and therefore its short-range interactions with other defects, are less sensitive and can be described with acceptable accuracy. This can be tested by simulating systems of increasing size in order to reach convergence of results with the desired accuracy. The size is usually limited for the practical reason that the computing (CPU) time required is proportional to the number of atoms. The need to achieve results in reasonable time limits atomic-scale modelling in two other ways. First, the CPU time per atom is determined by the time needed to compute the forces between, and energy of, atoms using an interatomic potential. Thus, to minimise the CPU time, the potential should have a range as short as possible. The potentials currently used in the ﬁeld are empirical potentials obtained by different realisations of the Embedded Atom Model (EAM) [9]. Their empirical parameters are based typically on ﬁts to properties of the metal such as elastic constants, phonon spectra, cohesive energy, stacking fault energy, and surface and point defect energies, and, increasingly frequently, to ab initio data. For reviews see Ref. [8].1 Second, the total CPU time is proportional to the number of ‘iterations’ required to complete the simulation. ‘Iterations’ here has one of two meanings, depending on the simulation method. In molecular statics (MS), a crystal at temperature T ¼ 0 K is modelled, i.e. the kinetic energy of the atoms is maintained equal to zero, and the system achieves equilibrium when the potential energy, computed by summing the interatomic potential energy of the atoms minus the work of forces applied to the system, is minimised. This state is found from a trial starting conﬁguration by a series of iterations in which the atoms are moved repeatedly. (See Ref. [1] for examples.) In molecular dynamics (MD), kinetic energy is not zero and at equilibrium the average kinetic energy per atom equals 3kBT/2, where kB is the Boltzmann constant. At a given time t, the acceleration of every atom is calculated from the force on it due to its neighbours using Newton’s second law (force ¼ mass � acceleration). This equation of motion is solved numerically for all atoms to predict their position at time (t þ Dt), where Dt is the MD time-step, e.g. Refs [1,10]. This procedure is repeated to enable the trajectory of atoms to be followed for as long as is necessary to complete the process under investigation, the number of iterations in this case being the number of time-steps. To maintain accuracy, Dt is typically of the order of one to a few femtoseconds (1 fs ¼ 10�15 s).

1

For the references to the potentials used for the results presented in this chapter, the reader is referred to the original papers we cite.

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We give a few examples of model size and computing resource required in Section 2. For the moment, it is sufﬁcient to bear in mind that the models of interest here contain typically from a few hundred thousand to a few million atoms. Size in this range is usually sufﬁcient for treatment of the elastic ﬁeld of one dislocation and for it to move and interact with other defects without severe restriction by the model boundary conditions. In simulation of atomic dynamics by MD, the number of time-steps that can be accomplished within a reasonable CPU time is typically in the range 105–107, so that the total simulated time is of the order of nanoseconds (1 ns ¼ 10�9 s). Thus, the spatial and time scales of the work reviewed here are nanoscale. A model for simulating dislocation–obstacle interactions on these scales should satisfy the following criteria. (a) It should not artiﬁcially constrain the dislocation core structure and it should ensure that the displacement, u, of the atoms from their perfect crystal sites exhibits the discontinuity that deﬁnes the dislocation with Burgers vector b, i.e. I b ¼ du; (1) where the integral encircles the dislocation line. (b) It should be large enough to permit accurate simulation of the effects of the elastic distortion ﬁeld of the dislocation. (c) It should allow for dislocation motion to occur as a result of application of external action in the form of stress or strain: this motion should not be restricted by the model boundaries. (d) It should permit simulation of either static (T ¼ 0 K) or dynamic (TW0 K) conditions. (e) Methods should be incorporated for visualising the atoms in the vicinity of the dislocation core and obstacle during the process under investigation. Computer models that satisfy these criteria are described in the Section 2, with emphasis on a method that allows the construction of an inﬁnite, periodic glide plane for the mobile dislocation. This model is applied in the following sections for simulation of the interaction of dislocations with crystalline defects. Practical limits on model size restrict the size of these defects to a few nanometres. Much of the research presented here has been driven by the need to investigate the effect of radiation on the mechanical behaviour of metals in current and future nuclear power systems, for the defects of concern are of the order of a nanometre in size and their interactions with dislocations cannot be observed directly by experiment. The core components of reactors are subjected to irradiation by a ﬂux of fast neutrons produced by the nuclear reaction (e.g. Ref. [11]). The neutrons induce damage in metals that can change their mechanical properties. Indeed, fast neutrons (as well as ions) produce localised regions of defects by the displacementcascade process, in which an atom is given sufﬁcient energy by an irradiating particle that it can displace many of its neighbours in an avalanche of collisions.

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Displaced atoms that do not return to their lattice sites become self-interstitial atoms (SIAs), mainly at the periphery of a cascade, and a corresponding number of sites are left vacant in the central region (see Ref. [12] and references cited therein). A substantial fraction of these defects form clusters with their own kind, either during the cascade process itself, which has a lifetime B10 ps, or after diffusion in the material. SIAs usually cluster as tightly packed planar arrays of crowdions that are nascent dislocation loops with perfect Burgers vector parallel to the crowdion axis, i.e. b ¼ ½/1 1 0S in FCC, ½/1 1 1S or /1 0 0S in BCC and 1/3/1 1 2� 0S in HCP (e.g. Refs [13–15]). Faulted loops with b ¼ 1/3/1 1 1S also form in FCC metals. Vacancies cluster in the form of loops or, in FCC metals, in a speciﬁc dissociated structure called a stacking fault tetrahedron (SFT). Depending on the metal and irradiation conditions, they can also agglomerate to form voids and when He is present, as a result of either transmutation or direct injection, He-ﬁlled bubbles can arise. Plasticity is strongly affected by such clustered defects. The yield stress is usually increased, the work-hardening rate and ductility reduced and ﬂow localisation by dislocation channelling can occur at high levels of cluster density (e.g. Refs [16–18]). In the latter case, deformation is inhomogeneous and localised in bands of intense plastic shear that appear to be clear of irradiation defects when observed in a transmission electron microscope (TEM) after the deformation, e.g. Ref. [16]. Linear elasticity theory can provide a description of dislocation interaction with obstacles. However, approximations have to be made for processes that are controlled by atomic mechanisms, in particular when the interaction involves direct contact between the dislocation and obstacle, for the core of the dislocation is involved in the interaction process. On the other hand, the defects produced under irradiation have sizes in the nanometre range and since their density is high, B1023 m�3 (e.g. Refs [19–21]), their average separation is on the order of a few tens of nanometres. They are amenable to atomic-scale simulations. The aim of the research presented below has been to study both the elementary mechanisms of interaction between one dislocation of deﬁnite character (edge or screw) with one particular defect and measure directly from the simulation the pinning effect due to the defect on the dislocation. In a multiscale approach, such information can be then used in dislocation dynamics (DD) simulations to simulate the more statistical problem of the glide of dislocations in populations of defects [22]. After a description of the methods in Section 2, we consider in Section 3 the interaction of dislocations with the crystal itself (exempliﬁed by the Peierls stress and DD) and solute atoms (interaction that controls solid solution hardening (SSH)). We then review interaction with voids and precipitates in Section 4. Obstacles with dislocation character are considered in Section 5, i.e. dislocation loops and SFTs. Because of the possibility of dislocation reactions, more complex and varied reactions are observed with this category of obstacle. The cases that have been studied extensively are those for FCC and BCC metals: HCP metals have received far less attention, mainly because of the absence of reliable interatomic potentials suited for the study of dislocations.

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2. Structure of models used to simulate dislocations at the atomic level 2.1. Rigid boundary model The simplest approach, which was used in early atomic-level modelling of dislocations [23–29], is to generate the atom coordinates of a perfect crystal of the required structure, orientation and size and impose on them the dislocation displacement ﬁeld given by linear elasticity theory. In order to prevent a return to the perfect crystal state when relaxation occurs, boundary conditions have to be applied to the model. This is achieved by creating a layer of atoms (FR) ﬁxed in their unrelaxed position around the outside of the inner region of mobile atoms (MR), i.e. rigid boundary conditions (RBCs) are used. This arrangement is shown schematically in Fig. 1(a) as a cross-section of a cylinder containing an edge dislocation. Invariance of the dislocation ﬁeld along its line allows periodicity along the y-axis and this is readily achieved by adding the translation 7Ly, i.e. the length of the model in the y direction, to the y coordinate of atoms in region MR. The thickness of the rigid and periodic layers has to be larger than the range of the interatomic potential in order that atoms in the inner region have a full set of neighbours. The RBC method with this conﬁguration has been used to study dislocation properties such as core structure and energy, and led to valuable insights in early work on the core properties of screw dislocations in BCC metals [29,30]. However, region MR must be sufﬁciently large for relaxation of the atoms in the vicinity of the dislocation core to be unrestricted by the rigid boundaries. This condition is particularly important when the dislocation can dissociate, i.e. the core is wide. For instance, the dissociation width of a screw dislocation in a model of Ni saturates

Fig. 1. (a) Representation of rigid boundary model showing regions of ﬁxed (FR) and mobile (MR) atoms. (b) Representation of ﬂexible boundary model showing the continuum (CR), Green’s function (GFR) and mobile (MR) atom regions.

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to the true value (B2 nm) only when the distance from the centre of the dislocation core to the ﬁxed boundary is W6.3 nm (B25b) [31]. The initiation of dislocation motion in a RBC model can be studied by applying increasing homogeneous shear strain, e, in small increments and relaxing the position of the mobile atoms at each increment. The critical value, eP, at which the dislocation moves from its initial position gives the Peierls stress tP ¼ GeP, where G is the elastic shear modulus. However, the rigid boundaries oppose this motion because their atom coordinates correspond to the initial position of the dislocation. This produces a conﬁguration force on the dislocation that increases as it moves nearer the boundary. This leads to an overestimation of the Peierls stress [32], which can be up to almost an order of magnitude in low Peierls stress crystals (see, e.g. Ref. [31]). Also, stress cannot be applied with this model.

2.2. Flexible boundary model Green’s function boundary conditions (GFBCs) offer a more sophisticated technique, for they allow ﬂexible boundaries to be simulated according to the elastic and/or lattice properties of the crystal, thereby enabling the boundaries to distort in response to the dislocation. (The displacement at point x due to an inﬁnitesimal force F at point xu is u ¼ G(x � xu)F, where G is the Green’s function.) Two-dimensional (2D) [33] and three-dimensional (3D) [31,34,35] realisations of GFBCs have been suggested and applied in studies of cracks and dislocations. The simulation cell consists of three regions shown schematically in Fig. 1(b). The linear elastic displacement ﬁeld of the dislocation is initially applied to the whole model, after which atoms in region MR are relaxed with their forces derived from an interatomic potential, whilst their neighbours in the Green’s function region (GFR) and continuum region (CR) are held ﬁxed. This results in non-zero forces on atoms in the GFR. These forces are then relaxed by displacing atoms in both the GFR (displacements calculated from the lattice Green’s function) and the outer CR (displacements calculated via the elastic Green’s function). The process is repeated until forces in the GFR fall below a chosen value. In practice, up to 10 iterations may be necessary to achieve reasonable accuracy for dislocation core structure and energy. For details see Refs [31,35]. The GFBC technique has several advantages over simpler RBC methods. It allows a signiﬁcant reduction in the minimum size of the inner region required to reproduce the correct core structure and Peierls stress [31]. Also, for the same number of atoms in the inner region, the dislocation can move further without strong interference from the boundaries, although a dislocation–boundary distance of typically B15–20a0 is required for reasonable results. The method is particularly attractive for simulations where the number of atoms has to be kept small because calculation of interatomic forces is computationally time-consuming, e.g. ab initio or many-body interactions [36–38]. However, self-consistent convergence with GFBCs requires many force calls and calculations of long-range Green’s functions, and results in lower computational efﬁciency than the RBC method for problems

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involving a large number of atoms. Examples of two simulations are given in Section 2.6.

2.3. Periodic array model The GFBC model is not suited to simulation of dynamic conditions (temperature TW0 K). Both the RBC and GFBC methods suffer from additional limitations. First, as already mentioned, the boundaries are not transparent and the dislocation cannot travel over long distances. Second, they are compatible with application of only external strain and not stress: when the dislocation moves under applied strain, the stress, which arises from the elastic part of the strain only, decreases and constant applied stress simulations cannot be performed. A way to circumvent these problems is to use a periodic array of dislocations (PAD), as proposed initially by Daw et al. [39]. The simulated crystal containing an initially straight dislocation has PBCs in the dislocation glide plane, i.e. not only in the dislocation line direction (y) but also in the glide direction (x), thereby creating an inﬁnite array of inﬁnitely long, parallel dislocations. The dislocation still experiences model-size effects due to interaction with its periodic images but the effect is constant throughout the simulation cell, irrespective of the dislocation position relative to the cell boundaries. Effects of external loading can be studied by applying either stress or strain, and the dislocation can glide over a long (in principle, inﬁnite) distance because of transparency associated with PBCs. Models based on a PAD are computationally efﬁcient for a large number of atoms and, as will be shown later, can be used for not only qualitative but also quantitative studies of DD and mechanisms, and for parameterisation of the processes linking the atomic and continuum approaches. To illustrate the method, we consider ﬁrst in Section 2.3.1 a general approach for creating periodicity in the glide plane of a dislocation and then present two particular examples of edge (Section 2.3.2) and screw dislocations (Section 2.3.3). Treatment of the BCs in the direction normal to the glide plane and ways in which external loading is applied are discussed in Section 2.4. 2.3.1. Formal description of creating periodicity The following approach provides a formal prescription for generating PBCs along both the dislocation line (y-axis) and its direction of motion (x-axis) for a dislocation of any character. It follows from the fact that the Burgers vector b of a perfect dislocation is a translation vector of the lattice, usually the shortest lattice vector. It can therefore be described as the difference between two other lattice vectors. The prescription is as follows. (i) Construct two half crystals, which we label l (upper) and m (lower), with the same orientation. (ii) Select lattice translations vectors tl and tm in l and m such that b ¼ (tl � tm).

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(iii) Strain l and m by el and em, respectively, to bring tl and tm into coincidence so that after straining tl ¼ tm ¼ t*, where t* is to be the translation vector for periodicity, Lx, along x. (iv) Glue the half crystals together and relax the atoms. The model now contains a dislocation along y with b ¼ (tl � tm). The procedure is shown schematically in Fig. 2 for positive edge and right-handed screw dislocations. The left-hand sketch in each case shows the two half-crystals (in cross-section in (a) and plan view in (b)) and the translation vectors tl and tm whose difference gives b. The right-hand ﬁgures show the models after relaxation when viewed in the þy direction (�y out of the paper): the displacements (with respect to þy) associated with the shear strains el and em used in (b) are indicated at the outer corners of the screw ﬁgures. All models have periodicity with translation vector t*. The RH/FS convention (e.g. Ref. [2]) is used to deﬁne the sense of b, i.e. with positive line sense out of the paper, b is as indicated by the arrow in (a) and the arrow head in (b). The strains used to bring tl and tm into coincidence account for the stress-free distortion introduced in the simulation cell by the dislocation. They are minimised by making Lx as long as possible.

Fig. 2. Procedure for creating (a) positive edge and (b) right-handed screw dislocations by straining, joining and relaxing half crystals l and m. The dislocation line lies along the y-axis and slips in the x direction in all cases. As explained in Section 2.4, shaded regions at the 7z faces here and in Fig. 3 are used to apply stress or strain to the dislocated model.

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A similar method can be used to create PAD models for interfacial dislocations such as twinning dislocations. In this case, l and m have different orientation and the dislocation forms a step in the twin boundary ([40], see p. 173 of Ref. [2]). Nevertheless, by appropriate choice of tl and tm such that (tl � tm) equals b of the twinning dislocation, periodicity can be exploited to study the long-range motion of the boundary (e.g. Ref. [41]). Practicalities for the edge and screw dislocations in a single crystal are considered in more detail in the following two sections. 2.3.2. Model for an edge dislocation The initial stress-free structure consists of l and m with N and (N � 1) y–z lattice planes with spacing b, respectively, as in Fig. 3(a). They are strained by �1/2N and 1/2(N � 1) to have the same length, Lx, in the x direction before being joined along the slip plane x–y, as in Fig. 3(b). During relaxation to form the edge dislocation, l and m are constrained so that they retain the same size Lx along x. The dislocation thus formed is not a ‘misﬁt dislocation’, which arises between half-crystals with different natural (stress-free) lattice parameters and results in an interface with zero dislocation content (see, e.g. p. 181 of Ref. [2] and the appendix of Ref. [42]). The strains used to remove the size difference b between l and m can be distributed in different ways, but the most obvious, which results in the lowest energy, has the misﬁt distributed approximately equally between them. It amounts to changing the cell length in direction x by þb/2 and �b/2 in the upper and lower halves of the simulation cell respectively. Imposition of periodicity in the rectangular cell shown in Fig. 3(b) results in application of a bending moment to the dislocated crystal because, as recalled in Fig. 4(a), an edge dislocation bends an unconstrained crystal. Forcing the crystal into the rectangular cell of Fig. 4(b), which is equivalent to Fig. 3(b), causes

Fig. 3. Schematic illustration of construction of an edge dislocation.

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Fig. 4. (a) Unconstrained and (b) constrained crystal containing an edge dislocation.

additional stress in the model. However, the bending moment decreases as Lx/Lz increases. Furthermore, if the dislocation slip plane coincides with the neutral axis, i.e. is in the middle plane of the crystal, the unbending stress is zero. The internal stress distribution in the vicinity of the dislocation core is then close to that of a dislocation in an unconstrained medium [42]. The contribution of the additional stress is maximum for atoms near the top and bottom of the model, well away from the dislocation core (see, e.g. Fig. 7 of Ref. [42]). Another consequence of periodicity in the x direction is that the dislocation in the computational cell experiences a shear stress sxz (and hence a glide force) due to its image in all the periodic cells. The largest value (due to one of its nearest-neighbour images) is of the order Gb/Lx. However, the net shear stress is zero for a straight dislocation because sxz due to an image on the right is opposite in sign to that due to the equivalent one on the left. It is non-zero for curved dislocations, but remains small. Also, as in the RBC model, the Peierls stress increases when Lx decreases. The variation is slow but may be of importance for low Peierls stress crystals. 2.3.3. Model for a screw dislocation The procedure follows that for the edge dislocation, but with important differences. The x–z planes perpendicular to the chosen direction of the dislocation line are sketched for the perfect crystal in Fig. 5(a). When the atoms are displaced by the screw dislocation displacement ﬁeld given by isotropic elasticity theory, i.e. z b arctan , (2) uy ¼ 2p x the cell is transformed to that in Fig. 5(b). The x–z planes form a helicoidal surface of pitch b and intersect the central x–y plane with a shift7b/2 between the xo0 and xW0 surfaces.

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Fig. 5. Creation of a left-handed screw dislocation (b ¼ [0, b, 0] and positive line sense �y) showing the effect on the atomic planes perpendicular to the y-axis. (a) Perfect crystal, (b) crystal with elastic displacements for an inﬁnite medium and (c) effect of imposing PBCs in the x-direction.

Periodicity along y is ensured by the independence of uy on y, but periodicity in the x direction is not achieved as easily as in the edge model because of the 7b/2 shifts in the y direction across the 7x surfaces. Some authors (e.g. Refs [43,44]) have avoided this difﬁculty by treating the boundaries in the 7x direction as free surfaces, i.e. free space in the regions |x|WLx/2. However, this solution suffers from the same limitations as the RBC method, i.e. the dislocation experiences an image force (attractive in this case) that increases as it moves away from the central position x ¼ 0. Also, there is no possibility of long-range motion. A PAD with PBCs in the x direction can be created, however, by adding a displacement 7b/2 to the y coordinate of atoms that cross the 7x boundaries in order to put the x–z planes near the central x–y plane into coincidence. The periodic cell on the right (xWLx/2) is generated by adding Lx and b/2 to the x and y coordinates, respectively, of atoms in the inner region: the corresponding displacements have opposite sign for the periodic cell on the left. This is equivalent to using a cell which is not a parallelepiped. This method has been used by several authors, e.g. Refs [45–48].

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In both the edge and screw models, shifts 7b/2 are added across the 7x surfaces in the Burgers vector direction. As seen above, in the edge case, it amounts to an expansion of the cell along with a bending moment. In the screw case, it results in shearing the cell. Also, inspection of Fig. 5(b) reveals that the x–z planes are not vertical in the7x surfaces but are inclined with angles of opposite sign in the xo0 and xW0 surfaces. Imposition of PBCs forces these planes to become vertical, as shown in Fig. 5(c), which is equivalent to applying a torque in the7x surfaces. The inﬂuence of this torque is difﬁcult to evaluate but it decreases with the ratio Lz/Lx. A PAD model for a mixed dislocation can be created by mixing the edge and screw shifts. 2.4. Boundary conditions in the z-direction and loading techniques Boundary conditions at the 7z surfaces are important because action due to external loading is applied there. It is clear from Figs 3(b) and 5(c) for the edge and screw dislocations, respectively, that PBCs cannot be used, and free BCs can only be employed when external load effects are not of interest. Two choices for the 7z boundaries are possible: either rigid or pseudo-free. In the former case, an atomic block (of thickness exceeding the range of the interatomic potential) is created at each boundary with atoms ﬁxed in their initial, strained position [42]. These are represented by the shaded regions in Figs 2 and 3. A second possibility is to create free space in the regions |z|WLz/2. The 7z surfaces can be either fully free [49,50] or constrained to 2D dynamics by allowing surface atom motion only in directions x and y [51,52]. This condition is compatible only with stress-controlled simulations, whereas rigid boundaries can be used for either stress- or strain-controlled simulations. Consider the model shown schematically in Fig. 3. A dislocation lies along the y axis perpendicular to the paper and was formed by the process described in Section 2.3, wherein atoms in the inner region, MR, are free to move in static or dynamic simulations. Blocks A and B at the 7z surfaces are used to apply external action and atoms in them are either ﬁxed in their initial position relative to each other using RBCs or free to move in any direction using free boundary conditions. The primary glide plane is the plane z ¼ 0. The glide force per unit length of dislocation line is given by F ¼ tb, where t is the component of stress resolved on the glide plane in the direction of b, i.e. sxz and syz for the edge and screw, respectively. F due to external action can be generated in two ways, as follows. (i) Shear strain e applied. Either one or both of the blocks A and B can be used to apply a strain, but for simplicity we assume that A is ﬁxed and the dislocation moves when actions are applied to B. Application of shear displacement u ¼ [ux, uy, 0] produces shear strain components �xz ¼

uy ux and �yz ¼ . Lz Lz

(3)

Rigid boundaries are required for A and B in order to maintain this strain during the relaxation of the mobile atoms. The corresponding applied shear stress components sxz and syz, which create a glide force on the edge

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and screw components of the dislocation, respectively, are calculated from the total force F B ¼ ½F Bx ; F yB ; F zB � on all the atoms in B (which at equilibrium is equal to the total force, FA, on A) by the relations: sxz ¼ �

F By F xB and syz ¼ � , Axy Axy

(4)

where Axy ¼ LxLy is the x–y cross-section area of the crystallite of mobile atoms. (ii) Shear stress t applied. For stress loading, a shear force F ¼ [Fx, Fy, 0] is applied either to the atoms in block B, which is allowed to move rigidly while block A is held ﬁxed, or as F applied to B and �F to A if A and B are either free or both allowed to move rigidly. The force components are chosen such that the required stresses sxz ¼ Fx/Axy and syz ¼ Fy/Axy are applied. The shear strain components resulting from the applied stress are calculated from u as in eq. (3). In the case of free boundary conditions, ux and uy are the average differences of displacement of atoms in A and B. Note that in order to apply a strain, e.g. exz, only the average difference in displacement in direction x between blocks A and B has to be ﬁxed while the atomic displacements in directions y and z can be unconstrained. As a result, another possibility to apply a strain is to constrain only the position of the centre of gravity of the blocks A and B and not the position of every atom [53]. This is achieved by subtracting at every simulation step the average force on a block from the forces on atoms within it, i.e. for atoms in B, Fx ¼ Fx � /FxSB and similarly for atoms in A. There are several differences between application of external action for simulation by MS (T ¼ 0 K) and MD (TW0 K). In MS, increasing strain can be applied in increments, with potential energy minimisation performed after every increment. In case (ii) the potential energy is the internal energy computed with the interatomic potential from the position of the atoms minus the work of the applied stress ( ¼ F(uB � uA)). With free boundaries, uA and uB are the average displacements in the two blocks. In MD simulations, both forms of external loading can be used. For scenario (i), the model crystal has to be equilibrated at the chosen temperature and shear strain then applied at a given rate by imposing velocity v ¼ [vx, vy, 0] on B, with stress calculated from FB (eq. (4)). In scenario (ii), constant shear stress or stress rate can be applied by imposing constant or increasing shear force F ¼ [Fx, Fy, 0] after equilibration. With RBCs, B is free to move under the combined inﬂuence of external force F and internal force FB and has to be treated as a separate super-particle of mass MB with an equation of motion coupled with those of all mobile atoms. The corresponding shear strain rate at any instant is calculated from v/Lz. (Again, with free boundary conditions v is the average atomic velocity in B.) A feature of PAD models pointed out in Ref. [42] is that dislocation motion obeys the Orowan relationship between mobile dislocation density, rD, dislocation velocity, vD, and the resulting plastic shear

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strain rate (see, e.g. Ref. [2]). After integration over time, t, with constant rD, this law takes the form Z b � D ¼ x� , (5) � ¼ vD brD dt ¼ xbr Lx Lz where x� is the mean distance of dislocation motion.

2.5. Restrictions on model parameters Available computing power limits the size and complexity of atomic models that can be simulated. The total CPU time required is proportional to the product of the number of atoms and the number of MS relaxation iterations or MD time-steps. With the models presented in Sections 2.1–2.3, the boundary conditions in the glide direction result in forces on the dislocation due to either periodic or image dislocations, but their inﬂuence can be made minimal if the model is large enough. The PAD method is particularly favourable because the periodic images remain at constant distance from the dislocation. Also, even with scalar computing, it is possible to simulate cell volumes that correspond to large but realistic dislocation density rD. For example, a model of a metal with dimensions Lx ¼ 25 nm, Ly ¼ 40 nm, Lz ¼ 25 nm containing one dislocation has rD ¼ 1/LxLz ¼ 1.6 � 1015 m�2, which is within the range found experimentally in heavily cold-worked metals. It contains approximately 2,000,000 mobile atoms, a number that can be simulated easily with a scalar computer and empirical interatomic potentials. With one obstacle to dislocation motion in the cell, i.e. a periodic obstacle spacing of 40 nm, the model is representative of many real situations. By using parallel computing, rD can be reduced by at least an order of magnitude. To illustrate the suitability of models of this size, Fig. 6 contains shear stress versus shear strain plots for MS simulation of glide of an edge dislocation through a periodic row of 2 nm voids in Fe at T ¼ 0 K [42]. (Here and throughout the text, we use Fe to denote a-Fe, the stable BCC phase of pure iron below 911 1C at atmospheric pressure.) The plots correspond to three models of different size, each containing one void. (The form of these plots is explained in Section 4.2.1.) The critical (maximum) shear stress is seen to be independent of Lx (C1 and C2) and to be halved when Lz is doubled (C3), as expected from elasticity theory (see Section 4.2.3, eqs (13) and (14)). The limitation on model dimensions imposed by computing power is therefore not signiﬁcant for the problems to be discussed in later sections. Restrictions on timescale are more serious, however, and lead to values of plastic strain rate, �_, in MD simulations that are high compared to experiment. To illustrate, consider glide of a dislocation across the 25 � 40 � 25 nm3 simulation cell above. The maximum MD time-step required to maintain accuracy is typically in the range 1–5 fs, depending on T, so a simulated time of 10 ns requires between 107 and 2 � 106 time-steps. Assuming the computer code runs at 10�6 s of CPU time per atom per time-step, the total CPU time for the simulation is in the range 46–231 days. For the dislocation to glide the distance Lx in this time, its velocity, vD, has to

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Fig. 6. Stress–strain curves for glide of an edge dislocation through a row of 2 nm voids in Fe for models of different size. Lx ¼ 30 nm, Ly ¼ 41.4 nm. Approximate number of atoms: C1 ¼ 2 � 106, C2 and C3 ¼ 4 � 106. (From Ref. [42].)

equal 2.5 ms�1 and the strain rate that has to be imposed on the model is, from eq. (5) with b ¼ 0.25 nm, �_ ¼ rD bvD ¼ 106 s�1 . In fact, most studies in the literature have used �_ in the range 106–108 s�1. These values are more than 10 orders of magnitude larger than the typical strain rate of 10�4 s�1 applied to macroscopic tensile specimens in laboratory tests. With such a difference, even massively parallel computing cannot bridge the gap between simulation and experiment. Note, however, that vD in MD simulations is realistic for dislocations in free ﬂight when the applied stress is tens to hundreds of MPa. Also, in real tensile tests dislocations spend most of the time immobile in contact with obstacles and not in free ﬂight, so that the applied strain rate mainly controls the contact time with obstacles and not the free ﬂight velocity. The high values of �_ place restrictions on the MD simulation of thermally activated processes. There may be several possible unpinning mechanisms for a dislocation blocked by an obstacle. If some can be thermally activated, the probability of their occurrence is directly proportional to the time the dislocation spends in contact with the obstacle, which is limited in simulation to times of the order of 10 ns. Now consider a typical experiment with �_ ¼ 10�4 s�1 , rD ¼ 1012 m�2 and distance between obstacles L ¼ 10�6 m. The apparent value of vD from Orowan’s law is 4 � 10�7 ms�1. If the time of free ﬂight between obstacles can be neglected compared with the waiting time at an obstacle, Dt, then Dt ¼ L/vD ¼ 2.5 s. Thus, the time a dislocation can spend in contact with an obstacle in an MD simulation is about 10 orders of magnitude shorter and the probability of a thermally activated process occurring is small. In other words, because �_ is high, the stress in the model increases rapidly, which decreases the activation enthalpy of the interaction processes and strongly favours athermal processes. Nevertheless, it will be seen in later sections that effects of T and �_ are observed in MD simulations of dislocation–obstacle interactions.

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2.6. Other practical issues The pros and cons of the methods reviewed here can be summarised as follows. RBC models are simple to implement and fast for even millions of atoms with shortrange empirical interatomic potentials. They are convenient for core structure studies and immobile dislocation–defect interaction simulations, but not for application of external stress or strain. GFBC models overcome some of the disadvantages of rigid boundaries. They are suitable for static conditions (T ¼ 0 K) and simulations where long-range elastic effects inﬂuence dislocation core structure and interaction energy with other defects, or when the interatomic force calculations are computationally expensive and the number of atoms is small. It is not sensible to use them when the dislocation bends strongly under applied strain and sweeps a large area, for such cases require a large inner region and demand large computational resource because of the long-range nature of the elastic Green’s function. The PAD technique has been developed for use with short-range empirical interatomic potentials and has been applied extensively for simulation of dislocation–obstacle interaction under a wide range of conditions, i.e. applied strain at T ¼ 0 K and applied strain or stress at TW0 K. It is intrinsically less accurate than the GFBC method, but this is more than offset by its computational efﬁciency for models with up to millions of atoms. To illustrate this, consider glide of an edge dislocation of the ½[1 1 1]ð1 1� 0Þ slip system in GFBC and PAD models of Fe containing approximately 0.5 � 106 atoms at 0 K [54]. It encounters a row of 2 nm spherical voids of spacing 20 nm and, under increasing applied strain, overcomes them. (Details of this are presented in Section 4.2.1.) The dislocation shape in the glide plane at the maximum stress when the dislocation breaks away is shown in Fig. 7(a). Both shapes are identical in the vicinity of the void, but forward motion of the dislocation between the voids has been restricted in the GFBC model by the boundaries, which are about 15a0 away, where a0 is the lattice parameter. The dislocation climbs in both cases by absorbing vacancies as it leaves the void, as shown in Fig. 7(b). The CPU time with the GFBC method was more than one order larger than with the PAD model, demonstrating that the latter offers similar accuracy with much smaller computational resource. A model-speciﬁc effect can arise when the PAD model is used to study the reaction of a screw dislocation with an obstacle. It will be seen in Sections 4 and 5 that if the dislocation absorbs point defects from an obstacle, it acquires a helical turn. This can glide along the dislocation line, i.e. in the direction of b, and, if L is short and �_ or t are low, it can move through a periodic boundary and reappear on the other side of the original obstacle, thereby restoring it. The dislocation usually changes its slip plane by double cross-slip in this process. We do not consider that this simulates a real reaction. Finally, since a strength of atomic-scale modelling is its ability to simulate processes in the dislocation core, it is important to visualise and identify structure in this region. For the edge dislocation with b ¼ ½[1 1 1] in the BCC metal considered above, a simple analysis of atomic disregistry in the core provides satisfactory

20

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Fig. 7. (a) Comparison of the line shape at the critical stress for a ½[1 1 1]ð1 1� 0Þ edge dislocation overcoming a row of 2 nm voids in Fe at 0 K simulated with GFBCs and PAD models. (b) [1 1 1] projection of line shape after dislocation breakaway showing climb by absorption of vacancies. Scale unit a0 ¼ lattice parameter. (From Ref. [54].)

results for MS and MD simulations [42]. Atomic positions in each ð1 1 2� Þ atomic plane perpendicular to the dislocation line are considered. The point of maximum relative [1 1 1] displacement between two nearest-neighbour atoms in the [1 1 1] direction and an atom in the adjacent ð1 1� 0Þ plane is taken as the intersection of the dislocation line with that ð1 1 2� Þ plane. By treating all ð1 1 2� Þ planes, one can determine the location of the whole dislocation line, including kinks and jogs. The line shapes in Fig. 7 were obtained in this way. The technique can be applied easily for the screw dislocation in cases when the core is compact. It is less suitable for metals where the dislocation dissociates. Other geometrical methods are available, based for example on the analysis of the centro-symmetry of atomic environments [1] or on the comparison of local atomic environments with perfect reference crystals [45]. In the latter case, 12 neighbours in FCC coordination in an FCC metal correspond to the perfect structure, whereas 9 indicates the HCP arrangement, i.e. a stacking fault, 11 or 10 ﬁrst neighbours occur at partial dislocations and lower coordination can occur in other crystalline defects. This method can be adapted easily for BCC [44] and HCP metals [55,56]. It has proved effective in detecting and distinguishing product dislocation segments formed in complex reactions in BCC metals by combining it with identiﬁcation of atoms with high potential energy [57]. Thermal ﬂuctuations at high T and �_ can cause noise. To exclude this, the analysis is repeated over a short period of time (every 10 time-steps during 100–200 time-steps) and only those atoms which remain in defective environments during this time period are shown. This procedure works well for T up to B600 K and �_ up to B108 s�1. More complicated and/or rigorous methods based on local structure analysis of

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neighbours [58] or Voronoi polyhedra analysis [59] can also be applied, but usually at a cost to CPU time.

3. Dislocation glide in pure metals and solid solutions 3.1. Glide in pure crystals The basic controllers of dislocation glide are the nature of the bonding between atoms and the crystal structure of the metal itself [60], for they determine the arrangement of atoms in the core region of a dislocation. The mobility of dislocations depends strongly on the ability of the core to spread. (See Refs [5,6] for recent reviews.) If a metastable stacking fault can form on a plane by shear displacement between atoms, this may lead to dissociation of a dislocation with perfect b into partial dislocations separated by a fault and hence a relatively wide core. Crystal symmetry guarantees an extremum on the stacking fault energy surface (also known as the g-surface) for some slip systems in metals [61], e.g. ½/1 1 0S{1 1 1} in FCC and 1/3/1 1 2� 0S(0 0 0 1) in HCP, but not others, e.g. ½/1 1 1S{1 1 0} or ½/1 1 1S{1 1 2} in BCC and 1/3/1 1 2� 0Sf1 0 1� 0g in HCP. Whether or not a stable fault occurs in the latter cases depends on the atomic bonding: ab initio calculations indicate that one does occur in the HCP metal zirconium [62] but not in the BCC metals [63]. We now consider dislocation motion against the intrinsic resistance of the lattice, remembering that the interatomic potential used for the simulation should provide a good description of the core structure, whether at rest or moving, in the metal being modelled. 3.1.1. Glide at 0 K: the Peierls stress Because of the periodic nature of the crystal lattice, the energy of a straight dislocation varies periodically as it glides, with minima (the Peierls valleys) separated by maxima (the Peierls barriers). In this chapter, the Peierls stress, tP, is taken to be the minimum applied shear stress resolved in the slip direction on the slip plane needed to overcome the Peierls barrier at 0 K. It can be as high as 0.5% of the shear modulus in BCC [64] and HCP [65] metals where screw dislocations have non-planar core structures, and in covalent semiconductors [66] where dislocation glide requires bond swapping. The Peierls stress can be determined from MS simulations by applying either increasing shear strain, e, and therefore stress, t, until the dislocation moves at t ¼ tP. In principle, all types of model boundary conditions described in Section 2 can be used but, as explained there, the resistance of the boundary with RBCs or GFBCs can inﬂuence dislocation glide, and hence the value of tP. This is particularly so when tP is low [32]. The Peierls stress depends on the metal and the interatomic potential. Consider the plots in Fig. 8, obtained for straight edge dislocations in PAD models under shear loading (Section 2.4) for (a) Fe [67] and (b) a-Zr [68]. In Fe, the two potentials predict similar elastic constants, no metastable stacking faults and almost identical core structures for perfect dislocations [57], yet the value of tP differs by a

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Fig. 8. Plots of t versus e for glide at 0 K of (a) ½[1 1 1]ð1 1� 0Þ edge dislocation in Fe (from Ref. [67]) and (b) 1/3h1 1 2� 0if1 0 1� 0g edge and screw dislocations in Zr (from Ref. [68]). The labels for the interatomic potentials are A97 [69] and A04 [70] for Fe, and A95 [71] and M07 [72] for Zr.

factor of more than three. (The elastic modulus, G, for [1 1 1]ð1 1� 0Þ shear in Fe is 73 GPa, so tP/G is approximately 0.3 � 10�3 and 1.1 � 10�3 for the A97 and A04 models, respectively.) The two potentials used for Zr also give the same elastic constants and lattice parameter ratio, and both predict stable intrinsic faults on the basal and prism planes, although the fault energy, g, is different in the two models [68]. Again, these different potentials result in markedly different tP values for the same slip system, as shown in (b). (G for /1 1 2� 0Sf1 0 1� 0g shear in a-Zr is 34 GPa, so tP/G is approximately 0.2 � 10�3 and 0.5 � 10�3 for the edge dislocation with the A95 and M07 potentials, respectively, and 1.3 � 10�3 and 0.7 � 10�3, respectively, for the screw dislocation.) The difference between results for tP for the same metal indicates that caution must be exercised in attaching weight to values of the Peierls stress obtained in this way. Nevertheless, some effects are seen in such simulations that are consistent with experiments and their interpretation. The edge dislocation tends to have a wider core and lower tP than the screw of the same slip system in the same metal. Edge dislocations dissociate and have low tP values (B10�4G) on the {1 1 1} planes of FCC metals and (0 0 0 1) planes of the HCP metals that have a low-energy, basal stacking fault. tP/G is higher for glide of the edge dislocation on any planes of BCC metals and on the prism plane of HCP metals. Ease of glide of screw dislocations is determined by the possible existence of a non-planar core structure [5]. tP is low in the FCC metals because dissociation on a {1 1 1} plane results in a planar core, and likewise in HCP metals with a low-energy basal stacking fault. It is higher for prism slip in HCP metals when the screw core spreads only on the basal plane or simultaneously on the prism and basal planes if faults occur with similar energy in both planes. Screw dislocations in BCC metals have high tP, usually above 1 GPa, because of their non-planar core. Two structures have been found, depending on the

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interatomic potential: a threefold degenerate core, proposed initially in Ref. [64] and predicted by early pair potentials (see Ref. [29]), and a non-degenerate compact core, ﬁrst observed in simpliﬁed MS simulations based on interaction potentials between rows of atoms [73] and later predicted by ab initio calculations [38,63] and the interatomic potential in [70]. However, tP is somewhat ill-deﬁned for the BCC screw. With some potentials that predict a degenerate core [30,44], the dislocation advances by one atomic distance at a lower critical stress and adopts a metastable sessile structure. It remains immobile until an upper critical stress is reached and motion then becomes unbounded. Also, tP is sensitive to non-Schmid components in the applied stress, particularly normal stress perpendicular to the glide plane [44,74]. (See also Refs [5,6].) 3.1.2. Glide at ﬁnite temperature The minimum applied resolved shear stress for dislocation glide at TW0 K is less than tP because of thermal activation. When the stress is high enough so that a dislocation no longer feels the Peierls stress, its free-ﬂight motion has a viscous character with steady-state velocity, vD, proportional to the resolved shear stress t vD ¼

tb , B

(6)

where B is a friction coefﬁcient. With increasing stress, vD falls below values given by eq. (6) and tends asymptotically to the transverse sound velocity of the material (Bfew kms�1). B has various origins in real materials [75], but in MD simulations is controlled solely by phonon damping. MD simulations using PAD models to determine vD as a function of t yield values of B in the range 1–100 mPas [52,76–78]. Examples of data obtained by simulation for vD versus t at 100 and 300 K for the edge dislocation in the two models of Fe used for Fig. 8(a) are presented in Fig. 9. It is seen that the potential from Ref. [70], which gives the higher tP, also results in lower vD for the same value of t. With log–log scales, the plots are linear with gradient equal to 1, as expected from eq. (6), and the value of B increases with increasing T due to increasing phonon damping. The relation between B and T was shown to be linear in FCC metals and Fe [52,77], in agreement with Leibfried’s theory of phonon damping [75]. As proposed initially by Seeger [79], a dislocation in a high Peierls stress crystal at TW0 K and low t spends most of its time aligned with the bottom of a Peierls valley until thermally activated nucleation of a kink-pair moves part of it into the next valley. Subsequent propagation of the kinks along the line transfers the rest of the dislocation to the new position. An early expression based on an analogy between a dislocation and a vibrating string was proposed by Friedel [80]: bL DHðtÞ exp � , (7) vD ¼ doD kB T lc lc

24

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Ch. 88

Fig. 9. Plots of vD versus t for edge dislocation glide at TW0 K on the ½[1 1 1]ð1 1� 0Þ slip system in Fe. The two potentials A97 [69] and A04 [70] are those used in Fig. 8(a).

where d is the distance between Peierls valleys, oD a characteristic frequency close to the Debye frequency, L the dislocation length and lc the size of the critical kink-pair nucleus. DH is the activation enthalpy of the kink-pair nucleus and is a decreasing function of t. This velocity relation has been checked at the atomic-scale by Rodney [53] from a comparison between MS and MD simulations for a Lomer dislocation in an FCC model of Al. The MS simulations used the Nudged Elastic Band method [81] to determine minimum energy paths between Peierls valleys at different values of t. The maximum enthalpy along a given path is DH(t). MD simulations were performed at constant applied strain rate, �_ ¼ 1:5 � 107 s�1 , and different temperatures. The examples of the stress–strain curves in Fig. 10(a) show that serrations occur with each jump of the dislocation from one Peierls valley to the next. Each jump produces a plastic strain, Dep, given by eq. (5) with x� ¼ d, which compensates part of the elastic strain and results in a stress drop Dt ¼ GDepE30 MPa, where G is the shear modulus. To extract the enthalpy–stress relation, the stress is taken to be the average jump stress and the corresponding DH* is computed from its linear relation with the thermal energy, a dependency well known from experiment [82,83]. Indeed, a characteristic enthalpy DH* is obtained from Orowan’s law in steady state and eq. (7): bL DH* . (8) �_ ¼ rD bvD ¼ rD bdoD 2 exp � kB T lc

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Fig. 10. (a) Curve of t versus e for thermally activated glide of a Lomer dislocation in Al at 100, 200 and 300 K with �_ ¼ 1:5 � 107 s�1 . (b) Comparison between kink-pair formation enthalpies obtained from static (open squares) and dynamic (ﬁlled diamond) simulations. A PAD model was used with free surfaces in the z direction. In the static simulations, the stress was applied by means of external forces in the7z surfaces. In the MD simulations, �_ was applied by controlling the velocity of the centres of gravity in the 7z surfaces. (From Ref. [53].)

26

D.J. Bacon et al.

Taking logarithms of both sides yields ! rD bdoD bL DH* ¼ kB T ln . l 2c �_

Ch. 88

(9)

Setting oD ¼ 5 � 1013 s�1, b ¼ d ¼ 0.258 nm, L ¼ 9.7 nm ( ¼ Ly of the model), rD ¼ 10�2 nm�2 and lc ¼ b, gives DH* ¼ CkBT with C ¼ 11.3. This proportionality coefﬁcient is signiﬁcantly smaller than that in experiments (C ¼ 25B30) because of the high applied strain rate. Fig. 10(b) compares the enthalpy curve with the result of the static simulations, proving the accuracy of the velocity law of eq. (8) for the range of T and �_ explored here. 3.1.3. Comparison with experiment A striking difference between modelling and experiment is that the Peierls stress predicted from atomic-scale simulations is several times larger than values deduced from experiment. For instance, with the Fe interatomic potentials used above, the Peierls stress for the ½/1 1 1S screw dislocation in Fe is above 1 GPa while the experimental value is 400 MPa [84]. A similar observation has been made for potassium [85]. High values have also been obtained with ab initio calculations [38], with the exception of molybdenum modelled within the Generalized Pseudopo tential Theory [37]. Various explanations have been put forward to explain the discrepancy, such as the effect of kink dynamics [38], stress concentrations [86] and collective behaviour of dislocations [87]. More research is needed to provide better understanding of this discrepancy. 3.2. Glide in solid solutions 3.2.1. Background The yield and ﬂow stress of a metal can be strongly inﬂuenced by the presence of solute atoms, an effect that is exploited in ‘alloy strengthening’. Whether in solution (SSH) or in precipitates of a second phase (‘precipitation strengthening’), the solute atoms make the metal inhomogeneous at the atomic scale and create a resistance to dislocation glide, e.g. Refs [88,89]. Simulations of solid solutions are described in this section and precipitates are considered in Section 4. There are two principal issues involved in attempting to model SSH and gain a quantitative estimate of the critical applied stress for glide. One is concerned with the short-range interaction of a dislocation with the solute obstacles, which we return to below. The other arises from the statistical nature of the distribution of obstacles in an alloy. Foreman and Makin [90] made the ﬁrst computer simulation study of this problem in the approximation of constant line tension to model glide of a dislocation in a ﬁeld of randomly positioned pinning points, and an analytical treatment was presented by Kocks [91]. More sophisticated approaches to the statistical nature of SSH have also been considered in semi-analytical treatments, e.g. in Refs [92–95]. (A review of some of the earlier models has been presented by Haasen [96].) It is not possible to address this issue quantitatively by MD simulation

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because the simulation cells required to obtain results with statistical signiﬁcance would be too large. Instead, atomic-scale modelling has mainly focused on the inter action between a dislocation and one or a few solute atoms near or within its core. Until recently, this interaction was derived from the long-range solution obtained by linear elasticity theory. Solutes can be modelled using several descriptions depending on their nature [2–4]. The simplest considers each solute atom to be a centre of dilatation due to its misﬁt in the solvent crystal. As a result of the associated volume change, a dislocation interacts with the solute if it has a non-zero pressure component in its stress ﬁeld. The interaction energy decreases as (r)�1, where r is the distance from the solute atom to the dislocation core, and can be negative (attractive) or positive (repulsive) depending on the position of the solute atom with respect to the dislocation. If the symmetry of the site occupied by the solute atom is lower than that of the perfect crystal, more than one distinct orientation of that atom must exist, i.e. its distortion ﬁeld is not spherically symmetric: consequently there is an interaction with the shear stress ﬁeld of a dislocation, again proportional to (r)�1. Substitutional solute atoms in metallic alloys do not lower the symmetry: examples treated by atomic-scale simulation include Cu and Cr in Fe, Al in Ni and Mg in Al (see Section 3.2.2). Interstitial solutes may lower the symmetry, the classic case being that of carbon (C) in Fe, for which the C atom occupies an octahedral site and creates a distortion with tetragonal rather than cubic symmetry (see Section 3.2.3). An induced interaction can also arise if the elastic properties of the solute atom are different from the host crystal. This interaction is of order (r)�2. These descriptions based on elasticity theory break down when r tends to zero, i.e. within the dislocation core, and atomic-scale simulations are adopted. Ab initio methods can be used to model single solute atoms or small clusters in the core [97], but, due to their computational cost, cannot be used to simulate dislocation motion in a solid solution. The PAD method and short-range potentials discussed in Section 2 are suitable. Although speciﬁc electronic effects of solute atoms cannot be represented by these potentials, properties such as size misﬁt or symmetrybreaking can. 3.2.2. Substitutional solute atoms Alloys simulated to date include solutions of Al in Ni (FCC), Mg in Al (FCC) and Cu or Cr in Fe (BCC). A signiﬁcant source of hardening comes from the shortrange interaction of the dislocation with particular conﬁgurations of solute atoms, speciﬁcally pairs of solute atoms across or along the dislocation glide plane, rather than more diffuse, long-range interaction as in classical SSH theories [96]. In the case of Al in Ni, nearest-neighbour pairs of Al atoms with one atom just below and one just above the dislocation glide plane are strong pinning centres. Rodary et al. [98] simulated the glide of an edge dislocation with PAD boundaries under constant applied shear stress for solute concentration, c, in the range 1–8 at.%. An example is shown in Fig. 11. With the interatomic potential set used, an Al atom is oversized with a volume misﬁt parameter of 3%. To ensure that the gliding dislocation encountered the most representative random arrangement

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Fig. 11. Dislocation gliding in a Ni–3at.% Al solid solution at 300 K under t ¼ 70 MPa. The dimensions of the computational cell are Lx ¼ 30 nm, Ly ¼ 43.12 nm, Lz ¼ 7.32 nm. The visualisation shows Al atoms and atoms in the Shockley partial cores which do not have FCC or HCP coordination. (From Ref. [98].)

compatible with overall computing time, the dislocation was allowed to make 20 sweeps of the model via the transparent x boundaries. Dislocation velocity, vD, was obtained from the average x coordinate of the core atoms. If the dislocation paused for at least 50 ps before completing 20 sweeps, vD was taken to be zero. Plots of vD versus t for pure Ni and 3, 5 and 8at.% alloys are presented in Fig. 12(a). vD tends to the limit of the transverse velocity of sound at high t, depends linearly on t at lower stress and falls to zero at the stress ts, which depends on solute concentration. The velocity extrapolates to zero at the stress td. Rodary et al. refer to ts and td as the ‘static’ and ‘dynamic’ threshold stresses, respectively. These stresses increase linearly with solute concentration, c, as shown by the data in Fig. 13(a). With increasing T, ts decreases due to thermally assisted motion of the dislocation over the strong part of the obstacle spectrum, but td is unaffected because it represents the effect of the weak part of the obstacle spectrum. The MD simulations showed that a signiﬁcant part of hardening was due to obstacles consisting of Al–Al pairs in nearest-neighbour coordination across the slip plane that are formed from second-neighbour pairs by the passage of the dislocation. This is because glide of a Shockley partial shifts atoms above the glide plane by 1/6/1 1 2S with respect to those below and thus changes their respective position. A nearest-neighbour Al dimer is highly repulsive in Ni–Al for two reasons. One is elastic in origin and due to the positive misﬁt. The other is chemical: second neighbour pairs of Al atoms are energetically favourable over ﬁrst neighbours because of the stability of the Ni3Al intermetallic phase, which is included in the interatomic potential. The resistance of Al dimers across the glide plane is not accounted for by elasticity since the interaction with the dilatation ﬁeld of the solute above the glide plane cancels (at ﬁrst order) that of the solute below. In more quantitative studies of SSH in this alloy, Proville et al. [100] and Patinet and Proville [101] measured by static simulations the pinning force and force range

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Fig. 12. Dislocation velocity versus t at 300 K in (a) Ni–Al (from Ref. [98]) and (b) Fe–Cu (from Ref. [99]).

for single solute atoms and pairs that may or may not cross the glide plane of either screw or edge dislocations. The interaction was found to be signiﬁcant only when the solutes are just above or below the glide plane. For such short-range conﬁgurations, the two Shockley partials of a dislocation interact separately with

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Fig. 13. Concentration dependence of ts and td at 300 K in (a) Ni–Al (from Ref. [98]) and (b) Fe–Cu (from Ref. [99]).

the solute atoms and the energy of the solute is always minimum in the stacking fault of the dislocation. As a result, the pinning force on the trailing partial is larger than that on the leading one. Also, since the partials of both edge and screw dislocations have edge components that interact with the dilatation ﬁeld of solute atoms, the pinning forces on screw and edge dislocations are of same magnitude. This short-range effect is contrary to assumptions in theories of SSH, in which interaction with screws is assumed to be weak because the edge components of their

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partials have opposite sign and cancel at large distance. The CRSS at T ¼ 0 K for both edge and screw dislocations increases linearly with c at the same rate, about 25–30 MPa/at.%, in agreement with experimental indentation hardness measure ments on Ni–Al [102]. The strongest conﬁguration for both edge and screw dislocations is a dimer of solute atoms in the plane just above the glide plane: the pinning force is about twice that for a solute atom in the same plane, which is the location for a single atom with the highest obstacle strength. The second strongest conﬁguration is the dimer that crosses the glide plane, as described above. Pinning forces found by simulation are smaller than estimates obtained by elasticity theory, showing that the latter does not account accurately for the short-range interactions. The interaction energy of edge and screw dislocations with single solute atoms in the Al–Mg system was computed using cylindrical simulation cells with RBCs and no loading in Ref. [103]. The interaction energy with the two partials is symmetrical in this case because the dislocation is at rest. As in Ni–Al, the strongest interaction is when the solute atom is in a plane adjacent to the glide plane. The importance of short-range interactions has also been conﬁrmed for the BCC alloy Fe–Cu, in which Cu is an oversized substitutional solute. MS simulation of motion of the ½/1 1 1Sf1 1� 0g edge dislocation under increasing strain shows that the critical stress, tc, is determined by individual obstacles in the form of either single Cu atoms or pairs of solute atoms adjacent to the glide plane [99]. For single atoms, tc is highest when the solute occupies a site in the plane immediately below the extra half-plane of the dislocation. It is the one that gives the maximum attractive interaction energy with the dislocation. For Cu–Cu doublets, the maximum tc is approximately 50% higher than that for one solute atom and occurs for the ﬁrst-neighbour pair in the plane immediately below the extra halfplane. This conﬁguration is equivalent to the strongest obstacle pair in Ni–Al, except that in that case the pair is above the glide plane. By varying the model size in the simulations of individual obstacles, tc was found to be approximately proportional to the reciprocal of the spacing of strong obstacles along the dislocation line, a result consistent with the continuum treatment of localised obstacle strengthening. MD simulation of edge dislocation glide in Fe–Cu solid solutions at TW0 K revealed similar behaviour to that in Ni–Al discussed above, i.e. smooth motion with constant velocity at high t and irregular motion with pauses and complete stops at low t [99]. Plots for vD versus t in Fe–Cu solutions at 300 K are shown in Fig. 12(b). They have similar form to those for Ni–Al in (a), but vD at a given stress is much less sensitive to change in c. The threshold stresses ts and td deﬁned in Fig. 12(a) are plotted as functions of c for Fe–Cu in Fig. 13(b). Both increase linearly with c, but, unlike those for Ni–Al, do not extrapolate to the same value at c ¼ 0. Tapasa et al. note that dislocation glide in the equivalent simulation of pure Fe occurs at tB1 MPa, so even a concentration as small as 0.5at.% of Cu in Fe has a signiﬁcant effect on ts. Rodary et al. [98] have pointed out that the linear dependence of ts on c is consistent with the importance of the strongest obstacles in the solute spectrum, because, according to Friedel statistics [80], the stress at which a dislocation moves

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across a ﬁeld of point obstacles is proportional to the square root of obstacle density, and the density of pairs of solute atoms is proportional to solute concentra tion. It is also consistent with interpretation of the experimental concentrationdependence of hardening in Cu–Mn solid solutions, according to which pairs of Mn atoms are responsible for the hardening [104]. 3.2.3. Interstitial solute atoms The important role of carbon interstitial solute in the phenomena of yielding and strain ageing in steel was ﬁrst treated theoretically via elasticity in [105]. The signiﬁcance of the interaction of carbon atoms with the dislocation core itself was identiﬁed, but it has been possible only recently to examine this within the scope of large-scale atomic simulations. Potentials that accurately reproduce all important properties of C in Fe have not been available, so Tapasa et al. [106] employed a combination of the EAM interatomic potential for Fe–Fe interaction from Ref. [69] and a pair potential for Fe–C interaction from Ref. [107], which models an atom having the same octahedral site and distortion ﬁeld as C in Fe and similar migration energy. Carbon in this low symmetry site creates a tetragonal distortion, for it repels its two ﬁrst-neighbour Fe atoms in a /1 0 0S direction and attracts its four second neighbours in the two transverse /0 1 1S directions. Thus, C solutes are located in either a ð1 1� 0Þ atomic plane parallel to the dislocation glide plane (with tetragonal distortion axis (TDA) in the [0 0 1] direction) or sites between two ð1 1� 0Þ planes (with [1 0 0] or [0 1 0] TDAs symmetric about b). Tapasa et al. [106] used MS to investigate the interaction energy of the edge dislocation with a C atom in sites near the slip plane. Sites half an interplanar spacing below the ð1 1� 0Þ plane at the bottom of the extra half-plane have the highest binding energy, Eb ¼ 0.68 eV. C atoms in sites within ð1 1� 0Þ atomic planes have a maximum Eb of 0.5 eV when located one below the extra half-plane. These values compare with the elasticity estimate of about 0.5 eV obtained in Ref. [105] and values in the range 0.5–0.8 eV calculated later when the inﬂuence of the tetrahedral distortion was taken into account, e.g. Refs [108–111]. This distortion gives rise to C interaction with the ½/1 1 1S screw dislocation, for which Eb is predicted to be approximately 40% of the value for the edge [111]. Interestingly, Becquart et al. [112] have recently developed a new potential for a C atom in Fe using ab initio data and ﬁnd that the maximum Eb for the screw dislocation is 0.41 eV. Thus, although maximum binding occurs inside the core region where linear elasticity is not strictly valid, the Eb value it gives for C in Fe is consistent with the atomic-level treatments. Simulations of solutions of C have not been made, but the critical stress, tc, at which an edge dislocation overcomes a row of single C atoms in sites of high binding and spacing L ¼ 3.51 or 11.23 nm at T ¼ 0 K and W0 K has been determined [106]. Examples of the critical line shapes are shown in Fig. 14. The tc values for these conditions are 466, 146 and 50 MPa, respectively. The critical stress at T ¼ 0 K has an L�1 dependence consistent with strengthening due to point obstacles with barrier energy Eb. The temperature-dependence of tc in the model with L ¼ 3.51 nm is plotted in Fig. 15 for three values of �_ . tc decreases strongly with

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Fig. 14. Edge dislocation shape (visualised by atoms in the core) at tc for a carbon atom (small sphere) in the site of maximum binding energy, Eb, in Fe crystals at (a, b) T ¼ 0 K for two different spacings, L, between carbon atoms and (c) T ¼ 300 K. The applied strain rate is �_ ¼ 5 � 106 s�1 . (From Ref. [106].)

increasing T at low T and increases with increasing �_ . It is small and largely independent of T above about 300 K. Carbon atom migration within the core occurs during the time the dislocation is pinned when TZ800 K, but extensive drag of solute does not occur at the applied strain rates accessible to MD simulation because the dislocation jumps forward by too large a distance as it unpins for a C atom to be recaptured. Tapasa et al. deduced the activation parameters for slip in this atomic-scale model with the following treatment. Dislocations slip by overcoming barriers, each of which exerts a resisting force with a proﬁle such that the area under the force– distance curve equals the total energy, Eb. (See, e.g. Refs [2,113,114].) At TW0 K, the energy required is partly provided as mechanical work by the applied load: it is written tcbLd* or tcV*, where d* and V* are the activation distance and volume, respectively. The remainder is thermal and has to overcome the free energy of activation DG* ¼ E b* � tc V*, where E b* is the total energy required between the dislocation states separated by d*. DG* is the Gibbs free energy change at constant T and �_ between those two states. The probability of DG* being provided by thermal ﬂuctuations is exp(�DG*/kBT) if DG*ckBT. Hence, from eq. (8) the

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Fig. 15. Temperature dependence of tc for a carbon atom in the site of maximum Eb with L ¼ 3.51 nm. The dashed line indicates qtc/qT in the region of strong temperature-dependence. (From Ref. [106].)

macroscopic plastic strain rate is �DH* �_ ¼ rD A exp , kB T

(10)

where A ¼ bDn. D is the glide distance between each obstacle overcome and n the effective attempt frequency. G* has been replaced by the enthalpy H* by taking the entropy term in �_ to be unity. In experiments, DH* and V* are determined from tensile tests at constant T and constant �_ . If a change in �_ from �_1 to �_2 causes a change Dtc in the ﬂow stress at constant T, then V* can be obtained from V* ¼

kB Tlnð�_2 =�_ 1 Þ . Dtc

(11)

The activation enthalpy is obtained from tests over a range of T at constant �_ by using the relation @tc DH* ¼ �V*T . (12) @T �_ Tapasa et al. applied these equations to the MD data in Fig. 15. Dtc ¼ 42 MPa when �_ decreases from 20 to 5 � 106 s�1 at T ¼ 100 K, which gives V* ¼ 2.9b3 from eq. (11). The values at 200 K and 300 K are 2.2b3 and 4.7b3, respectively. Since L is 3.51 nm (¼ 14b), d* is approximately 0.2–0.3b, which is reasonable for an interstitial atom. Use of eq. (12) with these V* values and the gradient of the dashed tc versus T line in Fig. 15 gives DH* ¼ 0.03, 0.05 and 0.14 eV for T ¼ 100, 200 and

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300 K, respectively, i.e. 3–6kBT. This energy is much smaller than Eb ¼ 0.68 eV that has to be overcome by the dislocation at T ¼ 0 K. This exposes a limitation of MD for simulation of thermally activated dislocation processes already mentioned in Section 2.5. The unavoidably high �_ limits the time available for activation and results in small values of DH*, as mentioned at the end of Section 3.1.2. This conclusion was also drawn by Domain and Monnet [43], who used MD simulation of glide of a ½/1 1 1S screw dislocation in Fe to reveal that DH* is B20kBT smaller than that realised in experiment at much smaller strain rate. 3.2.4. Extension to microscopic models An outstanding issue in relation to MD simulation of SSH is how to incorporate information gained from relatively small models into larger-scale analyses that recognise the statistical nature of dislocation behaviour. Rodary et al. [98] have provided a brief discussion of how results from their MD modelling of Ni–Al might be incorporated in a higher-level, micromechanical approach to predict macro scopic stress–strain curves. Proville et al. [100,101] used the pinning forces and force ranges computed from MS simulations to parameterise analytical expressions of CRSS proposed in various classical theories of SSH [96]. One difﬁculty is that the theories assume a single type of obstacle whereas simulations exhibit a range of pinning forces depending on obstacle type (single atom or dimer) and position with respect to the glide plane. A generalisation of the Mott–Nabarro model to include several types of obstacles was proposed in Ref. [100] and showed good agreement with MS simulations. Olmsted et al. [103] proposed a semi-analytical approach based on computation of a one-dimensional energy landscape for the motion of a straight edge dislocation through a random ﬁeld of Mg solutes in Al using the single solute energy values. The stress to unpin a straight edge dislocation trapped in a local energy minimum was estimated and compared with values obtained by MS for a dislocation moving under stress through a random array of solutes. A good agreement was obtained. The thermally activated rate of dislocation unpinning versus t and T was calculated semi-analytically, and again good agreement was found with results of MD simulations. A challenge for future work in this area is to provide more accurate information on the motion of dislocation cores containing solute atoms across wide ranges of T and vD, and build it into continuum-based models so that effects such as dislocation–dislocation interactions and dislocation climb can be simulated.

4. Voids and precipitates 4.1. Introduction We now turn to obstacles that are larger than the individual solute atoms considered in Section 3, but are still only up to a few nanometres in size. As noted in Section 1, much of the research using atomic-scale simulation has been driven by the need to investigate the effect of obstacles that are of importance for metals in

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current and future nuclear power systems. MD simulation of displacement cascades shows that a signiﬁcant fraction of the point defects are created in clusters with their own kind. SIA clusters are usually in the form of dislocation loops. The clustered vacancies resulting from radiation damage may have dislocation character, i.e. loops or SFTs. These vacancy clusters also form during annealing of the vacancy supersaturation created by rapid quenching of metals from temperatures just below the melting point. The obstacle properties of dislocation loops and SFTs are considered in Section 5. In many situations, vacancies cluster to form voids. Voids are common features of the microstructure after either irradiation at high temperature or ageing after quenching. When He atoms are present, e.g. as a result of a transmutation reaction with atoms in the metal, these cavities may contain He gas. Voids are usually of near-spherical shape, with facets on close-packed planes that have low surface energy. It will be seen below that they can offer strong resistance to dislocation motion as a result of direct contact between dislocation and void. The interaction is characterised by the following. It is energetically favourable for a dislocation to intersect a void because its core and strain energy is zero within the cavity. However, this is at least partially offset by the fact that the dislocation has to create a surface step (in the direction of b) by shear as it penetrates a void. These two effects determine the obstacle strength of the void, i.e. the magnitude of the applied stress that is necessary for the dislocation to break away. These important details, and the effect of the cavity on the dislocation energy, are not easily modelled using elasticity theory [115]. Precipitates also form a class of obstacle known to be important for the strength of alloys, e.g. Refs [88,89]. For many applications, alloy composition and heat treatment are deliberately chosen to develop a population of precipitates that offers optimum resistance to dislocation motion and is stable in the temperature range required for use. In terms of atomic-scale simulation, such alloys have received little attention because the size and dispersion of their precipitates are such that the strengthening mechanisms are quite well understood from conventional, continuum-based models. Computer simulation has tended to focus on alloy systems where the precipitates are small and research is required to understand the dislocation–precipitate mechanisms at the atomic level. The alloys in question are again important for current and future nuclear power systems. They are Fe–Cu and Fe–Cr, and are also considered in this section. If the dislocation penetrates and shears a precipitate, the features that determine the obstacle strength may be similar to those mentioned for a void, i.e. an interface step has to be created and the dislocation energy in the precipitate may be different from that in the surrounding matrix, in this case as a result of differences in properties such as the elastic constants and stacking fault energy. Furthermore, the dislocation may induce a change within the precipitate to a more favourable crystal structure, an effect discovered by computer simulation of Cu precipitates in Fe (see below). The features of strengthening are different, however, if the dislocation does not penetrate the precipitate, e.g. if the precipitate is incoherent with the matrix or if the stress state near it is sufﬁcient to induce the dislocation to bypass it by cross-slip.

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Important details that determine the critical applied stress and breakaway mechanism are not easily modelled using elasticity theory. For example, the surface step on a void (or interface step around a precipitate) increases in size to b as an edge dislocation is pulled from the obstacle, in contrast to the situation for a screw where it decreases to zero, and the energy of the step may not be known with precision. Furthermore, the effect of a small void or precipitate on the dislocation energy and stress ﬁeld can only be treated with approximation. Atomic-level simulation incorporates these effects naturally. We ﬁrst consider the obstacle resistance encountered by edge dislocations due voids in Fe and Cu (Section 4.2.1) and precipitates in Fe (Section 4.2.2) at T ¼ 0 K. In this MS modelling, shear strain e is applied in increments De and the potential energy, E, is minimised after every increment: the applied shear stress,t, is calculated from the reaction force exerted by the moveable atoms on the rigid outer blocks of atoms (Section 2.5). MS simulation neglects effects due to kinetic energy and provides for direct comparison with continuum modelling of these obstacles (Section 4.2.3). The effects of temperature on void and precipitate strengthening are presented in Section 4.3, and examples of simulations of bubbles and loose clusters of vacancies are described in Section 4.4.

4.2. Edge dislocation–obstacle interaction at T ¼ 0 K 4.2.1. Voids This interaction has been most widely studied for Fe using the interatomic potential from Ref. [69]. The computational cell and BCC crystallographic indices for the edge dislocation are shown schematically in Fig. 16(a). A spherical void of diameter D is placed with its equator on the dislocation slip plane. The periodic spacing of voids in the y direction is L. (From now on, we use L for the obstacle spacing of both edge and screw dislocations, i.e. for Ly and Lx respectively.) Results for 339 vacancy voids (D ¼ 2 nm) in a crystal of 2.07 M atoms (L ¼ 41.4 nm, Lx ¼ 29.8 nm, Lz ¼ 19.9 nm) are presented in Fig. 17(a) as curves of t and DE versus e, where DE is the change in E from the initial state when the straight dislocation is far from the row of voids. Four regions are indicated on the plots. The dislocation glides towards the void at the Peierls stress of 23 MPa in region I. It is attracted by, and pulled into, the void in region II: a step of length b is created on the entry surface of the void but E decreases because of the decrease in length of the dislocation. Stress becomes negative because the plastic shear strain due to dislocation motion is larger than the imposed strain: t does not return to zero until e increases and the dislocation becomes straight and coincident with the centre-to-centre line of the voids, i.e. the minimum energy conﬁguration is achieved. In region III the dislocation bows between the voids under increasing e until it is released at the critical resolved shear stress, tc. As shown in Fig. 7, visualisation of the atoms in the dislocation core demonstrates that on breaking away, the dislocation absorbs a few vacancies from a void and so acquires a pair of superjogs, i.e. jogs of more than one-atom height. Region IV corresponds to motion of the jogged dislocation. The dependence of the

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Fig. 16. Schematic illustration of the atomic cell and crystallography used to simulate interaction of dislocations with a row of obstacles in (a) BCC and (b) FCC metals. The sense of positive applied resolved shear stress, t, is shown. The centre-to-centre spacing of obstacles is Lx and Ly for the screw and edge dislocations, respectively. The tetrahedron in (b) is also used to identify the Burgers vectors and slip planes for the FCC case.

t–e plot on D for L ¼ 62 nm in Fe at 0 K is shown in Fig. 18. The maximum, tc, of each curve varies approximately logarithmically with D for a given L, as can be seen in Fig. 19. Furthermore, it is approximately inversely proportional to L, e.g. for the three values of L shown for the 2 nm void, tc varies in the ratio 1:1.40:1.99 while

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Fig. 17. Change in potential energy and applied stress as functions of applied strain in an Fe crystal at T ¼ 0 K containing an edge dislocation (L ¼ 41.4 nm) gliding through (a) a 2 nm void containing 339 vacancies and (b) a 2 nm precipitate containing 339 Cu atoms. (From Refs [42,116].) Note the difference in the scale of the ordinate axes between (a) and (b).

Fig. 18. Stress–strain curves obtained for edge dislocation–void interaction in Fe at 0 K using the interatomic potential from Ref. [69]. The model size and D are indicated. (From Refs [116–119].)

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Fig. 19. tc versus D (log scale) for rows of voids in Fe at 0 K and three different centre-to-centre spacings, L.

Fig. 20. Shape of the dislocation line at tc for 2 nm voids with different spacing, L, in Fe at 0 K. Black line: L ¼ 41.4 nm, tc ¼ 207 MPa; grey line: L ¼ 83.6 nm, tc ¼ 104 MPa. Dislocations are visualised by showing atoms in the core. (From Ref. [117].)

L�1 varies as 1:1.35:2.02. The shape of the dislocation line in the ð1 1� 0Þ slip plane at tc for crystals with L ¼ 41.4 and 83 nm is presented in Fig. 20. The radius of curvature of the line between the obstacles is smaller for the smaller L, i.e. larger tc. This, and the L�1-dependence, is as expected from the line tension model of elasticity. However, it should be noted that the dislocation conﬁguration in the vicinity of the voids is almost identical for the two L values, a feature not predicted by that model. Dislocation line shape in the slip plane at tc for voids of different size is presented in Fig. 21 by black lines. (Grey lines are the dislocation shape for Cu precipitate

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Fig. 21. Critical line shape in the ð1 1� 0Þ plane for a dislocation passing a row of either spherical voids or Cu precipitates of different size in Fe at 0 K. Spacing L ¼ 62 nm and D varies from 0.9 to 5 nm. Black line: shape for voids; grey line: shape for precipitates. The labels for each pair of shapes indicate: tc for void, D and tc for precipitate, in descending order. (From Ref. [120].)

obstacles discussed in Section 4.2.2). As seen in Fig. 18, small voids with D of up to about 1.5 nm in Fe give an almost linear dependence of t on e in region III. The gradient dt/de is slightly lower than the elastic shear modulus because of plastic strain due to the bowing dislocation. These small voids offer relatively weak resistance. Larger voids are relatively strong obstacles and the dislocation bows out more before breakaway at tc, resulting in decreasing dt/de as tc is approached. The dislocation bends sufﬁciently to form a screw dipole in the critical conﬁguration when DZ2 nm, as shown in Fig. 21. This shape is the same as that for an edge dislocation to overcome a row of impenetrable obstacles by the Orowan mechanism, although an Orowan loop is not left around the obstacle in the void case and the spacing of the dislocations in the dipole is less than D. The mutual attraction due to self-stress of the dislocation segments at the void surface assists their alignment in the screw dipole arrangement and results in the decrease of dt/de as the dipole is extended. The dipole length can reach several tens of nanometres for DW5 nm. The mechanism controlling dislocation breakaway is associated with the difﬁculty of formation of a step on the exit surface of the void, which is believed to be a consequence of the threefold structure of the ½/1 1 1S screw core. In fact, a simple shear step is not formed. Instead, the edge dislocation climbs by absorbing some vacancies from the void at breakaway (Fig. 7) and the dipole is annihilated by unzipping from the void surface rather than by cross-slip of the screw segments. This effect has been found with two different potentials for Fe [69,70], even though

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Fig. 22. Stress–strain curves obtained for edge dislocation–void interaction in Cu at 0 K using the interatomic potential from Ref. [121]. The model size and D are indicated. (From Refs [120,122].)

they reproduce different screw core structures [57]. The amount of climb depends on D (see examples in Ref. [116]). The model orientation for edge dislocation–void interaction in Cu is drawn in Fig. 16(b). The t–e plots for 0 K have a similar sequence of regions I–IV to those for Fe, as demonstrated for data for L ¼ 35 nm and D in the range 1–5 nm in Fig. 22. However, there are differences in region III as t rises from 0 to t due to the difference in the structure of edge dislocations in the two metals. In contrast with Fe, climb does not occur when edge dislocations overcome voids in Cu at 0 K. Furthermore, the t–e plots do not show a marked decrease in dt/de as tc is approached, and there are signiﬁcant drops in t for small D before eventual breakaway. These differences from Fe are due to the dissociated dislocation core structure in this FCC metal. For small voids (Dr2 nm), the ﬁrst drop in t occurs when the leading Shockley partial breaks from the void. The step formed by this on the exit surface is a partial step 1/6/1 1 2S and the stress required is small. Breakaway of the trailing partial determines tc. The line shapes in the (1 1 1) plane at these two maxima in t are shown in Fig. 23. For larger voids, tc is controlled by the leading partial and the dislocation breaks away as a whole. Creation of a pure screw segment is not possible and dt/de does not decrease by dipole formation. As a consequence of these effects, voids in Cu are weaker obstacles than those in Fe when small and stronger when large. The latter effect arises from the high Peierls stress of one of the partials [120]. The dissociation of the core into distinct partials prevents dislocation climb, but it is possible that in metals with high stacking fault energy, such as Al, constriction of the core could result in dislocation–void interaction more like that in Fe.

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Fig. 23. Critical line shape in the (1 1 1) plane for a dislocation passing a row of 2 nm voids with spacing 35 nm in Cu at 0 K. Left-hand ﬁgure: t ¼ 115 MPa; right-hand ﬁgure: t ¼ 118 MPa. The core region of Shockley partials is shown by black symbols and the stacking fault is grey. (From Ref. [122].)

4.2.2. Precipitates As noted in Section 4.1, atomic-level simulation of dislocation–precipitate interaction has concentrated on nanoscale precipitates that occur in metals used in nuclear power systems. Among these, Cu in Fe is important because Cu-rich precipitates of small size (Dofew nanometres) form during neutron irradiation of ferritic pressure vessel steels that contain small amounts (a few tenths of a percent) of Cu. Cu precipitates that nucleate during thermal ageing of Fe–Cu alloys transform martensically as they grow from the BCC crystal structure coherent with the Fe matrix to a twinned 9R form of the FCC structure. The size at which they transform from BCC to 9R falls in the range from about 4 to 10 nm, depending on the heat treatment (e.g. Ref. [123]). In neutron-irradiated steels, however, Cu precipitates remain small and BCC in structure. Together with radiation damage formed by vacancies and SIAs, they make a signiﬁcant contribution to irradiation hardening (e.g. Refs [124–126]). The interaction of an edge dislocation with spherical BCC Cu precipitates coherent with the Fe matrix at 0 K has been simulated for various D and L values using the interatomic potential from Ref. [69] with the model depicted in Fig. 16(a) [116–120,122,127–129]. As anticipated in Section 4.1, some features of the interaction are similar to that for voids. This is illustrated by noting the similar form of the DE and t versus e plots for rows of 2 nm voids and precipitates in Figs 17(a) and (b), respectively, although the reduction in E when the dislocation

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penetrates the obstacle and tc are smaller for the precipitate. These differences in magnitude arise from the fact that the precipitate is a region with non-zero modulus and the step on the Cu–Fe interface has lower energy than the free-surface step on a void. Consequently, the dislocation simply shears the precipitate without being pulled into screw orientation. For example, the critical included angle, jc, between the dislocation segments at the obstacle surface is 701 for the D ¼ 3 nm precipitate compared with 01 for the void of the same size (Fig. 21). Dislocation climb does not occur. More signiﬁcant differences between Cu precipitates and voids arise for D greater than about 3 nm. In these larger precipitates, the BCC Cu undergoes a dislocation-induced, partial transformation to a more stable FCC-like structure. This is demonstrated by the projection of atom positions in four atomic planes near the equator of a 4 nm precipitate after dislocation breakaway in Fig. 24. The {1 1 0} planes have a twofold stacking sequence in the BCC metals, as can be seen by the upright and inverted triangle symbols near the outside of the precipitate, but atoms represented by circles are in a different sequence. Atoms away from the Fe–Cu interface are seen to have adopted a threefold sequence characteristic of the {1 1 1} planes in an FCC metal. This transformation, ﬁrst found in MS simulation of a screw dislocation penetrating a precipitate [130,131], increases the obstacle strength and results in critical line shape (Fig. 21) and tc (Fig. 25) that are close to those of voids of the same size. It is also accompanied by formation of jogs on the dislocation as it leaves the precipitate, as seen on the right of Fig. 24. These atomic-level mechanisms are not predicted by continuum treatments, such as the line-tension

Fig. 24. Position of Cu atoms in four consecutive ð1 1� 0Þ planes through the centre of a 4 nm precipitate in Fe after dislocation breakaway at 0 K. The ﬁgure on the right shows the dislocation line in [1 1 1] projection after breakaway: climb to the left indicates absorption of vacancies whereas climb to the right is due to absorption of atoms. (From Ref. [117].)

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Fig. 25. Stress–strain curves obtained for interaction between an edge dislocation and Cu precipitates with diameter D and spacing L ¼ 42 nm in Fe at 0 K. The model size and D are indicated.

and modulus-difference approximations that form the basis of the Russell–Brown model of Cu precipitate strengthening of Fe [132]. 4.2.3. Comparison of atomistic and continuum results at 0 K The simulations of the preceding section neglect kinetic effects and are the atomiclevel equivalent of conventional linear elasticity modelling of dislocations. In the latter, the simplest approach to strengthening due to a dispersion of localised obstacles uses the line tension approximation. For a row of obstacles of spacing L, the dislocation breaks away when the forward force tbL on length L overcomes the resistance of one obstacle, i.e. when the obstacle can no longer withstand the line tension force 2Gcos(j/2) imposed by the two segments of the bowing line at the obstacle: here G is the line tension (equal to approximately Gb2/2 in the constant line tension approximation) and j is the included angle between the segments, as shown for a 3 nm precipitate in Fig. 21. The critical stress is reached when j is reduced to the critical angle jc and so tc ¼ a

Gb , L

(13)

where a ¼ cos(jc/2). A factor of approximately 0.8 has to be introduced to account for randomness in the obstacle arrangement [90,91]. It was seen in the data for voids above that the dependence of tc on L�1 holds for atomic-scale interactions, as it should since there is no reason to doubt the Peach–Koehler expression that the

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force per unit length on a dislocation is tb. The parameter a that characterises obstacle strength does not have a ﬁrm theoretical basis, however. In the literature where experimental tensile test data for, say, irradiated steels have been used to validate a theoretical expression for yield stress, eq. (13) has frequently been employed with a obtained by empirical ﬁtting to the test data. It generally falls in the range 0.1–0.5. Unfortunately, this approach is inadequate for quantitative and predictive modelling because, as seen in preceding sections, a depends sensitively on obstacle size, type and atomic mechanisms. For example, the 3 nm precipitate in Fig. 21 has jc ¼ 701 and tc ¼ 145 MPa, yet tc given by eq. (13) with this angle and L ¼ 62 nm is approximately 210 MPa. Similarly, tc for the 2 nm void with this L is 161 MPa, but eq. (13) with jc ¼ 01 predicts 260 MPa. The line tension model over estimates obstacle strength because it neglects self-interaction between the branches of the dislocation across the obstacle, which tends to pull them into alignment. Self-stress interaction between different parts of a dislocation was ﬁrst included in modelling of strengthening due to impenetrable obstacles (Orowan strengthening) and voids using linear elasticity in Refs [115,133]. By computing the equilibrium shape of a dislocation bowing between obstacles under increasing stress, it was shown that the maximum stress for which equilibrium could be achieved ﬁtted the relationship tc ¼

Gb lnðD�1 þ L�1 Þ þ D , 2pAL

(14)

where G is the elastic shear modulus and D is an empirical constant; A equals 1 if the initial dislocation is pure edge and (1 � n) if pure screw, where n is Poisson’s ratio. It was shown in Refs [115,134] that eq. (14) holds for anisotropic elasticity if G and n are chosen appropriately for the slip system in question, i.e. if Gb2/4p and Gb2/4pA are set equal to the pre-logarithmic energy factor of the screw and edge dislocations, respectively. The value of G obtained in this way is 64 GPa for /1 1 1S{1 1 0} slip in Fe and 43 GPa for /1 1 0S{1 1 1} slip in Cu [135]. The explanation for the D- and L-dependence of tc is that voids and impenetrable Orowan particles are ‘strong’ obstacles to dislocation motion and so the dislocation segments at the obstacle surface are pulled into parallel, dipole alignment at tc by self-interaction [115,133]. For every obstacle, the forward force, tcbL, on the dislocation has to match the dipole tension, i.e. energy per unit length, which is proportional to ln(D) when D{L and ln(L) when L{D [2]. Thus, tcbL correlates with Gb2ln(D�1 þ L�1)�1. The correlation between tc obtained in atomic-scale simulation and the harmonic mean of D and L, as in eq. (14), is tested for voids and precipitates in Fig. 26. Values of tc obtained in the continuum modelling of Refs [115,133,134] were found to ﬁt eq. (14) with D equal to 0.77 for the Orowan process and, with a realistic estimate of void surface energy, 1.52 for voids. Lines for these two values of D are drawn on Fig. 26. It is seen that the void data for Fe ﬁt eq. (14) for D down to about 5b, i.e. just over 1 nm. tc for voids in Cu fall below the prediction of eq. (14) when D is small and above it when large, a transition that seems to be related to whether tc is controlled by release from the void of a single partial or both, as explained in

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Fig. 26. Critical stresstc (units Gb/L) versus the harmonic mean of D and L (unit b) for voids in Fe (circles) and Cu (triangles) and Cu precipitates in Fe (hexagons) for crystals at 0 K with various sizes (L: 41.4–83 nm; Lx: 20–60 nm; number of atoms: 2–8 � 106). Lines obtained in Refs [115,133,134] as best ﬁt to tc values obtained in continuum modelling for voids and impenetrable Orowan particles are also shown. (From Refs [116,118,120,127].)

Section 4.2.2. The data for Cu precipitates in Fe follows the continuum prediction for large D but falls well below it for small precipitates that are easily sheared. It is perhaps surprising at ﬁrst sight to ﬁnd that the atomic modelling data and the continuum results are in good agreement across the size range down to about Do2 nm for voids in Fe and 3–4 nm for the other obstacles. The explanation lies in the fact that in the atomic simulation, as in the earlier continuum modelling, large obstacles offer sufﬁcient resistance to dislocation glide at T ¼ 0 K to enable the dislocation line segments at the obstacle surface to adopt dipole alignment at tc. This shape would not be achieved at this stress in the line-tension approximation where self-stress effects are ignored. This work shows that eq. (14) can be used in continuum modelling for nanoscale obstacles that are sufﬁciently strong. The measure of strength and identiﬁcation of the atomic mechanisms involved require atomic modelling, however. The incorporation of MS data into large-scale DD simulations requires further study. As a start, Monnet [136] has shown how the results for t and DE from atomic simulation in Fig. 17(a) can be interpreted and analysed using linear R elasticity. The method decomposes the work done by the applied stress, i.e. V td� where V is the model volume, into three components. R One is the linear elastic strain energy in the model due to the stress, i.e. ðV=2GÞ tdt. Another is the work dissipated as the dislocation overcomes lattice friction. This is extracted from the system in the MS technique, i.e. energy minimisation prevents an increase in E as a dislocation moves

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at the Peierls stress, even though work is done. The third component is the work required for the dislocation to bow whilst pinned by the voids: the energy of the additional line length can be treated in the line-tension approximation. Monnet shows that it is possible to determine the dislocation–void interaction energy by subtracting the elastic and curvature components from DE. Shearing of the void by the dislocation leads to an increase of E equal to the formation energy of the steps created on the void surface and dislocation climb by absorption of vacancies does not seem to play an important role because the energy contribution is insigniﬁcant. 4.3. Temperature effects for voids and precipitates Numerous simulations by MD show that, in general, tc decreases and the dislocation line bows out less in the critical condition with increasing T (see Fig. 27). Mechanisms for some obstacles are similar to those at 0 K. For example, dislocation climb does not occur when an edge dislocation is released from a void in Cu at TW0 K and voids in Fe shrink due to climb at all temperatures [122]. The latter phenomenon was found using the interatomic potential for Fe from Ref. [69] and has now been observed in simulations [137,138] using more recent potentials [69,139]. There is, however, a considerable difference in the T-dependence of tc for

Fig. 27. Critical line shape for a dislocation passing a row of 2 nm voids with spacing 41 nm in Fe at 0, 100 and 300 K. (From Ref. [120])

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Fig. 28. Plot of tc versus T with �_ ¼ 5 � 106 s�1 for (a) voids of different size in Fe (L ¼ 41.4 nm) and Cu (L ¼ 35.4 nm), and (b) Cu precipitates of different size (triangles) and a 2 nm void (circles) in Fe (L ¼ 41.4 nm). (From Refs [116–120].)

voids in Fe and Cu. As shown in Fig. 28(a), tc for iron has a rather gradual decrease over the range from 0 to 600 K whereas it exhibits a signiﬁcant drop in Cu between 0 and 100 K and only a very weak decrease beyond that. (We will return to this feature of Cu in Section 5.5 on SFTs.) The same conclusions were drawn by Hatano and Matsui [140], who used a similar, but smaller MD model of Cu with an edge dislocation and voids in the size range D ¼ 0.6–5 nm with L ¼ 23 nm. They found no signiﬁcant variation of tc for T in the range 100–500 K, a result attributed to the absence of dislocation climb at breakaway. The data for breakaway stress of the leading and trailing partials at 300 K were analysed separately in Ref. [140] and the effects of dissociation were found to be the same as those at T ¼ 0 K discussed in Section 4.2.1. Hatano et al. [141] have reported several distinctive features for the interaction of a screw dislocation with voids with D ¼ 1–6 nm in Cu using an MD model with 920,000 atoms and L ¼ 20 nm. For low T equal to either 10 or 150 K, the t–e plot exhibits only one maximum as the two Shockley partials unpin from a void together, unlike those for the edge dislocation (see Fig. 23), for which the partials have larger spacing. The dependence of tc on D is as in eq. (14) but with A ¼ (1 � u), which is predicted from continuum modelling [133] because segments of the initially straight screw dislocation are pulled towards the edge dipole conﬁguration, for which the energy per unit is proportional to (1 � u)�1. However, the D-dependence of tc changes between 150 and 300 K, and the tc values are higher than expected from the low-T results. This is due to constriction and crossslip of the dissociated dislocation at the void surface. Cross-slip does not occur at lower T, presumably because the thermal energy is insufﬁcient to activate cross-slip at the high strain rate of the simulations. Cross-slip can result in more than one outcome and at 450 K a mechanism which involves double cross-slip and a

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formation of prismatic loop can be activated. This leads to the unusual effect of increased tc at high T. However, caution has to be exercised when simulating a screw dislocation because interactions between the obstacle and its images can occur through the periodic boundaries (Section 2.5). They can affect the breakaway process, particularly when T is high and L is as small as 20 nm. It will be shown in Section 5.5.2 that the behaviour of a screw dislocation between free surfaces in an FCC metal has speciﬁc features. The effects of temperature on dislocation interaction with Cu precipitates in Fe are more complex, for the stability of BCC Cu within a precipitate is dependent on T and D. The free energy difference between the FCC and BCC phases increases with decreasing T but the surrounding Fe matrix tends to stabilise the BCC phase, and so there is an interplay between precipitate size and temperature in the dislocation-induced transformation process. The variation of tc with T for D ¼ 1–5 nm is shown for the edge dislocation in Fig. 28(b). Data for a 2 nm void are included for comparison. tc is small for small precipitates. Their structure remains coherent with the Fe matrix across the temperature range and tc has a weak dependence on T above 100 K. The critical size for the BCC-FCC-like transformation increases with increasing T. Thus, for large precipitates the T-dependence of tc is strong in comparison with voids, to the extent that when T increases to 450 K, a 4 nm precipitate offers only about the same obstacle strength as a 2 nm void. The dislocation induces the structure change at low T but this is suppressed above 450 K and the cutting process is simple shear, as in the small precipitates. This is inferred from the climb proﬁles for D ¼ 4 nm in Fig. 29: the

Fig. 29. [1 1 1] projection of an edge dislocation with b ¼ ½[1 1 1] after breakaway from a 4 nm Cu precipitate in Fe at different temperatures. Climb down is due to absorption of atoms by the dislocation and creation of vacancies inside the precipitate, whereas deviation up is due to the opposite process.

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phase transformation is accompanied by the creation of vacancies and extra atoms in the precipitate, resulting in dislocation climb seen for To450 K. The size dependence of the Cu transformation mechanism has now been conﬁrmed for screw dislocations by MD simulation of screw–precipitate interaction at 10 K under constant t [142]. It was observed that the phase transformation occurred for DW1.8 nm and that when D exceeds 2.5 nm the dislocation bypass mechanism becomes Orowan looping due to the coherency loss of the precipitates. Values of tc were not obtained. The simulations suggest that the yield stress of an under-aged or neutronirradiated Fe–Cu alloy containing small, coherent precipitates should have a weak T-dependence, whereas the dependence should be stronger in an overaged or electron-irradiated alloy where the population of coherent precipitates has larger size [127]. More direct experimental evidence in support of the simulation predic tions has been obtained by Lozano-Perez et al. [143] by utilising the crystallography of the BCC-9R martensitic transformation. Planes of atoms in twinned 9R precipitates exhibit characteristic ‘herring-bone’ fringe contrast when viewed along a /1 1 1S direction of the Fe matrix in a high-resolution electron microscope (HREM). The angle, a, between fringes in neighbouring twin bands is approximately 1291 immediately after the transformation, but relaxes during annealing to 1211 to reduce the strain energy [123]. Samples of a Fe-1.3 wt%Cu alloy were aged at 550 1C before cooling to room temperature, in order that precipitates larger than about 5 nm would transform to 9R whilst smaller precipitates would retain the BCC structure. One set of samples was deformed by bending at room temperature and both sets were then annealed at 400 1C to allow transformed precipitates to relax. HREM foils were prepared at �60 1C so that any remaining untransformed precipitates should transform to 9R. Angle a was then measured for precipitates in both sets of samples. The results are shown in Fig. 30, from which it is seen that precipitates in the undeformed alloy have a relaxation threshold of about 4–5 nm, while all the precipitates in the deformed samples appear to be relaxed. High-chromium (Cr) ferritic/martensitic steels offer another alloy system where coherent precipitates harden the material during ageing and/or irradiation. Cr-rich au phase separates from the a phase and forms a ﬁne dispersion of nanoscale obstacles to dislocation motion, e.g. Refs [144–146]. Unlike Cu in Fe, Cr is stable in the BCC structure, but it has a larger elastic modulus than Fe and au precipitates are expected to repel dislocations, rather than attract them. Terentyev et al. [147] have simulated the interaction of an edge dislocation with a row of spherical Cr precipitates in a matrix of either pure Fe or Fe–10at.% Cr, the latter being used to compare the obstacle contributions of precipitates and a solid solution. The MS method (T ¼ 0 K) with the model in Ref. [42] described in Section 2.3 was used with the potential from Ref. [148]. The precipitate spacing was 14 nm for small precipitates (D ¼ 0.6 or 1.2 nm) and 28 nm for larger ones. Examples of the critical line shape and corresponding stress–strain curves obtained for several values of D are shown in Fig. 31. (The stress in the plot is the applied stress minus 80 MPa, which is the Peierls stress at 0 K for the edge

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Fig. 30. Angle a between herring-bone fringes in adjacent twin-related bands in Cu precipitates in Fe, plotted against D for deformed and undeformed samples. (Reprinted from Ref. [143] with permission from Taylor & Francis Ltd., http://www.informaworld.com.)

Fig. 31. Interaction of an edge dislocation with an au Cr precipitate in Fe. (a) Conﬁguration of the dislocation in the ð1 1� 0Þ slip plane at tc for precipitates of different size D, and (b) the corresponding t versus e plots. (Reprinted from Ref. [147] with permission from Elsevier Ltd. http://www.sciencedirect. com/science/.)

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dislocation in pure Fe with the potential used (Fig. 8(a)). It is seen that the dislocation is repelled by a precipitate but ﬁnally shears it at tc. No defects are formed inside a precipitate and the dislocation does not acquire jogs as a result of the shearing process. The stress reaches the critical value just as the dislocation enters the precipitate in both the pure Fe and Fe-10% Cr matrices (see, e.g. Fig. 3 in Ref. [147]). tc is plotted in Fig. 32(a) in the form predicted by continuum modelling of impenetrable particles with dislocation self-stress included (eq. (14)). The atomistic and continuum values are in good agreement at large D, for which the dislocation sidearms at the obstacle are pulled into near-screw alignment at tc, but not when D is small, i.e. when jcc01 (compare Fig. 31 and Fig. 21). Fig. 32(a) also contains estimates of tc derived from two theoretical treatments for shearable particles using the approximation of constant line tension (equal to Gb2/2). For the chemical strengthening (CS) mechanism, Terentyev et al. used MS to calculate the additional (compared with pure Fe) applied force required for the dislocation to shear a Cr precipitate by ½[1 1 1] on the ð1 1� 0Þ plane. The other estimate uses the shear modulus difference (SMD) between Fe and Cr, for which a formula from Ref. [149] was used since the Russell–Brown analysis applies to alloys in which the precipitate modulus is less than that of the matrix. Both treatments underestimate tc except when D is small. When the two contributions are added, however, the total tc they predict is much closer to the simulation data. Fig. 32(b) compares the dependence of tc on the harmonic mean of L and D for matrices of pure Fe and the Fe-10%Cr solid solution. tc is much higher in the latter case. The difference, labelled DtFeCr, is almost constant and independent of D. It is approximately 250 MPa and equals the stress required for continued edge dislocation glide in the Fe-10%Cr solution without precipitates. In other words, the two contributions

Fig. 32. Plot of tc (units Gb/L) versus Cr precipitate diameter at T ¼ 0 K. (a) Comparison of values obtained by simulation with theoretical estimates based on either eq. (14) or models of chemical strengthening (CS) and shear modulus difference (SMD). (b) The effect on tc of Cr in solution in the Fe matrix. (Reprinted from Ref. [147] with permission from Elsevier Ltd. http://www.sciencedirect.com/ science/.)

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of solution and precipitate strengthening combine by linear addition in this system. (This is consistent with conclusions on the superposition rule by some workers, but not others. For a recent discussion see Ref. [150].) In contrast to the shearable obstacles above, precipitates in some materials are not coherent with the surrounding matrix, e.g. in overaged or dispersionstrengthened alloys. In these, a dislocation either stays in its slip plane by bowing between the obstacle and leaving a loop around it (‘Orowan’ mechanism), or screw segments cross-slip to allow bypass on a different slip plane (‘Hirsch’ mechanism). (See, e.g. Refs [2,89,151].) The interaction between an edge dislocation and a rigid, impenetrable particle has been simulated by Hatano [152]. The particle was created in a Cu crystal by deﬁning a spherical region in which the atoms were immobile. The MD model used the interatomic potential from Ref. [121] and was strained at a constant rate of 7 � 106 s�1 at T ¼ 300 K. With reference to Fig. 3, the outcome was found to depend on whether strain was applied by moving either upper block B alone or A and B by equal but opposite amounts. The Hirsch mechanism was found to operate in the former method. In the sequence shown in Fig. 33: (a) the dislocation bows round the obstacle to form a screw dipole; (b) the screw segments cross-slip onto inclined ð1 1 1� Þ planes at tc; (c), (d) they annihilate by double cross-slip, allowing the dislocation, now with a double superjog, to bypass the obstacle; (e) a prismatic loop with the same b is left behind; and (f) the dragged superjogs pinch-off as the dislocation glides away, creating a loop of opposite sign to the ﬁrst on the right of the obstacle. tc varies with D and L as predicted by the continuum modelling that led to eq. (14), but is over three times larger in magnitude. Hatano argues that this could arise from either higher stiffness of a dissociated dislocation or a dependence of tc on the initial position of the dislocation. It is also possible that the requirement for the dislocation to constrict

Fig. 33. Visualisations of an edge dislocation bypassing an impenetrable particle of 3 nm diameter in Cu at 300 K. Atoms in the dislocation cores are shown, but those in the stacking faults between the partial dislocations are not visualised. See text for details. (Reprinted with permission from Ref. [152]. r (2006) by the American Physical Society.)

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and the absence of a component of applied shear stress on the cross-slip plane results in a high value of tc. When the model was loaded by displacing both blocks A and B, only Orowan looping occurred, although tc was almost the same. Hatano speculates that the presence of stress asymmetry in the vicinity of the obstacle as the strain pulse travels from the top of the model promotes cross-slip and the Hirsch mechanism. The modelling issues raised by this work require further study.

4.4. Bubbles and loose clusters of vacancies Despite the potential importance of He bubbles, i.e. He-ﬁlled cavities, for the mechanical properties of structural materials in fusion energy applications, there have been few atomic-level studies of dislocation–bubble interaction. This is explained in part by difﬁculties in developing suitable interatomic potentials for metal–He systems, for those in use have deﬁciencies in describing He-defect properties. Nevertheless, some MD simulations have been performed [137,153,154]. A small model of Fe containing about 0.5 � 106 atoms was used to study interaction between an edge dislocation and a row of 2 nm cavities containing He. He:vacancy ratios of up to 5 were simulated for T between 10 and 700 K. It was found that tc has a non-monotonic dependence on the He:vacancy ratio, dislocation climb increases with this ratio and interstitial defects are formed in the vicinity of the bubble. However, the equilibrium properties of He-ﬁlled bubbles were not investigated and interpretation of these results is not straightforward. Vacancies do not always agglomerate in compact arrangements such as voids or dislocation loops. Positron annihilation experiments on neutron-irradiated Fe indicate that vacancies cluster together in loose arrangements in the core of dis placement cascades [154] and this is supported by MD modelling [12,155,156]. Inter action between dislocations and vacancies in diffuse clusters is therefore relevant to the vacancy contribution to matrix hardening in irradiated Fe. The effect on tc and dislocation climb of changing the arrangement of 59 vacancies from a void with D ¼ 1 nm to a loose cluster has been investigated for the case L ¼ 41.4 nm by Osetsky and Bacon [120] using the interatomic potential for Fe from Ref. [70]. Clusters were created in spherical zones centred on the slip plane with vacancy concentration, cv, equal to 10, 15, 25 or 100%. The vacancies were placed randomly on lattice sites and three different arrangements were simulated for the 15% condi tion. The crystals were annealed at 600 K for 1 ns before shear strain was applied at a constant rate of either 2 or 5 � 106 s�1 at temperatures in the range from 1 to 600 K. tc versus T is plotted for all the cv values in Fig. 34(a). Although tc for a loose cluster is smaller than that for a void with the same number of vacancies, the difference is small for the higher cv values. For instance, the T-dependence of the plot for cv ¼ 25% follows that for the void with only a slight reduction in stress (15–20 MPa). The tc values for the three clusters with cv ¼ 15% show variability associated with the differences in the vacancy migration during interaction, but the trend in the mean value of tc again follows that for the void. The stress values for the 10% cluster are the lowest, although the unexpectedly high value for 450 K again

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Fig. 34. (a) Plots of tc versus T for an edge dislocation breaking away from a row of vacancy clusters in Fe. All clusters contain 59 vacancies at concentrations from 10 to 100%. (b) [1 1 1] projection of the shape of a ½[1 1 1] edge dislocation after breakaway at 300 K from the vacancy zones with cv values indicated. The extra half-plane is in the upper part of the ﬁgure in each case. (From Ref. [120].)

shows the signiﬁcance of the vacancy rearrangement. We conclude that although loose clusters of vacancies are weaker obstacles than voids, they still offer signiﬁcant resistance to slip and may make an important contribution to matrix hardening in the temperature range where they are stable. The vacancy absorption by climb of the edge dislocation seen on breakaway from a void (Section 4.3) also occurs for clusters. The visualisations of the core atoms in Fig. 34(b) show the shape of the ½[1 1 1] edge dislocation along the bottom of its extra half-plane after passing through a cluster at 300 K. It is clear from the extent of climb that vacancy absorption is a more efﬁcient process for zones where the vacancies are loosely packed. 4.5. Conclusions It is clear from the simulation studies of dislocation interaction with obstacles such as voids and precipitates reviewed in this section that a treatment based on linear elasticity can provide a good approximation of processes and strengthening effects that occur in some cases. In these, the elasticity approximation has been shown to be valid, even for obstacles only a nanometre or so in size. A key requirement for validity is that dislocation self-stress must be allowed for in the elasticity approach because interaction of nearby segments of the same dislocation can determine how an obstacle is overcome. Self-stress is incorporated in modern DD codes, see e.g. Ref. [1]. The agreement between continuum and atomistic modelling is obviously best for T ¼ 0 K, i.e. when minimum potential energy deﬁnes the equilibrium conﬁguration. The results of the two methods are also closer for the BCC structure, in which the dislocation is not dissociated, than in, say, Cu, where it is. This suggests

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that the dissociated nature of the dislocation core should be taken into account in DD modelling of many FCC metals. MS and MD simulations have revealed effects that were not foreseen in earlier continuum treatments. One is climb of an undissociated dislocation by absorption or emission of point defects as it leaves an obstacle. From simulations to date, this seems to be a general phenomenon for voids and clusters of vacancies. Another effect is strengthening due to a dislocation-induced phase transformation of a metastable precipitate. The predictions that have arisen from atomic-level simulation are now found to be consistent with accurate experimental measure ments on nanoscale Cu precipitates in Fe. This points to the power of simulations provided the interatomic potentials employed are sufﬁciently reliable.

5. Obstacles having dislocation character 5.1. Dislocation loops and SFTs Although loops and SFTs can be created as a result of dislocation reactions, this chapter is concerned with those that form in irradiated or quenched metals, i.e. those that nucleate and possibly grow in a supersaturation of point defects. SFTs have a vacancy nature, whereas dislocation loops can form from vacancies or SIAs. The shape, Burgers vector and nature of these extended defects have been widely studied over the past 50 years, particularly by transmission electron microscopy (TEM), and are reasonably well understood (e.g. Refs [2,3]). We denote the Burgers vector of a dislocation loop by bL. Perfect loops have bL equal to a lattice translation vector, e.g. ½/1 1 0S in the FCC structure and ½/1 1 1S or /1 0 0S in the BCC structure. If the metal has a stable stacking fault of low enough energy on the plane in which the loop forms, the loop may enclose a fault and have a partial Burgers vector. Stable faults are not known to exist in the BCC metals where all loops are perfect. In FCC metals with low fault energy on {1 1 1} planes, faulted (‘Frank’) loops with bL ¼ 1/3/1 1 1S are common. The relative stability of faulted and perfect loops depends on fault energy and loop size, with the result that Frank loops can unfault to become perfect loops when large enough. This process depends on the loop nature. Vacancy loops contain an intrinsic fault corresponding to a missing plane in the ternary stacking of {1 1 1} planes. It can be removed by a single Shockley partial sweeping across the loop, e.g. on the (1 1 1) plane: 1 1 1 ½1 1 1� þ ½1 1 2� � ! ½1 1 0�. 3 6 2

(15)

Frank loops formed by SIAs contain an extrinsic fault, i.e. an extra plane in the {1 1 1} stacking sequence, and this is removed when two Shockley partials sweep the planes adjacent to the fault, e.g. 1 1 � þ 1 ½1� 2 1� � ! 1 ½1 1 0�. ½1 1 1� þ ½2 1� 1� 3 6 6 2

(16)

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The partials may glide either together as a double core (a ‘D-Shockley’ partial) or separately. It will be seen in Section 5.3.2 that these unfaulting reactions can be caused by interaction with dislocations. Frank loops with lowest energy have hexagonal shape with side orientations that depend on the material, e.g. /1 1 0S directions in Cu [157] and /1 1 2S directions in austenitic steels and Ni [158,159]. The lowest energy shape of perfect ½/1 1 0S loops is a rhombus, with loop sides lying on the glide prism formed by the two sets of {1 1 1} planes that contain bL [13]. In metals such as Ag, Au and Cu with relatively low stacking fault energy, vacancy Frank loops may not unfault but dissociate into an SFT with its six sides consisting of stair-rod partial dislocations. The vacancy content of the loop is thereby redistributed over the four faulted {1 1 1} faces of the tetrahedron [160,161]. SFTs formed in this way are commonly observed after quenching and ageing [162– 164]. Nanometre-scale SFTs can also be created directly in displacement cascades in irradiated FCC metals, see, e.g. experimental observations [165] and MD simulations [166] on Cu. Large SFTs are also formed during fast deformation in Al, a material with a high fault energy [167]. The most commonly observed loops in irradiated BCC metals are interstitial in nature and have bL ¼ ½/1 1 1S (e.g. Ref. [19]), which is consistent with the fact that dislocation energy is proportional to b2 and ½/1 1 1S is the shortest lattice vector. These loops are also the most common type created in MD modelling of displacement cascades [12]. MS simulations indicate that the most stable loop shape is a hexagon with sides along /1 1 2S directions. Experiments on neutron-irradiated Fe and ferritic alloys show that interstitial loops with bL ¼ /1 0 0S also form in these materials. Their fraction of the loop population increases with irradiation temperature [21,168,169]. They also form in electron- and ion-irradiated Fe [170, 171]: their sides lie in /1 0 0S directions. /1 0 0S loops do not have the shortest bL and various explanations for their occurrence have been proposed. However, recent calculations based on anisotropic elasticity theory have shown that changes in the elastic constants of Fe as T is increased towards the a–g transformation reverse the order of stability of ½/1 1 1S and /1 0 0S loops, so that the latter are favoured at high T [172]. Thus, in order to account for the whole temperature range of interest, both types of loops are considered here. Perfect dislocation loops are glissile along their glide prism, in principle. Small loops containing up to B100 SIAs have the structure and properties of bundles of crowdions with their axis parallel to bL. Consequently they exhibit rapid thermally activated 1D motion. At high enough temperature, bL can ﬂip from one direction to another by thermal activation, leading to 3D diffusion composed of 1D motion separated by changes of direction [173–175]. The critical stress and atomic processes involved in the interaction of gliding dislocations with loops and SFTs have been simulated using the PAD model (Section 2.3) and the results, including effects of obstacle size, dislocation–obstacle geometry, T and �_ are presented in Sections 5.3–5.5. The geometry for loops in BCC metals (Section 5.3) is the same as that in Fig. 16(a) for obstacles in the form of voids and precipitates. The simulation cell used for the FCC metals is shown schematically in Fig. 16(b). In both cases, the ‘positive’ edge dislocation has its extra

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half-plane above the slip plane in the ﬁgure and the positive direction of the righthanded screw dislocation is the x direction. The Thompson tetrahedron [2,3] drawn in Fig. 16(b) will be used to label the Burgers vectors and slip and habit planes of the FCC simulations. The perfect dislocation has b ¼ AC with an initial line orientation that depends on its character. Upon initial relaxation, the dislocation dissociates on plane ABC into partials with Burgers vectors Ad and dC. The variety of reactions observed in simulations to date is too large to be described comprehensively in this review. However, it is possible to classify them in a qualitative way according to some common characteristics. Four main types of interaction have been identiﬁed, depending on the character of the dislocation and on whether the obstacle is unchanged, modiﬁed or absorbed by the dislocation. In addition, some reactions occur which are speciﬁc to the type of defect, i.e. SFT or loop. These reaction categories are summarised in Section 5.2 and examples for loops and SFTs are presented in Sections 5.3–5.5. 5.2. Classiﬁcation of main reactions 5.2.1. Reaction R1: the obstacle is crossed by the dislocation and both are unchanged In this, the defect is sheared by the dislocation but is fully reconstructed after the dislocation unpins. This reaction has been observed for both screw and edge dislocations with SFTs (see Fig. 51 below) and Frank loops (see Fig. 36 below) in FCC metals. It has also been observed for edge dislocations interacting with large ½/1 1 1S loops that have bL inclined to the glide plane in BCC metals (see Fig. 44 below). Steps corresponding to b form on the defect surface (1 step on loops, 2 ledges on SFTs) but are mobile and disappear at a border of the defect. The obstacle and the dislocation thus remain unchanged and tc is low. The requirements are that the dislocation slip plane should be close to an SFT apex or loop edge, T should be low and vD, i.e. �_, high. 5.2.2. Reaction R2: the obstacle is crossed and modiﬁed and the dislocation is unchanged This occurs in two instances. First, as a modiﬁcation of reaction R1 in FCC metals when the steps are stable because of a relatively long distance to either the SFT apex or the loop edge. Multiple reaction with dislocations on the slip plane shears the obstacle into separate parts (see Fig. 52 below for SFTs). Second, some /1 0 0S loops in BCC metals are transformed by an edge dislocation into mixed /1 0 0S/ /1 1 1S loops (see Fig. 48 below). 5.2.3. Reaction R3: partial or full absorption of the obstacle by an edge dislocation that acquires a double superjog This is seen when SFTs with their base above the slip plane of a positive edge dislocation are absorbed as a pair of glissile superjogs on the dislocation (see Fig. 53 below). The superjogs can thus be dragged away by the dislocation. The probability of this process increases with T and SFT size and decreasing distance between the

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base and slip plane. The reaction occurs for loops in two situations. First, glissile loops with bL inclined to the dislocation glide plane are attracted towards the dislocation. bL of small loops changes on contact with the dislocation and becomes that of the dislocation itself (see Fig. 41 below). Large loops react with the dislocation to form a common segment with different b, but this segment glides across the loop to convert bL to the Burgers vector of the dislocation (see Fig. 42 below). Second, sessile and glissile loops away from the glide plane with bL parallel to it may also ﬂip under the torque of the dislocation and be absorbed by it. This reaction occurs when the loops are small and close to the slip plane, and is favoured by high T and low vD. Another variant of R3 arises when the slip plane of an edge dislocation intersects either a Frank loop (Fig. 37) or a BCC /1 0 0S loop (see Fig. 47 below). The dislocation absorbs part of the loop after transforming the bL of this part into its own. 5.2.4. Reaction R4: temporary absorption of part or the entire obstacle into a helical turn on a screw dislocation This is the most common reaction for a screw dislocation (see Fig. 38 for Frank loops, Fig. 45 for BCC ½/1 1 1S loops and Fig. 54 for SFTs). The screw dislocation absorbs an SFT or a loop, either partially or completely, to form a helical turn by a succession of cross-slip events, a process promoted by high T and low vD. A helical turn can only glide in the direction of its b and so cannot be dragged away by the screw dislocation. Moreover, it is a very effective pinning agent because it is not localised but extended along the length of the screw dislocation. The turn has to close up in order for the dislocation to unpin, with the result that a loop with the Burgers vector of the screw dislocation is left behind. 5.2.5. Other reactions Other reactions associated with particular situations. (i) Reaction R1SFT: the dislocation is unchanged but an SFT is reduced in size and small clusters of vacancies form. This variant of R1 occurs when the slip plane of the dislocation (screw or edge) coincides with the base of an SFT. A small portion of the SFT is detached, usually as a chain of vacancies, without creating a superjog on an edge dislocation. An example of R1SFT can be seen in Fig. 11 of Ref. [176]. (ii) Reaction R3drag: the obstacle is dragged by the dislocation and both remain unchanged. This occurs in BCC and FCC metals when a glissile loop has bL parallel to the slip plane of an edge dislocation but does not intersect it. An SIA loop is pushed by the compressive stress of the dislocation above the slip plane or dragged by the tensile stress below. This results in a loop sweeping effect, ﬁrst analysed using elasticity theory in Refs [177,178]. MD simulations show that as applied stress increases a terminal velocity is reached at which the force due to the dislocation cannot overcome the loop friction and the loop is left behind. Another example of drag is seen in FCC metals when bL of a loop that intercepts the glide plane ﬂips to a direction in the glide plane

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but not that of b of the dislocation. The loop is then attached to the gliding dislocation by a junction segment (see Fig. 35 below) and is dragged away. Note that reactions R3drag, R3 and R4 are potentially important as mechanisms for clear band formation in irradiated metals because all or part of the defects are carried away by the dislocations. R3drag and R3 have a direct effect since the edge dislocation can either drag/push the point defect cluster or absorb it into a double superjog. R4 can move defects but only by the motion of the helical turn along a screw dislocation line. 5.3. Loops in FCC metals 5.3.1. Perfect interstitial loops In the ﬁrst detailed study of edge dislocation–loop interaction using large-scale MD, Rodney and Martin [51,76] used the interatomic potential from Ref. [179] to simulate Ni. Small glissile loops containing from 4 to 37 SIAs with bL ¼ BD (Fig. 16(b)) inclined to the dislocation glide plane were considered. A loop in this conﬁguration can glide along its glide cylinder and react with the dislocation by reactions R3 and R3drag above. There exists a critical distance of a few nanometres within which the glissile loop is spontaneously attracted to, and captured by, the dislocation. Capture occurs at one of the Shockley partials of the dislocation after a rotation of bL to a /1 1 0S direction in the glide plane. An example is shown in Fig. 35, where a 37-SIA loop is attached to the core of the Ad partial after bL changed from BD to BA, allowing for an energetically favourable junction reaction: BA þ Ad ¼ Bd.

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The reorientation is stochastic and in some cases bL ﬂips to AC and the loop is absorbed as a pair of superjogs. Loops absorbed in this way are not obstacles to the motion of the dislocation but are dragged in the direction of their Burgers vector by the moving dislocation. Drag of small dislocation loops by gliding edge dislocations has also been observed in MD simulations of Cu [77,180] for situations where the loop does not intersect the dislocation glide plane and has bL parallel to it (reaction R3drag).

Fig. 35. Visualisation of dislocation core atoms after trapping of a 37-SIA loop at one of the Shockley partials in Ni at 100 K. (From Ref. [51].)

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If placed within a few nanometres of the glide plane, such loops are trapped in the elastic ﬁeld of the passing dislocation and dragged at speeds of up to several 100 ms�1, until at high enough velocity the dislocation breaks free. The friction due to a dragged loop was found to add an extra contribution, BL, to the dislocation friction coefﬁcient, i.e. the coefﬁcient in eq. (6) becomes Beff ¼ B þ BL. For loop sizes of a few tens of SIAs, BL is of the order of 100 mPas and so larger than B (in the range 1–100 mPas). It increases with loop size, T and loop density 1/L. A physical model for BL was derived in Ref. [77] by noting that BLL equals the loop mobility, mL, and that mL ¼ (kBT)/(DLL), where DL is the loop diffusivity. By using data obtained by MD for DL of loops undergoing one-dimensional motion under the inﬂuence of temperature alone [174], the values of BL predicted by the model are consistent with those from the simulations of drag by a dislocation. 5.3.2. Interstitial Frank loops Several interaction conﬁgurations have to be considered, depending on the loop Burgers vector (bL ¼ aA, bB, cC or dD), and the dislocation character (edge or screw), Burgers vector direction (b ¼ AC or CA) and glide direction (the dislocation may approach the loop from one side or the other). Eight sets of non-equivalent interactions were simulated at T ¼ 600 K in Ref. [46]. The loops were hexagonal with /1 1 2S sides, in agreement with TEM observations of irradiated austenitic steels. The case of screw dislocations has also been studied at T ¼ 0 K and T ¼ 100 K for /1 1 2S- and /1 1 0S-sided loops [45,181]. Potentials for Ni [179,182,183] or Cu [183] were used to provide models with stacking fault energy within the range for austenitic steels. In the simulations in Ref. [46], the loop diameter, D, was 6 nm and L ¼ 50 nm, so that the defect density, N, corresponded to pﬃﬃﬃﬃﬃﬃﬃ values ﬃ observed experimentally, i.e. inter-loop distance in glide plane 50 nm � 1= ND with D ¼ 6 nm and NB5 � 1022 m�3. The interaction mechanisms found cover all reactions presented in Section 5.2. Loops are absorbed only after they have been unfaulted, bL usually being transformed into that of the dislocation. Unfaulting always starts by a cross-slip event on the dislocation and so edge dislocations, which can cross-slip only if bent to become screw, shear Frank loops in most conﬁgurations. Screw dislocations, on the other hand, unfault Frank loops in most conﬁgurations when cross-slip is promoted by high T (600 K). At low T (100 K), the shape of the Frank loops matters. Loops with /1 1 0S sides are more easily absorbed than those with /1 1 2S sides because in the former case, four of the six sides lie in {1 1 1} planes, allowing the dislocation to cross-slip and recombine with them, while in the latter case, no sides are in {1 1 1} planes. It has also been shown that the presence of a second dislocation in the simulation cell, which mimics an embryonic pile-up, favours cross-slip and unfaulting by increasing the resolved shear stress in cross-slip planes [46]. Three illustrative examples are now presented. R1/R2: loop shearing. Fig. 36 shows an AC edge dislocation gliding in the þx direction on the plane ABC and reacting with a cC Frank loop on plane ABD. The dislocation simply cuts through the loop at a low stress of 95 MPa, forming a stable step on the loop, visible in (b). When the edge dislocation re-enters through the

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Fig. 36. Interaction of an edge dislocation with a Frank loop in an attractive conﬁguration in Cu at 600 K leading to shear of the loop (reaction R2): snapshots shown at (a) 55 MPa, (b) 95 MPa, (c) 95 MPa and (d) 95 MPa. (From Ref. [46].)

periodic boundaries and shears the defect again, the double step that is formed becomes mobile in the loop surface and annihilates on the loop border, as in (c), thus reforming a faulted loop without damage seen in (d). R3: loop absorption and drag. The process shown in Fig. 37 has a negative edge dislocation (b ¼ CA) gliding in the þx direction to react with a cC Frank loop. The loop initially repels the dislocation, but when t ¼ 60 MPa the dislocation contacts the loop (a), then bends and constricts to acquire a screw segment that cross-slips (b). This segment creates a D-Shockley, i.e. two Shockley partials on adjacent {1 1 1} planes, by the reaction cC þ CA ¼ cA. The loop starts to unfault as the cross-slipped segment rotates around the upper part of the loop (c). The second dislocation arm acquires screw character and also cross-slips, and when the two cross-slipped segments meet, the upper part of the loop is absorbed by the line (d), which glides away at t ¼ 130 MPa, leaving the lower, faulted part behind as seen in (e, f).

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Fig. 37. Interaction of an edge dislocation with a Frank loop in a repulsive conﬁguration in Cu at 600 K leading to the partial loop absorption (reaction R3): snapshots shown at (a) 65 MPa, (b) 105 MPa, (c) 130 MPa, (d) 130 MPa, (e) 130 MPa and (f) 130 MPa with a different viewing angle. (From Ref. [46].)

R4: loop absorption into a helical turn. Fig. 38 shows six stages in the interaction of an AC screw dislocation gliding in the þy direction with an aA Frank loop with /1 1 0S sides as t is increased from 0 to 400 MPa at T ¼ 0 K. The dislocation gliding on plane ABC in (a) constricts at the point where it ﬁrst contacts the loop and then cross-slips in (b) onto ACD to combine with a loop side to form a D-Shockley partial by the reaction AC þ aA ¼ aC. This partial dislocation unfaults the loop by sweeping over its surface as the AC dislocation cross-slips at the remaining loop sides, see (c–e). This process converts the loop into a helical turn on the screw dislocation (e, f). As mentioned in Section 5.1, the helical turn is an obstacle for glide of the screw dislocation and it offers high resistance because, as seen in (f), it is not localised but expands over the length of the dislocation in order to minimise its line energy. Fig. 39 shows that with increasing t the dislocation escapes by the bowing of the segments dissociated in ABC planes, thereby forcing the helical turn to close on itself so that a perfect interstitial loop with bL ¼ AC is left behind. Absorption is controlled by cross-slip reactions on the screw dislocation. Thus, at high T (600 K),

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Fig. 38. Interaction of a screw dislocation with a /1 1 0S-sided Frank loop in Ni at T ¼ 0 K (reaction R4). (From Ref. [181].)

such events have a high probability and occur in all conﬁgurations with /1 1 0S- and /1 1 2S-sided loops except one conﬁguration (AC screw dislocation gliding in –x direction and reacting with an aA /1 1 2S-Frank loop). At 300 K loops with /1 1 0S sides are absorbed but /1 1 2S-sided loops are sheared because, as mentioned above, their sides do not belong to {1 1 1} planes. The critical stress tc for a screw dislocation to overcome rows of Frank loops with different values of spacing, L, at 100 K is plotted in Fig. 40. The data are compared with dashed lines showing the stress given by continuum modelling of the Orowan process, i.e. eq. (14) with A ¼ (1 � n), D ¼ 0.615 and effective size D close to the real size of the loops [45]. The good agreement shows that tc is determined by the condition that the screw dislocation has to bow out in order to unpin from these strong obstacles. We note that upon unpinning, the dislocation is re-emitted in a {1 1 1} plane parallel but different from its initial glide plane. This is due to the 3-D structure of the helical turn, which contains no segment in the initial glide plane of the dislocation. The process shown in Fig. 39 clearly depends on the periodic boundary conditions along the dislocation line, but dislocation re-emissions in new glide planes have also been observed with multiple loops that break the periodicity [46] as well as in DD simulations without periodic boundaries [22]. This effect is believed to play a role in the broadening of clear bands in irradiated metals since it allows screw dislocations to change glide plane through a mechanism analogous to double cross-slip [22,46].

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Fig. 39. Unpinning of a screw dislocation from a helical turn (reaction R4) at different times (in ps) in Ni under t ¼ 600 MPa at T ¼ 100 K: (a) 18, (b) 24, (c) 25.2 and (d) 27. (From Ref. [45].)

5.4. Interstitial loops in BCC metals 5.4.1. ½/1 1 1S loops Depending on the orientation of bL, two cases can be distinguished in the interaction of a ½[1 1 1]ð1 1� 0Þ edge dislocation with ½/1 1 1S loops. First, two of the four bL orientations, namely ½[1 1 1] and ½½1 1 1� �, are parallel to the dislocation glide plane and these loops can be dragged (reaction R3drag in Section 5.2). This situation was considered in Refs [180,184] for hexagonal loops containing either 37 or 61 SIAs, with centres between 3 and 9 nm below the glide plane of the positive edge dislocation. The process is as described in Section 5.3.1. The second case concerns loops with bL ¼ ½½1 1� 1� and ½½1� 1 1� inclined to the glide plane. (They are equivalent for the ½[1 1 1]ð1 1� 0Þ dislocation.) Hexagonal loops with /1 1 2S sides and centres placed initially below the glide plane have been simulated for a variety of numbers of SIAs: 37 and 331 [184], 99 [185] and 169 [186]. Sensitivity of results to the interatomic potential was examined by using the potential of Ref. [69] in Ref. [184] and that of Ref. [70] in Ref. [186]. As mentioned in Section 5.1, ½/1 1 1S loops move easily by one-dimensional glide, and if within a few nanometres of a gliding edge dislocation, slip on their glide prism under

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Fig. 40. Variation of tc, in units of G (taken to be 74.6 GPa), with b/L for hexagonal Frank loops with either /1 1 0S sides (triangles) or /1 1 2S sides (squares) in Ni at T ¼ 100 K. Dashed lines are the Orowan stress predicted from elasticity theory (eq. (14)). (From Ref. [45].)

attractive elastic interaction and react on contact with the dislocation. The outcome depends on both the defect size and, for large loops, the temperature. Small loops (37 SIAs) undergo an R3 reaction, for bL rotates spontaneously to that of the dislocation and the loop is thereby absorbed and moved away as a double superjog. tc falls from 51 MPa at 0 K to 13 MPa at 450 K for L ¼ 41 nm. Two stages in this process at T ¼ 0 K are shown in Fig. 41. These interstitial loops are weaker obstacles than voids with a similar number (27 or 59) of vacancies (Figs 18 and 28). Large loops (Z99 SIAs) are unable to change bL in this way, presumably because the energy barrier is too large. Two interaction mechanisms are observed, depending on T. An example for high T is provided in Fig. 42 for the case of a 331-SIA loop at T ¼ 300 K and �_ ¼ 20 � 106 s�1 . When the loop contacts the dislocation, a [0 1 0] junction segment forms by the energetically favourable reaction 1 1 ½1 1 1� � ½1 1� 1� ¼ ½0 1 0�, 2 2

(18)

and the remainder of the loop retains its ½½1 1� 1� Burgers vector, as shown in (b). The [0 1 0] segment is sessile in the ð1 1� 0Þ slip plane, but glissile in the inclined (1 0 1) plane. However, it is pinned at its ends by the junctions with the ½[1 1 1] and ½½1 1� 1� lines, and is too short to bow under the applied stress. As t increases, the

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Fig. 41. Visualisations of the spontaneous glide and absorption process of a 37-SIA loop on an edge dislocation in Fe at T ¼ 0 K (reaction R3). (Reprinted from Ref. [184] with permission from Taylor & Francis Ltd., http://www.informaworld.com.)

Fig. 42. Visualisation of the spontaneous glide and transformation process of a 331-SIA loop in Fe at T ¼ 300 K and �_ ¼ 20 � 106 s�1 (reaction R3). (Reprinted from Ref. [184] with permission from Taylor & Francis Ltd., http://www.informaworld.com.)

½[1 1 1] line segments pinned at the junctions are pulled into the screw orientation in (c) and, assisted by the applied force on the [0 1 0] segment, cross-slip on ð1 0 1� Þ planes, thereby allowing the [0 1 0] segment to move downwards on its (1 0 1) glide plane, as in (d). Slip of this segment transforms the Burgers vector of the remainder of the loop by the reaction 1 � 1 ½1 1 1� þ ½0 1 0� ¼ ½1 1 1�, 2 2

(19)

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and results again in the formation of a pair of superjogs on the gliding dislocation (e). Large ½½1 1� 1� loops at high enough temperature are thus absorbed and dragged as in reaction R3. tc is 220 MPa and the jogged dislocation line continues to glide at t ¼ 12 MPa. The same reaction and transformation process occurs for other strain rates at 300 and 450 K, and with the sense of the applied strain reversed, i.e. the dislocation approaching the loop from the other side. The dependence of tc on T and �_ is illustrated by the data plotted in Fig. 43. In contrast to this, the reaction with large ½½1 1� 1� loops at low T is of R1 type. The process is illustrated in Fig. 44 for a 331-SIA loop at 100 K, for which tc ¼ 290 MPa. A [0 1 0] segment again forms and a screw dipole is pulled out of the dislocation as at high T. However, the mobility of the [0 1 0] segment is strongly reduced at low T and the screw dipole reaches a longer extension and annihilates by cross-slip before the [0 1 0] segment becomes mobile. The ½½1 1� 1� loop is thus restored and left behind when the dislocation breaks away. The critical dipole length and tc increase when T decreases further – e.g. tc ¼ 530 MPa at 1 K for a 169-SIA loop – because of the low mobility of the BCC screw dislocation at low T. Unlike the situation for small loops, tc for large loops is higher than for voids containing a similar number of point defects, e.g. when L ¼ 41.4 nm, T ¼ 300 K and �_ ¼ 5 � 106 s�1 , tc ¼ 212 MPa for the 4.9 nm loop containing 331 SIAs compared with 150 MPa for a 2 nm void containing 339 vacancies (Fig. 28). This is probably due to the fact that the effective obstacle size is larger for the loop. The critical line shape consisting of parallel screw segments at the obstacle is the same in both cases,

Fig. 43. Temperature dependence of tc for 37-SIA and 331-SIA loops in Fe for various values of �_ in units of 106 s�1. The value of tc at 0 K for the small loop obtained by MS simulation is also shown. (Reprinted from Ref. [184] with permission from Taylor & Francis Ltd., http://www.informaworld.com.)

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Fig. 44. Visualisations of the interaction process for a 331-SIA loop in Fe at T ¼ 100 K and �_ ¼ 5 � 106 s�1 (reaction R1). (Reprinted from Ref. [184] with permission from Taylor & Francis Ltd., http:// www.informaworld.com.)

but the dipole spacing is larger for the loop and tc scales roughly as the logarithm of the obstacle size for this conﬁguration (eq. (14)). Less attention has been paid to the interaction of screw dislocations with ½/1 1 1S loops in BCC metals. Liu and Biner [187] have simulated the interaction with a ½½1 1� 1� hexagonal loop containing either 37, 127 or 271 SIAs in Fe at 100 or 300 K using the interatomic potential from Ref. [70]. Free boundaries were imposed in the ½1 1 2� � glide direction while a mixture of ﬁxed and rigid boundaries was used in the ½1 1� 0� direction to apply strain at a constant rate in the range 0.8–3 � 108 s�1. As with edge dislocations, the loop is attracted to just above the ð1 1� 0Þ slip plane as the dislocation approaches. Then, depending on the loop size, one of two mechanisms occurs. The bL of a small loop (37 SIAs) changes to ½[1 1 1] and the loop is absorbed on the line to form a helical turn (reaction R4) (see Fig. 3 of Ref. [187]), a similar conﬁguration to that found in MS simulations in Ref. [188]. Loops with 127 or 271 SIAs form a segment with b ¼ [0 1 0] by the favourable reaction of eq. (18). The dislocation eventually converts the other sides of the loop to b ¼ [0 1 0] before breakaway, as illustrated in Fig. 45 (reaction R2). 5.4.2. /1 0 0S loops The variety of mechanisms and obstacle strengths is wider for these. Some reactions end with complete absorption of the loop by the formation of superjogs (R3), whereas others result in almost no absorption (R2). There is no apparent correlation between tc and the amount of absorption. In some cases, the residual loop retains its original /1 0 0S Burgers vector while in others it is partially transformed into ½/1 1 1S type. Terentyev et al. [189] modelled a ½[1 1 1]ð1 1� 0Þ edge dislocation and a periodic row of interstitial dislocation loops with L ¼ 41 nm and bL ¼ [1 0 0], [0 1 0] or [0 0 1] in Fe at T ¼ 300 K under an applied strain rate of 107 s�1. The square loops had /1 0 0S or /1 1 0S sides and contained either 162 or 169 SIAs, respectively. Results were compared for two different interatomic potentials for Fe [69,70]. It was found that tc, varies from 30 to 220 MPa, depending on the orientation of bL and the

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Fig. 45. Snapshots from two perspectives of the interaction of a screw dislocation with a 127-SIA loop in Fe at 100 K at the times indicated (reaction R4). The process is shown schematically in the lower ﬁgures. (Reprinted from Ref. [187] with permission from Elsevier Ltd. http://www.sciencedirect.com/science/.)

position of the loop relative to the glide plane. In general, the passage of the dis location through a row of loops with bL ¼ [0 0 1] parallel to the dislocation glide plane requires the highest stress (100–220 MPa). The two other classes of loop with bL ¼ [1 0 0] and [0 1 0] offer less resistance to dislocation motion, tc varying from 30 to 190 MPa. Reactions requiring high tc for dislocation unpinning are mainly those where either part of the loop transforms to b ¼ ½/1 1 1S and the remaining /1 0 0S part is sessile (reaction R2) or the dislocation is strongly attracted to the /1 0 0S loop that cannot glide with it. Reactions resulting in low tc are those where a ½½1 1� 1� or ½½1� 1 1� segment is created and quickly propagates across the loop surface, converting it into b ¼ ½[1 1 1] superjogs on the dislocation line (reaction R3). The following three examples illustrate some effects. Of the three /1 0 0S loop vectors, only bL ¼ [0 0 1] results in a product segment that is glissile in the ð1 1� 0Þ slip plane. This case is illustrated in Fig. 46. The dislocation motion is from right to left and initially results in the formation of a

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Fig. 46. Visualisation of the interaction between a ½[1 1 1]ð1 1� 0Þ edge dislocation and a [0 0 1] loop in Fe at 300 K (reaction R3): tc ¼ 175 MPa. (From Ref. [189].)

short ½½1 1 1� � segment on contact with the ﬁrst corner of the loop (see (a)) by the reaction 1 1 ½1 1 1� þ ½0 0 1� � ¼ ½1 1 1� �. 2 2

(20)

Under increasing stress, the dislocation arms bow out until one touches the opposite corner, creating another short ½½1 1 1� � segment. At this stage the outer arms of the dislocation are pinned at the opposite corners, which are linked by a ½[1 1 1] segment that bisects the original loop, and as the line bows forward a screw dipole is formed (see (b)). At tc ¼ 175 MPa one of the two screw arms cross-slips down and then up on a V-shaped surface of ð1 0 1� Þ and ð0 1 1� Þ planes (see (c)), thereby converting bL below the slip plane by the reaction 1 1 ½0 0 1� � ½1 1 1� ¼ ½1� 1� 1�. 2 2

(21)

The dipole then detaches from the ﬁnal loop corner (see (d)) and glides away with a set of superjogs containing about 25% of the original interstitials. This leaves two conjoined triangular loops with a common bL ¼ ½[1 1 1] segment: the upper part is half of the pre-existing [0 0 1] loop and the lower has bL ¼ ½½1� 1� 1�. Fig. 47 shows three stages in the interaction with a [0 1 0] loop. The line is initially repelled by the loop but under increasing t makes contact in the middle of the two [0 1 0] sides (see (a)). The upper part of the loop converts spontaneously to b ¼ ½½1 1� 1� by the reaction 1 1 ½1 1 1� � ½0 1 0� ¼ ½1 1� 1� 2 2

(22)

and slips on its glide prism to below the dislocation glide plane (see (b)). The conﬁguration formed in this way is the same as that depicted in Fig. 42, in which a ½[1 1 1] dislocation and ½½1 1� 1� loop interact to form a product segment with b ¼ [0 1 0]. The remainder of the interaction is identical to Fig. 42 and the loop is eventually converted to a ½[1 1 1] superjog.

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Fig. 47. Visualisation of the interaction between a ½[1 1 1]ð1 1� 0Þ edge dislocation and a [0 1 0] loop in Fe at 300 K (reaction R3): tc ¼ 125 MPa. (From Ref. [189].)

Fig. 48. Visualisation of the interaction between a ½[1 1 1]ð1 1� 0Þ edge dislocation and a [1 0 0] loop in Fe at 300 K (reaction R2): tc ¼ 110 MPa. (From Ref. [189].)

The third reaction shown in Fig. 48 involves a [1 0 0] loop positioned initially with its lower [0 0 1] side lying in the dislocation glide plane. The dislocation is initially repelled by the loop, but under increasing t contacts the lower side ﬁrst at one corner and then the other (see (a)). The segment of the gliding dislocation between the corners bows backwards due to repulsion (see (b)). As the sidearms of the dislocation bow towards each other under increasing t (see (c)), the lower half of the loop is converted to b ¼ ½½1� 1 1� by the reaction 1 1 ½1 1 1� � ½1 0 0� ¼ ½1� 1 1�. 2 2

(23)

The arms meet and breakaway at tc ¼ 110 MPa (see (d)). The ﬁnal product is a ½[1 1 1] shear loop in the ð1 1� 0Þ slip plane sharing a (pre-existing) [1 0 0] segment with a ½½1� 1 1� loop formed by reaction (23). With regard to the transformation of /0 0 1S loops into mixed /0 0 1S//1 1 1S loops, we note that previous computer simulations have shown that such doubleloop conﬁgurations can form due to interactions between gliding interstitial clusters [190–192]. For example, the reaction product of Fig. 46(d) is also formed by a favourable reaction between two interstitial loops of similar size and Burgers vector [0 0 1] and ½½1� 1� 1�. MD simulations reveal that this double-loop product is stable and immobile for at least 10 ns at temperature up to 1000 K [192]. The two-loop

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complex of Fig. 48(d), which can form by reaction between ½[1 1 1] and ½½1� 1 1� loops, is unstable, however, for it transforms in the ﬁeld of another edge dislocation to the original perfect square loop with bL ¼ [1 0 0]. Concerning ½/1 1 1S screw dislocations, Marian et al. [193] have simulated the interaction with a 113-SIA square loop with bL ¼ [1 0 0] in Fe at 100 K using the interatomic potential from Ref. [69]. Free-surface conditions were imposed along the ½1 1 2� � glide direction of the dislocation and in the ½1 1� 0� direction and shear tractions were imposed on the ð1 1� 0Þ surfaces to apply stress. Under a constant stress t ¼ 750 MPa, the dislocation moved towards the loop by kink generation and, at the same time, two opposite sides of the loop transformed into two ½/1 1 1S segments by the reaction 1 1 ½1 0 0� ¼ ½1 1 1� þ ½1 1� 1� �. 2 2

(24)

This is not energetically favourable according to Frank’s rule for the energy change when parallel dislocations react and is presumably assisted by the stress and/or presence of the screw dislocation. The resulting conﬁguration is shown in Fig. 49(a). As the screw dislocation moves closer, a perfect ½[1 1 1] loop is emitted from the original loop (see (b) and (c)) and is absorbed on the line as a helical turn while the remaining [1 0 0] loop is trapped on the line by a common ½½1 1� 1� �

Fig. 49. Stages in the interaction between a 113-SIA [1 0 0] loop and a ½[1 1 1] screw dislocation in Fe at 100 K under t ¼ 750 MPa (reaction R4). (Reprinted from Ref. [193] with permission from Elsevier Ltd. http://www.sciencedirect.com/science/.)

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segment (see (d)). By increasing t in 50 MPa increments, it was found that the dislocation broke through the obstacle at t ¼ 1 GPa, an increase of 250 MPa over the stress for glide of the screw alone. It remains to be seen what the effects of loop size, side and bL orientation, and interatomic potential are on the interaction between screw dislocations and /1 0 0S loops.

5.4.3. Comparison of obstacle strength for voids and loops in iron On the basis of the simulations reviewed in this section, Terentyev et al. [189], have compared tc obtained under the same simulation conditions (L ¼ 41 nm, T ¼ 300 K, �_ ¼ 107 s�1 ) for interaction of a ½[1 1 1]ð1 1� 0Þ edge dislocation with a row of obstacles consisting of either spherical voids or ½/1 1 1S or /1 0 0S interstitial dislocation loops in Fe modelled with the interatomic potential in Ref. [70]. Each obstacle contained approximately the same number of point defects, namely 169 for voids and ½/1 1 1S loops, and 162 or 169 for /1 0 0S loops. The values of tc are plotted in Fig. 50, which also contains values for ½/1 1 1S loops and voids of smaller and larger size. The Burgers vectors of the loops considered are shown along the abscissa of the ﬁgure and the labels C1, etc. were used in Ref. [189] to denote the position of the /1 0 0S loops with respect to the dislocation glide plane and the orientation of their sides. Of the four orientations of loops with bL ¼ ½/1 1 1S, those plotted for bL inclined to the dislocation slip plane, i.e. ½½1� 1 1�

Fig. 50. Comparison of tc for /1 0 0S and ½/1 1 1S interstitial loops and voids in Fe modelled with the potential in Ref. [70] for the conditions L ¼ 41 nm, T ¼ 300 K, �_ ¼ 107 s�1 . The labels for the /1 0 0S loop conﬁgurations are those deﬁned in Ref. [189]: ﬁlled symbols for loops with /1 0 0S sides (162 SIAs), open symbols for /1 1 0S sides (169 SIAs). The number of SIAs in the ½/1 1 1S loops (37–361) and vacancies in the voids (59–339) are indicated against the data points. The values given by eq. (14) are labelled ‘Orowan stress’. (After Ref. [189].)

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or ½½1 1� 1�, are relatively strong obstacles. The void of 169 vacancies is a weaker obstacle than either the ½/1 1 1S loops or some of the /1 0 0S loops containing the same number of SIAs. It should be borne in mind that loops and voids containing 169 point defects are not fully representative of the size spectrum of point defect clusters in irradiated metals, and it is clear from the additional tc values for larger and smaller clusters plotted in Fig. 50 that the size-dependence of tc is different for loops and voids. The data for ½½1 1� 1� loops demonstrate a 10-fold increase in tc as the number of interstitials increases from 37 to 331. This is because smaller loops are readily absorbed as a pair of superjogs by spontaneous transformation of bL to ½[1 1 1]. Larger loops react with the dislocation so that one side forms a common segment with b ¼ [0 1 0], which is sessile in the ð1 1� 0Þ plane and presents a strong obstacle. The obstacle strength of voids, in contrast, has a much weaker variation with number of vacancies because the mechanism of edge dislocation cutting and unpinning is not strongly dependent on diameter. As D increases, the dislocation branches pinned at the void surface approach more closely to the screw orientation, and tc increases (see Section 4.2.1). As a result, voids are weaker obstacles than many loops when large, but stronger when small. Furthermore, the strength of voids has weaker T-dependence than that of ½/1 1 1S loops [118,186]. All the tc data obtained by MD simulation with T ¼ 300 K fall below the values predicted by eq. (14). The continuum approximation used for that equation mimics a crystal at T ¼ 0 K, i.e. no kinetic effects are considered. It was seen in Section 4.2.1 that tc obtained by atomic-level simulation of void strengthening approaches the critical value given by eq. (14) as T tends to 0 K if D is Z2 nm. Although the /1 0 0S loops that provide the strongest resistance, such as those labelled C1, C4, C6D and C6 in Fig. 50, cause the dislocation sidearms to bow to the screw orientation as applied stress increases, dislocation breakaway occurs before a stable screw dipole is drawn out. Thus, temperature (and possibly applied strain rate) affects the dislocation release mechanism and the measured obstacle strength falls below the theoretical limit represented by eq. (14).

5.5. Stacking fault tetrahedra 5.5.1. Particular reactions Interactions involving SFTs have been simulated extensively for both edge and screw dislocations and a wide range of interactions was observed [49,118,176,194– 199]. We present here examples of the main reaction types following the categories described in Section 5.2. First, however, it is necessary to identify SFT-speciﬁc parameters that arise from the geometry of the interaction. The Thompson tetrahedron shown in Fig. 16(b) shows an SFT conﬁguration. An SFT has two orientations (Face and Edge) relative to the approaching dislocation and two orientations of its apex (Up and Down) relative to the dislocation slip plane. Due to the symmetry, all other arrangements, including the dislocation line sense, are equivalent to these. Therefore, the conﬁguration shown in Fig. 16(b) can be

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described as ED/Edge/Up (edge dislocation approaching an edge of the SFT with apex up) or SD/Edge/Up in the case of a screw dislocation. Two other important parameters are the SFT height, H, and the distance, h, between the SFT base and dislocation slip plane: these are expressed in units of the spacing (a0/O3) between {1 1 1} planes (see Ref. [176] for explanation). In this notation the total number of vacancies, Nv, in a regular SFT is H(H þ 1)/2. For example, if an edge dislocation intersects an SFT of 36 vacancies (H ¼ 8) through its geometrical centre (h ¼ 4) this is denoted as ED/Edge/Up/4/8. Other important parameters such as T and �_ or t_ are also indicated in the notation below. R1: shear followed by complete restoration. In this, the SFT is sheared and ledges are formed on two faces. Fig. 51 shows the reaction (SD/Face/11/16, 100 K, 2.5 MPa ps�1) for Cu. As emphasised in Section 5.3, conditions for this are that the dislocation intersects the SFT close to its apex, i.e. h/HW0.5, and low T and high vD. This reaction has a low tc.

Fig. 51. Shear of a SFT by a screw dislocation, followed by full restoration (reaction R1). (Reprinted from Ref. [176] with permission from Taylor & Francis Ltd., http://www.informaworld.com.)

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Fig. 52. Shear of an SFT by an edge dislocation, followed by creation of stable ledges (reaction R2). Numbers indicate the number of dislocation passes through the simulation cell. (From Ref. [195].)

R2: shear with permanent damage. Here, the SFT is sheared by the gliding dislocation and two stable ledges are formed on its faces, sometimes accompanied by removal of one or more vacancies from an apex to an edge. This and other examples can be found in Ref. [176]. When an SFT with stable ledges is sheared by multiple dislocations, two situations arise: either the damage accumulates and the SFT is progressively sheared, or the ledges become unstable and the SFT is restored to its initial conﬁguration. The former process is illustrated in Fig. 52 for the case (ED/Up/6/12, 300 K, 2 � 107 s�1) in Cu when the dislocation passes through the simulation cell four times and separates apex and base parts. (Atoms in the stacking faults are omitted for clarity.) R3: partial absorption into edge dislocation. This is observed only for ED/Down and is favoured by high T, small h/H and large H, though the minimum H at which it occurs depends on the interatomic potential [176]. The reaction is illustrated in Fig. 53 by (ED/Down/6/12, 450 K, 2 � 107 s�1) in Cu, where the insets show the Burgers vectors involved. The dA Shockley partial of the edge dislocation comes into contact with the face of the SFT in (a), soon followed in (b) by the Cd partial. The dA partial slips through the SFT, while the Cd partial reacts with the ab stair-rod partial edge of the SFT in (c). Finally, the dislocation detaches from the SFT with the formation of a pair of superjogs that are clearly visible in (d). The jogs inherit the initial shape of the SFT, with the consequence that one lies in a {1 1 1} plane and is glissile, while the other is in a {1 0 0} plane and is a constricted segment of a Lomer dislocation having a low mobility. Further motion of the dislocation creates small vacancy clusters visible in (d). The corresponding interstitials are absorbed in the sessile jog to enable it to change shape and plane and become glissile. R4: temporary absorption of part of the SFT as a helix on a screw dislocation. This is seen only for the SD/Face arrangement. This conﬁguration was ﬁrst considered by Kimura and Maddin [200], who proposed a mechanism based on cross-slip of the screw dislocation on an SFT face that could lead to complete absorption of the SFT by formation of a helical turn. Simulation has now conﬁrmed this mechanism in general although complete absorption has not been seen in MD simulations. It is illustrated in Fig. 54 by the case (SD/Face/7/16, 100 K, 2.5 MPa ps�1) for Cu. When the leading partial comes into contact with the SFT face in (a), the dislocation spontaneously constricts. It then cross-slips and dissociates in the SFT face into partials bC and bA. The latter sweeps over the part of the SFT face below the glide plane and removes the stacking fault in this region, whereas the bC partial remains

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Fig. 53. Partial absorption of an SFT into an edge dislocation (reaction R3). (Reprinted from Ref. [176] with permission from Taylor & Francis Ltd., http://www.informaworld.com.)

Fig. 54. Partial absorption of an SFT into a screw dislocation (reaction R4). (Reprinted from Ref. [176] with permission from Taylor & Francis Ltd., http://www.informaworld.com.)

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immobile (see (b)). The bA partial then combines with the ba stair-rod on the righthand side of the SFT and forms an Aa Frank partial. This high-energy partial is unstable and dissociates into dislocations with Burgers vectors AC and Ca. The AC segment is a cross-slipped segment of the initial screw that moves away from the SFT on the right-hand side, as shown in (c) and (d). On the other side of the SFT, the screw dislocation cuts the SFT, forming a new stacking fault on the cut plane and thereby recreating a smaller tetrahedron. Thus, the screw absorbs part of the SFT in a helical turn dissociated in several {1 1 1} planes, as seen in (e). This mechanism has two differences from the one proposed in Ref. [200]. First, the authors supposed that both the bC and bA cross-slipped partials would sweep the whole face of the SFT and fully remove the fault, while in simulation only one partial is mobile and unfaults only the base part of the SFT. Second, it was also assumed that the Aa Frank partial would remain constricted, whereas it decomposes and forms a cross-slipped segment for the screw dislocation that leaves the SFT and forms the helical turn. As noted earlier, a helical turn is an obstacle for a gliding screw dislocation and unpinning requires the dislocation to bow out and force the turn to close onto itself. When the dislocation unpins, the vacancies that formed the turn are left as a vacancy cluster, separated from the remaining SFT along the AC direction by a distance that depends on the helix size and dislocation velocity. Examples are provided in Fig. 4 of Ref. [176]. 5.5.2. Other cases The wide range of interaction mechanisms and their dependence on T and SFT orientation and size complicates prediction of strengthening due to these obstacles. First, tc depends strongly on the interaction geometry. An example is provided by an edge dislocation cutting a 4.2 nm SFT at different levels, h. The interaction mechanism varies between R4, R2 and R1, and results in a large variation in tc, as seen in Fig. 55(a). Second, the picture is complicated by the effect of T. For some geometries, the reaction mechanism does not change and tc decreases with increasing T. This is shown in Fig. 55(b) for a 4.2 nm SFT in conﬁguration ED/Up with h/H ¼ 0.5. When the SFT orientation is inverted to apex-down, the reaction type is R1 at 0 and 100 K, R2 at 300 K and R3 at 450 K, resulting in the variation of tc with T shown in the ﬁgure. Regardless of the complexity, dislocation–SFT interactions provide a situation where atomic-scale modelling can be related directly to experiment. Dislocation– SFT interactions have been investigated by in situ straining in a TEM, e.g. Refs [199, 202–206]. The reactions observed can be separated loosely into three types. The ﬁrst occurs when a dislocation passes through an SFT without visible change of either defect, i.e. reaction R1. In the second, a dislocation cross-slips and the base portion of the SFT disappears, leaving a smaller defect. The third type is seen when a dislocation undergoes extensive cross-slip in the vicinity of the SFT, ultimately creating a superjog which persists until the experiment is terminated. The second and third types are similar to reaction R4, with the difference that in simulation the base part of the SFT is released back as a vacancy cluster in the vicinity of the

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Fig. 55. Variation of tc for an edge dislocation crossing a 4.2 nm SFT (_� ¼ 5 � 107 s�1 ) as a function of (a) distance between the edge dislocation glide plane and the SFT base [195] and (b) temperature [201]. In (a) the SFT apex is oriented up and in (b) cases of apex up and down are both shown.

original SFT. The reason for this difference is that a computational model with periodic boundary conditions along the dislocation line imposes conservation of the total number of vacancies. Thus, in order to reproduce experimental thin-foil condi tions, Osetsky et al. [198] modelled a screw dislocation, 50–80 nm long, with free surfaces perpendicular to the line (x-faces in Fig. 16(b)). They found that a fast mass transport effect to a surface occurs due to superjog glide, leading to the elimination of a large number of vacancies and the disappearance of the base part of the SFT. Fig. 56 shows a time sequence of this process for an SFT containing 820 vacancies, H ¼ 8.3 nm, h ¼ 6.1 nm and �_ ¼ 107 s�1 ; T ¼ 300 K and L ¼ 50 nm. Figs 56(a) and (b) show how the screw dislocation initially cross-slips in the SFT face and removes the part of the stacking fault between its glide plane and the SFT base. The right-hand segment then cross-slips back and is restored to its original glide plane (see (c)). The left-hand segment with mixed character in (c) achieves pure screw orientation in (d) on the plane coincident with the SFT upper face by annihilation of its edge component at the left-hand surface. This process represents vacancy transport to that surface of the simulation cell. The dislocation now contains a glissile edge superjog of height h. The left screw segment continues to bend under the applied stress, forcing the superjog to glide towards the right-hand surface, as in (e), where it disappears, as shown in (f). The screw dislocation is now restored, but on a plane h above its original slip plane, and glides away. In agreement with the experiments, most of the vacancies of the SFT (765 out of 820) are transferred to the surfaces, with the creation of surface steps, and a much smaller SFT (of 55 vacancies) remains at the tip of the original one. This mass transport occurred over a time of only 0.7 ns. The critical stress was 100 MPa, smaller than the value 170 MPa for an R1 reaction with the same dislocation length. The mechanism in Fig. 56 has variations depending on the position of the SFT relative to free surfaces, L, H and h, T and �_ [198].

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Fig. 56. Sequence of conﬁgurations during interaction of a screw dislocation with a 10.2 nm SFT (Lx ¼ 50 nm, T ¼ 300 K and �_ ¼ 107 s�1 ). The left-hand image in each pair is a plan view of the (1 1 1) slip plane; the right-hand image is a ½1 1� 0� view with the slip plane edge-on. The free x-surfaces are the left and right faces. (Reprinted from Ref. [198] with permission from Taylor & Francis Ltd., http:// www.informaworld.com.)

5.6. Conclusions The MD simulations described above have assisted in understanding reactions where obstacles have dislocation character. The possibility for screw segments to cross-slip is a key feature of some mechanisms. It allows for loop and SFT unfaulting and is systematically involved in the creation of helices on dislocation lines that can absorb, either partially or completely, obstacles and/or modify their structure (e.g. Burgers vector change). Such reactions are therefore sensitive to both T and stacking fault energy, g, in that the probability of cross-slip decreases with decreasing T and g. Strain- and stress-rates are also important since lower rates allow the dislocation to remain for a longer time in contact with the defect, which increases the probability of cross-slip. MD simulations have also shown that temperature and deformation rate can change the interaction mechanism itself, an effect previously unforeseen. The mobility of helices along a dislocation is a signiﬁcant mechanism for frag menting obstacles and transferring point defects over relatively long distances. These effects have not been considered to date in DD continuum modelling. The same comment can also be applied to climb of edge dislocations by superjog formation,

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which has been shown here to be able to remove a signiﬁcant fraction of dislocation obstacles. Climb is more effective for loops and SFTs than for voids and precipitates (Section 4), for which only a few defects are absorbed by a single dislocation. The ability of atomic-scale simulation to reveal new mechanisms and provide interpretation of experimental observations has been demonstrated in Section 5.5.2 by the example of in situ TEM deformation. It is now clear that some reaction mechanisms that occur in thin ﬁlms are speciﬁc to the conditions and cannot be applied for bulk materials. Finally, the dislocation obstacles considered here offer good examples where atomic-scale modelling can interact productively with DD modelling by providing details of mechanisms and quantiﬁcation of their parameters. The integration of atomic and continuum treatments is a requirement for the development of a true multiscale materials modelling approach to the effects of microstructure on mecha nical properties. This has been done recently for Frank loops [22] but simula tions involving all the details of the interaction mechanisms remain computationally expensive and more investigations are required on these issues.

6. Concluding remarks Signiﬁcant progress in applying atomic-scale simulation to problems of practical importance has been made over the past decade because of three factors. First, on the hardware side the dramatic increase of computing power per unit of investment has allowed simulation of much larger models and also provided researchers with facility to easily visualise the collective behaviour of hundreds or even thousands of atoms. Second, computer codes have been improved by the development of boundary conditions suitable for dislocations of different type and techniques for simulating external loading, and this has led to more realistic modelling. Third, many new interatomic potentials have been produced with the use of ab initio data, and this development has increased conﬁdence in the results of simulation. As a result of these changes, MD and MS simulations can now be performed for obstacle microstructures with realistic length scale, defect size and dislocation density. We have shown that the parameters of current computer models are well suited to replication of microstructure created by radiation damage, which is characterised by a high density of nanometre-sized defects. Furthermore, atomicscale simulations are now compatible with the scales of in situ deformation in a TEM, and so provide comparison between modelling and experiment, despite the short timescale in the former and resolution limit in the latter. The research reviewed here provides a broad view of individual dislocation interaction with obstacles. Atomic-scale mechanisms such as absorption of defects by edge dislocations that acquire superjogs, the formation of helical turns on screw dislocations with subsequent long-range mass transfer along dislocation lines, and cross-slip of screw dislocations (or screw parts on bowed edge dislocations) are key features of most interaction processes. The results obtained to date are sufﬁciently extensive for us to have attempted a qualitative categorisation based on similarities

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between the effects of different obstacles. These are most apparent for obstacles with dislocation character, i.e. perfect dislocation loops in FCC and BCC metals, and faulted Frank loops and SFTs in FCC metals. (Although we have not discussed them, effects such as loop absorption and transformation have also been found in the few simulations that have been reported for the HCP structure [55,56].) The similarities across crystal structures and defect types are consistent with changes observed in all metals due to radiation damage, for phenomena such as hardening, embrittlement and clear band formation are common. It is particularly encouraging that the well-established concepts of dislocation theory, based on linear elasticity, offer a valid interpretation of many of the effects observed for nanoscale obstacles, particularly with regard to the use of Frank’s rule for reaction between parallel dislocations and the inﬂuence of dislocation self-stress. This provides direct validation of continuum DD simulations that use algorithms based on these concepts. The same conclusion has been made as a result of simulation of junction formation between pairs of interacting dislocations [207–209]. There are still signiﬁcant problems, however, that limit successful application of atomic-scale techniques. � Interatomic potentials. The realism of large-scale atomic simulations is dependent on the validity of interatomic potentials, which have to be short-ranged for computational efﬁciency and reproduce interactions between atoms in conﬁg urations far from perfect. Validity remains limited for many metals. Potentials for relatively simple metals are perhaps reliable, but those for transition metals, where electronic effects in bonding and magnetism are strong, are more problematic. Conﬁdence in the simulated core structure of dislocations in some FCC metals is high, but the core of screw dislocations in BCC transition metals is hard to reproduce and the situation for HCP metals, in which the energy and stability of stacking faults on the basal and prism planes determines dislocation behaviour, is no better. The development of potentials for alloys, where it is important to replicate local chemistry and thermodynamic properties, is also at an early stage. Thus, it is wise to remember that most current potentials provide models that have some of the properties of the real metals in question, such as crystal structure, elastic constants, point defect energies, etc., but not all. By carrying out simulations with different potentials for the same metal, potentialspeciﬁc peculiarities may be identiﬁed. � Timescale. This is possibly the most serious limitation of investigations based on MD. Even with large supercomputers, the timescale accessible to MD simulation remains several orders of magnitude lower than that experienced in experiments, and this restricts the ability to study thermally activated processes. We have demonstrated that MD simulations reproduce thermal effects, such as a decrease of obstacle strength as T increases and an increase in the occurrence of cross-slip reactions when T increases and/or dislocation velocity decreases, i.e. when the strain rate decreases. Nevertheless, thermally activated dislocation processes that can be observed in MD simulations have activation enthalpies smaller than 0.1 eV and most of the energy to overcome an obstacle is provided by the applied stress.

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� Length scale. Despite clear successes in using atomic-scale models to study interactions between individual dislocations and small obstacles under a variety of loading conditions, the creation of computational approaches capable of predicting mechanical behaviour due to microstructure changes resulting from thermal or irradiation treatments is still a severe challenge. It demands not only atomic-level data based on longer times, but also knowledge of effects that operate over longer length scales. Examples include statistical effects of obstacle populations; cooperative dislocation effects such as pile-ups; mechanisms that affect the operation and blocking of dislocation sources; the inﬂuence of intergrain and interphase boundaries on dislocation motion; and the role of these boundaries in acting as sources and sinks for dislocations and other defects. We conclude that while application of massively parallelised, scalable models in the near future may increase times and lengths, and reduce strain rates, by a few orders of magnitude, atomic-level simulation of millisecond and millimetre scales will not be achievable. Developments such as more efﬁcient algorithms for accelerated dynamics and saddle-point search methods to give access to thermally activated processes compatible with experiment will help, but not solve, the problem. To exploit the power of atomic simulation, it will still be necessary to link it to continuum-based DD modelling by using MD simulation to generate the local rules that describe dislocation–obstacle interaction. This provides the route to study the effects mentioned in the previous paragraph. Although the variety of mechanisms already revealed by MD is large, we have seen that it should be possible to categorise them in such a way that they can be imported into DD codes.

Acknowledgements Much of the work described in this chapter was carried out with support of grants from a variety of sources, including grants GR/N23189/01, GR/R68870/01 and GR/ S81162/01 from the UK Engineering and Physical Sciences Research Council; grant F160-CT-2003-508840 (‘PERFECT’) under programme EURATOM FP-6 of the European Commission; and the Division of Materials Sciences and Engineering and the Ofﬁce of Fusion Energy Sciences, U.S. Department of Energy, under contract with UT-Battelle, LLC. The authors thank Prof. Y. Bre´chet, and Drs. G. Martin, Y. Mastukawa, R.E. Stoller and S.J. Zinkle for many fruitful discussions on theoretical modelling and experimental results, and Drs. T. Nogaret and D. Terentyev for their contributions to dislocation-loop simulations.

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CHAPTER 89

Dislocations in Shock Compression and Release M.A. MEYERS, H. JARMAKANI Materials Science and Engineering Program, Departments of Mechanical and Aerospace Engineering and Nanoengineering, University of California, San Diego, La Jolla, CA 92093-0418, USA and

E.M. BRINGA, B.A. REMINGTON Lawrence Livermore National Laboratory, Livermore, CA 94550, USA

r 2009 Elsevier B.V. All rights reserved 1572-4859, DOI: 10.1016/S1572-4859(09)01501-0

Dislocations in Solids Edited by J. P. Hirth and L. Kubin

Contents 1. 2.

3. 4. 5. 6.

7.

8.

Introduction 94

Early models for dislocations in a shock front 97

2.1. Smith’s model 97

2.2. Hornbogen model 98

2.3. Homogeneous dislocation nucleation model 98

2.4. Zaretsky model 103

2.5. Weertman mechanisms 104

2.6. The question of supersonic dislocations 105

Polycrystallinity effects 106

Dislocation structures generated in different metals 110

Stability of dislocation structure generated in shocks 113

Detailed characterization of shock-compressed metals 115

6.1. Explosively driven ﬂyer-plate impact 115

6.1.1. [0 0 1] copper impacted at 30 GPa 115

6.1.2. [2 2 1] copper impacted at 30 GPa 117

6.1.3. [0 0 1] copper impacted at 57 GPa 119

6.1.4. [2 2 1] copper impacted at 57 GPa 122

6.2. Laser shock compression of copper 122

6.2.1. TEM of pure copper 123

6.2.2. Pressure decay effects in pure copper 127

6.2.3. Copper–aluminum alloys 127

6.3. The slip–twinning transition in Cu and Ni 133

6.3.1. Modeling of the slip stress 133

6.3.1.1. Monocrystalline Cu and Cu–Al 133

6.3.1.2. Ni 135

6.3.1.3. Ni–W, 13 at.% 135

6.3.2. Modeling of the twinning stress 137

6.3.2.1. Grain-size and stacking-fault energy effects on twinning 138

6.3.2.2. Critical pressure for slip–twinning transition 140

6.4. Dislocation loop analysis: stacking-fault transition 142

6.5. Quasi-isentropic compression of metals 146

6.5.1. Gas-gun ICE setup 146

6.5.2. Laser ICE setup 147

6.5.3. TEM 148

6.5.3.1. Gas-gun ICE 148

6.5.3.2. Laser ICE 149

6.5.4. Twinning threshold modeling: ICE and shock 150

Molecular dynamics simulations of dislocations during shock compression 152

7.1. Computational methods 152

7.2. FCC single crystals 153

Comparison of computational MD and experimental results 163

8.1. Comparison of monocrystals and polycrystals 163

8.2. MD simulations of shocks in nanocrystalline nickel 167

8.3. Effect of unloading on nc Ni 174

9. Simulations of loading at different strain rates 176

10. Incipient spallation and void growth 178

10.1. Dislocation emission and void growth 180

11. Conclusions 190

Acknowledgment 192

References 192

1. Introduction The response of metals to very high strain rate deformation is reasonably well-understood. In particular, shock experiments have been carried out for over 60 years. Detonating explosives in direct contact with the metal were ﬁrst used [Fig. 1(a)], followed by the use of ﬂyer plates driven by explosives and gas-guns to create the compressive pulse in the material [Figs 1(b) and 1(c)], and pressures attained were on the order of tens of GPa with accompanying strain rates on the order of 107 s�1 with durations on the order of microseconds or fractions thereof. More recently, laser pulses have also been used to study shock compression in metals. The generation of shock pulses in metals from laser-pulse induced vaporization at the surface was ﬁrst demonstrated by Askaryon and Morez [1] in 1963. Shortly thereafter, White [2] and others [3–5] advanced this technique and postulated that lasers could be used to obtain Hugoniot data for a broad range of pressures. The use of surfaces covered by a laser-transparent overlay was introduced by Anderholm [6]; this enabled the conﬁnement of the vapor products resulting in an increase of the peak pressure of the shock incident on the metal. In 1963, Leslie et al. [7] reported dislocation structures in shock-compressed iron. Early experiments by Johari and Thomas [8] and Nolder and Thomas [9] investigated defect substructures generated in explosively deformed copper and copper–aluminum alloys and nickel. Table 1 presents the main reviews on the topic with the primary emphasis of the articles. The rapid heating and thermal expansion of the surface layers during laser irradiation generates a shock which propagates through the material. Shock pressures higher than those in planar impact setups can be achieved (up to 10,000 GPa), and the strain rates attained are as high as 109–1011 s�1. A basic difference is that the duration of the pulse in the laser shock is on the order of nanoseconds rather than microseconds. Fig. 1(d) shows several modes by which lasers can be used to shock compress materials: (i) the direct illumination conﬁguration, (ii) plasma conﬁning overlay, (iii) laser-driven ﬂyer plates, and (iv) the hohlraum conﬁguration, converting the laser pulse to X-rays. Shock amplitudes as high or higher than those generated by explosives or planar impact devices can be generated with a basic difference: the duration of the shock pulse is in the nanosecond range [10]. More accurate microstructural characterization is possible due to the self-quenching mechanism associated with laser shock. Some of the earliest experimental work on laser shock damage is from Armstrong and Wu [11], who carried out Berg-Barret X-ray diffraction experiments on laser damage in zinc. Laser shock and isentropic compression experiments (ICEs; discussed in detail in Section 6) are rapidly evolving as effective methods to explore the extreme pressure, strain rate, and temperature regimes inaccessible through other

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Fig. 1. (a) Conﬁguration for generating an HE-induced shock in metal; (b) Schematic of a shock recovery experiment performed by acceleration of a ﬂyer plate by an explosive charge; (c) gas-gun driven shock compression experiments; (d) Methods of laser shocking materials; (i) direct laser illumination at an intensity above the ablation threshold; (ii) laser irradiation through a transparent overlay to increase achievable pressures; (iii) laser accelerated ﬂyer plate; and (iv) laser generated X-rays through a hohlraum (indirect drive).

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Table 1

Review articles on dislocation effects in shock loading

Author

Year

Focus

Dieter [12,13] Zukas [14] Leslie [15] Davison and Graham [16] Meyers and Murr [10] Mogilevsky and Newman [17] Murr [18] Meyers [19] Gray [20,21] Remington et al. [22] Armstrong and Walley [23]

1961, 1962 1966 1973 1979 1981, 1983 1983 1988 1994 1992, 1993 2004 2008

First TEM

Physics of shock-induced defects Metallurgical effects Mechanisms of deformation TEM, mechanical effects Shock front models, TEM TEM Physics of laser shock compression High strain rate deformation of metals

techniques. Although laser shock compression does not yet have the temporal and spatial uniformity of pressure as plate impact experiments, it has a signiﬁcant advantage, especially from the point of view of recovery. The post-shock cooling is orders of magnitude faster than in plate-impacted specimens because of two key factors: (a) the short duration of the pulse and (b) the rapid decay, creating a selfquenching medium. More recently [24–26], pulsed X-ray diffraction has been used to obtain quantitative information of the lattice distortions at the shock front. These measurements can be used to resolve issues of dislocation generation and motion as well as lattice distortions at the shock front. Coupled with recovery experiments to examine the deformation substructures, laser shock experiments are being used to obtain an understanding of compressive shock defect generation and relaxation processes. In this chapter, recent work on the laser shock compression of copper, copper alloys, and nickel is reviewed, examining the effects of crystallographic orientation, pressure decay, and stacking-fault energy (SFE) on the deformation microstructure. There is speciﬁc emphasis on the slip to twinning transition. A new criterion for the transition from perfect to partial dislocation nucleation is proposed. This criterion explains the transition from cells to stacking faults, why for pure copper the cell structure gives rise to planar stacking faults above a critical pressure, and how this transition pressure decreases with a decrease in SFE. The response of dislocations to shock compression can be rationalized through the classic Orowan equation [27], proposed in 1940: g ¼ krbl�,

(1)

where g is the strain, r the mobile dislocation density, b the Burgers vector, l� the mean distance traveled by a dislocation, and k is a proportionality constant. By taking the time derivative: g_ ¼ kb

d � ðrl Þ. dt

(2)

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Thus, the strain rate has two components: g_ ¼ kbr

dl� dr þ kbl� . dt dt

(3)

Prescribed strain rates can be either accommodated by dislocation movement (at a velocity d�l=dt), the generation of dislocations (dr=dt), or by a combination of both. At the lower strain rates, the velocity of dislocations is not a limiting factor, but in shock compression, especially at higher pressures (above the HEL), the dr=dt term dominates. Thus, the generation and not the movement of dislocations determines the overall conﬁguration. Indeed, Zerilli and Armstrong [28] pointed out this aspect of the Orowan equation which cannot be overlooked, if one con siders the change in dislocation density due to the activation of sources. Armstrong and Elban [29] also note that heat generated by rapid dislocation motion in shockcompressed energetic crystals plays an important role in the detonation process. Armstrong et al. [30,31] addressed the relationship experimentally obtained by Swegle and Grady [32] and expressed it in terms of dislocation dynamics, incorpora ting nucleation and propagation of dislocations at the front. Speciﬁc mechanisms for the generation and high-velocity motion of these dislocations are presented in Section 2. There exist a number of reviews on defects generated in shock com pression. The most prominent are shown in Table 1. These works supplement the material presented here and provide a broad background.

2. Early models for dislocations in a shock front A number of models have been proposed for the generation of dislocations in shock loading. They will be reviewed next. The dislocation generation mechanisms operating under shock loading vary from the conventional ones operating at low strain rates, where the ﬁrst term in eq. (3) (dislocation motion) dominates to mechanisms uniquely associated with high strain rates, including the dominance of the second term in eq. (3) (dislocation generation). 2.1. Smith’s model Smith [21] made the ﬁrst attempt to interpret the metallurgical alterations produced by shock waves in terms of fundamental deformation modes. He depicted the interface as an array of dislocations that accommodates the difference in lattice parameter between the virgin and the compressed material. In this sense, the Smith interface resembles an interface between two phases in a transformation. Fig. 2(a) shows the interface if no dislocations were present; the deviatoric stresses cannot be relieved. This interface of dislocations would, according to Smith, move with the shock front, as shown in Fig 2(b). Since the density of dislocations at the front is, according to Smith, 103–104 times higher than the residual density, sinks and sources, moving at the velocity of the shock, were postulated.

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Fig. 2. (a, b) The Smith [33] and (c) Hornbogen [34] models for dislocation generation in shock loading.

2.2. Hornbogen model Hornbogen [34] modiﬁed Smith’s model because it could not account for the residual dislocation substructures found in shock-loaded iron, where screw disloca tions lying on /1 1 1S directions were found. Hornbogen’s explanation is shown schematically in Fig. 2(c). Dislocation loops are formed as the compression wave enters the crystal. The edge components move with the velocity of the shock front, so that their compression zone forms the wave; and the screw components remain and extend in length as the edge components advance. As will be seen in Section 5, the residual dislocation density in iron is, indeed comprised primarily of screw dislocations. 2.3. Homogeneous dislocation nucleation model The limitations of Smith and Hornbogen’s proposals led Meyers [35,36] to propose a model whose essential features are as follows: 1. Dislocations are homogeneously nucleated at (or close to) the shock front by the deviatoric stresses set up by the state of uniaxial strain; the generation of these dislocations relaxes the deviatoric stresses. 2. These dislocations move short distances at subsonic speeds. 3. New dislocation interfaces are periodically generated as the shock wave propagates through the material. This model presents, with respect to its predecessors, the following advantages: 1. No supersonic dislocations are needed. 2. It is possible to estimate the residual density of dislocations.

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An early version was proposed in 1978 [36]. Fig. 3 shows the progress of a shock wave moving through the material in a highly simpliﬁed manner. As the shock wave penetrates into the material, high deviatoric stresses effectively distort the initially cubic lattice into a rhombohedral lattice [Fig. 3(a)]. When these stresses reach a certain threshold level, homogeneous dislocation nucleation can take place [Fig. 3(b)]. This process of deviatoric stress buildup and relaxation through the homogeneous dislocation loop generation repeats itself [Figs 3(c) and 3(d)]. Fig. 4 shows an idealized conﬁguration of dislocation loops when a shock wave propagates through the lattice. The planes are (1 1 1) and the dislocations are edge dislocations. The screw components are not shown in the picture. As the shock-front advances, the dislocation interface is left behind. As this occurs, elastic deviatoric stresses build up.

Fig. 3. Shock front evolution according to a homogeneous dislocation nucleation model [36].

Plastically distorted region X2

h

X1

Elastically distorted

d2 O

Fig. 4. Stresses due to dislocations on a reference point 0 at shock the front (adapted from [37]).

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We reproduce below the early and more recent calculations and discuss the reasons for the difference. As shown in Fig. 4, successive layers of interfacial dislocation loops are generated and left behind by the shock front. The insertion of dislocations relaxes the deviatoric stresses that elastically distort an ideal lattice to rhombohedral. Hence, a reduced cubic lattice is restored by the insertion of dislocations in the near vicinity of the shock front. The dislocation spacing along the front required to accommodate this is d2 (Fig. 4). This situation is analogous to the epitaxial growth of ﬁlms, in which interface dislocations, creating a semi-coherent boundary, accommodate the disregistry. The dislocation spacing along the front is calculated from the ratios of the original and compressed lattices. The ratio between the initial and compressed speciﬁc volumes of the lattices, V0 and V, is: � �3 V bs ¼ , (4) b0 V0 where b0 is the original Burgers vector and bs the compressed Burgers vector. The spacing of dislocations at the front is given by the epitaxial growth equation: d2 ¼ k

b0 bs , b0 � bs

(5)

where k is an orientation factor. These interplanar spacing can be expressed in terms of dislocation densities. The dislocation density generated can be calculated from d2, and h, the spacing between successive dislocation loop layers nucleated. Since each distance d2 corresponds to two dislocations (on planes (1 1 1) and (1 �1 1)), the spacing d2/2 is taken. Thus, the dislocation density, r, is: r¼

2 . d 2h

(6)

At this point, the early and more recent calculations diverge. In the early version, h was taken equal to d2, as a ﬁrst approximation. This is reasonable and is based on the assumption that the stress ﬁeld of a dislocation has a radius equal to the dislocation spacing, d2. Thus: ! 2 r¼ . (7) d 22 The more recent model, developed with the important input of Ravichandran (private communication, 2002), uses a more detailed analysis. The spacing between dislocation loop layers is calculated by using the stress ﬁelds around dislocations and summing them at a generic point 0 at the front over the stress ﬁeld of all dislocations. The stress ﬁelds due to the dislocation arrays balance elastic distortion at the shock front. Thus, when the stresses at the front reach a critical level (at which homogeneous dislocation nucleation of loops can occur), a new layer of dislocations is formed. The shear stress at point 0 in the front (Fig. 4) can be

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estimated from the stress ﬁelds due to the last layer of dislocations. This is described by Meyers et al. [37]. Assuming edge dislocations only: pﬃﬃﬃ þ1 X Gb 2 2 1 s11 � ! ¼ 0, (8) n 2pð1 � nÞ nd 2 �1 s22

pﬃﬃﬃ þ1 X Gb 1 2 2 2 � ð�2h Þ ! ¼ 0, 3 3 2pð1 � nÞ n3 n d2 �1

s12

pﬃﬃﬃ þ1 X Gb 2 2 1 p4 � h ! ¼ , 2pð1 � nÞ n2 d 22 n2 90 �1

(9)

(10)

where n is Poisson’s ratio. Thus, the normal stresses due to dislocations at 0 are zero and one has only the shear stress. When this stress equals the stress required for the nucleation of a dislocation loop, a new dislocation is generated. This is discussed further in Section 6.4. The dislocation density can be obtained from the stress for homo geneous nucleation of dislocation loops. From Xu and Argon [38], the activation energy is zero at the critical stress for plastic ﬂow, which is considered as the stress at which the loops are generated. This stress was taken as [39]: s12 ¼ 0:04G.

(11)

For a more detailed discussion, see Section 4.1. The spacing between dislocation planes can be calculated by setting eq. (10) equal to eq. (11): h¼

0:8ð1 � nÞ 2 d 2. p2 b

(12)

The dislocation density is obtained from eq. (6): r¼

2p2 bs . 0:8ð1 � nÞd 32

(13)

The Rankine–Hugoniot equation connecting pressure P to speciﬁc volume V is [29]: P¼

C 20 ð1 � V=V 0 Þ . V 0 ½1 � Sð1 � V=V 0 Þ�2

(14)

where C0 is the ambient sound speed, and S is an equation of state (EOS) parameter. The application of eqs (4), (5), (7), and (14) leads to the dislocation density as a function of pressure (early model): P¼

C 20 f1 � ½1=1 þ kb0 ðr=2Þ1=2 �3 g V 0 f1 � S½1 � ½1=1 þ kb0 ðr=2Þ1=2 �3 �g2

.

(15)

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120

Pressure (GPa)

100

80

Experimental Trueb Meyers (1978) Murr and Kuhlmann-Wilsdorf (1978)

Meyers early (1978) Experimental Modified Meyers et al. (2003) With dislocation motion

60

No dis location motion

40

20

0 13 10

10

14

10

15

16

10

17

10

18

10

-2

Dislocation density (m )

Fig. 5. Experimental (from Murr [40], Meyers [36], Bernstein and Tadmor [41], Murr and KuhlmannWilsdorf [42], and Trueb [43]) and computed dislocation densities as a function of shock pressure for nickel.

These initial calculations of dislocation densities produced results orders of magnitude higher than the observed results [36]. They are shown in Fig. 5. A recalculation of the improved model by Meyers et al. (2003) [37], using a slightly modiﬁed approach, is shown below. By substituting eq. (4) into eq. (5) and the result into eq. (13) one obtains: � ��2=3 " � �1=3 #3 2p2 V V r¼ 1� . 2 3 V0 0:8ð1 � nÞk b0 V0

(16)

Solving eqs (14) and (16) by assigning different values V/V0o1, one obtains the dislocation density as a function of pressure. This is shown in Fig. 5. The difference between the original (1978) and improved models is clear from the two plots in Fig. 5 and by comparing eqs (7) with (13). Eq. (13) can be approximated as 35bs/d23. For low pressures, d2W35bs and the early model (1978) prediction gives higher dislocation densities. For PW32 GPa, d2o35bs and the 2003 model predicts a higher dislocation density. There is also a second case: moving dislocations. If the dislocations are assumed to move under the inﬂuence of the high residual stresses, they try to ‘‘catch up’’ with the front, the maximum of h2 is reached. This results in an increase in the spacing between dislocation arrays from h, given in eq. 12, to h2: � � kV d h2 ¼ h 1 þ , (17) Us

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where Vd is the dislocation velocity and k an orientation factor. When Vd ¼ 0, eq. (17) reduces to eq. 12. When the dislocation velocity equals the shear wave velocity Vsw (our maximum assumed velocity for dislocation velocities), h2 reaches a maximum: �

� kV sw h2 ¼ h 1 þ , Us

(17a)

The predictions based on the two values of h (eq. (12) for stationary dislocations and eq. (17a) for dislocations moving at the shear sound velocity) are shown in Fig. 5 and compared with experimental results. The calculations are lower, by a factor of 5–10, than the measured densities. However, this is much closer to the experimental results than the earlier model [36]. Evidently, the improved calculation predicts values that compare more favorably with dislocation densities measured from transmission electron microscopy. The approach was thought to predict realistically the currently observed results. However, recent molecular dynamics (MD) calculations predict results that are much closer to the original calculations (early model). One possible reason for this is that the ‘‘improved’’ model, based on stresses from dislocations, does not consider the stresses from the opposing dislocation in each loop. Only the dislocations in the last layer, facing the front, are included. If the dislocations in the backs of the loops were considered, there would be cancellation of the stress ﬁelds when the loop radius is small in comparison with h. As shown in Section 6.4, the nucleating loop radius, which is a function of the applied shear stress, is small at the high shear stresses imposed by shock compression. Thus, the early model might be a better representation of the generation of dislocations, and the spacing between adjacent dislocation layers in the shock wave propagation direction is closer to d2 than to h. This is further discussed in Section 8.

2.4. Zaretsky model Zaretsky [44] proposed a dislocation model based on multiplication and motion of partial dislocations (rather than perfect dislocations) bounding a stacking fault. In essence, their model extends the homogeneous dislocation nucleation model of Section 2.3 to partial dislocations. Stress-activated stretching of lateral branches of the partial dislocation bowed-out segment results in collapse of these branches with subsequent restoration of the ‘‘initial’’ dislocation half-loop and generation of a ‘‘fresh’’ partial dislocation loop, both capable to produce the next multiplication act. The multiplication results in the exponential increase of the concentration of both dislocations and stacking faults. The model explains the variations of X-ray diffraction patterns for material undergoing shock compression and the shockinduced formation of twins. Fig. 6 shows an illustration of the generation of stacking faults according to Zaretsky [44].

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Fig. 6. The Zaretsky [44] mechanism for partial dislocation loop formation; (a) partial loop expansion; (b) stacking-fault overlap; and (c) successive partial loops forming twins.

2.5. Weertman mechanisms Weertman [45] considered two regimes of shock-wave propagation: weak shocks (pressure small in comparison with bulk elastic modulus) and strong shocks (pressure on the same order of bulk elastic modulus). For weak shocks, Weertman and Follansbee [46,47] considered the front as composed of a superposition of plastic waves. They applied the Orowan equation [eq. (2)] eliminating the dr/dt term: _

dl g_ ¼ kbr . dt

(18)

Thus, conventional mechanisms of dislocation motion accommodate the plastic strain at the front. For strong shocks, Weertman [45] proposed a mechanism incorporating both supersonic dislocations in a Smith interface at the front and a homogeneous dislocation generation behind the front. This is a hybrid of the Smith and Meyers models, as can be seen in Fig. 7.

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Fig. 7. The Weertman mechanism for dislocations in a strong shock using supersonic dislocations at the front and subsonic dislocations behind the front (Weertman [45]).

2.6. The question of supersonic dislocations The existence of supersonic dislocations is fundamentally important and yet is an unresolved question. One of the earliest suggestions was a special conﬁguration postulated by Eshelby [48] to move dislocations supersonically. Weertman [49,50] developed the mathematical analysis of subsonic, transonic, and supersonic dislocations, using the relativistic theory originally proposed by Frank [51]. He divided the behavior into three regimes: subsonic (below the shear-wave velocity); transonic (between the shear and the longitudinal wave velocities); and supersonic (higher than the longitudinal wave velocity). Thus, these researchers accepted the possibility of supersonic dislocations. Gumbsch and Gao [52] carried out MD calculations in simple shear and obtained the velocities shown in Fig. 8 for an edge dislocation. Their atomistic simulations show that dislocations can move transonically and even supersonically if they are created as supersonic dislocations at a strong stress concentration and are subjected to high shear stresses. We note that the divergence in dislocation velocity at the Rayleigh velocity (edge dislocations) or shear-wave velocity (screw dislocations) is a consequence of assuming a compact dislocation core. MD simulations show a spreading core and these divergencies vanish. The topic is discussed further by Hirth and Lothe [39] and Hirth et al. [53]. Gilman [54] showed that the limiting speeds of moving dislocations are determined by inertial effects, or by viscous drag, at their cores. He developed simple expressions for the limiting speeds. He argued that the Frank proposal that the speeds are limited by the inertia of the elastic ﬁelds, accompanied by Lorentz contractions, is ﬂawed because it neglects the angular momentum of a moving dislocation; or, equivalently, because it assumes that the motion is steady if the velocity is constant, which is not possible because the motion

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Fig. 8. Edge dislocation velocity versus applied shear strain in tungsten. The different regimes of stable dislocation motion at constant velocity (solid circles) are connected by solid lines. Open circles mark substantially varying velocities or the (average) velocity of dislocations before stopping. Dashed lines indicate the transverse and longitudinal elastic wave velocities. From Gumbsch and Gao [52].

creates plastic deformation. He also discarded the Gumbsch and Gao [34] MD calculations predicting supersonic dislocations.

3. Polycrystallinity effects Meyers [55,56] and Meyers and Carvalho [57] proposed, in 1976, that the shock front was affected by the polycrystallinity of the material and acquired an irregular conﬁguration. This concept had been originally expressed in a qualitative manner in Meyers’ [55] doctoral dissertation. They performed simple calculations showing that the shock-front width increased with increasing grain size, for the same travel distance. Based on experimental results by De Angelis and Cohen [58] suggesting that grain rotation could occur in shock compression, Meyers et al. performed a number of experiments in aluminum [59], copper [60], and stainless steel [61]. For aluminum, Dhere et al. [59] varied the grain size from 26, 70, and 440 mm and subjected the systems to shock deformation at 5.8 GPa. However, after the shocks they could not detect any change in texture by X-ray diffraction, even though the cold rolled samples had signiﬁcant texture changes. They also looked at misorientations within the grains by Kikuchi lines. Braga [60] shocked a copper bicrystal and observed a higher dislocation density close to the interface, suggesting

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Fig. 9. Effect of polycrystallinity on the residual dislocation distribution in copper; (a) bicrystal close to the interface (grain boundary); (b) bicrystal far from the interface. From Braga [60].

an effect of polycrystallinity. Fig. 9 shows the dislocation cell structure in the monocrystalline region (away from the boundary) and close to the boundary. There is a deﬁnite difference. In stainless steels, Kestenbach and Meyers [61] identiﬁed changes in deformation mechanisms as a function of grain size. However, no grain rotation was observed. Later, experiments carried out with Murr, Hsu, and Stone [62] on polycrystalline and monocrystalline Fe–Ni–Cr alloys revealed some differences in the micro structure, the polycrystal exhibiting a slightly larger dislocation density. Systematic experiments with Murr [63,64] on nickel (monitoring the pressure pulse decay over distances of over 100 mm with samples having two widely different grain sizes) failed to reveal signiﬁcant changes. Diagnostics (manganin gauges) did not show any difference in the rate of attenuation of the shock wave. This research direction was discontinued in the 1970s for lack of more sensitive diagnostic tools. In 2006, atomistic simulations of shock wave propagation in nanocrystals were carried out by Bringa et al. [65]. The calculations demonstrate that the width of the wave is indeed a function of grain size, pressure, and time. The atomistic calculations match the analytical calculations of Meyers [56] and Meyers and Carvalho [57] for the width of the shock front for polycrystalline copper which, in turn, agreed with measurements of Jones and Holland [66] in the microcrystalline regime, as shown in Fig. 10. The MD simulations also reveal the details of the propagation of a shock wave through the nanocrystalline (nc) metal. Fig. 11 shows snapshots at two different

0.10

trise (µs)

0.08

0.06

0.04

experiment (Jones & Holland) calculated (Meyers) MD extrapolation × 3

0.02 0

10

20

30

40 d (µm)

50

60

70

80

Fig. 10. Experimentally measured (Jones and Holland, [66]), analytically calculated (Meyers, [56]), and MD predictions of the rise time of the shock wave in copper as a function of grain size, d, in the conventional grain-size domain (Adapted from Bringa et al. [65]).

Fig. 11. MD simulations of shocked polycrystalline Cu showing the wave front at two different times. Grainboundary atoms are shown as small black dots. d ¼ 5 nm, P ¼ 22 GPa, Up ¼ 0.5 km s�1 and strain ¼ 10%. Atoms are colored according to their kinetic energy (red, high – moving at Up; blue, low – unshocked). The upper frame shows a sharp front inside the grains, with some refraction due to orientation. Note that the energy levels track the GB, and that in frame (b) the front itself tracks the shape of one of the grains. Some of the stacking faults generated by the wave (emitted from grain boundaries) are marked with blue circles. From Bringa et al. [65].

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times. The original color illustrations can be found in Bringa et al. [65]. The simulations suggest that the effect of grain boundaries in the width of the wave front is small compared to the effect of anisotropy from crystal to crystal. This is the reason why the continuum model by Meyers [56,57] was able to predict the front dispersion due to polycrystallinity. The dispersion of the wave calculated by MD, represented by the shock-front width normalized to the grain size, Dz/d, versus grain size, at three shock pressures (22, 34, and 47 GPa) is shown in Fig. 12(a). This shows that the normalized shock-front thickness decreases with increasing grain size

22 GPa 34 GPa 47 GPa

∇

z/d

3

2

1 4

6

8

10

12 14 d (nm) (a)

16

18

20

0.7 0.6

∇

x/dca

0.5 0.4 0.3 0.2 Simulation Model: Late Time Model: Early Time

0.1

0

5

10 x/dca

15

20

(b) Fig. 12. (a) Results of MD simulations for shock front width, normalized to the grain size, versus pressure and grain size (from Bringa et al. [65]). (b) Normalized width of the shock front, Dx/d versus normalized propagation distance x/d (d ¼ grain size); analytical (Model) and MD simulations. From Barber and Kadau [67], Fig. 2, p. 144106.

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and with increasing pressure, whereas the absolute (un-normalized) shock-front width decreases with decreasing grain size. These simulations lead to a better understanding of the physics governing shock width. One of the possible applica tions of nanocrystalline (nc) metals, due to the greater sharpness of the shock front for small grain size, is as targets in the National Ignition Facility (NIF) at Lawrence Livermore National Laboratory [68]. The nc grain size would ensure that the ﬂuctuations in the shock front remain small, decreasing the level of undesired hydrodynamic instabilities, which degrade inertial conﬁnement fusion (ICF) capsule performance. Barber and Kadau [67] extended the Meyers–Carvalho [56,57] analysis and obtained a shock-front width that varies as the ½ power of the penetration distance. Fig. 12(b) shows the normalized width of the shock front, Dx/d versus normalized propagation distance x/d (d ¼ grain size). Both the analytical (Model) and MD simulations are shown. This result provides additional conﬁrmation of the effect of polycrystallinity on the shock-front conﬁguration.

4. Dislocation structures generated in different metals Dislocation structures generated by shock loading have been exhaustively investi gated. By far the most effective method of characterization is transmission electron microscopy (TEM). The ﬁrst detailed characterization is due to Dieter [12,13] followed by Leslie [7]. The dislocation structure shown in Fig. 13 for BCC iron subjected to a 7 GPa shock reveals straight screw dislocations aligned in two directions. This morphology led Hornbogen to propose his shock propagation mechanism (Section 2.2). As the pressure is increased above the a(BCC)-e(HCP) transition, a completely different structure results, with profuse debris from the phase transition. In FCC metals, on the other hand, one does not observe such a structure. Indeed, even Ta, a BCC metal, does not have such an aligned dislocation

Fig. 13. Dislocation structure in BCC iron subjected to 7 GPa pressure shock. From Leslie et al. [7], Fig. 3, p. 122.

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Fig. 14. Dislocation cells in shock-compressed (a) copper (5 GPa) and (b) nickel (10 GPa).

structure, being characterized by more random orientations of dislocation lines. Characteristic of high stacking fault energy (SFE) FCC metals are loose dislocation cells, illustrated in Fig. 14 for Cu and Ni. As the pressure increases the cell size decreases. Murr and co-workers [18,40,69–77] carried out extensive investigations on shock-recovered specimens. Murr and Kuhlmann-Wilsdorf [42] correlated the cell sizes to the shock pressures and proposed the relationship between the dislocation density, r, and pressure, P: r / P1=2 .

(19)

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Eq. (19) is a direct consequence of the Kuhlmann-Wilsdorf (K-W) relationship between dislocation cell size, l, and dislocation density (her Principle of Similitude): l / r�1=2 .

(20)

The effects of both pressure and pulse duration for shocked Ni are seen in the P�tp plot by Murr [73] (Fig. 15). He developed similar maps for different metals

Fig. 15. Effect of shock pressure and pulse duration on residual substructure of nickel. Courtesy of L. E. Murr.

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Fig. 16. Shock structures generated in intermetallic compounds (a) Ni3Al (14 GPa) and (b) NiAl (23.5 GPa). Courtesy of G. T. Gray, LANL; also, Ref. [20], Fig. 6.13, p. 204.

and alloys. For Ni, as the pressure is increased the cell size decreases. At 30 GPa a new mechanism sets in – deformation twinning. The effect of pulse duration is evident in the horizontal line of photomicrographs, at 25 GPa. The dislocation cells become more distinct as the pulse duration increases. As the SFE of FCC metals decreases (through alloying) the critical transition pressure for twinning also decreases. The dislocation cells are gradually replaced by planar dislocation arrays and stacking faults. This transition is treated in Section 6. Intermetallic compounds are also hardened by shock waves. The dislocation struc tures are more complex by virtue of the ordered nature of the structure. Fig. 16 shows the dislocations generated in Ni3Al and NiAl (Gray, private communication).

5. Stability of dislocation structure generated in shocks The loose residual dislocation cell structure often encountered after shock compres sion is not stable, since the equilibration time for recovery is minimal. Hence, upon plastic deformation at conventional strain rates (10�3–10�4 s�1), the residual dislocation structure often collapses into better deﬁned cells, with an associated stress drop. This phenomenon, called ‘‘work softening,’’ was ﬁrst observed in lowtemperature shock deformation of FCC metals followed by ambient temperature deformation (e.g., Longo and Reed-Hill [78,79]). The shock-induced structure after subsequent plastic deformation is shown in Fig. 17 [63]. One can see a large elongated cell that has formed and annihilated the smaller loose cells characteristic

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Fig. 17. TEM showing breakdown of shock-wave induced residual substructure in Ni shocked to 20 GPa and subsequently deformed in tension at ambient temperature until failure; TEM foil taken from neck portion of tensile specimen.

of shock compression (see Section 4). The quasi-static tensile response in the annealed condition, and after having been shocked is shown in Fig. 18(a). When the shocked specimen is deformed at ambient temperature, it necks immediately upon the onset of plastic deformation. However, when it is deformed at 77 K, the shockinduced structure continues to produce work hardening. This is the classical manifestation of the phenomenon of work softening as described by Longo and Reed-Hill [78]. This led Meyers [80] to propose that shock hardened Ni exhibited work softening. However, the results can be interpreted differently; Gray [81] suggested that the softening could be due to other causes and that only compressive response would identify the phenomenon incontrovertibly. This was successfully carried out by Lassila et al. [82] and is shown in Fig. 18(b). The compressive true stress–true strain curve for shock-compressed copper at ambient temperature shows a clear softening. In a similar fashion to the response of Ni, the 77 K response shows the characteristic hardening. The shock-compressed copper was mechanically tested in compression at a strain rate of 10�3 s�1 and temperature of 300 K; the conditions subjected to lower pressures (27 and 30 GPa) exhibited work softening, in contrast to the conventional

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20 GPa@77K

Annealed@298K 200

True Stress (MPa)

Engineering Stress

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600 500 30 GPa 300°K

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200 � = 10–4 s–1

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0.2 0.3 0.4 Engineering Strain (a)

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0.2 0.3 True Strain (b)

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Fig. 18. Stress–strain curves for shock-recovered nickel (under subsequent tension) and copper (under subsequent compression) samples: (a) Engineering stress versus engineering strain for annealed and shock-recovered Ni. [80] (b) True stress versus true strain for annealed and shock-recovered Cu [82].

work hardening response. This work softening is due to the uniformly distributed dislocations and the formation of loose cells, evolving, upon plastic deformation at low strain rates, into well-deﬁned cells, with a size of approximately 1 mm.

6. Detailed characterization of shock-compressed metals We focus next on two metals that we have investigated in detail: copper and nickel. This constitutes the doctoral dissertations of Schneider [83], Cao [84,85], and Jarmakani [86] carried out in collaboration with LLNL researchers. We used two techniques: ﬂyer-plate impact (by explosives and gas-guns) and laser-driven shock waves. 6.1. Explosively driven ﬂyer-plate impact We summarize here the work presented by Cao et al. [87,88] using the experimental setup shown in Fig. 1(b) and specimens pre-cooled to 90 K to minimize thermal effects. It is seen that, in spite of the care taken, there was extensive recrystallization for the higher-pressure (57 GPa) experiments. The specimens were monocrystalline Cu with two crystalline orientations: [0 0 1] and [2 2 1]. 6.1.1. [0 0 1] copper impacted at 30 GPa Fig. 19 shows the scanning electron microscope – electron channeling contrast (SEM-ECC) pictures from a 30 GPa post-shocked [0 0 1] Cu specimen. Fig. 19(a) reveals that the back surface of the sample was full of slip band traces. Fig. 19(b) provides a more detailed view of the area with slip band traces. The presence of two sets of lines, which are spaced almost exactly 901 apart, is clear evidence for {1 1 1}

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Fig. 19. SEM pictures of the residual microstructure from [0 0 1] Cu shocked at 30 GPa, driven by an HE-accelerated ﬂyer plate. (a) Back surface of Cu sample showing traces of slip bands; (b) detailed view of slip bands forming a 901 angle [87,88].

Fig. 20. (a) Stacking faults were observed in 30 GPa post-shocked [0 0 1] copper samples. (b) Two sets of perpendicular traces of the stacking faults were shown on the (0 0 1) plane when the TEM electron beam direction is B ¼ [0 0 1]; (c) detailed view of the stacking faults [87,88].

traces on the plane of observation, (0 0 1). The microstructure shown by TEM for the same specimen (not shown) conﬁrms that the deformation markings are slip bands and stacking faults. Fig. 20(a) shows traces of the stacking faults. The thin foil has straight ‘‘boundaries’’ resulting from fracture along the slip bands, which were also found by SEM [as in Fig. 20(a)]. Fig. 20(b) shows the two sets of stacking faults as [2 2 0] and [2 2 0] traces in the (0 0 1) plane when the TEM electron beam direction is B ¼ [0 0 1]. It seems that the stacking faults in [2 2 0] direction were formed before the [2 2 0] ones, because they are continuous, while the [2 2 0]

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stacking faults are segmented. The occurrence of stacking faults is comparable to that found by Meyers and co-workers [37] on laser-induced shock compression of monocrystalline copper. Very little work has been done using TEM to provide the three-dimensional (3D) picture of microstructural evolution during plate impact. Fig. 20(c) reveals the microstructure along the [1 0 0] crystal orientation shocked to 30 GPa. The [1 0 0] crystal orientation is perpendicular to the [0 0 1] shock direction ((0 0 1) shock-front plane). Stacking faults similar to the ones on the (0 0 1) plane were observed on the (1 0 0) plane [Figs 20(a) and 20(b)], which may indicate the stacking faults are distributed throughout the sample for the 30 GPa case. The traces of these stackingfault packets form an angle of 901, which is exactly the expected angle. Later, in Section 7 (Fig. 51(a)), a montage is presented. 6.1.2. [2 2 1] copper impacted at 30 GPa Fig. 21 shows a SEM–ECC picture for the [2 2 1] orientation. At higher magniﬁcation (not shown), the details are illustrated more clearly. Two traces of slip bands are present with an angle of 561. They are the traces of {1 1 1} planes on (2 2 1). Although the substructure of the [2 2 1] copper shocked at 30 GPa is full of bands, the morphology of these bands varies throughout the sample. The formation of similar bands in shocked samples has been described by Gray and Huang [89]. Microbands having widths of 20–30 nm were found within the larger bands. Fig. 22(a) shows the regular slip band morphology. In Fig. 22(b), slip bands were found inside some larger bands. The microstructure on the (1 1� 0) plane in the 30 GPa impacted [2 2 1] sample is shown in Fig. 22(c). Similar bands as shown in Fig. 22(a) were also observed. These bands align with ½1� 1 2� orientation, which indicates that they might be the traces of ð1 1 1Þ planes on (1 1 0). The basic difference with the [0 0 1] crystal is that two (or more) systems are simultaneously activated in the former, whereas

Fig. 21. SEM–ECC picture for 30 GPa shocked [2 2 1] Cu samples: front surface of the sample perpendicular to the shock propagation direction [87].

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Fig. 22. (a) Regular slip bands in Cu; (b) the formation of bands when electron beam direction is B ¼ [0 0 1]. Slip bands were formed inside those bands; (c) two sets of slip bands interact with each other [87].

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Fig. 23. SEM image of 57 GPa post-shocked copper sample with [0 0 1] orientation [87].

Figs 22(a) and 22(c) show primarily one trace. In contrast with [0 0 1], one slip system is highly activated with minor activity in the cross-slip system. 6.1.3. [0 0 1] copper impacted at 57 GPa SEM–ECC analysis shows that the shock-induced structures of the surface perpendicular to the shock propagation direction (Fig. 23) consists of a mix of recrystallized grains (area A in the picture), and bands (area B) with a width of 15–16 mm. TEM conﬁrms that the structure is not uniform. Microtwins, dislocation tangles, deformation bands, and slip bands are seen in the regions. The diversity of the post-shocked microstructures was induced by the high shock pressure and post-shock heating. Microtwins were observed throughout the sample [Fig. 24(a)]. The electron beam direction is [0 1 1]; they have a ð1� 1� 1Þ habit plane, whose perpendicular is marked in ﬁgure. The sizes of these microtwins vary from 80 to 180 nm. Murr [18] and Johari and Thomas [8] showed that twinning is a favored deformation mechanism under shock loading. This is treated in Section 6.3, where a formal criterion is presented. Fig. 25(a) shows the general view near the back surface of the specimen (foil parallel to shock-front plane). A shear band with a width of about a 1.5 mm crosses the foil. Compared with the slip bands around it, this shear band is larger and breaks the other slip bands. The microbands in Fig. 25(a) have distinct characteristics. The vertical bands are larger than the horizontal ones, whereas the number of horizontal ones is much higher than that vertical ones. Fig. 25(b) is a detailed image of these slip bands. Two sets of slip bands having a width of about 0.5 mm are shown. The direction of vertical slip bands was identiﬁed as ½1 1� 2�, which might be the trace of a ð1� 1 1Þ plane. The horizontal bands seem to be cut by the vertical ones and recovery effects appear in these bands. By measuring the distances between the repeated structures in both Figs 25(a) and 25(b), we found that they have the same width of around 500 nm. The periodicity of the features of both the dislocations and bands is remarkable. We speculate that these dislocation features are due to the recovered slip traces seen in Fig. 25(a).

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Fig. 24. Microtwins in 57 GPa post-shocked [0 0 1] copper samples: (a) image of microtwins; (b) microtwins with the habit plane of ð1� 1� 1� Þ shown at the electron beam direction of (0 1 1) [87].

Deformation bands are shown in Fig. 25; these localized shear bands undergo thermal recovery in places [Fig. 25(c) and 25(d)]. Between these bands, there are dislocation tangles and in some places the dislocation density is very high. The dislocation density was lower and the arrays were extended in the second thin foil along the shock direction. Mughrabi et al. [90] found some dislocation cell structures very similar to our observations, but they are quite unlike the cells observed by other investigators (e.g., Johari and Thomas [8]). Gray and Follansbee [91] concluded that increasing peak pressure or decreasing pulse duration (dwell

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Fig. 25. TEM of 57 GPa post-shocked [0 0 1] copper samples: (a) overview of the sample (�10 K) showing shear bands; (b) slip bands; (c) bands that underwent recovery; (d) detail of recovered band from (c) [87].

time of pulse) decreased the observed dislocation cell size and increased the yield strength. The dislocation cells were extended and, therefore, showed some deformation characteristics. Murr [18] measured the dislocation cell sizes in shock-compressed Cu and Ni. For a shock pressure of 57 GPa, one would expect cell diameters around 90 nm. Cell-like structures with poorly deﬁned cell walls are also observed in stainless steel [72]. If the shock-pulse duration is low, the

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substructures are more irregular because there is insufﬁcient time for the dis locations generated by the peak pressure (in the shock front) to equilibrate. Other studies conﬁrm substructure consisting of tangled dislocations in cellular arrays. 6.1.4. [2 2 1] copper impacted at 57 GPa The [2 2 1] copper samples shocked at 57 GPa were fully recrystallized. This recrystallization is consistent with post-shock cooling calculations conducted by Cao et al. [88]. For 57 GPa, the calculated residual temperature is 420 K. Although this is sufﬁcient for recrystallization at long times, the post-shock cooling effectively returns the temperature to 300 K in 20–40 s. This would most probably not be sufﬁcient for large-scale recrystallization. Cao et al. [88] proposed that shear localization can lead to temperature rises of up to 500 K above the predictions for shock compression/isentropic release. 6.2. Laser shock compression of copper The principal results obtained by Meyers et al. [37] and Schneider et al. [92,93] are summarized here. Two orientations of single-crystal copper were investigated: [1 0 0] and [1 3 4]. The experiments were done on the Omega laser at the Laboratory for Laser Energetics (LLE) at the University of Rochester in New York. The highpower laser pulse was used to launch a strong shock into the Cu sample, which evolved into a decaying blast wave. The results from radiation–hydrodynamic simu lations give the resulting pressure versus position in the sample, shown in Fig. 26.

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Pressure, GPa

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20

320 J 205 J 10 40 J

0 0

0.5 Distance, mm (a)

1

0.01

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Fig. 26. (a) Simulated pressure proﬁles as a function of distance from the energy deposition surface for a laser energy of 200 J; (b) Maximum pressure as a function of distance from the laser driven surface for three laser energies. In all cases, the laser wavelength was 351 nm, pulse shape was 3 ns square, and the laser spot size on target was B2.5 mm in diameter [37].

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Fig. 26(a) shows the decay of the shock pressure generated by an initial laser energy of 200 J, pulse shape of 3 ns square, and spot size (diameter) of B2.5 mm. As the pulse attenuates, its length increases. The initial duration of the pulse is approximately 10 ns. It can be seen that the pressure is not maintained during the propagation of the pulse; the decay of the maximum pressure for pulses with laser energies of 40, 205, and 320 J is shown in Fig. 26(b). The shock strength at the surface of the Cu crystal can be extracted from the laser energy, pulse length, spot size, using hydrocode calculations. This can be veriﬁed by VISAR measurements. Due to the short duration of the shock created by the 3 ns laser pulse, the decay in the specimen is very rapid as shown by calculated pressure proﬁle. Snapshots of these pressure proﬁles at various times up to a depth of 1 mm are shown in Fig. 26(a) for a laser energy of 200 J. The amplitude of the pressure wave in the sample decays substantially and the pulse duration broadens as a function of distance. Fig. 26(b) shows the decay of the maximum pressure in the specimens at these three laser energy levels. 6.2.1. TEM of pure copper For the [0 0 1] orientation, shock experiments at 12 and 20 GPa pressures create a cellular organization with a medium density of ½[1 1 0]-type dislocations. The average cell size is between 0.2 and 0.3 mm for the 20 GPa case. Qualitatively, these results conﬁrm previous observations, albeit at a pulse duration that is lower by a factor of 10–100 than that applied by Murr [18,70,73,94]. Fig. 27 shows a plot of that data. The predicted cell size from Murr’s data, at a pressure of 12 GPa, is 0.4 mm. One interesting feature is the observation of a large number of dislocation loops. For example, loops as small as 25 nm and as large as 250 nm are indicated in 35 Murr

PRESSURE, GPa

30

Gray Current results

25

20 15

10 5 0.1

0.2

0.3

0.4 0.5 CELL SIZE, um

0.6

0.7

0.8

Fig. 27. Cell size as a function of pressure for shock-loaded copper [37]. Adapted from Murr [40] and

Gray [20].

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Fig. 28(a). Given the density of loops observed, far greater than that observed in undeformed Cu, we suggest that loop nucleation is an essential component of laser-induced shock compression. This is consistent with the mechanism of plastic deformation presented in Section 2.3 and schematically shown in Fig. 28(b). This mechanism for dislocation generation at the shock front, based on the nucleation of dislocation loops and their expansion behind the front, is still evolving.

Fig. 28. (a) Observation of numerous loops in the 40 J shocked Cu specimens. The different sizes (l ¼ large; s ¼ small) and shapes (e ¼ elongated) of the high density of loops are indicated in B ¼ [1 0 1]. (b) Nucleation of dislocation loops behind the shock front [93].

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Fig. 29. Four sets of stacking faults (marked as A, B, C, D) are observed in [0 0 1] Cu at 200 J (20 GPa): Variant A exhibits the highest density of occurrence, g ¼ 200, B ¼ [0 0 1] [37].

A laser energy of 200 J (40 GPa initial pressure) created dense dislocation tangles and stacking faults. There are no readily discernible dislocation cells, but four variants of stacking faults are observed. These traces are analogous to previous observations by Murr [18,73]. The features are signiﬁcantly different than the dislocation cells observed at the lower energy. These traces have orientations /2 2 0S, as shown in Fig. 29. Single-crystal copper samples with [1� 3 4] orientation were shocked at energies of 70 and 200 J corresponding to initial pressures of 20 and 40 GPa. The speci mens shocked at 20 GPa contained a well-deﬁned cellular network comprised of 1/2/1 1 0S dislocations with a slightly larger (0.3–0.4 mm) average cell size (see Fig. 30), as compared to the [0 0 1] orientation. The dislocation density is on the order of 1013 m�2. The cells are comprised primarily of three dislocation systems: (1 1 1)[1� 0 1], (1 1 1)[1 1� 0], and (1� 1 1)[1 0 1].

At the higher energy of 200 J for the [1� 3 4] orientation, the deformation substructure continued to be cellular, albeit with a ﬁner (0.15 mm) average cell size and a signiﬁcantly higher dislocation density, 1014 m�2 [Fig. 30(b)]. This is in direct contrast to the mechanism change observed in [0 0 1]. Again, the three slip systems previously described dominate the deformation substructure. A large number of loops are also visible. These were found to contribute to the cell walls and were often commonly found within the cells. The difference observed between the defect substructure of the [0 0 1] and [1� 3 4] orientations is due to the number of activated slip systems. Because of the

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Fig. 30. (a) Defect substructure of monocrystalline copper with orientation ½1� 3 4�, shocked with a laser energy of 70 J. Probed on the TEM with beam direction [0 1 1]; g ¼ ½2 2� 2�; (b) defect substructure of monocrystalline copper with orientation ½1� 3 4�, shocked with a laser energy of 200 J. Probed on the TEM with beam direction [0 1 1]; g ¼ ½2� 2� 2� [92,93].

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symmetry of [0 0 1], multiple slip systems are activated, and interactions between dislocations are more common, which enable the defects to relax into a stacking fault-dominated substructure. The [1� 3 4] orientation, consisting of dislocations with limited mobility and interaction, continues to form cells as the relaxed substructure to higher-pressure levels. 6.2.2. Pressure decay effects in pure copper Figs 31(a)–31(c) show the dislocation cells for a laser irradiation energy of 200 J at three locations: A, C, and E, corresponding to distances into the sample of 0.25, 1.25, and 2.25 mm (see Fig. 3 in Ref. [92]). The decay in pressure, shown in Fig. 26, is accompanied as expected by an increase in cell size and decrease in dislocation density. The average cell sizes are: 0.14 mm for specimen A [Fig. 31(a)]; 0.22 mm for specimen B (at a distance of 0.75 mm from the driven surface, not shown); 0.41 mm for specimen C [Fig. 31(b)], 0.76 mm for specimen D (at a distance of 1.75 mm, not shown), and 1.43 mm for specimen E [Fig. 31(c)]. The dislocation densities decrease from 1014 m�2 at the front to 1011 m�2 at position E. In Fig. 32(a), the cell sizes as a function of distance from the laser irradiated surface are plotted for the three energies. The cell sizes vary consistently with the three energy levels. Fig. 32(b) shows that the cell size and pressure correlate at different locations within the specimen. 6.2.3. Copper–aluminum alloys As previously mentioned, the addition of small amounts of aluminum (o7 wt.%, the solubility limit) lowers the SFE of copper, which is approximately 78 mJ/m2. Early work by Johari and Thomas [8] demonstrated this effect on the defect substructure in copper–aluminum alloys. In this section, a detailed analysis of the effect of SFE on the threshold pressure for twinning is presented. Systematic differences were observed by transmission electron microscopy in the deforma tion substructures of the different compositions: copper–2 wt.% aluminum (Al) (4.2 atomic percent) and copper–6 wt.% Al (12.6 atomic percent). The experimen tally obtained stacking-fault energies of the Cu–2 wt.% Al and (Cu–6 wt.% Al) are 37 and 5 mJ m�2, respectively. Both pressure and crystal orientation signiﬁcantly affected the deformation substructures of laser-shocked Cu–2wt.% Al. The samples with [0 0 1] orientation shocked at 70 J (20 GPa initial pressure) had regular cells with an average size of 250 nm and cell wall thickness of 50 nm. The average dislocation line length was considerably longer, 150 nm, and the dislocation density was on the order of 1014 m�2. The dislocations were also observed to gather on the primary planes. The dislocation substructure for this condition (70 J) was made of the eight primary slip systems. In the Cu–2 wt.% Al with [0 0 1] orientation and shocked at 200 J, stacking faults were readily observed as the dominant defect substructure (see Fig. 20 in Ref. [87]). Because of the 2 wt.% addition of aluminum, the SFE is nearly half that of pure copper, and one would expect to observe twinning. However, this is not the case. Instead, four stacking-fault variants are observed. These stacking faults are similar to those observed for pure copper (Fig. 29). The faults are well-deﬁned with clean boundaries having a regular spacing of 250 nm.

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Fig. 31. TEM images of the defect substructures showing the pressure decay effects of ½1� 3 4� Cu at different distances from the laser-driven surface at ELaser ¼ 200 J (for each image, the TEM beam � (c) 2.25 mm, g ¼ ½2� 2 2� � [92]. direction B ¼ [0 1 1]): (a) 0.25 mm, g ¼ ½2 2 2� �; (b) 1.25 mm, g ¼ ½0 2 2�;

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Fig. 32. (a) Cell size as a function of distance from the laser driven surface for the ½1� 3 4� crystal orientation at three initial energies: 70, 200, and 300 J. Twinning is observed when dislocation cell sizes fall below an average size of 0.05 mm represented by the bottom line; (b) cell size as a function of estimated pressure for the three energy levels [92].

The Cu–2 wt.% Al with [1� 3 4] orientation was observed to have a substantially different defect substructure than pure copper or Cu–2wt.% Al with [0 0 1] orientation. The effects of the change in SFE were generally more pronounced. The dislocations were arranged in planar arrays. The defect substructure consisted of long dislocation lines as shown in Fig. 33. The dislocation line length averaged

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Fig. 33. Bright ﬁeld images of Cu–2wt.% Al shocked at ELaser ¼ 70 J (20 GPa) along the ½1� 3 4� direction, and imaged with B ¼ (0 1 1) and g ¼ ½0 2 2� � for all conditions: Specimen BB0.75 mm from the driven surface [93].

500 nm and the dislocation density was 1013 m�2. Obviously, for this condition there is one dominant slip system [1� 0 1](1 1 1); the two secondary systems are also observed, but in less proportions. A number of dislocation loops are also observed. Cu–2wt.% Al with [1� 3 4] orientation shocked at 200 J exhibited twinning (specimen A, B0.25 mm from the laser-driven surface, Fig. 34). Two variants are observed. It appears that the larger twin may act as the nucleation site for the second twin. The twins were found in a relatively low proportion, but are the systems predicted by Schmid factor calculations. The twins varied in size and proportion with the primary variant, (1 1 1)[2� 1 1], having an average length of 4 mm � occurred in greater and a width of 20–30 nm. The secondary variant, (1 1� 1)[1� 1� 2], numbers, but with shorter lengths with an average of 2 mm. We expected that a � would also be found, but the co-secondary twinning variant, (1 1� 1)[1 1� 2], occurrence of this system was relatively rare. This suggests that the sample may have been slightly misaligned from the [1� 3 4] loading axis, and thereby favored the two observed twinning systems having higher Schmid factors than calculations indicate. A high density of dislocations was also observed (not shown here). The defect substructure for all energies in Cu–6 wt.% Al with [0 0 1] orientation consisted of either stacking faults or dislocations since, for this system, the SFE is less than 5 mJ m�2. The dislocation structure consists of large planar arrays and regions of dislocation pileup since the low SFE inhibits cross-slip. Many of the � 1/6/1 1 2S. The defect dislocations observed were Shockley partials: {1 1 1} substructure was primarily made up of planar arrays of dislocations and had a dislocation density on the order of 1013 m�2 and a line length of 500 nm. The planar

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Fig. 34. Bright ﬁeld images of Cu–2wt.% Al with ½1� 3 4� orientation shocked at ELaser ¼ 200 J (40 GPa) imaged with B ¼ (0 1 1) and g ¼ ½0 2 2� � for all conditions: Specimen AB0.25 mm from the driven surface [93].

arrays were spaced at a distance of 1 mm. Stacking faults and stacking-fault tetrahedra were also observed. The fault spacing was equivalent to the distance between planar arrays (1 mm). For the Cu–6 wt.% Al with [0 0 1] orientation and laser shocked at 200 J, the defect substructure was predominantly stacking faults [Figs 35(a)–35(c)]. The stacking faults had a width of 100 nm, length of 1 mm, and spacing of 400 nm. The areal density was 0.84 � 105 m�1. Dislocations were also observed throughout the specimen, typically near the fault boundaries. For the [1� 3 4] orientation of the Cu–6 wt.% Al, three variants of stacking faults were observed in the 70 J condition with one system preferred. Dislocations were also observed. The stacking-fault width was 250 nm on average and the spacing 300 nm. The areal density was on the order of 0.1 � 105 m�1. Dislocations were arranged in planar arrays and tangles with a density of 1013 m�2. The dislocation line length and planar spacing was about 1 mm and 250 nm, respectively. The Cu–6 wt.% Al [1� 3 4] specimens shocked at 200 J contained a residual defect substructure similar to the 70 J specimens. Partial dislocations dominate the defect substructure, which comprised a dislocation density of 1013 m�2 and an average line length of nearly 1 mm. The dislocations are preferentially aligned along speciﬁc planes with a spacing of 1 mm and there is one primary slip system, [1� 0 1] (1 1 1). Some stacking faults were also observed with most being aligned to [2� 1 1](1 1 1). Stacking faults typically formed at high pressures and then were found to decay into either cells or planar arrays of dislocations as the pressure decayed through the sample. As expected, decreasing SFE enhanced the propensity to form stacking faults for both orientations. Similarly, cells and planar arrays became more clearly

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Fig. 35. Bright ﬁeld images of Cu–6wt.% Al with [0 0 1] orientation shocked at ELaser ¼ 200 J (40 GPa) imaged with B ¼ (0 0 1) and g ¼ [0 2 0] for all conditions: (a) Specimen AB0.25 mm from the shocked surface; (b) Specimen BB0.75 mm from the shocked surface; (c) Specimen CB1.25 mm from the shocked surface [93].

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deﬁned as the pressure decreased and pulse duration increased. Twinning was not readily observed in most of these conditions suggesting there may be some unresolved time dependence in twin nucleation, or more complex factors affecting twinning (Bernstein and Tadmor [41]). However, it is also possible that many of the stacking faults observed are actually nanotwins as the thickness of the twins could be so small that traditional transmission microscopy methods may be unable to resolve the changes in the structure. The experimental results are plotted in Figs 36(a)–36(c). The positions A–D were converted into pressures through the radiation–hydrodynamics simula tions. The transition from loose dislocations/cells to stacking faults/twins is approximately indicated in Fig. 36(a). Figs 36(b) and 36(c) show the change of dislocation densities and stacking-fault densities versus pressure, respectively. The energetics of loop nucleation for perfect and partial dislocations is discussed in Section 6.4. Both deformation twinning and stacking-fault formation are the direct consequence of partial dislocation loop nucleation and expansion. In the case of twinning, one has separated and prescribed arrays of partial dislocation loops on adjacent planes.

6.3. The slip–twinning transition in Cu and Ni The primary aim of this section is to provide a constitutive description of the onset of twinning in both copper and nickel. Copper–aluminum and nickel–tungsten are also modeled. The parameters affecting slip and twinning will be discussed ﬁrst, followed by modeling of the onset of twinning. Predictions of the model are compared to experimental work. 6.3.1. Modeling of the slip stress 6.3.1.1. Monocrystalline Cu and Cu–Al. The constitutive response for slip in FCC metals is well-modeled by the Zerilli–Armstrong constitutive description [28], which captures the essential physical phenomena. For monocrystalline Cu, the equation used is as follows: sS ¼ sG þ C 2 f ð�Þ expð�C 3 T þ C 4 T lnð�_ÞÞ þ ks d �1=2 ,

(21)

where sG is the athermal component of stress, e the strain, f(e) the work hard ening factor, d the grain size, T the temperature, ks the Hall–Petch slope, and C2, C3, and C4 are constants. sG, C3, and C4 are adopted from Ref. [28] and C2 ¼ 115 MPa. The work hardening f(e) was incorporated by taking a poly nomial representation of the stress–strain curve for single crystals with the [0 0 1] and [1� 3 4] orientations from Ref. [37]. This is the only manner by which three-stage response can be incorporated without excessive complexity. The [0 0 1] orientation is expected to have the lowest threshold pressure for twinning of all orientations,

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Fig. 36. Experimental results for laser-shocked Cu–Al alloys: (a) Experimentally observed transition from dislocation cells and planar arrays to stacking faults and twins as a function of composition and crystal orientation; (b) experimentally observed dislocation densities as a function of pressure and composition; (c) experimentally determined areal densities of stacking faults as a function of pressure and composition [93].

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whereas [1� 3 4] should have a substantially higher threshold pressure due to its more gradual hardening. The polynomials used in these calculations are: For [0 0 1]: f ð�Þ ¼ 19466:2 �6 � 18522:2 �5 þ 7332 �4 � 1582 �3 þ 189:5 �2 � 2:4 � þ 0:07. (22) � For [1 3 4]: f ð�Þ ¼ �6293 �6 þ 7441:4 �5 � 3163 �4 þ 515:65 �3 � 4 �2 þ 0:13 �1 þ 0:059. (23) The addition of small amounts of aluminum in copper not only lowers the SFE, but drastically inﬂuences the strength and hardness. In pure metals, dislocations are relatively mobile, but when solute atoms are added the dislocation mobility is greatly reduced. In these alloys, the solute atoms become barriers to dislocation motion and can have the effect of locking them. Substantial work has been done developing solid-solution theory for concentrated solid solutions [95–98]. The ﬂow stress of concentrated solid solutions increases with the atomic concentration of the solute. For many systems, the following proportionality is observed: s0 / ½CS �m ,

(24)

where s0 is the ﬂow stress, and CS the concentration of the solute, and m a parameter that is found to vary between ½ and 1. Copper–aluminum has been shown to follow this description [95] with m ¼ 2/3. Therefore, we incorporated this compositional term into the modiﬁed Z–A equation as shown below 2=3

ss ¼ sG þ C S C 2 f ð�Þ expð�C 3 T þ C 4 T lnð�_ÞÞ þ ks d �1=2 .

(25)

6.3.1.2. Ni. For Ni, sG ¼ 48.4 MPa, C2 ¼ 2.4 GPa, C3 ¼ 0.0028 K�1, C4 ¼ 0.000115 K�1, and ks ¼ 0.2 MN m�3/2 in eq. (21). A strain-hardening function is taken as f(e) ¼ en in the Z–A equation. The strain-hardening exponent, n, in the nc Ni regime was simply equated to 0 as determined by measurements carried out on the same material by Choi et al. [99]. The values of C3 and C4 used are those for copper [28] since data on Ni was not available. The nickel Hall–Petch slope for slip, ks, has been established by several researchers [100–103] and Asaro and Suresh [104] compiled hardness data for nickel spanning both the micrometer and nanometer regimes. A ks value of B0.2 MN m�3/2 was calculated from that set of data. Stress–strain plots of nickel with micrometer sized grains were utilized to establish C2. The current model predicts a yield strength of B1.9 GPa for Ni having a grain size of 30 nm, which is in good agreement with the literature [104]. 6.3.1.3. Ni–W, 13 at.%. Roth et al. [105] obtained the increase in yield stress in Ni as a result of alloying with different elements. They estimate that the ﬂow stress of Ni increases from 100 to B450 MPa due to the addition of 13 at.% W. A plot of the increase in ﬂow stress of Ni with tungsten content is shown in Fig. 37(a). The data

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600

Flow Stress (MPa)

500 400 300 200 100 0

0

2

4

6 8 10 Concentration (at. %)

(a)

12

14

16

500

Fe - 25% Ni (BCC - Nilles & Owen)

Fe - 3% Si (PX, Löhe & Vöhringer)

Twinning Shear Stress, MPa

400

Fe - 3.3% Si (SX, Narita & Takamura)

300 Fe - 2.5% Si (PX, Löhe & Vöhringer)

200

Fe (SX, Harding)

Fe (SX, 103 s–1, Harding)

Cu (SX, Thornton & Mitchell) Ag (SX, Narita & Takamura)

100

Zr (PX, Song & Gray)

Cu - 20% Zn (SX, Thornton & Mitchell)

Ag - 4% In (Narita & Takamura)

0

0

100

200 Temperature, K (b)

300

400

Fig. 37. (a) Slip stress of Ni as a function of the concentration of W (at.%). Adapted from Meyers et al. [106]; (b) twinning stress as a function of temperature for a number of metals – both mono and polycrystals. Adapted from Meyers et al. [106].

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was extracted from work carried out on Ni having a grain size between 100 and 300 mm. The effect of solid–solution addition to the yield stress increment is as follows: !m X 1=m K i Ci , (26) DsSS ¼ i

where m is B1/2, Ki is the strengthening constant for solute i, and Ci is the concentration of solute i (for W, Ki ¼ 977 MPa at. fraction�1/2). The Zerilli– Armstrong equation as a function of tungsten content is obtained by adding the solid–solution term into the athermal component of stress: !m X 1=m sslip ¼ sG þ K i Ci þ C 2 �n expð�C 3 T þ C4 T ln �_ Þ þ ks d �1=2 . (27) i

The strain-hardening exponent, n, for the nc Ni–W samples was again equated to 0. The Z–A model predicts a yield strength of B2.2 GPa for Ni–W with a grain size of 10 nm, very close to the 2.38 GPa value reported by Choi et al. [99]. We esti mated the Hall–Petch, ks, slope for Ni–W using yield strength data on Ni–W samples having grain sizes in the micrometer regime and microhardness measurements carried out on the nc Ni–W samples. A ks value of 0.1 MPa m�3/2 was estimated. 6.3.2. Modeling of the twinning stress In shock loading, the dislocation arrangements are more uniform than after quasistatic deformation of the material. High SFE materials often are found to twin above a threshold pressure during shock compression whereas they may never twin at quasi-static conditions except at very low temperatures. Twinning propensity, however, increases in both modes of deformation (quasi-static and high strain rate) when the SFE is decreased. SFE can be manipulated in materials by alloying. For example, in copper, which has a relatively high SFE (78 mJ/m2), the SFE is nearly cut in half by adding 2 wt.% Al. This effect can be correlated to the change in the electron to atom ratio (e/a) in an alloy as given by: e (28) ¼ ð1 � xÞZ 1 þ Z 2 ¼ 1 þ xdZ; a where x is the atomic fraction of the solute in the alloy, Z1 and Z2 are the number of valence electrons for the solute and solvent atoms, respectively, and dZ equals (Z1�Z2). Despite the fact that dislocation activity is directly associated with twinning, slip by dislocation motion is much more sensitive to strain rate and temperature [107–110], whereas twinning is much less sensitive to these parameters [106]. Fig. 37(b) shows the twinning shear stresses as a function of temperature for a number of metals, and clearly indicates that the twinning stress is temperature insensitive over the range considered. This trend is actually still subject to debate as results have been conﬂicting. In their review article on mechanical twinning, Christian and Mahajan [111] proposed that BCC metals have a negative

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dependence of twinning stress on temperature, whereas FCC metals have a weakly positive dependence. In the analysis on the onset of twinning that follows, it is assumed that the twinning shear stress is insensitive to temperature, pressure (except in Section 6.5.4) and strain rate. 6.3.2.1. Grain-size and stacking-fault energy effects on twinning. The effect of grain size on the twinning stress has been found to be greater than that on the slip stress for many metals and alloys [106,111]. A Hall–Petch relationship can, thus, be ascribed to the twinning stress: sT ¼ sT0 þ kT d �1=2 ,

(29)

where kT is the twinning Hall–Petch slope (higher than the ks slope for slip), sT0 is the initial twinning stress assumed for a monocrystal ðlimd)1 ðd �1=2 Þ ¼ 0Þ, and d is grain size. The normal twinning stress (sT) used in this calculation was 300 MPa for pure copper. We assume that this critical stress remains constant. Haasen [112] carried out low-temperature tensile tests on monocrystalline Ni and observed twinning at 4.2 K and 20 K at a shear stress considerably higher than that for copper. This shear stress was estimated to be equal to 250–280 MPa, which is equivalent to a normal stress, sT0 , of 500–560 MPa. Meyers et al. [113] conducted shock compression experiments on copper up to pressures of 35 GPa. They detected an abundance of twins for grain sizes between 100 and 300 mm, but found no traces of twinning at a grain size of B10 mm. Similar results were obtained by Sanchez et al. [74]. Vo¨hringer [95] established that the twinning Hall–Petch slope for copper, kT, is B0.7 MN m�3/2, which is signiﬁcantly higher than that for slip, ksB0.3 MN m�3/2. In the present modeling of nickel, we assume that kT for nickel is three times ks. Thus, a kT value of 0.6 MN m�3/2 is used for Ni. Solid–solution strengthening and SFE effects are incorporated into the slip– twinning model as a result of alloying with tungsten. The addition of solute atoms hinders the movement of dislocations, hence, creating a strengthening effect [114]. Alloying also signiﬁcantly reduces the SFE, gSF . For instance, it has been shown that the SFE of copper decreases by nearly 50% by the addition of 2 wt.% Al [115]. This effect is related to the change in the electron to atom ratio (e/a). Partial dislocations are under elastic equilibrium, where the repulsive forces between the bounding partials are balanced by the forces needed to minimize the stacking-fault area and maintain a minimum energy conﬁguration. Thermodynamically, alloying can alter the difference in the free energy between the HCP (stacking-fault ribbon) and FCC structures and, therefore, the energy of the ribbon between two partials as well as their separation. The twinning stress, tT , is shown to vary with SFE. Venables [116,117] and Vo¨hringer [118,119] performed extensive analyses on the twinning stress for a number of alloys and found that it varies with the square root of the SFE: � � gsf 1=2 tT ¼ k , (30) Gbs

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100 (134) (001)

Pressure (GPa)

80 60 40 20 0

0

1

2

3 4 5 Weight % Aluminum

6

7

Fig. 38. Calculated critical shock pressures for the transition from slip to twinning for Cu-Al alloys as a function of increasing aluminum concentration.

where k is a proportionality constant and G is the shear modulus and bs is the Burgers vector. A good ﬁt is obtained for copper and nickel alloys with a k value of 6 and 6.8 GPa, respectively. Recently, there have been attempts to obtain twinning stresses by atomistic methods. The atomistic studies point to the relevance of the SFE, but also to the need to take into account other variables, like the unstable stacking-fault and twinning energies (Berstein and Tadmor [41], Van Swygenhoven et al. [120], Ogata et al. [121], and Siegel [122]). Fig. 38 shows the critical twinning stresses for copper and copper–aluminum alloys. The following values for stacking-fault energies were used: Cu–2wt.% Al: 37 mJ/m2; Cu–4wt.% Al: 7 mJ/m2; Cu–6wt.%Al: 4 mJ/m2. The twinning stresses are calculated based on the calculated stacking-fault energies using eq. (30), neglecting the grain-size differences. For Ni–W, the shear modulus and the SFE, gSF, as a function of W were obtained by Tiearnay and Grant [123]. For Ni–13at.% W, G ¼ 88 GPa and the gSF ¼ 52.5 mJ/m2 (a decrease of 60% over pure Ni). Assuming a twinning Hall–Petch slope three times that of slip, we obtain a kT value for Ni–W equal to 0.3 MPa/m3/2. Just as in the case of pure Ni, a Hall–Petch behavior accounting for the effect of grain size on the twinning stress is adopted in predicting the critical twinning transition pressure in Ni–W (13 at.%). The following expression for the twinning stress was used: �

g sT ¼ k sf Gbs

�1=2

þ kTNiW d �1=2 .

(31)

For Ni–13 at.% W, k ¼ 6.8 GPa, kTNiW ¼ 0:3 MPa, gsf ¼ 52:5 mJ m�2 , G ¼ 88 GPa, bs ¼ 0.249 nm.

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6.3.2.2. Critical pressure for slip–twinning transition. In this analysis, we assume that the transition from slip to twinning occurs when the shear stress for twinning, tT, becomes equal to or less than the shear stress for slip, ts: tT � t s .

(32)

If one uses the same conversion parameters: sT � ss .

(33)

This is a reasonable approximation since both mechanisms are subjected to the same stress system at the shock front. Since the criterion described here is based on the critical shear stresses for slip and twinning, the pressure only enters insofar as it determines the shear stress and strain rate. We assume the twinning stress, sT, to be pressure and temperature independent. (Since the pressure dependence of both sslip and stwinning is generally taken to scale with the shear modulus, G(P)/G0, then it affects slip and twinning equally, and we can ignore pressure hardening in this slip–twinning analysis.) The dependence of shock pressure on strain rate for Ni, obtained through the Swegle–Grady relationship [124], is not available in the literature. As an approximation, the strain rate versus pressure behavior of copper is adopted. The reasoning for this approximation is that Al and Cu, both FCC metals, have a strain rate response to shock pressure that is very comparable even though the SFE of Al is much higher. One would expect that the behavior of Ni should not signiﬁcantly deviate from that of Al and Cu. Thus, the Swegle–Grady relationship for Ni is given as follows: �_ ¼ 7:84 � 10�33 � P4shock ,

(34)

where the pressure is in Pa and the strain rate in s�1. Two separate aspects have to be considered in the analysis: (a) plastic strain at the shock front and (b) shock heating. Both plastic strain by slip (and associated work hardening) and shock heating alter the ﬂow stress of a material by slip processes and need to be incorporated into the computation. The total (elastic þ plastic) uniaxial strain, e, at the shock front is related to the change in speciﬁc volume by: V ¼ e� . V0

(35)

The pressure dependence on strain, determined from the Rankine–Hugoniot equations, equation of state (EOS), and eq. (14) is expressed as follows [37,19]: Pshock ¼

C 20 ð1 � e� Þ . V 0 ½1 � Sð1 � e� Þ�2

(36)

The equations modeling the associated temperature rise in Cu and Ni as a func tion of shock pressure are represented below, which are second-order polynomials that were generated from thermodynamically calculated data in Ref. [19]: T shock Cu ¼ 10�19 P2 þ 2 � 10�9 P þ 295:55 K; T shock Ni ¼ 8 � 10�20 � P2shock þ 9 � 10�10 � Pshock þ 301:5 K;

(37)

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where the pressure is in Pa and the temperature is in K. For Ni–W, the temperature rise and strain associated with a given shock pressure are determined just as outlined in the case for pure Ni. Fig. 39(a) shows both the slip stress, ss (incorporating thermal softening, strain rate hardening, and work hardening) and sT as a function of pressure for nickel. The point at which the horizontal line determined by sT, eq. (31), intersects the slip stress at a given shock pressure, is deﬁned as the critical twinning transition pressure. This transition pressure for nickel having a grain size of 30 nm was found to be B78 GPa and is consistent with the fact that twins are not observed in experiments up to pressures of B70 GPa. The twinning transition pressure for nc Ni–W, 13 at.%, having a grain size of 10 nm is illustrated in the plot in Fig. 39(b).

7 6 σT

Stress (GPa)

5 4 3

σS

2 1 0

0

20

40 60 80 Shock Pressure (GPa) (a)

100

120

5 σS

Stress (GPa)

4 3

σT

2 1 0

0

10

20

30 40 50 60 Shock Pressure (GPa) (b)

70

80

Fig. 39. (a) Slip and twinning stress versus shock pressure for nanocrystalline (nc) nickel (grain size ¼ 30 nm); twinning threshold B78 GPa; (b) slip and twinning stress versus shock pressure for Ni–W (13 at.%) having a grain size of 10 nm; twinning transition takes place at B16 GPa.

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Twinning Transition Pressure (GPa)

120 100

Ni

80 60 40 Ni-W 13% 20 0 0.001

0.01

0.1 1 10 Grain Size (μm)

100

1000

Fig. 40. Calculated twinning transition pressure versus grain size for Ni and Ni–13 at.% W.

It is equal to 16 GPa, and is consistent with experiments where twins were observed at pressures of B38 GPa. The slip–twinning transition pressure as a function of grain size (micro to nanometer regime) was also calculated. The strain-hardening exponent was varied between n ¼ 0.5 in the micrometer regime (as determined by ﬁtting to stress–strain plots found in Andrade [103]) and n ¼ 0 in the nanometer regime. The result, seen in Fig. 40, clearly shows the much higher transition pressure in Ni as compared to Ni–W as well as the effect of grain size on the slip–twinning transition. 6.4. Dislocation loop analysis: stacking-fault transition The nucleation of dislocation loops was ﬁrst treated by Cottrell [125] and later further developed by Xu and Argon [126], Rice [127], and others. A mechanism was also proposed by Khantha and Vitek [128] for the generation of dislocations under extreme conditions. At pressures above 3–3.2 GPa, the activation energy for loop nucleation is lower than the thermal energy; thus, nucleation becomes thermally activated, whereas under conventional deformation at ambient tempera ture, it is not activated. As previously mentioned, Meyers [36] proposed in 1978 that dislocations in shock compression were homogeneously generated by loop expansion. Fig. 41(a) shows shear loops generated on {1 1 1} planes making an angle of 54.71 with the shock compression plane, (0 0 1). Whereas the nucleation and growth of perfect dislocation loops can lead to the formation of a cellular structure after multiple cross-slip and relaxation of the dislocation conﬁgurations, the stacking-fault packets observed in shock compression above 20 GPa cannot be accounted for by this mechanism. The corresponding nucleation of partial loops is shown in Fig. 41(b). The calculation introduced by Meyers et al. [19,129] for the energetics of nucleation of partial dislocation loops in copper was extended by Jarmakani et al. [130] to nickel. The analytical development is reproduced for the sake of clarity and continuity. The critical radius, rc (Fig. 41), can be found from the maximum of the

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Fig. 41. Nucleation of dislocation loops along {1 1 1} slip planes behind the shock front, which is in red (propagation along [0 0 1]): (a) perfect dislocations and (b) partial dislocations ([139]).

energy versus radius curve (Kan and Haasen [96] and Hull and Bacon [131]): dE ¼ 0. dr

(38)

The total energy of a perfect dislocation loop with radius r is the sum of the increase of the energy E1, due to a circular dislocation loop (assumed to be one-half edge and one-half screw), and the work W carried out by the applied stress t on the loop (assumed to be circular): � � � � 1 2 2�n 2r E ¼ E 1 � W ¼ Gb r ln � pr2 tb, (39) 2 1�n r0

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where n is Poisson’s ratio, G the shear modulus, b the Burgers vector, and t the shear stress. The critical radius is obtained by taking the derivative of eq. (39) with respect to r and applying eq. (38): � �� Gb 2 � n 2rc rc ¼ ln þ1 . (40) 8pt 1 � n r0 To obtain the total energy of the partial dislocation loop [Fig. 41(b)], both the energy of the stacking fault, E2, and work done by shear stress, W, have to be incorporated: E ¼ E 1 þ E 2 � W. Substituting the values of E1, E2, and W into eq. (41): � � � � 1 2�n 2r E ¼ Gb2p r ln þ pr2 gsf � pr2 tbp , 4 1�n r0

(41)

(42)

where gsf is the SFE and bp is the Burgers vector for a partial dislocation. The critical radius is obtained by the same method: pﬃﬃﬃ � ��

2�n 2rc

Gðb= 3Þ2 pﬃﬃﬃ ln þ1 . (43) rc ¼ r0 8p½ðtb= 3Þ � gsf � 1 � n For Ni, we have n ¼ 0.31, gsf ¼ 130 mJ m�2 , and G ¼ 76 GPa at zero pressure. G changes with pressure as follows [132]: G ¼ 76 þ 1:37 P ðGPaÞ.

(44)

The Burgers vector; b0, at P ¼ 0 is equal to 0.249 nm; it changes with shock pressure as: " sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ !#1=3 4PSV 0 2SðS � 1ÞV 0 P C 20 b¼ 1þ þ �1 b0 , (45) 2PS 2 V 0 C 02 C 20 where C0 is 4.581 km/s, S is 1.44, and V0 is the speciﬁc volume of Ni (m3/kg) at zero pressure. The shear stress, t, assuming elastic loading can be calculated from the shock pressure through: t¼�

1 � 2n Pshock . 2ð1 � nÞ

(46)

The calculated results are shown in the normalized plot of Fig. 42(a) (pressure and critical radius are divided by the shear modulus and Burgers vector, respectively). Evidently, the critical radius for perfect dislocations is lower than for partial dislocations at lower pressures; whereas with increasing pressure, partial dislocations become more favorable. The predicted transition pressure for Ni is B27 GPa, close to the experimentally observed twinning transition pressure,

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0.5 Partial-Ni Perfect-Ni Partial-Cu Perfect-Cu

0.4 Ni

P/G

0.3

0.2 Cu 0.1

0

0

5

10

15

20 r/b (a)

25

30

35

40

(001)

CELLS

STACKING FAULTS 1 μm (b) Fig. 42. (a) Theoretical result showing the critical radius of perfect and partial dislocations for Ni and Cu decreasing with shock pressure; (b) stacking faults and cells in the same TEM micrograph of laser driven, ramped compression of [0 0 1] Cu at Pmax ¼ 24 GPa, demonstrating that there is a critical value for the transition.

35 GPa [9,73,75,94], and about half the pressure at which stacking faults began to appear in our Ni MD study (Jarmakani et al. [130]). The predicted transition pressure for Cu, B5 GPa, is also signiﬁcantly lower than both MD and experi mentally observed results [92,93]. Experimental evidence for the cell-stacking-fault transition has been gradually amassing for copper, and the TEM micrograph of Fig. 42(b) is clear. For Ni, the transition pressure is much higher (27 GPa). This exceeds the critical pressure for twinning (P ¼ 16 GPa, calculated in Section 6) and

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is consistent with the absence of stacking-fault observations in shock-compressed nickel. Thus, one has the following defect regimes as P is increased: Cu: cells-stacking-faults-twins Ni: cells-twins. The TEM micrograph from Fig. 42(b) comes from a quasi-isentropic laser compression experiment at a nominal pressure of 24 GPa for a [0 0 1] copper monocrystal. One sees adjacent regions of stacking faults and dislocation cells, with a well-deﬁned discrete boundary. This was a fortuitous observation and the transi tion can be caused by pressure or strain rate. Nevertheless, it clearly illustrates the dual nature of the microstructure induced. 6.5. Quasi-isentropic compression of metals ICE is a shockless process where very high-pressure conditions can be accessed in ramp wave loading, and the accompanying temperature rise is much less severe than during shock experiments. The main motivation behind such a process is that the solid state of a material can be retained at higher pressures due to the lower temperatures experienced, and an understanding and characterization of the material response is, therefore, possible. In fact, quasi-isentropic experiments come very close to simulating conditions that occur in the core regions of planets [1 2 3]. ICE experiments in the early seventies were aimed at mimicking these conditions. Quasi-isentropic compression conditions can be achieved by various methods: gas-gun, laser, and magnetic loading. Early work on ICE with a gas-gun by Lyzenga et al. [133] used a composite ﬂyer plate with materials of increasing shock impedance away from the target material. Barker [134] placed powders of varying densities along a powder blanket and pressed the blanket to produce a pillow impactor having a smooth shock impedance proﬁle. Similarly, this current effort uses density-graded impactors. In the case of ICE via laser, McNaney et al. [135] used a shockless laser drive setup to compress and recover an Al alloy. A smoothly rising pressure pulse is generated by focusing a laser beam on a reservoir material (carbon foam), creating a plasma that ‘‘stretches out’’ through a vacuum and stagnates or piles up onto the sample. In magnetically driven experiments [136], the Z accelerator at Sandia National Laboratories (SNL) is capable of producing quasiisentropic compression loading of solids using magnetic pulses. An advantage of this method is that a smoothly rising pressure proﬁle can be generated without the initial spike at low pressures seen during impact experiments. Control over loading pressures and a rise time is also possible in graded density impactors to meet experimental requirements [137]. 6.5.1. Gas-gun ICE setup Quasi-ICEs via gas-gun were carried out on [0 0 1] copper and the recovered deformation substructure was analyzed. A two-stage gas-gun setup located at LLNL provided for the quasi-isentropic loading. It employs functionally graded

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Pressure (Mbar)

0.6

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0.5 0.4

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18 GPa C

0.2 0.1 0.0

Functionally Graded Material (FGM) Impactor (a)

Target

0

5

10 15 Time (μs)

20

25

(b)

Fig. 43. (a) Illustration of a functionally graded material (FGM) impactor hitting a target (darkness proportional to density); (b) pressure versus time proﬁle of gas-gun experiments using FGM impactors [137].

material (FGM) impactors designed with increasing density proﬁles (or shock impedance), as depicted in Fig. 43(a), to produce the smoothly rising pressure proﬁles. Three different FGMs were used, each providing a certain density range. A detailed description of the impactors can be found in Ref. [137]. Three experiments, A (52 GPa, 1700 m/s), B (26 GPa, 1260 m/s), and C (18 GPa, 730 m/s) are reported. The as-received samples belonging to each batch were in the form of cylindrical specimens having an average diameter and thickness of 6 and 3.6 mm, respectively. Two distinct pressure proﬁles were obtained using CALE, a hydro dynamics simulation code; one having a hold time of approximately 10 ms (A and C) and one having relatively no hold time (B), as shown in Fig. 43(b). It should be noted that A exhibited a spike or slight shock at the onset of the pulse duration due to the experimental setup and the likely effect on the microstructural deformation process is brieﬂy discussed in Section 6.5.4. Strain rates obtained via CALE were on the order of 104–105 s�1 lower than laser-driven ICE.

6.5.2. Laser ICE setup The Omega Laser System at the University of Rochester, NY, was used to generate a smoothly rising pressure pulse in the material. This pulse is created by focusing a laser beam on a reservoir material (carbon foam) facing the sample and separated from it by a necessary vacuum gap (B300 mm). The beam creates a plasma that ‘‘stretches out’’ through the vacuum and stagnates on the front face of the sample. The strain rates achieved with this setup were on the order of 107 s�1, three orders of magnitude higher than that of the gas-gun experiments. McNaney et al. [135] use the same shockless laser drive setup to compress and recover [0 0 1] copper and a more detailed description of the setup can be found in their publication. An illustra tion of the setup is provided in Fig. 44(a) accompanied by a typical pressure proﬁle modeled by CALE [Fig. 44(b)] [135]. The three peak pressures reported for the laser ICE experiments are 18, 24, and B59 GPa, very reasonably close to the

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30

Laser Beam

Material

Pressure (GPa)

Vacuum Gap

Reservoir Ablator

20

10 distance from 50 100 150 surface (µm) 25 75 125

0 (a)

0

20

40 60 Time (ns) (b)

80

100

Fig. 44. (a) Schematic of the laser isentropic compression experimental setup; (b) typical pressure versus time proﬁles obtained (B24 GPa).

Fig. 45. (a) Twinned regions in Cu B0.1 mm from surface, 52 GPa; (b) stacking faults at 1.3 mm running along [2 2 0], 26 GPa; (c) elongated and regular cells at 0.13 mm, 18 GPa.

gas-gun pressures. The pressure estimate of 59 GPa is more uncertain than the others because of the lack of benchmarking the data for the reservoir used in the experiment. An extrapolation from higher-pressure data was done instead. Section 9 describes some attempts to reproduce ICE loading using MD simulations. 6.5.3. TEM 6.5.3.1. Gas-gun ICE. TEM samples in Cu analyzed from A (52 GPa) revealed various deformation substructures. Dislocation activity was most abundant, however, other deformation features were found. At approximately 0.1 mm from the impact surface, some limited evidence of twining was found. Fig. 45(a) shows very clear twinned regions. At a TEM beam direction of B ¼ [0 1 1], both small and large twins were observed having ð1� 1� 1Þ twin habit planes. These microtwins are embedded within dislocated laths running along the same direction. The smallest twins measured had lengths of approximately 80 nm, and the longest twins were on

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the order of 1.5 mm. TEM images (not shown here) taken at the same depth with B ¼ [0 0 1] showed twins running along the ½2� 2 0� and [2 2 0] at 901 to each other. In certain areas of the sample, single lath variants and stacking faults with thicker features running along the ½2� 2 0� and ½2� 2� 0� directions were captured. We suggest that these substructures are due to thermal recovery. At 0.7 and 1.2 mm from the surface, heavily dislocated laths running along the [2 2 0] direction were observed having an average thickness of 0.6 and 0.7 mm. Twinning, conﬁrmed by a diffraction pattern, was evident at 1.2 mm. The average dislocation cell size at this depth was 0.15 mm. At 1.8 mm, dislocation cells with an average size of 0.2 mm were mostly abundant. Foils from B (26 GPa) mostly revealed dislocation cells, where the average cell size increased from 0.4 mm at 0.25 mm within the sample to 0.5 mm at 2.7 mm. At 0.9 and 1.3 mm from the impact surface, stacking faults were evident in a few isolated regions running along the [2 2 0] orientation [Fig. 45(b)]. Dislocated laths at 1.8 mm and elongated dislocation cells at 2.3 mm away from the impacted surface were observed stretched along the [2 2 0] direction. For experiment C (18 GPa), relatively large dislocation cells were the most abundant deformation substructure [Fig. 45(c)]. The average dislocation cell size varied from approximately 0.5 mm at 0.13 mm within the specimen to 0.6 mm at 2 mm. Elongated cells along the [2 2 0] direction were observed and some lath-like features were noticed in some regions, in particular closest to the impact surface at B0.1 mm within the sample. The elongated cells seem to have relaxed from the dislocated lath structures located at regions experiencing higher pressures closer to the impact surface. 6.5.3.2. Laser ICE. At the highest pressure of approximately 59 GPa for the laserICE experiments in Cu, a large number of faults/twins was observed [Fig. 46(a)]. They were preferentially oriented along the [0 2 2], identical to what has been reported in laser-shocked copper [92,93] and the gas-gun ICE experiments. They were found near regions of extremely high dislocation densities. Laths spaced at regular intervals of 500 nm (also their average width) were also observed with heavily dislocated regions in between. At a lower pressure of 24 GPa, stacking faults

Fig. 46. Deformation structures of laser isentropically compressed Cu: (a) Twins/laths at 59 GPa; (b) dislocation cells and stacking faults at 24 GPa; (c) dislocation cells at 18 GPa.

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were dominant. An interesting image, Fig. 46(b), was taken of a transitional substructure showing dislocation cells to the right and stacking faults to the left. The average dislocation cell size was 0.2 mm and the cells comprised of /1 1 0S type dislocations. The stacking faults were identical to the four variants observed in laser shock compression having a {1 1 1}1/6/1 1 2S nature. The average spacing was 650 nm with a width of nearly 150 nm. There was no visible difference in the material that contained cells and the area that contained stacking faults. The imaged area was taken from near the center of the sample and deepest part of the crater. This microstructure is also shown in Fig. 42(b). Dislocation cells, Fig. 46(c), similar to those observed in shock loading were the predominant mode of deformation for the samples loaded to 18 GPa. The defects were primarily ½/1 1 0S type dislocations which have relaxed into cells. The cell sizes measured in the isentropic specimens at this pressure were approximately 0.3 mm. One unique characteristic of the isentropic compression was the uniformity of the cell sizes at the given pressure. Unlike shock loading where there was substantial variance between cell sizes [37], the quasi-isentropically loaded specimens were very similar in size and shape. Also, the dislocation cells were more clearly deﬁned as compared to laser-shocked samples previously studied [37]. This is likely a result of the isentropic loading conditions. 6.5.4. Twinning threshold modeling: ICE and shock The Preston–Tonks–Wallace (PTW) [138] constitutive description was used by Jarmakani et al. [139] to determine the critical pressure for twinning in both laser and gas-gun quasi-isentropic compression, as it is very suitable for the very high strain rates in these experiments. It takes into account both the thermal activation and dislocation drag regimes. The instantaneous ﬂow stress in the thermal activation regime can be calculated from Eq. (7) in ref. [138], namely � � � �� t^ s � t^ y 1 t ¼t^ s þ ðs0 � t^ y Þ ln 1 � 1 � exp �p p s0 � t^ y � �� (47) �pyc � exp , ðs0 � t^ y Þ½expðpððt^ s � t^ y Þ=ðs0 � t^ y ÞÞÞ � 1� where t^ s and t^ y are the work hardening saturation stress and yield stress, respectively. Separate expressions modeling t^ s and t^ y in both the thermal activation regime and strong shock regime are provided by PTW (not given here for conciseness). The s0 parameter is the value of t^ s taken at zero temperature, c and y are the strain and work hardening rate, respectively, and p is a dimensionless material parameter. The ﬂow stress is normalized to the shear modulus, G (e.g., t^ y ¼ ty =G). Where appropriate the temperature dependence of the shear modulus was approximated as Gðr; TÞ ¼ G0 ðrÞð1 � aT^ Þ, where G0 ðrÞ is the zero temperature modulus as a function of density and a is a material constant. The pressure dependence of the model is due to the pressure dependence of the shear modulus. The model parameters were slightly modiﬁed to match the low strain rate work hardening behavior for /1 0 0S copper. In particular, the work hardening rate, y,

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2000 Shock Loading

Stress (MPa)

1500

Quasi-Isentropic Laser

Twinning Threshold

1000

500

0

Quasi-Isentropic Gas-Gun 0

20

40 60 Peak Pressure (GPa)

80

100

Fig. 47. Flow stress versus peak pressure for shock compression, gas-gun ICE and laser ICE experiments in Cu.

was adjusted to a value of 0.01 and saturation stress, s0, to a value of 0.0045. All other parameters are as given in Jarmakani et al. [139]. In the shocked region, the temperature and strain were taken from the simulations while the strain rates were determined from the Swegle–Grady relation [32]. Jarmakani et al. [139] also assumed that the ﬂow stress and twinning stress, being dependent on the atomic energy barrier, scale with the shear modulus, as is typical in high-pressure con stitutive models. This was not done in Section 6.3.2.1. Results of these calculations are presented in Fig. 47, where the ﬂow stress, as a function of peak drive pressure, for the shockless and shocked region are plotted for both quasi-isentropic gas-gun and laser compression. The twinning threshold was assumed to vary with pressure through the pressure dependence of G: sT ðPÞ ¼ s0T

GðT; PÞ , G0

(48)

where sT0 and G0 are the twinning threshold stress and shear modulus at ambient pressure, respectively. The quasi-isentropic gas-gun curve lies well below the twinning threshold curve at all pressures. Obviously, a slip–twinning transition is not predicted to occur during gas-gun loading, and a twinning threshold stress should, therefore, not be reached. This is inconsistent with experimental observations at Pmax ¼ 52 GPa, since twinning was observed at that pressure. The presence of the shock at the start of the shock pulse for this pressure condition creates a deviation from quasi-isentropic conditions and may be accountable for the presence of the twins observed. In the case of laser ICE, the threshold lies at Pmax ¼ 32 GPa, consistent with observations of the lack of twinning at 24 and 18 GPa, and their presence at 59 GPa. The steep shock loading curves in both cases arise due to the high strain rate dependence on both the shock pressure and ﬂow stress [Fig. 43(b)]. Shown in these ﬁgures are a detailed set of plots from MD simulations of shocked Cu and Ni at various shock strengths (30–171 GPa),

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propagation directions in the FCC lattice ([0 0 1, [2 2 1]), and in various presentation formats (szz(z), tshear(z), and pressure (z), at various time steps; tshear vs. szz, etc.).

7. Molecular dynamics simulations of dislocations during shock compression MD simulations of shocks have been carried out for decades, starting with shocks in unidimensional (1D) chains of atoms, [140] two-dimensional (2D) crystals (Mogilevsky [141,142]), and later leading to shocks in 3D crystals [143,144] and polycrystals [145]. Most simulations have been carried out [146] in crystals with fcc structure. However, there is a growing number of simulations for crystals with bcc and diamond structure (Zybin [147], C and Si), complex organic crystals (Strachan et al. [148] and even for quasicrystals [149]. MD simulations are ideally suited for comparison with laser-shock compression experiments because of similar time and length scales; thus, the combination of experiments and simulations provides valuable insight on the deformation processes involved. There are recent simula tions reaching sample lengths of up to several micrometers along the shock loading direction [150], comparable to the thickness of some experimental samples, but with much smaller simulated cross-sections, tens of nanometers on each side and using periodic boundary conditions. The difference between mono and polycrystals in MD simulations resides in the absence and presence of grain boundaries, respectively. A representative poly crystalline cross-section has to include several grains, and therefore the largest grain size simulated to date is only 50 nm [151], which is well-suited to model nanocrystals. 7.1. Computational methods We have carried out MD simulations using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) code [152,153] with the EAM potentials for Cu (Mishin et al. [154,155], ﬁtted to give a SFE of 45 mJ/m2) and Ni (Mishin et al. [155], ﬁtted to give a SFE of 125 mJ/m2 [156]). These potentials give a Hugoniot along the main symmetry directions, which agrees with experimental data. For better visualization, the ‘‘centro-symmetry’’ parameter is used to identify defective atoms (dislocation cores and stacking faults). It is of the form [152]: CSP ¼

6 X

j~ ri þ ~ r iþ6 j2 ,

(49)

i¼1

where ~ r i and ~ r iþ6 are the vectors from the central atom to the opposite pair of nearest neighbors (six pairs in an fcc system, i.e., the coordination number). Atoms in perfect fcc lattice positions have a CSP equal to zero, whereas atoms having faulty stacking will generate a non-zero CSP.

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In our MD simulations, two perfect fcc Cu crystals, [0 0 1] and [2 2 1], were shockcompressed at several pressures. Periodic boundary conditions were imposed on the lateral surfaces, and the surfaces normal to the shock-wave propagation direc tion were set as free surfaces. The [0 0 1] monocrystalline copper had dimension of B9 � 9 � 36 nm3 (25 � 25 � 100 fcc unit cells). This is sufﬁciently large to calculate the shock Hugoniot and study the early stages of shock-induced plasticity, given that much larger simulations produce similar results [142]. For [1 0 0] shock propagation, the three coordinate axes were [1 0 0], [0 1 0], and [0 0 1]. The [2 2 1] monocrystal had a dimension of B15.3 � 15.3 � 65 nm3 (42.42 � 42.42 � 180 fcc unit cells) along the three coordinate axes of ½1� 1� 4�, ½1 1� 0�, and [2 2 1]. These dimensions are required for periodic boundary conditions in the lateral directions. The shock waves were produced as described by Cao et al. [88] a piston applied to the material at a velocity Up. The velocity of the shock wave, Us, can then be calculated from the propagating front in our samples. The shock pressure can be calculated both from our MD simulations and from the Hugoniot relationship, once Us and Up are known. The [0 0 1] monocrystalline nickel sample consisted of 2 � 106 atoms and had dimensions of 17.6 � 17.6 � 70.4 nm (50 � 50 � 200 unit cells). The three coordinate axes were oriented in the [1 0 0], [0 1 0], and [0 0 1] directions. Two nanocrystalline (nc) samples were also shock-compressed in this study, one having a grain size of 5 nm and the other 10 nm. The 5 nm grain-sized sample consisted of 1,980,372 atoms (50 � 50 � 200 unit cells, 17.6 � 17.6 � 70.4 nm), and the 10 nm grain-sized sample had 7,942,605 atoms (100 � 100 � 200 unit cells, 35.2 � 35.2 � 70.4 nm). Prior to compression, the specimens were ﬁrst equilibrated to minimize their energy, and the initial temperature was set as 5 K. The velocity of the shock wave, Us, was measured by analyzing the shock-front propagation in the sample at different time steps, and the shock pressure was calculated from the following Hugoniot relation (see, e.g., [158]): Pshock ¼ r0 U s U p .

(50)

7.2. FCC single crystals MD simulations of shock phenomena in perfect fcc single crystals have been carried out for just over 25 years [143]. Most of the simulations to date have used the Lennard–Jones (L–J) 6–12 pair-potential [154,157,159,160] and the more realistic embedded atom method (EAM) many-body potentials for copper [85,156], Ni (Koci [161]). The EAM approach [85,156] allows an accurate description of elastic properties, EOS, and defect energies in metals, particularly for fcc metals like Cu, Ni, Al, etc. However, most EAM potentials are ﬁt to reproduce properties at ambient conditions and may lead to faulty results when used under shock conditions. Among the quantities that should be veriﬁed for an EAM potential which would be used for shock simulations are the EOS up to the desired simulation pressure, the shear stress versus pressure for uniaxial compression along the directions of interest, the elastic constants versus pressure, and the stacking fault and twinning energies [162].

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Holian et al. [145] and Germann and co-workers [154,159] used a L–J potential with zero SFE at ambient pressure and showed that, at shock strengths above the Hugoniot Elastic Limit (HEL), shock waves traveling along the [0 0 1] orientation resulted in the emission of intersecting Shockley partial dislocations that slipped along all the {1 1 1} close-packed planes. Stacking faults were formed since the trailing partial was never released. The large mobility of the partials at the shock front was such that the plastic wave was always overdriven (i.e., no elastic precursor was observed). This dislocation behavior is very similar to the model proposed by Smith [163], except that partial dislocation loops are emitted in MD simulations rather than perfect dislocations as outlined by the Smith model. Germann and co-workers [154,159] further studied shock propagation in the other [1 1 0] and [1 1 1] low-index directions, where they observed rather different behavior. An elastic precursor separated the shock front from the plastic region in the [1 1 1] case, and solitary wave trains were generated followed by an elastic precursor and a complex plastic zone in the [0 1 1] case. In both orientations, trailing partials were emitted leading to full dislocation loops bounded by thin stackingfault ribbons. These loops were periodically nucleated at the shock front, as proposed by Meyers [36], since they grew at a slower rate than the plastic shock velocity. The reader is referred to Figs 3 and 4, which present the basic elements of the homogeneous dislocation model. Bringa et al. [164] also studied the effect of crystal orientation on the shock Hugoniot along the low-index directions ([0 0 1], [0 1 1], and [1 1 1]) using two EAM potentials for copper. The plasticity in the three orientations was qualitatively similar to that of Germann et al. [154,159]. The molecular dynamics calculational procedure presented in Section 7.1 with the use of the centro-symmetry ﬁltering method was applied to copper and nickel and the results are presented in this section (Figs 48–56); the results by Cao et al. [88] and Jarmakani et al. [130] are summarized here. Cao et al. [88] investigated the non-symmetric [2 2 1] orientation of Cu, where a two-wave structure (elastic and plastic) was observed. The progression of the shock front through copper specimens is shown in Fig. 48; Fig. 48(a) corresponds to [0 0 1] and Fig. 48(b) to [2 2 1]. The defect structure is relatively unchanged during the advance of the front. For both orientations, we observe nucleation and growth of stacking-fault loops. Sequential snapshots of the ﬂow velocity of the atoms in the copper sample enable the calculation of the shockwave velocity for the two orientations. Fig. 49 shows the shock wave at three times for (a) the [0 0 1] and (b) the [2 2 1] orientations at Up ¼ 1 km/s. The wave front is in the right-hand side, and the rigid piston on the left side. Note that a plastic front exists for [0 0 1], but does not lead to a two-front structure. On the other hand, for the shock along [2 2 1], the front splits into an elastic precursor and a plastic front, as shown Fig. 49(b). Splitting of the elastic and plastic shock has been observed for [1 1 1] and [1 1 0] directions. Figs 50(a) and (b) show the pressure and the shear stress for the shock propagation along [0 0 1] (top) and [2 2 1] (bottom), for the three times shown in Fig. 49. The shear stress, ssh, was calculated as ssh ¼ 0.5[szz�0.5(sxx þ syy)], since the off-diagonal terms in the stress tensor were found to be negligible. For [2 2 1], the

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Fig. 48. MD simulation of the propagation of shocks in copper driven by a piston (particle) velocity of Up ¼ 1 km s�1 (Pshock ¼ 48.5 GPa) for (a) [0 0 1] and (b) [2 2 1] Cu at increasing times: (1) 2 ps; (2) 4 ps; (3) 6 ps. Light colors indicate stacking faults and dislocations [88].

decrease in shear stress [Fig. 50(d)] coincides with the pressure rise that leads to dislocation nucleation and the formation of a plastic front; the shear stress relaxes because of dislocation loop nucleation and growth at the plastic front. For [0 0 1] [Fig. 50(c)], this relaxation occurs within a region extremely close to the shock front. Figs 51 and 52 show the comparison of the computed and experimentally observed deformation features. The deformation features and shock Hugoniot obtained compared very well with experimental results. However, upon comparing the density of the deformation

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1.2 4 ps 5 ps 6 ps

1.0 0.8 0.6

Vz (km/s)

0.4

(100)

0.2

(a)

0.0 (b) 1.0 0.8 0.6 0.4

(221)

0.2 0.0 0

10

20

30

40

50

Z (nm) Fig. 49. Particle velocity versus distance for an MD simulation of shock propagation in (a) [0 0 1], and (b) [2 2 1] in single crystal Cu [88].

60 4 ps 5 ps 6 ps

50 40

20

4 ps 5 ps 6 ps

15

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10 (100)

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Shear (GPa)

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(221)

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0 0

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20 30 z (nm)

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Fig. 50. (a, b) Pressure proﬁles at three different times for (a) [0 0 1], and (b) [2 2 1] Cu monocrystals; (c, d) shear stress proﬁles at different times for (c) [0 0 1], and (d) [2 2 1] Cu monocrystals [88].

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Fig. 51. Defect structures in Cu: (a) MD simulation results for the propagation of a 48.5 GPa shock along the [0 0 1] direction; (b) MD simulation for the propagation of a 48.5 GPa shock along the [2 2 1] direction; (c) TEM micrographs for [0 0 1] Cu monocrystal shocked at 30 GPa; (d) TEM micrographs on [2 2 1] monocrystal of Cu shocked at 30 GPa [88].

features with experimental observations in recovered samples, they found that the dislocation densities in the simulations were several orders of magnitude higher. Two reasons were suggested by Cao et al. [88] for the difference: (a) the much shorter rise time in the MD simulations and (b) the post-shock relaxation and recovery processes that take place in real experiments. Kum [165] analyzed the deformation features in shock-compressed singlecrystalline Ni along the three low-indexed orientations. Two Morse-type pair potentials and one EAM potential were used in that work. However, the study is limited to one piston velocity and does not calculate the Hugoniot obtained from the MD simulations. The Hugoniot of Ni using one EAM potential, was calculated by Koci et al. [161], but their focus was the study of shock melting.

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Fig. 52. (a) Schematic illustration of traces of Cu {1 1 1} slip planes on the surface of a ð1 1� 0Þ plane; (b) MD simulation showing traces of the stacking fault slip systems on the surface of ð1 1� 0Þ shown by an MD simulation for shock propagation along [2 2 1] [88].

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Fig. 53. MD simulation showing dislocation structures at 8 ps in shocked [2 2 1] Cu as a function of particle/piston velocity: (a) 0.75 km/s (33.9 GPa); (b) 1 km/s (48.5 GPa); (c) 1.25 km/s (64.8 GPa); (d) 1.50 km/s (82.8 GPa); (e) 2.00 km/s (123.7 GPa); and (f) 2.5 km/s (171.3 GPa) [88].

The simulations show that, as the piston (equivalent to particle) velocity is increased, the defect density increases. The sequence of snapshots in Fig. 53 represents a range of pressures from 33.9 to 171.3 GPa for the [2 2 1] crystal. Note that the density of defects for P ¼ 171.3 GPa (Up ¼ 2.5 km/s) is extremely high and the material resembles a nearly amorphous material. The shock-melting pressure for this EAM copper was calculated as B200 GPa. Hence, the melting as determined from hydrodynamic calculations and the extreme dislocation density observed in the computation of Fig. 53(f) are consistent. A similar procedure was applied by Jarmakani et al. [130] to nickel, and the shock propagation proﬁles for [0 0 1] are shown in Fig. 54 for two pressures: 35 and 48 GPa. At 45 GPa, an elastic precursor is evident, which is absent at 35 GPa. The computations show that formation of dislocation loops behind the front, seen

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0.8 6 psec 8 psec 10 psec

0.7 0.6

Vz (km/sec)

0.5 0.4 0.3 0.2 0.1 0

0

10

20

30 40 50 Distance (nm) (a)

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6 psec 8 psec 10 psec

1.2

Vz (km/sec)

1

0.8

0.6

0.4

0.2

0 0

10

20

30 40 50 Distance (nm) (b)

60

70

80

Fig. 54. MD simulations for shocked single crystal Ni along [0 0 1]; (a) piston (particle) velocity at 6, 8, and 10 ps versus distance (below the HEL) for PB35 GPa; (b) piston (particle) velocity at 6, 8, and 10 ps versus distance (above the HEL) for PshockB48 GPa [130].

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Fig. 55. MD simulation of shock compression of Ni along [0 0 1]; Up ¼ 0.786 km/s: (a) Stacking faults, viewed along the longitudinal z direction; (b) plastic and elastic zone formation; notice the formation of dislocation loops; and (c) sketch of a dislocation interface in the homogeneous generation model (adapted from [36]).

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σzz Shear

Pressure (GPa)

80

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50 30 40 Distance (nm) (a)

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120

140

12

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Dislocation Nucleation

Shear Stress, τ (GPa)

8

6 Shear drop due to dislocations

4

2

0 0

20

40

60 80 σzz (GPa)

100

(b) Fig. 56. (a) Shear stress and szz versus sample depth for an MD simulation of single crystal Ni shocked along the [0 0 1] direction at UpB0.945 km/s; (b) shear stress versus szz.

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clearly in Fig. 55. The section in Fig. 55(a) shows the stacking faults on {111} planes whereas the 3D representation in which all atoms not belonging to dislocations were ﬁltered out is shown in Fig. 55(b). The partial dislocation loops are nucleated between the elastic precursor and the plastic wave front and expand, creating a very high dislocation density in the shock compressed region. The mechanism by which these loops nucleate and grow is shown schematically in Fig. 55(c). The computa tions, which start from a defectless crystal, predict the initiation of plastic ﬂow at exceedingly high pressures, that are indeed not realistic. This is shown in the simulation results shown in Fig. 56. Fig. 56(a) shows the normal and shear components of the stress. The shear component is shown in Fig. 56(b) as a function of increasing piston (or particle) velocity. Dislocation nucleation requires a shear stress of approximately 8 GPa in nickel (BG/10, since G ¼ 76 GPa). It can be seen that a pressure of B60 GPa is required to create a shear stress of 8 GPa. For lower pressures, there is no dislocation generation (tridimensional representation in left side of ﬁgure). At P ¼ 60 GPa, a shear stress of 8 GPa is reached and one sees the ﬁrst evidence of dislocations (top tridimensional ﬁgure). The shear stress decays beyond that, as the loops are nucleated and grow. This is accompanied by profuse partial dislocation activity in the slip planes. The inclusion of defects like dislocation loops in single-crystal simulations allows reasonable agreement with experimental HEL results (Kubota et al. [166]), and a more realistic plastic relaxation description (Bringa et al. [167]). Therefore, there is still need for simulation of crystals with defects, and for further systematic research on MD shock simulations. For instance, in contrast with simulations using L–J potentials, simulations, using a Cu EAM potential showed nucleation of some full dislocations in shocks along [0 0 1] (Bringa et al. [167]), and simulations using a Ni EAM potential showed homogeneous nucleation of partial loops as the shock propagated (Koci et al. [161]). In addition, simulations using ramp loading do show an elastic precursor for rise times of B50 ps or longer (Bringa et al. [167]) for EAM Cu loaded along [0 0 1].

8. Comparison of computational MD and experimental results 8.1. Comparison of monocrystals and polycrystals The defect spacing as a function of shock pressure was analyzed in order to quantify the induced plasticity [see Fig. 57(a)]. Clearly, the spacing between stacking-faults decreases as the shock pressure increases. Copper data from Cao et al. [84,87,88] are plotted as well. Holian [144] introduced two fundamental deformation parameters: shock-induced plasticity and shock strength. Shock-induced plasticity is deﬁned as a0/l, where a0 is the lattice parameter ( ¼ 0.352 nm for Ni), and l is the average lattice spacing between stacking faults. They deﬁned shock strength as the ratio between particle velocity and speed of sound in the material, Up/C0 (C0 ¼ 4.581 km/s for Ni). This shock-induced plasticity as a function of shock strength is plotted in Fig. 57(b). MD data on Cu from Cao [84,87,88] and on Ni from

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Jarmakani et al. [130], predictions from the homogeneous nucleation model of Meyers [36], and experimentally measured data from Murr [40] are also shown on the plot. For the results from Meyers [36] and Murr [40], the dislocation spacing, l, was extracted from the reported dislocation densities, r, using the equation

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pﬃﬃﬃﬃﬃﬃﬃﬃ l ¼ r�1 . The plasticity results from the MD simulations of Jarmakani et al. [130] and Cao [84,87,88] are consistent. These predictions also agree with computations by Holian [144]. Interestingly, the early analytical model by Meyers [168] is in agreement with these MD calculations. The experimentally determined shock plasticity of Ni from Murr [40] is, however, lower than the theoretical and MD results by an order of magnitude. This is shown in greater detail in Fig. 58. This suggests that relaxation processes are clearly at play in real experiments resulting in lower dislocation densities, as will be shown below. One possibility to explain the experimental discrepancy between Ni and Cu is that electron–phonon coupling in Cu is much larger than in Ni, resulting in shorter heating periods and therefore, shorter thermal relaxation times. The role of electron–phonon coupling has been explored in Ni shock melting by Koci et al. [161], using a two temperature model coupled to their MD simulations. We note that laser experiments have explored different relaxation scenarios as a function of electron–phonon coupling [169,170].

Fig. 58. Shock-induced plasticity calculated from experimental results for Ni: (a) Laser shock [37]; (b) plate impact [73].

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The shock-induced plasticity increases monotonically with shock strength and follows closely the total volumetric strain Up/Us. Given that the total lateral strain in our simulations is zero due to periodic boundary conditions, the elastic strain has the same magnitude as the plastic strain, given by Orowan’s equation [27]. The amount of dislocation motion needed to relax a given volumetric strain would be roughly the same for similar materials. Therefore, shock-induced plasticity would follow the total volumetric strain, even for different shock propagation directions and slightly different materials. Experimental measurements in the literature extracted from TEM images for laser-shocked copper monocrystals subjected to a broad range of pressures [83,91,92,171] were converted into shock-induced plasticity, and the corresponding pressures were converted to shock strengths. The spacing between stacking-fault packets for the laser shock experiments are plotted in Fig. 58(a). The same monotonic increase in shock-induced plasticity with shock strength as shown for the MD simulations in Fig. 57(b) is observed. However, there is a major difference: the experimental values are lower by a factor of 104. Plate impact [40,42,76,77,130,139,169,170,172–174] experiments on copper have been conducted since the seventies. Classic among these experiments are the sys tematic measurements made by Murr and co-workers [42,134–137] on inter-twin and inter-stacking-fault spacings. Fig. 57(b) shows the shock-induced plasticity calculated using the inter-twin spacings observed by Murr and co-workers [42,152,158,161,162], and the work to be reported in a paper by Cao et al. [172]. Jarmakani et al. [139] found similar results. The shock-pulse duration in Murr’s experiments [40,42,76,77,173] was B2 ms, which is in the same range as our work by Cao et al. (1.4–2 ms) [172]. Work by Andrade et al. [174] conﬁrms the twin spacing experimentally observed by Murr [40,42,76,77,169,173]. The calculated shockinduced plasticity from Murr’s data is on the order of 10�4, which is smaller than the experimental results in Cao et al. [172] (B10�3). These results, as well as the shockinduced plasticity in laser-shocked samples of B10�5 shown in Fig. 58(a) are compelling evidence for major effects that are not generally considered, leading to spacing between defects observed in simulations that is much smaller than that observed by TEM on recovered samples. There are several possible reasons for this discrepancy: (a) the higher strain rate in MD simulations; (b) simulation of only a small volume of perfect single crystal, without any initial defects [167]; and (c) the possibility that most defects are annealed out [167] and that TEM observations reveal a structure that is completely different from the one extant during shock compression. In our simulations, if we allow for the shock wave to reach the back of the sample and produce a rarefaction wave, most of the stacking-fault network disappears, making clear the important role that recovery can play in the TEM samples. Dynamic X-ray diffraction may be able to probe the dynamic dislocation generation seen in MD simulations in the future [167]. The effect of release (stress unloading) in the MD simulations was studied for comparison with experiments. The piston was released after 10 ps and the pressure (Ptot ¼ f(sxx, syy, szz)) was allowed to retract back to zero. Interestingly, almost all the partial dislocation loops disappear. The spacing between the few remaining

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stacking faults was measured, and the resulting residual plasticity was calculated. Fig. 57(b) shows the MD plasticity after release; an order of magnitude drop is evident, consistent with the experimental data by Murr [40]. The pressure rise due to compression and the accompanying drop due to release are shown in Fig. 59 for the case of Up ¼ 1.1 km/s. Only the defective atoms are shown. In experiments, the lower strain rate for unloading and the longer heating period will likely accelerate dislocation annihilation and lead to even lower residual dislocation densities than in our fast unloaded MD simulation. 8.2. MD simulations of shocks in nanocrystalline nickel Bringa et al. [167] carried out simulations of shocks in nanocrystalline (nc) Cu and found that partial dislocations and grain-boundary sliding dominated at a grain size of 5 nm, with full dislocations contributing more to plasticity as grain size increased up to 50 nm. However, even for these relatively large grains, the number of full dislocations was modest. Shocks in nc Ni, which has a much larger SFE, were expected to give a greater contribution from full dislocations, based on experimental results [175]. The 5 nm grain-sized sample was subjected to piston

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velocities between 0.2 and 1.3 km/s, and its Hugoniot was found to be very close to that of the monocrystalline sample. Fig. 60(a) provides an illustration of the shock wave for Up ¼ 0.67 km/s as it traverses the sample (average velocity vs. distance). The corresponding shock pressure within the sample is B38 GPa, which is at the HEL limit for the monocrystalline sample. Since grain boundaries (i.e., defects) exist in the sample, the HEL is signiﬁcantly lower than that in the single crystal. A single-wave structure is evident and not a two-wave structure as seen in the 0.8 6 psec 8 psec 10 psec

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single-crystalline results. This may be due to the fact that the particle velocities vary from grain to grain, introducing ﬂuctuations in the front that do not allow the plastic and elastic components to be resolved. In comparison with the single-crystal proﬁles shown in Fig. 54(a), the front thickness is increased from B2 to B10 nm. Fig. 60(b) shows the nc sample at 0 ps, after it has been relaxed to minimize its internal energy prior to shock propagation (left) and after the shock-wave has traveled for 10 ps (right). Grain boundaries act as sources and sinks for partial dislocations, leaving stacking faults behind as they travel through the grains. Two of these are marked for clarity. This defect conﬁguration is similar to the one observed by Van Swygenhoven et al. [176,177] in homogeneous tension simulations and by Bringa et al. [164,167] in shock simulations. Leading partials are mainly emitted from the grain boundaries, and trailing partials are seldom released. Limited evidence of twinning was also observed. Fig. 60(c) is a 3D view of the simulation shown in Fig. 60(b). A quantitative analysis of the deformation mechanisms was carried out on MD simulations of three samples of different grain size that were shocked using the same piston velocity of 0.67 km s�1 (B38 GPa): 5 nm Ni, 10 nm Ni, and 10 nm Cu. The three samples provide the means to study the effect of grain size and a different potential on the deformation behavior. We calculated the contributions to the effective strain introduced by shock compression from the various mechanisms of plastic deformation by determining the relative motion between nearest-neighbor pairs of atoms, and resolving this motion along the strain axis, as described by Vo et al. [178]. The procedure to quantify the dislocation contributions to the total plastic strain consists of three steps. The ﬁrst step locates nearest-neighbor atom pairs that have been sheared on glide planes and assigns local Burgers vectors responsible for the shearing. This step requires correction for the strain caused by atom pairs that are cut by multiple dislocations with different Burgers vectors. The second step distinguishes atoms in grain interiors that are cut by lattice dislocations from those that are involved in grain-boundary mechanisms. The third step evaluates the strain caused by the motion of the dislocations identiﬁed. Detailed procedures can be found in Refs [178,179]. Using this method, the contributions from partial dislocations, perfect dislocations, multiple dislocations on the same slip plane, and twinning can be identiﬁed. The difference between the total plastic deformation and these other contributions can then be attributed to grain-boundary sliding. Fig. 61 shows the three shocked samples. The color code is as follows: the blue atoms are not displaced and are in their original minimum energy state, the green atoms are displaced by the Burgers vector of a Shockley partial, the red atoms are displaced by a Burgers vector of a perfect dislocation, and the orange atoms are displaced by a Burgers vector larger than that of a perfect dislocation (due to grainboundary sliding). Original color illustrations are found in Jarmakani et al. [130]. In these, orange, green, red atoms can be distinguished. For the 5 nm Ni, the total shock strain in the sample was calculated to be B0.13. The total strain contribu tion due to dislocations (0.014) is dominated by partials, which makes up B60% of the total strain due to dislocations; perfect dislocations account for B10%.

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Fig. 61. Comparison of deformation stucture versus grain size for the same particle velocity of Up ¼ 0.67 mm/ms: (a) 5 nm Ni; (b) 10 nm Ni; (c) 10 nm Cu (the position of the shock front is marked for the three samples). Blue corresponds to atoms in their original minimum energy state, green to atoms displaced by the Burgers vector of a Shockley particial, red to atoms displaced by a Burgers vector of a perfect dislocation, and orange to atoms displaced by a Burgers vector larger than that of a perfect dislocation (implying grain-boundary sliding) (Original color illustrations in Jarmakani et al. [130]).

The contribution due to twinning is 26%. By subtracting the strain due to dislocations from the total strain, one obtains the strain due to grain-boundary sliding, 0.116; this represents approximately B90% of the total strain. In the case of the 10 nm samples, the strain contribution due to partials is 63% for Ni and 56% for Cu. Perfect dislocations account for 17.2% of the dislocation strain

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in Ni and 21% in Cu. The twinning contribution is greater in Cu, 19% as compared to 16% in Ni. The greater incidence of twins is to be expected since the SFE of Cu is signiﬁcantly lower. However, the difference in the contribution of full dislocations and twinning between Ni and Cu is marginal despite the fact that the SFE of Ni is nearly three times larger than in Cu. This points out to the complexity of dislocation nucleation in nanocrystals, as already pointed out by Van Swygenhoven et al. [177]. Grain-boundary sliding accounts for approximately 58% of the total shock strain in both 10 nm Ni and Cu in comparison with 90% for 5 nm Ni, signifying that it becomes more difﬁcult for larger grains to slide past one another under compression (Bringa et al. [151,180]). Note that the front portions of the 10 nm Cu and Ni samples do not show the grain boundaries highlighted in green. This is due to the fact that no grain-boundary sliding is taking place because the shock front has not yet traveled through that region. The contribution due to partials is comparable in the 5 and 10 nm grain-sized samples, but that from perfect dislocations is greater in the 10 nm samples, as expected. The twinning contribution is greater in the 5 nm grain-sized sample (5 nm Ni: 25.7%, 10 nm Ni: 15.7%), due to the fact that partial dislocations are more abundant, and therefore lead to more twinning as in the Zaretsky model [44] and as described in Shehadeh et al. for single crystals [181]. This result is also in agreement with the models proposed by Chen et al. [182] and Zhu et al. [183], where they show that the propensity for twinning increases with decreasing grain-size. This result for nc twinning behavior is in distinct contrast to the result shown in Fig. 40 for conventional grain sizes, where larger grains twin more readily. Our results are consistent with simulations of tensile deformation of nanocrystals, where grain-boundary sliding was considered crucial at dB5 nm (Schiøtz and Jacobsen [184]), mostly based on visual inspection of plasticity and on the observed rotation of certain grains. Recently, Vo et al. [185] carried out simulations of homogeneous uniaxial compression of nc Cu up to 20 nm grain size, and found similar results for the contributions of GB sliding and dislocations, with a cross-over to dislocation dominated plasticity at B15 nm, at a strain rate of 1010 s�1. Our analysis is the ﬁrst to quantify slip and GB sliding for shocked nc samples. In addition to the simulations of nanocrystals, laser shock compression experiments were carried out on nc Ni [175,186], with grain sizes between 30 and 50 nm. The samples in the experiments were prepared by electrodeposition at the Lawrence Livermore National Laboratory and were subjected to pressures between 20 and 70 GPa via laser irradiation [130]. The microhardness of the samples after shock compression was measured, and a 5–30% increase after being shocked was observed, clearly indicating dislocation storage. Fig. 62(a) shows a cross-section of a sample with microhardness measurements taken at ﬁve positions beneath the cratered surface. Fig. 62(b) shows the increase in residual hardness beneath the cratered surface where a maximum at position 3 occurs where laser intensity (i.e., deformation) is greatest. Fig. 62(c) shows the increase in residual hardness as a function of shock strength during shock compression of the samples. In congruence with the hardness data, TEM examination revealed heavy dislocation activity (rB1016 m2) due to these laser-induced shocks. Full dislocations

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were the main carriers of plasticity [see Fig. 63(a)]. Interestingly, deformation twins were not observed in any of the samples, even at pressures up to 70 GPa. This is discussed in Section 6. To further reduce the grain size to B10 nm, tungsten (13 wt.%) was added to the nickel electrolyte during electrodeposition as outlined by Schuh et al. [187,188].

Fig. 63. (a) TEM image of nc Ni with grain sizes of 30–50 nm shocked at B40 GPa showing dislocations; (b) TEM image of Ni–W (13 at.%) with grain sizes of 10–15 nm shocked at B40 GPa; deformation twins are evident (circles) (Courtesy Y. M. Wang; from Jarmakani et al. [130]).

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The Ni–W samples were loaded to pressures up to B38 GPa, and a shift in deformation mechanisms was observed. TEM analysis revealed that deformation twins were the predominant defect structures, indicated by circles in Fig. 63(b). A very low density of pre-existing annealing twins was observed in the as-prepared samples, and the twin density increased dramatically after shock loading. However, the addition of W lowers the SFE, and the increased twinning cannot be attributed to the decreased grain size alone. This discrepancy in dislocation behavior between MD simulations and actual experiments could be due to several factors. The samples in the experiments go through release, which leads to the annihilation and reabsorption of partials. There may be grain-size effects at play. Smaller grains favor partial dislocations, and one may have to go to larger grain sizes for perfect dislocations to be energetically favorable. For instance, Bringa et al. [167], showed signiﬁcant dislocation activity and dislocation junctions being formed only for grains above 20 nm. In addition, the MD potentials may not be very accurate in describing the stacking-fault and twinning energy surfaces, and the value of these surfaces under stress could change considerably. Another possibility may be that the time needed for the emission of full dislocations is much longer than the timescales simulated in MD. Loading and unloading in the laser-shock experiments take place over 1–10 ns, whereas the MD simulations are in the range of picoseconds, only capturing the initial stages of deformation, and at much higher strain rate. Warner et al. [189] recently showed that a full dislocation takes much longer than partials and twins to be emitted from a crack tip, and something similar may be happening in nc materials.

8.3. Effect of unloading on nc Ni In an analogous manner to the unloading MD simulations carried out on mono crystals (Fig. 59), the effect of unloading on the deformation structure of nc Ni was studied to provide a more realistic comparison with the experiments [130]. The sample shocked at 38 GPa, Up ¼ 0.67 km/s, was allowed to unload and the dislocation behavior within the grains was analyzed. Fig. 64(a) shows the average pressure within the sample as a function of time as it is loaded and unloaded. Fig. 64(b) shows the sample at 0 ps (before the shock), at maximum compression at 11 ps (ﬁrst picoseconds consisted of equilibration) before it is unloaded, and 18 ps after it has been unloaded to zero pressure. The principal features are stacking faults, which are mostly emitted from grain boundaries during compression. After unloading, B38% of the partials are reabsorbed. The light ellipses show regions where partials are reabsorbed and the dark ellipses indicate the partial dislocations that survive after unloading. Unloading at experimental, much slower, rates may lead to further reabsorption. The reabsorbtion of partials causes the contribution due to perfect dislocations to increase from 10.3% before unloading to 18.2% after unloading. As expected from the loading simulations of nc samples, grain-boundary sliding dominates the plasticity for d ¼ 5 nm. The phenomenon of reabsorption

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would explain the fact that partial dislocations are not observed in our experiments, only full dislocations. In addition, Bringa et al. [180] observed dislocation junctions forming inside grains with d ¼ 50 nm, but never forming for smaller grains. Such junctions in larger grains might survive unloading. For comparison, a smaller sample having B500,000 atoms and dimensions of 17.6 � 17.6 � 17.6 nm was compressed uniformly in uniaxial strain to a pressure of B38 GPa and then allowed to unload. The ﬁnal strain and strain rate applied were roughly the same as that experienced by the shocked sample, the principal difference being that there is no wave propagation in the latter simulations. Fig. 65(a) shows the average pressure within the sample as a function of time. The sample was compressed uniaxially for 4 ps to a strain of 0.13, held there for 10 ps, and released back to 0 strain within 4 ps. Fig. 65(b) shows the various stages of deformation. Partials are emitted and reabsorbed during this process. There are no major differences in defect distribution between uniform and shock compression. The percentage of strain corresponding to grain-boundary sliding is slightly decreased. Interestingly, approximately 39% of the partials disappear after unloading. Before unloading, grain-boundary sliding accounts for 79.2% of the total strain, in comparison with shock compression (90%).

9. Simulations of loading at different strain rates MD simulations of shocks generally deal with square loading pulses. This leads to extremely high strain rates at the shock front, B1010 s�1. In order to assess the effect of ramped pulses, Bringa et al. [167] carried out simulations with and without a ramp of 50 ps and a peak pressure of 35 GPa, slightly above the threshold for homo geneous nucleation of dislocations for the EAM Cu by Germann et al. [154]. They studied the inﬂuence of strain rate in the transition from a 1D compressed state to a quasi-3D compressed state, as shown in Fig. 66. Simulations found no difference between the ramped and unramped cases for a perfect single crystal. However, when dislocation sources were introduced, the ramp loading activated the sources before homogenous nucleation occurred, as shown in Fig. 67, and led to an early relaxation of the shear stress. This relaxation led in turn to a lower dislocation density in the region where homogeneous nucleation occurred once the peak pressure was reached. Despite the fact that the dislocation density was lower, stress relaxation was higher, because dislocation mobility was not impeded by the larger number of junctions produced at higher density. This is shown schematically in Fig. 68. In order to simulate larger systems for longer times, Shehadeh et al. [190] have carried out dislocation dynamics embedded into ﬁnite element simulations of shock loading, where plasticity is due to dislocation multiplication by pre-existing sources [190,191]. A criterion for homogeneous nucleation was added to model high pressure by ﬁtting MD results [181]. Results are shown in Fig. 69. They also found that lower strain rates lead to lower dislocation densities, as shown in Fig. 70, in agreement with experimental results by McNaney et al. [135]. Recent simulations by Hawreliak et al. [192], using a rise time of 300 ps and a sample of B4 mm along

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Fig. 66. Dynamic lattice compression during shock loading. The upper curve shows the particle velocity versus time behind a shock front moving to the right (z direction) at speed Us. The response of an initially cubic unit cell of side a0, is illustrated schematically by the blocks. The initial strain is uniaxial compression (1D), with azoa0 and ax ¼ ay ¼ a0 and large internal shear stress. The lattice responds by relaxing via volume-preserving dislocation ﬂow to a more 3D-symmetric compression, azBaxBay. The inset shows experimentally measured diffraction data with a peak corresponding to the unshocked material at zero strain and a peak corresponding to prompt (subnanosecond) 3D relaxation (azBax) for shocked copper at 3–4% strain, from Loveridge-Smith et al. [193]. Figure from Bringa et al. [167], where full colors are given.

the shock direction, showed that a large fraction of their Cu sample was kept solid at 3 Mbar, well above the shock-melting pressure of 2 Mbar, because of their offHugoniot loading.

10. Incipient spallation and void growth The study of the nucleation and growth of voids in ductile metals is of signiﬁcant interest for the understanding of failure under overall tensile loading. Such failure, for example, can occur upon reﬂection of tensile waves from a free surface of the shock-compressed plate. Material failure by void growth under dynamic loading conditions leads to spalling. Extensive analytical and computational research has been devoted to analyze ductile void growth and coalescence in various materials and under various loading conditions. Seitz [194] and Brown [195] postulated prismatic loops forming at the interface between a rigid particle and its matrix. In related work, Silcox and Hirsch [196] analyzed the dislocations that form the boundaries of stacking-fault tetrahedra in gold. These tetrahedra had sizes of approximately 35 nm. Later, Humphreys and Hirsch [197] analyzed coppercontaining small alumina particles and observed the formation of prismatic loops

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Fig. 67. Snapshot from our molecular dynamics (MD) simulation of a shock wave moving in the [1 0 0] direction of single crystal Cu, with a maximum pressure of 35 GPa and a load rise time of 50 ps. The shock propagates approximately toward the bottom right of the ﬁgure. The simulation includes 256 million atoms, but only defective atoms are shown with a coloring scheme following the centro-symmetry deviation values. This close-up view shows a dislocation source activated by the ramp, producing partial dislocation loops in several available {1 1 1} planes. The ‘‘butterﬂy’’ shapes result from loops colliding and forming sessile junctions that impede dislocation motion. The top of the loading ramp is strong enough to induce homogeneous nucleation of dislocations, and this is the front seen behind the source, with an extremely high dislocation density. From Bringa et al. [167].

by a cross-slip mechanism. This study involved primarily the interaction of existing dislocations with rigid particles. More recently, Uberuaga et al. [198] observed the direct transformation of vacancy voids to stacking-fault tetrahedra by MD. In the area of initiation and growth of voids under tensile loading, there are only a few dislocation-based mechanisms (Wolfer et al. [199]). This section describes an alternative mechanism of void growth by dislocation emission from the surface of the void (Lubarda et al. [200]). We show analytically for a 2D conﬁguration, that the imposed stresses in the laser shock experiments are sufﬁcient for emitting dislocations from the void surface. The critical stress for dislocation emission is found to decrease with an increasing void size, so that less stress is required to emit dislocations from larger voids. Fig. 71 shows SEM images of (a) the initial specimen and (b) the recovered specimen with the bulged bottom surface. The laser-induced shock was driven from the upper surface. The reﬂected

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Fig. 68. Dislocation structure resulting from MD simulations of shocked [1 0 0] copper. (a) Snapshot of a simulation with a shock drive of 0 ps rise time after tB100 ps, showing only dislocation atoms. (b) The same as (a) except that the shock drive had a 50 ps linear-ramp rise time. In both cases, the copper crystal included pre-existing dislocation sources. The three regions of dislocation activity – homogeneous, mixed, and multiplication – are marked. (c) The particle velocity, Vz, proﬁle for the ramped shock, where the ramp extends over z ¼ 0.29–0.43 mm. (d, e) The resulting dislocation density proﬁles for the 0 ps risetime case (d) and the ramped case (e). The locations of the pre-existing sources are illustrated by vertical arrows. From Bringa et al. [167].

tensile pulse at about 100 mm from the rear surface can be calculated from the decay of the shock pulse. It is equal in magnitude, but opposite in sign, to the shock pressure. The latter is about 5 GPa in magnitude. Voids were observed in the crosssection, ranging in size from 25–50 nm to 1 mm. Fig. 72 is a TEM image showing what is believed to be a void near the back surface of the shocked specimen. Its diameter is approximately 500 mm. It may be argued that electropolishing produced the void, but a larger number of perforations were found close to the back surface of the specimen, where void formation is expected. There is a light rim around the void, indicating an extremely high dislocation density, below the resolution where individual dislocations can be imaged. This void is similar to the one observed earlier by Christy et al. [201] using high-voltage transmission electron microscopy. In that experiment, the foil was not perforated and the same intense dislocation density was observed. The diameter of this work-hardened layer is approximately twice the void diameter. Thus, a much higher dislocation density characterizes the region surrounding the void compared to regions without observable voids. 10.1. Dislocation emission and void growth Void growth is indeed a non-homogeneous plastic deformation process. The plastic strains decrease with increasing distance from the void center. The far-ﬁeld strains

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Fig. 69. Multiscale dislocation dynamics plasticity (MDDP) simulation of a 35 GPa, 50 ps rise time shock wave, showing individual dislocations for a block of 0.25 � 0.25 � 10 mm3. The shock-front moves from left to right. Loops are homogeneously nucleated as the wave travels through the material, while those previously nucleated grow as the crystal relaxes. Dislocation–dislocation interactions become dominant at high r, leading to the development of a 3D pattern of intersecting loops in all available {1 1 1} slip planes, with large numbers of jogs and junctions. (a) 67 ps: Pre-existing loops that will act as dislocation sources can be seen ahead of the shock front, (b) 90 ps: Initial stages of heterogeneous nucleation from the sources, and (c) 122 ps: There is a region with both homogeneous and heterogeneous nucleation, showing lower dislocation density than the region with only homogeneous nucleation. From Shehadeh et al. [190,191].

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Fig. 70. Evolution of the average dislocation density within a thin slice calculated using MDDP simulations of 35 GPa shocks for 5 and 50 ps rise times. The initial dislocation microstructure affects the homogeneous nucleation (HN) process only at the early stages of the interaction process. The relaxed density of dislocations is affected by the shock wave rise time and the pre-existence of dislocation sources. From Shehadeh et al. [190].

are purely elastic, whereas plastic deformation occurs in the regions adjoining the surface of the void. Ashby [202] developed a formalism for the treatment of a non homogeneous plastic deformation by introducing the concept of the generation of geometrically necessary dislocations. Two different mechanisms were envisaged by Ashby [202], based on prismatic or shear loop arrays. The void growth situation is quite different from the rigid-particle model used by Ashby [202]. One can still postulate arrays of line defects to account for the non-homogeneous plastic deformation. Of critical importance is the fact that the shear stresses at 451 to the void surface are maximum, since the normal stresses are zero at the surface of the void. These shear stresses decay to zero at large distances due to the assumption of a far-ﬁeld hydrostatic stress state. Thus, the shear stresses are highest at the internal surface, triggering dislocation nucleation there. The mechanism of void growth derived by Lubarda et al. [200] for the emission of shear loops will be presented here. The shear loop mechanism involves the emission of dislocations along the slip plane S, and is shown in Fig. 73(a). These loops form preferentially at planes intersecting the void along a 451 orientation to the radius. Fig. 73(b) shows two slip planes intersecting the void surface at 451. This ensures a 451 angle between the slip plane S and the void surface, maximizing the driving force on the dislocation. The difference between this and the Ashby [202] loops is that the two opposite loops have dislocations of the same sign whereas Ashby’s [202] opposite loops have opposite signs (Figs 73 and 74). In the 2D case, four pairs of edge dislocations emitted from the surface of a cylindrical void under remote uniform tension give rise to an increase of the average void size by an amount approximately equal to the magnitude of the

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Fig. 71. A side view of the cylindrical specimen subjected to shock compression and subsequent tensile pulse reﬂection from the laser-induced shock wave: (a) Undeformed specimen, and (b) deformed specimen upon wave reﬂection with spall surface protruding in back.

dislocation Burgers vector. Other arrangements, involving more than four pairs of dislocations, can also be envisioned as giving rise to the expansion of the void [203]. After the void has grown a ﬁnite amount, the network of sequentially emitted dislocations may appear as depicted in Fig. 74. In an analytical treatment of the void growth by dislocation emission, we consider the emission of a single dislocation pair (shear loop) from the surface of a cylindrical void under far-ﬁeld biaxial tension. The critical stress required for the emission of a shear loop is calculated as a function of the material properties and the initial size of the void. The analysis is

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Fig. 72. Dark ﬁeld image of an isolated void and associated work-hardened layer near the rear surface of the specimen.

based on the criterion adopted from a related study of the crack blunting by dislocation emission (Rice and Thomson [204]). It is shown that the critical stress for dislocation emission decreases with increasing void size, so that less stress is required to emit dislocations from larger voids. At constant remote stresses, this would imply an accelerated void growth by continuous emission of shear dislocation loops. However, this is opposed by an increasingly thick work-hardened layer. The 2D problem was solved analytically by Lubarda et al. [200] and is presented here in a succinct fashion. Consider an edge dislocation near a cylindrical void of radius R in an inﬁnitely extended isotropic elastic body. The dislocation is at a distance d from the stress-free surface of the void, along the slip plane parallel to the x axis. The stress and deformation ﬁelds for this problem have been derived by Dundurs and Mura [205]. The interaction energy between the dislocation and the void is: E int

� � �� x2 x2 þ y2 Gb2 ¼ þ ln 2 ; 4pð1 � nÞ ðx2 þ y2 Þ x þ y2 � R2

R y ¼ pﬃﬃﬃ , 2

(51)

where b is the magnitude of the Burgers vector of the dislocation, G the elastic shear modulus, and n Poisson’s ratio of the material. The shear stress along the considered slip plane acting on a dislocation due to the pressure of the far-ﬁeld hydrostatic stress s is equal to: t¼

pﬃﬃﬃ 2s

x , ðx þ ð1=2ÞÞ2

where x ¼ x/R.

(52)

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S

S

(a)

45° 45°

109.5°

45° 45°

S

S

(b) Fig. 73. Shear loop mechanism for the growth of voids. (a) Emission of two pairs of dislocation shear loops from the void surface along the indicated slip planes, S. (b) Two slip planes intersecting the void surface at 451, the orientation that maximizes the force on the dislocation.

The total force on the dislocation due to both the applied stress and the interaction with the void (derivative of eq. (54)) is: ! pﬃﬃﬃ s F x ðxÞ x 1 b xðx4 þ ð1=4ÞÞ ¼ 2 2 � . (53) Gb G pð1 � nÞ R ðx4 � ð1=4ÞÞ ðx þ ð1=2ÞÞ2 The normalized force, Fx/Gb, versus the normalized distance, d/b, plot is shown in Fig. 75 for the case when R ¼ 10b, s ¼ 0.1 G, n ¼ 0.3 and d ¼ x�R/(20.5). The dislocation feels the maximum force of repulsion (from the void) at a position of dE2.11b. For d smaller than B1b, Fx/Gbo0 and the dislocation is attracted

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Fig. 74. Network of sequentially emitted shear loops.

0.04 0.03 0.02 0.01 Fx/Gb

0

–0.01

–0.02

–0.03

–0.04

–0.05

–0.06

0

2

4

6

8

10 d/b

12

14

16

18

20

Fig. 75. Normalized dislocation force Fx/Gb versus normalized distance from the void d/b, according to eq. (56), for R ¼ 10 b, s ¼ 0.1 G, and n ¼ 0.3. The dislocation feels a maximum force of repulsion (from the void) at a position of d/b ¼ 2.1.

to the void. For dWB1b, Fx/GbW0, and the dislocation is repelled from the void. In the limit d/b-N, the force on the dislocation vanishes since the dislocation is far from the void, and ﬁnds itself in the ﬁeld of uniform biaxial tension s. In the equilibrium dislocation position, the attraction from the void is balanced by the applied stress, so that the force Fx(x) in eq. (56) vanishes, that is, s b=R xðx4 þ 1=4Þ ¼ pﬃﬃﬃﬃﬃﬃ G 2pð1 � nÞ ðx4 � 1=4Þ

(54)

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This was done using the Rice and Thomson [204] criterion for the spontaneous emission of a dislocation from a crack tip. The stress required to emit a dislocation is: pﬃﬃﬃ scr b=R ð1 þ 2 rb=RÞ4 þ 1 pﬃﬃﬃ � pﬃﬃﬃ . (55) G 2pð1 � nÞ ð1 þ 2 rb=RÞ4 � 1 The plot of scr/G versus R/b is shown in Fig. 76 for a selected value of the material parameter, Rcore/b ¼ r ¼ 1.0. It should be noted that r is the ratio between the radius of the core and the Burghers vector and not the dislocation density, as in previous sections. The results are meaningful for sufﬁciently large sizes of voids, typically RW3rb (R W 3b–6b). The critical stress required for dislocation emission decreases both with increasing r and R/b. The smaller the dislocation width, the higher the applied stress must be to keep the dislocation in equilibrium near the void. It is noted that the force on the dislocation at a given equilibrium distance from the void due to a remote stress increases more rapidly with the ratio R/b than does the force due to attraction from the void surface. More involved dislocation models based on the Peierls–Nabarro concept, as used by Rice [206] and Rice and Beltz [207] to study the crack blunting by dislocation emission, or by Xu and Argon [126] in their study of the homogeneous nucleation of dislocation loops in perfect crystals, may be needed to further improve the analysis of the void growth by dislocation emission. The model agrees extremely well with MD simulation data, both for void growth [203,208], shown in Fig. 76, and for void collapse (Davila et al. [209]). Experimental data in the literature (e.g., Minich et al. [210]) indicate that the spall strength of high purity Cu single crystals is about 5 GPa. The spall strength of a polycrystalline Cu is about half that value, because of grain boundaries and intercrystalline defects, which promote void growth. Meyers and Zurek [211] and 0.35

Lubarda Model, Rcore = b Atomistic calc: Mean stress/G

Critical Stress, �cr /G

0.3 0.25 0.2 0.15 0.1 0.05 0 0

2

4

6

8 R/b

10

12

14

16

Fig. 76. Normalized critical stress scr/G required to emit a dislocation from the surface of the void versus normalized radius of the void R/b, according to eq. (56) (n ¼ 1/3, Rcore/b ¼ 1) and according to MD calculations. Adapted from Meyers et al. [203].

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Meyers [212] discuss the effects of polycrystallinity and grain size on void growth during spall experiments and reconcile the contradictory results. The higher spall strength observed for monocrystalline copper is due to different nucleation sites. In polycrystals, there is segregation of impurities at the grain boundaries, providing favorable initiation sites. In monocrystals, these sites are absent and initiation must occur from vacancy complexes. The 2D picture is somewhat more complex than the 1D one. Dislocation loops are emitted circumferentially around a void, along a plane intersecting it at 451. � Six loops, corresponding to the edge dislocations with directions ½1 1� 0�, ½1 0 1�, ½0 1 1� �, ½1� 1 0�, ½1� 0 1�, and ½0 1� 1�, are shown in Fig. 77(a). As they expand, their extremities approach each other and eventually react; this is energetically favorable under zero-stress conditions: Gb21 þ Gb22 � Gb27 .

(56)

The dislocations that form on reaction have a screw character. Successive loops can form on the same (1 1 1) plane, as shown in Fig. 77(c), or on adjacent planes, as the void grows. b3 = 2a [011]

b2 = 2a [101]

a [112] 2

a [121] 2

b7 = 2a [211]

b1 = 2a [110]

b4 = 2a [110]

a [121] 2

a [211] 2

b5 = 2a [101]

a [112] 2

b6 = 2a [011]

(111) Plane (a)

(111) Plane (b)

(111) Plane (c) Fig. 77. (a) Six edge dislocation loops forming at the intersection of the void surface and the (1 1 1) slip plan in a void subjected to hydrostatic expansion; (b) reactions between adjacent dislocation loops as they expand; (c) successive emission of loops. Adapted from Traiviratana et al. [203].

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The intersection of a void with eight {1 1 1} slip planes at 451 forming an octahedron, is shown in Fig. 78. The dislocation interactions become rather complex at that level. Traiviratana et al. [208] analyzed the more complex situation in which the perfect dislocations formed partials. In this case, one has to consider the reactions between the leading partials and the reactions between the trailing partials. Several researchers have simulated void growth using MD simulations [209,213– 219]. Rudd, Seppa¨la¨, and Belak [216–219] were primarily interested in void growth and did not focus on the dislocations. Potirniche et al. [214] used a uniaxial stress

Slip Plane Slip Plane Void Void

(a)

(b)

Slip Plane

Void

Top View (c) Fig. 78. (a) 3D model of a spherical void intercepting the eight slip planes that form an octahedron. (b) Another view of the 3D model of spherical void intercepting the eight slip planes. (c) Top view of the 3D model.

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Fig. 79. Emission of partial dislocation loops with trailing stacking faults (blue atoms) for three orientations of the tensile axis: (a) [1 0 0]; (b) [1 1 0]; (c) [1 1 1] [2 2 2].

conﬁguration, which led to necking and did not speciﬁcally analyze dislocation activity. Zhu et al. [215] modeled the void growth under shock loading and unloading conditions and obtained profuse evidence for shear loop emission. Marian, Knapp, and Ortiz [220,221] used the quasi-continuum simulation method and were indeed the ﬁrst to identify shear loops and some of their reactions as the strain increased. However, quasi-continuum calculations perform energy minimization of the system at zero temperature and may give results that differ from MD simulations. Davila et al. [209] modeled the inverse problem: the collapse of a void. We have also carried out MD simulations of void growth, as in the work of Traiviratana et al. [208]. Fig. 79 shows 3D visualizations of MD simulations of the early growth of voids for three directions of the tensile axis: [1 0 0], [1 1 0], and [1 1 1]. Dislocation loops are emitted in the three cases but their conﬁguration is more complex than in the simple 2D model of Section 8.2. The ﬁrst signiﬁcant difference is that partial dislocations are emitted instead of perfect dislocations assumed earlier. The blue atoms represent the stacking faults and the green atoms the surface of the voids. The original color illustrations can be found in Bringa et al. [222]. In Figs. 79(a) and 79(b) the trailing partial is also emitted while in Fig. 79(c) only one partial loop is seen. A second difference is that the loops combine to form biplanar (for [1 1 0], Fig. 79(b)) or triplanar (for [1 1 1], Fig. 79(c)) loops. It should be noted that none of our MD simulations showed mobile prismatic loops, thought by many to be the primary mechanism for loop expansion. Further analysis is provided by Meyers et al. [203].

11. Conclusions (a) Shock-wave compression produces extreme regimes characterized by a state of uniaxial strain, high strain rates, high pressures, and high temperatures. (b) The conditions are such that plastic deformation occurs primarily at the wave front and release portions and not homogeneously throughout the material, as is the case in conventional deformation.

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(c) Dislocation generation and motion in shock compression is governed by these extreme regimes. At lower pressures (below P/GB1) conventional dislocation sources are activated and multiplication processes of dislocations dominate the effects. At higher pressures (P/GW1) homogeneous nucleation of dislocation loops takes place with the expansion of the loops through subsonic and possibly, in some cases, supersonic dislocation motion. Several mechanisms of dislocation accommodation are discussed. (d) This profuse dislocation generation and motion leads to loose dislocation cells in FCC metals and a homogeneous dislocation distribution in other structures. (e) As the SFE of metals is decreased, the tendency for planar dislocation arrays and stacking faults increases. A criterion for the transition from perfect dislocation loops to partial dislocation loops is presented. (f) There is also a threshold for twinning, which is reached when the strain rate imparted by shock compression is such that the ﬂow stress by dislocation ﬂow becomes higher than the twinning stress. A criterion for the onset of twinning is presented, for the case in which the twinning stress is independent of strain rate and temperature whereas the ﬂow stress by dislocation motion is determined by thermal activation. (g) The results of MD calculations are presented. This is a powerful simulation technique that is helping to elucidate some of the thornier issues in shock compression. MD predicts an increasing dislocation density (and decreased spacing) with increasing pressure, in agreement with transmission electron microscopy characterization of recovered shocked specimens. (h) Comparison of experimental results (by TEM) and MD simulations reveals that the dislocation spacings are orders of magnitude different, being much smaller in the MD simulations. The MD simulations reveal that the annihilation rate of dislocations upon unloading is very high. This suggests that the dislocation density during shock compression might be higher than the residual dislocation density after release. (i) Controlled quasi-isentropic compression experiments-ICE (ramp-wave com pression, in which the loading is adiabatic), provides a strain rate that is lower than shock compression at the same pressure. MD simulations of ramp-wave compression, were also carried out, and characterized. The residual disloca tion densities are signiﬁcantly lower for quasi-isentropic (ramp wave) com pression than for shock compression, as predicted by McNaney et al. [135] and Shehadeh et al. [181]. (j) Upon being reﬂected at a free surface, a shock wave creates a tensile pulse which, if of sufﬁcient magnitude, generates fracture of the metal either by crack or void nucleation, growth, and coalescence. The initiation and growth of voids is shown to occur by the formation and expansion of shear loops from the void surface. These shear loops, originally postulated by Lubarda et al. [200], are shown to be the principal mechanism of void expansion.

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Acknowledgment This research was funded by UCOP under the ILSA program. The support by Dr. D. Correll is gratefully appreciated. The help by Mr. Y. Seki in manuscript preparation was essential to its completion. Laser compression experiments were carried out at the OMEGA (LLE, University of Rochester) and Jupiter (LLNL) laser facilities. Discussions with L. E. Murr, K. S. Vecchio, G. T. Gray, N. N. Thadhani, D. H. Kalantar, J. McNaney, V. Lubarda, J. S. Wark, and B. Kad were helpful in formulating the ideas expressed here. Parts of this chapter come from the doctoral dissertations of Buyang Cao and Matthew S. Schneider, whose dedicated work enriched our contribution. We also thank N. Q. Vo, for his help with the nc plasticity analysis, and S. Traiviratana for providing us the simulations shown in Fig. 79.

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CHAPTER 90

Mechanical Properties of Nanograined Metallic Polycrystals G. SAADA LEM, CNRS-ONERA, UMR 104 CNRS, 29 Av. de la Division Leclerc, BP 72, 92322 Chatillon Cedex, France and

G. DIRRAS Laboratoire PMTM, Institut Galile´e, 99 Av. Jean-Baptiste Cle´ment, Universite´ Paris 13,93430 Villetaneuse, France

r 2009 Elsevier B.V. All rights reserved 1572-4859, DOI: 10.1016/S1572-4859(09)01503-4

Dislocations in Solids Edited by J. P. Hirth and L. Kubin

Contents 1. Introduction 202 1.1. General remarks 202 1.2. Scope 205 2. Microstructures in as-prepared ng polycrystals 206 2.1. Introduction 206 2.2. Bulk ng polycrystals 207 2.2.1. General remarks 207 2.2.2. Elastic moduli 207 2.2.3. Grain boundaries and triple junctions 208 2.2.4. Dislocations, stacking faults and twins 208 2.2.5. Dislocation density, average grain size, grain size distribution, grain shape and crystallographic texture 209 2.2.6. Stability of the microstructure 213 2.3. Freestanding thin films 214 2.3.1. Single layers 214 2.3.2. Multilayers 214 2.3.2.1. fcc/fcc multilayers 214 2.3.2.2. bcc/fcc multilayers 215 3. Evolution of the microstructure during plastic flow 215 3.1. Introduction 215 3.2. Bulk ng polycrystals 216 3.2.1. Introduction 216 3.2.2. fcc metals 216 3.2.2.1. Dislocations and twinning activity 216 3.2.2.2. Internal elastic strains 217 3.2.2.3. GB activity 218 3.2.3. bcc metals 219 3.2.4. Conclusion 221 3.3. Freestanding multilayers 221 4. Mechanical behaviour of ng polycrystals 221 4.1. Introduction 221 4.2. Strain-controlled deformation and hardness 222 4.2.1. Introduction 222 4.2.2. The proof stress s0.2 and the Hall–Petch law 223 4.2.2.1. Experimental results at 300 K at quasi-static applied strain rate 223 4.2.2.2. The elastic–plastic transition and the HP law 223 4.2.2.3. The elastic–plastic transition in ng polycrystals 226 4.2.2.4. Model of the micro-deformation stage 227 4.2.3. Micro- and nano-indentation tests 228 4.2.4. Room temperature flow properties at low strain rates 230 4.2.5. Effect of strain rate 230 4.2.6. Effect of temperature 232 4.2.7. Kinematical constraints on dislocation mechanisms 232 4.3. Nanotwinned fcc metals 234 5. Molecular dynamics 236

6. Dislocation-mediated plasticity of ng polycrystals 237 6.1. General remarks 237 6.2. SAGBs 238 6.2.1. Structural information 238 6.2.2. Plastic flow associated with nucleation of perfect dislocations at SAGBs 239 6.2.3. Plastic flow associated with nucleation of partial dislocations at SAGBs 240 6.3. LAGBs 241 6.3.1. Structure of LAGBs 241 6.3.2. Plastic flow associated with nucleation of perfect dislocations at LAGBs 242 6.3.2.1. Possible emission mechanisms 242 6.3.2.2. Deformation mechanism 242 7. Conclusion 243 References 243

1. Introduction 1.1. General remarks The general principles underlying the processing of high strength and ductile metallic materials have been understood since the beginning of the development of dislocation theory. Ductility necessitates a reasonable mobility of the dislocations, while strengthening is related to the building of obstacles restricting their propagation. The latter (dislocations, solute atoms, second phase inclusions and grain boundaries) are introduced by alloying (structural hardening), plastic flow (strain hardening) and grain size refinement [1,2]. In the following we focus on the effect of grain size refinement. Metallic polycrystals whose average grain size d is larger than about 0.5–1 mm, will be further referred to as coarse-grained (cg) polycrystals. When cg polycrystals are uniaxially strained at a constant strain rate, their 0.2% proof stress, s02, varies linearly with the reciprocal of the square root of the grain size d. It is known as the Hall–Petch (HP) law [3,4], and the result is expressed as: s02 ¼ sm þ kd 1=2

(1)

Both k, known as the Hall–Petch constant, and sm, whose meaning is made explicit in Fig. 1, depend on the material and on the conditions of the experiment.

Fig. 1. Stress–strain curve of a cg metallic polycrystals up to the macroyield strain (the symbols are defined in the text).

y1.1

Mechanical Properties of Nanograined Metallic Polycrystals

203

Table 1 Values of sm (MPa) and k (MPa m1/2) measured at 300 K for 1.5 mmodo0.3 mm

sm k

Cu [6]

Al [6]

Ag [6]

Ni [7]

Nb [6]

Mo [6]

Fe [3]

Ti [6]

Zr [6]

Mg [6]

26 0.11

16 0.07

37 0.07

22 0.16

69 0.04

108 1.77

30 0.7

78 0.4

29 0.025

7 0.03

There is general agreement that, using the data of Table 1, eq. (1) holds quite satisfactorily in a wide domain (1.5 mmodo300 mm) of grain sizes for almost all metals and alloys, provided one compares materials of similar microstructure during plastic flow. The importance of the as-processed microstructure, even in cg polycrystals, is exemplified by the observation that the variation of s02 of Cu specimens prepared by rolling and annealing is well represented by eq. (1), for 4 mmodo178 mm, but with values of the constants significantly different from those given in Table 1 (sm ¼ 92 MPa, and k ¼ 0.4 MPa m1/2 Ref. [5]). Leaving aside, for simplicity sake, the various effects of impurity segregation, of twinning and of inhibition of glide systems due to lattice friction, microstructural observations reveal that in cg polycrystals deformed at room temperature, slip starts in some well oriented grains when the applied stress sa reaches a critical value sm, known as the microyield stress. The corresponding strain, em, is the microyield strain. The moving dislocations are blocked at the grain boundaries (GBs), where they accumulate and develop an internal stress. This internal stress increases with increasing plastic strain ep, which triggers the motion of dislocations in the neighbouring grains. When the plastic strain reaches a value eY large enough, known as the macroyield strain, the whole polycrystal may be considered to undergo plastic deformation. The yield stress sY is therefore the result of the strain hardening resulting from the plastic strain eY. The tangent modulus y, or strain-hardening rate, is defined by y¼

dsa dp

(2)

Typically, during elastic straining, y is equal to E*, the Young’s modulus E corrected for the effect of the stiffness of the machine. As soon as plastic flow begins, y decreases steadily. In cg polycrystals, at eY ¼ 0.2%, sY is identified with s02. The corresponding strain-hardening rate yY is of the order of 0.06 E* for dE1.5 mm. The grain size affects s02 and y in the same way. By extrapolating eq. (1) down to very small-grained materials, one expects an increase in strength up to values of the order of 1 GPa [8], which has been observed in various experimental conditions. This remark has stimulated the processing of polycrystals of smaller and smaller grain sizes, which have been classified, as microcrystalline (mc) when dE1 or a few micrometers, ultrafine crystalline (ufc) when 100 nmodo1000 nm and nanocrystalline (nc) when do100 nm [9]. Since the grain size alone neither defines the microstructure nor characterizes the mechanical

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Table 2 The s02 proof stress of some ng materials

d (nm) s02 (GPa) calculated s02 (GPa) experiment

Ni

Cu

Cu

Al

Al

Ti

20 1.150 1.6 [10]

26 0.71 0.78 [11]

40 0.58 1.25 [12]

40 0.35 0.29 [13]

150 0.18 0.44 [14]

100 1.34 0.95 [15]

Note: Comparison of experimental values and values given by eq. (1), using the data of Table 1.

behaviour of these very small-grained polycrystals, we consider them simply as cg polycrystals, materials with a grain size larger than one or a few micrometers, and as nanograined (ng) polycrystals, materials with a grain size smaller than about 500 nm. Various mechanical tests on these products have revealed a remarkable increase of strength, at the expense of ductility. As shown in Table 2, however, the measured values of the yield stress of ng polycrystals significantly differ from those obtained by extrapolating eq. (1). One reason for this discrepancy lies in an inaccurate description of the elastic– plastic transition. Indeed, due to the processing conditions, and to the grain size, both the macroyield stress and the macroyield strain of ng polycrystals differ from those of cg polycrystals. Despite the large amount of papers published in the last 30 years on the subject, a comprehensive description of the deformation and strengthening mechanisms of these materials is still lacking. Indeed our present understanding of the deformation mechanisms of cg polycrystals depends on: 1. Processing specimens with well designed and characterised microstructures. 2. Gathering consistent and systematic data through mechanical tests over a wide range of strain rates and in a large temperature interval. 3. Thoroughly observing the evolution of the microstructure during mechanical testing. 4. Establishing a consistent correlation of the mechanical behaviour with the microstructure by a relevant theoretical analysis. Instead, for ng polycrystals, the present situation is characterised by: 1. Unsatisfactory control and description of the as-prepared microstructure Due to their very large specific GB area, the processing of ng polycrystals imposes the use of highly irreversible thermomechanical processes, whose control, despite numerous improvements conducted worldwide, remains rather empirical, and far from satisfactory [9,16]. Up to now, indeed, there is no complete reproducibility of the mechanical behaviour of specimens, even when they are obtained by apparently identical routes. This is even true for asprocessed materials obtained from the same compact. Despite the amount of research on this problem, the design of a material presenting both high strength and ductility remains, in most cases, largely problematic.

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2. Incomplete information on the evolution of the microstructure The characterisation of the evolution of the microstructure during plastic flow is still incomplete. As a consequence, discrimination between the relevant deformation mechanisms, and comparison of their effect on different materials remains a challenge. 3. Inaccuracy of the measurement of the relevant mechanical parameters Due to the difficulty of processing sufficient amounts of samples with controlled microstructure, the preparation of ng polycrystals specimens of convenient shapes and sizes for uniaxial straining at constant strain rate is not always possible. For this reason, extensive use was made of micro- and nanoindentation, and of more indirect techniques such as rolling or creep. Due to the lack of stability of the products, systematic analysis of the effect of temperature on the mechanical behaviour is almost nonexistent. 4. Inadequate theoretical interpretation In the case of cg polycrystals, there is a general, and justified agreement on the relevance of the identification of the 0.2% proof strain flow stress s02 to the yield stress sY and of the latter to a fraction, generally 1/3, of the Vickers hardness HV, in application of the Tabor rule. The extension of these assertions to the case of ng polycrystals must be questioned.

1.2. Scope We do not consider, despite the interest in them, ng materials such as nanowires, nanopillars or nanospots. We rather focus on the mechanical behaviour of bulk ng polycrystals, and on those aspects of the mechanical behaviour of thin films directly related to this purpose. We start in Section 2 with a rapid review of the features of the as-prepared microstructure of available ng polycrystals. A survey of the observation of the evolution of the microstructure of bulk ng polycrystals and of freestanding multilayers during mechanical testing is developed in Section 3. Section 4 is devoted to the analysis of the observed mechanical properties of these materials, focusing on uniaxial strain-controlled deformation. Emphasis is put on the very strong constraints imposed by the smallness of the grain size of ng polycrystals on the mean free path of dislocations and their nucleation or multiplication rate. The small grain size of ng polycrystals, whose diameter is smaller than about 50 nm, allows performing molecular dynamics (MD) investigations that give useful hints on the elementary processes controlling plastic flow in ng polycrystals. They are discussed in Section 5. Lack of information prevents us from investigating all the possible competing mechanisms controlling plastic flow in ng polycrystals. There are however good reasons to think that dislocation nucleation and absorption at GBs are relevant mechanisms. Section 6 is devoted to their analysis. We conclude in Section 7.

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2. Microstructures in as-prepared ng polycrystals 2.1. Introduction Bulk ng polycrystals samples of pure and commercially pure metals, as well as alloys, have been successfully produced by means of various processing methods. For example, in the most common crystallographic structures: fcc: Cu [17–19], Ni [20], Al [14], Pd [21], Au [22], Ni-Fe [23], Al-Mg [24] bcc: W [25], Fe [26], Ta [27] hcp: Ti [28], Zn[29], Mg[30], Zr[31]. Co and its alloys (e.g. Co-P) are special in that they exhibit hcp [32], fcc [33] structures or a mixture of both phases depending on the processing conditions. For the sake of clarity and simplicity, only microstructures resulting from three major processing routes are presented and discussed in Section 2.2. 1. Electrodeposition (ED) is a one-step processing route which produces fully dense samples of elemental materials such as Ni [34], Co [32] and Cu [35] or alloys such as Ni-Fe and Ni-W [36], in the shape of metallic sheets up to about hundred of micrometers thick, with grain sizes down to tens of nanometres. 2. Powder metallurgy (PM) based methods are two-step processing techniques, which consist in the production of powder particles (elemental or alloyed), followed by their consolidation or compaction. The powder particles are obtained by various methods such as mechanical alloying, sol–gel synthesis, electro-explosion of wire and inert gas phase condensation. For a complete review on these processes see Ref. [16]. The consolidation/compaction of powders is usually carried out under extrusion [37], unidirectional or hot isostatic pressing, as well as by use of the emerging spark plasma sintering technique [37,38]. 3. The most used severe plastic deformation (SPD) techniques are: high plastic torsion straining (HPT), multiple forging, cold rolling and equal channel angular pressing (ECAP) or extrusion (ECAE). For a detailed review on these methods, see Ref. [39]. Due to the decisive role they play in the development of microelectronic devices, thin films are the centre of a wide area of research. Ascertaining their structural integrity in service conditions has stimulated a large number of studies. Recent work has shown that thin films are particularly well suited for the study of the mechanical properties of ng polycrystals. The last 20 years have seen indeed the development of techniques such as electron beam evaporation or magnetron sputtering on a glass substrate, allowing the processing of multilayers whose total thickness may vary between some tens of nanometres to a few micrometers. The spacing l of the individual layers (i.e. their repeat distance) may be as small as 10–50 nm and it scales with the grain size. Detailed information on their processing can be found in Ref. [40].

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Both bulk ng polycrystals and freestanding ng polycrystals thin films are adapted to classical mechanical analysis, particularly by strain-controlled deformation. Relevant elements of their microstructure are described in Sections 2.2 and 2.3.

2.2. Bulk ng polycrystals 2.2.1. General remarks Single component materials, are characterised by the degree of purity, the grain size d, the grain size distribution, the grain morphology, the texture, as well as by the nature of the GBs and the content in various types of crystal defects (dislocations, twins, stacking faults), in addition to surface defects. Besides the knowledge of the alloying components, the characterisation of alloys requires information about their state of dispersion: intragranular or in the GB, more or less ordered solid solution and coherent or incoherent inclusions. In as-prepared ng polycrystals, the average internal elastic strain, measured by X-ray diffraction (XRD) is of the order of 0.3%. The microstructural origin of this elastic strain is not identified. Depending on the processing route, the processed materials may contain uncontrolled amounts of impurities and contaminants [41,42] and exhibit porosity [39], as well as inhomogeneous particle distribution [43]. Nevertheless, artefact-free nc Cu23 (the symbol Xx defines a material X whose average grain size is x nm) samples have been produced by in situ consolidation of Cu powders [44]. 2.2.2. Elastic moduli The measurement of elastic moduli gives important information on the porosity of the material. From data obtained by quasi-static techniques (tensile, bending and hardness tests), the Young’s modulus, the bulk modulus and the shear modulus of ng polycrystals are usually lower than those of cg polycrystals materials. Consolidated materials present an extreme case with a 25–30% modulus depression. Similar results have been obtained for the elastic moduli of Fe, Cu and Cu-Ni alloys prepared by mechanical milling/alloying, with average grain sizes down to about 5 nm [45], as well as on twinned or untwinned ng polycrystals of electrodeposited Cu [46]. On the other hand, the elastic modulus of near fully dense nc iron (d in the range of 8–33 nm) has been shown, in a miniaturised disk bend test, to be about 8% higher than the coarse-grained iron counterpart [47]. The GB region has been proposed to be elastically softer than the grain interior, becoming increasingly softer with increasing tensile strain [48]. Although the low atomic density of GB regions [8] may play a role, porosity due to incomplete particles bonding [49] has a stronger impact on elastic properties than grain size reduction per se [50,51]. Gold nanowires of diameter varying between 40 nm and 250 nm have been synthesised electrochemically on alumina substrates, and deformed in bending using the atomic force microscopy (AFM) technique. The wire thickness has no effect on the measured Young’s modulus: the Young’s modulus of nanowires

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consisting in a single chain of gold atoms has been shown to be the same as that of the bulk [52]. Similar results are obtained by sound wave velocity measurements. The Young’s modulus of ng polycrystals of Fe, Ti and various Fe-C alloys obtained by ball milling and compaction, with dE20 nm, is about 0.98 times the Young’s modulus of the standard counterpart [53,54]. In the case of Cu, for similar grain sizes and porosity, the corresponding ratio is 0.88 [55]. This unexplained discrepancy might be due to a texture effect. Therefore, we postulate that the decrease of elastic moduli is a consequence, and a measure, of porosity rather than a genuine effect resulting from the GB structure. 2.2.3. Grain boundaries and triple junctions Both GBs and triple junctions constitute the so-called interphase component of the bulk polycrystals. They are critical features of the microstructure, which cannot be considered as fully understood. Triple junctions commonly exhibit a disordered open-like structure [56]. As a consequence, and similarly to GBs, they are more prone to contain impurities than grain interiors. High-resolution electron microscopy (HREM) observations indicate that the crystal structure of Cu [9], Ni [9] and Al [57] ng polycrystals is perfect right to the GBs, whose average thickness wGB is about 0.5 nm. Therefore, the ratio of the GB volume cGB to the intracrystalline volume is of the order of the ratio of the GB thickness wGB to the grain diameter. cGB

2pwGB d

(3)

For dE30 nm, cGBE0.1. Non-equilibrium GB structures are a typical feature of metals processed by means of SPD techniques. For example, HREM reveals that GBs of Cu and Ni prepared by ECAP [58], of sub-micrometer-grained Al-Mg [59] or of Ti [60] deformed by HPT exhibit zigzag configurations with irregular arrangements of facets, and steps. Dislocations walls whose misorientation is a few degrees, representing 30–40% of the total GB area, are frequently observed [61]. There is no reliable estimate of the density of extrinsic dislocations in GBs. 2.2.4. Dislocations, stacking faults and twins A high density of nanotwins has been observed in ultrafine-grained Ni-Co and Cu processed by electrodeposition [46,61,62]. As discussed in Sections 3 and 4, asgrown ng polycrystals containing a large density of twins exhibit a specific mechanical behaviour. Stacking faults have been observed by transmission electron microscopy (TEM), for example in Al [36,63,64] prepared by cryogenic ball milling and in HPT Cu [65]. Contrary to statements by the authors, this does not prove that the propagation of Shockley partials is the dominant deformation mechanisms for Al and Cu ng polycrystals. Shockley partials are seldom observed in ng polycrystals that have not

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been processed through intense plastic deformation. Individual dislocations are seldom observed and in situ TEM observations does not reveal any appreciable amount of dislocation activity. One reason for this is as follows. In order to observe a single dislocation of length d in each grain, the dislocation density must be of the order of d2, which, for dE100 nm, means a density larger than 1014 m2. Furthermore, some crucial information such as the Burgers vector of the observed dislocations is most often undetermined. Finally, due to the specific area of GBs, one expects an important amount of the dislocations to be too close to the GB, or even incorporated into it, to be detected by TEM. As a consequence the measurement of dislocation density is generally performed by XRD techniques, which are sensitive to the average grain size. 2.2.5. Dislocation density, average grain size, grain size distribution, grain shape and crystallographic texture Defining unambiguously the grain size is not simple and implies some arbitrariness because the microstructure is generally not homogeneous (especially after SPD processing). Besides, it is not easy to discriminate between the various types of boundaries: small angle (misorientation smaller than 0.1–0.2 radian) grain boundaries (SAGBs), large angle grain boundaries (LAGBs) and boundaries between coherent scattering domains. As a consequence, the fine-grained microstructures claimed to be obtained in the literature are often dubious because grain sizes were measured by various techniques which measure different physical quantities, and, not surprisingly, often produce different grain size values [66]. Orientation imaging microscopy coupled with electron back scattering diffraction allows one to extract both the nature of the boundaries and the misorientation between adjacent grains. Two neighbouring points are considered to belong to the same grain if their misorientation is smaller than a certain tolerance angle. The measured grain size distribution depends critically on this parameter [67]. Furthermore, the detection limit of orientation imaging microscopy is about 1–21. Thus, SAGBs with misorientations smaller than this limit cannot be detected. Therefore, there is some arbitrariness in fixing a limit from which a given boundary is considered as being a SAGB. XRD measures the size of coherent scattering domains notwithstanding whatever the exact structure of their boundaries. It allows, with the help of some assumptions, to calculate the dislocation density, crystallite sizes and two different averages of the grain size (over the volume and over the number of grains). This method has been successfully applied to SPD-processed materials. In a more local and less quantitative point of view TEM techniques provide analyses of local details such as grain morphologies, size and distribution, grain orientation and GB fine structure. The above comments are illustrated by the work of Zhilayev et al. [66]. These authors used peak profile analysis based on high-resolution XRD and TEM to measure the distribution of grain sizes and the dislocation density in ng polycrystalline nickel processed by different methods such as electrodeposition, equal channel angular pressing, cold rolling, high pressure torsion straining and their

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Table 3 Grain size measurement in Ni processed by various techniques [66]

XRD (nm) TEM (nm)

ECAP

ECAPþCR

HPT

ECAPþHPT

ECAPþCRþHPT

ED

71 350

38 300

42 170

48 140

46 100

24 35

Note: ECAP ¼ equal channel angular pressing; CR ¼ cold rolling; HPT ¼ high plastic torsion straining; ED ¼ electrodeposition; XRD ¼ X-ray diffraction; TEM ¼ transmission electron microscopy.

combinations. As can be seen in Table 3, the grain size measured by TEM is about two to five times larger than the grain size measured by XRD, whatever the processing technique. The reason for this discrepancy is related to the fact: (i) X-rays have better sampling statistics (X-ray studies about 108 grains, where TEM investigate only 102 to 103 grains). (ii) Both techniques are sensitive to small angle misorientations. But there are some dislocation walls (e.g. dipolar walls), which do not give contrast difference in TEM while they break coherency of X-ray scattering. (iii) the size of scattering domains equals the grain size in materials where there is no substructure inside the grains. This is no longer the case in SPDprocessed materials. Therefore, in this case, the discrepancy in crystallite size between X-ray and TEM measurements is larger. Nevertheless it should be noticed that for a complete microstructure determination, the information obtained by one technique could be refined and amended by the other [68]. From Table 3, the measured XRD (respectively TEM) grain sizes dXRD (respectively dTEM) are of the order of 50 nm (respectively 150 nm), while the dislocation densities quoted in Ref. [66] are of the order of 1015 m2. The number of dislocation segments of average length d in a grain is, therefore, of the order of 2–3 (respectively 20–30). This result is far from trivial since the ratio l/d of the average distance l between the intragranular dislocations to the grain size is critical in all description of the mechanical behaviour of polycrystals. A systematic investigation comparing the Williamson–Hall plots of computergenerated ng polycrystals of Al at the atomic scale, with a well defined twin or dislocation content, shows that an appreciable fraction of the dislocations lays in the GBs and in some cases cannot be identified via the observation of their core structure [69]. The same study reveals that twins cannot be detected unless their thickness is small compared to the grain size. We now discuss a feature of ng polycrystals that seems to have been overlooked, that is, the relevance of the average grain size to characterize the microstructure. Let N be the total number of grains in a given specimen, and n(d)dd the number of grains whose grain size lies between d and d þ dd and define: PðdÞ ¼

nðdÞ N

(4a)

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1

PðdÞdd ¼ 1

(4b)

0

The grain size distribution in ng polycrystals is known to be lognormal. P(d) is thus expressed as: " # 1 1 Lnðd=d 0 Þ 2 (5) PðdÞ ¼ pﬃﬃﬃﬃﬃﬃ exp 2 s sd 2p P(d) satisfies eq. (4b) and presents a maximum value Pm for d ¼ dm. d m ¼ d 0 exp s2 Pm ¼

expðs2 =2Þ pﬃﬃﬃﬃﬃﬃ d m s 2p

ð6Þ

Furthermore, we have 2 s /dS ¼ d 0 exp 2 /d 2 S ¼ d 20 expð2s2 Þ ¼ /dS2 expðs2 Þ 2 9s ¼ /dS3 expð3s2 Þ /d 3 S ¼ d 30 exp 2 D /d 2 S /dS2 ¼ d 20 ðexpð2s2 Þ expðs2 ÞÞ

ð7Þ (8)

d0 and s can be determined numerically from the measurement of dm and Pm, by means of eq. (6). Using eqs (7) and (8), d0 and s are expressed as functions of D and /dS: d0 ¼

s¼

/dS2 ðD þ /dS2 Þ1=2 1=2 D þ /dS2 Ln /dS2

ð9Þ

This result shows that eq. (44) of Ref. [70] is incorrect, likely as the result of a misprint. In a representative example, we have dm ¼ 25 nm pﬃﬃﬃﬃ and Pm ¼ 0.03. From eq. (6), one obtains s ¼ 0.48, d0 ¼ 31.5 nm, /dS ¼ 35 nm, D ¼ 18 nm and /d3S ¼ 2/dS3. The average volume V* of the grains whose size is smaller than an arbitrary value d* is given by: R d* 3 d PðdÞdd (10) V* ¼ 0R d* 0 PðdÞdd

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For our example taking d* ¼ 2dm ¼ 50 nm, we find that: The proportion of grains having a grain size smaller than 2dm is equal to 83%. These grains occupy 37% of the total volume. Their average volume is equal to 0.37/d3S. The proportion of grains having a grain size larger than 2dm is equal to 17%. These grains occupy 63% of the total volume. Their average volume is equal to 4 /d3S. Therefore, this material is somehow better represented by a distribution of large grains of average diameter 41/3/dSE1.6/dS embedded into small grains of average diameter 0.371/3/dSE0.7/dS, than by a distribution of grains of average size /dS. This result is a consequence of the width of the grain size distribution. Quite often, the actual grain size distribution shows some irregularities, which reinforce the tendency towards bimodality. For example, let us consider ED-processed materials, which present various textures. In particular, /1 1 1S or /1 1 0S out-of-plane fibre-like textures have been observed, while the in-plane texture is random [71]. Nevertheless, with such a processing method, the grain size distribution is relatively narrow; average grain sizes as small as 20 nm, with a maximum grain size of 80 nm are easily obtained. Generally speaking, the consolidation of nanopowders during spark plasma sintering or hot isostatic pressing produces a rather homogeneous microstructure with equiaxed grains, the interiors of which are generally dislocation-free. These methods are unique in that that they permit one to tailor the microstructure and thus the mechanical properties. For example, one can process a composite-like microstructure by controlling the volume fraction of larger grains in a fine-grained matrix, without an intermediate heat treatment, contrarily to the case of SPD microstructures. The materials so processed have a random crystallographic texture. Consolidation methods are able to produce materials with grain sizes down to tens of nanometres [49]. However, the microstructure of consolidated ballmilled powders is heterogeneous, due to its far from equilibrium state. The microstructure of materials processed by severe plastic deformation consists of grains with LAGBs comprising clusters of small subgrains with SAGBs. Nevertheless, homogeneous microstructures have been reported in materials processed by SPD methods such as high plastic torsion straining and ECAE, provided they have undergone an adequate number of revolutions or passes, respectively. Actually, most analyses reveal a complex evolution of the microstructure during SPD. The final product is strongly textured, with a texture varying within the thickness of the sample. In pure metals one may obtain a mean grain size of about 100–200 nm. In many metals or alloys processed by high plastic torsion straining consolidation of powders after ball milling, the grain size is typically smaller than after ECAP pressing and can be as low as 15–20 nm. Very generally, the probability of defects occurrence, as evaluated from XRD experiments, increases when plastic deformation is introduced during such processes like inert

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gas condensation or ball milling [56]. Dislocation densities of about 1015 m2 were reported in ECAP-processed Cu with a grain size of 150 nm [72]. 2.2.6. Stability of the microstructure 1. Samples processed by electrodeposition [37,73,74], as well as by powder metallurgy [43,75] exhibit stability concerns, which are illustrated in the following examples. The strength of electrodeposited nc Ni increases after annealing for 1 h at temperatures below 150 1C, with little change in the grain sizes or detectable impurity segregation [76,77]. The effect of annealing on as-processed electrodeposited Ni25 has been analysed through the evolution during isochronal annealing of the full width at half maximum (FWHM) of various diffraction spots, during in situ XRD. No change in the FWHM occurred at temperatures lower than 100 1C. An appreciable decrease of the FWHM, without change of grain size, is observed at temperatures between 100 1C and 200 1C. Grain growth is observed at temperatures higher than 200 1C [78]. Annealing at 200 1C induces a bimodal grain size distribution, which improves the ductility. Above 250 1C, impurity segregation to GBs induces a transition to brittle behaviour [76]. Segregation of impurities at GBs, particularly sulphur, along with abnormal grain growth, was reported during static annealing performed at temperatures above 400 1C [73,74]. The analysis of the evolution of electrodeposited Ni20 at temperatures varying between 393 1C and 693 1C shows that impurities play a significant role in the thermal stability. Annealing at a temperature lower than 493 1C (0.3Tm, where Tm is the melting temperature) induces only a reordering of GBs, while annealing at a higher temperature results in GB migration and grain coalescence [79]. 2. In PM-processed materials, impurities and inclusions resulting from contamination during consolidation may have the beneficial effect of stabilising the microstructure by preventing grain growth. For example, the cryomilling of Fe-10 wt.% Al powders in liquid Ar or N results in nc structures which are thermally stable at least up to 67% of the melting temperature of Fe. The enhanced thermal stability of these powders is attributed either to the formation of g-Al2O3 in liquid argon, or to the formation of oxynitrides, g-Al2O3 and AlN particles in liquid nitrogen [80]. In contrast, cryomilling of elemental Fe results in a nc structure, which grows to a sub-micrometer scale following annealing at 1223 K. The formation of Fe3O4 particles does not result in enhanced thermal stability. The same conclusion has been reached on ufg Fe produced by hot isostatic pressing [75]. 3. A similar situation has been observed in Ti processed by high pressure torsion straining. After a short time anneal at temperatures up to 300 1C, XRD analysis detects important changes in the lattice distortions without visible

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grain growth. This reveals a heterogeneous intragranular distribution of the internal stress, with a length scale smaller than the grain size [60]. Grain growth starts only at temperatures above 350 1C. 4. Stress-assisted grain growth resulting from the high driving force due to the combined effect of the small grain size and higher stress level during testing, was reported recently [13]. 2.3. Freestanding thin films 2.3.1. Single layers Very generally, the grain size depends on film thickness and texture depends on the processing conditions. Typical layers of Cu [81,82], Ag [81,82] and Al [81,83], prepared by evaporation or sputtering, consist of columnar grains. They show no evidence of porosity or impurity segregation. However, the level of entrapped H and O is not measured. The columnar grains present a strong crystallographic texture, with a {1 1 1} plane parallel to the surface. The GBs do not present any specific character and individual dislocations are seldom observed. In Ag and Cu [82,84], the in-plane diameter scales with the sample thickness while, in Al, it is of the order of 11 nm for 30 nm thick plates, and of 22.5 nm for 50 nm thick films [83]. Films of Ni of average grain size of about 30 nm [85], as well as films of Al of thickness 30–50 nm were obtained by sputtering [83]. The latter present a lognormal distribution of grain sizes with an average grain size of the order of half the thickness of the foil. The grains are mostly equiaxed, and contain no isolated dislocations and no pores. The thick films present a weak {1 1 1} texture in the growth direction. 2.3.2. Multilayers Alternate layers of total thickness of about 1–5 mm, with thicknesses in the range of 5–500 nm, are commonly prepared by sputtering or evaporation. Very generally, they show no evidence of pores or impurity segregation. Besides the grain size d, and the texture, other specific features characterising the microstructure are the film thickness w, the layer spacing l and the elastic moduli, which are the source of image forces on the dislocations, the mismatch at the interfaces, which is the source of coherency stress and the relative solubility, which controls the chemical sharpness of interfaces. 2.3.2.1. fcc/fcc multilayers. Typical examples are Cu/Ag [81,82,84], Cu/304 stainless steel (Cu/SS) [86], Ni/Cu [87][88–91], Ni/Au [91], Ni/Ag [91], whose relevant characteristics are given in Table 4. In general, vapour deposited films display intermixing of a few monolayers at the interface, even for non-miscible systems. The number of intermixed monolayers is estimated to be at most of the order of three in Ag/Cu, which may be nevertheless important when the multilayer thickness is smaller than 100 nm. In Cu/SS, the multilayer consists in fccCu/fccSS/bccSS layers, the fccSS layers being much thinner than the bccSS layers.

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Table 4 Some typical features of fcc/fcc multilayers System

mB/mA Mismatch (%) Texture

Cu/Ag [81,82,84] Cu/304SS [86] Cu/Ni [89–91] Ni/Au [91] Ni/Ag [91]

0.56 2 1.5 0.34 0.37

10.2 0.48 2.63 14.8 14.8

Matching

(1 1 1) Incoherent (1 1 1) Coherent (1 1 1) [91], Cube [90] Coherent, semi-coherent (1 1 1) Incoherent (1 1 1) Incoherent

Miscibility None None Yes Yes None

Very generally, the grains are columnar with a pronounced out-of-plane texture. The ratio of the average in-plane grain diameter dp to the layer thickness l increases from about 1 for lE1 mm to very large values for the smallest l values. In Cu/SS, for example dpE33l for lE10 nm. In Ag/Cu multilayers, the interface plane is a {1 1 1} plane and dp is of the order of 100–150 nm for a layer thickness in the range of 0.85–90 nm. Concerning the presence of dislocations, the situation is rather obscure. Dislocations have been detected in relatively thick layers (W100 nm). The strong variations of contrast observed in bright field TEM images of thinner layers can be attributed to dislocations, but no clear identification exists. GBs normal to the interfaces of the multilayer are observed, but no detailed information is available on their structure. 2.3.2.2. bcc/fcc multilayers. The most studied examples are Cu/Cr [87,89] and Cu/Nb [87,89,92–94]. One observes Cufcc/Crbcc interphases with Kurdjumov–Sachs matching whatever the layer thickness l [82]. Cufcc/Nbbcc with Kurdjumov–Sachs matching is observed for lW15 nm, while Cubcc/Nbbcc is observed for lo15 nm [92]. Dislocations are observed in deformed samples, but not in the as-grown ones [94]. The dependence of the grain size on layer thickness is as follows [87]: Cu=Cr :

Lnd ¼ 0:21 þ 0:82 Lnl;

(11a)

Cu=Nb :

Lnd ¼ 2:1 þ 0:45 Lnl,

(11b)

where d and l are expressed in nanometers. From eq. (11), for lE10 nm, dE0.8l in Cu/Cr and dE2.3l for in Cu/Nb, while for lE100 nm, dE0.6l in both cases.

3. Evolution of the microstructure during plastic flow 3.1. Introduction Specific difficulties limit the efficiency of classical observation techniques when applied to the evolution of the microstructure. For example, the grain size interferes with the specimen thickness in in situ TEM. The distributions of intrinsic and

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extrinsic dislocations in GBs, as well as the intragranular dislocation distribution close to the GBs are practically unknown. This section in which we examine successively the situation in freestanding multilayers (Section 3.2) and in bulk ng polycrystals (Section 3.3), consists therefore in a list of open questions rather than in a comprehensive account of the mechanisms of plastic flow in ng polycrystals.

3.2. Bulk ng polycrystals 3.2.1. Introduction Due to the uncertainty on the actual microstructure of the as-prepared samples, there do not exist reference specimens that would allow performing systematic investigations of the evolution of the microstructure during plastic deformation. Furthermore, the application of the techniques used for cg polycrystals meets specific difficulties when applied to ng polycrystals, as pointed out in Section 2.2.3. As a consequence, information is sketchy, not to say contradictory in some cases. We summarize the results for the most extensively studied fcc metals in the next section. Observations on bcc metals are summarised in Section 3.2.3. 3.2.2. fcc metals 3.2.2.1. Dislocations and twinning activity. In Ni ng polycrystals whose mean grain size is smaller than about 30 nm, some post mortem TEM observations report the absence of dislocation storage within grains after room temperature deformation, whatever the deformation mode. In parallel, no dislocations are observed in the vicinity of GBs, even after deformation by rolling [95–97]. Contrariwise, deformation by twinning and glide of Shockley partials is observed [98]. The flight time tf of a dislocation moving at the velocity v is of the order of d/v. For dE50 nm and assuming vW1 mm/s, tfo0.05 s, dislocations are therefore likely to cross the whole grain and be trapped at the GB before being seen. The possibility of TEM observations of either dislocation motion or of intragranular dislocation interactions leading to dislocation storage is very remote. Therefore, the lack of direct observation of perfect dislocations should not be interpreted as a proof that perfect dislocation glide is not a usual deformation mechanism in a given ng polycrystal. Indeed, the motion of partial dislocations leaves stacking faults or twin boundaries (TBs) while perfect dislocations leave no trace. In Ni25 deformed in uniaxial compression at 77 K under quasi-static (E3.103 1 s ) and dynamic (E2.6.103 s1) strain rates [99], one notices the following. In both cases, perfect dislocations are observed in about 15% of the grains. In samples deformed at high strain rates, twinning is observed in 45% of the grains, and stacking faults are observed in 2% of the grains. In samples deformed at low strain rates, twinning is observed in 22% of the grains, and stacking faults are observed in 16% of the grains. No defects (dislocations, stacking faults or twins) are found in 50% of the grains.

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Whether the large proportion of defect-free grains means a high degree of heterogeneity or reflects the impossibility of finding adequate contrast conditions is not clear [99]. For example, in Ni10 prepared by pulsed laser deposition, glide of perfect dislocations inside the grains is observed by in situ HREM [95,100], while in situ TEM [97] detects motion of perfect dislocations only close to cracks. In situ TEM has revealed groups of dislocations generated at a GB ahead of a crack tip in Cu23 prepared by ball milling, and both strength of about 1 GPa and 15% ductility [11]. When the grain size distribution is bimodal, for example in Cu prepared by electrodeposition [34], or inert gas condensation [101], dislocations are observed in the larger grains (dW200–500 nm), while the small grains appear to be almost dislocation-free. Dislocation debris, tangles and dislocation walls are found after uniaxial compression of commercial purity Al at room temperature [102]. The average grain size is of the order of 150 nm, but the debris are observed only in the grains whose size is larger than about 500 nm and which do not contain oxide particles. Deformation by twinning and by propagation of Shockley partials was observed in micro-indented or Al1535 manually ground at room temperature [57]. Abundant deformation twinning is also observed in Pd15 prepared by inert gas condensation and deformed by rolling [103]. Grain rotation provides indirect information on the existence of intragranular deformation [104]. Electrodeposited Cu20 deformed by rolling at room temperature at a strain rate of 103–102 s1 may exhibit a plastic strain of 5000%, without showing any grain growth. The main change in the microstructure consists in a large increase of the misorientation between the adjacent grains, from about 61 to 181, together with an increase of the microstrain up to 0.15% [105]. As shown in more detail in the next section, as-grown twins strongly influence the mechanical behaviour of ng polycrystals of Cu. For example, Cu400 exhibits a high TB density, which is defined by the ratio rT ¼ d/wT of the average grain size to the average twin spacing wT. It shows higher strength and ductility than Cu30, which does not contain twins [46,61]. For this material TEM analysis shows dislocation accumulations at the coherent TBs, as well as a displacement of the TBs resulting from the high strains produced by indentation through the creation of steps and jogs. TBs appear to be both strong barriers to dislocation motion and sources of dislocation nucleation and multiplication. 3.2.2.2. Internal elastic strains. The XRD technique has been used to follow the evolution of the internal elastic strain during the tensile deformation of electrodeposited Ni25 at room temperature and at 180 K, with applied strain rates of about 105–104 s1. The evolution of the order-dependent FWHM of X-ray line profiles [78,106,107] was measured in situ. 1. At room temperature: (a) The as-processed specimen shows no modification of the FWHM until the applied stress sa reaches a value of about 1.4 GPa. Then the FWHM increases steadily, showing an increase of the root mean square. Therefore,

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plastic flow induces sources of stress. Upon unloading, one observes a decrease of the FWHM down to a value smaller than the initial one. Leaving the specimen unloaded at room temperature induces a further decrease of the FWHM. Therefore, plastic flow may remove part of the sources of stress induced by processing. Reloading the same specimen results in an increase of the FWHM from the beginning of plastic deformation. (b) The FWHM does not increase when the maximum applied stress is sar400 MPa. It increases progressively with sa provided that saZ800 MPa, until a value saE1.4 GPa is reached. For saZ1.4 GPa, the FWMH is clearly irreversible upon unloading. Therefore, a minimum stress sm is necessary to produce a change in the sources of stress, but full plastic flow occurs only for a higher stress sY. This point is discussed in more detail in Section 4.2.2.2. (c) The initial value of the FWHM of annealed specimens is smaller than that of as-processed specimens, all the more as the temperature is high. The FWHM of samples deformed in the plastic regime is clearly irreversible. For a specimen annealed at 140 1C, some features observed for as-processed specimens are found, but with less intensity. For a specimen annealed at 180 1C, these features disappear. Therefore, annealing removes part of the sources of stress induced by processing. 2. At 180 K: Loading of as-processed specimens at 180 K shows a behaviour similar to the one observed at room temperature. But an increase of the FWHM is observed upon unloading. Successive loading–unloading cycles produce a cumulative, irreversible increase of the FWHM. Therefore, the mechanical behaviour is thermally activated. We further discuss these points in Sections 4.2.6 and 6.3.2.1. 3.2.2.3. GB activity. GB sliding is observed in Al120 for a deformation at 300 1C [14,106]. There is no direct evidence of GB sliding due to room temperature straining, although the relative rotation of lattice fringes in different grains and the existence of nanocavities in deformed crystals could be interpreted as resulting from this mechanism [107]. Grain growth has been observed by TEM in various instances in Cu [97,100,106], in Al [108], in Ni deformed by indentation [92] and in a fcc Co-P12 alloy deformed in tension at room temperature [33]. In situ TEM tensile deformation of Ni9.7 has revealed GB migration resulting in appreciable grain growth [109]. There is, however, some controversy about the relevance of this observation [110,111]. Freestanding samples of Al50 prepared by magnetron sputtering have been deformed in tension and examined by post mortem TEM [13], in order to eliminate the possible effect of the overall thickness of the sample, as is the case for in situ TEM, as well as the effect of deformation gradients, as is the case in indented materials. Grain growth does occur, but not in all samples. When it does, it does

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from the beginning of plastic deformation. The grain growth is discontinuous and appreciable. The diameters of the grown grains may attain 500 nm. The resulting grain size distribution is bimodal, which has the effect of enhancing ductility. The growth mechanism is not understood, although it seems to be stress-assisted. In situ TEM reveals GB motion and rotation in sub-micrometric grains (dE200– 500 nm), as well as in nanometer-sized grains [108] of Al samples. Stress-assisted discontinuous grain growth is observed in thin films of Al deformed in tension at a strain rate of 5 105 s1 at room temperature [112]. The grain growth is revealed by both TEM and XRD analyses. Furthermore, in the absence of grain growth, the ductility of the samples strongly decreases. 3.2.3. bcc metals The plastic deformation in compression of ball-milled and consolidation-processed Fe with grain sizes varying between 80 nm and 1 mm is localised in shear bands oriented at 451 from the compression axis. Little change in the microstructure (i.e. grain size and shape, crystallographic texture) is detected outside these bands. Inside the bands, intragranular dislocations are observed as well as strong grain elongations. Cavitation at triple nodes is observed, which suggests the possibility of grain rotations [26,113]. At room temperature, a transition of the deformation mode, from strain hardening to strain softening, occurs for grain sizes below 1 mm, reflecting the transition from homogeneous deformation to a deformation localised into shear bands. The homogeneous deformation is found to be carried out by lattice dislocations, while the deformation within shear bands involves lattice dislocations as well as boundary-related mechanisms. The latter are probably accommodated by boundary opening [113]. There is a lack of microstructure characterisation and information is missing about its evolution upon high temperature straining. However, there is one report that at higher temperatures (350–650 1C), cg polycrystals of Fe (5 mmodo1 mm) deform by lattice dislocations; electron beam scattering diffraction experiments reveal the presence of a strong fibre-like crystallographic texture. Contrariwise, the ng polycrystals counterpart (grain size: 500–250 nm) deforms mostly via boundaryrelated mechanisms. This is confirmed by both a higher strain rate sensitivity of the flow stress (E0.5 at 650 1C) and a random crystallographic texture. Plastic straining in this domain of temperature does not induce grain growth. This is due to the high temperature processing of the samples (about 700 1C). The evolution of the distribution of the grain sizes and the misorientations due to plastic straining of Fe at various temperatures is shown in Figs 2 (cg polycrystals of Fe) and 3 (ng polycrystals of Fe). For the cg polycrystals at room temperature and at 350 1C, the grain size distribution is shifted to the left (decrease of the grain/crystallite size) and the frequency of the boundary misorientations is higher at small angles (less than 101). This indicates a strong activity of lattice-based dislocations, which results in the creation of SAGBs. For the ng polycrystals, at 350 1C, the average grain size stays almost the same, the fraction of new SAGBs being relatively small. This suggests a moderate intragranular plastic activity.

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Fig. 2. The grain size distributions (a) and misorientations (b) across the boundaries in Fe with an average grain size of 5 mm, after compression tests at 350 and 650 1C. The distribution of the as-processed material is also shown.

Fig. 3. The grain size distributions (a) and misorientations (b) across the boundaries in Fe with an average grain size of 0.25 mm. The grain size distribution of the as-processed material is also shown.

The situation is even more marked for straining at 650 1C as shown in Fig. 3. The frequency of boundary misorientations follows fairly well that of a randomly misoriented bcc material. Thus, given the strain rate sensitivity value and the absence of marked crystallographic texture, grain growth and intragranular plasticity, we are left only with the possibility that in the ng polycrystals strained at 650 1C, GB mechanisms accommodate the deformation. Under dynamic loading, V100 hot-consolidated after ball milling behaves like ng polycrystals of Fe or amorphous metallic glasses [114]. Localised deformation occurs, which eventually develops into cracks that lead to a load drop in the dynamic stress–strain curves. Under dynamic loading by a compressive Kolsky bar system at a strain rate of about 5000 s1, ng polycrystals of refractory W150–500 processed by severe plastic deformation deform mainly via localised adiabatic shear bands, which reduces the load bearing capacity of the tested samples [115].

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The indentation of Ta thin films with d varying between 10 nm and 30 nm shows the development of different twin systems [116]. 3.2.4. Conclusion Although GB motion has been observed, there is little doubt that slip by perfect or partial dislocations and twinning are the most active deformation processes at room temperature and, very likely, below. Indeed, there are strong indications that the possible deformation processes depend not only on intrinsic parameters such as grain size, grain size distribution, GB structure, but also on constraints imposed by the strain path. Indeed, the deformation mechanisms appear to be a complex interplay of lattice dislocations-based and GBs-based mechanisms. More detail is given in Section 6. 3.3. Freestanding multilayers Very few observations of dislocation activity in freestanding multilayers and single layers are available. Observations show slip lines parallel to the {1 1 1} plane in Cu/Ni multilayers [90], as well as dislocations crossing the interfaces and exhibiting opposite curvatures in Cu/Nb multilayers with layer thickness in the range of 7–17 nm [94]. Dislocation glide apparently generated at a junction between GBs has been observed in Au300 [117]. Cu/Nb multilayers with a thickness of about 15 mm and an initial layer thickness l varying from 30 nm to 75 nm have been rolled at room temperature in order to decrease l down to 30 nm [118]. The objective was to discriminate between the intragranular straining effect, the layer thickness, the grain size d and the texture. The initial grains are columnar with dEl with no evidence of dislocations. The texture indicates a {1 1 0}Nb//{1 1 1}Cu interface. Rolling produces an extension of the in-plane grain size without changing the out-of-plane grain size. Observation does not reveal any modification of either the interior of the grains or of the GBs. The true plastic strain ep is estimated by measuring the thicknesses of the layers after rolling; it varies from 0.01 to 0.92. Due to the absence of grain rotation, one can conclude that plastic flow results from dislocation glide. In these experiments there is no significant evolution of the microstructure.

4. Mechanical behaviour of ng polycrystals 4.1. Introduction As pointed out in Sections 2 and 3, the control, knowledge and characterisation of the as-processed microstructure, as well as of its evolution during mechanical testing, are incomplete. Because of the limited amount of material for ng polycrystals, their mechanical properties have been historically investigated using micro-indentation and nano-indentation tests. Also, most studies regarding

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mechanical behaviour measurements were devoted to determining whether or not ng polycrystals obey the HP law, when plotting the 0.2% proof stress s02 in straincontrolled tests, or the Vickers hardness HV as a function of d1/2. Finally, the interpretations of the experimental results are often presented in terms of models relying on assumptions applicable to cg polycrystals, whose relevance in the case of ng polycrystals should not be taken for granted. Due to the lack of systematic experimental investigations, we only attempt to give here a synthetic description of the mechanical behaviour of bulk ng polycrystals.

4.2. Strain-controlled deformation and hardness 4.2.1. Introduction No universal type of stress–strain curve exists to describe the mechanical behaviour of ng polycrystals. However, the curves depicted in Fig. 4 represent qualitatively the most observed types of behaviour. In Fig. 4(a), the stress–strain curve presents three domains. In domain 1, the tangent modulus y decreases steadily from E* down to a value y1 of the order of or smaller than 0.05E*. Then, in domain 2, the tangent modulus decreases more or less steadily down to 0. Finally, in domain 3, y is negative, plastic flow is unstable and the specimen deforms until fracture. E* is smaller than the Young modulus E corrected from the stiffness of the machine when the ng polycrystals specimens contain porosity [19]. In Fig. 4(b), the extent of domain 2 is very small. This type of curve has been observed, for example in ng bcc metals deformed in compression. In Fig. 4(c), the specimen fractures before reaching domain 2. The reference data are s02, the maximum or ultimate stress sU, the corresponding strain eU and the ductility measured by the fracture strain ef. Contrary to what is observed in cg

Fig. 4. Typical aspects of the stress–strain curves of a nanograined metallic polycrystal. (a) Existence of a plateau of appreciable extension. (b) Existence of a maximum followed by softening. (c) Fracture after a small plastic strain.

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polycrystals, the tangent modulus y02 is in most cases very large, about E*. For eU oep oef, plastic flow is unstable. It is worth pointing out some features of the mechanical behaviour of ng polycrystals, which are out of control for the time being and will not be discussed further: the effect of impurities [10], the tension–compression asymmetry [19,21,107] and the influence of the specimen size [10,33,119]. 4.2.2. The proof stress s0.2 and the Hall–Petch law 4.2.2.1. Experimental results at 300 K at quasi-static applied strain rate. Many authors who compiled the proof stress s02 as a function of d1/2 found that part of the points were distributed close to a straight line. For example Fig. 1 of Ref. [120] collects results for Cu, showing that some points are aligned, but not all. Alternatively, one can represent this variation by a relation similar to that of eq. (1), in restricted intervals, and using various coefficients sm and k. The discrepancy between the measurements and the extrapolation of eq. (1) down to grain sizes smaller than 100 nm is illustrated in Table 2. Of course, the grain size is not always the relevant scale. For example, Cu processed by a pulsed electrodeposition technique contains a large density of nanotwins, with only one type of twin in each grain. Typical grain sizes are of about 400–600 nm, while the average twin spacing wT may be much smaller. In these materials, s02 increases from 490 MPa to 860 MPa when wT decreases from 100 nm to 15 nm [46,121–123]. Hence, more relevantly, one can extrapolate eq. (1) down to 15 nm with the same constants by substituting the average twin spacing wT for the grain size d [123]. The TBs appear to be obstacles of strength comparable to that of GBs. The strong strengthening induced by a high twin density is not restricted to ng polycrystals. The tensile plastic deformation at 4.2 K of pure Cu polycrystals with a grain size of 29 mm induces a strength of the order of the theoretical strength. This strength results from the development of a dense dislocation network and of very thin twinned lamellae. In both these two last examples, the critical length controlling the strength is the twin lamellae thickness, not the grain size [124]. A similar situation occurs in Cu/Nb multilayers deformed by rolling (see Section 3.2). Indentation tests show that the flow stress of the as-prepared multilayer of thickness l ¼ 30 nm is of the order of 1.3 GPa. The stress–strain curve shows a stress of about 1.5 GPa for ep ¼ 0.01, increasing to 1.7 GPa for ep ¼ 0.92 [118], which is the manifestation of an appreciable strain hardening. This is a strong indication that the relevant scale is that of the layer thickness. 4.2.2.2. The elastic–plastic transition and the HP law. As pointed out in the introduction, the yield stress of polycrystals results from the hardening of the material during the micro-deformation stage. To define its extent, let _ a ; s_ a and _p be the applied total strain rate, the corresponding applied stress rate and the average plastic strain rate. One has s_ a ¼ E*ð_a _p Þ

(12)

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From eqs (2) and (12) the tangent modulus y is: _ p dsa ¼ E* 1 y da _ a

Ch. 90

(13)

These equations are valid whatever the elementary mechanism controlling plastic flow, provided the averaging procedure is consistent. Although plastic strain is localised on the glide planes swept by the gliding dislocations, we consider here its average at the specimen scale. During the elastic stage, _p is nil and the tangent modulus y is equal to E*. When the applied stress sa reaches the value sm, the corresponding (elastic) strain being em, plastic deformation starts. The plastic strain rate _p increases, y starts decreasing and sa keeps increasing. When the plastic strain reaches a value eY, for a value sY of the applied stress, the whole crystal undergoes plastic straining. Early theories of the yield stress in cg polycrystals assume that during the microstraining stage, slip remains localised in one plane and is blocked at the GB in the form of a planar dislocation pileup. The stress sah at a distance x ahead of a pileup pﬃﬃﬃﬃﬃﬃﬃﬃﬃ of length D scales as ðsa sm Þ D=x [125]. Identifying sm with the yield stress of the single crystal and D with the grain size, one easily recovers eq. (1). Despite the fact that these theories are still very popular, one may believe that they are inadequate for the following reasons. 1. The quantitative agreement is not very good. 2. There is no reason to describe the plastic flow process as planar. 3. Most importantly, the pileup models cannot explain the effect of grain size on the strain-hardening rate. By saying this, we do not mean that pile-ups do not exist, or that they play no role. Considered alone, however, they do not represent adequately the situation. Reproducing a scaling law is a necessary condition for a model to be relevant, it is not a sufficient condition. Models implying the multiplication of intragranular dislocations seem to be more representative of the processes occurring during plastic flow in cg polycrystals. They can be summarised as follows [126,127]. A given plastic strain ep corresponds to the development, in each grain, of an average number n / p d=b of dislocation loops of diameter d and Burgers vector of length b, which corresponds to a total length of dislocations / nd / p d2 =b. Assuming that part of these dislocations is left inside the grains, the intragranular 3 / =bd, which induces a stress opposing the dislocation density r / nd=d pﬃﬃﬃ pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ p dislocation glide / r / p =bd. With reasonable assumptions one obtains for the variation of the applied stress: rﬃﬃﬃﬃﬃﬃﬃﬃﬃ ap b (14) sa s0 þ E* d

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Then, rﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ aY b sY ¼ s0 þ E* d

(15)

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ dsa ab yY ¼ E* d p ¼Y 2dY

(16)

Here, a is a number measuring the ratio of the dislocation length generated inside the grains to the dislocation length locked at the GBs [128]. Eq. (1) is recovered by setting pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ (17) k ¼ E* abY Eqs (16) and (17) capture the experimental result that the grain size influences in approximately the same way the flow stress and the strain-hardening rate using orders of magnitude corresponding reasonably to the values given in Table 1. When aE1 and eYE0.002, eq. (14) yields rﬃﬃﬃ b yY 16E* (18) d In cg polycrystals, dW1.5 mmE6000 b, and yYo0.2E*. In ng polycrystals, on the other hand, the above analysis gives inconsistent results. For example, the application of eq. (18) with dE50 nmE200 b yields yYWE*. Remarks: 1. Despite various attempts, no consistent calculation of k exists so far. The above model does not give any insight on the origin of the development of dislocations in the neighbouring grains. It has been proposed [129] that ledges at the GBs concentrate stress, which triggers glide in neighbouring grains. This should increase the dislocation density, but there is no direct evidence for this mechanism. 2. Assuming that plastic flow results from dislocations that sweep completely the individual grains and are blocked at the surface of GBs, the internal stress opposing plastic flow is the stress due to surface dislocations at GBs. Consider at first a situation in which a single grain (1) plastically deforms. When it is swept by n1 dislocations loops, the back stress in this grain is pbn1/d. If a neighbouring grain deforms plastically by the sweeping of n2 dislocations loops, the back stress in grain (1) is pb (n1n2)/d. Assuming a Gaussian distribution of the positions pﬃﬃﬃof the moving dislocations, one finds that the back stress in each grain is / n, wherep nﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ is the average number of dislocations having swept each grain. Since n / bp =d one recovers eq. (15).

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Whatever the exact mechanism, it is clear that the validity of the HP law relies on the assumption that sY is measured at the end of the micro-straining stage. 4.2.2.3. The elastic–plastic transition in ng polycrystals. Eq. (6) allows one to determine the plastic strain rate by measuring the variation of the tangent modulus with the applied stress. This has been done for freestanding Ag/Cu multilayers, with a repeat distance l varying between 46 nm and 1390 nm. The deformation was carried out in tension up to a total strain of about 1.2%, at an imposed plastic strain rate of 105 s1 [130]. It is verified that _p is nil up to a stress sm and further increases linearly with sa. The variation of _p is well represented by: _ p ¼ _a

sa sm sY sm

sa sm

(19)

sm is measured on the curve y(s); it was shown to vary with the repeat distance l as: sm ¼ 0:4

mb l Ln l b

(20)

It is reasonable to identify sm to the microyield stress and to propose that it corresponds to the stress necessary to activate dislocation sources, whose length scales with l. The stress sY is determined by fitting the experimental curve. The stress–strain curve is well represented by: a m , (21) sa ¼ sY ðsY sm Þ exp C sY sm where C is the biaxial modulus. One can identify sY with the macro-elastic limit. Its variation with l is expressed as: rﬃﬃﬃ b sY ¼ sY0 þ sY1 , (22) l with sY0 ¼ 278 MPa and sY1 ¼ 9570 MPa. This experiment brings three important results: 1. The extent of the micro-deformation stage may be as large as 1%. 2. The macroyield stress scales with the reciprocal of the square root of the thickness of the layer. But the value of sY1Ob and sY0 differ from the values of s0, and k valid for cg polycrystals. 3. The microyield stress depends on the grain size. Similar results are observed on freestanding thin films of thickness of 1–3 mm, in which the grain size is of the order of the film thickness. The large extent (E0.7%) of the micro-deformation stage in bulk ng polycrystals was mentioned in Section 2.2.6. This result was confirmed in an analysis of the evolution of internal stresses by XRD in Ni30 deformed in tension at room temperature [131].

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The large extent of the micro-deformation stage up to values definitely larger than 0.2% is also observed in films on substrates deformed by thermal cycling [130]. The situation is however more involved, and is not discussed in more detail here. 4.2.2.4. Model of the micro-deformation stage. Simple scaling arguments show that the existence of a micro-deformation stage of the order of 1% is quite general. Let gp1 be the average elementary plastic shear undergone by a grain swept by a single dislocation of Burgers vector b. By definition, Gpn is the corresponding average plastic shear of a polycrystal of volume V made of N grains, of average volume d3, in which n grains have undergone an identical elementary plastic shear gp1. One has gp1

b d

(23)

nb (24) Nd The corresponding plastic elongation epn of a polycrystal deformed in uniaxial straining is: Gpn

pn ¼ jGpn j

nb , Nd

(25)

where j depends on the averaging process. Depending on the specific model, j is identified to either the Taylor factor (about 1/3) or the Von Mises factor (about 1/O3). The plastic elongation epN of a polycrystal in which each grain has been sheared by one dislocation is thus: b (26) d With bE0.25 nm and dE25 nm, epNE102 jE(0.3–0.6) 102. A lower bound of the macroyield strain is obtained by identifying eY to epN. This result is far from trivial since almost all the results published so far concerning the mechanical behaviour of ng polycrystals in uniaxial strain-controlled deformation identify the macroyield stress sY to s02. Due to the large value of the tangent modulus during the micro-deformation stage, the information contained in the published results, not to say their exact meaning, particularly as concerns the strainhardening rate, must be questioned. Notice, for example that, since the stress s02 corresponds to emoepoeY, its value combines sm and sY. Some expressions departing from eq. (1) have been indeed proposed. Systematic experiments are, therefore needed, in order to describe accurately the elastic–plastic transition and to measure the extent of the microplastic deformation stage in ng polycrystals. These results are easily generalised to the two following relevant situations. pN j

1. Plastic flow resulting from nucleation and glide of partial dislocations. The above equations apply upon substituting the length of the partial Burgers vector bP to b. The above results remain qualitatively correct.

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2. Plastic flow resulting from twinning. Let bT and wT be the Burgers vector of the twin dislocation and the distance between twinning planes. Applying the reasoning developed above, the plastic strain corresponding to the creation of one twinned lamella of width hT is: pT

g¼

jghT d

bT wT

(27a)

(27b)

with gEO3 in fcc metals.

4.2.3. Micro- and nano-indentation tests A representative example of a plot of the Vickers hardness HV as a function of d1/2 is given in Fig. 2 of [132] in the case of Ni. This figure gathers results obtained by micro-indentation and by nano-indentation. There is no doubt that many experimental points are aligned. For these points one may write: HV ¼ 0:330 þ 1:1d nm 1=2 (28) 3 GPa However, many points obviously do not agree with the line representing eq. (28). Besides, identifying HV/3 to s02, one may verify that the constants s0 and k, appearing in eq. (28) significantly differ from those quoted in Table 1. On the other hand, it was also claimed that for do10–15 nm, the hardness of electrodeposited Ni saturates [133] and that the hardness of ng polycrystals of Cu and Pd prepared by inert gas condensation either saturates, or even decreases with decreasing grain size [134]. Although there is no doubt that the room temperature hardness of ng polycrystals increases as the grain size decreases down to 10–15 nm, the existence of a HP law valid for 15 nmodo300 mm is in contradiction with the experimental data as can be seen in Table 2. For do15 nm, the existence of the saturation of the hardness, or of its decrease with decreasing grain size is questionable. Part of these contradictions or discrepancies arises from difficulties in modelling and (or) analysing correctly the complexity of the mechanics of contact [135–138]. In micro-indentation tests, the Vickers hardness HV is computed as the ratio of the load to the projected imprint size. Due to the inaccuracies that may occur in micro-hardness testing, hardness measurements of ng polycrystalline materials are now routinely investigated using instrumented nano-indentation (or depth-sensing indentation). In that case, the displacement of a sharp indenter probe of various shapes is recorded continuously as a known load is applied and further removed. In addition, the possibility to control the loading or the loading–unloading cycles allows performing creep tests or strain rate sensitivity measurements.

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Nano-hardness numerical results were shown to be 10–30% larger than the micro-hardness data [139]. According to this study, the discrepancy may have two origins, the assumption that the contact is purely elastic and the use of the projected area at peak instead of the residual projected area. It is generally well accepted that the factors influencing the accuracy of the results, are: the actual tip shape and surface quality, the specimen surface finish, the correct setting of the initial point of contact, the modulus of the indenter which is used to determine the modulus of the sample under investigation and the occurrence of sink-in collapse of the area surrounding the indenter or pile-up (increase of the contact area). The influence of this last factor on the calculated hardness value requires in some cases to image the imprint for obtaining an exact measurement of the surface contact. Finite element analysis shows that in systems that exhibit the pile-up phenomenon, the difference between calculated and actual contact areas can be as high as 60% [140]. Overestimations in hardness up to 100% have been reported [141]. In addition, the stress–strain diagrams extracted from the analysis of indentation data with the help of various assumptions and models, often significantly differ from those obtained from strain-controlled, tensile stress–strain curves [142]. Yang and Vehoff [143] used a nano-indentation technique combined with highresolution AFM to probe grain size effects on the mechanical properties of ng polycrystalline nickel. This material is produced by pulsed electrodeposition; it has a nominal grain size of about 20 nm and is subsequently heated up to produce controlled grain sizes. It is shown that the indent position greatly influences the hardness measurement. In particular, indents on GB triple points, GBs or in the centre of grains give different results. Unfortunately, no value of the corresponding hardness is shown in that study. This observation is nevertheless interesting, particularly in the case where the indent size is made smaller than the grain size to probe the response of a single grain. The dislocation–GB interaction is also probed by adjusting the indent size relative to the indented grain. When the indent size is smaller than the grain size, the hardness increases (from 2.5 GPa to 4.2 GPa for grain sizes of about 2000 nm to 750 nm, respectively), because the size of the plastic zone scales with the grain size. But when the indent size is of the order of, or larger than the grain size, the plastic zone size spreads over the surrounding grains, and the hardness decreases (from 4.2 GPa to 2.8 GPa for grain sizes in the range of about 100–750 nm). Finally, the identification of the flow stress to one-third of the Vickers hardness HV, relies on the assumption that the material exhibits a perfect elastic–plastic behaviour, despite the complexity of the stress and strain distributions around the indent [144]. This is certainly not the case in most, if not all, ng polycrystalline materials at small strains, in the conditions where the flow stress is measured. In summary, there is no doubt that depth-sensing indentation is a powerful tool for the characterisation of mechanical properties of small volumes of materials, but the physical meaning of the results is still uncertain. For our purpose, we conclude that the available results obtained by indentation techniques must be considered as hints rather than as physical facts.

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4.2.4. Room temperature flow properties at low strain rates The room temperature tensile ductility of ng polycrystals with grain size smaller than about 30 nm is generally very low, of the order of 2–3%, when the as-processed material does not contain a large twin density. There are, however, a few noticeable exceptions whose characteristics are summarised in Table 5. These exceptions very likely result from a specific, but not identified, microstructure. The ductility of nanotwinned Cu processed by pulsed electrodeposition increases with the reciprocal of the twin lamellae density, or equivalently with the decrease of their average width. A ductility as large as 14% to 18% is obtained for wT E 15 nm, together with sUE1–1.2 GPa. Another interesting characteristic value is that of the strain-hardening rate, which may reach 2 GPa in domain 2 [46,121,123]. Remark: Electrodeposited Cu20 has been successfully cold-rolled to a strain of 5000% with a microstrain measured by XRD of about 0.16%. The hardness increases from about 1 GPa for the as-processed material to about 1.2 GPa in the deformed material [105]. This example of very large ductility is, for the time being, unique. 4.2.5. Effect of strain rate The strain rate sensitivity m is defined by: @ ln s m¼ @ ln _ ;T

(29)

The strain rate sensitivity is measured by strain rate jumps on a given stress– strain curve, or less accurately by a comparison of the stress at a given level of plastic strain on the stress–strain curves of materials deformed at various strain rates. Whenever one may consider that plastic flow may be described as a thermally activated process, the strain rate sensitivity is related to the apparent activation

Table 5 Some examples of ng metals presenting appreciable ductility (ef is the final plastic strain, s02 the 0.2% proof stress, sU the ultimate stress and y the strain hardening rate)

Cu26 Al-5%Mg26 Ni44 Ni-15%Fe9 Ti100 Co30

ef (%)

s02 (GPa)

sU (GPa)

y (GPa)

Processing

15 8 8 6.7 13 10

0.8 0.6 0.5 1.2 0.95 0.95

1.1 0.75 1 2.3 1 1.6

1 2 Parabolic Parabolic o0.5 Parabolic

IGC [11] ISCMA [112] ED [145] ED [145] SPD [15] PED [146]

Note: IGC ¼ inert gas condensation; ISCMA ¼ in situ consolidation mechanical alloying; ED ¼ electrodeposition; SPD ¼ severe plastic deformation; PED ¼ pulsed electrodeposition.

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volume Va by the relation [147,148]: V a ¼ bV

kB T msa

(30)

Here kB is the Boltzmann constant, T the absolute temperature and bV a numerical constant (bVEO3). In Fe, the strain rate sensitivity is much smaller for ng polycrystals than for the cg polycrystals counterpart. Typically mE0.005–0.01 for do100 nm and m increases more or less linearly up to 0.06 for dE10 mm. The measurements have been made for a strain rate varying between 4.104 s1 and 3.103 s1. Similar results are found for Ta [27,149]. In both cases, the increase of strain rate induces a decrease in ductility. The apparent activation volume obtained by using eq. (30) is of the order of 26 b3, and 12 b3, for Fe and Ta respectively [27]. The situation is somewhat different in fcc metals, as documented in the case of Ni and Cu. In Ni29, when the strain rate is lower than about 1 s1, m is of the order of 0.015–0.02 at room temperature, which can be compared to the value m ¼ 0.005 for cg polycrystals of Ni. The strain rate sensitivity increases very rapidly with increasing strain rate above 1 s1. In Ni21, sU is insensitive to the applied strain rate when the latter is smaller than 1 s1, while it increases from about 1.4 GPa to about 2.5 GPa when the strain rate increases from 1 s1 to 103 s1 ðD ln sU =D ln _ 0:6Þ [10]. From RT room temperature measurements, mE0.02 for Cu300 [27] and 0.026 for Cu23. The increase of the strain rate from 104 s1 to 102 s1 results in a decrease of the ductility from about 10% to 6%. In Cu which does not contain nanotwins and for strain rates lower than about 1 s1, m varies from 0.01 to 0.06 when d decreases from 190 nm to 10 nm [96]. Similar results are obtained by quasi-static hardness tests [150]. When the strain rate varies between 103 s1 and 104 s1, the strain rate sensitivity measured by strain rate jumps is D ln s15% =D ln _ 0:15 [151]. In nanotwinned Cu, m decreases from 0.037 down to 0.015 when wT increases from 15 nm to 100 nm [121], with a plastic strain rate lower than about 1 s1. In another example, the ductility increases from 12% to 15% when the strain rate increases from 104 s1 to 101 s1 [122], which is puzzling. The apparent activation volume is also measured by relaxation tests [152–154]. One method consists in plotting Lnðdsa =dtÞ as a function of Lnsa. Indeed, starting from eq. (6), and taking into account that during a relaxation test da =dt ¼ 0, one has: dp ds ¼ dt E*dt

(31)

The method was used for Ni30 and gives m ¼ 0.02 at room temperature [77]. Repeated relaxation tests on Ni25 [155] show that the activation volume decreases with decreasing applied stress. At room temperature, Va decreases from about 100 b3 for saE0.5 GPa to 15 b3 for saE2 GPa. At 180 K, Va decreases from about 250 b3 for saE0.5 GPa, to 50 b3 for saE2 GPa [78]. These values must be considered as trends, or hints, because they compare results obtained on ng polycrystals processed by different techniques, as well as strain rate

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sensitivity values measured at different strains. Furthermore, whatever the technique used, the strain rate sensitivity and the strain-hardening rate are in fact coupled quantities, which are considered as decoupled in most of the articles quoted in this section. The relative error is of the order of the ratio y/E*, which is not negligible [154]. From eq. (30), at room temperature, for Cu and Ni: Va

0:4 b3 msa ðGPaÞ

(32)

In Ni30, saE 1.1 GPa, mE0.03, and VaE12 b3, a somewhat similar situation is found in nanotwinned Cu with a high density of twins. The strain dip test was used to separate the contribution of the long-range stress si and the short-range stress s* stress in electrodeposited Ni30. It is found that for 2%oeo4%, siE1.3 GPa and the ratio s*/si is about 0.3. The apparent activation volume is found to be E10 b3 [85]. These results are a hint, not a proof that plastic flow proceeds by thermally activated glide, with an activation area of the order of a few b3. 4.2.6. Effect of temperature A decrease in the deformation temperature has a dramatic effect in ng polycrystals, compared to what is observed in cg polycrystals. Nanotwinned Cu deformed in tension a strain rate of 104 s1 shows a large increase in sU (from 700 MPa at 300 K to 900 MPa at 77 K), as well as in ductility from 12% at 300 K to 32% at 77 K. At this last temperature, the strain-hardening rate may be as large as 2 GPa [122]. For Ni30 produced by pulsed electrodeposition and deformed in tension, the ductility ef decreases from 4%, at 300 K, to 2.7% at 77 K. The ratio s02 ð77 KÞ= s02 ð300 KÞ is about 1.5. In hcp Co, a tensile test at 77 K gives ef E 6%, s02 E 1700 MPa and sU E 2500 MPa [146]. Deforming Ni30 in tension at a strain rate of 104 s1 at 77 K induces an increase in strength of about 30%. Furthermore, the strain rate sensitivity decreases from 0.02 at 300 K to 0.002 at 77 K [77]. In parallel, m rapidly increases with temperature above 300 K. Whether this increase is genuine, or results from an instability of the microstructure is not clear. 4.2.7. Kinematical constraints on dislocation mechanisms The above observations indicate that, in most cases, plastic straining is the result of the glide of intragranular perfect or partial dislocations, and (or) twinning. The factors controlling the competition between these two mechanisms are unknown. This justifies the proposition, prompted by MD simulations, that room temperature plasticity results from dislocation nucleation, fast propagation to GBs, followed by absorption in the latter [156,157]. The grain size imposes constraints on the kinematics of glide, which are analysed in the following. In a grain of diameter d, let r and v be the mobile dislocation density and the average dislocation velocity. One has very generally: _p ¼ j_gp ¼ jrbv

(33)

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j is defined in eq. (26). Substituting into eq. (13), one obtains y ¼ jrbv _ p ¼ _a 1 E*

233

(34)

The flight time tf of a dislocation in a grain is: tf

d v

(35)

As pointed out in Section 3.2.2.1, tf is very small about a few 101 s. We assume now that dislocations of average length d are created at a rate ng per second, in a grain, at a GB, or at the edges of the latter and propagate until they are absorbed to the next GB. Then, the intragranular mobile dislocation density satisfies the following differential equation: dr ng rv ¼ dt d 2 d

(36)

After integration: r ¼ r0 exp

t ng t 1 exp þ tf vd tf

(37)

Here r0 is the intragranular dislocation density at the time t ¼ 0. Dislocations nucleation and propagation can be considered as uncoupled, provided the flight time of the dislocations is negligible compared to the nucleation time. This condition is written as: v (38) ng o d When this condition is satisfied: ng r vd Substituting into eq. (34): y d ng ¼ 1 _a E* jb

(39)

(40)

Assuming yEE*/3 at yield and jE1/3, ng ¼

2d _a b

(41)

Substituting into eq. (39): _ a o

bv 2d 2

(42)

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For bE0.25 nm and dE25 nm, eq. (42) gives: _a o2:105 v ðm=sÞ

(43)

The average dislocations velocity is not precisely known. It is clear, however, from eqs (41–43) that condition (38) is fulfilled for plastic flow at quasi-static imposed strain rates, while this is not true for plastic flow in dynamic conditions ð_a 4108 s1 Þ. There are indeed strong indications (see Section 3.2.1) that the processes of plastic flow under large imposed strain rates (about 103 s1) differ from those observed for strain rates smaller than 1 s1 [99]. Serrations are observed for large imposed deformation rates in the range of 3.103 s1–4.103 s1 [31]. Since, for practical reasons, MD approaches are restricted to plastic strain rates larger than 106 s1, they are not adapted to the description of deformation mechanisms at low (o1 s1) applied strain rates. If the specimen deforms by twinning, let ngT be the creation rate per unit grain of twinned lamellae of width hT0. The corresponding plastic strain rate is: _ pT jngT

ghT d

(44)

Eq. (40) is transposed as: ngT j_a

ghT d

(45)

4.3. Nanotwinned fcc metals We consider now a heavily twinned ng fcc metal. Each grain is made of matrix (M) lamellae and twin (T) lamellae, all of approximately the same width wT, separated by TBs. A qualitative analysis, detailed below, shows that such a material can plastically deform without generating large stress concentrations. We notice at first that an fcc single crystal submitted to an applied tensile stress may deform plastically whatever the orientation of the tensile axis, since fcc crystals have eight independent glide systems. We now consider successively: 1. A fcc single crystal CM, made of matrix lamellae of the same orientation, and of thickness wT, whose boundaries are impassable obstacles to gliding dislocations, as shown in Fig. 5a. When submitted to an applied tensile stress sa, such a crystal deforms plastically. One may assume that the average plastic strain of each lamella is equal to the average plastic strain /CM S (cf. Fig. 5a). 2. A fcc single crystal CT in twin relation with the matrix, made of twin lamellae of thickness wT, submitted to the same applied tensile stress sa as shown in Fig. 5b. CT deforms plastically.

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Fig. 5. Matching between lamellar crystals. (a) Single crystal made of identical matrix lamellae M. (b) Single crystal made of identical twin lamellae T. (c) Nanotwinned single crystal MTMT.

The average plastic strain of each lamella is equal to the average plastic strain /CT S. For a given value s*a of sa, one has generally /CT Sa/CM S. Said differently, the yield stress sYM of the matrix differs from the yield stress sYT of the twin. Both sYM, and sYT increase with decreasing thickness of the lamellae. We may assume, without lack of generality, that sYMosYT. 3. A sample made of twin-related M and T lamellae of same thickness wT, submitted to a tensile applied stress sa [Fig. 5(c)]. The matrix starts to deform as soon as the applied stress reaches sYM and generates dislocations that are blocked at the TBs. These dislocations exert a back stress on the dislocations of CM and a forward stress on the dislocations of CT. Increasing the applied stress increases the average stress in CT, until it reaches sYT. Under further increase of sa, the whole crystal deforms. Since both the matrix and twin phases have eight independent glide systems one does not expect the creation of a large stress concentration at the TBs. However, the plastic strain is not homogeneous. The heterogeneity manifests itself either as heterogeneous dislocations distributions at, or in the vicinity of the TBs or even as kinks or ledges along the latter. But this heterogeneity is at a sufficiently small scale not to affect the ductility. 4. A ng fcc metal heavily twinned. The difference in this case, is the possibility of plastic strain heterogeneities localised at the vicinity of the intersections of twin and GBs. Due to their relatively small scale, these heterogeneities do not induce large stress concentrations. In contrast, in an untwinned polycrystal, the incompatibility generated at GBs scales with the grain size.

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5. Molecular dynamics Computer simulations of material structure and properties can be divided naturally into two categories: continuum simulations using finite element methods and atomistic simulations where the material is regarded as an aggregate of atoms interacting through forces that are defined by potential functions. The strengths and weaknesses of numerical modelling, especially finite element methods and MD simulations are discussed in a recent overview [39]. In what follows, without going into details, general pictures of mechanical properties and deformation mechanisms extracted from MD are pinpointed and commented. As discussed in the above paragraphs, mechanical properties and related deformation mechanisms of ng polycrystals with grain size in the range 5–20 nm are not definitively established due to experimental difficulties in preparing dense and clean materials with uniform grain sizes. This is not the case for MD simulations where a range of grain sizes can be generated and numerically tested in the same conditions. Following Van Swygenhoven and coworkers [158–160], these computations have greatly simulated the quest for understanding intrinsic properties of ng polycrystals with microstructures belonging to above range of grain sizes. MD simulations particularly provide a detailed picture of some atomic-scale processes during plastic deformation of ng polycrystals. By varying the grain size between 5 nm and 50 nm, it has been shown that the hardness of Cu5–50 exhibited a maximum (2.3 GPa) for grain sizes in the range of 10– 15 nm [161]. This behaviour was ascribed to a shift in the microscopic deformation mechanism from dislocation-mediated plasticity in the coarse-grained materials, with dislocation nucleation at GBs, to GB sliding in the nc region for the smaller grain sizes. A similar mechanism has also been reported for nc Al [162]. These simulations give a flow stress value for the 15 nm grain-sized material, which is two to three times higher than experimental data [21]. In parallel, atomic-scale analysis of the mechanisms behind the plastic deformation of the simulated behaviour reveals that in large grains (d ¼ 12–50 nm) dislocations were generated at a GB, propagated though the grain and were absorbed at the opposite GB. No dislocation tangles were observed in accordance with the lack of strain hardening in the stress–strain curves. In a subsequent study on the same Cu5–50, it was found that dislocations, were constantly created at GB sources, propagated through the grains and vanished at the opposite GBs until a number of dislocations were absorbed. This resulted in the build up of a back stress that prevented further absorption events. In such a case, dislocation pile-ups of five to six dislocations were formed, with a pile-up length of 35–40 nm [163]. Van Swygenhoven et al. [164] have studied the atomic mechanism for dislocation emission from nanosized GBs in Ni. Samples of 5, 12 and 20 nm average grain sizes have been studied under tensile test at 300 K, under a strain rate of about 107 s1. While no dislocation activity was observed for the 5 nm grain-sized sample, an increased dislocation activity was present when the grain size increased from 12 nm to 20 nm. In this case, GBs were found to emit a partial dislocation by local atomic shuffling and stress-assisted free volume migration. The free volume was often emitted

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or absorbed in a neighbouring triple junction. However, the contribution of the partials to the observed plasticity was minor compared to the contribution from GB sliding triggered by the atomic activity. No perfect dislocations were observed [164]. The nucleation of extended dislocations from the GBs in ng polycrystals of aluminium, with grain sizes up to 70 nm, was studied in MD simulations of 2D columnar networks of Al nanograins [165]. It was shown that the magnitude of the dislocation splitting distance deq scaled by the grain size d represents a critical length scale controlling the low-temperature mechanical behaviour. For deqWd, the leading partials nucleate from the boundaries, glide across the grains and become incorporated into the opposite boundaries, leaving behind a grain sheared by a stacking fault. By contrast, for deqod, two Shockley partials connected by a stacking fault are emitted consecutively from the boundary, leading to a deformation microstructure similar to that of coarse-grained aluminium. In addition to the discussion by Meyers et al. [39], and particularly when comparing results obtained from different interatomic potentials and (or) different methods for generating the model polycrystal, one should keep in mind the following. Not accounting for the high level of the applied stress, the very short time scale (300 ps) during MD simulations may not allow for the build up of the necessary conditions for nucleating the trailing partial from a GB. This may explain why numerous stacking faults are observed within the simulated microstructure. Notice that the possibility of reduced stacking fault energy at the nc scale has been suggested. This would lead effectively to the nucleation of partial dislocations. Further studies are needed to clarify this point. A recent experimental study did show the presence of perfect dislocations by HREM investigation of post mortem Ni25 tensile-tested at room temperature [166]. MD results should be considered with care since they analyse the mechanical behaviour at very high deformation rates, which differs from quasi-static behaviour, as pointed out in Section 4.2.7. Furthermore, MD simulations reveal only the core structure of dislocations. It is well known that a dislocation line cannot end in a crystal. In the MD simulations, the part of the dislocation that is in the GB is often not revealed. Details such as these are essential for one to give a proper description of the glide mechanism. Therefore, simulating defect-related behaviour such as GBs and nanograins necessitates a multi-scale simulation that links continuum mechanics and MD. Simulations should therefore be regarded as a source of inspiration and not as a mean to validate or disprove the existence of a mechanism [9].

6. Dislocation-mediated plasticity of ng polycrystals 6.1. General remarks There is little doubt that GBs play a major role in the physical processes controlling plastic flow of ng polycrystals. For example, the mechanical behaviour of ng polycrystals with average grain size of about 10 nm is consistent with Coble creep

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[167]. On the other hand, superplastic behaviour is consistent with large scale GB motion. Stress-assisted boundary motion was also observed recently. Despite their interest, we shall not discuss these mechanisms. We shall rather focus on the relation between GBs and dislocation glide. At least in fcc metals, MD studies strongly suggest that, during plastic flow, discrete dislocations, (perfect, partial or twinning dislocations) are emitted from GBs, cross the entire grain and are absorbed at the next GB [164]. As pointed out in Sections 2 and 3, the information available on the structure of GBs, as well as on their behaviour under an applied stress is rather poor. We attempt, however, to propose some assumptions on the possible mechanisms that are consistent with experimental observations and with our general knowledge of dislocation theory. Despite their important structural differences that greatly influence the mechanisms of emission and absorption of dislocations under the effect of an applied stress, it is often difficult to decide whether a given GB is a SAGB or a LAGB (Section 2.2.5). We therefore discuss them successively in Sections 6.2 and 6.3.

6.2. SAGBs 6.2.1. Structural information An ideal infinite SAGB is a periodic planar discrete dislocation network, whose spacing is at least of the order of a few b, typically larger than 1–2 nm, whose long-range stress field decreases with the distance x1 from the boundary as expð2pox1 =bÞ. The misorientation o between the adjacent grains bounded by a SAGB is smaller than 0.2 radian. In practice, however, one observes quite often networks made of finite SAGBs bounded by edges (or triple junctions). Two situations must be distinguished: The SAGB network consists of tilt boundaries, made of periodic arrays of discrete edge dislocations parallel to their intersection (Fig. 6). Their long-range stress field is that of wedge disclinations located along the triple junctions [168]. These boundaries are obstacles to dislocation motion; therefore, they generally

Fig. 6. Tilt boundaries and wedge disclinations. (a) The incomplete tilt boundary is equivalent to a wedge disclination dipole of rotation o. (b) At their intersection, two incomplete tilt boundaries are equivalent to one wedge disclination.

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contain extrinsic dislocations that may cross the triple junctions, thus satisfying the continuity of the Burgers vectors. In all other cases, the SAGBs consist of periodic discrete dislocations distributions of various Burgers vectors, which cross the edges at nodes, where the dislocations satisfy the continuity of the Burgers vectors. Their long-range stress field can be described as that of a defect localised on the edge. This defect is not a disclination in general [169]. In contrast to infinite SAGBs, finite arrays of periodically distributed dislocations generate long-range stress fields. The latter can be shown to be the sum of stress fields whose source is located on the edges of the boundaries. In the isotropic elastic case, the stress field of a given edge has been shown to decrease with the distance r to the edge as Ln(r/R), where R is some cut-off radius [169]. For the anisotropic elastic case, the stress field varies as rn, with 0ono1 [170]. The following examples illustrate the fact that the state of stress inside the grains resulting from the existence of SAGBs cannot be determined without a precise knowledge of the whole SAGBs structure. The resolved shear stress t on the glide plane of a finite tilt boundary of height h made of equidistant edge dislocations increases, passes through a maximum and decreases with the distance x1 to the boundary [171]: t¼

mbhx1 2pð1 nÞ x21 þ ðh2 =4Þ l

(46)

The resolved shear on the glide plane of a finite number of equidistant screw dislocations decreases monotonously with the distance from the boundary: t¼

mb h Arctg pl 2x1

(47)

The resolved shear stress on the glide plane of an incomplete twist SAGB made of two sets of perpendicular screw dislocations, one of them infinite, the other one finite, increases with the distance from the boundary: mb 2 h (48) 1 Arctg t¼ 2l p 2x1

6.2.2. Plastic flow associated with nucleation of perfect dislocations at SAGBs To be extracted from a SAGB, a perfect dislocation must satisfy the following conditions: Its Burgers vector must belong to the lattice of one of the adjacent grains. If it is created from a GB dislocation, another GB dislocation with Burgers vector (very

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Fig. 7. Extraction of a dislocation from a SAGB. The dislocation can glide in the grain which contains its Burgers vector: the glide of dislocation 1 (respectively 2) is restricted to grain 1 (respectively 2).

small in the SAGB case) equal to the difference between the other two must be left in the boundary. The dislocation line must lie at the intersection of a glide plane with the plane of the SAGB. This may require a rearrangement since short-range interactions between the dislocations may change the orientation of the dislocations in the GB. The stress must act in the correct direction. In the example of Fig. 7, dislocations 1 and 2 are compelled to move in different subgrains. A simple summation of the stress field of each dislocation constituting the boundary, reveals that the emission of dislocations constituting the SAGB requires a local stress of the order of 0.2 mo, where o is the rotation associated with the SAGB. This stress may result, for example from an incoming dislocation [172]. This process is consistent with the stresses observed during plastic flow. However, the activation distance necessary to extract the dislocation from the SAGB is of the order of 0.2–0.4 l, where l is the average spacing of the dislocations [172]. This is contradictory with the observation that the activation volume is of the order of 10–20 b3. Therefore, the extraction of perfect dislocations from a SAGB may contribute to plastic flow, but does not control it. Once extracted, a dislocation segment of length d shears the whole grain before being absorbed, producing a plastic strain about b/d, and creating a length 3d of dislocations incorporated in the opposite SAGBs. 6.2.3. Plastic flow associated with nucleation of partial dislocations at SAGBs The process of extraction is equivalent to the process of nucleation of a dislocation loop at the boundary (Fig. 8). Due to the order of magnitude of the stress during plastic flow, the extraction of a Shockley partial can occur in a way similar to that described for perfect dislocations. However, the activation distance is quite large even in this case. Therefore, the extraction (or nucleation) of partial dislocations at SAGBs is unlikely to control plastic flow.

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Fig. 8. Activation of dislocation glide at a SAGB. (a) A dislocation segment of Burgers vector b is extracted from the SAGB and propagates in one of the adjacent grains. (b) A dislocation loop of Burgers is nucleated close to the SAGB. Part of it is absorbed in the SAGB.

6.3. LAGBs 6.3.1. Structure of LAGBs Despite our lack of knowledge on the structure of LAGBs, one can propose reasonable assumptions relying on more or less direct information: 1. The heavy plastic deformation imposed on the material during severe plastic deformation processing results in the creation of a large density of dislocations. Only a small part of them is observed inside the grains, which suggests that most of them are absorbed in the GBs. MD simulations reveal the existence of centres of elastic strain in the GBs, akin to a distribution of dislocations [69]. Indeed, it is known from observations made on bicrystals, that dislocations may be incorporated in GBs either as secondary discrete dislocations of small Burgers vector [173,174], or as a distribution of infinitesimal quasi-dislocations. The latter are not observed; their existence is deduced from the application of the conservation of the Burgers vector. Modifications of the dislocation core structure of dislocations incorporated in GBs have also been observed [175]. 2. A dislocation dipole of Burgers vector 7b, of width p d ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ generates a strain field whose root mean square is of the order of ðb=dÞ ð1=2pÞLnðd=bÞ. For dE100 b, the root mean square is of the order of 0.01. This is of the order of magnitude of the measured RMS (E3 103) in as-prepared ng polycrystals. This implies that LAGBs harbour more or less well defined dislocations distributions that are sources of stress. 3. We do not mean, however, that a GB consists only of a dislocation distribution. There are good reasons to believe that other structural elements contribute to the structure of the GBs. Some of them only have been identified [176], other kinds of unidentified disorder are very likely to occur.

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Fig. 9. Localised shear of intensity B on GB 1 creates a (quasi) dislocation of Burgers vector B at the triple node. B is not a lattice vector. This dislocation may emit dislocations in any of the three grains.

6.3.2. Plastic flow associated with nucleation of perfect dislocations at LAGBs 6.3.2.1. Possible emission mechanisms. One may imagine various possible emission mechanisms. Due to the lack of information we focus on the following. 1. Nucleation of dislocation loops at the boundary, which can spread into the grain. The nucleation is assisted both by the stress field associated to the boundary and by the rearrangement of the dislocations distribution in the boundary. 2. Nucleation of dislocation loops as a result of inhomogeneous GB sliding (Fig. 9). Inhomogeneous GB sliding may occur anywhere along a given boundary, nucleating a dislocation. The dislocation may either remain in the GB or generate a lattice dislocation in either grain limited by the boundary. This mechanism is more likely to occur at triple nodes. Despite the lack of direct evidence, the mechanisms proposed above are supported by observations on the behaviour of electrodeposited Ni. The XRD peak broadening shows a time dependent recovery when the specimen is deformed in tension at 300 K beyond the microplastic regime [69]. On the other hand, tensile deformation at 180 K produces a non-reversible peak broadening, which disappears after annealing at 300 K [177]. This supports the assumption that plastic deformation results in the injection of dislocations in the GBs, which can be rearranged by heating. The above remarks also apply to the nucleation of partial dislocation loops. 6.3.2.2. Deformation mechanism. The measurements of the activation volumes agree on the fact that, after some plastic flow, the activation volume is of the order of 10–20 b3 (see Section 4.2.5). This is not consistent with a purely mechanical model. These results suggest rather that the critical process is the rearrangement of the defects inside the GBs. As seen in Section 3.2.2, plastic deformation at room-temperature of as-processed materials results in a decrease of the RMS of the stress. The same can be said for

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annealing at relatively low temperature (To180 1C for Ni). This decrease may have two origins: (a) As-processed materials contain intragranular dislocations, which are relatively mobile. Plastic deformation or thermal annealing should displace them towards the GBs where they can rearrange themselves, cancelling part of the internal stress. (b) The initial distribution of defects (discrete or infinitesimal, dislocation distribution, other types of defects) in the GBs is also unstable and may be rearranged by plastic deformation or thermal annealing.

7. Conclusion In this chapter we have refrained from attempting to discuss quantitative modelling of the mechanical behaviour of ng metals, since the present attempts, although they might be stimulating, do not rely on a consistent set of experimental results. The reasons for this situation are well known: incomplete control of the processed and deformed materials, absence of standard procedures for mechanical testing, unjustified transfer to ng polycrystals of concepts whose validity has been tested for cg polycrystals. This last point is illustrated by the following examples: 1. The average grain size is the starting point of almost all analyses. As pointed out in Section 2.2.5, the accuracy and the exact meaning of the experimental observations are questionable. The width of the grain size distribution is so large that a bimodal description would certainly be more accurate. 2. The values of the yield stress published in the literature refer essentially to measurements made at an unidentified stage of the microplastic domain (Section 4.2.2). There is a definite need for a proper description of the elastic– plastic transition and of the strain-hardening mechanisms. 3. Standard procedures for mechanic testing should be defined in order to allow comparing the observations made by different laboratories. These procedures should include a combination of annealing treatments and prestraining. 4. The emission of dislocations from a GB, followed by intragranular glide is the main identified mechanism controlling plastic flow at relatively low temperatures (To180 1C for Ni). This mechanism is likely to depend on changes in the structure of the GBs.

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CHAPTER 91

Hydrogen Effects on Plasticity I.M. ROBERTSON, H.K. BIRNBAUM Department of Materials Science and Engineering, University of Illinois, Urbana, IL 61801, USA and

P. SOFRONIS Department of Mechanical Science and Engineering, University of Illinois, Urbana, IL 61801, USA

r 2009 Elsevier B.V. All rights reserved 1572-4859, DOI: 10.1016/S1572-4859(09)01504-6

Dislocations in Solids Edited by J. P. Hirth and L. Kubin

Contents 1. Introduction 251 2. Experimental observations 257 2.1. Macroscopic measurements of plastic flow properties 257 2.2. Summary of observations 267 2.3. Dislocation densities and distributions 267 3. In situ TEM studies of dislocation behavior 269 4. Thermal activation parameters for dislocation motion 273 5. Discussion 274 5.1. Hydrogen effects on dislocation mobility 274 5.2. Shear localization 276 5.3. Elastic shielding of stress centers 279 5.4. Temperature and strain rate effects 285 6. Summary 289 Acknowledgments 290 References 290

1. Introduction The very dramatic and deleterious effects of hydrogen on fracture have motivated many studies of the influence of hydrogen on the mechanical properties of metals. Hydrogen is a ubiquitous solute and is present in metals, in many cases inadvertently, as a result of processing, corrosion, etc. In the course of attempts to understand the mechanisms of hydrogen embrittlement [1], a wide range of observations have been made on the interaction of hydrogen with dislocations and on changes it induces in the macroscopic stress–strain response of metals. Many of these studies suffer from a number of shortcomings. Since hydrogen embrittlement presents a serious technical problem, many studies were carried out on commercial alloys, the complexity of which mitigates against the development of a mechanistic understanding of the phenomena observed. In addition, many studies were not designed to focus on the mechanism(s) by which hydrogen affects plastic behavior, in that they did not take the ‘‘special’’ properties of solute hydrogen into account. In this paper, we review observations of the effects of hydrogen on the plastic deformation of metals reported in the literature, choosing those that contribute to an understanding of the basic hydrogen–dislocation interactions as these dictate the plastic response of metals. We do not attempt to review the voluminous literature in detail [2]. Hydrogen differs from other solutes in several very important respects and these need to be kept in mind as we attempt to understand the effects of H on dislocations and plasticity. Many of these parameters enter into the understanding of the diverse behavior of different systems. What follows is a brief description of some of these parameters. Hydrogen resides in interstitial octahedral and tetrahedral sites in bcc, fcc, and hcp metals. In bcc systems, the occupied site depends on the system and can be either the tetrahedral or the octahedral site, although for the systems for which site occupancy has been determined the tetrahedral site is favored [3]. As an example, in the V–H system the site occupied depends on the phase being considered and at high temperatures in the disordered a-phase it occupies the tetrahedral site whereas in the ordered b-phase it occupies the octahedral site [3]. In Nb and Nb-alloys, H occupies the tetrahedral site except in the Nb0.5V0.5 alloy in which it occupies both sites [4]. In fcc metals, H occupies predominantly the octahedral site (Pd [5], Ni, FCC YHx [6], austenitic fcc Fe–Cr–Mn–Ni steels [7]), although there are reports of H occupying the tetrahedral site (FCC TiHx, TiDx, and YHx [6]) – both sites have cubic symmetry. In hcp metals, H occupies the tetrahedral site [8–10]. The octahedral and tetrahedral sites posses a tetragonal symmetry and assuming a coordinate system with the tetragonal axis along the 3-direction, the strain field

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can be expressed as: 2 3 0 11 0 6 7 ½ ¼ 4 0 22 0 5 0 0 33

I.M. Robertson et al.

with

11 ¼ 22 .

Ch. 91

(1)

Despite the tetragonal interstitial site symmetry, the distortion field associated with H in bcc systems has, based on diffuse neutron scattering [11,12] and Huang diffuse X-ray scattering [12,13], cubic symmetry. This indicates that 11 ¼ 22 ¼ 33 . This cubic symmetry is an indication that the interstitial has strong interactions with more than first nearest-neighbor sites and that these longer-range force constants reduce the tetragonality of the distortion field [12]. Despite the fact that its atomic size is small, hydrogen has a large partial molal volume of solution, VH, and hence a large distortion field. Curiously, VH is a constant fraction of the atomic volume in most bcc systems and is constant, independent of composition, H/M, over the complete range of solid solutions in fcc systems [12]. For bcc systems, VHB0.17O, where O is the atomic volume. For fcc systems, VH/O will vary by about a factor of 3, while the actual value of VH is approximately constant at VH ¼ 2.9 Å3. The importance of the distortion field around the H interstitial in determining its effect on the mechanical properties stems from the fact that the primary interaction with dislocations appears to be that of an elastic stress center. Thus, the observation that the distortion field has cubic symmetry suggests that the interaction of H with screw dislocations is not due to the deviatoric components of the distortion field but may be attributed to a second-order interaction caused by the local elastic moduli change close to the H interstitial. On the other hand, the interaction with edge dislocations would result from the dilatational components of the distortion field as well as from the local moduli changes. Hydrogen has a very significant effect on the acoustic elastic moduli of those systems in which measurements have been made. The effects are different for H in bcc and in fcc systems. High-frequency measurements over a very wide range of H/M and temperature [14,15] have shown that in Nb, Ta, and V, H increases the bulk modulus B ¼ (C11þ2C12)/3 and the shear modulus C44, and decreases the shear constant, Cu ¼ (C11C12)/2. Hydrogen effects on the higher phonon frequencies show that for bcc metals, H increases all phonon branch frequencies in the three principal directions [16]. Since the transverse frequency in the /100S, T100 ¼ C44, and the longitudinal frequency in the /110S, L110 ¼ (C11þC12þ2C44)/2, are both increased by H, these effects are consistent with the lower frequency acoustic measurements. However, the transverse mode T2 ¼ (C11C12)/2 ¼ Cu also increases with H/M in contrast with the lower frequency acoustic measurements in which Cu decreases with H/M. This dispersion with frequency has not been explained, although it may be evidence for an H Snoek effect (a relaxation phenomenon associated with solute redistribution) – an effect that has not otherwise been seen. The effects of H on the elastic properties indicate that changes in the electronic structure of the solids play an important role in determining these properties. Since the lattice expansion that accompanies solution of H would

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normally be expected to lead to a decrease in the phonon frequencies, the increases seen in the bcc systems are clearly due to electronic effects. Phonon dispersion measurements in the Pd–H system [16] do show the expected decrease in the frequencies associated with the lattice expansion. Another way in which H differs from other solutes is its very high mobility [17]. While diffusion of H in metals results from phonon-assisted tunneling at temperatures below B20 K [18], in the temperature range important to hydrogen effects on plasticity, the temperature dependence of the diffusivity can generally be described by an Arrhenius relation. The diffusivity of H is extremely high in the vicinity of 300 K and its behavior is characterized by very small activation enthalpies (see Table 1). In addition to the high H diffusivities, the mobility of H is very dependent on trapping at lattice defects [19]. As a consequence, the effective diffusivity, Deff is described by: Deff ¼ D

K . K þ ðaN T Þ=ðbN L ÞðyT =yL Þ

(2)

In eq. (2), D is the lattice diffusivity; NT, the density of traps; a, the number of hydrogen atoms that can be accommodated per trap; NL, the density of the solvent atoms; b, the number of lattice interstitial sites per solvent atom; yT and yL, the occupancies of the trap and lattice sites, respectively; and K ¼ exp(DE/kT), the equilibrium constant with DE being the trapping enthalpy, k, Boltzmann’s constant, and T, the absolute temperature. Equilibrium between H in traps and lattice sites is given by: yT yL DE ¼ exp . (3) 1 yT 1 yL kT Experimentally it is observed that H is trapped in a large variety of defects and some examples are given in Table 2 [20–30]. Most traps are ‘‘saturable,’’ i.e., they achieve an equilibrium H concentration, and the remainder of the H is in interstitial Table 1 Hydrogen diffusivity in metals (see Ref. [17]) System

Diffusivity at 300 K (m2 s1)

Activation enthalpy (eV)

Pd (fcc) Ni (fcc) Fe (bcc) Steels (bcc)a Stainless steels (fcc)a Nb (bcc) Ta (bcc) V (bcc)

3.9 1011 6.0 1014 1.6 108 1.5 109 1.7 1016 8.2 1010 1.9 1010 5.4 109

0.230 0.420 0.069 0.083 0.561 0.106 0.140 0.045

a

H diffusivities in these systems depend on the particular compositions.

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Table 2 Examples of hydrogen traps System

Trap

Trapping enthalpy (eV)

Ref.

‘‘pure Fe’’ ‘‘pure Fe’’ ‘‘pure Fe’’ ‘‘pure Fe’’ ‘‘pure Fe’’ Ferritic steels Ferritic steel fcc stainless steel Niobium Niobium Niobium Nickel Nickel Nickel Aluminum Aluminum Palladium Palladium

Dislocations Vacancies Ti solutes C interstitials N interstitials Ti–C precipitates Fe3C boundaries gu boundaries N interstitials O interstitials H–H pairs Substitutional solutes Vacancies Grain boundaries Vacancies Grain boundaries Vacancies Dislocations

0.62 0.63 0.19 0.03 0.13 1.0 0.11 0.10–0.15 0.12 0.09 0.06 0.08–0.12 0.44 0.12 0.52 0.15 0.23 0.6

[20] [22] [22] [22] [22] [22] [22] [22] [22] [26] [26] [25] [22] [24,31] [22] [22] [22] [22]

lattice sites. The experimental values of the binding enthalpies indicate very strong trapping at defects in the temperature range of interest for plasticity and dislocation studies. In systems such as ferritic steels and stainless steels, the H diffusivity is greatly reduced by the presence of traps [17,19] making it difficult to categorize the effects of H on plasticity without taking this into account. In the vicinity of 300 K, the range of Deff in different steels is about four orders of magnitude lower than in pure metals as a result of trapping effects. As discussed in greater detail in the body of this paper, the mobility of H affects dislocations and plasticity in several ways. These include: a. the kinetics of formation of H atmospheres around dislocations and other elastic stress centers, b. the response of these atmospheres to changes in the local chemical potentials and stresses, c. the amount of H available for the formation of these atmospheres, d. the stress gradients and damage created when H is introduced into the lattice, and e. the formation of second phases such as hydrides before and during deformation. A good deal of the controversy over the effects of H on plasticity and the stress– strain response of solids has arisen due to neglect of the kinetic factors associated with the interaction of H with dislocations. In the temperature range where these

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studies have been carried out, the kinetic aspects of the interactions must be considered. This is in contrast to most other solutes that in the temperature range of interest are much less mobile than H and hence can be treated as fixed pinning points. Since H is an ‘‘incidental’’ or ‘‘accidental’’ solute addition to most metals, the interest in its effect on dislocations and plasticity stems from its connection with ‘‘hydrogen embrittlement.’’ In many systems, hydrogen embrittlement is a misnomer and the failure is the result of plastic failure processes, albeit on a very local scale [1]. This was first suggested by Beachem [32] to account for the presence of ductile features on fracture surfaces of hydrogen embrittled steels. This was a significant and noteworthy departure from all prior explanations of the presence of these features on fracture surfaces [33,34] as it suggested that plastic processes were impacted directly by hydrogen and were not simply a consequence of the hydrogenenhanced fracture event. Since that early suggestion, a large number of observations have supported the conclusion that in many systems, hydrogen embrittlement results from locally ductile failure. In situ environmental cell TEM deformation and fracture studies have shown that the basic processes of hydrogen-related fracture are local plastic failure in regions in front of the crack tip [35–46]. These enhanced plastic processes are not restricted to the locale of the crack tip but occur at distances up to 1 106 m ahead of it. A critical feature is that the enhanced plasticity processes are confined to a specific and narrow region [47,48] that ultimately is manifested in the appearance of the fracture surfaces. A unique aspect of this type of fracture is that the presence of hydrogen in the environment, or in solid solution, results in the dynamic maintenance of a high concentration of solute hydrogen close to the crack tip. As clearly shown in the in situ TEM deformation studies, hydrogen causes local plastic deformation at stresses below that required in the absence of H or when H is uniformly distributed in solid solution. Dislocation velocities near the crack tips are greatly increased when the environmental cell atmosphere is changed from vacuum or inert gas to hydrogen gas. Similar results were obtained using water vapor in the case of Al and Al alloys [38,39]. These results, discussed in greater detail below, caused the crack to proceed by ductile failure process at or immediately in front of the crack tip. Since the plasticity proceeds at significantly lower stresses than that required to cause deformation of the volume away from the crack tip (where the H concentration was lower), the failure occurs in the absence of macroscopic deformation – hence giving the impression of ‘‘embrittlement.’’ These observations formed the basis for the HELP (hydrogen enhanced localized plasticity) mechanism of hydrogen embrittlement [1]. Of course, the HELP mechanism for hydrogen embrittlement, while observed for bcc [35,36], fcc [37,39,49], and hcp [40] pure metals, and alloys [38,42–44,46,50] is not the only hydrogen embrittlement mechanism. Clear evidence, by direct observation as well as a posteriori studies, indicates that other mechanisms of hydrogen embrittlement also are operative, with the actual mechanism depending on the circumstance. In systems where hydrides can be formed, e.g., Nb, V, Ta, Zr, Mg and Ti, stress-induced hydride formation and cracking is generally the dominant environmental fracture mechanism [51]. Even in these studies, hydrogen-enhanced

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dislocation motion is observed but the failure occurs because the hydride fails by cleavage. For example, in hcp a-Ti [40] failure occurs at a low stress intensity and slow crack growth rates by the formation and fracture of a stress-induced hydride. However, at higher imposed stress intensities and faster crack growth rates, failure is by the HELP mechanism since the crack is driven faster than the formation and growth rate of hydrides ahead of the crack. In systems in which both mechanisms are viable, the one that dominates is determined by the kinetics of hydride formation or H solute segregation to dislocations at the crack tip. Both are competitive with the alternative of general ductile fracture that will occur if failure due to either H-related mechanism does not precede it. As the HELP and the stress-induced hydride formation and cleavage mechanisms depend on H mobility, they offer an intrinsic explanation for why the embrittlement disappears at low and at high temperatures. At low temperatures, the mobility of H is reduced and general ductile failure occurs before the HELP or the hydride mechanism can intervene. At low temperatures in hydride-forming systems, hydride precipitation occurs in the absence of external stress stabilization and the hydrides act as any other brittle phase cracking under stress, but the propagation of the fracture between hydrides is by normal ductile failure mechanisms. At high temperatures in hydride-forming systems, stress stabilization of the hydrides is not sufficient and the material behaves in a ductile manner. As was discussed in some detail [1], the HELP mechanism also does not lead to failure at temperatures at which the H is uniformly distributed in the lattice rather than in dislocation atmospheres. Although hydrogen-enhanced fracture is not the emphasis of this paper it should be noted that an alternate explanation is associated with hydrogen reducing the effective lattice cohesive strength – the so-called decohesion model [52]. This reduction is a consequence of interstitial hydrogen changing the local and global atomistic [53] and electronic structure [54,55], as well as increasing the separation distance between atoms. Liang and Sofronis [56,57] studied intergranular failure of Ni-based alloys by considering the effect of hydrogen on fracture in terms of how it influences the cohesive properties of the grain boundary and the grain boundary carbide/matrix interfaces, the failure of which is resisted by cohesive tractions. The determination of the effect of hydrogen on the cohesive tractions [58] was carried out on the basis of the fast separation limit of the Hirth and Rice thermodynamic theory of interfacial decohesion [59]. Such a cohesive element approach was also undertaken by Serebrinsky et al. [60] to study hydrogen-assisted crack propagation in bcc iron systems. The fundamental idea in this model is that hydrogen reduces the ideal work of fracture along {1 1 0} planes as determined by density functional theory [61,62]. On the basis of these ab initio calculations, the authors predict that a sufficient amount of hydrogen is available to fracture transgranularly a system such as bcc iron. The difficulty with these cohesive element approaches is achieving and more importantly maintaining the critical hydrogen concentration ahead of a propagating crack. More recently, through modeling intergranular crack propagation in IN903 superalloy, Dadfarnia et al. [63] pointed to several shortcomings of the decohesion theory in predicting the experimentally measured fracture response.

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These investigators used a traction-separation law derived from the thermodynamic theory of interfacial decohesion of Mishin et al. [64]. Unlike the Hirth and Rice theory, the Mishin et al. approach addresses quantitatively interfacial separation at stages between the fast and slow separation limits.

2. Experimental observations 2.1. Macroscopic measurements of plastic flow properties Early studies of the effects of solute H on the deformation of solids often gave conflicting results. A review of the literature suggests that this confusion resulted primarily from two aspects of the experiments; namely, the method of introducing hydrogen and its effect on the defect population, and the strain rate of the test. Hydrogen often was introduced from high-fugacity sources, either from the gas phase or from electrolytic solutions by cathodic charging, and at temperatures where the H diffusivity was somewhat limited. A consequence of high-fugacity charging is ‘‘near surface damage,’’ such as the formation of voids and pressurized H2 gas bubbles. When these defects form a decrease in the flow stress, termed softening, is often observed [65,66]. Another consequence of high-fugacity charging is that the large lattice expansion that accompanies the high surface concentrations generates a stress gradient [67,68] and, in addition to the effects of residual stresses, causes high dislocation densities in the near surface region that lead to surface hardening [69]. In unstable stainless steels, this ‘‘near surface damage’’ can include stress-induced martensitic phases [67,70,71] and these act as barriers to dislocation egress through the surface. The high dislocation densities and the residual stresses generally lead to an increase in the flow stress. Thus it is immediately evident that different effects can result from high-fugacity charging and this is system dependent. Hydride-forming systems have been less well studied with respect to the effects of H on their deformation behavior. In these systems attention has focused on the stress-induced hydride formation and cleavage mechanisms which leads to brittle behavior in a limited temperature range [51]. Hydrides formed during charging can have several roles in influencing the deformation processes. Cathodic charging of systems such as Ni [72], Pd, Ti, etc. can cause the formation of a very hard, brittle hydride at the surface. In the case of Ni, this ‘‘hydride’’ (actually the b-phase solid solution) forms a continuous coating on the surface with the attendant formation of a high compressive stress and a barrier to the egress of dislocations [73]. Deformation during charging results in the cracking of this hydride and the decrease in the cross section of the specimen, followed by further ‘‘hydride’’ formation. The process causes an increase in the apparent flow stress of the specimen and a decrease in the strain to failure. In other systems, the formation of the hydride plays a dual role. Small hydride particles can act as barriers to dislocation motion leading to hardening. However, due to the very large volume of formation of hydrides, their formation and growth is accompanied by the

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generation of high densities of accommodation dislocations [40,74] and these can serve as dislocation sources leading to softening. Again in these systems, H charging can lead to a duality of behaviors. A clear demonstration of the importance of avoiding damage during the introduction of H was provided by Kimura and Birnbaum [73] who studied the effects of cathodic charging during deformation at rates between 2 106 and 8 105 s1 in Ni and Ni–C alloys. At high cathodic current densities, decreases in the flow stress were observed during charging and increases once charging ceased. At low current densities, small transient increases in flow stress were observed. The effects were largest in the temperature range where significant H diffusion into the specimen can be expected. In all cases, cathodic precharging caused insignificant increases in flow stress and was shown to be the result of formation of surface hydrides. These results are consistent with the creation of lattice damage in the surface region; e.g., the introduction of dislocations in the near surface region during the dynamic charging experiments. Such an increase in the mobile dislocation density would appear at the macroscale as softening. Precharging as well as simultaneous charging during the test can produce a thin surface ‘‘hydride’’ (the b-phase) which hardens the surface and thus prevents the egress of dislocations. This results in an increase in flow stress once the charging is stopped and no further dislocation injection occurs. The second factor that can lead to seemingly disparate results is the strain rate at which the mechanical property tests are conducted. Strain rate effects are coupled to temperature effects and will be discussed in detail later. However, at this point it should be noted that interstitial H can act as a solute pinning point and impede the motion of dislocations leading to hardening when tests are performed at strain rates and temperatures under which H is immobile. At low strain rates, when H can move in response to the changing stress field of a passing dislocation, solute H can lead to decreases in the flow stress, i.e., softening. This fact was not recognized in the early studies as they were often carried out at relatively high strain rates. Interestingly, the strain rate and temperature effects observed for the plastic response in the presence of H are closely related to those observed for H effects on fracture. In the following discussion, we focus attention on the effects of H on plastic deformation in those cases where the H charging does not cause structural damage e.g., void formation and H bubble formation. In the early observations of the stress– strain response of ferritic and stainless steels, when H was added to the material the yield and flow stresses increased, as did the work hardening rate [75]. These experiments were carried out on precharged specimens at temperatures between 300 and 200 K and at strain rates of the order of 104 s1 using a wide range of grain sizes. Similar increases in the yield and flow stress were reported for 99.8% Ni polycrystals and single crystals, that were either cathodically precharged or charged during the tensile tests, were obtained at strain rates of about 104 s1 [76–80]. These tests also showed serrated yielding at room temperature and at lower temperatures, consistent with dislocation pinning by H. In the absence of the formation of voids, gas bubbles, or fissures, severe charging with H has generally been observed to lead to hardening, i.e., an increase in the

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flow stress for both single crystals and polycrystalline specimens of Fe [65,81,82], ferritic, and stainless steels. Increases in the flow stress can result from the introduction of a high near surface dislocation density and surface compressive stress due to the H concentration gradient developed during charging, or due to the formation of more effective point defect pinning points from C–H or N–H clusters. Measurements of the flow stress in H-charged Fe as a function of grain size have been interpreted using the Hall–Petch relation: sf ¼ s0 þ K f d 1=2

(4)

In eq. (4) sf is the flow stress; s0, the lattice frictional stress; Kf, the Hall–Petch slope that measures the strength of the grain boundaries to slip propagation; and d, the grain size. In H precharged Fe, Adair [83] observed low yield points and found that the Hall–Petch analysis indicated an increased friction stress and a decreased Hall–Petch slope, whereas in tests conducted with simultaneous H charging, Bernstein [84] observed the opposite effect – a decreased s0 and increased Kf. While these disparate observations may reflect differences in material purity or charging conditions, it is difficult to make a definitive decision on the basis of what is known. However, the results are consistent with the injection of dislocations in the case of deformation and simultaneous charging [84] and the possible formation of small gas bubbles in the precharged specimens [83] where the cathodic charging conditions were considerably more severe. Experiments by Lunarska and Wokulski [85] on Fe whiskers are of interest as these single crystals contained a relatively low as-grown dislocation density. Tensile tests at strain rates between 2 and 11 104 s1, high rates for hydrogen effects, were carried out with simultaneous cathodic charging under high H-fugacity conditions. Once plastic deformation began these whiskers exhibited significant decreases in flow stress when H charging was started and rapid recovery of the flow stress when H charging ceased. These observations can be interpreted as due to the injection of dislocations during the charging periods. Slip lines in the absence of H charging were ‘‘wavy’’ and localized to the region near the failure ‘‘neck’’ whereas those in the H-charged specimens were planar and more uniformly distributed along the length of the specimen, again consistent with the injection of dislocations during the H charging. High-fugacity H charging causes structural damage and has a major effect on the deformation properties of high-purity Al (99.999% purity) [86]. Introduction of large amounts of H in Al can be achieved either by cathodic charging or by etching in NaOH aqueous solutions. This resulted in very large softening of the tensile flow stress and severe slip localization (Figs 1 and 2). While this was originally interpreted as due to solute H, neutron scattering, TEM and SEM studies [87] have shown that the hydrogen is present as gas bubbles ranging in size from about 2 nm to 35 106 m (Fig. 3). Apparently, the presence of a large number of gas bubbles resulted in the softening behavior as well as shear localization. To demonstrate that the macroscopic property changes depended on the conditions under which hydrogen was introduced, a low-energy Hþ ion plasma

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Fig. 1. Stress–strain curves for high purity Al (99.999% purity) polycrystalline specimens in the uncharged state and with Al/HB2 103 H added by cathodic charging. Each curve is the average of 25 tests with the error bars determined from the repeated measurements. Reprinted from Ref. [86] with permission from Elsevier.

was employed to introduce hydrogen to thin specimens of pure iron (99.99% purity) [88]. Due to the high-energy reference state of the Hþ ions a high-fugacity condition is obtained and significant H/M values can be achieved without the concomitant near surface damage. When this charging method was used during tensile deformation, significant decreases in the flow stress of Fe were observed over the temperature range from 77 to 300 K [88]. A maximum decrease in the flow stress was observed in the vicinity of 200 K. This softening is believed to be due to dislocation–hydrogen interactions and to neither surface damage nor dislocation injection, as judged by comparing the effects in the H plasma with those observed in inert gas plasmas. As can be seen from the above discussion, the situation in metals of relatively high purity and in steels is complex because of competing factors. In another system of interest, stainless steels, the possibility of H and stress-induced phase transitions must be considered too [67,71,89]. Altstetter et al. have measured the tensile properties of stable 310 and unstable 304 stainless steels at strain rates in the range from 103 to 106 s1 and at temperatures between 77 and 295 K [90,91]. The yield and flow stresses of both alloys were increased by the addition of solute H. In both steels, high H concentrations caused the formation of a g*-‘‘hydride’’ [67,71,92] and in the unstable steels H stabilized the formation of au and e martensite. These induced phases provide barriers to dislocation motion. In the 310 SS, increases in the flow stress were observed at H/M values well below that required for the formation of the g*-phase. Altstetter et al. proposed that in view of the fact that the g*-phase has a large volume of formation it can be stress stabilized in dislocation

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Fig. 2. Micrograph showing coarse slip lines due to shear localization in high purity Al (99.999%) cathodically charged with H (H/M ¼ 2 103). Reprinted from Ref. [86] with permission from Elsevier.

cores at low H/M [93–95]. Thus, the increase in the flow stresses observed for stainless steels on adding solute H could be caused by stress-enhanced phase transitions at relatively low H/M. However, severe strain localization is observed in both stainless steels [90,91] and this can also result in an increase of the tensile flow stress. Another approach to examining the effect of H on plasticity was pursued using cathodic charging at current densities of 1.0 A m2 during creep measurements [96]. These experiments revealed an initial decrease in the transient creep strain and strain rate as the H/M value increased. At longer times, the transient creep rates in the presence of H remain high and hence the creep strain at long times with H is greater than in uncharged specimens. Consistent with these measurements, lowpressure hydrogen atmospheres were observed to markedly decrease the creep rupture lifetime of Fe–Ni alloys at elevated temperatures [97]. This effect was not due to the formation of gas bubbles of H2 or CH4 or any other structural changes. The steady-state creep rate was not affected by H, nor was the extent of grain boundary sliding. The extent of steady-state creep (Stage II) was markedly shortened in the presence of the H and the specimens failed by entering tertiary creep (Stage III) at a significantly lower total creep strain. In contrast to these

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Fig. 3. TEM image of high purity Al (99.999%) charged with H to H/M ¼ 2 103. The charged Al showed a range of H2 bubbles with sizes between about 2 to 35 mm (Image courtesy of R. Pickerill and I. M. Robertson).

measurements in non-hydride forming systems, the addition of H to a-Ti alloys [98] resulted in a large increase in the transient creep strain once a critical H/M was exceeded. This increase was attributed to the formation of hydrides and to the dislocations that accommodate the volume changes associated with hydride formation acting as generators of dislocations [40,74]. Perhaps the strongest evidence for decreases in the tensile flow stress due to H charging is provided by the series of papers by Kimura et al. [99–103] on Fe of various purities. Zone-refined Fe having a residual resistivity ratio, RRR ¼ (r273/r4.2), of about B5500 did not show any evidence for hydrogen damage (surface blistering) after cathodic charging at 20 A m2 at temperatures around 295 K, while less-pure specimens did show permanent surface damage. In slow strain rate tests, _ ¼ 8 105 s1 , with concurrent cathodic charging an immediate decrease in the flow stress was observed. On ceasing the charging there was an almost immediate increase in the flow stress which continued with a high work hardening rate. This decrease on charging and increase on cessation of charging could be repeated during the course of the tensile test. The decrease in the flow stress was greatest at about 200 K and decreased as the strain rate increased. These

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effects were less pronounced in lower purity specimens (RRR ¼ 1800) where small increases in flow stress preceded the decreases. Cathodic precharging of H into the high-purity Fe with a current density of 100–300 A m2 (considerably higher than was used in the experiments where the H charging occurred during the straining) resulted in decreases in the flow stress [100]. The largest effects were observed in the highest purity specimens and with the shortest aging time after charging. As with the experiments in which charging occurred during deformation, the decreases in flow stress were greatest at test temperatures of about 200 K. The response of high-purity Fe (RRRB5000) single crystals to hydrogen were studied with similar techniques and they showed decreases in the flow stress, particularly in the early parts of the stress–strain curves and at temperatures of about 200 K. However, in the experiments by Kimura et al. [99–103], at temperatures below 190 K the decrease in flow stress on beginning cathodic charging was temporary and then it increased as deformation and charging continued. No structural characterization to determine the extent and nature of the damage due to the charging was reported. To discover the induced-damage state, Tabata and Birnbaum conducted a TEM investigation on similarly charged material. They found that whereas cathodic charging at the low-current densities did not cause any near surface damage at room temperature, at a charging temperature of about 200 K, small H2 bubbles were formed in the near surface region. An example of these bubbles is shown in the image presented in Fig. 4. If these bubbles were present in the samples used in the studies by Kimura et al. [99–103] they would serve as obstacles to dislocation motion and thus would provide an explanation for the observed increase in flow stress on charging below 190 K. Oguri et al. [102] studied Fe–C alloys prepared from high-purity zone-refined Fe. Under cathodic charging conditions, using current densities of 10 A m2, they observed both softening and hardening in the temperature range 170–295 K at a strain rate of 8 10–5 s1. In these alloys, small amounts of C gave solid solution softening [104] while at higher C concentrations, solid solution hardening was observed. In the softening range of C concentrations, the addition of H by cathodic charging during straining resulted in decreases of the flow stress with a maximum effect at about 200 K and hardening at about 273 K. At higher C concentrations, the addition of H generally caused increases in the flow stress, with the exception of a small decrease at about 200 K. While the results of Kimura et al. [99–103] seem to clearly support decreases of the flow stress when H is added to high-purity Fe at temperatures above about 190 K, there are a number of independent factors that enter into the totality of interpretation needed to explain their results. One cannot obtain a definitive mechanistic explanation based on the information presented by their experiments. In the case of hydrogen, Kimura et al. [99–103] proposed that the observed softening was due to modification of the Peierls potential and its effect on screw dislocation mobility and the hardening at low temperatures to hydrogen at the dislocation core pinning kinks on screw dislocations. It was argued that hydrogen impacted screw dislocations only. Atomistic simulations of hydrogen effects on

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Fig. 4. TEM image of the near surface region of zone-refined Fe cathodically charged with H at 20 A m2 at 200 K following the procedure used by Kimura et al. [78–82]. The defects shown are small H2 gas bubbles as shown by dark field imaging experiments. These defects were not seen in specimens charged in the vicinity of 295 K (T. Tabata and H. K. Birnbaum, unpublished results).

kink-pair nucleation and mobility suggest that enhanced nucleation occurs when H transitions from a weaker to a stronger binding site and hardening when hydrogen– kink pair interactions occur [105]. However, the observations reported in Fig. 4 suggest that a simpler mechanism based on introduction of dislocation obstacles in the form of small bubbles may account for the low-temperature hardening. Thus, it appears that decreases in the flow stress by the addition of H may occur over the entire temperature range studied, if the superimposed hardening due to the H2 bubbles is taken into account. In the Fe–C alloys, the hardening and softening were attributed by Kimura et al. [102] to the effects of the C and H on kink nucleation and motion on screw dislocations. In the case of C, the solid solution softening was attributed to elastic stresses imposed by the C atom on the screw dislocation which influenced kink-pair nucleation. The explanation proposed by Kimura et al. [99– 103], requires what these authors considered to be a questionable set of arguments and are unable to explain all the observed results. However, at this time the importance or impact of H effects on kink nucleation and motion in bcc systems can be neither supported nor discarded definitively. The influence of H on the properties of polycrystalline Ni having various levels of purity and C interstitial concentrations has been studied. Here H was introduced by rapid cooling from elevated temperatures in an H2 gas atmosphere or by exposure to H2 gas during deformation [47]. Both charging methods produced no irreversible

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damage and the high-temperature charging method introduced large amounts of H (H/M ¼ (3–7) 104). Significant softening was observed in the early stages of plastic deformation when specimens were tested at 300 K and at very low strain rates (_ ¼ 107 s1 ) in low-pressure H2 atmospheres or when solute hydrogen introduced by high-temperature charging was present (Fig. 5). The decreases in the flow stress at low strains were greatest when C interstitials acted as barriers to dislocation motion. At higher strain rates (Fig. 6) there was no significant decrease in flow stress due to H, and at still higher strain rates, increases in the flow stress due to H were observed. These results clearly show the dependence of the material response to strain rate, with the level of hydrogen-induced softening decreasing with increasing strain rate. During the many studies of the effects of H on plasticity, there have been a number of reports of serrated yielding [76,78,79,106–108]. Serrated yielding is generally attributed to the repeated pinning and breakaway of mobile dislocations from solute atoms [109]. The critical strain rate, _ c , below which serrated yielding occurs is given by: _ c ¼ rm bvc ¼ 4brm

DkT . rc DE

(5)

7

Fig. 5. Stress–strain curves for Ni containing 1200 appm C strained in H2 ( ) or He (K) gas at 100 kPa pressure at 300 K and at a strain rate of 107 s1. The specimens were 25 mm in thickness. Each curve is the average of at least seven repeated tests and the error bars indicate the standard deviations at the indicated strains. These results show a significant decrease in the flow stress when the tests were conducted in gaseous H2. Data points omitted for clarity. Reprinted from Ref. [47] with permission from Elsevier.

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7

Fig. 6. Stress–strain curves for Ni containing 1200 appm C strained in H2 ( ) or He (K) gas at 100 kPa pressure at 300 K and at a strain rate of 7 105 s1. The specimens were 25 mm in thickness. Each curve is the average of at least seven repeated tests and the error bars indicate the standard deviations at the indicated strains. No significant decreases in the flow stress due to H were seen. Data points omitted for clarity. Reprinted from Ref. [47] with permission from Elsevier.

Here rm is the mobile dislocation density; b, the magnitude of the Burgers vector of the dislocation; vc, the velocity at which the dislocation breaks away from the H atmosphere; D, the diffusivity; rc, the effective radius of the atmosphere; and DE, the strength of the dislocation–solute interaction. For each strain rate, serrated yielding occurs within a specific temperature range with a lower bound being the temperature below which a hydrogen atmosphere cannot reform on the dislocation during the deformation process and the upper bound by the temperature at which the atmosphere keeps pace with the mobile dislocation to which it is attached i.e. T lc ð_ÞoToT uc ð_Þ. In Ni [108], Tuc was independent of _ (Tuc ¼ 227 K) and was consistent with the temperature below which hydrides would form in the stress field of the dislocation core [93–95]. The lower critical temperature, Tlc, exhibited an Arrhenius temperature dependence with _, as expected from eq. (5), with an activation enthalpy of 0.57 eV at the lower strain rates and 0.25 eV at the higher strain rates. The interpretation of these values is that Tlc is determined by diffusion in the core of the dislocations at the high strain rates where hydrides cannot form and in the hydrides at the core of the dislocations at the low strain rates where the dislocations are moving slow enough to allow them to form.

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2.2. Summary of observations In view of the complexity of the observations relating to the effects of H on macroscopic plastic deformation, we summarize the general conclusions that can be drawn from the experiments. Introduction of H from high-fugacity environments causes irreversible damage in the near surface region in the form of voids, gas bubbles, high compressive stresses, high dislocation densities and in some systems ‘‘hydrides’’ or second phases. These effects are particularly significant in systems and at temperatures where the H diffusivity is low. Voids and H2 gas bubbles decrease the measured tensile flow stress for deformation by decreasing the cross section of the specimen. They can also result in severe slip localization. Small bubbles can cause hardening by providing dislocation pinning points. Second phases, if formed, provide impediments to the egress of dislocations and thus cause hardening. Enhanced near surface dislocation densities can act to increase the flow stress when the specimens are precharged, due to the formation of a ‘‘work-hardened’’ surface layer. When H charging is carried out during the deformation, the dynamic injection of dislocations can lead to softening. Similarly, the dynamic formation of small hydrides during the deformation can inject dislocations into the specimen, also leading to softening whereas the static surface hydride layer formed during precharging can cause hardening. Intrinsic effects of H on the plastic properties of metals have been established in high-purity metals using H-charging methods that do not cause permanent structural damage. In very high purity Fe, H causes a decrease of the flow stress at temperatures above about 190 K and an increase below; the latter effect may be a consequence of small gas bubbles formed during the initial cathodic charging. In Ni, softening caused by H is seen at very low strain rates. Serrated yielding in Ni–H alloys below 227 K demonstrates the hardening effects of H at low temperatures and is interpreted as being caused by the repeated breakaway of dislocations from the H atmospheres (or dislocation core hydrides) and the reforming of these atmospheres.

2.3. Dislocation densities and distributions Since solute H has been shown to affect the flow stress of metallic systems, one expects differences in dislocation structures and densities. Several studies have attempted to discern this difference and we review those that deal with the effects of solute H. Dislocation structures associated with the accommodation of the volume expansion related to hydride formation and decomposition have been reported [40,74]. These dislocation structures are distinct and, as they do not pertain to the current discussion, they are not considered further. Matsui et al. [100]

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reported a higher dislocation density, as determined by a TEM study, in Fe cathodically charged at 200 K; this was the condition that yielded the greatest degree of softening. The density increases were small, changing from 3 108 to 6 109 cm2 in uncharged material to between 3 109 and 6 109 cm2 in the charged material. Although Matsui et al. concluded this difference was significant, the normal variability in dislocation density within and between TEM samples questions the validity of this conclusion. In Ni–H alloys, several TEM studies have reported no statistically significant difference in the dislocation densities for H-softened or hardened specimens [110,111]. Resistivity measurements [112] have indicated that there is a small increase in the generation of dislocations in Fe deformed in H2 gas compared to those deformed in inert atmospheres. Although the dislocation densities do not appear to be greatly changed by the presence of H in deformed specimens, there are very significant differences in the dislocation distributions caused by the presence of H. After deformation, the dislocations exhibit a high degree of slip localization and tangled dislocation structures and in the case of the steels, slip localization was associated with carbide particles [113,114]. Additional evidence for H-induced shear localization is obtained from slip-line studies in ferritic steels [115–118], in the ferrite phase of duplex stainless steels [119], and in the austenitic stainless steels 21Cr–6Ni–9Mn, stable 310 SS [90,91], and unstable 304 SS [90,91]. An example of hydrogen-enhanced shear localization is shown in Fig. 7, in which slip traces as a function of hydrogen content are compared in 310 SS [90] and in the austenitic stainless steel 21Cr–6Ni–9Mn [120]. High-purity Al (99.999% purity), that normally exhibits very fine slip, showed highly localized, coarse shear bands after high-fugacity charging with H (Fig. 2) [121].

Fig. 7. (a–d) Slip traces on the surface of 310 SS tensile tested to failure at 295 K and a strain rate of 5.5 105 s1. (a) H/M ¼ 0.0; (b) H/M ¼ 0.8 103; (c) H/M ¼ 1.8 102; and (d) H/M ¼ 2.7 102. Reprinted from Ref. [90] with permission from Springer. (e–h) Indentations made into a grain with a surface orientation of (018) prior to hydrogen charging (e) as well as after hydrogen charging (f) display slips with a similar general pattern of slip steps. The upper right regions, marked by the box in (e) and (f) are detailed in (g) and (h) and show that the number of slip steps is increased by charging with hydrogen B. Reprinted from Ref. [120] with permission from Elsevier.

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In the case of the Al–H system, the shear localization was probably the result of the introduction of gas bubbles during the H charging [87] (Fig. 3). In contrast, a decrease in shear banding has been reported in steels at low H fugacities [122].

3. In situ TEM studies of dislocation behavior A large number of direct observations of the effects of solute H on the behavior of dislocations have been carried out by conducting deformation experiments in situ in an environmental cell TEM [123,124]. This method allows observation of dislocation behavior in H2 gas at a fugacity of up to about 40 MPa [125] while the specimen is being stressed. Studies have been carried out on fcc [37,39,49], bcc [35,36,126], and hcp [40] systems, on pure metals, on alloys [38,42–44,46] and on intermetallic compounds [41], with the observations being independent of the metallic system. To understand the results of the in situ TEM experiments it is essential that the limitations be understood. Although detailed analysis of the dislocation behavior is possible, the stress applied to the specimen or the H/M value in the observed volume cannot be determined. The shape of a typical miniature tensile TEM specimen – an electron transparent region with a wedge-shaped profile within a much thicker region (100 106 m) – precludes measurement on the local stress in the observation region. The proximity of free surfaces does impact the dislocation mobility and deformation processes and, therefore, the relevance of conclusions drawn from such observations to deformation behavior in bulk specimens must be made cautiously and claims verified by alternate means. However, the conclusions discussed below are believed to be generally characteristic of bulk behavior. This statement is based on the fact that observations have been made in relatively thick specimens (B1 106 m thickness) using 1 MeV electrons [43] as well as at lower accelerating voltages and these microscopic observations have been coupled to studies in macroscopic specimens. In addition, observations of dislocations and sources moving completely within the volume of the TEM specimens were identical to those made on dislocations that terminated at the surfaces of the specimen. In the reported environmental cell TEM deformation studies, the specimen tensile rod was designed to maintain a constant load when the displacement applied to the specimen was stopped, i.e., it was designed as a ‘‘soft machine.’’ In some cases, the experiments were carried out in a ‘‘hard’’ tensile rod designed to maintain a constant displacement when the applied displacement rate was decreased to zero. The results in both cases were confirmatory. Although the H2 pressure in the environmental cell could be measured directly, the actual concentration in the specimen could not be. However, as the specimen thickness was about 100 nm in the regions studied, equilibration with the H2 gas is expected to take place in less than a second after the gas is introduced into the cell. The fact that the H2 gas molecules are dissociated and the H is ionized by the electron beam increases the H fugacity and assures that any surface oxides do not provide a barrier to H entry [125]. In some alloy systems [127], an evaporated film of Pd in the vicinity of the

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electron transparent regions was used to enhance entry of H [128]. Observations made with H2 atmospheres were unique to this environment and were not duplicated when an inert gas such as He was added to the cell. The major observations made with the environmental cell TEM apply to all systems studied and are: Introduction of H2 gas into the environmental cell, while the specimen was under constant applied stress, resulted in increased dislocation velocities. Dislocations that were moving under the stress moved faster and those that were immobile began to move. Removal of the gas resulted in a decrease in dislocation velocity. Reintroduction of the gas to the environmental cell and hence the metal increased the dislocation mobility. The effect of introducing and removing the gas on the dislocation velocity could be repeated provided the stress remained constant. An example of the hydrogen-induced enhancement of the dislocation mobility in Fe is shown in the time-sequence of images presented in Fig. 8; these dislocations were created by deforming the sample in vacuum, the applied load was then held constant and the dislocations allowed to come to rest before hydrogen gas was introduced to the environmental cell. The repeatability of the effect is demonstrated with the first introduction and removal of hydrogen occurring in images Figs 8(a)–8(e); note the similarities in dislocation configurations in images Figs 8(e) and 8(f) which shows the images are stationary. Hydrogen gas was introduced again in Fig. 8(g) and the dislocations begin to move again. The relative magnitude of the increase in velocity due to hydrogen can be determined by introducing gas while the dislocations are mobile under an applied load and then recording the change for specific dislocations. Examples of the results of such experiments in Fe [35,126] and a-Ti [40] are presented in Fig. 9.

Fig. 8. Hydrogen induced dislocation motion in iron. The time (seconds: hundredths of seconds) of the images is (a) 0, (b) 0:17, (c) 0:90, (d) 2:03, (e) 3:73, (f) 3:84, (g) 14:53, (h) 14:77, (i) 21:29, (j) 21:54, (k) 23:54, and (l) 26:34. (T. Tabata, unpublished work).

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Fig. 9. (a) Ratio of the dislocation velocity in H2 at the indicated pressures to that in vacuum for Fe of various purities as measured in the environmental cell TEM. Fe(H) is zone refined Fe with a RRR of B5500. Fe (I) is purified Fe with about 52 appm (C and N). Fe (II) is ‘‘pure’’ iron with about 130 appm (C and N) (Ref. [126]). (b) Ratio of the dislocation velocity in H2 at the indicated pressures to that in vacuum for a-Ti. Reprinted from Ref. [40] with permission from Elsevier.

Enhancements in velocity of 10 can be achieved even at modest pressures of H2. The effect of cycling the gas in and out of the environmental cell is shown for a-Ti in Fig. 9b. Hydrogen enhancement of the dislocation velocity was independent of material, crystal structure, and dislocation type. It was observed for dislocations moving on the {1 1 0}/1 1 1S and {1 1 2}/1 1 1S slip systems in bcc Fe, on {1 1 1}/1 1 0S systems in fcc metals, and on the (0 0 0 1) and f1 0 1 0g planes in hcp systems. These effects have been observed for perfect edge, screw and mixed dislocations, for isolated dislocations, for dislocations in tangles, for partial dislocations, and for grain boundary dislocations. Dislocation velocity enhancement by H2 was largest in materials that contained other elastic obstacles such as interstitial solutes. This was particularly evident in Fe [Fig. 9(a)] and in Ni. Experiments on gu-strengthened Inconel 718 [50] showed extreme enhancement of the dislocation velocity when H2 was added to the environmental cell. Here the velocity enhancement was sufficiently high to preclude capturing the motion of individual dislocations on the recording medium, which had a time resolution of 1/30th s. Enhanced dislocation velocities were reported for H2 gas and for water-saturated helium gas, but not for dry helium gas [38,39]. These observations confirmed that H and not simply pressure fluctuations were responsible for the enhanced dislocation velocities. Dislocation generation rates from sources such as grain boundaries, stress concentrations, etc., were increased by the presence of H2 gas in the environmental cell and decreased by the return to a vacuum environment.

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Data obtained on hydrogen-enhanced dislocation mobility, such as presented in Fig. 9a for Fe–H, can be used to estimate the resultant softening. The plastic strain rate, _, is given by: _ ¼ rm bvd ,

(6)

where vd is the average dislocation velocity and rm the mobile dislocation density. Dependence of the dislocation velocity on the resolved shear stress, t, can be described by: m t vd ¼ v0 , t0

(7)

where v0, t0, and m are empirical constants. The resistance to dislocation motion is contained in the parameter t0 and the stress exponent, m ¼ 2.6 [129]. The ratio of the flow stress in H to that in vacuum, tH/tv, can be calculated from eqs (6) and (7) and the data of Fig. 9(a) (obtained at constant applied stress) with the assumption that rm, v0, and m are not affected by H solutes. The results from such a calculation are presented in Fig. 10, which shows a softening of about 50% can be obtained in Fe with the low H/M values that are appropriate to the environmental cell TEM experiments.

Fig. 10. Calculated ratio of the flow stress in H to that in vacuum using the data of Fig. 6 and eqs (6) and (7).

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4. Thermal activation parameters for dislocation motion At present the direct study of the effects of H on dislocation behavior in bulk specimens is not possible. However, one can use macroscopic methods to determine H effects on the interactions of dislocations with defects that provide barriers to thermally activated dislocation motion [130]. We put forward such studies on highpurity Ni and Ni–C alloys using load relaxation, and temperature and strain rate change techniques [131]. The thermally activated strain rate can be expressed as: R DH 0 bA*ds* , (8) _ ¼ _ 0 exp kT where the parameter _0 is given by _ 0 ¼ nD rm Ab expðDS*=kÞ; nD, the dislocation attempt frequency in overcoming barriers; A*, the slip plane area swept by the dislocation per activated event and is a function of the effective stress s*, which is equal to the applied stress minus the internal stress opposing dislocation motion; DS*, the activation entropy for slip activation, and DH 0 *, the activation enthalpy for slip activation at zero stress. These measured activation parameters are related to the dislocation velocity by: DH* _0 exp , (9) vd ¼ rm b KT R where DH* ¼ DH 0 bA*ds*. The critical parameters that control the thermal activation of dislocations over barriers are the activation area, A*, and the activation enthalpy, DH*. In the stress relaxation experiments, interstitial C decreased the relaxation rate in Ni and H increased the relaxation rate in both pure Ni and Ni–C alloys [131]. Since the H was introduced from a relatively low-pressure H2 gaseous atmosphere, the changes in the relaxation rate cannot be attributed to irreversible damage [132]. Using eq. (8) and appropriate variations thereof to analyze the results of stress relaxation, strain rate change, and temperature-change experiments, one can calculate the influence of hydrogen and carbon solutes on the activation parameters. The results of this analysis are shown in Figs 11 and 12. Interstitial C increases A* and DH* over wide stress ranges consistent with the strengthening effect of C on the flow stress of Ni. Introduction of solute H decreases these activation parameters consistent with interstitial H causing a decrease in the obstacle strength. The softening effect of H is greatest at the lowest stresses at which the thermal activation over barriers dominates dislocation behavior. The effect of H on the velocity of dislocations can be calculated from this data using eq. (9); the results are presented in Table 3. In this table, sa is the applied stress, dDH* is the hydrogen-induced change in activation enthalpy, dA*/b2 is the hydrogen-induced change in activation area, and vH d =vd is the dislocation velocity in the presence of H relative to that in the absence of H as calculated from the measured dislocation parameters. The values of vH d =vd indicate that very significant enhancement of dislocation velocities are expected in Ni and Ni–C alloys on adding

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Fig. 11. Dependence of the activation area, A*, for dislocation motion on the applied stress for pure Ni and for Ni–H, Ni–C, and Ni–C–H alloys. Reprinted from Ref. [131] with permission from Elsevier.

solute H, particularly at the lower applied stresses. These results are consistent with the in situ TEM studies in which the stresses, while not directly measured, are expected to be in the range of 100–150 MPa and the measured dislocation velocity enhancements are of the order of 10. The largest amount of softening occurs at the lowest stresses, and hence at the lowest dislocation velocities, suggesting that the effect is dependent upon the motion of H with the dislocations. The significance of this observation is discussed further below. These results were obtained in relatively thick specimens and the agreement with the TEM observations obviates the criticism that the observed enhanced dislocation velocities are ‘‘thin film’’ artifacts.

5. Discussion 5.1. Hydrogen effects on dislocation mobility In the above presentation of the experimental data we showed that the enhanced dislocation velocity due to solute H could be used to estimate a decrease of the flow stress in Fe by about 50%. This estimate was made using eqs (6) and (7), assuming

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Fig. 12. Dependence of the activation enthalpy, DH*, for dislocation motion on the applied stress for pure Ni and for Ni–H, Ni–C, and Ni–C–H alloys. Reprinted from Ref. [131] with permission from Elsevier.

Table 3 Effect of hydrogen on the activation enthalpy, activation area, and dislocation velocity

Pure Ni Pure Ni Pure Ni Ni–C Ni–C Ni–C

sa (MPa)

dDH* (eV)

dA*/b2

vH d =vd

50 100 150 50 100 150

0.32 0.11 0.05 0.49 0.13 0.06

500 200 100 800 300 150

3 105 81 7 3 108 181 11

that rm, m, and v0 are independent of H, and that the effect of H is on t0 only. This estimate is for the stress range of the in situ TEM studies which is restricted by the technique and the time resolution of the camera to dislocation velocities of a few micrometers s1. Consistent with the TEM experiments, the dislocation activation parameters obtained from the relaxation experiments on Ni show that in both pure Ni and Ni–C alloys, H causes very large increases of dislocation velocities, particularly at low stresses (Table 3).

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Although these experiments clearly show that H-enhanced dislocation velocities should give rise to decreases in the flow stress these effects cannot by themselves explain all of the macroscopic deformation experiments. As previously discussed, solute H can cause both softening and hardening of the macroscopic flow stresses. While some of the experimental results can be explained by irreversible damage caused by hydrogen charging, it will be demonstrated below that in the absence of irreversible damage, both macroscopic hardening and softening can be observed due to interstitial H.

5.2. Shear localization In the stainless steel–H alloys, strain localization was accompanied by significant increases in the flow stress [90,91,133], whereas in the Al–H alloys, significant decreases in the flow stresses were observed [121]. In situ environmental cell TEM experiments also showed strain localization when H was added to the cell. Here it is worth noting that shear localization is not a unique phenomenon of hydrogen and is often observed in the deformation microstructure in systems containing precipitates [134] and in irradiated materials [135–138]. In hydrogen-charged steels [113,114], strain localization due to H has been correlated to the carbide precipitates. However, in other systems strain localization caused by H occurs in the absence of any second phases or voids and in such systems the mechanism by which H causes shear localization is not yet established. It has been suggested that slip planarity is increased when H is added through it decreasing the stacking-fault energy. This effect would increase the equilibrium separation distance of the bounding partial dislocations and thereby increase the force needed to create the constriction necessary for cross slip, making it less probable. A direct test of this idea was carried out by determining the stacking-fault energy in 310 SS from the curvature of dislocation nodes created by deforming samples in the TEM in vacuum and in hydrogen environments [44]. The change in node curvature was consistent with the stacking-fault energy decreasing from 36.9 mJ m2 to 29.7 mJ m2 when H was present. This 20% reduction is smaller than the 40% reduction reported by Pontini and Hermida [139] for a 304 steel; the difference may be related to the experimental techniques – TEM versus X-ray diffraction. Delafosse and coworkers [140–143] have proposed that the change in separation distance between the bounding partial dislocations is not necessarily or entirely attributable to a decrease in the stacking-fault energy but to the screening effect of hydrogen on the edge component of the partial dislocations; this screening mechanism is discussed further in Section 6.3. The impact of the screening effect on the equilibrium separation distance between the bounding partial dislocations and the work required to create the constriction necessary for cross slip is shown for nickel as a function of hydrogen content in Fig. 13; here it was assumed that the stacking-fault energy was independent of hydrogen concentration and constant at 100 mJ m2. An alternative explanation that is more general considers the initial distribution of hydrogen around a mixed character dislocation and the need for

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Fig 13. Relative change in equilibrium dissociation distance and work of constriction as a function of hydrogen concentration in nickel at 300 K. Reprinted from Ref. [140] with permission from Elsevier.

redistribution of hydrogen as the dislocation character shifts to pure screw for cross slip [49]. Since there is no driving force to cause the hydrogen to move from the atmosphere to the surrounding lattice, the nonuniform distribution of the hydrogen atmosphere around the dislocation prohibits it from cross slipping. It has yet to be determined which of these mechanisms is operative but what is clear is that the magnitude of the change in the equilibrium separation distance is unlikely to be of sufficient magnitude to have a profound effect on the ability of the dislocations to cross slip. The effects of strain localization caused by H on the increases or decreases of the macroscopic tensile flow stress can however be understood [144]. Clearly, the flow stress in the region of localization is reduced relative to the flow stress of the nondeforming volume (this is by definition shear localization). The effect of hydrogen-related decreases in the stress to move dislocations and shear localization can be to increase or decrease the measured macroscopic flow stress measured in a tensile test. This may account for the disparity of softening and hardening observations due to H in various experiments. We can distinguish three cases where H-caused shear localization occurs. Case I. In systems where H reduces the barriers to dislocation motion, an inhomogeneous distribution of H can cause shear localization, since the flow stress is lower where the H concentration is greatest. This may occur if H entry is facilitated at slip-line intersections with the surface and may be what occurs in front of propagating cracks. Case II. Shear localization can also occur when the specimen is hardened by the introduction of hydrogen (due to the formation of hydrides or H clusters), if the

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initial deformation locally reduces the stress for continued slip, e.g., by ‘‘cutting’’ through the hydrides. Removal of these clusters during the initial passage of dislocations leads to slip localization, as in other precipitation hardened systems. A related situation is when gas bubbles are formed during charging. Case III. Shear localization can occur in systems with a uniform hydrogen distribution, no stress concentrators and no second phases through the hydrogeninduced volume dilatation effect [145]. Relating the imposed displacement rate to the plastic strain rate in a macroscopic test such as a tensile test generally is based on the assumption that slip occurs uniformly over the gage length, lu. This assumption is inappropriate when shear localization occurs. With localization, slip is confined to those regions in which dislocations are active and the sum of these localized portions of the gage length then comprise the active gage length, ll. The imposed macroscopic displacement _ is the same regardless of the degree of slip localization and is given by the rate, u, condition: u_ ¼ l u ð_Þunlocalized ¼ l l ð_Þlocalized .

(10)

The effect of H is threefold – (1) shear localization is induced, (2) the velocity of the dislocations is increased at each stress level, and (3) the number of mobile dislocations is increased. Condition 10 can be combined with eqs (6) and (7) to yield: l v rmH voH ðtH =toH ÞmH ¼ . lH rmv vov ðtv =tov Þmv

(11)

Here the subscripts v and H denote deformation in vacuum (a relatively uniform distribution of slip along the gage length, lv ¼ lH) and in H-charged specimens (localized shear, ll ¼ lH). The mobile dislocation densities are given by rmv and rmH, mv and mH are the stress exponents, vov and voH and tov and toH are the corresponding constants in eq. (7). The stresses required for the imposed displacement rate in the two cases are tv and tH. The number of parameters in eq. (11) is too large to permit evaluation through analysis of the available data set. However, since the experimental observations indicate that the effect of H on dislocation behavior is to reduce the strength of obstacles to dislocation motion, a reasonable assumption is that solute H affects primarily t0 leading to: 1=m tH lv toH . (12) ¼ tv lH tov Thus measurement of the effects of H on the macroscopic flow stress alone is not a reliable means of determining dislocation behavior. In the general case, as localization occurs, the plastic strain rate in the active slip regions must increase to meet the imposed u_ and this normally would result in an increased flow stress [eqs (6) and (7)]. However, localization caused by solute H is accompanied by a decrease in the stress to move dislocations, an increased dislocation velocity at the

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imposed stress, and an increase in the dislocation density in the active shear regions. Thus, as seen in eq. (12), this can lead to a decrease of tH/tv, i.e., a decrease in the flow stress. Increases in the flow stress due to the introduction of H may result from shear localization if the increased dislocation mobility due to H does not balance the shear localization. 5.3. Elastic shielding of stress centers The H-caused increases in the dislocation velocity and the related flow stress changes were observed in fcc, bcc, and hcp systems, in pure metals, solid solutions, and in precipitation hardened systems. Its cause lies in the elastic interactions between dislocations and other stress centers, such as solute atoms and precipitates. The primary interaction between the stress centers, such as dislocations, and H is ð2Þ the elastic interaction energy, W int ¼ W ð1Þ int þ W int [146], which is expressed as the sum of the first-order dilatational interaction energy and the second-order modulusrelated interaction energy. The first-order dilatational interaction energy W ð1Þ int can be expressed as: 1 a W ð1Þ int ¼ skk Dv, 3

(13)

where sakk is the hydrostatic stress field of a defect and Dv is the unconstrained volume dilatation of the H solute. The second-order H-defect interaction energy, W ð2Þ int , derives from the H-induced modulus change and can be expressed as: 1 0 a 0 W ð2Þ int ¼ ij kl vs ðC ijkl C ijkl Þ. 2

(14)

In eq. (14) vs is the volume over which the solute atoms alter the elastic stiffness Cijkl, aij are strains caused by the applied stress saij in the absence of the solute atoms, 0ij are elastic strains inside the volume vs after the solute atom has been introduced in the lattice in the presence of the applied stresses which are held constant. The primed stiffnesses correspond to those characteristic of the local stiffness in the presence of the solute atoms. These interactions give rise to H ð2Þ ð2Þ atmospheres around edge dislocations (W ð1Þ int and W int ), screw dislocations (W int ), ð1Þ ð2Þ and point defects (W int and W int ). Since the local hydrogen concentration at each point in the stress field of every defect responds to the total stress at that point, the local H concentration at two interacting defects responds to the sum of the stresses due to all defects [147]. As the defects move relative to each other, the mobile H solutes respond and the H atmospheres act to minimize the total energy of the system, i.e., the H solute atmospheres act to shield the elastic interactions. Hydrogen atmospheres, treated as a set of cylindrical dilatational centers with a radial expansion of er0 and with the axis of their cylinders parallel to the length of the dislocation, generate stresses described by: srr ¼

mDa ; pr2

sff ¼

mDa . pr2

(15)

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where Da ¼ pro =ð1 nÞ ¼ V H =2N A ð1 nÞh, m is the shear modulus of the material, r, f, and x3 are the polar coordinates of the H dilatational cylinder with x3 parallel to the dislocation, NA the Avagadro’s number, and h the distance between H atoms in the dilatational cylinder. The interaction energy per unit length of this H cylinder with a stress field saij is: W ð1Þ int ¼

sa11 þ sa22 V H . 2 NA

(16)

If there are multiple sources of stress, e.g., the presence of several dislocations, the W ð1Þ int is determined from a linear superposition of the stresses. It is this interaction energy that determines the H/M at every point in the stress field. Setting DE ¼ W ð1Þ int and yL ¼ c0 =b in eq. (16) with c0 being the nominal concentration in the absence of the defect, we calculate the hydrogen concentration as C ¼ bNLyT measured in H atoms per unit volume. Since the stress field of the H solute is purely deviatoric, the interaction between the H cylinders is zero, however, the shear stress field of the H does cause a force on an adjacent dislocation. In the case of the two parallel edge dislocations 1 and 2 shown in the inset of Fig. 16, the shear stress due to H at the core of dislocation 2 resolved along the slip plane is given by: Z Z m V H 2p R sin 2f Cðr; fÞ drdf, (17) tH ¼ 2pð1 nÞ N A 0 r0 r where r0 is the inner cutoff radius of dislocation 2 and R is the outer cutoff radius of the atmosphere centered at the core of dislocation 2. The problem of the interaction between dislocations with hydrogen atmospheres was solved using the above relations and the application of both analytic and finite element solutions – with excellent agreement between the two methods [146–147]. All of the calculations corresponded to an equilibrium distribution of H atoms. Some results from these calculations are shown in Figs 14 and 15. Fig. 14(a) [147] shows the isoconcentration contours for H around an isolated edge dislocation in Nb (for which the required data set is most complete) and Fig. 14(b) [140] for a dissociated dislocation in nickel. These isoconcentration plots show that in cases for which the first-order interaction dominates, hydrogen is accumulated in the tensile side of an edge dislocation and for the partial dislocation the distribution is biased toward the edge component. The distribution, shape, and isoconcentration lines, of hydrogen are modified in the presence of another elastic obstacle. Fig. 15 shows the changes in the shape and concentrations that occur when a second identical edge dislocation is positioned coplanar and proximal to the first. Changes in the H concentrations are due to the presence of the stress fields of both dislocations interacting with the H. Integrating the forces on the dislocations due to the entire H distribution allows calculation of the effect of the solute atmosphere on the interactions between the two parallel dislocations. These results are shown in Fig. 16 for the shear stress on dislocation 2 due to the presence of dislocation 1 and the H atmospheres. In the absence of the atmosphere, the normalized shear stress, t/m, shows the expected (l/b)1 dependence with separation distance l along the slip

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Fig. 14. Contours of the normalized H concentrations, c/co, around: (a) An isolated edge dislocation in Nb. The average concentration in the lattice is co ¼ H/Nb ¼ 0.1 and the temperature is 300 K. Reprinted from Ref. [147] with permission from Elsevier. (b) A dissociated screw dislocation in Ni. Reprinted from Ref. [140] with permission from Elsevier.

plane. As shown in Fig. 16 the H atmospheres provide an attractive force that varies as (l/b)2 and decreases the repulsion between the dislocations. In effect, H ‘‘shields’’ the elastic interaction between the dislocations. Here it is important to note that the asymmetric distribution of hydrogen around an edge dislocation introduces a directional dependence to the effect of hydrogen on the stress-field distribution. For example, the normal stress associated with a dislocation array will actually be increased in the presence of hydrogen atmospheres and this will impact interactions with dislocations on other slip systems. Calculations using this methodology were carried out for the interactions between solutes and edge and screw dislocations in the presence of H atmospheres [147]. Some examples of these results are shown in Figs 17 and 18. For the case of dislocations interacting with interstitial C solutes, the results of the finite element calculations in the absence of H agreed very well with the analytic solutions of Cochardt et al. [148]. As an edge dislocation moves past a C atom lying close to the slip plane (x2/b ¼ 0.505) (Fig. 17) it experiences an interaction energy of –1.1 eV per distance ‘‘a’’ along the dislocation line, a being the lattice parameter. Adding H at a concentration of H/M ¼ 0.1 decreases the interaction energy, i.e., H shields the elastic interaction. The first-order volumetric interaction is relatively small – a decrease of the interaction energy of about 0.14 eV – whereas the decrease due to the volumetric and the second-order modulus effect is much larger – a decrease of the interaction energy of about 0.48 eV. Also shown is a calculation for the edge dislocation based on the formalism of Larche and Cahn [149] that accounts for the modulus change but assumes a uniform distribution of H. As seen in Fig. 17, elastic shielding due to H causes a very significant decrease in the interaction energies between an edge dislocation and an elastic pinning point such as a C interstitial. The shielding effect would be applicable for the interaction of edge dislocations with any elastic center. As seen in Fig. 17, the width of the interaction curve is also

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Fig. 15. Contours of the normalized H concentrations, c/co, around two parallel edge dislocation having the same Burgers vectors. The average concentration in the lattice is co ¼ H/Nb ¼ 0.1 and the temperature is 300 K. The parameters used in the calculation are those suitable for the Nb–H system. The grid lines are at distances equal to the Burgers vector. Separation of the edge dislocations are (a) 10b, (b) 8b, (c) 6b. Reprinted from Ref. [147] with permission from Elsevier.

decreased, an effect that is equivalent to a decrease in the activation area, A*. These results are in agreement with the determination of the effect of H on the thermal activation parameters for dislocations [131] (see Figs 12 and 13). These effects are somewhat more complex for the elastic shielding of screw dislocations [147] and these studies are ongoing. The effects are considerably smaller than for edge dislocations as the interaction is solely due to the secondorder modulus change effect and hence the H concentrations in the screw dislocation atmospheres are very small. Nonetheless, elastic shielding by H does decrease the barriers provided by interstitial solutes such as C to the motion of screw dislocations (Fig. 18). Elastic shielding of interacting stress centers by hydrogen provides a very natural explanation for the wide variety of observations made using the environmental cell TEM technique. The increase in dislocation velocity on adding H is a direct result of

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Fig. 16. The normalized shear stress, tH/m, due to the H atmospheres, tD/m, due to dislocation 1, and the net shear stress, (tHþtD)/m, at the core of dislocation 2 along the slip plane versus the normalized distance between the dislocations, l/b, at a temperature of 300 K and H/Nb ¼ 0.1, 0.01, and 0.001. The dislocation Burgers vectors are of the same sign. Reprinted from Ref. [147] with permission from Elsevier.

decreasing the effectiveness of elastic interactions between the moving dislocations and nearby stress centers, such as other dislocations and solute pinning points, by elastic shielding due to H. The fact that the increased dislocation velocities due to H is most prevalent in systems that have immobile interstitials acting as dislocation barriers (Fig. 10) is consistent with the decrease in the interaction energy by elastic shielding (Figs 17 and 18). Since the concept of elastic shielding is applicable to all crystal systems and in alloys as well as pure materials, this offers a natural explanation for the enhanced velocity observations in a wide range of systems. The reversibility of the enhanced velocity on adding and removing H [45] is also in agreement with the expectations of the elastic shielding model. The H-shielding model makes several predictions about the behavior of dislocations in the presence of hydrogen. For example, it predicts that the equilibrium spacing between dislocations in the pileup should be less in the presence than in the absence of hydrogen. Although as noted by Chateau et al. [140] this change in spacing does not necessitate an increase in the stress concentration at the head of the dislocation pileup. This prediction was verified by Ferreira

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Fig. 17. Plot of the interaction energy for an edge dislocation with a C solute in a bcc metal as a function of the normalized dislocation position, x1/b, for a C atom lying x2/b ¼ 0.505 below the slip plane and having its tetragonal axis along the [1 0 0]. This calculation is at 300 K and H/Nb ¼ 0.1. H-shielding results are shown for the volumetric and for the volumetric and modulus interactions (Ref. [147]) as well as for the Larche’ and Cahn formalism. (Ref. [149]). Reprinted from Ref. [147] with permission from Elsevier.

et al. [49] who showed that the spacing between dislocations in pileups in 310 stainless steel and in 99.999% pure Al decreased when hydrogen was introduced to the material. The increased density in the pileup indicates a decrease in the repulsive elastic interactions between dislocations due to elastic shielding caused by H. An example of this effect in 310 SS is shown in Fig. 19, which is a composite image created by superimposing a negative image of a dislocation pileup (the dislocations appear white) in hydrogen on the positions in vacuum (dislocations appear black). Clearly, all dislocations have moved closer to the obstacle and the separation distance between the dislocations has decreased. This latter effect is quantified for different gas pressures in Fig. 19(b). In the case of Al, this decrease was reversible on removal of the H, while in 310 SS there was a hysteresis in the motion, suggesting strong pinning of the dislocations. The fact that H atmospheres can affect the relative energies of edge and screw dislocation components by the formation of strong H atmospheres at the edge and weak atmospheres at the screw

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Fig. 18. Plot of the interaction energy for a screw dislocation with a C solute in a bcc metal as a function of the normalized dislocation position, x1/b, for a C atom lying x2/b ¼ 0.505 below the slip plane and having its tetragonal axis along the [0 1 0]. This calculation is at 300 K and H/Nb ¼ 0.1. H shielding results from the modulus interaction. Reprinted from Ref. [147] with permission from Elsevier.

components was also shown by a study of the cross slip of dislocations in Al using the environmental cell TEM [150]. Introduction of H2 gas during the cross slip process stabilized the edge components relative to the screw components and stopped the cross slip process. Removal of the H2 allowed cross slip to continue. This provides an intrinsic effect that prohibits cross slip from occurring.

5.4. Temperature and strain rate effects The response of dislocations to the presence of H atmospheres is dependent on the temperature and strain rate. For binding enthalpies (DE) of H to dislocations (see Table 2) of 0.1 eV (Ni) and 0.62 eV (Fe) the H concentrations at the dislocations are enhanced relative to the concentration in the lattice [eq. (3)] by factors given by exp(DE/kT) as shown in Table 4. At these high concentrations, the dislocation

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Fig. 19. (a) Effect of internal hydrogen on the dislocation separation distance in 310 SS. The black dislocations show the position in vacuum and the white dislocations the positions in a hydrogen environment. (b) The relative change in position as a function of hydrogen. Reprinted from Ref. [49] with permission from Elsevier. Table 4 Parameters describing dislocation breakaway Temp.

100 200 300 400 500 600

Nickel

Iron 2 1

Dislocation atmosphere (H/M)a

D (m s )

ec (s )

Dislocation atmosphere (H/M)a

D (m2 s1)

ec (s1)

1.0 3.3 102 4.8 103 1.8 103 1.0 103 6.9 104

4.5 1028 1.8 1017 6.0 1014 3.5 1012 4.0 1011 2.0 1010

1.5 1018 1.2 107 6.2 104 4.8 102 0.69 4.2

1.0 1.0 1.0 1.0 1.0 3.3 101

7.7 1011 4.4 109 1.6 108 3.1 108 4.7 108 6.1 108

2.6 101 2.9 101 1.7 102 4.3 102 8.0 102 1.3 103

1

a Values of H/M ¼ 1.0 in the dislocation atmosphere indicate a saturated atmosphere with Fermi-Dirac statistics.

atmospheres follow Fermi–Dirac statistics as all interstitial sites close to the dislocation are occupied. Also shown in Table 4 is the H diffusivity for these two metals and the calculated critical strain rate, _c , for breakaway of the dislocations from the H atmospheres as a function of temperature [eq. (5)]. The results of these calculations are also presented graphically in Figs 20 and 21. Hydrogen effects on the flow stress can be observed over the temperature range where significant H atmospheres are present at the dislocations and the mobility of H allows the atmospheres to move with the dislocations at the rate determined by the strain rate. Between these two limits, elastic shielding of the interaction of dislocations with obstacles can decrease the flow stress in the absence of excessive localization.

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Fig. 20. Concentration of H in the dislocation atmosphere and the critical breakaway strain rate for breakaway of the dislocation from the H atmosphere as a function of temperature for Fe with H/M ¼ 2 106.

As seen in Table 4 and Fig. 20, the high binding enthalpy for Fe allows it to have a significant H atmosphere over the temperature range 50–600 K with a Fermi– Dirac distribution below about 500 K. The high H diffusivity over the temperature range above 50 K allows the H atmosphere to move with the dislocations at reasonable tensile strain rates. Over the entire temperature range of interest, this high diffusivity and high binding enthalpy to the dislocations results in very high breakaway strain rates. Based on these calculations, H in Fe should result in enhanced dislocation velocities over the range 50–600 K. This situation would be markedly changed if the dislocation trapping enthalpy was significantly reduced or the presence of solute traps decreased the H diffusivity [eq. (2)]. Steels, having significantly reduced H diffusivities due to trapping [17], would not have H atmospheres that could move with the dislocations at low temperatures. Hence, the breakaway strain rates for steels would be greatly decreased at lower temperatures, and particularly at temperatures below 300 K. In high-purity Fe, H atmospheres can move with the dislocations over the entire temperature range 50–600 K and hence should decrease the flow stress over the same temperature range by the H-shielding effect discussed above. This is consistent with the observations by Kimura et al. of decreases in the flow stress on adding H to high-purity Fe in the absence of structural damage [17]. The increases in flow stress they observed below 190 K may be due to the formation of H2 bubbles during the low-temperature charging that

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may act as dislocation barriers. The hardening due to the barriers would be superimposed on the inherent softening due to H in solid solution. Alternatively, this low temperature hardening may reflect a decrease in the H diffusivity due to the remaining solute traps and hence a decrease in the ability of the atmospheres to move with the dislocations. For fcc systems, where the interaction enthalpy between dislocations and H is of the order of 0.1 eV and the H diffusivities are not as large as in Fe, the case of the Ni–H system (Table 4 and Fig. 21) is ‘‘typical.’’ Significant H atmospheres are present below about 500 K. At _ ¼ 106 s1 the atmosphere can move with the dislocations above about 225 K. Below 225 K hardening and serrated yielding is expected as a result of the lagging of the atmospheres behind the dislocations and the ‘‘breaking’’ away of the dislocations from the atmospheres as is observed [108]. At a strain rate of 106 s1 decreases in the flow stress may be expected between about 225 and 500 K due to elastic shielding. Again, these behaviors can be affected by the formation of dislocation core ‘‘hydrides’’ [108] and shear localization (see above). Strain rate effects on the H-related decreases in the flow stress are seen in the results on the Ni–C system [47] (Figs 5 and 6) in which a decrease in the flow stress is seen at _ ¼ 107 s1 but not at _ ¼ 105 s1 . In this case, the diffusivity D is expected to be significantly reduced by trapping at the C solutes. Since the critical strain rate for breakaway is linearly proportional to D [eq. (5)] the curve for the

Fig. 21. Concentration of H in the dislocation atmosphere and the critical breakaway strain rate for breakaway of the dislocation from the H atmosphere as a function of temperature for Ni with H/M ¼ 104.

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critical strain rate in Fig. 21 is shifted downwards by an amount equal to the decrease in diffusivity at each temperature.

6. Summary The effects of H on the plastic properties of metals is dependent upon many factors, chief among which are: (a) the manner in which H is added to the system, (b) the concentration of H, (c) the strain rate of the test, (d) the temperature of the test, (e) the ability or inability of the system to form ‘‘hydrides’’ or other second phases, and (f) the H diffusivity and the presence of traps that affect the diffusivity. Introduction of H from a high-fugacity source can cause formation of H2 bubbles or a high H concentration gradient in the near surface region with the concomitant increase of near surface dislocation density and the injection of dislocations into the solid or formation of near surface phase transitions. The effects can be summarized as follows: Formation of large H2 bubbles causes a decrease in the flow stress due to the decreased cross section of the specimen as well as shear localization. Formation of small H2 bubbles can cause increases in the flow stress due to pinning of dislocations. Similarly, the formation of a high near surface dislocation density, a surface ‘‘hydride layer’’ or near surface martensitic phases (in unstable stainless steels) can cause increases in the flow stress. Charging of H during deformation can result in the injection of dislocations from hydrides or from the concentration gradients in the near surface regions. These effects can result in decreases in the flow stress. Measurements made at ‘‘high’’ strain rates (‘‘high’’ as measured by the H diffusivity and by the ability of the H solutes to move with the dislocations) tend to result in increased flow stresses. Decreases in the flow stresses are generally observed at low strain rates. Solute H increases the mobility of edge, screw and mixed dislocations in fcc, bcc, and hcp systems as long as the H atmospheres can move with the dislocations. Consequently, solute H causes an intrinsic decrease in the flow stress over much of the temperature range of interest and at low strain rates. Hydrogen can cause shear localization when it is present as a second-phase hydride and when it is in solid solution. Strain localization tends to increase the macroscopic flow stress. Whether a macroscopic measurement exhibits a decreased or increased flow stress depends on whether or not the hardening due to shear localization exceeds the softening due to increased dislocation mobility. The increased mobility of dislocations caused by solute H can be explained on the basis of a theory of ‘‘elastic shielding’’ of the interactions between the dislocation and other elastic stress centers.

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Calculations made using the concept of ‘‘elastic shielding’’ are in general agreement with measurements of the parameters that describe dislocation interactions with elastic pinning points, e.g., C and with other dislocations.

Acknowledgments The authors acknowledge support from (a) the U.S. Department of Energy Grant GO15045; and (b) National Science Foundation Grant DMR 0302470. They also acknowledge the major contributions made by their students, research associates, and colleagues over the many years that this research has been carried out.

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Author Index Bacon, D.J. 4, 7, 11–13, 15–18, 20, 22–24, 27, 29–34, 36–37, 39–44, 46–49, 51, 54–55, 57–62, 66, 68–70, 72–80, 84, 143 Bacroix, B. 204, 206, 213, 217–219 Bahn, S.R. 208 Bahr, D.F. 268 Bailat, C. 7, 276 Bailey, R. 211 Baker, C. 253–254 Ballufﬁ, R.W. 241 Baluc, N. 7, 48, 55, 276 Bando, Y. 253 Baquera, M.J. 111, 145 Barashev, A.V. 22–24, 32–34, 41, 55, 66, 70, 75 Barber, J.L. 109–110 Barker, L.M. 146, 166 Barnes, D.E. 221 Baroˆ, M.D. 209–210 Baro, M.D. 209 Bartelt, M.C. 84 Basinski, Z.S. 26 Baskes, M.I. 5, 10, 61–62, 115, 135, 152–153, 171, 189, 208, 253–254, 256 Bassani, J.L. 23, 257 Baudelet, B. 217 Bauer, G. 252 Beachem, C.D. 255 Bechade, J.L. 7 Becker, R.Z. 137, 146–147, 166, 176, 187, 189–191 Becket, C.W. 152 Becquart, C.S. 32 Belak, J.F. 163, 166–167, 169, 174, 176, 178–180, 189 Bell, C.E. 94 Bellon, P. 169 Belov, Y. 239 Beltz, G.E. 187 Bencteux, G. 32 Benson, D.J. 187–190, 206–207, 236–237 Bentley, J. 58 Benyoucef, M. 213, 219 Bernard, F. 206 Bernstein, I.M. 259, 268, 276 Berre, B. 252

Abascal, J.L.F. 21 Abbaschian, R. 33 Abe, Y. 49 Abraham, D.P. 260–261, 268, 276 Abraham, F.F. 84 Ackland, G.J. 22–24, 32, 37, 39, 41–43, 48, 54–55, 66, 70, 74–75 Adair, A.M. 259 Agraı¨ t, N. 208 Ahmad, J. 206, 208 Alamo, A. 51 Alefeld, G. 252–254, 287 Aleksandrov, I.V. 206, 234 Alexander, H. 21 Alexandrov, I.V. 213 Allen, A. 178 Allen, M.P. 5 Alshits, V.I. 23 Altstetter, C.J. 257, 260–261, 268, 276 Anderholm, N.C. 94 Andersen, H.C. 21 Andrade, U.R. 138, 166 Angelo, J.E. 61–62 Aoki, M. 9 Aono, Y. 26 Arai, S. 58 Arakawa, K. 58 Aravas, N. 278 Argon, A.S. 26, 36, 101, 142, 187, 273 Armacanqui, M.E. 257 Armstrong, R.W. 94, 96–97, 133, 135 Arnold, W. 97 Arsenault, R.J. 263 Arsenlis, A. 80, 84 Asami, K. 206, 208 Asano, S. 257, 259, 264 Asaro, R.J. 135, 211, 216, 228, 231 Asay, J.R. 146, 166 Ashby, M.F. 182, 224, 273 Askaryon, G.A. 94 Auerbach, J.M. 110 Aul, F.W. 94, 110 Aust, K.T. 206, 213, 228 Averback, R.S. 169 Averty, X. 51 295

296

Author Index

Berstein, N. 102, 133, 139 Beudry, B.J. 251 Biener, J. 221 Bilby, B.A. 32 Billard, S. 204, 206, 218 Biner, S.B. 70–71 Birnbaum, H.K. 252–260, 262, 264–271, 273–277, 279–286, 288 Birringer, R. 203, 207, 238 Bisong, C. 253 Bitzek, E. 15, 23 Blakemore, J.S. 265 Boehly, T. 99, 101–102, 117, 122–123, 125, 133, 140, 150, 161, 165–166 Bolshakov, A. 229 Bond, G.M. 255, 269, 271 Bonnentien, J.L. 206 Bonnet, R. 208 Bonneville, J. 231–232 Bonny, G. 51–53 Bosch, C. 276 Bøttiger, J. 214–215 Bouchaud, E. 239 Boulanger, L. 58 Bowen, D.K. 8, 23 Brandstetter, S. 204, 210, 214, 218, 226, 232, 241–242 Brechet, Y. 27–31, 35 Bringa, E.M. 107–109, 115–116, 122, 152–159, 163, 165–167, 169, 171, 174, 176, 178–180, 187–191 Brown, L.M. 45, 178 Browning, N.D. 206, 218, 223 Brunner, D. 24 Buckley, C.E. 259, 269 Budrovic, Z. 242 Bulatov, V.V. 4–5, 9, 15, 20–21, 23, 26, 56, 76, 84 Bull, S.J. 228 Bullough, R. 8, 58 Bunzel, P. 238 Cady, C.M. 206, 234 Cahn, J.W. 281, 284 Cahn, R.W. 31 Cai, W. 4–5, 15, 20–21, 23, 56, 84 Caillard, D. 231 Calder, A.F. 22, 24, 32, 37, 39, 41, 43, 48, 55, 66, 70, 74 Canova, G.R. 24 Cao, B.Y. 115–122, 127, 142, 153–159, 163, 165 Cao, H.S. 208 Cao, W.Q. 213, 219 Carlsson, A.E. 9

Carnahan, B. 285 Caro, A. 107–109, 152, 165, 167, 169, 171, 176, 236 Carter, E.A. 256 Carter, R.G. 43 Carvalho, M.S. 106–107, 109–110 Caskey, G.R. 258 Caturla, M.J. 154, 169 Cazamias, J.U. 154, 169 Celli, V. 165–166 Chakraborty, S.D. 152 Champion, Y. 206 Chandler, E.A. 96 Chang, C.P. 217 Chang, J. 4, 21, 23 Chang, R. 8 Chaouadi, R. 43 Chateau, J.P. 276–277, 280–281, 283 Chauhan, M. 213 Chaussidon, J. 14, 20, 23 Chen, H. 204, 206 Chen, J. 231 Chen, M.W. 171, 208, 217, 229, 237 Chen, S.P. 171 Chen, W. 218 Chen, X. 208, 217 Cheng, S. 206, 213, 217–218, 223, 231 Cheng, X. 171, 189–190, 208, 217 Chevallier, J. 214–215 Chisholm, M.F. 216–218 Chiu, Y.L. 55 Cho, H.S. 212 Cho, K.C. 206, 220 Cho, Y.W. 51 Choi, I.S. 135, 137 Chokshi, A.H. 138, 166, 228 Choo, H. 206, 218, 223 Christian, J.W. 137–138, 154 Christy, S. 180 Clifton, R.J. 23 Cochardt, A.W. 281 Cohen, J.B. 106 Cohen, M. 32 Colvin, J.D. 96 Corbett, J.W. 253–254 Cotterill, R.M. 8 Cottrell, A.H. 32, 142 Curtin, W.A. 23, 31, 35, 174 Dai, Y. 7, 276 Dalla Torre, F. 204, 223, 231 Damask, A.C. 32 Danilkin, S.A. 251

Author Index Dao, M. 135, 137, 207–208, 216–217, 223, 230–231 Davis, R.C. 135 Davison, L. 96 Daw, M.S. 5, 10, 115, 152–153, 256 De Angelis, R.J. 106 De Carlan, Y. 51 De Diego, N. 7 De Hosson, J.T.M. 216, 231 De La Rubia, T.D. 15, 76, 84 De Novion, C.H. 51 de Prieto, E. 270 Dehm, G. 223 Delafosse, D. 251, 276–277, 280–281, 283 DeLeo, G.G. 253–254 Deng, X. 189–190 Derlet, P.M. 4, 48, 58, 139, 169, 171, 210, 226, 232, 236–238, 241–242 Detor, A.J. 135, 137 Devincre, B. 84 Dewald, D.K. 269 Dhere, A.G. 106 Diaz de la Rubia, T. 176, 181–182 Dickey, J.E. 8 Diehl, J. 24 Dienes, G.J. 32 Dieter, G.E. 96, 110 Differt, K. 276 Dirras, G.F. 204, 206, 212, 213, 217–219, 226–227 Dohi, K. 43 Domain, C. 14, 21, 32, 35, 51, 70 Dontje, T.D. 171 Dorn, J.E. 21 Dowding, R.J. 206, 220 Doyama, M. 8 Dremov, A.P. 152 Duchaineau, M. 163, 166–167, 169, 174, 176, 178–180 Dudarev, S.L. 48, 58 Duesbery, M.S. 8, 23, 26 Dundurs, J. 184 Da´vila, L.P. 187, 189–190 Eades, J.A. 269 Eastman, J.A. 206, 223, 236, 255, 264–266, 288 Ebrahimi, F. 206, 217, 230 Economou, S.J. 163 Edwards, D.J. 7, 58 Edwards, M.J. 146–147, 166, 176, 191 Efremov, V.V. 152–153 El Sherik, A.M. 228 Eldrup, M. 55

297

Elert, M.L. 152 Eliezer, D. 257, 260 Elkedim, O. 208 Embury, J.D. 214–215, 223 Englert, A. 8 English, C.A. 43, 58 Erb, U. 206, 213, 228 Erhart, P. 154, 169, 187, 189–190 Eshelby, J.D. 105, 224 Esquivel, E.V. 111, 145 Essmann, U. 276 Estreicher, S.K. 253–254 Evans, L. 96 Eyre, B.L. 58 Eyring, H. 231 Failor, B.H. 96 Fan, G.J. 206, 218, 223 Farkas, D. 62, 152, 169, 236 Farrell, K. 58 Ferber, M. 207 Ferreira, P.J. 208, 255, 269, 276–277, 283–284, 286 Ficalora, P.J. 268 Finnis, M.W. 4–5, 42, 54 Fivel, M. 7, 14, 20, 23, 65, 83 Flanagan, W.F. 135 Florando, J.N. 84 Foiles, S.M. 10 Follansbee, P.S. 104, 120, 166 Follstaedt, D.M. 217–218 Foltz, J.V. 111, 123 Fonde`re, J.P. 204, 206, 218 Foreman, A.J.E. 26, 45 Fouge`re, G.E. 207 Frank, F.C. 105, 224 Franc- ois, D. 203 Frederiksen, S.L. 21, 23 Freeman, A.J. 256 Freund, L.B. 206 Friedel, J. 23, 31, 238 Froseth, A.G. 139 Frøseth, A.G. 169, 171 Fu, C.L. 256 Fu, L.F. 206, 218, 223 Fuess, H. 251 Fukushima, H. 253–254 Fukuyama, S. 264 Furukawa, M. 206, 208 Gaffet, E. 206 Gao, H. 105–106 Garrison, K.E. 8

298

Author Index

Garruchet, S. 32 Gavriljuk, V.G. 251 Gehlen, P.C. 9 Geller, C.B. 256 Geng, W.T. 256 Geoffroy, G. 51 Gerard, R. 43 Germann, T.C. 152–154, 176 Gertsman, V.Y. 203 Gianola, D.S. 204, 214, 218 Gieseke, W. 32 Gil Montoro, J.C. 21 Gil Sevillano, J. 224 Gilman, J.J. 105 Girardin, G. 276 Gleiter, A. 203, 232 Gleiter, H. 203, 207, 228, 236–238 Goddard, W.A. 152 Goettert, J. 212 Goldenberg, T. 268 Golubov, S.I. 7, 9, 19–20, 58, 62 Gonzalez, B. 111 Goren, S. 251 Grady, D.E. 97, 140, 151 Graham, L.J. 8 Graham, R.A. 96 Grant, N.J. 139 Gray, G.T. 96–97, 113–114, 117, 120, 123, 166, 206, 234 Gregori, F. 99, 101–102, 117, 122–123, 125, 133, 140, 150, 161, 165–166 Greulich, F. 123, 145 Grifﬁn, A.J. 215, 218, 221 Groger, R. 26 Grosinger, W. 223 Gubicza, J. 209–210, 217–218 Guiu, F. 231 Gullett, P.M. 189 Gumbsch, P. 15, 23, 105–106 Gunderov, D.V. 208 Gue´rin-Mailly, S. 206 Guyot, P. 21 Haasen, P. 26–27, 35, 135, 138, 143 Hafez Haghighat, S.M. 48 Hafok, M. 207, 213 Hagi, H. 257 Hall, C.A. 146, 166 Hall, E.O. 202–203 Haller, E.E. 253–254 Hamilton, J.C. 152, 166 Hammerberg, J.E. 171 Hammon, D. 221, 223

Hamza, A. 213, 231–232 Han, S. 22–24, 41, 55, 66, 70, 75 Hanninen, H.E. 255, 269 Hanson, K. 26 Haouaoui, M. 204 Haque, M.A. 207, 214 Harry, T. 22, 24, 32, 37, 39, 41, 43–44, 48, 66, 70, 74 Hartwig, K.T. 204 Hasnaoui, A. 232, 236–238 Hatano, T. 49, 54 Hauer, A. 96 Hawreliak, J. 163, 166–167, 169, 174, 176, 178–180 Hay, J.L. 229 Haynam, C.A. 110 Hazzledine, P.M. 223 He, D.W. 208 He, H. 189–190 Hector, L.G. 23, 31, 35 Hellmig, R.J. 210 Hemker, K.J. 171, 204, 208, 212, 214, 217–218, 229 Hempelmann, R. 206, 251 Henkelman, G. 24 Herbert, E.G. 229 Hermida, J.D. 276 Hernandez, C. 9 Heubaum, F. 255, 264–266, 288 Hibbard, G.D. 213 Hiratani, M. 26, 84 Hirohata, N. 206, 213 Hirsch, P.B. 54, 58, 178 Hirth, J.P. 4, 9, 12, 27, 32, 57, 59, 101, 105, 214–215, 251, 256, 268, 279–280, 285 Hixson, R.S. 96–97 Hoagland, R.G. 9, 179, 214–215, 221, 223 Hoc, T. 84 Hodge, A.M. 165, 167, 171, 221 Hoffman, M. 203, 207 Hohler, B. 253–254 Holian, B.L. 152–154, 163–165, 171, 176 Holland, J.R. 107–108 Hommes, G. 84 Honeycombe, R.W.K. 138 Honeycutt, J.D. 21 Hoover, C.G. 171 Hoover, W.G. 152–153, 171 Horita, Z. 208 Hornbogen, E. 98, 106 Horner, H. 252 Horsewell, A. 7, 58 Horst, V. 206

Author Index Horstemeyer, M.F. 189 Horton, J.A. 204, 216–218, 230 Horton, L.L. 58 Hsiung, L.L. 84 Hsu, C.Y. 107 Hsu, K.C. 107, 113–114 Huang, B. 213 Huang, C.X. 115–121, 127, 163, 165 Huang, H. 9, 214–215 Huang, J.C. 117 Hugo, R.C. 217 Hull, D. 4, 11–12, 17, 27, 33, 46, 54, 57, 59, 143 Humphreys, F.J. 54, 178 Hunsinger, J.J. 208 Huntington, H.B. 8 Hwang, C. 268, 276 Hwang, S. 206 Hyde, J.M. 43 Hy¨tch, M. 206, 241 Ichinose, H. 253 Imura, T. 272 Indenbom, V.L. 23 Inoue, A. 206, 208 Ishida, I. 58 Ishida, Y. 253 Ishino, S. 66 Isshiki, M. 58 Ito, K. 23 Ivanov, A. 251 Ivanov, D.S. 153, 157, 163, 165–166 Ivanov, Y.F. 217 Iwasaki, H. 173 Jacobsen, K.W. 21, 23, 171, 208, 236 Jarmakani, H. 142–143, 150–151, 166–167, 171 Jax, P. 135 Jena, P. 253–254 Jenkins, M.L. 43, 51–52, 58 Jia, D. 206, 219, 231 Jiang, D.E. 256 Jiao, T. 220 Jin, M. 218–219 Jin, Z.H. 223, 230 Johari, O. 94, 119–120, 127 Johnson, H.H. 253–254 Johnson, N.M. 253–254 Johnson, Q. 96 Johnson, R.A. 32, 165–166 Jones, O.E. 107–108 Jonsson, H. 24 Josephic, P.H. 255, 273 Jumel, S. 70

299

Kad, B. 122, 124, 126, 130–132, 134, 142–143, 145, 149–151, 166 Kad, B.K. 99, 101–102, 115–123, 125–129, 133, 140, 145, 149–150, 161, 163, 165–166 Kadau, K. 109–110, 152 Kalantar, D.H. 96, 99, 101–102, 115–134, 140, 142, 145, 149–150, 161, 163, 165–167, 169, 174, 176, 178–180, 182, 184, 191 Kan, T. 135, 143 Kaneko, T. 49 Kao, P.W. 217 Karaman, I. 204 Karch, J. 228 Karimpoor, A.A. 206 Kaski, K. 76 Kayano, H. 51 Kazmi, B. 111, 166 Kecskes, L.J. 206 Kecsks, L.J. 220 Keeler, R.N. 96 Kehr, K.W. 253 Kelchner, C.L. 152, 166 Kelly, A. 202 Kelly, P.M. 53 Kenik, E. 122, 124, 126, 130–132, 134, 145, 149 Kestenbach, H.J. 106–107 Khantha, M. 142 Khatamian, D. 251 Kiener, D. 223 Kienle, W. 120 Kim, B.K. 209 Kim, W.W. 51 Kimura, A. 257–258, 260, 262–267, 284, 288 Kimura, H. 78, 80, 262–264, 267, 284 King, A.H. 221 Kirchheim, R. 253–254, 261, 266 Kiritani, M. 58 Kitajima, K. 26 Kitamura, A. 259, 264 Kizuka, Y. 58 Klaver, P. 73 Kle´man, M. 238 Knapp, J.A. 190, 217–218 Koch, C.C. 204, 206–207, 217, 219, 230 Koci, L. 153, 157, 163, 165–166 Kock, C.C. 207 Kocks, U.F. 26, 45–47, 49, 273 Koizumi, Y. 206, 213 Kojima, S. 58 Kong, D. 206, 217 Korn, C. 251 Kozlov, E.V. 217 Krasnochtchekov, P. 169

300

Author Index

Kratochvil, J. 60 Kratochvil, P. 135 Kress, J.D. 62 Kronmuller, H. 253–254 Krysl, P. 211 Kubin, L.P. 21, 24, 60, 84 Kubota, N.A. 163 Kuhlmann-Wilsdorf, D. 102, 111, 166 Kum, O.J. 157 Kumar, K.S. 203–204, 208, 216–218, 237 Kumar, M. 187, 189–190 Kumnick, A.J. 253–254 Kundson, M.D. 146, 166 Kung, H.H. 214–215, 217–218, 221 Kuramoto, E. 26 Kwon, S.C. 51 Labusch, R. 26, 135 Lagerpusch, U. 54 Lambard, V. 51 Landt, J.A. 94 Langdon, T.G. 206, 208–209 Langlois, C. 206 Langlois, P. 206 Larche, F.C. 281, 284 Lartigue-Korinek, Y. 206 Lasalmonie, A. 203 Lassila, D.H. 114–121, 127, 163, 165, 253–254 Latanision, R.M. 258, 264 Lavernia, E.J. 208, 213 Le, T.D. 268, 276 Lee, H.J. 14, 76, 80, 276 Lee, J.-L. 253 Lee, J.S. 206 Lee, J.-Y. 253 Lee, R. 96 Lee, T.C. 255, 269 Lee, T.D. 268 Legros, M. 204, 214, 218 Leslie, W.C. 94, 96, 110 Leveugle, E. 171, 176 Li, H. 230 Li, J.C. 4, 21, 23, 139, 225 Li, M. 229 Li, Y. 169 Lian, J. 217 Lian, K. 212 Liang, Y. 256, 278 Liao, X.Z. 171, 208 Liaw, P.K. 206, 218, 223 Lin, J.K. 269 Lin, Z.B. 165–166 Linga Murty, K. 204, 217, 230

Little, E.A. 58 Liu, G.C.T. 225 Liu, K.W. 221, 238 Liu, X.Y. 9, 70–71 Lomdahl, P.S. 152–154, 164, 176 Longo, W.P. 113–114 Lopez, H. 111 Lorenz, K.T. 146–147, 166, 176, 191 Lothe, J. 4, 27, 57, 59, 101, 105, 279–280 Louzguine, D.V. 206, 208 Loveridge, A. 96 Loveridge-Smith, A. 178 Lowe, T.C. 206, 234 Lozano-Perez, S. 51–52 Lu, C. 228 Lu, K. 207–208, 213, 217, 223, 230–232 Lu, L. 207–208, 216–217, 223, 230–232 Lu, Q.H. 223, 230–232 Lu, Y.C. 214–215, 218, 221 Lubarda, V.A. 136–138, 179, 182, 184, 187, 189–191 Lucas, B.N. 229 Lunarska, E. 259 Luppo, M.I. 7, 276 Ma, E. 171, 206, 208, 213, 216–217, 219–220, 223, 229–232, 237 Maab, R. 226 Maddin, R. 78, 80 Madec, R. 84 Magerl, A. 252 Magness, L. 206 Magnin, T. 251, 276–277, 280–281, 283 Mahajan, S. 137–138, 154 Mai, Y.W. 228 Maier, H.J. 204 Makenas, B.J. 258, 262, 267 Makin, M.J. 26, 45, 60 Malerba, L. 51–53, 66, 73, 76 Malow, T.R. 207 Mao, S.X. 217–218 Mareschal, M. 152–154, 176 Marian, J. 15, 73–74, 190 Markmann, J. 217, 238 Martin, G. 15, 23, 27–31, 35, 61 Martin, J.L. 214, 231–232 Martin, J.W. 26, 36, 54 Masters, B.C. 58 Masumura, R.A. 223 Mathon, M.H. 51 Matsui, H. 49, 262–264, 267, 284 Matsukawa, Y. 76, 78, 80–82 Matsumoto, T. 255, 264–266, 288

Author Index Matz, W. 214–215 Mazzolai, F.M. 252 McCormick, P.G. 206 McCrea, J.L. 213 McInteer, W.A. 268 McNaney, J.M. 142–143, 146–147, 150–152, 165–167, 171, 176, 191 McQueen, R.G. 153, 166 Mehl, M.J. 62, 152 Mencik, J. 228 Mendelev, M.I. 22–24, 41, 55, 66, 70, 75 Metzger, H. 252 Meyers, M.A. 94, 96, 98–99, 101–103, 106–111, 113–134, 136–138, 140, 142–143, 145, 149–151, 153–159, 161, 163–167, 171, 179–180, 182, 184, 187–191, 206–207, 236–237 Michalak, J.T. 94, 110 Milligan, W.W. 217–218, 223 Mimura, K. 58 Minamino, Y. 206, 213 Minkin, I.O. 152–153 Minor, A.M. 218–219 Mirshams, R.A. 213 Mishin, Y. 62, 152, 257 Mishra, A. 206–207, 236–237 Misra, A. 214–215, 221, 223 Mitchell, A. 96 Mitchell, T.E. 214–215, 218, 221 Mizubayashi, H. 206 Mizuuchi, K. 206, 213 Mogilevsky, M.A. 96, 152–153 Mohamed, F.A. 213 Mohles, V. 39, 42–43, 47, 49, 54 Moin, E. 111, 166 Monnet, G. 14, 35, 47 Monnet, I. 7 Monzen, R. 43, 51 Moody, N.R. 61–62 Moran, B. 152–153 Morez, E.M. 94 Mori, H. 58 Mori, M. 253 Mori, T. 58 Moriarty, J.A. 9, 26 Moriya, S. 262–264 Morris, C.E. 96–97 Morris, J.W. 26, 218–219 Mughrabi, H. 120, 202, 276 Mukherjee, A.K. 204, 206, 208, 214, 230, 232, 236 Munir, Z.A. 206 Mura, T.J. 184 Murphy, K.L. 207

301

Murr, L.E. 94, 96, 102, 107, 111, 119, 121, 123, 125, 138, 145, 164–167 Murty, G.S. 26 Murty, K.L. 219, 230 Myers, S.M. 253–254 Nabarro, F.R.N. 4, 27, 224, 265 Nagakawa, J. 58 Narang, P.P. 251 Narayan, J. 206 Narita, N. 257, 260 Nastasi, M. 214–215, 218, 221 Natter, H. 206 Nazarov, A.A. 217 Nembach, E. 54 Nemoto, M. 206, 208 Newman, P.E. 96 Ngan, A.H.W. 26 Nguyen, J.H. 32, 143, 150–151, 166 Nibur, K.A. 268 Nicholson, R. 202 Nieh, T.G. 173, 213, 221 Nieman, G.W. 207, 212 Niewczas, M. 223 Nishimura, C. 206 Noda, K. 272 Nogaret, T. 7, 14, 62–65, 83 Nolder, R.L. 94, 145 Nomoto, A. 66 Nordlund, K. 51 Nurislamova, G. 209–210 Odette, G.R. 74 Odunuga, S. 169 Ogasawara, Y. 58 Ogata, S. 139 Oguri, K. 262–264 Ohr, S.M. 276 Oliver, W.C. 229 Olmsted, D.L. 23, 31, 35 Olsson, P. 51, 73 Onimus, F. 7 Ono, K. 58 Onyewuenyi, O.A. 268 Opperhauser, H. 258, 264 Oriani, R.A. 253–257, 269, 273 Orlikowski, D. 143, 150–151, 166 Orowan, E. 96, 166 Ortiz, M. 190, 256 Osetsky, Y.N. 7, 12–13, 15–20, 23, 29–34, 39–40, 42–44, 47–49, 51, 55, 58, 60–62, 66, 68–70, 72–73, 75–82, 84

302

Author Index

Otsuka, R. 257, 259, 264 Owusu-Boahen, K. 221

Qu, S. 174 Quin, X.Y. 206

Padmanabhan, K.A. 238 Paidar, V. 4, 21–23 Painter, G.S. 256 Pak, H.R. 180 Palumbo, G. 206, 213, 228 Panarella, E. 94 Pande, C.S. 223 Papaconstantantopoulos, D.A. 62, 152 Paris, S. 206 Park, N. 107–109 Parthasarathy, T.A. 9 Patinet, S. 28, 35 Paton, N.E. 262 Patriarca, M. 76 Paul, G.L. 251 Pearton, S.J. 253–254 Peisl, H. 252 Perez, M. 32 Perez, R.J. 213 Perlado, J.M. 73–74 Perrin, R.C. 8 Pesch, W. 270 Petch, N.J. 202 Pettifor, D.G. 9 Pharr, G.M. 207, 228–229 Phillips, R. 9, 21 Phillpot, S.R. 232, 236–237 Pierce, T.G. 84 Pilvin, P. 7 Pineau, A. 203 Pink, E. 263 Pinto, H. 251 Pippan, R. 207, 213, 223 Plimpton, S.J. 152, 163, 166 Pointon, T.D. 146, 166 Pond, R.C. 12 Pontini, A.E. 276 Potirniche, G.P. 189 Pound, B.G. 253 Pratt, P.L. 231 Preston, D.L. 144, 150, 152 Priego, V. 58, 73 Prioul, C. 7 Proville, L. 27–31, 35 Putaux, J.L. 241

Ramesh, K.T. 206, 219–220, 231 Rao, S.I. 9, 23, 26 Raulot, J.M. 32 Ravelo, R.J. 152–154, 171, 176 Ravichandran, G. 96, 99, 101–102, 117, 122–123, 125, 133, 140, 150, 161, 165–166 Raynolds, J.E. 256 Reed-Hill, R.E. 33, 113–114 Remington, B.A. 96, 99, 101–102, 115–134, 140, 142, 145–147, 149–150, 152, 161, 163, 165–167, 169, 171, 174, 176, 178–180, 182, 184, 191 Re´vesz, A´. 209–210 Rhee, M. 84 Rice, J.R. 142, 184, 187, 256 Rice, M.H. 153, 166 Richter, D. 251 Robach, J.S. 76, 80, 276 Robertson, C. 7, 14, 62–65, 83 Robertson, I.M. 58, 76, 80, 255–256, 258, 262, 267–271, 276–277, 283–284, 286 Robino, K. 107, 113–114 Rodary, E. 27–31, 35 Rodney, D. 7, 14–16, 20, 23–25, 27–31, 35, 60–67, 76–80, 83–84 Rodrigues, J.A. 261, 266 Rodriguez, M.V. 268 Rong, Z. 23, 61–62, 66, 68–70 Rosen, A. 228 Rosolankova, K. 163, 166–167, 169, 174, 176, 178–180 Ro¨sner, H. 217, 238 Ross, C.A. 229 Roth, H.A. 135 Rozenak, P. 255, 257, 260, 269 Rubio-Bollinger, G. 208 Rudd, R.E. 154, 163, 166–167, 169, 174, 176, 178–180, 189 Rush, J.J. 251 Russell, K.C. 45 Ruste, J. 70

Qi, Y. 216–217, 234 Qian, L. 208, 217, 229 Qiao, D.C. 206, 218, 223

Saada, G. 225–227, 231–232, 239–240 Sabirov, I. 207, 213 Saif, M.T. 214 Saif, T.A. 207 Saintoyant, L. 14 Saka, H. 272 Sakai, S. 206 Salazar, M. 237 San Marchi, C. 268, 276

Author Index Sanchez, J.C. 111, 138 Sanders, P.G. 206, 222–223, 236 Sapozhnikov, P. 152 Sato, A. 58 Satoh, Y. 58 Satoy, Y. 58 Savic, P. 94 Scattergood, R.O. 36, 46–47, 49, 204, 206–207, 217, 219, 230 Schaublin, R. 7, 48, 55, 74, 276 Scho¨ber, T. 241 Schell, N. 214–215 Schiffgens, J. 8 Schiroky, G.H. 268, 276 Schiøtz, J. 171, 236 Schmatz, W. 252 Schmitt, B. 226, 242 Schneider, M.S. 99, 101–102, 115–134, 140, 142, 145, 149–150, 161, 163, 165–166, 179, 182, 184, 187, 189–191 Schober, T. 251, 270 Schoeck, G. 32, 231 Schoek, G. 281 Schuh, C.A. 135, 137, 167, 171, 173, 208 Schuster, B.E. 206, 220 Schuster, G.B.A. 261 Schwaiger, R. 135, 137, 207–208, 217, 223, 230 Schweitz, K.O. 214–215 Schwink, C.H. 32 Seeger, A.Z. 23, 137, 154 Seitz, E. 252 Seitz, F. 178 Seppa¨la¨, E.T. 189 Serebrinsky, S. 256 Sergueeva, A.V. 204, 206, 208, 214, 230 Serra, A. 7, 12, 58, 62, 73 Serruys, Y. 58 Server, W.L. 43 Shaked, H. 251 Shan, Z. 207–208, 217–218, 223, 230 Sharp, J.V. 7 Shehadeh, M.A. 171, 176, 181–182, 191 Shen, T.D. 114–115, 207, 214–215 Shen, Y.F. 208, 217, 223, 230–231 Shen, Y.G. 228 Sheng, H. 171, 208, 217 Shenoy, V.B. 9, 21 Shi, X. 229 Shih, D.S. 255–256, 258, 262, 267, 269–271 Shim, J.H. 14, 51 Shimomura, Y. 58 Shiraishi, H. 58 Siegel, D. 139

303

Siegel, R.W. 207, 212 Silcox, J. 58, 178 Simmons, J.P. 9 Sinclair, J.E. 9 Singh, B.N. 7, 55, 58, 61–62, 66, 76, 276 Sirois, E. 273–275, 282 Skeen, C.H. 94 Smirnova, M. 152 Smith, C.S. 98, 154 Smith, D.J. 208 Smith, G.C. 258, 264–265 Smith, R.F. 152, 165, 167, 171 Sofronis, P. 256–257, 278–285 Soisson, F. 58 Somerday, B.P. 268, 276 Soneda, N. 43, 66 Song, Z. 189–190 Spaczer, M. 169, 236 Spaepen, F. 214–215 Spa¨tig, P. 231–232 Spencer, J.A. 217–218, 223 Spielman, R.B. 146, 166 Springer, T. 252–253 Srinivasan, S.G. 171, 189, 208 Srolovitz, D.J. 22–24, 41, 55, 66, 70, 75 Stach, E.A. 217–219 Staehle, R.W. 258 Stassis, C. 251 Staudhammer, K.P. 111, 121, 138 Stavola, M.J. 253–254 Stergar, E. 242 Stgar, W.A. 146, 166 Stolken, J. 154, 169 Stoller, R.E. 7, 55, 58, 76, 78, 80–82 Stolyarov, V.V. 204, 206, 230 Stone, G.A. 107 Strachan, A. 152 Straub, G.K. 152–153 Strudel, J.L. 203 Suganuma, K. 51 Sui, M.L. 217, 223, 230–232 Suresh, S. 135, 137, 203–204, 206–208, 216–218, 223, 228, 230–231, 237 Surinach, S. 209–210 Sutton, A.P. 43, 51, 241 Suzuki, M. 58 Swadener, J.G. 214–215 Swegle, J.W. 97, 140, 151 Swygenhoven, H. 4 Szelestey, P. 76 Szommer, P. 217–218 Szpunar, J.A. 209

304

Author Index

Tabata, T. 255, 269–270 Tadmor, E.B. 102, 133, 139 Takahashi, A. 66 Takaki, S. 262–264 Takata, H. 58 Takeuchi, A. 206, 208 Takeuchi, S. 23 Tang, M. 24, 84 Tang, X. 268, 276 Tanguy, D. 152–154, 176 Tanimoto, H. 206 Tanushev, N. 154, 169 Taok, H. 58 Tapasa, K. 29–34 Taylor, K.N.R. 251 Terentyev, D.A. 51–53, 66, 70, 72–73, 75–76 Teter, D.F. 255, 269 Tewary, V.K. 9 Thibault, J. 241 Thomas, G. 94, 119–120, 127, 145 Thompson, A.W. 135, 179, 262, 268 Thomson, R.C. 8–9, 135, 184, 187 Tichy, G.I. 42, 54 Tichy, J. 210 Tiearney, T.C. 139 Tien, C.W. 261 Tildesley, D.J. 5 Titchmarsh, J.M. 51–52 Tjong, S.G. 204, 206 Tobe, Y. 259, 264 Toft, P. 7, 58 Tonks, D.L. 150 Torralba, B. 165 Torralva, B.R. 152, 167, 171 Traiviratana, S. 187–190 Trillo, E.A. 111, 145 Trinkle, D.R. 27 Trueb, L.F. 102 Tsai, D.H. 152 Tsui, T.Y. 207, 229 Tsuji, N. 206, 213 Tucker, R.P. 276 Tyson, W.R. 259, 264 Uberuaga, B.P. 179 Uchikoshi, M. 58 Udovic, T.J. 251 Ulmer, D.G. 260–261, 268, 276 Ungar, T. 120, 209–210 Valerio, O.L. 111 Valiev, R.Z. 204, 206, 208, 214, 217, 220, 230, 234

Valone, S.M. 179 van Duin, D. 152 Van Duysen, J.C. 70 Van Petegem, S. 204, 210, 214, 218, 226, 232, 241–242 Van Swygenhoven, H. 139, 152, 169, 171, 203–204, 208, 210, 214, 218, 223, 226, 231–232, 236–238, 241–242 Vecchio, K.S. 166 Vehoff, H. 229 Venables, J.A. 138 Verdier, M. 214–215, 226–227 Veyssie`re, P. 225 Victoria, M. 7, 58, 107–109, 152, 165, 167, 171, 204, 223, 231, 276 Vieira, S. 208 Vitek, V. 4, 8–9, 21–23, 26, 42, 54, 142, 241 Vo¨hringer, O.Z. 135–138 Volkl, J. 253–254, 287 Vorhauer, A. 207, 213 Voskoboinikov, R.E. 7, 20, 55, 58, 84 Voskoboynikov, R.E. 20, 84 Voter, A.F. 62, 171, 179 Vydyanath, H.R. 111, 123 Wagner, G.J. 189 Wagner, N.J. 171 Wallace, D.C. 144, 150 Wallenius, J. 51 Walley, S.M. 96 Walsh, J.M. 153, 166 Wang, H. 206, 214–215 Wang, J. 208 Wang, P. 216–218 Wang, Y. Wang, Y.M. 152, 165, 167, 171, 208, 213, 217, 221, 223, 230–232 Wark, J.S. 96, 99, 101–102, 117, 122–123, 125, 133, 140, 142, 150, 161, 163, 165–167, 169, 174, 176, 178–180 Warner, D.H. 174 Was, G.S. 6 Weber, S.V. 96 Wechsler, M.S. 276 Weertman, J.R. 104–105, 206–207, 212, 217, 222–223, 236 Wegner, P.J. 110 Wei, Q.M. 206, 213, 220, 231 Weissmu¨ller, J. 217, 238 Wen, M. 26, 264 Wenzl, H. 251 Wetscher, F. 207, 213 Whang, S.H. 213

Author Index White, C.T. 152 White, R.M. 94 Wiedersich, H. 281 Wiezorek, J.M.K. 217–218 Wilcox, B.A. 258, 265 Wilkens, M. 120 Wille, T.H. 32 Windle, A.H. 258, 264–265 Wipf, H. 251 Wirth, B.D. 14–15, 51, 73–74, 76, 80, 154, 169, 276 Wokulski, Z. 259 Wolf, D.P. 232, 236–237 Wolfer, W.G. 179 Woo, C.H. 9 Wooding, S.J. 22 Woodward, C. 9, 23, 26–27 Wu, B.Y.C. 208 Wu, C.C. 94 Wu, R. 256 Wu, X.L. 216–217, 234, 237 Xiao, C. 213 Xu, G. 101, 142, 187 Xu, Y.B. 115–121, 127, 163, 165 Yamakov, V. 232, 236–237 Yamamoto, N. 58 Yamasaki, T. 173 Yan, X. 218 Yang, B. 206, 229 Yang, H. 229 Yeske, R.A. 261 Yin, W.M. 213 Yip, S. 4, 9, 21, 23, 26, 84, 139 Yokogawa, K. 264

York, C.M. 94 Yoshihira, O. 206, 213 Yoshiie, T. 58 Youngdahl, C.J. 206, 217, 222–223 Youssef, K.M. 204, 207, 217, 219, 230 Yu, C.Y. 217 Yu, D.Y.W. 214–215 Zaiser, M. 26 Zaoui, A. 203 Zaretsky, E. 103–104, 171 Zbib, H.M. 105, 171, 176, 181–182, 191 Zelinski, J. 268 Zenji Horita, Z. 206 Zerilli, F.J. 97, 133, 135 Zevin, L.S. 257, 260 Zhai, Q. 206, 217 Zhang, H.T. 206 Zhang, K. 213 Zhang, X. 206, 214–215, 221, 223 Zhao, Y.G. 171 Zhao, Y.H. 208 Zhigilei, L.V. 165–166 Zhilyaev, A.P. 209–210 Zhou, F. 208 Zhou, S.J. 9 Zhou, Z. 229 Zhu, B. 211 Zhu, W. 189–190 Zhu, X.G. 206 Zhu, Y.T. 171, 206, 208, 216–217, 234, 237 Zinkle, S.J. 7, 58, 76, 80–82 Zocher, M.A. 152 Zukas, E.G. 96 Zurek, A.K. 187 Zybin, V. 152

305

Subject Index activation enthalpy 18, 24, 34 – parameters for slip 33 ageing 36, 58 Al 24, 27, 42 alloy strengthening 26 Al-Mg 31 au precipitates 51 a-Fe 17 a-Zr 21 anisotropic elasticity 46 – elasticity theory 58 applied strain 10 – rate 24 applied stress 10 athermal processes 18 atomic disregistry 19 atomic-scale simulation 4, 83 austenitic steels 62

CPU time 5, 17, 19, 21 Cr 27, 51 critical angle 45 critical line shape 44, 51, 69 critical stress 32 cross-slip 36, 49, 60, 62, 64, 68, 69, 78, 80, 82 crowdions 7, 58 CRSS 31 Cu 27, 42, 48, 49, 54, 61, 62, 77 Cu precipitates 47, 50 defect density 62 3-D diffusion 58 dimer of solute atoms 31 dislocation – channelling 7 – climb 19, 41, 48, 49, 51 – core structure 6 – density 16 – dipole 41, 47 – dissociation 4, 8, 21, 42, 49, 54 – dynamics (DD) 7, 31, 47, 56, 65, 82 – glide 21 – glide plane 10 – induced transformation 44, 57 – line shape 40 – loop 7, 36, 57 – mobility 21 – motion 6, 9 – non-degenerate compact core 23 – obstacle interactions 4 – precipitate mechanisms 36 – reactions 57 – self-stress 56 – velocity 16, 28 dispersion-strengthened alloys 54 displacement cascade 6, 36, 58 displacement ﬁeld 8, 13 distribution of obstacles 26 double cross-slip 19, 54 double-loop 73 drag 33, 60, 61, 63, 66 ductility 7

basal stacking fault 22 bending moment 12 body-centred cubic (BCC) 4, 7, 21, 56, 59, 66 boundary conditions 8 breakaway stress 49 bubbles 7, 55 Burgers vector 4, 6, 10, 57 carbon interstitial 27, 32 cavities 36 centre of dilatation 27 centro-symmetry 20 chemical strengthening 53 clear band formation 61, 84 climb 56–57 cluster density 7 coarse grained (cg) 202–205, 207, 216, 219, 222, 224–226, 231, 232, 243 coherency loss 51 coherent interface 43, 50 computational efﬁciency 9 constant line tension 53 constriction 42, 49 continuum modelling 47, 76 continuum scale 5 core structure 4, 9, 21 307

308

Subject Index

edge dislocation 12, 21, 22, 29, 32, 37, 54, 55, 60, 75 elastic constants 5 elastic shielding 253, 282, 284–287, 289, 291–293 elastic-plastic transition 204, 223, 226, 227, 243 Embedded Atom Model (EAM) 5 embrittlement 84 empirical interatomic potentials 19 energy factor 46 enthalpy-stress relation 24 expression 45 external loading 10, 15, 16 extrinsic fault 57 face-centred cubic (FCC) 4, 7, 21, 56, 57 fault energy surface 21 Fe 17, 19, 21, 27, 37, 48, 70, 75 Fe-C 32 Fe-Cr 36 Fe-Cu 31, 36, 43 ferritic alloys 58 ferritic pressure vessel steels 43 ferritic/martensitic steels 51 ﬂexible boundary model 9 ﬂight time 216, 233 ﬂow localization 7 ﬂow stress 26 Frank loops 57–60, 65 Frank partial 80 Frank’s rule 74, 84 free boundaries 70 free surfaces 14 free-ﬂight motion 23 free-surface conditions 74 friction 60 – coefﬁcient 23, 62 Friedel statistics 31 g-surface 21 generalized pseudopotential theory 26 Gibbs free energy 33 glide – cylinder 61 – force 15 – plane 4, 60, 61, 65, 66, 71 – prism 58, 66 glissile loops 60, 61 Green’s function 9, 19 – boundary conditions 9 hardening 28, 84 HCP 7, 21, 22, 84 He bubbles 55

helical turn 19, 60, 64, 65, 70, 74, 78, 80 helix 78 high-resolution electron microscope (HREM) 51 homogeneous dislocations, shock 98–104, 154 Hornbogen model 93, 99 hydrogen embrittlement 254, 258 hydrogen enhanced localized decohesion 255 image force 13, 14, 50 impenetrable obstacles 46, 54 in situ deformation 80, 83 incoherent interface 36 induced interaction 27 interaction energy 27, 48 interatomic potential 5, 8, 16, 19, 21, 66, 70, 84 interface step 36 interfacial dislocations 12 intermetallic 28 internal energy 16 interstitial loops 62, 66 interstitial solute atoms 32 intrinsic fault 57 irradiation 6, 36, 61, 76 – defects 7 – hardening 43 isotropic elasticity 13 jogs 44 junctions 67 kink dynamics 26 kink pair 23 laser shock 93, 95–97, 123, 132, 151, 166, 167, 172, 180 length scale 85 line tension 26, 40, 45, 47 linear elasticity 4, 7, 8, 27, 45 loading techniques 15 Lomer dislocation 24, 78 long-range interaction 27 /100S loop 58–60, 70, 75, 83 ½/111S loops 59, 60, 66, 74, 75 – absorption 63, 64 – mobility 62 – shearing 62 macroyield 202–204, 226, 227 many-body interactions 9 martensitic transformation 51 Meyers model 105 Mg 27 microcrystalline (mc) 203

Subject Index micro-deformation 223, 226, 227 microplastic 242, 243 micro-straining 224, 226 microyield 203, 226 misﬁt 27 – dislocation 12 misorientations 209, 210, 217, 219, 220, 238 molecular dynamics (MD) 5, 18, 93, 104, 153, 155, 180 molecular statics (MS) 5 Mott-Nabarro mode 35 multiscale modelling 4, 7, 83 nanocrystalline (nc) 203, 207, 213, 236, 237 nanocrystalline metals 107, 110, 141 nanograined (ng) 204–211, 216, 217, 219–223, 225–232, 234–237, 241, 243 nanopowders 212 nanoscale precipitates 43 nanotwinned 230–232, 234, 235, 238 nearest-neighbour pairs 27 neutron irradiation 43 neutron-irradiated Fe 55, 58 Ni 8, 27, 62 Ni-Al 31, 35 non-degenerate compact core 23 non-planar core structure 22 Nudged Elastic Band method 24 obstacle strength 18, 36 Orowan – law 16, 18, 24 – looping 41, 51, 55 – mechanism 41, 46, 54, 65 parallel computing 17 Peierls – barrier 21 – stress 7, 9, 13, 21, 23, 37, 51 – valley 21, 23 perfect loops 57, 61, 64 periodic – array of dislocations 10 – boundaries 50, 63, 65 – images 17 periodicity 10, 12, 14 phase transformation 51 pinning 7, 26–28, 31, 35 plastic shear strain 37 plastic strain rate 17 point defect energies 5 Poisson’s ratio 46

309

positron annihilation 55 precipitates 7, 35, 43, 48, 83 precipitation strengthening 26 prism plane 22 prismatic loop 54 processing 202–207, 209, 210, 212, 214, 218, 219, 230, 241 quasi-isentropic compression 93, 147, 151, 192 quenching 36, 58 radiation damage 84 relaxation 8 RH/FS convention 11 rigid boundary condition 8, 19 Russell-Brown model 45, 53 saddle-point search method 85 screw dipole 41, 54, 69, 72 screw dislocation 13, 19, 22, 30, 32, 49, 61, 70, 74, 78 self-interaction 46 self-interstitial atoms (SIAs) 7, 43 self-stress 46, 47 sessile jog 78 shear localization 253, 262, 264, 271, 272, 279–282, 291, 292 shear loops 143, 183, 186, 187, 191, 192 shear modulus 9, 41, 46, 53 shear strain 9, 15 – rate 17 shear stress 13, 16 shearable particles 53 Shockley partial dislocation 28, 29, 42, 49, 57, 58, 61, 63, 64, 78 short-range interaction 27 single layers 214 size dependence 51 slip-twinning transition 133–142, 151 Smith model 155 solid solution 26, 51 – hardening 7, 26 solute atoms 7, 26, 27 sound velocity 23 spalling 179 stacking facult 20, 21, 30, 57 – energy 5, 42, 62, 82 – tetrahedron (SFT) 7, 36, 57–60, 76, 78, 83 stacking sequence 44 stair-rod partial 58, 80 strain-controlled simulations 15 strength of alloys 36

310

Subject Index

stress concentrations 26 stress-controlled simulations 15 structural information 238 superjogs 37, 54, 59, 61, 67, 69–72, 76, 78, 80, 82, 83 surface step 36 tangent modulus 203, 222–224, 226, 227 temperature effects 48 tensile tests 18, 34 terminal velocity 60 tetragonal distortion 32 thermal activation 23 thermal ﬂuctuations 20 thermally-activated processes 18, 28, 35, 58, 85 Thompson tetrahedron 59, 76 three-fold degenerate dislocation core 23 threshold stress 28, 31 time step 5, 17, 20, 84 translation vector 11 transmission electron microscope (TEM) 7, 57, 80, 83 twin 216, 217, 221, 223, 228, 230, 232, 234, 235, 238

twinning 43, 51, 216, 217, 221, 228, 232, 234, 238 – dislocations 12 – stress 93, 137–142, 152, 192 ufg 213 ultra ﬁne crystalline 203 undissociated dislocation 57 unfaulting 82, 58, 62 vacancies 7, 19, 43 vacancy absorption 56 vacancy loops 57 vibrating string 23 visualisation 37 void growth, dislocation emission 179–190 voids 7, 17, 19, 35, 37, 46, 48, 75, 83 Voronoi polyhedra 21 Weertman model 104, 105 work-hardening 7 yield stress 7, 51 Zaretsky’s model 103, 104, 171