Interface defects, reference spaces and the Frank–Bilby equation

Progress in Materials Science 58 (2013) 749–823

Contents lists available at SciVerse ScienceDirect

Progress in Materials Science journal homepage: www.elsevier.com/locate/pmatsci

Interface defects, reference spaces and the Frank–Bilby equation J.P. Hirth a, R.C. Pond b,⇑, R.G. Hoagland c, X.-Y. Liu c, J. Wang c a b c

Materials Physics and Applications, MPA-CINT, Los Alamos National Laboratory, Los Alamos, NM 87545, USA University of Exeter, College of Engineering, Mathematics and Physical Sciences, North Park Road, Exeter EXA 4QF, UK Materials Science and Technology Division, MST-8, Los Alamos National Laboratory, Los Alamos, NM 87545, USA

a r t i c l e

i n f o

Article history: Received 19 July 2012 Accepted 3 October 2012 Available online 17 October 2012

a b s t r a c t The physical basis for the Frank–Bilby equation is considered. Dual descriptions in terms of interface physics and mechanics are introduced. Natural (NDP), commensurate (CDP) and rotated (RCDP) dichromatic patterns are introduced. Burgers vectors are defined by symmetry operations or circuits in the CDP and RCDP. Structures are described for misfit arrays, tilt arrays, twist arrays, disconnections and combinations of these defects. The concepts of partitioning of elastic distortions, array energies, node formation, and the lateral spreading of defects within interfaces are considered. Examples with analytical solutions, numerical solutions and iterative solutions are presented. We elucidate some principles that emerge from the solutions and present reasons why some results differ from other methods of analysis. Ó 2012 Elsevier Ltd. All rights reserved.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Frank–Bilby equation and reference spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. The Frank–Bilby equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Reference spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Natural dichromatic patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Commensurate dichromatic patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Rotated-commensurate dichromatic patterns. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4. Symmetry classification of dichromatic patterns . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

⇑ Corresponding author. Address: Coombe Cottage, Sandygate, Exeter EX2 7JL, UK. Tel.: +44 1392 874644. E-mail address: [email protected] (R.C. Pond). 0079-6425/$ - see front matter Ó 2012 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.pmatsci.2012.10.002

751 752 752 753 753 753 755 755

750

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

2.3. Methodology for determining the defect content of semi-coherent interfaces . . . . . . . . . . . . . 2.4. Reference lattices used by other researchers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3. Misfit dislocations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Misfit arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2. Strain partitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3. Strain field attenuation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4. Defect arrays producing rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Tilt walls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Tilt–misfit analogues. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. Tilting at interphase interfaces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4. Twist walls. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5. Asymmetric tilt interfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5. Disconnections, steps, and line-forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Disconnections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Steps and line-forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6. Partitioning and nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7. The general interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8. Energies of interface defect arrays . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9. Delocalized and diffuse boundaries. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10. Applications of the F–B eq. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.1. Misfit arrays. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2. Misfit with added tilt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.3. Superposition of pure rotations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.4. Conversion of edge array to screw array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.5. Three or more dislocation sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.6. The Nishiyama–Wasserman interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.7. Numerical solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.8. Phase transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10.9. Defects arising from mismatching point symmetries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11. Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12. Conclusions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix A. Topological theory of interfacial defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.1. Defect character in single crystals and bicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.2. Elementary defects in semi-coherent bicrystals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3. Circuit mapping. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.1. Characterisation of an elementary interfacial defect . . . . . . . . . . . . . . . . . . . . . . . . . . . A.3.2. Characterisation of the defect content of an interfacial defect array . . . . . . . . . . . . . . A.3.3. Location of the circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.4. Description of rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A.5. Rotation, misfit accommodation, and spacing defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix B. Matrix notation and coordinates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.1. Transformation matrices. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.2. Symmetry operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.3. Terrace and habit plane coordinate frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B.4. Burgers vectors expressed in terrace and habit coordinate frames . . . . . . . . . . . . . . . . . . . . . . B.5. The probe vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix C. The standard model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix D. Burgers vectors, line directions, and spacings of interface dislocations in the general case D.1. Distortions associated with the CDP and RCDP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.2. Line directions and spacing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D.3. Alternative solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

755 759 760 760 762 762 763 763 765 765 767 768 769 769 772 773 777 779 780 781 782 785 786 789 790 791 793 800 802 804 806 806 807 807 808 809 809 810 811 811 811 814 814 814 815 815 816 817 818 819 820 821 821

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

751

1. Introduction This work deals with the semi-coherent interfaces that are key structural features in a wide range of engineering materials. Their structures are based on coherent ‘‘terraces’’ which have low chemical energy, and the attendant coherency strains in the adjacent crystals are accommodated by a superposed network of dislocations. Sometimes these defect networks produce a rotation, tilting or twisting the crystals away from the initial orientation. Such interfaces arise, for example, in epitaxial layers [1], precipitation [2], and both diffusional and diffusionless displacive phase transformations [3]. Some grain boundary structures, those where the constituent grain boundary dislocations are well separated, can be regarded as a special case of this category of interfaces: here the terraces are naturally coherent, and the defect network produces only tilting and twisting. Some incoherent interfaces, e.g., those in large precipitates or massive transformation products, and some high angle grain boundaries can be described by limiting procedures of the present work, but core overlap of defects occurs and other physical descriptions may be more appropriate. The dislocation content of an experimentally observed, computer simulated, or specified hypothetical interface can be quantified by applying the Frank–Bilby equation (F–B eq.), which identifies the total Burgers vector, B, intersected by a probe vector lying in the interface [4–7]. Implementation of the F–B eq. requires the selection of a reference lattice: the Burgers vectors of the individual defects, b, whose sum is B, are defined in this reference. Mathematically, the choice of reference lattice is arbitrary, and, historically, various options have been taken. For example, one of the adjacent crystal lattices may be used [8,9], or a median lattice intermediate between these two [10]. However, recent advances in our understanding of interfacial defects [11] emphasize that the choice of reference is not arbitrary: it is the union of the adjacent crystal lattices. Such unions are referred to as dichromatic patterns (DPs) [12] on the basis that one crystal be designated ‘‘white’’ and the other as ‘‘black’’. Moreover, the elastic distortion field produced by a defect network is relevant: the constituent strains and rotations are partitioned between the crystals in a manner that depends on the equilibrium equations of elasticity [13]. Thus, the identification of an appropriate DP is not simply a geometrical procedure. The objective of the present work is to describe the construction of appropriate DPs, and to illustrate their use in the application of the F–B eq. to a range of semi-coherent interfaces. For simplicity, we primarily consider the planar bicrystal interface for cases where the elastic moduli of the adjacent crystals are the same (homogeneous) and isotropic: then elastic distortions are equally partitioned. The cases of dissimilar and anisotropic moduli and unequal layer thicknesses are treated more briefly. Also, we concentrate on interfaces where the coherency strains are fully accommodated, excluding the case of epitaxial films with sub-critical thickness [14], although the model extends to such cases [15]. Nonlinear elastic effects are considered in a limited manner. In all of these cases, analytical results are presented. We do not consider highly nonlinear elastic cases, which are so complex that they are better described by atomistic simulations. In Section 2 we reprise the F–B eq., and demonstrate its relation to the formal theory of interfacial defects [16] which takes into account all the symmetry operations in the black and white space groups, as opposed to only translation symmetry as in the original formulation. We also introduce three types of DPs which are appropriate for bicrystals where (i) no coherency strains are present, (ii) coherency strains but no rotations exist, and (iii) both coherency strains and rotations arise. These DPs are designated natural (NDP), commensurate (CDP), and rotated (RCDP), the latter introduced in the present work. In most cases the CDP corresponds to one-to-one lattice site matching at the terrace plane, in which case the CDP is also referred to as coherent. Finally, we discuss how to express the F–B eq. in a manner compatible with the use of these three reference spaces. In Section 3 we focus on interfaces between misfitting crystals: we consider a crystal dislocation, initially remote from the interface, climbing to the interface where it contributes to the accommodation of misfit. We show that the Burgers vectors, b, of perfect dislocations are defined in the NDP, i.e. they are translation vectors of either the natural black or white lattices. However, after penetrating the short-range distortion field of the interface, their Burgers vectors are modified, becoming misfit dislocations with b characteristic of the CDP. Section 4 is a discussion of the character of individual defects in interfaces where a rotation, but no misfit, is present: low-angle tilt and twist grain boundaries are simple examples. Here, B is

752

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

comprised of combinations of black and white translation vectors, as would result from the reaction of crystal dislocations reaching the interface by glide through their parent crystal. For this special case the CDP is a single crystal lattice, and the combinations of black and white translation vectors are defined in the RCDP. These ideas are extended to the situation where both misfit and rotation are present. Disconnections are interfacial line defects with both dislocation and step character [17]. Where these occur in networks, their distortion fields can accommodate misfit and may also produce rotations. However, their step character simultaneously rotates the overall interfacial plane, or habit, away from the terrace plane. Thus, removal of the coherency strain field from such a rotated habit requires further consideration, and this is reviewed in Section 5. As outlined earlier, partitioning of an interface’s elastic distortion field between the adjacent crystals is pertinent in the present context: the partitioning of strains is well established [14], but partitioning of rotations has only been explicitly elucidated recently [18,19]. The current understanding is outlined in Section 6, where we show that B can itself be regarded as being partitioned. Strain and rotation components of a distortion field are most concisely described using tensor notation, recalled in Section 7. When expressed in an appropriate coordinate frame, these strains and rotations can be quantified in terms of the ratio jbj/L of a parallel array of dislocations with spacing, L. The energies of segmented interface defect arrays are considered in Section 8. Section 9 treats cases where the interface spreads over several planes. In Section 10, the F–B eq. is applied to various semi-coherent interfacial structures: simple cases are described first, before progressing to cases of greater crystallographic complexity. Section 11 provides some discussion and Section 12 is a summary of our main conclusions. Four appendices deal with some important details underlying the present approach.

2. The Frank–Bilby equation and reference spaces In this section we reprise the classical derivation of the F–B eq. In their original formulation, Bilby et al. [5,6] imagined the dislocations at an interface to be continuously distributed, referring to such configurations as ‘‘surface dislocations’’. In this view, there is no inhomogeneous distortion field near the interface, and the interface is an abrupt transition from one undistorted crystal to the other. On the other hand, experimental observations using TEM [2,20–24] show that the interfacial dislocation content is usually organised into arrays of discrete defects, and these exhibit a locally variable distortion field that penetrates into both adjacent crystals. Thus, to apply the F–B eq. to real interfaces, one must demonstrate the equivalence of the overall defect content in these two visualisations. Here, we develop parallel notations to emphasize this equivalence for the continuous and discrete models. For the characterisation of discrete defects, we employ the topological theory of interfacial defects [16], which has been reviewed comprehensively by Sutton and Balluffi [25], and is summarised in Appendix A; the matrix notation used is set out in Appendix B.

2.1. The Frank–Bilby equation The F–B eq. quantifies the total Burgers vector, B, intercepted by a probe vector v in an interface of interest [4–7]. Inspired by the procedure for characterising defects in single crystals, we can choose the reference space to be a single Bravais lattice. This reference can be distinct from either of the crystal lattices: then, Appendix A, the reference is created from the (white) P crystal lattice by the affine deformation, refAP, and from the (black) Q by refAQ. Similarly, the natural crystal lattices are created from the reference by PAref and QAref. The result of such deformations operating on the probe vector is given by

B ¼ ððP Aref Þ1  ðQ Aref Þ1 Þv sym ¼ ðref AP  ref AQ Þv sym :

ð1Þ

Christian [9] showed that Eq. (1) can be understood in terms of circuit mapping: an initially closed circuit in the bicrystal of interest that includes segments in both crystals maps to an open circuit with closure failure B in the reference lattice. The vectors vsym and vsym correspond to the irreducible

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

753

descriptions of the white and black circuit segments and become refAPvsym and refAQvsym after mapping into the reference. From this viewpoint, the distortion matrix, (refAP  refAQ), is a topological tool that transforms vsym into the corresponding closure failure in the reference lattice. As shown in Appendix A, the coordination with topology theory requires that the probe vector used in the F–B Eq. be opposite in sign to vsym. The sense of v also must be chosen consistently with the RH/FS convention [10], and the protocol adopted in the present work is set out in Appendix B. In all following equations we replace vsym by v. We emphasize that the closure failure, B, represents a continuous distribution of dislocations, even though it is quantified in a discrete lattice. The homogeneous distortion matrix QAP evidently is independent of the choice of reference lattice [26], whereas the components refAP and refAQ do so depend, with the consequences discussed at the end of this section. Alternatively, one can formulate the discrete defect content using the topological theory. However, we stress that the choice of reference space is then not arbitrary for meaningful descriptions of discrete defect content, and the relevant choices are introduced next. 2.2. Reference spaces According to the topological theory, the character of an elementary discrete interfacial defect is described by a Volterra operation, Vint, which can be expressed as a proper combination of symmetry operations, one from each of the crystals: expressed in generic form, Vint = WPWQ, Eq. (A2). Thus, reference spaces must represent all the symmetry operations of both crystals: hence the use of DPs which are the unions of the space groups of the white and black crystal lattices. Three types of DP are required for bicrystals where distortions involve both strains and rotations, as introduced next. 2.2.1. Natural dichromatic patterns Natural dichromatic patterns, or NDPs, are formed by the union of the space groups of the natural white and black crystal lattices. In general, the two unit cells would have different lattice parameters and orientations. A simple case, for illustration, is where two primitive-cubic crystals with monatomic nQ motifs have parallel unit cells but misfitting lattice parameters, anP 0 and a0 , Fig. 1a, where the superscript ‘‘n’’ implies the specific natural state discussed in Appendix B.1. An idealised bicrystal can be created from such a NDP: the interface plane is taken to be a notional dividing plane, and white atoms are located at white lattice sites above this plane, and similarly for black atoms below this plane. Fig. 1b shows the (0 0 1) idealised bicrystal formed in this manner. We note the absence of any inhomogeneous distortion field in idealised bicrystals: this corresponds to the visualization of Bilby et al. [5–7] and provides an important link between the continuous and discrete defect models. Generally, nQ for interphase interfaces the ratio of lattice parameters, anP 0 =a0 , is irrational, so there is no periodicity in the Vernier atomic arrangement at the bicrystal interface. Exceptionally, periodic NDPs occur, for example in ‘‘lattice-matched’’ epitaxial systems [27]. More complex NDPs can be envisaged, for example where the natural P and Q crystals have different Bravais lattices and orientations. In these cases, spacings along the interface are designated anP and anQ: in general, these spacings do not correspond to translation vectors of either lattice. The symmetry operations exhibited by the crystals are designated WnP and WnQ when expressed in their crystal frames, and can be re-expressed in the orthogonal coordinate frame x, y, z, as set out in Appendix B. 2.2.2. Commensurate dichromatic patterns The NDP described above is transformed into a commensurate dichromatic pattern, or CDP, Fig. 1c if the white and black lattices are homogeneously deformed by either a uniaxial or biaxial distortion parallel to the eventual terrace plane. Thus, the modified crystals in this case exhibit tetragonal symcQ c metry, and their lattice parameters parallel to the terrace plane are equal, acP 0 ¼ a0 ¼ a , where the superscript ‘‘c’’ implies the commensurate (coherent) state. However, the lattice parameters normal cQ to the terrace plane retain the original values, i.e. ccP ¼ anP ¼ anP 0 , and c 0 . Fig. 1d is the coherent bicrystal formed when the biaxial displacements applied to crystals P and Q in Fig. 1a are equal and opposite. To a very good approximation in this case [19], the magnitudes of the primitive translation  nQ vectors in the coherent terrace become the mean value ac ¼ ha0 i ¼ anP =2. This is an example of 0 þ a0

754

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 1. Schematic illustration of (a) natural dichromatic pattern, NDP, (b) ideal bicrystal formed from the NDP, (c) commensurate dichromatic pattern, CDP, (d) ideal coherent bicrystal formed from the CDP, and (e) the rotated-CDP, RCDP, and (f) ideal bicrystal formed from the RCDP. The terrace frame, x, y, z, and the habit frame, x0 , y0 , z0 , are indicated in (d) and (f) respectively: the terrace frames of crystals P and Q are shown rotated by ±u/2 in (f).

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

755

equipartitioning of strains. Again, we emphasize that there is no inhomogeneous distortion field in Fig. 1d. Translation symmetry operations in the two crystals are designated (I, tcP) and (I, tcQ), those parallel to the terrace plane being common to the two crystals, i.e. having the same components parallel to x and y. In the general case, in contrast to the case depicted in Fig. 1, these common translations may not be primitive. However, z components of the vectors tcP and tcQ retain the magnitudes of the corresponding vectors tnP and tnQ in the natural state. 2.2.3. Rotated-commensurate dichromatic patterns Rotated-commensurate dichromatic patterns, or RCDPs, Fig. 1e, are modifications of CDPs where a rotation has been introduced, changing the relative orientation of previously commensurate crystals. Fig. 1f illustrates the instance where the tilt has been introduced into the idealised coherent bicrystal depicted in Fig. 1d. Here, the total rotation, u, is partitioned equally between the two crystals with respect to the initial terrace plane. The bicrystal configuration illustrated is an idealised structure in which no atomic relaxations are considered. The rotation reorients all the translation vectors in the CDP, including previously commensurate terrace vectors tcP and tcQ: these are re-designated as, trcP and trcQ respectively in the RCDP, Fig. 1e. In Fig. 1f, u represents a tilt rotation, i.e. the rotation axis is parallel to the terrace. However, u may also be a twist rotation, i.e. where the rotation axis is perpendicular to the terrace, or a tilt/twist combination. Rotations arise as a consequence of the particular defects in a misfit-accommodating network: for example, there may be some residual Burgers vector content perpendicular to the final interface plane. Alternatively, some supplementary rotations may be present in order to reduce the total energy of the interface. 2.2.4. Symmetry classification of dichromatic patterns DPs with 3-D periodicity have symmetries belonging to the classical space groups: these DPs are known as coincidence-site lattices, CSLs, in the grain boundary literature [8]. DPs with 0, 1 or 2-D periodicity belong to point, rod and layer groups respectively [28]. Bicrystals are entities exhibiting a unique plane: those with 0, 1 or 2-D periodicity in that plane belong to rosette, band and layer groups respectively [28]. All possible bicrystal groups have been tabulated by Pond and Vlachavas [29]: as an example, the coherent bicrystal in Fig. 1d exhibits the layer space group p4mm, where ‘‘p’’ represents the primitive square array of translation operations, and the point symmetry operations of 4 mm are common to the two crystals and leave the interface orientation invariant. The point symmetries of DPs and bicrystals depend on the relative position of the components, but their translation symmetry is independent of this [29]. An additional notation is useful for specifying the translation symmetry of 2-D coherent bicrystals. A commensurate bicrystal is designated Cm/n when a primitive translation vector parallel to the terrace plane is formed by straining the natural vector mtnP to match ntnQ, where m and n are integers. The simple case illustrated in Fig. 1d is an example of a C1/1 case, since, parallel to y; tcP ¼ 0; acP 0 ;0   and tcQ ¼ 0; acQ ; 0 , so m = n = 1. Examples of cases where m – 1,n – 1, described as domain epitaxy, 0 are given in [30]. 2.3. Methodology for determining the defect content of semi-coherent interfaces In order to introduce the utility of DPs in combination with the F–B eq., we discuss the defect content of the semi-coherent (0 0 1) interface illustrated in Fig. 1, where no long range strains persist. In other words, the P and Q crystals exhibit their natural lattice parameters in regions remote from a real interface. Our methodology is as follows [18]. Step I: one uses the F–B eq. to determine the continuous defect content of the coherent bicrystal, Fig. 1d, using the natural bicrystal, Fig. 1b, as reference. We follow the traditional approach in interface physics [6,9,12,25] of starting with the deformation matrix cPn, which operates on a point xnP to transform it to a point xcP = cPnxnP. When the elastic distortion is uniform, cPn corresponds to the displacement matrix of points defining a unit volume of P, and transforms the natural to the commensurate lattice (see Appendix B). If no rotations are present, the distortions reduce to elastic strains and the deformation matrix has the form,

756

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

0 c Pn

B ¼@

1 þ ecP xx

0

0 0

1 þ ecP yy 0

0

1

C 0 A; 1

ð2Þ

c cP with ecP xx and eyy having equal magnitudes in this case. Formally, the coherency strain parameter e , extensively used in the literature, e.g. [1–12], is the difference between coherency strains in P and cP cQ cP cQ cP Q, e.g. , ec ¼ ecQ xx  exx . A similar expression can be written for cQn, with exx ¼ exx , and eyy ¼ eyy . The F–B eq. then has the form,

B c ¼ ð c Pn  c Q n Þ v ¼ c E n v ;

ð3Þ

where Bc is the content of infinitesimal ‘‘coherency’’ dislocations and cEn is the matrix with compocQ nents of the form ecP ij  eij . As a topological tool, cEn corresponds to the transformation which maps v from the natural bicrystal into the coherent dichromatic reference space, the CDP. However, mathematically, it is identical to the terrace coherency strain matrix defined below, i.e. the total strain to align P and Q planes adjoining at the interface. Below, we exploit this duality in the visualisation of the meaning of cEn. We regard Bc as being distributed continuously along the bicrystal interface, producing an elastic field that identically reproduces the coherent bicrystal, Fig. 1d, from the natural one, Fig. 1b. Thus, for a probe vector v,

Bc ¼

Z

v

ginf ðvÞdv;

ð4Þ

0

where ginf(v) is the Burgers vector of the coherency dislocations per unit length parallel to v. For example, if v is parallel to y, the integrand becomes ginf(y)dy. To first order, we have

0 c En

ec

B ¼@ 0

0

0 ec 0

0

1

C 0 A;

ð5Þ

0

 nP   nP  nQ cP where ec ¼ ecQ = a0 þ anQ . xx  exx ¼ 2 a0  a0 0 Step II: using the CDP, one identifies the character of discrete interfacial dislocations admissible in the coherent bicrystal, and uses the F–B eq. to determine the density of these defects necessary to compensate the coherency defect content, i.e. produce a total misfit strain parameter em = ec. AnalomP gous to the coherency case, em ¼ emQ xx  exx . The simplest case is when the misfit is fully accommodated by dislocations with, in this case, bg = bm parallel to the terrace: the superscript g indicates generic interface/grain boundary dislocation (Appendix A) and m indicates a misfit dislocation. For such an equilibrium array, the necessary discrete defect content, designated B, is simply

B ¼ Bc ;

ð6Þ

and, in this case, neither B nor Bc contain quantities related to rotation. Combining Eqs. (3) and (6), we have

B ¼ c En v ¼ n Ec v :

ð7Þ

In real bicrystals, misfit dislocations have localised cores, although, as developed in Section 3, to demonstrate the formal equivalence of continuous and discrete defect contents, it is pertinent to regard both B and Bc as being distributed continuously. In the array giving Eq. (3), the total defect content of such continuous defects, B + Bc, is zero [31], corresponding to the removal of all coherency strains at long-range. In actuality B comprises an array of discrete edge dislocations with bm defined in the CDP. For the case illustrated in Fig. 1, a square array of defects is necessary: one set has line direction parallel to x, and bm = tcP  tcQ = [0, by, 0], and a similar set parallel to y. Such dislocations could reach a nominally equilibrium interface by climbing through crystal P, say: as explained in Section 3, the bs of such dislocations are initially characterised by translation vectors of the nP lattice, bP = tnP, and change to bm = tcP on approaching the interface: Burgers vector is conserved in this process through the agency of a ‘‘spacing defect’’, discussed in Appendix A.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

757

Alternatively, the misfit could be accommodated by the glide of P crystal dislocations on planes inclined to the interface, designated ‘‘slant’’ dislocations. As for the case above, their bs change from NDP to CDP vectors on approaching the interface: these latter have components with magnitude equal to ac parallel to the terrace, and anP 0 perpendicular to this plane which produces a tilt by an angle, u. Step III: when a rotation is present, the RCDP is the appropriate pattern. The total infinitesimal defect content in the ideal coherent bicrystal (derived from the CDP) now includes both coherency disR v inf locations, Bc, and continuous infinitesimal tilt dislocations defined by Bt ¼ 0 f ðvÞdv ¼ c Xrc v . A specific example is shown in Fig. 8d. The formation of the rotated-coherent bicrystal (based on the RCDP) from the natural bicrystal (derived from the unrotated NDP specified in Appendix B) can be visualised in two stages. First, it entails the addition of coherency dislocations to create the coherent bicrystal, and then the subsequent removal of the tilt dislocations, which produces the rotation associated with the rotational function rcXc. This removal is equivalent to the addition of rotational dislocations that also can be imagined to be smeared into a continuous distribution Br = Bt = rcXcv. The rotational function rcXc, Appendix A, is equally partitioned between the two crystals, so we have rc Xc

¼



p rc Xc

  rc XQc :

ð8Þ

Hence the Burgers vector content corresponding to the formation of the rotated-coherent bicrystal from the natural bicrystal is

Binf ¼ Bc  Bt ¼ Bc þ Br ¼ ðrc En þ rc Xc Þv ¼ rc Dn v :

ð9Þ

Also, the total Burgers vector of the discrete misfit relieving network can be written as

B ¼ ðn Erc þ c Xrc Þv ¼ n Drc v :

ð10Þ

Here, nDrc is the elastic distortion matrix: separation of nDrc into nErc and cXrc is valid only for the standard model of Appendix C. Eq. (6) is still valid for this case, but both B and Bc contain rotational contributions. There are also the subsidiary conditions nErc = rcEn and cXrc = rcXc. Importantly, these relations also reveal that n Drc

¼ rc Dn ;

n Dc

¼ c Dn :

ð11Þ

Eq. (11) expresses the duality mentioned above: rcDn is regarded as the topological mapping matrix, while nDrc is expressed by the long-range elastic strains and rotations produced by sets of discrete dislocations. Of course, as emphasized above, nDrc can be represented by hypothetical, continuous, infinitesimal misfit dislocations. As discussed in Section 6, Eqs. (6), (11) and the above subsidiary relations are thermodynamic results, true generally, including inhomogeneous and nonlinear elastic cases, Thus, B removes the long range distortions, as in the misfit case. When the magnitude of u is sufficiently small, one can use the linear-elastic approximation and superpose the tilt boundary on the misfit-relieved configuration obtained in step II. With this approximation, the relevant bs of the slant dislocations are still defined in the CDP. However, for larger values of u, one must use a non-linear geometric treatment, in accord with the standard model, involving bs defined in the RCDP [15]. In the RCDP the white translation vectors become trcP ¼ rc XPc tcP , and similarly for the black ones. The rigorous formulation of Eqs. (9) and (10) is set out in Eqs. (A10) and (A11). Since the final interface orientation depends on the partitioning of the rotation between the two crystals, we see that the selection of a probe vector, v, in a hypothetical interface is not arbitrary, see Appendix B. For example, in steps I and II above, the interface remains parallel to the (0 0 1)P/Q terrace, so v can be chosen in this plane. However, if the misfit is accommodated by an array of slant dislocations and the resulting rotation is equally partitioned, the final interface plane, designated x0 , y0 , z0 , rotates away from (0 0 1)P/Q by (u/2), v must lie in this plane, and both B and Bc are defined in this plane. The x0 , y0 , z0 coordinates are also applicable for asymmetric grain boundaries and interphase interfaces. This topic is discussed further in Sections 4 and 5. Eq. (10) and its reduced form for the pure misfit case, equation (7), encapsulate our implementation of the F–B method. The distortion rcDn, determined from the homogeneous deformation and distortion components c Pn ; c Q n ; rc XPc and rc XQc , gives the continuous defect content intersected by v, and enables

758

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

the discrete defect content, B, to be determined for however many sets i of discrete dislocations as are present in a particular interface. Eq. (6) becomes

Ri Bi ¼ Bc ;

ð12Þ

and the subsidiary Eq. (11) becomes

Ri ðn Drc Þi ¼ rc Dn :

ð13Þ

The spacings and line directions of the dislocations in each set are adjusted so Eq. (7) or (10) is satisfied. Our methodology is summarized in the flow diagram, Fig. 2, which is an elaboration of steps I, II and III above for the case where misfit and rotation are included. Each box in Fig. 2 represents a bicrystal, and contains a brief description of the state of rotation, u, coherency strain, ec, and misfit dislocation strain, em. The circles represent the action of adding dislocations or smearing discrete defects into infinitesimal ones. The bottom box represents the ideal natural bicrystal, such as depicted in Fig. 1b: there are no inhomogeneous distortions present, and this state also corresponds to the Bilby bicrystal. The introduction of infinitesimal rotational dislocations, finf, creates the unrotated natural bicrystal where the rotation present initially, u, has been removed. Next, the misfit, em, is removed, i.e. coherency strain ec is added, by the introduction of infinitesimal coherency dislocations, ginf, thereby creating an ideal coherent bicrystal like that depicted in Fig. 1d. Then, discrete dislocations with Burgers vectors bg are introduced in accord with Eq. (10). These relieve the coherency strains and restore the rotation: we show these steps separately in Fig. 2 by the introduction of the misfit and rotational components, bm and br, respectively. The real bicrystal is thus created, exhibiting an inhomogeneous distortion field near the interface, but no long-range strain, and the rotation u. In the final step, the discrete defect content is smeared into infinitesimal defects, so the natural bicrystal is regained, where the total dislocation content (finf + br) + (ginf + bm) is zero. The ideal bicrystals in Fig. 2 are obtained from DPs. We summarize the relation of these DPs to the steps I, II and III in Section 2.3, and emphasize the duality in the interpretation of the distortions nDc and nDrc as follows. The quantity [(tcP  tnP)  (tcQ  tnQ] gives the difference in displacements nuc, and

Fig. 2. Flow chart of the F–B methodology. Boxes represent bicrystals, and circles represent the action of introducing infinitesimal rotational dislocations, finf, infinitesimal coherency dislocations, ginf, discrete misfit dislocations, bm, and discrete rotational dislocations, br, or the spreading of the latter two discrete types to become infinitesimal.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

759

similarly for nurc, see Appendix D. The gradients rnuc and rnurc produce the elastic distortions nDc and nDrc, and their inverses, that comprise the strains and rotations in Eqs. (2) and (8) as well as in the description in Appendix D. The distortions can be represented as the fields of continuous infinitesimal dislocations. The interface in the coherent bicrystal separates the adjoining terrace planes, with a similar separation in the real bicrystal for either the misfit or the misfit plus rotation cases. Discrete dislocations are added to the ideal coherent or rotated bicrystals to eliminate long range strains and to produce long-range rotations. The Burgers vectors of these dislocations are vectors of either the CDP or the RCDP. The condition for the elimination of long-range strains is the F–B eq., expressed in terms of the dislocation content of real and infinitesimal dislocations intersected by a probe vector, Eq. (10), or as a balance of distortion fields, Eq. (11). Because vectors in the analysis are those of both the NDP and the CDP (or RCDP), superscripts n, c, and rc on vectors and subscripts on distortions and distortion components are added to avoid confusion. In applications, Section 10, the relevant DPs or ideal bicrystals are evident from the context and the sub-superscripts are dropped. Throughout this work, we treat equilibrium interfaces where the displacements and distortions presented above are elastic. The same equations can be used to describe the plastic distortions (transformation distortions in the phase transformation case), that are accomplished by the motion of interface defects. These plastic distortions are treated, using the same approach as in the present work, in [3]. Eq. (13) provides an alternative procedure to Eq. (12) for determining the i sets of defects necessary to remove long-range strain which is more amenable mathematically than use of probe vectors, Appendix D. In interface coordinates x0 , y0 , z0 , with z0 normal to the interface, coordinates x0, y0, z0 are fixed on the interface dislocations, with z0 normal to the interface and x0 parallel to n. As discussed in Section 6, the screw components in an array only produce long-range strains e0xy and rotational 0 0 0 functions x0xy ¼ x0yx ¼ bx =4L0 : edge components by have only long-range strains e0yy ¼ by =2L0 : edge 0 0 0 0 0 components bz have only long-range rotational functions xyz ¼ xyz ¼ bz =4L . Hence the distortions (nDrc)i are known in x0i coordinates and can be expressed in x0i coordinates by the coordinate transformation in equation (B7),

ðn D0rc Þ ¼ x0 Ax0 ðn D0rc Þðx 0 Ax0 Þ1 :

ð14Þ

Eq. (10) then gives the solution. 2.4. Reference lattices used by other researchers DPs are used as reference spaces in this work for the reasons set out above. They incorporate aspects of reference spaces used previously by other workers, but differ in important respects. To emphasize these differences, we briefly contrast our current procedure with previous usage of monochromatic reference states. An early example is Frank’s suggestion of a ‘‘median’’ lattice for grain boundaries, symmetrically disposed between the white and black lattices [32]. Presumably, the underlying notion here is that the b of grain boundary defects are intermediate between those of the dislocations in the adjacent crystals. Such a notion would apply for pure symmetric tilt, pure symmetric twist or pure misfit boundaries, with equally partitioned strains and rotations. However, it would not apply in general: for example when steps are present or for asymmetrical boundaries. A widely used choice of reference lattice is one of the crystal lattices [9]. This choice leads to simple forms of Eq. (1):

B ¼ ðI  ðQ AP Þ1 Þv sym ;

ð15Þ

for the white lattice, or

B ¼ ððP AQ Þ1  IÞv sym ;

ð16Þ 1

for the black, where (PAQ) = (refAP) (refAQ): these also could be expressed in terms of elastic distortions as in Eq. (10). However, as pointed out by Knowles and Smith [33], for example, application of Eq. (15) to a given interface structure gives a different result compared with that determined from Eq. (16) and both differ from the accurate equation (9), even for the isotropic, homogeneous case, the differences

760

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

becoming more marked when anisotropic elasticity or rotations are present. Hence, Eqs. (15) and (16) are not used in the present work. We do note that extreme partitioning could lead to the use of these equations. Eq. (16) would apply when P is a thin film on a thick Q substrate, and (15) would apply if P were infinitely hard. Christian [9] showed that the formalism of Eqs. (15) and (16) is equivalent to Bollmann’s ‘‘O’’ construction [8], which defines the moiré relationship of two superposed unstrained lattices. For latticematched systems, the 2-D periodic ‘‘O’’ lattice is equivalent to a DP with 2-D translation symmetry (for such cases the NDP and CDP constructions are equivalent), and both approaches would give the same results in the isotropic, homogeneous case, but not for the more general cases treated here. CSLs are an important concept in the literature on grain boundary structures, and Bollmann’s DSC ‘‘lattice’’ [8] is closely related to the special case of DPs with 3-D translation symmetry (as above, the NDP and CDP constructions are equivalent for naturally commensurate lattices). In general, 3-D CSLs do not arise for interphase interfaces, but some authors [34] propose that the adjacent crystals be strained homogenously, thereby creating a constrained-CSL. However, for semi-coherent interfaces, our view is that strains are only necessary to ensure coherency at terraces. Periodicity normal to the terrace is not required, and hence CDPs in the present work exhibit natural lattice parameters in that direction. Of course, homogeneous strains parallel to the terrace would induce Poisson strains in the normal direction. When the misfit is fully accommodated, these strains are automatically compensated [18]. 3. Misfit dislocations The concept of misfit dislocations relieving coherency strain [35] is well known in dislocation theory. Dislocations within the body of a single crystal produce residual elastic strains that are small and oscillating in sign [36]. Then, instead of a description by a Burgers circuit in the strained crystal by the SF/RH convention, giving a local Burgers vector, a more useful approach is by the FS/RH convention giving the true Burgers vector in the reference lattice [37]. A coherent interface separates crystals containing elastic coherency strains that are of the same magnitude but opposite signs and which can be large. In Section 2 and Appendix A we have shown how the Burgers vectors of defects in such interfaces can be characterized by mapping into a DP. Here, we consider the simple case where an array of dislocations with b parallel to the interface, Fig. A3, accommodates misfit. 3.1. Misfit arrays We consider the same bicrystal as depicted in Fig. 1, i.e. where the two crystals have parallel but misfitting unit cells. Following a micromechanical argument, Leibisch [38] and Niggli [39], one can imagine beginning with two separate crystals, Fig. 1b and Fig. 3a, and bonding them to form the ideal coherent bicrystal in Fig. 1c and Fig. 3b: forces would be needed on the surfaces normal to y to maintain the uniform coherent state. For simplicity, some of the discrete atomic planes in Fig. 1 are represented as lines in Fig. 3. The same coherency strains arise for continuous infinitesimal dislocations g(y)dy uniformly distributed on the interface of the ideal bicrystal in Fig. 3b. The Burgers vector content, Bc, is defined by Eq. (3): per unit length this is equal to the total misfit strain factor, em, for any probe vector, v. The real bicrystal is depicted in Fig. 3c, where misfit dislocations have been added to remove longrange elastic strains. The Burgers vectors of these defects, defined in the CDP (see Appendix A.4), have the magnitude jbmj = ac. The same long-range strains persist if the misfit dislocations are also smeared into infinitesimal, continuous dislocations, distributed on the interface, creating the ideal ‘‘Bilby bicrystal’’ [5,6] of Fig. 3d, where all elastic strains are collapsed into the interface, as envisioned in the original development of the F–B eq. [4–6]. As discussed in Section 2, all long-range strains are removed when B = Bc for any probe vector. The spacing of defects in a regular misfit array is L. Local interactions generally require that the spacing L be an integer number m of repeat distances in the boundary [40] with a Peierls-type energy barrier separating them, and the analysis in terms of dichromatic spaces is simplified if L has this form. Hence if the mean spacing is 4.1 natural repeat distances,

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

761

Fig. 3. (a) Schematic illustration of the separate nP and nQ crystals. (b) Surface tractions applied to create the cP and cQ crystals, followed by bonding to create the ideal commensurate bicrystal. The coherency strain is represented by a continuous array of infinitesimal dislocations ginf(y)dy with Burgers vector content Bc equal to ec = em per unit length. (c) Length L of misfitrelieved bicrystal. (d) ‘‘Bilby bicrystal’’ representation, with misfit dislocations also smeared into a continuous array with density B = Bc.

the actual spacings could be nine successive spacings of 4 and one of 5: this is likely the lowest energy configuration but other arrangements are possible. In turn this array can be represented as a boundary with L = 4 superposed on a boundary with L = 37, i.e., a boundary with mostly L = 4 but with L = 5 every 37 units. A general mean spacing can be represented as an array of superposed spacing changes with ever increasing spacing. Therefore, a convenient analysis is to treat the regular boundary with L = 4 and to correct it with superposed arrays. At high temperatures, where the Peierls barrier is smeared out, one can have any L value, rational or irrational. To emphasize this point, bm, the Burgers vector of a misfit dislocation, is fixed once the CDP [41] is established: this is a great advantage of the DP approach. Then different arrangements of L are possible, provided that the average value hLi is constant. Hence an array can be uniform with L = 6 say, or it could have alternating L = 5 and L = 7. The longrange field would be the same. The difference would be that the distance zcrit, where the strains effectively vanish, would be larger in the second case. Similarly one could have L = 10 for 12 successive spacings, followed by one interval with L = 11, the pattern repeating. Or one could have all dislocations spaced at L = (131/13) if the Peierls barrier were unimportant. The two situations would give the same far field result, but again zcrit would differ in the two cases. Consider the introduction of a single misfit dislocation in an otherwise coherent bicrystal, Fig. 4a. Here there are no surface tractions and the strain fields gradually decrease with jzj and diminish to zero in regions remote from the interface. A misfit dislocation could be formed by an edge dislocation climbing in from crystal Q (lower) or P (upper). Mapping a circuit around such a defect into its parent crystal’s lattice would give different Burgers vectors for the two cases, i.e. either bg = tnP or bg = tnQ. However, this is nonsense since the interface dislocation is the same in the two instances. Instead, mapping into the CDP in the isotropic elastic case and for a bicrystal with equal thicknesses, gives bg = tcP, or, equivalently, bg = tcQ: these have the same magnitude jbgj = ac. When the defect is remote from the interface, circuits mapped to the nP or nQ lattice in the NDP give the true Burgers vectors bP or bQ. As a dislocation climbs in from the upper or lower crystal in Fig. 4a, its local Burgers vector expands or contracts but mapping into the nP or nQ lattice in the NDP still gives bP or bQ. At the interface, the circuit is mapped into the CDP and the Burgers vector is bg = bm. If the dislocation is created by adding a (0 1 0)P plane to P and punching it in at the surface, as in Fig. 4b, the initial unstrained parameter anP is expanded to ac to maintain a coherent terrace locally, tantamount to adding a spacing defect, and the punched-in half plane becomes the misfit dislocation bm. Also, if a representative area element is sheared by a gliding misfit dislocation, Fig. 4c, the shear offset corresponds to the Burgers vector bm.

762

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 4. (a) Coherent bicrystal without surface tractions: unit cell dimensions approach their natural values for large jzj. (b) (0 1 0)P plane punched in at the surface to create a misfit dislocation bm. (c) Offset of a fiducial marker produced by glide of a misfit dislocation.

3.2. Strain partitioning As is well known for the misfit case [35], the coherency strain ec is partitioned between the crystals, as is the strain em associated with arrays of misfit dislocations. These are total strains relating P and Q: as we now show they are partitioned between the two crystals. Partitioning is unequal in a true strain description [15]. However, to an excellent approximation, the total terrace plane strain can be represented by the coherency strain parameter ec = 2(anP  anQ)/(anP + anQ), or ec = Da/hai. In the isotropic elastic limit for the present case, the remote strain fields are given by ePxx ¼ ePyy ¼ ec =2 and eQxx ¼ eQyy ¼ ec =2. When the equilibrium array of misfit dislocations is superposed, equal and opposite strains are superposed in the remote region jzj  L, where the net strain becomes zero. One can readily verify that this equality is only true with bm as the Burgers vector in accord with the standard model described in Appendix C. One consequence of this theory is that the appropriate equilibrium dislocation spacing L is expressed in embedded coordinates. If a misfit dislocation is inserted into a length of coherent interface, L0 = nac, with n an integer, the added half-plane of the dislocation causes the length to increase by ac/2 to L = (n + 1/2)ac; adding two dislocations of opposite sign, one climbing in from P and one from Q, increases the length by ac, so each contributes ac/2. If these two dislocations coincide they add a length ac to the terrace, thereby comprising a spacing defect. In the case of Fig. 3c, for the equilibrium array that completely removes misfit, bm/L = em = ec. One can readily verify that this equality is only true with bm as the Burgers vector. 3.3. Strain field attenuation The convergence of the strain fields to zero for z > zcrit, following St. Venant’s principle, is illustrated in Fig. 5, which represents the strain fields of an equilibrated interface in an atomistic simulation of a {1 1 1}–{1 1 1} interface between Cu and Ag. The details of the simulation are given in [13]. In addition to an array of misfit dislocations, some disconnections are also present: these latter also contribute a small amount to misfit relief and a small tilt rotation. [11,15,18]. The system was constrained to relieve misfit only in the y direction. The colored regions at the interface represent the strain energy density associated with misfit dislocations and disconnections with a misfit-relieving component together with coherency strains near the interface. The Burgers vector of the disconnection is smaller than that of the misfit dislocations and it has a correspondingly smaller strain field.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

763

Fig. 5. Relaxation of coherency strains parallel to y in an interface, between Cu and Ag, with {1 1 1}–{1 1 1} (xy) terraces [13].  0 and z//[1 1 1]. The elastic shear strain eyz is represented by color, zero strain for the background, blue for negative Here y==½1 1 shear strain dominated by dislocations and red for positive shear strains dominated by coherency. The Burgers vectors of the misfit dislocations are larger than those of the disconnections, and hence their energy is correspondingly larger.

The physical considerations in this Section show unequivocally that the Burgers vectors of misfit dislocations are translation vectors of the CDP. Hence, the Burgers vectors are defined directly by circuit operations or by symmetry theory as described in detail in Appendix A. 4. Defect arrays producing rotations The various representations of misfit dislocations, including their representation in terms of discrete defects, smearing into infinitesimal arrays, and the use of DPs to define Burgers vectors are discussed in Section 3. Pure tilt and twist boundaries have also been much discussed in the wider literature, e.g. [25]. However, since little attention has been directed to tilts or twists superposed on misfit arrays, they are considered in this section. Although these are more difficult to envision than the pure misfit counterparts, their representation in terms of DPs and ideal bicrystals is completely analogous to the pure misfit case. For simplicity, we discuss symmetrical tilt and twist interfaces in homophase materials initially, and subsequently consider asymmetric and heterophase interfaces. 4.1. Tilt walls Two symbolic representations of a tilt wall as arrays of edge dislocations are shown in Fig. 6a and b. In (a), the boundary is regarded as an alternating sequence of P and Q crystal dislocations, bP and bQ, while (b) is an array of grain boundary dislocations [42–44]. In the latter, the y0 components of the crystal dislocation Burgers vectors are regarded as having cancelled, so only the z0 components remain. The Burgers vectors of the defects in Fig. 6b have half the magnitude and half the spacing of bg depicted in Fig. A2a. These two models are consistent with both the F–B eq. and the simpler formulation introduced by Frank [4,10] (see Fig. 19-13 in [42]), and are in accord with the standard model in Appendix C. According to Frank, the mean separation, L, between dislocations in the boundary is given by P

L ¼ b =½2 sinðu=2Þ:

ð17Þ P

Q

Here, u is the tilt angle, and b or, equivalently, b , is the magnitude of a Burgers vector of a perfect dislocation in either crystal nP or nQ.

764

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 6. Schematic representation of a symmetrical tilt boundary as (a) an array of crystal dislocations, (b) an array of grain boundary dislocations. The NDP/RCDP for two hexagonal lattices rotated about [0 0 0 1] case is depicted in (c), and an ideal bicrystal in (d). (e) is an atomic scale simulation of a [0 0 0 1]/13.2° bicrystal in hcp metal [45].

The DP for such cases can be constructed as follows. Since we are considering a homophase case, we take the white and black spacegroups to be coincident initially: we refer to this reference space, where P and Q are naturally coherent, as a grey-CDP. Next, the white and black spacegroups are rotated by ±u/2 respectively, forming the RCDP, as illustrated in Fig. 6c for hexagonal lattices. For homophase cases, the RCDP is the same construction as the NDP. In ideal bicrystals created from such NDPs/ RCDPs, Fig. 6d, corresponding planes (but not necessarily lattice sites) in crystals P and Q meet regularly along the boundary. Such a correspondence is seen in Fig. 6d, and the interplanar spacing acP = acQ projected onto the boundary is designated, arcP = arcQ. The interface in Fig. 6c is periodic, but the same matching occurs for aperiodic boundaries. In general, interfaces are only periodic for particular values of u corresponding to a 1, 2 or 3D CSL. As discussed for misfit dislocations in Section 3, a non-periodic array can consist of lengths of periodic arrays separated by a single period of differing length. The translation vectors tnP and tnQ (or, equivalently, trcP and trcQ) can be identified with bP (and bQ) in Eq. (17). Now, according to equation (B5), (or (B7), admissible interfacial dislocations have Burgers vectors equal to tnP  tnQ, (or trcP  trcQ), which is perpendicular to the interface with magnitude 2bP cos(u/2), as illustrated in Fig. A2a. Such dislocations can dissociate into pairs with Burgers vectors perpendicular to the interface and equal magnitudes, br = bP cos(u/2) [42], where the superscript r indicates rotation with tilt character in this case: these are the defects illustrated in Fig. 6b. Alternatively, each of the dislocations bP and bQ in Fig. 6a can dissociate into a tilt component, br, and a misfit component, bm. Strictly, these are partial interfacial defects in the sense that they are not contained in the set of perfect defects, Vint, equation (A2). Furthermore, the misfit components of bP and bQ have opposite signs, and can therefore mutually annihilate, again creating the configuration in Fig. 6b. Such annihilation can be imagined to occur in different ways (dependent on the relative motion of the two misfit components and the localization of their cores), each leading to a different short range distortion

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

765

field. However, the far-field is independent of these details, so the tilt component, br, assumes primary significance in this context. Hence, one can re-express Eq. (17) in terms of this parameter, i.e. r

L ¼ b = sinðuÞ;

ð18Þ

or, to linear order r

L ¼ b =u:

ð19Þ

In real tilt boundaries, equilibrium is achieved when the rotational distortion field is symmetrically partitioned with respect to the final interface plane, x0 , y0 , z0 [13]. Such a structure, obtained by atomicscale computer simulation is shown in Fig. 6e: here, two hcp crystals are tilted by u = 13.2 about [0 0 0 1]P/Q [45]. With respect to the x0 , y0 , z0 coordinate frame, the symmetrical form of the rotational  1 0 0Þ distortion field and the curvature of the near vertical ð1 P=Q planes are evident: this would also be the case for the model structures in Fig. 6a and b. The cores of the interfacial dislocations are indicated  0Þ planes, and, although these resemble those of 1=3½1 1 2  0 in Fig. 6e by the pairs of terminating ð2 2 4 dislocations in crystals P and Q, they are better represented as grain boundary dislocations. In other words, the model shown in Fig. 6b is more appropriate than 6a. A perfect interfacial dislocation with Burgers vector 2br could arise, for example, if a crystal disloP  0 glides along ð1  1 0 0Þ to the interface, cuts the boundary, and appears as a cation with b ¼ 1=3½1 1 2 P P Q 1  2 0 . If the emerging dislocation were bQ ¼ 1=3½1 1 2  0 instead, the crystal dislocation, b ¼ 1=3½1 Q Q defect left behind would be a perfect disconnection with Burgers vector bg, similar to the disconnection illustrated in Fig. A1 but with bg parallel to the interface. 4.2. Tilt–misfit analogues The pure-misfit case, Section 3, and the present pure-tilt case are analogous in various respects. For example, an analysis in terms of DPs is easiest if L is an integer number of planes matching at the interface in the real bicrystal. Hence if the mean defect spacing is, say, 8.2 spacings, the defects could be analyzed for a crystal with 8 spacings and a more widely spaced defect set would be introduced, etc. as in the misfit case, to give a mean spacing of 8.2. The absence of long-range strains for the misfit and tilt cases can also be described by analogous dislocation models. In the misfit case, relative to the natural bicrystal, the total dislocation content of the accommodated interface is zero, corresponding to the superposition of the coherency content, Bc, and the misfit defect content, B, where B = Bc as illustrated in Fig. 3. Similarly, for the tilt case, the total defect content with respect to the natural bicrystal can be shown to be zero, as follows. Here, one begins with a ‘‘coherent bicrystal’’ in which the white and black crystals have the same orientation (i.e. a single crystal derived from the grey CDP). This bicrystal is notionally obtained by applying surface torques to the free surfaces of a tilted bicrystal derived from the NDP (or, equivalently in this case, the RCDP), as illustrated in Fig. 7a. Alternatively, Fig. 7b, the ‘‘coherent bicrystal’’ can be created from the tilted bicrystal by the addition of continuous infinitesimal dislocations finf(v), whose integral over R v inf v is the rotational dislocation content, Bt ¼ Binf ¼ 0 f ðvÞdv. Next, one finds the discrete defect conr tent, B needed to rotate the crystals back to the real boundary, Fig. 7c, or the equivalent Bilby bicrystal, Fig. 7d. In the real bicrystal, the discrete grain boundary dislocations, br, have net Burgers vectors, Br = Bt, and are superposed on the continuous array of Fig. 7b. For the ideal Bilby bicrystal, the discrete dislocations, br, can also be represented as continuous infinitesimal tilt dislocations whose inteRv r gral over v is Br ¼ 0 f ðvÞdv, Fig. 7d. We emphasize that the discrete dislocation content Br is the physically significant quantity determined by the F–B eq., representing the defect content of the tilted bicrystal with respect to the coherent one. 4.3. Tilting at interphase interfaces If the tilt wall is at an interphase interface also containing misfit dislocations, the sequence of types of bicrystals is illustrated in Fig. 8. The analysis follows the scheme set out in Fig. 2, although Fig. 8a and b are combined as one box in Fig. 2. Fig. 8a is the natural bicrystal which is equivalent to the ideal

766

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 7. (a) Perfect crystal. Arced arrows indicate that this crystal is rotated relative to the real tilted grain boundary: (b) Continuous dislocation content of crystal (a) with respect to the tilted bicrystal: (c) ideal bicrystal corresponding to RCDP: (d) tilted Bilby bicrystal.

Bilby bicrystal, Fig. 8b: it differs from the simple tilt wall case, Fig. 7d, because now acP – acQ. Here, the total defect content is zero, as represented by the superposition of four sets of continuous dislocations in Fig. 8b: the rotational dislocations, finf, the coherency dislocations, ginf, and, after discretizing, the discrete rotational dislocations, br and the misfit dislocations bm. On adding finf, the unrotated natural bicrystal of Fig. 8c is obtained, and, by subsequently adding ginf, one obtains the ideal coherent bicrystal, Fig. 8d. Vertical planes in the coherent bicrystal meet at the interface with the commensurate spacing ac. In Fig. 8e, the discrete rotational defects are added, thereby restoring the tilt. Their Burgers vectors are obtained from the RCDP which, as for the grain boundary example, contains the admissible Burgers vectors bg = 2br as well as bP and bQ. Coherency strain is then imagined to be accommodated by introducing the discrete misfit dislocations: their Burgers vector, bm, is obtained from the CDP, and its resolved interface component is defined in the RCDP. In the real crystal, these last two actions are accomplished simultaneously by the insertion of a single set of discrete dislocations with Burgers vector bg comprising the components br and bm. Although not represented explicitly in Fig. 8, the discrete defect array produces an inhomogeneous short range displacement field. In order to complete the circuit, one removes the short range distortions in Fig. 8f by spreading the discrete dislocation content, thereby regaining the natural/Bilby bicrystal, Fig. 8b. We append Fig. 8g to emphasize the consequences of partitioning. Let the interface in the coherent bicrystal contain a marker vector, v0. In Eulerian coordinates, after superposing the tilt, the far-field (where strains vanish) vectors v P0 and v Q0 in the true bicrystal are rotated by ±u/2, but local strains remain near the interface. These local strains vanish in the ideal Bilby bicrystal, the ‘‘far field’’ extends to the interface, and hence v P0 and v Q0 can be transposed there: these vectors are rotated by ±u/2 relative to a new vector, v, in the tilted and misfit relieved real bicrystal. Because the rotations are symmetric, the final interface contains v, parallel to y0 . This partitioning is evident also for the pure tilt wall as in Fig. 7c and d. If a crystal dislocation, bP (or bQ), is added to a tilt array, it exhibits a tilt component, br, together with a misfit component bm. If these components appear as separate and localized decomposition products, the former would be a partial tilt defect, br, and the latter a partial misfit defect, bm. Similarly, if a dislocation is removed from a tilt array it leaves a spacing defect with analogous, opposite sign, partials.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

767

Fig. 8. Two-phase bicrystals with both misfit and tilt present. Bicrystals (a) natural/Bilby obtained from the NDP; (b) the defect content of the natural Bilby bicrystal; (c) the unrotated natural bicrystal; (d) the coherent bicrystal obtained from the CDP; (e) the rotated coherently strained bicrystal obtained from the RCDP; and (f) the real bicrystal. (g) Marker vectors in the coherent, 0 0 real and Bilby bicrystals, reflecting the symmetrical tilting with respect to the final interface frame, x , y , z0 .

4.4. Twist walls Twist boundaries are treated in a manner analogous to tilt cases. One begins by defining the relevant DPs: for homophase cases, the starting point is the grey-CDP, and one then rotates the P and Q space groups by ±g/2 respectively, with g replacing u. As before, a RCDP so formed is the same

768

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 9. Burgers vectors of perfect and partial dislocations in a twist boundary. (a) Pure twist wall. (b) Projection of far-field crystals.

construction as the NDP, and Fig. 6c is an example for two hexagonal lattices rotated about [0 0 0 1]P/Q. An ideal bicrystal can subsequently be created from this pattern by selecting the dividing plane and then locating atomic motifs at P and Q lattice sites on either side of this plane. Thus, an (0 0 0 1) twist bicrystal could be obtained from Fig. 6c if the dividing plane is parallel to (0 0 0 1). If the real bicrystal has a semicoherent interface structure, it comprises coherent segments where the atomic arrangement is characteristic of the CDP, separated by a network of interfacial screw dislocations. In the far-field, however, the atomic arrangements are characteristic of the RCDP/NDP. The Burgers vectors of admissible interfacial defects belong to the set Vint, Eq. (A2). Fig. 9b shows cross grids of perfect screw dislocations bP and bQ in sets III and IV. The corresponding perfect interfacial dislocations in the RCDP have vectors with magnitude 2bP cos(g/2). However, analogously to the tilt case, these decompose into cross grids I and II of partials br with Burgers vectors having half this magnitude, i.e. br = bP cos(g/2), Fig. 9a. Here the superscript implies rotation with twist character. The projection of the vectors bP and bQ in Fig. 9 b onto the interface reflects the NDP, with vectors in sets III and IV, and, dashed lines, the CDP, with vectors in sets I and II. The twist angle, g, is related to the defect spacing by P

r

L ¼ b =½2 sinðg=2Þ ¼ b = sinðgÞ:

ð20Þ

Each set of screws in Fig. 9 produces a rotation g/2 partitioned to g/4 in each crystal, as shown for one set in the insert. Both sets together produce the rotation g. For (0 0 0 1) twist boundaries, the interfacial array comprises three sets of br dislocations. Each set produces a rotation xxy = xyx = g/3, and each of these is partitioned equally in magnitude between crystals P and Q. In the interphase interface case the Burgers vectors, bg, would be given by equation (A2), or, graphically, by the CDP or RCDP methods analogous to those depicted in Figs. 6 and 7. 4.5. Asymmetric tilt interfaces A more complex, asymmetrical low-angle tilt wall separating simple cubic crystals is shown in Fig. 10. The white and black space groups are superimposed to form the grey-CDP, and the interface plane chosen is inclined to the notional terrace plane by an angle h: this anticipates the analogous situation for disconnections, Section 5, except that there is no coherency, ec, strain present here. Two sets of crystal dislocations with orthogonal Burgers vectors bP and bQ are present. In the final bicrystal, the terrace planes are inclined to the grain boundary by h  u/2 for crystal Q and h + u/2 for P. As shown next, long-range strains are removed when the (partial) misfit components of bP and bQ intersected by a probe vector v along y0 sum to zero [46], i.e. Rbm(P) = Rbm(Q), or

RbP sinðh þ u=2Þ ¼ RbQ cosðh  u=2Þ:

ð21Þ r(P)

P

r(Q)

Q

The tilt components of the two dislocations are b = b cos(h + u/2) and b = b sin(h  u/2) respectively, corresponding to tilt angles uP = 2 sin1(br(P)/2LP) and uQ = 2 sin1(br(Q)/2LQ). The requirement

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

769

Fig. 10. Asymmetrical low-angle tilt boundary comprising two sets of edge dislocations: the interface plane is inclined to the terrace plane, and the rotation is equipartitioned. Angles refer to Q. Inserts show interface Burgers vectors in x0i coordinates.

for equal partitioning of rotation, shown but not discussed in the classical presentation [46], means that uQ = uP = u/2, where u is the total tilt rotation, i.e.

u ¼ xyz ¼ uQ þ uP ¼ 2 sin1 ðbrðPÞ =2LP Þ þ 2 sin1 ðbrðQÞ =2LQ Þ:

ð22Þ

With respect to Fig. 8b, the initial vector v0 is inclined to the cube plane by h, and after superposition of the tilt wall, the imbedded vector rotates to h ± u/2. The habit plane is parallel to y0 , and contains the vector v.The imbedded crystallographic cube planes are inclined to the habit plane by ±u/2. All of the foregoing examples are somewhat special in that simple arrays are considered. As discussed in detail in [47], the arrays in Fig. 6a, for example, could be produced by glide of equal numbers of crystal dislocations with Burgers vectors bP and bQ into the interface. In general these glide systems need not be symmetric, and they may not intersect the boundary along parallel lines [48]. The purpose of the F–B eq., is to take all this into account. It is then the overall discrete Burgers vector content, B, that both satisfies removal of coherency strains, and that is the content of the Bilby bicrystal with infinitesimal continuous dislocations as in Figs. 7d and 8a. Similarly, for Fig. 10, B relates to average Burgers vectors separated by average distances as indicated. More generally, other crystal systems may entail perfect crystal dislocations inclined to the terrace plane by arbitrary angles. Yet the components in the x0i coordinates still determine the misfit compensation and tilt. That is, the component, B0z for example, would partition and contribute to the overall tilt rotation, possibly causing interface rotation as in Fig. 10. Similarly, net B0y components would contribute to the relief of coherency stresses. For both cases, distortions associated with equal and opposite components, such as the misfit components in Fig. 10, would cancel, yielding no contribution to B but contributing to local nonlinear elastic 0 fields. For example, if a final by component of one dislocation were to combine with an opposite sign 0 by component of a dislocation from another set, a spacing defect would form. For simple tilt walls and misfit arrays, the separation of Binf into coherent and rotational portions gave physical insight into the interface configuration. For a less symmetric boundary like that in Fig 10, all components are resolved into the interface (habit plane) coordinates, and the concept of coherency in that plane does not have a physical connotation. Similarly, although the Burgers vectors of each set of defects are still defined in the CDP or RCDP coordinates, only the rotated Burgers vector components physically sum to relieve coherency strains. Yet the Bilby bicrystal is still equivalent to the ideal natural bicrystal. In such a case we identify the resolved strains and Burgers vectors in the 0 interface coordinates, e.g. 0yy or bz without designating these as coherency, misfit or tilt dislocations. inf Analogously, we consider B without splitting it into components. 5. Disconnections, steps, and line-forces 5.1. Disconnections Disconnections are defects that produce phase transformations, although they can also be formed at interfaces as a consequence of plastic deformation. For a static interface, they can also serve to remove misfit [19]. Many authors have considered defects of this type, as reviewed in [3]. Later, they

770

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

were formally described in symmetry theory as being composed of a step of height h and a Burgers vector bg belonging to the set Vint [17,18,49,50]. Fig. 11a shows an array of such defects together with the lattice invariant deformation (LID), comprising slip or twinning, that is also needed to completely remove misfit. The average habit plane, x0 y0 z0 , is shown in Fig. 11b. As for the defects presented in Sections 3 and 4, the disconnections separate coherent terraces, which contain coherency dislocations g(y). The disconnection Burgers vector, bg, has an in-terrace component by that produces the transformation as it moves. The component bz is another coherency dislocation that can move conservatively along with the disconnection to complete the transformation, provided certain conditions are met for the atomic densities in the two phases [51]. In the present context, once the Burgers vectors are transformed to coordinates x0i , fixed on the ha0 bit plane, Fig. 11b, all of the foregoing considerations apply. Components by serve to relieve misfit; 0 0 0 components bz contribute to tilt, and bx produce shear strains exz as well as twist. These parameters are depicted in the xi and x0i coordinates in Fig. 12a and b, respectively, for the simpler case where 0 bx ¼ 0. Now, it is necessary to define the angle h = tan1(h/L), which defines the inclination of the habit plane with respect to the initial terrace orientation. In the final configuration, the terrace planes immediately at the interface retain their initial orientation, but those beyond the distortion field have rotated in crystals P and Q by ±u/2 relative to this plane, as shown in Fig. 12c. Hence the angles between the habit plane and the far-field terrace planes are h + u/2 for P and h  u/2 for Q (note that u is positive for the resultant Burgers vector parallel to z0 shown in Fig. 12c). The variation of the terrace plane orientation through the interfacial distortion field described above is analogous to the simpler case of the tilt wall, Fig. 6e, where no disconnections are present, i.e. h = 0. Then, the terrace plane immediately at the interface (the habit plane) remains parallel to the initial terrace plane orientation, but, beyond the distortion field, the terrace planes are further rotated in crystals P and Q by ±u/2. Some alternative descriptions to Fig. 12b are presented in Appendix B in [52]. Disconnections have both a kinematic purpose, producing the phase transformation and its accompanying transformation (plastic) strain, and a structural role, relieving misfit strain and producing rotation for the final equilibrium habit plane. This dual nature of disconnections was qualitatively suggested by Christian [9], has been extensively discussed by Aaronson and coworkers [53,54] for diffusional phase transformations, and is quantitatively expressed for the topological model (TM) in [3]. All remote strains and rotations produced by the arrays must be partitioned to the two phases as in the prior examples. The complication relative to the other cases considered so far is that the steps alone

Fig. 11. (a) Arrays of disconnections (bg, h) and crystal dislocations (b, 0) accommodating coherency strains at an interface between matrix P (upper crystal) and product Q (lower crystal). (b) The average habit plane, x0 y0 z0 .

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

771

Fig. 12. Equilibrium habit plane viewed along the disconnection lines, x, x0 : Burgers vectors are expressed in (a) the terrace frame xyz, and (b) the habit frame x0 y0 z0 . (c) Angular relationships of the far field P and Q terrace planes with respect to the habit plane. (d). Equilibrium habit plane when bz = 0. (e) Tilt wall of extrinsic dislocations, bez0 , to be superposed on (a).

produce a rotation h of the interface and the physical rotation u is superposed on this already rotated interface. General nonlinear equations for the habit plane and orientation relationship are given in [19], together with reduced forms in the linear limit. For the case where em ¼ em yy is accommodated by disconnections, and em xx is negligible, the key equations for crystal P are

em ¼

½by þ bz tanðh  u=2Þ tan h ; h

ð23Þ

772

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

and

u ¼ 2sin1



 ½ðbz cosðh  u=2Þ  by sinðh  u=2Þcosðh  u=2Þtanh em sinðh  u=2Þcosðh  u=2Þ :  2h 2 ð24Þ

Here, for simplicity, strains, Burgers vectors and h are expressed in terrace plane coordinates, xyz. In the phase transformation case, the rotation u/2 is usually so small that its effect on bg and h is negligible, i.e., cos(u/2)  1. Hence, the Burgers vectors are given by a circuit mapped into the CDP of Fig. A1. The step height h is the smaller of hcP = n  tcP and hcQ = n  tcQ, where n is the unit normal parallel to z in the coherent terrace plane pointing toward P, Appendix A. Analogous to the results for the asymmetric grain boundary in Fig. 10, the rotations and coherency strains are partitioned equally when expressed in the current Eulerian interface coordinates, as reflected by the presence of the angle u/2 in Eqs. (23) and (24) and Fig. 12. However, the habit plane is almost always irrational. When the TM is applied to the martensite case, the transformation strain generated by the motion of dislocations gives a result identical to the so-called shape strain of the phenomenological theory (PTMC) [55–57], as demonstrated in [3]. However, since the PTMC does not generally incorporate partitioning of rotations, the predicted habit planes in the two models differ by u/2 in the general case. Here, we consider only the disconnections. The nature of the partitioning is illustrated in Fig. 12a and b as discussed above. In order to relate to the PTMC, we consider the case illustrated in Fig. 12d where bz = 0, i.e. hcP = hcQ. The misfit component of bg, namely by, cancels the coherency component bc and hence there is no total long range strain eyy. Both by and bc resolve into the x0i coordinates in the same 0 way so, there also is no long-range strain e0yy . Similarly, bz ¼ 0, so u = 0, there is no rotational partition0 ing and the TM and PTMC predictions for the habit plane agree exactly. In general, bz is finite, as in Fig. 12a. However, one can imagine superposing a tilt wall of extrinsic dislocations with Burgers vector 0 bez , thereby producing a net tilt equal to u, Fig. 12e. With no net z0 component of Burgers vector, Fig. 12c again applies, and the interface is defined by the x0i coordinates and the probe vector v0 in the interface plane. One now removes the extrinsic tilt wall, which is equivalent to superposing the tilt wall u on a pure step interface. The situation described in Fig. 8f then applies, where the imbedded probe vector v0 rotates symmetrically by ±u/2 becoming v P0 and v Q0 , each vector being inclined by the angle u/2 relative to the new vector v in the habit plane. Then, Fig. 12a, the habit plane at the interface, just as in Fig. 12c, remains inclined at an angle h with respect to the terrace planes. The rotational portion of the dislocation distortions are localized to the dislocations. However, the far field planes with the same crystallographic indices as the terrace plane are rotated by h ± u/2 relative to the habit plane, Fig. 12b. This analysis is consistent with theoretical expectation [18,19] and with results of an atomistic simulation [13]. Examples of the analysis of interfaces by the TM disconnection model are reviewed for simple metals in [3], extended to simple inorganic compounds in, e.g. [3,58,59], and to more complex compounds in, e.g. [60]. Symmetry theory has been used to describe admissible defects at transformation interfaces and to portray the role of disconnections in accommodation of transformation strain, but these topics are beyond the scope of the present work. Analogous to the asymmetrical tilt wall in Fig. 10, all components are resolved into interface (habit plane) coordinates, and the concept of coherency in that plane does not have a physical connotation. Similarly, although the Burgers vectors of each set of defects are still defined in the CDP or RCDP coordinates, the Burgers vector components do not physically relate to coherency in those coordinates, although the Bilby bicrystal still physically resembles Fig. 8e. Hence, we identify the resolved strains 0 and Burgers vectors in the interface coordinates, e.g. e0yy or bz , without designating these as coherency, inf misfit or tilt dislocations. Analogously, we consider B without splitting it into components. 5.2. Steps and line-forces Disconnections can form as a consequence of crystal growth, phase transformation, or cutting of a grain boundary or interface by glissile dislocations. For the cutting case, one can form interface dislocations, as discussed in Sections 3 and 4, or disconnections. In the limit, disconnections become pure

773

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

dislocations with h = 0, or, rarely, pure steps with bg = 0. Pure steps are often macroscopic, as discussed in [3]. There is one other possibility. As shown in [61], the cutting of an interface screw dislocation array by glissile dislocations can create a line-force defect. In twist grain boundaries, there is no long-range line-force field since the fields of the two cut orthogonal arrays cancel, but long-range line force fields can exist at other interfaces. If present, these line force arrays are associated with relative rotations of the two crystals, and must be considered in addition to any rotations associated with tilt or twist dislocation arrays. 6. Partitioning and nonlinearity The partitioning of strain is well-known for the misfit case [14,35], where it is required to satisfy the equilibrium equations. Less well-known is the corresponding partitioning of rotations, generally complete when the distance z from the interface exceeds the defect spacing, [19]. Obviously, in Figs. 7– 10 and 12, the rotation is equally partitioned to an amount u/2 or g/2 in each grain. Of importance relative to the Frank–Bilby eq., the partitioning can be envisioned in another way. The elastic fields of parallel arrays of equally spaced dislocations are given, for example, in Sections 19-5 of [42]. For a rh screw array parallel to the x-axis, n = [1, 0, 0], br = [br, 0, 0], the distortion field is r

@ux b sinð2py=LÞ ¼ ; @z 2L coshð2pz=LÞ  cosð2py=LÞ

ð25Þ

r

@ux b sinhð2pz=LÞ ¼ : @y 2L coshð2pz=LÞ  cosð2py=LÞ

ð26Þ

Remote from the interface in P, at large z, @ux/@z vanishes and @ux/@y = br/2L, and similarly @ux/ @y = br/2L in Q. This gives remote strains exy = ±br/4L and rotational functions xxy = ±br/4L in the P and Q crystals, respectively. The fields in Eqs. (25) and (26) are based on the linear elastic fields of the dislocations, so the remote rotations are as described in Appendix A.4. A rh screw array parallel to the y-axis produces the remote field exy = br/4L and a function xxy = ±br/4L when the result is transformed into the coordinates of Eq. (26). Thus, for the square array, the total strain exy is zero and the remote rotational functions in P and Q are ±br/2L, giving a total rotation of P relative to Q of 2 tan1(br/2L). If, in contrast to the above case, the second set of screws in the square array is left-handed, the total rotation is zero and the total remote shear strains in P and Q are exy = ±br/2L. Consistent with this analysis, a circuit around one of the above defects gives br as the Burgers vector. Of course, as in Eq. (19), the result can be expressed in terms of the Burgers vectors of crystal dislocations, bP or bQ. The analysis for the tilt array is more complex. However, the result is the same: for a simple tilt wall, the rotation is partitioned equally to the two grains [19,42]. Analogous to the above analysis for screws, the edge array of Fig. 6 gives remote rotational functions in P, xyz = xzy = br/2L, again in agreement with Eq. (19) for small rotations because u is negative in this case. Extending the analysis to interfaces where the Burgers vectors differ in the two phases, one sees that the partitioning remains the same, analogous to the tilt wall in Fig. 10. Hence, for the tilt array formed from dislocations such as those in Fig. 10, the tilt angle in either phase, relative to final x0i coordinates fixed on the habit plane, is given by rQ

rP

½ðb =2LQ Þ þ ðb =2LP Þ=2 ¼ sinðu=2Þ; rQ

Q

ð27Þ rP

P

Q

P

subject to the subsidiary condition (b /2L ) = (b /2L ). That is, the spacings L and L also differ in this example. This result would follow directly from the Frank–Bilby eq. discussed subsequently. In order to connect to Section 2, we reiterate that brP is the z0 component of bP. Another aspect of partitioning is that the partitioning factor j can differ from 0.5 when either layer thicknesses differ, or anisotropic elastic constants differ, or the average elastic constants in the isotropic approximation differ for the two phases. As can be deduced from the ideal Bilby bicrystal in Fig. 3, the partitioning is the same for coherency strains and dislocation strains in the misfit case. Superposition of individual dislocation fields shows that the same is true for rotations. Both statements are also true for the cases where elastic constants differ. Hence, the preceding analysis applies for both

774

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

cases in the sense that the superposition of coherency strains and misfit dislocation strains cancels the long-range stresses. However, the degree of partitioning differs [62,63]. The details of the analysis of unequal partitioning are too extensive to be included here, but we can comment on the form of the partition and on limiting cases. We first consider unequal layer thicknesses in the isotropic, homogeneous, linear elasticity case. In the limit that the thickness of one layer becomes relatively small, the compliance of the thicker layer forces most of the strain into the thin phase in a well-known way [14]. In the limit, all of the strain is in the thin phase. As shown in Fig. 13, the dislocation arrays on the two interfaces of the thin film cancel remote from the thin film, but add within the thin film. Within the thin film, provided that the film thickness Z exceeds L, the coherency strain becomes jem, while that in the thicker layer becomes (1  j)em, where j varies between 1 and 0.5, the latter applying for equal thicknesses. With this known interface state, the F–B analysis proceeds in a straightforward way by simply incorporating the factor j into the equations in the preceding analysis, see Appendix D. Similarly, when the elastic constants differ, the remote strain is greater for the material with softer elastic constants for both the anisotropic elastic and inhomogeneous elastic cases. For linear anisotropic or inhomogeneous isotropic elasticity, the strain partitioning can be described again by the factor j, now varying from 0 to 1, Appendix D. Several examples, together with ways to approximate the anisotropic results [64,65], are given in [66]. Again, once the partitioning is known, the F–B analysis proceeds straightforwardly in the linear elastic case. In these elastically inhomogeneous examples, the modified fields can be considered to arise from image forces associated with image dislocations at the interface, simple to envision in the isotropic, inhomogeneous case [42]. Effectively, to first order, the field in the softer crystal sees a larger Burgers vector equal to bg plus that of the image dislocation, while the harder crystal sees a smaller Burgers vector equal to the difference between bg and the image vector. Since the long-range distortion field is proportional to the Burgers vector, the partitioning of strains or rotations is the same as the partitioning of the Burgers vectors. Hence, one can either consider an actual interface array or, alternatively, a single interface dislocation to determine the partitioning ratio. In the anisotropic elastic case, uniform line force distributions may be present, differing from those discussed in Section 5.2, causing added rotations that must be treated [62]. When the partitioning is unequal, the methodology in Appendix A and Section 2 is unchanged. The only difference is that the translation vectors in P and Q become unequal, one scaling with j and the other with (1  j). In all examples, the final interface can be envisioned as being created from a CDP or RCDP, which obviously satisfies both compatibility (no displacement discontinuities) and equilibrium (no total

Fig. 13. Misfit dislocation arrays at the interfaces of an included thin film. Both the dislocation array fields and coherency fields add at A, cancel at B.

775

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

forces or torques acing across any plane within the bicrystal or at free surfaces). In ideal Bilby bicrystals, the same elasticity conditions are valid. In real bicrystals the same is true of the long-range fields. As shown for 1/rk fields, with k an integer >1, in perturbation theory, higher-order terms, while strongly nonlinear, also satisfy the equilibrium conditions and are incompatible only in the core region. The actual value of the partitioning parameter j is that which both satisfies the incompatibility and equilibrium conditions. The latter ensures a minimum total elastic energy, except when layer thicknesses approach the thin film limit where there is a critical thickness for dislocation formation in a coherent interface [14]. The linear elastic, isotropic, homogeneous model is often used because of the complexity of anisotropic or inhomogeneous elasticity and/or nonlinear effects. We give one comparison of the different approximations to indicate the accuracy of the simpler method. Disconnections at a Cu–Ag interface with {1 1 1} terraces have been studied in an atomistic simulation [13]. The potentials used and simulation methods are presented in that work. A representative disconnection array is shown in Fig. 12a and, in the Cu–Ag example, misfit dislocations are also present. Results were obtained for five cases, presented in Table 1, for an interface with disconnections as the only defects present. Case 1 is the isotropic elastic result using Voigt average elastic constants. Case 2 is an MD (molecular dynamics) simulation with specially modified Cu and Ag potentials. The lattice constants are the same as true Cu and Ag, but the potentials are adjusted so that the elastic constants of Cu and Ag are the same. Case 3 is the inhomogeneous isotropic elastic result with each crystal separately described by average anisotropic elastic constants [63], derived from the stress functions presented earlier for such a case [62,64]. Case 4 is the linear, anisotropic elastic prediction [65,66]. Case 5 is the result of an MD simulation [13] with potentials matched to true Cu and Ag. The homogeneous cases strongly support the theoretical analysis presented previously: in the isotropic, homogeneous case misfit strains and rotations are partitioned equally relative to the x0i coordinates of the habit plane. Copper and silver have moderately high anisotropic elastic ratios (see Appendix A in Ref. [42]). Hence, the results indicate that it is a fair approximation for typical metal pairs or other interface pairs with similar elastic constants, to use the simpler isotropic elastic approximation. The partitioning is unequal in the other three cases, but, encouragingly, they group closely together relative to the isotropic cases. This shows that the inhomogeneous isotropic case, which has relatively simple analytical expressions and therefore entails a briefer analytical analysis than the last two cases, can be used as an improved approximation if one wishes to go beyond the isotropic homogeneous case. Only in special cases would one need to use the general anisotropic elastic or atomistic calculations that are more complex to implement. One cannot say whether the anisotropic elastic or MD case is better, since the former is still linear elastic, while the latter entails some approximation in the potentials used. The presentation in Appendices A and B, together with Sections 1–5, are all applicable in the framework of the standard model of Appendix C. Simulations like the MD calculations just mentioned do not conform to the standard model in that nonlinear stress–strain–displacement relations are present. Then the uniform strain assumption leading to Eq. (2) does not apply and the distortions cannot be separated into superposed strains and rotations. Also, there are cross terms between edge and screw components of dislocations. Hence Eqs. (2)–(9) are inapplicable. Yet the final form of Eqs. (10)–(12) still apply:

B ¼ c Dn v :

ð28Þ

Table 1 Results for partitioned rotation for Cu–Ag {1 1 1} interface with a superposed tilt wall for several cases. Here L = 0.867 nm and angles are in degrees. Case

Source

u (Cu)

u (Ag)

u

u (Ag)/u (Cu)

1. 2. 3. 4. 5.

Eq. (9) [13] [62,63] [65] [13]

0.932 0.934 0.701 0.604 0.730

0.932 0.934 1.092 1.014 1.052

1.864 1.868 1.743 1.618 1.782

1.00 1.00 1.56 1.68 1.44

Homogeneous Hom. ‘‘Cu–Ag’’ Inhomogeneous Anisotropic Elastic MD

776

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Eq. (2) is traditional in the classic works on phase transformation and is valid for the terrace-defect structure discussed here where the standard model applies. Of course, in general, the F–B eq. applies to the nonlinear case via Eq. (28) and the Bilby bicrystal model applies as well, extrapolating far-field linear elasticity to the interface. However, difficulty arises in associating the fields with local defect arrays because of the nonlinearity. Similarly, for arrays with high angle tilt components or misfit defects with near atomic spacing, the F–B eq. and Bilby bicrystal still apply but identification of specific defects is difficult. One could use Eq. (28) at the outset and avoid Eqs. (2)–(10), and that is the approach in Appendix D. However, for many examples where the standard model applies, Eq. (10), with its superposition of strain and rotation is convenient. With nonlinearity, the inclusion of higher order elastic constants complicates the analysis. Similarly, for deformation matrices, the computation of inverses is complicated because of the many high-order terms in the determinants. Yet Eqs. (6), (11) and (28) are still exact because elastic strain is a state variable. The isothermal, reversible strain energy is a free energy. Hence, if one can construct a reversible cycle from state S to F to S, then the change in any state variable A is FAS = SAF. Fig. 2 depicts a closed cycle that hypothetically can be reversible. Therefore, in general, B = Binf, and similarly for the related quantities in Section 2. Simulations of the dissociation of crystal dislocations in fcc metals [67], indicate that nonlinear effects become important at defect spacings L 1 nm. In this regime, before high nonlinearity dominates, one could model boundaries and interfaces by including core interactions, also important if jogs or kinks are present on interface dislocations. There is a promising new treatment [68] of core energies and interaction energies on a parallel with the linear fields of dislocations. There are also phenomenological models entailing continuous distributions of dislocations: the Peierls model (see [42]), the standard core model [69], and the tube [70]. Finally, there are several empirical cutoff models, e.g. [71–73], some of which may contain extraneous line-force fields. As demonstrated in [73], in order to prevent the introduction of erroneous core terms, one must not mix any of these models. Another effect that appears for L 1  2 nm and with relatively low stacking-fault energies is the increased extent of partial dislocations. For large L, the interaction energy between perfect dislocations in a network are small and the extent of dislocation extension is small, occurring locally at nodes and minimally for straight dislocations; large areas of stacking fault are energetically unfavorable. In contrast, for small L, the interaction energy dominates, and the partial dislocations become dominant because of their lower line energies. Fig. 14 depicts a {1 1 1}  {1 1 1} twist boundary for a Cu–Ag

Fig. 14. {1 1 1}/{1 1 1} twist boundary at a Cu–Ag interface [74]. Dark areas are unfaulted, light areas faulted.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

777

bicrystal. The segment spacing is 3.1 nm. For large L, a hexagonal network of perfect dislocations, slightly extended at nodes, would be the favored configuration. Instead the minimum energy twist boundary for small L is the depicted triangular lattice of partial dislocations 1/6h112i. Nodes of this type have been (infrequently) observed in TEM, see Figs. 175 and 176 in [24]. In a related way, parallel dislocations in a given set have an interaction force proportional to 1/L [44]. Hence for a three dislocation network with in-interface Burgers vectors, the repulsive force between the like sign parallel segments can become so large that the network converts to two intersecting sets of continuous, straight dislocations when the spacings become small. An example of such straight dislocation sets is given in Section 10.7. In considering these or other refinements to the F–B solutions, one must consider whether they are significant compared to assumptions such as that of isotropic elasticity, if employed. 7. The general interface The above developments have been presented for simple single, square or hexagonal arrays that can be treated in two dimensions in order to develop concepts of reference spaces and interface defect properties. We now turn to the general interface, retaining the emphasis on defects separated by coherent terraces, This can be a general misfitting and rotated interface or one containing disconnections. As in the preceding analysis, we select coordinates in the current Eulerian interface with z0 as the normal to the interface. In the F–B eq. as formulated in Eq. (10), we are concerned with a distortion matrix, D0ij ¼ @u0i =@x0j :

2

D0xx

D0xy

6 0 D0ij ¼ 6 4 Dyx

D0xz

7 D0yz 7 5:

D0yy

D0zx

3

D0zy

ð29Þ

D0zz

In either the linear elastic case or the standard model of Appendix C, the distortion matrix can be  0 0 0 0 0 separated into asymmetric portion, the strain matrix, eij ¼ 1=2 @ui =@xj þ @uj =@xi and the rotation matrix x0ij ¼ 1=2 @u0i =@x0j  @u0j =@x0i . In a more general nonlinear case, the strains and rotations couple, but for the examples considered here, and in the standard model of Appendix C, they are separable. cP There are three components of the coherency strain parameter ecij ¼ ecQ ij  eij that must be specified as degrees of freedom. Hence the relevant strain components are

2

e0cP e0cP xx xy 6 0cP 0cP Eij ¼ 4 exy e0cP yy 0

0

0

3

7 0 5;

ð30Þ

e0cP zz

and similarly for Q. The consideration of accommodation [52] shows that there are no strains e0cxz or e0cyz . Also examination of results like those in Eqs. (25) and (26) reveals that no regular interface array of 0m dislocations can produce strains e0m xz or eyz . This arises because of the cancellation of equal and opposite strain fields in the arrays and is obvious for discrete arrays: limiting forms also show that this is also true for arrays of coherency dislocations. These terms are absent in either the isotropic or anisotropic elastic descriptions. Hence these two components can be set equal to zero in Eq. (30). The term e0cP zz is zero for the equilibrium bicrystal, where any Poisson terms from the misfit dislocations exactly cancel those of the coherency dislocations. Only for a non-equilibrium interface, where the Poisson terms do not cancel, or when accommodation incompatibilities arise [52], is this term non-zero. There are also three components of the rotational function that are independent degrees of freedom in general:

2

0

6 0cP 6 X0cP ij ¼ 4 xxy x0cP xz

x0cP xy 0 x

0cP yz

3

x0cP xz 7 7 x0cP yz 5; 0

ð31Þ

778

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

and similarly for Q. Thus, the distortion matrix, Eq. (29) has all nine components in general, but Eqs. (30) and (31) show that there are seven terms in D0ij . However, since e0cP zz is zero, there are six conditions or degrees of freedom that must be specified. A general solution to the F–B eq. therefore requires constraints entailing the specification of six independent variables. An analogy is single crystal slip, where five independent slip (or deformation) systems are required for a general plastic strain. As discussed in detail in Appendix D, the independent variables for a given set of dislocations contain the factors bg, L and the sense vector n. In the most general case, with bg specified and n fixed, six sets with six independent variables would be needed. In the analog mentioned above, for plastic deformation, with constancy of volume, five strain components are required for a general deformation, and, in the von Mises criterion, rotations are assumed to be unconstrained [75]. However, the Basinskis [76] have shown that accommodation constraints exist for the general deformation of a copper crystal, so that nine slip systems were needed and observed. When tilt components or disconnections are 0 present, two sets with Burgers vector components, bz , normal to the plane are present and the rotations x0xz and x0yz are present as added degrees of freedom. For these edges, rotation in the glide plane would require jog formation and at least conservative climb if not general climb. Hence this factor would tend to constrain n to be fixed. For twist walls, kinks are required for in-plane rotation. An example where kink formation, and hence change of line direction, is unlikely arises where a slant 0 twinning dislocation intersects the interface: its bx component thereby contributes to twist, but its 0 bz component would have to climb. Thus edge, screw, or mixed dislocations can be constrained against rotation within the interface in the general case and six sets are needed. Analogous to the Basinski observation [76], nine sets could be needed with rotational accommodation suppressed. In many cases, the number of degrees of freedom are reduced. If the edge dislocations with com0 ponents bz are free to change line direction by forming jogs, i.e. if they can arbitrarily assume any line direction, one set of edge dislocations, with the two independent variables bg and n, suffices to fix both rotations and the needed sets reduce to five. If the dislocation sets are free to undergo twist rotation in the interface by kink formation, two sets with the variables bg and n would provide four total independent variables, reducing the number of needed sets to three, the minimum for the general case with arbitrary symmetry. With high symmetry a further reduction is possible. For the case where there is no tilt component present, the minimum number of needed sets is one, although two are usually needed, see Section 10. When the distortion matrix is rotated to coordinates fixed on the interface, the number of independent variables does not change, although the form of the matrix obviously changes. F–B solutions then involve, in general, sets of simultaneous equations with up to six variables. In addition, there are three local degrees of freedom, corresponding to rigid translations of the interface in the three orthogonal directions. These do not change the general solution, which is the same, independent of these translations. They do determine the minimum energy state: for in-plane translations, the motions map the equivalent of the minima for the Peierls energy for defects in the interfacial plane. Examples of the independent dislocation sets are presented in Section 10. As an example, for the case where all Burgers vectors lie in the interface plane and are free to rotate, and where x0xy ¼ x0yx , two sets of dislocations provide a solution but there are an infinite number of three-set solutions. The reason for so many solutions is that there are three sense vectors n and three spacings L which over-determine the equilibrium, long-range, stress-free requirement. Hence added criteria are needed to enable the prediction of a favored set. There is an analog in the selection of slip systems that satisfy the von Mises criterion for arbitrary deformation of a single crystal [75] or the requirement for four independent slip systems distributed in the two grains for compatible plastic deformation of a bicrystal [77,78]. With many available independent slip systems in cubic crystals, for example, the von Mises requirement is highly over-determined [79–81] and various criteria are used to limit the number of systems in a given case [81]. For the present F–B solutions, one possibility is the determination of the minimum energy sets [82]. Usually, the minimum energy configuration entails the relaxation of the intersecting sets into a network connected at nodes [24]. Another related possibility would be to select the sets that satisfy a balance of line tension forces at the nodes [42]. One expects the number of required systems to vary for interfaces formed by different processes. For thin films or multilayers formed by deposition processes, or for multilayer structures formed by rolling, the resolved shear stress in the interface plane is either zero, for high symmetry cases, or, more

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

779

generally, small compared to that on slant glide systems [83]. Hence, Burgers vectors from the slant systems are invariably found in the early stages of coherency strain relief [14]. Burgers vectors lying in the interface form later as a consequence of dislocation interactions [14,84]. Thus, interfaces in deposited thin films are expected to be of the most general type. For interfaces formed by displacive (martensitic) transformations or diffusive–displacive (bainitic, platelike) transformations, the disconnections producing the transformation almost always have a tilt component [3,18] and can have a twist component [47]. Moreover the LID is also likely to entail slant systems for the same reasons as for the deposition case. Lastly, added systems are needed to accommodate pileup-type incompatibilities at plate tips. Hence phase transformation interfaces are also expected to be of the general type. With diffusional bonding, the interfaces are created by interface relaxation, without the need for deformation of either crystal. In many MD simulations, two crystals are directly bonded. Hence the needed sets of defects in the latter two cases are less and may entail only Burgers vectors lying in the interface. Low-angle polygonized grain boundaries formed by the recovery of deformed crystals also can be of this simpler type, while those formed by recrystallization are more general.

8. Energies of interface defect arrays For a given selection of terrace plane, there are often several possible networks of defects that satisfy the condition of removal of long-range strains, in which case the interface with the lowest total energy should be favored. Interfacial energy is the combination of the elastic energy of the component defect arrays in a network and the chemical energy of the interface. The latter is comprised of the core energy of dislocations, including spacing defects as a second order effect, and the excess energy of terraces. Since chemical terms are similar for alternative defect networks, we discuss only the differences in elastic energy. In addition to possible kinematic constraints from pinning and mobility considerations, six factors are involved. For a dislocation in unit area of interface, its self energy scales as b2f(n)ln(L/r0). Here f(n) depends on dislocation character and r0 is a core parameter. For an array of parallel dislocations, the interaction energy of a dislocation with the array scales as b2f(n)ln(L/r0)/L. The interaction energy in the parallel array is positive, i.e. the interaction force is repulsive. The first, most important factor in these relative energies is the magnitude of the Burgers vector, since it is squared. Almost always, observed dislocations at interfaces have the smallest one or two Burgers vectors available. Second is the dislocation spacing, L: the total dislocation contribution to the elastic energy decreases monotonically with increasing L. Hence the most widely spaced among a number of possible configurations has the lowest elastic energy. Third, is the dislocation character: dislocation line energy almost always decreases with increasing screw character, i.e. f(n) decreases with an increase in screw character. Fourth are possible dissociations at intersections into a new dislocation linking threefold nodes, possibly extended. As mentioned in Section 6, the self energy tends to favor network formation with nodes, while the interaction energy tends to suppress network formation. Fifth is the core energy, which becomes important when L is small: core energy tends to increase with increasing screw character. Sixth is the spreading of the interface over several planes, forming a diffuse interface, Section 9. For the particular case of a displacive phase transformation involving both disconnections and LID, the disconnections must form first, followed by lattice invariant slip or twinning. In such cases, the energy of the original disconnection array may fix the disconnection sense vectors, and thereby constrain the overall defect network required to remove long-range strain. For arrays with small L and networks containing nodes, the node energy contribution to the interfacial energy is usually negligible except for dissociation effects. The splitting of a dislocation into two others may reduce interfacial energy, but does not change the overall long-range distortion field associated with a network. Following [85], for the case of no rotation, x0ij ¼ 0, (the general form is given in Section 19-2 in [42]), we define reciprocal vectors in the boundary by Ni = Ni(n n) and Ni = 1/Li, where n is a unit vector normal to the interface and pointing toward P. Thus, a network of defects comprising parallel arrays, Ni, which satisfies the F–B eq. continues to do so if Ni splits into N0i and N00i , provided that Ni ¼ N0i þ N00i . Similarly, faceting of a dislocation line, as can occur in the anisotropic elastic case, or local curvature, caused for example by torques at

780

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

nodes, do not affect the solution provided that the overall mean line direction does not rotate. The reverse of this statement demonstrates that an array of edge dislocations can fix both x0xz and x0yz , as discussed previously. Node formation at the intersections of two non-parallel arrays results in a network of segments comprising three arrays, such as a hexagonal array, connected at three (or possibly four) nodes. However, one can analyze such a network as three arrays of continuous straight dislocations with smaller Burgers vectors, meeting at sixfold nodes. As proved in [44], a single set of regularly spaced, collinear segments with Burgers vector, bg, has the same long-range field as a single continuous dislocation with Burgers vector Xbg, where X is the fraction of the original line that contains dislocation segments. g Thus, the preceding analysis can be used for the three-array networks if the Burgers vectors bi are reg placed by the reduced values, X i bi , see Fig. 21d. This is useful in calculating the energies of, for example, hexagonal networks, comprising piecewise straight defect segments. Rather than using the more complex equations for piecewise segments, e.g. Chapter 6 in [42], one can use the simpler equations for straight dislocations, albeit with reduced Burgers vectors. Another application of reduced Burgers vectors is to formulate the condition for no twist rotation, x0xy ¼ 0, namely

X X i bgs i

Li

i

¼ 0;

ð32Þ

gs

where bi represents the screw component of the dislocations in the ith array. 9. Delocalized and diffuse boundaries In the preceding treatment, most interfaces were sharp planar structures, or exhibited stepped planar segments separated by defects. There are a number of ways that boundaries can become less localized, as we now discuss. We show that, provided the standard model applies, the long-range fields are unchanged by interface delocalization. Hence, if one determines the net dislocation content as in Section 4.5, the F–B analysis still applies. The first example is an interface with planar segments or faults extending over multiple planes. A simple example is a fcc–fcc (1 1 1) interface containing disconnections with partial dB Burgers vectors separating intrinsic-type and extrinsic-type terraces. The extrinsic-type faults extend over two planes and are bounded on successive planes by a zonal dislocation comprising Ad and Cd partials that have long-range fields corresponding to dB: see Fig. 10-8 in [42]. The partial pair is the simplest zonal disconnection with h = 2h0, with h0 equal to the terrace interplanar spacing, i.e. h = ih0 with i > 1. The motion of a disconnection involves both shear and shuffle. Most non-basal dislocations in hcp crystals are zonal, extending over up to four planes [42] and shuffling is included in their motion. Disconnections in such cases have step heights, h > h0. Another example is for disconnections in high R twin boundaries which also have i > 1 [86,87]. In all cases the long-range fields can be analyzed by the F–B eq. to give a mean habit plane that has the spread defects clustered about it. An additional example is that of extended dislocation nodes. As shown in [88], alternate nodes in {1 1 1} twist boundaries in fcc crystals can extend with partials bounding extrinsic faults. Again, the bounding partials would extend over several planes. Dislocations in fcc grain boundaries can dissociate into stair-rod partials in the boundary and 1/6h1 1 2i partials extending into the adjacent crystals [25]. A second example is the spreading of an interface by the formation of jogs on interface dislocations.  0 ==½1 1  1 Atomistic MD simulations of Cu a–Nb b interfaces with Kurdumov–Sachs orientations, ½1 1 a b and (1 1 1)a//(1 1 0)b, have revealed three types of interfaces [89]. An example is shown in Fig. 15. Terrace segments are coherent and bounded by interface misfit dislocations. In the first type, KS1, the Cu and Nb retain their crystal structures up to the planar boundary, which contains the misfit dislocations. In the second, KS2, the first Cu layer, a0 , at the interface has a nominal bcc structure matching the Nb. Second and higher Cu layers have fcc a structure and the misfit dislocations separate the first and second Cu layers. In the third, KSmin, some dislocations climb from the KS1 interface to the plane between the 1st and 2nd Cu layers. In the vicinity of these dislocations the local interface has effectively formed a local KS2 structure. The mechanism of climb is by jog-pair formation [90]. Such jogs are important sites for point-defect trapping [91,92]. The minimum energy configuration, KSmin,

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

781

Fig. 15. Kurdumov–Sachs interfaces in a Cu–Nb bicrystal [90].

represents a minimization of several energy terms. For KS2 the self-energy of dislocations could be lower, relative to KS1, if the Cu were ‘‘softer’’, because more of the elastic field is in the Cu. One cannot be decisive on this point because c11 and c44 are larger in Nb but c12 is larger for Cu and nonlinear constants are important. The interfacial energy between a0 and Nb differs from that between a and Nb. The repulsive core–core interaction energy at nodes is decreased. Finally, there is a larger coherency strain in the first layer. Of importance here, the jog formation spreads the interface over two planes. Again, provided that the standard model is applicable, the F–B analysis still applies to the mean interface. There is an interesting analogue of the KS2 structure in thin film deposition by the Bauer mechanism [93], where the first monolayer deposited onto a foreign substrate assumes the crystal structure of the substrate. An early example is the deposition, observed in a field-ion microscope, of Ag onto a W field-ion tip [94]. The first Ag monolayer was pseudomorphic with the underlying bcc structure. Subsequent fcc Ag grew by forming islands on the pseudomorphic layer. Another example is the vapor deposition of films of Cu embedded in Nb [95]. When the film thickness is less than 1.2 nm, the Cu has a bcc structure. A somewhat analogous situation arises when there is a large difference between the elastic constants of the two crystals abutting the interface. In such a case the misfit dislocations can ‘‘stand-off’ from the coherent interface into the softer crystal by several layers [96,97]. An example observed in TEM is the interface between close-packed planes in Ni and Al2O3, where TEM reveals the misfit dislocations reside two planes into the Ni [98]. Profuse vacancy or antisite point defect concentrations could also create a diffuse boundary at high temperatures. 10. Applications of the F–B eq. This section presents some simple solutions of the F–B eq. to illustrate various trends. Some of the final examples are more complex. We have seen that the solution of the F–B eq., Eq. (10), can involve up to six independent variables. Appendix D presents a form of the F–B eq. that is amenable to a numerical solution. In special cases, such as phase transformations, even though the relevant equations, e.g. (25) and (26) are transcendental, the defect spacing is so large that one can solve the F–B eq. by iteration, achieving an accurate solution after one or two iterations [15,47]. In the simplest solutions, the geometry is such that one can solve the problem by inspection.

782

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

In these examples the sense vectors n are fixed. In 10-7 general solutions are presented with n varying. Some of the simple solutions presented may not be minimum energy configurations. However, symmetry and energetic considerations indicate that they at least represent local minima. 10.1. Misfit arrays Our first example is the primitive cubic (P) – primitive orthorhombic (Q) case illustrated in Fig. 16. The natural translation vectors parallel to the terrace plane, tnP, tnQ, are shown: these are equal in the x direction, but misfit in the y direction. Following the deformations cPn and cQn, the translation vectors parallel to y adopt the commensurate value tcP = tcQ. Thus, the CDP exhibits rectangular translation symmetry in the unique plane combined with mirror symmetry perpendicular to x, y and z: the layer space group [12,27–29] is therefore, pmmm. For convenience, the Miller indices of the Burgers vectors, planes, etc. are referred to one of the reference P or Q single crystals, and it is implicit that the magnitudes and directions of these crystallographic quantities are those of the CDP, or, in the subsequent examples, the RCDP. The total coherency strain parameter can be represented as



nQ ec ¼ 2ðanP  anQ Þ=ðanP þ anQ Þ ¼ 2 bnP 0  b0

. nP nQ b0 þ b0 ;

ð33Þ nP

where anP and anQ are reference lengths in the general case, which for this special case become b0 and nQ cP cQ cP b0 the lattice parameters of P and Q parallel to y. Also, ec ¼ ecQ yy  eyy , where eyy and eyy are the coherency strains of Q (negative) and P (positive) respectively. When these are equally partitioned we have cP ec =2 ¼ ecQ yy ¼ eyy :

ð34Þ c

mP Similarly, the misfit strain parameter is em ¼ emQ yy  eyy , opposite in sign to e . That is, when the coherency strain is fully relieved by the misfit strain, the total strain is zero and ec + em = 0. Hence

mP em =2 ¼ emQ yy ¼ eyy ;

where

ð35Þ

mP emQ yy and eyy are the equally partitioned misfit strains. Thus, we have

0

0 0 B ec c En ¼ n Ec ¼ @ 0 0

0

1 0 C 0 A:

ð36Þ

0

In the formalism set out in Section 2, the probe vector illustrated in Fig. 16 is

Fig. 16. Equilibrium interface for primitive cubic (0 0 1) P and (0 0 1) primitive orthorhombic Q. Misfit is only present in the y direction. A section of the CDP at the interface is also shown in plan view.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

v0 ¼ v0 j:

783

ð37Þ

In Fig. 16, we select the sense of v, and the accompanying parameters n, t and b, in accord with the convention in Appendix B. (One could use the opposite sense by reversing the signs of these vectors, giving the same final result. In later more complex cases, the appropriate sign for the components of v must be used in the accompanying equations.) By substituting into Eq. (10), we obtain the discrete and coherency dislocation content intersected by v0,

0

B ¼ Bc ¼ n Ec v 0

0 B ¼ @0

ec

10 1 0 0 CB C 0 A@ v0 A ¼ ec v0 j:

0

0

0

0

ð38Þ

0

Recalling that ec is negative, Eq. (38) shows that Bc is parallel to j, corresponding to tensile coherency strain in P, as depicted schematically in Fig. 3, and that B is parallel to j. For defects with n = i, this misfit can be accommodated by admissible dislocations with Burgers vectors equal to g

m

b ¼ b ¼ tcP ¼ ½0 1 0;

ð39Þ

corresponding to the ‘‘extra half planes’’ being in the P crystal: as defined in Appendix A, P is the crystal in the region z > 0. Based on the physical understanding of misfit arrays [42], the strains associated with these misfit dislocations are m mP emQ yy ¼ eyy ¼ b =2L0 ;

ð40Þ

giving the total misfit strain parameter as m mP c emQ yy  eyy ¼ e ¼ b =L0 :

ð41Þ

To obtain the defect spacing, L, from Eq. (38) we use the reciprocal vector N = N(n n), where N is defined in Section 8 as the number of defects per unit length, 1/L0. Thus, Eq. (38) can be written (for the one set of defects in this case), m

m

B ¼ ðb  NÞv0 j ¼ b Nv0 j ¼ b ðv0 =L0 Þj:

ð42Þ

Hence, for a probe vector of unit length, we obtain

em ¼ bm =L0 ;

ð43Þ

which agrees with the physical model and Eq. (41). Now, let v be selected differently, as shown in Fig. 17, so its magnitude is v0 sec a. Thus,

v ¼ vx i þ vy j ¼ ðv sin aÞi  ðv cos aÞj ¼ ðv0 tan aÞi  v0 j;

ð44Þ

and the equivalent of Eq. (38) is

0

0

B B ¼ Bc ¼ n Ec v ¼ @ 0 0

0

0

10

B ec 0 C A@ 0

v0 tan a

0

v0

1 C A ¼ ec v0 j;

ð45Þ

0

which is identical to (41). B becomes indeterminate in the limit of a = p/2, when v is parallel to n. The equivalent of expression (42) is m

m

m

B ¼ b Nvj ¼ b ðv=LÞj ¼ b ðv0 =L0 Þj;

ð46Þ

v=L ¼ v0 =L0 ;

ð47Þ

since

where L is the spacing of the dislocations along v. Thus, the solution is independent of the inclination of v. Equations.(45)–(47) are the key findings demonstrating the advantage of the F–B method. Formal proofs of Eqs. (46) and (47) on the basis of matrix algebra are provided in [4–6,26] and in Appendix D.

784

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 17. Same bicrystal as in Fig. 16 but with different v inclined to y by an angle a.

Fig. 17 can also be used to illustrate the number of dislocation sets needed for a given problem. In both of the examples depicted in Figs. 16 and 17, one set of dislocations suffices to provide a solution. Suppose that Fig. 17 applies for a bicrystal where bg is inclined at an angle to the y axis. The edge components of the dislocations can relieve the misfit strain. However, the screw components would give rise to uncompensated rotations xxy and strains exy. Hence, one or two additional dislocation sets would be required to remove these added distortion components. This is true independent of the orientation of the line direction n of the first set. An example similar to the case illustrated in Fig. 17 is possible for an a  b interface in Ti, where the  1Þ  ==ð1 1  0 0Þ . The principal strain is in the y direction ½1 1 1 k½ 1 12  0 , and is 30 terrace plane is ð2 1 b a b a  ==½0 0 0 1  . In principle, the smaller strain could be adtimes larger than that in the x direction, ½0 1 1 b a justed to zero by solid solution alloying. Such a pseudo-bicrystal model with zero minor strain has been analyzed in [50] for the habit plane of the martensite a0  b interface. There, the misfit was compensated by a disconnection array that arises naturally as a consequence of the transformation mechanism. Here, we treat the same hypothetical bicrystal and assume that the interface is created by diffusional bonding [99], so that no steps are present. na nb The lattice parameters [50] are b0 ¼ 0:295 nm, and b0 ¼ 0:328 nm, and the coherency strain c parameter parallel to y on the terrace plane is e = 0.1059 With equal partitioning of the strains,  0 and the translation vector parallel to y in the coherent bicrystal is tca = tcb, derived from 1=3½1 1 2 a 1/2[1 1 1]b respectively: equations (A3) and (B8) then give the Burgers vector of the crystal dislocations, bg = bm, that can accommodate the misfit and has magnitude 0.312 nm. The dislocation spacing necessary for removal of long range strain is L0 = bm/em = 2.95 nm. This dislocation spacing is larger than the spacing of the disconnections in the phase transformation case, 1.26 nm, because the Burgers vector is much smaller for the disconnections. Treatments of simple misfit usually include partitioning. In more complex cases where rotation or stepped interfaces are analyzed, a frequent assumption is to set the Burgers vector bg equal to the NDP vectors, bP or bQ, instead of the partitioned values from the CDP. For the present case, the use of bP or bQ would cause an error of 2.6% in L0. In addition, if the average Burgers vector were bP or bQ, long range stresses and strains would exist unless extrinsic dislocations were added. All early TEM observations of interfaces, formed for example by diffusional bonding [95], indicated that a semi-coherent structure of defects separating low energy commensurate terrace planes was favored over a Vernier type structure. As L0 approaches defect core sizes, however, a more uniform, incoherent structure should eventually be favored [20,100–102]. Atomistic simulations of the extension of perfect dislocations in fcc metals [67] show that core overlap begins for L0 1 nm. Below such a spacing, structural units [103,104], with some local periodicity, could be present, but with L0 small enough, no discrete defects would be discernible. In this regard, atomistic simulations revealing a

785

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

terrace-defect structure for L0 approaching 1 nm [13,89] as well as HRTEM observations of similar structures, e.g. [105], are interesting and significant. Energetically, the condition for a terrace-defect structure is

cinc L0 > cL0 þ K lnðL0 =r0 Þ;

ð48Þ

where cinc and c are the interfacial energies of the incoherent interface and terrace, respectively, r0 is the defect core radius, and K is the standard defect energy prefactor. Principle 1. The Burgers vectors of interface misfit dislocations are vectors of the CDP. Principle 2. In order to reflect the partitioning of strain and rotation, the F–B Burgers vector content B can be considered to partition to the two crystals, B = BP + BQ. 10.2. Misfit with added tilt To illustrate this case, we treat the same bicrystal as in 10.1 above, except that the P crystal is taken to be fcc rather than primitive. Hence the layer space group for the CDP is still pmmm. The translation vectors in the section of the CDP parallel to the terrace are unchanged. A pure misfit case, like that in Fig. 16, could be achieved by diffusional bonding, producing misfit dislocations, as before, with bm (derived from [0 1 0]P). However, if the bicrystal was grown by vapor deposition of P on a substrate Q, there would be no resolved coherency shear stress on potential pure misfit dislocations. Instead, formation of dislocations with Burgers vector bP, inclined to the surface so that a large resolved coherency stress acted on them, would be favored. At lateral surfaces, shear stresses would exist parallel to the interface but this would be an edge effect, confined to the lateral surface region. Observations of thin films of Ni vapor deposited on (0 0 1) Cu [106] verify the above postulates. For thin films, where the driving P forces for dislocation nucleation are small, slant dislocations, bP, with a misfit component by but also P with a component bz normal to the interface, were observed. Only for thicker films, with larger driving forces and where a larger density of dislocations was present, were the energetically favored Lomer dislocations formed in an increasing proportion. These dislocations were not nucleated directly, however, but formed as a consequence of interaction of slant dislocations with differing Burgers vectors. Thus, as a consequence of the tilt, Fig. 18 represents interface x0i RCDP coordinates, and v is defined in these coordinates. Let us consider one set of slant dislocations with bP derived from 1/2[0 1 1]P in the CDP: we can regard this defect as having a component by = 1/2[0 1 0]P, and a component, bz = 1/2[0 0 1]P. If we ignore bz, the treatment of Section 10.1 would apply, with ec negative. The addition of the bz components introduces a tilt wall rotation u in addition to the misfit array, as depicted schematically in Fig. 8. r 0 m 0 Thus, Burgers vector components b ¼ bz ¼ bz cosðu=2Þ and b ¼ by ¼ by secðu=2Þ are now defined in the RCDP, rather than the CDP. For crystal dislocations acting as misfit dislocations, we have trcP ¼ rc XPc tcP . Moreover the RCDP coordinates are rotated relative to the CDP coordinates by u/2 in P and u/2 in Q. The coherency strain in these rotated coordinates is related to that expressed in the natural CDP coordinates, ec, by the standard transformation, equation (B3). As shown in Appendix A, the (physical) partitioned rotation matrix is independent of such a rotational coordinate change. Here xPyz ¼ xPzy ¼  tanðu=2Þ and similarly for Q with a change in sign. Hence

0

rc En

0

0

B ¼ n Erc ¼ @ 0 ec cos2 ðu=2Þ 0

0

1

0

C A;

2

ð49Þ

e sin ðu=2Þ c

0

and

0 P rc Xc

0

B ¼ @0

0 0

0 tanðu=2Þ

0

1

C  tanðu=2Þ A; 0

0 Q rc Xc

0

B ¼ @0

0 0

0  tanðu=2Þ

0

1

C tanðu=2Þ A: 0

ð50Þ

786

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 18. Similar bicrystal as in Fig. 16, but with slant dislocations having components of their Burgers vectors parallel to y and z.

Thus,

c Xrc

¼

n

 P 1

rc Xc

0 0 o  1 B  rc XQc ¼ @0

0



0

0

0 2 tanðu=2Þ

1

C 2 tanðu=2Þ A:

ð51Þ

0

Now, we can substitute into Eq. (10),

B ¼ ðn Erc þ c Xrc Þv ¼ n Drc v :

ð52Þ

This result is now expressed in interface coordinates

0

0

0

0

10

0

x0i

1

where v =

0

v0y j

giving

B C 0 0 ec cos2 ðu=2Þ 2 tanðu=2Þ C A@ v0y A ¼ ec cos2 ðu=2Þv0y j þ 2 tanðu=2Þv0y k : 2 c 0 0 2 tanðu=2Þ e sin ðu=2Þ

B B ¼ @0

ð53Þ

Moreover, L = by sec3(u/2)/ec. The sin2 term is almost always negligible. The strain and rotational components of the distortion rcDn are assumed to be equally partitioned, so we may write B = BP + BQ. Another example of this type is given in [59] for the case of the monoclinic/orthorhombic martensite interface [103,107]. In that case the rotation u is sufficiently small that cos (u/2) 1. Examples with larger rotations are given in [19]. With misfit and superposed twist rotation, the result is analogous, described by Eqs. (49)–(53) with the following changes. The differences are that xPyx and xQyx , which were zero in the pure misfit case, are replaced, e.g for a rh screw, by xPyx ¼ tanðg=2Þ in P and xQyx ¼  tanðg=2Þ in Q. The results for the relatively simple single set of defects in the example of Fig. 18 anticipate the more general treatments of Appendix D and the twist plus misfit cases in Section 10.7 for multiple sets of defects. In the large L, small u limit, cos(u/2) 1, sin(u/2) is negligible, and the result reduces to a simple superposition of misfit and rotation as in prior work on disconnections [18], and as demonstrated in Section 10.8. Principle 3. With tilt rotation, u, present, Burgers vector components and dislocation spacings, all in the coherent terrace, are increased by a factor sec (u/2): e.g., bm becomes bm sec(u/2): Burgers vector components normal to the interface are decreased from br to br cos(u/2). For a twist wall the equivalent spacings in the interface and Burgers vectors br are decreased by a factor cos(g/2). 10.3. Superposition of pure rotations Matthews [108] demonstrated the possibility of lowering the energy of an array when a rotation is superposed. For a simple example, one could imagine a single array of lh screw dislocations in a crystal

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

787

(with long-range stresses). In the bicrystal context, we imagine shear misfit strain between a simple cubic crystal and a simple monoclinic crystal as illustrated in Fig. 19a. The layer space group for this CDP is p112/m. For this example, we consider large spacings so that the rotational functions and shear strains can be linearized and the difference between xi and x0i coordinates can be ignored. The section of the CDP parallel to the interface is depicted in Fig. 19a. To first order, the coherency strain cP c ecyy is negligible. The coherency strain parameter for set I is ecxy ¼ ecQ xy  exy ¼ bx =L, where xy is positive in this case, and the rotation of P relative to Q is given, in analogy to the previous case, by g0/2 = (bx/2L), where g0 is negative in this case. This is analogous to the misfit plus tilt example of the preceding section, with the equivalent of Eq. (52) given (in a form to cover both examples in Fig. 19) as

Fig. 19. (a). Misfit in shear between a simple cubic crystal and a simple monoclinic crystal. P and Q rotate by g0/2 relative to the CDP. A lh screw array cancels the misfit strain. (b) A rh screw wall with the same magnitude Burgers vector superposed on the array in (a). The spacing of both sets in (b) is twice the spacing in (a).

788

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

1 vx C CB 0 0 A@ vy A 0 0 0 0 h i h i  c 2 c 2 g0 ¼ exy cos ðg0 =2Þ þ g0 vy i  exy cos þ g0 vx j ¼ BP þ BQ : 2 0

0 B B ¼ @ ecxy cos2 ðg0 =2Þ þ g0

ecxy cos2 ðg0 =2Þ  g0 0

10

ð54aÞ

Thus, an array of lh screw dislocations, Fig. 19a, accommodates the coherency strain: for this case,

ecxy cos2 ðg0 =2Þ þ g0 ¼ 0, so the second bracketed term in Eq. (54a) is zero. The alternative array, Fig. 19b, also accommodates the misfit and can be created from that of Fig. 19a by superposing a pure twist wall of rh screws with lines along x and y, with spacings 2L. The rh screws with n in the x direction annihilate one half of the lh screws in Fig. 19a, changing their spacing to 2L. Those with n in the y direction remain with spacing 2L. The rotation fields of the lh screws and rh screws in Fig. 19b cancel so B then has no rotational component. The values of the coherency strains for each set would be ecxy =2 Thus, the two sets would sum to give

B ¼ ecxy vy i  ecxy vx j;

ð54bÞ

the same as the limiting form of Eq. (54a) when g0 = 0. That is, the misfit is still removed, but the superposed twist wall produces a rotation g0 that cancels the rotation g0 of Fig. 19a. The total line length is the same in both figures, but the line energy of the dislocations changes from (lb2/ 4p)ln(L/r0) for Fig. 19a to (lb2/4p)ln(2L/r0) for Fig. 19b, where l is the shear modulus (or the average shear modulus if elastic inhomogeneity is included in the analysis). Hence, if there is no constraint to rotation, the array of Fig. 19a would be favored. If there are compatibility constraints to rotation, the array ofFig. 19b would be favored. We emphasize that ecxy and g0 in Eq. (54) are partitioned to cP ecxy =2 ¼ ecQ xy  exy =2 and ±(g0/2) in the two crystals. Thus, for alternative equilibrium arrays like those in this example, the array with the smaller L spacing (smaller b) would be favored because the total energy of an array scales as (b2/L)ln(L), or since, with em xy fixed b / L, as L ln(L). The limit occurs when a smaller b would cause a high energy fault to appear on one or the other side of a defect. There are two degrees of freedom for both arrays in Fig. 19. For that in Fig. 19a the two principal strains are fixed by one set of dislocations and the other is unconstrained since the rotation is free. For that in Fig. 19b, the rotation is fixed to be zero, and two dislocation sets are needed. For the screw array in Fig. 19b, obviously the rotations produced by the arrays are equal and opposite. With multiple arrays and with mixed dislocations present, one selects the same v0 for each of the i sets of dislocations present and the screw components bs for each. The rotation is as follows.

g¼

rh lh X X 2 tan1 ðbs =2LÞi  2 tan1 ðbs =2LÞi ; i

ð55Þ

i

or, to linear order

g¼

rh lh X X ðbs =LÞi  ðbs =LÞi : i

ð56Þ

i

The rotation g is zero when the two sums are equal and opposite, i.e., the total rh contribution to the rotation in B is equal and opposite to the total lh contribution. Principle 4. The addition of pure rotations to an array that removes misfit, changes b and L and hence raises or lowers the energy of the array of interfacial defects. Trend 1. For arrays of defects with the same screw-edge character, small b and L values are favored energetically. In the example shown in Fig. 19b, the angles between the line directions of sets I and II deviate slightly from 90°. Thus, the strains do not cancel exactly and either another set of defects or a slight variation in the n values would be needed make the strains zero. The corrections are small here, and we postpone such refinements to Sections 10.7 and 10.8.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

789

10.4. Conversion of edge array to screw array The conversion of arrays [109] is relevant to the above Matthews model [108]. Consider two face centered orthorhombic crystals rotated with respect to one another by 90° as in Fig. 20a. The CDP has square symmetry in the plane of the interface, with layer space group p4/mmm. The coherency strain cP is pure shear with ecP xx ¼ eyy and similarly for Q. The solution is similar to that of Section 10.1 applied in both the x and y directions. In this example, the coherency strain parameter, ecxx ¼ ecI , is positive, and ecyy ¼ ecII , is negative. The total discrete Burgers vector content is

Fig. 20. Two orthorhombic crystals with the same lattice parameters rotated by 90° with respect to one another. The projected CDP has square symmetry in the interface plane. (a) Misfit strain accommodation by two sets of edge dislocations: extra half planes are in z > 0 for set I, in z < 0 for set II. (b). Misfit strain accommodation by rh and lh sets of screw dislocations.

790

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

0

ecI

B B¼@ 0 0

0

0

10

vx

1

B C ecII 0 C A@ vy A ¼ ecI vx i  ecII vy j: 0

0

ð57Þ

0

Hence, one possible solution is the orthogonal array of edge dislocations in Fig. 20a. These have II  0 0, where the former set of edges has its added plane in P, Burgers vectors bI = [0 1 0] and b ¼ ½1 I I mI the latter set in Q. Also, L = b /e and similarly for LII. III  0 Another possibility is the array of lh and rh screws in Fig. 20b with Burgers vectors b ¼ 1=2½1 1 IV 0 and b ¼ 1=2½1 1 0; respectively. These produce pure shear strains in coordinates xi fixed on the dislocations, but when rotated into the xi coordinates, the strain matrix is identical to that in Eq. (57). Also when the x and y components of (b/L) are summed, the resultant also is the same as Eq. (55). Hence the solutions both relieve misfit and both have zero rotations from the CDP orientation. One solution can convert to the other by local reactions at nodes [109]. Hence intermediate networks with mixed dislocation character are also possible. The ratios of screw to edge dislocation spacings and Burgers p vectors are Ls/Le = bs/be = 2/2. The ratio of energies of the arrays is

Gs ð1  mÞ lnðLs =r 0s Þ ¼ pffiffiffi ; Ge 2 lnðLe =r 0e Þ

ð58Þ

where Poisson’s ratio m arises because of the difference in the energy prefactors for screws and edges. Here the character factor (1  m) and the Ls/Le spacing ratio both favor the screw array energetically. However, r0 is usually smaller for screws. Thus in the limit of small L, where the core terms become dominant, the edge array can have lower energy. Another similar case would arise for two simple orthorhombic crystals. Again there would be screw p and edge arrays but now Le/Ls = 2. Now only the (1  m) character factor favors the screw array, and the edge array can be favored over a larger range of L values. Trend 2. For screw versus edge arrays with the same L, screw arrays usually have lower elastic energy. This trend can be reversed in the limit of small L, where core terms become dominant. 10.5. Three or more dislocation sets Fig. 21 shows CDPs for three different solutions that produce a pure twist boundary in between two fcc crystals. As already mentioned, n values are assumed, so other solutions are possible as shown in Section 10.7. Here we consider the large L limit so that line tension at the nodes dominates the configuration. When superposed, the strain components cancel in each case. For Fig. 21a, the Burgers vectors bI, bII and bIIIare all of the type 1/2h1 1 0i. As described in Section 10.1, the Burgers vectors are referred to the single crystal values and it is implicit that their lengths change in the RCDP. The Set I comprises rh screws. The edge components of sets II and III cancel, so they are tantamount to a single p array of rh screws, set IV, with Burgers vector bIV = ( 3/2) bII and spacing LIV. With (bIV/LIV) = (bI/LI) the strains of the two arrays cancel and one has a pure total rotation g(a) = 4 tan1(bI/2LI). This rotation has a contribution 2 tan1(bI/2LI) from each set and, in turn, this rotation is partitioned to a rotation ±tan1(bI/2LI) in each crystal P or Q. For Fig. 21b, there are three equivalent sets of screws with b of the type 1/2h1 1 0i. The total rotation in this case is g(b) = 6 tan1(bI/2LI). For the set in Fig. 21c, node reactions for (b) proceed to form the final, equilibrium, hexagonal network in (c). Each segment is a pure rh screw and the Burgers vectors are again of the type 1/2h1 1 0i. As discussed in Section 8 and shown in Fig. 21d, a parallel group of segments from one set is equivalent to an array like one of those in Fig. 21b but with Burgers vectors bIV = (1/6)h1 1 0i since X = 1/3 in this case. Hence, in this example, the rotation is 6 tan1(bIV/2LIV) = 6 tan1(bI/6LI). In order to decide which array is favored, we set the three rotations equal and compare energies. That for array (a) is complicated by the different Burgers vectors, so we give an approximate result to facilitate a comparison. Because all sets have screw character, the energies depend only on b and L. Here, all b’s are normalized to b = 1/2h1 1 0i and all L values are normalized to L = LI in (c). The L values scale as L(a):L(b):L(c) = 2:1.5:1. From lowest to highest energy, the result is G(c):G(a):G(b) = (b2/ p 3L)ln(L/r0): (b2/L)ln(2L/r0): (b2/L)ln( 3L/2r0). The dominant factor is the small Burgers vector for

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

791

Fig. 21. Three twist arrays producing a pure rotation at the interface with parallel {1 1 1} planes in P and Q in a fcc crystal.

(c). The energies of (a) and (b) are close and essentially are only distinguished by the logarithmic term. The possibility of forming networks with discontinuous line lengths is the basis for creating the small effective Burgers vectors that lead to low energies. The preponderance of three set networks in TEM observations [24] is undoubtedly related to this possibility. Trend 3. This large L result indicates that solutions with two dislocation sets are not always favored over three sets. Two sets would tend to be favored if the shortest Burgers vectors in the CDP are close to being orthogonal. Three dislocation sets would tend to be favored if the shortest vectors are close to 60°. 10.6. The Nishiyama–Wasserman interface As a final simple example, we treat the Nishiyama–Wasserman, (0 1 1)a/(1 1 1)c interface for a ferrous system. The interface projection of the CDP is presented in Fig. 22, with Q and P the a and c phases, respectively. Other orientations, such as KS, are treated by superposing the appropriate twist rotation, so that the Burgers vectors of admissible defects are then obtained from the relevant RCDP, Section 10.7. Again, the indexing of the a and c phases is referred to the perfect Q and P crystal pre 0 ; ½0 1 1  ==½1 1 2  , and [0 1 1]a//[1 1 1]c. The NW CDP exhibits the p2/m11 layer cursors, ½1 0 0a ==½1 1 c a c

792

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 22. Centered-rectangular translation symmetry of the CDP for the NW interface between Q, bcc a, and P, fcc c. The unit cell  0 , ½0 1 1  ==½1 1 2  , and [0 1 1]a//[1 1 1]c. indices referred to bcc or fcc are ½1 0 0a ==½1 1 c a c

space group [12,27–29]. The symbol 2/m represents symmetry elements common to the two phases:  1 0 =½1  0 0 , and the mirrors are perpendicular to this. The symbol ‘‘p’’ the 2-fold axis is parallel to ½1 c a means 2-D translation symmetry in the x, y plane, and the common translations, tcP = tcQ, form a centered rectangular array. A typical coherency strain parameter for ferrous systems [47] is ecxx ¼ 0:1254 II and ecyy ¼ 0:0772. A solution is represented in Fig. 23 with emII ¼ ecxx ¼ b =LII and III I mIII mI c ðe þ e Þ ¼ eyy ¼ by =L . The screw components of sets III and I are equal and opposite so there is no net screw component and no rotation. Hence, analogous to the previous example, sets III and I  1 , i.e., with Burcan be replaced by virtual edge sets V and VI, not shown, with Burgers vectors 1=2½0 1 a gers vectors equal to the edge components of bIII and bI. Sets V and VI would still have the spacing LIII. Set IV represents partial dislocations not discussed here, but used in Section 10.8, where they correspond to components of disconnections. Represented in the fcc and bcc natural crystals, the partial  1 1 or 1=12½3 1  1 , notations often used in phase transformations, e.g., [2,3]. is 1=12½2 c a Here, v points down and to the left. The solution is the superposition of two solutions like that in Eq. (57), although here ecxx is negative and ecyy is positive.

0

ecII

B B¼@ 0 0

0

ecIII 0

10

1 vx CB C 0 A@ vy A ¼ ecII vx i  ecIII vy j: 0 0 0

Fig. 23. Misfit dislocation array that relieves the coherency strains in Fig. 22.

ð59Þ

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

793

There could also be a three pure edge solution analogous to the screw equivalent in the preceding example. We mentioned earlier that resolved shear stress could cause an energetically less favorable interface to form. The dislocations in sets I and III in Fig. 23 could glide into the interface as screw dislocations and then glide within the interface into the mixed configuration. If they were to form a three edge configuration, the glide within the interface could be deterred by the nodes that form or by some nonzero resistance to glide in the interface. Also, for small L, the self-energy of the added nodes could disfavor the three set case. Like sign screws or edges with added planes in the same direction relative to the interface repel one another at nodes. Any associated standoff would lead to jog/kink formation which could act as pinning sites. The opposite case, e.g., opposite sign screws, attract. The junctions could lead to pinning in this case, although kink formation can be enhanced at such nodes [112]. All of above are ‘‘simple’’ in the sense that the character of dislocations is fixed in each case. The next examples are more complex in that n is a variable. Many previous principles apply to these examples. 10.7. Numerical solutions All previous examples were simple in the sense that pure screws, pure edges, or mixed dislocations with fixed values of n, i.e., fixed screw:edge ratios, were considered. We now turn to more complicated cases where the n vectors are variable. The Cu/Nb, c/a, system is chosen to provide numerical examples in both the NW and KS ORs. Interfaces in layered composites in this system, produced by PVD and by rolling diffusion-bonded products, have been an important focus of experimental work [113–115]. Indeed, interfaces with KS OR have been observed in PVD composites and interfaces with NW and KS coexisting have been observed in rolled diffusion-bonded composites. Burgers vectors, determined from the RCDP, depend on twist rotation, but the twist rotation that converts the NW OR to KS in this system is so small, that, as we show below, the Burgers vectors in the RCDP are virtually the same as in the CDP. However, for larger twist rotations, solutions for misfit line directions and spacings should, correctly, involve Burgers vectors in the RCDP, not those derived from the CDP or even the NDP. To demonstrate the magnitude of errors involved in using the non-partitioning reference states of Eqs. (15) and (16) we also present results for a hypothetical system where the twist rotation is relatively larger. We begin with the NW OR [110,111] with [1 1 1]ck[0 1 1]a as the interface normal, and coordinates as in Fig. 22, but with coordinates rotated as in Fig. 24 for convenience in the use of numerical pro ==½1 1 2  and y==½1  0 0 ==½1  1 0 . grams. The interface coordinates are z==½0 1 1a ==½1 1 1c ; x==½0 1 1 a c a c In the following, the crystals P and Q correspond to Nb and Cu, respectively. The NW NDP projected onto the interface plane for this system is shown in Fig. 25a. The difference between the sizes of the [1 1 1]c face and the [1 1 0]a face have been exaggerated for clarity. Partitioning of the strains between the two lattices creates the CDP whose projection along [1 1 1]ck [1 1 0]a is shown in red in Fig. 25a. The layer space group of the CDP is p1 m2 1. The translation vectors, tcP = tcQ, in the unique plane, are the Burgers vectors of admissible dislocations, including bI, bII and bIII, depicted in Fig. 24. The m2 symmetry  =½0 0 1 and the mirror planes perpenalong y arises because of the coincident 2-fold axes along ½0 1 1 c a dicular to this which are not broken by the strain nEc. For the KS relationship, the P and Q lattices are further misoriented by the rotations  g2 about [1 1 1]ck [1 1 0]a, where g 5.26°. When this additional rotation is applied to the NW CDP, we obtain the KS RCDP, Fig. 25b. This breaks the NW CDP translation symmetry: now, the Burgers vectors of perfect interfacial dislocations are given by combinations of rotated translation vectors consistent with equation (A2), which we write in general form as bg = trcP  trcQ. As discussed in Section 4.3, these can decompose into partial dislocations with br = bg/2: these Burgers vectors are very close in magnitude (in the RCDP the Burgers vectors are reduced by cos(g/2) = 0.9989) and have the same direction as the vectors of the CDP, bI, bII and bIII, depicted in Fig. 25a. In addition to breaking translation sym As a consequence, two variant KS RCDPs metry, the point symmetry of the KS RCDP is reduced to 1. arise, inter-related by the broken 2-fold and mirror symmetry of the NW CDP: these are equivalently inter-related if the sense of the additional rotations ±g/2 applied to P and Q is changed. The strain tensors needed for equal partitioning (j = 0.5, Appendix D) are, for the P (bcc) and Q (fcc) crystals,

794

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 24. Coordinates for NW CDP.

Fig. 25. A portion of the dichromatic patterns projected onto the interface plane for Cu (blue) and Nb (black). (a) The NW OR with the CDP shown as red. (b) The KS OR where a twist has been applied to each crystal in the CDP. The RCDP is red. Burgers vectors, br, correspond to the sides of each polyhedron.

2 P c En

6 ¼ c EQn ¼ 4

0:02647 0 0

0

0

0

0

3

7 0:12922 0 5:

ð60Þ

Clearly, the coordinates chosen here correspond to principal strain axes, and the coherency strain matrix becomes

2

3 0:05294 0 0 6 7 P Q 0 0:25844 0 5: c En ¼ c En  c En ¼ 4 0 0 0

ð61Þ

795

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Perfect Burgers vectors can now be chosen from the translation vectors in the projected CDP and we select

2

3 0:227385 6 7 I b ¼ 4 0:146424 5;

2

3 0 6 7 II b ¼ 4 0:292847 5;

0

ð62Þ

0

where the dimensions, here and throughout the section, are in nm. Having defined the Burgers vectors, we can now determine the line directions, n, and spacings, L, for the M sets of interface dislocations, I, II, III, considered following Appendix D. The solution of the F–B eq. in Appendix D, given by equations (D12), is convenient for computational purposes and is used in this section. In particular, as equations (D12) reduce to four equations when all Burgers vectors are contained in the interface plane, as in this case, they can be solved in closed form for M = 2, i.e., when there are only four unknowns, nI, nII, LI, and LII. Two distinct M = 2 solutions, given as solution 1 and 2 in Table 2, with v as the angle between the x-axis and the line direction and b as the angle between the Burgers vector and the sense vector, Fig. D1b. For solution 1, the Burgers vectors are bI and bII. For solution 2, the Burgers vectors are bI and bIII = bI  bII. The line directions and spacings are shown schematically in Fig. 26a and b for solutions 1 and 2, respectively. In both solutions there are line directions that do not align with low index crystallographic directions. Such a configuration could have a greater energy than one with larger screw components, because the latter has lower line energies than for edge components, and so may be energetically unfavorable in a real fcc/bcc bicrystal. In addition, the arrangement in solution 2 would certainly be unstable if the L spacings were large: there is not a balance of line tensions and the two sets of dislocations would react at the nodes forming a third set of pure edge dislocations aligned parallel to the x-axis, with all angles among the dislocations closer to 120° as suggested in Fig. 26c. However, as discussed in Section 6, when L spacings approach nm lengths as for Cu–Nb, interaction forces among the dislocations become important and the configuration of Fig. 26b can become stable. Reactions to form nodes for Fig. 26a are less likely. As discussed in Section 8, solutions such as that in Fig. 26c are tantamount to M = 3 solutions. For M = 3, given three Burgers vectors obtained from the CDP, there are six degrees of freedom, so equations (D13) in Appendix D have many solutions (literally an infinite number). Additional conditions, e.g., invoking line energies, would be needed to obtain line directions and spacings as discussed in Section 7. However, we present two examples where we ‘‘guess’’ two of the unknowns and solve for the remainder. The Burgers vectors are bI, bII, and bIII as in solutions 1 and 2. Solutions 3 and 4 in Table 2 show results for solution 3 with vI = vIII = p/2. In solution 4 line directions are assumed to be vI = 32.78° and vIII = 147.23°. The line directions assumed in both solutions 3 and 4 correspond to low-index crystallographic directions in the CDP. The line directions, spacings, and Burgers vectors for solutions 3 and 4 are shown schematically in Fig. 27. The dislocation content in solution 3 is about one half of that in solution 4, and both have a smaller content, and hence lower elastic energies, than the two M = 2 solutions in Fig. 26. That, in addition to the fact that the dislocations in Fig. 27 lie along low-index directions, would tend to favor M = 3 relative to M = 2.

Table 2 Dislocation spacing and line directions for various M = 2 and M = 3 solutions in the NW OR (solutions 1–4) and KS OR (solutions 5– 7) described in the text. The superscript in each case identifies the dislocation set. Solution

LI (nm)

LII (nm)

LIII (nm)

vI (°)

vII (°)

vIII (°)

bI (°)

bII (°)

bIII (°)

1 2 3 4 5 6 7

4.296 1.123 8.591 4.651 1.241 2.142 _

1.123 _ 1.133 1.425 _ 0.906 1.241

_ 1.123 8.591 4.651 0.906 _ 2.142

90.00 7.514 90.00 32.78 32.30 150.05 _

7.514 _ 0 0 _ 10.31 32.30

_ 172.49 90.00 147.23 169.69 _ 150.05

57.22 139.71 57.22 114.35 114.92 2.83 _

97.51 _ 90.00 90.00 _ 79.69 57.70

_ 40.28 122.78 65.55 22.47 _ 62.73

796

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 26. Schematic diagram of the M = 2 sets of dislocations for the two solutions for the NW OR in Cu/Nb. The n are the sense vectors of the dislocation lines and the small colored arrows Burgers vectors. (a) Solution 1. Set I is red and set II is green. (b) Solution 2. Set I is red and set III is blue. (c) Modification of solution 2 that shows the likely reaction between sets I and III.

Finally, the KS OR derives from applying a rotation that twists one crystal by g/2 and the other by g/2 relative to NW. As noted above, rotations in the ±g sense are energetically and topologically equivalent [52]. Adding the rotation to the coherency strain matrix in Eq. (61) gives the distortion matrix

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

2

797

3

0:052935 0:091881 0 6 7 D ¼ 0:091881 0; 25844 0 5: 4 rc n 0

0

ð63Þ

0

The projected RCDP is that in Fig. 25b. Burgers vectors, given by bg = (trcP  trcQ)/2, are smaller by cos(g/2) in the RCDP than in the CDP because of rotation, but, because g 5.26° is small, the differences in the components of Burgers vectors are quite small, 104 nm, in this case. Here, again, solutions for two sets of dislocations are obtained directly from equations (D12). In contrast to the NW OR, there are now three distinct M = 2 sets of solutions. The results are given as solutions 5–7 in Table 2. Schematic views of these three sets of M = 2 dislocation arrangements are in Fig. 28. Solution 6 is very similar to that found previously in MD simulations of an atomistic model of Cu/ Nb with KS OR [89]. The measurements of both the line orientations and the spacing of set I are nearly identical to values in Table 2. However, in [89] the computed spacing for set II of 0.89 nm was based on the Q reference state instead of the CDP used here, i.e., Eq. (16) was used, which accounts for the result being smaller than the 0.91 nm in Table 2. We now consider two hypothetical examples involving a much larger twist rotation than in the NW to KS case given above, but wherein the Burgers vectors remained essentially unchanged. In previous

Fig. 27. Schematic of the M = 3 sets of dislocations for solutions 3 and 4 for the NW case. In each case, two line directions were assumed as described in the text. (a) Solution 3. Set I is red, set II is green and set III is blue. (b) Solution 4. Set I is red, set II is green and set III is blue.

798

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. 28. Schematic diagram of the M = 2 sets of dislocations for the three solutions in the KS OR listed in Table 2 with sets I, II and III shown as red, green and blue, respectively. (a) Solution 5. (b) Solution 6. (c) Solution 7.

studies [50,55–57,89,116–118], estimates of dislocation arrangements assumed the Burgers vectors to be those of the CDP or even the NDP, and here we also compare the consequences of those assumptions with results involving Burgers vectors extracted from the RCDP. Again three sets of M = 2 dislocations arrangements are obtained for the case of an NW CDP plus an imposed twist rotation. For a twist rotation of g = 20° the Burgers vectors in the RCDP are significantly different than in the CDP, e.g.,

799

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

2

3

0:223933 6 7 I b ¼ 4 0:114200 5;

2

3

0 6 7 II b ¼ 4 0:288401 5;

0

ð64Þ

0

and bIII = bI  bII, as before. In Table 3, solutions 8A (using bI and bIII), 9A (using bI and bII), and 10A (using bII and bIII) are the three M = 2 solutions for the correct Burgers vectors in the RCDP. Solutions 8B, 9B, 10B are for equivalent Burgers vectors in the NW CDP, Eq. (62), while solutions 8C, 9C, and 10C are for results using the NDP vectors. Neither of the latter two solutions correctly treats partitioning. For solutions B, The Burgers vectors have the wrong length, and for C, they have both a less accurate length and the incorrect initial values for n. The differences in L between A (RCDP), B (CDP), and C (NDP) solutions are significant, roughly up to 10%. Also, compared to Table 2, the L values for the 20° rotation are much smaller, as expected. The line directions and Burgers vector orientations are the same to four places for solutions A and B in this orientation but quite different than in KS, Table 2, solutions 5–7. For the C solutions, the Burgers vector orientations are quite different but the line directions are only slightly different. The net rotation produced by the screw parts of the dislocations, Eq. (55), is also computed and found to be 20° to four significant figures, consistent with the imposed conditions. The major feature is that the deviation of 10% between the A and C solutions is larger than for the KS case of Table 2. This arises essentially because the solution is not an equilibrium solution. Physically, the differences relate to the changes in Burgers vector length and character. Table 4 presents three solutions for M = 2 dislocation sets for the case of an NW CDP plus an imposed 40° twist rotation. As before, solutions A are for Burgers vectors in the RCDP, B are for Burgers vectors in NW CDP and C results using the NDP vectors, in fcc Cu. From this RCDP the Burgers vectors are

Table 3 Dislocation spacing and line directions for various M = 2 solutions (8–10) for an NW CDP plus a twist rotation of g = 20°. The superscript in each case identifies the dislocation set. Solution (M)

LI (nm)

LII (nm)

LIII (nm)

vI (°)

vII (°)

vIII (°)

bI (°)

bII (°)

bIII (°)

8A 8B 8C 9A 9B 9C 10A 10B 10C

0.743 0.755 0.667 0.628 0.638 0.621 _ _ –

_ _ – 0.497 0.505 0.476 0.743 0.755 0.667

0.497 0.505 0.476 _ _ – 0.628 0.638 0.621

85.36 85.36 86.71 171.46 171.46 171.69 _ _ –

_ _ – 33.27 33.27 36.96 85.36 85.36 86.71

146.73 146.73 143.04 _ _ – 171.46 171.46 171.69

61.86 61.86 63.29 24.24 24.24 21.69 _ _ –

_ _ – 56.73 56.73 53.04 4.64 4.64 3.29

0.49 0.49 6.96 _ _ – 41.32 41.32 38.32

Table 4 Dislocation spacing and line directions for three M = 2 solutions (11–13) for an NW CDP plus a twist rotation of g = 40°. The superscript in each case identifies the dislocation set. Solution (M)

LI (nm)

LII (nm)

LIII (nm)

vI (°)

vII (°)

vIII (°)

bI (°)

bII (°)

bIII (°)

11A 11B 11C 12A 12B 12C 13A 13B 13C

0.348 0.370 0.327 0.293 0.312 0.303 – _ –

– – – 0.274 0.292 0.269 0.348 0.370 0.327

0.274 0.292 0.269 – – – 0.293 0.312 0.303

105.43 105.43 104.41 175.84 175.84 175.95 – _ –

– – – 43.66 43.66 47.23 105.43 105.43 104.41

136.34 136.34 132.77 – – – 175.84 175.84 175.95

41.79 41.79 45.59 28.62 28.62 25.95 – – –

– – – 46.34 46.34 42.77 15.43 15.43 14.41

10.88 10.88 17.23 – – – 36.94 36.94 34.05

800

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

2

0:213674

3

6 7 I b ¼ 4 0:137594 5; 0

2

0

3

6 7 II b ¼ 4 0:275189 5:

ð65Þ

0

The differences in L among the A–C solutions are again significant, up to 10%. Compared to Table 2, the L spacings for the 40° rotation are much smaller. Indeed, the spacings are of the order of core dimensions and core overlap would dominate the atomic structure. Line orientations of the C solutions are hardly different from the A/B results, but, of course, very different from those of the three KS OR cases. The combinations of pairs of translation vectors used to determine the Burgers vectors in the CDP of NW, and in the RCDP’s of KS, g = 20°, and g = 40° cases, were identical. However, for even larger rotations they may not be the same. As mentioned in Appendix C, symmetry limits the maximum rotation angle for which a given Burgers vector is meaningful [37,119]. To see this, consider, a simple twist rotation of an fcc crystal about a h111i axis. At g = 60°, the two crystals are in a twinned orientation, and the interface is a twin fault but can be viewed as containing no discrete dislocations or as containing partials every atomic spacing (see the tilt analog in Fig. 23-3 in [42]). For rotations larger than 60°, misfit dislocations would have finite spacing but their Burgers vectors would derive from a different combination of pairs of translation vectors than those for g < 60°, i.e., at 60°, there is a transition in the combination of pairs of translation vectors determining the Burgers vectors because h1 1 1i is a 3-fold rotation axis. For the Cu/Nb system, there is a similar, but somewhat more complex geometry as an examination of Fig. 25 shows. In this system, a transition occurs at a somewhat smaller twist angle, g = 57.22°, and at this orientation the interface separates the two crystals in a pseudotwin, probably high-energy, orientation. In the fcc analogue, the perfect crystal is achieved at g = 120°, but in Cu/ Nb, another, pseudotwin orientation occurs at g = 114.44°, a second transition angle. Finally, at g = 180°, the Cu/Nb system is returned to the CDP configuration. 10.8. Phase transformations Section 10.7 presented exact numerical solutions for several cases within the approximation of the standard model. The results showed that for rotations of less than 5° and with defect spacings L greater than several nm, effects such as the change of the CDP values for b and L when twist rotation is superposed are very small. For tilt rotations the effects are much larger. However, for phase transformations, the disconnection tilts are small, usually 1° or less e.g. [107]. If one neglects these changes, more complex solutions can still be derived to a good approximation by manual iteration, with rapid convergence. Typically, for phase transformations, these approximations give accurate results. The following example of martensite in steel illustrates the method. We present the example of the (5 7 5)c habit plane of lath martensite in steel [47], where the array of defects is comprised of disconnections and dislocations. We begin with the NW CDP, shown in Fig. 22: as before, this CDP exhibits the layer space group p2/m11. For convenience, we employ the same indices for these translations in the CDP as in the NDP, although of course, they are slightly different. The principal coherency strain matrix cEn in these coordinates is

0 B c En ¼ @

0:12526 0 0

0

0

1

C 0:07717 0 A: 0 0

ð66Þ

The Burgers vectors of admissible dislocations (lattice-invariant deformation, or LID) and the Burgers vectors and step heights of disconnections were determined from this NW CDP. Again, we describe Burgers vectors with crystallographic indices for the undeformed NDP crystals. Among the g  1 or bIII in Fig. 22: its comformer is the crystal dislocation with in-plane Burgers vector, bLID ¼ 12 ½1 1 a ponents expressed in nm in the x, y, z frame are [0.135, 0.211, 0]. The disconnection with Burgers vecg 1  0 and tcQ ¼ 1 ½1 1  1  , tor, bD , is derived from the difference of two out-of-plane translations tcP ¼ 12 ½1 c a 2  g  g      which has components [0.135, 0.141, 0.004], so b ¼ 0:78 b . In Fig. 22, this vector is shown as D

LID

801

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

t cQ t cP bgD

g

Fig. 29. Schematic illustration of the formation of the disconnection: only the small edge component of bD is visible in this 1  1 projection. The step character is clearly seen. The open symbols represent the . . .ABC. . . stacking along [1 1 1]c and  0 1 =½1 ½1 c a the filled ones the . . .AB. . . stacking along [0 1 1]a.

IV

g

b ¼ bD . The overlap step height is one (0 1 1)a inter-planar spacing with downwards sense when traversing along y from left to right. These features are illustrated schematically in Fig. 29, which, like Fig. A1a, shows the stepped c and a surfaces before bonding to form coherent interfaces on either side  0 1 =½1 1  1 , is almost in screw orienof the defect. The disconnection line direction, nD, taken to be ½1 c a g

tation in the figure and hence only the small edge component of bD is visible. This disconnection can move conservatively across the terrace plane, as is essential in martensitic transformations [18]. We analysed two set, M = 2, arrays using Eq. (10), taking additional rotations c XPrc and c XQrc to have magnitudes ±g/2, which spanned the range of misorientation relationships from NW to KS. For such small supplementary rotations, we assumed that the Burgers vectors remain those for the CDP, rather than the slightly different values that pertain in the RCDP. Solutions to Eq. (10) were obtained iterag tively. In the first step, the component bz of bD was suppressed and the step character of the disconnections was ignored temporarily. Thus, at this stage, the habit plane is taken to coincide with the terrace plane. The network predicted using the F–B eq. for the case where g = 2.5°, i.e. about midway between NW and KS, is depicted in Fig. 30. The disconnections are close to rh screw orientation, resembling the situation shown in Fig. 19, and the crystal dislocations are close to lh screw orientation. The disconnection and LID dislocation spacings, LD and LLID respectively, are also shown.

Fig. 30. Network of disconnections and dislocations for g = 2.5°.

802

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Subsequently, the above solution was refined as follows. First, the disconnection step character was taken into account, rotating the interface by h1 = tan1(h/LD) about nD. The new coordinates, x0 , y0 , z0 , were defined with x0 parallel to nD and z0 perpendicular to the habit plane. We imposed equipartitioning of tilt, uD, so that the habit plane was inclined to the terrace plane by h ± uD/2, as explained in Section 5.1. Similarly, the tilt uL associated with the LID dislocations was determined. Coherency strains and rotations, rc Dinf n , for the up-dated bicrystal were then calculated, and an up-dated defect network found using the F–B eq. This procedure was repeated until the resulting changes to network parameters became small: the solutions thus obtained were found to converge relatively rapidly, and a full account is given elsewhere [120]. As a final check, the distortion matrix rcDn was calculated (Section 2 and Appendix D), and shown to be virtually identical to rc Dinf n . For g = 2.5°, the tilts introduced by the LID and disconnections were 0.54° and 0.14° respectively, and the habit plane was (0.5043, 0.7007, 0.5043)c, very close to the experimentally observed habit [121]. The small values of uD and uL justify the small angle approximation in the iteration. This defect network resembles that discussed in Section 10.4, where the combination of lh and rh near-screw defects accommodates principal strains of opposite sign, and a superposed twist-producing array modifies the initial defect spacings and orientations. Sandvik and Wayman [121], using TEM to investigate  1 and spacings beFeNiMn alloys, observed an array of crystal dislocations with Burgers vector 12 ½1 1 a tween 2.6 nm and 6.3 nm oriented between 12° and 15° from screw, consistent with Fig. 30. Moreover, Moritani [122] used high-resolution TEM to view the disconnection structure of this alloy along  0 1 =½1 1  1 , and observed an array consistent with Figs. 29 and 30. ½1 c a There are further TM solutions for martensites in Fe alloys [47]: these invoke combinations of other disconnections and LID, and correspond to experimentally observed habits. We mention one other possibility in order to connect to Section 8. In this case both the disconnections and LID are in near edge orientation, which would give a solution corresponding to a small tilt from the NW OR. For example, the LID could be the Burgers vectors bIII as described above but now in edge orientation. Similarly, the disconnection vector could be bIV, also in edge orientation. As discussed in Section 8, an edge disconnection solution of this type should be more efficient in removing coherency strains, although other factors would favor screw arrays. An edge solution of this type would be possible for any bcc/fcc martensite and, in a pole figure, would give a habit plane rotated along a great circle from [111]c toward one of the h1 1 2ic directions in the (1 1 1)c plane. Yet experiments reveal a variety of habit plane orientations, including the one described in Fig. 30 and usually deviating from that of the edge solution. This indicates that there are kinematic constraints that mitigate against the edge solution. In particular, resolved coherency strains favor LID on slant glide planes, Section 8, that would perforce involve rotations. Finally, for lenticular plate martensite, the transformation disconnections are present as loops, with all orientations appearing around the loop. Hence, several variants with different defect sets are possible in different regions of a plate [52] and the edge solution could appear even though not the most favored case. 10.9. Defects arising from mismatching point symmetries In this section, we illustrate one example of an interfacial defect that arises from a mismatch of point symmetries between crystals P and Q. For simplicity, we choose a lattice-matched epitaxial sys tem: thus the two cubic lattices (symmetry Fm3mÞ have identical parameters and orientations, so the CDP is a grey cubic lattice (the NDP is degenerate with this). An ideal (0 0 1) bicrystal would be perfectly coherent, with layer space group p4mm. Let the substrate crystal, Q, be Si, which has the non symmorphic space group Fd3m, and the epi-layer, P, be NiSi2, which exhibits the symmorphic space  group Fm3m. Now, the substrate surface can exhibit demi-steps, as illustrated schematically in Fig. 31a: here, the surfaces separated by the step are inter-related by the 4-fold screw operation parallel to [0 0 1], designated 4þ 1 . The two surfaces are energetically degenerate, with structures relatively rotated about [0 0 1] by p/2, and the step height is hQ = 1/4[0 0 1]. By contrast, the epi-layer surface cannot exhibit complementary demi-steps because its space group does not include 4þ 1 . Thus, to accommodate such a substrate demi-step, the epi-layer may grow continuously across it, as indicated schematically in Fig. 31a. In consequence, an interfacial dislocation separating energetically degenerate regions of interface is formed, as depicted schematically in Fig. 31b.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

803

Fig. 31. Schematic view along [110] of a dislocation formed between lattice-matched epitaxial NiSi2 grown on a [0 0 1]Si substrate. (a) The substrate exhibits a demi-step, while the epilayer is flat. (b) After bonding, energetically degenerate regions of 1  1.  interface are separated by an interfacial dislocation with b ¼ 1=4½1

The character of the resulting defect is given by equation (A2): the operation WP is the 4-fold rotation, [4+/0] and (WQ)1 is the 4-fold screw rotation, [4+/w]. The origin for both crystals is chosen to be a centre of symmetry, and then, w (which defines the screw displacement and location of the axis rel1  1Þ:  in other words, the ative to the origin) is equal to 1/4[1 1 1]. Thus, the product W P W Q ¼ ðI; 1=4½1    defect is a perfect interfacial dislocation with b ¼ 1=4½1 1 1. Moreover, the overlap step height is zero since hP is zero. Relating to the presence of the demi-step, this defect could be called a demi-dislocation and the equivalent with an overlap h a demi-disconnection. Since the defect line separates different interface domains, an interface domain line (IDL) could be regarded as being superposed on the dislocation. This IDL is the 2D analog of the 3D inversion domain boundary (IDB) separating different domains in 3D.

804

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Our purpose in mentioning this is that nonlinear strain fields including fields with dislocation or lineforce character, could be associated with the IDL, or other lines associated with the breaking of symmetry. This example serves to show that lattice matching is a necessary but not sufficient condition for avoiding interfacial defects. Several other examples have been discussed elsewhere [16]. Group theo1  1  dislocations only arise on {hk0} interfaces, or delineate juncretical arguments show that 1=4½1 tions between {hkl} facets, consistent with experimental observations [16]. For the defect illustrated above, the origins of the two crystals are aligned: at interfaces where a rigid-body displacement breaks such alignment, the Burgers vector is augmented by an additional component.

11. Discussion In the F–B eq., there is a duality in the description of bicrystals in terms of homogeneous transformations from one crystal to the other or in terms of the elastic distortion fields of interface defects. The reference coherent, or rotated coherent, interface can be described as a set of continuous infinitesimal coherency dislocations and rotational dislocations (or disconnections). At equilibrium the long-range field of these defects is cancelled by the fields of discrete misfit and tilt dislocations. Since we are concerned with the equilibrium interface, the discrete defects can be replaced by continuous infinitesimal defects to produce the ideal Bilby bicrystal. The F–B eq. is then described in two complementary ways. First, the defect content is determined in terms of homogeneous deformations, in turn describable by symmetry theory or by circuit methods. Second one can equate the elastic distortion fields of the two infinitesimal arrays in the Bilby bicrystal. We have presented several types of solutions to the F–B eq. that describe the defect content of equilibrium interfaces between bicrystals in these two ways. As with most treatments of the problem, all of the detailed solutions are for the homogeneous, isotropic elastic approximation of coherency and defect fields. We show that such solutions are consonant with the standard model for dislocation energy: i.e., nonlinear geometric effects are included in the analysis but not nonlinear elasticity. The standard model is implicit in early treatments of dislocation theory but had not been explicitly stated. For simple cases with one or sometimes two defect sets present, the F–B eq. can be solved analytically or by a rapidly converging iteration. For other cases there are up to six or seven degrees of freedom and we present a numerical solution. Few complex solutions are presented, but we discuss cases that deviate from the standard model, i.e., treatments incorporating anisotropic elasticity, inhomogeneous isotropic elasticity, or nonlinear elasticity. Examples are presented that indicate the degree of approximation entailed in the use of the standard model. The analytical treatments that go beyond the standard model are so complex that they are rarely used, so the degree of approximation is an important consideration. We show that the defect Burgers vectors and step heights (if steps are present) are defined in DPs, either CDPs if misfit is present, but not tilt or twist rotation, or RCDPs if misfit and rotations are present. These Burgers vectors always differ, sometimes only slightly, from the Burgers vectors of either crystal adjoining the interface in its perfect state. This finding is consistent with the implicit results for early treatments of misfit [32], but has not been explicitly incorporated conceptually for either of the above cases. A second important concept is the partitioning of both strains and rotations at an interface. Again, this notion was included implicitly in early treatments of both misfit [3] and tilt rotation [10,46]. More recently, partitioning is included in most treatments of misfit or pure tilt or twist, but it has been ignored in most treatments of the interfaces resulting from phase transformations. When defects are widely spaced, the partitioning is generally negligibly small. Partitioning is not included explicitly and is not implicit in all applications of the phenomenological theory of martensite transformation (PTMC) [55–57]. The topological model (TM) [18] does include partitioning explicitly. Consequently there is a difference of u/2 in the habit plane predictions in the two models [18,19]. Here u, the tilt rotation across the interface, is the same as the orientation relationship (OR) in phase transformation theory [55–57]. Analogously, the line orientations in a twist wall differ by g/2 when partitioning is ignored. Examples in Section 10.7 demonstrate these differences as well as added differences

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

805

associated with Burgers vector length changes accompanying rotation. Often, but not always [3], u or g is less than 1°, whereupon the differences in habit plane or dislocation line orientations become negligible compared to experimental scatter. The factors associated with Burgers vector magnitude changes are negligible for angles u or g less than a few degrees as specifically shown in Section 10. As a proviso concerning partitioning, we emphasize that equipartitioning is only valid for the planar interface in a bicrystal that is infinite in the plane of the boundary and with remote free surfaces normal to the interface; as well as in the homogeneous isotropic model. This is the usual assumption for interface models [1–13,25,55–57]. For other geometries, differing partitioning is likely as summarized in [52]. As an example, for small equiaxed precipitates, almost all of the elastic distortions are forced into the precipitate by the compliance of the matrix. Also, as demonstrated in Section 6, the partitioning differs for either anisotropic or nonlinear elastic models. The O-lattice [8] has been developed as an alternative description of equilibrium interfaces, and elegant models have connected Bollmann’s original work to reciprocal crystallographic space [123,124]. Also, a near CSL-lattice, with the interface coherently strained to produce a CSL has been used [125]. As mentioned in Section 2, these coincide with the present development in terms of DPs only for the misfit case, and then only in the homogeneous, isotropic approximation: this latter case has been termed a 2D O-lattice [25]. Contrary to comparisons between the TM and the O-lattice in [116], the models agree exactly when Burgers vectors are properly defined. For cases involving rotations, such as an interface containing disconnections, the models do not agree except in special cases. The O-lattice approach includes rotation but does not include rotational partitioning for the general case. This is demonstrated in [117], where a martensite habit plane calculated for the O-lattice approach differs from the TM result and agrees with the PTMC result, which also ignores rotational partitioning. Hence the Burgers vectors in the O-lattice approach are not vectors of the RCDP as required by the symmetry theory incorporated in the TM. One could modify the O-lattice approach by incorporating procedures from the TM. The key reason for the difference between the PTMC and the TM models of martensitic transformations is that the former entails an invariant plane (IP) in two dimensions, or an invariant line (IL) in three dimensions. This terminology arises from the description of the transformation as an invariant plane strain, clearly shown in Fig. 7.2 in [57]. Thus, as discussed in Sections 4 and 5, when the transformation dislocations contain only edge Burgers vector components by lying in the terrace plane, there is exact agreement between the PTMC and the TM: the transformation distortion is an invariant plane strain. However, for the transformation case when there is also a Burgers vector component normal to the terrace plane, the distortion is planar but it is not an invariant plane strain: it contains rotational components. This result is demonstrated in Fig. 8g. An interface vector v0 in the terrace plane, corresponding to the CDP, rotates separately to become the two vectors v P0 and v Q0 in the two crystals. This is a consequence of rotational partitioning. Kelly [118] has suggested that something is amiss in the TM because it disagrees with the PTMC in the general case. Physical descriptions have been presented to refute this contention [13,15,126], but the essential reason for the difference is that the transformation distortion is not an invariant plane strain in the general case and treating it as such implicitly excludes partitioning. Again, one could modify the PTMC by incorporating procedures from the TM. There is also agreement between the TM and PTMC for the special case of a pure tilt grain boundary, where an invariant line can again be defined [127]. While there is general agreement on the designation of misfit dislocations, there have been a variety of descriptors for defects with step character, as reviewed in [3,128]. The disconnection definition [17,49] provides a formal crystallographic definition of the step height h, the Burgers vector b, and, more generally, quantities describing steps with disclination or dispiration character. Of these, structural ledges [129], growth ledges [130] and moiré ledges [131] (for those defects where h is a multiple of terrace plane spacings) correspond to disconnections of various types [3,130,132]. The overall habit plane has been employed mostly for shear-type plate transformations such as those forming bainite and martensite. Excluding LID, this entails finding a stepped interface that contains only the defects that cause the transformation and whose distribution relieves misfit in one direction (a broader case is considered here in Section 10.8). In this category are O-lattice models [8,120], already discussed; invariant line (IL) models [133]; the PTMC [55–57]; and edge-to-edge models [127,134]. The present TM model coincides with the IL and PTMC models when the dislocation portion of the disconnections

806

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823 0

has no bz component, as discussed above. Otherwise, there is a disagreement in the habit plane orientation by an angle u/2, often small. Implicit, but not always specifically discussed, the O-lattice, IL, PTMC and edge-to-edge models all treat ideal (Bilby) interfaces where the far field strain-free structure extends to the interface: this corresponds as well to the NDP in the present work. In such an ideal interface, local strains associated with discrete interface defects are not considered in the direct analysis of habit planes and orientation relationships. However, many have interpreted the habit plane by means of discrete defects [128,135], as is done here with the TM. The edge-to-edge models involve matching at the interface of planes of the two crystals that have equal interplanar projections in the interface [4,134,136].There are differences in the models, such as the degree of rotation required to give matching [136–138]. With no misfit there is edge-to edge matching for a tilt wall but not for a twist wall. Similarly, there is edge-to-edge matching for a CDP, Fig. 1c, or an RCDP, as shown in the equivalent ideal bicrystals in Fig. 6c, Fig. 7d and Fig. 8d. However, with any misfit component present, there is never precise edge-to-edge matching in a true equilibrium bicrystal, Fig. 8f. This conclusion is in accord with the discussion in [139]. One can, of course, construct a number of interfaces that have edge to edge matching for a given bicrystal, but they would not be equilibrium interfaces containing only disconnections. Thus, the edge-to-edge models can be used to estimate habit plane orientations, and one form is consistent with the PTMC [134], but they are exact only in special cases or when rotation angles are negligibly small. The classical F–B method entails determining the number of defects intersected by an interface probe vector. This method is convenient for simple solutions to the F–B eq. and we present several solutions. We also present a new F–B solution in which the interface defects are described directly in terms of their distortion fields. This procedure is more convenient for numerical calculations, important when there are three or more sets of defects in the interface. With three sets, for arrays where the Burgers vectors lie in the interface, or with four or more sets when there are also out-ofplane Burgers vectors, the solution is over-determined and there are an infinite number of possible solutions. Analogous to the von Mises solution for the number of deformation systems for the arbitrary plastic strain of a single crystal, energy criteria are needed to select the most favorable set. 12. Conclusions 1. Analytical solutions of the F–B eq., describing the equilibrium structure of a bicrystal interface, apply for the standard model of dislocation theory, including geometric nonlinearities but not nonlinear elasticity. 2. The model applies for anisotropic elasticity or inhomogeneous isotropic elasticity, but is most facile in the case of homogeneous isotropic elasticity. 3. The geometric properties of interface defects, Burgers vectors and step heights, are described in dichromatic patterns: natural, commensurate, or rotated commensurate. 4. Strains and rotations are partitioned between the adjoining crystals comprising a bicrystal. 5. Up to six degrees of freedom are present for a general interface, requiring six sets of dislocations. In general, numerical methods are needed for a solution to the F–B eq., although simpler analytical methods can be employed when only one or two sets of defects are present. A new solution for the F–B eq. is presented which is more convenient than the classical solution when many sets of defects are present in an interface. 6. For cases other than simple, high-symmetry interfaces, there are an infinite number of solutions to the F–B eq. Energy criteria are needed to select the most favorable solution.

Acknowledgments This research was supported by the US Department of Energy, Office of Sciences. Office of Basic Sciences. The authors are grateful for support and beneficial input from A. Misra; for specific contributions from M.J. Demkowicz and J. Lothe; for helpful comments from R.W. Balluffi, R. Bullough and C.N. Tomé; and for stimulating questions from P.M. Kelly.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

807

Appendix A. Topological theory of interfacial defects Topology theory enables the characterisation of line defects in any form of ordered material [140]. This appendix summarises the aspects of the topological theory of interfacial defects relevant to bicrystals with semi-coherent interfaces. A.1. Defect character in single crystals and bicrystals The topological nature of line defects in continua was first developed by Volterra, who showed that dislocations and disclinations are characterised by translation and rotation operations respectively [141]. For the case of single crystals, the Volterra operations, Vcrys, characterising perfect defects must be proper symmetry operations of the underlying crystal (even though those symmetries are broken in the defective structure). In terms of the Seitz notation of the International Tables for Crystallography [142], symmetry operations are expressed W = (W, w). Thus, for a dislocation

V crys ¼ W ¼ ðI; tÞ;

ðA1Þ

where t is a translation operation of the crystal (equal to b), and I represents the identity matrix. For a disclination, Vcrys = (R, 0), where R is a symmetry rotation operation. Dispirations can also arise, characterised by rotation-screw operations. An analogous procedure for defects in bicrystals is the basis of the topological theory [29]. Here, the white crystal is designated P, and the black Q. Admissible defects are ‘‘perfect’’ in the sense that they separate energetically and structurally degenerate regions of interface (terraces), although partials can also be defined. Now, the set of Volterra operations for the bicrystal case, Vint, characterising admissible defects has the form

V int ¼ W P W Q ;

ðA2Þ

where the superscripts indicate the relevant crystal. Eq. (A2) is a generalised expression of ‘‘mismatch’’ between two crystals, showing that admissible interfacial defects are characterised by proper combinations of symmetry operations, one from each of the crystals. The diversity of interfacial defects predicted by Eq. (A2) depends on the extent to which the adjacent crystals exhibit common symmetry: for common symmetry elements, WP = WQ, and hence the combination WP(WQ)1 = (I, 0), so no defect character arises for that particular pairing. The reader is referred to the original works for extensive illustrations of interfacial defects [11,16]. Eq. (A2) makes clear that, in a comprehensive treatment of defects, it is not sufficient to consider only the translation symmetry of the crystals. For example, two improper symmetry operations, WPWQ, may combine to give a proper operation, Vint: an example is the case of interfacial dislocations that arise when a white mirror plane is aligned with a black mirror-glide plane [25]. An example of such a more general case is given in Section 10.9: this more general application of symmetry theory is one of its major advantages. Other methods can be used to describe the simpler translation deformations that are the primary cases treated here, but the results are consistent with symmetry theory as shown here. Another facility of the topological theory is that it elucidates the equivalence of a priori characterisation by specifying the Volterra operation, Vint, and a posteriori characterisation by circuit mapping around an observed defect [11]: the former corresponds to the irreducible description of the circuit, as shown in Section A.3. Moreover, equation (A2) can be used to characterise not only individual interfacial defects, but also arrays of defects: in the latter case, we show that the F–B eq. is a particular case of Eq. (A2). The set of Volterra operations defining admissible interfacial defects, Vint, Eq. (A2), includes the set of Volterra operations for single crystal defects, Vcrys, Eq. (A1). For example, crystal dislocations originating in the white crystal correspond to g

V int ¼ V crys ¼ W P W Q ¼ ðI; tP ÞðI; 0Þ ¼ ðI; b Þ;

ðA3Þ

and similarly for dislocations from the black crystal. Thus, all defects that are admissible within the P and Q crystals are also admissible in the interface.

808

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

A.2. Elementary defects in semi-coherent bicrystals Equations (A2) and (A3) are written in generic form: the symmetry operations WP and WQ pertain to the P and Q crystals in their current reference state. Introduction of an individual defect will not significantly modify the initial reference state, but introducing an array would do so, and this, in turn, modifies the exact form of WP and WQ to be substituted into Eq. (A2). In the present context, we are particularly concerned with defects in coherent terraces. We therefore consider the introduction of an elementary defect into a coherent terrace, and the consequences of a rotation produced by an array of defects pre-existing in the terrace. As illustrated in Fig. 1d and f respectively, and defined in Appendix B, the terrace coordinate frame is defined by x, y, z, and the final habit frame, when a rotation is present, is x0 , y0 , z0 . Thus, as outlined in Appendix B, if the current reference frame is the CDP for example, the crystal symmetry operations, WP and WQ, must be re-expressed in the terrace frame as WcP and WcQ before substitution into Eq. (A2). Similarly, they must be re-expressed as WrcP and WrcQ if the current reference is a RCDP. Consider the introduction of a defect into an initially flat coherent terrace. First the two crystals in their natural states are deformed by cPn and cQn to create the coherent state so that 2-D periodicity results at the terrace after bonding and the relevant reference space is the CDP. The pre-subscript indicates the final state and the post-subscript indicates the initial state: i.e. cPn and cQn act as operators. The reverse deformations nPc and nQc are homogeneous transformations from cP to nP and cQ to nQ respectively. In Section 2 we also see that nPc, for example corresponds to the transformation that creates the long range deformation of the equilibrium bicrystal. The set of admissible defects in this interface are characterised by Vint = WcPWcQ, and we consider a disconnection for which Vint = (I, tcP)(I, tcQ)1. One can imagine the creation of this defect by juxtaposing surfaces of the P and Q crystal with incompatible steps, as depicted schematically in Fig. A1a. We emphasize that this visualisation is a topological construct where the P and Q components are extracted from the CDP. The surface structures on either side of the white step are identical, and similarly for the black ones: an observer at the foot of the P-step is transported to the top by the elementary white translation operation, (I, tcP), and, similarly, from the foot to the top of the Q step by (I, tcQ). When the two surfaces to the left of the steps are bonded to form the terrace, one can carry out a Volterra operation, Vint, to bring the two right-hand surfaces together, thereby forming the identical interface structure after bonding. In this way, a disconnection is created separating energetically degenerate interfacial regions, characterised a priori by g

V int ¼ ðI; b Þ ¼ ðI; tcP  tcQ Þ:

ðA4Þ

Eq. (A4) illustrates the usefulness of a CDP as graphical indicator of the b of admissible defects: if the pattern is constructed with the relative position of the P and Q lattices so that lattice sites in the terrace plane are coincident, admissible bgs are vectors joining black sites to white. This notion was first discussed by Bollmann [8] in the context of DSC dislocations. The heights of the surface steps are hcP = n  tcP and hcQ = n  tcQ, where n is a unit vector, normal to the coherent terrace plane and pointing in the z direction. The ‘‘overlap’’ step height, h, is the smaller of hcP and hcQ [17]. The use of the CDP implicitly means that partitioning of distortions (strains and rotations) is included in the analysis. This is further discussed in the following section. For the general case, it is convenient to choose the origin of tcP and tcQ in a CDP as a coincident point in the unique plane, as in Fig. 1c. If a defect such as that shown in Fig. A1 is introduced into the terrace when a pre-existing array of disconnections or dislocations with a tilt component of their Burgers vectors is present, thereby producing a rotation, R, the current reference state would be the RCDP rather than the CDP. Similarly, if a tilt wall is superposed on a perfect crystal or on a coherent bicrystal, the reference state would be a RCDP. In the RCDP, the white crystal lattice is defined by rc XPc þ c Pn , and the black by rc XQc þ c Q n . With respect to this reference, and with equal partitioning of the total rotation, the Burgers vector of the disconnection would be g

V int ¼ ðI; b Þ ¼ ðI; trcP  trcQ Þ: g

ðA5Þ

Eq. (A5) shows that b changes continuously with variation of R, and such changes must be taken into account unless the rotation is very small. As for the CDP, a RCDP has the useful graphical property

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

809

(a)

(b) Fig. A1. (a). Schematic illustration of P and Q crystals before bonding to form the coherent terrace. The steps exhibited on each surface are incompatible, and lead to the formation of a disconnection on bonding. (b) After bonding, a circuit which is clockwise about the defect line direction, n//x, and closed in the distorted bicrystal as indicated.

whereby admissible bgs are vectors joining black sites to white. Also, h would now be the smaller of hrcP and hrcQ. A.3. Circuit mapping A.3.1. Characterisation of an elementary interfacial defect Next, we consider the a posteriori characterization of the elementary disconnection in Fig. A1, i.e. by mapping a closed circuit constructed around the defect, Fig. A1b, into the CDP. A more detailed account is given in [52,143]. We use the FS/RH convention [42]: an observer [143] traces out a clockwise circuit with respect to the defect line-direction, n, here parallel to x. The closed circuit passes through ‘‘good’’ material, starting at S, proceeding through the P crystal to J, and thence through the Q crystal back to F. (Since the interface structures are identical at S/F and J, the observer is transported by equal and opposite displacements across the interface at these points, and hence, in combination, they contribute zero to the total circuit, and can be disregarded henceforth). This circuit is then mapped into the CDP, where it fails to close. In the formal topological theory, the mapped P circuit segment is expressed as a sequence of k white symmetry operations, and, similarly, l operations for the black segment [11,142,144]. The combined sequence of operations is designated the ‘‘circuit operator’’, (Vcct)1, given by cQ cQ cP cP cP ðV cct Þ1 ¼ W cQ l . . . W2 W1 Wk . . . W2 W1 :

ðA6Þ

810

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

If the circuit comprises only translation operations, as in Fig. A1b, (Vcct)1 displaces the observer from point S to F in the (open) circuit after mapping into the CDP: therefore, the operation transporting the observer from F to S is Vcct. Here, Vcct is the irreducible description, obtained by cancellation of operations in the circuit. We consider the cancellation of circuit elements after mapping in two stages. First, the P segment reduces to the vector SJ(P), and the Q segment to JF(Q). At this stage, the circuit has been reduced to SF = SJ(P) + JF(Q). After further cancellation of common translations along the interface, SJ(P) reduces to tcP, and JF(Q) to tcQ. Thus, finally, the irreducible closure failure in the CDP is FS = tcP  tcQ. This result is identical to the a priori form, i.e. g

V cct ¼ V int ¼ ðI; tcP  tcQ Þ ¼ ðI; b Þ;

ðA7Þ

thus characterizing the defect by the same operation as the Volterra displacement, equation (A4). A.3.2. Characterisation of the defect content of an interfacial defect array The circuit described above encircles an elementary defect, and ‘‘shrinks’’ to the irreducible description when mapped into the CDP. A similar procedure can be followed when an array is encircled, entailing a length of interface FJ, except that the closed circuit S, J, F must interrogate a larger portion of interface than the single defect spacing, and the reference space will be a RCDP if the array induces a rotation. After mapping, the closure failure FS can be expressed in terms of FJ(Q) and JS(P), but these now correspond to translations in the RCDP, trcQ and trcP, respectively, that are large in magnitude relative to the spacing of a set of dislocations in the interface. Hence, we can write,

V cct ¼ ðI; trcP  trcQ Þ ¼ ðI; BÞ;

ðA8Þ

where B is the Burgers vector content in the length FJ. For a single dislocation in a crystal, equations (A7) and (A8) yield true Burgers vectors [42]. The opposite mapping from the reference lattice to the real crystal gives a local Burgers vector influenced by local strain fields. Here, we deal with ideal Bilby bicrystals, e.g., Fig. 3d and Fig. 7d, where there are no local strain fields. In this sense, an ideal natural bicrystal can be viewed as an array of continuous infinitesimal misfit dislocations with Burgers vectors finf superposed on infinitesimal coherency dislocations ginf, see Fig. 2. Thus, one can also map in the SF/ RH sense from the c- or r–c state to the n-state and identify B as the misfit Burgers vector content in the length FJ. Alternatively, one can map a closed circuit in the c-state to the n-state in a FS/RH sense, with the closure vector FS = Bc, where Bc is the Burgers vector content of coherency dislocations. Furthermore, we can identify the vector JS(P) in the closed n-bicrystal circuit with the probe vector vsym and FJ(Q) with vsym. Thus the mapping of vsym from the n- to the c-state gives a closure failure

B ¼ ðc Pn  c Q n Þv sym ¼ c En v sym :

ðA9Þ

Moreover, Section 2, the difference (cPn  cQn) is equal to the coherency strain produced in a conversion from the n-state to the c-state. This duality pervades the entire paper. The form in equation (A9) has been presented extensively in symmetry theory as applied to arrays of disconnections or dislocations [3,18,25,51,52]. However there is an anomaly in the dual description relating symmetry theory to the theory of elastic distortions, coupled here for the first time. In topology theory [3,18,52], as discussed above, b is described in terms of n and translation vectors t. In Appendix B, we show that this leads to the definition of a probe vector v, to be used in the F–B Eq., that is opposite in sign to vsym. Thus, we have

B ¼ ðn Pc  n Q c Þv ¼ c En v ¼ n Ec v ;

ðA10Þ

B c ¼ ð c Pn  c Q n Þ v ¼ c E n v :

ðA11Þ

For the more general case, on mapping these into the RCDP, Binf replaces Bc, and

  Binf ¼ ðc Pn  c Q n Þ þ ðrc XPc  rc XQc Þ v ¼ ðrc En þ rc Xc Þv ¼ rc Dn v ;

ðA12Þ

where rc XPc and rc XQc are the rotational functions of the P and Q crystals formed after rotation from the NDP (or, equivalently, the CDP) to the RCDP, rcXc is the difference in these rotational functions, and rcDn is the difference in elastic distortions.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

811

  The opposite mapping gives B, where, for example, n Pc þ c XPrc is the opposite homogeneous transformation from the commensurately strained and rotated P state to the natural P crystal in the NDP reference and similarly for crystal Q. Written for a general matrix A, where symmetry operations other than translation can be present, one finds

B ¼ ðP Aref  Q Aref Þv ¼ ððref AP Þ1  ðref AQ Þ1 Þv ¼ ðref AP  ref AQ Þv :

ðA13Þ

Here the last form follows from the physical condition, B = Bc, which also corresponds to a thermodynamic result, Section 8. In terms of the general matrix A, we can also write

Bc ¼ ðref AP  ref AQ Þv ¼ ððP Aref Þ1  ðQ Aref Þ1 Þv ¼ ðP Aref  Q Aref Þv :

ðA14Þ

Either form, Eq. (A13) or (A14) can be regarded as the F–B eq. Eq. (A14) shows that matrix inverses are equivalent to negative matrices. Both are simple in linear elasticity. In nonlinear elasticity, the inverses are complicated [146], and the negative matrices are simpler. A.3.3. Location of the circuit For a specified interface defect, F, S and J must lie at the interface for a closed circuit. Equivalently, in the reference DP, the origin of the translation vectors, say trcP and trcQ for the RCDP case, is at a coincident point on the interface. For an array, the interface vector v corresponds to the circuit in P collapsed to the interface. This condition is analogous to the Frank circuit rule for a partial dislocation [32,37]. Contrariwise, any vector pair in the CDP or RCDP is a possible Burgers vector for an interface defect. The translation vectors can have very large z components for disconnections, with correspondingly large step heights. A.4. Description of rotation The rotational functions relate to components xij = 1/2 [(@ui/@xj)  (@uj/@xi)] of the elastic distortions Dij. The standard model of Appendix C leads to one difficulty in the form of the rotational function. The dislocation array fields in Section 6, for a pure tilt wall, are the sums of linear elastic fields. They give a long-range distortion, expressed in tensor coordinates as Dyz = Dzy = (br/2L0), where L0 is the perfect crystal spacing before the introduction of the dislocations. In the standard model, Frank’s formula leads to the result Dyz = Dzy = (br/2L), where L is the final spacing in accord with the standard model. The difference in the ratio of the two expressions, L0/L = cos(u/2) is a consequence of the use of the nonlinear geometric standard model, in contrast to a purely linear elastic model. We correct this by using L instead of L0 (or d instead of d0 for disconnections) in expressing the distortions in Section 6. In the F–B model, Appendix D, we deal separately with the contributions of P and Q while only their r difference equates to D above. Hence, we have directly, xPyz ¼ DPyz ¼ xPzy ¼ DPzy ¼b =2L ¼  tanðu=2Þ, and similarly, with opposite sign, for Q. The total distortional function is rcXc = 2 tan(u/2). We use this form for consistency with symmetry theory. For the rigid rotation of either P or Q in the ideal bicrystal of Fig. 1f, the rotation ±u/2 is evidently constant. Hence Frank [10] defined such a rotation as the vector R = (Rx, 0, 0) = nu, again obviously constant. Hence both u and Dyz are independent of a coordinate transformation, as can be proved by operating on, say, rc XPc , as in equation (B3). In order to distinguish these quantities, we refer to X as the rotational function and R, or u, or the twist g as rotations. In linear elasticity, sin (u) = u and all of these quantities are often called rotations, in contrast to the nomenclature here. A.5. Rotation, misfit accommodation, and spacing defects Commonly, one refers to defects in terms of the function they perform in an interface: in the present context tilting/twisting and misfit accommodation are important. The case of pure tilting occurs in a symmetrical tilt grain boundary, as illustrated schematically in Fig. A2a. Here, the P and Q surface

812

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. A2. (a) Schematic illustration of the formation of an interfacial ‘‘pure tilt’’ dislocation, (b) a mixed misfit and tilt case., and (c) a slant P dislocation. The dashed lines link pairs of bonding points, a–a0 , o–o0 and b–b0 .

steps have equal heights but opposite senses, hcP = n  trcP = hcQ = n  trcQ, so that the overlap step height is h = 0, and equation (A8) gives g

r

b ¼ 2b ¼ trcP  trcQ ;

ðA15Þ

813

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

which is perpendicular to the interface. An array of such defects with spacing 2L leads to a symmetric boundary structure where each crystal is rotated by u/2 = tan1(jtrcPj/2L), as deduced by Frank [32], and jbgj = 2jtrcPj cos(u/2). For small values of u, the energy of symmetric tilt boundaries may be reduced by a rigid-body displacement of Q with respect to P by L, thereby creating mirror-glide symmetry across the boundary rather than mirror symmetry. In this special circumstance, one may regard the dislocations as having dissociated into two ‘‘partial’’ dislocations (indicated by italic symbols) each with br = bg/2, although, because of the symmetry, they would separate energetically degenerate regions of interface. Alternatively, this configuration can be viewed as an array of slant dislocations with alternating Burgers vectors [44]. In general, a displacement with magnitude other than half of the boundary period, 2L, creates a fault at the interface, and the bicrystal would exhibit neither mirror symmetry nor mirror-glide symmetry. The y components of trcP and trcQ correspond individually to partial misfit dislocations in the interface. However, in the combined defect of Fig. A2a they mutually annihilate to form a spacing defect of length s equal to the overlap between the two y components, jtrcPj = jtrcQj for this special case. For twist boundaries, the above relations become g/2 = sin1(jtrcPj/2L) and jbgj = 2jtrcPjcos(g/2). Thus, again, one may regard the dislocations as having dissociated into two ‘‘partial’’ dislocations) each with br = bg/2, separating energetically degenerate regions of interface. Equation (A15) still applies to this case, with br referring to the twist case. The general tilt case entails an interface containing a defect with mixed misfit and tilt character, Fig. A2b. Here, trcP and trcQ have opposite senses and different magnitudes, bg = trcP  trcQ and the overlap step height is zero. Physically, this dislocation is likely to dissociate as in the previous example, but it is the smallest perfect dislocation in this case. The partial misfit and tilt components are bg = bm + br, m r rcQ rcP rcQ where jb j ¼ t rcP y  t y , and jb j ¼ t z  t z . If the crystals in Fig. A2b were the same phase and the step riser was rotated, the bicrystal of Fig. A2b could deform to coincide with that in Fig. A2a. If the dislocation were to dissociate, the misfit and rotation for each would become the smaller values: e.g., trcP and t rcP for the P dislocation. y z The slant dislocation in Fig. A2c is a perfect dislocation of the white crystal and is not a white-black vector as discussed above. Hence, its Burgers vector can be determined directly in the white lattice portion of the CDP. The circuit in this case need not correspond to the S-F-J circuit, but it can be so described, with the displacement fields partitioned as described above. The bicrystal in Fig. A2b can be imagined to change into two grains of the same phase and the step rotated so that it conforms to that of Fig. A2a: in this notional topological distortion the translation vectors rotate and deform. In a similar way the translation vectors can be envisioned as rotating into an antiparallel configuration lying in the interface. In this limit the dislocation would become the perfect misfit dislocation g

m

b ¼ 2b ¼ tmcP  tmcQ :

ðA16Þ

However, this defect would certainly dissociate. In the limit that the translation vectors lie in the interface, there is a corresponding degeneracy. In that limit, white and black vectors coincide in the interface and tmcP = tmcQ. Analogous to the example of Fig. A2c, a crystal dislocation Burgers vector bm becomes a perfect vector of either the white or the black lattice in the CDP. Thus it can be directly determined by a circuit in either the white or black crystal and, as illustrated in Fig. A3, g

m

b ¼ b ¼ tmcP ¼ tmcQ :

ðA17Þ P

u

For the pure spacing defect case, consider an edge dislocation, b ¼ ½0; b cos 2 ; b sinðu=2Þ. Let this defect climb towards a symmetrical tilt boundary comprising an array of grain boundary dislocations with Burgers vectors br = [0, 0, b cos(u/2)]. Also, let a dislocation bQ = [0, b cos(u/2), b sin(u/2)] climb into the boundary at the same location. The misfit components of the two dislocations mutually annihilate, and the tilt components interact with the grain boundary dislocations and annihilate, thereby disturbing the interface periodicity. Such defects are known as a ‘‘spacing defects’’ [61] because they have local elastic fields that are the same as a dislocation because they correspond to a missing dislocation density in the original boundary. Similarly, dislocations that annihilate at semicoherent interfaces also lead to spacing defects, as outlined in Section 3.

814

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Fig. A3. Schematic illustration of a pure misfit dislocation.

Appendix B. Matrix notation and coordinates Topological arguments are developed in the present work using matrix notation, as summarised below. However, tensor notation is also employed in certain sections in order to conform to conventional micromechanics usage, as in Section 7. B.1. Transformation matrices Transformations and symmetry operations are expressed using the Seitz notation as in the International Tables for Crystallography [142]. For example, the orthogonal part of the transformation inter-relating the nP and cP lattices is designated cPn, where the post-subscript indicates the initial state and the pre-subscript the final state. Similarly, the nQ lattice is transformed into cQ by cQn. Thus, translation vectors are inter-related by

tcP ¼ c Pn tnP ;

ðB1Þ

tcQ ¼ c Q n tnQ :

ðB2Þ

and

Throughout, the pre- and post-subscripts c and rc indicate the reference commensurate (coherent), CDP, and rotated commensurate, RCDP, states, respectively. As developed here, the pre- and postsubscipt n refers to the specific unrotated NDP corresponding to the unrotated natural bicrystal in Fig. 2. This unrotated NDP is that for which two planes, one from each crystal, become coherent terrace planes in the CDP after pure misfit strains are applied.: an example is shown in Fig. 8c. This is convenient in describing elastic distortions and is consistent with the standard model of Appendix C. Other NDPs would be rotated, i.e. RNDPs in the present context, relative to the above NDP. We identify these by the subscript rn: Fig. 8b is the ideal bicrystal equivalent of such a rotated RNDP. However, RNDPs are not used as a reference state in the analysis and rn does not appear in the text. B.2. Symmetry operations Symmetry operations, written W = (W, w) [16,29], have a simple form when expressed in the host crystal’s coordinate frame since the elements of W for point symmetry operations are either ±1 or zero. Thus, a symmetry operation, W, relates a position vector, x, expressed as a 3 1 column matrix, to an equivalent point located at Wx + w. Translation operations are written W = (I, t), where t is a translation vector comprising integer combinations of a0, b0 and c0, and I represents the identity

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

815

matrix. P translation operations in NDPs, CDPs and RCDPs are expressed WnP = (I, tnP), WcP = (I, tcP) and WrcP = (I, trcP), respectively, and similarly for Q translations. Other symmetry operations, rotations, mirrors, screw-rotations and mirror-glide operations, are important in some applications, but less relevant in the present context. B.3. Terrace and habit plane coordinate frames In the present work, it is often necessary to express the resultant operation produced by a combination of two sequential symmetry operations, one from the P crystal, and one from the Q crystal. For example, Volterra operations, Vint, Eq. (A2), are of this nature. One therefore must choose a coordinate frame in which to express such a resultant operation, and, in the present context, the terrace frame, x, y, z, and the habit plane, x0 , y0 , z0 , are most convenient. In the former, x and y define the terrace plane, and z its normal, as depicted for the natural and coherent bicrystals in Fig. 1b and d respectively. The latter frame is illustrated in Fig. 1f for the case of equally partitioned tilt, i.e. where the crystal terrace planes in crystals P and Q are rotated by ±u/2 with respect to the habit plane. Furthermore, when an array of disconnections is present, the step character of these defects causes an additional rotation of the habit plane by h, although this is not partitioned [18]. Thus, the final habit plane in such cases, Section 5, is rotated by h ± u/2 with respect to the terrace planes of the two crystals. In the context of Appendix C, both coordinate sets, x, y, z, and x0 , y0 , z0 , are in accord with the standard model for ideal bicrystals created from the CDP and RCDP reference lattices respectively. That is, the misfit dislocations and the coherency dislocations are both smeared into continuous, infinitesimal distributions and the local strains of the true misfit dislocations near the interface are suppressed. B.4. Burgers vectors expressed in terrace and habit coordinate frames Equation (A2) can be formulated in an appropriate coordinate frame of reference by re-expressing symmetry operations in that frame. For example, an operation WnP is transformed from the natural P crystal frame into the terrace frame by the similarity transformation,

ðW nP ÞT ¼ T AnP W nP ðT AnP Þ1 ;

ðB3Þ

and similarly for Q crystal operations. Also, for disconnections or when tilt is present, one can formally re-express P and Q symmetry operations in the habit plane using the transformations H AnP and H AnQ . Transformation matrices invoked in such coordinate transformations (rotations) are designated using script font, A , to distinguish them from physical transformations: rotations of this nature are represented by R. The notation used in the present work for transformation matrices and symmetry operations indicates the relevant coordinate frame. Thus, for example, the dislocation character of a defect defined by equation (A2) expressed in the CDP is Vint = (I, tcP)(I, tcQ)1 = (I, tcP  tcQ). However, to simplify the nomenclature for Burgers vectors, we use the convention set out below. (a) Perfect dislocations: a Burgers vector of a dislocation belonging to the set Vint is designated bg. The superscript indicates a ‘‘generic’’ interface/grain boundary defect. The reference space in which it is defined is evident from the context. For a slant crystal dislocation that reaches the interface of a coherent bicrystal by gliding through the P crystal, see equation (A3) and Fig. A2c, we have g

b ¼ tcP :

ðB4Þ

Similarly, for a slant dislocation in the Q crystal, we have g

b ¼ tcQ :

ðB5Þ g

In the special case of a pure misfit dislocation, b is parallel to the interface, as illustrated in Fig. A3, and we have g

b ¼ tcP ¼ tcQ :

ðB6Þ

816

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

For defects defined in a RCDP, g

b ¼ trcP  trcQ :

ðB7Þ

An illustration of the formation of a defect in a symmetrical tilt boundary is depicted in Fig. A2b: bg, trcP and trcQ are shown explicitly. Equation (B7) is expressed in general form and applies to both tilt and twist cases. An illustration for the symmetric twist boundary is shown in Fig. 9 of Section 4. In order to compare this diagram with Fig. A2b, we imagine that the two bicrystals are obtained from the same RCDP. If the interface is the x, y plane in the tilt case, it corresponds to the z, y plane in the twist case. Then, the two specific translations depicted in Fig. 9 are trcP and trcQ so equation 0 01 bx 0 g (B7) again applies. The Burgers vector of such a dislocation has the explicit form b ¼ @ by A in the 0 bz 0 0 0 x , y , z coordinate frame. Discrete defects are not considered to exist in a natural bicrystal like Fig. 1b. However, an NDP is an important reference space for defining the infinitesimal dislocation content, Bc as described in Section 2: differences in translation vectors are important in defining the relative displacements u between lattice points that lead to the distortion matrix in Section 2 and Appendix D.

u ¼ tnP  tnQ :

ðB8Þ

For defects that have the functions of purely accommodating misfit the superscript ‘‘g’’ is replaced by ‘‘m’’, and similarly by ‘‘r’’ for defects which purely produce rotation: thus, in the former case, bg is writ0 1 0 1 0 0 m r m r ten, b , and in the latter, b . For defects with line direction x; b ¼ @ by A, and b ¼ @ 0 A where the bz 0 0 1 bx r rotation is a tilt, and b ¼ @ 0 A for twist, and similarly for defects in the x0 , y0 , z0 coordinate frame. 0 (b) Imperfect dislocations: Burgers vectors are indicated using italic symbols. Important examples arise in symmetrical tilt and twist grain boundaries, where bg = 2br. In addition, it is sometimes useful to decompose perfect Burgers vectors, bg, into partial misfit components, bm, and partial tilt/twist components, br. B.5. The probe vector In the F–B eq. the probe vector v lies in the interface. The selection of its direction, since it is related to circuits, is not arbitrary but is coupled to n [11,61]. We show in Section 10 that the results of the F–B eq. are independent of the rotation of v, verifying results in [26]. Moreover for each set of dislocations, the F–B eq. can be solved for each set independently and the results then summed in a common coordinate system. Hence a convention, convenient since it relates directly to L, is to choose the specific vector v0 equal to N(v0/N), where, Section 8 and Appendix D, N is the reciprocal interface vector of interface theory [24,42,85]. The – sign on N is necessary because the sense vector in grain boundary theory [42] is selected opposite to that used here (and generally in interface theory). Since v0 is directly related to n via the reference circuit here, it has the opposite sign from N. Hence, N = 1/L, with L the defect spacing in the interface. Since N = N(n n), then also v0 = v0(n n). Therefore, v0 obeys the same axioms [42] as b with respect to n. The same solution is obtained if v is rotated relative to v0, but if the rotation exceeds ±p/2, one must change the sign of n so that it remains parallel to (n v). The interrelation among the interface terms can be understood from Fig. B1. This figure shows the interface vectors, misfit field and coherency field. In this simple example there is only a y component of misfit and the misfit dislocation is pure edge. Derived quantities include the misfit strain difference mP mQ cP cQ n Ec ¼ eyy  eyy , the coherency strain difference c En ¼ eyy  eyy . These differences are interrelated by the thermodynamic condition nEc = cEn. The misfit dislocation content, Section 2, is then m

m

B ¼ ðb  NÞv0 j ¼ ðb =LÞv0 j:

ðB9Þ

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

817

Fig. B1. (a) convention for the orientations of the vectors n, n, v0 and N, (b) the long-range misfit strains and sign of the misfit dislocation Burgers vector, (c) the coherency strains and sign of the coherency dislocation Burgers vector content.

The coherency dislocation content is

Bc ¼ n Ec v 0 ¼ c En v 0 :

ðB10Þ

The F–B eq. is then solved by the self-consistent inter-relation of these quantities

B ¼ Bc ¼ n Ec v 0 :

ðB11Þ

Study of Fig. B1 shows that a self-consistent solution only exists when the above conventions for v and N are employed. The extension to more general examples is presented in the main text. The same equations hold more generally if bg replaces bm,v replaces v0, and the coordinates become those, x0i , fixed on the final interface. Then equations (B9)–(B11) relate to the actual implementation of the F– B eq. Knowing bm, one can solve equation (B11) theoretically for N and n, for as many defect sets as necessary, and compare the results with experiments or atomistic simulations. The interrelation of v and N implies that a dislocation network is treated. The Burgers vector of a single dislocation, where N is irrelevant, can also be determined in terms of v. Appendix C. The standard model Nonlinear elasticity [145,146] entails nonlinear stress and strain, the extension of Hooke’s law, and nonlinear strain energy as well as geometric factors. The latter is associated with the use of embedded Eulerian coordinates versus laboratory Lagrangian coordinates. An example is that initial and final coordinates after elastic straining are independent in linear elasticity, i.e., there is no distinction between Eulerian and Lagrangian coordinates. Hence, the distortion matrix can be split into independent symmetric (strain) and antisymmetric (rotation) portions. These submatrices are interdependent in nonlinear elasticity. Within dislocation theory, fully nonlinear treatments are complex and are little used other than in perturbation theory [147,148]. Yet dislocation theory does include some aspects of nonlinear elasticity. While not specifically elucidated, traditional dislocation theory [32,149] entails linear stress strain relations but nonlinear geometric factors: we call this the standard model. The model applies in both the isotropic and anisotropic elastic cases. The field of a single dislocation is treated as linear elastic and the significantly nonlinear portions of the field lie in a core region, defined by a core radius. A subtle example of geometric nonlinearity is that Burgers circuits and their analogs are defined in ‘‘good’’

818

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

material [10] which lies outside the core region in a real material and are referred to an ideal perfect reference crystal to define the true Burgers vector. Hence local strains do not affect the Burgers vector. Interactions between dislocations, image interactions, dislocation energies, etc. are all consistent with this definition of the Burgers vector. For example, in computer simulations of a dislocation in an infinite medium [150], a plot of the energy within a cylinder of radius R centered on the dislocation as a function of ln R has a slope consistent with the true Burgers vector when R is greater than the core radius. Since the long-range fields of the defects are linear elastic, the fields of arrays of dislocations can be added linearly in accord with the principle of superposition [151,152]. Thus, fields of pileups, multipoles and boundaries are found by such superposition. Provided that the arrays are regular and provided that Eulerian coordinates are used, one can describe long-range fields even when core overlap occurs. An example is Frank’s formula for a grain boundary. With Eulerian coordinates, with true Burgers vectors, and with superposition, the rotation of a large angle grain boundary is given by Eqs. (9) and (10) as can be verified by simple plane counting as in Fig. 19-13 in [42]. There is the proviso that these equations apply only within a span of rotations limited by symmetry [37,119]. For example for tilt walls with rotation angle u, produced by rotations from [100] about [0 0 1] in a simple cubic crystal, Eq. (1) applies with b=[1 0 0] when referred to a perfect single crystal provided-p/4 < u < p/4. For p/4 < u < 3p/4, fourfold symmetry applies and the relevant Burgers vector becomes b = [0 1 0]. Thus, Frank’s formula applies in the present work either for grain boundaries or for tilt or twist components of interface dislocation arrays. Moreover with the use of Eulerian coordinates in the sense of describing an interface in its current configuration, the distortion matrices can still be separated into independent strain and rotation portion for large tilt or twist angles in the standard model. For a grain boundary or interface, the Bilby–Bullough [6,7] model of replacing actual Burgers vectors by an array of continuous infinitesimal dislocations with the same net Burgers vector per unit area extends the linear elastic field to the interface itself. This array is also consistent with Frank’s formula [32]. Finally, the CDP and RCDP reference spaces and the mapping of the translation vectors among them are obviously done in Eulerian coordinates as described in Appendix A. The principal result of perturbation theory is that 1/r2 terms arise in the strain fields. These have the form of the field of a right circular cylinder placed in a cylindrical hole [153] or of line force dipoles or multipoles [148]. These fields have little effect on internal interactions but superpose to give image stresses. For example the Seeger–Haasen fields superpose to give a volume change of a crystal containing dislocations. For perfect interface defects separating identical terraces, the sum of such fields produces uniform dilatation of the interface in the z direction or uniform shears in the x and y directions, but no added dislocation content and no added force on the defect. Only for partial defects, where terrace structure differs on either side of a defect, can small differences in the dilatation or shear fields arise and small added Burgers vector content then appears in the relevant defect. As another example, Simmons and Bullough [154] described dislocation cores in terms of arrays of line force dipoles. They specifically noted that final Eulerian coordinates should be used in the dipole description.

Appendix D. Burgers vectors, line directions, and spacings of interface dislocations in the general case The purpose of this appendix is to expand upon Eqs. (12) and (13) to provide a description of interfaces in which misfit is removed by arrays of dislocations. Components of the Burgers vectors of these dislocations lie in the interface plane and are effective in removing misfit. Out of plane components contribute to tilt rotations and other in-plane components contribute to twist components. A subsidiary aim is to develop equations that can be solved numerically for any number of sets of parallel dislocation or disconnection arrays. In addition, we derive the starting point, Eq. (2), of the development in Section 2 and prove, in a different manner than earlier proofs [2–6,26], the independence of the solution of the F–B eq. on the probe vector v. The possible Burgers vectors for perfect dislocations in the terrace plane are the set of vectors between various pairs of lattice sites in the CDP or RCDP, equations (B10) and (B11), which have

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

819

corresponding ideal bicrystals with all distortions confined to the two planes abutting the interface. Distortions are created from the lattice sites at the interface in the NDP, that can be made coincident through combinations of translation, rotations (tilts and twists), and strains. As discussed in Section 7, all components of the distortion matrix are present in general, but some become zero in special cases. D.1. Distortions associated with the CDP and RCDP When a lattice is subjected to a homogenous distortion, the displacement vectors, u, of lattice points, t, are given by

u ¼ Dt;

ðD1Þ

where the columns of u and t are the displacement vectors and corresponding position vectors, and D is the distortion matrix

D ¼ $u;

ðD2Þ

written explicitly in Eqs. (29)–(31). For all cases of interest, the terms in the distortion matrix are independent of position. After distortion, the new positions of the lattice sites are

t0 ¼ t þ u ¼ ðI þ DÞt;

ðD3Þ

where I is the identity matrix and the term (I + D) is the homogenous deformation matrix, such as cPn in section (2), where, as in Appendix A, the subscript c indicates final state and n the initial state. If displacement vectors are known for three points in the solid, the distortion matrix is

D ¼ ut1 :

ðD4Þ

1

The inverse, t , is nonsingular if the three points are noncollinear. The positions of the remaining sites in the solid are given by Eq. (D3). In order to compute the distortion matrix for a DP, one first translates the nP and nQ lattices so that one lattice site is coincident, this site becoming the origin of coordinates. Except for the origin, P and Q lattice sites do not coincide in general. Various degrees of coincidence, Cm=n , can be considered, as in section 2.2.4. From equation (D3), the difference in displacements of the nP and nQ crystals, each subject to an arbitrary homogeneous deformation into a new, primed configuration, is P c un

 c uQn ¼ ðt0P  t0Q Þ  ðtnP  tnQ Þ: nP

ðD5Þ

nQ

Now define t and t to belong to the set of translation vectors in the nP and nQ crystals that we in0 0 tend to become commensurate, and, to form the CDP, we require that t Q = t P, so that the difference in displacements of lattice points that become commensurate in the two crystals becomes c un

¼ c uPn  c uQn ¼ ðtnQ  tnP Þ:

ðD6Þ

The property of interface dislocations, determined from Burgers circuits is such that cun is constant for a given system. We assume that the displacements of each crystal from the natural state to the CDP are partitioned linearly such that they sum to cun, i.e., c uPn ¼ jc un , and c uQn ¼ ðj  1Þc un . Also, the distortions, expressed as strain matrices, to be applied to the P and Q crystals to achieve this are P c En

¼ jc un ðtÞ1 ;

Q c En

¼ ðj  1Þc un ðtÞ1 ;

ðD7Þ

where j is a dimensionless parameter that varies from 0 to 1, and t ¼ tnP ð1  jÞ  tnQ j, the matrix of partitioned interface translation vectors. The prefixed and postfixed subscripts in Equations (D6) and (D7) identify the final and initial lattices respectively, Appendix B. By setting j = 0.5, displacements from nP to cP and from nQ to cQ,, are equipartitioned, and, also, the strains are equipartitioned, which satisfies equilibrium in the homogeneous isotropic linear elastic approximation. In that case t ¼ ðtnP  tnQ Þ=2 ¼ tcP ¼ tcQ , the average of those translation vectors originating at the interface that define interface Burgers vectors: i.e. t equals ±bm, appendix B. In the more general case involving elastic anisotropy and/or large strains, Section 6, one does not know, a priori, the appropriate choice for j.

820

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

Nevertheless, an approximate but useful CDP can be created by setting j = 0.5, consistent with the standard model of Appendix C. The matrix, Eq. (2), of coherency strains that align crystals P and Q into the CDP is c En

¼ c EPn  c EQn ¼ c Pn  c Q n :

ðD8Þ

Here we have used the fact that the difference in distortion matrices, defined in equation (D1), is the same as the difference in deformation matrices, defined in Eq. (D3), applied to the two crystals. Also important are the distortion matrices needed to create the RCDP from the CDP by twist and/or tilt with rotation, Eq. (A12) for the standard model. When added to equation (D8) these give the total distortion rcDn in Eq. (13). Thus, if rc XPc and rc XQc are the rotations of the P and Q lattices from the CDP to the RCDP, then the total distortion, equation (A12), is rc Dn

¼ ðc Pn  c Q n Þ þ



P rc Xc

  rc XQc ¼ rc En þ rc Xc :

ðD9Þ

For example, if P and Q are an fcc and bcc lattice in a Nishiyama–Wasserman OR, then equation (D9) provides the distortion matrices for a twist of P relative to Q into a Kurdjumov–Sachs OR. The net rotation of crystals P and Q from the CDP to the RCDP is rcXc, and we assume the rotations of the P and Q lattices are also linearly partitioned as jrcXc, and (j  1)rcXc. Then, with the inclusion of rotations, equation (D7) still applies but with distortions D replacing the strains E. D.2. Line directions and spacing The Burgers vectors are available from the reference state, the CDP or the RCDP. Each set of dislocations with Burgers vectors bI consists of a parallel array with spacing LI. A vector v lying in the interface of interest, that is otherwise arbitrarily oriented and has length v, intersects v  NI dislocations of set I, where NI is the reciprocal vector introduced in Section 8 (25, 42, ), NI = NI(n nI) and NI = 1/LI, where n is a unit vector normal to the interface and pointing toward the upper crystal. Written explicitly, v  NI = v sin cI/LI, and cI is the angle between v and the line direction nI of the Ith set. Then we find, from the Frank–Bilby relation, regarding v as the probe vector, and using the relation nDrc = rcDn

X bI sin cI LI

0

¼ n Drc i ;

ðD10Þ

where i0 is a unit vector parallel to v, and the sum is over all sets of misfit dislocations. We now rearrange equation (D10) as follows:

cos q Djx þ

I M X bj I¼1

LI

! sin vI

 sin q Djy þ

I M X bj I¼1

LI

! cos vI

¼ 0;

ðD11Þ I

where Djk is the jkth term in nDrc, each summation is over M distinct sets, and bj is the jth component of Burgers vector in the Ith set. Recognizing that each of the two terms in parentheses is independent of q, and, therefore, must be zero, gives M equations, where M is seven in the most general case, Section 7. I M X bj I¼1

LI

sin vI ¼ Djx ;

I M X bj I¼1

LI

cos vI ¼ Djy

ðD12Þ

for j = x, y, or z. For a twist boundary at a bicrystal interface there are only in-plane components of BurI I gers vectors ðbx ; by ; 0Þ and distortion components given by

2 n Drc

Dxx

6 ¼ 4 Dyx 0

Dxy Dyy 0

0

3

7 05 0

for the reasons given in Section 7. Hence, the number of terms in (D12) is reduced to four.

ðD13Þ

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823

821

Fig. D1. Coordinates for dislocation line n and (a) probe vector v and (b) Burgers vector b.

D.3. Alternative solution Eqs. (12) and (13) provide another possible solution to the F–B eq. that is amenable to numerical analysis. That is, distortions are treated directly and no probe vector is involved in the solution. This solution is not used here, but provides a means to check the solutions of Eq. (D12). As shown in Fig. D1b, coordinates x0, y0, z0 are fixed on the interface dislocations, with z0 normal to the interface and x0 parallel to n. As discussed in Section 6 and in Section 19-7 of (42), the screw components in an array only produce 0 0 0 long-range strains e0xy ¼ bx =2L0 ¼ bs =2L0 and rotational functions x0xy ¼ x0yx ¼ tanðbz =2L0 Þ: edge com0 0 0 0 0 ponents by have only long-range strains eyy ¼ by =2L0 ¼ be =2L0 ; and edge components bz have only long0 range rotations x0yz ¼ x0zy ¼ tanðbz =2L0 Þ ¼ tanðbn =2L0 Þ. Explicitly, 0

0

bs =4L0 ¼ b  n=4L0 ;

0

0

be =2L0 ¼ b  N=2;

0

bn =2L0 ¼ b  n=2L0 :

ðD14Þ

x0i

Hence the distortions nDrc are known in coordinates for the Ith set and can be expressed in xi coordinates by the coordinate transformation in Eq. (14). Solutions for however many sets are needed are then inserted into Eq. (D14), giving M terms corresponding to the left side of Eq. (D12). Just as for equations (D12), with n free, the equations can have too many degrees of freedom. Indeed, Eq. (D12) and (D14) are essentially equivalent. References [1] Mathews JW. In: Matthews JW, editor. Epitaxial growth. New York: Academic; 1975. p. 559 [Part II]. [2] Aaronson HI. In: Zackay VF, Aaronson HI, editors. Decomposition of austenite by diffusional processes. New York: Wiley Interscience; 1962. p. 387. [3] Howe JM, Pond RC, Hirth JP. Prog Mater Sci 2009;54:792. [4] Frank FC. Acta Metall 1953;1:15. [5] Bilby BA. In: Report of the conference on defects in crystalline solids. London: Physical Soc; 1955. p. 124. [6] Bilby BA, Bullough R, Smith E. Proc Roy Soc (Lond) 1955;A231:263. [7] Bullough R, Bilby BA. Proc Phys Soc 1956;B69:1276. [8] Bollmann W. Crystal defects and crystalline interfaces. Berlin: Springer-Verlag; 1970. [9] Christian JW. Trans JIM 1976;Suppl. 17:211. [10] Frank FC. In: Report of the symposium on the plastic deformation of crystalline solids. Pittsburgh: Carnegie Institute of Technology; 1950. p. 150. [11] Pond RC, Hirth JP. Solid State Phys 2004;47:288. [12] Pond RC, Bollmann W. Philos Trans Roy Soc (Lond) 1979;A292:449. [13] Wang J, Hirth JP, Pond RC, Howe JM. Acta Mater 2011;59:241 [2010]. [14] Freund LB, Suresh S. Thin film materials. Cambridge: Cambridge University Press; 2003. [15] Pond RC, Hirth JP. Philos Mag 2010;90:805. [16] Pond RC. In: Nabarro FRN, editor. Dislocations in solids, vol. 8. North-Holland: Amsterdam; 1989. p. 1. [17] Hirth JP, Pond RC. Acta Mater 1996;44:4749. [18] Pond RC, Ma X, Chai YW, Hirth JP. In: Nabarro FRN, Hirth JP, editors. Dislocations in solids, vol. 13. Amsterdam: Elsevier; 2007. p. 225. [19] Hirth JP, Pond RC. Philos Mag 2010;90:3129. [20] Nicholson RB, Thomas G, Nutting J. J Inst Metals 1958–1959;87:429. [21] Patterson RL, Wayman CM. Acta Metall 1964;12:1306. [22] Christian JW. In: Zackay VF, Aaronson HI, editors. Decomposition of austenite by diffusional processes. New York: Wiley Interscience; 1962. p. 371. [23] Howe JM, Dahmen U, Gronsky R. Philos Mag 1987;A56:31.

822 [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40] [41] [42] [43] [44] [45] [46] [47] [48] [49] [50] [51] [52] [53] [54] [55] [56] [57] [58] [59] [60] [61] [62] [63] [64] [65] [66] [67] [68] [69] [70] [71] [72] [73] [74] [75] [76] [77] [78] [79] [80] [81] [82] [83] [84] [85] [86] [87] [88] [89]

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823 Amelinckx S. Solid State Phys 1964;Suppl. 6. Sutton AP, Balluffi RW. Interfaces in crystalline materials. Oxford: Oxford University Press; 1995. Knowles KM. Philos Mag 1982;46A:951. Pond RC. J Cryst Growth 1986;79:946. Shubnikov AV, Koptsik VA. Symmetry in science and art. New York: Plenum Press; 1977. Pond RC, Vlachavas DS. Proc Roy Soc Lond 1983;386A:95. Narayan J, Larson BC. J Appl Phys 2003;92:278. Bonnet R. Acta Metall 1981;29:437. Frank FC. Philos Mag 1951;42:809. Knowles KM, Smith DA. Acta Cryst 1982;38A:34. Bonnet R, Durand F. Philos Mag 1975;32:997. Frank FC, van der Merwe JH. Proc Roy Soc (London) 1949;198:205. Gibeling JG, Nix WD. Acta Metall 1980;28:1743. Hirth JP, Balluffi RW. Acta Metall 1973;21:929. Liebisch Th. Physikalische kristallographie. Leipzig: Veit and Co.; 1891. Niggli P. Lehrbuch der mineralogie, 3rd ed. Part 1. Gebruder Borntraeger. Berlin; 1941. Ashby MF, Spaepen F, Williams S. Acta Metall 1978;26:1647. Pond RC, Medlin DL, Serra A. Philos Mag 2006;86:4667. Hirth JP, Lothe J. Theory of dislocations. 2nd ed. Melbourne (FL): Krieger; 1992. Bullough R. Philos Mag 1965;12:1139. Hirth JP, Pond RC, Lothe J. Acta Mater 2006;54:4237. Khater HA, Serra A, Pond RC, Hirth JP. Acta Mater 2012;60:2007. Read Jr WT. Dislocations in crystals. New York: McGraw-Hill; 1953. Ma X, Pond RC. J Nucl Mater 2007;361:313. Clarke WAT, Wagoner RH, Shen ZY, Lee TC, Robertson IM, Birnbaum HK. Scripta Metall 1992;26:203. Hirth JP. J Phys Chem Solid 1994;55:985. Pond RC, Celotto S, Hirth JP. Acta Mater 2003;51:5358. Pond RC, Sarrazit F. Interface Sci 1996;4:99. Hirth JP, Pond RC. Prog Mater Sci 2011;56:586. Aaronson HI, Hall MG. Metall Mater Trans 1994;25A:1797. Aaronson HI, Howe JM. Metall Trans 1991;22A:1137. Wechsler MS, Lieberman DS, Read TA. Trans AIME 1953;197:1503. Bowles JS, Mackenzie JK. Acta Metall 1954;2:129. Wayman CM. Introduction to the crystallography of martensitic transformations. New York: Macmillan; 1964. Hirth JP, Mitchell TE. Acta Mater 2008;56:5701. Pond RC, Ma X, Hirth JP, Mitchell TE. Philos Mag 2007;87:5289. Heinz NA, Ikeda T, Snyder GJ, Medlin DL. Acta Mater 2011;59:7724. Hirth JP, Pond RC, Lothe J. Acta Mater 2007;55:5428. Dundurs J, Mura T. J Mech Phys Solids 1964;12:177. Hirth JP, Barnett DM, Lothe J. Philos Mag 1979;40A:39. Chou YT. Scripta Metall 1976;10:331. Barnett DM, Lothe J. J Phys F 1974;4:1618. Ting TCT. Anisotropic elasticity. Oxford Univ. Press; 1996. Henager CH, Hoagland RG. Scripta Mater 2004;50:1091. Clouet E, Ventelon L, Williams F. Phys Rev Lett 2009;102:055502. Lothe J. In: Indenbom VL, Lothe J, editors. Elastic strain fields and dislocation mobility. North-Holland: Amsterdam; 1992. p. 175. Eshelby JD, Laub T. Can J Phys 1967;45:887. De Wit R. Solid State Phys 1960;10:249. Cai W, Arsenlis A, Weinberger R, Bulatov V. J Mech Phys Solids 2006;53:561. Lothe J, Hirth JP. Phys Status Solidi 2005;242b:836. Hoagland RG, Demkowicz MJ. Unpublished research. Kocks UF, Tomé CN, Wenk H-R. Texture and anisotropy. Cambridge: Cambridge University Press; 1998. Basinski ZS, Basinski SJ. Philos Mag 2004;21:213. Livingston JD, Chalmers B. Acta Metall 1957;5:322. Hook RE, Hirth JP. Acta Metall 1967;15:1099. Taylor GI. J Inst Metals 1938;62:307. Hosford WF. The mechanics of crystals and textured polycrystals. Oxford: Oxford University Press; 1993. Chap 7. Leffers T. Deformation textures: simulation principles. In: Kallend J, Gottstein G, editors. Proc of the 8th int conf on textures of materials (ICOTOM-8). Warrendale (PA): TMS; 1988. p. 273. Vattre A, Demkowicz MJ. Unpublished research. Al Fadhala K, Tomé CN, Beaudoin AJ, Robertson IM, Hirth JP, Misra A. Philos Mag 2005;85:1419. Mitlin D, Misra A, Mitchell TE, Hirth JP, Hoagland RG. Philos Mag 2005;85:3379. Read WT, Shockley W. In Imperfections in nearly perfect crystals. New York: Wiley; 1952. p. 352. Merkle KL, Thompson LJ, Phillipp J. Philos Mag Lett 2002;82:589. Kurtz RJ, Hoagland RG, Hirth JP. Philos Mag 1999;79:683. Carter CB, Foll H. In: Ashby MF, Bullough R, Hartley CS, Hirth JP, editors. Dislocation modeling of physical systems. Oxford: Pergamon; 1981. p. 554. Demkowicz MJ, Hoagland RG, Wang J. In: Hirth JP, editor. Dislocations in solids, vol. 14. Amsterdam: Elsevier; 2008. p. 141.

J.P. Hirth et al. / Progress in Materials Science 58 (2013) 749–823 [90] [91] [92] [93] [94] [95] [96] [97] [98] [99] [100] [101] [102] [103] [104] [105] [106] [107] [108] [109] [110] [111] [112] [113] [114] [115] [116] [117] [118] [119] [120] [121] [122] [123] [124] [125] [126] [127] [128] [129] [130] [131] [132] [133] [134] [135] [136] [137] [138] [139] [140] [141] [142] [143] [144] [145] [146] [147] [148] [149] [150] [151] [152] [153]

823

Demkowicz MJ, Hoagland RG, Hirth JP. Phys Rev Lett 2008;100:136102. Demkowicz MJ, Hoagland RG. Int J Appl Mech 2009;1:421. Demkowicz MJ, Thilly L. Acta Mater 2011;59:7744. Venables JA, Spiller GDT, Hanbucker. Rep Prog Phys 1984;47:399. Moazed KL, Pound GM. Trans Metall Soc AIME 1964;230:234. Kung H, Griffin AJ, Nastasi M, Mitchell TE, Embury JD. Appl Phys Lett 1997;71:2103. Kamat SV, Hirth JP, Carnahan B. Scripta Metall 1987;21:1587. Nix WD. Mater Sci Eng 1997;234A:37. Kamat SV, Hirth JP, Carnahan B. Mater Res Soc Symp Proc 1988;103:55. Schober T, Balluffi RW. Philos Mag 1970;21:109. Wolf D, Merkle KL. Material interfaces. London: Chapman and Hall; 1992. Howe JM. Interfaces in materials. New York: Wiley; 1997. Howe JM, Aaronson HA, Hirth JP. Acta Mater 2000;48:3977. Bishop GH, Chalmers B. Scripta Metall 1968;2:133. Frost HJ, Ashby MF, Spaepen F. Scripta Metall 1980;14:1052. Howe JM, Tsai MM, Csontos AA. In: Kiowa M, Otsuka K, Miyazaki T, editors. Proc conf on solid–solid phase transformations II (PTM’99). Sendai: Japan Inst. Metals; 1999. p. 1136. Mitlin D, Misra A, Mitchell TE, Hoagland RG, Hirth JP. Appl Phys Lett 2004;85:1686. Chen I-W, Chiao Y-H. Acta Metall 1985;33:1827. Matthews JW. Philos Mag 1974;29:797. Hirth JP. J Mater Res 1993;8:1572. Nishiyama Z. Sci Rep Tohoku Imp Univ 1934;23:637. Wasserman G, Grewen J. Texturen metallischer Werkstoffe. Berlin: Springer; 1962. Cai W, Bulatov VV, Chang J, Li J, Yip S. Phys Rev Lett 2001;86:5727. Misra A, Hirth JP, Hoagland RG. Acta Mater 2005;53:4817. Mara NA, Bhattacharyya D, Dickerson P, Hoagland RG, Misra A. Appl Phys Lett 2008;92:231901. Zhang KY, Embury JD, Han K, Misra A. Philos Mag 2008;88:2559. Zhang W-Z, Yang X-P. J Mater Sci 2011;46:4135. Gu XF, Zhang W-Z, Qiu D. Acta Mater 2011;59:2011. Kelly PM. In: Olson GB, Lieberman DS, Saxena A, editors. Proceedings of ICOMAT 2008. TMS: Warrendale (PA); 2010. p. 115. Cahn JW, Mishin Y, Suzuki A. Acta Mater 2006;54:4953. Ma X. Ph.D. Thesis, University of Liverpool, 2007. Sandvik BPJ, Wayman CM. Metall Trans 1983;14A:835. Moritani T. Ph.D. Thesis, Kyoto University, 2003 Zhang W-Z, Purdy GR. Philos Mag 1993;68A:291. Zhang W-Z. Philos Mag 1998;78:913. Balluffi RW, Brokman A, King AH. Acta Metall 1982;30:1453. Pond RC, Hirth JP. In: Olson GB, Lieberman DS, Saxena A, editors. Proceedings of ICOMAT 2008. Warrendale (PA): TMS; 2010. p. 123. Kelly PM. Personal communication, 2012. Nie JF. Acta Mater 2004;52:795. Hall MG, Aaronson HI, Kinsman K. Surf Sci 1972;31:257. Aaronson HI, Furuhara T, Hall MG, Hirth JP, Nie JF, Purdy GR, et al. Acta Mater 2006;54:1227. Purdy GR, Hirth JP. Philos Mag Lett 2006;86:147. Dahmen U. Acta Metall 1982;30:63. Zhang M-X, Kelly PM. Acta Mater 1998;46:4617. Dahmen U. Scripta Metall 1987;21:1029. Nie JF. Scripta Mater 2005;52:687. Kelly PM, Zhang M-X. Scripta Mater 2005;52:679. Qui D, Zhang W-Z. Scripta Mater 2005;52:683. Christian JW. The theory of transformations in metals and alloys. Oxford: Pergamon; 2002. Mermin D. Rev Mod Phys 1951;42:809. Volterra V. Ann Sci Ecole Norm Sup (Paris) 1907;24:401. Hahn T, editor. International tables for crystallography. Dordrecht: Reidel; 1983. Pond RC. Int Sci 1995;2:299. Lay S, Loubradou M, Johansson SA, Wahnström G. J Mater Sci 2012;47:1588. Murnahan FD. Introduction to applied mathematics. New York: Dover; 1963. Borg SF. Matrix–tensor methods in continuum mechanics. New York: van Nostrand; 1963. Teodosiu C. Elastic models of crystal defects. Berlin: Springer-Verlag; 1982. Seeger A, Teodosiu C, Petrasch C. Phys Status Solidi 1975;67b:207. Nabarro FRN. Theory of crystal dislocations. Oxford: Pergamon; 1967. Gehlen PC, Hirth JP, Hoagland RG, Kanninen MF. J Appl Phys 1972;43:3921. Love AEH. The mathematical theory of elasticity. New York: Dover; 1944. Sokolnikoff IS. Mathematical theory of elasticity. New York: McGraw-Hill; 1956. Seeger A, Haasen P. Philos Mag 1958;3:470. Simmons JA, Bullough R. In Simmons JA, deWit R, Bullough R, editors. Fundamental aspects of dislocation theory, Spec Publ 317. Washington (DC): Nat Bur Stds; 1970. p. 89.