The role of disconnections in phase transformations

Progress in Materials Science 54 (2009) 792–838

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

The role of disconnections in phase transformations J.M. Howe a, R.C. Pond b, J.P. Hirth c,* a

Department of Materials Science and Engineering, 140 Chemistry Drive, University of Virginia, Charlottesville, VA 22904-4745, USA University of Exeter, School of Engineering, Computing and Mathematics, North Park Road, Exeter, EX4 4QF, UK c 114 E, Ramsey Canyon Rd., Hereford, AZ 85615, USA b

a r t i c l e

i n f o

a b s t r a c t The topological model of phase transformations is described in terms of disconnections. Disconnection motion is shown to produce a variety of phase transformations with accompanying plastic strain. These strains, together with elastic strains associated with equilibrium arrays of interface defects, define expected habit planes and orientation relationships. Non-equilbrium defect arrays resulting from kinetic constraints are discussed. Example applications of the topological model are presented for several types of martensitic transformations and several examples of diffusional transformations. Ó 2009 Elsevier Ltd. All rights reserved.

Contents 1. 2.

3.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Structural properties of defects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1. Dislocation and step character of interfacial defects. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2. Coherency strain relief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1. Reference states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2. Dislocation networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3. Ancillary misorientations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.4. Dislocation–disconnection networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.5. Energy of defect networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3. Compatibility effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4. Terraces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5. Implementation of the topological model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kinetic aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1. Disconnection motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

* Corresponding author. E-mail address: [email protected] (J.P. Hirth). 0079-6425/$ - see front matter Ó 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.pmatsci.2009.04.001

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3.2. 3.3.

4.

5.

6.

Transformation strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diffusional flux . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1. Composition change at interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.2. Diffusional fluxes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4. Defect intersections. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5. Diffusional drag effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6. Large steps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7. Special effects in the martensite case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8. Kinetic influence on variant selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observations of martensitic transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1. Transmission electron microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2. Ti–Nb alloys . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3. (575)c habits of lath martensite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Observations of diffusional transformations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1. Preliminary remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2. Diffusive–displacive transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1. The Al–Ag system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2. The Ti–Al system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3. Disconnections in coherent precipitates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4. Defects accommodating angular incompatibilities between facets . . . . . . . . . . . . . . . . . . . . . . 5.5. Scaling (oxidation) reactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.6. Hydride Formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Discussion and summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

793

805 806 806 807 808 808 809 809 810 810 810 813 814 820 820 821 821 823 825 827 832 833 834 835

1. Introduction Pertinent to the theme of this volume, i.e., how the work on dislocations by Nabarro impacted other fields, one of the earliest references on structural effects in phase transformations is his famous analysis of the role of elastic strain energy in determining the equilibrium shapes of precipitates, [1]. Other early work on the action of dislocations in a phase transformation is on the a to c transformation in Al–Ag by Barrett and Geisler [2], where the authors say, ‘‘thin lamellae of the hexagonal close-packed phase . . . could result from local regions in which a (1 1 1) plane glides over the plane next to it as in crystallographic slip”. A significant paper on the nature of ‘‘misfit dislocations”, a term that came into common use in phase transformations, appeared in 1949 [3]. The Burgers vector content of such dislocation networks accommodating a given coherency strain was quantified in the classical topological theories of Frank [4], Bilby [5] and Bullough and Bilby [6]. By 1953, dislocations were clearly recognized as being responsible for the crystal structure changes in both diffusional and martensitic transformations, as evidenced by papers on the interfacial structure of martensite by Frank [7] and on the mechanisms of several diffusional transformations by Barrett [8]. The term ‘‘transformation dislocation” also appeared in the literature just a few years prior to this [9]. In the late 1950s, transmission electron microscopy (TEM) began to reveal the transformation and misfit dislocations previously proposed, see e.g. [10,11] so that by the early 1960s these concepts were widely accepted in the field of phase transformations [12,13]. The transformation dislocations were recognized as accomplishing the lattice variant, or homogeneous, component of phase transformations (or twinning), and the misfit dislocations were recognized as accommodating the lattice-invariant, or inhomogeneous, component of phase transformations. The transformation dislocations were also recognized to constitute steps, which moved along the interface to cause the crystal structure change. These concepts and TEM investigations continued well into the 1990s. In the mid-1980s high-resolution TEM (HRTEM), became available to reveal the atomic structure of transformation interfaces, in particular the transformation dislocation steps associated with both diffusional and martensitic transformations [14,15], and shortly thereafter, in situ heating studies revealed how such steps propagated along the interfaces [16]. Many of these aspects of structures are described in several reviews [17–20].

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The terrace-ledge-kink model for growth at surfaces by Burton, Cabrera and Frank [21] was initially applied to solid–solid transformations by Aaronson [13]. This was followed by a number of kinetic analyses, see e.g. [22–26]. Almost all of these analyses were based on finding solutions to flux balances across a step or train of steps, without regard to the dislocation character of the step, although the latter has been considered [27,28]. By the 1990s, many different approaches for determining interfacial structure and habit planes were in use and interfacial defects were being given different names based on their functionality (e.g., the misfit dislocations). Often, one or the other of the dislocation or step character was emphasized. Over the past two decades or so, refinement of these notions has led to the concept of a defect, the disconnection, that has both dislocation and step character [29,30]. This concept, combined with quantitative application of topological methods, has led to improved dislocation models of both diffusional and diffusionless transformations. In parallel with these developments, continuing improvements in experimental techniques, particularly TEM, have provided direct observations of dislocation processes in phase transformations. The steps in phase transformation theory are synonymous with ledges in the fields of surface science and crystal growth. In those areas, the terrace-ledge-kink model of Kossel, Stranski and others is widely used [18,21,31–33]. Here, we use the step terminology. Ironically, the combined defect has been considered before and identified with one of its components as either a transformation dislocation [9] or growth ledge [13]. While the development of these ideas [34–37] makes clear that both strain relief and transformation are associated with the defects so defined, the vector field definition of the disconnection reveals other quantitative details of the defect properties. The classical topological works mentioned earlier [4–6] considered the translation symmetries of the two crystal lattices abutting at an interface. When the complete space-group symmetries of these crystals are taken into account [38], a comprehensive understanding of the dislocation and step character of admissible interfacial defects is obtained. In the context of phase transformations, two subcategories with distinct topological natures arise. The first category comprises transformation disconnections [30], which exhibit combined dislocation and step character, and are hence represented by the parameters (b, h), where b is the Burgers vector and h is the step height. By virtue of its step character, disconnection motion along an interface causes the transfer of material from one phase to the other. In addition, its dislocation character produces a deformation. Thus, disconnection motion couples deformation with interface migration and is the elementary mechanism underlying displacive transformations [39]. Defects in the second subcategory do not exhibit step character and are hence characterised by the pair (b, 0); this category includes perfect, partial and twinning dislocations that originated in the bulk of the adjacent crystals (in the terminology of the Phenomenological Theory of Martensitic Crystallography [17], these correspond to lattice-invariant deformation modes; the former category corresponding to the lattice-variant component). Arrays of such bulk defects can accommodate coherency strains, but their motion along an interface cannot produce transfer of material between the phases. On the other hand, disconnections can perform both primary functions. Motion of bulk defects or disconnections along an interface is either by glide or climb, and the magnitude of long-range diffusion accompanying climb can be quantified in terms of the component of b perpendicular to the interface, the step height, h, and the difference in density between the adjacent crystals for each species present [30]. This advance has been critically important for clarifying the special circumstances necessary for martensitic transformations [40]. An important conclusion is that glissile interfaces must comprise coherent terraces with a network of appropriate disconnections and bulk defects superimposed to relieve any coherency strains [41]. In addition, this quantification of mass transfer during defect motion elucidates the relationship between martensitic transformations (conservation of all (substitutional) atomic species present) and diffusional–displacive transformations (conservation of (substitutional) sites, but not species) [42,43]. Concomitant interstitial diffusion can be readily incorporated into such transformation models. Section 2 is a review of the properties of interfacial defects relevant to structural aspects of phase transformations. It defines the topological character, b and h, of individual defects, and summarises their ability to accommodate coherency strain when organised into networks. This discussion includes the additional considerations that arise in applying the Frank–Bilby equation when disconnections, with their attendant step character, form part of a network. In addition, the energy stored by a defect

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array is outlined using crystal plasticity theory [44]. Section 3 deals with kinetic aspects of disconnection models of transformations. These aspects are the magnitude of diffusional fluxes, constraints on diffusionless processes, and kink formation and motion. Experimental observations of martensitic transformations are presented in Section 4 and those of disconnection processes in diffusional transformations are presented in Section 5. These findings are discussed in view of the theoretical understanding reported in Sections 2 and 3. Discussion and a summary are set forth in Section 6. 2. Structural properties of defects 2.1. Dislocation and step character of interfacial defects Consider a bicrystal where a planar surface of one crystal, designated ‘‘white” (k), abuts the surface of a ‘‘black” (l) one. The topological character of an admissible defect that can be superimposed on this reference structure is given by a combination of symmetry operators, one from each of the crystals [38]. Using the matrix formalism for symmetry operators set out in the International Tables for Crystallography [45], we find that the operator characterising a defect is given by

Qij ¼ WðkÞi W ðlÞ1 j ;

ð1Þ

where WðkÞi and WðlÞj are the relevant operators and the asterisk implies that WðlÞj has been expressed in the k coordinate frame. The set of defects defined by expression (1) includes dislocations that have reached the interface from either of the crystals and also a range of other defects peculiar to the interface. In the former case, a perfect white crystal dislocation is represented by Qij ¼ WðkÞi ¼ ðI; tðkÞi Þ, where I represents the identity operation and t(k)i is the translation vector equal to the dislocation’s Burgers vector, b. An interfacial defect arises, for example, when dislocations from both crystals coincide at the interface; then Qij ¼ ðI; tðkÞi  t ðlÞj Þ, or, in other words, the defect exhibits dislocation character with b ¼ tðkÞi  t ðlÞj . Hirth and Pond [30] have defined the ‘‘overlap” step height, h, associated with such a defect to be the smaller of n  tðkÞi and n  t ðlÞj , where n is the unit normal to the interface pointing into the k crystal. When h is finite, the defect is a disconnection, characterised by (b, h), as depicted schematically in Fig. 1. Disconnections are perfect, when they separate energetically degenerate regions of interface, and partial otherwise; a subset of perfect disconnections separate degenerate regions inter-related by rotation or mirror operations [38]. Since there is no overlap step when only one crystal dislocation is present, such a defect is characterised by (b, 0).The description of the disconnection in terms of b and h is the only plausible choice. The long-range elastic field is that of the dislocation, while the overlap step produces no long-range elastic field but does produce an offset of the interface. There are other types of defects of this type, including line-force disconnections and spacing defects [46,47], but these are more relevant for grain boundaries than for interphase interfaces and are not considered here in detail. Also, for disconnections with large

Fig. 1. Schematic illustration of a perfect disconnection formed by bonding incompatible surface steps on the k and l crystals [38]. If the k and l surfaces to the left of the defect are bonded first, a Volterra operation Qij is needed to correctly juxtapose the surfaces on the right hand side before joining to form a degenerate interface.

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step heights, the corners of the steps have elastic fields that extend over a distance of the order of the step height [48]. For disconnections with step heights of one or a few interplanar distances, these fields merge into the core field of the defect and need not be considered separately. Similarly, the component of the overlap step that is parallel to the interface can have spacing defect character and the dislocation can undergo core spreading on the step [47]. The local fields associated with these effects also merge into the non-linear core fields. Hence, for arrays of disconnections, the interface can be described by the superposition of the fields of the individual disconnections and the steps can be considered to lie normal to the adjoining terraces. 2.2. Coherency strain relief Coherent interfaces between distinct phases generally require the adjacent crystals to be strained, and such strains can be removed by the superposition of an appropriate network of line defects [3]. Here, we show how to determine such network configurations when these comprise both dislocations (b, 0) and disconnections (b, h). 2.2.1. Reference states For quantitative modelling, one must define the bicrystal reference structure employed since this, in turn, determines the parameters (b, 0) and (b, h) of the defects in the strain-accommodating network. In the present context the use of two references, the natural and the coherent, is convenient. In the natural reference the two crystals exhibit their unstrained unit cell dimensions (at the chosen temperature), and the crystals are relatively disposed with selected misorientation and interface plane. We refer to the latter as the terrace plane. Later, for actual interfaces containing steps, the average interface plane is defined as the habit plane, but the planar regions between defects will still correspond to the terrace plane. In the coherent reference both crystals are homogeneously strained to produce a coherent terrace. An illustration of natural and coherent reference states is depicted in Fig. 2. (The term ‘‘reference state” is used here to mean either a reference bicrystal structure, as above, or a reference dichromatic pattern – i.e. the pattern where the k and l lattices interpenetrate [49]).

Fig. 2. Schematic view of the (1 0 0)o/(1 0 0)m terrace plane between orthorhombic and monoclinic ZrO2 [41]. (a) The interface is incoherent and the crystals exhibit their natural lattice parameters (the monoclinic cell shown comprises two crystal unit cells). (b) The interface is coherent following uniaxial strains of the two crystals.

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The defect content of the coherent with respect to the natural state, Bc, can be formally quantified using the Frank–Bilby equation. Let c Kn and c Mn be matrices representing the homogeneous deformations necessary to transform the natural k and l crystals respectively into their coherent forms. Then 1 Bc ¼ ðc K1 n  c Mn Þv ;

ð2Þ

where v is a probe vector in the coherent structure. (The Frank–Bilby equation is equivalent to expression (1), where WðkÞi and W ðlÞj represent the probe vector after mapping into the k and l crystals, respectively, of the natural reference [50].) The defects in the coherent interface, Fig. 2b, can be regarded as smeared into a distribution of continuous, infinitesimal defects [51]; we refer to this as the ‘‘coherency” dislocation content, as signified by the superscript Bc . 2.2.2. Dislocation networks In real bicrystals of significant size, coherency strains are accommodated by networks of discrete defects [44]. Thus, superimposing a network of localised defects with total content Bc onto the coherent reference would remove coherency strain at long-range from the interface. Then, the total dislocation content with respect to the natural reference state would be zero, and only short-range strains would remain. In practice, it may not be possible to construct networks comprising admissible defects in the coherent reference that sum exactly to Bc. However, networks can be identified where any excess Burgers vector content corresponds to the introduction of low-angle tilt or twist boundaries. Thus, small angular deviations from the reference state may arise, but the k and l crystals retain their natural cell dimensions at long-range. For clarity, we consider first dislocation networks where no ancillary tilts or supplementary twists arise, i.e., any Burgers vectors normal to the habit plane are suppressed and can be superposed later, together with and twists.  associated tilts  1 . The defect content necessary to The terrace coherency strain is expressed as n Ec ¼ c K1 n  c Mn remove coherency strain, B = Bc, is given by

B ¼ c Ec v ;

ð3Þ

where v = [x, y, 0]. Furthermore, one can write

0 n Ec

exx

B ¼@ 0

0

0

0

0

0

1

eyy 0 C A;

ð4Þ

where exx and eyy are the principal strains in the (x, y) terrace plane. The components exz, eyz, and ezz are free because of the suppression of defect components bz, so two or three sets of independent defects are required to fix the other three components as described below. The total defect content of the strain relieving network, B, is a sum of Burgers vectors of admissible defects in the coherent reference; one can obtain these Burgers vectors either directly from expression (1) with the coherent k and l unit cell dimensions, or graphically from the coherent dichromatic pattern. Here, we consider first some simple examples of networks consistent with Eq. (3) for dislocations having no component of b normal to the terrace plane. Depending on the multiplicity of feasible Burgers vectors in a dichromatic pattern and their orientation with respect to the principal axes, there may be more than one network capable of relieving the strain, n Ec . Eq. (3) shows that coherency strains where both exx and eyy are finite can be accommodated by two-array networks if both sets of dislocations have Burgers vectors of the form b = [bx, by, 0] and their line directions and spacings are unconstrained. The condition exy = 0 is satisfied if the two-dislocation network has no twist component. Two-array networks are important in phase transformations and are treated further in terms of Eq. (3) below. However, there are circumstances where a single array may suffice, and other situations where three arrays are needed and simple examples are outlined next. In the case where exx – 0 and eyy = 0 and an admissible dislocation exists with b = [bx, 0, 0], a single array of defects in edge orientation with spacing bx/exx would accommodate the coherency strain. In the case of balanced biaxial strain, exx = eyy, as occurs for example at epitaxial {1 1 1} interfaces between fcc crystals, a hexagonal network of three arrays of edge dislocations with b=1/2h1 1 0i is needed [44]. The principal axes are indeterminate in this case, so x may be arbitrarily chosen parallel

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to one of the three h1 1 0i directions. Thus, one edge array with b = [bx, 0, 0] accommodates exx while the other two, in combination, accommodate eyy. Of these three sets, only two are independent [52], in accord with the statement following Eq. (4). If the Peierls structure of the interface constrains d to not quite equal bx/exx, for example, sets of defects with increasing spacings can be arranged to remove coherency strains on average, analogous to the technique for describing an irrational number as a series of rational fractions. Returning to the geometry of two-array networks, we let the Burgers vectors of the dislocations be bp and bq. One selects a unit probe vector vp parallel to the p-disconnection line, so that only the qdislocation array is cut; Eq. (3) leads to the dislocation density, Bq = nEcvp. Similarly, the dislocation density, Bp = nEcvq, can be established using a unit vector parallel to the q-dislocation line-direction, vq. One can either solve these equations to find vp and vq assuming bp and bq, or vice-versa. In the former case one has, p

vq ¼

ðn Ec Þ1 b p

jðn Ec Þ1 b j

;

ð5Þ

:

ð6Þ

and q

vp ¼

ðn Ec Þ1 b q

jðn Ec Þ1 b j

The line-senses of the dislocations, np and nq, are either parallel or anti-parallel to vp and vq, respectively; in order to be consistent with the RH/FS convention np  vq and nq  vp must be parallel to n p p (i.e. point toward the (upper) k crystal). Furthermore, since Bp ¼ ðb =d Þ sinðhp  hq Þ and Bq ¼ q q p q p q p q ðb =d Þ sinðh  h Þ, where d and d are the array spacings and h and h are the angles subtended by np and nq from a common direction, we have p

p

d ¼

jb j sinðhp  hq Þ; jðn Ec Þv q j

ð7Þ

and q

q

d ¼

jb j sinðhp  hq Þ: jðn Ec Þv p j

ð8Þ

To illustrate the use of Eqs. (5)–(8), consider the case where exx = eyy. Several authors have discussed this case [53,54]. Matthews [53] considered the instance of two orthorhombic crystals; the natural and coherent reference states are shown projected along [0 0 1] in Fig. 3. The matrix 1 1 n Ec ¼ ðc Kn  c Mn Þ takes the form

Fig. 3. Reference states for two orthorhombic crystals. The [0 0 1] projection direction is normal to the interface plane. The natural k and l crystal unit cells are depicted by dashed and full lines, respectively; the coherent unit cell is depicted by bold lines.

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Fig. 4. Dislocation networks accommodating equal and opposite principal strains; (a) edge dislocations, and (b) screw dislocations (not to scale).

0 n Ec

e

0

0

0

0

0

1

B C ¼ @ 0 e 0 A

ð9Þ

with e ¼ ð1 þ exx ðkÞÞ  ð1 þ exx ðlÞÞ ¼ ð1 þ eyy ðkÞÞ  ð1 þ eyy ðlÞÞ, where the strains exx(k), etc. are defined in Fig. 3. Taking the pair vp = [1 0 0]coh and vq = [0 1 0]coh, substitution into Eqs. (5) and (6) shows q  0 0 with nq ¼ ½0 1  0 . Using Eqs. (7) and (8) we that bp = [0 1 0]coh with np = [1 0 0]coh, and b ¼ ½1 coh coh obtain the spacings of these orthogonal edge dislocation arrays, dp = dq = be/e, where be = |bp| = |bq|. The resulting network is depicted in Fig. 4a; note that the extra half-planes of the q-dislocations are pffiffiffi located p inffiffiffi the l and the p-dislocations in the k crystal. Alternatively, taking v p ¼ 1= 2½1 1 0coh and q p p q   v q ¼ 1= 2½1 1 0coh ; we obtain bp = ffiffiffi [1 1 0]coh with n = [1 1 0]coh, and b ¼ ½1 1 0coh with n ¼ ½1 1 0coh . Also, dp = dq = bs/e, where bs ¼ 2be . In this network the p-dislocations are right-handed and the orthogonal q-dislocations are left-handed screws, as depicted in Fig. 4b. Either configuration in Fig. 4 fully accommodates the strains exx and eyy. 2.2.3. Ancillary misorientations The Burgers vectors of the dislocations discussed above are parallel to the terrace plane. However, in the context of phase transformations, network defects sometimes include edge components normal to the interface, bz, that do not contribute to strain accommodation in the terrace plane. Such an array of defects produces a tilt array, introducing a rigid-body rotation between the crystals about an axis parallel to the defects’ line-direction, thereby modifying the misorientation relative to the reference coherent state [40]. The field of this array can be superposed on the in-interface array in the terrace plane, thereby compensating coherency without introducing cross-terms in the total elastic distortion field. The rotation is partitioned between the two crystals in a manner depending on their relative

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elastic compliance [55]: in the isotropic elastic case the tilt rotation u is partitioned equally between the two crystals. The defect content producing the ancillary tilt rotation can be incorporated into Eq. (3), as follows:

B ¼ ðn Ec þ RÞv ;

ð10Þ

R ¼ ðRk Þ1  ðRl Þ1 ;

ð11Þ

and

k

l

where R and R , each equal in magnitude to u/2, are the partitioned rotations of the k and l crystals. The matrix (nEc + R), defines the closure failure of a Burgers circuit associated with the vector v. If two or more arrays of defects are present in a network, ancillary tilts may be associated with each one. There are several observations of such tilts in the literature, especially in epitaxial systems [56]. In the phase transformation case twist arrays can similarly be added if they lower the energy of the interface. The Burgers vectors and rotation would still be given by Eqs. (10) and (11) and u would be the total rotation about the normal to the interface plane partitioned between the two crystals. Note that the interface remains coherent in the regions between the dislocations in the final network. Matthews considered the case depicted in Fig. 4b; superposition of a twist array of right-handed screw dislocations results in a decrease of the p-dislocation spacing and an increase in that of the q-dislocation spacing [53]. In the limit a purely p-dislocation network is possible. Tilt and twist arrays may be present at an interface, separately or together. 2.2.4. Dislocation–disconnection networks Further considerations are necessary when one of the defect arrays in a network comprises disconnections because of their step character, h. An array of disconnections causes the resultant interface plane to deviate from the terrace orientation. We designate the resultant interface plane the ‘‘habit” plane; clearly, the deviation of the habit plane from the terrace depends upon the line-direction and spacing of the disconnection array. In other words, a probe vector, v, to be used in the Frank–Bilby equation is not independent of B because of the coupled topological properties, b and h, of disconnections. Also, one must consider whether the line-direction of the dislocations, nL (the superscript L signifies lattice-invariant deformation, LID), or disconnections, nD, are free parameters. For example, if the dislocations in an array glide to the habit plane from their crystal of origin, and are sessile therein, nL will be fixed along the intersection of the habit and glide planes, nG. If conditions permit dislocation climb, nL may not be constrained in this way. Also if the LID defects are twinning dislocations (disconnections), they will be sessile whether or not climb can occur (any motion in the interface plane would create a high-energy fault). A practical procedure taking these factors into account [41] is to first find a provisional solution to Eq. (3), treating nD and nL as unconstrained parameters, and temporarily suppressing the z components of bD and bL. The step height, h, of the disconnections is also temporarily set to zero, and a provisional terrace network determined. Using the current network parameters, one can 0 find an updated habit plane by introducing the disconnection steps. Also, the updated components bz 0 0 0 of the defects (the coordinate frame based on the habit plane is x y z ) and the current line-directions can be computed. The procedure is repeated iteratively. Using crystal plasticity theory, one can determine the residual coherency strain at any stage until an acceptable value is reached. However determined the equilibrium habit plane is described by Eqs. (3), (4), and (10) in the rotated (x0 , y0 , z0 ) coordinates fixed on the habit plane. All long-range coherency stresses are relieved by the equilibrium set of disconnections and dislocations, with their relevant b0 components. There are no net long-range elastic strain components, only distortion components associated with rotations. Because the habit plane is rotated relative to the terrace plane, strains such as ezz need not be zero and often are not. The in-terrace plane Burgers vector components are those of the coherently strained reference lattice. Out of plane components are those of the unstrained reference. The step heights are also those of the unstrained reference lattice. If the defect spacings are not the equilibrium values, there are residual coherency stresses and strains, including Poisson contributions to the out-ofhabit-plane components, and there are small differences in h and b0 . Generally, these modifications are very small compared to corrections associated with elastic anisotropy and non-linear elasticity and can be neglected. Hence, the treatment of the equilibrium habit plane is exact within the isotropic

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Fig. 5. Schematic illustration showing the disconnection content of an interface, with Burgers vector components resolved in the terrace (upper) and habit plane (lower) frames [40]. The terrace plane is inclined at an angle h to the horizontal habit plane. Coherency strain is represented by the equivalent ‘‘coherency” defect content, bc. The x-axis points out of the page.

linear elastic approximation, but the treatment is also an accurate approximation for the non-equilibrium case. Of course, these considerations apply to the bicrystal case where any stresses r0zz are relaxed to zero. If there are constraints in the z direction, plastic strains e0zz build up as the 0 disconnections with bz components move and these cause elastic incompatibility strains to evolve as well. Such incompatibility effects are treated in Section 2.3. By way of illustration, consider a simple case where exx = 0 and the finite strain eyy is accommodated by a single array of disconnections with nD parallel to x for which bD = [0, by, bz]coh, Fig. 5. For this case [40,41] an exact answer can be obtained without an iterative procedure. The equilibrium habit plane is inclined to the terrace plane by the angle h, obtained from 1

eyy ¼ ðby tan h þ bz tan2 hÞh :

ð12Þ

The ancillary rotation about x, uD, is equal to 1

/D ¼ 2 sin ½ðbz cos h  by sin h  eyy h cos hÞ sin h=2h:

ð13Þ

Because of the equal partitioning of uD between the crystals, the apparent habit plane inclination for an observer in the k crystal remote from the interface, unable to resolve the near-field elastic strains of the discrete defects (or an experimentalist using optical microscopy and X-ray diffraction), is h + uD/2. Conversely, for an observer in the l crystal, it is h  uD/2. If exx is not zero, it can be accommodated by an array of dislocations from, say, the l crystal with components of bL = [bx, 0, bz]coh. Formally, the iterative refinement procedure outlined above is needed to find the final configuration on the habit plane, and this is shown schematically in Fig. 6. The terraces remain coherent between the defects, and the distortion diminishes with distance from the interface. An ancillary tilt of the phases about nL arises as outlined above since bz – 0. We have emphasized simple solutions to Eq. (3) which correspond to some examples of phase transformations in the literature. In general, with bz still suppressed, Eq. (3) can be satisfied by two independent arrays (disconnection plus LID dislocations) plus a twist rotation, as demonstrated for martensite in Fe [57] or by three independent arrays plus the condition of no twist rotation. Sargent and Purdy [58] showed that three in-plane Burgers vectors are required for a general misfit case. Here we indicate that this is also true for three in-plane Burgers vector components, since the normal components produce pure rotations. The reduction of the three array requirement to two is analogous to the reduction of a two array requirement to one when a twist rotation is added in the Matthews example previously discussed. For example, if one array is sessile (single twin or glide LID on a plane inclined to the terrace plane) then three independent arrays will be required except for a special line-direction of the LID in the terrace plane. Also the iteration previously described when steps are introduced into the solution can lead to components of b that vary with step spacing, which again

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Fig. 6. Schematic illustration of an interphase interface showing the terrace segments and defect arrays [41]. Coherently strained terraces are reticulated by arrays of disconnections (b, h) with spacing dD and crystal slip dislocations (b, 0) in the (lower) l crystal. The terrace and habit (primed) coordinate frames are shown and the line directions of the disconnections, nD, and dislocations, nL, are parallel to x and close to y0 , respectively.

can lead to the need for more independent systems. In this case, however, the ancillary tilt associated with the introduction of steps is small, so the third array would likely be widely spaced and simply act as a perturbation of the two array result. An analogue of the requirement for three independent systems in general is the general deformation of a bicrystal, which requires four independent systems distributed between two crystals that are free to rotate, as verified experimentally [59,60]. Finally, even more independent systems would be required if there are constraints on tilt or twist rotations. As an example, for deformation of a single crystal of Cu with rotational constraints, Basinski has observed eight independent glide systems instead of the usual five (von Mises) when rotations are unconstrained [61]. 2.2.5. Energy of defect networks The energy per unit length of an array of dislocations, W, is the product of the line energy W/L of a single dislocation and the number of dislocations per unit length (1/d) where d is the dislocation spacing. Thus W has the form

W¼

" # 2 A 2 d be d ; bs ln þ ln d C s ð1  mÞ C e

ð14Þ

where the subscripts s and e stand for screw and edge, respectively, A contains the shear modulus and other constants, C is the core parameter and t is Poisson’s ratio (when d is small the argument of the logarithm in Eq. (15) becomes a hyperbolic function [44]; for the cases considered here b/d is sufficiently small that the approximation in Eq. (14) is valid). According to simulations as well as the Peierls model the core energy is larger for screws, represented in Eq. (14) by a smaller core radius (see Section 8-3 in [44]), i.e. Cs < Ce. For large d, the (1  m) factor dominates and Eq. (14) indicates that screw arrays have a smaller line energy than edges. However, for small d, the core parameter can dominate and the edges can have smaller line energy than screws. Since e = b/d, expression (14) also shows that W / b, implying that defects with smaller Burgers vectors accommodate coherency strain more efficiently. Thus for the pure shear case discussed in Section 2.2.2, see e.g., Fig. 4b, the screw solution would be favoured for large d, while the edge solution could be favoured for small d. We discussed

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earlier that in some cases twist rotations can be introduced into a network. For example, the orthogonal array of screw dislocations discussed in Section 2.2.2 comprise orthogonal right-handed and lefthanded arrays; Matthews [53] pointed out that superimposing a twist array of right-handed screw dislocations has the effect of increasing the spacing of one array while decreasing the spacing of the other. Furthermore, he indicated that the interfacial energy may decrease as a consequence, reaching a minimum when the array of left-handed screws is annihilated. This conjecture is supported by the analysis of Eq. (14). For the minimum energy case, the superposition conserves the initial number of screw dislocations in the zero-twist array, converting to a single array of screws with half the initial spacing d/2. This is energetically favourable provided that d is not too small. An added factor can enter when the interface dislocations can dissociate into partials separated by a fault analogous to a stacking fault. When d is very small the partial interactions dominate and the configuration of the interface dislocations can change drastically from the large d case. The overall interfacial energy is comprised of the dislocation energies of the disconnection and LID arrays, and their interaction energy, as well as contributions from the interfacial free energies of the terraces and steps. Both the disconnections and the LID dislocations have some spreading width, so the coherent terrace area decreases as d decreases. In the limit that d ? b, the interface becomes incoherent with a higher energy. 2.3. Compatibility effects To this point, we have dealt with local regions for an ideal equilibrium structure of a single interface that is free of long-range stresses. For actual plates, there are end-effect stress concentrations that could influence the spacing of interface defects and lead to local curvature of the plate interface or vice-versa. We mention these briefly for completeness although a detailed treatment is beyond the scope of the present work. As shown in subsequent examples, there are regions removed from the plate tips where the interface arrangement is near the ideal one previously described. In displacively formed plates containing disconnection arrays, there are always pileup arrays at plate or lath tips for both disconnections and LID defects, with associated strain/stress concentrations. The long-range fields of such pileups are equivalent to those of a superdislocation [44,62]. These stresses decrease rapidly, in both plate and matrix, with distance from the tip because of interactions between opposite sign pileups at opposite sides of the plate. In some cases the pileups are equivalent to superdislocation dipoles and the stresses decrease even more rapidly with distance from the tip. In addition the stress concentrations can be relieved by added LID in the matrix. The stress relief would be analogous to that relieving the pileup tip stresses for a mode II crack. As another class of accommodation, the elastic pileup stresses can be minimized by plate–plate interactions. One example is the self-accommodation seen and described by Miyazaki et al. [63], with four plates in a ‘‘window-frame” configuration in a cubic matrix. Another is the offset stack of GP II plates seen and described by Brown et al. [64] in the Al–Cu system. More subtlely, accommodation strains can be associated with rotations arising from ancillary tilts and twists. The compliance of the matrix tends to suppress the rotation of the matrix arising from partitioning of these tilts/twists. This leads to back stresses tending to oppose the rotation, an effect particularly important for plates with small aspect ratios. This could also lead to added LID in the matrix. The back stress would also tend to suppress the transformation. This effect could tend to favor symmetric LID systems for which there are neither ancillary tilts nor twists relative to those that have rotations, even though other factors would favor the latter. Finally, we mention the possibility of an internal accommodation within a plate. In Section 2.2 we emphasized that there is no unique defect network for accommodating a given coherency strain; for example, Matthews [53] showed that the networks depicted in Fig. 4a and b can coexist either as contiguous domains or by interaction to a hybrid form. Alternative networks could therefore appear in different regions of the same macroscopic plate; if these comprise dislocations and disconnections, they would in general exhibit local habit planes and ancillary rotations. Of course new defect-containing boundaries would be present to separate these regions. For example, very complex assemblies of self accommodating regions and twins have been observed in single martensitic plates in uranium alloys [65].

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2.4. Terraces The terrace planes considered so far are either naturally periodic in 2D (a rare event), or can be so constrained by modest coherency strains. In this section we consider further the crystallography of 2D coherent terraces, and also introduce the idea of terraces exhibiting only 1D periodicity. The energy of a 2D coherent terrace, such as that depicted in Fig. 2, varies with the relative displacement of the two crystals, represented by a vector p parallel to the interface. The vectors p of all possible translation states lie within the cell of non-identical displacements, or ‘‘cnid” [66], which is the Wigner–Seitz cell of the lattice defined by the two smallest magnitude independent vectors in the coherent reference, tcx and tcy . Fixing p, but allowing local relaxation, and calculating the energy of the corresponding bicrystal enables one to construct the interfacial c-surface (surface energy as a function of p [66]) also related to the generalized stacking-fault energy [67] and the unstable stacking-fault energy [68]. When a coherent bicrystal exhibits symmetry the cnid is partitioned into regions representing symmetry related structures. Also, other points in a cnid represent the structures obtained when k- and/or l-lattice planes parallel to the terrace are removed, followed by displacement parallel to z to conserve the initial density [66]. Such c-surfaces may exhibit principal and subsidiary minima corresponding to bicrystals with varying degrees of metastability. In a manner analogous to stacking-faults in single crystals, partial interfacial line defects may arise on a terrace separating regions with different p. Two such bicrystal structures may be either distinct, with different energies, or crystallographically equivalent, with degenerate energies [38]. The defects defined by expression (1) always separate degenerate structures, either identical or symmetry related [38]. One can envision the atomic arrangements in the two crystals juxtaposed at the interface (without local relaxation) by locating atomic bases at each k and l lattice site. In the simplest bicrystal structures, there is a one-to-one atomic correspondence across the interface; here the k and l atomic planes parallel to the terrace have equal atomic densities. In more complex structures, the ratio of the atomic densities of these planes is m/n, where m and n are integers. These correspondences relate to the extent of diffusional flux and shuffling accompanying disconnection motion, as discussed in Section 3. The preceding treatments of strain accommodating defect networks are based on reference structures coherent in 2D. Experimental observations and atomistic simulations [56] show that strains up to about 15% are accommodated by localised interfacial defects in heteroepitaxial interfaces. However, interfaces can become incoherent (or incommensurate) in one dimension when the strains become so large that localization into defects involves core overlap, so that discrete defects and defect strain fields cannot be identified [69]; for sufficient misfit, the strains spread evenly along the interface. The resulting terraces, coherent in one direction and incoherent in the other, can still appear in habit planes that contain the dislocation and disconnection defects discussed in Section 2.2. The cnids for such interfaces become a line parallel to the coherent direction with length tc . The nature of favourable 1D coherent terraces is an active area of research [70,71]; some experimental observations suggest that they arise when the k atomic plane is opposed by a higher index l plane, or vice-versa [71,72]. While the defects in such interfaces may be topologically similar to those in 2D coherent interfaces, the mobilities of 1 and 2D interfaces in response to a driving force may differ significantly [71,73]. 2.5. Implementation of the topological model We now summarise the implementation of the topological model to find the defect structure of a two-dimensional transformation interface. One first seeks a matched terrace plane with small misfit; these display energy minima [74,75], and, as explained in Section 3, 2-D coherency is essential for diffusionless transformations. Then the crystals adjoining the terrace plane are uniformly strained to achieve coherence at the interface. The crystal lattices are interpenetrated to form a dichromatic pattern. This is the coherent reference state. Inspection of the reference state allows one to select candidate disconnections (b, h) and dislocations (b, 0). For simple systems, one can then deduce the correct disconnection and LID spacings and line-directions to relieve all long-range coherency strains, and thus determine the orientation of the habit plane and the ancillary rotation u. For less simple systems

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one can proceed to solve Eq. (3) to determine the spacings and line directions. In any case, tilt or twist rotations can be superposed if needed, partitioning the rotations between the two crystals. The concept of the habit plane as an invariant–plane interface in the phenomenological theory of martensite crystallography [17] is a matching condition in rigid crystal modelling. The topological model, based on dislocation theory, shows that this is a sufficient but not necessary condition for the existence of glissile semi-coherent interfaces with short-range strain fields. The phenomenological and topological theories give identical predictions for habit plane, orientation relationship and transformation displacement only in cases where no ancillary rotations arise [40,41]; otherwise, small discrepancies between the habit plane predictions arise because of the partitioning of ancillary tilts in the topological model. Also, additional solutions inconsistent with the invariant plane notion are predicted using the topological model, as in the case of ferrous, alloys for example [57]. 3. Kinetic aspects 3.1. Disconnection motion The nucleation of a new plate of martensite is clearly heterogeneous [76], with the possible exception of driving forces so large that the mechanism approaches one akin to spinodal decomposition [77,78], a case not considered here that may even occur by direct shuffling rather than defect motion [79,80]. Diffusional plates are also likely to be heterogeneously nucleated. There is ample evidence of precipitate nucleation on dislocations in Al alloys, for example, and grain boundaries are another heterogeneous site, as evidenced by the observations of sideplates in both grains adjoining the interface [81]. The equations for nucleation are complex and site specific, so we do not address the details of the nucleation process here. Once nucleated, there are several sources for continued formation of disconnections at a source. These include pole-type spiral mechanisms [65], strain fields at plate facets [64,82], precipitate intersections [83], and repeated nucleation from grain boundaries [84,85]. Hence, transformation proceeds by a train of moving disconnections. These defects have been called transformation dislocations [9,86] or transformation disconnections [87] to connote their role in causing the transformation. There are three classes of behaviour. For fast martensite, such as that forming in quenched steels, the driving forces are so large that the disconnections athermally overcome any Peierls resistance and move at velocities approaching the speed of sound, as indicated by sonic emissions and velocity measurements [88,89]. The motion is likely to be phonon damping controlled with a possible inertial contribution and the disconnections should remain nominally straight as they move, by analogy to dislocation motion at such high velocities [90]. For slow martensite, such as that in shape memory alloys, motion by kink-pair nucleation and lateral propagation should be the operative mechanism. The disconnections should nominally remain parallel to Peierls valleys, although locally they would deviate from such parallelism because of the presence of kinks. Lateral kink propagation could occur by a thermally activated diffusional motion or by a drift mechanism involving phonon drag [91]. The effective Peierls barrier should be greater, leading to slower disconnection velocity, in non-metallic crystals and for disconnections that are extended zonally and require shuffles as they move. For diffusional transformations, again the mechanism should be one of kink-pair nucleation and propagation [92]. In this case the kink motion is diffusion controlled as the components have to interchange to produce the composition change in the two phases that accompanies the transformation. 3.2. Transformation strain For the static equilibrium of Section 2, the elastic strain fields of the disconnections were important in relieving coherency strains. Moving disconnections also serve to produce the plastic transformation strain. A moving dislocation sweeping an area A produces an engineering strain per unit volume C = bDAt. Here, A = An with n a unit normal to A, and the superscript indicates a row vector. Hence a straight disconnection of length ‘ moving by dy in the y direction in the terrace plane coordinates

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sweeps an area A = ‘dy, numerically equal to dy for a defect of unit length. This motion produces an engineering strain

0

1 0 1D 0 0 cxz bx B C C C ¼ @ 0 0 cyz A ¼ dyB @ by A ð 0 0 nz Þ; 0 0 ezz bz

ð15Þ

with nz = 1. Here by and bz are edge components and bx is the screw component. This strain matrix can also be regarded as a tensor strain Et plus a rotation R

0

0

0

C ¼ Et þ R ¼ B @ 0

0

exz eyz

1

0

1

0

1

D 0 0 /xz exz bx B C B 0 /yz C eyz C Aþ@ 0 A ¼ dy@ by A ð 0 0 nz Þ /xz /yz 0 ezz bz

ð16Þ

Here, for example, the component exz = cxz/2 = bxdy/2 and uxz = bxdy/2. For a train of disconnections as in Fig. 6, with 1/dD defects per unit length in the y direction, the strain produced by the train is C/dD. Uniform motion of unit area of the train by dy transforms a volume numerically equal to hdy/dD from k to l and translates the habit plane in the z0 direction by dz0 = dy sin h. Thus, knowing b, one can simply write the total engineering strain components produced by motion of the disconnection train, CDm , as

0

0 1D bx dy C B C CDm ¼ B @ 0 0 cyz A ¼ D @ by A ð 0 0 nz Þ: d 0 0 ezz bz 0 0 cxz

1

ð17Þ

In this formalism, an additional coherency strain eyy arises because of the motion along z of the coherency dislocations with Burgers vector bcy and line direction x. When the disconnection array moves by dy, the resulting strain is

0

C

CD m

0 B ¼ @0 0

0

eyy 0

0 1 0 C C dy B 0 A ¼ D @ bcy Að 0 ny d 0 0 0

1

0 Þ:

ð18Þ

Using expression (2.8) of reference [41], one can readily confirm that the ‘‘climb” of these dislocations is conservative. We emphasize that these are plastic strains and rotations. There are also rotations associated with the elastic distortion fields of the defects as discussed in Section 2.2.3; these are associated with the interface structure, which is unchanged on average as the train of defects moves. Thus the rotations connected to the elastic distortions are unchanged by the motion dz0 of the habit plane. The LID defects are also associated with the interface structure and have long-range rotations, as discussed in Section 2.2.3, which are also unchanged by the motion dz0 of the habit plane. The LID defects translate in their glide/twinning plane when the disconnections move laterally by dy, producing an additional plastic engineering strain

0

1L bx B C g C ¼ D L @ by A ð x d d sin v bz L m

dy

gy

g z Þ;

ð19Þ

where [gx, gy, gz] is the unit normal of the glide/twinning plane and v is angle between the glide and terrace planes. An additional strain exx arises through the conservative motion along z of coherency dislocations with Burgers vector bcx and line direction y. 3.3. Diffusional flux 3.3.1. Composition change at interface To calculate the diffusive flux of material associated with the motion of a dislocation or disconnection along an interface one must take into account any differences of composition and density between the two phases. Let the interface be the plane x, y in a Cartesian coordinate frame, and

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consider a defect parallel to x moving in the direction y. For a length of defect, ‘, moving a distance dy the number of (substitutional) atoms of species A, dNA, that must diffuse to/away (+ve/ve) from the defect’s core is given by

dNA ¼ ‘ðhDX A þ bz X A Þdy;

ð20Þ

where DXA = X(k)A  X(l)A is the difference between the number of A atoms per unit volume in the k and l crystals, XA is either X(k)A or X(l)A (that with the larger of n  tðkÞi and n  t ðlÞj ), and bz is the component of b normal to the interface plane [30]. Conservative motion of disconnections occurs only when dNi = 0 zero for each species present. When dNi is non-zero, long-range diffusion must occur to remove or add this amount as the defect moves. For the case of disconnections in interphase interfaces, this only arises for motion along coherent interfaces (i.e. where the interface exhibits 2-D periodicity) [39]. In these circumstances, atoms can then be transferred from k to l in a relatively orderly manner by, in general, a combination of shear and shuffling. While the number of species present is thereby conserved, the volume occupied by them may not be since the crystal densities are not usually identical. Thus, interfacial coherency is seen to be important not only from an energetic point of view [74,75], but also from a mechanistic perspective in site-conserving transformations [42,43]. Eq. (20) can be quantised in terms of elementary kink jumps. Consider the two independent translation vectors with smallest magnitude on a coherent terrace; for simplicity, let these be orthogonal, tcx and tcy : Thus, a y-kink on a disconnection lying along x will have length tcy ; overall disconnection motion parallel to y will occur by kink jumps of tcx . In such an elementary motion the g(l)A atoms of species A in the cell with volume v ðlÞ ¼ tcx  tcy  tðlÞ will be replaced by g(k)A atoms in the corresponding cell, v ðkÞ ¼ tcx  tcy  tðkÞ; where t(k) and t(l) are defined by Eq. (1). The difference dgA ¼ gðkÞA  gðlÞA ¼ v ðkÞA XðkÞA  v ðlÞA XðlÞA is related to dNA by dgA ¼ dNA ðtcx tcy =‘dyÞ. Thus, for a unit jump we have

dgA ¼ t cy ðhDX A þ bz X A Þtcx :

ð21Þ

3.3.2. Diffusional fluxes Taking the time derivative of Eq. (20), we find that the current of atoms produced at a moving disconnection is

IA ¼ ‘ðhDX A þ bz X A Þv D ; D

ð22Þ

where v is the steady state velocity of a disconnection. As shown in Eq. (22), when treated in the moving frame of reference, this current equals the diffusion flux, JA, of A multiplied by the area enclosing the disconnection through which the flux is passing. The rate of transformation, expressed as a velocity v normal to the habit plane is given by v = vD sin h, and tan h = h/dD, see Figs. 5 and 6. Here dD need not equal the ideal equilibrium spacing since it can be kinetically controlled. Thus the rate of transformation is given by a solution of the diffusion equations for JA, together with a kinetic analysis for dD. We first consider cases where local equilibrium boundary conditions are maintained at the interface and consider variations from this assumption in Section 3.5. The basic diffusion process is a hemispherically symmetric diffusion flux around each moving kink. However, by analogy to climb kinetics [65] and crystal growth [21], we expect that the mean kink spacing is typically much less than d, so that the hemispherical fields overlap, and one has essentially semicylindrical diffusion fields around each disconnection with some overlap possible for small dD (if the hemispherical fields are important, the behaviour qualitatively resembles that discussed for the semicylindrical case) [22–25]. In addition, enhanced pipe diffusion along the disconnection core would also favour the disconnection acting as a cylindrical source. Analogous fields occur for step motion in crystal growth [21] where the solutions are Bessel functions or hyperbolic functions. These solutions have the form that ov/od is positive and o2v/od2 is negative and decreasing with increasing d. Thus a regularly spaced array tends toward an asymptotic d with an array that is relatively stable with respect to a local fluctuation. For d less than the asymptotic value, the array is unstable with respect to fluctuations, tending to undergo successive pair-wise bunching [93]. In the present case, the elastic repulsion between like-sign disconnections suppresses the bunching tendency and provides stability for a regularly spaced array on average, as verified in computer simulations [28]. Even for a pure step, there are weak line-force dipole interactions that lead to elastic repulsion [94].

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In the pure-step crystal growth, d tends to vary inversely with the thermodynamic driving force. In the present case, the spacings at the source would tend to vary in this manner as well [26]. However, the form of the equation would cause the spacing to converge to near the ideal spacing because of the elastic interaction. As shown subsequently this supposition is supported by experimental results. 3.4. Defect intersections Interactions between defects in interfaces are subject to the overall conservation of b and the crystal step heights hðkÞ ¼ n  tðkÞi and hðlÞ ¼ n  t ðlÞj [39]. When two defects, (bp, hp) and (bq, hq) intersect, they mutually step one another and an additional length of dislocation normal to the interface, with Burgers vector bp + bq, is created along their mutual step. This intersection defect may be sessile, even if one or both of the interacting defects are glissile. Glissile intersections can arise when a glissile disconnection, (bp, hp), intersects a crystal dislocation, (bq, 0), provided the crystal dislocation can reach the terrace by glide, i.e. bq has a component bz [39]. This property is important in martensitic transformations where it is necessary for an array of disconnections, such as those in Fig. 6, to remain glissile despite the presence of an intersecting array of crystal dislocations. Under these circumstances, interface motion is conservative by the mechanism of synchronous lateral motion of the disconnection array across the interface. In the limit of very large driving forces for the transformation of martensite, non-conservative motion could occur with little constraint with rows of point defects emanating from the sessile intersections (Eqs. 16-16 and 16–20 in [44]). There is another local effect associated with defect intersections. Intersecting like-sign screw dislocations or disconnections locally repel one another [95]. Similarly edges with the same sign of b  n repel one another. Opposite sign defects attract. Hence, for the repulsion case, there can be local standoffs of the defects [96], unless the intersection rearranges, for example into an extended node. Another manifestation of the repulsion was revealed in an atomistic study of a near Kurdjumov–Sachs interface between Cu and Nb, where the misfit is large, 20% [97]. The intersecting arrays were found to lie on separate abutting planes instead of lying on the same plane. 3.5. Diffusional drag effects One limit of behaviour is the case of the diffusional growth of a plate-shaped product with accompanying shear at high temperatures and low driving forces. The requirement for diffusion to enable the composition change accompanying the transformation dominates the kinetics [26]. The diffusional requirement can be thought of as exerting a large (osmotic) drag force on the moving disconnections that is nominally balanced by the thermodynamic driving force and any elastic Peach–Koehler forces that might be present. The defects at the interface act as sites where the equilibrium boundary conditions are maintained and the resulting kinetics are given by standard diffusion solutions [24,98–101]. Because the velocities of disconnections are very low (v  D/X, where D is diffusivity and X is a characteristic diffusion distance), moving boundary conditions are negligible, as has been verified experimentally [26,101]. For lower temperatures (far from equilibrium), as for lower bainitic transformations, the situation is not as clear [102,103]. Diffusion becomes so slow that part of the free energy dissipation can occur at the defect site, with the local composition undergoing some concentration change but not to the equilibrium composition. In one limit, the disconnection motion is almost without compositional partitioning, with the diffusion necessary to establish the equilibrium concentrations occurring subsequently [103]. In the other limit, the equilibrium concentrations would nearly be obtained with diffusional drag as in the high temperature case [102]. An intermediate case can be envisioned where some diffusional drag is present but the interface concentration deviates from local equilibrium [104]. There is some experimental support for both views, but the issue has not been definitely resolved [105]. Another effect that can operate at lower temperatures when long-range diffusion cannot occur is the drag of disconnections associated with Cottrell or Snoek atmospheres of solute atoms. The diffusion is short range for these atmospheres and there is an attendant diffusion controlled drag on the disconnections.

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A final effect that can occur at lower temperatures is the breakdown of the Darken equations [106,107]. When vacancy sources/sinks are remote from the growing interface, the assumption of local vacancy equilibrium, inherent in the Darken equations [108] breaks down. The effective diffusivity is then not the value measured for bulk crystals. Instead it is depressed to approximately equal the self diffusivity of the slower diffusing component. This effect would also depress the growth rate and could be a factor in explaining rates slower than those analogous to the high temperature case. Pieraggi et al. [109,110] have emphasized that interfacial dislocations may be needed to act as vacancy sources/sinks and permit local equilibrium and the use of the Darken equations. In terms of Fig. 6, these can be interfacial dislocations with b = [0 0 bz] or with b = [0 by 0]. The latter dislocations thus can have three disparate roles: instruments for misfit relief; sources for pinning, necessitating climb and tending to retard the transformation; and sources for vacancy equilibration, tending to enhance the transformation. The former dislocations are probably more important for vacancy equilibration as they do not interact with the coherency stresses. 3.6. Large steps There are many observations of diffusional growth where the disconnections have unit or a small multiple of atomic step heights. In the absence of extrinsic drag effects, elastic interactions would favour the maintenance of nominal equal spacings between unit disconnections of one or a few atomic step heights. Also, from an irreversible thermodynamic viewpoint, the maintenance of such an array would maximize the growth rate and maximize the rate of free energy dissipation [111,112]. However, analogous to the motion of growth steps at free surfaces [113] if extrinsic drag effects are present because of impurity adsorption at a disconnection or because of intersections, pileups can occur when one disconnection is selectively pinned relative to its neighbors. For the crystal growth case, step bunching occurs but discrete spacings are maintained for the surface steps since there is no long-range interaction among them. For the disconnection case, there are long-range elastic repulsions between defects but short-range elastic attractions. These local interactions between the dislocation components of the disconnections favour the collapse of the pileup into a defect with a sharp superstep of height qh, and Burgers vector qb, where q is an integer and h and b are the relevant quantities for a unit disconnection. For brevity and to contrast this case with those mentioned next, this disconnection is called type A. As verified in computer simulations [28], q can be in the range 5–15 depending on the elastic properties and magnitudes of b. Another example of a disconnection with a superstep, type B, can occur when the disregistry across the step is such that m atomic planes of say the l phase nearly match n planes of the k phase, with m and n different integers. It is then favourable for the disconnection to exhibit minimal bz by opposing m k-planes against n l-planes across the step. Disconnections separating crystallographically related (energetically degenerate) terraces in the Al–Cu system show such behaviour [114]. Late in the stages of plate growth, disconnections with supersteps having very large q (qh up to 0.1 lm), have been observed. These defects have been called growth ledges [13]. They can form if the strain fields of the defects of type A (or B) are cancelled by the injection or emission of lattice dislocations with a net Burgers vector opposite to that of the type A disconnection [115,116]. It would be rare for the disregistry to be such that the complete Burgers vector component of the large-step-height defect to be zero; hence, strictly the defect would be another disconnection with a superstep, type C. However the average Burgers vector content would be so small relative to the step height that the type C defect would be tantamount to a pure step as implied by the definition of a growth ledge [13]. Unit steps/disconnections have been observed on such supersteps [112,117]. This suggests that the motion of these defects occurs by unit disconnection motion on the superstepriser, although direct attachment is also possible if there is not a clearly defined structure on the superstep. The motion of a type C superstep must also involve the absorption/emission of unit disconnections on the adjoining terraces [115]. 3.7. Special effects in the martensite case For the special case of a martensite transformation, which may involve only one chemical component, Eq. (20) has dNi = 0 for all atomic species. That is, bz = hDXi/Xi for each species. In this circum-

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stance the components bz of the dislocations can move conservatively along with the disconnection. In other words, the number of atoms of each species present is the same in the elemental cells v(k) and v(l), even though these cells may have different volumes. Climb is not required. For the case of slow martensite transformations, defects would be subject to elastic interactions in the same way as described for diffusional transformations in Section 3.3.2 and one would again expect these to comprise a regular array with near the ideal equilibrium spacing. For the fast martensite case there is less influence of the disconnection interaction because of relativistic effects when the velocity is near the sound velocity ([44], Chapter 7). Here one would expect defect configurations determined by source formation rates. However, once a plate has stopped growing, or during subsequent annealing, the spacing would tend to approach the equilibrium spacing. Without annealing one would expect a broader distribution of spacings because of the stochastic nature of the inertial and damping events. This greater spread is found in TEM studies of the interface of martensite in a low alloy steel [118]. 3.8. Kinetic influence on variant selection For the diffusional and slow martensite cases, the kinetics should exert no direct effect on the selection of the variant as discussed in Section 2, although the applied stress might via the Peach–Koehler force that adds to the thermodynamic driving force. However, for the fast martensite case there are two factors that can influence the selection. One cause of such an effect is that the disconnections certainly must precede the formation of the LID defects since the driving force for LID only arises once the transformation by disconnections (to a partially strain-relieved state) has occurred. The formation of edge disconnections in this partially transformed state, if favourably oriented, could be more favourable because they more efficiently remove the largest normal coherency strain. Hence if they are then locked in, the LID might lead to a different interface structure than in, say, a diffusional case where mixed or screw disconnections might be more favourable. The other factor is the relative mobility for the LID. Perforce, fast martensite tends to form at relatively low temperatures where diffusion is suppressed. At these temperatures screw dislocations have non-planar cores in most metal structures with the exception of fcc metals, as well as for many nonmetallic structures [119]. The non-planar cores lead to much lower mobility than for edge or mixed dislocations. Thus networks requiring screw LID would tend to be suppressed in all but fcc martensites. 4. Observations of martensitic transformations 4.1. Transmission electron microscopy The objective of this section is to discuss experimental observations of parent-martensite interfaces, particularly their defect structures. The most effective experimental technique used is TEM, so it is appropriate to briefly review its capabilities in this field. According to the topological model, the distortion field of a static interface can be written as

0

1 0 1 0 /0xy /0xz e0xx e0xy e0xz C B 0 0 e0yy e0yz A þ @ /xy 0 /yz C A: 0 0 e0xz e0yz e0zz /xz /yz 0

0 C ðx0 ; y0 ; z0 Þ ¼ B @ exy s

ð23Þ

The strain and rotational distortion components of Cs(x0 , y0 , z0 ) vary from point to point, describing both the short- and long-range fields produced by superposition of the coherency strain and the elastic field of the defect network. At short-range all strains are partitioned between the phases in a manner depending on their relative elastic compliances. In the case where the coherency strain is completely accommodated by the network, all of the component strains except e0zz vanish at long-range. At shortrange, these strains alternate in sign along x0 and y0 respectively, in anti-phase across the habit plane, and diminish to zero when z0 zd is approximately equal to the relevant defect spacing, dD or dL [44]. The magnitude of e0zz will be finite at long-range in the ‘‘single interface” case, where no constraint by the matrix is present. For z0 > zd the rotational distortions are constant and equal to the ancillary tilts, /0xz and /0yz ; and twist, /0xy .

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Fig. 7. Inverse Fourier transform of a HRTEM image of martensite (upper)–austenite (lower crystal) interface in Fe–30.5%Ni– 10%Co–3%Ti (mass%), possibly showing the partitioned near-field strains (Ogawa and Kajiwara [121]). The incident beam is  0 1 =½1 1  1 and the horizontal interface (approximate position indicated by white bars) is parallel to ð1 2 1Þ : parallel to ½1 c a c

TEM can be used to measure ancillary rotations accurately; when sharp Kikuchi lines are present in electron diffraction patterns, rotations down to one tenth of a degree can be measured [120]. Measurements of local strains for z0 < zd are more problematic. Fig. 7 is an inverse Fourier transform [122] of a HRTEM image of the near field region of a parent-martensite interface in Fe–30.5%Ni–10%Co–3%Ti (mass%) from the work of Ogawa and Kajiwara [121] who have attempted to determine these strains. These authors describe the near-field as a ‘‘transition lattice”, a notion consistent with strain partitioning. Defect networks are studied using two image formation techniques, diffraction contrast and lattice imaging [120,123]. In the former method, contrast is formed when the electron beam is scattered by a defect’s strain field. Hence the method is most useful for relatively widely spaced defects, and impracticable for spacings below about 2 nm. For example, Sandvik and Wayman [124] used the g  b = 0 method to carry out a Burgers vector determination for a lattice-invariant dislocation array with dL in the range 2.6–6.3 nm in lath martensite formed in an Fe–Ni–Mn alloy. Chen and Chiao [125] resolved a disconnection array with dD 4.2 nm in the orthorhombic to monoclinic transformation in ZrO2. The transformation rate was sufficiently slow in this case for the lateral motion of disconnections to be observed in situ. In general, the magnitudes of dD encountered in martensitic transformations are less than those of L d , primarily because |bD| < |bL|, so that lattice imaging is needed to resolve typical disconnection arrays [15]. Instrumental resolution has been sufficient to detect such arrays with dD < 1 nm since the 1980s, provided the specimen is very thin, the electron beam is incident simultaneously along low index directions in both crystals and the interface is edge-on [123]. Moreover, for reliable image interpretation, disconnections must be parallel to the incident beam; an example from the work of Klenov [126] is shown in Fig. 8. The components of bD in the plane of the image can be determined from such images by circuit mapping, and h can be measured directly [39]. This array is in excellent agreement with the model configuration depicted in Fig. 5 [40]. Small ancillary rotations were also apparent in this study, consistent with the topological model; for example a rotation about an axis parallel to the disconnections is present, i.e. /0yz 0:5 from the reference Burgers orientation relationship. These considerations show that the magnitudes of the coherency strains in a particular transformation govern the efficacy of TEM in the study of defect networks. Naturally coherent terraces only occur rarely. Examples of coherent (1 1 1)c/(0 0 0 1)e martensites have been observed in austenite with low stacking-fault energy, such as certain alloy steels [127]. No strain-accommodating network need be superimposed on such terraces. However, disconnections are required for growth; in Co–32Ni  and h ¼ [128,129] for example, irregular configurations of disconnections with b ¼ 1=6h112i 2dð1 1 1Þ are observed. 1-D coherency strains may arise naturally on a terrace plane, as for example in the case of (1 0 0)o/(1 0 0)m terrace in the orthorhombic to monoclinic transformation in ZrO2 depicted in Fig. 9(a). A single array of disconnections is then predicted as in Fig. 5 [41], in excellent

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Fig. 8. Transmission electron micrograph of a parent–martensite interface in Ti–10wt.%Mo [40,126]. The habit exhibits a terrace/disconnection structure.

Fig. 9. Scale drawing of the terrace planes in (a) ZrO2 and (b) Ti. Full lines represent the martensite and dashed lines the parent crystals, and bold lines depict the coherent state after equal and opposite straining, exx and eyy, of each phase [41].

agreement with the observations of Chen and Chiao [125] mentioned above. Most transformations  0 0Þ terrace in the cubic  1Þ  =ð1 1 involve terraces with 2-D coherency strains, as for the case of the ð2 1 b a

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to hexagonal transformation in Ti illustrated in Fig. 9b. Klenov [126], using lattice-imaging, resolved the disconnection array in Ti–10Mo, Fig. 8, and this was found to be in excellent agreement with topo 1Þ twinning, logical predictions [40]. The small strain in the x-direction was accommodated by ð1 0 1 a as observed earlier by Hammond and Kelly [130]. In the following sections, two examples of martensitic transformations are chosen for more detailed analysis; the first in Ti–Nb alloys relates to the systematic change of interface structure with increasing coherency strain. The second, in Fe-based alloys, emphasizes differences between the topological and phenomenological approaches. 4.2. Ti–Nb alloys Chai et al. [131] have recently investigated the structure of austenite–martensite interfaces in a series of Ti alloys with 20, 22 and 24 atomic% Nb, with transformation temperatures 220 °C, 140 °C and 65 °C respectively. The crystallography of these bcc (b)-orthorhombic (a00 ), transformations is very similar to that illustrated in Fig. 9b for Ti with the hexagonal lattice distorted to become orthorhombic.  1Þ kð1 1 0Þ ; The reference orientation relationship is as follows: x,y-terrace plane ð2 1 b a  kx and ½1 1 1  k½1 1  0 ky. The variation of the coherency strains exx and eyy with Nb con 1  k½0 0 1 ½0 1 b a b a tent are depicted in Fig. 10; note that exx = 0 for Ti–20Nb and increases with Nb content, whereas eyy decreases with Nb content. These coherency strains were found to be accommodated by a disconnection array with nD parallel to x, as in Fig. 5, and lattice-invariant type-I twinning on ð1 1 1Þa00 with nL close to y. The latter are L shown in Fig. 11; the twinning dislocations have b ¼ ½bx ; 0; bz a00 and accommodate exx. Virtually no twinning was observed in Ti–20Nb, Fig. 11a. Fine twins (ranging from a few nm to tens of nm wide) were observed in Ti–22Nb, as arrowed in Fig. 11b, often terminating within the a00 plates. In Ti–24Nb, twins up to 60 nm wide were observed, arrowed in Fig. 11c. The authors showed that the observed extent of twinning was consistent with accommodation of exx for each alloy; furthermore, they calculated the anticipated ancillary rotations, uxz, to be 0°, 0.02° and 0.08° for the three alloys. Chai et al. [131] used lattice-imaging to resolve the disconnection arrays accommodating the strain eyy. Fig. 12a is a schematic illustration of the formation of a disconnection on the ð2 1 1Þb kð1 1 0Þa00 terrace in the manner depicted in Fig. 1. The translation vectors t(k) and t(l), i.e. t(b) and t(a00 ), that define bD = [0, by, bz] through Eq. (1) are depicted in the figure, and the components by and bz are shown in the inset. This disconnection is glissile despite its finite component bz, as can be confirmed by application of Eq. (20); Fig. 12b shows that the number of atoms is conserved during motion, even though volume

Fig. 10. Variation of the coherency strains, [131].

exx and eyy ; on the ð2 1 1Þb ð1 1 0Þa00 terrace interface with respect to the Nb content

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Fig. 11. TEM images showing the variation of lattice-invariant twinning in (a) is Ti–20%Nb, (b) Ti–22%Nb and (c) Ti–24%Nb  00 , SAD pattern obtained from the region circled in (b), showing the ð1 1 1Þ 00 twinning relationship [131]. alloys. (d) is a ½1 0 1 a a

is not. HRTEM images of the disconnection arrays for the 22%Nb and 24%Nb alloys are shown in  1Þ kð1 1 0Þ 00 terrace segments of slightly variable Fig. 13. These are seen to separate coherent ð2 1 b a length. The variation of by, bz and h with Nb content is shown in Fig. 14a. Substituting these parameters into Eqs. (12) and (13) one can determine h and uD, and hence the habit plane orientation (with respect to b) xT = h + uD/2. The values of uD were calculated to be 0.2°, 0.17° and 0.13° for the Ti– 20%Nb, Ti–22%Nb and Ti–24%Nb alloys, respectively, which is consistent with the experimental value of 0.2°. Fig. 14b shows that the calculated and observed habit plane orientation and disconnection spacings agree within experimental error for all three alloys. 4.3. (575)c habits of lath martensite Lath martensite forms with habit planes close to f575gc and a narrow range of orientations centred on the Greninger–Troiano relationship in low-carbon and interstitial-free alloy steels [132–134]. The optimal candidate terrace plane is (1 1 1)c/(0 1 1)a because the coherency strains, although large, are acceptable and glissile disconnections can arise. The principal coherency strains are depicted in Fig. 15a when the crystals adopt the Nishiyama–Wasserman orientation relationship [135]. The reference coherent bicrystal exhibits 2-D centred-rectangular periodicity in the terrace plane defined by  0 =½1 0 0 ; and [0 1 0]c, parallel to y and derived from [1 0 0]c, parallel to x and derived from 1=2½1 1 c a  ; Fig. 15b. The projected Burgers vectors of two glissile disconnections, bD and bD ;  =½0 1 1 1=2½1 1 2 1 2 c a L L and two crystal dislocations, b1 and b2 ; are indicated in Fig. 15b. Expressed in the coherent reference, D D these vectors are b1 ¼ ½0; 1=6; bz c ; b2 ¼ ½1=2; 1=3; bz c , where bz = h(c)  h(a) = 0.004 nm, L L  b1 ¼ 1=2½1 1 0c and b2 ¼ ½1 0 0c , Tables 4.1 and 4.2. A schematic illustration of the formation of the D D disconnections b1 and b2 is shown in Fig. 16. Note that the line directions of the defects, nD1 and nD2 1  0 ½1  0 1 , so that only the small components are taken to be parallel to the viewing direction, ½1 c c D D of b1 and b2 perpendicular to this are evident in the figure. These have opposite sign for the two deD D fects. The overlap step height h is positive for b1 and negative for b2 ; and both have magnitude hðaÞ in this case.

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 1Þ ==ð1 1 0Þ 00 terrace. The enlargement on the right indicates the Fig. 12. (a) Schematic illustration of a disconnection on the ð2 1 b a Burgers vector ð1 1 1Þc =ð0 1 1Þa components, by and bz, and the overlap step height, h. (b) Schematic diagram showing the two 00 atoms per unit transforming volume in the b and a phases [131].

Fig. 13. HRTEM images of the b/a00 interface structure in (a) Ti–22%Nb and (b) Ti–24%Nb. Athermal omega particles (x) are observed in the b phase [131].

One can determine the provisional network accommodating the coherency strains on the terrace by first substituting the parameters exx and eyy into expression (4) to obtain n Ec and then solving Eqs. (5)–(8), assuming values for bD and bL. Because of the multiplicity of potential disconnection/dislocation combinations available in the Fe case, several networks are feasible in principle. To discriminate between them one would need to use Eq. (14) to calculate their energies, and also consider the

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Fig. 14. (a) Variation of by, bz and h for the disconnection illustrated in Fig. 12 as a function of Nb content. (b) Comparison of the D theoretical and experimental values of the average disconnection spacing, d , and the habit plane inclination, xT ¼ h þ /D =2 [131].

generation of the defects involved. The simplest network, geometrically speaking, comprises an array D D of b1 disconnections in edge orientation, i.e. nD1 k½1 0 0c , with d1 ¼ by =eyy ¼ 1:1 nm, and an orthogonal L  , with dL ¼ by =exx ¼ 2:15 nm. This network is array of b2 dislocations in edge orientation, i.e. nL2 k½010 2 c analogous to the orthogonal network depicted in Fig. 4a and the networks in Ti–Mo and Ti–Nb disL cussed in Section 4.2. However, the array of b2 dislocations is sessile in this case and could only move with an advancing interface by climb. It is possible that the lattice-invariant deformation occurs by  1  crystal dislocations on separate glide planes; these defects would slip of 1/2[1 1 1]a and 1=2½1 1 a be glissile in the interface, but their reaction there would produce sessile defects. An alternative

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Fig. 15. (a) Scale drawing of the rhombic atomic arrangements in the (1 1 1)y/(0 1 1)a terrace plane of a ferrous alloy. Full lines represent the ‘‘natural” martensite crystal and dashed lines the austenitic parent; bold full lines represent the coherent state. (b) Rectangle defining the centred 2-D cell of the coherent reference state with the projected Burgers vectors of two glissile disconnections and two crystal dislocations [57].

Table 4.1 Topological parameters of candidate disconnections.

D

b2 D b1

bx (nm)

by (nm)

bz (nm)

|b| (nm)

h (nm)

t c ðcÞ

tc ðaÞ

0.13504 0

0.14072 0.07036

0.00375 0.00375

0.19507 0.07046

0.20294 +0.20294

1  2 ½1 1 0c 1 2 ½0 1 1c

1  2 ½1 1 1a 1  2 ½1 1 1a

Table 4.2 Topological parameters of candidate dislocations.

L

b1 L b2

b (a-frame)

bx (nm)

by (nm)

bz (nm)

|b| (nm)

Possible slip plane

1  2 ½1 1 1a

0.13504 0.27008

0.21108 0

0 0

0.25058 0.27008

 0 1Þ ð1 a ð0 1 1Þa

½1 0 0a

D

network, which is glissile and analogous to that depicted in Fig. 4b, comprising b2 disconnections and L b1 dislocations is shown in Fig. 17a; the disconnections are relatively close to screw orientation, similarly to the view in Fig. 16b, and the dislocations are also very close to screw orientation. Recall from Section 2.2.3 that additional twist misorientation has the effect of decreasing the spacing of one array

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D 1  0 ½1  0 1 . Fig. 16. Schematic illustration of (a) and (b) b2 disconnections in the coherent reference interface viewed along ½1 c c The defects have screw components (not seen) of the same sign and edge components with opposite sign, and steps with opposite senses. Lateral motion of such defects would cause transformation in a conservative manner. The symbols represent site levels along ½1 1 1c =½0 1 1a , not the projection direction [57].

and increasing that of the other and modifying n for both. Networks modified by ancillary twist can be predicted by using Eq. (10) rather than (3), where Rc and Ra are the matrices representing partitioned rotations of c by uxy/2 and a by uxy/2, respectively. Ma and Pond [57] determined the provisional network configurations for twist angles uxy up to 5.26°, i.e. the orientation range from Nishiyama–Wasserman, through Greninger–Troiano, to Kurdjumov–Sachs [136]. The Greninger–Troiano network for D L uxy = 2.5° is shown in Fig. 17b. Note that d2 has diminished while d1 increased relative to their values

for /xy ¼ 0 ; and both arrays are now close to screw orientation. The next stage of iteration for the near twist network solutions described above is to introduce the step character of the disconnections and hence define the habit plane. The normal to this plane is determined by rotating the normal to the terrace plane by the angle w = tan1 h/dD about an axis par

allel to nD2 : The configuration depicted in Fig. 17b has w ¼ 9:18 which corresponds to a habit plane (0.505 0.700 0.505)c. Further refinement would require the bz components of the disconnections to be re-instated and the network adjusted using probe vectors in the updated habit plane. At equilibrium, the misfit along this plane is accommodated by the defect network, and further small adjustments of defect line directions and separations may be needed. Any defect content with finite

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L

819

D

Fig. 17. Schematic illustration of defect networks with b1 lattice-invariant dislocations and b2 disconnections for (a) /xy = 0° (Nishiyama–Wasserman) and (b) /xy = 2.5° (Greninger–Troiano) [57]. D

L

z0 -component, bz0 or bz0 , perpendicular to the final habit plane does not affect misfit-relief but acts as a low-angle tilt boundary thereby introducing an ancillary change in u. The defect arrays in Fig. 17b produce ancillary tilts, uL  0.54° and uD  0.14° about nL1 and nD2 , respectively; these contributions modify the relative orientation, slightly misaligning the (1 1 1)c and (0 1 1)a planes for example. Although the habit plane structures reported above are not fully refined, the provisional structures show good agreement with experimental observations in the literature. For example, Sandvik and Wayman [132] studied lath martensite in an Fe–Ni–Mn alloy using TEM, Fig. 18. They observed an ar 1 lattice-invariant dislocations with dL in the range 2.6–6.3 nm, nL varying between 10° ray of 1=2½1 1 1 2 a and 15° from screw orientation in a (575)c habit plane, and uxy ranging between 0.16° and 3.16°. This observation resembles closely the network predicted here for uxy = 2.5°, Fig. 17b, namely: L d1 ¼ 3:77 nm; nL2 oriented 12.99° from screw orientation and habit plane very close to (575)c. Moritani et al. [118] reported similar observations. Observations by HRTEM of disconnection arrays in lath martensite have been reported by Moritani  1  0 ½101 et al. [118], as shown in Fig. 19. The projection direction is ½1 c c ; and the defects are consisD tent with being the b2 disconnection depicted in Fig. 16b. Only the components of b perpendicular to the viewing direction can be determined from the micrograph. Similar disconnection arrays have been revealed by HRTEM for plate-shaped martensite in Fe–7.9%Cr–1%C (at.%) alloy [15]. The coherency of the (1 1 1)c/(0 1 1)a terraces is evident in this image, and the authors report a spread of ancillary twists

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L  1 1=2½0 1  1 in Fe–20%Ni–5%Mn (mass%) Fig. 18. Weak-beam image of lattice-invariant dislocations with b1 1=2½1 1 a c  7 , closely resembling the predicted near-screw array in Fig. 17. [132]. The habit plane is ð5 7 5Þc and nL ¼ ½0 5 c

Fig. 19. HRTEM image of disconnections in Fe–20.2%Ni–5.4%Mn (mass%) lath martensite. The defects are viewed end-on along 1  0 ½1  0 1 ; a Burgers circuit which, after mapping into the coherent reference state, indicates that the observed defects are ½1 c c consistent with being the disconnections depicted in Fig. 16b. The local habit plane is close to ð1 2 1Þc [118].

around the Greninger–Troiano orientation relationship [134]. This observation conforms to the topological model since any twist is partitioned between the adjacent phases; the terraces are always coherent, a particular orientation relationship describing the correlation of a and c unit cells beyond the short-range distortion field. 5. Observations of diffusional transformations 5.1. Preliminary remarks In martensitic transformations the absence of long-range diffusion imposes severe limitations not only on the character of active dislocations and disconnections, but also on feasible interface structures and orientation relationships between the adjacent phases. These transformations are based on coherent terraces where certain admissible disconnections, defined by Eq. (1), are glissile according to Eq. (20). For transformations where long-range diffusion occurs, these restrictions are relaxed and a

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much broader variety of defects and transformation crystallography arises. Many observations have been reported in the literature, and the approach adopted here is to discuss selected examples in distinctive cases. Therefore, we begin this section by discussing ‘‘diffusional–displacive” transformations where balanced diffusion of different species leads to the conservation of substitutional sites, thereby producing crystallographic characteristics in common with the martensitic case. Subsequently, various transformations where resultant diffusional fluxes arise are considered. In certain cases, the compatibility of crystallographically distinct interfaces is accommodated at their junctions by a crystal dislocation mechanism. Finally, transformations involving interstitial diffusion are discussed. In all cases, the structure of the interface is one of terraces bounded by dislocations or disconnections. We do not consider reactions involving incoherent interfaces, as can occur in massive transformations, for example. 5.2. Diffusive–displacive transformations 5.2.1. The Al–Ag system An interesting situation arises when the number of substitutional sites is conserved in a transformation, but the occupancy of these sites changes through balanced diffusive fluxes of the species present [43]. On the basis of the topological arguments advanced earlier, one would expect the characteristic features of martensite transformations, e.g. shear and surface relief, to arise but with concomitant balanced diffusion. That diffusional transformations can be accompanied by well-defined shears in this manner was recognized a number of years ago [137], and several examples of such diffusive–displacive transformations have been reviewed by Muddle et al. [42]. One example from this class of paramilitary transformations is the precipitation of the hexagonal c phase Ag2Al in a Al [42,43,137]. This is representative of a number of such transformations that can be easily described by the motion of disconnections. Formal proof of the conservation of sites on the basis of Eq. (20) is postponed until Section 5.2.2; here we concentrate on the transformation shear. The lattice parameters are a(a) = 0.40495 nm (with near neighbor spacing equal to 0.2863 nm) and a(c) = 0.2871 nm, c(c) = 0.4662 nm values are for the Ag-rich side of the c phase and yield the maximum c/a ratio).  0i parallel to h1 1 2  0i . The periodicity of the coherent The terrace plane is ð1 1 1Þa kð0 0 0 1Þc with h1 1 a c reference state exhibits 3-fold symmetry with ac = 0.2867 nm and the biaxial coherency strain is ex  x = eyy = 0.28% (x and y are 1=2½1 1 0a and 1=2½1 1 2a , respectively). The three disconnections with smallest magnitude Burgers vectors are obtained by substitution of tðcÞ ¼ ½0 0 0 1c and 1  2  ; 1=2½1 2  1  or 1=2½2 1  1  into Eq. (1). These have downward sense steps, so movement tðaÞ ¼ 1=2½1 a a a rightwards along the interface effects growth of c; their step height is h ¼ cðpcffiffiffiÞ ¼ 0:4662 nm [30]. The D first disconnection has a Burgers vector pffiffiffi b1 ¼ ½0; 1=3; bz c , with by ¼ ac = 3 ¼ 0:1655 nm (derived  from 1=6½1 1 2a ), and bz ¼ ð2aðaÞ= 3  cðcÞÞ ¼ 0:0014 nm. The other two disconnections are D D D b2 ¼ ½1=2; 1=6; bz c and b3 ¼ ½1=2; 1=6; bz c . Thus, for disconnections with n parallel to x, b1 D D defects are pure edge, whereas b2 and b3 are 30° defects with edge components by =2 and equal  0. The maximum possible and opposite screw components, 1=2½1; 0; 0c derived from 1=4½1 1 D transformation strain, corresponding to a train of b1 disconnections, is cyz ¼ by =h ¼ 0:355 and ezz ¼ 0:003. All of these quantities are in excellent agreement with experimental observations. D D D At elastic equilibrium the spacing of b1 disconnections would be d1 ¼ 59:1 nm. An array of b2 and D b3 disconnections have edge components with the wrong sign to accommodate misfit. On the other D D D hand, alternating b2 and b3 disconnections, spaced d2=3 ¼ 29:6 nm apart would accommodate misfit, but rightward motion would effect growth of a rather than c. In both cases, the spacings are quite large because of the relatively low coherency strain eyy . The coherency strain exx can be relieved L  0 spaced at by crystal dislocations with n parallel to y and b ¼ ½1; 0; 0c derived from 1=2½1 1 a L d = 102.4 nm. These spacings are so large that the tilt angle associated with the bz component, Eq. (13), is negligible. The system has been observed in in-situ TEM studies [27,92] and found to proceed by disconnection motion via a kink-pair nucleation and propagation mechanism. Indeed this was the first system and one of only a few where single-atom kinks have been observed on the disconnections [14], Fig. 20.  0i and all disconnections and kinks (unit or superkinks)  0i =h1 1 2 The Peierls valleys are parallel to h1 1 a c closely follow these directions. Topologically, the transformation proceeds by a train of equally spaced

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Fig. 20. HRTEM image at the edge of a c plate viewed perpendicular to the ð1 1 1Þa kð0 0 0 1Þc habit plane, showing a series of  0i atom rows visible in the c single-atom kinks (arrows) in disconnections at the edge, which align along the close-packed h1 1 2 c phase [14].

 0 k½1 2  1 0 [14]. The a circuit Fig. 21. HRTEM image of the edge of a c precipitate plate viewed parallel to the close-packed ½1 1 a c 1  2  1=4½1 1  0 and the c segment XF =23½0 0 0 1,  giving a resultant Burgers vector, segment STUVWX =23  2=3½1 1 1 þ 1=12½1 D D D FS =23½0 0  bz c þ 1=12½0 1 0c , where 1=12½0 1 0c corresponds to one b1 and either a b2 or b3 disconnection.

D

D

D

parallel b1 ; b2 and b3 disconnections in nominally equal numbers, Fig. 21 [14], analogous to observations of the martensite transformation in Co [139], Co–Ni [140] and Fe-based shape-memory alloys D D D [141]. The sum of the Burgers vectors for such a distribution of (b1 þ b2 þ b3 ) triplets is zero, favorable because it minimizes the elastic energy of the train. When the closed circuit encircling the plate end in Fig. 21 is mapped into the reference state, the closure failure is 23½00  bz c þ 1=12½0 1 0c . Thus the end D D of the plate, which is 46 ð1 1 1Þa =ð0 0 0 2Þc planes in thickness, corresponds to seven sets of (b1 ; b2 and D D D D b3 ) triplets in combination with one additional b1 and a b2 or b3 disconnection. In the z direction, the small component 23bz = 0.03 nm, producing misfit on the end of the plate, is uncompensated. The Burgers vector cancellations cause the plastic strain cyz to be lowered from its upper bound value of 0.355 to near zero. Experimentally, the disconnection spacing on well-developed c plates was found as 20– 150 nm, depending on the alloy composition and aging conditions [138,142]. This implies that motion of the observed disconnection content can produce the transformation strain, but it contributes only minimally to accommodation of the 0.28% coherency strain. The equal spacing of the unit-step height disconnections essentially arises from an irreversible thermodynamic repulsive force between the disconnections. The rate of diffusional growth, and hence the rate of dissipation of free energy, is maximized for such disconnection arrays [21]. In contrast, if

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D

D

the transformed product were annealed, one would expect sets of b1 ; b2 and b3 disconnections to clump into a step of height 3 h with residual dislocation character, b ¼ ½0; 0; 3bz c . Accordingly, the elastic fields of the disconnections would be largely removed, decreasing the energy of the system, and continued transformation would occur by a shuffle-based motion of the compound defects. The c phase has some degree of order, with ð0 0 0 2Þc planes being alternately Ag and a more random mixture of Ag and Al in accordance with the wide range of stoichiometry for this compound [143,144]. This produces a change in the crystal symmetry but does not change the admissible disconnections as described here, although it could affect kink formation and motion energies. Hence no modification of the above description is needed. 5.2.2. The Ti–Al system A second example is the formation of lamellar c-TiAl and a2-Ti3Al microstructures. This reaction is similar to that in the Al–Ag system, and has nominally the same disconnections, but it is more complicated because of the presence both of misfit on the terraces and of ordering. c-TiAl exhibits the L10 structure and a2-Ti3Al has the ordered DO19 configuration. Ignoring the chemical differences between species, one sees that the sites of the former fall on f.c.c. sites (actually a small tetragonality is present) and h.c.p. sites for the latter. The relative orientation of the c and a2 crystals closely approximates the ð1 1 1Þ=ð0 0 0 1Þ configuration for the previously discussed Al–Ag2Al reaction. The compound c-TiAl is primitive tetragonal with a two-atom motif. For simplicity, we assume that the atomic motifs occupy the reference f.c.c. sites, i.e., the primitivepffiffiffiunit cell is defined by the translation vectors  0 and ½0 0 1 with ðc=aÞ ¼ 2, and the motif of the structure is Ti at 0, 0, 0 and 1=2½1 1 0c ; 1=2½1 1 c c c Al at 1/2, 0, 1/2. The densities of the atomic sites are X(c)Ti = X(c)Al = 1/2q, where q is the atomic site density of the reference f.c.c. structure, 4/a3, and a is the cubic lattice parameter. The compound a2Ti3Al exhibits the DO19 structure and again, for simplicity, we assume that the atomic motifs decorate the lattice sites of the reference ideal h.c.p. structure. The translation vectors defining the unit cell are  21  0 and ½0 0 0 1 , and the motif comprises eight atoms. The atomic site densities 1  0 ; 2=3½1 2=3½2 1 a a a are X(a2)Ti = (3q)/4 and X(a2)Al = q/4. D  which is observed commonly in We consider a disconnection with b derived from 1=6½1 1 2 c experimental studies of such interfaces [145–151]. A high-resolution electron micrograph of such a disconnection in the ð1 1 1Þc kð0 0 0 1Þa TiAl–Ti3Al interface of a low-misfit quaternary alloy is presented in Fig. 22. Its topological parameters (b, h), deduced from mapping of the circuit shown in the figure, 1  2   ; 2dð1 1 1Þ ); the translation vectors to be substituted into Eq. (1), are tðcÞ ¼ 1=2½1 are (1=6½1 1 2 c c c  , and, since the magnitudes of the step heights h(c) and h(a2) are equal, their difand tða2 Þ ¼ ½0 0 0 1 a

ference, bz, is zero. (Note that the beam direction in Fig. 22 is opposite to that in Fig. 21; alternatively stated, the c lattices of the TiAl and Al(Ag) crystals are relatively rotated by p about ½1 1 1. Thus, nD for the disconnection in Fig. 22, taken to be outward from the micrograph, is opposite to that discussed for Fig. 21. With the same nD assigned as for the Ag2Al defects, the present disconnection would be char-

 ; 2dð1 1 1Þ Þ obtained with beam direction parallel to Fig. 22. HRTEM image of a disconnection with ðb; hÞ ¼ ð1=6½1 1 2 c c 21  0 from ð1 1 1Þ kð0 0 0 1Þ lamellar c=a2 interface in low misfit Ti–44Al–4Zr–4Ta–02Si (at.-%) alloy [145].  1 0 k½1 ½1 c a c a

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acterised b1 with a positive sense step height.) Motion of this disconnection rightwards effects the growth of Ti3Al. Moreover, it conserves the number of atomic sites, as can be seen by suppressing the species differences and substituting X(c) = X(c)Ti + X(c)Al = q, and X(a2) = X(a2)Ti + X(a2)Al = q into Eq. (20); it follows that DX and bz are simultaneously zero and hence dN = 0. However, in practice these sites are unequally partitioned between the atomic species in the two crystals (DXTi = q/4 and DXAl = q/4 for the stoichiometric phases), and hence equal and opposite fluxes of Ti and Al atoms arise during defect motion. This can be seen by the following substitutions into expression (20) with bz = 0,

1 ldð1 1 1Þ qdy; 2 1 dNAl ¼ lhDX Al dy ¼  ldð1 1 1Þ qdy: 2 dNTi ¼ lhDX Ti dy ¼

ð24aÞ ð24bÞ

Thus, motion of this disconnection leads to simple shear deformation but requires a flux of material, analogous to the Al–Ag case described in the previous section. In this way a displacive–diffusive transformation should exhibit the same crystallographic features as a displacive analogue if a source of these disconnections were to operate [42,43,137]. Only 18 of the atoms are now sheared into the correct position, compared with 12 for the single species case. Chemical ordering of the c and a2 phases does not affect the analysis of diffusional fluxes described above as this requires only short-range atom jumps. Disconnections are perfect only when the vectors substituted into Eq. (1) are true translation vectors of the ordered structures, otherwise they separate  is clearly a translainterfaces with different energies. For the defect shown in Fig. 22, tða2 Þ ¼ ½0 0 0 1 c 1  2  , which can be decomposed into tion vector of the a2 phase. In the c phase, tðcÞ ¼ 1=2½1 c  , is also a translation vector, as can be confirmed from Fig. 22, which shows the 1  0 þ ½0 0 1 1=2½1 c c alternating Ti and Al (0 0 2) planes edge-on (parallel to AB). Perfect disconnections with opposite D  1 1 . Imperfect disconsense steps of the same height, h = þ2dð1 1 1Þc , are observed to have b ¼ 1=3½2 c nections were not observed in this quaternary system. The second consequence of tetragonal ordering is to break the 3-fold symmetry of the interface. The coherent reference state in the present case exhibits 2-D rectangular translation symmetry with  1 0 and ½0 1 0 parallel to ½1 1  2 . For the quaternary alloy, tetragonality leads to a ½1 0 0c parallel to ½1 c c c coherency strain eyy ¼ 0:5%, although this was not observed to be accommodated by an array of defects in the experimental study [152], in contrast to observations in the binary system mentioned below (the reasons for this difference are not known, although a possible reason could be a reduction of the nucleation rate of disconnections in the presence of the added solutes). It also causes dð1 1 1Þc to differ by 0.5% from dð0 0 0 2Þa , so the Burgers vectors of the disconnections acquire a component bz. Breaking the 3-fold symmetry implies that the Peierls barriers for perfect disconnections with b parallel to  1 and ½2  1 1 , if vestigial order persists. This  should differ from those with b parallel to ½1 2 ½1 1 2 c c c supposition is supported by calculations for single crystals, which show that the TiAl phase has a deep L  edge dislocations and shallow Peierls valleys for those with Peierls valley for b1 ¼ 1=6½1 1 2 c L  1 and bL ¼ 1=3½2  1 1 30° dislocations [153]. The height of the kinks could also be double b2 ¼ 1=3½1 2 3 c c in such a case and this has been shown experimentally by HRTEM for the binary alloy, Fig. 23. In this figure, the mechanism of disconnection motion was by kink propagation analogous to the Ag2Al case, with the disconnections and kinks aligned along the h1 1 0i Peierls valleys. D  ; bD0 ¼ 1=6½1 2  1 and Trains of disconnections with all three Burgers vectors, b1 ¼ 1=6½1 1 2 c 2 c D0  1 1 were observed in the binary alloy. For a disordered system, all these defects would b3 ¼ 1=6½2 c be perfect disconnections, whereas, in a fully ordered system, only one would be perfect and the other D0 D0 two partial (signified b2 and b3 ). Perfect disconnections in the ordered system would have Burgers D D  1 and b ¼ 1=3½2  1 1 . We suppress the components bz in this discussion of vectors b2 ¼ 1=3½1 2 3 c c D D0 D0 coherency. For arrays containing equal numbers of b1 ; b2 and b3 , the total Burgers vector would sum to zero, analogous to the Al/Ag2Al case. Arrays comprising these three disconnections were observed, implying that the retained vestigial order, or its effect, was less in this case. The misfit caused by the tetragonality of the c phase is eyy 2% for the binary. Thus one expects that the ratio of edge to 30° partials would differ from the 2:1 of Al/Ag2Al with an excess of one type removing coherency. The D0 D0 observed ratio was 3:1, implying that pairs of b2 and b3 partials, each of which has a component  2i, accommodated coherency. For complete removal of coherency, the spacing of misfit1=12h1 1

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 1 ) at the edge of a c/a2 plate interface [150]. Fig. 23. ½1 1 1c =½0 0 0 1a HRTEM image of two-atom high kinks (along ½0 1 c

relieving disconnections would be 4.1 nm. The observed spacing was about 15 nm. The excess one-infour disconnections that relieve misfit would be spaced at 60 nm. Thus, the observed trains and ratio do not give complete coherency strain removal. As with the Al/Ag2Al case, one would expect annealing D D0 D0 to produce clumping of b1 ; b2 and b3 triplets into three-high steps. The excess defects would remain as unit disconnections accommodating strain or could combine with the steps to form four-high D0 D0 imperfect disconnections with a Burgers vector equal to b2 or b3 . These results are pertinent to the issue of whether strain can precede interdiffusion in such transformations. Such a bifurcation is certainly not the case in the late stages of precipitation where large steps form, as discussed in Section 3.6, or when disconnections move by climb, as in the Al–Cu case of Section 5.3. However, for the present case, strain preceding complete interdiffusion is possible. If the equilibrium concentrations are achieved within a few atomic distances from the kink, this would have a negligible effect on the diffusion kinetics. Consistent with this interpretation, in-situ HRTEM and energy-loss spectroscopic observations [154] of disconnection motion in the Al–Cu, Al–Ag and Ti–H systems at low velocities (on the order of a nanometer per second or less), demonstrate that the compositional partitioning that accompanies the transformations is complete within a few atomic distances of the disconnections and that the latter move at rates in accord with diffusion solutions of the moving line defects [22–26] with local equilibrium boundary conditions at the line defects. However, if the distance over which concentration relaxation occurs is of the order of the disconnection spacing, as might be the case at relatively low transformation temperatures and concomitant high driving forces, the diffusion kinetics would be altered. The present case of an order-order transformation is typical of other ordered examples in that there are more complex possibilities that are beyond the scope of the present work but which we mention for completeness. There are eight possible minima in the cnid (see Section 2.4) for the terrace planes so partial disconnections are possible, e.g. the separation of a 1/3 h1 1 2i edge into two 1/6 h1 1 2i partials. In addition, one of these structures can be converted into another by near-neighbor diffusional exchange (this would not occur in the bulk because it would translate the fault plane but would not change its energy). That is, two adjacent rows of atoms in the correct structural but wrong compositional positions can convert to the correct positions by short-range diffusional exchange. This process could occur near the core of a partial disconnection and hence could accompany kink motion. Enhanced atomic motion at disconnections due to the presence of kinks (and presumably associated vacancies) has been observed experimentally by in-situ HRTEM [16], and this motion could facilitate such short-range compositional rearrangements. The observation of disconnections with 1/3 h1 1 2i Burgers vectors and the observation of double kinks show that such partial disconnection and local diffusional effects are not complete in the present case. Remanent effects from the ordering persist as disconnections move in the Ti3Al/TiAl transformation. 5.3. Disconnections in coherent precipitates The formation of h0 -CuAl2 precipitates in a-AlðCuÞ is an interesting example of diffusional transformation by lateral disconnection motion [114,155–158]. Several disconnections are observed experi-

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mentally, some involving motion with balanced diffusional fluxes and others with unbalanced fluxes. Since these precipitates exhibit coherent f1 0 0ga interfaces, disconnections are not required for misfit accommodation. Consequently, feasible defects have no component of b parallel to the interface and hence no transformation shear arises. Another special feature of this precipitation system is that, according to Eq. (1), admissible disconnections arise due to non-symmorphic symmetry elements (mirror-glide planes or screw-rotation axes) in the h0 -CuAl2 crystal, the structure of which is depicted in Fig.  24. For example, consider the case illustrated schematically in Fig. 25 where WðkÞi ¼ 4þ ; ½0 0 1k , i.e. a rotation of 90° about ½0 0 1k and translation of tðkÞi ¼ ½0 0 1 in a-AlðCuÞ, is

Fig. 24. The unit cell of h0 -CuAl2 , space group P42 mc; the 42 axis is parallel to ½0 0 1 and inter-relates the Cu layers parallel to ð0 0 1Þ.

Fig. 25. Schematic illustration of the formation of a perfect disconnection formed by joining surfaces exhibiting an a-AlðCuÞ step and a h0 -CuAl2 demi-step.

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  aligned with WðlÞj ¼ 4þ ; 12 ½0 0 1l , i.e. a rotation of 90° and a screw-displacement of 12 cðlÞ where c is the lattice parameter in the [0 0 1] direction of h0 -CuAl2 . The ð0 0 1Þ surfaces of the two crystals exhibit incompatible steps; the arrowed lines and crosses in the figure represent surface features inter-related by 2-fold and 4-fold rotation axes respectively. Substitution of the 4 and 42 operators into expression (1) shows that the resulting disconnection has dislocation character with b ¼ ½0 0 1k  12 ½0 0 1l . The interface structures on either side of this defect are not identical, but are relatively rotated by 90°; nevertheless, the two interface structures are crystallographically equivalent and hence energetically degenerate. The step height h can be defined as in Section 2.1. The surface step on the h0 -CuAl2 crystal is known as a demi-step [38] to distinguish this type from steps like the a-AlðCuÞ one which are characterised by a translation vector. Designating the unit cell heights, m for a and n for h0 , the defect illustrated schematically in Fig. 25 has m = 1 and n = 0.5 and b ¼ ½0 0 1a  0:5½0 0 1h0 is large, corresponding to 44% misfit on the step riser, and hence the elastic strain energy is therefore also very large. Moreover, since the number of atoms per unit cell is 4 for a and 6 (4Al+2Cu) for h0 , application of expression (20) shows that motion of this defect involves climb. Instead, disconnections with multiple step heights are formed. The favoured cases, corresponding to those observed experimentally, are m = 3, n = 2 and m = 5, n = 3.5. The former case has residual misfit of 4.3%, much smaller than the previous case; an early HRTEM observation from the work of Dahmen and Westmacott [157] is shown in Fig. 26. The latter has an even smaller misfit of 0.45% on the step riser, as illustrated schematically in Fig. 27. Application of expression (20) demonstrates that motion of the m = 3, n = 2 defect involves balanced fluxes and conservation of substitutional sites, whereas the m = 5, n = 3.5 defect climbs with unbalanced fluxes. 5.4. Defects accommodating angular incompatibilities between facets Precipitates must be bounded by low energy interfaces; in any particular case there may be few feasible structures. Furthermore, the ideal orientation relationships for two such bicrystals may not be identical. Thus, for two such interfaces to coexist forming a facet junction in precipitates of reasonable size, any angular incompatibility must be accommodated. Recently, a mechanism involving a source of crystal dislocations at a junction followed by climb of these defects along the interfaces has been studied in stainless steel and titanium aluminide [159,160]. As illustration, we concentrate on the observations in commercial stainless steel with composition Fe–24.4%Cr–6.8%Ni–3.7%Mo (wt.%). Several other workers have investigated similar systems such as Ni–Cr, for example [161–174]. Samples were solution treated at 1400 °C for 30 min, water quenched and aged at

Fig. 26. HRTEM image of a disconnection (m = 3, n = 2) in the interface between a-AlðCuÞ and #0 -CuAl2 [157].

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Fig. 27. Schematic illustration of the surface step configurations for the m = 5, n = 3.5 disconnection in the a–h0 interface.

1000 °C for periods up to 30 min. This treatment produced acicular Ni-rich c particles in a Cr-rich a matrix, with a(c)/a(a) = 1.255; the particles were typically 30 lm long with approximately rhombic 1  1 =½1  0 1 and their cross-sections 2 lm on edge. The long axis of the particles was about 5° from ½1 a c  0 1 and   overall orientation relationship with the matrix was close to Kurdjumov–Sachs; ½111a k½1 c  1Þ . More detailed measurements using convergent-beam diffraction revealed small anguð1 0 1Þa kð1 1 c lar deviations from this, varying in sense and magnitude somewhat from particle to particle. The mean orientation, determined from measurements of over 100 precipitates, was equal to a rotation by 1°  0 1 and 1  1 k½1 away from Kurdjumov–Sachs toward the Pitsch orientation relationship; ½1 a c  1 0Þ kð0 1 0Þ [159]. The magnitudes of the measured rotations about this axis were up to 2° with ð1 a c a mean value of 1°, as shown in Fig. 28. Fig. 29 is a diffraction contrast image with the beam direction approximately parallel to the long axis, showing the two principal facets, designated I and II. The index of I is ð6; 11; 7Þc , i.e. close to  , and II is close to ð1 0 1Þ . Arrays of crystal dislocations oriented parallel to the long axis are ð121Þ c c

Fig. 28. Schematic plots of the orientation distributions (number of particles, N, as a function of misorientation angle about  0 1 ) for a titanium aluminide alloy (TiAl) and stainless steel (SS) [159]. The former is a delta function and the latter is 1  1 k½1 ½1 a c Gaussian.

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1  1 k½1  0 1 showing facets I and II; the insets show crystal Fig. 29. Cross-sectional TEM image taken approximately along ½1 a c dislocation arrays in both facets [159].

1  1 k½1  0 1 of (a) facet I; D indicates end-on c crystal dislocations, and SF the stacking-fault of a Fig. 30. HRTEM images along ½1 a c dissociated dislocation, and (b) facet II with end-on c crystal dislocations indicated by arrow heads. The f1 1 1gc lattice-plane spacing is 0.21 nm [159].

evident in both facets. HRTEM images of facets I and II are shown in Fig. 30a and b, respectively, again showing the c-crystal dislocation arrays seen end-on in both.  1Þ terrace, with ½1 1  1 k½1  0 1 , i.e. where the adjacent Facet I is based on the coherent ð1 0 1Þa =ð1 1 c a c crystals ideally exhibit the Kurdjumov–Sachs orientation relationship. Here, the misfit parallel to  0 1 is 2.6%, but the defect structure accommodating this coherency strain was not investi1  1 k½1 ½1 a c

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1  1 k½1  0 1 , indicating the misfit relieving disconnection structure Fig. 31. HRTEM image of a type I facet viewed along ½1 a c [159].

gated. However, the defects accommodating the 6.12% coherency strain along ½1 2 1c were studied. An D array of disconnections similar to b2 (see Section 4.3) is consistent with the experimental observation, 1  1  Fig. 31. The b of these disconnections is obtained from Eq. (1) with tðaÞ ¼ 1=2½1 a and  0 1  , giving edge components of 0.04 nm parallel to ½1 2 1 and 0.005 nm normal to the tðcÞ ¼ 1=2½1 c c 1  1 of the disconnections in facet (it is assumed that, on average, the screw components 1=6½1 a the array cancel). In fact, the observed disconnection array slightly over-compensates the coherency strain (see the effect of superimposed crystal dislocations below). The value of h obtained from Eq.  1Þ . Also, the bz components  2Þ ð1 2 (12) is consistent with the overall facet inclination, i.e. close to ð3 1 a c of the disconnection array lead to a tilt, /D , Eq. (13), equal to 0.6°, in good agreement with the experimental values. The sense of this deviation from Kurdjumov–Sachs is away from Pitsch, and is indicated on Fig. 28 by Css.  0 1 were observed in all type I facets. However 1  1 k½1 Arrays of crystal dislocations lying along ½1 a c their spacings were somewhat variable from one precipitate to another, with an average value of  1 0 (i.e. with an uncer or 1=2½1 9.5 nm. Circuit mapping showed that their b was either 1=2½0 1 1 c c  0 1 ). The edge components of these arrays exactainty in the sign of their screw components, 1=4½1 c erbate the misfit accommodation parallel to ½1 2 1c , i.e. have opposite sign to the edge components of 1  1 k½1  0 1 , u, towards Pitsthe disconnections, and the bz edge components cause a rotation about ½1 a c ch. Thus, for the average dislocation spacing, u  1.3°, so the total angular deviation from Kurdjumov– Sachs (due to dislocations and disconnections) is 0.7° towards Pitsch, consistent with experimental observations. The bicrystal structure underlying the type II facet is obtained by abutting the ð1 1 2Þa plane against ð1 0 1Þc with Pitsch orientation relationship. There is evidence that the energy of such interfaces, formed by abutting relatively dense atomic planes, can be relatively low [66]. In the present case  0 1 is 2.6%, as for facet I, but it is 11.9% parallel to ½1  1 0 =½0 1 0 . Since 1  1 k½1 the misfit along ½1 a c a c no defect structure was observed accommodating this latter misfit, the interface is considered to be a one-dimensionally commensurate interface [69,70,73].  1 0 =½0 1 0 were observed in all type II facets, with an Arrays of crystal dislocations parallel to ½1 a c average spacing of 1.6 nm. The b of these defects were found by circuit mapping to be consistent with  1 0 =½0 1 0 were observed to spread 1=2½1 1 0c or 1=2½0 1 1c . The edge components parallel to ½1 a c somewhat along the interface. However, the bz components of these defects remained localised at the interface, and, for the average spacing, caused a rotation u  5.1° from Pitsch towards Kurdjumov–Sachs.

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Fig. 32. Schematic illustration of the accommodation mechanism at incompatible facet junctions. (a) c and a crystal surfaces at the I and II facets before being brought into contact by the formation of disclinations; the disclination angle, x is equal to /a  /c , where u is the angle between the I and II surfaces in the relevant crystal. (b) Dislocation arrays accommodating the junction incompatibilities [160].

The mechanism for accommodating the angular incompatibility at a type I–II facet junction can now be summarised with the aid of Fig. 32. For very small particles, the incompatibility can be accommodated by disclinations located at each of the four facet junctions, Fig. 32a. As a particle enlarges, the disclination quadrupole is expected to transform into a dislocated configuration, Fig. 32b, with smaller strain energy [44]. Fig. 32b is a simplification in so far as actual precipitate junctions were observed to be rounded, comprising interleaved I and II facet configurations. In the actual case, there are still some residual corner strains in the configuration of Fig. 32b that can be reduced by some non-uniform climb rearrangement of the dislocation arrays near the corners. We presume that the relevant crystal dislocations are generated at the junctions and subsequently climb along the facets I and II. Climb rates are likely to depend inversely on the magnitude of bz and directly on the interfacial diffusion rate in the type I and II facets. Since bz is larger for the dislocations in the I facets and diffusion rates are probably faster in the II facets, one expects a denser array to arise in the II facets, as is observed experimentally. This mechanism is also consistent with the observed spread of precipitate orientations and the bias towards the Kurdjumov–Sachs orientation, Fig. 28. For stainless steel, the angular incompatibility

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between the orientation of the misfit relieved I facet and Pitsch is 0.6° + 5.6° = 6.2°. This value should be compared with the sum of the tilts due to the observed crystal dislocations superimposed on the type I and II facets, namely 1.3° + 5.1° = 6.4°. With the observed partitioning of the tilts, the average precipitate orientation is expected to be 1.4° from Kurdjumov–Sachs towards Pitsch. This compares well with the observed value of 1.0°. Interestingly, observations of precipitate orientations in a titanium aluminide alloy, with almost identical crystallography, show a different behaviour [160]. Here, all the crystal dislocations were found to be present in the type II facets, giving a single observed orientation relationship corresponding to the misfit relieved type I facet, CTiAl, Fig. 28. 5.5. Scaling (oxidation) reactions Disconnections can have a role in scaling reactions, although there are interesting variations in behaviour depending on the mechanism of scaling. The role of interface defects in scaling reactions has been discussed by several authors [175,176], and the role of disconnections has been analyzed [176–179]. The defect mechanisms vary markedly, depending on whether the mechanism is one of cation diffusion or anion diffusion. As examples, we consider the oxidation of nickel by a cation vacancy mechanism and sulfidization of molybdenum by an anion vacancy mechanism. The solubility of oxygen or sulphur in metals is so small that it can be neglected. Hence, the terrace structure and the degree of coherency can be deduced from a dichromatic pattern of the metal and of the cation sublattice of the oxidant. For the case of oxidation by cation vacancy diffusion, we consider Ni oxidizing to NiO [175]. The terrace plane is ð0 0 1ÞNi =ð0 0 1ÞNiO with h1 0 0i directions parallel in the interface. Since the lattice parameters are a(Ni) = 0.3524 nm and a(NiO) = 0.4195 nm, the misfit is exx = eyy = 17.4%. Terrace/defect structures have not been observed directly in this system, but the observed oxide orientations and the effect of rare-earths in pinning the oxidation [179] are consistent with such a structure. Expressed in the coherently strained reference frame, disconnections with D the smallest magnitude Burgers vectors have b1 ¼ ½0; 0; bz c with bz = 0.034 nm and step height L h = 0.176 nm. These can combine with crystal misfit dislocations, b ¼ h1=2; 1=2; 0ic with magnitude D 0.273 nm, to form another disconnection b2 . The growth proceeds at the free surface of the scale. There, oxygen from the vapor ionizes and adds to a kink site, a Ni cation joins a kink site and creates a cation vacancy, two electronic holes are created, and the kink on a surface ledge translates in the elementary growth step. There is no strain barrier for the kinetics at the kink sites and the activation energies are those for surface diffusion and kink formation [21]. The cation vacancies and holes diffuse to the metal/scale interface. There a metal atom jumps into the cation vacancy and ionizes, annihilating the holes. The oxygen sublattice is unchanged at the metal–scale interface, which does not move relative to the oxide but serves as a conduit for metal vacancy transfer. The metal vacancy so formed either diffuses into the metal and annihilates at a sink (dislocation, grain boundary, or void), or annihilates at an interface dislocation in the Ni. The latter can be a misfit dislocation with its extra half plane in the metal (climbing away from the interface and returning in the Pieraggi–Rapp mechanism [179]) or a tilt dislocation (one with a component of its Burgers vector normal to the interface [176]). Thus, disconnections can exist at the interface but they do not move laterally along the interface as scale growth occurs. Hence, they are not needed for growth and misfit can be relieved by misfit dislocations. Two orthogonal arrays of misfit dislocations of the type bL spaced at 1.59 nm would remove the coherency strain in this example. In contrast, when an oxidation reaction proceeds by anion vacancy diffusion, growth proceeds at the metal/scale interface. As an example we consider the sulfidization of a Mo to b MoS2, a reaction studied by Lee and Rapp [180]. Mo has a lattice parameter of a(a) = 0.314 nm and the hexagonal C7 structure b has lattice parameters a(b) = 0.314 nm and c(b) = 1.230 nm. The terrace plane is  0 0Þ kð0 0 1Þ . The periodicity of the coherent reference state is analogous to the 2-D centred rectð1 1 b a  0 , and ½0 1 0 derived from angular one shown in Fig. 15b with ½1 0 0c ¼ ½1 0 0a ¼ 1=3½1 1 2 b c ½0 4 0a k½0 0 0 1b . Thus the coherency strains are exx ¼ 0 and eyy ¼ 2:09%. Disconnections with the smallest magnitude Burgers vectors, which we obtain by substituting the vectors derived from 1  1 into Eq. (1), have bD ¼ ½0; 1=8; bz  with bz = 0.115 nm, and  1 0 and tðaÞ ¼ 1=2½1 tðbÞ ¼ 1=3½1 2 1 b a c D step height h = 0.157 nm. As indicated, the y component of b1 can be of either sign. A train of these D defects with positive y components would accommodate eyy if nD ¼ ½1 0 0c and d 7:43 nm; a tilt

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/D 0:9 would then arise. Pairs of defects of opposite sign could combine to form disconnections b2 with double height steps and net Burgers vectors [0, 0, 2bz]c, with the shear replaced by a glide shuffle. D Also disconnections b2 with opposite sign steps and Burgers vectors could annihilate during growth, so overall growth could proceed by formation and annihilation of pairs of opposite sign disconnections, analogous to the advance of a surface step by kink pair formation and annihilation. Single height disconnections or misfit dislocations would still be needed to relieve the coherency strain but they could be arranged to remove all or part of the tilt. Mechanistically, two anions add to a kink site on one of these disconnections and create two anion vacancies, a metal atom joins a kink site and ionizes, creating four electrons, and the sulphide disconnection y-kink moves an elemental distance along x, as described by Eq. (21) in Section 3.3.1). A sim 1 0 tilt dislocation in the sulphide. The electrons and ilar process could occur at a jog on a 1=3½1 2 b anion vacancies diffuse to the free surface. There, sulphur from the vapor annihilates both an anion vacancy and two electrons. In contrast to the cation diffusion case, the kink kinetics for disconnections and jog tilt dislocations in the interface involve elastic energy and elastic interactions so the interface kinetics are more complex. The disconnection can serve in this case both to relieve misfit and to act as the translating defect that propagates the transformation. The other defects cannot serve both roles, misfit dislocations relieving coherency strain but not propagating the transformation and tilt dislocations propagating the transformation but not relieving misfit. Since the elastic energy of either disconnection is less than that of either type of dislocation, the translation defect type is favored energetically. However, in the Mo–MoS2 case the cation density is one-half the Mo atom density at the terrace plane. Thus, Eq. (20), one-half the Mo flux enters the sulphide and one half must flow to another sink to maintain conservation of mass. One way to achieve this would be for equal numbers of disconnections and a tilt dislocations to be present, with the latter acting as sinks for the added Mo flux. Another mechanism would be for the added Mo at disconnections to correspond to an annihilation flux, with metal vacancies diffusing in the metal from sites such as grain boundaries or lattice dislocations to the disconnections and annihilating the excess Mo atoms there. Scales growing by interstitial diffusion would behave analogously. The anion interstitial case would also involve growth proceeding at the metal scale interface and the cation interstitial case would involve growth at the free surface of the scale. Thus there is a dramatic difference in behaviour between cation and anion diffusing scales for all cases. 5.6. Hydride Formation The formation of hydrides occurs by mechanisms closely resembling the oxidation cases. Surface hydrides form when hydrogen diffuses interstitially through the hydride, with growth at disconnections at the hydride/metal interface. Internal hydride precipitation, the more studied case, involves hydrogen diffusing through the metal and forming the hydride at disconnections. In the simplest case, the metal atoms act as a template and the hydrogen fills interstitial sites. An example is the formation of b PdHx in a Pd with x = 0.6–0.7 [181] and H in octahedral sites. Both phases are fcc. At x = 0.6, the lattice parameter of b is 0.4025 nm while that of a is 0.3580. The terrace plane is (0 0 1) with exx ¼ eyy ¼ 11:7%, and the probable disconnection is two interplanar spacings high, h = 0.3580 nm, with a Burgers vector bD = [0, 0, bz] and bz = 0.0445 nm. In other cases, the metal sublattice undergoes a change in crystal structure when the hydride forms. An example is the formation of c titanium hydride of nominal composition TiH in a Ti, studied by HRTEM in [182,184]. The c phase is face-centred tetragonal with H ordering in tetrahedal sites  0Þ=ð1  1 0Þ with ½2 1 1  0k½1 1 0. There is essentially peraligned along [0 0 1]. The terrace plane is ð0 1 1 fect matching of Ti atoms in the terrace plane so there are no coherency strains. The entire volume change of 15.3% is accommodated by extension normal to the terrace plane. The corresponding D disconnections have h = 0.29 nm, dð0 1 1 0Þ in a, and b1 ¼ ½0; by ; bz c with by ¼ aðaÞ=2 and bz ¼ 0:044 nm. In addition to these a second disconnection formed by the combination of two disconnections with opposite sign by components to form a double height disconnection h = 0.58 nm and D b2 ¼ ½0; 0; 2bz c , Fig. 33 [184]. Burgers circuit analysis of the defect in the micrograph gives a closure D  0 and the c segfailure equal to b2 ; alternatively stated, the a circuit segment reduces to tðaÞ ¼ ½1 0 1 D  ment to tðcÞ ¼ ½1 1 0; substitution of these vectors into Eq. (1) gives b2 .

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D

Fig. 33. HRTEM image of a b2 disconnection in a Ti–c TiH viewed along ½0 0 0 1=½0 0 1 [184].

Since there is no coherency strain, there is no tendency for disconnections to remain on the terraces. Instead they were observed to move to the plate tips [183,184]. There the accumulation of like sign Burgers vectors produced incompatibility strains. Interestingly, the accommodation mechanism to relieve these strains was often observed to be nucleation of another hydride platelet near the tip [182]. 6. Discussion and summary All of the quantitative treatments have been made in the isotropic elastic, bicrystal, planar interface assumptions. The planar interface assumption is exact for lath type plates. In Sections 2 and 3, we discussed how end effects can be treated for lenticular plates and showed that much of the interface can be approximated as being planar. For thin plates, the modifications of the elastic coherency fields are straightforward as are the definitions of the misfit strains [185]. The major effect is that for a thin plate sandwiched by thick plates, the analogue of a plate within a matrix, the strains and rotations tend to partition to the thinner plate. For the inhomogeneous, isotropic elastic case, the coherency fields are also straightforward [3,185]. The elastic fields of dislocations and disconnections are more complex but are available [186]. For the anisotropic elastic case, coherency strain and rotation fields are known [55], as are the fields of dislocations and disconnections [187,188]. The major feature of these treatments is that both strains and rotations tend to partition to the softer elastic phase. For cases where both the reactant and product phases are metals or intermetallic phases, the elastic constants do not differ markedly and use of these more complicated models is probably not needed in view of other approximations in the topological model or other theories for transformations. When covalent or ionic phases form within metals, however, the elastic properties can differ by factors of up to about five, and the more complex treatments would be justified. In some systems the topological model accurately predicts the observed equilibrium spacing of defects that would remove all coherency strain, examples being thermal martensite in zirconia [125,189], athermal martensite in Ti–Mo [39,40,126,130] and Ti–Nb [131] and a diffusional transformation in a Ti–Cr alloy [190,191]. Equivalently, the topological model accurately predicts the {5 7 5}c martensite habit plane in Fe alloys, Section 4.3, implying that the equilibrium spacing is achieved. In

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other examples, such as the Ag2Al/Al interface, Section 5.2.1, the observed spacing of misfit relieving defects is less than that which would relieve misfit strain completely. In many of these cases, including Ag–Al, the misfit strain is small. These observations parallel those for misfit dislocations in thin films and multilayers, where variations in film thickness change the coherency strain in the thin layers. The arrays that completely relieve strain are achieved only at strains much greater than that at the critical thickness where misfit dislocations become stable [192]. The extra strain is required to overcome nucleation barriers, frictional resistance to glide, and solute or impurity pinning. The disparity for disconnections likely arises from the same causes. Thus, in cases where the topological model does not predict the observed habit plane, the reason in many cases is that equilibrium strain relief was not achieved in the observed specimens. In all cases, the disconnections observed, in HRTEM, are consistent with theoretical expectation [38]. This includes both disconnections that glide and produce the shear transformations and disconnections that contain dislocation components in climb configuration to lower elastic energy, as for the Al–Cu system, Section 5.3. In the Ti/TiH interface in Fig. 33, the tilt rotation associated with the bz component of a disconnection can be discerned. As expected from dislocation theory [46], the rotation is partitioned predominantly to the softer phase, Ti in this case [193]. Hence, for an array of such disconnections, the tilts of the individual disconnections sum to give the tilt wall rotation in the Fe alloy shown in Fig. 7. Obviously, the disconnections also account for the transformation shear strain and normal strain perpendicular to the terrace planes. We have treated a variety of cases where the interface comprises well-defined terraces separated by dislocations or disconnections. As mentioned in Section 2, such terraces are ubiquitous because they exhibit pronounced energy minima [74,75] and are essential for diffusionless processes. It will be interesting to further apply the topological model to well-defined terraced interfaces formed during diffusional transformations, where potentially a greater variety of defect arrays can obtain at the interface, particularly with regard to the lattice-invariant component of the transformation [12,105,162]. The topological model may apply to other transformations that have not been analyzed by the present methodology, such as some cases of massive transformation or transformations proceeding by diffusion-induced boundary migration. However, there may be other cases, such as some massive transformations or the growth of spherical intragranular precipitates, where the interfaces are incoherent [69,71,72,194] and the topological model will be inapplicable. Acknowledgments The authors thank G.R. Purdy, R.A. Rapp and B. Pieraggi for their helpful discussions. JMH gratefully acknowledges support of this research by the National Science Foundation under Grant DMR0554792. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16]

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