MATERIALS 兴 HIENGE& ENGIEERING ELSEVIER Materials Science and Engineering A 438-440(2006)109-112 www.elsevier.com/locate/msea Kinematic and topological models of martensitic interfaces R C. Pond a,*X. Maa. J. P. Hirth Department of Engineering, University of Liverpool. Brownlow Hill, Liverpool L69 3BX, UK Received 19 April 2005; received in revised form 12 January 2006: accepted 2 February 2006 A new model of martensitic transformations has been presented recently where the habit plane consists of segments of coherent terraces reticulated by an array of line defects. These defects accommodate any coherency strains present and the transformation is effected by lateral motion of disconnections across the terraces. Provided the terraces and line defects satisfy certain criteria, the mechanism of transformation is onservative. Moreover, the topological parameters of the defects( Burgers vectors and step heights) can be determined rigorously enabling the overall habit plane inclination and orientation relationship to be determined The objective of the present paper is to compare predictions for the habit planes according to this topological method with those of the classical theories in the case of transformations in ZrO2 and Ti. The two approaches lead to disparate solutions, and the origin and magnitude of the disparities are elucidated. Unlike habit planes in the classical model, which are invariant planes, topological ones are misfit-relieved semi-coherent o 2006 Elsevier B v. All rights reserved Keywords: Martensitic phase transformation; Interface structure; Martensitic crystallography 1. Introduction that coherency strains may be partitioned unequally between the A new model of martensitic transformations was presented Clearly, the physical basis of the TM is different from that recently [1]; this model is known as the topological model (TM) of the Phenomenological Theory of Martensite Crystallogra- because the interface structure is comprised of coherent terrace phy (PTMC)[6, 7]. In general, predictions for the HP, orien- gments reticulated by an array of line defects. In general, this tation relationship(OR), and transformation displacement are array has two sets of defects, crystal or twinning dislocations not expected to be identical according to the two approaches (lattice-invariant deformation or LID)and transformation dis- [1]. The objective of the present paper is to elucidate the ori- locations or disconnections [2]. The array accommodates any gin of discrepancies between HP orientations predicted by the coherency strains and also provides the mechanism of transfor- TM and PTMC. For clarity, we consider transformations where mation through synchronous lateral motion of disconnections one of the principal strains is zero, which simplifies the analy- across the terraces. The overall interface plane or habit plane sis because no LId need be invoked [5]. We demonstrate that (HP)deviates from the terrace plane(TP) because disconnec- the two approaches lead to the same HP prediction in a sp tions, unlike LID, exhibit"overlap"step heights, h [3]. Dis- cial crystallographic circumstance, referred to as the congruent connection motion in this manner is diffusionless provided the case, but differ systematically otherwise. Martensitic transfor- TP and Burgers vector/step height couple of the disconnections, mations in ZrO2 [8] and Ti [9] are used for illustration; one b, h, satisfy certain criteria [4]. In addition, it also leads to a principal strain is zero in the former and is small in the latter transformation displacement and is hence consistent with exper- Transmission electron microscopy (TEM) has been used to study imental observations of martensitic transformations in metals the parent/martensite interface structure in these materials [1, 8 d ceramics [5]. Moreover, the TM is adaptable in the sense and both were observed to consist of terraces and disconnec tions as imagined in the tM. Furthermore the magnitude of the coherency strain on the terrace plane is very similar in the two Corresponding author. Tel. +44 151 794 4660: fax: +44 151 4675 materials, whereas the b, h values for the disconnections differ, E-lmail address: r.c. pond @liv. ac uk(RC. Pond) and hence comparison of the two cases is valuable 0921-5093/S-see front matter e 2006 Elsevier B doi:10.1016/msea.200602.132
Materials Science and Engineering A 438–440 (2006) 109–112 Kinematic and topological models of martensitic interfaces R.C. Pond a,∗, X. Ma a, J.P. Hirth b a Department of Engineering, University of Liverpool, Brownlow Hill, Liverpool L69 3BX, UK b 114 E. Ramsey Canyon Road, Hereford, AZ 85615, USA Received 19 April 2005; received in revised form 12 January 2006; accepted 2 February 2006 Abstract A new model of martensitic transformations has been presented recently where the habit plane consists of segments of coherent terraces reticulated by an array of line defects. These defects accommodate any coherency strains present and the transformation is effected by lateral motion of disconnections across the terraces. Provided the terraces and line defects satisfy certain criteria, the mechanism of transformation is conservative. Moreover, the topological parameters of the defects (Burgers vectors and step heights) can be determined rigorously enabling the overall habit plane inclination and orientation relationship to be determined. The objective of the present paper is to compare predictions for the habit planes according to this topological method with those of the classical theories in the case of transformations in ZrO2 and Ti. The two approaches lead to disparate solutions, and the origin and magnitude of the disparities are elucidated. Unlike habit planes in the classical model, which are invariant planes, topological ones are misfit-relieved semi-coherent configurations. © 2006 Elsevier B.V. All rights reserved. Keywords: Martensitic phase transformation; Interface structure; Martensitic crystallography 1. Introduction A new model of martensitic transformations was presented recently [1]; this model is known as the topological model (TM) because the interface structure is comprised of coherent terrace segments reticulated by an array of line defects. In general, this array has two sets of defects, crystal or twinning dislocations (lattice-invariant deformation or LID) and transformation dislocations or disconnections [2]. The array accommodates any coherency strains and also provides the mechanism of transformation through synchronous lateral motion of disconnections across the terraces. The overall interface plane or habit plane (HP) deviates from the terrace plane (TP) because disconnections, unlike LID, exhibit “overlap” step heights, h [3]. Disconnection motion in this manner is diffusionless provided the TP and Burgers vector/step height couple of the disconnections, b, h, satisfy certain criteria [4]. In addition, it also leads to a transformation displacement and is hence consistent with experimental observations of martensitic transformations in metals and ceramics [5]. Moreover, the TM is adaptable in the sense ∗ Corresponding author. Tel.: +44 151 794 4660; fax: +44 151 794 4675. E-mail address: r.c.pond@liv.ac.uk (R.C. Pond). that coherency strains may be partitioned unequally between the parent and product crystals, or incompletely accommodated [4]. Clearly, the physical basis of the TM is different from that of the Phenomenological Theory of Martensite Crystallography (PTMC) [6,7]. In general, predictions for the HP, orientation relationship (OR), and transformation displacement are not expected to be identical according to the two approaches [1]. The objective of the present paper is to elucidate the origin of discrepancies between HP orientations predicted by the TM and PTMC. For clarity, we consider transformations where one of the principal strains is zero, which simplifies the analysis because no LID need be invoked [5]. We demonstrate that the two approaches lead to the same HP prediction in a special crystallographic circumstance, referred to as the congruent case, but differ systematically otherwise. Martensitic transformations in ZrO2 [8] and Ti [9] are used for illustration; one principal strain is zero in the former and is small in the latter. Transmission electron microscopy (TEM) has been used to study the parent/martensite interface structure in these materials [1,8] and both were observed to consist of terraces and disconnections as imagined in the TM. Furthermore, the magnitude of the coherency strain on the terrace plane is very similar in the two materials, whereas the b, h values for the disconnections differ, and hence comparison of the two cases is valuable. 0921-5093/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.msea.2006.02.132
R.C. Pond et al. /Materials Science and Engineering A 438-440(2006)109-112 5139 11364 Fig. 1. Schematic illustration of parent and product lattices viewed along eg: (a)orthorhombic and(double)monoclinic cells for ZrO2, and(b) bcc and hcp cells for Ti. Unit sphere and deformation ellipsoids for(c)ZrO2 and(d) Ti; the principal axes and habit plane inclinations are shown 2. The phenomenological and topological models transferred from one crystal to the other, some being sheared into their new locations and the remainder shuffling [5]. How The central proposition of the PTMc is that the ever, because the densities of the two crystals are not generally parent-martensite interface is an invariant plane(IP)of the total equal, this process does not correspond to b being parallel to the shape transformation [5-7]. The classical theories are algorithms TP analogously to the case of dislocations in single crystals 3] for finding this plane given the lattice deformation, correspon- In the cases of Zro2 and Ti, the horizontal planes in Fig. I(a and dence and mode of LID. In other words, no strain arises in either b), respectively, are candidate TPs when coherently strained, as of the crystals because all vectors parallel to the HP remain depicted in Fig. 2(a and b). Moreover, disconnections with b invariant in the transformation. The IP can be found by iden- as indicated in Fig. 2, with magnitudes defined in the reference tifying any two non-parallel invariant vectors. If one princi coherent interface, would traverse these terraces conservatively strain of the lattice deformation is zero, the associated Eigenvec- a more detailed impression of the formation of the disconnec tore constitutes one invariant vector, so the solution for the IP tion for Ti, indicating the step height h, is shown in Fig. 3. The becomes the 2-D problem of finding an invariant vector vi per- coordinate frame used in Figs. 2 and 3 is referred to as the terrace endicular to ei after the requisite rigid-body rotation Ris carried frame out The IP solutions for ZrO2 and Ti are depicted in Fig. 1; the Now the spacing of disconnections necessary to accom- unit cells of the parent(orthorhombic)and product(monoclinic) modate the misfit strains can be calculated. Following Olson lattices are shown in Fig. 1(a) for ZrO2 projected along ei, andand Cohen [10], the coherency strain can be modelled as similarly for Ti(B and a). The corresponding deformation ellip- an array of dislocations, as indicated in Fig. 4(a); the total soids are depicted in Fig. I(c and d); for ZrO2 the inclination Burgers vector of these defects per unit length of ter- of the IP to the horizontal is 13.529 and r(with respect to the race, bey, is equal numerically to the total strain which is onfiguration shown in Fig. I(a))is a clockwise rotation(of the given by (e(a)yy -e()yy)(l+e(a)yy +e(B)yy+e(a)yye(B)yy) monoclinic crystal) about ei by 0.035, while for Ti these are where e(B)yy represents the 11.364 and 0.528(of a)anticlockwise, respectively. orthorhombic/B crystal parallel to y, and e(a)yy signifies the Using the TM, one first has to identify a candidate TP and compression of the monoclinic/o crystal in this direction: the disconnection that satisfy the criteria for diffusionless trans- total misfit strain on the terrace parallel to y is hereafter written formation [4]. In fact, the criteria are rarely satisfied for dis- as Eyy. When both crystals are equally strained, the magnitude of onnections in general inter-phase interfaces, but solutions can Eyy is 3. 84 and 3.80% for ZrO2 and Ti, respectively. As shown in be found, for example, in coherent interfaces. When a discon- Figs. 2 and 3, disconnections have Burgers vector components nection moves conservatively across a terrace plane, atoms are b, and bx, in addition to their step character. Thus, introducing
110 R.C. Pond et al. / Materials Science and Engineering A 438–440 (2006) 109–112 Fig. 1. Schematic illustration of parent and product lattices viewed along ei; (a) orthorhombic and (double) monoclinic cells for ZrO2, and (b) bcc and hcp cells for Ti. Unit sphere and deformation ellipsoids for (c) ZrO2 and (d) Ti; the principal axes and habit plane inclinations are shown. 2. The phenomenological and topological models The central proposition of the PTMC is that the parent–martensite interface is an invariant plane (IP) of the total shape transformation [5–7]. The classical theories are algorithms for finding this plane given the lattice deformation, correspondence and mode of LID. In other words, no strain arises in either of the crystals because all vectors parallel to the HP remain invariant in the transformation. The IP can be found by identifying any two non-parallel invariant vectors. If one principal strain of the lattice deformation is zero, the associated Eigenvector ei constitutes one invariant vector, so the solution for the IP becomes the 2-D problem of finding an invariant vector vi perpendicular to ei after the requisite rigid-body rotation R is carried out. The IP solutions for ZrO2 and Ti are depicted in Fig. 1; the unit cells of the parent (orthorhombic) and product (monoclinic) lattices are shown in Fig. 1(a) for ZrO2 projected along ei, and similarly for Ti ( and ). The corresponding deformation ellipsoids are depicted in Fig. 1(c and d); for ZrO2 the inclination of the IP to the horizontal is 13.529◦ and R (with respect to the configuration shown in Fig. 1(a)) is a clockwise rotation (of the monoclinic crystal) about ei by 0.035◦, while for Ti these are 11.364◦ and 0.528◦ (of ) anticlockwise, respectively. Using the TM, one first has to identify a candidate TP and disconnection that satisfy the criteria for diffusionless transformation [4]. In fact, the criteria are rarely satisfied for disconnections in general inter-phase interfaces, but solutions can be found, for example, in coherent interfaces. When a disconnection moves conservatively across a terrace plane, atoms are transferred from one crystal to the other, some being sheared into their new locations and the remainder shuffling [5]. However, because the densities of the two crystals are not generally equal, this process does not correspond to b being parallel to the TP analogously to the case of dislocations in single crystals [3]. In the cases of ZrO2 and Ti, the horizontal planes in Fig. 1(a and b), respectively, are candidate TPs when coherently strained, as depicted in Fig. 2(a and b). Moreover, disconnections with b as indicated in Fig. 2, with magnitudes defined in the reference coherent interface, would traverse these terraces conservatively; a more detailed impression of the formation of the disconnection for Ti, indicating the step height h, is shown in Fig. 3. The coordinate frame used in Figs. 2 and 3 is referred to as the terrace frame. Now the spacing of disconnections necessary to accommodate the misfit strains can be calculated. Following Olson and Cohen [10], the coherency strain can be modelled as an array of dislocations, as indicated in Fig. 4(a); the total Burgers vector of these defects per unit length of terrace, bcy, is equal numerically to the total strain which is given by (e()yy − e()yy)(1 + e()yy + e()yy + e()yye()yy) −1, where e()yy represents the uniaxial expansion of the orthorhombic/ crystal parallel to y, and e()yy signifies the compression of the monoclinic/ crystal in this direction: the total misfit strain on the terrace parallel to y is hereafter written as εyy. When both crystals are equally strained, the magnitude of εyy is 3.84 and 3.80% for ZrO2 and Ti, respectively. As shown in Figs. 2 and 3, disconnections have Burgers vector components by and bz, in addition to their step character. Thus, introducing
R C. Pond et al/ Materials Science and Engineering A 438-440(2006)109-112 Fig. 2. Schematic view of coherently strained terraces for(a)ZrOz and(b) Ti: disconnections with the indicated b could move conservatively across these terraces. per unit length is equal(and opposite) to bcy will quench the strain parallel to y on the terrace plane as well as parallel to y on the hp in addition , r will be null since there is no residual Burgers vector content parallel to z or z. In real systems b, is generally finite, so different HP predic tions are anticipated using the PtMc and TM. However, since Fig 3. Schematic illustration of the formation of a disconnection in the ti ter- e. In the terrace coordinate frame (see Fig. 4a). the disconnections b has omponents b, and b sufficient disconnections so that the sum of b, is equal(and oppo- site)to bey will not remove all the coherency strains because the staggered array of b components remains. Pond et al. have set out the necessary procedure [1], and interested readers should refer to that work for full details. As indicated in Fig. 4(b), dis- onnections must be introduced onto the terrace with spacing A so that the residual Burgers vector components parallel to y/in b the primed HP frame(Fig. 4(b)) vanish. Residual components parallel to z produce a rotation R relative to the reference struc tures in Fig. 1, but this does not produce a long-range strain field. The spacing of the disconnections A and their step heights h determine the hP inclination 3. Congruent and disparate predictions of habit plane orientation approaches agree only when the component b or, in other words, when the inter-planar spacing of the ter race planes is the same in the parent and product, as shown Fig 4. Schematic illustrations of (a) coherency strain visualised as an aray dislocations with total Burgers vector equal to bey per unit length, (b)config in Fig 4(c). In this case, no misfit arises on disconnection risers, ration of disconnections (b, h)superimposed on the'coherency'array, bey,and and a disconnection array with spacing such that the sum of by (c)the congruent case where b,=0
R.C. Pond et al. / Materials Science and Engineering A 438–440 (2006) 109–112 111 Fig. 2. Schematic view of coherently strained terraces for (a) ZrO2 and (b) Ti; disconnections with the indicated b could move conservatively across these terraces. Fig. 3. Schematic illustration of the formation of a disconnection in the Ti terrace. In the terrace coordinate frame (see Fig. 4a), the disconnection’s b has components by and bz. sufficient disconnections so that the sum of by is equal (and opposite) to bcy will not remove all the coherency strains because the staggered array of bz components remains. Pond et al. have set out the necessary procedure [1], and interested readers should refer to that work for full details. As indicated in Fig. 4(b), disconnections must be introduced onto the terrace with spacing λ so that the residual Burgers vector components parallel to y in the primed HP frame (Fig. 4(b)) vanish. Residual components parallel to z produce a rotation R relative to the reference structures in Fig. 1, but this does not produce a long-range strain field. The spacing of the disconnections λ and their step heights h determine the HP inclination. 3. Congruent and disparate predictions of habit plane orientation The two approaches agree only when the component bz = 0, or, in other words, when the inter-planar spacing of the terrace planes is the same in the parent and product, as shown in Fig. 4(c). In this case, no misfit arises on disconnection risers, and a disconnection array with spacing such that the sum of by per unit length is equal (and opposite) to bcy will quench the strain parallel to y on the terrace plane as well as parallel to y on the HP. In addition, R will be null since there is no residual Burgers vector content parallel to z or z . In real systems bz is generally finite, so different HP predictions are anticipated using the PTMC and TM. However, since Fig. 4. Schematic illustrations of (a) coherency strain visualised as an array of dislocations with total Burgers vector equal to bcy per unit length, (b) configuration of disconnections (b, h) superimposed on the ‘coherency’ array, bcy, and (c) the congruent case where bz = 0
R.C. Pond et al. Materials Science and Engineering A 438-440(2006)109-112 HPs in crystalline solids are not invariant planes they are inter- facial terraces reticulated by an array of line defects that, with the PTMC appropriate spacing and orientation, lead to no long-range strain field. Moreover, where the TPs and defect parameters conform to certain criteria [4], the interface can migrate conservatively and effect a transformation displacement. One can impose a change from the TM to the PTMC habit plane by adding extrinsic dis- locations to the interface [1]. However, the HP would then be sessile and require climb to advance; only the tM habit plane can move conservatively. The two approaches are congruent when disconnection step risers exhibit no misfit. i.e. when b=0. In this circur -0150.1 0.05 disconnections act asanti-coherency defects as discussed by Olson and Cohen [10]any finite b, component would have Fig. 5. Plots showing the variation of HP inclination vs b /h(a)according to to be regarded as a'coherency'defect using this terminology) the TM and the ptmc for ZrO, and Ti: the arrows indicate the actual values ball(a)for the t When b2=0, the congruent habit plane inclination, 0, is equal to tan-(hEy/by)which is 13.513 for ZrO and 11.3790for the b2 component corresponds to an expansion normal to the Ti. The lower value for Ti arises because b /h is larger for this habit plane, its magnitude may be small for metals and ceram- material so fewer disconnections are needed to accommodate ics. Thus, relatively small discrepancies may be present for suc the coherency strain(for ZrO2, b, =0. 162 nm and h=1014 naterials. Nevertheless, it is instructive to artificially vary the and for T1, by=0.048 nm and h=0.256 nm). We note that the agnitude of b to reveal its role, especially if the misfit the terrace is held constant. As shown in Fig. 3. b is egual to defects, 2b, 2h, but this would not affect the inclination of the h(B)-h(a), where h(B) and h(a)are the orthorhombic/B and interface or the transformation displacement. Also, the discon- monoclinic/a crystal surface step heights, respectively. The vari- ation of hp inclination according to the tm and the ptmc is he magnitude of by is almost four times that for Ti. plotted for ZrO2 and Ti in Fig. 5 as a function of the dimen The disparity between the TM and PTMC solutions Sionless parameter b / h(a)(n. b. the magnitude of h(B)has been as b2 I increases. For positive values of b /h(a), the component of b. resolved onto the hp assists in the accommodation of wo predictions for each material are congruent when b, =0 and coherency strain along this plane, leading to lower values of disparate for finite values. The value of b h(a)for the actual HP inclination being predicted by the TM than the PTMC,as ZrO and Ti systems are-0002 and 0.047, respectively; thus seen in Fig. 5. Conversely, accommodation of coherency strain the HP discrepancy is only 0.041 for the ceramic but -1.0320 is exacerbated for negative values for the metal, as indicated by the arrows in Fig. 5 References 4. Discussion [1R. C. Pond, S. Celotto, J. P. Hirth, Acta Mater. 51(2003)5385. This study has revealed the origin of discrepancies between (2)J. P Hirth, J Phys. Chem. Sol. 55(1994)985 3] J. P Hirth, R C. Pond, Acta Mater. 44(1996)4749 the HP predictions for ZrO2 and Ti using the TM and the PTMC for transformations where one principal strain is zero. In the [5] J.W. Christian, The Theory of Transformations in Metals and Alloys, Perg- latter theory the two materials are effectively considered to be amon Press. Oxford. UK. 2002 ontinua, so'matching'of irrational vectors at the HP is admissi- [6] M.S. Wechsler, D.S. Lieberman, T.A. Read, Trans. AIME 197(1953) ble For the TM the atomic nature of the materials is represented [7 J.S. Bowles, J.K. MacKenzie, Acta Metall. 2(1954)129, 138, 224 and'matching'along the TP is quantified in terms of coherency [81-w. Chen. Y.-H. Chiao, Acta Metall. 33(1985)1827 strain and rigorously determined values of the b, h parameters of [9] C. Hammond, PM. Kelly, Acta Metall. 17(1969)869 disconnections. Thus, we conclude that, in general, martensitic [10] G.B. Olsen, M. Cohen, Acta Metall. 27(1979)1907
112 R.C. Pond et al. / Materials Science and Engineering A 438–440 (2006) 109–112 Fig. 5. Plots showing the variation of HP inclination vs. bz/h() according to the TM and the PTMC for ZrO2 and Ti; the arrows indicate the actual values of bz/h() for the two materials. the bz component corresponds to an expansion normal to the habit plane, its magnitude may be small for metals and ceramics. Thus, relatively small discrepancies may be present for such materials. Nevertheless, it is instructive to artificially vary the magnitude of bz to reveal its role, especially if the misfit on the terrace is held constant. As shown in Fig. 3, bz is equal to h() − h(), where h() and h() are the orthorhombic/ and monoclinic/ crystal surface step heights, respectively. The variation of HP inclination according to the TM and the PTMC is plotted for ZrO2 and Ti in Fig. 5 as a function of the dimensionless parameter bz/h() (n.b. the magnitude of h() has been varied to change bz, while h() was held fixed). As expected, the two predictions for each material are congruent when bz = 0 and disparate for finite values. The value of bz/h() for the actual ZrO2 and Ti systems are −0.002 and 0.047, respectively; thus the HP discrepancy is only 0.041◦ for the ceramic but −1.032◦ for the metal, as indicated by the arrows in Fig. 5. 4. Discussion This study has revealed the origin of discrepancies between the HP predictions for ZrO2 and Ti using the TM and the PTMC for transformations where one principal strain is zero. In the latter theory the two materials are effectively considered to be continua, so ‘matching’ of irrational vectors at the HP is admissible. For the TM the atomic nature of the materials is represented, and ‘matching’ along the TP is quantified in terms of coherency strain and rigorously determined values of the b, h parameters of disconnections. Thus, we conclude that, in general, martensitic HPs in crystalline solids are not invariant planes; they are interfacial terraces reticulated by an array of line defects that, with the appropriate spacing and orientation, lead to no long-range strain field. Moreover, where the TPs and defect parameters conform to certain criteria [4], the interface can migrate conservatively and effect a transformation displacement. One can impose a change from the TM to the PTMC habit plane by adding extrinsic dislocations to the interface [1]. However, the HP would then be sessile and require climb to advance; only the TM habit plane can move conservatively. The two approaches are congruent when disconnection step risers exhibit no misfit, i.e. when bz = 0. In this circumstance, disconnections act as ‘anti-coherency’ defects as discussed by Olson and Cohen [10] (any finite bz component would have to be regarded as a ‘coherency’ defect using this terminology). When bz = 0, the congruent habit plane inclination, θ, is equal to tan−1(hεyy/by) which is 13.513◦ for ZrO2 and 11.379◦ for Ti. The lower value for Ti arises because by/h is larger for this material so fewer disconnections are needed to accommodate the coherency strain (for ZrO2, by = 0.162 nm and h = 1.014 nm, and for Ti, by = 0.048 nm and h = 0.256 nm). We note that the disconnections observed in TEM studies of Ti [4] are ‘double’ defects, 2b, 2h, but this would not affect the inclination of the interface or the transformation displacement. Also, the disconnection spacing λ would be considerably larger in ZrO2 because the magnitude of by is almost four times that for Ti. The disparity between the TM and PTMC solutions increases as |bz| increases. For positive values of bz/h(), the component of bz resolved onto the HP assists in the accommodation of coherency strain along this plane, leading to lower values of HP inclination being predicted by the TM than the PTMC, as seen in Fig. 5. Conversely, accommodation of coherency strain is exacerbated for negative values. References [1] R.C. Pond, S. Celotto, J.P. Hirth, Acta Mater. 51 (2003) 5385. [2] J.P. Hirth, J. Phys. Chem. Sol. 55 (1994) 985. [3] J.P. Hirth, R.C. Pond, Acta Mater. 44 (1996) 4749. [4] R.C. Pond, S. Celotto, Int. Mater. Rev. 48 (2003) 225. [5] J.W. Christian, The Theory of Transformations in Metals and Alloys, Pergamon Press, Oxford, UK, 2002. [6] M.S. Wechsler, D.S. Lieberman, T.A. Read, Trans. AIME 197 (1953) 1503. [7] J.S. Bowles, J.K. MacKenzie, Acta Metall. 2 (1954) 129, 138, 224. [8] I.-W. Chen, Y.-H. Chiao, Acta Metall. 33 (1985) 1827. [9] C. Hammond, P.M. Kelly, Acta Metall. 17 (1969) 869. [10] G.B. Olsen, M. Cohen, Acta Metall. 27 (1979) 1907.