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)1907112 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 ceram￾ics. 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 vari￾ation of HP inclination according to the TM and the PTMC is plotted for ZrO2 and Ti in Fig. 5 as a function of the dimen￾sionless 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 admissi￾ble. 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 inter￾facial 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 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 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 discon￾nection 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, Perg￾amon 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.