S. Deville et al. Acta Materialia 52(2004 )5709-5721 energy change related to the transformation has been ex- The total area of t/m interfaces in the partially trans pressed in the following way for a microcrystal of radius formed arrangement can also be expressed by Apartial(hunt)=1.385(hit +hunt) 4 △Gm=3(△Ghcm+△Gd+△Gm) Equation (1) becomes thus +4πr ∑ (1)△Gm=0416(-hm)△Gdm where AGa-m is the total Gibbs energy change, AGchem +1.385(/a+h2a)△Shm is the volumic free energy variation, A Gail and AGshr The only remaining terms are the chemical free en are the free energy variations associated to the dilata ergy variation (in volume and surface). Hence, it is pos tional(dil) and shear(shr)components of the trans- sible quantifying the total free energy change as a formation, 2ASsurface represents the sum of the unction of the variants height, by using values from terms taking into account the surface effects(surface he literature for the different terms of equation (4). free energy variation, creation of twin boundaries △ Gch=-285.100×(1-m/1448)Jm-3[27]and and creation of interfaces between the t and m ASchem =0.36 J m-2[28]. The plot of the total free en phases), and Ac,ev corresponds to the energy spent ergy change as a function of the variants height provides in the formation of new surface due to microcracks the expected critical size(Fig. 11). The value of ASche opening (Ac is the ted by the cracks, 7c the was measured in the case of incoherent precipitates, fracture energy per unit area and V the microcrystal and a critical size for the variants(htot)of 6 nm(for volume).Based on the current observations, at least hunt =0)is obtained here in that case. Though no values for transformation occurring at the surface, this ap- were measured for the case of coherent precipitates, we proach is not very realistic. The present analysis leads could expect the surface free energy to decrease of one indeed to considerable modifications of this equation, range of order for the case of coherent precipitates, by in particular when the first stages of the transforma comparison with the behavior of metallic materials. In tion are considered at case, the critical size falls down to l nm (Fig. I1) These values should be compared with the ones ob- The ABCI correspondence choice leads to the cancel lation of the terms related to the dilatation and shear not accommodated. Since the contribution of these com omponents of the transformation strains, since all ponent was very important, the expected critical size these transformations strains are accommodated and relaxed by the relief displacement outside of the dependent of the temperature), was greater by several ranges of order (typically 1-10 um)[8], and in good with the Transformation does not occur preferentially at the By using the outputs of our analysis, it was possible boundaries, since the nucleation stage is con- modifying the classical thermodynamic theory to take trolled by the crystallographic orientation of the to account the strain accommodation the effect of grains with respect to the free surface. All the created he free surface and the nature of the t-n interfaces interfaces (junction planes, habit planes)are therefore The calculated critical size is much lower than the ones coherent, at least during the first stages, which does obtained previously, and found so low that no critical not lead to the formation of additional stresses. all the terms related to surface effects, but the surface free 3,E18 coherent No microcrack is formed during the first stage of the transformation Finally, the transformed parts of the volume are not pherical at all, but present a pyramidal configura tion. The respective volume and surface area of a par tially transformed part can be calculated exactly with calculations outputs. he 1E18 Let us consider hunt as being the height of the transformed zone(see Fig. I(b)), and h,ot the height variants height (m) of the transformed zone. The volume Partial of a par Fig. Il. Total free energy change of a stack of four accommodated tially transformed part can be calculated by simple geo- variants as a function of the variants height. The change in energy was metric considerations as estimated for the case of coherent and incoherent f-m interface. The V partia (hunt )=0.416(hiot-hunt) critical size for transformation is much smaller in the case of a coherent interfaceenergy change related to the transformation has been ex￾pressed in the following way for a microcrystal of radius r: DGq–m ¼ 4 3 pr3 ð Þ DGchem þ DGdil þ DGshr þ 4pr2XDSsurface þ AcccV ; ð1Þ where DGq–m is the total Gibbs energy change, DGchem is the volumic free energy variation, DGdil and DGshr are the free energy variations associated to the dilata￾tional (dil) and shear (shr) components of the trans￾formation, PDSsurface represents the sum of the terms taking into account the surface effects (surface free energy variation, creation of twin boundaries and creation of interfaces between the t and m phases), and AcccV corresponds to the energy spent in the formation of new surface due to microcracks opening (Ac is the area created by the cracks, cc the fracture energy per unit area and V the microcrystal volume). Based on the current observations, at least for transformation occurring at the surface, this ap￾proach is not very realistic. The present analysis leads indeed to considerable modifications of this equation, in particular when the first stages of the transforma￾tion are considered: The ABC1 correspondence choice leads to the cancel￾lation of the terms related to the dilatation and shear components of the transformation strains, since all these transformations strains are accommodated and relaxed by the relief displacement outside of the free surface. Transformation does not occur preferentially at the grain boundaries, since the nucleation stage is con￾trolled by the crystallographic orientation of the grains with respect to the free surface. All the created interfaces (junction planes, habit planes) are therefore coherent, at least during the first stages, which does not lead to the formation of additional stresses. All the terms related to surface effects, but the surface free energy change, are therefore cancelled. No microcrack is formed during the first stage of the transformation. Finally, the transformed parts of the volume are not spherical at all, but present a pyramidal configura￾tion. The respective volume and surface area of a par￾tially transformed part can be calculated exactly with calculations outputs. Let us consider hunt as being the height of the untransformed zone (see Fig. 1(b)), and htot the height of the transformed zone. The volume Vpartial of a par￾tially transformed part can be calculated by simple geo￾metric considerations as V partialð Þ¼ hunt 0:416 h3 tot
h3 unt
: ð2Þ The total area of t/m interfaces in the partially trans￾formed arrangement can also be expressed by Apartialð Þ¼ hunt 1:385 h2 tot þ h2 unt
: ð3Þ Equation (1) becomes thus DGq–m ¼ 0:416 h3 tot
h3 unt
DGchem þ 1:385 h2 tot þ h2 unt
DSchem: ð4Þ The only remaining terms are the chemical free en￾ergy variation (in volume and surface). Hence, it is pos￾sible quantifying the total free energy change as a function of the variants height, by using values from the literature for the different terms of equation (4), i.e. DGchem =
285.106 · (1
T/1448) J m
3 [27] and DSchem = 0.36 J m
2 [28]. The plot of the total free en￾ergy change as a function of the variants height provides the expected critical size (Fig. 11). The value of DSchem was measured in the case of incoherent precipitates, and a critical size for the variants (htot) of 6 nm (for hunt = 0) is obtained here in that case. Though no values were measured for the case of coherent precipitates, we could expect the surface free energy to decrease of one range of order for the case of coherent precipitates, by comparison with the behavior of metallic materials. In that case, the critical size falls down to 1 nm (Fig. 11). These values should be compared with the ones ob￾tained when the shear and dilatational components are not accommodated. Since the contribution of these com￾ponent was very important, the expected critical size (dependent of the temperature), was greater by several ranges of order (typically 1–10 lm) [8], and in good agreement with the experimental observations. By using the outputs of our analysis, it was possible modifying the classical thermodynamic theory to take into account the strain accommodation, the effect of the free surface and the nature of the t–m interfaces. The calculated critical size is much lower than the ones obtained previously, and found so low that no critical -1,E-18 0,E+00 1,E-18 2,E-18 3,E-18 0,E+00 2,E-09 4,E-09 6,E-09 variants height (m) ∆Gt-m (J) incoherent coherent hc hc Fig. 11. Total free energy change of a stack of four accommodated variants as a function of the variants height. The change in energy was estimated for the case of coherent and incoherent t–m interface. The critical size for transformation is much smaller in the case of a coherent interface. S. Deville et al. / Acta Materialia 52 (2004) 5709–5721 5719