Y Wang, A G Khachaturyan/Materials Science and Engineering A 438-440(2006)55-63 The microscopic phase field model has been applied to study the energy and sliding resistance of Peierls grain boundaries (small angle grain boundaries consisting of regular dislocation networks)[16] as a function of misorientation. It has also been applied to study the shearing of y precipitates[66]in an effort to +Layer-1 understand the effect of microstructural characteristics on defor- (variant 1) o For a sinmple fcc-hcp MT, the microscopic phase field model ation mechanisms observed in Ni-based superalloy 12 of dislocations with ab initio data of the GSF energy is bein applied directly to study the nucleation mechanisms associated with dissociation of a group of pre-existing lattice dislocations as suggested by Olson and Cohen [54]. Fo (variant 2) r more co MTs such as the fcc-bcc lattice rearrangment, however, the faulting mechanisms proposed for the formation of a marten- volume site embryo involve multiple sets of dissociated dislocations distributed among multiple adjacent atomic planes [55]. While ig. 3. Relaxed shapes of mi ocapon oops at iteren ixed vo umes tod a scrutiny of atomistic simulations, more straightforward descrip tions have been proposed [17](a 2D attempt using the Element Free-Galerkin method can be found in (59). In particular, a new Bain path as model inputs. The elastic energy of the mislocation elementary defect at the atomic scale called mislocation was is calculated using the KS theory [1,6,7]. The total energies of introduced [17. A mislocation is defined as the smallest ele eneous (single variant) and a heterogeneous(two twin- mentary defect for MTs and further decomposition of it into a related variants) two-layer fault of the Bain mislocations are multiple set of dislocations is unnecessary. Forexample, the Bain shown in Fig. 7. It can be seen clearly that the activation energy for the formation of an internally twinned embryo is much lower listortion could be regarded as a mislocation of the fcc- b than that of a single variant embryo. The relaxed shapes of transformation whose core contains both dilatational and shear strain components. The concept of mislocations may provide a the mislocation loops at fixed volumes for the heterogeneous powerful way of describing martensite nucleation, growth and two-layer fault are shown in Fig. 8, which seems to be dif impingement using the simple laws of interactions and reactions ferent from the oblate spheroidal embryo assumed in literature between these elementary defects The core profile of a mislocation, the activation barrier of the minimum energy pathway, and the morphology of a critical 4 Summary nucleus consisting of mislocations have been investigated [17] by using the microscopic phase field model [13, 16], with ab Phase field modeling capabilities developed for marten- tio calculations of the crystalline energy corresponding to the sitic transformations at multiple length scales are reviewed. At the mesoscopic length scales, the models are shown to be able to describe arbitrary multivariant and multiphase strain- accommodating morphological patterns produced by marten- sitic transformations of arbitrary stress-free transformation strains without any a priori assumptions on the transforma- tion paths. Homogeneous nucleation under large undercooling collective process or a correlated process assisted by autocatal- ysis. The autocatalytic effect also plays an important role in the rowth process leading to a herringbone structure consisting of djacent internally twinned plates of invariant plane habits. The models ability to describe martensitic transformations in a poly- crystalline material under external stresses is also demonstrated At the microscopic level, the newly developed phase field model of dislocation core structure and phase field model of miso- 20253035 cation allow for quantitative characterization of the minimum energy pathways and critical nucleus configurations with state- of-the-art ab initio and atomistic calculations as inputs. These Fig.7. Total energy of a homogeneous(single variant)(+)and a heterogeneous multi-scale modeling capabilities developed within a common loops calculated by the microscopic phase field method [171. Here u is shear phase field framework could offer an opportunity to develop modulus, d is the interplanar distance of (110)twin plane. The effective radius unprecedented new understanding of martensitic transforma- is given as R=Vv/ad, where Vis the volume of the nucleus tons62 Y. Wang, A.G. Khachaturyan / Materials Science and Engineering A 438–440 (2006) 55–63 The microscopic phase field model has been applied to study the energy and sliding resistance of Peierls grain boundaries (small angle grain boundaries consisting of regular dislocation networks) [16] as a function of misorientation. It has also been applied to study the shearing of γ precipitates[66] in an effort to understand the effect of microstructural characteristics on defor￾mation mechanisms observed in Ni-based superalloys. For a simple fcc–hcp MT, the microscopic phase field model of dislocations with ab initio data of the GSF energy is being applied directly to study the nucleation mechanisms associated with dissociation of a group of pre-existing lattice dislocations as suggested by Olson and Cohen [54]. For more complicated MTs such as the fcc–bcc lattice rearranegment, however, the faulting mechanisms proposed for the formation of a marten￾site embryo involve multiple sets of dissociated dislocations distributed among multiple adjacent atomic planes [55]. While the activation energy of such a faulting pathway is awaiting for scrutiny of atomistic simulations, more straightforward descrip￾tions have been proposed [17] (a 2D attempt using the Element￾Free-Galerkin method can be found in [59]). In particular, a new elementary defect at the atomic scale called mislocation was introduced [17]. A mislocation is defined as the smallest ele￾mentary defect for MTs and further decomposition of it into a multiple set of dislocations is unnecessary. For example, the Bain distortion could be regarded as a mislocation of the fcc→bcc transformation whose core contains both dilatational and shear strain components. The concept of mislocations may provide a powerful way of describing martensite nucleation, growth and impingement using the simple laws of interactions and reactions between these elementary defects. The core profile of a mislocation, the activation barrier of the minimum energy pathway, and the morphology of a critical nucleus consisting of mislocations have been investigated [17] by using the microscopic phase field model [13,16], with ab ini￾tio calculations of the crystalline energy corresponding to the Fig. 7. Total energy of a homogeneous (single variant) (+) and a heterogeneous (two twin-related variants) (×) two-layer fault consisting of Bain mislocation loops calculated by the microscopic phase field method [17]. Here μ is shear modulus, d is the interplanar distance of (110) twin plane. The effective radius ¯ is given as R = V/πd, where V is the volume of the nucleus. Fig. 8. Relaxed shapes of mislocation loops at different fixed volumes for a heterogeneous two-layer fault simulated by the microscopic phase field model [17]. Bain path as model inputs. The elastic energy of the mislocation is calculated using the KS theory [1,6,7]. The total energies of a homogeneous (single variant) and a heterogeneous (two twin￾related variants) two-layer fault of the Bain mislocations are shown in Fig. 7. It can be seen clearly that the activation energy for the formation of an internally twinned embryo is much lower than that of a single variant embryo. The relaxed shapes of the mislocation loops at fixed volumes for the heterogeneous two-layer fault are shown in Fig. 8, which seems to be dif￾ferent from the oblate spheroidal embryo assumed in literature [67]. 4. Summary Phase field modeling capabilities developed for marten￾sitic transformations at multiple length scales are reviewed. At the mesoscopic length scales, the models are shown to be able to describe arbitrary multivariant and multiphase strain￾accommodating morphological patterns produced by marten￾sitic transformations of arbitrary stress-free transformation strains without any a priori assumptions on the transforma￾tion paths. Homogeneous nucleation under large undercooling generates multivariant polytwinned embryos through either a collective process or a correlated process assisted by autocatal￾ysis. The autocatalytic effect also plays an important role in the growth process leading to a herringbone structure consisting of adjacent internally twinned plates of invariant plane habits. The model’s ability to describe martensitic transformations in a poly￾crystalline material under external stresses is also demonstrated. At the microscopic level, the newly developed phase field model of dislocation core structure and phase field model of mislo￾cation allow for quantitative characterization of the minimum energy pathways and critical nucleus configurations with state￾of-the-art ab initio and atomistic calculations as inputs. These multi-scale modeling capabilities developed within a common phase field framework could offer an opportunity to develop unprecedented new understanding of martensitic transforma￾tions