Effect of elastic interaction on nucleation: II. Implementation of strain energy of nucleus formation in the phase field method

In paper I of this series, the coherency strain energy associated with nucleus formation in an elastically anisotropic crystal of an arbitrary pre-existing microstructure was presented in a close form as a linear function of the nucleus volume, which can be incorporated directly into the classical nucleation theory. In this paper we discuss incorporation of the coherency strain energy into the explicit nucleation algorithm of the phase field method, a formulation based on the classical nucleation theory, with an emphasis on the difference between conserved and non-conserved fields. In general, the local nucleation driving force is a function of both local field(s) and local stress, but for a conserved system with linear dependence of transformation strain on concentration the local nucleation driving force reduces to a function of local concentration only. To compare quantitatively the explicit nucleation algorithm with the Langevin force approach, a cubic → tetragonal transformation is considered in two dimensions. With the same material parameters and starting microstructure, a spatial correlation between nuclei and the pre-existing misfitting particles is achieved consistently in the two approaches on the same time and length scales. With the effect of local stress on nucleation incorporated, the explicit nucleation algorithm provides an efficient means for studying stress-induced nucleation phenomena in coherent solids, which are responsible for the formation of many self-assembled morphological patterns during coherent transformations.