On phase-field modeling with a highly anisotropic interfacial energy

We report on phase-field approaches that allow for anisotropies sufficiently high so that the interface develops sharp corners due to missing crystallographic orientations. This implies the necessity of a regularization that enforces local equilibrium at the corners, and we use the method of Eggelston et al. (Physica D 150, 91 (2001)), generalized to arbitrary crystal symmetries and rotations of the crystalline axes. Two different anisotropic phase-field formulations are presented and discussed: The classical model that allows the interface to vary with orientation, and another more recent formulation that has a constant interface width. We develop an explicit finite-difference scheme that combines a two-step differentiation with a stagnation grid formulation. The presented numerical implementation is stable and accurate enough to account for odd crystal symmetries and high angle rotations of the initial crystalline orientation. Even in the case of highly anisotropic interfacial energies, both formulations show excellent agreement with the well-known Wulff construction of the equilibrium shape of a particle embedded in a matrix.