Regularized Anisotropic Motion-By-Curvature in Phase-Field Theory

The kinetic equation for anisotropic motion-by curvature, largely used to model the dynamics of grain boundaries and interfaces, is ill-posed when the surface energy is strongly anisotropic. In this case, corners or edges are present on the Wulff shape, which span a range of missing orientations. In the sharp-interface problem the surface energy is augmented with a curvature-dependent term that rounds the corners and regularizes the dynamic equations. This introduces a new length scale in the problem, the corner size. In phase-field theory, an approximation of the Willmore energy is often added to regularize the model. We discuss the convergence of the Allen-Cahn version of the regularized phase-field model toward the sharp-interface theory for strongly anisotropic motion-by-curvature, in two and three dimensions, for large corner size with respect to interface thickness. We also study the opposite limit, for corner size smaller than the interface width, and show that the shape of corners differs from the sharp-interface picture. However, we find that the phase transition at the interface is preserved and presents the same properties than the classical problem. This demonstrates that regularization also operates in the intermediate regime, i.e. for corner size and interface width of same order, that is numerically more attractive and allows for simulations in three-dimensions. As an illustration, we investigate the dynamics of the faceting instability, when initially unstable surfaces decompose into stable facets. Finally, we study nucleation of crystal surfaces in a two-phase system.