Evolution of crystal morphologies to equilibrium by surface diffusion with anisotropic surface free energy in three dimensions

We present a mathematical model for the evolution of finite crystal surface by surface diffusion with smooth anisotropic surface free energy. Employing this model we study the entire evolution path of a single crystal to equilibrium. We examine examples of simple cubic crystals with different levels of anisotropic surface free energy and different initial crystal configurations. We find that with a mildly anisotropic surface free energy, the crystal morphology is smooth in evolution and evolves to a unique equilibrium crystal shape. There are rapid reductions of the total surface energy and fast changes in the crystal morphology in the early evolution. In the early evolution, the amount of energy reduction with time follows closely to a power law with the exponent 14 when the initial surface is piecewise planar. With a severely anisotropic surface free energy, depending on the initial crystal configuration and surface free energy, edges, corners and faceting by hill-and-valley structures on crystal surface may occur. The equilibrium crystal shape approaches a polyhedron Wulff shape as the anisotropy in the surface free energy becomes more severe.