Interface evolution in three dimensions with curvature-dependent energy and surface diffusion: Interface-controlled evolution, phase transitions, epitaxial growth of elastic films

When the interfacial energy is a nonconvex function of orientation, the anisotropic-curvature-flow equation becomes backward parabolic. To overcome the instability thus generated, a regularization of the equation that governs the evolution of the interface is needed. In this paper we develop a regularized theory of curvature flow in three dimensions that incorporates surface diffusion and bulk-surface interactions. The theory is based on a superficial mass balance; configurational forces and couples consistent with superficial force and moment balances, a mechanical version of the second law that includes, via the configurational moments, work that accompanies changes in the curvature of the interfaced a constitutive theory whose main ingredient is a positive-definite, isotropic, quadratic dependence of the interfacial energy on the curvature tensor. Two special cases are investigated: (i) the inter-face is a boundary between bulk phases or grains, and (ii) the interface separates an elastic thin film bonded to a rigid substrate from a vapor phase whose sole action is the deposition of atoms on the surface.