Homogenization of composite vicinal surfaces: Evolution laws in 1+1 dimensions

We apply classical homogenization to derive macroscopic relaxation laws for crystal surfaces with distinct inhomogeneities at the microscale. The proposed method relies on a formal multiscale expansion in one spatial coordinate. This approach transcends the coarse graining applied previously via Taylor expansions. Our work offers an extension of the static homogenization formulated in a brief report [D. Margetis, Homogenization of reconstructed crystal surfaces: Fick’s law of diffusion, Phys. Rev. E 79 (2009) 052601] to account for surface evolution. The starting point is the Burton–Cabrera–Frank (BCF) model for the motion of line defects (steps) separating nanoscale terraces. We enrich this model with sequences of distinct material parameters, i.e., disparate diffusivities of adsorbed atoms (adatoms) across terraces, kinetic sticking rates at step edges, and step energy parameters for elastic-dipole interactions. Multiscale expansions for the adatom concentration and flux are used, with a slow diffusive time scale consistent with the quasi-steady regime for terrace diffusion. The ensuing macroscopic, nonlinear evolution laws incorporate averages of the microscale parameters.