Fick’s law for the diffusion of adatoms on crystal surfaces, below the roughening transition, was generalized so as to account for surface reconstruction. In this case, the material parameters varied spatially at the micro-scale, and the coarse graining for crystalline steps, via Taylor expansion, was not strictly applicable. By using elements of the theory of composites in one independent space dimension, the micro-scale description was homogenized so as to deduce the macroscopic adatom flux from step kinetics. This approach relied upon a multi-scale expansion for the adatom density. The effective surface diffusivity was determined by applying appropriate discrete averages of the micro-scale kinetic parameters.

Homogenization of Reconstructed Crystal Surfaces - Fick's Law of Diffusion. D.Margetis: Physical Review E, 2009, 79[5], 052601