Spinodal decomposition of an ABv model alloy: Patterns at unstable surfaces

We develop mean-field kinetic equations for a lattice gas model of a binary alloy with vacancies (ABv model) in which diffusion takes place by a vacan cy mechanism. These equations are applied to the study of phase separation of finite portions of an unstable mixture immersed in a stable vapor. Due t o a larger mobility of surface atoms, the most unstable modes of spinodal d ecomposition are localized at the vapor-mixture interface. Simulations show checkerboard-like structures at the surface or surface-directed spinodal w aves. We determine the growth rates of bulk and surface modes by a linear s tability analysis and deduce the relation between the parameters of the mod el and the structure and length scale of the surface patterns. The thicknes s of the surface patterns is related to the concentration fluctuations in t he initial state.