Interface dynamics in a mean-field lattice gas model

A mean-field approximation of stochastic lattice gas models with attractive nearest-neighbour interaction allows to write generalized Cahn-Hilliard equations. We use such equation to perform numerical simulations in two dimensions of the growth of regular "snowflakes." It naturally shows curvature and kinetic effects at the interface as assumed by the classic phenomenological equations of dendritic growth. In addition, we find solute trapping. The dendrite tips are stabilized by the Gibbs-Thomson boundary condition. We calculate the surface tension and show that it has the expected angular variation. We compare our model to other microscopic growth models and the phase-field models and discuss the influence of noise. Stationary interface states advancing with constant velocity are also investigated. We present numerical results as well as a continuum approximation which gives an analytic expression for the shape correction in the limit of not too large interface velocities. We observe departure from local equilibrium at the interface and solute trapping. The associated kinetic coefficients are calculated. The two effects are found to be related. We finally give an expression for the interface mobility and derive a relation between this mobility and the kinetic coefficients. We also examine droplets and stripes of unstable A-B compounds with vacancies (ABv model) immersed in a stable vapor. We observe the appearance of ordered structures starting from the surface and propagating into the bulk. We calculate the characteristic lengths and the propagation velocities of these patterns using a linear stability analysis. We show that the thickness of the ordered layer depends on the model parameters and the strength of the initial noise.