Growth modes and models for smooth and rough surfaces

A brief survey of general growth modes and models for smooth vs. rough surfaces is given, with special reference to the growth rate in the different regimes (continuous growth, layer growth, step flow, spiral growth, dendritic growth). In particular, two models, mapping surfaces onto the 6-vertex model, are studied. Under one mapping the surface steps are identified with the lines occurring in the line representation; the other mapping is that of van Beijeren. Master equations are written for finite-size versions of the models, and solved analytically (for extremely small sizes), or by Monte Carlo simulation, using Glauber's rules for the growth, evaporation and diffusion probabilities. For the first model the steady growth rate is shown to be smaller than the initial growth rate for an equiprobable ensemble of configurations. In the second model both the growth mode and the surface structure depend explicitly on temperature T and supersaturation Δμ. The surface is rough at all temperatures, but the nature of the roughness changes: at high T, the roughness is logarithmic if Δμ is small, but a cross over takes place to a power-law behaviour when Δμ increases.