Gibbs-Thomson formula for small island sizes: Corrections for high vapor densities

In this paper we report simulation studies of equilibrium features, namely, circular islands on model surfaces, using Monte Carlo methods. In particular, we are interested in studying the relationship between the density of vapor around a curved island and its curvature. The ''classical'' form of this relationship is the Gibbs-Thomson formula, which assumes that the vapor surrounding the island is an ideal gas. Numerical simulations of a lattice gas model, performed for various sizes of islands, do not fit very well to the Gibbs-Thomson formula. We show how corrections to this form arise at high vapor densities, wherein a knowledge of the exact equation of state (as opposed to the ideal-gas approximation) is necessary to predict this relationship. By exploiting a mapping of the lattice gas to the Ising model, one can compute the corrections to the Gibbs-Thomson formula using high field series expansions. The corrected Gibbs-Thomson formula matches very well with the Monte Carlo data. We also investigate finite size effects on the stability of the islands both theoretically and through simulations. Finally, the simulations are used to study the microscopic origins of the Gibbs-Thomson formula. It is found that smaller islands have a greater adatom detachment rate per unit length of island perimeter. This is principally due to a lower coordination of edge atoms and a greater availability of detachment moves relative to edge moves. A heuristic argument is suggested in which these effects are partially attributed to geometric constraints on the island edge.