  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 vapour
around a curved island and its curvature-the Gibbs-Thomson formula. Numerical
simulations of a lattice gas model, performed for various sizes of islands,
don't fit very well to the Gibbs-Thomson formula. We show how corrections to
this form arise at high vapour densities, wherein a knowledge of the exact
equation of state (as opposed to the ideal gas approximation) is necessary to
predict this relationship. 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. 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. A heuristic argument is suggested in which it is partially attributed
to geometric constraints on the island edge.
