  We study the excess free energy due to phase coexistence of fluids by Monte
Carlo simulations using successive umbrella sampling in finite LxLxL boxes with
periodic boundary conditions. Both the vapor-liquid phase coexistence of a
simple Lennard-Jones fluid and the coexistence between A-rich and B-rich phases
of a symmetric binary (AB) Lennard-Jones mixture are studied, varying the
density rho in the simple fluid or the relative concentration x_A of A in the
binary mixture, respectively. The character of phase coexistence changes from a
spherical droplet (or bubble) of the minority phase (near the coexistence
curve) to a cylindrical droplet (or bubble) and finally (in the center of the
miscibility gap) to a slab-like configuration of two parallel flat interfaces.
Extending the analysis of M. Schrader, P. Virnau, and K. Binder [Phys. Rev. E
79, 061104 (2009)], we extract the surface free energy gamma (R) of both
spherical and cylindrical droplets and bubbles in the vapor-liquid case, and
present evidence that for R -> Infinity the leading order (Tolman) correction
for droplets has sign opposite to the case of bubbles, consistent with the
Tolman length being independent on the sign of curvature. For the symmetric
binary mixture the expected non-existence of the Tolman length is confirmed. In
all cases {and for a range of radii} R relevant for nucleation theory, gamma(R)
deviates strongly from gamma (Infinity) which can be accounted for by a term of
order gamma(Infinity)/gamma(R)-1 ~ 1/R^2. Our results for the simple
Lennard-Jones fluid are also compared to results from density functional theory
and we find qualitative agreement in the behavior of gamma(R) as well as in the
sign and magnitude of the Tolman length.
