Simulation of vapor-liquid coexistence in finite volumes: A method to compute the surface free energy of droplets

When a fluid at a constant density rho in between the densities of coexisting vapor (rho(v)) and liquid (rho(l)) at temperatures below criticality is studied in a (cubic) box of finite linear dimension L, phase separation occurs in this finite volume, provided L is large enough. For a range of densities, one can observe a liquid droplet (at density rho(l)' slightly exceeding rho(l)) coexisting in stable thermal equilibrium with surrounding vapor (with density rho(v)' > rho(v), so in the thermodynamic limit such a vapor would be supersaturated). We show, via Monte Carlo simulations of a Lennard-Jones model of a fluid and based on a phenomenological thermodynamic analysis, that via recording the chemical potential mu as function of rho, one can obtain precise estimates of the droplet surface free energy for a wide range of droplet radii. We also show that the deviations of this surface free energy from the prediction based on the "capillarity approximation" of classical nucleation theory (i.e., using the interfacial free energy of a flat liquid-vapor interface for the surface free energy of a droplet irrespective of its radius) are rather small. We also study carefully the limitation of the present method due to the "droplet evaporation/condensation transition" occurring for small volumes and demonstrate that very good equilibrium is achieved in our study, by showing that the radial profile of the local chemical potential from the droplet center to the outside is perfectly flat.