Effective Hamiltonian of liquid-vapor curved interfaces in mean field

We analyze a one-component simple fluid in a liquid-vapor coexistence state, which forms an arbitrarily curved interface. By using an approach based on density functional theory, we obtain an exact and simple expression for the grand potential at the level of mean field approximation that depends on the density profile and the short-range interaction potential. By introducing the step-function approximation for the density profile, and using general geometric arguments, we expand the grand potential in powers of the principal curvatures of the surface and find consistency with the Helfrich phenomenological model in the second order approximation.