We determine the steady-state structures that result from liquid-liquid demixing in a
free surface film of binary liquid on a solid substrate. The considered model corresponds
to the static limit of the diffuse interface theory describing the phase separation
process for a binary liquid (model-H), when supplemented by boundary conditions at the
free surface and taking the influence of the solid substrate into account. The resulting
variational problem is numerically solved employing a Finite Element Method on an adaptive
grid. The developed numerical scheme allows us to obtain the coupled steady-state film
thickness profile and the concentration profile inside the film. As an example we
determine steady state profiles for a reflection-symmetric two-dimensional droplet for
various surface tensions of the film and various preferential attraction strength of one
component to the substrate. We discuss the relation of the results of the present diffuse
interface theory to the sharp interface limit and determine the effective interface
tension of the diffuse interface by several means.