Surface free energy of a finite system without interfaces at low temperatures

We derive the specific surface free energy for a rather general system at low temperatures that can be rewritten as a gas of non-interacting contours (polymers). To this end, we use a standard cluster expansion series for the systemʼs partition function. A specific regime of 'weak' boundary conditions is assumed to ensure that no interfaces or large droplets occur in the system. We illustrate the general results, using a simple lattice-gas model.