Derivation of a non-local interfacial Hamiltonian for short-ranged wetting:: I.: Double-parabola approximation

We derive a non-local effective interfacial Hamiltonian model for short-ranged wetting phenomena using a Green's function method. The Hamiltonian is characterized by a binding potential functional and is accurate to exponentially small order in the radii of curvature of the interface and the bounding wall. The functional has an elegant diagrammatic representation in terms of planar graphs which represent different classes of tube-like fluctuations connecting the interface and wall. For the particular cases of planar, spherical and cylindrical interfacial ( and wall) configurations, the binding potential functional can be evaluated exactly. More generally, the non-local functional naturally explains the origin of the effective position-dependent stiffness coefficient in the small-gradient limit.