Statistical mechanics of fluids adsorbed in planar wedges: finite contact angle

I consider the statistical mechanics of inhomogeneous fluids applied to fluids adsorbed in planar wedges. Exact results are described that belong to an infinite subset of models defined as the intersection of any two identical semi-infinite planar wall-fluid potentials. This geometry is interesting as a generic example of adsorption onto structured interfaces and of interfacial phase transitions controlled by the substrate geometry. Previously described virial theorems are extended to the case of a general wall-fluid model. This enables the consideration of wedge filling when Young's contact angle far from the wedge apex is finite. The virial theorems generate two important relations: the wedge sum rules. The first sum rule links the interfacial free energy far from the wedge apex to the structure induced at the apex. The second sum rule links the free energy of the apex region to the structure induced by the apex. When Young's contact angle at the wedge walls is finite these relations further yield an exact result for the macroscopic contact angle in terms of the nanoscopic structure at the three-phase contact line (the intersection of the liquid-vapor surface with a wedge wall): the contact angle sum rule. These exact results are of direct relevance to computer simulation studies of adsorbed films. In addition, they take on special significance in the vicinity of continuous interfacial phase transitions: an approach to complete filling and the filling transition at bulk liquid-vapor coexistence.