Solvable models of corner wetting in two and three dimensions

Corner wetting is studied in systems with short-range interactions using a solid-on-solid approximation to the nearest-neighbor Ising model. Complete wetting models are described by the generating functions for linear [two dimensions (2D)] and plane [three dimensions (3D)] partitions. The wetted area (volume in 3D) is found to behave as $〈A〉\sim \frac{1}{H^2}$ (2D) and $〈V〉\sim \frac{1}{H^3}$ (3D), as a function of the bulk field $H$. In the presence of surface pinning potentials on two edges of an $n\times n$ square array, we find that corner wetting occurs in two stages. The edge with weaker pinning wets first via a first-order transition, and at a second, higher temperature, the edge with stronger pinning wets via a second-order (Abraham) transition.