Complete wetting near an edge of a rectangular-shaped substrate

We consider fluid adsorption near a rectangular edge of a solid substrate that interacts with the fluid atoms via long range (dispersion) forces. The curved geometry of the liquid-vapour interface dictates that the local height of the interface above the edge $\ell_E$ must remain finite at any subcritical temperature, even when a macroscopically thick film is formed far from the edge. Using an interfacial Hamiltonian theory and a more microscopic fundamental measure density functional theory (DFT), we study the complete wetting near a single edge and show that $\ell_E(0)-\ell_E(\delta\mu)\sim\delta \mu^{\beta_E^{co}}$, as the chemical potential departure from the bulk coexistence $\delta\mu=\mu_s(T)-\mu$ tends to zero. The exponent $\beta_E^{co}$ depends on the range of the molecular forces and in particular $\beta_E^{co}=2/3$ for three-dimensional systems with van der Waals forces. We further show that for a substrate model that is characterised by a finite linear dimension $L$, the height of the interface deviates from the one at the infinite substrate as $\delta\ell_E(L)\sim L^{-1}$ in the limit of large $L$. Both predictions are supported by numerical solutions of the DFT.