Does the roughness of the substrate enhance wetting?

Abstract

We consider a semi-infinite three-dimensional Ising system with a rough wall to describe the effect of the roughness $r$ of the substrate on wetting. For sufficiently low temperature, we show that the difference of wall free energies $\ensuremath{\Delta}\ensuremath{\tau}\left(r\right)$ of the two phases behaves like $\ensuremath{\Delta}\ensuremath{\tau}\left(r\right)\ensuremath{\approx}r\ensuremath{\Delta}\ensuremath{\tau}\left(1\right)$, implying that roughness enhances wetting for $\ensuremath{\Delta}\ensuremath{\tau}\left(1\right)g0$ and drying for $\ensuremath{\Delta}\ensuremath{\tau}\left(1\right)l0$.