Does the Roughness of the Substrate Enhance Wetting?

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 $\Delta \tau \left(r\right)$ of the two phases behaves like $\Delta \tau \left(r\right)\approx r\Delta \tau \left(1\right)$, implying that roughness enhances wetting for $\Delta \tau \left(1\right)>0\mathrm{}$ and drying for $\Delta \tau \left(1\right)<0\mathrm{}$.