An equilibrium lattice model of wetting on rough substrates

We consider a semi-infinite 3-dimensional Ising system with a rough wall to describe the effect of the roughness r of the substrate on wetting. We show that the difference of wall free energies Delta tau(r) = tau(AW)(r) - tau(BW)(r) of the two phases behaves like Delta tau(r) similar to r Delta tau(1), where r = 1 characterizes a purely flat surface, confirming at low enough temperature and small roughness the validity of Wenzels law, cos theta(r) approximate to r cos theta(1), which relates the contact angle theta of a sessile droplet to the roughness of the substrate.