Rigorous generalization of Young's law for heterogeneous and rough substrates

We consider a SOS type model of interfaces on a substrate which is both heterogeneous and rough. We first show that, for appropriate values of the parameters, the differential wall tension that governs wetting on such a substrate satisfies a generalized law which combines both Cassie and Wenzel laws. Then in the case of an homogeneous substrate, we show that this differential wall tension satisfies either the Wenzel' s law or the Cassie s law, according to the values of the parameters.