Contact angles on heterogeneous surfaces: A new look at Cassie's and Wenzel's laws

We consider a three-dimensional liquid drop sitting on a rough and chemically heterogeneous substrate. Using a novel minimization technique on the free energy of this system, a generalized Young's equation for the contact angle is found. In certain limits, the Cassie and Wenzel laws, and a new equivalent rule, applicable in general, are derived. We also propose an equation in the same spirit as these results but valid on a more "microscopic" level. Throughout we work under the presence of gravity and keep account of line tension terms.