Contact Angles of Liquid Drops Subjected to a Rough Boundary

The contact angle of a liquid drop on a rigid surface is determined by the classical theory of Young-Laplace. For chemically homogeneous surfaces, this angle is a constant. We study the minimal-energy configurations of liquid drops on rough surfaces. Here the actual angle is still constant for homogeneous surfaces, but the apparent angle can fluctuate widely. A limit theorem is introduced for minimal energy configuration, where the rigid surface converges to a smooth one, but the roughness parameter is kept constant. It turns out that the limit of minimal energy configurations correspond to liquid drop on a smooth surface with an appropriately defined effective chemical interaction energy. It turns out that the effective chemical interaction depends linearly on the roughness in a certain range of parameters, corresponding to full wetting. Outside this range the most stable configuration corresponds to a partial wetting and the effective interaction energy depends on the geometry in an essential way. This result partially justifies and extends Wenzel and Cassie's laws and can be used to deduce the actual inclination angle in the most stable state, where the apparent one is known by measurement. This, in turn, may be applied to deduce the roughness parameter if the interfacial energy is known, or visa versa.