Wetting properties of well-structured heterogeneous substrates

We study numerically the wetting properties of model heterogeneous flat substrates. The shapes of three-dimensional liquid drops in equilibrium with such substrates in the framework of the classical capillary theory are obtained. The numerical method used for minimizing the free energy is based on the local variations approach. It has been extended here to treat chemically heterogeneous substrates with "mesa" defects, i.e., sharp boundaries between surface patches with different surface tensions. The method allows inclusion of the gravity and the line tension of the contact line as well as different constraints, e.g. the constant volume constraint. We discuss the implications for the standard and modified Cassie equations and for the interpretation of the experimental data. Two different situations arise regarding the averaging of the contact angle for the two types of substrates considered: that with radial symmetry and that without (the dart board substrate).