Mechanical derivation of the Wenzel and Cassie equations using a statistical interpretation of drop dispensation

The global mechanical equilibrium condition of a liquid on a rough and chemically heterogeneous surface was derived for three-dimensional situations from a statistical outlook of dispensation of many drops and the assumption of local mechanical equilibrium. Unlike the conventional thermodynamic derivations from variational methods, the current proof is based on vector algebra rather than differential geometry. The mechanics-based derivation becomes less intricate although the minimum energy condition is not established. An effective contact angle is computed from the directional sampling of three-phase lines after local drop dispensations. The final expression is a combined mechanical version of the Wenzel and Cassie equations.