The actual contact angle on a heterogeneous rough surface in three dimensions

A general equation is presented for the actual contact angle on a solid surface in a three-dimensional setting. The solid surface may be rough or heterogeneous or both. The effects of the existence of line tension and its variation with the position of the contact line are also included. It is shown that when line tension can be ignored, the actual contact angle at each point on the solid surface always equals the intrinsic contact angle (which is given in this case by the Young equation). However, when line tension is significant, the actual contact angle deviates from the Young contact angle by a term proportional to the geodesic curvature of the contact line and a term depending on the directional derivative of the line tension. Various situations are presented and discussed. Of particular interest is the example of a drop on a sphere, for which it is shown that the actual contact angle equals the Young contact angle when the contact line coincides with the equator of the sphere.