Droplet profiles under the influence of van der Waals forces

We study the effect of van der Waals forces on globally energy minimizing profiles for liquid droplets which lie on a solid substrate in a vapour atmosphere and which are assumed to have a uniform cross-section. We prove that for repulsive van der Waals forces as well as for certain short range repulsive-long range attractive forces, there exists a unique globally minimizing profile. Although this profile necessarily contains vertical bounding segments, the height A of the vertical bounding segments can often be demonstrated to be order of magnitude smaller that the overall height B of the droplet. This is the case, in particular, when the droplet is sufficiently large, the Hamaker constant is sufficiently small, and the attractive forces are sufficiently mild. In the presence of repulsive forces only, A is on the order of angströms when B is on the order of millimeters, for realistic parameter values. Moreover, conditions are prescribed under which Young's law is satisfied to leading order despite the appearance of the vertical segments, when the contact angle is measured via an inscribed circle construction at a distance ξ0 from the edge of the droplet, where A [Lt ] ξ0 [Lt ] B.