Stability of rectilinear contact lines: I. Single contact line theory

Abstract

Theory applicable to one-dimensional models of rough surfaces is developed. A partly submerged solid disturbs the surface of a large pool of liquid. Part of the solid's surface is a cylinder with horizontal rulings. Any section of the cylinder by a plane Q orthogonal to its rulings is the curve C B (which need not be circular and may in fact be any piecewise smooth curve). The section of the liquid-vapor interface by Q is the curve C LV which meets C B at the three-phase contact point P. C LV belongs to a family F of solutions of the Laplace capillarity equation with the same horizontal asymptote. The asymptote meets C B at O which is the origin of a coordinate s along C B. A point on C B with coordinate s intersects a member of F with contact angle θ( s). Along C B there is a prescribed piecewise continuous equilibrium contact angle distribution θ e( s). A stable contact point with coordinate s 1 implies in general at s 1: θ = θ e (which is a generalization of Young's equation) and dθ ds ⩽ dθ e ds . Exceptions occur at jumps and corners, where a generalization of the (modified) Gibbs conditions is derived. Sufficiency criteria are also proved.