Deformation near a liquid contact line on an elastic substrate

The equilibrium configuration of a liquid drop on a solid is determined by local energy balance. For a very stiff substrate, energy balance is represented by Young's equation. The equilibrium configuration near a line separating three fluids, in contrast, is determined by a balance of forces—their surface tensions—which is represented graphically by Neumann's triangle. We argue that these two are limiting cases of the more general situation of a drop on an elastic substrate in which both configurational energy balance and force balance must be satisfied independently. By analysing deformation close to the contact line of a liquid drop on an elastic substrate, we show that the transition from the surface tension-dominated regime to the elasticity-dominated regime is controlled by a dimensionless parameter: the ratio of an elasto-capillary length to the characteristic length scale over which surface tension acts. Because of the influence of substrate elasticity, the contact angle is not necessarily given by Young's equation. For compliant solids, we show that the local deformation and stress fields near the contact line are governed by surface tensions. However, if surface tension happens to be different from surface energy, configurational energy balance may not be consistent with force balance.