Equilibria of liquid drops on pre-stretched, nonlinear elastic membranes through a variational approach

Abstract

We study the equilibrium of planar systems consisting of sessile and pendent drops on pre-stretched, nonlinear elastic membranes. The membrane experiences large deformations due to both capillary forces and the drop's weight. The membrane's surface energies are allowed to depend upon stretches in the membrane. We minimize the free energy of the system to obtain the governing equations. This recovers all equations found by force balance, in addition to an extra condition that must hold at the triple point. The latter closes the system's mathematical description and defines a unique equilibrium given the membrane's material and pre-stretch, and the properties of drop's fluid and its volume. The extra condition simplifies to continuity of stretches at the triple point when the surface energies are strain-independent. We then solve these coupled nonlinear equations to obtain the global equilibria of the drop–membrane system. We report the effects of drop's volume and membrane's pre-tension on the system's geometry and tension distribution in the membrane. Through this, we align the theory closely with experiments, which will then allow the use of the present system both as an elastocapillary tension probe and as a device to measure solid surface energies.