Dynamics of droplets on cones: self-propulsion due to curvature gradients

We study the dynamics of droplets driven by a gradient of curvature, as may be achieved by placing a drop on the surface of a cone. The curvature gradient induces a pressure gradient within the drop, which in turn leads to spontaneous propulsion of the droplet. To investigate the resulting driving force we perform a series of experiments in which we track a droplet's displacement, s, from the apex of a cone whose surface is treated to exhibit near-zero pinning effects. We find an s ∼ t1/4 scaling at sufficiently late times t. To shed light upon these dynamics, we perform an asymptotic calculation of the equilibrium shape of a droplet on a weakly curved cylinder, deriving the curvature-induced force responsible for its propulsion. By balancing this driving force with viscous dissipation, we recover a differential equation for the droplet displacement, whose predictions are found to be in good agreement with our experimental results.