Spontaneous Movement of a Droplet on a Conical Substrate: Theoretical Analysis of the Driving Force

Experiments and simulations have shown that a droplet can move spontaneously and directionally on a conical substrate. The driving force originating from the gradient of curvatures is revealed as the self-propulsion mechanism. Theoretical analysis of the driving force is highly desirable; currently, most of them are based on a perturbative theory with assuming a weakly curved substrate. However, this assumption is valid only when the size of the droplet is far smaller than the curvature radius of the substrate. In this paper, we derive a more accurate analytical model for describing the driving force by exploring the geometric characteristics of a spherical droplet on a cylindrical substrate. In contrast to the perturbative solution, our model is valid under a much weaker condition, i.e., the contact region between the droplet and the substrate is small compared with the curvature radius of the substrate. Therefore, we show that for superhydrophobic surfaces, the derived analytical model is applicable even if the droplet is very close to the apex of a conical substrate. Our approach opens an avenue for studying the behavior of droplets on the tip of the conical substrate theoretically and could also provide guidance for the experimental design of curved surfaces to control the directional motion of droplets.