Stochastic Relaxation of the Contact Line of a Water Drop on a Solid Substrate Subjected to White Noise Vibration: Roles of Hysteresis

Abstract

Relaxation of the three phase contact line of a sessile drop of water on a low energy surface is studied by subjecting it to a white noise vibration. While a spring force acts on the contact line whenever the contact angle deviates from its equilibrium value, it is opposed by hysteresis. The drop, therefore, remains pinned at a metastable state. With an appropriate amount of vibration, the drop can reach a global equilibrium state irrespective of its initial state, be it advanced or retreated. While the end state is free of hysteresis, the current study sheds light on the dynamics of relaxation that is analyzed in conjunction with a modified Langevin equation. Instead of exhibiting a smooth relaxation as predicted by the Langevin equation with a smooth background potential, stepwise relaxation is observed in most cases. These stepwise relaxations can be explained if the background potential is made slightly corrugated that signifies the existence of metastable states of a drop on a surface. The fluctuation of the displacement of the contact line is highly non-Gaussian. It is shown that an exponential distribution of the displacement fluctuation arises due to the nonlinear hysteresis term in the Langevin equation. The observations of stick-slip motion, the large time of relaxation, and the anomalous displacement fluctuation suggest that hysteresis is present during the relaxation process of the drop even though the final state reached by the drop is free of hysteresis. Finally, we compare the displacement fluctuations of the contact line on two different surfaces: a silicone rubber and a fluorocarbon monolayer. Although the displacement fluctuation is exponential in both cases, the later surface exhibits a greater variance of the distribution than the former plausibly due to differences in hysteresis. This result indicates that the fluctuation of displacement may be used as a tool to study the surface property of a low energy substrate.