Dynamics and Statics of Liquid-Liquid and Gas-Liquid interfaces on Non-Uniform Substrates at the Micron and Sub-Micron Scales
Droplets and bubbles are ubiquitous motifs found in natural and industrial processes. In the absence of significant external forces, liquid-liquid and gas-liquid interfaces form constant mean curvature surfaces that locally minimize the free energy of a given system subject to constraints. However, even for sub-micron bubbles and droplets free of hydrodynamic and hydrostatic stresses (small Capillary, Weber, and Bond numbers), non-equilibrium at the contact line of sessile bubbles and droplets can influence geometries and dynamics. First, the wetting of micron-sized ellipsoidal particles was considered. In the space of axially symmetric interfaces, it is found that multiple constant mean curvature surfaces can satisfy volume and contact angle constraints. Partial encapsulation may be preferred even when the droplet's volume is sufficient to fully engulf the particle. The co-existence of multiple equilibrium states suggests possible hysteretic encapsulation behavior. Secondly, motivated by electron microscopy observations of sub-micron bubbles in a liquid cell, a small mobile and growing bubble confined between two weakly diverging plates is considered theoretically. Scaling analysis suggests that observed bubbles move by continuously wetting and de-wetting the substrates onto which they are adhered. 2D and 3D models are constructed incorporating the Blake-Haynes mechanism, which relates the dynamic contact angle to contact line velocity; partial pinning of the contact line is also considered. In 2D, the system is fully described by a set of non-linear ordinary differential equations that can be readily solved. In 3D, the non-linear PDE system and constraints were solved using a pseudo-spectral method. Both 2D and 3D models predict that in order for a doubly confined bubble to grow in a super-saturated solution it must first increase its curvature; this is in contrast to a free-floating bubble whose curvature always decreases with the addition of mass/volume. Since the surface concentration is proportional to the internal pressure of the bubble, this geometric change temporarily regulates the growth of the bubble. The model predicts growth rates like those observed experimentally that are several orders of magnitude lower than predictions made by classical mass transfer driven growth theory developed by Epstein and Plesset.