Unified framework for mapping shape and stability of pendant drops including the effect of contact angle hysteresis

Abstract

Solution of the Young-Laplace equation for fluid-fluid interface shapes is crucial in several applications related to colloids, capillarity, and surfaces. Despite significant progress in analysis and computation, the solution framework in literature is quite sparse with different norms for non-dimensionalisation and often limited to simplistic boundary conditions. These problems can be overcome and a unified solution framework can be developed if the relevant parameters are non-dimensionalised with respect to the capillary length, and collapsed onto a single graphical chart in the form of axes and contours. Here we use vector parameterized cubic splines to represent axisymmetric droplet profiles, which are then evolved quasi-statically to get to the equilibrium shapes using a novel thermodynamic free energy minimization-based algorithm. The generated dataset for a wide range of geometric and surface parameters is validated and then compiled in the form of a graphical chart. The chart traces the complete profile of axisymmetric pendant drops, predicts their evolution and stability on any surface, all without running any computer simulation. Most importantly, the contact angle contours on this chart offer the capability to incorporate the effect of contact angle hysteresis, a situation which is practically relevant, but has yet not been modelled.