Spline based modeling of two-dimensional droplets on rough and heterogeneous surfaces

Abstract

Interactions between liquids and solids are ubiquitous in our physical environment. A sessile droplet on a homogenous and smooth surface is known to minimize its liquid-vapor interface area and assume a spherical cap shape. However, practical engineering applications involve fluid flow over rough and heterogeneous surfaces and the curvature varies along the liquid-vapor interface. Despite significant efforts over the past few decades, a comprehensive model, that, while capturing the complex dynamics at the three-phase contact line, is also able to predict the apparent contact angle values typically observed in experiments, is missing. In this work, we use vector valued parameterized cubic spline-based representation to model the droplet shapes on rough and heterogeneous surfaces. Thermodynamics driven minimization of the free energy via the minimization of the liquid-vapor interface area in three dimensions is reduced to the minimization of the perimeter of the liquid-vapor interface in this case. We show that the minimization of the perimeter is mathematically equivalent to the minimization of the spread in the curvature along the liquid-vapor interface. The shape of the resulting interface near the three-phase contact line is shown to resemble the shapes typically observed during high resolution microscopic contact angle experiments. Moreover, the contact angles estimated by approximating the obtained droplet profiles using circular arcs are found to match the average contact angle values in experiments. We believe that this simple and computationally inexpensive approach can be used to understand nature’s design and extend it to enable fine fluidic manipulation capability in biological, manufacturing, microfluidic, and thermal management applications.