Equilibria of Pendant Droplets: Spatial Variation and Anisotropy of Surface Tension

TL;DR

This paper investigates the equilibrium shapes of pendant droplets, revealing that surface tension varies spatially and anisotropically, which challenges the traditional assumption of constant, isotropic surface tension.

Contribution

It analytically and experimentally demonstrates that equilibrium states of droplets can have spatially varying and anisotropic surface tensions, extending the classical Young-Laplace model.

Findings

Surface tension varies with position on the droplet surface.

Equilibrium states can have surface tensions less than the surface energy.

Surface tension is anisotropic in pendant droplets.

Abstract

An example of capillary phenomena commonly seen and often studied is a droplet of water hanging in air from a horizontal surface. A thin capillary surface interface between the liquid and gas develops tangential surface tension, which provides a balance of the internal and external pressures. The Young-Laplace equation has been historically used to establish the equilibrium geometry of the droplet, relating the pressure difference across the surface to the mean curvature of the surface and the surface tension, which is presumed constant and isotropic. The surface energy per unit area is often referred to as simply surface energy and is commonly considered equal to the surface tension. The relation between the surface energy and the surface tension can be established for axisymmetric droplets in a gravitational field by the application of the calculus of variations, minimizing the total potential energy. Here it is shown analytically and experimentally that, for conditions of constant volume of the droplet, equilibrium states exist with surface tensions less than the surface energy of the water-air interface. The surface tensions of the interface membrane vary with position and are anisotropic.