Contact angles of liquid drops subjected to a rough boundary

TL;DR

This paper analyzes how surface roughness affects the apparent contact angle of liquid drops, introducing a limit theorem that links roughness to effective chemical interactions and extends classical wetting laws.

Contribution

It introduces a limit theorem for minimal-energy configurations of drops on rough surfaces, connecting roughness to effective chemical energy and extending Wenzel and Cassie's laws.

Findings

Effective chemical interaction depends linearly on roughness in full wetting regime.

Outside this regime, partial wetting occurs with geometry-dependent effective energy.

The results enable deducing surface roughness or interfacial energy from contact angle measurements.

Abstract

The contact angle of a liquid drop on a rigid surface is determined by the classical theory of Young-Laplace. For chemically homogeneous surfaces, this angle is a constant. We study the minimal-energy configurations of liquid drops on rough surfaces. Here the actual angle is still constant for homogeneous surfaces, but the apparent angle can fluctuate widely. A limit theorem is introduced for minimal energy configuration, where the rigid surface converges to a smooth one, but the roughness parameter is kept constant. It turns out that the limit of minimal energy configurations correspond to liquid drop on a smooth surface with an appropriately defined effective chemical interaction energy. It turns out that the effective chemical interaction depends linearly on the roughness in a certain range of parameters, corresponding to full wetting. Outside this range the most stable configuration corresponds to a partial wetting and the effective interaction energy depends on the geometry in an essential way. This result partially justifies and extends Wenzel and Cassie's laws and can be used to deduce the actual inclination angle in the most stable state, where the apparent one is known by measurement. This, in turn, may be applied to deduce the roughness parameter if the interfacial energy is known, or visa versa.