The validity of Cassie's law: A simple exercise using a simplified model

TL;DR

This paper derives a modified Cassie's law for contact angles on heterogeneous substrates using an interface displacement model, showing it aligns with previous formulas and is supported by recent experimental data.

Contribution

The paper introduces a modified Cassie's law derived from an interface displacement model, providing a more accurate description of contact angles on heterogeneous surfaces.

Findings

Modified Cassie's law aligns with Marmur's previous formula.

Recent experiments support the validity of the modified law.

The model considers constant volume and vapor pressure conditions.

Abstract

The contact angle of a macroscopic droplet on a heterogeneous but flat substrate is studied using the interface displacement model which can lead to the augmented Young-Laplace equation. Droplets under the condition of constant volume as well as constant vapor pressure are considered. By assuming a cylindrical liquid-vapor surface (meniscus) and minimizing the total free energy of the interface displacement model, we derive an equation which is similar but different from the well known Cassie's law. Our modified Cassie's law is essentially the same as the formula obtained previously by Marmur [J.Colloid Interface Sci. {\bf 168}, 40 (1994)]. A few consequences from this modified Cassie's law will be briefly described in the following sections of this paper. Several sets of recent experimental results seem to support the validity of our modified Cassie's law.