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

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.