 Solution

Advance the projected contact line by $dx$ per unit transverse length without wetting the gaps. A fraction $f$ converts solid–vapor interface into solid–liquid interface, contributing $f({\gamma }_{SL}-{\gamma }_{SV})dx$. The remaining fraction creates liquid–vapor interface above trapped gas, costing $(1-f){\gamma }_{LV}dx$. The outer droplet surface contributes ${\gamma }_{LV}\cos {\theta }^{\ast }dx$. ThusSubstituting Young's equation givesThis is the ideal Cassie–Baxter state relation. Its $f$ is projected solid-contact fraction, not the roughness factor of Wenzel wetting. A suspended interface over the gaps is essential; invasion of the texture changes the area bookkeeping.