Equilibrium contact angle and adsorption layer properties with surfactants

Abstract

The three-phase contact line of a droplet on a smooth surface can be characterized by the Young-Dupré equation. It relates the interfacial energies with the macroscopic contact angle ${\theta }_e$. On the mesoscale, wettability is modeled by a film-height-dependent wetting energy $f(h)$. Macro- and mesoscale description are consistent if $\gamma \cos {\theta }_{\mathrm{e}}=\gamma +f(h_{\mathrm{a}})$ where $\gamma$ and $h_{\mathrm{a}}$ are the liquid-gas interface energy and the thickness of the equilibrium liquid adsorption layer, respectively. Here, we derive a similar consistency condition for the case of a liquid covered by an insoluble surfactant. At equilibrium, the surfactant is spatially inhomogeneously distributed implying a non-trivial dependence of ${\theta }_{\mathrm{e}}$ on surfactant concentration. We derive macroscopic and mesoscopic descriptions of a contact line at equilibrium and show that they are only consistent if a particular dependence of the wetting energy on the surfactant concentration is imposed.This is illustrated by a simple example of dilute surfactants, for which we show excellent agreement between theory and time-dependent numerical simulations.