Equilibrium contact angle and adsorption layer properties with surfactants

TL;DR

This paper derives a consistency condition linking macroscopic contact angle and mesoscale wettability for surfactant-covered liquids, supported by theoretical analysis and numerical simulations.

Contribution

It introduces a novel consistency condition for equilibrium contact angles in the presence of insoluble surfactants, bridging macroscopic and mesoscale descriptions.

Findings

Derived a new relation for surfactant-influenced contact angles.

Showed that mesoscale and macroscale models agree under specific conditions.

Validated the theory with numerical simulations of dilute surfactants.

Abstract

The three-phase contact line of a droplet on a smooth surface can be characterized by the Young-Dupr\'e 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.