Quantifying surfactant adsorption at fluid interfaces by combining X-ray reflection and simulation

Adsorption of surfactants to fluid interfaces occurs in daily-life and technological contexts like dish washing and oil spill remediation. The surfactant surface coverage $\Gamma$ governs interface characteristics like tension $\gamma$, viscoelastic properties, and the stability of thin foam films. Directly measuring $\Gamma$ as a function of the bulk concentration $c$ is highly desirable but challenging, particularly for non-ionic surfactants that lack easily detectable labels. Here, we propose a generic approach to deduce the adsorption isotherm $\Gamma(c)$: As a first step, we use atomistic molecular dynamics simulations of surfactant-loaded air/water interfaces with known $\Gamma$ to obtain interfacial electron density profiles. From these profiles, we then compute theoretical X-ray reflectivity curves, which we compare with experimental measurements to find the matching $c$. We focus on two non-ionic surfactants (C$_{12}$EO$_6$} and $\beta$-C$_{12}$G$_2$}) with previously verified force fields to demonstrate how this combined approach of experiments and simulations can determine the adsorption isotherm. By using the equation of state $\gamma(\Gamma)$ from simulations, our results replicate the measured surface tension isotherms $\gamma(c)$.