Vertical concentration gradients of soluble surfactants in the rupture of thin liquid films

Hypothesis: Surfactant-laden thin liquid films can rupture due to van der Waals forces, and being able to accurately predict the rupture time is important for applications involving coatings, foams, and emulsions. A common simplification in modeling film rupture is to assume that diffusion along the film thickness is so rapid that the surfactant concentration can be replaced by an averaged value. However, we hypothesize that vertical concentration gradients can develop as a result of surfactant adsorption at the interface, potentially rendering the vertical-averaging (VA) approximation inaccurate under certain conditions. Simulations: We assess the accuracy and limitations of this approximation by performing calculations with a lubrication-theory-based model that explicitly accounts for surfactant concentration gradients along the film thickness for a film on a horizontal solid substrate. Linear stability analysis and nonlinear simulations are performed to understand the role of vertical concentration gradients on film rupture. Findings: Results show that when surfactant diffusion is slow relative to advection and adsorption, substantial surfactant vertical concentration gradients can emerge. These gradients slow down adsorption and increase stabilizing Marangoni stresses, leading the VA approximation to underestimate the rupture time. Significant deviations in predicted rupture time are also observed when the initial bulk and surfactant concentrations are not in equilibrium, which is common in industrial applications.