Long-Scale Evolution of Thin Liquid Films Bounded by a Viscous Phase with Diffusing Charged Surfactants

Abstract

The long-scale evolution of a thin liquid film between a viscous bounding phase and a fixed substrate is addressed. Positively charged surfactants are allowed to diffuse along the free water/bounding-phase interface and the substrate has fixed electric charges of opposite sign. We assume that the free surface deformation occurs on a length scale that far exceeds the film thickness. In that case, the Navier–Stokes and continuity equations with appropriate boundary conditions lead to a system of two nonlinear coupled equations, for the film thickness and for the concentration of the diffusing charged surfactant. The direction of Marangoni flows depends on the relationship between the viscosities of the film and of the bounding phase. Because the surfactants are electrically charged, the film dynamics accounts for the competition between diffusive flux and ion migration. The numerical integration of the coupled equations makes it possible to follow the time evolution of the film subject to an initial random perturbation. Numerical results are compatible with linear analysis calculations. The model predicts the formation of steady patterns both for the film thickness and for the distribution of the charged surfactant.