Effect of soluble surfactants on vertical liquid film drainage

Abstract

A mathematical model is established to investigate the gravity-driven drainage of vertical films containing a soluble surfactant by considering the coupling effect of surface elasticity, adsorption coefficient, and surfactant solubility. The lubrication theory is applied to derive the four coupled nonlinear partial differential equations describing the film thickness, surface velocity, and surfactant concentration on the surface and in the bulk. Simulated results showed that the surface elasticity, adsorption coefficient, and surfactant solubility are indispensable factors in the drainage process of a liquid film containing a soluble surfactant. In the initial stage of the drainage, the initial film thickness increases with increasing surface elasticity and the film surface tends to be more rigid. With further drainage, the liquid film exhibits different notable features for high and low elasticity. For low surface elasticity, the surfactant distribution produces a positive Marangoni effect, which counteracts gravity. However, for high surface elasticity, the film surface exhibits a reverse Marangoni effect, which accelerates the drainage and leads to an unstable film. As the solubility decreases, both the film stability and initial surface elasticity enhance. The surface elasticity gradually approaches a limiting dilational elasticity modulus owing to the film thinning. For a large Ks, the film surface is insufficient to produce a strong Marangoni effect and then the liquid film tends to easily destabilize. For a small Ks, the soluble surfactant is similar to an insoluble surfactant, and the film is much thicker in the later stage of the drainage.