Effect of surface elasticity on drainage process of vertical liquid film with soluble surfactant

Abstract

The aim of the present paper is to investigate the gravity-driven draining process containing soluble surfactant when considering the coupling effects of surface elasticity and surfactant solubility. A nonlinear coupling evolution equation including liquid film thickness, surface velocity and surfactant concentration (both on the surface and in the bulk) is established based on the lubrication theory. Assuming that the top of liquid film is attached to the wireframe and the bottom is connected to a reservoir, the drainage evolution is simulated with the software called FreeFem. The effects of surface elasticity and solubility on liquid film draining are discussed under their coupling. The simulation results show that the surface elasticity is an indispensable factor in the process of liquid film drainage with soluble surfactant, and the surfactant solubility also has an important influence on the process. At the initial stage of liquid draining, the initial thickness of liquid film increases with increasing surface elasticity, and the surface tends to be more rigid; with the drainage proceeding, the liquid film with high and low elasticity illustrate different notable draining features:in the case of low surface elasticity, the distribution of surfactant forms a surface tension gradient from top to bottom on the film surface, leading to positive Marangoni effect that counteracts gravity. However, in the case of high elasticity, the film surface presents a surface tension gradient from bottom to top, resulting in a reverse Marangoni effect, which accelerates the draining and makes the film more susceptible to instability. The solubility of surfactant dominates the number of adsorbent molecules on the film surface, which affects the surface elasticity. When the solubility of the surfactant is great (β → 0), the film is extremely unstable, and it breaks down quickly. As the solubility decreases (namely, β increases), the stability of the film increases, and the initial surface elasticity also rises. The surface elasticity gradually approaches to the limiting dilational elasticity modulus due to the film being thinner.