Effect of concentration-dependent disjoining pressure on drainage process of vertical liquid film

For the drainage under the gravity of a vertical foam film containing insoluble surfactant, an improved concentration-dependent disjoining pressure model is formulated based on the published experimental results. The lubrication theory is used to establish the evolution equations of the film thickness, the surface concentration of insoluble surfactant, and the surface velocity, and the evolution characteristics of the film under different disjoining pressures are simulated numerically. The results show that the drainage process of a vertical liquid film generally undergoes two stages: the first stage is the thick film stage and the gravity plays a leading role in the drainage process; the subsequent stage is the thin film stage, the effects of capillary pressure and disjoining pressure increase gradually, and the disjoining pressure dominates the evolution of the film. The disjoining pressure effect is closely related to surfactant type and the correlation strength between the surfactant concentration and electrostatic repulsion force of disjoining pressure. For the ionic surfactant, electrostatic repulsion force increases with the increase of the surfactant concentration, but it is opposite for the nonionic surfactant. It is likely that the free hydroxide ions, which are considered to render the surface negatively charged, are partly adsorbed by the nonionic surfactant. So the surface charge of the foam film decreases as the concentration of the nonionic surfactant increases, resulting in a decrease in electrostatic repulsion. Therefore, some ionic surfactants can improve the stability of liquid film drainage and slow down the drainage process, while the effects of some nonionic surfactants are opposite. When the disjoining pressure is positively correlated with surfactant concentration, with the increase of correlation strength coefficient alpha, the thinning and drainaging processes of the film tend to slow down, hence the stability of the film is enhanced. When the disjoining pressure is negatively correlated with surfactant concentration, with the increase of the absolute value of alpha, the drainage process of the film is accelerated and the risk of film rupture is augmented. The results obtained in this paper are consistent with some of the experimental results, indicating that the concentration-dependent disjoining pressure is indeed an important factor in maintaining the stability of foam film containing some certain anionic or nonionic surfactants. The improved concentration-dependent disjoining pressure model established in this paper could not explain the phenomena of parts of cationic nor non-ionic surfactant film in drainage experiments. It can be inferred that the structure of surfactant molecule, the more detailed disjoining pressure model and the coupling of the disjoining pressure and surface elasticity should be considered in the future work.