Adsorption–Desorption at Anomalous Diffusion: Fractional Calculus Approach

A mathematical model of the anomalous diffusion of surfactant and the process of adsorption–desorption on an interface is analyzed using a fractional calculus approach. The model is based on time-fractional partial differential equations in the bulk phases and the corresponding time-fractional description of the flux bulk–interface. The general case, when the surfactant is soluble in both phases, is considered under the assumption that the adsorption–desorption process is diffusion-controlled. Some of the most popular kinetic models of Henry, Langmuir, and Volmer are considered. Applying the Laplace transform, the partial differential model is transformed into a single multi-term time-fractional nonlinear ordinary differential equation for the surfactant concentration on the interface. Based on existing analytical solutions of linear time-fractional differential equations, the exact solution in the case of the Henry model is derived in terms of multinomial Mittag–Leffler functions, and its asymptotic behavior is studied. Further, the fractional differential model in the general nonlinear case is rewritten as an integral equation, which is a generalization of the well-known Ward–Tordai equation. For computer simulations, based on the obtained integral equation, a predictor–corrector numerical technique is developed. Numerical results are presented and analyzed.