Fractional calculus approach to models of adsorption: Barrier-diffusion control

TL;DR

This paper develops a fractional calculus-based mathematical model for surfactant adsorption under barrier-diffusion control, providing exact solutions, asymptotic analysis, and a numerical method to better understand adsorption kinetics and surface tension behavior.

Contribution

It introduces a fractional differential model for adsorption, transforming complex integral equations into simpler forms with fewer parameters and deriving exact solutions and asymptotes.

Findings

Exact Mittag-Leffler function solutions for Henry model

Asymptotic behavior of other models derived from Henry model

Numerical predictor-corrector method for simulation

Abstract

The mathematical model of surfactant adsorption under mixed barrier-diffusion control is analyzed using techniques from fractional calculus. The kinetic models of Henry, Langmuir, Frumkin, Volmer and van der Waals are considered. First, treating the Ward-Tordai integral equation as a fractional order one, the partial differential model is transformed into a single fractional ordinary differential equation for the adsorption. A transformation of the obtained equation is proposed that reduces the number of parameters to two dimensionless groups (at Frumkin and van der Waals models a third parameter appears). In the simplest case of Henry adsorption isotherm the fractional differential model depends on a single dimensionless group and an exact solution exists, represented in terms of Mittag-Leffler functions. Based on this solution, second order asymptotes (at small values of the adsorption) are derived for the other models. The asymptotes of the adsorption result in a higher order asymptotes for the surface pressure (surface tension). For small surface coverage, all considered models converge to the Henry model's predictions, making it a universal first-order approximation for the surface tension. Next, the fractional differential model is written as an integral equation %of fractional order that can be considered as a generalization of the well-known Ward-Tordai equation to the case of barrier-diffusion control. For computer simulation of the obtained integral equation a predictor-corrector numerical method is developed and numerical results are presented and discussed.