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

TL;DR

The study addresses surfactant adsorption under mixed barrier-diffusion control by formulating adsorption kinetics as a composite fractional differential equation with orders $1$ and $1/2$, yielding a generalized Ward-Tordai framework. The approach reduces the problem to a small set of dimensionless groups, enabling exact Henry-model solutions in terms of the Mittag-Leffler function $E_{1/2}$ and systematic second-order asymptotics for adsorption and surface tension across Henry, Langmuir, Frumkin, Volmer, and van der Waals isotherms. It also derives fractional-integral representations that generalize Ward-Tordai for mixed control and develops a predictor-corrector numerical scheme for efficient simulation of the resulting Volterra-type equations. Overall, the paper provides a unified, low-parameter framework that captures diffusion- and barrier-dominated dynamics, yields accurate short- and intermediate-time predictions, and offers practical tools for numerical exploration of adsorption kinetics under varied kinetic models. The methodology and results are broadly applicable to other interfacial transport problems where mixed kinetic controls and anomalous diffusion are relevant, including different geometries and regimes.

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.

Figures (10)

• Figure 1: Barrier-diffusion controlled adsorption $\Gamma^*_H(\tilde{t})$ (eq.(\ref{['SBDCAH1']}) - solid lines) for Henry kinetic model at different ${ Ba}$. The limiting cases of diffusion control and barrier control are given by dashed and dash-dotted lines respectively.
• Figure 2: Comparison of the short-time (dashed lines) and second order (eq. (\ref{['SSOADC']}), dash-dotted lines) asymptotes with numerical results (solid lines). The presented results are in the case of diffusion-controlled adsorption for Langmuir and Volmer kinetic models at different values of $\Gamma_\infty^* .f_e$.
• Figure 3: Comparison of the numerical results (solid line) with the asymptotes, first (eq. (\ref{['FOABDC']}), dashed lines) and second order (eq. (\ref{['SSOABDC1']}), dash-dotted lines) for van der Waals model at ${ Ba}=10$ and different values of $\tilde{\beta}$. For the values of $\tilde{\beta}$ (0; 2; 4) the corresponding values of $\Gamma_\infty^*$ are: (2.06; 1.54; 1.31) at $\Gamma_\infty^* .f_e=5$ and (11.7; 3.35; 1.585) at $\Gamma_\infty^* .f_e=1.2$
• Figure 4: Comparison of the second order asymptotes (dash-dotted lines) with numerical results (solid line) at ${ Ba}=10$ and different $\Gamma_\infty^* .f_e$ in the case of Langmuir $\Gamma^*_L$ (blue) and Volmer $\Gamma^*_V$ (red). The first order asymptotes (eq. (\ref{['FOABDC']}), dashed lines) as well as the solution $\Gamma^*_H$ of Henry model (eq. (\ref{['SBDCAH1']}), dotted line) are also given. The values of $\Gamma_\infty^*$ corresponding to the values of $\Gamma_\infty^* .f_e$ are these from Fig. \ref{['Fig2']}.
• Figure 5: Comparison of the second order asymptotes of the adsorption $\tilde{\Gamma}(\tilde{t})$ in the case of van der Waals model (dashed line) and for Frumkin model (dash-dotted line) with numerical results (solid line) at ${ Ba}=10, \ f_e=10$ and different $\tilde{\beta}$. The asymptote $\tilde{\Gamma}_V(\tilde{t})$ (dashed line) is the exact solution in case of Henry isotherms.