Computation of dynamic adsorption with adaptive integral, finite difference, and finite element methods

doi:10.1016/S0021-9797(02)00096-6

Journal of Colloid and Interface Science

Volume 258, Issue 2, 15 February 2003, Pages 310-321

https://doi.org/10.1016/S0021-9797(02)00096-6 Get rights and content

Abstract

Analysis of diffusion-controlled adsorption and surface tension in one-dimensional planar coordinates with a finite diffusion length and a nonlinear isotherm, such as the Langmuir or Frumkin isotherm, requires numerical solution of the governing equations. This paper presents three numerical methods for solving this problem. First, the often-used integral (I) method with the trapezoidal rule approximation is improved by implementing a technique for error estimation and choosing time-step sizes adaptively. Next, an improved finite difference (FD) method and a new finite element (FE) method are developed. Both methods incorporate (a) an algorithm for generating spatially stretched grids and (b) a predictor–corrector method with adaptive time integration. The analytical solution of the problem for a linear dynamic isotherm (Henry isotherm) is used to validate the numerical solutions. Solutions for the Langmuir and Frumkin isotherms obtained using the I, FD, and FE methods are compared with regard to accuracy and efficiency. The results show that to attain the same accuracy, the FE method is the most efficient of the three methods used.

Introduction

Dynamic surface tension behavior at fluid/fluid interfaces is important to many applications involving foams [1], coatings [2], lung surfactants [3], and drop and jet breakup [4]. The relation between the adsorbed surface density of surfactant and the dynamic surface tension at the interface permits studying interfacial transport by dynamic surface tension measurements. For a nonionic surfactant below its critical micelle concentration (cmc), the dynamic surface tension of a newly created surface is governed by a two-step process [5]. First, the surfactant monomers in the bulk solution diffuse to the subsurface layer. Then, the surfactant is adsorbed from the subsurface layer to the surface layer to affect the surface tension. To model the two-step process in a spatially one-dimensional problem, the diffusion in the bulk solution is described by a transient diffusion equation coupled with a time-dependent surface density. The general solution for the surface density is an integral [6] relating the surface density and the subsurface concentration. To complete the mathematical statement of the problem, the relationship between the surface density and the subsurface concentration, which depends on the adsorption/desorption mechanism, is needed. Often, a local equilibrium assumption or a so-called “diffusion controlled” mechanism applies [6], [7], [8], [9]. The adsorbed surface density is related to the subsurface concentration by a dynamic isotherm, which usually has the same functional form as the equilibrium adsorption isotherm [10], [11]. Once the dynamic adsorption isotherm is specified, the problem is well posed and can be solved explicitly for the surface density and the bulk concentration profile.

The problem of diffusion-controlled adsorption with a linear adsorption isotherm was first formulated and solved analytically by Sutherland [7]. However, since most of the isotherms are nonlinear for high surface tension reduction, numerical solutions to the problem are needed. The often-used numerical method is applying the trapezoidal rule approximation to the Ward and Tordai equation [12], [13]. Another approach is based on the idea that over small increments of time, the relationship between the adsorbed surface density and the sublayer concentration is nearly linear [14]. This approach then builds a solution by linearization of the dynamic isotherm that appears in the Ward and Tordai equation over a large number of successive small intervals in time. However, an analytical solution with the linearized isotherm is then required at each step. Thus, the solution procedure is more complicated compared to the situation in which the isotherm is linear and not easily formulated.

Finite difference (FD) methods for this problem have also been used to directly solve the differential equations governing the dynamic adsorption problem without using the Ward and Tordai method. For diffusion-controlled adsorption in a spatial domain of infinite extent, Miller developed an FD algorithm based on the Crank–Nicholson method [8]. The time-step size was variable and increased by 1% in each step. The spatial grid was uniform but empirically adjusted in time. For dynamic adsorption within a finite spatial domain, Chang and Franses developed a fully implicit FD scheme using the control volume method [11]. This scheme was further extended to pulsating area conditions in spherical coordinates [15].

Although several numerical methods for solving the one-dimensional diffusion-controlled adsorption problem are available, the accuracy and efficiency of various methods have heretofore not been determined. To date, the convergence of the trapezoidal rule approximation to the Ward and Tordai equation could only be ensured by decreasing the time-step size. In the published FD schemes, previous authors either used small time steps of uniform size or empirically adjusted them, but did not guarantee the accuracy of computations. Clearly, what are needed are methods that incorporate algorithms for systematically generating spatial grids for minimizing the number of grid points used, employ adaptive time-integration techniques, and provide precise estimates of computational accuracy (or error).

The aim of this article is to investigate the accuracy and efficiency of key computational methods for analyzing one-dimensional diffusion-controlled adsorption. First the integral (I) method, which entails solving a Ward and Tordai type integral equation when the extent of the spatial domain is arbitrary, is put on a firmer foundation. The time truncation error of the trapezoidal rule approximation for the I method is evaluated, which allows time-step sizes to be chosen adaptively to ensure solution accuracy and computational efficiency. Next, an improved FD scheme and a new finite element (FE) approximation are developed. In both methods, a stretched grid is used for spatial discretization, which makes it possible to obtain mesh-insensitive solutions with drastically fewer grid points. Moreover, time integration with both the FD and FE methods is carried out using a predictor–corrector method and time-step sizes are determined adaptively. Finally, solutions obtained with the I, FD, and FE methods for nonlinear isotherms are compared with respect to their accuracy and efficiency. The new algorithms should be of interest for several reasons to researchers who work on problems of diffusion and adsorption of surface-active substance at interfaces. First, many researchers who work on such problems use primarily the I and FD methods but without evaluating computational errors in or optimizing the efficiency of their algorithms. Second, the FE method, while a popular technique for solving ordinary and partial differential equations, has not been used in past studies of such problems.

Section snippets

Governing equations

For dynamic adsorption of a nonionic surfactant below its cmc, molecules are transported through a diffusion layer of length l and adsorb onto the interface. For a stagnant medium, l would be practically infinite. However, because of mixing, convection in the bulk phase can increase the rate of adsorption by decreasing the effective diffusion layer thickness and maintaining a nearly constant bulk concentration at x=l, where x is the coordinate measured from the interface. A model with a finite

Code validation with the Henry isotherm

The codes implementing the algorithms based on the I, FD, and FE methods have been written in FORTRAN. The calculations have been carried out on a Sun Ultra 60 workstation at Purdue University. For code validation and studying the effects of time-step sizes and the number of mesh points on solution accuracy, the codes have been tested for the limiting case of a linear, or Henry, isotherm for which an exact solution exists [16], $θ(τ)=1−2N_{l} ∑ n=1 ∞ e^{−β_{n}^{2}τ/N_{l}^{2}} β_{n}^{2} +N_{l} +N_{l}^{2},$ where the β_n's are the roots

Conclusions

Three methods for solving the problem of one-dimensional diffusion-controlled dynamic adsorption in planar coordinates are analyzed. A technique for error estimation in the integral (I) method with the trapezoidal rule approximation is developed. Error estimation is then used to choose adaptively time-step sizes in the I method. The analytical solution of the linear Henry isotherm, used for code validation and convergence tests, shows that the I method with adaptive time-step sizes is more

Acknowledgements

This research was supported in part by grants from the National Science Foundation (Grant CTS 96-15649, CTS 96-15649, and 0135317) and the National Institutes of Health (Grant HL 54641-02) to Elias I. Franses, and a grant from the Basic Energy Sciences Program of the US Department of Energy (Grant DE-FG02-96ER14641) to Osman A. Basaran.

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Computation of dynamic adsorption with adaptive integral, finite difference, and finite element methods

Abstract

Introduction

Section snippets

Governing equations

Code validation with the Henry isotherm

Conclusions

Acknowledgements

Colloids Surf. A

Colloids Surf.

Colloids Surf. A

Chem. Eng. Sci.

J. Colloid Interface Sci.

Adv. Colloid Interface Sci.

Biotechnol. Bioeng.

Ind. Eng. Chem. Res.

Ann. Biomed. Eng.

Phys. Fluids

J. Chem. Phys.

Aust. J. Sci. Res. Ser. A

Colloid Polym. Sci.