(651c) Analysis of Finite Difference Schemes for Diffusion in Spheres With Variable Diffusivity

The partial differential equation (PDE) in spherical coordinates for mass transport by diffusion (Fick's second law) and for heat transport by conduction with a constant diffusivity is readily solved to closed form analytical solutions for common boundary conditions [1, 2]. In the most general case of variable diffusivity with an arbitrary, nonlinear functional form, the PDE is not separable and must be solved numerically. If the diffusivity is constant, the well-known second-order centered finite difference discretization scheme can numerically approximate for the spatial derivatives of the PDE [1, 3]. However, analogous discretization schemes for the general case of variable diffusivity have not been analyzed previously in the literature. Variable diffusivity is important in physical systems such as drying or wetting of food products (e.g., cereals, legumes, and tomatoes) that have been observed to have concentration-dependent water diffusivity [4,5] , diffusion of acid through chemically-amplified resist materials undergoing photolithography where the diffusivity depends on the extent of the chemical reactions [6], and diffusion of medicines through drug-loaded, biodegradable polymer microspheres that have internal porosity that changes with chemical degradation thus modifying the effective diffusivity of the drug [7-9]. Both the food product wetting/drying and polymer microsphere drug delivery examples involve diffusion through a spherical domain.

The method of lines can be used to reduce the PDE to a system of ordinary differential equations (ODEs) by discretizing the radial dimension onto a finite grid using some finite difference discretization scheme [10]. The resulting system of semi-discrete ODEs for the species concentration at each grid point can be solved using a standard ODE solver such as RADAU5, an implicit, 4-5th order Runge-Kutta solver with adaptive time-stepping [11]. The challenge for solving the PDE numerically by this technique is in the choice of finite difference discretization scheme to handle the variable diffusivity term accurately and conservatively.

Here, we present and compare three finite difference discretization schemes for numerically approximating the spatial derivatives of the diffusion equation in spherical coordinates with a variable diffusivity term. Five diffusivity cases are treated with the schemes: 1) constant diffusivity, 2) time-dependent diffusivity, 3) spatially-dependent diffusivity, 4) concentration-dependent diffusivity, and 5) implicitly time-dependent and spatially-dependent diffusivity. Analytical or semi-analytical solutions are available for verification of the first four cases. The fifth case shows the performance of the schemes on the complex case of variable diffusivity dependent on the dynamic porosity of the sphere, which in turn depends on the concentration of a different reacting species distributed throughout the sphere [8]. The results of this analysis yield the recommendation for using one of the discretization schemes as the preferred finite difference method for numerically solving the diffusion equation in spheres with variable diffusivity.

References

[1] J. Crank, The Mathematics of Diffusion, Oxford University Press, Oxford, 2nd edition, 1975.

[2] H. S. Carslaw, J. C. Jaeger, Conduction of Heat in Solids, Oxford University Press, New York, 2nd edition, 1986.

[3] M. N. Özişik, Heat Conduction, JohnWiley & Sons, New York, 2nd edition, 1993.

[4] K. H. Hsu, A diffusion model with a concentration-dependent diffusion coefficient for describing water movement in legumes during soaking, J. Food Sci. 48 (1983) 618–622.

[5] G. Xanthopolous, S. Yanniotis, A. G. Boudouvis, Numerical simulation of variable water diffusivity during drying period of peeled and unpeeled tomato, J. Food Sci. 77 (2012) E287–E296.

[6] J. S. Petersen, C. A. Mack, J. L. Sturtevant, J. D. Byers, D. A. Miller, Nonconstant diffusion coefficients: Short description of modeling and comparison to experimental results, Proc. SPIE 2438 (1995) 167–180.

[7] A. N. Ford Versypt, D. W. Pack, R. D. Braatz, Mathematical modeling of drug delivery from autocatalytically degradable PLGA microspheres—A review, J. Controlled Release 165 (2013) 29–37.

[8] A. N. Ford Versypt, Modeling of Controlled-Release Drug Delivery from Autocatalytically Degrading Polymer Microspheres, Ph.D. Dissertation, University of Illinois at Urbana-Champaign, Urbana, IL, 2012.

[9] A. N. Ford, D. W. Pack, R. D. Braatz, Multi-scale modeling of PLGA microparticle drug delivery systems, in: E. N. Pistikopoulos, M. C. Georgiadis, A. C. Kokossis (Eds.), 21st European Symposium on Computer Aided Process Engineering: Part B, Interscience, New York, 2011, pp. 1475–1479.

[10] R. J. LeVeque, Finite Difference Methods for Ordinary and Partial Differential Equations: Steady-State and Time-Dependent Problems, SIAM, Philadelphia, 2007.

[11] E. Hairer, G. Wanner, Solving Ordinary Differential Equations II: Stiff and Differential-Algebraic Problems, Springer, New York, 2nd edition, 1996.