Mathematical models of diffusion through membranes from spatially distributed sources

The objective was to numerically simulate diffusion through a membrane from regularly spaced sources distributed on the membrane surface. Two examples in which this physical situation arise are chemicals applied to the membrane as powdered materials or deposited in a volatile solvent that evaporates leaving behind a residue that partly covers the surface. Transient and steady-state finite element models in 2D and 3D were constructed to simulate diffusion from chemical sources distributed in a regular network on the surface of a membrane subject to either zero concentration or no flux on the membrane surface opposite to the surface in contact with the source. We assumed that local equilibrium was established with the membrane surface in direct contact with the chemical sources of constant concentration and that chemical did not enter or leave the membrane surface in the regions with no chemical contact. We calculated solutions for linear and square chemical sources as a function of the distance between these sources relative to the membrane thickness and as a function of the fraction of the membrane surface covered, including surface fractions that were smaller than have been considered previously. When sources are closely spaced relative to the membrane thickness, they interact such that flux from a spatially distributed source cannot be distinguished from a source that uniformly covers the membrane surface. When the distance between sources is large compared to the membrane thickness, there is no interaction between sources, and the effects of source regions are simply additive. The lag time associated with diffusion across the membrane, when plotted as a function of distance between source regions, has a maximum value that corresponds to the onset of interaction between source regions. The steady-state flux from line and square sources are similar when they cover more than about 25% of the membrane surface, but as the surface area in contact with the source decreases below 25%, the flux from line sources is increasingly greater than from squares. The differences between lines, squares and also circles of the chemical source covering the same fraction of the membrane surface can be explained by differences in the perimeter of the source. An algebraic equation for steady-state flux that was fit by regression to the finite difference solutions of linear and square sources covering 20% or more of the surface reported by Itoh et al. [N. Itoh, T.H. Wu, K. Haraya, Two- and three-dimensional analysis of diffusion through a dense membrane supported on a porous material, J. Membr. Sci. 99 (1995) 175–183] is inaccurate when the area fraction covered is smaller than 20%.