In this study membrane permeation was measured from finely divided pure powder and saturated aqueous solutions of two test chemicals with low vapor pressure, methyl paraben and 4-cyanophenol, through silicone rubber (polydimethylsiloxane) using diffusion cells operated to insure the concentration was negligible at the interface between the membrane and the receptor solution. The steady-state flux from the pure powder was substantial and similar to that from a saturated aqueous solution, but smaller by an amount that was statistically significant. However, there was no statistically significant difference in the ratios of the fluxes from the pure powder and saturated water for MP and CP (0.66±0.10 and 0.72±0.07, respectively), suggesting a physical rather than chemical cause for the different flux results for powder and saturated water. One explanation is that diffusion proceeds from the entire membrane surface for the saturated solution, but only from the membrane surface in direct contact with the pure powder for these nearly non-volatile chemicals. In contrast to these new results, an earlier study saw no difference in the rate or amount of absorption of 3- and 4-cyenophenol into silicone rubber membranes mounted on an attenuated total reflectance Fourier transform infrared (ATR-FTIR) crystal. The results from mathematical model simulations of non-volatile chemical sources covering only a fraction of the membrane surface in the diffusion cell and ATR-FTIR experimental systems suggest that the measurement variability was large enough to make the earlier study insensitive to differences in the powder and saturated solutions. From a comparison of the mathematical model simulations to the new diffusion cell results, we estimate that less than 5% of the membrane surface had direct contact with chemical in the powder experiments.

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%.