(380ae) Modeling Pressure-Induced Diffusion of Water in Dense Polymer Membranes

Understanding the molecular mechanisms that govern transport through polymeric membranes is critical for the rational design of novel materials and modelling of processes. Pressure-driven membrane separations of solvents and dissolved solids, such as reverse osmosis and organic solvent nanofiltration, are widely employed in industry. Despite this, there is still disagreement over the mechanism by which solvent transport occurs in response to a hydraulic driving force. Models are broadly classified by whether they assume that solvent is transported down a hydraulic pressure gradient, i.e., pore-flow, or if solvent is induced to diffuse across the membrane due to the equilibrium sorption at the feed and permeate faces of the membrane, i.e., solution-diffusion. One key prediction of solution-diffusion models is the existence of a non-linear relationship between pressure and flux leading to a ceiling, or limiting, flux at very high transmembrane pressure. However, this effect is often viewed as too small to be readily observed for water.

In our work, we demonstrate a non-linear relationship between water flux and transmembrane pressures of up to 3700 psig for two rubbery polymer membranes, Nafion 117 and crosslinked poly(ethylene glycol diacrylate). We additionally observe a linear relationship for a glassy polymer, cellulose acetate. The non-linearity we observe is much greater in magnitude than is typically expected for water permeation. To explain this, we independently measure the water sorption isotherm for these materials, as well as estimate the partial molar volume of sorbed water in the membrane phase from the hydrated density. By combining these measurements with plausible thermodynamic assumptions, flux can be described well for both the rubbery and glassy polymers with only the water diffusion coefficient as a fitting parameter. This suggests that the large degree of non-linearity we observe is a thermodynamic effect which can be rationalized within the solution-diffusion framework, rather than a mechanical effect such as compaction.

Additionally, we are able to rationalize the diffusion of water in the membranes by employing the Mackie-Meares model to predict water diffusion coefficients as a function of composition. The Mackie-Meares model is based on statistical arguments and requires no adjustable parameters yet has been observed to be effective in predicting diffusion coefficients in ion exchange membranes and PEGDA-based hydrogels. In conjunction with the aforementioned thermodynamic model, we achieve a nearly-quantitative parameter-free prediction of solvent flux through both XLPEGDA and Nafion 117.

These results suggest the validity of the assumptions underlying the solution-diffusion model for describing transport in water-swollen membranes. We demonstrate that the stronger than expected non-linearity in the pressure-flux relationship can be parsimoniously attributed to the thermodynamics of the water-polymer interaction and does not necessarily imply a mechanical effect. Simultaneously, we demonstrate that the lack of such non-linearity similarly does not necessarily disprove the solution-diffusion hypothesis in favor of pore-based explanations. Rather, additional information is required to infer the transport mechanism from transport experiments.