(144h) 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, such as reverse osmosis, are widely employed in industry. Despite this, there is still disagreement over the mechanism by which hydraulic solvent transport occurs. The solution-diffusion (SD) model proposes that the membrane comprises a homogeneous fluid phase through which penetrants diffuse and is widely believed to describe the transport of penetrants in dense polymer membranes. In the SD model, the application of a transmembrane hydraulic pressure difference, ∆P, primarily acts to establish a concentration gradient of penetrant across the membrane, leading to diffusion. However, in recent literature, several studies have claimed that solvent transport in dense polymer membranes occurs via “pore-flow”, i.e., via a pressure gradient through the membrane, which is modeled as a porous solid.

In this study, we demonstrate a that the SD model accurately describes the hydraulic water flux (J_w) through several different polymers. Generally, J_w is observed to be a linear function of ∆P for reverse osmosis membranes. The SD model can be simplified to yield this result, where J_w is proportional to the hydraulic permeance. However, in the SD model, flux is proportional to the transmembrane concentration difference, rather than ∆P, and this has been shown to manifest as a non-linear relationship between flux and ∆P for some organic solvent penetrants. In this study, we measure the relationship between Jw and ∆P at pressures up to ~240 bar. This relationship is linear for dense films of cellulose acetate, a glassy polymer. However, it is highly non-linear for Nafion 117 (in the sodium counterion form) and crosslinked poly(ethylene glycol diacrylate) (PEGDA) films, which are rubbery under the conditions of this study. We measure the water sorption isotherm for all materials and show that J_w is linearly proportional to the transmembrane concentration difference, as predicted by the SD model. We show that the relationship between J_w and ∆P is described well via the SD model and can even be predicted, with nearly quantitative accuracy and without the use of any fitting parameters, by employing the Mackie-Meares model to estimate the water diffusion coefficient in each polymer.

These results suggest the validity of the SD model for water transport in dense polymer membranes, in contrast to pore-flow models. However, we also show that the simplified result of the SD model, that J_w is proportional to the hydraulic permeance, is not valid for these materials, even when the relationship between J_w and ∆P is linear. Specifically, the transmembrane concentration difference induced by pressure is larger than that predicted by the simplified SD result, by as much as five times, for the materials considered here, and more generally for materials with non-linear sorption behavior. This result highlights the importance of understanding the thermodynamic relationship between penetrant activity and sorption when evaluating the accuracy of the predictions of the SD model.