Rigorous calculations of permeation in mixed-matrix membranes: Evaluation of interfacial equilibrium effects and permeability-based models

We present rigorous calculations of single-component permeation in mixed-matrix membranes (MMMs), and show their importance in developing a more reliable understanding of MMM permeation behavior. We first develop methods for the construction of detailed and large-scale 3D mixed-matrix membrane (MMM) models, which are then solved by finite-element methods. Our models explicitly account for the effects of matrix-filler interfacial equilibrium in addition to the differences in Fickian diffusivity between the two phases. Analytical equations (e.g., Maxwell model) can only predict the MMM permeability under an implicit assumption that the interfacial equilibrium constant K and the diffusivity ratio of the filler and the matrix (Df/Dm) can be lumped into a single parameter, the permeability ratio Pf/Pm=KDf/Dm. It is shown here that the individual values of K and Df/Dm, and not the combined permeability ratio Pf/Pm, determine the MMM permeability. Our simulations also indicate that an ideal MMM shows no significant direct effect of filler particle size. We fit our computational data to an empirical correlation that can be easily and accurately used to calculate ideal MMM permeabilities, given equilibrium and diffusion data for the matrix and filler. We also examine some current issues regarding interpretation of MMM permeation behavior. For example, CO2 solubilities and diffusivities in representative MOF filler and polymer matrix materials are used to rigorously compare the 'exact' predictions with permeability-based models. The rigorous calculations show non-monotonic behavior of the MMM permeability as a function of the matrix permeability, which cannot be predicted by permeability-based models. Also, the 'apparent' CO2 permeability of ZIF-8 fillers extracted with Maxwell and Lewis-Nielsen models from the computational MMM permeation data, varies by 3 orders of magnitude depending upon the matrix polymer. Though the ZIF-8 filler maintains a constant permeability of ~3000 Barrer, the permeability models would require postulation of (spurious) non-idealities such as matrix-dependent filler behavior or interfacial rigidification to explain the results. Overall, this work provides a method for more reliable use of models to understand and design MMMs, as well as to better interpret the large and growing body of experimental data on these membranes.