(496e) Rigorous Calculations of Permeation in Mixed-Matrix Membranes

Polymeric membranes face an intrinsic trade-off between the permeability and selectivity. A widely-taken approach to overcome this trade-off is to incorporate higher-performance nanoporous particles (e.g., zeolites, metal-organic frameworks) as fillers into polymeric membranes. Such membranes are also referred to as ‘mixed-matrix’ membranes (MMMs), and have been shown to yield enhanced separation performance (higher permeability, higher selectivity, or both).

Several analytical models have been developed to understand and predict the effective permeability and selectivity of MMMs, such as the Maxwell, Bruggeman, Pal, Lewis-Nielsen, and other models. The most significant – yet least discussed – limitation of the above models is that none of them consider the effects of the adsorption equilibrium at the polymer/filler interface on the effective permeability. This can lead to qualitatively and quantitatively erroneous interpretations of permeability data from MMMs when analyzed with the above ‘permeability-based’ models, since most polymer/filler interfaces will not have an interfacial equilibrium constant of unity. Furthermore, the models often cannot successfully interpret experimental MMM permeation data, or reconcile data from different sources, without the postulation of ‘non-idealities’ such as the presence of interfacial voids or rigidified polymeric regions at the interface with the filler. These hypotheses introduce additional fitting parameters and are difficult to verify by independent characterization.

With the advent of a large new class of nanoporous MOF materials as MMM fillers, a large quantity of permeation data is emerging that is increasingly difficult to interpret. A fundamental difficulty is that the permeability-based models do not allow one to separate the effects of interfacial equilibrium from the effects of hypothesized non-idealities. Thus, there is a requirement for rigorous predictions of permeation in ideal MMMs that capture the true dependence of the effective permeability on the diffusivities of the matrix and filler phases, the adsorption equilibrium at the polymer/filler interface, and the volume fraction of the filler. Such predictions can then be used for reliable interpretation of MMM data, assessment of the effects of the intrinsic material properties versus those originating from possible non-idealities, and selection of matrix and filler materials for desired separation properties.

We will discuss rigorous calculations of single-component permeation in mixed-matrix membranes (MMMs), and show their value in developing a reliable understanding of permeation behavior in these membrane architectures. 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 interfacial equilibrium between the matrix and the filler, in addition to the differences in molecular diffusivity between the two phases. Analytical models (e.g., Maxwell model) can only predict the MMM permeability when the interface equilibrium constant K=1. Most real MMMs do not satisfy this condition. It is shown that the individual values of the equilibrium constant K and the diffusivity ratio D_f/D_m, and not the combined permeability ratio P_f/P_m = K.D_f/D_m, determine the MMM permeability. We then use our ‘exact’ predictions to examine some of the current speculations regarding MMM permeation behavior. We use CO₂ solubility and diffusivity data on example filler and polymer materials to show that the use of permeability-based models to analyze MMM permeation data leads to spurious results. Our simulations reveal that the incorrect permeabilities obtained from the Maxwell (and other) models are likely to mislead researchers to hypothesize non-idealities such as matrix-dependent filler behavior or rigidification of the matrix polymer near the interfaces. Our simulations also indicate clearly that an ideal MMM shows no significant direct effect of filler particle size. Finally, we fit our computational data to an empirical correlation that can be used to calculate MMM permeabilities given adsorption and diffusion data for the matrix and filler phases.