Rigorous calculations of permeation in mixed-matrix membranes: Evaluation of interfacial equilibrium effects and permeability-based models Tanya Singh, Dun-Yen Kang, Sankar Nair n School of Chemical & Biomolecular Engineering, Georgia Institute of Technology, 311 Ferst Drive NW, Atlanta, GA 30332-0100, USA article info Article history: Received 16 June 2013 Received in revised form 2 August 2013 Accepted 5 August 2013 Available online 12 August 2013 Keywords: Mixed matrix membranes Permeation Modeling Finite-element Metal-organic frameworks abstract 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 (D f /D m ) can be lumped into a single parameter, the permeability ratio P f /P m ¼KD f /D m . It is shown here that the individual values of K and D f /D m , and not the combined permeability ratio P f /P m , 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, CO 2 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’ CO 2 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 perme- ability 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. & 2013 Elsevier B.V. All rights reserved. 1. Introduction Membranes offer an energy-efficient alternative to traditional thermodynamically-controlled separations [1]. Polymeric mem- branes provide a range of molecular transport properties, rela- tively easier processing techniques, and a low membrane fabrication cost per unit membrane area. However, polymeric membranes face an intrinsic trade-off between the permeability and selectivity [2]. A widely-taken approach to overcome this trade-off is to incorporate higher-performance nanoporous parti- cles (zeolites, metal-organic frameworks, or nanoscale materials such as porous layers or nanotubes) as fillers into polymeric membranes. Such membranes are also referred to as ‘mixed- matrix’ membranes (MMMs) [3–6]. These membranes have been shown to yield enhanced separation performance (higher perme- ability, higher selectivity, or both), and can preserve to a large extent the good processibility characteristics of polymeric membranes. Several analytical models, such as the Maxwell, Bruggeman, Pal, Lewis–Nielsen, and other models, have been developed to under- stand and predict the effective permeability and selectivity of MMMs. These analytical models are described in detail in previous reviews [7,8]. A significant – yet rarely 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 effec- tive permeability. Effectively, they employ the permeabilities of the Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/memsci Journal of Membrane Science 0376-7388/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.memsci.2013.08.010 n Corresponding author. Tel.: þ1 404 894 4826; fax: þ1 404 894 4200. E-mail address: sankar.nair@chbe.gatech.edu (S. Nair). Journal of Membrane Science 448 (2013) 160–169