Rigorous calculations of permeation in mixed-matrixmembranes: Evaluation of interfacial equilibrium effectsand 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

a r t i c l e i n f o

Article history:Received 16 June 2013Received in revised form2 August 2013Accepted 5 August 2013Available online 12 August 2013

Keywords:Mixed matrix membranesPermeationModelingFinite-elementMetal-organic frameworks

a b s t r a c t

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 permeationbehavior. We first develop methods for the construction of detailed and large-scale 3D mixed-matrixmembrane (MMM) models, which are then solved by finite-element methods. Our models explicitlyaccount for the effects of matrix-filler interfacial equilibrium in addition to the differences in Fickiandiffusivity between the two phases. Analytical equations (e.g., Maxwell model) can only predict theMMM permeability under an implicit assumption that the interfacial equilibrium constant K and thediffusivity ratio of the filler and the matrix (Df/Dm) can be lumped into a single parameter, thepermeability ratio Pf/Pm¼KDf/Dm. It is shown here that the individual values of K and Df/Dm, and notthe combined permeability ratio Pf/Pm, determine the MMM permeability. Our simulations also indicatethat an ideal MMM shows no significant direct effect of filler particle size. We fit our computational datato 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 issuesregarding interpretation of MMM permeation behavior. For example, CO2 solubilities and diffusivities inrepresentative 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 ofthe MMM permeability as a function of the matrix permeability, which cannot be predicted bypermeability-based models. Also, the ‘apparent’ CO2 permeability of ZIF-8 fillers extracted with Maxwelland Lewis–Nielsen models from the computational MMM permeation data, varies by 3 orders ofmagnitude 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-idealitiessuch as matrix-dependent filler behavior or interfacial rigidification to explain the results. Overall, thiswork provides a method for more reliable use of models to understand and design MMMs, as well as tobetter 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 traditionalthermodynamically-controlled separations [1]. Polymeric mem-branes provide a range of molecular transport properties, rela-tively easier processing techniques, and a low membranefabrication cost per unit membrane area. However, polymericmembranes face an intrinsic trade-off between the permeabilityand selectivity [2]. A widely-taken approach to overcome thistrade-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 polymericmembranes. Such membranes are also referred to as ‘mixed-matrix’ membranes (MMMs) [3–6]. These membranes have beenshown to yield enhanced separation performance (higher perme-ability, higher selectivity, or both), and can preserve to a largeextent the good processibility characteristics of polymericmembranes.

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 ofMMMs. These analytical models are described in detail in previousreviews [7,8]. A significant – yet rarely discussed – limitation of theabove models is that none of them consider the effects of theadsorption equilibrium at the polymer/filler interface on the effec-tive permeability. Effectively, they employ the permeabilities of the

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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

two phases (filler and polymer matrix) to describe the overallmembrane permeability. The interfacial adsorption equilibrium isonly implicitly included in the sense of a ‘bulk’ quantity, within thefiller and matrix permeabilities. As pointed out in a recent work [9],this produces a situation equivalent to assuming an interfacialequilibrium constant K�1. This can lead to qualitatively andquantitatively erroneous interpretations of permeability data fromMMMs when analyzed with the above ‘permeability-based’models,since most polymer/filler interfaces will not have an interfacialequilibrium constant of unity. Furthermore, the models oftencannot 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 rigidifiedpolymeric regions at the interface with the filler. These effects aretaken into account by modifications of the permeability-basedmodels to include intermediate regions at the interfaces betweenthe matrix and the filler, as exemplified by investigations of zeoliteA/polymer and carbon molecular sieve/polymer MMMs [10–12].However, these modifications – generally involving the inclusion ofan ‘interphase’ structure between the filler and the matrix [13] –introduce additional fitting parameters into the models such as theinterphase thickness and interphase permeability, which are diffi-cult to verify by independent characterization. With the advent of anew class of nanoporous metal-organic framework (MOF) materialsas MMM fillers, a large quantity of permeation data is emerging [8]that is increasingly difficult to interpret, e.g., the case of the MOFZIF-8 [14,15] considered later in this paper.

Although non-idealities may actually exist in several types ofMMMs, a reliable assessment of their contribution to the permea-tion behavior would first require an accurate understanding of theexpected intrinsic permeation behavior in an ideal (defect-free)MMM. A fundamental difficulty is that the permeability-basedmodels do not allow one to reliably separate the effects of the idealpermeation behavior from the effects of hypothesized non-idealities. The effective medium theory (EMT) [16,17] explicitlyaccounts for interfacial equilibrium effects in addition to theFickian diffusivity in both phases, and has been recently consid-ered in detail [9] to illustrate the importance of matrix-fillerinterfacial adsorption in MMMs. However, it was also found tohave several drawbacks (Section 3.2) that prevent its general usefor quantitative predictions or fitting of permeation data fromcomposite membranes [9]. Boom et al. [18] developed a simple 2Dnumerical simulation of a single filler particle placed in a matrix toqualitatively explain trends in experimentally observed pervapora-tion behavior of zeolite/polymer MMMs. Although the predictionsfrom that study could not be directly extended to 3D MMMscontaining randomly distributed fillers, the simulation resultsindicated that the filler/matrix interfacial adsorption equilibriumplayed a large role in determining the permeation behavior in amanner that cannot be explained by the permeability-basedanalytical models. It has also been found that functional MMMsincorporating MOFs are often free of obvious defects such asinterfacial voids or poor dispersion, due to their generally bettercompatibility with polymeric phases [8]. Hence, it also becomesimportant to accurately describe the intrinsic effects of interfacialadsorption and diffusion in MMM systems in order to predict andselect filler and matrix materials for high performance in targetseparations. Thus, there is a requirement for ‘rigorous’ predictionsof permeation in ideal MMMs that capture the true dependence ofthe permeability on the diffusivities of the matrix and filler phases,the polymer/filler interface adsorption equilibrium, and thevolume fraction of the filler. Such predictions can then be usedfor reliable interpretation of MMM data, assessment of the effectsof the intrinsic material properties versus those originating frompossible non-idealities, and selection of matrix and filler materialsfor desired separation properties.

Towards the above purpose, the objectives of this paper arethree-fold. Firstly, we develop techniques to construct large (up to30 μm thickness�225 μm2 surface area), three-dimensional (3D),statistically valid, models of MMMs with different sphericalparticle sizes and loadings, and then obtain single-componenttransport properties of such membranes via computational finite-element methods within the COMSOL Multiphysics simulationpackage. These predictions can be considered as ‘exact’ to thelimits of accuracy of the numerical simulation methods. Single-component transport is described by Fick's law and the adsorptionequilibrium at the filler–matrix interfaces is explicitly invoked as aboundary condition. Secondly, we use our predictions to conduct adetailed investigation of the effective permeability of MMMs as afunction of filler loadings, particle size, and the properties of theindividual polymeric and filler phases. We compare the ‘exact’computational results with those of existing analytical models,and discuss the implications of our findings on the interpretationof MMM permeation data with permeability-based models. Thisdiscussion also considers examples of real filler and matrixmaterials, although a more comprehensive comparison of the‘exact’ predictions with experimental data is a study in itself andwill be reserved for a separate work. Thirdly, we introduce anempirical correlation that satisfactorily fits our data over a largerange of adsorption and diffusion parameters and thereby allowseasier application of our results.

2. Computational methods

2.1. Model assumptions and boundary conditions

Finite element modeling of mass transport was employed for a3D heterogeneous composite membrane system under single-component Fickian diffusion of penetrant molecules and includingadsorptive equilibrium at the filler–matrix interfaces. Since a mainpurpose of the present paper is the direct comparison to analyticalpermeability-based models, the diffusion coefficients for the filler(Df) and matrix (Dm) phases were not taken to be functions of thelocal penetrant concentration in this paper, although theconcentration-dependence of the Fickian diffusivity can be incor-porated into the computational model without difficulty todescribe nanoporous fillers like zeolites or MOFs in more general-ized terms. Adsorptive equilibrium was assumed to be establishedat the matrix–filler interfaces, and was described by aconcentration-independent equilibrium constant. Again, thisallows ease of definition of the permeabilities and direct compar-ison with analytical models, although the concentration-dependence of the interface equilibrium can be incorporated intoour computational model by a more detailed adsorption isotherm.The objective of the simulation is to calculate the ‘effectivediffusivity’ (Deff) of the MMM. It should be noted that Deff is nota microscopic diffusivity and cannot be used in a differentialequation such as Fick's law. It is a macroscopic membrane trans-port quantity that, when multiplied by a linear concentrationgradient between the feed and permeate side, conveniently givesthe flux through the MMM in the same manner as the fluxcalculation for a pure polymeric membrane having a constantdiffusivity Dm and the same thickness and concentration differ-ential as the MMM. The ratio of these two fluxes therefore givesDeff/Dm. Considering the macroscopic definition of Deff above, theratio Deff/Dm exactly equals the permeability ratio Peff/Pm, since theadsorption equilibria are the same for the MMM and the purepolymer membrane at the fluid–membrane interfaces on the feed-side and permeate-side.

The boundary conditions used in our simulations are given inFig. 1, which is a 2D projection of the matrix–filler system with

T. Singh et al. / Journal of Membrane Science 448 (2013) 160–169 161

only one filler shown. The concentrations at the membrane feedand permeate sides were set to 1 mol/m3 and 0 mol/m3 respec-tively. To allow permeation to occur only in the x direction,insulation boundary conditions were employed at the edges ofthe membrane model in the y and z directions. Given the largephysical size of the models, the insulation boundary conditionsproduced results practically identical to models with periodicboundary conditions in the y and z directions. The equilibriumconstant K at the matrix–filler interfaces was set as the ratio ofconcentrations (C2 and C1) of the diffusing species at the interfaceon the filler and the matrix sides respectively.

2.2. Model construction and numerical methods

COMSOL Multiphysics was used to create and simulate 3Dcomposite membrane models. Initially, we constructed modelswith uniformly distributed fillers to determine the model sizesrequired for reliable results that are free of any simulation box sizeartifacts. A single ‘filler in a box’ building block was first con-structed. For a given filler, the box size was modified to achievedifferent filler loading fractions (by volume). The interfacialboundary condition of adsorption equilibrium was manually setat the surface of each filler particle using the COMSOL userinterface. This building block was then replicated to construct full3D models of varying width, thickness and height. The concentra-tion boundary conditions were added at the feed and permeatefaces of the 3D model, and insulation (i.e., zero flux) conditionswere specified at the other four faces. Models of varying sizes weresimulated to determine the appropriate dimensions for obtainingresults free of any numerical artifacts.

Physically realistic modeling of MMMs requires simulationswith membranes containing statistically random dispersions offillers. The creation of such models was significantly more com-plicated. The 3D models with randomly dispersed fillers werecreated via the COMSOL feature that allows linking with MATLABprograms. A MATLAB program was developed to perform threekey functions. For any given filler size, filler volume fraction, andoverall 3D model box size, it first calculates the number of fillerparticles required and places them in random, non-overlappingpositions by means of a Metropolis Monte Carlo method imple-mented with a hard-sphere model of particle interactions. Sec-ondly, the program creates a numerical description of the entire3D model, including the coordinates of the center-of-mass of eachfiller, the coordinates of each filler–matrix interface, and theadsorption boundary conditions at the interfaces. Thirdly, theprogram provides all the above information to COMSOL, whichthen creates the finite-element discretization of the membranemodel and solves the mass transport equations. Therefore, unlikethe uniformly distributed filler models, the randomly distributed

models involve programming in all the features of an arbitrarilylarge membrane model instead of manually creating or selectingthem from the COMSOL user interface. This non-trivial, automated,approach to the model construction speeds up the generation ofany given model by many orders of magnitude, and allows us tocreate a variety of detailed models that can be used to interrogatemass transport phenomena in composite membranes. For thefinite element solution, the volumetric mesh type selected was‘tetrahedral mesh’. Fig. 2 shows the tetrahedral meshing per-formed on a 15�15�15 μm3 model with fillers of size 2 μm at30 vol% loading. The mesh density was varied at different resolu-tions between the ‘extremely fine’ and ‘normal’ limits in COMSOL.The results were accepted as converged with respect to the meshdensity, when a further increase in mesh density produced nosignificant change in the value of Deff/Dm (to third decimal place).For example, the 15�15�15 μm3 model with 2 μm fillers at30 vol% loading (Fig. 2) has an average mesh density of 112 ele-ments/mm3 in the filler phase and 72 elements/mm3 in the matrixphase. The iterative solver used the damped version of Newton'smethod. Typically, 10–20 solver iterations were sufficient toconverge the total membrane flux to within a fractional accuracyof 10�4. The fractional accuracy of convergence of the localconcentrations was better than 10�6.

2.3. Processing and analysis of the numerical solutions

After the convergence of the finite element simulation, the rawnumerical data were post-processed to obtain the effective Fickiandiffusivity (Deff) of the composite membrane. The data post-processing involved the calculation of the integrated molar flowrate of yz-slices normal to the permeation direction (x), fromwhich the flux was calculated via division by the area of the slice.Several check protocols were applied to all models. These proto-cols included the following steps: (1) the flux in different yz-slicesof the membrane (taken at different x-positions) was calculatedand checked to be the same, as required under steady-stateconditions, (2) the concentration ratios between the filler andthe matrix were checked at the filler–matrix interfaces to ensurethat the adsorption equilibrium condition was correctly enforced,and (3) for all the constructed models, the effective diffusivity(Deff) was checked to determine whether Deff¼Df¼Dm whenboth the matrix and the filler have the same diffusivity and an

Fig. 1. Schematic 2D projection of MMM model showing one filler in the matrix.

Fig. 2. Tetrahedral mesh representation of a 15�15�15 μm3 MMM model withrandomly dispersed fillers of size 2 μm at 30 vol% filler loading.

T. Singh et al. / Journal of Membrane Science 448 (2013) 160–169162

interfacial equilibrium constant of unity. The MMM flux was thendivided by the flux obtained from a model containing only thematrix phase, to obtain Deff/Dm which is the same quantity as theeffective permeability ratio Peff/Pm estimated by the permeability-based models. Fig. 3a and 3b show example concentration profilesobtained for the model of Fig. 2, with the diffusivity ratio Df/Dm

and equilibrium constant K both equal to 1. Fig. 3a shows theconcentration profiles in both the matrix and filler domains,whereas Fig. 3b shows the concentration profile in the fillers only.

3. Results and discussion

3.1. Mass transport with interface equilibrium constant of unity

The Maxwell model is commonly employed to fit or predictmixed-matrix membrane permeability data [7,8,19,20]. It is anadaptation of the expression originally derived to predict electricalconduction in heterogeneous media. The fitting of MMM perme-ability data to the Maxwell model has an implicit assumption ofnear-continuity of the species concentration at the filler–matrixinterfaces [9], corresponding to K¼1 in our simulations. Fig. 4ashows the comparison of the effective diffusivity (Deff/Dm) fromthe ‘exact’ computational model with the predictions from the

Maxwell model, for a membrane with 2 μm fillers and differentvalues of Df/Dm. It is clear that the Maxwell model makes a fairlygood prediction of the membrane permeability at low volumefiller volume fractions (ϕ) and low values of Df/Dm. The Maxwellmodel is inaccurate at higher volume fractions and for high-permeability fillers. Fig. 4b shows Deff/Dm versus the filler volumefraction for cases with Df/Dm¼10 and 100. The 3D MMM modelcontained randomly distributed fillers of 2 μm size, and theequilibrium constant was set to unity. The ‘exact’ answers are

Fig. 3. Concentration profiles of (a) both matrix and filler domains, and (b) fillerdomain only, of a membrane model with randomly distributed fillers. Length unitsare in μm and concentration in mol/m3.

Fig. 4. (a) Comparison of COMSOL simulations (symbols) with Maxwell model(solid lines) for Deff/Dm (effective diffusivity) versus ϕ (volume fraction), for Df/Dm¼10 (black), 100 (green), 1000 (blue), and 10,000 (red); with K¼1.(b) Comparison of COMSOL simulations (symbols) with analytical predictions ofMaxwell (black lines), Bruggeman (green lines), Pal (blue lines) and Lewis–Nielsen(red lines) models for Deff/Dm versus ϕ for Df/Dm¼10 (red symbols and solid lines)and 100 (green symbols and dashed lines); with K¼1. The 3D composite modelcontained randomly distributed fillers of 2 μm size. Error bars for each data pointare obtained by averaging results from three independent Monte Carlo-generatedmodels with the same filler volume fraction. (For interpretation of references tocolor in this figure legend, the reader is referred to the web version of this article.)

T. Singh et al. / Journal of Membrane Science 448 (2013) 160–169 163

compared with the predictions from a selection of permeability-based models, namely the Maxwell, Bruggeman, Pal, and Lewis–Nielsen models [7,8]. The Bruggeman model was originally devel-oped to predict the dielectric constant of a composite medium atlow filler loading, and also takes into effect the random distribu-tion of spherically shaped fillers [21]. However, the model must besolved numerically. The Lewis–Nielsen model is an adaptation ofthe expression developed originally to predict the elastic moduli ofcomposites [22,23]. The Pal model is also solved numerically, andhas been employed to predict the effective diffusivity in mixedmatrix membranes [24]. While all the analytical models showqualitative trends in agreement with the simulations, the Brugge-man and Lewis–Nielsen models are found to most closely predictthe permeability over a large range of Df/Dm and the filler loadingfraction.

3.2. Mass transport with non-unity interface equilibrium constants

Here we consider rigorously the effects of non-unity equili-brium constants on Deff/Dm as a function of the volume fraction.Simulations were carried out for different combinations of Kand Df/Dm, chosen such that the same permeability ratio Pf/Pm¼KDf/Dm¼10 was maintained. It is clearly observed in Fig. 5 that theindividual values of K and Df/Dm, and not the overall permeabilityratio Pf/Pm, determine the effective diffusivity (or effective perme-ability) through the membrane. It is also clear that the equilibriumconstant K has a stronger effect than the diffusivity ratio Df/Dm, if itis desired to obtain high values of the effective permeability inrelation to the base polymeric material. The lowest values of Deff/Dm are obtained when K¼1. The above observations rigorouslyconfirm that the behavior of MMMs can deviate widely from thatpredicted by permeability-based models, when the interfaceequilibrium constant deviates from unity. The permeability-based models do not contain a physically realistic mechanism toseparate the effects of interfacial equilibrium and diffusivity ratios.Comparing the data for the case of K¼1, Df/Dm¼10 (plotted asblack circles in both Figs. 4a and 5) with the Maxwell modelprediction (shown as the black line in Fig. 4a), it is seen that theMaxwell model can predict the data well at low loadings. How-ever, Fig. 5 shows that the effective MMM permeability deviates

from the Maxwell model for Ka1, even though the permeabilityratio is fixed at Pf/Pm¼10.

Fig. 5 also compares the ‘exact’ simulation results to thepredictions of the effective medium theory (EMT) for permeabilitybehavior in composite membranes [9,16,17]. The EMT is useful tothe present analysis because it explicitly separates the effects ofinterfacial equilibrium and diffusivity. The lines in Fig. 5 show thepredictions from the EMT for the same combinations of K and Df/Dm considered previously. While the EMT correctly predicts thatthe individual values of K and Df/Dm (and not the permeabilityratio Pf/Pm) determine the effective membrane permeability, itspredictions are quantitatively unrealistic. The large deviation ofthe EMT from the ‘exact’ predictions is due to its mode oftreatment of the two phases. In particular, the EMT does notconsider the filler as being a true discontinuous phase. Theinaccuracy in its predictions increases with K, since the fillerphase is effectively treated as a second continuous phase. On theother hand, the Maxwell model considers the filler as a truediscontinuous phase, and the subscripts of the matrix and fillerphases are not interchangeable. Furthermore, the EMT cannotpredict the effective permeability below a percolation thresholdof ϕ�0.3 at higher values of K (41) [9], a limitation that does notexist in the present computational approach. The effect of the

Fig. 5. Effects of varying equilibrium constants and diffusivity ratios for a constantpermeability ratio Pf/Pm¼KDf/Dm¼10. The combinations shown are K¼1, Df/Dm¼10 (black circles); K¼2, Df/Dm¼5 (green); K¼5, Df/Dm¼2 (blue); and K¼10,Df/Dm¼1 (red). The 3D model contained randomly distributed fillers of 2 μm size.The lines represent the corresponding Effective Medium Theory (EMT) predictions.Error bars are obtained as in Fig. 4. (For interpretation of references to color in thisfigure legend, the reader is referred to the web version of this article.)

Fig. 6. Concentration profiles of the permeant in a cross-sectional 2D slice of themembrane for two of the parameter combinations shown in Fig. 5, (a) K¼2,Df/Dm¼5 and (b) K¼10, Df/Dm¼1, at filler volume fraction ϕ¼0.4. Length units arein μm and concentration in mol/m3.

T. Singh et al. / Journal of Membrane Science 448 (2013) 160–169164

interface equilibrium constant can be further understood fromFig. 6, which shows the concentration profile of the permeant in across-sectional 2D slice of the membrane for two of the parametercombinations shown in Fig. 5, i.e., K¼2, Df/Dm¼5 (Fig. 6a) andK¼10, Df/Dm¼1 (Fig. 6b) at filler volume fraction ϕ¼0.4. Due tothe large concentrations in the filler particles relative to thematrix, the permeant concentration gradients inside the fillerscannot clearly be seen in the figure. However, it is clear that as Kincreases, the fillers act as local regions of high concentration inthe matrix, which effectively increase the driving force for trans-port and lead to a higher flux.

3.3. Effects of filler size

Tantekin-Ersolmaz et al. presented experimental work indicat-ing that the permeability of an MMM system decreased withdecreasing particle size, under conditions of constant filler volumefraction [25]. We examine this situation via predictions using threedifferent particle sizes (Fig. 7). For the cases of K¼1 (Fig. 7a) aswell as non-unity K values (Fig. 7b), our results show that there is

no direct effect of filler size (and hence interfacial area per unitvolume) on the value of Deff/Dm. Our results indicate that theexplanation of the experimental observations [25] requires con-sideration of other indirect non-idealities caused by the differ-ences in surface area or filler size. For example, a large interfacialarea may alter the packing of a significant fraction of polymerchains and hence alter its diffusivity or solubility. The transportproperties of nanoscale fillers may also be different from those oflarge particles. The above example also illustrates that rigorouspredictions of permeation in MMMs allow more reliable attribu-tion (or ruling out) of the roles of the intrinsic adsorption,diffusion, and morphological properties of the MMM in explainingexperimentally observed behavior.

3.4. Back-calculation of filler permeabilities

The permeability-based models can also be used to back-calculatevalues of the filler permeability Pf from experimental MMM data. Thevalue of the matrix permeability (Pm) used in such a calculation isobtained from a measurement with a pure polymer membrane. Theback-calculated value of the filler permeability (Pf) can then becompared with independent data on the filler material (from otheradsorption and diffusion measurement techniques), and used tohypothesize non-idealities in the MMM or in the filler itself [14].Although the validity of such a procedure was questioned very earlyin the MMM literature [18], it continues to be carried out in manystudies [8]. The results of the present computational approach do notsuffer from possible uncertainties in experimental data related tonon-ideal MMMmorphologies. Hence we use them to rigorously testthe reliability of the back-calculation approach using independentsolubility and diffusion data from seven materials (one filler and sixmatrix polymers), treating our computational predictions as ‘exact’permeation data from ideal MMMs made with these materials.Taking an example of recent interest, Table 1 summarizes CO2

adsorption and diffusion properties of a filler material (the MOFZIF-8) and six matrix polymer materials (PDMS, 6FDA-DAM-15%,6FDA-DAM:DABA-4:1, Matrimid 5218, Ultem 1000, and Torlon) at298–308 K and 2 atm, obtained from the literature [26–35]. Althoughthe precise values listed in Table 1 for each material are not criticallyimportant to this analysis, they are well within the accepted rangesfor the properties of these materials. The solubilities and diffusivitiesare used to obtain the values of K and Df/Dm (Table 1) for MMMsmade with ZIF-8 loaded into each of the six polymers. This informa-tion is used to obtain computational predictions of Deff/Dm¼Peff/Pm,which are treated as equivalent to ‘exact’ experimental data from adefect-free MMM, and are then used to back-calculate the ‘apparent’permeability of the same filler (ZIF-8) in each of the six MMMs viathe Lewis–Nielsen and Maxwell models. Two filler volume fractions(ϕ¼0.3 and 0.5) are considered.

Fig. 7. Effective membrane diffusivity versus volume fraction for filler sizes of 4 μm(red), 2 μm (blue), and 0.2 μm (green) under the following conditions: (a) K¼1, andDf/Dm ¼10 (open circles) or 100 (closed circles); and (b) Df/Dm ¼1, and K¼10 (opencircles) or 100 (closed circles). Error bars are negligible, as seen in Figs. 4 and 5. (Forinterpretation of references to color in this figure legend, the reader is referred tothe web version of this article.)

Table 1Fickian diffusivity and solubility parameters for CO2 at 298–308 K and 2 atm innominal filler and matrix materials [26–35].

D(10�8 cm2/s)

S (10�2 cc STP/cc-cm Hg)

P¼D� S(Barrer)

K¼Sf/Sm Df/Dm

Filler (f)ZIF-8 200 15.3 3060 – –

Matrix (m)PDMS 2000 1.5 3000 10.2 0.16FDA-DAM-15% 60 8.1 486 1.89 3.336FDA-DAM:DABA – 4:1

6.45 33.8 218 0.45 31.0

Matrimids 5218 0.98 9.2 9 1.66 204.1Ultems 1000 0.18 7.2 1.3 2.13 1111.1Torlons 0.076 6 0.46 2.55 2631.6

T. Singh et al. / Journal of Membrane Science 448 (2013) 160–169 165

Fig. 8 shows the apparent ZIF-8 filler permeability (Pf) back-calculated from the Maxwell and Lewis–Nielsen models, as afunction of the matrix polymer permeability. Both models fail tocorrectly back-calculate the permeability of ZIF-8. The apparentpermeability of ZIF-8 shows a very large variation as a function ofthe polymer permeability. When ZIF-8 is loaded into a highlypermeable polymer like 6FDA-DAM or PDMS, a spuriously high(5000–50,000 Barrer) ZIF-8 permeability is back-calculated byanalytical models, as seen from Fig. 8 for both the Maxwell andLewis–Nielsen models at ϕ¼0.3 and 0.5. Conversely, when ZIF-8 isloaded into polymers with intermediate (6FDA-DAM:DABA 4:1) orlow (Matrimid, Ultem, or Torlon) permeability, a spuriously lowvalue for the ZIF-8 permeability is obtained. This is shown in Fig. 8for all the six polymers via the Lewis–Nielsen model at ϕ¼0.5.Furthermore, for the three polymers with the lowest permeabil-ities, the Maxwell model (at ϕ¼0.3 and 0.5) and the Lewis–Nielsen model (at ϕ¼0.3) predict non-physically negative ZIF-8permeabilities, and hence the corresponding lines in Fig. 8 are notextended to all the six polymers. The above effects have beenrecently seen for CO2 permeation in experimental studies of ZIF-8/polymer MMMs [14,15], wherein apparent ZIF-8 permeabilities inthe range of 10–10,000 Barrer were obtained using data fromMMMs containing ZIF-8 in various polymers ranging from Torlonto 6FDA-DAM (see Supplementary Information for experimentaldata from Ref. [15]). The authors hypothesized that the process ofmembrane formation leads to changes in the structure of the ZIF-8filler, and that the changes are dependent on the matrix char-acteristics (e.g., fractional free volume, density, Tg). Another exam-ple of similar behavior was observed very early in the MMMliterature by Duval et al., particularly for CO2 permeation in zeoliteMFI/polymer MMMs [36]. Both the above literature examples werebased upon experimental data, which could be argued as resultingfrom potential non-idealities in the MMM microstructures. How-ever, the present analysis clearly shows that ‘apparent’ fillerpermeabilities calculated from the Maxwell or similar modelsresult simply from their failure to account for interface equilibriumeffects. Considering the very large variations in apparent perme-ability observed here as well as in experimental data [14,15,18,36],

we conclude that: (1) filler permeabilities obtained from thepermeability-based models will only be close to the true valuesunder fortuitous circumstances (e.g., ZIF-8 in 6FDA-DAM at ϕ¼0.5using the Lewis–Nielsen model, Fig. 8), and (2) such data shouldnot be used to hypothesize non-idealities such as voids, rigidifiedregions around the filler, or compression/densification of the fillerby the matrix. The contribution (if any) of hypothesized non-idealities should first be separated from the intrinsic behaviorexpected in a defect-free MMM, calculated by correct inclusion ofthe effects of K and Df/Dm as shown in this work. Furthermore, theexistence of any non-idealities should be independently verifiedby other characterization techniques.

Fig. 9 shows the MMM permeability enhancement (Peff/Pm)predicted by the rigorous calculations as well as the Lewis–Nielsenand Maxwell models, for the matrix polymers and ZIF-8 fillerlisted in Table 1. Both models are in qualitative agreement withthe ‘exact’ predictions for relatively low-permeability polymers atlow filler loadings. However, they cannot predict two importantobservations. Firstly, both models show a monotonic decrease inpermeability enhancement as the polymer permeability increases.They cannot predict the dip in permeability enhancement for theZIF-8/6FDA-DAM:DABA – 4:1 MMM, found from the rigorouscalculations (Fig. 9). This behavior is caused by the value of theinterface equilibrium constant for the ZIF-8/6FDA-DAM:DABA –

4:1 system (Table 1), which is lower than that of all the otherpolymer materials and is also less than unity. Secondly, the modelsare unable to predict the permeability enhancement occurring inMMMs made with high-permeability polymers such as PDMS. Thecase of ZIF-8 in PDMS is specifically interesting, because Table 1shows that Pf/Pm�1 (due to a fortuitous combination of K¼10.2and Df/Dm¼0.1). The Lewis–Nielsen and Maxwell models predictthat no significant CO2 permeability enhancement should occur insuch a MMM. However, the rigorous calculations (Fig. 9) showlarge enhancements of Peff/Pm¼1.96 (2.97) at ϕ¼0.3 (0.5). Thelarge interface equilibrium constant overcomes the effect of lowerfiller diffusivity. As shown in Fig. 6, high local concentrations ofCO2 exist in the filler phase and result in a significantly enhancedpermeability over the pure polymer. This enhancement will occur

Fig. 8. 'Apparent' CO2 permeabilities of ZIF-8 as a function of polymer matrix permeability, back-calculated by applying the Maxwell and Lewis–Nielsen models to the ‘exact’Peff/Pm data obtained from the computational MMM model. The dashed horizontal line represents the true permeability of the ZIF-8 filler as listed in Table 1. The curves areonly a guide to the eye.

T. Singh et al. / Journal of Membrane Science 448 (2013) 160–169166

even in MMMs wherein the polymer has a higher permeabilitythan the filler (i.e., Pf/Pmo1), if the interface equilibrium isfavorable enough (also see Section 3.5). Such permeation results,

when observed experimentally, should first be understood interms of the intrinsic adsorption and diffusion properties, beforeconsidering any contributions of defects or non-idealities.

Fig. 10. Surface plot comparison of effective MMM diffusivity versus the interface equilibrium constant and the diffusivity ratio as obtained from the computational model,for filler volume fractions (a) 0.2, (b) 0.3, (c) 0.4, and (d) 0.5.

Fig. 9. Effective CO2 permeability enhancement in defect-free MMMs containing ZIF-8, as a function of polymer matrix permeability. The predictions from the computationalmodel, the Maxwell model, and the Lewis–Nielsen model are shown. The adsorption and diffusion parameters are as listed in Table 1. The lines are only a guide to the eye.

T. Singh et al. / Journal of Membrane Science 448 (2013) 160–169 167

3.5. Overall trends and an empirical correlation

Fig. 10a–d shows the overall behavior of the effective mem-brane permeability Deff/Dm as a function of the interface equili-brium constant (K) and diffusivity ratio (Df/Dm) for four differentvolume fractions (ϕ¼0.2–0.5). Both (K) and diffusivity ratio (Df/Dm) have a large effect on the effective membrane permeability.Apart from the other effects already discussed in this work, wenote that there exist K and Df/Dm regions for which Deff/Dmo1(41) even though Pf/Pm41 (o1). These phenomena occur whenthe diffusing species is poorly (strongly) adsorbed in the fillerphase even though it diffuses rapidly (slowly) in the filler phase.These cases would appear counter-intuitive from the viewpoint ofthe permeability-based models and therefore assigned to thepresence of non-idealities, but they can be understood as arisingfrom local depletion (or increase) of the concentration drivingforces inside the membrane when Ko1 (or 41). Furthermore,Fig. 10 shows that a large variety of MMM behavior exists outsideof that which can be explained by the permeability ratio alone. Forany given Pf/Pm value and filler loading ϕ, the permeability-basedmodels only predict a single possible value of Deff/Dm, which lieson (or very close to) the single curve along the Df/Dm axis at K¼1in Fig. 10. However, the actual value lies on a locus of points on theDeff/Dm surface corresponding to different combinations of K andDf/Dm that lead to the same Pf/Pm, and could be very different inmagnitude depending on the individual values of K and Df/Dm.Finally, Fig. 10 shows that all the surfaces reach plateaus at largevalues of K and Df/Dm for a fixed filler volume fraction. This isexpected in all MMM systems. Similarly, the surfaces reach valleysfor very small K and Df/Dm (corresponding to nearly impermeablefillers).

We provide two methods for easier use of our results in thedesign and selection of MMM materials. Firstly, we show 2D plotsof Deff/Dm versus K with Df/Dm as a parameter, over a large range ofϕ¼0.05–0.5 (Supplementary Information). This allows lookup andinterpolation of predicted exact values of Deff/Dm for known valuesof the other variables, or conversely of Df/Dm (or K) using knownvalues of the other variables. Secondly, we have fit all of ourcomputational data to an empirical correlation that can be easilyused as a mathematical tool for selection of candidate MMMmaterials and for analysis of experimental MMM data. We noticethat the behavior of Deff/Dm (Supplementary Information) followsthe shape of a sigmoidal function. The 2D plots (SupplementaryInformation) for different ϕ values differ in the values of the upperand lower asymptotes, with the slopes of the curves beingessentially the same in the non-asymptotic region. Furthermore,for a given ϕ the curves shift to the left or right with changes in Df/Dm, but the lower and upper asymptotes remain the same. Allthese observations strongly suggest the generalized logistic func-tion [37] as a suitable fitting function. The following form of thelogistic function is used here:

Def f

DmðK;Df =Dm;ϕÞ ¼ b1ðϕÞþ

b2ðϕÞ�b1ðϕÞ½1þb3ðDf =Dm;ϕÞexpð�b4log 10KÞ�b5

ð1Þ

As noted earlier, the curves at different ϕ differ in the lower andupper asymptotic values b1 and b2 respectively, which are onlyfunctions of ϕ. By making a preliminary plot of the asymptoticvalues b1 and b2 as functions of ϕ, we find that these parametersare very well represented by a linear and a cubic polynomialrespectively:

b1ðϕÞ ¼ 1þb11ϕ ð2aÞ

b2ðϕÞ ¼ 1þb21ϕþb22ϕ2þb23ϕ

3 ð2bÞThe K-dependence of the correlation is already shown in Eq. (1).The Df/Dm dependence of Eq. (1) arises from the parameter b3,

which shifts the curves horizontally as a function of Df/Dm whilemaintaining the same asymptotes. Preliminary analysis showedthat an exponential function represented this behavior accurately:

b3ðDf =DmÞ ¼ b31expð�b32log 10Df =DmÞ ð3ÞThe parameter b31 is a strong function of ϕ, and was wellrepresented by a quadratic polynomial:

b31ðϕÞ ¼ b311þb312ϕþb313ϕ2 ð4Þ

The other two parameters in Eq. (1), b4 and b5, did not show asignificant dependence on K, Df/Dm, or ϕ, and are treated asconstants. Overall, this leads to a ten-parameter correlation model,which is fitted by nonlinear least-squares regression to 1210 datapoints from the rigorous 3D simulations represented by the plotsshown in the Supplementary Information. The ranges of the threeindependent variables are: 0.1r(K, Df/Dm)r31.6 and 0rϕr0.5.The regression was implemented with the GRG Nonlinear solver inMicrosoft Excel 2010, which incorporates an automated methodfor regression with multiple trial parameter sets and practicallyensures convergence to a global minimum. As mentioned above,the initial guesses for the parameters were obtained from pre-liminary fitting with a few representative data sets and theasymptotic values. The final values of the correlation parametersare given in the Supplementary Information, and should be usedas is (with the given number of significant figures) for best results.The correlation gives an excellent fit of the data. It predicts Deff/Dm

with an average accuracy of 1.6% and a least-squares residual of2.0%. This high level of accuracy makes it useful as a mathematicaltool for calculating permeability and ideal selectivity of MMMs,without having to perform the rigorous 3D simulations. In theSupplementary Information, we provide example plots at fourvalues of ϕ, showing the excellent agreement between therigorous 3D simulation data and the empirical correlation. TheMaxwell model predictions are also included in these plots, and ingeneral are not seen to provide physically reliable guidance forMMM material selection except under fortuitous circumstances.For the trivial case of ϕ¼0, the correlation returns exactlythe expected Deff/Dm value of unity. For the other trivial case ofK¼Df/Dm¼1, it accurately reproduces the expected Deff/Dm value ofunity, with actual Deff/Dm values in the range of 0.98–1.01 depend-ing on the value of ϕ. For very large (or very small) values of K andDf/Dm, it accurately predicts the asymptotic behavior of Deff/Dm.

4. Conclusions

In the present paper, it has been shown that rigorous calcula-tions of permeation are important for a reliable understanding oftransport behavior in MMMs. We have developed methods forautomated construction of detailed and large-scale 3D MMMmodels, which are then solved by finite-element methods usingthe COMSOL Multiphysics package. Our models explicitly accountfor the effects of interfacial equilibrium between the matrix andthe filler, in addition to the differences in molecular diffusivitybetween the two phases. The computational results are consideredto be ‘exact’ to the limits of accuracy of the numerical method. Wehave shown that analytical models based on permeability values(e.g., Maxwell model) only predict the MMM permeability withreasonable accuracy when the interface equilibrium constant K isclose to unity. Most real MMMs do not satisfy this condition. It isshown that the individual values of the equilibrium constant K andthe diffusivity ratio Df/Dm, and not the combined permeabilityratio Pf/Pm¼KDf/Dm, determine the MMM permeability. The EMTmodel, which correctly predicts that the individual values of K andDf/Dm are of importance, does not quantitatively predict thepermeation behavior correctly. We have used our ‘exact’

T. Singh et al. / Journal of Membrane Science 448 (2013) 160–169168

predictions to examine some of the current speculations on MMMpermeation behavior. Our simulations indicate that an ideal MMMshows no significant direct effect of filler particle size, and hencethe particle size dependence experimentally observed in someworks must be due to other indirect effects of particle size(e.g., on the matrix polymer packing due to the enhanced inter-facial interactions). Furthermore, we use available CO2 solubilityand diffusivity data for example filler and polymer materials totest the performance of permeability-based models. The experi-mentally observed behavior actually derives from the combinationof interfacial equilibrium and diffusion in the two-phase mem-brane. However, the interpretation of this behavior usingpermeability-based models may create spurious results and leadresearchers to hypothesize different types of non-idealities in theMMM. The intrinsic transport and adsorption behavior in the idealMMM should first be correctly described before additional con-tributions from non-idealities are invoked. In addition to thecapability to construct realistic MMM systems and simulate themby finite-element techniques, we also provide extensive data plotsand an accurate empirical correlation to quickly obtain reliablepredictions of MMM permeation behavior. It is recommended thatthe empirical correlation be used in conjunction with adsorptionand diffusion data for the filler and matrix phases, since the ratioPf/Pm used in permeability-based models does not appear to be areliable quantitative measure of MMM behavior.

Acknowledgment

This work was partially supported by the National ScienceFoundation (CBET-0848546) and an Air Products Fellowship.

Appendix A. Supplementary materials

Supplementary data associated with this article can be found inthe online version at http://dx.doi.org/10.1016/j.memsci.2013.08.010.

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