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Journal of Membrane Science 325 (2008) 125–129

Contents lists available at ScienceDirect

Journal of Membrane Science

journa l homepage: www.e lsev ier .com/ locate /memsci

inite element assessment of the potential of platelet-filled polymersor membrane gas separations

lga Gusevaa,b,∗, Andrei A. Gusevc

Materials Simulations GmbH, Bluntschlisteig 1, P.O. Box 624, 8027 Zürich, SwitzerlandEmpa, Swiss Federal Laboratories for Materials Testing and Research, Überlandstrasse 129, 8600 Dübendorf, SwitzerlandInstitute of Polymers, Department of Materials, ETH Zürich, 8093 Zürich, Switzerland

r t i c l e i n f o

rticle history:eceived 4 March 2008eceived in revised form 1 July 2008ccepted 4 July 2008vailable online 18 July 2008

a b s t r a c t

We use the finite element method to analyze the role of filler aspect ratio and volume loading on theeffective permeability and selectivity of gas-separation membranes consisting of a polymer matrix filledwith platelet-shaped molecular sieving particles. On the basis of direct 3D finite element estimates, wedevelop and validate a quick arithmetic procedure for predicting the effective permeability and selectivityof platelet-filled systems. We use this procedure to show that it is feasible to obtain mixed matrix platelet-

eywords:as separationermeabilityermselectivityixed matrix membrane

filled membranes with effective permselectivities considerably exceeding the upper Robeson’s bounds.© 2008 Elsevier B.V. All rights reserved.

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olymereolite

. Introduction

In the past decades, polymeric membranes have becomencreasingly important for gas and vapour separations [1,2].owever, they are known to exhibit a trade-off between theermeability and selectivity as represented by the upper boundependencies developed by Robeson [1]. Although molecular siev-

ng materials such as zeolites and carbon molecular sieves canxceed the upper bound, these materials are expensive, and its also difficult to fabricate them as membranes [2]. In order tobtain an advanced membrane exceeding the upper bound, onean load high-performance particles into a flexible polymer matrixith good mechanical properties and possibly high selectivity

3,4]. A number of publications have recently appeared wherene demonstrated the possibility of manipulating the size andhape of zeolite particle using structure-directing agents [5,6] or

icroemulsions [7]. Depending on the organic structure-directing

gents used, different particle morphologies and crystal structuresf zeolite (siliceous ZSM-5, [Si96O192]–MFI) were obtained: coffin-haped and oval shaped crystals with aspect ratios up to 4 and flat

∗ Corresponding author at: Empa, Swiss Federal Laboratories for Materials Testingnd Research, Überlandstrasse 129, 8600 Dübendorf, Switzerland. Tel.: +41 44 823361; fax: +41 44 823 4015.

E-mail address: olga.guseva@empa.ch (O. Guseva).

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376-7388/$ – see front matter © 2008 Elsevier B.V. All rights reserved.oi:10.1016/j.memsci.2008.07.030

latelets with a maximal aspect ratio of about 10 [5,6]. Ref. [7]emonstrated the possibility of using non-ionic microemulsionso manipulate the shape and size of silicalite-1 materials, growingoth spheres (3–10 �m in diameter) and high-aspect-ratio platelets>50 �m in diameter) which were robust aggregates of small sub-

icron particles that were stable to calcinations. High-aspect-ratioanoporous particles can be achieved by certain layered zeolite pre-ursors. For example, materials with three-dimensional porosityithin layers such as MCM-22 [8], a strontium silicalite (AMH-3)

9] or borogermanate, K4[B8Ge2O17(OH)2] [10] have been recentlybtained. Nanocomposite membranes with substantial enhance-ent in selectivity have been fabricated using exfoliated layers of

uch materials [11–13].Previous theoretical work was largely focused on low-aspect-

atio spherical particles [3]. In a recent work [14], we havesed the finite-element method to elucidate the possibility ofsing high-aspect-ratio hollow carbon nanotubes for obtainingigh-performance composite membranes for gas separations. Wehowed that one can favorably combine the unusually fast trans-ort properties of carbon nanotubes with the intrinsic separationoefficients of polymer matrices.

In this work, we consider filler particles of platelet shape andnalyze the influence of the platelets’ aspect ratio and volume load-ng on the effective permeability and selectivity of mixed matrix

embranes. We conduct numerical finite element predictions foromposite membranes with filler volume fractions up to 0.4. In

1 embrane Science 325 (2008) 125–129

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26 O. Guseva, A.A. Gusev / Journal of M

ractice, such membranes can still have good mechanical prop-rties comparable to those of the unfilled polymer matrix. Forxample, in Ref. [15] a successful fabrication of glassy fluorinatedolyimide zeolite-mixed (modified ZSM-2 zeolite) membranes waseported. It was found that membranes with 20 wt.% of filler weref good mechanical quality whereas 50 wt.% filler membranes werelready too brittle.

. Two-phase composites with lamellar morphology

It is obvious that for a given two-phase material, the highestossible selectivity can be achieved using the lamellar morphologyith the volume fraction of the high-selective phase of 1 (100%).ith the solute flux perpendicular to the lamella’s interface, the

ffective permeability P is given by the series model:

= PmPi

(1 − f )Pi + fPm(1)

here Pm and Pi are the permeabilities of the matrix and the second,igh-selective phase, which we, for convenience, will further callinclusion”, and f is the volume fraction of this inclusion phase. Ithould be noted that the lamellar morphology can be viewed as aimiting case of a fully-aligned platelet morphology as the platelet’sspect ratio a → ∞.

Let us consider the O2/N2 Robeson selectivity–permeability plot1]. As was pointed out by Koros and Mahajan in Ref. [2], a com-

ercially attractive region for such system involves systems withn O2 permeability more than 1 Barrer and O2/N2 selectivity above. Assuming a matrix permeability and selectivity of 1 Barrer and, respectively, and taking an inclusion phase with permeability of0 Barrers and selectivity of 30, the following results for the effec-ive permeability P and selectivity ˛ as a function of the volumeraction of the second phase f (which is proportional to the lamellahickness) are obtained (see Fig. 1a).

Fig. 1a shows that even using advanced fillers with Pi/Pm � 1nd ˛i/˛m � 1, one still needs to consider very high filler load-ngs to noticeably improve the composite’s P and ˛ compared tohose of the matrix. High composite’s selectivity values (˛ ≈ 10 and

ore) can be achieved only at rather high fractions of the secondhase of above 0.8. By using a less permeable second phase, onean, however, obtain more appealing results. For example, withpermeability and selectivity of the second phase of 0.1 Barrer

nd 30, respectively, one obtains the results depicted in Fig. 1b.ere, already at a volume fraction of less than 0.3, one can achieveuite substantial composite’s selectivity values. By increasing theermeability of the first phase from 1 to 5 Barrers, one obtains aembrane with permselectivity properties exceeding the Robe-

on’s upper bound, already at a fairly small filler volume fraction of.03 (Fig. 1c).

This preliminary analysis of permselectivity properties of two-hase lamellar morphology indicates that there is a potential toxceed the upper bound. However, in practice it is difficult to imple-ent such lamellar morphology. Real molecular sieving materials

an rather be fabricated and processed in a form of discrete parti-les of different aspect ratios [5–7] and then be put into a polymeratrix to yield a mixed matrix membrane. Accordingly, we nowant to investigate if and to which extent the potential seen with

he lamellar morphology can be approached using perfectly alignedlatelets of finite aspect ratios, randomly distributed in a polymeratrix.

. Two-phase composites with platelet morphology

We have conducted direct finite-element permeability cal-ulations, assuming periodic multi-inclusion computer models

biato

ig. 1. Effective permeability (solid-lines) and selectivity (dashed-lines) of two-hase lamellar composites predicted assuming different parameters for the phases’ermeability (Barrer) and selectivity.

omprised of a random dispersion of non-overlapping plateletsith aspect ratios (defined as the ratio of the platelet diameter to

he thickness) ranging from 1 (sphere) to 80. As it was demon-trated above with the lamellar morphology, the most promisingermselectivity properties should be expected when the matrixermeability (Pm) considerably exceeds that of the inclusions (Pi).ccordingly, in the numerical analysis we have considered Pi/Pm

atios of 0.001, 0.01, 0.024 and 0.1.We generated three-dimensional periodic computer models

omprised of a random dispersion of 25 perfectly aligned identi-al round platelets. The models were meshed into unstructuredorphology-adapted meshes using the Gridder/Palmyra software

16] (see Fig. 2).Permeability calculations were performed as described in Ref.

19]. Based on unstructured meshes, we solved the governingaplace’s equation for the local chemical potential �:

iv P(r) grad � = 0 (2)

ssuming a local permeability coefficient P(r) of 0.001, 0.01, 0.024r 0.1 inside the platelets and 1 everywhere in the matrix.

In general, the gas permeability through a material is described

y a 3 × 3 symmetric permeability tensor. Here we considersotropic matrices so the local permeability tensor can be writtens P(r)·ıik, where P(r) is the local permeability coefficient and ıikhe Kronecker tensor with indices i and k varying from 1 to 3. Theverall, effective permeability coefficients were calculated based on

O. Guseva, A.A. Gusev / Journal of Membrane Science 325 (2008) 125–129 127

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For validation, we compared results of such quick arithmeticcalculations with direct finite element calculations of permeabilityand selectivity for a particular system, namely for that of zeo-lite platelets randomly distributed in a PDMS poly(imide siloxane)

ig. 2. Random-microstructure computer model with 25 non-overlapping fully alignic periodic boundary conditions are employed. The orthorhombic box stands fornstructured tetrahedral-based mesh used to predict the effective, composite pe14,17–20]).

linear-response relation between the overall flux and the exter-al chemical–potential gradient applied. Tetrahedral elements withubic-order polynomial interpolation were employed in calcula-ions. In this work we were interested in the overall permeabilitycross the membrane. Accordingly, we have focused on the perme-bility coefficient P relating the z-components of the overall fluxnd the chemical potential gradient.

We define the following decay function F, characterizing theeviation of the effective permeability of system of platelet mor-hology from that of the corresponding lamellar morphology:

= P − Plamella

Pm − Plamella(3)

here P is predicted permeability of a mixed matrix membraneith platelets of aspect ratio a and volume fraction f, Pm and Pi

tand for assumed permeabilities of the matrix and the inclusions,nd Plamella is calculated using Eq. (1) with the same values of Pm,i and f.

We have chosen to approximate numerical estimates of F by atretched exponential master-form function written in terms of theroduct x = af (see Fig. 3):

= exp

[−(

x

x0

)ˇ]

(4)

arameters ˇ and x0 were determined by least-square fitting ati/Pm ratios of 0.001, 0.01, 0.024 and 0.1. Fig. 4 shows that bothand x0 exhibit a linear dependence on the ratio Pi/Pm:

ˇ = −3.418256Pi + 0.741575

Pm

x0 = −15.668589Pi

Pm+ 2.708506

(5)

he stretched (fractional) exponential decay function has been fre-uently used to fit experimental and numerical data. Compared

Feo

entical platelets of aspect ratio 5 dispersed at a volume fraction of 0.16. Orthorhom-eriodic unit cell. The sketch on the right shows a periodic, morphology-adaptiveility. For more methodological details and related application studies (see Refs.

o a one-parameter exponential function (Eq. (4) with ˇ = 1), thetretched exponential form has two fitting parameters x0 and ˇhich characterize the position and the width of the characteristicecay interval. Here, we consider Eq. (4) as an expedient fitting formnd do not attempt to further elaborate on the possible physicaleaning of the two fitting parameters.One can now readily perform estimation of the effective perme-

bility P of any composite material filled with fully aligned plateletsf aspect ratio a and volume fraction f in a quick arithmetic calcu-

ig. 3. Numerically predicted values of decay function F (symbols) and stretchedxponential function (line) fitted to the numerical predictions. Here, results for Pi/Pm

f 0.01 are shown.

128 O. Guseva, A.A. Gusev / Journal of Membrane Science 325 (2008) 125–129

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atrix. The O2 permeability of PDMS41 matrix of 32.24 Barrers and2/N2 selectivity of 2.7 were taken from Ref. [21]; for the Zeolite 4A

nclusions, the input parameters of P(O2) = 0.77 Barrer and selectiv-ty ˛(O2/N2) = 37.0 were taken from Ref. [3]. Direct finite elementalculations of permeabilities were performed as described above.he platelet aspect ratios were varied from 1 to 80 and volumeraction from 0.02 to 0.32. Comparison of the results of direct calcu-ations with those obtained using the quick arithmetical proceduref Eq. (5) is shown in Fig. 5.

Fig. 5 shows a good agreement of both sets of data, thus vali-ating the proposed procedure for quick estimation of the effectiveermeability.

Assuming the same values of O2- and N2-permeabilities of theatrix (Pm) and inclusions (Pi), we have also estimated the com-

osite permeability (P) using semi-empirical equation of Cussler22]:

Pm

P= 1 − f + 1

Pi/(Pmf ) + 4(1 − f )/a2f 2(6)

here aspect ratio a is defined as the ratio of the platelet diametero its thickness.

Fig. 6 shows that the Cussler’s predictions become less accu-ate upon decreasing effective permeability. It has been alreadyeported elsewhere [19] that for fully impermeable platelets, bothussler and modified Cussler–Aris formulas provide poor estimatesf the effective permeability.

ig. 5. Comparison of the effective permeability of O2 (solid-circles) and N2 (open-ircles) gases calculated using the quick arithmetic procedure with those calculatedirectly using the finite element method.

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Fst

ig. 6. Comparison of the effective permeability of O2 (solid-circles) and N2 (open-ircles) gases calculated using the Cussler’s semi-empirical equation with thosealculated directly using the finite element method.

Using the quick procedure, one can now readily assess the poten-ial of gas separation with such mixed matrix membranes for anrbitrary parameter set. As an example, we present here results forhe effective permeability and selectivity of a mixed matrix zeo-ite/PDMS membranes assessed assuming a set of aspect ratios of0, 50, 100 and 150 and zeolite volume fractions 0.1, 0.2, 0.3, 0.4 and.5 (see Fig. 7). The same as above permeabilities and selectivitiesere assumed for the PDMS41 matrix and the Zeolite 4A inclusions.

One can see that by using platelets with an aspect ratio of 50nd volume fractions of above approximately 0.22, one can alreadyxceed the Robeson’s upper bound. However, with a platelet aspectatio of 50 the potential of lamellar morphology (open symbols inig. 7) is still far from realization. In order to considerably exceedhe upper bound, aspect ratios of 100 and more are needed (seeig. 7). The permselectivity of lamellar morphology can be prac-ically reached with platelets of aspect ratio of 100 at a volumeraction of 0.5, whereas with an aspect ratio of 150 already at a vol-me fraction of 0.3. On contrary, with low-aspect-ratio inclusions

say, 10 or less, including spheres) one cannot realize the overallermselectivities above the upper bound at the platelet loadingselow 0.5. This conclusion is in agreement with a previous work oforos et al. [3], who employed the Maxwell’s model to calculate the2/N2 permselectivity of mixed matrix membranes with spherical

ig. 7. Predicted zeolite-platelets/PDMS matrix membrane performance. For eachymbol kind, the point on the right corresponds to a volume fraction of 0.1, then tohe left the volume fraction increases as 0.2, 0.3, 0.4 and 0.5 (the left point).

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O. Guseva, A.A. Gusev / Journal of M

ller particles. They also reported that in order to exceed the upperound, a volume fraction of spherical particles of more that 0.5hould be used. In the current work, perfectly aligned platelets wereonsidered. This morphology maximizes the separation effect ands therefore of most interest in applications. Moreover, it was foundxperimentally in Ref. [15] that during the membrane fabrication,any of zeolite particles appear to orient in such a way that their

argest face (hexagonal face) becomes parallel to the membraneurface. Using morphologies with well-aligned high-aspect-ratiolatelets, it seems feasible to realize the relatively large platelet

oadings required for exceeding the Robeson’s upper bound (seeig. 7). However, it may prove a serious technological challenge tovoid platelet agglomeration.

In calculations, we assumed that the presence of the plateletsoes not induce permeability changes in the matrix. While thisremise seems to be appropriate for microscopic platelets, one canhallenge its validity when dealing with materials involving exfoli-ted atomically thin sheets of layered minerals. In such situations,ne can use atomistic simulations (either molecular dynamics [23]r transition-state-theory kinetic Monte–Calro [24,25]) to quan-ify the effect of such molecular-level changes [26]. For a recentnteresting computational study, see Ref. [27].

In this work, we assumed concentration-independent transportoefficients. This assumption is appropriate for low-pressure gasransport. However, at elevated pressures non-Fickian effects mayecome significant. One can use for example the Maxwell–Stefanramework to describe such concentration-dependent effects [28].t seems then straightforward to generalize our finite elementpproach to incorporate such effects.

. Conclusion

In this work, we have identified the role of geometric factorn the effective permeability of mixed matrix membranes com-osed of a polymer matrix filled with perfectly aligned randomlyistributed platelets of varying aspect ratio. Based on prelim-

nary analysis conducted with lamellar morphology, we haveocused on the situations where the matrix permeability con-iderably exceeded that of the platelets. We proposed a quickrithmetic procedure to calculate the effective permeability andhe selectivity of such mixed matrix membranes. By using thisrocedure, we have analyzed the trade-off between the perme-bility and selectivity of mixed matrix membranes composed ofeolite platelets dispersed in a PDMS matrix. We have demon-trated that by using platelets of aspect ratio 100 and more, it iseasible to achieve permselectivity values considerably exceedinghe Robeson’s upper bound even at already small platelet volumeoadings.

cknowledgement

The work was in part supported by the European Commis-ion 6th Framework Program Project MULIMATDESIGN “Com-uter aided molecular design of multifunctional materials with

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rane Science 325 (2008) 125–129 129

ontrolled permeability properties”, contract number: NMP3-CT-005-013644.

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