Basis of Permeability/Selectivity Tradeoff Relations in Polymeric GasSeparation Membranes

Benny D. Freeman†

Department of Chemical Engineering, North Carolina State University, Campus Box 7905,Raleigh, North Carolina 27695-7905

Received September 14, 1998; Revised Manuscript Received November 30, 1998

ABSTRACT: Gas separation properties of polymer membrane materials follow distinct tradeoff rela-tions: more permeable polymers are generally less selective and vice versa. Robeson1 identified the bestcombinations of permeability and selectivity for important binary gas pairs (O2/N2, CO2/CH4, H2/N2, etc.)and represented these permeability/selectivity combinations empirically as RA/B ) âA/BPA

-λA/B, where PA

and PB are the permeability coefficients of the more permeable and less permeable gases, respectively,RA/B is selectivity ()PA/PB), and λA/B and âA/B are empirical parameters. This report provides a fundamentaltheory for this observation. In the theory, λA/B depends only on gas size. âA/B depends on λA/B, gascondensability, and one adjustable parameter.

Introduction and Background

Polymer membranes are used commercially to sepa-rate air, to remove carbon dioxide from natural gas, andto remove hydrogen from mixtures with nitrogen orhydrocarbons in petrochemical processing applications.2For a given pair of gases (e.g., O2/N2, CO2/CH4, H2/N2,etc.), the fundamental parameters characterizing mem-brane separation performance are the permeabilitycoefficient, PA, and the selectivity, RA/B. The permeabilitycoefficient is the product of gas flux and membranethickness divided by the pressure difference across themembrane. Gas selectivity is the ratio of permeabilitycoefficients of two gases (RA/B ) PA/PB), where PA is thepermeability of the more permeable gas and PB is thepermeability of the less permeable gas in the binary gaspair.

Polymers with both high permeability and selectivityare desirable. Higher permeability decreases the amountof membrane area required to treat a given amount ofgas, thereby decreasing the capital cost of membraneunits. Higher selectivity results in higher purity productgas. Over the past 25 years, the gas separation proper-ties of many polymers have been measured, and asubstantial research effort in industrial, government,and university research laboratories has resulted inpolymers that are both more permeable and moreselective than first generation materials.3

A rather general tradeoff relation has been recognizedbetween permeability and selectivity: Polymers that aremore permeable are generally less selective and viceversa.1,3 On the basis of an exhaustive literature survey,Robeson1,4 quantified this notion by graphing the avail-able data as shown in Figure 1, which presents hydro-gen permeability coefficients and H2/N2 separationfactors for many polymers. Materials with the bestperformance would be in the upper right-hand cornerof this figure. However, materials with permeability/selectivity combinations above and to the right of theline drawn in this figure are exceptionally rare. Thisline defines the so-called “upper bound” combinations

of permeability and selectivity of known polymer mem-brane materials for this particular gas pair. Lines suchas the one shown in Figure 1 were constructed on anempirical basis for many gas pairs using publishedpermeability and selectivity data. The upper boundperformance characteristics were best described by thefollowing equation:1

which indicates that as the permeability of an upperbound polymer to gas A, PA, increases, selectivity of thepolymer for gas A over gas B, RA/B, decreases. Robesonreports values for λA/B and âA/B for many common gaspairs.1,4

The reason for this tradeoff has been widely discussed,but no theoretical justification of the form of eq 1 orfundamental predictions of λA/B and âA/B have beenoffered. Robeson noted an excellent empirical correlation

† Tel (919)515-2460, fax (919)515-3465. E-mail benny_freeman@ncsu.edu. URL: http://www.che.ncsu.edu/membrane.

Figure 1. Relationship between hydrogen permeability andH2/N2 selectivity for rubbery (O) and glassy (b) polymers andthe empirical upper bound relation.1

RA/B ) âA/B/PAλA/B (1)

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10.1021/ma9814548 CCC: $18.00 © 1999 American Chemical SocietyPublished on Web 01/05/1999

between λA/B and the difference between the kineticdiameters of the penetrant molecules, dB - dA.1 Thetheory presented in this article gives excellent predic-tions of λA/B with no adjustable parameters and suggeststhat the slope of the upper bound is a natural conse-quence of the strong size-sieving nature of the stiff chainglassy polymeric materials whose properties generallydefine the upper bound. Moreover, the theory providesgood estimates of âA/B with only one adjustable param-eter. The theory is developed for amorphous polymersand does not account for the influence of penetrantconcentration on permeation properties.

Theory

As Sir Thomas Graham proposed,5 the transport ofgases in dense, nonporous polymers obeys a solutiondiffusion mechanism. Under the driving force of apressure difference across a membrane, penetrantmolecules dissolve in the upstream (or high pressure)face of a membrane, diffuse across the membrane, anddesorb from the downstream (or low pressure) face ofthe membrane. Diffusion is the rate-controlling step inpenetrant permeation. The rate-controlling process indiffusion is the creation of gaps in the polymer matrixsufficiently large to accommodate penetrant moleculesby thermally stimulated, random local segmentalpolymer dynamics.3,6 Based on this mechanism, thepermeability coefficient of a polymer to gas A can bewritten as

where SA is the solubility coefficient and DA is thediffusion coefficient of gas A.

The selectivity of a polymer for gas A over gas B, RA/B,is given by

The diffusion of small molecules is an activated process,and at temperatures away from thermal transitions inthe polymer (e.g., glass transition, melting, etc.), theArrhenius equation is obeyed:7

where DoA is a front factor, EDA is the activation energyfor diffusion, R is the gas constant, and T is the absolutetemperature. Barrer8 and Van Amerongen9 observed asimple correlation between the front factor and activa-tion energy:

where a and b are independent of gas type. Additionally,a is independent of polymer type and has a universalvalue of 0.64.10 b has a value of -ln(10-4 cm2/s) ) 9.2for rubbery polymers and -ln(10-5 cm2/s) ) 11.5 forglassy polymers.11 Equation 5 is often referred to as a“linear free energy” relation. Similar relations between

front factors and activation energies are observed forviscosity of organic liquids, molten salts, and metals12

and for first-order chemical reaction kinetics,13 whichare also activated processes described by the Arrheniusequation.

Two explanations have been proposed to justify eq 5.First, Barrer noted that eq 5 may be the result of arather simple “compensation effect”.10 The range of lnDA in eq 4 is always small relative to that of EDA/RT.10

Therefore, to a first approximation in many cases, lnDA is almost constant over the ranges of temperaturetypically explored. The parameters EDA and DoA in eq 4are determined by curve fitting the data to this equa-tion. Any uncertainty in the value of EDA (determinedfrom the curve fit) that acts to increase the apparentEDA is compensated for by an increase in the apparentvalue of DoA and vice versa. Similar arguments havebeen advanced to account for the relationship betweenfront factors and activation energies for viscosity.14 Onthis basis, eq 5 is simply a mathematical artifact.

Second, within the framework of the Eyring’s acti-vated state theory, ln DoA is proportional to the entropychange of activation.15 In this context, eq 5 indicatesthat activation entropy is directly proportional to acti-vation energy. A linear free energy relation in chemicalkinetics is interpreted to mean that the reactionmechanism is similar among a related series of reac-tions.13 By analogy, the mechanism of small moleculediffusion in polymers is taken to be similar among avariety of polymers when eq 5 is obeyed. However,mechanistic differences may exist between glassy andrubbery polymers since b is substantially higher forglassy polymers than for rubbery polymers. At a fixedactivation energy and assuming the penetrant jumplength is not strongly different in rubbery and glassypolymers, the activation entropy is an order ofmagnitude lower in glassy polymers than in rubberypolymers.16 Interestingly, the polymers that define theupper bound relation in Figure 1 and in related plotsfor other light gas pairs are stiff-chain, amorphousglassy polymers.1

On the basis of continuum mechanics models for thecreation of holes or vacancies in materials, Lawson16 andKeyes17 suggested that the ratio of activation entropyto activation energy should be approximately 4 timesthe thermal expansion coefficient, which has beenobserved in some cases for diffusion of small moleculesin polymers. Alternatively, according to Barrer’s “zone”theory of diffusion, a penetrant executes a diffusionjump in an activated region (or zone) comprising seg-ments of polymer molecules near the diffusing penetrantmolecule. The size of the activated zone determines bothEDA and activation entropy or, equivalently, ln DoA.8 Thelarger the size of the activated zone required fordiffusion, the higher is the energy, EDA, required tocreate it and the larger is the entropy change associatedwith the creation of the activated state.

In addition to eq 5, the theory requires a relationbetween activation energy and penetrant size. VanKrevelen reports that activation energy of diffusioncorrelates best with the square of penetrant diameterfor a wide range of polymers and does not provideseparate correlations for rubbery and glassy polymers.11

Meares found that the activation energy of gas diffusioncoefficients in poly(vinyl acetate) increased linearly withpenetrant collision diameter squared both above andbelow the glass transition temperature.18 Meares

PA ) SADA (2)

RA/B )PA

PB)

SA

SB

DA

DB(3)

DA ) DoAexp(-

EDA

RT) (4)

ln DoA) a

EDA

RT- b (5)

376 Freeman Macromolecules, Vol. 32, No. 2, 1999

developed a qualitative macroscopic explanation of theseresults based on the notion that the activation energyis proportional to the volume of the activated state,which is proportional to the product of penetrantdiameter squared and penetrant jump length.18 Othermore recent models, such as Brandt’s model, alsoprovide support for the notion that activation energydepends on the penetrant diameter squared for stiff-chain glassy polymers such as those that define theupper bound relations.19 In glassy polymers, activationenergy vs penetrant diameter squared plots usuallyextrapolate to activation energies of zero for finitesized penetrants.6,20 Brandt’s model ascribes thisobservation to the existence of a finite interchainseparation in the nonactivated or equilibrium state.19

Therefore, the effect of penetrant size on activationenergy is modeled as follows:

where c and f are constants which depend on thepolymer, and dA is the penetrant diameter. For lightgases, the kinetic diameter, which characterizes thesmallest zeolite window through which a penetrantmolecule can fit,21 is the most appropriate measure ofpenetrant size for transport property correlations.6,11

Therefore, in this work, dA is taken to be the kineticdiameter of the penetrant. Van Krevelen reports cvalues from 250 for extremely flexible poly(dimethyl-siloxane) to approximately 1100 cal/(mol Å2) for stiff-chain, glassy poly(vinyl chloride).11 Polymers thathave high diffusivity selectivity should have high valuesof c. In this regard, for a high-performance glassypolyimide (synthesized from 3,3′,4,4′-biphenyltetra-carboxylic dianhydride and 4,4′-diaminodiphenyl ether),the value of c may be as high as 2400 cal/(mol Å2).22

The value of f ranges from 0 for rubbery polymers andlow-performance glassy polymers to approximately 14 000cal/mol for the polyimide prepared by Haraya et al.22

The ratio xf/c is a crude measure of the averagedistance between polymer chains. For the polyimide justdescribed, xf/c is 2.5 Å.

Equation 6 should apply to polymers with interchainspacings on the order of or smaller than the size of thepenetrant molecules so that thermally activated motionof polymer chain segments controls penetrant diffusion.The theory is not expected to be valid if the interchainspacings are significantly larger than the size of thepenetrant molecules because, in this case, the rate-limiting step for penetrant diffusion would not neces-sarily be controlled by polymer chain motion. The theoryis restricted to light gas molecules, such as He, H2, N2,O2, CO2, CH4, and the like, where gas kinetic diameterprovides a good measure of penetrant size as it relatesto transport properties. For larger penetrants (e.g., dA> 4.4 Å for poly(vinyl acetate)20) the activation energyversus penetrant diameter squared relation becomesconcave to the penetrant size axis. Moreover, for largerpenetrants, kinetic diameter is not a good estimate ofpenetrant size important for transport properties.23 Inthese cases, eq 6 is not strictly obeyed, and the theorywill not be valid.

Combining eqs 4, 5, and 6 gives the following expres-sion for the diffusion coefficient:

Thus, within the scope of this theory, the logarithm ofgas diffusivity decreases in proportion to penetrantdiameter squared. This scaling of diffusion coefficientwith penetrant size is consistent with the empiricalcorrelations of Van Krevelen11 and Teplyakov andMeares.24 Based on eqs 2 and 7, the permeabilitycoefficient is

and selectivity is

Combining eqs 8 and 9 yields

Equation 10 is the primary result of this theory and canbe used to gauge selectivity changes as one exploresvarious polymers with different permeability coeffi-cients. This equation is based on the four hypothesesembodied in eqs 2, 4, 5, and 6.

For a wide range of polymers, solubility and solubilityselectivity change much less than either selectivityor permeability coefficients.1 Therefore, for a particulargas pair and a fixed value of the parameter f, the termin curly brackets in eq 10 changes little from onepolymer to another. In this case, the logarithm ofselectivity should decrease linearly as the logarithm ofpermeability increases, consistent with the empiricalupper bound line in Figure 1. Equation 10 has thesame mathematical form as the empirical relationreported by Robeson if λA/B and âA/B in eq 1 are identifiedas follows:

and

Using kinetic diameters for dA and dB in eq 11, theslopes of the ln RA/B vs ln PA plots were calculated andcompared with the empirically determined slopes fromRobeson1,4 in Figure 2. The kinetic diameters used forthis calculation are given in Table 1. In general, theagreement between the calculated and observed slopesis strikingly good given the approximate nature of theslopes and the fact that the predictions are based on atheory with no adjustable parameters.

Robeson observed an excellent empirical correlationof λA/B with the difference between the kinetic diameters(in angstroms) of the gas molecules:1

If the expression for λA/B in eq 11 is rewritten as follows

EDA) cdA

2 - f (6)

ln DA ) -(1 - aRT )cdA

2 + f(1 - aRT ) - b (7)

ln PA ) -(1 - aRT )cdA

2 + f(1 - aRT ) - b + ln SA (8)

ln RA/B ) ln(SA/SB) + ln(DA/DB) ) ln(SA/SB) +

(1 - aRT )c(dB

2 - dA2) (9)

ln RA/B ) -[(dB

dA)2

- 1] ln PA + {ln(SA

SB) - [(dB

dA)2

- 1]× (b - f(1 - a

RT ) - ln SA)} (10)

λA/B ) (dB/dA)2 - 1 (11)

âA/B )SA

SBSA

λA/B exp{-λA/B[b - f(1 - aRT )]} (12)

λA/B ) dB - dA (13)

Macromolecules, Vol. 32, No. 2, 1999 Polymeric Gas Separation Membranes 377

then eq 13 will provide as good a correlation for λA/B aseq 11 if the term in square brackets is constant. Forthe penetrant pairs in Figure 2, the term in squarebrackets is approximately 0.8 and exhibits littlevariation. For example, the lowest value of the term insquare brackets is 0.60 for O2/N2, and the highest valueis 0.95 for He/N2. In contrast, λA/B values for these gaspairs vary by more than a factor of 7. Thus, the goodagreement that Robeson observed using eq 13 isconsistent with theory since the term in square bracketsin eq 14 varies little among the gas pairs considered.

To estimate âA/B, solubility and solubility selectivityvalues for different gases are required. Additionally, theparameter f is not prescribed by this treatment and istreated as an adjustable parameter. Like gas dissolutionin liquids, penetrant dissolution in polymers is regardedas a two step thermodynamic process: (1) condensationof the gaseous penetrant to a liquidlike density and (2)mixing of the pure compressed penetrant with thepolymer segments.6 The first step is governed bypenetrant condensability, and the second dependson polymer-penetrant interactions. For light gaspenetrants that do not undergo specific interactionswith the polymer, the first effect is often dominant, andas a result, penetrant solubility in polymers typicallyscales with convenient measures of penetrant condens-ability, such as penetrant boiling point, critical temper-ature, or Lennard-Jones temperature, ε/k.6 For thelatter, k is Boltzmann’s constant, and ε is the potentialenergy well depth parameter in the Lennard-Jonespotential energy function. A model of penetrantsolubility in polymers, developed from classicalthermodynamics, gives the following relation betweenpenetrant Lennard-Jones temperature and gassolubility in amorphous polymers where strong specificpolymer-penetrant interactions (e.g., hydrogenbonding) are not important:24,25

where M and N are parameters. For a variety of liquids,rubbery polymers, and glassy polymers, N is 0.023K-1.6,11 M is sensitive to polymer-penetrant inter-actions and, consequently, varies somewhat frompolymer to polymer. However, for simplicity, M isconstrained to a constant value. Accordingly, VanKrevelen recommends a value of -9.84 (with solubilityin units of cm3 (STP)/(cm3 cmHg)).11 Equation 15 canbe used to calculate solubility selectivity: ln(SA/SB) )N(εA/k - εB/k). As the parameter N has the same valuefor many materials, predictions of solubility selectivityshould be more accurate than predictions of solubility.

Equation 15 was used in eq 12 to estimate âA/B asshown below:

These values are compared in Figure 3 with the âA/Bvalues reported by Robeson. The Lennard-Jonestemperatures are recorded in Table 1. Equation 16contains one adjustable parameter, f. For simplicity, this

parameter was constrained to be a constant for all gaspairs, and its value was determined by a least-squaresminimization procedure assuming that the experimenttemperature, T, was 298 K. If some of the data weredetermined at other temperatures, then the value of fwould be adjusted accordingly. The best value of f wasapproximately 12 600 cal/mol.

(dB

dA)2

- 1 ) [dB + dA

dA2 ](dB - dA) (14)

ln SA ) M + N(εA/k) (15)

âA/B ) N(εA

k-

εB

k )(M + NεA

k )λA/B

×

exp{-λA/B[b - f(1 - aRT )]} (16)

Figure 2. Comparison of slopes of ln RA/B vs ln PA plots, λA/B,reported by Robeson1,4 with theoretical prediction (solid line).Gas pairs are listed as A/B (i.e., O2/N2 implies that O2 ) Aand N2 ) B). dA and dB are the kinetic diameters of penetrantsA and B, respectively, taken from the tabulation by Breck.21

Figure 3. Comparison of front factors of ln RA/B vs ln PA plots,âA/B, reported by Robeson1,4 with theoretical prediction (solidline). The best fit of the data to the theory is obtained withthe parameter f set to 12 600 cal/mol. The units of âA/B are[cm3 (STP) cm/(cm2 s cmHg)]λA/B.

Table 1. Penetrant Size and Condensability Parameters

penetrantkinetic

diam (Å)21ε/k

(K)11 penetrantkinetic

diam (Å)21ε/k

(K)11

He 2.69 10.2 O2 3.46 107H2 2.8 60 N2 3.64 71CO2 3.3 195 CH4 3.87 149

378 Freeman Macromolecules, Vol. 32, No. 2, 1999

The agreement of the theory with Robeson’s valuesis impressive considering that only one adjustableparameter is used to describe the upper boundpermeability/selectivity behavior of these 11 gas pairs.However, for some gas pairs, there are significantdeviations between the calculated front factors andthose reported by Robeson. The assumption that M andf are constant for all polymers that might lie on theupper bound for all of these gas pairs is clearly a strongapproximation which could be relaxed if one wereinterested in predictions related to specific gas pair/polymer combinations. Also, the upper bound lines weredrawn somewhat subjectively, and some disagreementwith the theory is not surprising.

Implications and ConclusionsEquation 10 can be written as

This form of eq 10 highlights the impact of changes inpenetrant diffusion coefficients from polymer to polymeron the upper bound selectivity. The solubility selectivityterm in eq 17 should change little from polymer topolymer, and the term λA/Bb is a constant for a givengas pair and a given polymer class (i.e., rubbery orglassy). This result, that diffusivity should play a moreimportant role than solubility in determining upperbound selectivity values, is consistent with theconclusions of Robeson.1 The theory also provides arationale for the following heuristic that has emergedas a result of many years of systematic experimentalstructure/property studies:3,6 for these light gas pairs,the most productive route discovered to improvepermeability/selectivity properties is to modify polymerchemical structure to increase polymer backbonestiffness (i.e., increase c, which increases RA/B (cf. eq 9))while simultaneously disrupting interchain packing toincrease f, thereby increasing diffusivity and, in turn,permeability (cf. eq 8).

Based on the agreement of the theory with theexperimental data for many gas pairs, the fundamentalcharacteristics of polymers with outstanding gasseparation properties for one pair of gases are sharedby polymers having excellent separation properties forother gas pairs. Indeed, Robeson noted that some highglass transition temperature, rigid backbone, amor-phous polymers with relatively large interchain spacings(i.e., high fractional free volume) lie on or near the upperbound lines for many gas pairs.1 Of course, the vastmajority of polymers do not lie on the upper bound lines.Rubbery polymers generally sieve penetrants weaklybased on size, so the parameter c in eq 6 is low or theactivation energy is proportional to only the first powerof penetrant diameter. Both of these characteristicsmove polymers away from the upper bound lines.Additionally, if interchain spacing is low, f will be lower,thereby decreasing diffusion coefficients (cf. eq 7) and,in turn, permeability coefficients.

Within the scope of this simple theory, there is noinfluence of polymer structure on the slope of the upperbound or tradeoff curves, λA/B, as this parameterdepends only on penetrant size ratio. If this is true, thenthe slopes of the upper bound lines are unlikely tochange with further polymer development efforts. In

contrast, âA/B contains variables that can be systemati-cally tuned by rational polymer structure manipulationto simultaneously improve permeability and selectivitycharacteristics. In this regard, the most fruitful pathwayfor development of higher performance polymericmembranes for the separations discussed in this paperis to systematically increase âA/B, either through solubil-ity selectivity enhancement and/or increases in chainstiffness (i.e., increasing c), while simultaneouslyincreasing interchain spacing (i.e., fractional freevolume), to increase selectivity while maintaining orincreasing permeability. Increasing interchainseparation to increase permeability without sacrificingselectivity should only be effective as long as theinterchain separation is not so large that penetrantdiffusion coefficients are no longer governed bythermally stimulated polymer segmental motions.Materials that are beyond this limit may already exist.Poly(1-trimethylsilyl-1-propyne), the most permeablepolymer known and the polymer which has the highestfree volume of all hydrocarbon-based polymers, also hasthe lowest selectivities for permanent gases (e.g., O2/N2) of all polymers and exhibits many permeationcharacteristics similar to those of microporousmaterials.26-28

Simply increasing polymer backbone stiffness (to theextent that this increases c in eqs 7 and 9) results inincreased selectivity but lower diffusivity and, therefore,permeability. So backbone stiffness increases should becoupled with increases in interchain separation toachieve both higher permeability and higher selectivity.These simple considerations suggest that simultaneouschain stiffness and interchain separation increases canbe used to systematically improve separation perfor-mance until the interchain separation becomes largeenough that the polymer segmental motion no longergoverns penetrant diffusion. As indicated above, unlesssignificant enhancement in solubility selectivity couldbe achieved, this limit would represent the asymptoticend point in the performance of polymeric membraneswhose separation properties are dictated by the hypoth-eses embodied in eqs 2, 4, 5, and 6. To achieve stillhigher selectivity/permeability combinations, materialsthat do not obey these simple rules would be required.

References and Notes

(1) Robeson, L. J. Membr. Sci. 1991, 62, 165.(2) Henis, J. M. S. Commercial and Practical Aspects of Gas

Separation Membranes; CRC Press: Boca Raton, FL, 1994.(3) Stern, S. A. J. Membr. Sci. 1994, 94, 1.(4) Robeson, L. M.; Borgoyne, W. F.; Langsam, M.; Savoca, A.

C.; Tien, C. F. Polymer 1994, 35, 4970.(5) Graham, T. Philos. Mag. 1866, 32, 401.(6) Petropoulos, J. H. Mechanisms and Theories for Sorption and

Diffusion of Gases in Polymers. In Polymeric Gas SeparationMembranes; Paul, D. R., Yampol’skii, Y. P., Eds.; CRCPress: Boca Raton, FL, 1994; pp 17-81.

(7) Barrer, R. M. Trans. Faraday Soc. 1939, 35, 644.(8) Barrer, R. M. Trans. Faraday Soc. 1942, 38, 322.(9) vanAmerongen, G. J. J. Appl. Phys. 1946, 17, 972.

(10) Barrer, R. M.; Skirrow, G. J. Polym. Sci. 1948, 3, 549.(11) Van Krevelen, D. W. Properties of Polymers: Their Correla-

tion with Chemical Structure; Their Numerical Estimationand Prediction from Additive Group Contributions; Elsevi-er: Amsterdam, 1990.

(12) Barrer, R. M. Trans. Faraday Soc. 1943, 39, 48.(13) Leffler, J. E. J. Org. Chem. 1955, 20, 1202.(14) Barrer, R. M. J. Phys. Chem. 1957, 61, 178.(15) Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate

Processes; McGraw-Hill Book Co., Inc.: New York, 1941.(16) Lawson, A. W. J. Chem. Phys. 1960, 32, 131.(17) Keyes, R. W. J. Chem. Phys. 1958, 29, 467.

ln RA/B ) -λA/B ln DA + {ln(SA

SB) - λA/B(b - f(1 - a

RT ))}(17)

Macromolecules, Vol. 32, No. 2, 1999 Polymeric Gas Separation Membranes 379

(18) Meares, P. J. Am. Chem. Soc. 1954, 76, 3415.(19) Brandt, W. W. J. Phys. Chem. 1959, 63, 1080.(20) Brandt, W. W.; Anysas, G. A. J. Appl. Polym. Sci. 1963, 7,

1919.(21) Breck, D. W. Zeolite Molecular Sieves; Wiley-Interscience:

New York, 1974.(22) Haraya, K.; Obata, K.; Hakuta, T.; Yoshitome, H. Maku

(Membrane) 1986, 11, 48.(23) Tanaka, K.; Taguchi, A.; Hao, J.; Kita, H.; Okamoto, K. J.

Membr. Sci. 1996, 121, 197.

(24) Teplyakov, V.; Meares, P. Gas Sep Purif. 1990, 4, 66.(25) Michaels, A. S.; Bixler, H. J. J. Polym. Sci. 1961, 50, 393.(26) Srinivasan, R.; Auvil, S. R.; Burban, P. M. J. Membr. Sci.

1994, 86, 67.(27) Pinnau, I.; Toy, L. G. J. Membr. Sci. 1996, 116, 199.(28) Pinnau, I.; Casillas, C. G.; Morisato, A.; Freeman, B. D. J.

Polym. Sci., Polym. Phys. Ed. 1996, 34, 2613.

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380 Freeman Macromolecules, Vol. 32, No. 2, 1999