Journal of Membrane Science 341 (2009) 178185

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Journal of Membrane Science

journa l homepage: www.e lsev ier .com

An emp ndand its

L.M. Roba Lehigh Univerb Department o

a r t i c l

Article history:Received 30 AReceived in reAccepted 3 JunAvailable onlin

Keywords:Gas permeability correlationPermselectivity correlationKinetic diametersUpper bound

s for cwher>1.0. Tdiffateri

gas pair was modest over the entire range of permeability. It was found that n correlates with the kineticdiameter of the specic gases of the pair by a relationship: n 1(dj/di)2 1 in agreement with theory.Correlations exist between n and k for the noted relationship and nu and ku of the upper bound relationshipof Pi = kunuij where ij = Pi/Pj. The experimental values of n 1 enable the determination of a new set ofkinetic diameters showing excellent agreement between theory and experimental results. The value of

1. Introduc

The vastture allowsof gas diffucorrelationsious structureceiving corelationshipthe permea

The initiing minor sthe uniqueseparation.sus Pi wher

CorresponE-mail add

(B.D. Freeman

0376-7388/$ doi:10.1016/j.m

Downloaded 1/k was found to be virtually an exact t with the relationship developed by Freeman in predicting thevalue of ku for the upper bound relationship using the new set of kinetic diameters where the calculationswere constrained to minimize the error in (n 1) = (dj/di)2 1. The signicance of these results includesa new set of kinetic diameters predicted by theory and agreeing with experimental data with accuracysignicantly improved over the zeolite determined diameters previously employed to correlate diffusionselectivity in polymers. One consequence of this analysis is that the kinetic diameter of CO2 is virtuallyidentical to that of O2. Additionally, the theoretical relationship developed by Freeman for the upper boundprediction is further veried by this analysis which correlates the average permeability for polymericmaterials as compared to the few optimized polymer structures offering upper bound performance.

2009 Elsevier B.V. All rights reserved.

tion

amount of permeability data now present in the litera-for signicant correlation of the basic characteristicssion, solubility and permeability in polymers. Manyare already well documented in the literature for var-reproperty relationships. One of these correlationsnsiderable interest is referred to as the upper boundcorrelating the separation factor, ij(ij = Pi/Pj), with

bility of the more permeable gas, Pi.al references [1,2] have been recently updated [3] show-hifts in the upper bound except for specic cases wheregas solubility in peruorinated polymers dominates theThe relationship noted involved a loglog plot of ij ver-e the upper bound line (above which virtually no data

ding author. Tel.: +1 610 481 0117.resses: lesrob2@verizon.net (L.M. Robeson), freeman@che.utexas.edu), drp@che.utexas.edu (D.R. Paul), rowe@che.utexas.edu (B.W. Rowe).

points exist) is ij expressed by the relationship:

Pi = kunuij (1)

where nu is the upper bound slope and ku is referred to as the frontfactor and is equal to Pi where ij = 1. The value of nu was foundto correlate with the kinetic diameter difference in the gas pairswith the relationship: 1/nu dj di where dj and di represent thegas diameters of the lower permeable gas and the higher perme-able gas, respectively. The prediction of the empirical upper boundrelationship has been presented by Freeman [4] employing funda-mental relationships to correlate both nu and ku. It was noted thatthe slope, nu, can be expressed by

1nu

=(

djdi

)2 1 = ij (2)

Since (dj/di)2 1 = [dj + di/d2i ](dj di) and the term in square

brackets is approximately constant for the gas pairs of interest, thetheoretical correlation and the empirical correlation show reason-able agreement. Freeman also correlated the front factor, ku, with

see front matter 2009 Elsevier B.V. All rights reserved.emsci.2009.06.005irical correlation of gas permeability atheoretical basis

eson a,, B.D. Freeman b, D.R. Paul b, B.W. Rowe b

sity (adjunct), 1801 Mill Creek Rd., Macungie, PA 18062, United Statesf Chemical Engineering, University of Texas at Austin, Austin, TX 78712, United States

e i n f o

pril 2009vised form 3 June 2009e 2009e 12 June 2009

a b s t r a c t

A large database of permeability valuein the following correlation: Pj = kPniare chosen such that the value of n isorders of magnitude implying solutionexisting in known dense polymeric mfrom http://www.elearnica.ir/ locate /memsci

permselectivity in polymers

ommon gases (He, H2, O2, N2, CO2 and CH4) has been employede Pi and Pj are the permeabilities of gases i and j; the indiceshe plots of log Pi versus log Pj show linear behavior over nine

usion behavior persists over the entire range of permeabilitiesals. The scatter of data around the linear correlation for each

L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178185 179

solubility relationships, the upper bound slope, and fundamentalconstants related to the diffusion of gases in polymers as will bediscussed later. While the upper bound slope correlated reason-ably well with the kinetic diameter difference (based on zeolitedetermined[5]), some sof signicantions in theThis was adVariable Mediameters wgas diametefor the six gequivalent t

Several gdict the perunits. The based on poume ideas tunits of thepredicts anability. Witseparation,volume wasities to predof aromaticbasic equat

P = An exp(

where An anmer fractionper mole ounit. In ordfree volumempirical fafrom a perm

AnotherPaul methocontributiolarge data bchain [9,10both permecomprisedapplicationlates, polysketones), povation that trelationshipthe permeaThis grouption:

ln P =n

i=1

where i isrepeat unitThe choicegroups wasgroup arouof aromaticmeability. Asubstitutionity. The struthe main ch

Yampolskii et al. presented a group contribution method forprediction of gas permeability and diffusion coefcients of glassypolymer based on the chemical structure of the repeat units withthe groups chosen to be specic atoms and their bond congura-

1,12utio

lity fe ander coa rel

on anden

role oand

lity ay. Paunship

then to

nts alecuent w

mmad bylyakoefceab

f four dif

s mo[21]

d2i

c anlecutionactio

dataed thdata

n O2plo

rsusexpe

per bres f

ults:

perwh

sted253ifferees wr of sd atlassycontationta arhere

nikinetic diametersalso referred to as Breck diameterscatter existed at least partially due to a small numbert gures reported for Breck diameters as well as varia-

zeolite data versus values expected in polymer diffusion.dressed recently by Dal-Cin et al. [6] where the Error inthod (EVM) was applied to predict a new set of kinetichich allowed for better correlation of 1/nu and ther relationships. The Dal-Cin sum of kinetic diametersases of interest (He, H2, CO2, O2, N2, CH4) was virtuallyo the Breck diameter sum.roup contribution methods have been applied to pre-

meability of gases in polymers based on their structuralrst method noted in the literature was by Salame [7]lymer cohesive energy density and fractional free vol-hat assigned parameter values based on the repeatingpolymer structure. This approach for O2/N2 separationincreasing separation factor with decreasing O2 perme-h the increasing interest in aromatic polymers for gasa group contribution method based on fractional freereported by Park and Paul [8] with quite good capabil-ict permeability and permselectivity for a wide varietypolymers of interest. This approach started with the

ion:

BnFFV

)(3)

d Bn are constants for a particular gas, FFV is the poly-al free volume (FFV = (V V0)/V), V0 is the dense volume

f repeat unit and V is the volume per mole of repeater to improve the predictive capability, the fractional

e was varied for the gases for a specic polymer. Thectors for 41 different structural units were determinedeability database of 102 polymers.approach with similar characteristics to the Park andd was published in the same time frame. This groupn approach was based on solving a least squares t of aase of polymers containing aromatic units in the main

]. This approach was primarily directed at predictingability and permselectivity of aromatic polymers whichmany of the polymers of interest for gas separations. These polymers included polycarbonates, polyary-ulfones, polyimides, aromatic polyamides, poly(arylly(aryl ethers). This approach was based on the obser-he permeability of copolymers followed a logarithmeticinvolving the volume fraction of the comonomers and

bility of the homopolymers comprising the copolymer.contribution method was based on the following equa-

iln Pi (4)

the volume fraction of a specic group i of the polymerand Pi is the permeability contribution of that group.of subdividing the polymer repeat units into specicconsidered critical in cases where symmetry of the

nd the main chain such as the iso versus para linkagegroups was noted to be a major factor in polymer per-lso aromatic group substitution such as alkyl or halide(mono- or di-) also greatly inuenced the permeabil-

ctural units were thus chosen to be subdivided aroundain bond.

tions [1contribmeabivolum

Othwherediffusienergyof the[15]. Jiameabidensitrelatiosize offunctiosegmegas moconsist[4], suoutline

Tepsion coto permment o(two fothe gaBrandt

Edi = c

wheregas mocorrelawith fr[22].

Theincludber ofdata owas emted veresultsthe upstructu

2. Res

Theeraturelms tedata iswith doratorinumbedirectethus, gwhilecorrelaThe daaxis) w

Pj = kP]. A similar approach using molecular connectivity groupn has been noted by Bicerano [13] to predict oxygen per-rom a correlation with cohesive energy density, molar

van der Waals volume.rrelations worth noting include that of Meares [14],

ationship was noted between the activation energy ofd the square of the gas diameter times the cohesive

sity. This method was recently reviewed in a discussionf the cohesive energy density on diffusion in polymersXu [16] showed a correlation between the log of the per-

nd the molar free volume divided by the cohesive energyl and DiBenedetto [1719] analyzed in more depth thebetween the activation energy for diffusion and the

gas molecule using a Lennard-Jones (612) potentialdescribe the energy of interaction between polymer

s they move apart to accommodate a diffusion jump of ale. In the end, the result of this more complex analysis is

ith the mathematical approximation used by Freemanrized by Eq. (5) below, and the conceptual frameworkMeares [14].v and Meares described a method for predicting diffu-

ients and solubility constants for various gases leadingility predictions [20]. This procedure involved assign-r temperature-dependant parameters for each polymerfusion and two for solubility). The relationship betweenlecule size and the activation energy, Edi, was noted bywith the relationship:

f (5)

d f are constants related to the polymer and di is thele size (also referred to as the kinetic diameter). Theof the activation energies of permeability and diffusionnal free volume was demonstrated by Yampolskii et al.

base employed for the recent upper bound paper [3]e published literature since 1991 and had a large num-points for each gas pair of interest. For example, the

/N2 included over 1000 different values. This databaseyed for the analysis to be discussed where Pi is plot-Pj on a loglog plot. This approach, therefore, yieldscted for typical polymeric materials as compared withound results which correlate polymers with optimizedor maximum separation factors.

permeability correlation

meability database employed data reported in the lit-ere the gas pair values were determined on the samein the same laboratories. The temperature range for this5 C. Correlation of data on the same polymer structurent preparation methods and/or tested in different lab-ould not be as accurate. It should be noted that a largeamples where gas pairs were reported involved studiesachieving values approaching upper bound properties;polymers dominate the dataset. In essence, the dataset,ining a few rubbery polymers, primarily represents afor polymers below their glass transition temperature.e presented as loglog plots of Pi (y-axis) versus Pj (x-the basic equation:

(6)

180 L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178185

describes thequation fo

log Pi = l

The value oPi = 1.0 Barrein the uppePi and compbetween th

1nu

n 1

Mathematiclog Pi versusto the suggeP(N2) is shoshown on tand that ofclose

1nu

= 0.17

It is inteear behaviosuggests thrange of peRef. [3] follospread of daest and coveconstant N2binations o(15 pairs ofgas pairs ar

All gas ptionship wCO2/O2 as tequal. The Bas will be smuch less. Tgas pairs ar

The corrting n versu

Fig. 2. CO2/CH4 permeability correlation.

Fig. 1. Correlation of P(O2) and P(N2) data.

e results when i and j are chosen such that n > 1.0. Ther the plot on loglog coordinates is given by

og kn

+ log Pjn

(7)

f k has units of (Barrers)1n. The value of k is Pj wherer. Note that k and n are not the same values as employedr bound relationship noted above. Solving Eq. (1) foraring with Eq. (7) shows an approximate relationship

e upper bound slope nu and n of Eq. (6) expressed as

(8)

ally, this would be an exact relationship if in plots oflog Pj the upper bound relationship were a line parallel

sted correlation line, i.e., Eq. (6). The plot of P(O2) versuswn in Fig. 1. The upper bound line (from Ref. [3]) is alsohe plot. In this case, the slope of the upper bound linesthe correlation lines, as suggested by Eq. (8), are quite

65 and n 1 = 0.1576 (9)resting to note that the loglog relationship shows lin-r over nine orders of magnitude in permeability. Thisat the solutiondiffusion mechanism is valid over thisrmeabilities. As expected, the upper bound slope fromws the upper range of values on the plot in Fig. 1. Theta below and above the correlation slope is quite mod-rs less than an order of magnitude of O2 permeability atpermeability. The same correlation exists for all com-

f the gas pairs of the list of He, H2, O2, N2, CO2, and CH4ij combinations where Pi > Pj). Additional plots of othere illustrated in Figs. 29.airs, except CO2/O2, also exhibit an upper bound rela-

here 1/nu > 0. This apparently does not occur withhe true kinetic diameters of this gas pair are virtuallyreck diameter difference for this pair is 0.16 ; however,hown, the predicted kinetic gas diameter difference ishe n and k values (Eq. (6)) from Figs. 19 and the other

e listed in Table 1.elation of n with the kinetic diameter is shown by plot-s (dj/di)2 1 and gives the best results when the Dal-Cin

Fig. 3. CO2/N2 permeability correlation.

Table 1n and k values from Eq. (5).

Gas pair n k (Barrers)1n (dj/di)2 1 (Dal-Cin values[6])O2/N2 1.1576 0.176 0.1305H2/N2 1.5242 0.003408 0.6025He/N2 1.8290 0.00123 0.9718H2/CH4 1.7480 0.00145 0.8760CO2/N2 1.1212 0.03419 0.0961He/H2 1.1609 0.6228 0.2305He/CH4 2.133 0.0002938 1.3085N2/CH4 1.1386 0.9625 0.1701CO2/O2 1.017 0.223 0.0304He/O2 1.5937 0.01376 0.7442H2/O2 1.3297 0.04334 0.4175He/CO2 1.5961 0.0667 0.7988H2/CO2 1.3500 0.139 0.4619CO2/CH4 1.3034 0.0176 0.2833O2/CH4 1.3096 0.1173 0.3235

L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178185 181

Fig. 4. CO2/O2 permeability correlation.

kinetic diameters are used. This is shown in Fig. 10 where the corre-lation coefcient has an R value of 0.992. Using the Breck diametersfor (dj/di)2 1 gave an R value of 0.977. When dj di was employedusing the Breck diameters the R value was 0.971. The Dal-Cin EVManalysis shothat of O2. Tkinetic diamexaminatiolation whenis in the dediameter isgases.

The n vahundreds totion of thesappropriateare based o

Fig. 7. N2/CH4 permeability correlation.ws the CO2 kinetic diameter to be slightly greater thanhe plot for CO2/O2 with n = 1.017 indicates that the O2eter should be slightly higher than that of CO2. A careful

n of Fig. 10 shows the data points to be above the corre-CO2 is in the numerator and below the line when CO2

nominator also indicating that the Dal-Cin CO2 kineticslightly higher than it should be relative to the other

lues from the log Pi versus log Pj plots comprise literallygreater than 1000 data points. Intuitively, the correla-

e n values with the kinetic diameters should be morethan those derived from the upper bound slopes thatn a few polymers optimized for permselectivity. TheFig. 5. He/H2 permeability correlation.

Dal-Cin kinupper boun

Kineticanalysis th(n 1) = (djnite intercBreck and Ddj/di, not th

Table 2Comparison o

Gas Breck()

He 2.6H2 2.89CO2 3.3O2 3.46N2 3.64CH4 3.8Fig. 6. H2/N2 permeability correlation.etic diameters offer an excellent t but are based on thed slope analysis.diameters were determined using a least squaresat constrains the values to conform to the form

/di)2 1 with the line going through the origin, i.e., noept. The results are shown in Table 2 along with theal-Cin diameters. Because this analysis was based on

e absolute values of dj and di, an appropriate constraint

f kinetic diameters.

diameter L-J collisiondiameter ()

Dal-Cindiameter ()

CurrentAnalysis ()

2.556 2.555 2.6442.928 2.854 2.8754.07 3.427 3.3253.46 3.374 3.3473.71 3.588 3.5683.817 3.882 3.817

182 L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178185

Fig. 8. O2/CH4 permeability correlation.

was needed to generate a single set of diameters. The primary rea-son that the kinetic diameters for diffusion in polymers might bedifferent than the simple diameter determined from other infor-mation (e.gthe molecuand CH4 arity in CH4 dwas xed atsecond viraing settingthe Breck dwith variatvery slightlkinetic diamon xing th

Fig. 11 shdiameters linear regretional t (R

Fig. 10. Correlation of n employing the Dal-Cin kinetic diameters.

then the slope becomes closer to one (1.0076) without an apprecia-ble change in the R value. The data point with the largest variationfrom the line of Figs. 11 is for He/CH4 separation. The He/CH4 plot

i versus log Pj also had the largest spread of data around thecorrelation thus yielding the largest potential for error in theinatmajEVM

2 dias anabe exvolumhe mnor dt thaly idy uni., viscosity or second viral coefcients) would be thatles are not spherical. Among the gases of interest, Hee the closest to spherical. Based on the small variabil-iameter reported in the literature, its kinetic diameter3.817 , as reported by Hirschfelder et al. [23] based on

l coefcients. Another constraint was also tested involv-the sum of the kinetic diameters equal to the sum ofiameters. This assumption yielded diameters very closeions only in the third signicant gure and an R valuey lower. However, based on the discussion above, theeter results used in the following discussion are based

e CH4 diameter at 3.817 .ows the relationship between n 1 and the new kinetict to the theoretical expression derived by Freeman [4]. Assion shows the intercept is nearly (0,0) with an excep-= 0.998). If the intercept is xed at the (0,0) intercept,

of log Plineardeterm

TheDal-Cinthe COsquaremighton thealong tthe misuggesvirtual(ideallFig. 9. H2/CO2 permeability correlation.

Fig. 11. Corrtionship (kin(n 1) = (dj/di)ion of n.or difference between the least squares analysis and the

diameters is for CO2 and O2. The Dal-Cin results showmeter to be modestly larger than O2 whereas the leastlysis shows O2 to be marginally higher. Intuitively, CO2pected to have a larger kinetic diameter than O2 based

e comparison of the molecules; however, CO2 is planarajor axis and possibly slightly more compact than O2 iniameter of the molecules. At any rate, the data stronglyt the effective kinetic diameters of these two gases areentical. Another difference is that the slope of the linety) is closer to unity for the least squares analysis data.elation of n 1 with the theoretical kinetic diameter rela-etic diameters determined from least squares t constraining2 1.

L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178185 183

Fig. 12. Correlation of upper bound slope values with permeability gas pair slopes.

The comparison of 1/nu versus (n 1) is shown in Fig. 12. Asexpected, the upper bound slope correlation values are all abovethe linear line indicating linear proportionality from the expres-sion noted in Eq. (8). The slope of the line is 1.0681 indicating thatthe upper bound curves modestly deviate more from the averagepolymer da

Anotherby polymerand polypydata plotteddata on allresults are iimide datadata (Tabledata for polmers, the oseparation cthe polyimithan when

Fig. 13

Table 3Data comparison of values of Eq. (10).

Gas pair k1/nuu (Barrers)1/nu 1/k (Barrers)n 1 k1/nuu

a 1/kb

O2/N2 9.1992 5.6818 0.1736 0.1508H2/N2 1237 293.4 0.00358 0.001679He/CH4 51060 3404 9.56 109 1.592 108H2/CH4 3334 689.7 1.849 105 2.280 105CO2/CH4 115.04 56.82 0.04824 0.05254He/H2 3.9978 1.6056 0.03835 0.03959He/O2 673 72.670 1.2765 105 7.950 105H2/O2 99.860 23.07 0.004059 0.011640He/N2 10002 813 1.7234 106 4.1734 106He/CO2 216 14.99 1.3732 106 1.6402 105H2/CO2 38.9 7.194 0.0002667 0.0022275

a (cm3(STP)cm/cm2s cmHg)1/nu .b (cm3(STP)cm/cm2 s cmHg)(n1).

3. Correlation of front factors

The values of ku and k are related by the following

k1/nuu 1k

(10)

provided the relationship given by Eq. (8) is strictly valid as may beseen by combining Eqs. (1) and (6). This relationship can be testedby comparing the values of the front factors from the upper bound,Eq. (1), versus those from the correlation line from Eq. (6) for thedifferent ga

datae pr

ad shiquey Fre

o thewn iicalby Erisonm thku, w

= SiSjta as the kinetic diameter difference increases.data le has been compiled listing permeability dataclass. One le with signicant data includes polyimides

rrolones. An interesting comparison involves the O2/N2as log P(O2) versus log P(N2) and compared with the

polymers from the upper bound correlation le. Thellustrated in Fig. 13. The values for Eq. (6) for the poly-are n = 1.1343 and k = 0.1483 compared to all polymer1) of n = 1.1576 and k = 0.176 Barrers(1n). As most of theyimides are on or above the correlation line for all poly-bservation that polyimides generally yield better O2/N2haracteristics is correct. Also to be noted is the spread ofde data around the correlation line is somewhat lowerall polymers are employed (see Fig. 1).

Theonly thdata hmer unmade beters sare shoa graphgestedcompa

Frofactor,

k1/nuu. Comparison of polyimide data with typical polymer results.s pairs.for k1/nuu were chosen from Table 12 of Ref. [3] using

ior upper bound data since the newer upper boundifts of several gas pairs based on peruorinated poly-gas solubility relationships. In addition, the predictionseman [4] were based on the older upper bound param-ir choice facilitates that comparison. These parametersn Table 3 using two different sets of units. Fig. 14 showscomparison where the solid line is the relationship sug-q. (10). As can be seen, there is general agreement in this

of the two front factors.e theoretical model developed by Freeman [4], the frontas predicted to be as follows

Sij exp{

ij[b f

(1 aRT

)]}= ij (11)Fig. 14. Correlation of 1/k1/nuu with 1/k.

184 L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178185

where Si and Sj are the respective gas solubilities. a and b are deter-mined from the free energy relationship between the activationenergy of diffusion, Edi and D0i:

ln D0i = aE

R

where a isglassy polymthe relationsize of theadjustable vt of the upto12,600 cashown to beTb, or the Ltions:

ln Si = m +

ln Si

where m, xrelationship(specicallyrinated polshown in Esolubility rebility is in ufor k1/nuu w

Since theapproximatthe value shlation betwin Fig. 12), tto the value1/k1/nuu infall about th

1k

= SiSj

Sij e

Using the aand employij were cal(14) were ca

ij =(

djdi

)

using the vashown in Ta

The plotmental resuused by Freeshowing thabove discuCO2, the datator and beare removeexact. Thisto the otherJones solubanalysis as wuniversal vain Fig. 14. Th

primarily d

to P(0.093697i

is in units o

Comparison of predicted results with experimental data with (the units of/k are [cm3(STP)cm/(cm2 s cmHg)]ij where ij for the predicted results werened from the constrained least squares analysis results and ij = n 1 for theental results determined from the data).

Comparison of typical polymer data and group contribution results forand O2/N2 gas pairs in the log Pi versus log Pj plot.

comparison of the group contribution data with the log Pilog Pj data is shown in Fig. 16 for O2 and N2 group values from] and CO2 and CH4 group values from Ref. [10]. The lines forfor typical polymer data are taken from Fig. 1 and for CO2/CH4ig. 2. The group contribution data show good correlation withical polymer data as would be expected.

clusions

large database of gas permeability of polymers, primarily inssy state, used for the recent upper bound analysis [3] was

d here to correlate the polymer permeability results for var-s pairs chosen from the list He, H2, O2, N2, CO2, CH4. Plots ofersus log Pj showed linear behavior over the entire perme-range, i.e., Pj = kPni , where i and j were chosen such that n > 1,h of the 15 separate gas pair values. The upper bound analy-di

T b (12)

0.64 and b has values of 9.2 and 11.5 for rubbery anders, respectively. The parameter, f, is determined from

ship between activation energy of diffusion and thepenetrant molecule given by Eq. (5) where c and f arealues dependant upon the specic polymer. The bestper bound data with theory yields a value of f equal

l/mol [4]. The solubility coefcients for gases have beenrelated to the critical temperature, Tc, the boiling point,ennard-Jones parameter (/k) by the following equa-

0.027Tb, ln Si = x + 0.016Tc,

= y + 0.023(

ik

)(13)

and y have unique values for each polymer. Theses appear to work well for a wide range of polymersaliphatic and aromatic polymers) except for peruo-

ymers where the slope values are different than thoseq. (13). For the analysis by Freeman, the Lennard-Joneslationship was chosen with y set to 9.84 when solu-nits of cm3(STP)/(cm3-cmHg). For this model, the unitsere (cm3(STP)cm/cm2s cmHg)

1/nu .proportionality shown in Eq. (10) appears to be a good

ion, as shown in Fig. 14, 1/k should be proportional toown in the right side of Eq. (11). Since there is a corre-

een 1/nu and (n 1) (as shown in Eq. (8) and illustratedhen the values of the right side of Eq. (11) can be relateds of 1/k and n 1 of Eq. (6). The comparison of 1/k andFig. 14 shows close to an equivalence since the valuese equal value line; thus:

xp{

ij[b f 1 a

RT

]}= ij (14)

and b (b chosen for glassy polymers) values noted aboveing the Lennard-Jones solubility relationship, values ofculated and compared with 1/k. The values of ij in Eq.lculated from:

2

1 (15)

lues of dj and di from the least squares analysis resultsble 2.of log of ij (predicted values) versus log(1/k) (experi-lts) is shown in Fig. 15 for f = 12,600 cal/mol (same valueman in the upper bound theory [4]). The t is quite good

e validity of the Freeman theory as expected from thession. It is interesting to note that with data involvinga points are above the line if CO2 is in the gas pair numer-low the line if CO2 is in the denominator. If the CO2 datad from the analysis, the t is even better and virtuallyindicates that the CO2 solubility relationships relative

gases may not be properly predicted by the Lennard-ility relationship. As the value of 12,600 cal/mol ts this

ell as the upper bound analysis, it appears that it has alue. This is not unexpected based on the results showne ratio of ij (upper bound)/ij (average value) is, thus,

etermined by the ratio of P1/nui

/P(n1)i

which is equal+1.0681(n1))(n1) from the correlation in Fig. 12; note Pif cm3(STP)cm/cm2 s cmHg.

Fig. 15.ij and 1determiexperim

Fig. 16.CO2/CH4

TheversusRef. [9O2/N2from Fthe typ

4. Con

Thethe glautilizeious galog Pi vabilityfor eac

L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178185 185

sis, employing the equation: ij = kuPnui where ij = Pi/Pj, describesthe relationship for the polymers with optimized separation capa-bilities, in essence, the exception rather than the rule. There is nobasis in the literature that demonstrates that the average (expected)performance of a polymeric material should have a relationshipwith the polymers exhibiting optimized performance. This analy-sis shows that there is indeed a relationship although not an exactequivalence. The data in this analysis allow for a determinationof the kinetic diameters expected for polymer diffusion selectiv-ity correlation showing an excellent t of experimental data withthe theoretical expectation. This analysis yields an improved t ofexperimental results over the zeolite determined diameters pre-viously noted as the best set of kinetic diameters to t diffusionselectivity. The major difference between these diameters and thezeolite determined diameters is the comparison of CO2 and O2where the new set shows almost identical values (O2 larger by0.022 ) whereas the zeolite diameters show O2 larger by 0.16 .While that difference may seem insignicant, it has a major effecton the calculated results.

Another interesting result of this analysis involves the use ofthe value of ij developed to correlate the value of ku in the upperbound equation. Using the same basic relationship but employingthe kinetic diameters calculated from the average permeability gaspair relationship, an exact t of 1/k with ij was observed. The valueof f in theemployed tof f = 12,600in ij valuevalues residas correlate(Fig. 12).

Comparidata with avery modesdocumentegenerally hvalues for trelations w

The corrdatabase (2for correlatoretical anasolubility corelationshipnoted in therature.

References

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Teplyalities

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is paper would be a important contribution to the lit-

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An empirical correlation of gas permeability and permselectivity in polymers and its theoretical basisIntroductionResults: permeability correlationCorrelation of front factorsConclusionsReferences