An empirical correlation of gas permeability and permselectivity in polymers and its theoretical basis

Journal of Membrane Science 341 (2009) 178–185

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An empirical correlation of gas permeability and permselectivity in polymers and its theoretical basis L.M. Robeson a,∗ , B.D. Freeman b , D.R. Paul b , B.W. Rowe b a b

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

a r t i c l e

i n f o

Article history: Received 30 April 2009 Received in revised form 3 June 2009 Accepted 3 June 2009 Available online 12 June 2009 Keywords: Gas permeability correlation Permselectivity correlation Kinetic diameters Upper bound

a b s t r a c t A large database of permeability values for common gases (He, H2 , O2 , N2 , CO2 and CH4 ) has been employed in the following correlation: Pj = kPin where Pi and Pj are the permeabilities of gases i and j; the indices are chosen such that the value of n is >1.0. The plots of log Pi versus log Pj show linear behavior over nine orders of magnitude implying solution–diffusion behavior persists over the entire range of permeabilities existing in known dense polymeric materials. The scatter of data around the linear correlation for each gas pair was modest over the entire range of permeability. It was found that n correlates with the kinetic diameter of the specific 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 relationship of Pi = ku ˛niju where ˛ij = Pi/ Pj . The experimental values of n − 1 enable the determination of a new set of kinetic diameters showing excellent agreement between theory and experimental results. The value of 1/k was found to be virtually an exact fit with the relationship developed by Freeman in predicting the value of ku for the upper bound relationship using the new set of kinetic diameters where the calculations were constrained to minimize the error in (n − 1) = (dj /di )2 − 1. The significance of these results includes a new set of kinetic diameters predicted by theory and agreeing with experimental data with accuracy significantly improved over the zeolite determined diameters previously employed to correlate diffusion selectivity in polymers. One consequence of this analysis is that the kinetic diameter of CO2 is virtually identical to that of O2 . Additionally, the theoretical relationship developed by Freeman for the upper bound prediction is further verified by this analysis which correlates the average permeability for polymeric materials as compared to the few optimized polymer structures offering upper bound performance. © 2009 Elsevier B.V. All rights reserved.

1. Introduction

points exist) is ˛ij expressed by the relationship: Pi = ku ˛niju

(1)

The vast amount of permeability data now present in the literature allows for significant correlation of the basic characteristics of gas diffusion, solubility and permeability in polymers. Many correlations are already well documented in the literature for various structure–property relationships. One of these correlations receiving considerable interest is referred to as the upper bound relationship correlating the separation factor, ˛ij (˛ij = Pi /Pj ), with the permeability of the more permeable gas, Pi . The initial references [1,2] have been recently updated [3] showing minor shifts in the upper bound except for specific cases where the unique gas solubility in perfluorinated polymers dominates the separation. The relationship noted involved a log–log plot of ˛ij versus Pi where the upper bound line (above which virtually no data

−

∗ Corresponding author. Tel.: +1 610 481 0117. E-mail addresses: [email protected] (L.M. Robeson), [email protected] (B.D. Freeman), [email protected] (D.R. Paul), [email protected] (B.W. Rowe).

Since (dj /di ) − 1 = [dj + di /di2 ](dj − di ) and the term in square brackets is approximately constant for the gas pairs of interest, the theoretical correlation and the empirical correlation show reasonable agreement. Freeman also correlated the front factor, ku , with

where nu is the upper bound slope and ku is referred to as the front factor and is equal to Pi where ˛ij = 1. The value of nu was found to correlate with the kinetic diameter difference in the gas pairs with the relationship: −1/nu ∼ dj − di where dj and di represent the gas diameters of the lower permeable gas and the higher permeable gas, respectively. The prediction of the empirical upper bound relationship has been presented by Freeman [4] employing fundamental relationships to correlate both nu and ku . It was noted that the slope, nu , can be expressed by 1 ∼ = nu

 2 dj

− 1 = ij

di 2

0376-7388/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.memsci.2009.06.005

(2)

L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178–185

solubility relationships, the upper bound slope, and fundamental constants related to the diffusion of gases in polymers as will be discussed later. While the upper bound slope correlated reasonably well with the kinetic diameter difference (based on zeolite determined kinetic diameters—also referred to as Breck diameters [5]), some scatter existed at least partially due to a small number of significant figures reported for Breck diameters as well as variations in the zeolite data versus values expected in polymer diffusion. This was addressed recently by Dal-Cin et al. [6] where the Error in Variable Method (EVM) was applied to predict a new set of kinetic diameters which allowed for better correlation of −1/nu and the gas diameter relationships. The Dal-Cin sum of kinetic diameters for the six gases of interest (He, H2 , CO2 , O2 , N2 , CH4 ) was virtually equivalent to the Breck diameter sum. Several group contribution methods have been applied to predict the permeability of gases in polymers based on their structural units. The first method noted in the literature was by Salame [7] based on polymer cohesive energy density and fractional free volume ideas that assigned parameter values based on the repeating units of the polymer structure. This approach for O2 /N2 separation predicts an increasing separation factor with decreasing O2 permeability. With the increasing interest in aromatic polymers for gas separation, a group contribution method based on fractional free volume was reported by Park and Paul [8] with quite good capabilities to predict permeability and permselectivity for a wide variety of aromatic polymers of interest. This approach started with the basic equation: P = An exp

 −B  n

FFV

(3)

where An and Bn are constants for a particular gas, FFV is the polymer fractional free volume (FFV = (V − V0 )/V), V0 is the dense volume per mole of repeat unit and V is the volume per mole of repeat unit. In order to improve the predictive capability, the fractional free volume was varied for the gases for a specific polymer. The empirical factors for 41 different structural units were determined from a permeability database of 102 polymers. Another approach with similar characteristics to the Park and Paul method was published in the same time frame. This group contribution approach was based on solving a least squares fit of a large data base of polymers containing aromatic units in the main chain [9,10]. This approach was primarily directed at predicting both permeability and permselectivity of aromatic polymers which comprised many of the polymers of interest for gas separation applications. These polymers included polycarbonates, polyarylates, polysulfones, polyimides, aromatic polyamides, poly(aryl ketones), poly(aryl ethers). This approach was based on the observation that the permeability of copolymers followed a logarithmetic relationship involving the volume fraction of the comonomers and the permeability of the homopolymers comprising the copolymer. This group contribution method was based on the following equation: ln P =

n 

i ln Pi

(4)

i=1

where i is the volume fraction of a specific group i of the polymer repeat unit and Pi is the permeability contribution of that group. The choice of subdividing the polymer repeat units into specific groups was considered critical in cases where symmetry of the group around the main chain such as the iso versus para linkage of aromatic groups was noted to be a major factor in polymer permeability. Also aromatic group substitution such as alkyl or halide substitution (mono- or di-) also greatly influenced the permeability. The structural units were thus chosen to be subdivided around the main chain bond.

179

Yampolskii et al. presented a group contribution method for prediction of gas permeability and diffusion coefficients of glassy polymer based on the chemical structure of the repeat units with the groups chosen to be specific atoms and their bond configurations [11,12]. A similar approach using molecular connectivity group contribution has been noted by Bicerano [13] to predict oxygen permeability from a correlation with cohesive energy density, molar volume and van der Waals volume. Other correlations worth noting include that of Meares [14], where a relationship was noted between the activation energy of diffusion and the square of the gas diameter times the cohesive energy density. This method was recently reviewed in a discussion of the role of the cohesive energy density on diffusion in polymers [15]. Jia and Xu [16] showed a correlation between the log of the permeability and the molar free volume divided by the cohesive energy density. Paul and DiBenedetto [17–19] analyzed in more depth the relationship between the activation energy for diffusion and the size of the gas molecule using a Lennard-Jones (6–12) potential function to describe the energy of interaction between polymer segments as they move apart to accommodate a diffusion jump of a gas molecule. In the end, the result of this more complex analysis is consistent with the mathematical approximation used by Freeman [4], summarized by Eq. (5) below, and the conceptual framework outlined by Meares [14]. Teplyakov and Meares described a method for predicting diffusion coefficients and solubility constants for various gases leading to permeability predictions [20]. This procedure involved assignment of four temperature-dependant parameters for each polymer (two for diffusion and two for solubility). The relationship between the gas molecule size and the activation energy, Edi , was noted by Brandt [21] with the relationship: Edi = cdi2 − f

(5)

where c and f are constants related to the polymer and di is the gas molecule size (also referred to as the kinetic diameter). The correlation of the activation energies of permeability and diffusion with fractional free volume was demonstrated by Yampolskii et al. [22]. The database employed for the recent upper bound paper [3] included the published literature since 1991 and had a large number of data points for each gas pair of interest. For example, the data on O2 /N2 included over 1000 different values. This database was employed for the analysis to be discussed where Pi is plotted versus Pj on a log–log plot. This approach, therefore, yields results expected for typical polymeric materials as compared with the upper bound results which correlate polymers with optimized structures for maximum separation factors. 2. Results: permeability correlation The permeability database employed data reported in the literature where the gas pair values were determined on the same films tested in the same laboratories. The temperature range for this data is 25–35 ◦ C. Correlation of data on the same polymer structure with different preparation methods and/or tested in different laboratories would not be as accurate. It should be noted that a large number of samples where gas pairs were reported involved studies directed at achieving values approaching upper bound properties; thus, glassy polymers dominate the dataset. In essence, the dataset, while containing a few rubbery polymers, primarily represents a correlation for polymers below their glass transition temperature. The data are presented as log–log plots of Pi (y-axis) versus Pj (xaxis) where the basic equation: Pj = kPin

(6)

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

Fig. 2. CO2 /CH4 permeability correlation.

describes the results when i and j are chosen such that n > 1.0. The equation for the plot on log–log coordinates is given by log Pi = −

log Pj log k + n n

(7)

The value of k has units of (Barrers)1−n . The value of k is Pj where Pi = 1.0 Barrer. Note that k and n are not the same values as employed in the upper bound relationship noted above. Solving Eq. (1) for Pi and comparing with Eq. (7) shows an approximate relationship between the upper bound slope nu and n of Eq. (6) expressed as −

1 ∼n − 1 nu

(8)

Mathematically, this would be an exact relationship if in plots of log Pi versus log Pj the upper bound relationship were a line parallel to the suggested correlation line, i.e., Eq. (6). The plot of P(O2 ) versus P(N2 ) is shown in Fig. 1. The upper bound line (from Ref. [3]) is also shown on the plot. In this case, the slope of the upper bound lines and that of the correlation lines, as suggested by Eq. (8), are quite close −

1 = 0.1765 nu

and n − 1 = 0.1576

(9)

It is interesting to note that the log–log relationship shows linear behavior over nine orders of magnitude in permeability. This suggests that the solution–diffusion mechanism is valid over this range of permeabilities. As expected, the upper bound slope from Ref. [3] follows the upper range of values on the plot in Fig. 1. The spread of data below and above the correlation slope is quite modest and covers less than an order of magnitude of O2 permeability at constant N2 permeability. The same correlation exists for all combinations of the gas pairs of the list of He, H2 , O2 , N2 , CO2 , and CH4 (15 pairs of ij combinations where Pi > Pj ). Additional plots of other gas pairs are illustrated in Figs. 2–9. All gas pairs, except CO2 /O2 , also exhibit an upper bound relationship where −1/nu > 0. This apparently does not occur with CO2 /O2 as the true kinetic diameters of this gas pair are virtually equal. The Breck diameter difference for this pair is 0.16 Å; however, as will be shown, the predicted kinetic gas diameter difference is much less. The n and k values (Eq. (6)) from Figs. 1–9 and the other gas pairs are listed in Table 1. The correlation of n with the kinetic diameter is shown by plotting n versus (dj /di )2 − 1 and gives the best results when the Dal-Cin

Fig. 3. CO2 /N2 permeability correlation.

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

n

k (Barrers)1−n

(dj /di )2 − 1 (Dal-Cin values[6])

O2 /N2 H2 /N2 He/N2 H2 /CH4 CO2 /N2 He/H2 He/CH4 N2 /CH4 CO2 /O2 He/O2 H2 /O2 He/CO2 H2 /CO2 CO2 /CH4 O2 /CH4

1.1576 1.5242 1.8290 1.7480 1.1212 1.1609 2.133 1.1386 1.017 1.5937 1.3297 1.5961 1.3500 1.3034 1.3096

0.176 0.003408 0.00123 0.00145 0.03419 0.6228 0.0002938 0.9625 0.223 0.01376 0.04334 0.0667 0.139 0.0176 0.1173

0.1305 0.6025 0.9718 0.8760 0.0961 0.2305 1.3085 0.1701 −0.0304 0.7442 0.4175 0.7988 0.4619 0.2833 0.3235

L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178–185

181

Fig. 6. H2 /N2 permeability correlation.

Fig. 4. CO2 /O2 permeability correlation.

kinetic diameters are used. This is shown in Fig. 10 where the correlation coefficient has an R value of 0.992. Using the Breck diameters for (dj /di )2 − 1 gave an R value of 0.977. When dj − di was employed using the Breck diameters the R value was 0.971. The Dal-Cin EVM analysis shows the CO2 kinetic diameter to be slightly greater than that of O2 . The plot for CO2 /O2 with n = 1.017 indicates that the O2 kinetic diameter should be slightly higher than that of CO2 . A careful examination of Fig. 10 shows the data points to be above the correlation when CO2 is in the numerator and below the line when CO2 is in the denominator also indicating that the Dal-Cin CO2 kinetic diameter is slightly higher than it should be relative to the other gases. The n values from the log Pi versus log Pj plots comprise literally hundreds to greater than 1000 data points. Intuitively, the correlation of these n values with the kinetic diameters should be more appropriate than those derived from the upper bound slopes that are based on a few polymers optimized for permselectivity. The

Fig. 7. N2 /CH4 permeability correlation.

Dal-Cin kinetic diameters offer an excellent fit but are based on the upper bound slope analysis. Kinetic diameters were determined using a least squares analysis that constrains the values to conform to the form (n − 1) = (dj /di )2 − 1 with the line going through the origin, i.e., no finite intercept. The results are shown in Table 2 along with the Breck and Dal-Cin diameters. Because this analysis was based on dj /di , not the absolute values of dj and di , an appropriate constraint Table 2 Comparison of kinetic diameters.

Fig. 5. He/H2 permeability correlation.

Gas

Breck diameter (Å)

L-J collision diameter (Å)

Dal-Cin diameter (Å)

Current Analysis (Å)

He H2 CO2 O2 N2 CH4

2.6 2.89 3.3 3.46 3.64 3.8

2.556 2.928 4.07 3.46 3.71 3.817

2.555 2.854 3.427 3.374 3.588 3.882

2.644 2.875 3.325 3.347 3.568 3.817

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L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178–185

Fig. 8. O2 /CH4 permeability correlation.

was needed to generate a single set of diameters. The primary reason that the kinetic diameters for diffusion in polymers might be different than the simple diameter determined from other information (e.g., viscosity or second viral coefficients) would be that the molecules are not spherical. Among the gases of interest, He and CH4 are the closest to spherical. Based on the small variability in CH4 diameter reported in the literature, its kinetic diameter was fixed at 3.817 Å, as reported by Hirschfelder et al. [23] based on second viral coefficients. Another constraint was also tested involving setting the sum of the kinetic diameters equal to the sum of the Breck diameters. This assumption yielded diameters very close with variations only in the third significant figure and an R value very slightly lower. However, based on the discussion above, the kinetic diameter results used in the following discussion are based on fixing the CH4 diameter at 3.817 Å. Fig. 11 shows the relationship between n − 1 and the new kinetic diameters fit to the theoretical expression derived by Freeman [4]. A linear regression shows the intercept is nearly (0,0) with an exceptional fit (R = 0.998). If the intercept is fixed at the (0,0) intercept,

Fig. 9. H2 /CO2 permeability correlation.

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

then the slope becomes closer to one (1.0076) without an appreciable change in the R value. The data point with the largest variation from the line of Figs. 11 is for He/CH4 separation. The He/CH4 plot of log Pi versus log Pj also had the largest spread of data around the linear correlation thus yielding the largest potential for error in the determination of n. The major difference between the least squares analysis and the Dal-Cin EVM diameters is for CO2 and O2 . The Dal-Cin results show the CO2 diameter to be modestly larger than O2 whereas the least squares analysis shows O2 to be marginally higher. Intuitively, CO2 might be expected to have a larger kinetic diameter than O2 based on the volume comparison of the molecules; however, CO2 is planar along the major axis and possibly slightly more compact than O2 in the minor diameter of the molecules. At any rate, the data strongly suggest that the effective kinetic diameters of these two gases are virtually identical. Another difference is that the slope of the line (ideally unity) is closer to unity for the least squares analysis data.

Fig. 11. Correlation of n − 1 with the theoretical kinetic diameter relationship (kinetic diameters determined from least squares fit constraining (n − 1) = (dj /di )2 − 1.

L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178–185

183

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

ku−1/nu (Barrers)

O2 /N2 H2 /N2 He/CH4 H2 /CH4 CO2 /CH4 He/H2 He/O2 H2 /O2 He/N2 He/CO2 H2 /CO2

9.1992 1237 51060 3334 115.04 3.9978 673 99.860 10002 216 38.9

a b

−1/nu

1/k (Barrers)n − 1

ku−1/nu a

1/kb

5.6818 293.4 3404 689.7 56.82 1.6056 72.670 23.07 813 14.99 7.194

0.1736 0.00358 9.56 × 10−9 1.849 × 10−5 0.04824 0.03835 1.2765 × 10−5 0.004059 1.7234 × 10−6 1.3732 × 10−6 0.0002667

0.1508 0.001679 1.592 × 10−8 2.280 × 10−5 0.05254 0.03959 7.950 × 10−5 0.011640 4.1734 × 10−6 1.6402 × 10−5 0.0022275

−1/nu

(cm3 (STP)cm/cm2 s cmHg) . (cm3 (STP)cm/cm2 s cmHg)(n−1) .

3. Correlation of front factors The values of ku and k are related by the following Fig. 12. Correlation of upper bound slope values with permeability gas pair slopes.

ku−1/nu ≈

1 k

(10)

The comparison of −1/nu versus (n − 1) is shown in Fig. 12. As expected, the upper bound slope correlation values are all above the linear line indicating linear proportionality from the expression noted in Eq. (8). The slope of the line is 1.0681 indicating that the upper bound curves modestly deviate more from the average polymer data as the kinetic diameter difference increases. Another data file has been compiled listing permeability data by polymer class. One file with significant data includes polyimides and polypyrrolones. An interesting comparison involves the O2 /N2 data plotted as log P(O2 ) versus log P(N2 ) and compared with the data on all polymers from the upper bound correlation file. The results are illustrated in Fig. 13. The values for Eq. (6) for the polyimide data are n = 1.1343 and k = 0.1483 compared to all polymer data (Table 1) of n = 1.1576 and k = 0.176 Barrers(1−n) . As most of the data for polyimides are on or above the correlation line for all polymers, the observation that polyimides generally yield better O2 /N2 separation characteristics is correct. Also to be noted is the spread of the polyimide data around the correlation line is somewhat lower than when all polymers are employed (see Fig. 1).

provided the relationship given by Eq. (8) is strictly valid as may be seen by combining Eqs. (1) and (6). This relationship can be tested by comparing the values of the front factors from the upper bound, Eq. (1), versus those from the correlation line from Eq. (6) for the different gas pairs. The data for ku−1/nu were chosen from Table 12 of Ref. [3] using only the prior upper bound data since the newer upper bound data had shifts of several gas pairs based on perfluorinated polymer unique gas solubility relationships. In addition, the predictions made by Freeman [4] were based on the older upper bound parameters so their choice facilitates that comparison. These parameters are shown in Table 3 using two different sets of units. Fig. 14 shows a graphical comparison where the solid line is the relationship suggested by Eq. (10). As can be seen, there is general agreement in this comparison of the two front factors. From the theoretical model developed by Freeman [4], the front factor, ku , was predicted to be as follows

Fig. 13. Comparison of polyimide data with typical polymer results.

Fig. 14. Correlation of −1/ku−1/nu with 1/k.

ku−1/nu =

Si  S ij exp Sj





−ij b − f

 1 − a 

RT

= ˇij

(11)

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L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178–185

where Si and Sj are the respective gas solubilities. a and b are determined from the free energy relationship between the activation energy of diffusion, Edi and D0i : ln D0i = a

Edi −b RT

(12)

where a is 0.64 and b has values of 9.2 and 11.5 for rubbery and glassy polymers, respectively. The parameter, f, is determined from the relationship between activation energy of diffusion and the size of the penetrant molecule given by Eq. (5) where c and f are adjustable values dependant upon the specific polymer. The best fit of the upper bound data with theory yields a value of f equal to12,600 cal/mol [4]. The solubility coefficients for gases have been shown to be related to the critical temperature, Tc , the boiling point, Tb , or the Lennard-Jones parameter (ε/k) by the following equations: ln Si = m + 0.027Tb ,

ln Si = x + 0.016Tc ,

ln Si = y + 0.023

ε  i

(13)

k

where m, x and y have unique values for each polymer. These relationships appear to work well for a wide range of polymers (specifically aliphatic and aromatic polymers) except for perfluorinated polymers where the slope values are different than those shown in Eq. (13). For the analysis by Freeman, the Lennard-Jones solubility relationship was chosen with y set to −9.84 when solubility is in units of cm3 (STP)/(cm3 -cmHg). For this model, the units −1/nu for ku−1/nu were (cm3 (STP)cm/cm2 s cmHg) . Since the proportionality shown in Eq. (10) appears to be a good approximation, as shown in Fig. 14, 1/k should be proportional to the value shown in the right side of Eq. (11). Since there is a correlation between 1/nu and (n − 1) (as shown in Eq. (8) and illustrated in Fig. 12), then the values of the right side of Eq. (11) can be related to the values of 1/k and n − 1 of Eq. (6). The comparison of 1/k and 1/ku−1/nu in Fig. 14 shows close to an equivalence since the values fall about the equal value line; thus: 1 ∼ Si  = S ij exp Sj k





−ij b − f

1−a RT



= ˇij

Fig. 15. Comparison of predicted results with experimental data with (the units of ˇij and 1/k are [cm3 (STP)cm/(cm2 s cmHg)]ij where ij for the predicted results were determined from the constrained least squares analysis results and ij = n − 1 for the experimental results determined from the data).

(14)

Using the a and b (b chosen for glassy polymers) values noted above and employing the Lennard-Jones solubility relationship, values of ˇij were calculated and compared with 1/k. The values of ij in Eq. (14) were calculated from:

 2

ij =

dj

di

−1

(15)

using the values of dj and di from the least squares analysis results shown in Table 2. The plot of log of ˇij (predicted values) versus log(1/k) (experimental results) is shown in Fig. 15 for f = 12,600 cal/mol (same value used by Freeman in the upper bound theory [4]). The fit is quite good showing the validity of the Freeman theory as expected from the above discussion. It is interesting to note that with data involving CO2 , the data points are above the line if CO2 is in the gas pair numerator and below the line if CO2 is in the denominator. If the CO2 data are removed from the analysis, the fit is even better and virtually exact. This indicates that the CO2 solubility relationships relative to the other gases may not be properly predicted by the LennardJones solubility relationship. As the value of 12,600 cal/mol fits this analysis as well as the upper bound analysis, it appears that it has a universal value. This is not unexpected based on the results shown in Fig. 14. The ratio of ˛ij (upper bound)/˛ij (average value) is, thus, −(n−1)

primarily determined by the ratio of Pi1/nu /Pi

(0.093697+1.0681(n−1))(n−1) from the correlation to Pi is in units of cm3 (STP)cm/cm2 s cmHg.

which is equal

in Fig. 12; note Pi

Fig. 16. Comparison of typical polymer data and group contribution results for CO2 /CH4 and O2 /N2 gas pairs in the log Pi versus log Pj plot.

The comparison of the group contribution data with the log Pi versus log Pj data is shown in Fig. 16 for O2 and N2 group values from Ref. [9] and CO2 and CH4 group values from Ref. [10]. The lines for O2 /N2 for typical polymer data are taken from Fig. 1 and for CO2 /CH4 from Fig. 2. The group contribution data show good correlation with the typical polymer data as would be expected. 4. Conclusions The large database of gas permeability of polymers, primarily in the glassy state, used for the recent upper bound analysis [3] was utilized here to correlate the polymer permeability results for various gas pairs chosen from the list He, H2 , O2 , N2 , CO2 , CH4 . Plots of log Pi versus log Pj showed linear behavior over the entire permeability range, i.e., Pj = kPin , where i and j were chosen such that n > 1, for each of the 15 separate gas pair values. The upper bound analy-

L.M. Robeson et al. / Journal of Membrane Science 341 (2009) 178–185

sis, employing the equation: ˛ij = ku Pinu where ˛ij = Pi /Pj , describes the relationship for the polymers with optimized separation capabilities, in essence, the exception rather than the rule. There is no basis in the literature that demonstrates that the average (expected) performance of a polymeric material should have a relationship with the polymers exhibiting optimized performance. This analysis shows that there is indeed a relationship although not an exact equivalence. The data in this analysis allow for a determination of the kinetic diameters expected for polymer diffusion selectivity correlation showing an excellent fit of experimental data with the theoretical expectation. This analysis yields an improved fit of experimental results over the zeolite determined diameters previously noted as the best set of kinetic diameters to fit diffusion selectivity. The major difference between these diameters and the zeolite determined diameters is the comparison of CO2 and O2 where the new set shows almost identical values (O2 larger by 0.022 Å) whereas the zeolite diameters show O2 larger by 0.16 Å. While that difference may seem insignificant, it has a major effect on the calculated results. Another interesting result of this analysis involves the use of the value of ˇij developed to correlate the value of ku in the upper bound equation. Using the same basic relationship but employing the kinetic diameters calculated from the average permeability gas pair relationship, an exact fit of 1/k with ˇij was observed. The value of f in the ˇij relationship was the same for this analysis as that employed to fit the upper bound data. This implies that the value of f = 12,600 cal mol−1 may be a universal value and the difference in ˇij values between the upper bound and the average polymer values resides wholly in the difference in values of −1/nu and (n − 1) as correlated with the equation: −1/nu = 0.093697 + 1.0681(n − 1) (Fig. 12). Comparison of a database of only polyimide and polypyrrolone data with all the polymers in the upper bound database show only very modest differences, with the conclusion similar to the welldocumented observation in the literature that polyimides exhibit generally higher gas separation values. The group contribution values for the O2 /N2 and CO2 /CH4 gas pairs also yield similar correlations with the typical polymer permeability results. The correlation is only valid for the temperature range of the database (25–35 ◦ C). Insufficient data in the literature do not allow for correlation at lower or higher temperatures. However, a theoretical analysis of the effect of the diffusion coefficient and the solubility constant as a function of temperature on the upper bound relationship and the correlation for the average polymer values noted in this paper would be a important contribution to the literature.

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References [1] L.M. Robeson, Correlation of separation factor versus permeability for polymeric membranes, J. Membr. Sci. 62 (1991) 165. [2] L.M. Robeson, W.F. Burgoyne, M. Langsam, A.C. Savoca, C.F. Tien, High performance polymers for membrane separation, Polymer 35 (1994) 4970. [3] L.M. Robeson, The upper bound revisited, J. Membr. Sci. 320 (2008) 390. [4] B.D. Freeman, Basis of permeability/selectivity tradeoff relations in polymeric gas separation membranes, Macromolecules 32 (1999) 375. [5] D.W. Breck, Zeolite Molecular Sieves, John Wiley & Sons, New York, NY, 1974, p. 636 (Chapter 8). [6] M.M. Dal-Cin, A. Kumar, L. Layton, Revisiting the experimental and theoretical upper bounds of light pure gas selectivity-permeability for polymeric membranes, J. Membr. Sci. 323 (2008) 299. [7] M. Salame, Prediction of gas barrier properties of high polymers, Polym. Eng. Sci. 26 (1986) 1543. [8] J.Y. Park, D.R. Paul, Correlation and prediction of gas permeability in glassy polymer membrane materials via a modified free volume based group contribution method, J. Membr. Sci. 125 (1997) 23. [9] L.M. Robeson, C.D. Smith, M. Langsam, A group contribution approach to predict permeability and permselectivity of aromatic polymers, J. Membr. Sci. 132 (1997) 33. [10] D.V. Laciak, L.M. Robeson, C.D. Smith, Group contribution modeling of gas transport in polymeric membranes, in: B.D. Freeman, I. Pinnau (Eds.), Polymer Membranes for Gas and Vapor Separation, ACS Symposium Series 733, American Chemical Society, Washington, DC, 1999, p. 151. [11] Y. Yampolskii, S. Shishatskii, A. Alentiev, K. Loza, Group contribution method for transport property predictions of glassy polymers: focus on polyimides and poynorbornenes, J. Membr. Sci. 149 (1998) 203. [12] A.Y. Alentiev, K.A. Loza, Y.P. Yampolskii, Development of the methods for prediction of gas permeation parameters of glassy polymers: polyimides as alternating copolymers, J. Membr. Sci. 167 (2000) 91. [13] J. Bicerano, Prediction of Polymer Properties, 3rd ed., Marcel Dekker, Inc., New York, 1996. [14] P. Meares, The diffusion of gases through polyvinyl acetate, J. Am. Chem. Soc. 76 (1954) 3415. [15] A.Y. Alentiev, Y.P. Yampolskii, Meares equation and the role of cohesion energy density in diffusion in polymers, J. Membr. Sci. 206 (2002) 291. [16] L. Jia, J. Xu, A simple method for prediction of gas permeability of polymers from their molecular structure, Polym. J. 23 (1991) 417. [17] A.T. DiBenedetto, D.R. Paul, An interpretation of gaseous diffusion through polymers using fluctuation theory, J. Polym. Sci.: A 2 (1964) 1001. [18] D.R. Paul, A.T. DiBenedetto, Diffusion in amorphous polymers, J. Polym. Sci.: C 10 (1965) 17. [19] D.R. Paul, A.T. DiBenedetto, Thermodynamic and molecular properties of amorphous high polymers, J. Polym. Sci.: C 16 (1967) 1269. [20] V. Teplyakov, P. Meares, Correlation aspects of the selective gas permeabilities of polymeric materials and membranes, Gas Sep. Purif. 4 (1990) 66. [21] W.W. Brandt, Model calculation of the temperature dependence of small molecule diffusion in high polymers, J. Phys. Chem. 63 (1959) 1080. [22] Y. Yampolskii, S. Shishatskii, A. Alentiev, K. Loza, Correlations with and prediction of activation energies of gas permeation and diffusion in glassy polymers, J. Membr. Sci. 148 (1998) 59. [23] J.O. Hirschfelder, C.F. Curtiss, R.B. Bird, Molecular Theory of Gases and Liquids, John Wiley & Sons, New York, NY, 1954, pp. 1110–1111.