Explanation of a Selectivity Maximum as a ... - ACS Publications

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Ind. Eng. Chem. Res. 2004, 43, 5942-5949

SEPARATIONS Explanation of a Selectivity Maximum as a Function of the Material Structure for Organic Gas Separation Membranes Ryan L. Burns,† Keisha M. Steel,† Sean D. Burns,† and William J. Koros*,‡ Georgia Institute of Technology, Atlanta, Georgia 30332, and The University of Texas at Austin, Austin, Texas 78712

Many previous studies have examined structure-property relationships of both polymeric and carbon molecular sieve gas separation membranes, which show a widely known tradeoff between permeability and selectivity. This work presents an explanation of a lesser-known trend of a selectivity maximum as a function of the material structure for certain rigid polymers and a variety of carbon molecular sieving materials. Interestingly, this surprising trend of a maximum in selectivity over a range of materials is observed for certain gas pairs and not others over the same range of material structures. First-order modeling efforts, which treat molecular sieving membranes as effective heterogeneous composite materials comprising a distribution of selective entities, are used to explain and understand this trend of a maximum in selectivity. Several composite models are considered, and specific examples are demonstrated using both a parallel model and effective medium theory. The modeling results reflect trends in experimental data and suggest that subtle changes in the distribution of selective entities or the size of gas molecules can result in vastly different overall separation properties. Introduction A well-known challenge for materials development of gas separation membranes involves overcoming the natural tradeoff that occurs between flux and selectivity. In general, through chemical or physical processes, a material can be altered to provide a higher permeability at the expense of selectivity and vice versa. For example, there are many instances in the literature where an adjustment to the chemical structure of a polymeric membrane provides a decrease in permeability and an increase in selectivity, and a few are provided here for reference.1-4 Likewise, for carbon molecular sieving materials, there are many instances where changing a process variable in the material formation (e.g., final heat treatment temperature) results in a similar decrease in permeability and an increase in selectivity, and again a few are provided here for reference.5-7 These cases are familiar and for the most part well understood. A lesser-known challenge for materials development of gas separation membranes is recognizing and understanding cases where a chemical or physical change in material processing results in a decline in both flux and selectivity. This results in a selectivity maximum as a function of the material structure. Interestingly, this maximum in selectivity may be observed for certain gas pairs and not others over the same range of material structures. These cases are not widely familiar, are not well understood, and are the topic of this paper. * To whom correspondence should be addressed. E-mail: [email protected]. † The University of Texas at Austin. ‡ Georgia Institute of Technology.

We first noticed this trend with a series of poly(pyrrolone-imide) copolymers.8 Measurements with the material 6FDA-TAB/DAM showed a decrease in permeability and an increase in selectivity with increasing TAB/DAM ratio for O2/N2 and CO2/CH4 separations (shown in Figure 1a,b). Somewhat surprisingly, measurements for the C3H6/C3H8 separation showed a maximum in selectivity over the same range of copolymers, exhibiting very low flux and low selectivity for the material 6FDA-TAB (shown in Figure 1c). It was also demonstrated that the trends in permselectivity were due to trends in diffusivity selectivity and not solubility selectivity.8 These results raise an interesting question: How is it possible that one membrane material (6FDA-TAB) can possess such desirable separation characteristics for one gas pair separation (O2/N2 or CO2/CH4) and yet possess extremely undesirable separation properties for another gas pair (C3H6/C3H8) when the penetrant molecules are larger by only fractions of an angstrom? An explanation for this inquiry is the subject of the present work, but before proceeding, it is useful to point out where these trends have occurred previously within the literature. In the carbon molecular sieving literature, this surprising trend of a maximum in selectivity as a function of the material structure has been observed for a variety of gas pairs. Steel has observed this trend for C3H6/C3H8 separation as a function of the pyrolysis temperature, shown in Figure 2.9 In this case, the carbon materials were formed from the polyimide precursor, 6FDA/ BPDA-DAM, via vacuum pyrolysis. Varying the temperature of the heat treatment changes the material structure, similar to changing the monomer stoichiometry for the poly(pyrrolone-imide) copolymers. It would

10.1021/ie049800z CCC: $27.50 © 2004 American Chemical Society Published on Web 07/23/2004

Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004 5943

Figure 2. Carbon molecular sieving materials pyrolyzed from a 6FDA/BPDA-DAM precursor. Transport results, as a function of the heat treatment in the material formation, show a selectivity maximum.9

Figure 1. Permeability and selectivity results for the different copolymers of the family 6FDA-TAB/DAM: (a) O2/N2; (b) CO2/ CH4; (c) C3H6/C3H8. Results in this case show a surprising maximum in selectivity over the range of copolymers. The upper line here is the polymeric upper bound defined previously.26

be expected that an increased heat treatment would provide a smaller average critical ultramicropore size, resulting in a decrease in permeability and an increase in selectivity based on previous observations.5,10 Clearly, this is not the observed trend for the C3H6/C3H8 case, where a decrease in both permeability and selectivity from a 550 °C heat treatment to an 800 °C heat treatment is observed (Figure 2). Interestingly, O2/N2 and CO2/CH4 show the expected trend of a decrease in permeability and an increase in selectivity (not shown here),9 which corresponds to the poly(pyrrolone-imide) case outlined earlier. These two examples show a selectivity maximum for only C3H6 and C3H8 (the largest molecules tested); however, an extensive search of the

carbon literature provides evidence of the phenomena for many different gas pair separations. Okamoto et al. have observed a selectivity maximum for C3H6/C3H8 for carbons as a function of the pyrolysis temperature using a polyimide precursor.11 Ogawa and Nakano have observed a selectivity maximum for CO2/ CH4 as a function of the gelation temperature.12 Kusuki et al. have observed a maximum in selectivity for H2/ CH4 as a function of the heat treatment.13 Kane et al. have observed an O2/N2 selectivity maximum as a function of the synthesis temperature for carbon molecular sieving materials.14 Hayashi et al. have observed a maximum in selectivity for both O2/N2 and CO2/N2 as a function of the chemical vapor deposition time (which qualitatively controls the pore-size distribution).15 In a separate publication, Hayashi et al. observe a He/N2 selectivity maximum as a function of the carbonization temperature.16 It is possible that other examples of this trend exist in the carbon literature, but the examples cited here are intended to show that the observation of a maximum in selectivity is not uncommon in carbon materials for many different gas separations. However, this observed trend remains largely underdiscussed within the literature. In this paper, we make the assumption that molecular sieving materials can be described as effectively heterogeneous composite materials comprising discrete distributions of selective entities. Various composite models are considered with a focus on the parallel model and effective medium theory (EMT). These models will be used to show that considering the distribution of selective entities (instead of merely the average size) can account for the selectivity maximum phenomena observed for certain gas separations. It should also be noted that these models are intended to demonstrate the most probable explanation for the selectivity maximum. We recognize that other explanations may be possible, although we are unaware of any that can achieve this description at the present time. We believe the hypotheses offered herein are currently the most credible descriptions for the complex phenomena observed experimentally.

5944 Ind. Eng. Chem. Res., Vol. 43, No. 18, 2004

Understanding the Selectivity Maximum Phenomena Conventionally, it is believed that the diffusion coefficient of a given probe molecule decreases as the average size of the selective regions within a material decrease. Such selective entities could be a critical pore constriction in a carbon material or the transiently generated local perturbation in the chain spacing of a rigid polymer. The following analysis will attempt to demonstrate why this trend does not necessarily hold true in all cases. We believe that an overlooked assumption in the preceding notion is that the average size of the selective regions is the only important variable. More rigorously, it is valuable to consider the distribution of size-selective entities, either the distribution of pore sizes in a carbon material or the distribution of size-selective chain spacings within a rigid polymeric material (which relate to the distribution of free volume). Modeling Distributions of Selective Entities. For simplicity, it is useful to first consider a molecular sieving porous carbon material. Later it will be possible to connect these notions with the rigid polymeric materials. Various researchers have characterized the micropore distribution of carbon materials. Ogawa and Nakano have characterized the micropore distribution of carbonized membranes prepared by gel modification from 3.3 Å to greater than 5.0 Å.12 Steel has characterized the micropore distribution of carbon materials for pores ranging in size from 4 to 11 Å.9 Hayashi et al. have characterized pore distributions using sorption isotherms and the Dubinin-Astakhov equation.15 These studies reflect the fact that carbon molecular sieving materials are comprised of discrete micropore distributions. Therefore, it is useful to consider modeling work in an effort to correctly describe the effect of these pore distributions on the resulting transport properties and attempt to explain possible causes of the selectivity maximum phenomena. To begin, the assumption is made that a bulk carbon molecular sieving membrane can be considered as a composite material made up of an assembly of connected multiple discrete pore sizes. Each characteristic domain is envisioned as possessing an intrinsic permeability for an individual gas molecule, which would be the effective permeability if the membrane were composed of only that particular pore size. Ideally, these different local domains with distinct characteristic permeability properties can be combined by some “mixing rule” to yield the effective properties of the bulk medium. Initially, a parallel model for gas permeability is chosen to represent the effective permeability through a material of n characteristic domains: n

Ρeff )

∑1 Piφi

(1)

This equation can also be viewed in terms of probabilities, where the number fraction of pore i, φi, represents the probability of the penetrant molecule finding that pore and the permeability, Pi, represents the probability of transporting across that pore. It can further be assumed that the equilibrium solubility within the material is the same throughout because the solubility is related to the size of the larger cavities and the condensability of the penetrant but not necessarily

Figure 3. Illustration of a selectivity maximum based on a hypothetical pore-size distribution and a parallel permeability model. Membrane A is 99% pore1, with the balance an equal amount of pore2 and pore3. Membrane B is 99% pore2, and so on. While membrane C is 99% pore3, its properties vary significantly from a material that is 100% pore3. Table 1. Assigned Permeability Values of Penetrants A and B for pore1, pore2, and pore3 pore

permeability A (barrer)

selectivity A/B

pore1 pore2 pore3

400 40 4

5 50 500

the size of the critical selective micropores. The solubility coefficient can then be factored out of the previous equation to provide a parallel model for the effective diffusivity n

Deff )

∑1 Diφi

(2)

which can be used to demonstrate why the selectivity maximum phenomena are observed for some gas separations yet not for others over the same range of material structures. As a first-order analysis, a hypothetical distribution comprising domains characterized by 1 of 3 idealized pore sizes is considered, labeled pore1, pore2, and pore3. These domains are assigned permeability values that might be observed in a medium comprised only of such pores for two penetrants A and B, shown in Table 1. Model membrane materials can then be created with different number fraction distributions of pore1, pore2, and pore3. For simplicity, these materials can be labeled membrane A, membrane B, and membrane C. For illustration, membrane A is composed of 99% pore1 and the balance an equal amount of pore2 and pore3. Membrane B is 99% pore2, and so on. The results are shown in Figure 3, using eq 1 to calculate the permeability of penetrant A and the selectivity (PA/PB) for each membrane. The properties of membranes A and B are as expected, very near to those of 100% pore1 and very near to those of 100% pore2, respectively. On the other hand, membrane C (which is 99% pore3) does not have properties similar to those of 100% pore3. In fact, membrane C shows a selectivity of only 16, although 99% of the material has an overall selectivity of 500. This hypothetical situation is simply intended to illustrate how the selectivity maximum phenomena could arise. It is useful, however, to understand the drawbacks of this analysis in order to extend these ideas in a more critical manner.


Figure 4. Diffusivity selectivity versus diffusivity for three hypothetical separations of O2/N2, CO2/CH4, and C3H6/C3H8. 0 represents the pure pore. b represents membrane A, B, or C. The solid lines are trends for the membranes A-C.

The first obvious shortcoming of the previous analysis is the designation of permeability values. While the values were assigned based on expected trends, this designation is still somewhat arbitrary, and there is value in making it more rigorous. Furthermore, the experimental data have shown that these phenomena can exist for certain separations, yet not others, over the same range of material structures [this has been observed for the poly(pyrrolone-imide) copolymers and the carbon materials]. It is important to be able to explain these results over multiple data sets. Teplyakov and Meares have demonstrated an empirical relationship between the diffusion coefficient and molecular size of the gas molecule:17 2

D ) K1 exp[-K2(σeff) ]

(3)

where σeff is the effective size of a molecule and K1 and K2 are empirically fit coefficients. This expression can be used as a method to assign diffusion coefficients to specific selective entities (or pores). The choice of this expression is designed to provide structure to the assignment of the diffusion coefficient for each pore. Table 2 lists the effective diameter of six gas molecules of interest and the subsequent diffusion coefficient (arbitrary units) in hypothetical pores1-3, based on eqs 2 and 3. These values are then used, along with the parallel model (eq 1), to calculate the results for membranes A-C, which are defined just as they were in the last discussion. The results of this exercise are shown in Figure 3. It can be seen that in all cases membranes A and B closely match the results of pure pore1 and pure pore2. Membrane C begins to show deviations from the pure pore3 in all cases, and this is because pore3 does not necessarily dominate transport

Table 2. Diffusivity of Pores Based on the Effective Diameter and Equation 3a diffusivity penetrant effective diameter (Å) CO2 O2 N2 CH4 C3H6 C3H8

3.30 3.44 3.66 3.81 4.68 5.06

pore1

pore2

pore3

4318 489 55.4 2694 253 23.7 1234 84.7 5.81 704 38.6 2.12 17.5 0.219 0.00275 2.76 0.0165 0.00010

a K ) 1 × 106; K ) -0.5, -0.7, and -0.9 for pore1, pore2, and 1 2 pore3, respectively. Constants are chosen to result in a reasonable span in the order of magnitude of the diffusivity. Diffusion coefficients are in arbitrary units.

even though it is by far in the largest percentage. The term in eq 1 that determines which pore or entity dominates transport is Piφi, meaning that a pore can completely dominate transport even if it is the pore in the lowest percentage of the membrane. From the results in Figure 4, it can be seen that, for O2/N2 and CO2/CH4, pore1 begins to have some influence on the overall transport, which causes membrane C to deviate considerably from the results of pure pore3. However, there is still a linear relationship between membranes A-C on a log/log plot of diffusivity and diffusivity selectivity for O2/N2 and CO2/CH4, which matches conventional experimental observations. The C3H6/C3H8 case deviates from this trend because now pore1 completely dominates transport, so membrane C shows both a low diffusivity and a low diffusivity selectivity, thereby producing the selectivity maximum phenomena. Figure 5 shows the diffusivity for all six gas penetrants for each membrane. Clearly, as the size of the pentrant increases, the trend deviates from the exponential decline in the diffusion coefficient.


Figure 6. Comparison of the effective permeability as a function of the pore-size distribution using the parallel, series, Maxwell, weighted combination, and EMT models for a hypothetical binary composite material. Figure 5. Diffusivity for membranes A-C over the range of penetrants, created using the Meares equation and the parallel model.

This discussion is simply intended to demonstrate how it is possible for the selectivity maximum phenomena to occur for one gas pair and yet not another. In practice, the constants K1 and K2 could vary depending on the particular pore or selective entity, and this would explain why the selectivity maximum phenomena have been seen for a variety of gas pair separations (observed experimentally in the literature for carbon molecular sieving membranes). The final point of this exercise is that membrane C appears to be very poor for the C3H6/ C3H8 separation, yet the reason is not because it lacks selective entities but that these entities are not being utilized (not dominating transport for the particular separation of interest). This realization is important because other processing variables (such as temperature and pressure) might possibly be intelligently varied to change this fact. The parallel model for gas permeability (or diffusion) is only one model that can describe transport behavior in composite materials. It is useful to consider additional models to understand any limitations of the parallel model. Some additional models include the series model

Peff )

1 n

(4)

φi

∑1 P

]

Pd + 2Pc - 2Φd(Pc - Pd) Pd + 2Pc + Φd(Pc - Pd)

( )

1 φ + (φ1P1 + φ2P2)φ1 φ1 φ2 2 + P1 P2

(5)

where the subscript c denotes the continuous phase and the subscript d denotes the dispersed phase. Figure 6 shows these different models for a composite of two pore sizes, pore1 having a permeability of 400 barrer and pore2 having a permeability of 4 barrer. In this case, the Maxwell model is plotted assuming that pore1 is the continuous phase. Clearly, the parallel and series models differ strongly in their prediction of the overall transport permeability. The Maxwell model relatively follows the parallel model, and it can be shown that if pore2 was taken as the continuous phase, the Maxwell

(6)

This model is plotted in Figure 6, illustrating a fairly appropriate combination of both the parallel and series models. While this model more closely resembles the expected experimental results, it is currently limited because it cannot be easily extended to multiple pore sizes. EMT was created to describe conduction effects in a nonhomogeneous mixture. This theory assumes that the material is a homogeneous effective medium with local fluctuations in conductivity. Davis and co-workers extended this theory to apply to transport in composite materials, which can be represented by18,19

Rm - Ri

∑1 ΦiR

and the Maxwell model

[

Peff )

n

i

Peff ) Pc

model would relatively match the series model prediction. However, experimental reality should lie somewhere between the parallel and series models. At low fractions of pore1, it follows that pore2 would be the continuous phase and the series model would be more accurate. The converse it also true, and at higher percentages, the parallel model would be more accurate. A weighted combination model can be introduced that incorporates this concept:

i

+ 2Rm

)0

(7)

where R represents the conductivity, which is simply a proportionality constant relating a flux, J, and a driving force, ∇µ:

J ) -R∇µ

(8)

The analogy can be made to Fick’s law in order to extend the theory to the diffusion coefficient of a medium: n

Dm - Di

∑1 ΦiD

i

+ 2Dm

)0

(9)

Assuming that the equilibrium sorption of the pore cavities is constant throughout the medium (as before


Figure 7. EMT predictions of diffusivity selectivity over the range of pore distributions. Values for the diffusivities of pore1, pore2, and pore3 were obtained from Table 2. Solid lines represent constant pore3 compositions. The overall pore distribution is determined from the relationship ∑φi ) 1.

in eqs 1 and 2), this can also be applied to the effective permeability: n

Peff - Pi

∑1 ΦiP

i

+ 2Peff

)0

(10)

For the circumstances described here, EMT is the most useful model because it appropriately weighs the percentage of a pore within the material (similar to the weighted combination model shown in Figure 6) and it can be applied to an infinite number of selective entities. Because the effective permeability for EMT cannot be solved explicitly, a Fortran program was created in order to generate a selectivity “map” for a material of three pores over the range of possible distributions. The algorithm solves for the effective diffusivity (or permeability) using the series model as an initial guess and converges using Newton’s method. The input parameters to the model are the diffusivity (or, alternatively, permeability) of both gases for all three pores, and in this case, the diffusivity values generated in Table 2 for CO2, O2, N2, CH4, C3H6, and C3H8 were used. The results are shown in Figure 7 with diffusivity selectivity as a function of the number fraction of pore1. The solid lines represent constant pore3 curves. In this manner the entire distribution of the three pores can be represented on a two-dimensional plot. The first noticeable trend is that the selectivity decreases in all cases with increasing pore1 up to ∼ 60% pore3, which is somewhat intuitive because pore1 is slowly replacing the more selective entity, pore2. Surprisingly, the slopes of the curves change at ∼70-80% pore3, which indicates an increasing selectivity upon the addition of a less selective pore (exchanging pore1 for pore2). Physically, the reason

for this is the difference in diffusivity between pore1 and pore3. Essentially, gas molecules are able to pass easily through pore1 relative to the smaller pore3, and this allows pore3 to dominate transport. Another way of viewing the situation is that pore1 provides less competition with pore3 than does pore2. Essentially, for the hypothetical O2/N2 and CO2/CH4 case, this is a subtle effect occurring at 80% pore3, with only a slight increase in selectivity. On the other hand, for the hypothetical C3H6/C3H8 case, the effect occurs earlier (between 60 and 70% pore3) and more intensely, as evidenced by the large gap in selectivity between the 60% and 70% curves. This demonstrates how a subtle alteration in the material composition in this range produces a much larger effect on slightly larger molecules (in this case, the C3H6/C3H8 separation relative to O2/N2 and CO2/CH4). Figure 8 illustrates this concept. Upon shifting of the pore-size distribution to an average larger pore size (left to right), the O2/N2 and CO2/CH4 selectivities decrease as expected while the C3H6/C3H8 selectivity extends through a maximum. It should also be noted that the x-axis spacing between distributions in Figure 8 is somewhat arbitrary, and therefore the concavity of the curves is also subjective. However, the objective is only to illustrate how relatively larger molecules may show surprising and more dramatic responses to subtle shifts in the distribution of selective entities. Free-Volume Distributions in Polymeric Materials. The concepts involving the distribution of pore sizes in carbon materials can also be analogous to the distribution of critical transient chain spacings regulating diffusional jumps within a polymer (i.e., the distribution of free volume). Similar to the carbon materials, as the average transport regulating transient segmental


Figure 8. Diffusivity selectivity as a function of the pore-size distribution, predicted using EMT.

results, it is the third lifetime, the o-positronium, that is understood to correlate with free-volume “holes” in polymeric materials. For the polypyrrolone 6FDA-TAB (as well as other polypyrrolones), the third lifetime, τ3, as well as the intensity, I3, were unusually depressed. Evidence has shown that these aromatic ultrarigid polymer segments act as Lewis bases and are strong electron donors.25 This is consistent with the fact that the polymers are insoluble in typical organic solvents because they have strong intermolecular interactions. It is believed that the o-positronium is annihilated more rapidly in these “electron-rich” free-volume areas of the polypyrrolone and therefore provides ambiguous results as to the interpretation of the accessible free volume of the matrix. For these reasons, further study with PALS was not attempted on the poly(pyrrolone-imide) materials in this work. Although free-volume distributions could not be directly probed using PALS, a shift in the distribution is still expected with alteration of TAB/DAM stoichiometry. A visual example of this expected shift is demonstrated in Figure 9. This anticipated change is based on X-ray diffraction data, three-dimensional Hyperchem models, and diffusion coefficient data all published previously.8 It should be noted that the size and shape of the segmental distributions in Figure 9 are currently unknown and are only drawn to provide an example of the expected shift in the free-volume distribution with the material structure. Again, the analogy to the modeling of the carbon materials can be used to understand the reason for the unexpected trend in the diffusion coefficient for the larger C3H8 molecule (shown in Figure 9). Conclusions

Figure 9. Expected shift in chain spacing distributions with alteration of the TAB/DAM ratio. The size and shape of the distributions are unknown; however, a “shift” is expected based on X-ray diffraction data, Hyperchem modeling, and the diffusion coefficient data, published previously.

spacing becomes smaller, it would be expected that the overall free-volume distribution would also be shifted toward smaller distances. Clearly, this would be the expected trend as the TAB/DAM ratio is increased for the copolymer family discussed in this work. The trends in the diffusion coefficients shown in Figure 9 can be understood through analogy to the above modeling arguments for the microporous carbon membranes. The only known way to directly probe the free-volume distribution in polymeric materials is by positron annihilation lifetime spectroscopy (PALS). Many literature papers describing this technique for polymer membranes are available, and a few are provided here for reference.20-23 Zimmerman attempted this technique for polypyrrolone membranes, with somewhat ambiguous results.24 In the analysis of the positron annihilation

A surprising trend of a maximum in selectivity has been observed for certain gas pairs and yet not others over a range of material structures for both rigid copolymers and carbon molecular sieving materials. An extensive search of the literature demonstrated that this trend has been observed for carbon materials for a variety of gas separations although the trend does not appear to be previously discussed or well understood. When the assumption is made that the pore-size distribution in carbon materials is analogous to the freevolume distribution in rigid polymeric materials, it is possible to obtain a physical picture and a consistent mathematical description of the selectivity maximum in both types of molecular sieving materials. A unique assumption was made that considers using models for composite materials as a means to account for different selective entities as separate elements of a bulk material. Five different composite models were considered for further examination, and both the parallel and EMT models were used for additional study. Both of these latter models were able to demonstrate how the selectivity maximum may occur for relatively larger gas molecules and not smaller ones for the same system of materials under consideration. This lends insight into why certain membrane materials are “undesirable” only in certain cases and how they may be engineered to obtain improved transport properties under alternate operating conditions. Acknowledgment The authors acknowledge bp and the Separations Research Program at the University of Texas for fund-


ing support. The authors also thank John Wind, Rajiv Mahajan, and De Vu for helpful discussions regarding this work. Literature Cited (1) Zimmerman, C. M.; Koros, W. J. Polypyrrolones for membrane gas separations. I. Structural comparison of gas transport and sorption properties. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 1235. (2) Tanaka, K.; Taguchi, A.; Hao, J. Q.; Kita, H.; Okamoto, K. Permeation and separation properties of polyimide membranes to olefins and paraffins. J. Membr. Sci. 1996, 121, 197. (3) Walker, D. R. B. Synthesis and Characterization of Polypyrrolones For Gas Separation Membranes; The University of Texas at Austin: Austin, TX, 1993. (4) Tanaka, K.; Okano, M.; Toshino, H.; Kita, H.; Okamoto, K. I. Effect of Methyl Substituents on Permeability and Permselectivity of Gases in Polyimides Prepared from Methyl-Substituted Phenylenediamines. J. Polym. Sci., Part B: Polym. Phys. 1992, 30, 907. (5) Singh-Ghosal, A.; Koros, W. J. Air separation properties of flat sheet homogeneous pyrolytic carbon membranes. J. Membr. Sci. 2000, 174, 177. (6) Jones, C. W.; Koros, W. J. Carbon Molecular-Sieve Gas Separation Membranes. 1. Preparation and Characterization Based on Polyimide Precursors. Carbon 1994, 32, 1419. (7) Steel, K. M.; Koros, W. J. Investigation of porosity of carbon materials and related effects on gas separation properties. Carbon 2003, 41, 253. (8) Burns, R. L.; Koros, W. J. Structure-property relationships for poly(pyrrolone-imide) gas separation membranes. Macromolecules 2003, 36, 2374. (9) Steel, K. Carbon Membranes for Challenging Gas Separations; The University of Texas at Austin: Austin, TX, 2000. (10) Vu, D. Q. Formation and Characterization of Asymmetric Carbon Molecular Sieve and Mixed Matrix Membranes for Natural Gas Purification; The University of Texas at Austin: Austin, TX, 2001. (11) Okamoto, K.; Kawamura, S.; Yoshino, M.; Kita, H.; Hirayama, Y.; Tanihara, N.; Kusuki, Y. Olefin/paraffin separation through carbonized membranes derived from an asymmetric polyimide hollow fiber membrane. Ind. Eng. Chem. Res. 1999, 38, 4424. (12) Ogawa, M.; Nakano, Y. Separation of CO2/CH4 mixture through carbonized membrane prepared by gel modification. J. Membr. Sci. 2000, 173, 123. (13) Kusuki, Y.; Shimazaki, H.; Tanihara, N.; Nakanishi, S.; Yoshinaga, T. Gas permeation properties and characterization of asymmetric carbon membranes prepared by pyrolyzing asymmetric polyimide hollow fiber membrane. J. Membr. Sci. 1997, 134, 245.

(14) Kane, M. S.; Goellner, J. F.; Foley, H. C.; DiFrancesco, R.; Billinge, S. J. L.; Allard, L. F. Symmetry breaking in nanostructure development of carbogenic molecular sieves: Effects of morphological pattern formation on oxygen and nitrogen transport. Chem. Mater. 1996, 8, 2159. (15) Hayashi, J.; Mizuta, H.; Yamamoto, M.; Kusakabe, K.; Morooka, S. Pore size control of carbonized BPDA-pp′ODA polyimide membrane by chemical vapor deposition of carbon. J. Membr. Sci. 1997, 124, 243. (16) Hayashi, J.; Yamamoto, M.; Kusakabe, K.; Morooka, S. Simultaneous Improvement of Permeance and Permselectivity of 3,3′,4,4′-Biphenyltetracarboxylic Dianhydride-4,4′-Oxydianiline Polyimide Membrane by Carbonization. Ind. Eng. Chem. Res. 1995, 34, 4364. (17) Teplyakov, V.; Meares, P. Correlation aspects of the selective gas permeabilities of polymeric materials and membranes. Gas Sep. Purif. 1990, 4, 66. (18) Davis, H. T.; Valencourt, L. R.; Johnson, C. E. Transport processes in composite media. J. Am. Ceram. Soc. 1975, 58, 446. (19) Davis, H. T. The effective medium theory of diffusion in composite media. J. Am. Ceram. Soc. 1977, 60, 499. (20) Hong, X.; Jean, Y. C.; Yang, H. J.; Jordan, S. S.; Koros, W. J. Free-volume hole properties of gas-exposed polycarbonate studied by positron annihilation lifetime spectroscopy. Macromolecules 1996, 29, 7859. (21) Ito, Y.; Mohamed, H. F. M.; Tanaka, K.; Okamoto, K.; Lee, K. Sorption of CO2 in polymers observed by positron annihilation technique. J. Radioanal. Nucl. Chem. 1996, 211, 211. (22) Shantarovich, V. P.; Kevdina, I. B.; Yampolskii, Y. P.; Alentiev, A. Y. Positron annihilation lifetime study of high and low free volume glassy polymers: Effects of free volume sizes on the permeability and permselectivity. Macromolecules 2000, 33, 7453. (23) Yuan, J. P.; Cao, H.; Hellmuth, E. W.; Jean, Y. C. Subnanometer hole properties of CO2-exposed polysulfone studied by positron annihilation lifetime spectroscopy. J. Polym. Sci., Part B: Polym. Phys. 1998, 36, 3049. (24) Zimmerman, C. M. Advanced Gas Separation Membrane Materials: Hyper Rigid Polymers and Molecular Sieve-Polymer Mixed Matrices; The University of Texas at Austin: Austin, TX, 1998. (25) Jenekhe, S. A.; Johnson, P. O. Complexation-Mediated Solubilization and Processing of Rigid-Chain and Ladder Polymers in Aprotic Organic Solvents. Macromolecules 1990, 23, 4419. (26) Burns, R.; Koros, W. Defining the Challenges for C3H6/ C3H8 Separation Using Polymer Membranes. J. Membr. Sci. 2002, 211, 299.

Received for review March 14, 2004 Revised manuscript received June 1, 2004 Accepted June 14, 2004 IE049800Z

Explanation of a Selectivity Maximum as a ... - ACS Publications

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