Ind. Eng. Chem. Res. 1996, 35, 1231-1234
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Significance of Entropic Selectivity for Advanced Gas Separation Membranes Anshu Singh and W. J. Koros* Department of Chemical Engineering, University of Texas at Austin, Austin, Texas 78712
Commercial polymeric gas separation membranes exhibit a trade-off between productivity and selectivity. However, on this same trade-off curve molecular sieving materials like zeolites and carbon molecular sieves (CMS) lie above the upper bound for polymers. A comparison of the gas transport properties of these three different classes of materials highlights the importance of “entropic selectivity” which has hitherto not been focused on for membrane-based separation of gases. A whole new generation of polymeric membrane materials may result from enhancement of entropic selectivity of polymeric membrane materials. Introduction Air separation into oxygen enriched “permeate” and nitrogen enriched “reject” streams typifies current applications of gas separation membranes (see Figure 1). Practical gas separation membranes need to be highly permeable to one of the mixture components while significantly rejecting the other component. Current knowledge permits tailoring glassy polymer structures to address these dual requirements by introducing packing-inhibiting bulky groups on the polymer backbone to simultaneously hinder both segmental motion and intersegmental packing. The benefits achievable using this simple concept seem to be approaching a limit as shown in the rather extensive trade-off curve for gas permeability and selectivity in Figure 2. This is an adaptation of a similar figure published in 1991 (Robeson, 1991). In fact, the “upper bound” line drawn in Figure 2 still applies almost 5 years later. Molecular sieving materials like zeolites and carbon molecular sieves (CMS) lie above the upper bound for polymers on this trade-off curve. Understanding why polymeric membranes have been unable to achieve selectivities of the same order as zeolites or CMS is both scientifically and technologically interesting. Huge markets for O2/ N2 separations alone justify focusing on approaches to achieve gas separation by advanced materials. Although zeolites and molecular sieving carbons are useful indicators of achievable limits for separation performance, practical hollow fiber membrane modules are likely to rely upon polymeric substrates due to convenience, robustness, and compatibility with existing manufacturing systems. The challenge, therefore, is to identify and introduce the key missing elements in polymeric materials to boost performance above the upper bound and approach that of the molecular sieving materials.
Figure 1. Membrane-based separations.
Figure 2. Trade-off curve of oxygen permeability and oxygen/ nitrogen selectivity.
Background Typically nonporous, polymeric CMS and zeolite media transport gases by a so-called sorption-diffusion mechanism. Permeability of a penetrant in a membrane, defined as a pressure and thickness normalized flux, can be expressed as a product of a kinetic factor D, the diffusion coefficient, and a thermodynamic factor S, the sorption coefficient:
PA ) DASA
(1)
0888-5885/96/2635-1231$12.00/0
The sorptivity, S, is measured as the secant slope of the sorption isotherm at the upstream pressure conditions when the downstream pressure of the membrane is negligible. The diffusivity, D, is determined by taking the ratio of penetrant permeability and the sorption coefficient. Following from eq 1, the permselectivity, RA/B, of practical gas separation membranes can be interpreted as a product of two terms: one related to the ratio of diffusivities of the two gases in the membrane and the second determined by the ratio of sorp© 1996 American Chemical Society
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Ind. Eng. Chem. Res., Vol. 35, No. 4, 1996
q q Table 1. Comparison between Materials at 35 °C: DO2/DN2 ) exp[-∆EO /RT] exp[∆SO /R] 2,N2 2,N2
material 4A zeolite carbon molecular sievea upper bound polypyrrolone a
DO2/DN2 at 35 °C
EdqO 2 (kcal/mol)
q ∆EO 2,N2 (kcal/mol)
q /RT] exp[-∆EO 2,N2 energetic effects
q exp[-∆SO /RT] 2,N2 entropic effects
104 25-45 5.1
4.5 5.5 5.3b
-1.2 -1.0 -0.90
7.1 5.1 4.35
14.7 4.9-8.8 1.2
Karger and Ruthven, 1992. b Costello, 1994.
tivity of these gases in the membrane.
RA/B )
PA DA SA ) P B DB S B
(2)
D0 ) eλ2
[
] [
entropic selectivity
(3)
where Eqd is the energy of activation for the gas molecule to execute a diffusive jump from one cavity to
]
(Sdq A - Sdq B) -(Edq A - Edq B) DA ) exp exp DB R RT
(5)
}
D ) D0 exp[-Eqd/RT]
(4)
where λ is the average diffusive jump length in the diffusion medium, Sqd is the activation entropy of diffusion, and k and h are Boltzmann’s and Planck’s constants, respectively. For the O2/N2 pair, the difference in the kinetic diameters of the two molecules is very small; hence, λ can be considered similar for both gases. The diffusivity selectivity for such a gas pair can be given by the equation
Comparison of Molecular Sieving and Polymeric Materials for Gas Separations Comparing the gas transport properties of the three different classes of materials shown in Figure 2, namely, zeolite 4A, CMS, and an upper bound polymer, revealed to us the importance of a phenomenon that is probably best referred to as “entropic selectivity”. To date, entropic selectivity has not been considered quantitatively in membrane separation of gases. Zeolite 4A and CMS have dissimilar chemical properties but have similar correlations between diffusivity and gas molecular diameters, thereby indicating that the barrier to diffusion is entirely due to repulsive forces involved in passing through the constricted regions of the micropores (Karger and Ruthven, 1992). A rough analogy to the molecular sieving process can be drawn from the theory of diffusion of gases in glassy polymer matrices. In this case, as noted above, small gas molecules diffuse through openings occurring in the nonporous polymer matrix due to thermally induced local torsional motions of the polymer backbone (Koros et al., 1992). Since diffusion is an activated process in both molecular sieving and polymeric media, the diffusion coefficient can be written as an Arrhenius relationship, viz.
[]
Sqd kT exp h R
}
The sorptivity selectivity, SA/SB, has proven difficult to adjust without seriously off-setting the diffusivity selectivity for nonporous, polymeric membrane materials (Koros et al., 1992). On the other hand, adjustment of the diffusivity selectivity has been the major tool used in moving from the 1980 limit to the 1991 (and current) limit depicted in Figure 2. For O2/N2 separation, all of the materials that were compared showed a sorption selectivity only between 0.7 and 2. It is the diffusivity selectivity which accounts for the vast differences in the permselectivity values in Figure 2. A penetrant sorbed in a nonporous polymer matrix diffuses by executing a size-dependent jump. These jumps are moderated by the activation energy needed to create transient gaps of sufficient size to enable the jump to occur. Smaller penetrants require the localization of less activation energy. Size-dependent diffusion selectivity, therefore, favors the smaller of the two penetrants. This energetically biased selection process can be called “energetic selectivity”. Most attention in the past has been focused on this energetic selectivity due to a better understanding of how to measure and tailor activation energies of diffusion in typical adsorbent and membrane materials.
another, D0 is the temperature-independent preexponential term, T is the absolute temperature, and R is the universal gas constant. As observed in Table 1, there are relatively few differences in the exponential energy terms for polymers, zeolites, and CMS; hence, the preexponential term D0 in eq 3 accounts for the substantial difference in the diffusivity selectivity for the three classes of materials. From the transition state theory the preexponential factor is (Glasstone et al., 1941):
enthalpic selectivity
or
[ ] [
]
q q ∆SA,B -∆EA,B DA ) exp exp DB R RT
(6)
In eq 6 the ∆ terms refer to differences in activation quantities for oxygen and nitrogen. The diffusivity selectivity has been divided into an entropic and an enthalpic selectivity. From the values in Table 1, it can be seen that both energetic and entropic factors contribute significantly to the high values of diffusivity selectivities observed for zeolites and carbons. However, for the polymeric materials, the entropic factor is close to unity and its contribution to diffusivity selectivity is negligible. This observation is significant, since the glassy polymer considered in this study is one of the very best available in this generation and actually lies on the upper bound line at an overall O2/N2 selectivity of 6.5 and an oxygen permeability of 7.5 barrer (1 barrer ) 10-10 cm3(STP) cm/cm2 s cmHg). Entropic Selectivity Based on the Transition State Theory Applying the transition state theory to the specific rate constant for diffusion, the diffusion coefficient for
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Figure 3. Idealized schematic cross section of 4A zeolite crystals and micropores in CMS. Zeolite 4A has large cavities with average diameter of 11.2 Å (marked S ) and constricted windows of diameter 3.8 Å (marked S ′). CMS comprises graphitic planes of carbon, with packing imperfections forming the narrow slit-like pores (marked S ′).
a penetrant through any medium can also be described by the following equation (Glasstone et al., 1941):
[ ]
-Eqd kT Fq D ) λ2 exp h F RT
(7)
where Fq represents the partition function for the gas molecule in the transition state and F is the partition function for the same molecule in the normal state. The transition state occurs as the gas molecule passes through the constricted “windows” of molecular dimensions (region S ′ in Figure 3), while molecules in the large cavities (region S ) are referred to as being in the normal state. The partition function may be expressed as the product of translational, rotational, and vibrational contributions (see the appendix for partition function calculations).
F ) FtransFrotFvib
(8)
However, the partition function in the activated state, Fq, does not contain the translational partition function in the direction of gas diffusion. The factor kT/h accounts for this degree of freedom in eq 7. Combining eqs 4 and 7, the entropic selectivity term can also be written as
[
exp
]
Sdq A - Sdq B R
(Fq/F)A ) (Fq/F)B
(9)
Diatomic molecules like O2 and N2 have spherocylindrical structures. Experimental data for second virial coefficients at several temperatures provide estimates of the constants used for calculating interatomic potentials based on advanced models such as the Kihara potential (Hirschfelder et al., 1954; Bussery and AubertFrecon, 1991). These parameters provide estimates of molecular size and shape which have been confirmed by molecular charge density distribution calculations (Bader et al., 1967). In these calculations, the charge density contour encapsulating 95% of the charge is taken to be the size of the molecule. The lengths and widths of O2 and N2 molecules obtained from these two methods are shown in Figure 4. In light of these molecular dimensions, one can apply the transition state theory to diffusion of the O2/N2 gas pair through the structurally different pores of zeolites
Figure 4. Molecular sizes of oxygen and nitrogen molecules: (a) Kihara’s modified potential for spherocylindrical molecules fitted to second virial coefficient data. Nitrogen values from Hirschfelder et al., 1954. Oxygen values from Bussery and Aubert-Frecon, 1991. (b) Molecular charge density distribution calculations (95% charge encapsulation) confirm the above calculated values (Bader et al., 1967).
and CMS. A possible explanation for the difference in entropic contributions to selectivity emerges from this analysis. The size of the eight-sided window in the 4A zeolite is 3.8 Å. The gas molecules are considered to have 2 degrees of rotational freedom about the two axes of symmetry and 3 degrees of translational freedom in the normal state. This assumption is reasonable even for nitrogen with its higher quadrapole, since it has been established from entropy values that nitrogen exists in a freely rotating state at temperatures above 20 °C for zeolites similar to type 4A (Takaishi et al., 1974). The oxygen molecule can pass through the constricted windows rotating about either of the two axes of rotation, but the nitrogen molecule, which is longer, cannot. Nitrogen will pass through only when it has given up both of its degrees of rotational freedom and is aligned along its long axis, as shown in Figure 5. The nitrogen molecule may have two additional degrees of vibrational motion in the plane of the window due to the rocking of the molecules around its two axes of symmetry in its restricted environment. For diffusion through zeolites, the entropic selectivity is given by the following equation:
( )
exp
∆Sdq A,B R
1 )
Ftrans3
|
A
|
Ftrans3 Frot2 q 2 (Fvib )
(10) B
Calculation of the partition coefficients indicates an entropic selectivity of 14.6, which compares remarkably well to the experimentally observed value of 14.7. A similar analysis can be carried out for CMS. The micropores in the carbons are slit-shaped, having been formed by imperfectly packed microcrystals of graphite constricted into narrow regions, where the size of the openings are similar to the size of the constricting windows in zeolites. The oxygen molecule can pass through these pores rotating about either of its axes of rotation. On the other hand, the nitrogen molecule may pass through rotating about only one axis as illustrated in Figure 5. In this case, both oxygen and nitrogen molecules may have an additional degree of translational freedom along the graphitic planes, transverse to the direction of diffusion. The nitrogen molecule may or may not have an additional vibrational degree of freedom in the activated state depending on the aspect ratio of the slit-shaped pores. If an additional degree of vibration is present, in lieu of the degree of rotation which has been lost for the nitrogen molecule, the entropic selectivity is calculated to be 3.7, however, if there are no additional degrees of freedom, which might occur in the case of the slit width being very close to
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Appendix Partition Functions.
Translational Ftrans ) (2πmkT/h2)n/2an Rotational Frot ) (8π2IkT/h2)n/2 Vibrational Fvib )
Figure 5. View of zeolites and CMS in the direction of diffusion to illustrate degrees of freedom of oxygen and nitrogen. Oxygen can pass through the zeolite window rotating about either axes of symmetry, but nitrogen loses its rotation about both axes. In the slit-shaped pores of CMS, oxygen and nitrogen have a translational degree of freedom parallel to the graphitic planes also. Oxygen can again pass through rotating about either axes, while nitrogen can rotate around only one axis in the long narrow pores.
the width of the nitrogen molecule, the entropic selectivity is calculated to be 9. These values encompass the experimentally observed range of 4.9-8.8. The entropic contribution for conventional polymeric materials, which is essentially equal to unity, suggests that both oxygen and nitrogen molecules have the same number of degrees of freedom in the transition state. This presumably reflects the inability of the still slightly mobile polymer segments in current generation polymers to restrict rotational degrees of freedom of nitrogen as effectively as CMS and zeolites. Conclusion Diffusivities of gases in three different classes of materials, namely, zeolite 4A, CMS, and an upper bound polymer for gas separation membranes, have been compared. The comparison illustrated the significant advantage that the molecular sieving materials have over current polymeric materials in entropic selectivity. This factor is primarily responsible for the higher separation efficiencies along with higher throughputs observed for actual molecular sieving materials and estimated for potential materials. The entropic selectivity arises from oxygen retaining more degrees of freedom in the transition state than nitrogen. The polymer chains which are in constant thermal motion are incapable of selectively restricting the motion of nitrogen in the transition state, thus resulting in a complete loss of entropic selectivity. The above observations imply that advanced membranes for O2/N2 separation must have markedly lower motions of the polymer chains than are present in the best available materials in the current generation. Work is in progress to determine the entropic contributions of other membrane materials for several gas pairs and to establish a structureproperty relationship for entropic selectivity in polymeric materials.
]
n
where m is the mass of the molecule, I its moment of inertia, ν its frequency of vibration at 300 K, a the length of the cubic cavity in which the molecule is confined, and n the number of degrees of freedom for each type of motion of the molecule. For calculating values of partition functions, the following molecular parameters were used: parameter
oxygen
nitrogen
ref
θr ) h2/8π2Ik 2.07 K 2.88 K McQuarrie, 1973 2.0 × 1012 s-1 2.6 × 1012 s-1 Barrer, 1978 νa a (in normal state) 11.2 Å 11.2 Å Karger and Ruthven, 1992 a (in transition 3.8 Å 3.8 Å values of entropic state) (assumed) selectivities calculated are insensitive to this number a Freq. of vib. for the additional vib. degrees of freedom in the transition state were estimated by Barrer from entropy of sorption calculations for zeolites similar to zeolite A.
Literature Cited Bader, R. F. W.; Henneker, W. H.; Cade, P. E. Molecular Charge Distributions and Chemical Binding. J. Chem. Phys. 1967, 46, 3341-3363. Barrer, R. M. Zeolites and Clay Minerals as Sorbents and Molecular Sieves; Academic Press Inc.: New York, 1978. Bussery, B.; Aubert-Frecon, M. Semi-empirical investigation of the angular dependence of the interaction energy between two ground-state oxygen molecules. Chem. Phys. Lett. 1991, 179, 393-397. Costello, L. M. Temperature Dependence of Gas Sorption and Transport Properties in Glassy Polymers. Ph.D. Thesis, University of Texas at Austin, Austin, TX, 1994. Glasstone, S.; Laidler, K. J.; Eyring, H. The Theory of Rate Processes, 1st ed.; McGraw-Hill Book Co., Inc.: New York, 1941. Hirschfelder, J. O.; Curtiss, C. F.; Bird, R. B. Molecular Theory of Gases and Liquids; John Wiley & Sons. Inc.: New York, 1954. Karger, J.; Ruthven, D. M. Diffusion in Zeolites and Other Microporous Solids; Wiley-Interscience Publication: New York, 1992. Koros, W. J.; Coleman, M. R.; Walker, D. R. B. Controlled Permeability Polymer Membranes. Annu. Rev. Mater. Sci. 1992, 22, 47-89. McQuarrie, D. A. Statistical Thermodynamics; Harper & Row Publishers, Inc.: New York, 1973. Robeson, L. M. Correlation of Separation Factor versus Permeability for Polymeric Membranes. J. Membr. Sci. 1991, 62, 165185. Takaishi, T.; Yusa, A.; Ogino, Y.; Ozawa, S. Motional State of Sorbed Nitrogen in Mordenite. Trans. Faraday Soc. 1974, 70, 671-684.
Received for review September 5, 1995 Accepted December 4, 1995X IE950559L
Acknowledgment The authors gratefully acknowledge the support of the Department of Energy’s Office of Basic Energy Science under Grant No. DE-FG03-95ER145386.
[
exp(-hν/2kT) 1 - exp(-hν/kT)
Abstract published in Advance ACS Abstracts, February 15, 1996. X
