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Sorption Isotherms of Gases by Polymer Membranes in the Glassy State: An Explanation Based on the Nonequilibrium Thermodynamics
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
204, 135–142 (1998)
CS985554
Sorption Isotherms of Gases by Polymer Membranes in the Glassy State: An Explanation Based on the Nonequilibrium Thermodynamics Susanne Motamedian,* Wolfgang Pusch,* Akihiko Tanioka,* ,1,2 and Friedrich Becker† *Max-Planck-Institute fu¨r Biophysik, Kennedy Allee 72, D-60596 Frankfurt am Main 72, Germany; and †Institut fu¨r Physikalische und Theoretische Chemie der Universitaet Frankfurt P.O. Box 11 19 32, D-60439 Frankfurt am Main, Germany Received December 19, 1997; accepted March 26, 1998
Sorption isotherms of glassy polymers are concave to the pressure axis, and the absolute sorption levels are almost an order of magnitude higher than that of rubbery polymers on the relatively low-pressure side. There are several models to interpret this behavior, and the dual sorption model is the most widely accepted one among them. In the present work, sorption isotherms were first derived from permeation data with several gases for four polymer membranes which might be representative of the dual sorption model: two of them were in a glassy state (cellulose acetate and polyamide), the third one was a block copolymer which was composed of a glassy polymer and a rubbery polymer at the measuring temperature, and the fourth one was a Nafion membrane which was taken as a model membrane because of its channel structure with adsorption sites of charged groups supported by a rubbery polymer. These results did not necessarily support the dual sorption model. Subsequently, the validity of the underlying assumptions of two other sorption models for glassy polymers, that is, the matrix model and the deformation model, were examined and a new equation for the sorption isotherm with an ordering parameter was derived, which implied that the glassy polymer was in a nonequilibrium state and changed from the glassy to the rubbery state by absorbing the gas. q 1998 Academic Press Key Words: sorption isotherm of gas; glassy polymer; nonequilibrium thermodynamics; ordering parameter; dual sorption model; matrix model; deformation model.
1. INTRODUCTION
Transport and sorption of small molecules, i.e., gases and vapors, in glassy polymers have been of great interest scientifically and technologically (1–4). The main reason is that the sorption isotherms of glassy polymers are concave to the pressure axis and the absolute sorption levels are of almost an order of magnitude higher than those of rubbery polymers, 1
To whom correspondence should be addressed. Present address: Department of Organic and Polymeric Materials, Tokyo Institute of Technology, 2-12-1 Ookayama Meguro-ku, Tokyo 152, Japan.
though the rubbery polymers show a linear or convex relationship to the pressure axis (5, 6). In order to interpret this behavior, several models were developed which were based on different ideas. The most acceptable theoretical model is the dual sorption model, which is based on the assumption of two modes for sorption in the polymer (7). The polymer includes a dense region as one side, and the others are regions of the lower polymer density, that is, microvoids which are proposed to be formed in the polymer when it is cooled down below its glass transition temperature (8, 9). The dual sorption model is expressed by the following formulation (10): c Å cD / cH Å kD p /
cÅ
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[1]
where c Å total sorbed gas concentration, cD Å concentration of gas in a Henry’s law-type site, cH Å concentration in a Langmuir-type site, kD Å Henry’s law coefficient, C *H Å hole saturation parameter, b( ú0) Å hole affinity parameter, and p Å applied pressure. The first term of the equation follows Henry’s law and represents the sorption in the denser regions of the polymer, while the second term is of the Langmuir type and is the result of adsorption around the microvoids. The dual sorption model describes well the sorption phenomena of many glassy polymers. The validity of the underlying assumptions will be discussed in the following part of this paper. Contrary to the dual sorption model, the gas–polymer matrix model (11) assumes that only one population of sorbed gas molecules exists in the polymer at any given pressure. Based on the observation of changes in the physical properties, i.e., changes in the cooperative main-chain motions of polymers, changes in viscoelastic relation and Tg , in the presence of a sorbed gas, the authors propose that these changes arise from gas–polymer interactions. The model describes the sorption isotherm by a two-parameter equation:
2
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C *H bp , 1 / bp
(1 / 4aS0 p) 1 / 2 0 1 , 2a
[2]
0021-9797/98 $25.00 Copyright q 1998 by Academic Press All rights of reproduction in any form reserved.
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MOTAMEDIAN ET AL.
S Å S0 exp( 0ac),
[3]
where S Å solubility parameter, S0 Å solubility parameter in the zero-concentration limit, a Å conformation parameter, and c Å total sorbed gas concentration. Contrary to those two models which describe the sorption process by empirical equations, the deformation model by Lipscomb (12) is a thermodynamic model. The sorption of penetrants into a solid is treated as two discrete processes. Solid deformation in the first step is followed by mixing with the penetrant. The model is an extension of the Flory– Huggins (13) relationship of penetrant-induced swelling of rubbery materials, and the dependence of the gas pressure and its concentration in the polymer is given by pÅ
SD F
pr tr(sr ) £s exp t 3RT
GS S
D FS D D S D G
c£s exp 1 / c £s
/x
1 1 / c £s
2
/
1 0 cp £s 1 / c £s
B0£s c £s RT
[4]
where p Å applied pressure, pr Å reference pressure, c Å total sorbed gas concentration, cp Å polymer concentration, £s Å partial molar volume of gas, x Å Flory–Huggins interaction parameter, t Å gas external phase activity coefficient, and B0 Å bulk modulus. The difference between this relationship and the Flory–Huggins equation is given only by the stress relation terms, i.e., the pre-exponential factor involving the stress factor and the bulk modulus term in the exponent. The experiments described in the following should be the basis for a discussion of the validity of the underlying assumptions of the three models and the obtained knowledge should lead to improvement of the models.
methylbenzoate sulfonic acid tetrabutylammonium salt) and is in a glassy state at the measuring temperature (15). 2.2. Estimation of Sorbed Gas from Permeation Measurements (14)
The sorption isotherms were received from permeation experiments by the time lag method. This is not an exact procedure for determining the amount of sorbed gas. However, it has already been confirmed that we can examine the shape of the isotherm; that is, a concave curve, convex curve or straight line, depending on the gas–polymer system. The permeation equipment and the procedure were the same as already described. The membrane samples which had an effective area of about 10.5 cm2 were inserted between a lowpressure and a high-pressure compartment in a permeation cell made from stainless steel. The membrane samples were supported by a CONIDUR screen topped with an appropriate membrane filter. The permeation cell was a modification of a common hyperfiltration cell (16–19). The closed cell was placed into a water bath, and the low pressure compartment was evacuated to a pressure of about 0.05 mbar using a Leybold vacuum pump. After evacuation overnight, the feed pressure (p * ) of 1– 90 bar was adjusted in the high-pressure compartment, and the pressure increase in the low-pressure compartment was recorded as a function of time (t) employing MKS pressure gauges and measuring devices. The diffusion coefficients of the gases were obtained from time-lag and the permeabilities from the slope of the asymptote of the curve p * versus t. The solubility Sg of the gas in the membrane was obtained by the relationship Pg Å Sg Dg . The amount of the sorbed gas (Cg ) is calculated from Cg Å Sgp *. H2 , D2 , N2 , He, and CO2 were employed as test gases. Their purity was 99.99%. The permeation experiments were done at 25 or 357C at the pressures from 1 bar up to 90 bar. The measurement error was within 5%.
2. EXPERIMENTAL 3. RESULTS AND DISCUSSION
2.1. Materials
Two cellulose acetate membranes with different acetyl contents were examined; the acetyl content of the Bayer Cellit K-700 membrane was about 39.1%, and that of the Eastmann Kodak CA E 320 was about 32%. The membrane preparation procedure is described elsewhere (14). The second membrane was an acrylonitrile–styrene–butadiene block copolymer (ABSB) which was produced at Buna AG Merseburg, and the polymer was partly in the glassy and partly in the rubbery state at the measuring temperature (15). The Nafion membrane was a commercially available ion exchange membrane composed of perfluorinated carbon from Du Pont, and its thickness varied from 170 to 190 mm. The polyamide membrane obtained from CIBA-GEIGY AG was composed of a mixture of two diamines (4,4 *-diamino2,2 *-dimethyldiphenylmethane and 3,5-diamino-2,4,6-tri-
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Figures 1–4 exhibit the sorption isotherms which were calculated from the data of the permeation experiments. As can be seen in Fig. 1, the sorption isotherms of the Nafion membrane are linear and seem to follow Henry’s law (20). The same behavior is exhibited in the sorption isotherm of the polyamide membrane for Ar and N2 as shown in Fig. 2 and also for Ar and N2 in the ABSB membrane, whereas the sorption isotherm of CO2 appears linear only at a low pressure up to p õ 12 bar as shown in Fig. 3. At a higher pressure, a nonlinear increase with increasing pressure is found. This indicates a strong increase in the sorption capacity for CO2 . This behavior is observed often for CO2 and can be interpreted as CO2 acting as a softener for the membrane, which means that the dissolved gas changes the membrane properties irreversibly. Sorption isotherms of H2 and
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GAS ISOTHERM SORPTION BY POLYMER MEMBRANES
137
FIG. 1. Sorption isotherms of Ar ( s ), N2 ( , ), and He ( h ) gases for a Nafion 117 membrane at 357C.
FIG. 3. Sorption isotherms of CO2 ( , ), Ar ( s ), and N2 ( h ) gases for an ABSB membrane at 257C.
D2 by cellulose acetate membranes are shown in Figs. 4 and 5, respectively, which are concave relative to the pressure axis. The dual sorption model predicts that the sorption isotherms for glassy polymers are concave to the pressure axis, which means that only the sorption isotherms of cellulose acetate can be interpreted by this model. This represents well the experimental data for the cellulose acetate membrane, but the b parameters in Eq. [1] are negative values which are physically nonsensical as shown in Table 1.
The Nafion membrane was examined as a model membrane for a discussion on the validity of the underlying assumption of the dual sorption model. It is a copolymer with the following chemical repeating units: the structure consists of a (CF2 )n backbone with SO 03 groups placed inside of the channels (21). The former contributes to the sorption by Henry’s law, and the latter should provide an ideal site for adsorption. Therefore, the sorption isotherms should be concave to the pressure axis if one follows the dual sorption concept. Because the sorption
FIG. 2. Sorption isotherms of Ar ( s ) and N2 ( h ) gases for a polyamide membrane at 257C.
FIG. 4. Sorption isotherms of H2 gas for a cellulose acetate membrane at 257C and the application of Eq. [1].
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also requires a third state variable, a so-called ordering parameter ( j ) for full characterization of its state (24). This ordering parameter is defined by the affinity (A) of the actual state of the polymer membrane where A is equal to the chemical potential difference of the polymer in its actual state mp (T, p, j ) minus the chemical potential of an equilibrium state mp (T, p, j Å 0) of the polymer: A Å mp (T, p, j ) 0 mp (T, p, j Å 0).
[5]
The sorption isotherm model is based on the following observations and assumptions:
FIG. 5. Sorption isotherms of D2 gas for a cellulose acetate membrane at 257C and the application of Eq. [1].
isotherms are linear for all the examined gases, it can be concluded that the reason for the concave sorption isotherms is not a result of the two modes for sorption. This is a confirmation of the assumption of other authors (11, 12) who also called the existence of the two modes for sorption into question. It should be noticed that NMR and X-ray scattering measurements of polymer structure (22) could not provide conclusive evidence of the existence of two sorption sites either. The gas– polymer matrix model is based on the assumption of gas– polymer interactions which lead to a change in the physical properties of the polymer. This assumption could be provided by NMR measurements in which the influence of CO2 sorption on the cooperative main-chain motion of poly(vinyl chloride) was analyzed. Nevertheless, the matrix model predicts concave sorption isotherms for glassy polymers which means only that, like the dual sorption model, only the sorption isotherms for cellulose acetate can be interpreted by this model. The standard derivations for calculated curves are nearly the same as these calculated by the dual sorption equation as shown in Figs. 4 and 5. The most important thing is to notice that the gas– polymer matrix model expresses the sorption behavior by empirical parameters which do not provide physical insight into the sorption process. In the following, a new sorption isotherm is derived, which has a thermodynamic basis implying that the glassy polymer is in a thermodynamically nonequilibrium state. 4. SORPTION ISOTHERM WITH ORDERING PARAMETER BASED ON NONEQUILIBRIUM THERMODYNAMICS (15, 23)
The state of a glassy polymer is not determined only by the state variables, pressure (p) and temperature (T ), but
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(a) We cannot differentiate the state of the gas molecules in the polymer from each other; that is, there is only one population of sorbed gas molecules in the polymer at any given pressure (11). (b) With increasing gas concentration in the polymer, the physical properties change; that is, the cooperative mainchain motion of the polymer increases and the glass transition temperature changes to lower temperatures (5, 25–27). (c) By absorbing gas, the state of the polymer changes to a more disordered state, which means a change from the glassy state to the rubbery state. The change in the polymer state can be expressed by the change in the ordering parameter (28–30). For the transition from the glassy state to the rubbery state, the boundary conditions for the ordering parameter are 0°j°1
j Å 1, glassy polymer, j Å 0, rubbery polymer.
[6]
when the Gibbs–Duhem relationship for a glassy polymer– gas system is applied, the following relation is held (31): cp dmp (T, p, j ) / c dms Å 0,
[7]
where c and cp are the gas and polymer concentrations, reTABLE 1 Parameters of the Dual Sorption Model in the Cellulose Acetate Membranes (K-700 and E-320)–H2 and D2 Systems Calculated from Eq. [1]
Membranes
Gas
KD (cm3 (STP)/ cm3Mrbar)
E-320 K-700 E-320 K-700
H2 H2 D2 D2
0.0935 0.119 0.0898 0.107
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CH* (cm3 (STP)/ cm3M)
b (bar01)
30.9 38.9 31.4 27.6
00.00167 00.00186 00.00162 00.00181
GAS ISOTHERM SORPTION BY POLYMER MEMBRANES
139
spectively, and ms is the chemical potential of a gas in the polymer. Equation [7] indicates that the chemical potential of the gas is also a function of the ordering parameter, ms Å ms {T, p, j},
[8]
where j may vary from one membrane sample to another. ms is equilibrated to the chemical potential mg of the gas in the external phase to introduce sorption isotherm in the glassy polymer as a function of the ordering parameter. With T Å constant and the assumption that there is ideal behavior in the gas phase and in the dissolved gas phase, the chemical potential in both phases can be expressed in the following form: mg Å m0g / RT ln(p/p 0 ), ` s
0
ms Å m / RT ln(c/c ),
[9] [10]
where m0g and m`s are the standard potentials in the gas and dissolved gas phases, respectively, p 0 is the standard pressure in the gas phase, and c 0 is the standard gas concentration in the polymer. In the equilibrium state, mg Å ms .
[11]
From Eqs. [9] and [10], with Eq. [11] it follows that
S D
m0g 0 m`s cp 0 Å ln 0 RT c p
or c/c 0 Å (p/p 0 )exp[( m0g 0 m`s )/RT] Å S(p/p 0 ),
FIG. 6. Temperature dependence of the chemical potential of polymers.
p max is the maximum gas pressure where the phase conversion from glassy state to rubbery state is completed. At any temperature the chemical potential of the glassy polymer is higher than that of the rubbery polymer as seen in Fig. 6 because the glassy polymer is not in thermodynamic equilibrium. Therefore, it can be concluded with Eq. [7] that the chemical potential of the gas in a rubbery polymer is higher than the chemical potential of the gas in glassy polymers, which corresponds to the higher gas solubility in glassy polymers. The chemical standard potentials for j Å 1 and j Å 0 are assumed to be
[12]
m`s (1) å m`s,gl , m`s (0) å m`s,ru , Dm`s å m`s,ru 0 m`s,gl ¢ 0.
where S å exp[( m0g 0 m`s )/RT] (T Å constant).
[13]
Equation [12] is valid for glassy as well as for rubbery polymers. Because the ordering parameter j changes with increasing gas concentration in the polymer, it can be concluded that it is a function of gas pressure p according to the assumptions (a), (b), and (c). At T Å constant, the ordering j(0 ° j ° 1) can be defined as j(p Å p 0 ) Å (p max /p 0 0 p/p 0 )/(P max /p 0 0 1)
AID
If ms ( j ) is expanded with j in quadratic equation, the dependence of the chemical potential of the dissolved gas on the ordering parameter at T Å constant can be expressed by following equation with a constant as seen in Fig. 7: ms ( j ) Å m`s,ru 0 j[ Dm`s / a(1 0 j )],
[18]
where the slope is given by Ìms / Ìj Å 0 Dm`s 0 a(1 0 2j ),
[14]
with the boundary conditions
[17]
[19]
and at the boundary,
j(p Å p 0 ) Å 1, glassy polymer,
[15]
Ìms / Ìj ( jÅ0 ) Å 0 Dm`s 0 a,
[20a]
j(p Å p max ) Å 0, rubbery polymer.
[16]
Ìms / Ìj ( jÅ1 ) Å 0 Dm`s / a.
[20b]
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ms (c, j ) Å m`s,ru 0 jRT ln k 0 j[ Dm`s / a(1 0 j )] / RT ln(c/c 0ru ).
[25]
If the ordering parameter j is expressed by p/p 0 with Eq. [14], the explicit form of the function c/c 0 Å f ( p/p 0 ) is obtained by equilibrating Eqs. [9] and [25], with the maximum pressure p max as a parameter. Finally, the following equation is derived for p 0 Å 1 bar: c/c 0ru Å pk ( p
max0p ) / ( p max0 1 )
1 exp{[ m0g 0 m`s,ru / (p max 0 p)/(p max 0 1) 1 [ Dm`s / a(1 0 (p max 0 p)/(p max 0 1))]]/RT}.
[26]
FIG. 7. Dependence of ordering parameter on the chemical potential calculated by Eq. [18], where a Å 0.2 and a Å 0 with Dms` Å 0 and ` ms,ru Å 1.
For example, in the case of a ú 0 the slope of ms ( j ) becomes steeper as j r 0 and more gentle as j r 1 than the slope for the case of a Å 1. For the concentration dependence of the chemical potential of the dissolved gas ms with the assumption of ideal behavior and T Å constant, the following relationship for glassy polymers can be obtained: ms,gl (c, 1) Å m`s,gl / RT ln(c/c 0gl ),
[21]
ms,ru (c, 0) Å m`s,ru / RT ln(c/c 0ru ).
[22]
Equation [26] is applied to the experimental results for cellulose acetate–H2 and D2 gas systems in Figs. 4 and 5 with k Å 1.1 and a/RT Å 00.1, respectively, and nonlinear regressions are performed to determine the three parameters of p max /p 0 , Dm`s /RT, and ( mg0 0 m`s,ru )/RT. The calculated results are shown in Fig. 9 for H2 gas and in Fig. 10 for D2 gas, and the three parameters are listed in Table 2 with k and a/RT. It can be seen that the sorption behavior of H2 or D2 in a cellulose acetate membrane can be well described by the equation based on the ordering parameter. The linear sorption isotherms of the other membranes mean that the ordering parameters of these membranes do not change much during the experiment. In case of the Nafion membrane, the sorption in the channels is the reason why the ordering parameter of the membrane does not change. The state of the ABSB mem-
c 0gl and c 0ru are the standard concentrations in glassy and rubbery polymers, respectively, at p Å p 0 . The relationship between the two standard concentrations can be expressed as c 0gl Å kc 0ru with k ¢ 1.
[23]
The parameter k represents the higher gas solubility in glassy polymers. The dependence of the standard concentration c 0 as a function of the ordering parameter j can be described as c 0 ( j ) Å k jc 0ru
[24]
and is shown in Fig. 8. The chemical potential of the dissolved gas ms (c, j ) which is dependent on the concentration and ordering parameter given by the combination of Eq. [18] with Eq. [24]:
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FIG. 8. Change in standard concentration with ordering parameter calculated by Eq. [24], where k Å 2 and c 0ru Å 1.
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TABLE 2 Parameters of Eq. [24] in the System of a Cellulose Acetate Membrane–H2 and D2 Systems
FIG. 9. Sorption isotherms of H2 gas for a cellulose acetate membrane at 257C and the application of Eq. [26].
Gases
pmax/p0
Dms`/RT
` (m0g 0 ms,ru )/RT
a/RT
k
H2 D2
632 567
0.04 0.05
01.74 01.15
00.1 00.1
1.1 1.1
the glassy to the rubbery state by absorbing the gas. Those phenomena were already confirmed in the system of CO2 gas–polymer (28–30) and H2O vapor–hydrophilic polymer (32, 33) systems. The results in this study suggest that the sorption isotherm in a nonequilibrium state may also be explained by introducing the ordering parameter, and almost all kinds of gases have the possibility of changing the polymer state from glass to rubber. ACKNOWLEDGMENT
brane is already mainly disordered because the membrane is at measuring temperature already partially in the rubbery state composed of styrene–butadiene copolymer. The sorption isotherms of gas and vapor by glassy polymer membranes were concave to the pressure axis, and the absolute sorption levels were almost an order of magnitude higher than that of rubbery polymers on the low-pressure side. The high absolute sorption level at low pressure can be explained by the excess free energy in the glassy polymer compared with the rubbery polymer at the constant temperature and pressure. The curvature of the sorption isotherm can be explained from the reason why the polymer state changes from
FIG. 10. Sorption isotherms of D2 gas for a cellulose acetate membrane at 257C and the application of Eq. [26].
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This is a memorial article for Prof. Dr. Wolfgang Pusch and a part of the dissertation of Dr. Susanne Motamedian under his direction. Just before Dr. Motamedian finished her Ph.D. thesis, Prof. Pusch was killed in a horrible accident on 22 June 1992. This is a very sad memory and a great loss for membrane scientists all over the world.
REFERENCES 1. Barrer, R. M., and Barrie, J. A., J. Polym. Sci. 23, 331 (1957). 2. Stannett, V. T., Koros, W. J., Paul, D. R., Lonsdale, H. K., and Baker, R. W., Adv. Polym. Sci. 32, 69 (1979). 3. Haward, R. N., Ed., ‘‘The Physics of Glassy Polymers.’’ Applied Science Publishers, London, 1973. 4. Whyte, T., You, C., and Wagner, E., Eds., ‘‘Industrial Gas Separations,’’ ACS Symposium Series 223. American Chemical Society, Washington, DC, 1983. 5. Crank, J., and Park, G. S., ‘‘Diffusion in Polymers.’’ Academic Press, London, 1968. 6. Petropoulos, J. H., Adv. Polym. Sci. 64, 93 (1985). 7. Vieht, W. R., Howell, J. M., and Hsieh, J. H., J. Membr. Sci. 177 (1976). 8. Fechter, J. M., Hopfenberg, H. B., and Koros, W. J., Polym. Eng. Sci. 21, 925 (1981). 9. Koros, W. J., and Paul, D. R., J. Polym. Sci., Polym. Phys. Ed. 19, 1655 (1981). 10. Vieth, W. R., and Sladek, K. J., J. Colloid Sci. 20, 1014 (1965). 11. Raucher, D., and Sefcik, M. D., ‘‘Industrial Gas Separations’’ (T. Whyte, C. You, and E. Wagner, Eds.), ACS Symposium Series 223, Chapter 6. American Chemical Society, Washington, DC, 1983. 12. Libscomb, G. G., AIChE J. 35, 1506 (1990). 13. Flory, P. J., ‘‘Principles of Polymer Chemistry.’’ Cornell Univ. Press, Ithaca, NY, 1969. 14. Pusch, W., and Tanioka, A., Desalination 46, 425 (1983). 15. Motamedian, S., Ph.D. Dissertation, Johann Wolfgang Go¨the Universita¨t, Frankfurt am Main, Germany, 1993. 16. Pusch, W., Ber. Bunsenges. Phys. Chem. 81, 854 (1977). 17. Koros, W. J., and Paul, D. R., J. Polym. Sci., Polym. Phys. Ed. 16, 1947 (1978).
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18. Koros, W. J., and Paul, D. R., J. Polym. Sci., Polym. Phys. Ed. 16, 2171 (1978). 19. Pusch, W., Desalination 59, 105 (1986). 20. Chiou, J. S., and Paul, D. R., Ind. Eng. Chem. Res. 27, 2161 (1988). 21. Gierke, T. D., and Hsu, W. Y., ACS Symp. Ser. 180, 283 (1982). 22. Assink, R. A., J. Polym. Sci., Polym. Phys. Ed. 13, 1665 (1975). 23. Tanioka, A., Motamedian, S., and Pusch, W., Abstracts of ICOM’93, 2.20, 1993. 24. Motamedian, S., Pusch, W., Sendelbach, G., Tak, T.-M., and Tanioka, A., Proceedings of ICOM’90, 841, 1990. 25. Stern, S. A., Fang, S.-M., and Jobbins, R. M., J. Macromol. Sci.— Phys. B5(1), 41 (1971). 26. Stern, S. A., and Saxena, V., J. Membr. Sci. 7, 47 (1980).
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27. Stern, S. A., and Frisch, H. L., Annu. Rev. Mater. Sci. 11, 523 (1981). 28. Kamiya, Y., Mizoguchi, K., Naito, Y., and Hirose, T., J. Polym. Sci., Pt. B, Polym. Phys. 24, 535 (1986). 29. Kamiya, Y., Hirose, T., Mizoguchi, K., and Naito, Y., J. Polym. Sci., Pt. B, Polym. Phys. 24, 1525 (1986). 30. Kamiya, Y., Mizoguchi, K., Hirose, T., and Naito, Y., J. Polym. Sci., Pt. B, Polym. Phys. 27, 879 (1989). 31. Prigogine, I., and Defay, R., ‘‘Chemical Thermodynamics,’’ 4th ed., p. 71. Longmans, Green and Co., London, 1967. 32. Tanioka, A., Jojima, E., Miyasaka, K., and Ishikawa, K., J. Polym. Sci., Polym. Phys. Ed. 11, 1489 (1973). 33. Tanioka, A., Tazawa, T., Miyasaka, K., and Ishikawa, K., Biopolymers 13, 753 (1974).
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CS985554
Sorption Isotherms of Gases by Polymer Membranes in the Glassy State: An Explanation Based on the Nonequilibrium Thermodynamics Susanne Motamedian,* Wolfgang Pusch,* Akihiko Tanioka,* ,1,2 and Friedrich Becker† *Max-Planck-Institute fu¨r Biophysik, Kennedy Allee 72, D-60596 Frankfurt am Main 72, Germany; and †Institut fu¨r Physikalische und Theoretische Chemie der Universitaet Frankfurt P.O. Box 11 19 32, D-60439 Frankfurt am Main, Germany Received December 19, 1997; accepted March 26, 1998
Sorption isotherms of glassy polymers are concave to the pressure axis, and the absolute sorption levels are almost an order of magnitude higher than that of rubbery polymers on the relatively low-pressure side. There are several models to interpret this behavior, and the dual sorption model is the most widely accepted one among them. In the present work, sorption isotherms were first derived from permeation data with several gases for four polymer membranes which might be representative of the dual sorption model: two of them were in a glassy state (cellulose acetate and polyamide), the third one was a block copolymer which was composed of a glassy polymer and a rubbery polymer at the measuring temperature, and the fourth one was a Nafion membrane which was taken as a model membrane because of its channel structure with adsorption sites of charged groups supported by a rubbery polymer. These results did not necessarily support the dual sorption model. Subsequently, the validity of the underlying assumptions of two other sorption models for glassy polymers, that is, the matrix model and the deformation model, were examined and a new equation for the sorption isotherm with an ordering parameter was derived, which implied that the glassy polymer was in a nonequilibrium state and changed from the glassy to the rubbery state by absorbing the gas. q 1998 Academic Press Key Words: sorption isotherm of gas; glassy polymer; nonequilibrium thermodynamics; ordering parameter; dual sorption model; matrix model; deformation model.
1. INTRODUCTION
Transport and sorption of small molecules, i.e., gases and vapors, in glassy polymers have been of great interest scientifically and technologically (1–4). The main reason is that the sorption isotherms of glassy polymers are concave to the pressure axis and the absolute sorption levels are of almost an order of magnitude higher than those of rubbery polymers, 1
To whom correspondence should be addressed. Present address: Department of Organic and Polymeric Materials, Tokyo Institute of Technology, 2-12-1 Ookayama Meguro-ku, Tokyo 152, Japan.
though the rubbery polymers show a linear or convex relationship to the pressure axis (5, 6). In order to interpret this behavior, several models were developed which were based on different ideas. The most acceptable theoretical model is the dual sorption model, which is based on the assumption of two modes for sorption in the polymer (7). The polymer includes a dense region as one side, and the others are regions of the lower polymer density, that is, microvoids which are proposed to be formed in the polymer when it is cooled down below its glass transition temperature (8, 9). The dual sorption model is expressed by the following formulation (10): c Å cD / cH Å kD p /
cÅ
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[1]
where c Å total sorbed gas concentration, cD Å concentration of gas in a Henry’s law-type site, cH Å concentration in a Langmuir-type site, kD Å Henry’s law coefficient, C *H Å hole saturation parameter, b( ú0) Å hole affinity parameter, and p Å applied pressure. The first term of the equation follows Henry’s law and represents the sorption in the denser regions of the polymer, while the second term is of the Langmuir type and is the result of adsorption around the microvoids. The dual sorption model describes well the sorption phenomena of many glassy polymers. The validity of the underlying assumptions will be discussed in the following part of this paper. Contrary to the dual sorption model, the gas–polymer matrix model (11) assumes that only one population of sorbed gas molecules exists in the polymer at any given pressure. Based on the observation of changes in the physical properties, i.e., changes in the cooperative main-chain motions of polymers, changes in viscoelastic relation and Tg , in the presence of a sorbed gas, the authors propose that these changes arise from gas–polymer interactions. The model describes the sorption isotherm by a two-parameter equation:
2
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C *H bp , 1 / bp
(1 / 4aS0 p) 1 / 2 0 1 , 2a
[2]
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S Å S0 exp( 0ac),
[3]
where S Å solubility parameter, S0 Å solubility parameter in the zero-concentration limit, a Å conformation parameter, and c Å total sorbed gas concentration. Contrary to those two models which describe the sorption process by empirical equations, the deformation model by Lipscomb (12) is a thermodynamic model. The sorption of penetrants into a solid is treated as two discrete processes. Solid deformation in the first step is followed by mixing with the penetrant. The model is an extension of the Flory– Huggins (13) relationship of penetrant-induced swelling of rubbery materials, and the dependence of the gas pressure and its concentration in the polymer is given by pÅ
SD F
pr tr(sr ) £s exp t 3RT
GS S
D FS D D S D G
c£s exp 1 / c £s
/x
1 1 / c £s
2
/
1 0 cp £s 1 / c £s
B0£s c £s RT
[4]
where p Å applied pressure, pr Å reference pressure, c Å total sorbed gas concentration, cp Å polymer concentration, £s Å partial molar volume of gas, x Å Flory–Huggins interaction parameter, t Å gas external phase activity coefficient, and B0 Å bulk modulus. The difference between this relationship and the Flory–Huggins equation is given only by the stress relation terms, i.e., the pre-exponential factor involving the stress factor and the bulk modulus term in the exponent. The experiments described in the following should be the basis for a discussion of the validity of the underlying assumptions of the three models and the obtained knowledge should lead to improvement of the models.
methylbenzoate sulfonic acid tetrabutylammonium salt) and is in a glassy state at the measuring temperature (15). 2.2. Estimation of Sorbed Gas from Permeation Measurements (14)
The sorption isotherms were received from permeation experiments by the time lag method. This is not an exact procedure for determining the amount of sorbed gas. However, it has already been confirmed that we can examine the shape of the isotherm; that is, a concave curve, convex curve or straight line, depending on the gas–polymer system. The permeation equipment and the procedure were the same as already described. The membrane samples which had an effective area of about 10.5 cm2 were inserted between a lowpressure and a high-pressure compartment in a permeation cell made from stainless steel. The membrane samples were supported by a CONIDUR screen topped with an appropriate membrane filter. The permeation cell was a modification of a common hyperfiltration cell (16–19). The closed cell was placed into a water bath, and the low pressure compartment was evacuated to a pressure of about 0.05 mbar using a Leybold vacuum pump. After evacuation overnight, the feed pressure (p * ) of 1– 90 bar was adjusted in the high-pressure compartment, and the pressure increase in the low-pressure compartment was recorded as a function of time (t) employing MKS pressure gauges and measuring devices. The diffusion coefficients of the gases were obtained from time-lag and the permeabilities from the slope of the asymptote of the curve p * versus t. The solubility Sg of the gas in the membrane was obtained by the relationship Pg Å Sg Dg . The amount of the sorbed gas (Cg ) is calculated from Cg Å Sgp *. H2 , D2 , N2 , He, and CO2 were employed as test gases. Their purity was 99.99%. The permeation experiments were done at 25 or 357C at the pressures from 1 bar up to 90 bar. The measurement error was within 5%.
2. EXPERIMENTAL 3. RESULTS AND DISCUSSION
2.1. Materials
Two cellulose acetate membranes with different acetyl contents were examined; the acetyl content of the Bayer Cellit K-700 membrane was about 39.1%, and that of the Eastmann Kodak CA E 320 was about 32%. The membrane preparation procedure is described elsewhere (14). The second membrane was an acrylonitrile–styrene–butadiene block copolymer (ABSB) which was produced at Buna AG Merseburg, and the polymer was partly in the glassy and partly in the rubbery state at the measuring temperature (15). The Nafion membrane was a commercially available ion exchange membrane composed of perfluorinated carbon from Du Pont, and its thickness varied from 170 to 190 mm. The polyamide membrane obtained from CIBA-GEIGY AG was composed of a mixture of two diamines (4,4 *-diamino2,2 *-dimethyldiphenylmethane and 3,5-diamino-2,4,6-tri-
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Figures 1–4 exhibit the sorption isotherms which were calculated from the data of the permeation experiments. As can be seen in Fig. 1, the sorption isotherms of the Nafion membrane are linear and seem to follow Henry’s law (20). The same behavior is exhibited in the sorption isotherm of the polyamide membrane for Ar and N2 as shown in Fig. 2 and also for Ar and N2 in the ABSB membrane, whereas the sorption isotherm of CO2 appears linear only at a low pressure up to p õ 12 bar as shown in Fig. 3. At a higher pressure, a nonlinear increase with increasing pressure is found. This indicates a strong increase in the sorption capacity for CO2 . This behavior is observed often for CO2 and can be interpreted as CO2 acting as a softener for the membrane, which means that the dissolved gas changes the membrane properties irreversibly. Sorption isotherms of H2 and
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137
FIG. 1. Sorption isotherms of Ar ( s ), N2 ( , ), and He ( h ) gases for a Nafion 117 membrane at 357C.
FIG. 3. Sorption isotherms of CO2 ( , ), Ar ( s ), and N2 ( h ) gases for an ABSB membrane at 257C.
D2 by cellulose acetate membranes are shown in Figs. 4 and 5, respectively, which are concave relative to the pressure axis. The dual sorption model predicts that the sorption isotherms for glassy polymers are concave to the pressure axis, which means that only the sorption isotherms of cellulose acetate can be interpreted by this model. This represents well the experimental data for the cellulose acetate membrane, but the b parameters in Eq. [1] are negative values which are physically nonsensical as shown in Table 1.
The Nafion membrane was examined as a model membrane for a discussion on the validity of the underlying assumption of the dual sorption model. It is a copolymer with the following chemical repeating units: the structure consists of a (CF2 )n backbone with SO 03 groups placed inside of the channels (21). The former contributes to the sorption by Henry’s law, and the latter should provide an ideal site for adsorption. Therefore, the sorption isotherms should be concave to the pressure axis if one follows the dual sorption concept. Because the sorption
FIG. 2. Sorption isotherms of Ar ( s ) and N2 ( h ) gases for a polyamide membrane at 257C.
FIG. 4. Sorption isotherms of H2 gas for a cellulose acetate membrane at 257C and the application of Eq. [1].
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also requires a third state variable, a so-called ordering parameter ( j ) for full characterization of its state (24). This ordering parameter is defined by the affinity (A) of the actual state of the polymer membrane where A is equal to the chemical potential difference of the polymer in its actual state mp (T, p, j ) minus the chemical potential of an equilibrium state mp (T, p, j Å 0) of the polymer: A Å mp (T, p, j ) 0 mp (T, p, j Å 0).
[5]
The sorption isotherm model is based on the following observations and assumptions:
FIG. 5. Sorption isotherms of D2 gas for a cellulose acetate membrane at 257C and the application of Eq. [1].
isotherms are linear for all the examined gases, it can be concluded that the reason for the concave sorption isotherms is not a result of the two modes for sorption. This is a confirmation of the assumption of other authors (11, 12) who also called the existence of the two modes for sorption into question. It should be noticed that NMR and X-ray scattering measurements of polymer structure (22) could not provide conclusive evidence of the existence of two sorption sites either. The gas– polymer matrix model is based on the assumption of gas– polymer interactions which lead to a change in the physical properties of the polymer. This assumption could be provided by NMR measurements in which the influence of CO2 sorption on the cooperative main-chain motion of poly(vinyl chloride) was analyzed. Nevertheless, the matrix model predicts concave sorption isotherms for glassy polymers which means only that, like the dual sorption model, only the sorption isotherms for cellulose acetate can be interpreted by this model. The standard derivations for calculated curves are nearly the same as these calculated by the dual sorption equation as shown in Figs. 4 and 5. The most important thing is to notice that the gas– polymer matrix model expresses the sorption behavior by empirical parameters which do not provide physical insight into the sorption process. In the following, a new sorption isotherm is derived, which has a thermodynamic basis implying that the glassy polymer is in a thermodynamically nonequilibrium state. 4. SORPTION ISOTHERM WITH ORDERING PARAMETER BASED ON NONEQUILIBRIUM THERMODYNAMICS (15, 23)
The state of a glassy polymer is not determined only by the state variables, pressure (p) and temperature (T ), but
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(a) We cannot differentiate the state of the gas molecules in the polymer from each other; that is, there is only one population of sorbed gas molecules in the polymer at any given pressure (11). (b) With increasing gas concentration in the polymer, the physical properties change; that is, the cooperative mainchain motion of the polymer increases and the glass transition temperature changes to lower temperatures (5, 25–27). (c) By absorbing gas, the state of the polymer changes to a more disordered state, which means a change from the glassy state to the rubbery state. The change in the polymer state can be expressed by the change in the ordering parameter (28–30). For the transition from the glassy state to the rubbery state, the boundary conditions for the ordering parameter are 0°j°1
j Å 1, glassy polymer, j Å 0, rubbery polymer.
[6]
when the Gibbs–Duhem relationship for a glassy polymer– gas system is applied, the following relation is held (31): cp dmp (T, p, j ) / c dms Å 0,
[7]
where c and cp are the gas and polymer concentrations, reTABLE 1 Parameters of the Dual Sorption Model in the Cellulose Acetate Membranes (K-700 and E-320)–H2 and D2 Systems Calculated from Eq. [1]
Membranes
Gas
KD (cm3 (STP)/ cm3Mrbar)
E-320 K-700 E-320 K-700
H2 H2 D2 D2
0.0935 0.119 0.0898 0.107
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CH* (cm3 (STP)/ cm3M)
b (bar01)
30.9 38.9 31.4 27.6
00.00167 00.00186 00.00162 00.00181
GAS ISOTHERM SORPTION BY POLYMER MEMBRANES
139
spectively, and ms is the chemical potential of a gas in the polymer. Equation [7] indicates that the chemical potential of the gas is also a function of the ordering parameter, ms Å ms {T, p, j},
[8]
where j may vary from one membrane sample to another. ms is equilibrated to the chemical potential mg of the gas in the external phase to introduce sorption isotherm in the glassy polymer as a function of the ordering parameter. With T Å constant and the assumption that there is ideal behavior in the gas phase and in the dissolved gas phase, the chemical potential in both phases can be expressed in the following form: mg Å m0g / RT ln(p/p 0 ), ` s
0
ms Å m / RT ln(c/c ),
[9] [10]
where m0g and m`s are the standard potentials in the gas and dissolved gas phases, respectively, p 0 is the standard pressure in the gas phase, and c 0 is the standard gas concentration in the polymer. In the equilibrium state, mg Å ms .
[11]
From Eqs. [9] and [10], with Eq. [11] it follows that
S D
m0g 0 m`s cp 0 Å ln 0 RT c p
or c/c 0 Å (p/p 0 )exp[( m0g 0 m`s )/RT] Å S(p/p 0 ),
FIG. 6. Temperature dependence of the chemical potential of polymers.
p max is the maximum gas pressure where the phase conversion from glassy state to rubbery state is completed. At any temperature the chemical potential of the glassy polymer is higher than that of the rubbery polymer as seen in Fig. 6 because the glassy polymer is not in thermodynamic equilibrium. Therefore, it can be concluded with Eq. [7] that the chemical potential of the gas in a rubbery polymer is higher than the chemical potential of the gas in glassy polymers, which corresponds to the higher gas solubility in glassy polymers. The chemical standard potentials for j Å 1 and j Å 0 are assumed to be
[12]
m`s (1) å m`s,gl , m`s (0) å m`s,ru , Dm`s å m`s,ru 0 m`s,gl ¢ 0.
where S å exp[( m0g 0 m`s )/RT] (T Å constant).
[13]
Equation [12] is valid for glassy as well as for rubbery polymers. Because the ordering parameter j changes with increasing gas concentration in the polymer, it can be concluded that it is a function of gas pressure p according to the assumptions (a), (b), and (c). At T Å constant, the ordering j(0 ° j ° 1) can be defined as j(p Å p 0 ) Å (p max /p 0 0 p/p 0 )/(P max /p 0 0 1)
AID
If ms ( j ) is expanded with j in quadratic equation, the dependence of the chemical potential of the dissolved gas on the ordering parameter at T Å constant can be expressed by following equation with a constant as seen in Fig. 7: ms ( j ) Å m`s,ru 0 j[ Dm`s / a(1 0 j )],
[18]
where the slope is given by Ìms / Ìj Å 0 Dm`s 0 a(1 0 2j ),
[14]
with the boundary conditions
[17]
[19]
and at the boundary,
j(p Å p 0 ) Å 1, glassy polymer,
[15]
Ìms / Ìj ( jÅ0 ) Å 0 Dm`s 0 a,
[20a]
j(p Å p max ) Å 0, rubbery polymer.
[16]
Ìms / Ìj ( jÅ1 ) Å 0 Dm`s / a.
[20b]
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ms (c, j ) Å m`s,ru 0 jRT ln k 0 j[ Dm`s / a(1 0 j )] / RT ln(c/c 0ru ).
[25]
If the ordering parameter j is expressed by p/p 0 with Eq. [14], the explicit form of the function c/c 0 Å f ( p/p 0 ) is obtained by equilibrating Eqs. [9] and [25], with the maximum pressure p max as a parameter. Finally, the following equation is derived for p 0 Å 1 bar: c/c 0ru Å pk ( p
max0p ) / ( p max0 1 )
1 exp{[ m0g 0 m`s,ru / (p max 0 p)/(p max 0 1) 1 [ Dm`s / a(1 0 (p max 0 p)/(p max 0 1))]]/RT}.
[26]
FIG. 7. Dependence of ordering parameter on the chemical potential calculated by Eq. [18], where a Å 0.2 and a Å 0 with Dms` Å 0 and ` ms,ru Å 1.
For example, in the case of a ú 0 the slope of ms ( j ) becomes steeper as j r 0 and more gentle as j r 1 than the slope for the case of a Å 1. For the concentration dependence of the chemical potential of the dissolved gas ms with the assumption of ideal behavior and T Å constant, the following relationship for glassy polymers can be obtained: ms,gl (c, 1) Å m`s,gl / RT ln(c/c 0gl ),
[21]
ms,ru (c, 0) Å m`s,ru / RT ln(c/c 0ru ).
[22]
Equation [26] is applied to the experimental results for cellulose acetate–H2 and D2 gas systems in Figs. 4 and 5 with k Å 1.1 and a/RT Å 00.1, respectively, and nonlinear regressions are performed to determine the three parameters of p max /p 0 , Dm`s /RT, and ( mg0 0 m`s,ru )/RT. The calculated results are shown in Fig. 9 for H2 gas and in Fig. 10 for D2 gas, and the three parameters are listed in Table 2 with k and a/RT. It can be seen that the sorption behavior of H2 or D2 in a cellulose acetate membrane can be well described by the equation based on the ordering parameter. The linear sorption isotherms of the other membranes mean that the ordering parameters of these membranes do not change much during the experiment. In case of the Nafion membrane, the sorption in the channels is the reason why the ordering parameter of the membrane does not change. The state of the ABSB mem-
c 0gl and c 0ru are the standard concentrations in glassy and rubbery polymers, respectively, at p Å p 0 . The relationship between the two standard concentrations can be expressed as c 0gl Å kc 0ru with k ¢ 1.
[23]
The parameter k represents the higher gas solubility in glassy polymers. The dependence of the standard concentration c 0 as a function of the ordering parameter j can be described as c 0 ( j ) Å k jc 0ru
[24]
and is shown in Fig. 8. The chemical potential of the dissolved gas ms (c, j ) which is dependent on the concentration and ordering parameter given by the combination of Eq. [18] with Eq. [24]:
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FIG. 8. Change in standard concentration with ordering parameter calculated by Eq. [24], where k Å 2 and c 0ru Å 1.
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TABLE 2 Parameters of Eq. [24] in the System of a Cellulose Acetate Membrane–H2 and D2 Systems
FIG. 9. Sorption isotherms of H2 gas for a cellulose acetate membrane at 257C and the application of Eq. [26].
Gases
pmax/p0
Dms`/RT
` (m0g 0 ms,ru )/RT
a/RT
k
H2 D2
632 567
0.04 0.05
01.74 01.15
00.1 00.1
1.1 1.1
the glassy to the rubbery state by absorbing the gas. Those phenomena were already confirmed in the system of CO2 gas–polymer (28–30) and H2O vapor–hydrophilic polymer (32, 33) systems. The results in this study suggest that the sorption isotherm in a nonequilibrium state may also be explained by introducing the ordering parameter, and almost all kinds of gases have the possibility of changing the polymer state from glass to rubber. ACKNOWLEDGMENT
brane is already mainly disordered because the membrane is at measuring temperature already partially in the rubbery state composed of styrene–butadiene copolymer. The sorption isotherms of gas and vapor by glassy polymer membranes were concave to the pressure axis, and the absolute sorption levels were almost an order of magnitude higher than that of rubbery polymers on the low-pressure side. The high absolute sorption level at low pressure can be explained by the excess free energy in the glassy polymer compared with the rubbery polymer at the constant temperature and pressure. The curvature of the sorption isotherm can be explained from the reason why the polymer state changes from
FIG. 10. Sorption isotherms of D2 gas for a cellulose acetate membrane at 257C and the application of Eq. [26].
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This is a memorial article for Prof. Dr. Wolfgang Pusch and a part of the dissertation of Dr. Susanne Motamedian under his direction. Just before Dr. Motamedian finished her Ph.D. thesis, Prof. Pusch was killed in a horrible accident on 22 June 1992. This is a very sad memory and a great loss for membrane scientists all over the world.
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27. Stern, S. A., and Frisch, H. L., Annu. Rev. Mater. Sci. 11, 523 (1981). 28. Kamiya, Y., Mizoguchi, K., Naito, Y., and Hirose, T., J. Polym. Sci., Pt. B, Polym. Phys. 24, 535 (1986). 29. Kamiya, Y., Hirose, T., Mizoguchi, K., and Naito, Y., J. Polym. Sci., Pt. B, Polym. Phys. 24, 1525 (1986). 30. Kamiya, Y., Mizoguchi, K., Hirose, T., and Naito, Y., J. Polym. Sci., Pt. B, Polym. Phys. 27, 879 (1989). 31. Prigogine, I., and Defay, R., ‘‘Chemical Thermodynamics,’’ 4th ed., p. 71. Longmans, Green and Co., London, 1967. 32. Tanioka, A., Jojima, E., Miyasaka, K., and Ishikawa, K., J. Polym. Sci., Polym. Phys. Ed. 11, 1489 (1973). 33. Tanioka, A., Tazawa, T., Miyasaka, K., and Ishikawa, K., Biopolymers 13, 753 (1974).
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