Correlation and prediction of gas permeability in glassy polymer membrane materials via a modified free volume based group contribution method

Journal of Membrane Science 125 Ž1997. 23–39

Correlation and prediction of gas permeability in glassy polymer membrane materials via a modified free volume based group contribution method J.Y. Park, D.R. Paul Department of Chemical Engineering and Center for Polymer Research, The UniÕersity of Texas at Austin, Austin, TX 78712, USA Received 21 December 1995; accepted 25 January 1996

Abstract Over the past decade or more an extensive amount of data on the permeation of gases such as helium, hydrogen, oxygen, nitrogen, methane, and carbon dioxide in a wide array of glassy polymers has been published. Much of this work has been motivated by the search for materials with high permeability and high selectivity for potential use as gas separation membranes. This paper attempts to develop a method for correlating this data in a way that permits prediction of permselectivity behavior of other polymer structures. The method used involves an empirical modification of a free volume scheme that has been used in the past with some success. The previous method requires an experimental density of the polymer and an estimate of occupied volume from a group contribution method developed by Bondi. The present method actually predicts the density and uses a refined estimate of occupied volume specific to each gas. The parameters in the model were deduced from a database including over one hundred polymers. The new method significantly improves the accuracy of correlation and of prediction. Keywords: Diffusion; Gas separation; Glassy polymer membrane; Group contribution method

1. Introduction Over the past decade or more, an extensive body of experimental data on permeation of gases has been generated for a wide array of glassy polymers w1–34x. Much of this work has been done in these laboratories in an attempt to learn the principles that govern the relationship between gas permeability and polymer repeat unit structure because of an interest in developing better gas separation membranes. There is an important need to develop simple means to quantitatively correlate this information; such a predictive tool may be useful for guiding the development of new polymers.

A group contribution method seems ideally suited to achieve the stated objective, and several such approaches have been used for the correlation or prediction of gas permeability coefficients of polymers w35x. Salame has published extensively on the so-called ‘‘Permachor’’ approach for predicting the permeability of gases such as oxygen in barrier-type polymers w36–38x. The Permachor parameter for a given polymer is calculated from empirically derived factors for each chemical group in the polymer repeat unit. A similar methodology has been proposed more recently by Bicerano w39x that also considers the cohesive energy, packing, and rotational degrees of freedom of the polymer. Such schemes have been

0376-7388r97r$17.00 Copyright q 1997 Elsevier Science B.V. All rights reserved. PII S 0 3 7 6 - 7 3 8 8 Ž 9 6 . 0 0 0 6 1 - 0

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J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

devised largely to predict gas permeation behavior of barrier polymers, i.e. low permeability, and do not include the kind of chemical groups that have been found useful for achieving the high permeability and selectivity needed in membrane materials. The latter usually have aromatic, rigid backbones while barrier polymers often have more flexible carbon–carbon backbones with highly polar pendant units. These methods are not based on any fundamental theory and generally do not give accurate predictions to reliably learn much about selectivity of permeation to different gases; the prospect of extending these methods to achieve the current objective does not appear attractive. Another approach originally suggested by Lee w40x has been used with some success to correlate gas permeability coefficients in a variety of glassy polymers of the type that are of interest for gas s e p a r a tio n m e m b r a n e m a te r ia ls w 3 – 6,9,11,12,14,15,20,21,24,29 –34,41x. It employs the concept of free volume which does have some fundamental basis for correlation or prediction of transport properties w42–49x. However, the difficulty with free volume methods lies in the operational definition or quantitative evaluation of free volume. Lee proposed using the specific free volume defined as Ž V y Vo . where V is the polymer specific volume computed from a density measurement and Vo is the specific occupied volume which he calculated via a scheme developed by Bondi w46x that employs the van der Waals volume of the groups contained in the polymer repeat unit and a universal empirical factor. There are other group contribution methods that can be used to estimate the occupied volume w47–49x. When proper care is exercised in selecting high quality permeability and density data, reasonably good correlations can be developed for specific polymer families using such schemes w5–34x; however, the quality of the correlation achieved is still not fully adequate for the broader objective outlined above. At least a part of residual scatter in such correlations is believed to stem from the accuracy of published van der Waals volumes for chemical units of interest here and the type of universal packing parameter introduced by Bondi. This paper develops a more accurate scheme for predicting the permeability of six common gases in glassy polymers with rigid backbones by an empiri-

cal extension of the free volume concept. An extensive database of specific volume and permeability coefficients for over one hundred glassy polymers is used to evaluate the various empirical group contribution factors in the model using matrix or array methods ŽPC-MATLAB.. The database only includes materials whose properties are believed to have been determined with good accuracy. The resulting correlation allows more accurate prediction of gas permeability coefficients, compared to other available methods, for polymer structures that can be represented by the various groups defined here without the need for any density information.

2. Background The permeability coefficient, P, is comprised of both kinetic and thermodynamic factors which in principle depend on different aspects of the gasrpolymer pair, i.e. P s DS

Ž 1.

However, for a given gas the diffusion coefficient, D, varies from polymer to polymer a great deal more than does the solubility coefficient. While the diffusion coefficient may depend on many issues, the free volume of the polymer is among the most important. Solubility coefficients also depend on this parameter as well w14,32,33x. Thus, it should not be surprising that the permeability coefficient for a given gas in a series of polymers can be reasonably well correlated in terms of free volume. Indeed, extensive work from this laboratory and other groups has shown the utility of an expression of the following form P s A exp Ž yBrFFV.

Ž 2.

where A and B are constants for a particular gas. The fractional free volume, FFV, has been defined as FFV s Ž V y Vo . rV

Ž 3.

Here, V is the volume of the polymer Žmay be expressed as volume per unit mass or mole of repeat unit. which is obtained from experimental measurement of the polymer density at the temperature of interest Žtypically 308C.. The term Vo is the volume occupied by the polymer chains. A variety of ap-

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

proaches may be used to obtain this quantity, but all involve assumptions and various degrees of approximation. A common approach is Bondi’s group contribution method w46x where the occupied volume is computed from the van der Waals volumes, Ž V W . k , of the various groups in the polymer structure by K

Vo s 1.3

Ý Ž VW . k

Ž 4.

ks1

where K is the total number of groups into which the repeat unit structure of the polymer is divided. The factor of 1.3 was estimated by Bondi from the packing densities of molecular crystals at absolute zero and accounts for the fact that this volume is greater than the molecular volume; a major approximation is that a single universal value of 1.3 is assumed to apply for all groups and structures. Using this scheme, plots of log P versus 1rFFV have been constructed for various gases in a wide variety of polymer types. When the correlation is limited to a specific family of polymers, e.g. polysulfones w11–14x, polyarylates w31–34x, or polyesters w30x, a reasonably good correlation is obtained for each gas. However, when the correlation is broadened to include a wider range of polymer types, the correlation shows much more scatter. This is illustrated in Fig. 1 for CO 2 using the database described

25

later which includes 105 glassy polymers representing a wide variety of structural types. It goes without saying that care must be taken to ensure that accurate permeability and density data are used and that a consistent and appropriate set of parameters for each group are employed in the correlation. It should be pointed out that some recent papers have not taken this care with the result being extremely scattered plots. Even when extreme care is taken, there is scatter in these plots that go beyond any experimental errors in the permeability and density data. This may stem from at least two origins. Clearly, the notion of free volume may not capture all of the factors that affect the permeability. Also, there may be errors in the values of Ž V W . k available in the literature and, of course, the factor 1.3 in the Bondi’s method is only a crude approximation for the packing of polymer chains. What we seek here is a group contribution method that hopefully can transcend some of these issues and lead to more accurate predictions.

3. Development of the new correlation We start from the framework outlined above and extend it in some purely empirical ways. We retain the forms of Eqs. Ž2. and Ž3.. However, we do not assume that the effective FFV is the same for all gases in a given polymer or that Vo is given by Eq. Ž4.. Instead, we define for gas n

Ž FFV. n s V y Ž Vo . n rV

Ž 5.

K

Ž Vo . n s

Ý gn k Ž V W . k

Ž 6.

ks1

Fig. 1. CO 2 permeability coefficient as a function of fractional free volume determined by the Bondi method for polymers listed in Table 4. Line shown is least squares fit which defines A and B in Eq. Ž2.; the values shown in Table 2 were determined in this way for various gases using the amended database Žexcludes polymers 38, 89, and 91..

Here the gn k represents a set of empirical factors to be determined that depend on gas n and group k. We retain Ž V W . k or the values of the van der Waals volumes for group k as listed in the tables by van Krevelen w35x. However, this is only a convenience and does not really affect the outcome of this approach in any way. The notion of a different FFV in a given polymer for each gas makes physical sense because the size or structure of the gas molecule may influence the amount of space this molecule can access inside the polymer. Indeed, molecular simulations of the diffusion process also imply this w50x.

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J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

Table 1 Empirical factors obtained from the new group contribution method

a

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39 Table 1 Žcontinued.

a

NP , number of polymers contained in the database for b , g ŽCO 2 . and g ŽCH 4 .. V W , van der Waals volume Žcm3rmol..

27

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

28

A problem with the free volume scheme, as currently used, is that it requires knowledge of the polymer density in order to obtain the volume V used in Eqs. Ž3. or Ž5.; this information may not always be available and prevents this from being a truly predictive scheme. Various schemes for predicting density or V using group contribution methods have been proposed. For example, van Krevelen w35x suggests K

Vsb

Ý Ž VW . k

Ž 7.

ks1

with b s 1.55 at 258C for glassy polymers. We propose that a better prediction can be had if b is allowed to depend on the type of group k and this solves any problems with errors in values of Ž V W . k . Thus, we use here the following modification K

Vs

Ý bk Ž VW . k

Ž 8.

ks1

to develop a correlation for V for glassy polymers of the type of interest as gas separation membranes. The correlations presented here were initially developed using a database of 105 glassy polymers whose density Žat 308C. and permeability to various gases Žat 358C. are believed to be known very well. Many of these measurements were made at The University of Texas at Austin. Table 4 provides the chemical name and acronym, the density, the FFV calculated from Eq. Ž3., the permeability coefficients for six gases Žwhere available., and the reference where this information is reported. In the interest of brevity, chemical structures of the various polymers are not shown; this can be found in the references shown. In some cases, the repeat unit is isomeric

Fig. 2. Example of division of a polymer into groups.

Fig. 3. Comparison of polymer density calculated from Eq. Ž8. and b k from Table 1 with experimental density values for polymers in Table 4.

with another polymer, e.g. meta versus para phenylene ring connections, and these cases are so noted. After a preliminary analysis of the data for all 105 polymers in Table 4 it was concluded that the data for three of these materials Žthe polymers numbered 38, 89, and 91. fall well outside the correlation for all the remaining polymers. Thus, these three materials were dropped from the database for the final correlation presented in what follows. The data for these three polymers need to be examined again. We have defined 41 chemical groups that appear in these 105 polymers Žsee Table 1.. Fig. 2 shows an example of dividing a polymer Žtetramethyl bisphenol-A polysulfone. into chemical groups. The division of the repeat unit into groups is a somewhat arbitrary procedure, but experience can guide some of the choices. For example, in Fig. 2 one might define the aromatic rings and the methyl groups attached to them as separate groups. However, in most cases using larger groups, i.e. using the division shown in Fig. 2, leads to better results. Table 1 lists the number of polymers in the database that contain each group, the value of V W for each group as obtained from tables in van Krevelen w35x, and the values of b k and gn k obtained here. Least-squares type regression analyses using matrix and array operations available in the software package PCMATLAB ŽThe MathWorks, Inc.. were employed to calculate the ‘‘best’’ values of the empirical factors b k and gn k from the experimental density and gas

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

29

Table 3 Average percent errors ŽAPE. of polymer density and gas permeability predicted by group contribution methods. APE s n

w

Ý <Ž Pexpt y Pcalc .r Pexpt
Pexpt , experimental value of

is 1

gas permeability. Pcalc , calculated value of gas permeability Average percent error Ž%.

P ŽHe. P ŽH 2 . P ŽCO 2 . P ŽO 2 . P ŽN2 . P ŽCH 4 . r

Total number of polymers

Bondi’s method

New method a

47.4 39.5 77.1 73.4 85.8 121.5 4.55

21.6 26.1 37.3 34.8 38.7 39.1 0.701

b

79 63 102 101 101 102 102

a

Using the density calculated by the new group contribution method. b Van Krevelen’s method Ž b s1.55 for all chemical groups..

polymers in the database. The elements of this matrix denote how many of each of the 41 groups Žsee Table 1. are present in the repeat unit for each polymer. The van der Waals volumes of each of the 41 groups are also represented as a matrix defined as VW . Žb. The 1 = 41 matrix b Želements are the parameter b k for each group. for predicting polymer volume according to Eq. Ž8. was evaluated from the experimental values of V Žor density. via the following equation

Fig. 4. Histogram comparing experimental permeability values for various gases with those calculated from Eq. Ž2. with Vo determined by Ža. the Bondi method, Eq. Ž4., and Žb. the new method, Eq. Ž6..

b s Vr Ž V W .) Ng . permeability coefficients for the polymers in the database. The following procedure was used: Ža. A matrix Ng is defined to represent the molecular structure of the repeat units of each of the

Ž 9.

where V is a one dimensional matrix whose elements are the experimental values of the volume of each polymer expressed per mole of repeat unit; the .) denotes the array operation of element-by-ele-

Table 2 Values of A and B evaluated from the amended database using Bondi’s method to determine FFV. P s A expŽyBrFFV. ŽBarrer. CH 4 Number of polymers A B a b

At 10 atm, 358C. At 2 atm, 358C.

a

105 114 0.967

N2

b

104 112 0.914

O2

b

104 397 0.839

CO 2

a

105 1750 0.860

H2

b

65 1070 0.643

He

a

81 1800 0.701

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J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

ment multiplication ŽPC-MATLAB, The MathWorks, Inc... Žc. The constants A n and Bn in Eq. Ž2. for a

particular gas n were determined from plots of log P versus 1rFFV Žsee, for example, Fig. 1. using FFV calculated from Bondi’s group contribution method,

Fig. 5. Prediction of permeability coefficients using the new group contribution method Žsolid line.; solid points are experimental data from Table 4: Ža. helium, Žb. hydrogen, Žc. carbon dioxide, Žd. oxygen, Že. nitrogen, and Žf. methane.

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

31

4. Results and discussion

Fig. 6. Comparison of O 2 rN2 selectivity calculated by the new group contribution method with experimental values.

Eqs. Ž3. and Ž4., using the amended database in Table 4; Table 2 shows the values of A n and Bn obtained. Žd. Values of the new fractional free volumes for a particular polymer and gas n, in the sense defined by Eq. Ž5., can be obtained from these A n and Bn parameters using the experimental gas permeability coefficients by inverting Eq. Ž2.. Using PCMATLAB these operations can be conveniently executed and organized in matrix or array notation to give the one dimensional matrix FFVn . Že. The matrix of predicted volumes Žper mole of repeat unit. XV can be calculated as follows XV s b ) Ž V W .) Ng .

Table 1 lists the values of b k and gn k obtained by analysis of the database using the procedures outlined above. Fig. 3 shows a plot of the polymer density obtained from Eq. Ž8. using these b k values versus the experimental value for each of the glassy polymers in the amended database. The average percent error or APE Žas defined in Table 3. is 0.7% for this new method versus 4.55% using van Krevelen’s method, i.e. Eq. Ž7.. The predictive capability is considerably improved. As seen in Table 1, the values of b k vary considerably from group to group as opposed to a constant value Ž1.55 for glassy polymers. implied by the van Krevelen method. Permitting b k to vary from group to group simply recognizes that all groups may not be characterized by a single packing factor and possibly corrects for any errors in the tabulated van der Waals volumes of groups. The values of gn k listed in Table 1 vary a great deal from group to group in strong contrast to a fixed value of 1.3 as implied by Eq. Ž4.. The variation of gn k with gas type, however, is relatively minor; nevertheless, these small variations do result in some effect of gas type on the fractional free volume calculated for each polymer by this new method. As expected, this new group contribution method is able to predict the permeability coefficients of gases in

Ž 10 .

Žf. The matrix of occupied volumes Ž Vo . n for a particular gas n can be calculated from the polymer volume matrix V Žexperimental. or XV Žpredicted. as follows

Ž Vo . n s V y V.)FFVn

Ž 11 .

Žg. The matrix of empirical factors gn Ž1 = 41 matrix. for a particular gas n can be calculated as follows

gn s Ž Vo . nr Ž V W .) Ng .

Ž 12 .

In this step the Moore–Penrose pseudoinverse of Ng is used because this is a singular matrix.

Fig. 7. Comparison of CO 2 rCH 4 selectivity calculated by the new group contribution method with experimental values.

32

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

33

34

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

35

36

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

37

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J.Y. Park, D.R. Paul r Journal of Membrane Science 125 (1997) 23–39

polymers, where all the required values of A n and Bn ŽTable 2. and gn k ŽTable 1. are known, with much greater accuracy than the simple Bondi approach that assumes g s 1.3 for all polymer structural groups and all gases. This can be seen by comparing the average percent error, APE, between the calculated and experimental Žwhere available. permeability coefficients for each gas type for every polymer in the amended database, see Table 3. In general, the APE for the new method is about onehalf that for the Bondi method. Fig. 4 compares the accuracy of the predictive capability Žrelative to the experimental data. of the two methods in the form of histograms. The distribution is bell-shaped in both cases. For the Bondi method, the calculated permeability coefficient may be as much as one order of magnitude too low or high for some cases. For the new method, such extremes are eliminated and the predicted values lie much closer to the experimental values. Fig. 5 shows the experimental permeability data Žfrom Table 4. plotted versus the inverse fractional free volume, as defined by Eq. Ž5., for each gas. The solid lines were calculated from the values of A n and Bn in Table 2. The improvement in the correlation compared to using the fractional free volume obtained by the Bondi method is clear. Fig. 6 and Fig. 7 compares the ideal selectivity for the O 2rN2 and CO 2rCH 4 pairs computed by this new method with the experimental values. The agreement is quite good.

5. Conclusion The approach described here represents a considerable improvement in the correlation of gas permeability coefficients for glassy polymers compared to that provided by the analogous approach where fractional free volume is calculated by the method proposed by Bondi. This is not unexpected since new parameters are introduced and their values are deduced empirically from permeability data. However, the framework used for this scheme appears to be fundamentally sound at least to the extent that any simple mathematical construction can represent such a complex phenomenon as permeation. No doubt refinements in the analysis are possible and will lead

to improved correlations in the future. An especially noteworthy benefit of the method described here is the ability to predict the density of such glassy polymers rather accurately. Permeability coefficients calculated by the current scheme using experimental densities are, in general, slightly more accurate than those obtained using predicted densities.

Acknowledgements This research was supported by the Separations Research Program of the University of Texas at Austin and the Department of Energy, Basic Sciences Program. The authors express their appreciation to L.M. Robeson for helpful discussions.

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