Solubility in Glassy Polymers: Correlations through the Nonequilibrium

The representation of solubility isotherms of gases and vapors in glassy polymers has been analyzed through the use of a thermodynamic model, ...
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Ind. Eng. Chem. Res. 2001, 40, 3027-3037

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Solubility in Glassy Polymers: Correlations through the Nonequilibrium Lattice Fluid Model Marco Giacinti Baschetti, Ferruccio Doghieri,* and Giulio C. Sarti Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali, Universita` di Bologna, viale Risorgimento 2, 40136 Bologna, Italy

The representation of solubility isotherms of gases and vapors in glassy polymers has been analyzed through the use of a thermodynamic model, nonequilibrium lattice fluid, recently introduced. The possible use of the model as a predictive tool for gas solubility in glassy polymers has been discussed in previous works for the special cases in which the polymer density in pseudoequilibrium conditions is known from experimental measurements. In this paper, a different use of the model is presented in which the polymer density at different sorption conditions is not known from direct experimental measurements and is thus treated as a correlation parameter, following the typical linear dilation isotherm. The solubility isotherms thus obtained by the model have been compared with several experimental data, obtaining a rather satisfactory representation of the sorption isotherms, through use of two fitting parameters, at most. Introduction The description of the solubility of gases and vapors in glassy matrixes is relevant for many practical applications ranging from membrane separations to packaging and the development of barrier materials. The most widely used correlation for the sorption isotherms is offered by the dual-mode sorption model (DMSM),1,2 which is typically rather accurate and simple to use. According to its formulation, the model contains three parameters which are endowed with a physical meaning, albeit in a rather qualitative way. In any case these parameters are treated only as adjustable variables, and their physical meaning is used essentially to test the internal consistency; therefore, the minimum number of experimental information needed by the dual-mode model is given by three data points of the sorption isotherm. Several attempts have been made in recent years to develop different models which overcome the limits of DMSM. Some of them have empirical bases, such as the so-called continuous site,3 gas-polymer matrix,4 or concentration-temperature superposition models,5 lead to representations of the sorption isotherms which are often less accurate than those offered by DMSM, and require the use of a pair of adjustable parameters which do not have precise physical meaning. Some other models have been derived on the basis of a thermodynamic approach which includes the representation of stress development in the polymer matrix from gas sorption in the limit of small deformations6 or focuses on the interpretation of lattice compressibility through an equation of state7 or an activity coefficient model.8 A fair comparison of these models should consider the strength of the thermodynamic approach they rely on, which is different from one case to another, and it is not attempted in this work; however, it can be said in general terms that, even if the model parameters have specific physical meanings, acceptable representations * To whom correspondence should be addressed. Tel.: +39 0512093142. Fax: +39 051581200. E-mail: ferruccio.doghieri@ mail.ing.unibo.it.

of sorption isotherms are obtained only in the case in which a minimum of two adjustable parameters is considered which are determined from a best fit of the experimental data. A more detailed comparison of the correlations obtained from these models for gas solubility in glassy polymers can be found in a work by Barbari and Conforti.9 More recently, a new model has been developed,10-12 based on a thermodynamic analysis of nonequilibrium polymeric glasses; the present work is focused on the use of this model as a correlation tool for the sorption isotherms of swelling penetrants. Such a model considers a nonequilibrium lattice fluid (NELF) description for the glassy phase and proved rather successful in predicting solubility isotherms of gases and vapors in glassy matrixes. For the solubility prediction through the NELF model, necessary input information are equilibrium P-V-T properties of both polymeric and penetrant species and, more significantly, the polymer density at each sorption pressure. When the proper information is available, sorption isotherms can be reliably predicted over a rather broad pressure range, for different temperatures.10,11 On the contrary, when the polymer density is known only for the pure unpenetrated matrix, the corresponding solubility isotherms for swelling penetrants can be reliably calculated by the NELF model just in the low-pressure range,12 and good solubility predictions over a wide pressure range can be obtained only for the gases which induce negligible volume dilations. Our present aim is to test the use of the NELF model as a correlation tool for the most commonly encountered situations, in which the polymer dilation during sorption is not available from direct experimental measurements. The main motivation relies on the fact that in the presence of all relevant information the NELF model is indeed entirely predictive and is actually physically sound and robust; on the basis of its use, we are thus expecting to obtain good and robust correlations, which require a reduced number of experimental data points and are also more reliable to extrapolate solubility data outside the range of experimental investigation.

10.1021/ie000834q CCC: $20.00 © 2001 American Chemical Society Published on Web 05/03/2001

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Basics of the NELF Model Generalities. Several basic questions should be carefully considered in order to represent the thermodynamic properties of nonequilibrium phases. First of all, one is faced with the nontrivial problem of determining the set of state variables which affect properties in nonequilibrium conditions. Once the proper set of state variables is determined, one needs to derive thermodynamic relations which hold true among the thermodynamic properties of the material, in the resulting domain of nonequilibrium states. It is worthwhile to notice that the well-known tools of both classical and statistical thermodynamics are not applicable to that aim, because they invariably consider equilibrium conditions and cannot be simply applied to describe outof-equilibrium phases. The NELF model recently proposed by Doghieri and Sarti10-12 addresses the problem of representing thermodynamic properties for glassy polymeric mixtures, for the case of homogeneous, amorphous, and isotropic materials. The description of different nonequilibrium states of a glassy phase, at a given temperature, pressure, and composition, is accounted for by considering the polymer mass per unit volume, F2, as the only parameter measuring the departure from equilibrium conditions of the system. Therefore, the description of nonequilibrium glassy phases, containing a solvent in a polymeric matrix, can be obtained through the following set of variables: temperature T, pressure p, solvent mass fraction ω1, and polymer mass per unit mixture volume F2. Then, the corresponding specific Gibbs free energy G of such a system is given by an expression of the following form:

G ) G(T,p,ω1,F2)

(1)

For nonequilibrium thermodynamic analysis,11 the above assumption directly requires the introduction of a convenient equation for the time rate of change of the polymer mass per unit volume; the latter, indeed, can be obtained through a proper bulk rheology model. In the development of the NELF theory,11 a simple Voigt model is used, resulting in a kinetic equation for F2 of the following type:

dF2/dt ) f(T,p,ω1,F2)

(2)

On the basis of this equation, by applying the second law of thermodynamics, one can derive the general constraints for the expressions of the Helmholtz free energy and of relevant thermodynamic properties, as a function of state variables in the nonequilibrium domain. So, it is finally possible to calculate the out-ofequilibrium Helmholtz free energy for a given glassy system, starting from the simple knowledge of its expression in the domain of equilibrium states. In particular, when the bulk rheology model is expressed by eq 2, the polymer mass per unit volume becomes an internal state variable for the system under consideration; thus, from the general results for thermodynamic systems endowed with internal state variables, one finally obtains the relevant relationships among equilibrium and nonequilibrium thermodynamic properties.11 The main conclusions of the thermodynamic analysis outlined above can be summarized as follows:

(i) An extension of the specific Helmholtz free energy a to the entire domain of nonequilibrium states can be obtained directly from its equilibrium expression aEQ as a function of temperature, composition, and polymer mass per unit volume, according to the following relation:

a ) a(T,p,ω1,F2) ≡ aEQ(T,ω1,F2)

(3)

The corresponding expression for the Gibbs free energy in nonequilibrium conditions is then derived in a straightforward way as

G ) G(T,p,ω1,F2) ) aEQ(T,ω1,F2) + pv

(4)

where v is the specific volume of the system in actual nonequilibrium conditions, i.e.

v ≡ 1/F ) (1 - ω1)/F2

(5)

(ii) The chemical potential (on a mass basis) for the solvent species µ1 in the polymeric mixture can be obtained, for a general nonequilibrium state, from the following relationship:

µ1 )

( ) ∂(mG) ∂m1

(6)

T,p,m2,F2

where m1 and m2 are the solvent and polymer masses, respectively, and m is the total mass of the system. From eq 6, the nonequilibrium chemical potential of the solvent in the mixture turns out to be a function of the temperature, composition, and polymer density (or, equivalently, the specific volume); the latter quantity represents a nonequilibrium parameter and thus must be considered with particular attention, because its actual value is different from its “thermodynamic equilibrium” value, calculated for the same conditions of temperature, pressure, and composition. Remarkably, the results reported in eqs 3-6 allow one, in general, to obtain expressions of the nonequilibrium chemical potential for the solvent species in solvent-polymer mixtures, simply based on expressions for the Helmholtz free energy of the system under equilibrium conditions (aEQ). Phase Equilibria Involving a Glassy Phase. Once an expression for the solvent chemical potential in solvent-polymer mixtures µ1(s) is established for nonequilibrium conditions, it is possible to calculate the solvent solubility in the system, provided the polymer mass per unit volume F2 is known. For example, for a glassy phase of known polymer density F2, which is in contact with a pure solvent fluid phase at temperature T and pressure p, the solvent mass fraction ω1 can be calculated by solving the following phase equilibrium equation:

µ1(s)(T,p,ω1,F2) ) µ10(T,p)

(7)

where µ10 is the equilibrium chemical potential of a pure penetrant phase, at temperature T and pressure p. When the asymptotic pseudo-equilibrium conditions for the glassy phase are of interest, one has to consider that the polymer density eventually reaches a steady value, say F2∞, which must then be used in eq 7 in place of the generic value F2, to calculate the pseudo-equilibrium solubility. It must be emphasized that the pseudo-

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Figure 1. Asymptotic value for the pseudo-equilibrium polymer density as a function of pressure, reworked from refs 10 and 11.

equilibrium polymer density F2∞ is not an equilibrium quantity and depends on the temperature, pressure of the pure solvent phase, and the history of the polymeric sample. For the case of a simple isothermal sorption experiment, when the swelling induced by the penetrant is negligible, the asymptotic steady value F2∞ to be used in eq 7 practically coincides with the density F20 of the pure glassy polymer phase, at the beginning of the sorption experiment.12 In the general case, on the contrary, one should consider that in an isothermal sorption experiment, the polymer density F2 decreases as the penetrant pressure p in the fluid phase increases, from the original value F20 to the final asymptotic value F2∞; thus, in the phase equilibrium equation, the proper pseudo-equilibrium value F2∞ must be considered as function of p:

µ1(s)(T,p,ω1,F2∞(p)) ) µ10(T,p)

(8)

The analysis of several experimental measurements on polymer dilation16-19 indicates that in isothermal sorption experiments the pseudo-equilibrium polymer density F2∞ follows a linearly decreasing function of penetrant pressure p. Few examples of pseudo-equilibrium polymer density data, as measured by Koros and co-workers for the sorption of swelling agents in a family of polycarbonates, are shown in Figure 1 and follow a linear behavior. In view of the latter consideration, it is convenient to introduce the swelling coefficient k so that the pressure dependence of the pseudo-equilibrium polymer density F2∞ can be expressed as

F2∞(p) ) F20(1 - kp)

(9)

According to the cited observation, for the case of glassy polymers the coefficient k can be considered as independent of the pressure, over a relatively wide pressure range. It should be clear, however, that the swelling coefficient k, just like the pure polymer density F20, is a nonequilibrium property and not only depends on the temperature and polymer-penetrant pair but also is affected by the history of the specific polymer sample considered. It is known, in particular, that k has different values for sorption and for desorption processes.

By considering F2∞ as a linear function of the penetrant pressure, on the basis of eqs 8 and 9, one concludes that the NELF model allows for a simple representation of the solubility isotherm for any given penetrant-glassy polymer pairs, even for the case of swelling penetrants. The key parameters for this procedure are the nonequilibrium properties represented in terms of pure polymer density F20 and swelling coefficient k. The solubility values obtained from the solution of eq 8 are highly affected by the value of polymer mass per unit volume F2∞ and thus ultimately by those of F20 and k. A correct estimation of these two quantities is thus essential in order to obtain reliable representations of the sorption isotherms, as will be discussed in detail in the following sections.

Model Parameters and Their Evaluation Following the NELF model procedure sketched above, different expressions for the penetrant chemical potential in any nonequilibrium state are obtained, corresponding to the different equilibrium models which may be chosen to describe the equilibrium free energy aEQ. In the previous developments of the model10-12 and in the calculations which follow, the lattice fluid theory due to Sanchez and Lacombe13-15 (SL model) has been used to express the equilibrium free energy. SL Model and Its Characteristic Properties. After consideration of the expression for the equilibrium Helmholtz free energy of a polymer solvent binary mixture obtained by Sanchez and Lacombe, the following expression for the pseudo-equilibrium solvent chemical potential is derived,10,11 according to eqs 4 and 6:

( )

{[

M1P1* ω1F2 µ1(s) ) ln 1+ RT ω2F1* F1* RT1*

(

) ] (

)

F2 T1*P* T1*P* ω2F* ln 1 + -1 + T*P1* F2 ω2F* T*P1*

[ (

( )

F2 T1* P* F* 1+ - ω2 ω2F* T P1* F2*

2

)]}

∆P12* P1*

+ 1 (10)

Please notice that, in writing eq 10, some misprinting present in the homologous equation (48) of ref 11 has been corrected and the number of chain segments per molecule in the pure-component phase and in the mixture has been reported in terms of the corresponding characteristic temperature, pressure, and density. As is apparent, the following pure-component parameters appear in eq 10: characteristic density Fi*, corresponding to the close-packed density of pure component i; characteristic pressure Pi*, corresponding to the square of the familiar Hildebrand’s solubility parameter at 0 K; characteristic temperature Ti*, directly related to the molar bond energy i* for the monomer through the equation i* ) RTi*. Corresponding mixture characteristic properties appearing in the same equation are obtained from the pure-component parameters by means of proper mixing rules:

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P* ) F*

(

ω1 ω2 1 ) + F* F1* F2*

P1*ω1 P2*ω2 ω1ω2 + + F*∆P12* F1* F2 * F2*F1*

T* )

P* P2*ω2 P1*ω1 + F* T1*F1* F2*T2*

(

)

) (11)

The quantity ∆P12* in eq 11 is the only binary parameter involved, and it represents a measure of the interaction energy between segments of two unlike species in the mixture. Pure-component parameters can be obtained by fitting the pertinent P-V-T data with the results of the SL equilibrium equation of state. Experimental values of density as a function of temperature and pressure in the rubbery region for polymeric components, as well as the saturation density and pressure as a function of temperature for solvent components, are typically used to this aim. The binary interaction term ∆P12* for a given polymer-penetrant pair can be estimated from the comparison of the equilibrium model predictions with specific binary data for the mixture in the rubbery phase; typical data to that aim are, for instance, solubility or heat of mixing values. However, in the lack of specific binary data, a first-order approximation of the binary parameter can be obtained in terms of purecomponent lattice fluid properties, according to what was suggested by Sanchez and Lacombe themselves:15

∆P12* ) (xP1* - xP2*)2

(12)

The nonequilibrium chemical potential expression in eq 10 is the key result of the NELF model; when its use is made in the phase equilibrium condition, eq 8, assuming that eq 9 holds for the pseudo-equilibrium polymer density F2∞, it allows one to calculate the penetrant mass fraction ω1(s) in the polymeric phase. The penetrant solubility in the glassy phase is then given through an implicit relationship, which, for the sake of discussion, can be ultimately reduced to an explicit expression of the following type:

ω1(s) ) Ω(T,p;F1*,T1*,P1*,F2*,T2*,P2*,∆P12*;F20,k) (13) In eq 13, all of the parameters inherited from the SL model (Fi*, Pi*, Ti*, and ∆P12*) depend on the nature of the penetrant (subscript 1) or of the polymer (subscript 2), respectively, or are associated with the polymerpenetrant pair (∆P12*); they are all independent of temperature and pressure. On the contrary, the values of the pure glassy polymer density, F20, and of the swelling coefficient, k, which also appear on the righthand side of eq 13, are specific nonequilibrium properties of the polymeric matrix and depend on the temperature as well as on the thermal and mechanical prehistory of the polymer sample. It should be also noticed that, even if the pure components and binary characteristic properties of the SL model are intended to be independent of the temperature, pressure, and composition, different values for such properties have been reported in the literature, for

quite different temperature and pressure ranges.21 This fact suggests the use of, for the evaluation of such parameters, P-V-T data in a range of pressure and temperature close to the range of interest for the sorption process to be described. For a proper evaluation of the pure-component lattice fluid parameters, suitable for our present aim, it is largely recommended to use P-V-T data in a relatively large pressure range and at temperatures close to the glass transition. With respect to the binary parameter ∆P12*, it should be observed that nonequilibrium calculations presented in a previous work10 allow one to conclude that the predicted solubility of gases and vapors in glassy polymers usually is only slightly influenced by the precise value of the binary interaction parameter, at least for temperatures significantly lower than the glass transition temperature. For this reason, in the following calculation, this parameter has always been calculated through eq 12, thus avoiding the need of specific experimental data for the binary systems considered. Hence, through the procedure described above, all of the lattice fluid parameters for the pure components (polymer and penetrant) as well as for the binary mixture are estimated from pure P-V-T properties and are obtained independently of the solubility isotherm to be represented by the model. Nonequilibrium Parameters: Their Direct Measurements or Fitting to Data. Two further parameters remain to be discussed, to complete the analysis of the quantities affecting the solubility calculated by the NELF model; they are the dry polymer density F20 and the swelling coefficient k. Contrary to the characteristic SL model parameters, the densities of the dry polymer F20 and the swelling coefficient k cannot be obtained from equilibrium data because they both depend on the thermal and sorption history of the polymeric sample, and k depends also on the penetrant considered and not only on the polymeric species. As a consequence, they need to be evaluated through direct measurements on the sample of interest. Both F20 and k have a precise physical meaning, which allows for a straightforward measurement independent of the solubility data. The density of the pure glassy polymer, at the conditions of interest, can be measured through relatively simple and reliable procedures in most cases, without performing any sorption experiment, while the same is not true for the swelling coefficient. When these values are known, the NELF model can be used in a completely predictive way to calculate the sorption isotherm. On the other hand, when one or both of the numerical values of F20 and k are not experimentally available for the case of interest, the NELF model can be used in a simple correlation mode, using the unknown variable(s) as fitting parameter(s) to represent the experimental solubility isotherm. It is worth noting at this point that even the correlation use of the NELF model appears rather interesting; indeed, in the worst case in which both parameters F20 and k are not known and must be retrieved from a fitting procedure, virtually only two experimental data points of the sorption isotherm are required. More interestingly, when only one of them is not available from direct measurements, typically the swelling coefficient k, then one single data point of the sorption isotherm represents sufficient information for a correlation use of the NELF model. Explicit examples of the above procedures will be presented in the next section.

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One-Parameter Correlation. It is thus interesting to consider first of all the very frequent problem of representing the sorption isotherm for those cases in which the pure glassy polymer density F20, as well as the lattice fluid parameters for the two components, is known, while the swelling coefficient is unknown. In this respect, it is worthwhile to observe that, although F20 can vary only in a rather limited range, the solubility calculations may be rather sensitive to its value, which thus needs to be given with high accuracy. Then, the NELF model can be used as a correlation tool for the representation of the sorption isotherm; the swelling coefficient can be determined, for example, from one single solubility datum at a high pressure, say pH, by using the phase equilibrium condition at that pressure:

µ1(s)(T,p,ω1,F2∞(pH)) ) µ10(T,pH)

(14)

From eq 14 one derives the pseudo-equilibrium polymer density F2∞(pH) at penetrant pressure pH, based on the knowledge of the corresponding solute mass fraction ω1. The swelling coefficient is then obtained as follows:

k)

[

0

∞

]

F2 - F2 (pH) 1 pH F20

(15)

Once the swelling coefficient is known, the pseudoequilibrium polymer density F2∞(p) at any pressure p is then given by eq 9. Then the solubility at the same temperature and at any pressures p, different from pH, is calculated from the NELF model through the solution of the phase equilibrium condition given in eq 8. Several examples of the latter procedure, hereafter referred to as one-parameter correlation, are given in the following section, where the solubility isotherms for different polymer-penetrant systems are compared with the model correlation. Two-Parameter Correlation. Let us then consider the case in which both the pure polymer density and the swelling coefficient are unknown. In such cases a representation of the solubility isotherm can be obtained from the NELF model by using as few as two experimental solubility data to retrieve the nonequilibrium model parameters (two-parameter correlation). Two different procedures are of interest for that purpose. The first is indicated as two-point correlation and starts with the selection of two different penetrant pressure values, pL and pH, conveniently chosen in the low- and high-pressure range, respectively, for which the corresponding pseudo-equilibrium penetrant mass fractions ω1L and ω1H are known from experimental measurements. When phase equilibrium conditions at pressure pL and pH are written in the form of eq 8 and the pseudoequilibrium polymer density is assumed to linearly depend on pressure, as indicated in eq 9, then the following set of two equations is obtained:

[

µ1(s)(T,pL,ω1L,F20 - F20kpL) ) µ10(T,pL) µ1(s)(T,pH,ω1H,F20 - F20kpH) ) µ10(T,pH)

(16)

in which the two unknowns are F20 and k. The pure polymer density and swelling coefficient are then determined from the simultaneous solution of the above

equations. Then the entire sorption isotherm is calculated at all pressures p in a predictive way, through eqs 8 and 9. A different procedure to determine F20 and k from experimental solubility measurements, hereafter referred to as the S0 correlation, allows one to evaluate separately the two nonequilibrium parameters in two subsequent steps. The procedure is based on the recognition that the pure polymer density F20 is the only nonequilibrium parameter affecting the solubility coefficient in the low-pressure limit, S0, defined as follows: def

S0 ) lim pf0

c1 p

(17)

where c1 is a measure of the solute concentration in the polymeric mixture. In particular, when the latter is expressed as the standard penetrant volume per unit volume of dry polymer, the following relation is obtained for S0 from the NELF model:

S0 )

(

{

[ ) ] ( ) (

Tst M1P1* exp 1+ Tpst F1* RT1*

)

F20 T1*P2* F2* T1*P2* - 1 0 ln 1 -1 + + T2*P1* F2* T2*P1* F2 F20T1* (P * + P2* - ∆P12*) F2*P1*T 1

}

(18)

where Tst and pst are the standard temperature and pressure, respectively. The above relation allows for the direct calculation of the dry polymer density, from the measured value of the infinite dilution solubility coefficient S0, provided the pure-component SL characteristic values for both polymer and penetrant are known, together with the binary parameter ∆P12*. Consequently, the remaining nonequilibrium parameter k of the model can be estimated from the solubility at one high penetrant pressure pH, just as described above for the one-parameter correlation. Once the pure polymer density and swelling coefficient are determined from either the two-point or S0 correlation, the solubility at any arbitrary pressure p is obtained from the NELF model through a pure predictive procedure. To test the model robustness, in the following section the analysis is completed by applying the two-parameter correlation presented above, also to the solubility data already examined through the one-parameter correlation. In all cases, the resulting solubility isotherms are compared with the experimental data, as well as the values for the pure polymer densities and swelling coefficients obtained from the fitting procedure. Fitting Results In this work we focus on the NELF model as a correlation tool, to use in the rather common conditions in which a limited amount of solubility data are available and the dilation of the polymer matrix, induced by the penetrant sorption, is unknown. To obtain a more complete comparison and to ascertain the actual agreement between the retrieved values of the fitting parameters (k and possibly F20) and their

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Table 1. Lattice Fluid Parameters of the Relevant Species lattice fluid parameters component

T* (K)

P* (MPa)

F* (g/cm3)

ref

PMMA PS PC HFPC TMPC CO2 C2H4

695 750 755 716.5 761.6 300 295

560 360 534 446 446.4 630 345

1.270 1.099 1.275 1.6184 1.1739 1.515 0.68

11 11 10 17 17 10 11

physical meaning (swelling coefficient and pure polymer density), the correlation procedures will be applied to solubility data sets for which experimental dilation data are also available; the fitting parameters obtained from the correlation procedures, based on limited solubility data, will then be compared with the corresponding values resulting from direct dilation measurements. To that aim, of particular interest are data by Koros and co-workers17,18 presenting the solubility of different gases in three polycarbonates, the bisphenol A polycarbonate (PC), tetramethylpolycarbonate (TMPC), and hexafluoropolycarbonate (HFPC), insofar as they accurately measure, besides the solubility isotherms, also the volume of pure polymers and of penetrant-polymer mixtures. In particular, the sorption data of ethylene in PC and of carbon dioxide in all three polymers, at a temperature of 35 °C, will be examined hereafter. Those data allow for an accurate evaluation of the dry polymer density and of the swelling factor for the polymers used in sorption experiments and will be considered first in analyzing the reliability of the correlation procedures based on the NELF model. Also the solubility of carbon dioxide in poly(methyl methacrylate) (PMMA) at 33 °C and in polystyrene (PS) at 35 °C, obtained by Wissinger and Paulaitis,16 will be considered to test the NELF model as a correlation tool. In such cases, dilation data are available from the original work, but the corresponding values for the pure polymer density are not reported; therefore, the experimental values for the glassy density, F20, of pure PS and PMMA were substituted with the typical values available in the technical literature.22 For all of the systems considered, the pure-component characteristic SL parameters were already available from the literature and their values are reported in Table 1. One-Parameter Correlation. We refer here to the procedure discussed in the previous section in which one solubility datum at high pressure pH is chosen to retrieve the value of the swelling coefficient k, while the pure polymer density F20 is known. In determining the more convenient value of pH, one must consider that more reliable calculations are obtained when higher pressure values are used, corresponding to higher swelling, provided the system still remains a glass and did not turn into a rubbery phase. In all of the cases examined in the following, pH has been chosen as the highest pressure value in the experimental data set, for which reliable solubility data in the glassy phase were available. In Figure 2a experimental solubility isotherms for swelling penetrants such as C2H4 and CO2 in PC are compared with those calculated through the NELF model, according to the one-parameter correlation. The value of the glassy polymer density used for the calculation is reported in Table 2 and was measured by Koros

Figure 2. (a) Comparison with experimental data from refs 10 and 11 of the solubility isotherms obtained with the one-parameter correlation for C2H4 and CO2 in PC at 35 °C. (b) Comparison with experimental data from ref 17 of the solubility isotherms obtained with the one-parameter correlation for CO2 in TMPC and HFPC at 35 °C. (c) Comparison with experimental data from ref 16 of the solubility isotherms obtained with the one-parameter correlation for CO2 in PS at 35 °C and PMMA at 33 °C.

and co-workers.17,18 A very good agreement can be observed for the solubility at all pressures, for the cases both of ethylene and of carbon dioxide; clearly, in the model correlations the solubility exactly matches the experimental value at the highest pressure point, i.e., at pH, because of the procedure used.

Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3033 Table 2. One-Parameter Correlation: Comparison with Experimental Values of the Swelling Coefficients Retrieved, Based on the Reported Pure Polymer Density Values, and Average Absolute Error in the Solubility Isotherm systems inspected PC-CO2 PC-C2H4 TMPC-CO2 HFPC-CO2 PMMA-CO2 PS-CO2

k (MPa-1) exptl F20 (kg/L) correlation exptl 1.20018 1.20017 1.08317 1.47817 1.18120 1.0420

0.0118 0.0109 0.0230 0.0215 0.0218 0.0097

0.011418 0.012017 0.002917 0.018017 0.024316 0.012116

average error on the solubility (%) 2.27 4.2 11.57 7.32 5.73 4.48

The swelling coefficient values, retrieved for the experimental solubility data, are compared in Table 2 with the swelling coefficients obtained from direct measurement in refs 17 and 18 for C2H4 and CO2 in PC, respectively. Remarkably, the calculated values of k differ from the experimental data by less than 10%, in complete accord with the actual physical meaning of the variable used as a correlation parameter. The experimental solubility isotherms for CO2 in TMPC and HFPC are compared with those obtained from the one-parameter correlation in Figure 2b. The experimental values of glassy polymer density used in the calculation for TMPC and HFPC are reported again in Table 2. Solubility results from the model are fairly close to the experimental data for the case of HFPC, while slightly higher differences appear for TMPC. The same is true for the swelling dilations resulting from the correlation procedure when they are compared with the experimental values (see Table 2). The underestimation of the solubility of CO2 in TMPC at low pressure, calculated on the basis of the measured polymer density, and the relevant overestimation of the swelling coefficient obtained from high-pressure solubility data are two different manifestations of the same modest accuracy of the NELF model in this case. Finally, in Figure 2c the one-parameter correlation is compared with experimental solubility values for CO2 in PS and PMMA as reported in ref 16. The pure polymer density values used in these calculations were estimated from data reported in the literature22 and are indicated in Table 2. Quite good comparisons with experimental values result both for the solubility isotherms in Figure 2c and for the swelling coefficients obtained from the correlation procedure (see Table 2). Two-Parameter Correlation. The same solubility data sets considered above for the one-parameter correlation have been discussed in terms of the two-point correlation and the S0 correlation introduced in the previous section, and the results are briefly reported in the following. In all of the cases examined, the lowpressure pL and high-pressure pH values have been selected as the lowest and highest pressures, respectively, in the data set considered. In parts a and b of Figure 3, the solubility isotherms of C2H4 in PC and CO2 in PC, TMPC and HFPC, as measured by Koros and co-workers,17,18 are compared with the corresponding curves obtained from the NELF model, using the two-parameter correlations. In Figure 3c the same is done with solubility isotherms of CO2 in PS and PMMA, as determined by Wissinger and Paulaitis.16 The resulting values of the pure polymer density, F20, and swelling coefficient, k, for all of the cases above are finally compared with the experimental values, in Table 3 for the two-point correlation procedure and in Table 4 for the S0 correlation procedure.

Figure 3. (a) Comparison with experimental data from refs 10 and 11 of the solubility isotherms obtained with the two-parameter correlation for C2H4 and CO2 in PC at 35 °C (s, two-point correlation; ‚‚‚, S0 correlation). (b) Comparison with experimental data from ref 17 of the solubility isotherms obtained with the twoparameter correlation for CO2 in TMPC and HFPC at 35 °C (s, two-point correlation; ‚‚‚, S0 correlation). (c) Comparison with experimental data from ref 16 of the solubility isotherms obtained with the two-parameter correlation for CO2 in PS at 35 °C and PMMA at 33 °C (s, two-point correlation; ‚‚‚, S0 correlation).

From the comparisons reported in Figure 3a-c, it is self-evident that a very good representation of the solubility isotherm is obtained in all cases from the twopoint correlation and also for the sorption isotherm of CO2 in TMPC, which was not so satisfactorily described by the one-parameter correlation mode.

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Table 3. Two-Parameter Correlation: Comparison with Experimental Values of the Parameters Retrieved with the Two-Point Correlation Procedure, and Average Absolute Error in the Solubility Isotherm systems inspected

correlation

exptl

correlation

exptl

average error on the solubility (%)

PC-CO2 PC-C2H4 TMPC-CO2 HFPC-CO2 PMMA-CO2 PS-CO2

1.195 1.195 1.054 1.457 1.185 1.041

1.20018 1.20017 1.08317 1.47817 1.18120 1.0420

0.0112 0.0099 0.0119 0.0107 0.0221 0.0099

0.011418 0.012017 0.002917 0.018017 0.024316 0.012116

1.62 2.19 4.2 1.11 5.47 3.52

F20 (kg/L)

k (MPa-1)

Table 4. Two-Parameter Correlation: Comparison with Experimental Values of the Parameters Retrieved with the S0 Correlation Procedure, and Average Absolute Error in the Solubility Isotherm systems inspected

correlation

exptl

correlation

exptl

average error on the solubility (%)

PC-CO2 PC-C2H4 TMPC-CO2 HFPC-CO2 PMMA-CO2 PS-CO2

1.206 1.207 1.075 1.485 1.198 1.047

1.20018 1.20017 1.08317 1.47817 1.18120 1.0420

0.0126 0.0122 0.0199 0.0253 0.0231 0.0108

0.011418 0.012017 0.002917 0.018017 0.024316 0.012116

6.94 10.24 7.22 9.81 10.74 4.65

F20 (kg/L)

k (MPa-1)

A certain underestimation of solubility data is often obtained with the S0 correlation. When the fitting parameters F20 and k obtained from the two-point correlation procedure are compared with the experimental values, one finds an excellent agreement with the experimental data for both quantities, with the exception of only the CO2-TMPC system (see Table 3). In the latter case the correlation of the sorption isotherm is obtained by using a polymer glassy density appreciably lower and a swelling coefficient definitely higher than those experimentally measured in ref 18. Typically, higher pure polymer densities and swelling coefficients are obtained from the S0 correlation with respect to the two-point correlation in the cases examined. Discussion Let us consider first the one-parameter NELF model correlation, which is suitable for the very common situation in which the pure glassy polymer density F20 has been measured, while the polymer dilation, and thus the swelling coefficient k, is unknown. The procedure offers, in general, a very good representation of the sorption isotherm, also for highly swelling penetrants such as CO2; in several cases, such as the sorption of CO2 in PC, PS, and PMMA, the correlation method gives excellent results with average absolute deviations smaller than 6% in a wide pressure range, exceeding also 60 bar. In such systems, the volume dilation isotherm followed during sorption is very well represented by a straight line also by the NELF model. The corresponding swelling coefficient k, retrieved by the model, has actually the same value, independently of the particular point of the sorption isotherm chosen for the correlation procedure, apart from the data points at low pressure (below 5 bar), for which the dilation is still small and reasonably leads to a higher error in k. The typical behavior is clearly shown in Figure 4a. For CO2 sorption in HFPC and in TMPC, the one-parameter correlation is still rather satisfactory, with an average absolute deviation smaller than 11.6%. For these poly-

Figure 4. (a) Swelling coefficient values retrieved for C2H4 in PC by the one-parameter correlation, as a function of the pressure of the solubility data point considered. (b) Swelling coefficient values retrieved for CO2 in HFPC by the one-parameter correlation, as a function of the pressure of the solubility data point considered.

mers, the pure-component characteristic parameters used were taken from the literature and no direct comparison with the P-V-T equilibrium data was available. In such cases the lower precision is also reflected in a wider spread for the swelling coefficient k obtained when different solubility data points are used as input for the correlation (Figure 4b). We can say, in general, that the one-parameter correlation procedure is rather satisfactory. For its reliable application over a wide range of pressures, one needs to know, beyond the pure polymer characteristic parameters, the density of the pure glassy polymer, either from direct measurements or from suitable literature information, as it was done here for the cases of PS and PMMA; in addition, one needs to rely on the knowledge of the solubility value at one pressure, suitably chosen within the high-pressure domain. When the pure glassy polymer density is not reliably available, the only possibility left is the two-parameter correlation procedure, in either of the versions presented in the previous section. The S0 correlation mode seems to some extent simpler to apply, because the two fitting parameters are calculated separately from one another. However, as is clearly apparent from Figure 3 and Tables 3 and 4, this procedure leads systematically to a larger deviation from experimental data than the twopoint correlation mode. In particular, the S0 correlation offers always an underestimation of the sorption iso-

Ind. Eng. Chem. Res., Vol. 40, No. 14, 2001 3035 Table 5. Sensitivity Factors on S0 systems inspected

sensitivity factor ψ0 of the pure polymer density on S0

sensitivity factor φ0 of the binary mixing parameter on S0

PC-CO2 PC-C2H4 TMPC-CO2 HFPC-CO2 PMMA-CO2 PS-CO2

-36.9 -47.7 -22.7 -20.6 -33.2 -31.2

-0.042 -0.312 -0.165 -0.164 -0.022 -0.403

therm experimentally observed. In all of the cases inspected, the two-point correlation represents very well the sorption isotherm, with an absolute average deviation smaller than 5.5%. The solidity and robustness of the procedure is in all cases appreciated by comparing the values retrieved from the correlation, for the pure polymer density F20 and for the swelling coefficient k, with the corresponding experimental data (Table 3). The values obtained for F20 are indeed almost coincident with the measured data of PC, for both ethylene and carbon dioxide, of PMMA and PS; the largest deviation shown is about 3% for the density of pure TMPC. The comparison between experimental and retrieved values is much more stringent for the swelling coefficient, which is the slope in the volume isotherm; indeed, errors in the slopes of a function are, in general, much larger than errors in the values of the function itself. Nonetheless, also for this quantity the comparison is rather satisfactory for PC, PS, and PMMA, while larger deviations are encountered for HFPC and TMPC. The behaviors discussed above embody special evidences of the particular relevance of the polymer density F2 in affecting the penetrant solubility calculated by the NELF model. The polymer density is the only physical quantity representing the departure from equilibrium of the glassy phase, and because the solubility in the glass is typically much higher than that at true thermodynamic equilibrium at the same temperature and pressure, its role is really crucial at all pressures. It was also proved that the differences in the polymer density experienced between sorption and desorption cycles are responsible for the observed solubility hysteresis.10,11 A clear indication of the effect of the polymer density on the solubility isotherm is obtained by considering how the solubility coefficient is sensitive to variations of the pure glassy polymer density F20. To that aim, the sensitivity factor ψ0 defined as def

ψ0 )

( ) ∂ ln S0

(19)

∂ ln F20

is calculated from eq 18. One has

ψ0 )

{ (

) (

)

M1P1* F2* T1*P2* F20 - 1 ln 1 - 0 F1* RT1* F2 * F2 T2*P1*

[ (

F20/F2* 0

1 - F2 /F2*

1+

)]

F2* T1*P2* -1 F 0 T2*P1* 2

+

F20T1* (P * + P2* - ∆P12*) F2*P1*T 1

}

(20)

The sensitivity factors for the systems inspected are tabulated in Table 5. For the system CO2-TMPC, ψ0 is -22.7; in other words, a deviation on the polymer

density of as low as 1% propagates on S0 in an error of about 23%. The above system does not have the highest sensitivity on the pure polymer density F20, as is clear from Table 5, but it is explicitly mentioned because it shows the largest deviations between isotherms evaluated with different procedures; it also offers a justification of the fact that for the same system large deviations were observed between the density retrieved through the model and the experimental value. Remarkably, the deviations in density are not particularly large, lower than 3%, but enough to obtain sensible differences in the solubility values. The same arguments indicate why the systems which offer the best agreement between experimental and estimated values of F20, in the correlations applied in the previous section, also show the best fitting of experimental data and a good stability of results in changing the procedure used. Similar comments can be made, of course, for the case of the swelling coefficient. The agreement between the results retrieved from the fitting procedure and the experimental data is generally good, again with the exception of only the CO2-TMPC system. For this penetrant-polymer pair, indeed, the model leads to large deviations in volume swelling, because the calculated k values are very different from those obtained from experimental dilation data. The model shows, however, an internal consistency, because the calculated values are similar to each other. In the remaining systems, the NELF model proves to be able to describe satisfactorily different thermodynamic properties of nonequilibrium glassy systems. The above comments can also clarify why the S0 correlation mode systematically leads to an underestimation of the solubility isotherm, with respect to both the actual data and the two-point correlation mode. In actual practice, S0 is evaluated from the solubility isotherm by taking the slope of the line connecting the origin to the point at the lowest available pressure. In all of the cases inspected, in which the isotherm is concave to the pressure axis, that procedure leads to an underestimation of the true slope at zero pressure, thus the resulting S0 value is typically underestimated with respect to its real value. Correspondingly, the F20 value calculated from eq 18 is larger than its true value; the pure polymer is thus estimated to be denser than it really is and, consequently, the resulting isotherm is underestimated. Similar problems, of course, are not encountered with the two-point correlation mode. Use of the infinite dilution solubility coefficient to estimate the pure polymer density may be interesting when several low-pressure sorption data are available, so that the limit value of the solubility coefficient at vanishing pressure can be reliably estimated or when a negligible swelling effect is produced by the penetrant. In such cases, the S0 correlation procedure allows one to estimate the pure polymer density F20 from homogeneous data, thus minimizing the error. If the available data do not allow for a reliable extrapolation of the solubility coefficient at zero pressure, the two-point correlation procedure should be preferred. In closure, we remind that in all of the calculations reported in the previous sections, the binary parameter ∆P12* has been calculated through the first-order approximation, eq 12, consistently with the assumption that ∆P12* has only a minor effect on the solubility obtained by the NELF model, at least in comparison

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with the effect of the polymer density. That is indeed a reasonable assumption when the temperature is far from the glass transition, and thus the so-called excess free volume frozen in the polymeric mixture is relatively high. The effect of the polymer density on the pseudoequilibrium solubility calculation is usually dominant with respect to the energetic parameter when F20 is significantly different from the corresponding equilibrium value. The effect of the binary mixing parameter on the zero pressure solubility coefficient can be obtained through its sensitivity factor calculated as

(

)

F20T1* ∂ ln S0 φ0 ) )∂ ln ∆P12* F2*P1*T def

(21)

where the last equality is obtained from eq 18. The corresponding values for the systems considered are reported in Table 5; their absolute values are between 0.04 and 0.40, so that an error in ∆P12* as large as 100% would result in a much smaller error on S0, that is, 4% for the case of CO2 in PC or 17% in the case of CO2 in TMPC. However, the effect of the binary interaction parameter on predicted sorption isotherms at higher temperatures deserves some attention which is now left to a future work. In any case, the NELF model seems to be an excellent tool for the regression of solubility data, because it allows for a good description of the sorption isotherm using no more than two experimental data points, and quite often even one single data point only. The minimum information required is thus significantly reduced with respect to the dual-mode model. Besides, the parameters have a clear physical meaning so that the model allows also a more reliable extrapolation of the data. Finally, it is worthwhile to notice that from the fitting procedure previously described, the dilation coefficient associated with the sorption of swelling agents in glassy polymer is correctly predicted, so that the NELF analysis of solubility data also offers an estimation method for the swelling properties in penetrant-glassy polymer systems.

to a series of polymer penetrant pairs such as CO2 in PC, PS, HFPC, and TMPC at 35 °C and in PMMA at 33 °C, as well as C2H4 in PC at 35 °C. The results obtained show that the agreement between the experimental data and the NELF regression is always satisfactory if not very good and documents the effectiveness of the procedure proposed, and of the NELF model, as a correlation tool. In the second case, to obtain the complete representation of the sorption isotherm through the NELF model, two experimental solubility data need to be used to retrieve the two nonequilibrium parameters. To that aim, two different procedures have been discussed. The first (two-point correlation) simply consists of solving for the two unknown parameters a set of two algebraic equations, representing the phase pseudo-equilibrium for two solubility data points. The second (S0 correlation) is based on a preliminary estimation of the infinite dilution solubility coefficient from an experimental datum at a low pressure, and from it the unpenetrated polymer density is calculated; the swelling coefficient is then obtained from a second solubility data point at higher pressure. Also these procedures have been applied to the system above considered, resulting in a satisfactory representation of experimental sorption isotherms in all cases. Significantly, both the values of pure polymer density F20 and swelling coefficient k obtained from the regressions were in good agreement with those obtained through direct experimental measurements. The general validity of the model as a correlation tool for the case in which dilation data in sorption condition are not available has thus been demonstrated. On the other side, it has also been shown that NELF analysis of experimental sorption isotherms of swelling agents in glassy polymers allows one to reliably estimate the volume dilation induced by the sorption process. Acknowledgment This work was financially supported by the Italian Ministry for University and Scientific and Technological Research (ex 40%).

Conclusions

Literature Cited

The use of the NELF model as a correlation tool to describe sorption isotherms of gases and vapors in glassy polymers has been analyzed. This application of the NELF model relies on the observation that for most penetrant-glassy polymer systems the polymer mass per unit volume is a linear function of the penetrant pressure. In that case, (i) if the initial density F20 of the pure unpenetrated glassy polymer is known, all of the necessary information about the nonequilibrium dilation behavior of the system is embodied in one parameter, here identified in the swelling coefficient k, and (ii) if the initial density F20 of the pure unpenetrated glassy polymer is not specifically known either from direct experimental data or from the literature, the representation of the relevant nonequilibrium information requires the determination of two parameters, F20 and k. In the first case, the unknown model parameter k can be determined if at least one single solubility data point is available: thus, given a single solubility data at high pressure, the entire sorption isotherm can be reliably represented through the model, even for swelling penetrants. This procedure has been applied in this work

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Received for review September 22, 2000 Revised manuscript received December 18, 2000 Accepted December 19, 2000 IE000834Q