Correlations between Penetrant Properties and Infinite Dilution Gas

Jul 7, 2007 - Maria Grazia De Angelis , Riccardo Gaddoni , and Giulio C. Sarti ... Matteo Minelli , Karel Friess , Ondřej Vopička , Maria Grazia De ...
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Ind. Eng. Chem. Res. 2007, 46, 7645-7656

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Correlations between Penetrant Properties and Infinite Dilution Gas Solubility in Glassy Polymers: NELF Model Derivation Maria Grazia De Angelis, Giulio C. Sarti,* and Ferruccio Doghieri Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali (DICMA), UniVersita` di Bologna, Viale Risorgimento 2, 40136 Bologna, Italy

An analysis of the empirical correlations that relate the solubility coefficient of gases at infinite dilution in glassy polymers to penetrant properties has been performed, by considering gas sorption as a succession of three steps: condensation of the penetrant, dissolution in the equilibrium polymeric phase, and departure of the system from equilibrium. The infinite dilution solubility coefficient is obtained from the contributions of the three steps, the first two of which involve standard equilibrium thermodynamics, while the latter requires nonequilibrium considerations and is described by using the nonequilibrium lattice fluid (NELF) model. The various terms have been evaluated for several gases in polycarbonate, polysulfone, poly(phenylene oxide), and poly(methyl methacrylate). It has been found that the largest term in the expression of solubility is due to condensation, which explains the observed correlation between solubility and critical temperature, while the out of equilibrium contribution to solubility, which generally gives numerical values lower than the condensation term, is in turn more strongly correlated to the penetrant critical volume. Introduction Gas solubility in liquids and in polymers is often correlated with measures of gas condensability such as normal boiling temperature, TB,1 Lennard-Jones potential energy constant, /k,2,3 and critical temperature, TC.1,4 In particular, one finds empirically that the logarithm of the solubility coefficient is a linear function of the critical temperature of the gas:

ln(S0) ) a + bTC

(1)

The intercept a and the slope b in eq 1 are adjustable constants; the values of b vary between 14 × 10-3 and 19 × 10-3 K-1 for organic liquids.5 The values of the parameter b found in rubbery and glassy polymers are similar to those observed in liquids3,6-9 and vary generally between 17 × 10-3 and 18 × 10-3 K-1 for rubbers,3,6 while for glasses at 35 °C, one finds experimental values of 19 × 10-3 K-1 for polycarbonate (PC), 20.9 × 10-3 K-1 for polysulfone (PSf), 13.7 × 10-3 K-1 for poly(phenylene oxide) (PPO), and 20.8 × 10-3 K-1 for poly(methyl methacrylate) (PMMA).10 A generalized value of 0.017 K-1 at 25 °C for glassy polymers can be estimated from the empirical correlation proposed by van Krevelen.2 Other authors have considered solubility data at different temperatures for many penetrants, with TC values varying over a broad range, for rubbery and glassy polymers, and have suggested, alternatively, that the logarithm of the solubility coefficient is a linear function of the reciprocal reduced temperature squared (TC/T)2.11-13 This correlation is justified by the fact that the enthalpy of condensation of penetrants is linear with the square of the critical temperature over a broad range of TC values.11,14 An alternative correlation between gas solubility in glassy polymers and penetrant properties was proposed by Yampolskii et al.,15 who related the solubility of gases to the molecular surface area of the dissolved species; such a correlation is also * To whom correspondence should be addressed. E-mail: [email protected].

consistent with eq 1 because the molecular surface area and the critical temperature are to a good approximation linearly proportional to one another, for the most common penetrants considered.10 For rubbery polymers, Gee16 provided a simple theoretical framework for eq 1, by considering gas sorption as a two-step process involving condensation of the gas to a pure liquid, followed by the mixing of the pure component in the polymer matrix; this approach will be recalled in more detail in the following. For the solubility of gases and vapors in glassy polymers it is a known fact that the correlation given in eq 1 holds, even though a conclusive theoretical explanation for it has not been presented thus far. In fact, in a previous work10 we have addressed the same problem, showing that the NELF model is indeed able to predict to within a good approximation the infinite dilution solubility coefficient of various penetrants in different glassy polymers and is consequently consistent with the known empirical correlation between ln S0 and TC. In that work ln S0 is seen as the combination of an entropic and an energetic contribution, both of which revealed a linear dependence on penetrant TC, as a global result of a series of combined effects of penetrant properties. In the present work, a more direct explanation of the fact that the best scaling parameter for the solubility coefficient is the critical temperature is provided, by isolating the penetrant-dependent terms: such analysis is performed by considering the sorption process as the sum of consecutive physical steps. In particular, we will extend Gee’s approach16 to the case of glassy polymers, by considering the gas sorption in a glassy polymer as a sequence of three steps: condensation and mixing in the equilibrium polymeric phase, as in the case of rubbery polymers, followed by the departure of the polymeric phase from the equilibrium state, required to reach the glassy phase properties. The lattice fluid (LF) equation of state (EoS) by Sanchez and Lacombe17-19 is adopted for the evaluation of the mixing term, and its nonequilibrium version, the NELF model, is used to describe the out of equilibrium contribution of the glassy phase.20-26 For the sake of internal consistency the same LF EoS is also used to calculate the condensation contribution, even though several alternative ways

10.1021/ie070304v CCC: $37.00 © 2007 American Chemical Society Published on Web 07/07/2007

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are clearly available to calculate this step. The various terms will be evaluated separately and related to the penetrant critical temperature and size, to put in evidence the relative role of pure penetrant properties entering the various contributions. The penetrants considered for the detailed calculations are the most common gases and vapors typically encountered in relevant applications, namely, Ar, N2, O2, CO2, CH4, C2H4, C2H6, C3H8, n-C5H12, n-C6H14, SO2, and benzene; heavier components have not been considered because they rarely intervene in the gaseous phases in the most common applications, while lighter gases such as H2 and He, though particularly interesting for their special behavior in all generalized correlations, were not included in the list for the lack of sufficient experimental information about their solubility in glassy polymers. By comparing the relative importance of the different terms on the value of the solubility coefficient, we will offer a theoretical support for the observed empirical correlations between gas solubility and penetrant properties for glassy polymers. The present work will thus integrate and overcome the approach followed in ref 10, where the correlations between enthalpic and entropic terms and critical temperature are simply the result of the overall process which includes penetrant and polymer properties and their interactions and do not allow us to point out clearly why the penetrant critical temperature plays a dominating role on ln S0. Although the procedure presented may appear standard and straightforward, it has never been applied to glassy phases before. Correlation between ln(S0) and TC Rubbery Polymers. In Gee’s analysis16 the condensation step is considered to take place at the vapor pressure at temperature T, and the mixing process is described by using Flory-Huggins theory. By applying the phase equilibrium condition for the penetrant in the rubber, Gee obtained the following expression for the solubility coefficient σ, defined as the volume of gas in cubic centimeters (STP) dissolved in 1 cm3 of polymer under the pressure of 1 atm:16

TC ln(σ) = -(χ + 4.5) + 6 T

(2)

In deriving eq 2 the following reasonable assumptions were considered: (i) the reference gas is pure at 1 atm; (ii) the temperature dependence of vapor pressure is given by ClausiusClapeyron equation, considered valid also above TC; (iii) the molar entropy of vaporization of the penetrant at its normal boiling point has a value of approximately 20 cal/(mol‚K) for all substances; (iv) the ratio between the standard molar volume of an ideal gas and the molar volume of the penetrant in the liquid phase is a constant, and its logarithm is equal to 6.5, for all penetrants; (v) the Guldberg-Guye rule, relating normal boiling temperature TB and critical temperature, holds, that is, TB ≈ 0.6TC; and (vi) pressure variations do not affect the properties of the polymer mixtures. Equation 2 leads to a value of b in eq 1 equal to 0.019 K-1 at 35 °C, in good agreement with the value observed in rubbery polymers. Because the χ contribution is often negligible, eq 2 indicates that for all gases in many rubbery polymers there is a single master line for ln(σ) versus TC; that is, for rubbery polymers the dependence of gas solubility on critical temperature is a direct consequence of the condensation term and is not affected by the mixing term. A similar procedure was also followed to describe gas solubility in liquids by Prausnitz and Prausnitz and Shair,27,28

considering more specifically the volume of hypothetical pure liquid state; such a method is recalled in some detail in Appendix 1. Glassy Polymers. The above development suggests an interesting and rigorous procedure for the solubility of gases in glassy polymers, which allows the contributions of condensation, of equilibrium mixing, and of the nonequilibrium properties frozen into the glass to be elucidated. To that aim, the dissolution process is decomposed into three simple steps as follows: (i) First is the condensation of the pure gas to the hypothetical liquid state endowed with a molar volume equal to the partial molar volume existing in the equilibrium condensed mixture at infinite dilution, V h 1,eq, at the same temperature and pressure (see also Appendix 1); the corresponding variation of penetrant chemical potential is ∆µ1,cond and can be calculated on the basis of pure component properties alone, as long as V h 1,eq at infinite dilution is actually polymer-independent. (ii) The second is dissolution of the condensed gas into the polymer, leading to an equilibrium rubbery mixture. The corresponding variation of penetrant chemical potential, ∆µEq 1,mix, can be evaluated on the basis of a suitable equilibrium equation of state for polymer mixtures. (iii) The third is evolution from a rubbery equilibrium mixture to a nonequilibrium glassy phase characterized by the same temperature, pressure, and composition but with a different density given by the actual density value of the glass; the corresponding variation of penetrant chemical potential is ∆µNE 1 . The terms relative to steps i and ii are standard for all the cases of gas solubility in a condensed equilibrium phase. For the case of glassy polymers a new contribution comes into play, represented by the term ∆µNE 1 of step iii, associated with the departure from equilibrium frozen into the glassy phase. Its expression can be obtained by using the NELF model whose main features are briefly recalled hereafter. The NELF model properly extends the LF EoS17-19 valid for amorphous phases at equilibrium to the nonequilibrium state typical of glassy polymers. The model uses the same characteristic parameters as the LF theory (P*, T*, F*) to describe the pure components behavior and the same mixing rules to estimate the mixture properties. The extension to nonequilibrium states is given by the nonequilibrium thermodynamics of glassy polymers (NET-GP) approach, in which the state of a glassy polymer-penetrant mixture can be described by the typical variables used for equilibrium states, for example, temperature, composition, and pressure, plus the polymer density F2, which accounts for the departure from equilibrium present in the glass.20-26 The model also assumes that F2 is an internal state variable for the system so that the specific thermodynamic relations for systems endowed with an internal state variable can be applied.29 The phase equilibrium (or, more properly, pseudo-equilibrium) between a pure external gas (g) and a solid glassy phase (s) requires NE Eq(g) µNE(s) (T,p,ωNE (T,p) 1 1 ,F2 ) ) µ1

(3)

The value of the penetrant chemical potential in the glassy mixture can be derived and specialized in the limit of infinite dilution. The term ∆µNE 1 can thus be explicitly evaluated, as it represents the difference between the chemical potential of the penetrant in the glassy polymeric structure and in the hypothetical rubbery polymer structure at the same temperature and pressure, in the limit of infinite dilution.

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The phase equilibrium condition between gaseous penetrant and glassy polymer is represented by

∆µNE ∆µ1,cond ∆µEq 1,mix 1 + + )0 RT RT RT

(4)

As it is shown in Appendix 1, eq 4 allows to derive an explicit expression for the infinite dilution penetrant solubility coefficient in the glassy phase, S0, in which the terms containing the mixture composition, the terms containing the gaseous properties, and the terms containing the nonequilibrium glassy properties can be clearly identified:

ln(S0) ) ln

( )

TSTP + ΦNE + ΦEq mix + Φcond PSTPT

(5)

In eq 5 the contribution of the condensation term to ln S0, Φcond, is defined as

(

Φcond ≡ - ln(F˜ V1 ) +

)

∆µ1,cond RT

By adopting the LF EoS for mixtures, ΦEq mix is evaluated from the following expression

ΦEq mix ) -1 - F˜ eq 2

[

0

[[

r1 ln

Φcond ≡

T˜ 1

{

-(F˜ V1

-

F˜ L1 )

(

L-1)

[

L-1)

]}

(7)

In the above equation, the superscripts V and L label vaporand liquid-like phase properties, respectively, and a superimposed tilde indicates the following LF reduced variables for the pure penetrant:

F˜ 1 )

F1

ΦNE ≡ ln(F˜ NE 2 )(8)

p P/1

which are obtained by introducing the pure component characteristic density F/1, temperature T/1, and pressure p/1, respectively.17-19 The quantity r10 represents the number of lattice cells occupied by the penetrant molecule, and one has

r10 )

P/1M1 RT/1F/1

)

M1 F/1V/1

)

V1 V/1V˜ 1

(9)

where M1 is the penetrant molar mass, V1* is the volume occupied by a mole of penetrant lattice sites, V1 and V˜ 1 are the molar and reduced volume of the penetrant at the conditions of interest, respectively. The contribution of the equilibrium mixing term to ln S0, ΦEq mix, is defined as

ΦEq mix ≡ ln(φ1) -

∆µEq 1,mix RT

(12)

ω1/F/1 + ω2/F/2

The quantity Ψ represents the only binary parameter of the mixture entering the mixing rules. In the absence of specific binary data, a first-order approximation of the binary parameter is represented by Ψ ) 1, which is equivalent to assuming a geometric mean mixing rule for the binary characteristic pressure, P/12. The nonequilibrium contribution to ln S0, ΦNE, is defined as

F/1

T T˜ 1 ) / T1 p˜ 1 )

ω1/F/1

∆P* ≡ P/1 + P/2 - 2P/12 ) P/1 + P/2 - 2Ψ xP/1P/2 (13)

V-1)

(1 - F˜ L1 )(1/F˜

(11)

The term ∆P* in eq 11 represents the binary characteristic interactions, whose value can be obtained from its definition:

T˜ 1 1 1 + p˜ 1 V - L - 0 ln(F˜ L1 ) + F˜ 1 F˜ 1 r1 T˜ 1 ln

r10

-

and is independent of composition. In eq 11 F˜ eq 2 is the dimensionless density of the polymer in its hypothetical equilibrium state that satisfies the equilibrium EoS for the polymer; F˜ L1 is the dimensionless penetrant density in its hypothetical liquid state at temperature T. Penetrant composition is given by the penetrant volume fraction in the closed packed state, φ1, which appears in the LF EoS and is given by

φ1 )

(1 - F˜ V1 )(1/F˜ 1

˜ L1 ) ln(F˜ eq 2 /F

+

(1 - F˜ L1 )(1/F˜ 1

(6)

)

eq-1)

(1/F˜ 2 (1 - F˜ eq 2 )

and can be evaluated from the LF model for the pure penetrant:

r10

( )] ] ]

(F˜ eq M1 ∆P* ˜ L1 ) p˜ 1 1 2 -F 1 0 r + - L 1 eq / RT F1 T˜ 1 T˜ 1 F˜ 2 F˜ 1

(10)

∆µNE 1 RT

(14)

in which the reduced density, F˜ NE 2 , corresponds to the value of the pure polymer density in the glassy phase, say FNE 2 . The value of ΦNE can be evaluated on the basis of the NELF model expression for the chemical potential of the penetrant in the glassy phase, as follows:

Φ

NE

)

ln(F˜ Eq 2 )

- r1 ln 0

[

]

NE)-1

(1/F˜ 2 (1 - F˜ NE 2 )

Eq)-1

(1/F˜ 2 (1 - F˜ Eq 2 )

-

[

]

1/F˜ 2 r10V/1 V/ (1 - F˜ NE 2 ) NE 0 1 2 F ˜ ) + r ln ΨxP/1P/2 (F˜ Eq 2 2 1 1/F˜ 2Eq RT V/2 (1 - F˜ Eq 2 ) (15) NE

As it may be noticed, the above expression, valid for the infinite dilution case, is independent of composition; it contains the binary parameter Ψ and the pure polymer densities in the rubbery and glassy states; ΦNE becomes zero when the polymer ˜ Eq is at equilibrium, that is, when F˜ NE 2 ) F 2 . In the absence of appropriate models for the bulk rheology of glasses, the value of glassy polymer density to use in eq 15 must be taken from experimental data. Remarkably, the out of equilibrium term could as well be written as a function of the polymer fractional

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Table 1. Values of r10V/1, Critical Volume V1C, and V h 1,eq Partial molar volume at infinite dilution in hypothetical rubbery polymers with V h 1,eq at 35 °C with LF EoS and Ψ ) 1 (cm3/mol) Penetrant

r10V/1 (cm3/mol)

V1C (cm3/mol)

PC

PPO

PSf

PMMA

Ar N2 O2 CO2 CH4 C2H4 C2H6 C3H8 n-C5H12 n-C6H14 C6H6 SO2

28.54 29.70 24.81 29.05 32.02 41.25 46.98 63.90 96.33 111.20 78.58 36.88

74.90 89.80 73.40 93.90 99.20 130.40 148.30 203.00 304.00 370.00 259.00 122.20

30.14 25.68 27.70 43.17 35.94 48.26 54.56 72.74 106.44 122.09 86.23 44.01

31.09 27.74 28.49 43.17 36.83 49.07 55.47 73.71 107.11 122.75 86.79 44.89

28.8 23.28 26.57 41.59 34.51 46.75 52.89 70.89 104.64 120.20 85.02 42.68

29.97 24.07 27.67 43.17 36.18 48.95 55.32 73.70 107.80 123.63 86.91 44.33

free volume, that, in the LF model, can be estimated as

f)

F/2 - F20

(16)

F/2

where the characteristic density of the polymer, F/2, represents the closed packed density at 0 K. Thus the fractional free volume calculated from eq 16 measures the departure of the polymer volume from the hypothetical volume occupied by the crystal at absolute zero and is consistent with the theoretical framework considered here, although it is different from what is often calculated based on Bondi’s group contribution method.30 The derivation of eqs 5-7, 10, 11, 14, and 15 above is presented in some detail in Appendix 1. NE their expressions By substituting for Φcond, ΦEq mix, and Φ given by eqs 7, 11, and 15, respectively, one demonstrates that eq 5 collapses to the concise expression already presented in ref 10, where ln S0 was derived directly by equating the nonequilibrium chemical potential of the penetrant in the glassy polymer, at infinite dilution, to the pure penetrant chemical potential in the gas phase; more specifically one obtains

ln(S0) ) ln

( ) {[ ( ) ]

TSTP V/1 1 + r10 1 + / - 1 NE ln(1 - F˜ NE 2 )+ PSTPT V2 F˜ 2 V/1

}

T/ NE 1 2 - 1 + F˜ 2 Ψ P/P/ T P/ x 1 2 V/ 2

1

(17)

As already shown in ref 10, eq 17 offers a good estimate of ln S0 and is suitable to point out the effects of the entropic and enthalpic factors present in the expression of the Gibbs free energy of the mixture, and through that it may present a useful insight of physical factors influencing S0. However, eq 17 per se is not suitable to point out the role of penetrant critical temperature or penetrant size, and to this aim we need to use eq 5 instead. Calculation of the Different Contributions to ln S0 The evaluation of the different terms in eq 5 requires the knowledge of the independent characteristic parameters r10, T/1, and P/1, for the penetrants, and of the characteristic pressure, density, and volume (P/2, F/2, and ν/2) for the polymer, whose values for the penetrants and polymers here considered are taken from ref 10. The LF parameters might be found also in specific works and collections.19,31,32 To calculate the condensation term from eq 7, we need to determine the value of the reduced liquid density, F˜ L1 , that can

be estimated with the first expression in eq 8, where the molar volume of the liquid is taken equal to the partial molar volume h 1,eq. The latter has been in the equilibrium rubber: VL1 ) V estimated from the usual intercept rule, applied to the curve representing the molar volume of the equilibrium hypothetical rubbery mixture versus the penetrant mole fraction; the equilibrium molar volume has been calculated by using the LF EoS predictively and considering the value of the binary interaction parameter Ψ ) 1. The values of V h 1,eq obtained in this way at 35 °C for all the penetrants considered are reported in Table 1. The infinite dilution partial molar volumes of the penetrants at equilibrium strictly depend on the particular polymer matrix considered. Therefore, with that selection of VL1 , the penetrant condensation term would depend also on the polymer in which one considers the solubility, and this is not so consistent with the common idea of a condensation term associated to the penetrant properties only. Interestingly, however, the values of V h 1,eq calculated at 35 °C for all the penetrants and polymers considered are linearly proportional, with correlation factors R2 >0.98, to r10ν/1, which is the close-packed volume occupied by a penetrant mole in the lattice; thus, the values of r10ν/1 for all penetrants are also listed in Table 1. The average value of the proportionality constant is 1.116 ( 0.017. One can thus assume

V h 1,eq = 1.116r10V/1 for all polymers

(18)

Therefore, the F˜ L1 value for the hypothetical liquid state of the penetrant, to use in eqs 7 and 11, may be obtained as

h 1,eq ) 0.896 for all penetrants and polymers F˜ L1 ) r10V/1/V (19) The condensation term may thus be estimated with a fixed value of F˜ L1 for all penetrants and polymers. Of course, the same value will be used also for evaluating the mixing term. It is worth noting that the value of r10V/1, whose meaning is confined within the framework of the LF model, can be related to the critical molar volume of the penetrants, the values of which are also listed in Table 1. Indeed, by plotting the experimental values of the molar critical volume V1C versus r10 V/1 for all the penetrants considered in the present work, we obtain the following nice linear correlation:

V1C(exp) ) 3.220r10V/1 (R2 ) 0.994)

(20)

The empirical correlation represented by eq 20 will then be considered to hold true. (It must be noticed, however, that a

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NE evaluated from eqs 7, 11, and 15, respectively, for PC (a), PPO (b), PSf (c), and PMMA (d), as a function of Figure 1. Values of Φcond, ΦEq mix, and Φ penetrant critical temperature.

strict application of the LF EoS to the critical penetrant properties would in general require the following result:17

V1C ) V˜ 1Cr10V/1 ) (1 + xr10)r10V/1

(21)

where V˜ 1C is the reduced critical molar volume of the penetrant.) Therefore, after eqs 18 and 20, one may write the following approximate relationship between the penetrant critical molar volume and the partial molar volume of the penetrant, at infinite dilution in the hypothetical rubbery state of the polymer:

V h 1,eq = 0.346V1C

(22)

Remarkably, from eq 22 one obtains also the indication that the penetrant critical molar volume V1C, which is linearly proportional to the penetrant partial molar volume V h 1,eq in the polymer, can be conveniently considered as a scaling parameter for a correlation between solubility and penetrant size. That point will be further inspected and discussed in more detail in the following. The only adjustable binary parameter, Ψ, appearing in the NE expression for ΦEq mix and Φ , is set equal to unity in all the numerical calculations performed in the following so that the model is used in an entirely predictive way. Moreover, we need to know the value of the glassy polymer density FNE 2 at the

experimental conditions of interest, which appears in the nonequilibrium term ΦNE. It is clear that the same glassy polymer may exhibit different values of density at any temperature, depending on the particular history experienced. In the following calculations we adopt the same values of glassy density used in ref 10, where the estimation of FNE 2 , when needed, was also discussed. NE to ln S The specific contributions Φcond, ΦEq 0 mix, and Φ were calculated at 35 °C for the penetrants frequently considered in glassy polymer and listed in Table 1, for the four polymers selected, by using eqs 7, 11, and 15. We then considered the correlation between each contribution and pure penetrant properties as critical temperature and critical molar volume. Results (a) Correlation with Penetrant TC. By plotting Φcond, NE as a function of T , we obtain the correlations ΦEq C mix, and Φ represented in Figure 1; the values of slope and intercept of the interpolating lines for the different polymers are reported in Table 2. For any solute, the condensation term, Φcond, is the same for all polymers because the reduced liquid density of each penetrant is the same for all matrices, in view of the empirical correlation represented by eq 19. The value of Φcond increases linearly with

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Table 2. Correlation of Various Contributions with Penetrant TC Nonequilibrium Term

ΦNE

ΦNE ) aNE + bNETC PC PPO PSf PMMA

aNE

bNE (K-1)

R2

0.11 0.37 0.18 0.13

0.0040 0.0070 0.0068 0.0042

0.6744 0.6638 0.6755 0.6655

NE Mixing in the Glass Term, ΦEq mix + Φ NE ) a ΦEq NE+mix + bNE+mixTC mix + Φ

PC PPO PSf PMMA

aNE+mix

bNE+mix (K-1)

R2

-2.13 -1.71 -2.31 -2.06

0.0047 0.0078 0.0070 0.0047

0.9225 0.8036 0.9595 0.9229

Solubility10 ln S0 ) a + bTC PC PPO PSf PMMA

a

b (K-1)

R2

-5.0 -4.6 -5.2 -4.9

0.0222 0.0253 0.0246 0.0222

0.9904 0.9642 0.9906 0.9925

Solubility in the Rubber ln S0 rubber ) arubber + brubberTC PC PPO PSf PMMA

arubber

brubber (K-1)

R2

-5.11 -4.94 -5.37 -5.06

0.0182 0.0183 0.0178 0.0180

0.9803 0.9879 0.9557 0.9737

the critical temperature, and in particular we have for the list of penetrants chosen

Φcond ) -2.73 + 0.0175TC (R2 ) 0.982)

(23)

It is worth noting that the slope of the straight line interpolating Φcond and TC is very close to the value of 0.019 K-1 estimated by Gee16 using the procedure recalled previously. The intercept has a negative value, indicating that, for poorly condensable penetrants (e.g., TC < 156 K), the condensation term results in a lower solubility in the polymer. Remarkably, the linear relationship given in eq 23 may also be obtained in rather general terms under the more rough approximations required to derive eq A4 of Appendix 1; if one considers atmospheric pressure as indicative of relatively low gas pressure, TB appears in the expression of ∆µ1,cond in eq A4 and, through the usual relationship between the normal boiling temperature and TC, one obtains a linear relationship between Φcond and TC in more general terms, without using a specific EoS. The equilibrium mixing term, ΦEq mix, is in all cases negative; its value is essentially constant with penetrant critical temperature, and it is equal to -2.04 ( 0.45 for PC, to -1.82 ( 0.31 for PPO, to -2.43 ( 0.66 for PSf, and to -2.05 ( 0.52 for PMMA. Such values of ΦEq mix correspond to a multiplying factor on S0 (see eq 5) varying between 0.088 and 0.16. The out of equilibrium term, ΦNE, is always positive, thus it increases the solubility with respect to the value obtained in the corresponding hypothetical rubbery polymer. This behavior is consistent with the common intuitive argument associated to the fact that the nonequilibrium free volume is always higher than the corresponding value at equilibrium. The value of ΦNE increases with TC, as it can be seen in Figure 1 and in Table 2,

but the correlation is more scattered with respect to the one observed for the condensation term, with values of R2 equal to about 0.67 for all polymers. The value of ΦNE increases much more weakly than the condensation term with increasing the penetrant TC, the slope of the ΦNE versus TC curve varying between 0.004 K-1 and 0.007 K-1 for the four polymers considered. The intercept of such a curve seems to be related to the fractional free volume f of the polymer, being equal to 0.11, 0.18, and 0.13 for PC, PSf, and PMMA, respectively, while it is equal to 0.366 for PPO, the polymer characterized by the higher value of f. The value of the slope of the correlation line is substantially higher for the condensation term than for the out of equilibrium term, while the intercept is much larger for ΦNE. This result implies that for the less condensable penetrants we have that ΦNE > Φcond. The values of critical temperature at which the condensation term equals the nonequilibrium term are 210, 295, 271, and 215 K for PC, PPO, PSf, and PMMA, respectively. In other words, the sorption of less condensable penetrants in glassy polymers is more enhanced by the excess free volume of the glassy polymer than by the low value of the condensation term. One can also compute the values of ln S0 and correlate them to TC, obtaining the plots shown in Figure 2 and the values of a and b listed in Table 2, already presented in ref 10. Interestingly, the value of the correlation coefficient R2 for ln S0 versus TC is higher than each one of the coefficients of correlation obtained interpolating linearly the values of the single terms ΦNE, ΦEq mix, and Φcond with TC and is always higher than 0.99, except for the case of PPO, for which R2 is equal to 0.96. This behavior is due to the fact that, for some penetrants, the values of ΦNE and ΦEq mix show deviations from the straight line that are opposite in sign, as it can be noticed from Figure 1, so that such terms compensate by considering the sum of the contributions ΦNE and ΦEq mix. Following this result, we found it appropriate to evaluate the overall mixing term, ΦNE + ΦEq mix, representative of the dissolution of the penetrant from the pure hypothetical liquid state into the glassy polymer, and correlate it with TC. The intercept and slope of this relationship are also listed in Table 2: noticeably, the value of the correlation coefficient is rather high (R2 ) 0.8-0.96), as a result of the cancellation of terms that originate significant scattering from the straight line of the NE evaluated separately. two contributions ΦEq mix and Φ It is also interesting to examine the effect on solubility of the polymer free volume in excess over the equilibrium structure. To that aim, in Figure 2 we have plotted versus TC the actual value of ln S0 in the glassy polymer, as well as the value that the infinite dilution solubility coefficient would have in the hypothetical equilibrium rubber, ln S0 rubber. Clearly, such a quantity may be evaluated by subtracting the value of ΦNE from ln S0. Beyond the obvious result that the solubility in the nonequilibrium glass is always higher than the corresponding value in the equilibrium rubber, we also find that the solubility in a nonequilibrium glass is more strongly correlated to the penetrant critical temperature than the solubility in the equilibrium phase. This behavior is associated to the fact that the additional term ΦNE, present in the dissolution into a glassy phase, has a certain correlation with TC, as discussed above. The value of the slope of the correlation between ln S0 rubber and TC, brubber, is very similar for all polymers and equal to about 0.018 K-1, which is just slightly higher than what one would obtain by considering the condensation term as the only contribution to ln S0 rubber (0.0175 K-1).

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Figure 2. ln S0 and ln S0 rubber at 35 °C for PC (a), PPO (b), PSf (c), and PMMA (d), as a function of penetrant critical temperature.

A final consideration is in order on the experimental observation, made by different authors,11-13 that the value of ln S0 in polymers may rather be considered a linear function of the square of the critical temperature, TC2, when isothermal solubility data are examined, or of the square of the reciprocal of the reduced temperature, (TC/T)2, for non-isothermal data. Such a behavior has been reported for various rubbery and glassy polymers. For the list considered here of common penetrants and polymers of frequent interest in many applications, it has been shown that there is no serious argument to prefer a linear correlation between ln S0 and (TC/T)2 over the linearity with TC.10 (b) Correlation with Penetrant Molar Volume. In this section, we explore the possible effect on the infinite dilution solubility coefficient of penetrant properties other than critical temperature, considering in particular the penetrant molecular size. This property is also related to the penetrant condensability, and it affects the polymer-penetrant interactions, as well as the number of configurations available for the penetrant in the polymer matrix. Therefore, it is expected to have an important effect on the gas solubility in solid polymers and may lead to alternative correlations. As a measure of the penetrant size we have considered the value of the critical molar volume V1C that, as indicated by eq 22, is linearly proportional to the partial molar volume, V h 1,eq, of the penetrant at infinite dilution in the various polymers in the hypothetical equilibrium rubbery state. NE for the various penThe values of Φcond, ΦEq mix, and Φ etrants in PC, PPO, PSf, and PMMA versus values of V1C at 35

°C are plotted in Figure 3a-d. The values of the slopes A and of the intercepts B of the interpolating lines are reported in Table 3, together with the values of the correlation coefficient R2. Interestingly, one finds an excellent correlation between the out of equilibrium term, ΦNE, and the critical molar volume of the penetrant, with values of R2 higher than 0.99 for all the penetrants considered. This is associated to the fact that the out of equilibrium term is mainly related to volume effects. For larger penetrants, the term ΦNE becomes more important, and its value becomes comparable or even larger than the mixing term, which on the contrary is not correlated in any way to the critical volume. Moreover, also the condensation term Φcond may be represented through a linear dependence on V1C, according to the following correlation:

Φcond ) -0.893 + 0.0231V1C (R2 ) 0.748)

(24)

However, the value of the correlation coefficient is much lower than that obtained when plotting the same values versus TC (R2 ≈ 0.99). This correlation finds reasonable justification in the fact that the values of V1C and TC are with some approximation related linearly to one another with a value of the correlation coefficient equal to 0.709 for the list of penetrants inspected, as it may be seen in Figure 4. By plotting directly the logarithm of solubility, ln S0, versus the critical volume, one obtains the behavior represented by the open circles in Figure 5a-d. In Table 3 we report the values of

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NE evaluated from eqs 7, 11, and 15 respectively, for PC (a), PPO (b), PSf (c), and PMMA (d), as a function of Figure 3. Values of Φcond, ΦEq mix, and Φ penetrant critical volume.

Table 3. Correlation of ΦNE and ln S0 with Penetrant Critical Volume V1C ΦNE ) ANE + BNEV1C PC PPO PSf PMMA

ANE

BNE (mol/cm3)

R2

0.192 0.493 0.324 0.206

0.0074 0.0129 0.0125 0.0078

0.9960 0.9947 0.9962 0.9951

ln S0 ) A + BV1C PC PPO PSf PMMA

A

B (mol/cm3)

R2

-2.66 -2.34 -2.63 -2.54

0.0292 0.0359 0.0325 0.0289

0.7518 0.8525 0.7607 0.7371

the slope and intercept, A and B, in the following possible correlation:

ln(S0) ) A + BV1C

(25)

However, the value of R2 in such a correlation is substantially lower (0.73-0.76) than the one obtained by correlating ln S0 to TC with eq 2, with the exception of PPO, for which the value of R2 is reasonably high and equal to 0.85. This behavior is a

Figure 4. Correlation between V1C and TC for all the penetrants.

direct consequence of the fact that, numerically, the most important term in eq 5 is due to the condensation contribution, which is more closely related to the critical temperature of the penetrant rather than to its size. In summary, the mixing term, ΦEq mix, is in all cases negative and its value does not depend on the penetrant properties. The

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Figure 5. Values of ln S0, Φ(H), and Φ(S) evaluated from eqs 17, 26, and 27, respectively, for PC (a), PPO (b), PSf (c), and PMMA (d), as a function of penetrant critical volume.

condensation term, Φcond, increases linearly with the penetrant critical temperature, as one may also expect based on eq 11 or considering eq A4 of Appendix 1 and the relation between TB and TC. The condensation term might also be related to the penetrant critical volume, through eq 24, even though with less accuracy than using the linear correlation with critical temperature: this is a consequence of the fact that more condensable penetrants are also endowed with larger critical volumes. On the other hand, the nonequilibrium term, which is associated to the excess free volume of the glass with respect to the corresponding equilibrium state, is correlated very well to the critical volume of the penetrant (see Table 3), and that correlation is stronger than that with critical temperature, toward which a larger scatter is observed. We may thus conclude that the logarithm of infinite dilution solubility of gases in the polymers inspected is much better correlated to the penetrant critical temperature than to its critical molar volume, even though the nonequilibrium contribution is more satisfactorily correlated to the penetrant critical volume. That is a consequence of the larger effect associated to the condensation term over the nonequilibrium term for the systems of common interest, here considered. (c) Correlations for the Enthalpic and Entropic Contribution to Solubility. The present analysis is completed by evaluating the effect of penetrant size, represented by V1C, on the enthalpic and entropic contributions to solubility, appearing

in the expression for ln S0 given by eq 17, and defined as follows:10

Φ(H) ≡ r10F˜ NE 2

Φ

(S)

≡ r1

0

T/1 2 Ψ P/P/ T P/ x 1 2

{[ ( ) ]

1

V/1

(26)

( )}

V/1 1 NE 1 + / - 1 NE ln(1 - F˜ 2 ) + / - 1 V2 F˜ 2 V2

(27)

so that we have

ln(S0) ) ln

( )

TSTP + Φ(S) + Φ(H) PSTPT

(28)

Their correlation with penetrant TC was already studied previously.10 The values of Φ(H) and Φ(S) have been plotted versus V1C in Figure 5a-d. The value of the entropic term Φ(S) decreases linearly with C V1 , with a slope varying from -0.0624 mol/cm3 to -0.0465 mol/cm3 and a correlation coefficient R2 around 0.98 for all polymers (Table 4). Such a correlation is therefore much more accurate than the one that can be observed between Φ(S) and TC, for which the correlation coefficient R2 was found equal to 0.82 at most.10 The above considerations imply that the entropic contribution to solubility is, for a given polymer, a decreasing

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Table 4. Correlation of Φ(H) and Φ(S) with Penetrant Critical Volume V1C Φ(H) ) A(H) + B(H)V1C PC PPO PSf PMMA

A(H)

B(H) (mol/cm3)

R2

1.29 1.18 1.38 1.30

0.0907 0.0824 0.0964 0.0913

0.9369 0.9369 0.9369 0.9369

Φ(S) ) A(S) + B(S)V1C PC PPO PSf PMMA

A(S)

B(S) (mol/cm3)

R2

-3.82 -3.38 -3.87 -3.71

-0.0615 -0.0465 -0.0639 -0.0624

0.9831 0.9738 0.9842 0.9851

function of the critical volume, or equivalently of the volume occupied by a penetrant molecule in the polymeric phase. This is consistent with the lower probability encountered to accommodate a larger molecule inside the polymer matrix. The value of the enthalpic term Φ(H) increases linearly with C V1 values with a slope varying from 0.0824 mol/cm3 to 0.0964 mol/cm3 and values of the correlation coefficient, R2, that are equal in all cases to 0.94 (Table 4). This observation is consistent with the fact that (a) larger penetrants are generally more condensable than smaller ones, as it can be seen also from Figure 4, and that (b) a larger volume permits a larger number of contacts and energy interactions between penetrant and polymer molecules. The values of R2 that can be obtained by plotting the values of Φ(H) versus V1C are even higher than the ones obtained for a linear correlation between Φ(H) and TC, in which case R2 ≈ 0.90 for all polymers.10 By evaluating the solubility with eq 17, one observes (Table 3) that the value of ln S0 increases linearly with V1C. This is due to the fact that the enthalpic term is, in absolute value, higher than the entropic one in all cases. However, the solubility coefficient has a weaker correlation with the penetrant critical volume than each of the two terms separately, which correlate well with V1C but have opposite slopes. The reason for this behavior lies in the scattering from the linear trend induced by the summation of terms with opposite signs. A somewhat opposite behavior was observed by considering the correlations with the critical temperature of the type of eq 2; in that case the dependence of the logarithm of the solubility coefficient versus the critical temperature was much closer to linearity than that of the entropic and enthalpic terms considered separately.10 Conclusions The infinite dilution solubility coefficients of several gases were examined in commonly used glassy polymers, to offer a fundamental explanation of the well-known correlation between ln S0 and penetrant condensability. The gas dissolution in a glass at temperature T and pressure p has been considered as a three-step process, namely, (i) condensation to the hypothetical liquid state at T and p and at a molar volume equal to the partial molar volume at infinite dilution in the equilibrium rubber, (ii) isothermal and isochoric mixing into an equilibrium rubbery mixture, and (iii) isothermal and isobaric change in density to the nonequilibrium volume of the glassy polymer. For each step, the change in penetrant chemical potential at infinite dilution was explicitly calculated by using an EoS approach both for the equilibrium transformations (LF model) and for the nonequilibrium step (NELF model), thus obtaining explicit expressions of the specific additive

contributions to the value of ln S0, namely, Φcond, ΦEq mix, and ΦNE, respectively. Those contributions were numerically evaluated for the gases and vapors typically of interest in glassy polymers, dissolved in PC, PPO, PSf, and PMMA at 35 °C. The main results obtained indicate the following: (1) The infinite dilution values of the partial molar volumes in the equilibrium hypothetical rubbery mixtures, V h 1,eq, for all the penetrants and polymers considered are related to pure penetrant properties only through eq 18, so that the reduced penetrant density in the hypothetical liquid state, F˜ L1 , is the same for all penetrants and is equal to 0.896. Correspondingly, the term Φcond depends only on the pure penetrant properties and increases linearly with penetrant TC, with a rather high correlation coefficient (R2 > 0.98) and a slope b ) 0.0175 K-1 at 35 °C. That behavior is consistent with the expectation that Φcond must be correlated to a measure of penetrant condensability such as TC. The value of Φcond is negative for TC smaller than approximately 160 K, indicating that this term results in a reduction of the solubility coefficient S0 for poorly condensable penetrants. (2) The mixing term ΦEq mix contains equilibrium polymer properties as well as pure penetrant properties, and it is always negative and has a value equal to about -2 for all the penetrants and polymers considered; therefore, this term always has the effect of lowering the solubility S0. Moreover, its value has no correlation with the penetrant parameters examined. (3) The nonequilibrium term ΦNE is independent of composition and is a function of the departure from equilibrium of the system. It has a small but always positive value: as a consequence (a) the solubility in the glass is always higher than that in the hypothetical rubbery polymer, and (b) for TC < 210-295 K, ΦNE is larger than Φcond, contributing to enhance the solubility of poorly condensable penetrants in the glassy matrix. The value of ΦNE also increases approximately linearly with critical temperature, although the value of R2 is rather low (R2 ≈ 0.67), while as a result of the summation of terms that originate scattering, the correlation becomes much better by reporting versus TC the sum of the two terms ΦNE and ΦEq mix, representative of the dissolution in the glassy polymer. (4) The nonequilibrium term ΦNE is strongly related in a linear way (R2 ≈ 0.99) to the penetrant V1C, which is linearly proportional to the infinite dilution partial molar volume of the penetrant. The high correlation factor is due to the fact that ΦNE originates from the excess free volume of the glass with respect to the rubber, and therefore it is related to the volume occupied by the penetrant inside the matrix, to which the critical molar volume is linearly proportional. (5) By considering a possible correlation versus V1C for the other terms contributing to ln S0 it was observed that the condensation term may be considered a linearly increasing function of V1C, but with a correlation coefficient R2 ≈ 0.78 much lower than for the dependence of Φcond versus the critical temperature (R2 >0.98), which is clearly to be preferred. (6) The critical temperature is the best scaling parameter for the condensation term Φcond, while the critical volume is the best scaling parameter for ΦNE due to the different nature of the two contributions, one related to penetrant condensability the other to the excess free volume. (7) By considering the alternative decomposition of ln S0 into an enthalpic and entropic term and their relation with penetrant critical volume, it was pointed out that the entropic term Φ(S) is indeed a linear decreasing function of V1C, with a high correlation coefficient (R2 > 0.98), as a result of the fact that

Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007 7655

equilibrium mixture can be decomposed into two steps: isothermal compression to the pressure π where the molar volume h 1,eq (step a), followed by the hypothetical isothermal is V1L ) V and isochoric expansion to pressure p. The former contribution ∆µI1 to the molar Gibbs free energy is given by the Helmholtz free energy term plus the term ∆(pV) ) πV1L - pV1V. By using the LF model for the pure penetrant and π˜ 1 ) π/P /1, one has,

{

(

()

)

T˜ 1 ∆µI1 p˜ 1 π˜ 1 F˜ V1 r10 V L -(F˜ 1 - F˜ 1 ) + V - L + 0 ln L + )RT T˜ 1 F˜ 1 F˜ 1 r1 F˜ 1 T˜ 1 ln

Figure 6. Schematization of the gas sorption in rubbery polymers.

larger penetrants have lower degrees of freedom inside the matrix. On the other hand, the enthalpic term Φ(H) is linearly increasing with penetrant critical volume with R2 ≈ 0.93, because of the higher condensability and higher number of penetrant-polymer interactions of the larger penetrants. However, for the logarithm of the infinite dilution solubility coefficient, the best scaling parameter is the penetrant critical temperature, rather than the critical molar volume, likely because the condensation term is numerically more relevant than the nonequilibrium one, except for the very poorly condensing penetrants. Acknowledgment The financial support of EU Strep NMP3-CT-2005-013644 (MultiMatDesign) is gratefully acknowledged. Appendix 1 Prausnitz and Shair27,28 followed an approach similar to that presented by Gee16 for determining a generalized correlation for gas solubility in liquids. To that aim gas dissolution was considered as a three step process containing transformations between equilibrium states only (see Figure 6): (a) compression of the gas 1 from pressure p to pressure π where the gas reaches the liquid like molar volume V1L; (b) isochoric and isobaric mixing of the liquid-like penetrant with the liquid solvent; and (c) expansion of the solution from pressure π to pressure p. Step a is greatly unfavorable to gas dissolution due to the high pressure π required to reach V1L, and its effects are somewhat attenuated by step c in which the condensed mixture is brought back to the actual operating pressure p. The numerical value of the contribution of step a is thus not so meaningful per se, and a more realistic estimate of the effect of condensation is rather obtained by combining steps a and c into the single step d, virtually associated to the penetrant alone by introducing for the gaseous penetrant the concept of hypothetical liquid state as the liquid state at the temperature T, pressure p, and liquidlike molar volume V1L. Thus the calculation path considered is reduced to a two-step process formed by condensation to the hypothetical liquid state and mixing of the hypothetical liquid into the condensed solvent at the pressure p, labeled as step e in Figure 6. As already discussed by Prausnitz and Shair,27,28 gas condensation to the state of hypothetical liquid at the same pressure and at a molar volume equal to the partial molar volume in the

[

]}

V-1

(1 - F˜ V1 )1/F˜ 1

L-1

(1 - F˜ L1 )1/F˜ 1

(A1)

The isochoric expansion step, vice versa, introduces a ∆µII1 contribution which contains only the term ∆(pV) ) V1L(p - π)

r10 ∆µII1 ) (p˜ 1 - π˜ 1) RT T˜ F˜ L

(A2)

1 1

Therefore, by adding the two contributions for ∆µI1 and ∆µII1 given by eqs A1 and A2, respectively, the variation of chemical potential for the penetrant, from the gaseous state to the hypothetical liquid state at the same temperature and pressure, becomes

{

(

)

()

r10 T˜ 1 F˜ V1 ∆µ1,cond 1 1 )-(F˜ V1 - F˜ L1 ) + p˜ 1 V - L + 0 ln L + RT T˜ 1 F˜ 1 F˜ 1 r1 F˜ 1 T˜ 1 ln

[

]}

V-1

(1 - F˜ V1 )1/F˜ 1

L-1

(1 - F˜ L1 )1/F˜ 1

(A3)

The pure component condensation free energy term might also be estimated more simply as

(

)

Teb(p) ∆µ1,cond(T,p) ∆S˜ VAP(p) 1= RT R T

(A4)

In deriving eq A4 the gaseous phase was considered an ideal gas, the enthalpy of vaporization was taken as constant with temperature, and the pressure dependence of the liquid Gibbs free energy was neglected, in the range between the actual pressure and the saturation pressure at temperature T. Although derived using more approximate assumptions, eq A4 is useful to point out immediately the linear relationship of ln S0 versus TC. Indeed, by considering p )1 atm as indicative of the lowpressure values for gas solubility, one has Teb(p) ) TB ≈ 0.6TC, in view of the Guldberg-Guye rule. In any event, for internal consistency with the other Gibbs free energy terms calculated for steps ii and iii we prefer to consider the condensation term calculated from the LF EoS in eq A3. It is worth pointing out that the procedure considered for the condensation term applies equally well to penetrants at subcritical or supercritical temperatures. For the mixing term of step ii we consider the difference between the expressions of the penetrant chemical potential in the equilibrium mixture and the value in its hypothetical pure liquid state, as given by the LF theory; from eq 43 of ref 19, considering the infinite dilution limit one immediately obtains

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Ind. Eng. Chem. Res., Vol. 46, No. 23, 2007

[

F˜ eq ∆µ1,mix M1 ∆P* ˜ L1 2 -F 0 ) ln(φ1) + 1 + F˜ eq + r + 2 / 1 RT T˜ 1 F RT

(

p˜ 1 1 1 T˜ 1 F˜ eq F˜ L 2 1

)]

{[

+ r10 ln

1

]

eq-1

1/F˜ 2 (1 - F˜ eq 2 )

L-1

(1 - F˜ L1 )1/F˜ 1

+

}

ln(F˜ eq ˜ L1 ) 2 /F r10

(A5)

The value of the penetrant chemical potential in the glassy mixture can be derived from the NELF model18-22 and specialized in the limit of infinite dilution: the term ∆µ1NE representing the difference between the chemical potential of the penetrant in the glassy polymeric structure and the hypothetical rubbery polymer structure at the same temperature and pressure, in the limit of infinite dilution, can thus be evaluated as follows:

() [

NE

]

(1/F˜ 2 )-1 ∆µNE F˜ NE (1 - F˜ NE 1 2 2 ) ) ln Eq + r10 ln + (1/F˜ 2Eq)-1 RT F˜ 2 (1 - F˜ Eq ) 2

[

1/F˜ 2 V/ (1 - F˜ NE r10V/1 2 ) NE 0 1 F ˜ ) r ln 2 ΨxP/1P/2(F˜ Eq 2 2 1 / 1/F˜ 2Eq RT V2 (1 - F˜ Eq 2 )

]

NE

(A6)

Finally, to relate the various terms to solubility, one can apply the phase equilibrium condition represented by eq 4. The infinite dilution solubility coefficient S0, expressed in cm3(STP)/(cm3pol atm) can be conveniently expressed by using LF and NELF parameters as follows:

ln(S0) ) ln

( ) ( ) TSTP F˜ NE 2 φ1 + ln PSTPT F˜ V1

(A7)

Finally, eqs 4 and A7 can be combined to obtain the relationship between S0 and the three terms associated to the different steps of the dissolution process:

( ) ( ( ( )

)

∆µNE TSTP 1 NE + ln(F˜ 2 ) + ln(S0) ) ln PSTPT RT ln(φ1) -

≡ ln

) (

∆µEq 1,mix RT

- ln(F˜ V1 ) +

TSTP + ΦNE + ΦEq mix + Φcond PSTPT

)

∆µ1,cond RT

(A8)

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ReceiVed for reView February 28, 2007 ReVised manuscript receiVed May 16, 2007 Accepted May 17, 2007 IE070304V