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Solubility of gases and vapors in glassy polymers modelled through non-equilibrium PHSC theory
Fluid Phase Equilibria 241 (2006) 300–307
Solubility of gases and vapors in glassy polymers modelled through non-equilibrium PHSC theory Ferruccio Doghieri, Maria Grazia De Angelis, Marco Giacinti Baschetti, Giulio C. Sarti ∗ DICMA-Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali, Universit`a di Bologna viale Risorgimento 2, 40136 Bologna, Italy Received 10 November 2005; received in revised form 23 December 2005; accepted 23 December 2005 Available online 3 February 2006
Abstract Well established equation-of-state (EoS) models from Perturbed Hard Sphere Chain Theory have been adopted within the framework of non-equilibrium thermodynamics for glassy polymers (NET-GP) to describe the solubility of small penetrants in glassy polymers. The procedure presented here parallels the one already applied to the Lattice Fluid EoS, obtaining the Non-Equilibrium Lattice Fluid (NELF) model. The solubility is calculated as pseudo-equilibrium solute content in the glassy polymeric phase at fixed temperature and solute fugacity, and is compared to the experimental data. The NE-PHSC model predictions require pure component and binary EoS parameters, together with the value of the glassy polymer density during sorption. Several common glassy polymers, as well as a high free volume fluoropolymer, have been considered to test the model ability to represent the thermodynamic properties of glassy gas–polymer mixtures. A comparison between the performance of two NE-PHSC models, endowed with different expressions for the interaction potential, is also presented, offering interesting conclusions on their accuracy in the representation of the thermodynamic properties of relatively high-density systems. The results obtained in this work show that the NET-GP approach provides a reliable description of the properties of homogeneous glassy mixtures. © 2006 Elsevier B.V. All rights reserved. Keywords: Solubility; Glassy polymers; Non-equilibrium thermodynamics
1. Introduction Vapor–liquid or vapor–solid equilibria for polymeric solutions in melt or rubbery state are commonly calculated using well established thermodynamic tools to represent the properties of both vapor and condensed phases; in particular, for the equilibrium polymeric phase one can adopt either expressions for the excess Gibbs free energy and activity coefficients [1–5], or equations of state for the mixtures [6–14]. Such models offer valuable correlations for the solubility of low molecular weight species in rubbery polymers, which can be confidently used even for predictive purposes, based only on pure component data and on some information about the thermodynamic properties of the binary mixture. In contrast, the pseudo-equilibrium gas and vapor solubility in amorphous glassy polymers cannot be represented by means
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of the same equilibrium thermodynamic expressions, which indeed fail to describe even qualitative aspects of the observed trends. Empirical or semi-empirical tools are still widely used to correlate data of pseudo-equilibrium solute content in glassy polymers, the most popular and successful of which is, by far, the Dual Mode Sorption model (DMS) [15]. While this model allows to obtain good correlations and useful insights into the polymeric structure when specific experimental data are available, it cannot be used in a predictive mode to evaluate gas or vapor apparent solubility in glassy polymer phases. The main difficulty in building a predictive model for glassy phases lies in the fact that the thermodynamic properties of polymeric systems below the glass transition temperature are not only affected by the values of temperature, pressure and solute fugacity imposed through the external phase. Indeed, different values of apparent solubility in a given glassy polymer can be observed, at the same temperature pressure and solute fugacity, depending on the specific pre-treatments experienced such as quenching, annealing or preswelling. The non-equilibrium nature of glassy polymer phases needs thus to be properly considered in order to
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develop a model suitable to describe the relevant thermodynamic properties. Qualitatively, we can say that the departure from the equilibrium state of a glassy polymer results from the hindered mobility of polymer chains which does not allow the system to access all possible microstates and ultimately prevents it from reaching the free volume that would lead to the absolute minimum of its Gibbs free energy, i.e. the equilibrium free volume. The glassy polymer volume is thus a non-equilibrium quantity, whose value must be obtained from direct measurement or from bulk rheology models if available, and differs from the equilibrium value given by the equation of state. The nonequilibrium state of glassy phases does not allow direct use of well known results from classical and statistical thermodynamics of mixtures, and inspired the development of several specific approaches [16–18]. It is beyond the scope of this work to offer a detailed comparison of the different models cited. We simply remind that remarkable results were obtained using the Non-Equilibrium Lattice Fluid (NELF) model to predict and/or correlate pseudo-equilibrium gas solubility in glassy polymers under many different conditions [19–22]. The NELF model is a special application of the non-equilibrium thermodynamics for glassy polymers (NET-GP) [19–25], obtained from the proper extension of an existing equilibrium model, i.e. the Lattice Fluid (LF) theory. NET-GP is not restricted to any particular equation of state, but rather indicates the general relationships existing between the thermodynamic properties above and below the glass transition temperature and can be applied to all equations of state suitable to describe polymeric systems at equilibrium. In the present work we like to present an application of the NET-GP approach based on the use of Perturbed Hard Sphere Chains (PHSC) Theory developed by professor Prausnitz [6–9], that we would like to acknowledge for his major contributions to this field, his remarkable work, results and methods from which we have all learned. Our research group would also like to express gratitude for the knowledge and enthusiasm that professor Prausnitz transmitted to the students during his memorable lectures at the University of Bologna in Spring 2004. Indeed PHSC EoS has been rather successful to describe not only the behavior of mixtures of low molecular weight species but also that of polymeric mixtures under equilibrium conditions, i.e. above the glass transition temperature, and is thus a promising candidate to represent also the properties of nonequilibrium glassy phases through NET-GP theory. We will also compare the results obtained by using different versions of the PHSC models, that adopt different expression for the dispersion forces term in the residual Helmohltz free energy. In addition, PHSC EoS offers the advantage to consider pure component parameters directly related to molecular properties, which can therefore possibly be predicted based on molecular modeling tools. Examples of predictions and correlations are presented and discussed hereafter for the solubility of light gases in different conventional glassy polymers, such as bisphenol-A poly(carbonate) (PC) and poly(ethylmethacrylate) (PEMA), as well as a commercial high free volume perfluorinated glassy
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polymer, poly(2,2-bistrifluoromethyl-4,5-difluoro-1,3-dioxoleco-tetrafluoroethylene) (Teflon® AF2400). 2. Theoretical background 2.1. Non-equilibrium thermodynamics for glassy polymers The basic assumptions and general results of NET-GP approach, developed to represent thermodynamic properties of glassy polymer phases, have been reported in details in previous publications [19–25]. The main results which are relevant to obtain solubility isotherms in glassy polymers are briefly summarized hereafter for clarity sake. The NET-GP model relies upon the assumption that glassy polymers are homogeneous, isotropic and amorphous phases whose properties depend not only on composition and externally imposed conditions such as temperature and pressure, but also on convenient order parameters which describe the departure from equilibrium of the system. Several order parameters can be introduced, in principle, to properly describe the non-equilibrium state Σ of glassy polymer systems; however, the NET-GP approach focuses the attention on one single parameter which is meant to represent the departure of the glassy polymeric volume from the corresponding equilibrium value. The complete set of independent state variables required for the description of the thermodynamic properties of non-equilibrium glassy states is chosen as follows: = {T, p, Ωsol , ρpol } (1) where T, p and Ωsol are temperature, pressure and solute-topolymer mass ratio, while ρpol is the polymer mass per unit volume. In Eq. (1) ρpol is thus the order parameter which allows to measure the departure from equilibrium conditions when EQ compared to the corresponding equilibrium value ρpol . The latter quantity depends, of course, on system composition, temperature and pressure only, and it can be calculated through the condition of minimum Gibbs free energy for the system: ∂G EQ =0 (2) ρpol = ρpol (T, p, Ωsol ) ⇔ ∂ρpol T,p,Ωsol The second key assumption of the model is that the order parameter ρpol evolves in time according to a rate that depends only on the state of the system, i.e. the following equation holds: dρpol = f (T, p, Ωsol , ρpol ) dt
(3)
Eq. (3) can be readily derived for the system when the bulk rheology of the polymeric phase follows a Voigt model. In view of the assumption represented by Eq. (3), the order parameter ρpol plays the role of an internal state variable for the system, and the basic thermodynamic relations of the NET-GP model are directly derived by applying well established thermodynamic results for systems endowed with internal state variables [26,27]. The most important consequences of the above procedure lie in the fact that the Helmholtz free energy density, aNE , and solute chemical potential, µNE sol , in the non-equilibrium glassy phase depend on composition, temperature and polymer mass
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density only, and their values are not affected by the pressure of the system [20]: NE ∂a =0 (4) ∂p T,Ωsol ,ρpol µNE sol
=
∂aNE ∂ρsol
(5) T,p,ρpol
In view of the above conclusions, the following relationships are easily derived between aNE (or µNE sol ), in a general non-equilibrium state, and the corresponding properties aEQ (or EQ µsol ) evaluated on the true equilibrium curve: aNE (T, p, Ωsol , ρpol ) = aEQ (T, Ωsol , ρpol ) EQ
µNE sol (T, p, Ωsol , ρpol ) = µsol (T, Ωsol , ρpol )
(6) (7)
EQ
In Eqs. (6) and (7) aEQ and µsol indicate the equilibrium values of the Helmholtz free energy density and solute chemical potential in the polymer–solute mixture at a given temperature and species densities. Eqs. (6) and (7) express a simple, nontrivial and relevant result indicating that the free energy and solute chemical potential can be evaluated in all accessible nonequilibrium states of the system, starting from knowledge of their values on the restricted equilibrium domain, identified in terms of temperature, composition and polymer density. It is important to notice that the above equations represent a general non-equilibrium thermodynamic result, and any reliable expression for the equilibrium properties of amorphous phases could be specifically used in Eqs. (6) and (7) to finally describe the non-equilibrium thermodynamic properties of the solute–polymer mixture in the glassy phase. In this work we will consider the non-equilibrium free energy functions which can be obtained from Eq. (6) when Perturbed Hard Sphere Chain Theories are considered for the equilibrium properties. 2.2. PHSC(vdW) and PHSC(SW) EoS Perturbed Hard Sphere Chain Theory has been introduced by Prausnitz and co-workers and several specific forms of the free energy expression as function of system volume, temperature and composition have in fact been developed over the years [6–9], which essentially differ from one another in the expression used for the pair interaction potential between chain segments. As it is well known, in all PHSC models, the residual Helmholtz free energy is expressed as the sum of a reference term, accounting for chain connectivity and a hard sphere interaction and perturbation term, which represents the contributions of mean-field forces: ares = aref + apert
(8)
The first PHSC model here considered is the simplified Perturbed Hard Sphere Chain Theory by Song et al. [8] in which the reference term uses the modified Chiew equation of state, while the perturbation term is of the van der Waals type. In what follows, this model is referred to as PHSC(vdW) equation of
state. The pure component parameters involved in the expression for the Helmholtz free energy are sphere diameter σ, mass per segment M/r, and characteristic energy for the pair interaction potential ε, beyond the species molar mass. For the case of mixtures, the free energy expressions are obtained through the introduction of mixing rules and the binary parameters contained therein. We will here consider only the binary interaction parameters kij appearing in the expression of the characteristic energy for the interaction between pairs of unlike segments: √ (9) εij = (1 − kij ) εii εjj In addition to the specific form of the PHSC(vdW) model, a second version of the PHSC equation of state will be considered in the following, which was introduced by Hino and Prausnitz [9]. The latter version was obtained by replacing the simple van der Waals perturbation term with a second order perturbation term for a square well potential of variable width, following Chang and Sandler [28]. This will be referred to as “square well version” of the Perturbed Hard Sphere Chain Theory, PHSC(SW). For the case of pure fluids, a complete expression of the equilibrium residual free energy density as a function of temperature and specific volume is obtained from PHSC(SW) using four parameters, beyond molar mass, which represent sphere diameter σ, sphere mass M/r, characteristic energy ε and reduced well width λ, respectively. For the sake of simplicity, in this work a fixed value of λ equal to 1.455 was used for all components analyzed. This value was chosen based on the work by Hino and Prausnitz [9]. 2.3. NE-PHSC(vdW) and NE-PHSC(SW) The NET-GP results embodied by Eqs. (6) and (7), along with the expressions for the equilibrium free energy obtained by the Perturbed Hard Sphere Chain Theories, allow for the calculation of the pseudo-equilibrium solute content in a glassy polymer, which is kept in contact with a pure gaseous phase at temperature T and pressure p. The corresponding equations derived in that way from PHSC(vdW) or PHSC(SW) will be referred to as NEPHSC(vdW) and NE-PHSC(SW) models, respectively. PE resulting in The solute mass per unit polymer mass Ωsol pseudo-equilibrium conditions when the glass has been exposed to a pure gas at given values of temperature and pressure, is calculated from the phase equilibrium equation, which requires an equal value of the solute chemical potential in the gaseous and in the glassy phases: NE(s)
EQ(g)
PE PE µsol (T, p, Ωsol , ρpol ) = µsol
(T, p)
(10)
In Eq. (10) superscript NE(s) and EQ(g), respectively, label the chemical potential of the solute in the solid phase, calculated from NET-GP model, and in the gaseous phase, evaluated through the equilibrium EoS. The solute chemical potential in the solid glassy phase explicitly depends on the pseudoPE , and thus the pseudo-equilibrium solubility is equilibrium ρpol strongly dependent on the pseudo-equilibrium polymer density. It may be worth to notice that Eq. (10) is derived from the requirement that, under pseudo-equilibrium conditions, the Gibbs free
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energy G reaches a minimum at constant temperature and pressure, while the order parameter of the system is frozen to a value PE in which its time rate of change is no longer appreciable, ρpol EQ
even if it is far from the value ρpol which corresponds to the absolute minimum of the Gibbs free energy G. Detailed knowledge of bulk rheological properties for the polymer–solute mixture would allow to obtain an expression for the evolution law of density (function f in Eq. (3)), and thus to estimate the polymer density that is reached at pseudoequilibrium from known initial conditions, at assigned external constraints. Unfortunately, there are not many data useful to this aim in the open literature, not even for the most common solvent–polymer pairs. Thus, the pseudo-equilibrium polymer PE must be estimated or directly measured for the condensity ρpol ditions of interest. On the other hand, the knowledge of polymer density is rather immediate in the case of small solute concentrations (up to few weight percent): in this case, the effective partial molar volume of solute species in the glassy polymer is negligible with respect to close-packed molar volume of the pure solute component, and polymer density is almost equal to the mass density of the pure unpenetrated polymer at the same temperature and pressure. Therefore, in the limit of low penetrant concentrations, use of PHSC models, either in PHSC(vdW) or PHSC(SW) version, reduces the phase equilibrium equation, Eq. (10), to the following specific form: M M NE(s) PE 0 µsol T, p, Ωsol , ρpol ; σs , , εs , σp , , εp , ksp r s r p M EQ(g) = µsol T, p; σs , (11) , εs r s In Eq. (11) the explicit dependence on pure component PHSC parameters of the solute (σ s , (M/r)s , εs ) and of the polymer (σ p , (M/r)p , εp ), are clearly indicated, in addition to the solute–polymer pair interaction parameter ksp . Eq. (11) can be simply used to calculate directly the pseudo-equilibrium solubility isotherm, once (i) the pure component equilibrium parameters for both polymer and solvent have been determined from independent PVT data, (ii) the binary interaction parameter has 0 is known. It been evaluated and (iii) the polymer density ρpol 0 is a nonis important to remind that dry polymer density ρpol equilibrium property of the polymeric sample which, in general, depends not only on temperature and pressure, but also on the thermo-mechanical history of the sample itself and can thus be different for different samples of the same polymer, prepared through different protocols. This consideration is confirmed by the experimental observation that samples of the same polymer characterized by different thermal, mechanical and sorption histories can accommodate rather different amounts of penetrants, at the same temperature and solute fugacity [29]. 2.4. Solubility calculation and comparison with experimental data Several solute–polymer pairs have been considered, for which apparent solubility data were found in the literature, in
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different conditions of temperature and pressure. In many cases the solubility data are available in the limit of zero pressure, and are presented in the form of infinite dilution solubility coefficient, S0 : S0 = lim
C
p→0 P
(12)
where C is the concentration of gas in the polymer and p is the external gas pressure. The experimental solubility data considered in this work are: (1) Infinite dilution solubility coefficients, S0 , for the CO2 –PC system, in a broad temperature range (70–350 ◦ C) which includes both glassy and rubbery states (Tg of PC = 150 ◦ C). (2) Low pressure CH4 solubility isotherms in PEMA at three temperatures below Tg (25, 35, 45 ◦ C), in order to test the ability of the model to describe correctly the effect of temperature and pressure on apparent solubility. (3) Infinite dilution solubility coefficients S0 , of various gases at 35 ◦ C in a high free volume glassy polymer, Teflon® AF2400. These data allow to evaluate the ability of NEPHSC models to describe the solubility of gases of rather different chemical nature in high free volume glassy polymers. In all the examples here considered, the PHSC EoS parameters for the pure polymers were calculated from the polymer volumetric data in rubbery–melt phase, in relatively large intervals of temperature and pressure, taken from specific collections [30]. The PHSC EoS parameters for the pure solute were calculated based on saturated liquid density and vapor pressure data at different temperatures, measured by Vargaftik [31]. The PHSC parameters σ, M/r and ε found in this way and adopted in the calculation of solubility through the NE-PHSC model, are listed in Tables 1a and 1b, for PHSC(vdW) and PHSC(SW), respectively. The results of the fitting procedure of PVT data with the PHSC EoS used are given in Fig. 1; the specific volume of PC at different temperatures and pressures [30] is compared with the values calculated through the PSCH(vdW) EoS (Fig. 1(a)) and PSCH(SW) EoS (Fig. 1(b)), using the pure component parameters listed in Tables 1a and 1b, which have been chosen to obtain an optimal representation of measured specific volume in the rubbery phase. Experimental data are compared with PHSC model representation also below the glass transition temperature, in order to put in evidence the rather different results obtained for the expected equilibrium polymer volume from the two different versions of PHSC EoS. Indeed, the two equations of state give similar results for specific volume, isothermal compressibility and cubic dilation coefficient, in the equilibrium rubbery phase. In contrast, below the glass transition temperature, close to the experimental value considered in pseudo-equilibrium conditions, the PHSC(vdW) version gives an “equilibrium specific volume” of PC much larger than PHSC(SW) and, correspondingly, the departure from equilibrium resulting from the use of NE-PHSC(vdW) is much smaller than the one obtained by using NE-PHSC(SW).
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Table 1a Pure component parameters for PHSC(vdW) equations of state PHSC(vdW)
˚ σ (A)
M/r (g/mol)
ε/k (K)
AF2400 PC PEMA CH4 C 2 H6 C 3 H8 CO2 N2
4.2 3.707 3.595 4.126 3.916 3.998 2.346 3.85
68 25.74 22.47 16.01 17.75 20.71 10.96 25
257 393 317.5 182.1 206.3 219 128.6 112
Table 1b Pure component parameters for PHSC(SW) equations of state PHSC(SW)
˚ σ (A)
M/r (g/mol)
ε/k (K)
λ
AF2400 PC PEMA CH4 C 2 H6 C 3 H8 CO2 N2
4.055 3.333 3.45 3.672 3.449 3.505 2.484 3.52
98 32.05 32.78 16.01 17.79 20.63 16.26 27.62
240 321 290.5 164.9 164.9 199.6 145.11 108
1.455 1.455 1.455 1.455 1.455 1.455 1.455 1.455
2.5. Infinite dilution CO2 solubility coefficient in PC Experimental data for CO2 infinite dilution solubility coefficient, S0 , in PC, in the temperature range from 70 to 350 ◦ C,
Fig. 2. Infinite dilution solubility coefficient for CO2 in PC. Experimental data from ref. [32] are compared with model correlations in equilibrium and nonequilibrium versions for (a) PHSC(vdW) and (b) PHSC(SW).
Fig. 1. Results from fitting of rubbery volumetric data for PC with PHSC(vdW) EoS (a) and PHSC(SW) EoS (b): 0.1 MPa (); 59 MPa (); 118 MPa (); 177 MPa ().
are shown in Fig. 2 as measured by Wang and Kamiya [32]. The temperature dependence of S0 varies remarkably between the glassy and the rubbery region. Indeed, solute–polymer mixing is a highly exothermic process below Tg , while above Tg , in the rubbery region, the gas solubility is less affected by temperature. In Fig. 2 experimental data are compared with EoS correlations from PHSC(vdW) (Fig. 2(a)) and PHSC(SW) (Fig. 2(b)); the pure component parameters listed in Tables 1a and 1b have been used and the CO2 –PC binary interaction parameter has been chosen so to represent well the gas solubility coefficient at the glass transition temperature (Tg = 150 ◦ C) and has been kept constant at all temperatures. The binary parameter values obtained in this way are ksp = 0.05 and 0.075 for PHSC(vdW) and PHSC(SW), respectively. The temperature dependence of the solubility coefficients obtained from the two PHSC versions are essentially equivalent, although a slightly lower temperature sensitivity, closer to the experimental evidence in equilibrium phase, is obtained for the case of PHSC(vdW). As it is expected, the data calculated by the equilibrium EoS for temperatures below Tg fail to represent, even approximately, the temperature
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dependence experimentally observed. The effect of the extra free volume present in the glassy state on CO2 solubility can be obtained by using the NET-GP approach, through the use of NE-PHSC(vdW) or NE-PHSC(SW) by solving Eq. (11), with the same pure component parameters used for the calculation of solubility in the rubbery phase (Tables 1a and 1b) and the same value of the binary parameter ksp . The value of pure polymer 0 , at the various temperatures was taken from glassy density, ρpol the work of Zoller and Walsh [30]. Results of pseudo-equilibrium solubility calculations are shown by the solid line in Fig. 2. The two equations here considered show rather different predictions for the apparent solubility calculated in glassy PC. Indeed, remarkably good predictions are obtained from the use of NE-PHSC(SW), which well compares with experimental data, while poor predictions result from the use of NE-PHSC(vdW) model, which indeed shows smaller deviations for the solubility in the rubbery phase. In particular, the predictions based on NE-PHSC(vdW) model underestimate the effect that temperature has on S0 in the glassy region. 2.6. CH4 –PEMA solubility isotherms In Fig. 3 the experimental solubility isotherms are shown for CH4 in PEMA, at 25, 35 and 45 ◦ C [33], and the predictions obtained from NE-PHSC(vdW) and NE-PHSC(SW) models. The experimental solubility isotherms are concave towards the pressure axis and follow the typical trend observed in glassy polymers. In Fig. 3(a) the dashed and solid lines represent predictions of methane content in PEMA at 25 ◦ C from NEPHSC(vdW) and NE-PHSC(SW) models, respectively, and they are compared with experimental data. The pure polymer density value is taken for PEMA from Zoller and Walsh [30], and the pure component parameters for CH4 and PEMA are listed in Tables 1a and 1b; the default value of interaction parameter (ksp = 0) has been applied in both cases. Also for this polymer-penetrant pair the difference between the results obtained from NE-PHSC(vdW) and NE-PHSC(SW) is quite evident. Both non-equilibrium models predict the decrease of solubility coefficient with solute pressure, i.e. the negative concavity of the solubility isotherm. In contrast, a clear difference exists between the pseudo-equilibrium solute contents predicted by NE-PHSC(vdW) and by NE-PHSC(SW) models: the solubility calculated from NE-PHSC(SW) deviates from the experimental values by approximately 45%, while results from the NE-PHSC(vdW) model are three times lower than the experimental data. A more accurate representation of solute content for the pressure range considered can be obtained, of course, from both models when convenient values of binary interaction parameter ksp are introduced in non-equilibrium calculations. In Fig. 3(b) experimental data from ref. [33] show the pseudoequilibrium methane content in glassy PEMA up to 3.5 MPa, in the temperature range from 25 to 45 ◦ C. In the same figure, experimental data are also compared to non-equilibrium model correlations obtained by adjusting the binary interaction parameter ksp to a temperature independent best-fitting value, equal to −0.095 and to −0.040 for NE-PHSC(vdW) and NE-PHSC(SW), respectively. An acceptable representation
Fig. 3. (a) Solubility isotherm for CH4 in PEMA at T = 25 ◦ C. Comparisons are shown for experimental data from ref. [33] with predictions from NEPHSC(vdW) and NE-PHSC(SW) [ksp = 0 in both cases] and (b) solubility isotherms for CH4 in PEMA. Comparisons are shown for experimental data from ref. [33] with correlations from NE-PHSC(vdW) [ksp = −0.095] and NEPHSC(SW) [ksp = −0.040].
of the experimental solubility has been obtained by both nonequilibrium models considered, although a closer look reveals that the NE-PHSC(vdW) model predicts a lower temperature sensitivity of solubility with respect to NE-PHSC(SW) model and, therefore, a lower sorption enthalpy for the CH4 –PEMA mixture in this temperature range. The effect of temperature on apparent solubility is underestimated by both models; however, the change with temperature of the pseudo-equilibrium solute content predicted by NE-PHSC(SW) is closer to the experimental values than the one calculated by NE-PHSC(vdW), and shows a maximum deviation smaller than 20%. 2.7. Infinite dilution gas solubility coefficient in Teflon® AF2400 Pseudo-equilibrium solubility coefficient for N2 , CH4 , C2 H6 and C3 H8 in AF2400 at 35 ◦ C are shown in Fig. 4 after De Angelis et al. [34] and are compared with the predictions obtained by equilibrium and non-equilibrium PHSC equations of state.
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ficients calculated from NE-PHSC(vdW) and NE-PHSC(SW) simply based on the first order approximation for the binary parameter, ksp = 0, are already very close to the experimental values. Quite satisfactory predictions, also in quantitative terms, are obtained by NE-PHSC models with ksp = 0, for the solubility coefficients of light gases, while the same procedure clearly overestimates the gas solubility of alkane vapors in the fluorinated polymer matrix. A clear justification of the latter behavior can be reasonably found in the mixing rule for the characteristic energy, which in its default version (ksp = 0) is not adequate to represent the specific interaction between alkane solute molecules and perfluorinated polymer segments [35]. It is also relevant to observe that no significant difference is evident in this case between the predictions obtained from the two different versions of the NE-PHSC theories. 3. Discussion and conclusion
Fig. 4. Infinite dilution solubility coefficient for N2 , CH4 , C2 H6 and C3 H8 in AF2400 at T = 35 ◦ C. Experimental data from ref. [34] are compared with predictions from (a) PHSC(vdW) EoS and NE-PHSC(vdW) and (b) PHSC(SW) EoS and NE-PHSC(SW) [ksp = 0 in all cases].
Specific volume data for AF2400 above the glass transition temperature were taken from ref. [34] and have been used here to retrieve the pure component PHSC parameters, which are indicated in Tables 1a and 1b. The pure polymer density of glassy AF2400 at 35 ◦ C is also reported in ref. [34] and its value was used to calculate the pseudo-equilibrium solubility coefficient. In all calculations performed in this case, only the default value of the binary interaction parameter (ksp = 0) has been considered. Differences between experimental solubility coefficients and predictions from the equilibrium EoS are quite evident. By using the equilibrium PHSC(vdW) and PHSC(SW) equations the predicted solubility coefficients are almost two orders of magnitude lower than the experimental values. This remarkable additional capacity of AF2400 to accommodate sorbed gas molecules in excess over the thermodynamic equilibrium value is indeed a consequence of the high extra free volume which is frozen in the glassy structure of AF2400. More remarkably, from Fig. 4 one may also notice that pseudo-equilibrium gas solubility coef-
Application to Perturbed Hard Sphere Chain Theories of the non-equilibrium thermodynamic approach for glassy polymers has been presented and the resulting models have been used to calculate and correlate gas solubility in different glassy polymers. Similarly to the case of the NELF model developed earlier along the same lines, the NE-PHSC(vdW) and NE-PHSC(SW) models allow for calculation of low pressure pseudo-equilibrium gas solubility in glassy polymers based on pure component equilibrium PVT data for pure polymers and pure solutes, and using the available values for the unpenetrated polymer density in the glassy state. The PHSC based non-equilibrium models are indeed able to interpret in details the thermodynamic properties of polymer–solute systems in the glassy phase. In addition, they offer the advantage of having pure component parameters strictly related to specific molecular properties, making it possible, in case of need, to obtain their estimates based on the knowledge of the species chemical structure. The comparison between model calculations and experimental data presented in this work represents the typical behavior which is observed also for other polymer-penetrant pairs not explicitly shown for brevity sake. The examples reported confirm the ability of the model to represent peculiar temperature and pressure dependence of gas solubility coefficient in glassy polymers. As indicated by the NE-PHSC(SW) calculation of CO2 solubility coefficient in PC, as function of temperature, correct representation of temperature effect on gas solubility below the glass transition temperature can be obtained when the binary interaction parameter between solute and polymer species is retrieved from equilibrium (T > Tg ) solubility data. Under equilibrium conditions, above Tg , both PHSC(vdW) and PHSC(SW) offer good predictions for the solubility of low molecular weight gases in the polymer phase. Below the glass transition temperature, however, the corresponding non-equilibrium models are not equivalent. For NE-PHSC(vdW) a smaller departure from equilibrium volume is calculated, which is likely smaller than the actual value, albeit not subject to experimental verification; that leads generally to an underestimation of the pseudo-solubility in the glassy phase. On the contrary, NE-PHSC(SW) apparently offers a better estimation of the departure from equilibrium and
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thus, better evaluations of gas solubility in the glassy polymer are obtained. It is also relevant to observe that the high gas solubility coefficient measured in high free volume glassy polymers can be correctly represented by NE-PHSC models, by using the actual polymer density; that is sufficient to account for the excess volume, over the true equilibrium polymer volume, available to accommodate larger quantities of solute molecules. Finally, differences exist between the predictive ability of the two versions of PHSC theory here examined. Indeed, the square well version of the EoS offers satisfactory representations of thermodynamic properties of glassy states in all the cases considered; in contrast, in most of the conditions examined, poor results were obtained from the use of the PHSC version based on van der Waals type of interaction potential for the dispersion term. Remarkably, in the rubbery state the representations of thermodynamic properties of polymeric phases resulting from both PHSC EoS are substantially equivalent, while below Tg better results indeed come from use of the square well form of interaction potential. Most likely, such a difference is associated to better free energy estimates offered for much denser phases by PHSC(SW) with respect to PHSC(vdW). The latter conclusion is also consistent with the fact that comparable results are obtained by both NE-PHSC(vdW) and NE-PHSC(SW) models when relatively low density phases are considered, as it is the case of a high free volume glassy polymer as AF2400. Acknowledgments This research has been partially supported by the Italian Ministry for Education, University and Research, PRIN 2004, and by the EU Strep MULTIMATDESIGN (Contract No. NMP3CT-2005-013644). References [1] [2] [3] [4]
P.J. Flory, J. Chem. Phys. 9 (1941) 660–661. M.L. Huggins, J. Chem. Phys. 9 (1941) 440. M.M. Abrams, J.M. Prausnitz, AIChE J. 21 (1975) 116–128. T. Oishi, J.M. Prausnitz, Ind. Eng. Chem. Proc. Des. Dev. 17 (1978) 333–339. [5] H.S. Elbro, A. Fredenslund, P. Rasmussen, Macromolecules 23 (1990) 4707–4714.
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[6] S. Beret, J.M. Prausnitz, AIChE J. 21 (1975) 1123–1132. [7] Y. Song, S.M. Lambert, J.M. Prausnitz, Macromolecules 27 (1994) 441–448. [8] Y. Song, T. Hino, S.M. Lambert, J.M. Prausnitz, Fluid Phase Equilib. 117 (1996) 69–76. [9] T. Hino, J.M. Prausnitz, Fluid Phase Equilib. 138 (1997) 105–130. [10] D. Patterson, Macromolecules 2 (1969) 672–677. [11] P.J. Flory, Discuss. Faraday Soc. 49 (1970) 7. [12] I.C. Sanchez, R.H. Lacombe, Macromolecules 11 (1978) 1145–1156. [13] S.H. Huang, M. Radosz, Ind. Eng. Chem. Res. 29 (1990) 2284– 2294. [14] J.W. Kang, J.H. Lee, K.P. Yoo, C.S. Lee, Fluid Phase Equilib. 194 (2002) 77–86. [15] A.S. Michaels, W.R. Vieth, J.A. Barrie, J. Appl. Phys. 34 (1963) 1–12. [16] R.G. Wissinger, M.E. Paulaitis, J. Polym. Sci. Polym. Phys. Ed. 25 (1987) 2497–2510. [17] T.A. Barbari, R.M. Conforti, J. Polym. Sci. Polym. Phys. Ed. 30 (1992) 1261–1271. [18] D. Boudouris, C. Panayiotou, Macromolecules 31 (1996) 7915–7920. [19] F. Doghieri, G.C. Sarti, Macromolecules 29 (1996) 7885–7896. [20] G.C. Sarti, F. Doghieri, Chem. Eng. Sci. 53 (1998) 3435–3447. [21] F. Doghieri, M. Canova, G.C. Sarti, ACS Symp. Ser. 733 (1999) 179–193. [22] M. Giacinti Baschetti, F. Doghieri, G.C. Sarti, Ind. Eng. Chem. Res. 40 (2001) 3027–3037. [23] F. Doghieri, M. Quinzi, D.G. Rethwisch, G.C. Sarti, ACS Symp. Ser. 876 (2004) 74–90. [24] M. Giacinti Baschetti, M.G. De Angelis, F. Doghieri, G.C. Sarti, in: M.A. Galan, E. Martin del Valle (Eds.), Chemical Engineering: Trends and Developments, Wiley, Chichester (UK), 2005, pp. 41–61. [25] F. Doghieri, M. Quinzi, D.G. Rethwisch, G.C. Sarti, in: B.D., Freeman, Y. Yampolskii, I. Pinnau (Eds.), Materials Science of Membranes for Gas and Vapor Separation, Wiley, Chichester (UK), in press. [26] B.D. Coleman, M.E. Gurtin, J. Chem. Phys. 47 (1967) 597–613. [27] G. Astarita, Thermodynamics, Plenum Press, New York, USA, 1989. [28] J. Chang, S.I. Sandler, Mol. Phys. 81 (1994) 735–744. [29] G.K. Fleming, W.J. Koros, Macromolecules 23 (1990) 1353–1360. [30] P. Zoller, D. Walsh, Standard Pressure–Volume–Temperature Data for Polymers, Technomic, Lancaster, 1995. [31] N.B. Vargaftik, Handbook of Physical Properties of Liquids and Gases: Pure substances and Mixtures, second ed., Hemisphere Publishing, Washington, 1975. [32] J.-S. Wang, Y. Kamiya, J. Polym. Sci. Polym. Phys. Ed. 38 (2000) 883–888. [33] J.S. Chiou, D.R. Paul, J. Membr. Sci. 45 (1989) 167–189. [34] M.G. De Angelis, T.C. Merkel, V.I. Bondar, B.D. Freeman, F. Doghieri, G.C. Sarti, Macromolecules 35 (2002) 1276–1288. [35] R.S. Prabhakar, M.G. De Angelis, G.C. Sarti, B.D. Freeman, M.C. Coughlin, Macromolecules 38 (2005) 7043–7055.
Solubility of gases and vapors in glassy polymers modelled through non-equilibrium PHSC theory Ferruccio Doghieri, Maria Grazia De Angelis, Marco Giacinti Baschetti, Giulio C. Sarti ∗ DICMA-Dipartimento di Ingegneria Chimica, Mineraria e delle Tecnologie Ambientali, Universit`a di Bologna viale Risorgimento 2, 40136 Bologna, Italy Received 10 November 2005; received in revised form 23 December 2005; accepted 23 December 2005 Available online 3 February 2006
Abstract Well established equation-of-state (EoS) models from Perturbed Hard Sphere Chain Theory have been adopted within the framework of non-equilibrium thermodynamics for glassy polymers (NET-GP) to describe the solubility of small penetrants in glassy polymers. The procedure presented here parallels the one already applied to the Lattice Fluid EoS, obtaining the Non-Equilibrium Lattice Fluid (NELF) model. The solubility is calculated as pseudo-equilibrium solute content in the glassy polymeric phase at fixed temperature and solute fugacity, and is compared to the experimental data. The NE-PHSC model predictions require pure component and binary EoS parameters, together with the value of the glassy polymer density during sorption. Several common glassy polymers, as well as a high free volume fluoropolymer, have been considered to test the model ability to represent the thermodynamic properties of glassy gas–polymer mixtures. A comparison between the performance of two NE-PHSC models, endowed with different expressions for the interaction potential, is also presented, offering interesting conclusions on their accuracy in the representation of the thermodynamic properties of relatively high-density systems. The results obtained in this work show that the NET-GP approach provides a reliable description of the properties of homogeneous glassy mixtures. © 2006 Elsevier B.V. All rights reserved. Keywords: Solubility; Glassy polymers; Non-equilibrium thermodynamics
1. Introduction Vapor–liquid or vapor–solid equilibria for polymeric solutions in melt or rubbery state are commonly calculated using well established thermodynamic tools to represent the properties of both vapor and condensed phases; in particular, for the equilibrium polymeric phase one can adopt either expressions for the excess Gibbs free energy and activity coefficients [1–5], or equations of state for the mixtures [6–14]. Such models offer valuable correlations for the solubility of low molecular weight species in rubbery polymers, which can be confidently used even for predictive purposes, based only on pure component data and on some information about the thermodynamic properties of the binary mixture. In contrast, the pseudo-equilibrium gas and vapor solubility in amorphous glassy polymers cannot be represented by means
∗
Corresponding author. Tel.: +39 0512093142; fax: +39 051581200. E-mail address: [email protected] (G.C. Sarti).
0378-3812/$ – see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2005.12.040
of the same equilibrium thermodynamic expressions, which indeed fail to describe even qualitative aspects of the observed trends. Empirical or semi-empirical tools are still widely used to correlate data of pseudo-equilibrium solute content in glassy polymers, the most popular and successful of which is, by far, the Dual Mode Sorption model (DMS) [15]. While this model allows to obtain good correlations and useful insights into the polymeric structure when specific experimental data are available, it cannot be used in a predictive mode to evaluate gas or vapor apparent solubility in glassy polymer phases. The main difficulty in building a predictive model for glassy phases lies in the fact that the thermodynamic properties of polymeric systems below the glass transition temperature are not only affected by the values of temperature, pressure and solute fugacity imposed through the external phase. Indeed, different values of apparent solubility in a given glassy polymer can be observed, at the same temperature pressure and solute fugacity, depending on the specific pre-treatments experienced such as quenching, annealing or preswelling. The non-equilibrium nature of glassy polymer phases needs thus to be properly considered in order to
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develop a model suitable to describe the relevant thermodynamic properties. Qualitatively, we can say that the departure from the equilibrium state of a glassy polymer results from the hindered mobility of polymer chains which does not allow the system to access all possible microstates and ultimately prevents it from reaching the free volume that would lead to the absolute minimum of its Gibbs free energy, i.e. the equilibrium free volume. The glassy polymer volume is thus a non-equilibrium quantity, whose value must be obtained from direct measurement or from bulk rheology models if available, and differs from the equilibrium value given by the equation of state. The nonequilibrium state of glassy phases does not allow direct use of well known results from classical and statistical thermodynamics of mixtures, and inspired the development of several specific approaches [16–18]. It is beyond the scope of this work to offer a detailed comparison of the different models cited. We simply remind that remarkable results were obtained using the Non-Equilibrium Lattice Fluid (NELF) model to predict and/or correlate pseudo-equilibrium gas solubility in glassy polymers under many different conditions [19–22]. The NELF model is a special application of the non-equilibrium thermodynamics for glassy polymers (NET-GP) [19–25], obtained from the proper extension of an existing equilibrium model, i.e. the Lattice Fluid (LF) theory. NET-GP is not restricted to any particular equation of state, but rather indicates the general relationships existing between the thermodynamic properties above and below the glass transition temperature and can be applied to all equations of state suitable to describe polymeric systems at equilibrium. In the present work we like to present an application of the NET-GP approach based on the use of Perturbed Hard Sphere Chains (PHSC) Theory developed by professor Prausnitz [6–9], that we would like to acknowledge for his major contributions to this field, his remarkable work, results and methods from which we have all learned. Our research group would also like to express gratitude for the knowledge and enthusiasm that professor Prausnitz transmitted to the students during his memorable lectures at the University of Bologna in Spring 2004. Indeed PHSC EoS has been rather successful to describe not only the behavior of mixtures of low molecular weight species but also that of polymeric mixtures under equilibrium conditions, i.e. above the glass transition temperature, and is thus a promising candidate to represent also the properties of nonequilibrium glassy phases through NET-GP theory. We will also compare the results obtained by using different versions of the PHSC models, that adopt different expression for the dispersion forces term in the residual Helmohltz free energy. In addition, PHSC EoS offers the advantage to consider pure component parameters directly related to molecular properties, which can therefore possibly be predicted based on molecular modeling tools. Examples of predictions and correlations are presented and discussed hereafter for the solubility of light gases in different conventional glassy polymers, such as bisphenol-A poly(carbonate) (PC) and poly(ethylmethacrylate) (PEMA), as well as a commercial high free volume perfluorinated glassy
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polymer, poly(2,2-bistrifluoromethyl-4,5-difluoro-1,3-dioxoleco-tetrafluoroethylene) (Teflon® AF2400). 2. Theoretical background 2.1. Non-equilibrium thermodynamics for glassy polymers The basic assumptions and general results of NET-GP approach, developed to represent thermodynamic properties of glassy polymer phases, have been reported in details in previous publications [19–25]. The main results which are relevant to obtain solubility isotherms in glassy polymers are briefly summarized hereafter for clarity sake. The NET-GP model relies upon the assumption that glassy polymers are homogeneous, isotropic and amorphous phases whose properties depend not only on composition and externally imposed conditions such as temperature and pressure, but also on convenient order parameters which describe the departure from equilibrium of the system. Several order parameters can be introduced, in principle, to properly describe the non-equilibrium state Σ of glassy polymer systems; however, the NET-GP approach focuses the attention on one single parameter which is meant to represent the departure of the glassy polymeric volume from the corresponding equilibrium value. The complete set of independent state variables required for the description of the thermodynamic properties of non-equilibrium glassy states is chosen as follows: = {T, p, Ωsol , ρpol } (1) where T, p and Ωsol are temperature, pressure and solute-topolymer mass ratio, while ρpol is the polymer mass per unit volume. In Eq. (1) ρpol is thus the order parameter which allows to measure the departure from equilibrium conditions when EQ compared to the corresponding equilibrium value ρpol . The latter quantity depends, of course, on system composition, temperature and pressure only, and it can be calculated through the condition of minimum Gibbs free energy for the system: ∂G EQ =0 (2) ρpol = ρpol (T, p, Ωsol ) ⇔ ∂ρpol T,p,Ωsol The second key assumption of the model is that the order parameter ρpol evolves in time according to a rate that depends only on the state of the system, i.e. the following equation holds: dρpol = f (T, p, Ωsol , ρpol ) dt
(3)
Eq. (3) can be readily derived for the system when the bulk rheology of the polymeric phase follows a Voigt model. In view of the assumption represented by Eq. (3), the order parameter ρpol plays the role of an internal state variable for the system, and the basic thermodynamic relations of the NET-GP model are directly derived by applying well established thermodynamic results for systems endowed with internal state variables [26,27]. The most important consequences of the above procedure lie in the fact that the Helmholtz free energy density, aNE , and solute chemical potential, µNE sol , in the non-equilibrium glassy phase depend on composition, temperature and polymer mass
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density only, and their values are not affected by the pressure of the system [20]: NE ∂a =0 (4) ∂p T,Ωsol ,ρpol µNE sol
=
∂aNE ∂ρsol
(5) T,p,ρpol
In view of the above conclusions, the following relationships are easily derived between aNE (or µNE sol ), in a general non-equilibrium state, and the corresponding properties aEQ (or EQ µsol ) evaluated on the true equilibrium curve: aNE (T, p, Ωsol , ρpol ) = aEQ (T, Ωsol , ρpol ) EQ
µNE sol (T, p, Ωsol , ρpol ) = µsol (T, Ωsol , ρpol )
(6) (7)
EQ
In Eqs. (6) and (7) aEQ and µsol indicate the equilibrium values of the Helmholtz free energy density and solute chemical potential in the polymer–solute mixture at a given temperature and species densities. Eqs. (6) and (7) express a simple, nontrivial and relevant result indicating that the free energy and solute chemical potential can be evaluated in all accessible nonequilibrium states of the system, starting from knowledge of their values on the restricted equilibrium domain, identified in terms of temperature, composition and polymer density. It is important to notice that the above equations represent a general non-equilibrium thermodynamic result, and any reliable expression for the equilibrium properties of amorphous phases could be specifically used in Eqs. (6) and (7) to finally describe the non-equilibrium thermodynamic properties of the solute–polymer mixture in the glassy phase. In this work we will consider the non-equilibrium free energy functions which can be obtained from Eq. (6) when Perturbed Hard Sphere Chain Theories are considered for the equilibrium properties. 2.2. PHSC(vdW) and PHSC(SW) EoS Perturbed Hard Sphere Chain Theory has been introduced by Prausnitz and co-workers and several specific forms of the free energy expression as function of system volume, temperature and composition have in fact been developed over the years [6–9], which essentially differ from one another in the expression used for the pair interaction potential between chain segments. As it is well known, in all PHSC models, the residual Helmholtz free energy is expressed as the sum of a reference term, accounting for chain connectivity and a hard sphere interaction and perturbation term, which represents the contributions of mean-field forces: ares = aref + apert
(8)
The first PHSC model here considered is the simplified Perturbed Hard Sphere Chain Theory by Song et al. [8] in which the reference term uses the modified Chiew equation of state, while the perturbation term is of the van der Waals type. In what follows, this model is referred to as PHSC(vdW) equation of
state. The pure component parameters involved in the expression for the Helmholtz free energy are sphere diameter σ, mass per segment M/r, and characteristic energy for the pair interaction potential ε, beyond the species molar mass. For the case of mixtures, the free energy expressions are obtained through the introduction of mixing rules and the binary parameters contained therein. We will here consider only the binary interaction parameters kij appearing in the expression of the characteristic energy for the interaction between pairs of unlike segments: √ (9) εij = (1 − kij ) εii εjj In addition to the specific form of the PHSC(vdW) model, a second version of the PHSC equation of state will be considered in the following, which was introduced by Hino and Prausnitz [9]. The latter version was obtained by replacing the simple van der Waals perturbation term with a second order perturbation term for a square well potential of variable width, following Chang and Sandler [28]. This will be referred to as “square well version” of the Perturbed Hard Sphere Chain Theory, PHSC(SW). For the case of pure fluids, a complete expression of the equilibrium residual free energy density as a function of temperature and specific volume is obtained from PHSC(SW) using four parameters, beyond molar mass, which represent sphere diameter σ, sphere mass M/r, characteristic energy ε and reduced well width λ, respectively. For the sake of simplicity, in this work a fixed value of λ equal to 1.455 was used for all components analyzed. This value was chosen based on the work by Hino and Prausnitz [9]. 2.3. NE-PHSC(vdW) and NE-PHSC(SW) The NET-GP results embodied by Eqs. (6) and (7), along with the expressions for the equilibrium free energy obtained by the Perturbed Hard Sphere Chain Theories, allow for the calculation of the pseudo-equilibrium solute content in a glassy polymer, which is kept in contact with a pure gaseous phase at temperature T and pressure p. The corresponding equations derived in that way from PHSC(vdW) or PHSC(SW) will be referred to as NEPHSC(vdW) and NE-PHSC(SW) models, respectively. PE resulting in The solute mass per unit polymer mass Ωsol pseudo-equilibrium conditions when the glass has been exposed to a pure gas at given values of temperature and pressure, is calculated from the phase equilibrium equation, which requires an equal value of the solute chemical potential in the gaseous and in the glassy phases: NE(s)
EQ(g)
PE PE µsol (T, p, Ωsol , ρpol ) = µsol
(T, p)
(10)
In Eq. (10) superscript NE(s) and EQ(g), respectively, label the chemical potential of the solute in the solid phase, calculated from NET-GP model, and in the gaseous phase, evaluated through the equilibrium EoS. The solute chemical potential in the solid glassy phase explicitly depends on the pseudoPE , and thus the pseudo-equilibrium solubility is equilibrium ρpol strongly dependent on the pseudo-equilibrium polymer density. It may be worth to notice that Eq. (10) is derived from the requirement that, under pseudo-equilibrium conditions, the Gibbs free
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energy G reaches a minimum at constant temperature and pressure, while the order parameter of the system is frozen to a value PE in which its time rate of change is no longer appreciable, ρpol EQ
even if it is far from the value ρpol which corresponds to the absolute minimum of the Gibbs free energy G. Detailed knowledge of bulk rheological properties for the polymer–solute mixture would allow to obtain an expression for the evolution law of density (function f in Eq. (3)), and thus to estimate the polymer density that is reached at pseudoequilibrium from known initial conditions, at assigned external constraints. Unfortunately, there are not many data useful to this aim in the open literature, not even for the most common solvent–polymer pairs. Thus, the pseudo-equilibrium polymer PE must be estimated or directly measured for the condensity ρpol ditions of interest. On the other hand, the knowledge of polymer density is rather immediate in the case of small solute concentrations (up to few weight percent): in this case, the effective partial molar volume of solute species in the glassy polymer is negligible with respect to close-packed molar volume of the pure solute component, and polymer density is almost equal to the mass density of the pure unpenetrated polymer at the same temperature and pressure. Therefore, in the limit of low penetrant concentrations, use of PHSC models, either in PHSC(vdW) or PHSC(SW) version, reduces the phase equilibrium equation, Eq. (10), to the following specific form: M M NE(s) PE 0 µsol T, p, Ωsol , ρpol ; σs , , εs , σp , , εp , ksp r s r p M EQ(g) = µsol T, p; σs , (11) , εs r s In Eq. (11) the explicit dependence on pure component PHSC parameters of the solute (σ s , (M/r)s , εs ) and of the polymer (σ p , (M/r)p , εp ), are clearly indicated, in addition to the solute–polymer pair interaction parameter ksp . Eq. (11) can be simply used to calculate directly the pseudo-equilibrium solubility isotherm, once (i) the pure component equilibrium parameters for both polymer and solvent have been determined from independent PVT data, (ii) the binary interaction parameter has 0 is known. It been evaluated and (iii) the polymer density ρpol 0 is a nonis important to remind that dry polymer density ρpol equilibrium property of the polymeric sample which, in general, depends not only on temperature and pressure, but also on the thermo-mechanical history of the sample itself and can thus be different for different samples of the same polymer, prepared through different protocols. This consideration is confirmed by the experimental observation that samples of the same polymer characterized by different thermal, mechanical and sorption histories can accommodate rather different amounts of penetrants, at the same temperature and solute fugacity [29]. 2.4. Solubility calculation and comparison with experimental data Several solute–polymer pairs have been considered, for which apparent solubility data were found in the literature, in
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different conditions of temperature and pressure. In many cases the solubility data are available in the limit of zero pressure, and are presented in the form of infinite dilution solubility coefficient, S0 : S0 = lim
C
p→0 P
(12)
where C is the concentration of gas in the polymer and p is the external gas pressure. The experimental solubility data considered in this work are: (1) Infinite dilution solubility coefficients, S0 , for the CO2 –PC system, in a broad temperature range (70–350 ◦ C) which includes both glassy and rubbery states (Tg of PC = 150 ◦ C). (2) Low pressure CH4 solubility isotherms in PEMA at three temperatures below Tg (25, 35, 45 ◦ C), in order to test the ability of the model to describe correctly the effect of temperature and pressure on apparent solubility. (3) Infinite dilution solubility coefficients S0 , of various gases at 35 ◦ C in a high free volume glassy polymer, Teflon® AF2400. These data allow to evaluate the ability of NEPHSC models to describe the solubility of gases of rather different chemical nature in high free volume glassy polymers. In all the examples here considered, the PHSC EoS parameters for the pure polymers were calculated from the polymer volumetric data in rubbery–melt phase, in relatively large intervals of temperature and pressure, taken from specific collections [30]. The PHSC EoS parameters for the pure solute were calculated based on saturated liquid density and vapor pressure data at different temperatures, measured by Vargaftik [31]. The PHSC parameters σ, M/r and ε found in this way and adopted in the calculation of solubility through the NE-PHSC model, are listed in Tables 1a and 1b, for PHSC(vdW) and PHSC(SW), respectively. The results of the fitting procedure of PVT data with the PHSC EoS used are given in Fig. 1; the specific volume of PC at different temperatures and pressures [30] is compared with the values calculated through the PSCH(vdW) EoS (Fig. 1(a)) and PSCH(SW) EoS (Fig. 1(b)), using the pure component parameters listed in Tables 1a and 1b, which have been chosen to obtain an optimal representation of measured specific volume in the rubbery phase. Experimental data are compared with PHSC model representation also below the glass transition temperature, in order to put in evidence the rather different results obtained for the expected equilibrium polymer volume from the two different versions of PHSC EoS. Indeed, the two equations of state give similar results for specific volume, isothermal compressibility and cubic dilation coefficient, in the equilibrium rubbery phase. In contrast, below the glass transition temperature, close to the experimental value considered in pseudo-equilibrium conditions, the PHSC(vdW) version gives an “equilibrium specific volume” of PC much larger than PHSC(SW) and, correspondingly, the departure from equilibrium resulting from the use of NE-PHSC(vdW) is much smaller than the one obtained by using NE-PHSC(SW).
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Table 1a Pure component parameters for PHSC(vdW) equations of state PHSC(vdW)
˚ σ (A)
M/r (g/mol)
ε/k (K)
AF2400 PC PEMA CH4 C 2 H6 C 3 H8 CO2 N2
4.2 3.707 3.595 4.126 3.916 3.998 2.346 3.85
68 25.74 22.47 16.01 17.75 20.71 10.96 25
257 393 317.5 182.1 206.3 219 128.6 112
Table 1b Pure component parameters for PHSC(SW) equations of state PHSC(SW)
˚ σ (A)
M/r (g/mol)
ε/k (K)
λ
AF2400 PC PEMA CH4 C 2 H6 C 3 H8 CO2 N2
4.055 3.333 3.45 3.672 3.449 3.505 2.484 3.52
98 32.05 32.78 16.01 17.79 20.63 16.26 27.62
240 321 290.5 164.9 164.9 199.6 145.11 108
1.455 1.455 1.455 1.455 1.455 1.455 1.455 1.455
2.5. Infinite dilution CO2 solubility coefficient in PC Experimental data for CO2 infinite dilution solubility coefficient, S0 , in PC, in the temperature range from 70 to 350 ◦ C,
Fig. 2. Infinite dilution solubility coefficient for CO2 in PC. Experimental data from ref. [32] are compared with model correlations in equilibrium and nonequilibrium versions for (a) PHSC(vdW) and (b) PHSC(SW).
Fig. 1. Results from fitting of rubbery volumetric data for PC with PHSC(vdW) EoS (a) and PHSC(SW) EoS (b): 0.1 MPa (); 59 MPa (); 118 MPa (); 177 MPa ().
are shown in Fig. 2 as measured by Wang and Kamiya [32]. The temperature dependence of S0 varies remarkably between the glassy and the rubbery region. Indeed, solute–polymer mixing is a highly exothermic process below Tg , while above Tg , in the rubbery region, the gas solubility is less affected by temperature. In Fig. 2 experimental data are compared with EoS correlations from PHSC(vdW) (Fig. 2(a)) and PHSC(SW) (Fig. 2(b)); the pure component parameters listed in Tables 1a and 1b have been used and the CO2 –PC binary interaction parameter has been chosen so to represent well the gas solubility coefficient at the glass transition temperature (Tg = 150 ◦ C) and has been kept constant at all temperatures. The binary parameter values obtained in this way are ksp = 0.05 and 0.075 for PHSC(vdW) and PHSC(SW), respectively. The temperature dependence of the solubility coefficients obtained from the two PHSC versions are essentially equivalent, although a slightly lower temperature sensitivity, closer to the experimental evidence in equilibrium phase, is obtained for the case of PHSC(vdW). As it is expected, the data calculated by the equilibrium EoS for temperatures below Tg fail to represent, even approximately, the temperature
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dependence experimentally observed. The effect of the extra free volume present in the glassy state on CO2 solubility can be obtained by using the NET-GP approach, through the use of NE-PHSC(vdW) or NE-PHSC(SW) by solving Eq. (11), with the same pure component parameters used for the calculation of solubility in the rubbery phase (Tables 1a and 1b) and the same value of the binary parameter ksp . The value of pure polymer 0 , at the various temperatures was taken from glassy density, ρpol the work of Zoller and Walsh [30]. Results of pseudo-equilibrium solubility calculations are shown by the solid line in Fig. 2. The two equations here considered show rather different predictions for the apparent solubility calculated in glassy PC. Indeed, remarkably good predictions are obtained from the use of NE-PHSC(SW), which well compares with experimental data, while poor predictions result from the use of NE-PHSC(vdW) model, which indeed shows smaller deviations for the solubility in the rubbery phase. In particular, the predictions based on NE-PHSC(vdW) model underestimate the effect that temperature has on S0 in the glassy region. 2.6. CH4 –PEMA solubility isotherms In Fig. 3 the experimental solubility isotherms are shown for CH4 in PEMA, at 25, 35 and 45 ◦ C [33], and the predictions obtained from NE-PHSC(vdW) and NE-PHSC(SW) models. The experimental solubility isotherms are concave towards the pressure axis and follow the typical trend observed in glassy polymers. In Fig. 3(a) the dashed and solid lines represent predictions of methane content in PEMA at 25 ◦ C from NEPHSC(vdW) and NE-PHSC(SW) models, respectively, and they are compared with experimental data. The pure polymer density value is taken for PEMA from Zoller and Walsh [30], and the pure component parameters for CH4 and PEMA are listed in Tables 1a and 1b; the default value of interaction parameter (ksp = 0) has been applied in both cases. Also for this polymer-penetrant pair the difference between the results obtained from NE-PHSC(vdW) and NE-PHSC(SW) is quite evident. Both non-equilibrium models predict the decrease of solubility coefficient with solute pressure, i.e. the negative concavity of the solubility isotherm. In contrast, a clear difference exists between the pseudo-equilibrium solute contents predicted by NE-PHSC(vdW) and by NE-PHSC(SW) models: the solubility calculated from NE-PHSC(SW) deviates from the experimental values by approximately 45%, while results from the NE-PHSC(vdW) model are three times lower than the experimental data. A more accurate representation of solute content for the pressure range considered can be obtained, of course, from both models when convenient values of binary interaction parameter ksp are introduced in non-equilibrium calculations. In Fig. 3(b) experimental data from ref. [33] show the pseudoequilibrium methane content in glassy PEMA up to 3.5 MPa, in the temperature range from 25 to 45 ◦ C. In the same figure, experimental data are also compared to non-equilibrium model correlations obtained by adjusting the binary interaction parameter ksp to a temperature independent best-fitting value, equal to −0.095 and to −0.040 for NE-PHSC(vdW) and NE-PHSC(SW), respectively. An acceptable representation
Fig. 3. (a) Solubility isotherm for CH4 in PEMA at T = 25 ◦ C. Comparisons are shown for experimental data from ref. [33] with predictions from NEPHSC(vdW) and NE-PHSC(SW) [ksp = 0 in both cases] and (b) solubility isotherms for CH4 in PEMA. Comparisons are shown for experimental data from ref. [33] with correlations from NE-PHSC(vdW) [ksp = −0.095] and NEPHSC(SW) [ksp = −0.040].
of the experimental solubility has been obtained by both nonequilibrium models considered, although a closer look reveals that the NE-PHSC(vdW) model predicts a lower temperature sensitivity of solubility with respect to NE-PHSC(SW) model and, therefore, a lower sorption enthalpy for the CH4 –PEMA mixture in this temperature range. The effect of temperature on apparent solubility is underestimated by both models; however, the change with temperature of the pseudo-equilibrium solute content predicted by NE-PHSC(SW) is closer to the experimental values than the one calculated by NE-PHSC(vdW), and shows a maximum deviation smaller than 20%. 2.7. Infinite dilution gas solubility coefficient in Teflon® AF2400 Pseudo-equilibrium solubility coefficient for N2 , CH4 , C2 H6 and C3 H8 in AF2400 at 35 ◦ C are shown in Fig. 4 after De Angelis et al. [34] and are compared with the predictions obtained by equilibrium and non-equilibrium PHSC equations of state.
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ficients calculated from NE-PHSC(vdW) and NE-PHSC(SW) simply based on the first order approximation for the binary parameter, ksp = 0, are already very close to the experimental values. Quite satisfactory predictions, also in quantitative terms, are obtained by NE-PHSC models with ksp = 0, for the solubility coefficients of light gases, while the same procedure clearly overestimates the gas solubility of alkane vapors in the fluorinated polymer matrix. A clear justification of the latter behavior can be reasonably found in the mixing rule for the characteristic energy, which in its default version (ksp = 0) is not adequate to represent the specific interaction between alkane solute molecules and perfluorinated polymer segments [35]. It is also relevant to observe that no significant difference is evident in this case between the predictions obtained from the two different versions of the NE-PHSC theories. 3. Discussion and conclusion
Fig. 4. Infinite dilution solubility coefficient for N2 , CH4 , C2 H6 and C3 H8 in AF2400 at T = 35 ◦ C. Experimental data from ref. [34] are compared with predictions from (a) PHSC(vdW) EoS and NE-PHSC(vdW) and (b) PHSC(SW) EoS and NE-PHSC(SW) [ksp = 0 in all cases].
Specific volume data for AF2400 above the glass transition temperature were taken from ref. [34] and have been used here to retrieve the pure component PHSC parameters, which are indicated in Tables 1a and 1b. The pure polymer density of glassy AF2400 at 35 ◦ C is also reported in ref. [34] and its value was used to calculate the pseudo-equilibrium solubility coefficient. In all calculations performed in this case, only the default value of the binary interaction parameter (ksp = 0) has been considered. Differences between experimental solubility coefficients and predictions from the equilibrium EoS are quite evident. By using the equilibrium PHSC(vdW) and PHSC(SW) equations the predicted solubility coefficients are almost two orders of magnitude lower than the experimental values. This remarkable additional capacity of AF2400 to accommodate sorbed gas molecules in excess over the thermodynamic equilibrium value is indeed a consequence of the high extra free volume which is frozen in the glassy structure of AF2400. More remarkably, from Fig. 4 one may also notice that pseudo-equilibrium gas solubility coef-
Application to Perturbed Hard Sphere Chain Theories of the non-equilibrium thermodynamic approach for glassy polymers has been presented and the resulting models have been used to calculate and correlate gas solubility in different glassy polymers. Similarly to the case of the NELF model developed earlier along the same lines, the NE-PHSC(vdW) and NE-PHSC(SW) models allow for calculation of low pressure pseudo-equilibrium gas solubility in glassy polymers based on pure component equilibrium PVT data for pure polymers and pure solutes, and using the available values for the unpenetrated polymer density in the glassy state. The PHSC based non-equilibrium models are indeed able to interpret in details the thermodynamic properties of polymer–solute systems in the glassy phase. In addition, they offer the advantage of having pure component parameters strictly related to specific molecular properties, making it possible, in case of need, to obtain their estimates based on the knowledge of the species chemical structure. The comparison between model calculations and experimental data presented in this work represents the typical behavior which is observed also for other polymer-penetrant pairs not explicitly shown for brevity sake. The examples reported confirm the ability of the model to represent peculiar temperature and pressure dependence of gas solubility coefficient in glassy polymers. As indicated by the NE-PHSC(SW) calculation of CO2 solubility coefficient in PC, as function of temperature, correct representation of temperature effect on gas solubility below the glass transition temperature can be obtained when the binary interaction parameter between solute and polymer species is retrieved from equilibrium (T > Tg ) solubility data. Under equilibrium conditions, above Tg , both PHSC(vdW) and PHSC(SW) offer good predictions for the solubility of low molecular weight gases in the polymer phase. Below the glass transition temperature, however, the corresponding non-equilibrium models are not equivalent. For NE-PHSC(vdW) a smaller departure from equilibrium volume is calculated, which is likely smaller than the actual value, albeit not subject to experimental verification; that leads generally to an underestimation of the pseudo-solubility in the glassy phase. On the contrary, NE-PHSC(SW) apparently offers a better estimation of the departure from equilibrium and
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thus, better evaluations of gas solubility in the glassy polymer are obtained. It is also relevant to observe that the high gas solubility coefficient measured in high free volume glassy polymers can be correctly represented by NE-PHSC models, by using the actual polymer density; that is sufficient to account for the excess volume, over the true equilibrium polymer volume, available to accommodate larger quantities of solute molecules. Finally, differences exist between the predictive ability of the two versions of PHSC theory here examined. Indeed, the square well version of the EoS offers satisfactory representations of thermodynamic properties of glassy states in all the cases considered; in contrast, in most of the conditions examined, poor results were obtained from the use of the PHSC version based on van der Waals type of interaction potential for the dispersion term. Remarkably, in the rubbery state the representations of thermodynamic properties of polymeric phases resulting from both PHSC EoS are substantially equivalent, while below Tg better results indeed come from use of the square well form of interaction potential. Most likely, such a difference is associated to better free energy estimates offered for much denser phases by PHSC(SW) with respect to PHSC(vdW). The latter conclusion is also consistent with the fact that comparable results are obtained by both NE-PHSC(vdW) and NE-PHSC(SW) models when relatively low density phases are considered, as it is the case of a high free volume glassy polymer as AF2400. Acknowledgments This research has been partially supported by the Italian Ministry for Education, University and Research, PRIN 2004, and by the EU Strep MULTIMATDESIGN (Contract No. NMP3CT-2005-013644). References [1] [2] [3] [4]
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