C2H6 Mixtures in

Article pubs.acs.org/IECR

Equation of State Modeling of the Solubility of CO2/C2H6 Mixtures in Cross-Linked Poly(ethylene oxide) Matteo Minelli,† M. Grazia De Angelis,*,† Marco Giacinti Baschetti,† Ferruccio Doghieri,† Giulio C. Sarti,† Claudio P. Ribeiro, Jr.,‡ and Benny D. Freeman‡ †

Department of Civil, Chemical, Environmental, and Materials Engineering (DICAM), Alma Mater StudiorumUniversità di Bologna, I-40131 Bologna, Italy ‡ Department of Chemical Engineering, Center for Energy and Environmental Resources, University of Texas at Austin, Austin, Texas 78712, United States S Supporting Information *

ABSTRACT: The solubility of gaseous mixtures of CO2 and C2H6 in cross-linked poly(ethylene oxide) (XLPEO) has been modeled over a wide range of temperatures, pressures, and compositions with the Sanchez−Lacombe lattice fluid equation of state (LF EoS). The model binary gas−polymer parameters have been determined from pure gas solubility data and kept constant with temperature and composition. Multicomponent solubility is then calculated with no additional adjustable parameters, using a fully predictive procedure. The LF EoS model describes well the mixed gas solubility behavior data over wide ranges of temperature (from −20 to 35 °C), pressure (0−20 atm), and composition. The model also allows the representation of the solubility selectivity behavior of XLPEO at different conditions, permitting an a priori determination that ethane solubility is significantly enhanced by the presence of CO2, with a consequent reduction in CO2/C2H6 solubility selectivity relative to the pure gas value. In this regard, the behavior of this rubbery polymer is opposite that of many glassy systems, in which competitive sorption is often dominant, and the solubility of the less soluble gas (i.e., C2H6 in this case) is decreased by the presence of CO2, and the CO2 solubility selectivity is enhanced with respect to the pure gas value.

1. INTRODUCTION The knowledge of the solubility of fluid mixtures in polymers is essential for several industrial applications, although at the moment only a very few, time-consuming procedures have been established to measure mixed gas or vapor sorption in polymers.1−12 Compared to pure gas solubility, the number of experimental data sets available in the literature for multicomponent sorption in polymers is rather limited. Most of the studies in this field focus on removal of solvent mixtures from polymers, and the data are obtained by inverse gas chromatography (IGC) over limited ranges of operating conditions, often with little industrial interest.11−13 Indeed, the temperatures explored in these cases are often far above those of interest for most of the polymers considered, the pressures investigated are rather low (usually atmospheric), and the range of compositions studied are very narrow, since at least one of the penetrants is present in trace amounts in the gaseous phase. For these reasons, it is important to develop mathematical models that allow the calculation of solubility coefficients of multicomponent gas mixtures in polymers with a minimal amount of experimental information, possibly based on pure component properties and on pure gas solubility only. Several thermodynamic models have been developed to evaluate the solubility of gases and gaseous mixtures in rubbery polymers. Such models are based on the activity coefficient approach,14−17 or proper equations of state.18−25 Molecular simulations can also be applied to model the solubility of gaseous mixtures in polymers.26 © 2015 American Chemical Society

The present study focuses explicitly on application of the Sanchez−Lacombe lattice fluid (LF)21,22 equation of state (EoS) to describe the sorption of multicomponent gas mixtures in rubbery polymers and to predict the solubility of gas mixtures based only on pure component and binary experimental data. This approach is limited to rubbery polymers, which are at thermodynamic equilibrium and can be treated as high molecular weight liquids. The case of glassy polymers, which was treated previously,27 can be approached with a tool specifically built for glassy polymers, the nonequilibrium LF model (NELF), and adopts the lattice fluid representation of pure substances and their mixtures.28−32 This model was successfully applied to predict the solubility of gaseous mixtures, such as CO2/CH4 in poly(2,6-dimethyl-1,4phenylene oxide) (PPO) and CO2/C2H4 and CO2/N2O in poly(methyl methacrylate) (PMMA),27 as well as to predict pure component sorption in polymer blends.33,34 In membrane science, mixed gas solubility data in polymeric materials suitable for separations are required to determine realistically the gas separation performance of polymeric membrane materials based on the solution-diffusion mechanism.1,35−42 According to this approach, gas permeability in a dense polymeric membrane is the product of a solubility and a diffusivity contribution. In general, however, mixed gas permeability is somewhat easier to measure than mixed gas Received: Revised: Accepted: Published: 1142

September 27, 2014 December 23, 2014 January 5, 2015 January 5, 2015 DOI: 10.1021/ie5038215 Ind. Eng. Chem. Res. 2015, 54, 1142−1152

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Industrial & Engineering Chemistry Research

mixed gas solubility. Indeed, the material considered in this study is rather lightly cross-linked, based on equilibrium water uptake values in excess of 60%, so it is likely that the gases considered in this study were not able to sorb into the polymer sufficiently to cause the cross-links to substantially restrain the polymer swelling.53 In this study, we consider a XLPEO film sample prepared by photopolymerization of a mixture of poly(ethylene glycol) diacrylate (PEGDA) (30 wt %) and poly(ethylene glycol) methyl ether acrylate (PEGMEA) (70 wt %).47 This particular copolymer exhibits high CO2 permeabilities and good pure-gas selectivity values, and it can be used as a membrane material to purify natural gas.54 Ethane was chosen as the model light gas component, since it is the second major component of natural gas after methane; additionally, it forms a maximum pressure azeotrope with CO2 that can complicate effective natural gas treatment.55,56 Moreover, due to the similar condensabilities of CO 2 and C 2 H 6 , this mixture provides an interesting fundamental study of gas solubility in rubbery polymers. Four different gas mixtures (10, 25, 50, and 70 mol % CO2) and five operating temperatures, from −20 to 35 °C, are considered at pressures up to approximately 20 atm. The experimental data in this study are taken from previous papers;48,49,57 in these articles, the multicomponent Flory− Huggins model14,15,58 was used to describe the pure- and mixed-gas solubility data. However, this model, with composition-independent interaction parameters, leads to a poor description of the experimental results, especially at lower temperatures. Consequently, an empirical expression, accounting for the dependence of the binary parameters on the solute volume fractions, was introduced to effectively represent the mixed gas data. In view of these results, in the present study, the LF EoS model is applied to the experimental multicomponent gas sorption data in XLPEO. This model is expected to have a greater predictive power than the Flory− Huggins model, require fewer adjustable parameters, and, thus, be a more effective tool in the design of membrane separation processes.

solubility and diffusivity values. Therefore, most literature studies that report mixed gas data present mixed gas permeability coefficients rather than mixed gas diffusivity and solubility coefficients.43−45 Mixed gas solubility coefficients in polymers are usually determined by a manometric device coupled with a gas chromatograph, which measures the pressure and composition of the gaseous phase in the headspace of a chamber containing the polymer sample in equilibrium with the mixed gas phase. The amount of gas absorbed in the polymer can be evaluated from the initial and final amounts of gas present in the fluid mixture inside the chamber. This procedure can be accompanied by significant uncertainties in gas sorption levels, which require a careful choice of experimental conditions.1 The accurate determination of multicomponent solubility data is particularly useful when deviations between pure and mixed gas solubilities are expected. According to the literature, polymeric solutions often exhibit such deviations. For example, in the polybutadiene/benzene/cyclohexane system, the addition of either penetrant to the vapor phase enhanced the sorption of the other.7 Raharjo et al. verified that the presence of n-butane increased the solubility of methane in poly(dimethylsiloxane) (PDMS).8 Schabel et al. reported an increase in methanol vapor solubility in poly(vinyl acetate) (PVAc) at 40 °C in the presence of toluene,3 while Yurekli and Altinkaya found that toluene solubility decreased in the presence of methanol at higher temperature (100 °C).10 The infinite dilution solubilities of different vapors (i.e., methanol, isopropyl alcohol, methyl acetate, benzene, vinyl acetate, ethyl acetate, toluene, ethylbenzene, p-xylene) in both PVAc and polystyrene decreased with increasing concentrations of either CO2 or C2H4 dissolved in the polymer.11,12 Zielinski et al. verified that small amounts of water dissolved in the polymer can significantly lower the infinite dilution solubility of toluene in PVAc at 80 °C.11 Vopicka et al. noticed that the presence of CO2 strongly reduces the sorption of methane in a high free volume glassy polyacetylene, poly(1-trimethylsilyl-1-propyne) (PTMSP).46 These results were ascribed to competitive sorption between the penetrants, which favors sorption of the more soluble penetrant (i.e., CO2). Similar behavior was observed for the same gas mixture sorbing in another glassy polymer, poly(2,6-dimethyl phenylene oxide) (PPO).30 In this study, the solubility of C2H6/CO2 mixtures in crosslinked poly(ethylene oxide) (XLPEO) is modeled at various temperatures, pressures, and gas compositions.47−49 PEO-based membranes are of interest for CO2 removal from light gases, due to their high solubility selectivity toward CO2. Therefore, accurate measurement and modeling of the solubility of CO2containing mixtures in such materials is of interest. The specific case of PEO-based materials and mixtures containing CO2 is complicated by specific interactions between acid gases, such as CO2, and the ether groups in the polymer chain.50,51 Such interactions cannot be properly modeled with simple thermodynamic tools designed for apolar substances. Moreover, XLPEO is a cross-linked polymer, so its rigorous thermodynamic description would require accounting for the elastic constraints provided by the polymer network in the free energy expression.52 Such an approach is not followed in the present paper to avoid increasing the number of adjustable parameters in the model. At least in the range of temperatures, pressures, and compositions considered in this study, the introduction of a specific term accounting for polymer crosslinking is not necessary to accurately describe both pure and

2. MODEL DESCRIPTION The Sanchez−Lacombe model treats polymer chains as a set of beads on a lattice, in which polymer chains are mixed randomly with penetrant molecules. Unlike the Flory−Huggins model,14,15 the Sanchez−Lacombe model considers configurations with empty sites in the lattice, so free volume exists in the polymer−penetrant mixture, and volume changes upon mixing penetrant and polymer molecules are allowed. Each component of the mixture is completely characterized by three independent parameters: the characteristic pressure p*i , which is the hypothetical cohesive energy density of component i in the close-packed state (liquid at 0 K), the corresponding mass density ρ*i , and the characteristic temperature T*i , related to the depth of the potential energy well. The volume occupied by 1 mol of lattice sites of pure substance v*i can then be estimated as v*i = RT*i /p*i . The set of the three parameters can be determined from experimental pure component data. The lattice fluid theory provides the following dimensionless expression for the equilibrium total Helmholtz free energy of the mixture, A: 1143

DOI: 10.1021/ie5038215 Ind. Eng. Chem. Res. 2015, 54, 1142−1152

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Industrial & Engineering Chemistry Research ⎡⎛ 1 ⎞ A 1 = −ρ ̃ + T̃ ⎢⎜ − 1⎟ ln(1 − ρ ̃) + ln(ρ ̃) ⎢⎣⎝ ρ ̃ r ⎠ rNRT * +

∑ i

⎤ ln(ϕi)⎥ ⎥⎦ ri

polymer is determined, at fixed temperature and pressure, by satisfying the equation of state (eq 11) for the gas phase and for the polymeric phase and by equating the chemical potential of each penetrant i (eq 12) in both the penetrant and polymer phases.

ϕi

(1)

⎡ ⎛ ρ 2̃ + p ̃ + T̃ ⎢ln(1 − ρ ̃) + ρ ̃⎜⎜1 − ⎢⎣ ⎝

In eq 1, ϕi is the volume fraction of the component i, while r is the total molar average number of lattice sites occupied in the mixture, which can be calculated as p* Mi r = ∑ xiri = ∑ xi i RTi*ρ * i

i

i

T T* ρ ρ̃ = ρ* p̃ =

(2)

− ρ̃

∑ ϕipi* −

(3)

(5)

(6)

i

T* =

1 2

∑ ϕi ∑ ϕjΔpij* i

j≠i

T* p* p*v* v* ∑ xiri i = * * r R v p i i i

Np + 1

∑ j=1

ϕj(p*j − Δpi*, j )]

(12)

3. RESULTS AND DISCUSSION 3.1. Determination of Polymer Parameters. Modeling gas sorption in binary mixtures of XLPEO and CO2 begins with a determination of the pure component characteristic parameters for the LF EoS. This procedure is rather simple for gaseous penetrants, for which databases of EoS parameters are available in the literature. There is no complete set of LF parameters available for the polymer phase. For XLPEO, it is possible to employ the common procedure to determine EoS polymer parameters from the polymer specific volume at different pressures and temperatures. For the present polymer, the thermal expansion coefficient was only reported between −20 and 40 °C at 1 atm.48 However, such data allow a determination of the values of T* and ρ*, as illustrated in Figure 1. The change in specific volume of XLPEO with pressure (i.e., the isothermal compressibility) was not reported, so the value of characteristic pressure p* could not be determined. However, in the lattice fluid theory, the characteristic pressure is an energetic parameter representing the monomer cohesive energy density in the close-packed state.

The LF characteristic parameters of the mixture, ρ*, p*, and T*, depend on the composition of the mixture according to the following mixing rules:21 ω 1 =∑ i ρ* ρi* (7) i p* =

* rv i i [p* + RT i

The mathematical simplicity of the model allows the implementation of the calculation in a Microsoft Excel spreadsheet; the Excel solver tool is then used to obtain the numerical solution of the given set of nonlinear equations.

(4)

p p*

(11)

⎡ v* ⎛ 1⎞ 1⎤ = ln(ρϕ ̃ i) − ri ln(1 − ρ ̃)⎢ ⎜1 − ⎟ + ⎥ − ri ρ̃ ⎠ ρ̃ ⎦ RT ⎣ vi* ⎝

in which ωi is the mass fraction of component i. The dimensionless temperature T̃ , density ρ̃, and pressure p̃ are defined as

T̃ =

i

ϕi ⎞⎤ ⎟⎥ = 0 ri ⎟⎠⎦⎥

μi

where xi is the mole fraction of the ith component of the system and ri is the molar number of lattice sites occupied in the mixture by molecules of species i. As shown in eq 2, ri can be related to the LF characteristic parameters, the universal gas constant R, and the component molecular weight Mi. The volume fraction of each component i in the mixture is then conveniently expressed as the fraction of lattice sites occupied by the Ni molecules of species i in the mixture: ωi /ρi* rN ϕi = i i = ∑j rjNj ∑j (ωj /ρj*)

∑

(8)

(9)

where v* is the close-packed mer volume of the mixture. In eq 8, the quantity Δpij* characterizes the interaction energy between species i and j and is expressed as follows: Δpij* = pi* + p*j − 2(1 − kij) pi* p*j

(10)

in which each quantity kij represents the binary parameter associated with interactions between an i−j pair of species in the mixture. The LF model provides equations of state for pure components and mixtures as well as chemical potential expressions for each species, considered to describe both the gaseous and polymeric phases. The solubility of each gas in a

Figure 1. Volumetric data of XLPEO at atmospheric pressure48 and LF EoS prediction. (inset) pVT properties for un-cross-linked PEO: experimental data59 and LF EoS calculations. 1144


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were reported versus gas fugacity. Although fugacity is more appropriate than (partial) pressure to quantify the driving force for gas sorption in the polymer, in this work we use (partial) pressure, since it is an easily measurable quantity commonly used in gas separation process design. Figure 2 reports the experimental CO2 and C2H6 pure gas solubility isotherms in XLPEO,48 together with predictions

Therefore, this parameter should not be significantly affected by the presence of cross-links, so the p* value for XLPEO should be similar to the p* value of the same polymer without crosslinks (i.e., poly(ethylene oxide) PEO). The value of p* for pure PEO was obtained from pVT data published by Zoller and Walsh59 (see inset in Figure 1), and already reported in a previous work.60 Consistent with the above analysis of volumetric data, thermodynamic calculations performed in this work refer to the limiting case of infinite molar mass of polymeric species. For the sake of completeness, pure component characteristic parameters for the various penetrants and polymers are listed in Table 1. Table 1. Pure Component Characteristic Parameters of the Sanchez−Lacombe LF EoS CO2 C2H6 PEO XLPEO

T* [K]

p* [MPa]

ρ* [g/cm3]

Mw [g/mol]

ref

300 320 590 595

630 330 620 620

1.515 0.640 1.218 1.241

44 30 ∞ ∞

27 61 60 this work

3.2. Determination of Gas/Gas Binary Interaction Parameter. The binary CO2/C2H6 interaction parameter is obtained from the analysis of vapor−liquid equilibrium data reported by Brown et al. for this mixture at temperatures ranging from 210 to 270 K.62 If kij is set equal to 0.08, the Sanchez−Lacombe model provides a good representation of the experimental data, and the average deviation between the data and model calculation is less than 5%. The resulting value, along with the interaction parameter values for each gas with the polymer, is reported in Table 2. Table 2. Binary Interaction Parameters for the Sanchez− Lacombe LF EoS kij CO2−XLPEO C2H6−XLPEO CO2−C2H6

0.017 0.034 0.08

It is noteworthy that, in previous analyses of the solubility of gaseous mixtures in glassy polymers,28,30 the gas/gas binary interaction parameter has been considered equal to 0 (Berthelot’s rule).21 Indeed, the value of polymer/gas and gas/gas kij has a limited influence on the gas solubility in a glassy polymer, because it affects only the energetic contribution of solubility, that is less relevant than the entropic one in such structures, due to their excess of free volume. Conversely, when dealing with equilibrium states (rubbery polymers), the correct representation of any energetic interaction is crucial for the correct description of the thermodynamic state of the ternary mixture, and even the gas/gas binary parameter has to be accounted for. 3.3. Pure Gas Solubility. Binary interaction parameters kij for polymer/CO2 and polymer/C2H6 systems can be retrieved by fitting the model to pure gas solubility isotherms, minimizing the overall deviations at all temperature values. A unique value of kij is sufficient to describe the solubility of every gas/polymer couple, independent of temperature and composition. The resulting values are recorded in Table 2. All the solubility isotherms presented below are reported on a pressure basis, while in the original experimental work49 they

Figure 2. CO2 (a) and C2H6 (b) solubility isotherms in XLPEO at −20, −10, 0, 25, and 35 °C: experimental data48 together with LF EoS calculations.

from the Sanchez−Lacombe LF EoS obtained using the pure component parameters reported in Table 1 and the binary interaction coefficients for CO2/XLPEO and C2H6/XLPEO in Table 2. As one can see, the model provides an accurate description of the experimental trends over the pressure and temperature ranges considered. The average relative deviation between the model and experimental sorption values,48,49 calculated over the entire pressure interval, are reported in Table 3. In Table 3, a positive deviation means that the model overestimates the penetrant concentration in the polymer matrix, and a negative deviation indicates that the model underpredicts the penetrant concentration in XLPEO. Slight deviations between experimental data and model calculation are visible in Figure 2b. The model underestimates the C2H6 solubility at −20 °C and overestimates it at 35 °C; the discrepancies, however, are within a few percent. The use of a temperature-dependent binary interaction coefficient would 1145


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Table 3. Relative Average Deviation between Experimentally Measured49 Values of Gas Concentration in the Polymer and Those Calculated Using the LF EoS Model, Averaged over the Experimental Pressure Range Considered, for Each Composition and Temperature Data Set temperature (°C) −20 yCO2/yC2H6 0/100 10/90 25/75 50/50 75/25 100/0 a

a

−10

0

25

35

CO2

C2H6

CO2

C2H6

CO2

C2H6

CO2

C2H6

CO2

C2H6

− −0.05 −0.03 +0.03 +0.06 +0.08

−0.10 +0.46 +0.52 +0.53 +0.28 −

− −0.01 −0.01 +0.01 − +0.02

−0.01 +0.27 +0.27 +0.16 − −

− −0.01 +0.01 −0.02 +0.01 −0.03

+0.02 −0.01 +0.15 +0.09 +0.08 −

− +0.13 +0.06 +0.02 +0.04 +0.02

+0.12 +0.05 +0.13 +0.09 +0.07 −

− +0.06 +0.04 +0.05 +0.05 +0.02

+0.05 +0.10 +0.06 +0.10 +0.05 −

Gas mole fraction.

improve the agreement between the model calculation and the experimental data, although not significantly; very small variations of kij with temperature (less than 10%) would be required to minimize the difference between model and experimental data. Given the goodness of the model description of the experimental data and for the sake of simplicity, we adopted constant values of the binary interaction parameters to describe the solubility data at all temperatures. Interestingly, the CO2/XLPEO binary interaction parameter (kij = 0.017) matches exactly the value obtained when fitting the LF EoS model to the solubility of CO2 in PEO,60 already discussed in a previous work.63 The kij coefficient acts on the energetic interactions between the monomers of two unlike species and is reasonably not affected by the presence of crosslinks in the polymeric network. Therefore, the LF EoS model is able to provide an accurate representation of the experimental pure gas solubility data over a wide range of temperature and pressure, with average deviations below 12% (see Table 3). 3.4. Mixed Gas Solubility. Once the binary systems are appropriately parametrized, the calculation of CO2/C2H6 mixed gas sorption isotherms in XLPEO using the Sanchez−Lacombe LF EoS is entirely predictive. The experimental data are thus analyzed at different temperatures, from −20 to 35 °C, and total pressures up to approximately 25 atm. Several molar compositions of CO2/C2H6 gaseous mixtures are investigated, namely 10/90, 25/75, 50/50, and 75/25. In Table 3, the relative deviations between model calculations and experimental solubility data,48,49 averaged over the total pressure range, are reported at all temperatures and compositions of the gas phase. In Table 3, values of nominal compositions of the gaseous phase are reported for the sake of convenience; such values may differ from the actual equilibrium compositions (reported in the figure legends) due to uptake of gases inside the polymer. In Figures 3 and 4, experimental and calculated solubility isotherms for CO2 and C2H6 at 35 and 0 °C at different gas mixture compositions are reported; data belonging to the same curve in the plots correspond to equilibrium compositions that differ by less than 1%. Data obtained at other temperatures are analyzed similarly, although they are not reported here for the sake of brevity. However, they are included in the Supporting Information. As one can see from Table 3 and Figures 3 and 4, the model predictions accurately describe the experimental multicomponent solubility data of CO2 over the whole range of pressures, compositions and temperatures investigated experimentally, with deviations always well below 10%. Good agreement is

Figure 3. Mixed gas CO2 and C2H6 solubility isotherms in XLPEO at 35 °C at different compositions: experimental data49 together with LF EoS predictions.

observed also for C2H6 solubility, with deviations below 10% above 0 °C, and somewhat higher, up to +50%, at lower temperatures. In this range, the amount of CO2 in the polymer is high, and the second component effects might be stronger, thus stressing the assumptions in the modeling. In general, however, the prediction of multicomponent solubility by the model is satisfactory and validates the model assumptions, in particular the following: (i) the characteristic pressure p* of XLPEO is equal to that of pure un-cross-linked 1146


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Figure 4. Mixed gas CO2 and C2H6 solubility isotherms in XLPEO at 0 °C at different compositions: experimental data49 together with LF EoS predictions.

Figure 5. Pure and mixed gas CO2 (a) and C2H6 (b) solubility isotherms in XLPEO at 0 °C as a function of CO2 or C2H6 partial pressure for various gas phase compositions. The experimental data are from Ribeiro et al.,49 and the model predictions are from the LF EoS.

PEO; (ii) the presence of cross-links does not significantly affect the polymer free energy over the experimental range investigated; (iii) the mixed gas solubility values can be obtained with no additional adjustable parameters beyond the binary ones. This last finding, in particular, is a major improvement over the Flory−Huggins model, which requires an adjustable, concentration-dependent binary interaction parameter to accurately represent data from the ternary system. 3.5. Analysis of Mixed Gas Solubility. Comparison of the experimental results of Ribeiro et al.49 with the model predictions demonstrated the ability of the Sanchez−Lacombe LF EoS to describe the solubility of this gaseous mixture in cross-linked PEO. Therefore, it is possible to use the model to analyze in greater detail the thermodynamic properties of the ternary system. If the uptake of the gases is reported as a function of their partial pressure, rather than total pressure, as is done in parts a and b of Figure 5 for CO2 and C2H6, respectively, it is possible to evaluate the effect of the second component on the sorption of each penetrant. The experimental CO2 solubility data indicate a very limited effect of the presence of C2H6 on CO2 sorption, which is consistent with the modeling analysis, at least over the investigated pressure and composition ranges. Indeed, the experimental data points lie essentially all on the same trend line, as shown in Figure 5a.

Conversely, the solubility of C2H6 is influenced by the presence of the other component, as illustrated in Figure 5b, especially when the CO2 concentration becomes high enough to induce significant swelling of the polymer. At low CO2 content, ethane solubility seems to be practically equal to, or slightly lower than, pure C2H6 sorption. In contrast, at higher CO2 concentrations in the gas mixture, the C2H6 solubility is enhanced by the presence of CO2, with values up to 30% larger than those observed in pure gas sorption. Hence, in light of the good affinity of carbon dioxide with XLPEO, rather high values of CO2 solubility are achieved (up to almost 20% on a weight basis), which causes substantial volume dilation of the polymer matrix (up to 25%)49 that, in turn, enhances the solubility of the less soluble gas (C2H6 in this case). It is also of interest to analyze the swelling induced by the gaseous mixtures to the polymer matrix, and to compare the values predicted by the LF EoS with those measured experimentally and reported by Ribeiro and Freeman.49 To this aim, the polymer dilation (ΔV/V0) at 0 °C induced by CO2/C2H6 gaseous mixtures of different compositions is reported in Figure 6, together with the model predictions. The model indeed provides the volumetric behavior of the polymer/penetrant mixture without any further parameter, and as one can see, a very good agreement with the experimental 1147


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Figure 6. Dilation of the XLPEO matrix during sorption of CO2 and C2H6 mixed gas at 0 °C as a function of total pressure for various gas phase compositions. The experimental data are from Ribeiro et al.,49 and the model predictions are from the LF EoS.

data is observed. A similar trend is also observed at the other temperatures and is not reported here for the sake of brevity. 3.6. Model Prediction of Mixed Gas Effects. In the preceding sections, the LF EoS model has been successfully validated, and the values of pure component and binary parameters for the mixed gas CO2/C2H6 solubility in XLPEO in a range of temperature and compositions are conveniently retrieved. The availability of a reliable modeling tool allows for a comprehensive investigation of the thermodynamic behavior of the ternary system, providing useful information for both theoretical and practical purposes. In the present section, the LF EoS is employed to analyze in detail some peculiar aspects of mixed gas solubility in XLPEO, such as the extent of solubility enhancement or suppression due to the presence of a second penetrant and its relationship to temperature, pressure, and composition. Furthermore, the solubility selectivity, which is of interest for gas separation purposes, is also analyzed. In Figure 7, the effect of the second component (C2H6) on CO2 solubility in the polymer is represented by reporting values calculated at constant partial pressure of one gas component and variable partial pressure of the second one, as it was done experimentally in the studies by Koros and co-workers in the 1980s and 1990s.31,33−37,39,40 In particular, in Figure 7a the ratio between the mixed and pure gas solubilities of CO2 (cCO2(mix)/cCO2(pure)) at the same partial pressure is plotted vs C2H6 partial pressure, at fixed CO2 partial pressure, pCO2. As one can see in Figure 7a, CO2 solubility seems to follow a nonmonotonic trend with ethane partial pressure, possibly indicating the existence of two competing phenomena having opposite effects on the solubility. However, the extent of solubility deviation from pure gas conditions is very limited, below 5%, and a meaningful physical explanation of the observed trends is not reliable. On the other hand, the corresponding ethane solubility is always enhanced by the presence of CO2, as clearly illustrated in the inset in Figure 7a. The case of constant partial pressure of C2H6 and increasing CO2 partial pressure is illustrated in Figure 7b. The C2H6 solubility is enhanced by up to 30% in the presence of CO2, which swells significantly the polymer matrix. This effect becomes relatively more marked at the lowest

Figure 7. Second component effects in CO2/C2H6 solubility in XLPEO at 0 °C, evaluated from the LF EoS. (a) Ratio of mixed gas to pure gas solubility of CO2 at different CO2 partial pressures, versus C2H6 partial pressure. The corresponding C2H6 solubility is included in the inset. (b) Ratio of mixed gas to pure gas solubility of C2H6 at different C2H6 partial pressures, versus CO2 partial pressure. The corresponding CO2 solubility is included in the inset.

ethane partial pressure, whereas no significant variation can be observed in the CO2 solubility behavior (shown in the inset of Figure 7b). 3.7. Model Prediction of XLPEO Separation Performance. An evaluation of the relative performance of polymer membranes for gas separation applications is often performed using the so-called Robeson plot, which shows the existence of an empirical trade-off between permeability selectivity among a gas pair and the permeability of the most permeable penetrant.64−66 In the same manner, we examined the correlation between the CO2 solubility coefficient and the solubility selectivity, defined as the ratio of solubility coefficients of the two gases (evaluated at mixed gas conditions at the same partial pressure): αS =

SCO2 SC2H6

=

cCO2 /pCO

2

cC2H6/pC H 2

6

(13)

Figure 8 reports the CO2/C2H6 solubility selectivity as a function of the CO2 solubility coefficient, SCO2, at different temperatures, in the range from −25 to 40 °C, for a 50/50 1148


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function of CO2 solubility coefficient in XLPEO at 25 °C and at total pressures from 1 to 25 atm. Varying both the gas composition and total pressure, very slight changes in CO2/ C2H6 selectivity values are observed, of the order of a few percent, showing complex nonmonotonic behaviors. On the other hand, the solubility coefficient SCO2 varies more significantly, mainly as a function of the CO2 mole fraction of the gas phase. 3.8. Model Prediction of Mixed Gas Effects in Rubbery and Glassy Polymers. Finally, it is interesting to analyze the departure of mixed gas CO2/C2H6 solubility selectivity from pure gas (i.e., ideal) behavior, in order to assess the error that arises from using pure gas data to estimate mixed gas separation behavior in the absence of multicomponent data. In this regard, Figure 10 reports the ratio between solubility selectivity

Figure 8. Effect of temperature and pressure on the CO2/C2H6 solubility selectivity versus CO2 solubility coefficient in XLPEO. Calculations from LF EoS for a 50/50 gas mixture, at temperatures from −25 up to 40 °C and total pressures ranging from 0 to 20 atm.

mixture and at pressures up to 20 atm, as obtained from the Sanchez−Lacombe LF EoS. As one can see, the calculated points lie on the same master curve and the CO2 solubility coefficient decreases considerably at increasing temperature, changing by almost 1 order of magnitude from −25 to 40 °C. The solubility selectivity also decreases at increasing temperature, from a value of about 7 at −25 °C to 1.5 at 40 °C; indeed, as expected, the solubility selectivity of XLPEO membranes is highly favored at low temperatures. Interestingly, this solubility selectivity behavior is also observed for other gas compositions, which are not reported in Figure 8 for the sake of readability, and the same master curve is followed, as was also observed by Ribeiro et al.49 Furthermore, as one can see in Figure 8, increasing the total pressure enhances the CO2 solubility coefficient, and has variable effects on the solubility selectivity. The effect of composition was also addressed specifically in Figure 9, which reports the CO2/C2H6 solubility selectivity as a

Figure 10. CO2/C2H6 solubility selectivity in XLPEO: ratio between the actual mixed gas and ideal (i.e., pure gas) value as a function of the CO2/C2H6 molar ratio in XLPEO. The values in this figure were calculated using the LF EoS at 0 °C for gas phase CO2/C2H6 compositions ranging from 1/99 to 90/10 and total pressures in the range from 0 to 20 atm.

evaluated from mixed gas sorption and the corresponding value evaluated based on pure gas solubility coefficients at the same fugacities, αS(mix)/αS(pure), for total pressures between 1 and 25 atm, and gas phase molar compositions from 1/99 to 90/10 (CO2/C2H6). These values are reported as a function of the ratio of molar concentrations of the two penetrants in XLPEO (cCO2/cC2H6). As one can see, the calculated trends depend on the gas phase composition as well as on total pressure. However, the curves lie only in two quadrants of the plot. Indeed, the CO2/ C2H6 solubility selectivity decreases with increasing CO2/C2H6 molar ratio in the membrane. Moreover, the deviation from the value of the solubility selectivity estimated from pure gas data is higher than unity, if the concentration of CO2 is roughly lower than that of C2H6. Conversely, if the CO2 molar content in the membrane is larger than that of C2H6, the mixed gas solubility selectivity is lower than the value estimated based on pure gas sorption data. This second case is the most common one, since, due to its higher solubility in XLPEO, CO2 is often more abundant than ethane in the polymer. Therefore, in the majority of cases, the CO2/C2H6 solubility selectivity in gas mixtures is lower than its corresponding value calculated from pure gas sorption data due to the enhanced solubility of ethane

Figure 9. Effect of composition and pressure on CO2/C2H6 solubility selectivity versus CO2 solubility coefficient in XLPEO. Calculations are from the LF EoS at 25 °C for compositions ranging from 10/90 to 90/ 10 (CO2/C2H6) and total pressures in the range 0−25 atm. 1149


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in the case of the rubbery XLPEO, by the LF EoS and, in the case of glassy systems, with the corresponding nonequilibrium version (NELF),30 which can thus be used as reliable tools to describe the experimental findings.

in the presence of CO2 while the CO2 solubility is only slightly affected by the presence of ethane. It is interesting to compare the experimental behavior observed in mixed gas sorption of different gas mixtures in glassy and rubbery polymers. For CO2/CH4 sorption in PTMSP,46 PPO,30 and PIM-1,67 the analysis reveals a similar, but opposite, trend to that observed for CO2/C2H6 in rubbery XLPEO. Indeed, the complementary two quadrants of the plot are populated by the experimental points and by the model predictions. In particular, the solubility selectivity is higher than the ideal (i.e., pure gas) value when the molar concentration of CO2 exceeds that of the other gas. In the literature, there is only one example of mixed gas CO2/CH4 solubility in a rubbery polymer, and it was measured by Reijerkerk et al.9 The data are also reported in Figure 11, and they lie in the same quadrant as the data from the other rubbery polymer here considered, XLPEO, but for another gas mixture (CO2/C2H6).

4. CONCLUSIONS The solubility of CO2/C2H6 gas mixtures in a cross-linked poly(ethylene oxide) (XLPEO) polymer at various gas compositions, temperatures, and pressures was successfully modeled with the LF EoS model. In particular, the following were found: (i) The model does not require additional adjustable parameters to represent the ternary CO2/C2H6/XLPEO phase equilibrium, apart from the binary ones evaluated for the couples CO2/XLPEO, C2H6/XLPEO, and CO2/C2H6 from binary data. (ii) The binary parameters kij are independent of temperature and composition. (iii) The modeling analysis can be carried out using, for cross-linked PEO, the same values of energetic parameters, p* and kij, as those for un-cross-linked PEO, which is consistent with the physical meaning of such parameters. (iv) The effect of cross-linking on the free energy of the polymer mixture can be neglected, without losing the model predictive ability for solubility, at least over the ranges of temperatures, pressures, and compositions explored in this study. This result is likely due to the swelling induced by the gases in the polymer not being high enough to cause a significant elastic response by the matrix in the majority of cases considered. (v) It is not necessary to introduce additional parameters or terms accounting for the quadrupolar nature of CO2 and polar nature of poly(ethylene oxide), even though the LF EoS model does not account specifically for these features. Therefore, the LF EoS model is a more effective tool to represent multicomponent gas solubility data than the multicomponent Flory−Huggins model, which requires empirical concentration-dependent binary parameters to adequately describe the experimental data. The successful validation of the LF EoS model for this system, together with the set of parameters obtained, allowed a predictive estimation of the trends of solubility and solubility selectivity as a function of operating conditions. In particular, the model can be used to construct a solubility selectivity vs solubility coefficient plot for the system, which indicates that almost all data fall on the same master curve, with the low temperature values characterized by higher solubility coefficient and solubility selectivity than the high temperature ones. The model permits a systematic study of the effect of pressure and composition changes on the solubility selectivity vs solubility plot. Finally, the deviation of mixed gas behavior from pure gas behavior is analyzed, highlighting an interesting trend in XLPEO, which is opposite that observed in mixed gas sorption in glassy polymers: the CO2/C2H6 solubility selectivity is equal to or higher than the value computed from pure gas solubility coefficients if the molar content of CO2 in the polymer is lower than that of C2H6, while it is lower than the ideal value in the other cases. This behavior is likely related to the rubbery nature of the polymer; the swelling during sorption enhances the solubility of the less soluble component and lowers the solubility selectivity relative to the pure gas case. On the other hand, in glassy polymers, the competition between penetrants

Figure 11. Ratio between real (mixed) and ideal (pure) solubility selectivity αS,AB for some gas mixtures in glassy and rubbery polymers, versus the ratio between the molar concentrations of A and B. Data for A = CO2 and B = CH4 in PPO,39,40 PTMSP,46 PIM-1,67 (PEO-rPPO2500/T)7500;9 A = CO2 and B = C2H6 in XLPEO (this work); A = C4H10 and B = CH4 in PTMSP42 and PDMS.8

Interestingly, such trends are not limited to the case of CO2 containing mixtures but are also valid for n-C4H10/CH4: using the data collected by Raharjo et al. in rubbery PDMS8 and glassy PTMSP,42 one finds that the data for rubbery PDMS lie in the same quadrants as those of the other rubbery polymers and PTMSP data lie in the same quadrant occupied by data of other glassy polymers. The present observations seem to indicate that (i) the deviation of the solubility selectivity behavior from the ideal (i.e., pure gas) behavior is related to the molar ratio of the two components inside the polymer and (ii) such deviations are of opposite sign for glassy and rubbery polymers. In particular, the ratio between actual and ideal solubility selectivity for the gas pair A/B decreases as the molar ratio of A/B increases in the polymer for rubbery polymers, while it increases for glassy polymers. The reason for this behavior probably lies in the interplay between the different mechanisms that govern multicomponent effects during sorption in rubbery and glassy matrixes, namely swelling and competition. However, its rigorous description deserves further analysis that will be the subject of a future study. In particular, such phenomena can be properly described, 1150


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dominates the multicomponent sorption process, giving rise to a general depression of solubility, which is more marked for the less abundant component, resulting in lower solubility in the mixed gas state than in the pure gas one, thereby leading to higher-than-ideal solubility selectivity values in most cases.

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ASSOCIATED CONTENT

S Supporting Information *

Figures reporting experimental data of mixed gas CO2 and C2H6 solubility in XLPEO at constant temperature and different compositions: experimental data from Ribeiro and Freeman,48 together with predictions from the Sanchez− Lacombe LF EoS. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +39 (0) 51 2090410. Fax: +39 (0) 51 6347788. Notes

The authors declare no competing financial interest.

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REFERENCES

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