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Phase equilibria in systems with specific CO2–polymer interactions
Fluid Phase Equilibria 228–229 (2005) 487–491
Phase equilibria in systems with specific CO2–polymer interactions Ibrahim A. Ozkan, Amyn S. Teja∗ School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA
Abstract A lattice-based model for solutions with solute–solvent association is extended to the calculation of phase equilibria in polymer–CO2 systems in this work. The model is simple to use and requires a maximum of two adjustable parameters to correlate both cloud point and sorption data. Moreover, we show that one of these parameters can be obtained from spectroscopic data. Application of the modified model is demonstrated for the calculation of cloud point curves of seven polymer–CO2 systems. Good agreement with experiment was obtained, with average deviations between calculated and experimental data of 1.3%. The predictive capabilities of the model are demonstrated by calculating the sorption behavior of CO2 in polymers using parameters from cloud point data. © 2004 Elsevier B.V. All rights reserved. Keywords: CO2 ; Polymer; Cloud point; Sorption; Model
1. Introduction Polymer solutions in which CO2 is the solvent are of interest because of the advantages of supercritical CO2 as an environmentally benign solvent [1,2] and as a swelling agent [3,4] in polymer processing. CO2 has also been shown to exhibit specific Lewis acid–base interactions with a number of polymers containing electron-donating groups [5]. Such interactions have a significant effect on the phase behavior of polymer–CO2 systems, leading to an interest in developing models that can describe such behavior. There are two approaches to the development of models that can be used for systems which exhibit specific solute–solvent interactions at high pressures. Equation of state models that take into consideration the effect of pressure can be modified to account for specific interactions in solution. Alternatively, activity coefficient models that take account of specific interactions can be reformulated to incorporate the effect of pressure. This work adopts the latter approach, and incorporates pressure effects into the recently ∗
Corresponding author. Tel.: +1 404 894 3098; fax: +1 404 894 2866. E-mail address: [email protected] (A.S. Teja).
0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2004.09.035
proposed generalized Associative Reformulation of Thermodynamics (gART-L) model for associating polymer solutions [6–8]. Application of the modified gART-L model to the calculation of cloud point curves and sorption in CO2 –polymer systems using one or two adjustable parameters is discussed below.
2. The modified gART-L model The gART-L model was proposed by Sukhadia [6], Variankaval et al. [7,8] as a simple reformulation of the Flory–Huggins model to account for specific interactions between polymer segments and solvent molecules. The solution is assumed to consist of three types of species: complexes (“associated” polymer–solvent molecules), unassociated solvent and unassociated polymer molecules. The entropy of mixing in the Flory–Huggins model is then recalculated by restricting a certain number of solvent molecules to specific positions on the lattice because of their association with polymer segments. The remaining solvent molecules are distributed randomly on the lattice. Furthermore, the enthalpy of mixing is modified by defining two types of contacts
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due to association and dispersion, respectively. The resulting gART-L expression for the Gibbs energy of mixing Gm is given by [6–8]: Gm = αµΦ2 χa + Φ2 {1 − Φ2 (1 + αµ)}[(z − 2 − µ)α nRT Φ2 ln Φ2 + (1 − α)(z − 2)]χu + x 1 − Φ2 (1 + µα) + ln{1 − Φ2 (1 + µα)} 1 + µα − Φ1 ln
Φ1 µαΦ2 + µαΦ2 ln Φ1 − µαΦ2 Φ1 − µαΦ2
+ Φ2 (1 − α) ln(1 − α) + αΦ2 ln α
(1)
where z is the lattice coordination number (=10 in this and previous [6–8] work), R the gas constant, T the temperature, Φ1 and Φ2 the volume fractions of CO2 and polymer, and n the total number of moles. There are five characteristic quantities in Eq. (1): the solvent–solute binding ratio µ; the association ratio α; the number of segments x; and the specific interaction parameters for association χa and for dispersion, χu . The binding ratio is assumed to be 1 throughout this work because only one CO2 molecule is able to associate with a functional group in the polymers investigated in this work. Note that the binding ratio refers to complex formation between a polymer segment and µ solvent molecules according to P + µS ⇔ PSµ . The specific interaction parameter for association χa (which is related to the enthalpy of association Ha as discussed below) can be obtained from FTIR spectra of the polymer–CO2 mixture at the conditions of interest. The presence of CO2 under pressure causes a peak shift (ν) in the FTIR spectrum [5] due to Lewis acid–base interactions between CO2 and the associating functional group in the polymer segment. This peak shift is related [9] to the interaction parameter χa and hence the enthalpy of association Ha as follows: ∆Ha = 0.9874∆ν = RTχa
(2)
The enthalpy of association, in turn, is related to the equilibrium constant K for the association reaction (P + µS ⇔ PSµ ) as follows: K −Ha 1 1 ln (3) = − K0 R T T0 where K0 is the equilibrium constant for association at a reference temperature T0 (say, 300 K). In the above expression, Ha has been assumed to be independent of T over the range of temperatures of interest [5]. K can therefore be calculated at any temperature if Ha and K0 are known. Knowledge of the equilibrium constant K also yields the association ratio α as follows: (1 + K) − (1 + K)2 − 4Φ1 Φ2 K(1 + K) (4) α= 2Φ2 (1 + K)
Fig. 1. Cloud point behavior in PVAc–CO2 . Points represent experimental values, whereas lines represent calculations with x calculated from actual molar volumes of the pure components (solid line), and from molar volumes at ambient conditions (dashed line).
where Φ1 and Φ2 are volume fractions of the polymer and solvent, respectively. Note that K may also be determined from the relative amounts of associated and unassociated species present in a solution via spectroscopy. The dispersion parameter χu can be estimated from the solubility parameter of the non-polar analog (or homomorph) of the polymer [6–8]. The functional group in the polymer molecule which interacts strongly with CO2 is substituted with a non-polar group to obtain the homomorph. The solubility parameter is then estimated via group contributions as described elsewhere [10]. The segment number x is generally obtained from the ratio of the molar volumes of the solvent (CO2 ) and the solute (polymer) at ambient conditions [11]. This somewhat arbitrary ratio works well for liquids, since liquid volumes are not significantly affected by pressure. However, it is quite inadequate at supercritical CO2 conditions, because the molar volume of CO2 changes dramatically with pressure above its critical point. This has a significant effect on the cloud point curve, as shown for poly(vinyl acetate) (PVAc)–CO2 in Fig. 1. Note that a value of x calculated from the ratio of actual molar volumes (which depend on pressure and temperature) gives a much better fit of the cloud point data than the value of x obtained from the ratio of molar volumes at ambient conditions (a “liquid molar volume” of 55 cm3 /mol was assumed for CO2 at ambient conditions [12]). These calculations clearly show that the change in molar volume with T and P must be taken into account in order to apply the model to CO2 –polymer systems. Note also that x generally increases with pressure since the density of CO2 increases more rapidly with pressure than the density of the polymer. The effect of temperature is not as large and leads to only a modest increase in x. In the present work, the polymer molar volumes were obtained from the Tait equation [13] and CO2 properties from
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489
the Patel–Teja equation of state [14]. The modified gART-L model incorporating the pressure was then used to calculate phase equilibria and sorption behavior in CO2 –polymer systems as discussed below. Note that the original and modified gART-L models both require a knowledge of two characteristic parameters (Ha and K0 ) in the calculations. The remaining quantities in the models either have fixed values or they can be calculated from Ha and K0 . The number of parameters can be further reduced if Ha is obtained from FTIR spectra as shown below.
3. Cloud point and sorption calculations using the modified gART-L model Equilibrium between a polymer-rich phase (α) and a CO2 rich phase (β) can be described by β β
xiα γiα = xi γi
Fig. 2. Cloud point behavior in the PBMA–CO2 system. The solid lines represent calculations with Ha = −3.8015 kJ/mol and K0 = 7.536. The points represent experimental data for 5 wt.% polymer [15].
(5)
in which the activity coefficients γ i depend on the temperature, pressure, and composition. These equations may be β solved for any two of the quantities T, P, x2α and x2 . Thus, if the composition of one of the phases is known, then the pressure and composition of the other phase can be calculated at a given temperature using Eq. (5). Cloud point curves at constant concentration (isopleths) can then be calculated by repeating the calculations at different temperatures. In the present work, activity coefficients were obtained by differentiation of Eq. (1), and experimental cloud point data were fitted using either K0 , or K0 and Ha as adjustable parameters. The number of parameters employed in correlating cloud points depends on the availability of spectroscopic data (which yields Ha ) for the system. Values of the parameters obtained by such fitting are discussed in Table 1. In the case of sorption of CO2 in a polymer at low pressures, the equilibrium relation is (assuming negligible solubility of the polymer in CO2 ): xiα γiα = φi (T, P)P
(6)
where φi is the fugacity coefficient of CO2 . The fugacity coefficient was obtained using the Patel–Teja equation [14] for CO2 and Eq. (6) was used to calculate the amount of CO2 sorbed into a polymer. The parameters K0 and Ha from cloud point calculations and/or spectroscopic measurements were used in the calculations of γ i , so that no additional parameters are required in the calculations.
4. Results for cloud point behavior of polymer–CO2 mixtures Cloud point data of McHugh et al. [15] for the systems poly(butylmethacrylate) (PBMA)–CO2 and PVAc–CO2 were correlated using Ha values obtained from FTIR measurements [5]. A single adjustable parameter K0 was obtained by fitting these data. Cloud point data (see Table 1) [16] for a series of polyacrylate–CO2 systems were also correlated. However, two adjustable parameters (Ha and K0 ) were used in these calculations because spectroscopic measurements were not available. Average molecular weights and polydispersities of the polymers are available in the original references [15,16]. Experimental and calculated cloud point curves in the PBMA–CO2 system are shown in Fig. 2. The interaction energy Ha in this system was estimated to have a value of ∼4 kJ/mol from the peak shift in its FTIR spectrum [5]. As can be seen in the figure, the data are fitted well using only one adjustable parameter (K0 ). Cloud point curves for polymer Mw of 100,000 and 320,000 were calculated using the same value of the adjustable parameter, indicating that K0 does not apparently depend on the molecular weight. However, the molecular weight does affect x via the molar volume of the polymer. As a result, the cloud point curve for PBMA with Mw of 100,000 is shifted by as much as 45 K relative to the cloud point curve of PBMA with an Mw of 320,000. This indicates that higher temperatures would be needed for
Table 1 Cloud point calculations for PBMA–CO2 and PVAc–CO2 using the modified gART-L model Polymer
Mw (×1000)
Tg (K)
Temperature range (K)
P range (bar)
Ha * (kJ/mol)
K0
AAD%
PBMA PBMA PVAc
100 320 125
293 293 303
398.2–483.3 416.5–498.5 295.8–423.4
1300–3000 1430–3000 602.5–1046.3
−3.8 −3.8 −3.9
7.536 7.536 4.381
0.47 0.88 2.45
∗
From peak shift data in Ref. [5].
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I.A. Ozkan, A.S. Teja / Fluid Phase Equilibria 228–229 (2005) 487–491 Table 3 Association energies (Ha ) and equilibrium constants (K0 ) used in cloud point and sorption calculations System
Ha (kJ/mol)
K0
PMA–CO2 PEA–CO2 PBA–CO2 PEHA–CO2 PODA–CO2
−7.985 −5.624 −3.214 −2.861 −2.413
6.106 5.088 4.713 3.266 2.745
Fig. 3. Cloud point behavior in polyacrylate–CO2 systems. The solid lines represent calculations and points represent experimental data for 5 wt.% polymer [16].
CO2 to solubilize the higher molecular weight PBMA, which is typical for UCST type phase behavior. Similar behavior was encountered in the PVAc–CO2 system. Calculated results were in good agreement with experimental data using one adjustable parameter, although the average error of 2.45% was somewhat higher than the average error in the PBMA–CO2 system (0.88%). The model was also used to calculate cloud point curves in poly(acrylate)–CO2 systems using two adjustable parameters. Fig. 3 and Table 2 present comparisons between calculated and experimental [16] cloud point data for poly(methyl acrylate) (PMA), polyethylacrylate (PEA), poly(butyl acrylate) (PBA), poly(ethylhexyl acrylate) (PEHA), and poly(octadecyl acrylate) (PODA) in CO2 . Also shown in Table 2 are comparisons by McHugh et al. [16] using the SAFT equation to calculate cloud points in these systems. Both models are able to correlate the data well, using the same number of adjustable parameters in each case (and the same experimental data for cloud points). The fitted parameters for the modified gART-L model are shown in Table 3 and are of the same magnitude as the parameters obtained for mixtures of PBMA or PVAc with CO2 . Moreover, the values of the interaction energy are similar to the energies tabulated for Lewis acid–base interactions [17]. We may infer from these calculations that interactions between CO2 and the polymer are likely to be found in polyacrylate–CO2
Fig. 4. Sorption predictions for PVAc (Mw = 100,000)–CO2 systems. The squares, triangles and circles represent experimental sorption data at 313.15, 333.15, and 353.15 K, respectively [18].
systems. Note that the magnitude of the parameters decreases as the length of the alkyl tail on the acrylate increases. This could be due to a decrease in effective polarity as a result of shielding of the molecule by the alkyl tail, which limits CO2 access to the functional group on the polymer chain. Additionally, Tg values decrease with increasing alkyl tail length since the addition of longer alkyl chain affects the stiffness of the polymer. This leads to an increase in the magnitude of the parameters for the systems when the polymer component has a higher Tg .
5. Prediction of sorption behavior The solubility of CO2 in polymers was also predicted using the parameters obtained by fitting cloud point curves. The re-
Table 2 Cloud point calculations in polymer–CO2 systems using the modified gART-L and SAFT models Polymer
Mw (×1000)
Tg (K)
Temperature range (K)
P range (bar)
This work AAD%
SAFTa AAD%
PMA PBA PEA PEHA PODA
31 62 120 113 23
282 224 250 218 NA
296.2–473.7 355.7–469.0 319.2–468.7 430.2–483.0 486.5–536.9
1843–2242.5 1057.8–3000 1272.9–3000 1192.4–3000 1226.7–2663.8
2.29 0.59 0.84 1.57 1.21
2.51 2.66 1.80 4.43 1.47
a
Calculations of McHugh et al. [15].
I.A. Ozkan, A.S. Teja / Fluid Phase Equilibria 228–229 (2005) 487–491
sults of such calculations and comparisons with experimental data in the case of PVAc–CO2 are presented in Fig. 4. Reasonable agreement was obtained between predicted values and experimental sorption data (error in the range 5–20%). Note that there are no adjustable parameters in the sorption calculations, so the agreement may be regarded as reasonable.
6. Conclusions Cloud point curves in seven polymer–CO2 systems have been correlated using the modified gART-L model with one or two adjustable parameters depending on the availability of spectroscopic data. Good agreement with experimental data was obtained (average errors of 1.3%). The predictive capability of the model was also investigated by predicting the solubility of CO2 in polymers using parameters from cloud point data.
References [1] E.J. Beckman, J. Supercrit. Fluids 28 (2004) 121–192. [2] C.F. Kirby, M.A. McHugh, Chem. Rev. 99 (1999) 565–602. [3] M.A. McHugh, V.J. Krukonis, Supercritical Fluid Extraction— Principles and Practice, 2nd ed., Butterworths–Heinemann, 1994.
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[4] A.R. Berens, G.S. Huvard, R.W. Korsmeyer, F.W. Kunig, J. Appl. Polym. Sci. 46 (1992) 231–242. [5] S.G. Kazarian, M.F. Vincent, F.V. Bright, C.L. Liotta, C.A. Eckert, J. Am. Chem. Soc. 118 (1996) 1729–1736. [6] T. Sukhadia, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA, 1999. [7] N. Variankaval, M.E. Rezac, A.S. Abhiraman, ACS National Meeting, Paper PMSE 297, August 2000. [8] N. Variankaval, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA, 2001. [9] F.M. Fowkes, D.O. Tischler, J.A. Wolfe, L.A. Lannigan, C.M. Ademu-John, M.J. Halliwell, J. Polym. Sci., Part A: Polym. Chem. 22 (1984) 547–566. [10] R.P. Danner, T.E. Daubert, Physical and Thermodynamic Properties of Pure Compounds: Data Compilation, Hemisphere Publishing Corp., New York, 1990. [11] J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamics of Fluid-phase Equilibria, 3rd ed., Prentice-Hall, New Jersey, 1999. [12] J.M. Prausnitz, F.H. Shair, AIChE J. 7 (1961) 682–686. [13] P.G. Tait, Physics and Chemistry of the Voyage HMS Challenger, vol. II, Part IV, S.P.L.XI, 1888. [14] N.C. Patel, A.S. Teja, Chem. Eng. Sci. 37 (1982) 463–473. [15] M.A. McHugh, F. Rindfleisch, T.P. DiNoia, J. Phys. Chem. 100 (1996) 15581–15587. [16] M.A. McHugh, M. Lora, F. Rindfleisch, J. Appl. Polym. Sci. 73 (1999) 1979–1991. [17] M.D. Joesten, L.J. Schaad, Hydrogen Bonding, Marcel Dekker, NY, 1974. [18] C. Zhong, H. Masuoka, Fluid Phase Equilib. 144 (1998) 49–57.
Phase equilibria in systems with specific CO2–polymer interactions Ibrahim A. Ozkan, Amyn S. Teja∗ School of Chemical and Biomolecular Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0100, USA
Abstract A lattice-based model for solutions with solute–solvent association is extended to the calculation of phase equilibria in polymer–CO2 systems in this work. The model is simple to use and requires a maximum of two adjustable parameters to correlate both cloud point and sorption data. Moreover, we show that one of these parameters can be obtained from spectroscopic data. Application of the modified model is demonstrated for the calculation of cloud point curves of seven polymer–CO2 systems. Good agreement with experiment was obtained, with average deviations between calculated and experimental data of 1.3%. The predictive capabilities of the model are demonstrated by calculating the sorption behavior of CO2 in polymers using parameters from cloud point data. © 2004 Elsevier B.V. All rights reserved. Keywords: CO2 ; Polymer; Cloud point; Sorption; Model
1. Introduction Polymer solutions in which CO2 is the solvent are of interest because of the advantages of supercritical CO2 as an environmentally benign solvent [1,2] and as a swelling agent [3,4] in polymer processing. CO2 has also been shown to exhibit specific Lewis acid–base interactions with a number of polymers containing electron-donating groups [5]. Such interactions have a significant effect on the phase behavior of polymer–CO2 systems, leading to an interest in developing models that can describe such behavior. There are two approaches to the development of models that can be used for systems which exhibit specific solute–solvent interactions at high pressures. Equation of state models that take into consideration the effect of pressure can be modified to account for specific interactions in solution. Alternatively, activity coefficient models that take account of specific interactions can be reformulated to incorporate the effect of pressure. This work adopts the latter approach, and incorporates pressure effects into the recently ∗
Corresponding author. Tel.: +1 404 894 3098; fax: +1 404 894 2866. E-mail address: [email protected] (A.S. Teja).
0378-3812/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2004.09.035
proposed generalized Associative Reformulation of Thermodynamics (gART-L) model for associating polymer solutions [6–8]. Application of the modified gART-L model to the calculation of cloud point curves and sorption in CO2 –polymer systems using one or two adjustable parameters is discussed below.
2. The modified gART-L model The gART-L model was proposed by Sukhadia [6], Variankaval et al. [7,8] as a simple reformulation of the Flory–Huggins model to account for specific interactions between polymer segments and solvent molecules. The solution is assumed to consist of three types of species: complexes (“associated” polymer–solvent molecules), unassociated solvent and unassociated polymer molecules. The entropy of mixing in the Flory–Huggins model is then recalculated by restricting a certain number of solvent molecules to specific positions on the lattice because of their association with polymer segments. The remaining solvent molecules are distributed randomly on the lattice. Furthermore, the enthalpy of mixing is modified by defining two types of contacts
488
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due to association and dispersion, respectively. The resulting gART-L expression for the Gibbs energy of mixing Gm is given by [6–8]: Gm = αµΦ2 χa + Φ2 {1 − Φ2 (1 + αµ)}[(z − 2 − µ)α nRT Φ2 ln Φ2 + (1 − α)(z − 2)]χu + x 1 − Φ2 (1 + µα) + ln{1 − Φ2 (1 + µα)} 1 + µα − Φ1 ln
Φ1 µαΦ2 + µαΦ2 ln Φ1 − µαΦ2 Φ1 − µαΦ2
+ Φ2 (1 − α) ln(1 − α) + αΦ2 ln α
(1)
where z is the lattice coordination number (=10 in this and previous [6–8] work), R the gas constant, T the temperature, Φ1 and Φ2 the volume fractions of CO2 and polymer, and n the total number of moles. There are five characteristic quantities in Eq. (1): the solvent–solute binding ratio µ; the association ratio α; the number of segments x; and the specific interaction parameters for association χa and for dispersion, χu . The binding ratio is assumed to be 1 throughout this work because only one CO2 molecule is able to associate with a functional group in the polymers investigated in this work. Note that the binding ratio refers to complex formation between a polymer segment and µ solvent molecules according to P + µS ⇔ PSµ . The specific interaction parameter for association χa (which is related to the enthalpy of association Ha as discussed below) can be obtained from FTIR spectra of the polymer–CO2 mixture at the conditions of interest. The presence of CO2 under pressure causes a peak shift (ν) in the FTIR spectrum [5] due to Lewis acid–base interactions between CO2 and the associating functional group in the polymer segment. This peak shift is related [9] to the interaction parameter χa and hence the enthalpy of association Ha as follows: ∆Ha = 0.9874∆ν = RTχa
(2)
The enthalpy of association, in turn, is related to the equilibrium constant K for the association reaction (P + µS ⇔ PSµ ) as follows: K −Ha 1 1 ln (3) = − K0 R T T0 where K0 is the equilibrium constant for association at a reference temperature T0 (say, 300 K). In the above expression, Ha has been assumed to be independent of T over the range of temperatures of interest [5]. K can therefore be calculated at any temperature if Ha and K0 are known. Knowledge of the equilibrium constant K also yields the association ratio α as follows: (1 + K) − (1 + K)2 − 4Φ1 Φ2 K(1 + K) (4) α= 2Φ2 (1 + K)
Fig. 1. Cloud point behavior in PVAc–CO2 . Points represent experimental values, whereas lines represent calculations with x calculated from actual molar volumes of the pure components (solid line), and from molar volumes at ambient conditions (dashed line).
where Φ1 and Φ2 are volume fractions of the polymer and solvent, respectively. Note that K may also be determined from the relative amounts of associated and unassociated species present in a solution via spectroscopy. The dispersion parameter χu can be estimated from the solubility parameter of the non-polar analog (or homomorph) of the polymer [6–8]. The functional group in the polymer molecule which interacts strongly with CO2 is substituted with a non-polar group to obtain the homomorph. The solubility parameter is then estimated via group contributions as described elsewhere [10]. The segment number x is generally obtained from the ratio of the molar volumes of the solvent (CO2 ) and the solute (polymer) at ambient conditions [11]. This somewhat arbitrary ratio works well for liquids, since liquid volumes are not significantly affected by pressure. However, it is quite inadequate at supercritical CO2 conditions, because the molar volume of CO2 changes dramatically with pressure above its critical point. This has a significant effect on the cloud point curve, as shown for poly(vinyl acetate) (PVAc)–CO2 in Fig. 1. Note that a value of x calculated from the ratio of actual molar volumes (which depend on pressure and temperature) gives a much better fit of the cloud point data than the value of x obtained from the ratio of molar volumes at ambient conditions (a “liquid molar volume” of 55 cm3 /mol was assumed for CO2 at ambient conditions [12]). These calculations clearly show that the change in molar volume with T and P must be taken into account in order to apply the model to CO2 –polymer systems. Note also that x generally increases with pressure since the density of CO2 increases more rapidly with pressure than the density of the polymer. The effect of temperature is not as large and leads to only a modest increase in x. In the present work, the polymer molar volumes were obtained from the Tait equation [13] and CO2 properties from
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489
the Patel–Teja equation of state [14]. The modified gART-L model incorporating the pressure was then used to calculate phase equilibria and sorption behavior in CO2 –polymer systems as discussed below. Note that the original and modified gART-L models both require a knowledge of two characteristic parameters (Ha and K0 ) in the calculations. The remaining quantities in the models either have fixed values or they can be calculated from Ha and K0 . The number of parameters can be further reduced if Ha is obtained from FTIR spectra as shown below.
3. Cloud point and sorption calculations using the modified gART-L model Equilibrium between a polymer-rich phase (α) and a CO2 rich phase (β) can be described by β β
xiα γiα = xi γi
Fig. 2. Cloud point behavior in the PBMA–CO2 system. The solid lines represent calculations with Ha = −3.8015 kJ/mol and K0 = 7.536. The points represent experimental data for 5 wt.% polymer [15].
(5)
in which the activity coefficients γ i depend on the temperature, pressure, and composition. These equations may be β solved for any two of the quantities T, P, x2α and x2 . Thus, if the composition of one of the phases is known, then the pressure and composition of the other phase can be calculated at a given temperature using Eq. (5). Cloud point curves at constant concentration (isopleths) can then be calculated by repeating the calculations at different temperatures. In the present work, activity coefficients were obtained by differentiation of Eq. (1), and experimental cloud point data were fitted using either K0 , or K0 and Ha as adjustable parameters. The number of parameters employed in correlating cloud points depends on the availability of spectroscopic data (which yields Ha ) for the system. Values of the parameters obtained by such fitting are discussed in Table 1. In the case of sorption of CO2 in a polymer at low pressures, the equilibrium relation is (assuming negligible solubility of the polymer in CO2 ): xiα γiα = φi (T, P)P
(6)
where φi is the fugacity coefficient of CO2 . The fugacity coefficient was obtained using the Patel–Teja equation [14] for CO2 and Eq. (6) was used to calculate the amount of CO2 sorbed into a polymer. The parameters K0 and Ha from cloud point calculations and/or spectroscopic measurements were used in the calculations of γ i , so that no additional parameters are required in the calculations.
4. Results for cloud point behavior of polymer–CO2 mixtures Cloud point data of McHugh et al. [15] for the systems poly(butylmethacrylate) (PBMA)–CO2 and PVAc–CO2 were correlated using Ha values obtained from FTIR measurements [5]. A single adjustable parameter K0 was obtained by fitting these data. Cloud point data (see Table 1) [16] for a series of polyacrylate–CO2 systems were also correlated. However, two adjustable parameters (Ha and K0 ) were used in these calculations because spectroscopic measurements were not available. Average molecular weights and polydispersities of the polymers are available in the original references [15,16]. Experimental and calculated cloud point curves in the PBMA–CO2 system are shown in Fig. 2. The interaction energy Ha in this system was estimated to have a value of ∼4 kJ/mol from the peak shift in its FTIR spectrum [5]. As can be seen in the figure, the data are fitted well using only one adjustable parameter (K0 ). Cloud point curves for polymer Mw of 100,000 and 320,000 were calculated using the same value of the adjustable parameter, indicating that K0 does not apparently depend on the molecular weight. However, the molecular weight does affect x via the molar volume of the polymer. As a result, the cloud point curve for PBMA with Mw of 100,000 is shifted by as much as 45 K relative to the cloud point curve of PBMA with an Mw of 320,000. This indicates that higher temperatures would be needed for
Table 1 Cloud point calculations for PBMA–CO2 and PVAc–CO2 using the modified gART-L model Polymer
Mw (×1000)
Tg (K)
Temperature range (K)
P range (bar)
Ha * (kJ/mol)
K0
AAD%
PBMA PBMA PVAc
100 320 125
293 293 303
398.2–483.3 416.5–498.5 295.8–423.4
1300–3000 1430–3000 602.5–1046.3
−3.8 −3.8 −3.9
7.536 7.536 4.381
0.47 0.88 2.45
∗
From peak shift data in Ref. [5].
490
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Ha (kJ/mol)
K0
PMA–CO2 PEA–CO2 PBA–CO2 PEHA–CO2 PODA–CO2
−7.985 −5.624 −3.214 −2.861 −2.413
6.106 5.088 4.713 3.266 2.745
Fig. 3. Cloud point behavior in polyacrylate–CO2 systems. The solid lines represent calculations and points represent experimental data for 5 wt.% polymer [16].
CO2 to solubilize the higher molecular weight PBMA, which is typical for UCST type phase behavior. Similar behavior was encountered in the PVAc–CO2 system. Calculated results were in good agreement with experimental data using one adjustable parameter, although the average error of 2.45% was somewhat higher than the average error in the PBMA–CO2 system (0.88%). The model was also used to calculate cloud point curves in poly(acrylate)–CO2 systems using two adjustable parameters. Fig. 3 and Table 2 present comparisons between calculated and experimental [16] cloud point data for poly(methyl acrylate) (PMA), polyethylacrylate (PEA), poly(butyl acrylate) (PBA), poly(ethylhexyl acrylate) (PEHA), and poly(octadecyl acrylate) (PODA) in CO2 . Also shown in Table 2 are comparisons by McHugh et al. [16] using the SAFT equation to calculate cloud points in these systems. Both models are able to correlate the data well, using the same number of adjustable parameters in each case (and the same experimental data for cloud points). The fitted parameters for the modified gART-L model are shown in Table 3 and are of the same magnitude as the parameters obtained for mixtures of PBMA or PVAc with CO2 . Moreover, the values of the interaction energy are similar to the energies tabulated for Lewis acid–base interactions [17]. We may infer from these calculations that interactions between CO2 and the polymer are likely to be found in polyacrylate–CO2
Fig. 4. Sorption predictions for PVAc (Mw = 100,000)–CO2 systems. The squares, triangles and circles represent experimental sorption data at 313.15, 333.15, and 353.15 K, respectively [18].
systems. Note that the magnitude of the parameters decreases as the length of the alkyl tail on the acrylate increases. This could be due to a decrease in effective polarity as a result of shielding of the molecule by the alkyl tail, which limits CO2 access to the functional group on the polymer chain. Additionally, Tg values decrease with increasing alkyl tail length since the addition of longer alkyl chain affects the stiffness of the polymer. This leads to an increase in the magnitude of the parameters for the systems when the polymer component has a higher Tg .
5. Prediction of sorption behavior The solubility of CO2 in polymers was also predicted using the parameters obtained by fitting cloud point curves. The re-
Table 2 Cloud point calculations in polymer–CO2 systems using the modified gART-L and SAFT models Polymer
Mw (×1000)
Tg (K)
Temperature range (K)
P range (bar)
This work AAD%
SAFTa AAD%
PMA PBA PEA PEHA PODA
31 62 120 113 23
282 224 250 218 NA
296.2–473.7 355.7–469.0 319.2–468.7 430.2–483.0 486.5–536.9
1843–2242.5 1057.8–3000 1272.9–3000 1192.4–3000 1226.7–2663.8
2.29 0.59 0.84 1.57 1.21
2.51 2.66 1.80 4.43 1.47
a
Calculations of McHugh et al. [15].
I.A. Ozkan, A.S. Teja / Fluid Phase Equilibria 228–229 (2005) 487–491
sults of such calculations and comparisons with experimental data in the case of PVAc–CO2 are presented in Fig. 4. Reasonable agreement was obtained between predicted values and experimental sorption data (error in the range 5–20%). Note that there are no adjustable parameters in the sorption calculations, so the agreement may be regarded as reasonable.
6. Conclusions Cloud point curves in seven polymer–CO2 systems have been correlated using the modified gART-L model with one or two adjustable parameters depending on the availability of spectroscopic data. Good agreement with experimental data was obtained (average errors of 1.3%). The predictive capability of the model was also investigated by predicting the solubility of CO2 in polymers using parameters from cloud point data.
References [1] E.J. Beckman, J. Supercrit. Fluids 28 (2004) 121–192. [2] C.F. Kirby, M.A. McHugh, Chem. Rev. 99 (1999) 565–602. [3] M.A. McHugh, V.J. Krukonis, Supercritical Fluid Extraction— Principles and Practice, 2nd ed., Butterworths–Heinemann, 1994.
491
[4] A.R. Berens, G.S. Huvard, R.W. Korsmeyer, F.W. Kunig, J. Appl. Polym. Sci. 46 (1992) 231–242. [5] S.G. Kazarian, M.F. Vincent, F.V. Bright, C.L. Liotta, C.A. Eckert, J. Am. Chem. Soc. 118 (1996) 1729–1736. [6] T. Sukhadia, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA, 1999. [7] N. Variankaval, M.E. Rezac, A.S. Abhiraman, ACS National Meeting, Paper PMSE 297, August 2000. [8] N. Variankaval, Ph.D. Dissertation, Georgia Institute of Technology, Atlanta, GA, 2001. [9] F.M. Fowkes, D.O. Tischler, J.A. Wolfe, L.A. Lannigan, C.M. Ademu-John, M.J. Halliwell, J. Polym. Sci., Part A: Polym. Chem. 22 (1984) 547–566. [10] R.P. Danner, T.E. Daubert, Physical and Thermodynamic Properties of Pure Compounds: Data Compilation, Hemisphere Publishing Corp., New York, 1990. [11] J.M. Prausnitz, R.N. Lichtenthaler, E.G. de Azevedo, Molecular Thermodynamics of Fluid-phase Equilibria, 3rd ed., Prentice-Hall, New Jersey, 1999. [12] J.M. Prausnitz, F.H. Shair, AIChE J. 7 (1961) 682–686. [13] P.G. Tait, Physics and Chemistry of the Voyage HMS Challenger, vol. II, Part IV, S.P.L.XI, 1888. [14] N.C. Patel, A.S. Teja, Chem. Eng. Sci. 37 (1982) 463–473. [15] M.A. McHugh, F. Rindfleisch, T.P. DiNoia, J. Phys. Chem. 100 (1996) 15581–15587. [16] M.A. McHugh, M. Lora, F. Rindfleisch, J. Appl. Polym. Sci. 73 (1999) 1979–1991. [17] M.D. Joesten, L.J. Schaad, Hydrogen Bonding, Marcel Dekker, NY, 1974. [18] C. Zhong, H. Masuoka, Fluid Phase Equilib. 144 (1998) 49–57.