Ind. Eng. Chem. Res. 2006, 45, 365-371
365
A New Activity Model for Polymer Solutions in the Frame of Lattice Theory Qing Lin Liu* and Zhen Feng Cheng Department of Chemical and Biochemical Engineering, College of Chemistry and Chemical Engineering, Xiamen UniVersity, Xiamen, 361005, People’s Republic of China
A simplified activity model for polymer solutions is derived from the Gibbs-Helmholtz relation, in conjunction with the lattice theory in this work. The model includes not only combinatorial and residual terms but also a free-volume (FV) term. The calculated results differ significantly between the model equations with and without taking the FV effect into consideration. The validity of this new method is demonstrated by comparing the calculated activities from the proposed model equation with those by the Entropic-FV (EFV), UNIFACFV, and UNIFAC methods. The overall average absolute deviation (AAD) of solvent activities in 34 polymer solutions was 7.58% for the EFV method, 5.32% for the UNIFAC-FV method, 22.00% for the UNIFAC method, and 3.48% for the proposed model. It is shown that the new model can yield improved results over the other models and is able to predict the phase behavior for polymer solutions over a wide range of molecular weights, temperatures, and densities. comparison to the widely used UNIFAC, UNIFAC-FV, and Entropic-FV16 methods.
Introduction Understanding the phase behavior of polymer solutions is useful for polymer industrial processes, such as polymerization and polymeric membrane fabrication. It is also useful for the analysis of the swelling behavior of polymeric membranes, which is one important factor for studying the membrane’s separation mechanism. Laboratory determination of these data can be potentially expensive and time-consuming, so it is necessary to predict the solvent activities in polymer solutions using simple activity model equations. The most widely known models for polymer solutions can be grouped under two categories: (i) equation of state, based on the lattice-fluid theory of polymer solutions, such as the GC-Flory1 and GCLF2 models; and (ii) activity models based on group contribution, such as UNIFAC,3,4 UNIQUAC,5,6 and UNIFAC-ZM.7,8 Of these two type equations, the second one, which requires fewer model parameters and is comparatively simple, because it does not take the effect of pressure into consideration, can bring out reasonable solvent activity predictions. Therefore, this type of model expression is popular and can be taken as a good choice to describe the phase behavior for polymer solutions. These group contribution models, incorporated with the freevolume (FV) definition, have gained increasing attention and have been investigated extensively.9-12 The UNIQUAC-FV model is successfully applied to systems that exhibit both an upper critical solution temperature (UCST) and a lower critical solution temperature (LCST).13 UNIQAC-related models could be used to describe binary to ternary phase behavior for polymer solutions,14,15 although they can be used to characterize phase behavior for some polymer solutions successfully. FV-based models are generally making underestimations for solvent activity; thus, many modifications to the FV fraction expression have been made to improve the calculated results. The objective of this article is to develop an activity model for polymer solutions based on the Gibbs-Helmholtz relation, in conjunction with the cubic lattice theory. More than 30 polymer/solvent systems were used to demonstrate the validity of the proposed model by calculating the solvents activities with * To whom correspondence should be addressed. E-mail:
[email protected].
Model Development For algebraic simplifications, two component systems were taken as an example for the model development. According to the Gibbs-Helmholtz relationship ∆mixA ) ∆mixU - T∆mixS, we obtain
[ ] ∂
∆mixU )
( ) ∆mixA T 1 ∂ T
()
(1)
V,x
The Helmholtz energy of mixing is obtained by integrating the Gibbs-Helmholtz equation, using the Guggenheim’s athermal entropy of mixing17 as a boundary condition:
( )
∆mixA ∆mixA ) T T
+
(1/T)f0
()
(2)
()
(3)
∫01/T ∆mixU d T1
Dividing both sides of eq 2 by kNr yields
( )
∆mixA ∆mixA ) kNrT kNrT
+
(1/T)f0
∫01/T
∆mixU 1 d kNr T
At 1/T f 0, we assume that components 1 and 2 form an athermal mixture and we use the equation of Guggenheim for athermal mixtures of molecules of arbitrary size and shape; the mixed Helmholtz energy then can be expressed by18
( ) ∆mixA kNrT
(1/T)f0
)
φ1 φ2 ln φ1 + ln φ2 + r1 r2 φ2 θ2 z φ1 θ1 q ln + q2 ln (4) 2 1 r1 φ1 r2 φ2
[
]
where T is an absolute temperature, k is the Boltzmann’s constant, and the parameters ri, φi, and θi represent the relative molar volume, volume fraction, and surface fraction of component i, respectively. The variables φ1 and θ1 are defined by eqs 5 and 6, respectively:
10.1021/ie050761g CCC: $33.50 © 2006 American Chemical Society Published on Web 12/03/2005
366
Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006
φ1 ) θ1 )
N1r1 N1r1 + N2r2
(5)
N1q1 N1q1 + N2q2
(6)
( )
∆mixA ∆mixA ) kNrT kNrT )
where qi is the surface area parameter, as defined by
zqi ) zri - 2ri + 2
V*i ) NaVHri
(8)
where V* is the characteristic volume representing the hard core contribution and is obtained from the empirical expression.20 The internal energy of mixing for binaries on a cubic lattice21 was truncated for our purpose and can be written as follows:
( )]
()
∆mixU 1 1 ) ˜ 12G12 2φ1φ2 + φ12φ2 + φ1φ22 kNrT r2 r1
(9)
where the reduced interchange energy, ˜ 12, is defined by21
11 + 22 - 212 kT
˜ 12 )
(10)
where 11, 22, and 12 are interaction energy parameters for the 1-1, 2-2, and 1-2 pairs, respectively. 11 and 22 are calculated by the method reported by Yoo et al.;20 ij is obtained simply by
12 ) (11 × 22)1/2
(11)
G12 is calculated from the following equation:21
[
( )( )(
G12 ) 1 - 0.2911 1 -
)
1 1 2 11× r1 r2 r1r2
]
exp(0.6048r12 - 2.744r122) r12 )
()
1 1 5 + r1 r2 4r1r2
[
∆mixU 1 d kNr T
φ1 φ2 z φ 1 θ1 ln φ1 + ln φ2 + q1 ln + r1 r2 2 r1 φ1 φ2 θ2 1 q2 ln + ˜ 12G12 2φ1φ2 + φ12φ2 + r2 φ2 r2 φ1φ22
1 r1
(15)
So the activity of component 1 can be obtained as
( ) [ ] [
ln R1 ) ln φ1 + 1 -
r1 θ1 z φ2 + q1 ln + q1(θ1 - φ2) + r2 2 φ1
(
)]
r1φ2 1 1 1 (φ - θ1) + ˜ 12G12r1φ22 2 + + 2φ1 r2φ1 1 r1 r2 r1 (16)
q2
In the proposed model, we have assumed that the volume change upon mixing and the FV change are both negligible. However, the entropy of mixing is significantly affected by the FV change for some polymer solution systems. To take the effect of FV change into account for these systems, the first and second terms of eq 16 were modified, using Elbro et al.’s conceptual expression.16 Equation 16 was then rewritten as follows:
ln R1 ) ln
φFV 1
( ) [ () ] [ ( )]
+ 1-
VFV 1
VFV 2
φFV 2 +
θ1 z q ln + 2 1 φ1
r 1φ 2 (φ - θ1) + q1(θ1 - φ2) + q2 r2φ1 1 ˜ 12G12r1φ22 2 +
1 1 1 + 2φ1 r1 r2 r1
(17)
where the FV fraction is defined by16
φiFV )
xi(Vi - Viw)
∑j
(18) xj(Vj - Vjw)
-1
(12) (13)
()
() ( )]
where Vi and Viw are molecular volume and van der Waals volume of component i, respectively; the latter can be obtained using the Bondi method.22
Substituting eq 9 into eq 3 gives
∫01/T
(1/T)f0
∫01/T
(7)
where z is the lattice coordination number. A simple cubic lattice is used in this study (z ) 3). The parameter ri can be related to the characteristic volume by19
[
() [ () ( )] [ +
( )]
∆mixU 1 1 1 d ) ˜ 12G12 2φ1φ2 + φ12φ2 + φ1φ22 kNr T r2 r1 (14)
The Helmholtz free energy of mixing expression can be derived by substituting eqs 14 and 4 into eq 3:
Calculation and Discussion Parameter Determination. Area parameters, volume parameters, and the group interaction parameters required for UNIFAC, UNIFAC-FV, and Entropic-FV methods are obtained from the database.23 The two empirical constantssthe external degree of freedom parameter (c1) and the proportionality factor (b), which are required in the UNIFAC-FV methodsare set to values of c1 ) 1.1 and b ) 1.28, as originally used by Oishi and Prausnitz.9 The interaction parameter for the proposed model is adopted from the empirical expression.20 The coordination number used for this model is set to 3; for the other three models, it is set to 10. The densities of solvents are obtained from the handbook literature,24 and those of polymers are taken from the data series literature.25 The densities of polyisobutylene
Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 367 Table 1. Comparison of Solvent Activity Predictions between Different Activity Models Average Absolute Deviation, AAD (%) UNIFAC-FVa UNIFACa
system
temperature (K)
number of experimental points, N
EFVa
PIB(1000000)/butane PIB(1000000)/butane PIB(1000000)/butane PIB(40000)/cyclohexane PIB(50000)/cyclohexane PIB(50000)/cyclohexane
298.15 308.15 319.65 298.15 298.15 311.15
7 12 3 8 11 10
19.09 20.36 16.89 7.49 5.52 7.87
2.75 5.72 1.82 4.61 1.92 4.85
59.00 59.87 62.30 24.34 23.94 31.04
1.65 2.80 3.01 2.90 1.38 3.14
PIB(100000)/cyclohexane PIB(100000)/cyclohexane PIB(50000)/hexane
298.15 313.15 298.15
10 10 8
3.15 4.47 2.60
3.70 4.05 7.44
15.51 15.39 30.87
1.31 2.82 8.99
PIB(1170)/pentane PIB(1170)/pentane PIB(40000)/pentane PIB(1000000)/pentane PIB(2250000)/pentane PVAc(9000)/acetone PVAc(48000)/benzene PVAc(143000)/benzene PVAc(143000)/VAc PS(15400)/toluene PS(10300)/toluene PS(800)/benzene PS(63000)/benzene PS(500000)/benzene PS(900000)/benzene PS(154000)/cyclohexane PS(154000)/toluene PS(290000)/toluene PS(900000)/toluene PS(97200)/ethylbenzene PS(97200)/ethylbenzene PS(800)/chloroform PS(90000)/chloroform PS(290000)/chloroform PPO(500000)/benzene
298.15 308.15 298.15 319.65 298.15 303.15b 303.15 303.15 303.15 298.15b 321.65 298.15 303.15 293.15 318.15 303.15 298.15 298.15 298.15 283.15 308.15 298.15 298.15 298.15 333.35
6 6 9 5 12 5b 8 9 10 6b 9 12 8 16 7 11 11 12 5 7 4 12 4 11 11
10.85 7.09 14.74 20.38 4.01 3.16b 2.78 8.13 3.26 5.52b 6.05 1.61 2.92 7.14 1.91 3.24 0.77 9.84 2.71 1.05 1.18 22.39 15.30 13.39 0.92
11.18 4.83 8.41 7.73 3.81 1.90b 3.48 9.98 4.29 7.40b 7.09 15.36 3.87 7.74 3.59 2.26 0.92 10.34 3.29 0.85 1.00 1.08 11.23 9.25 3.21
24.81 24.39 38.11 59.34 12.44 21.92b 11.37 17.87 13.06 10.47b 20.85 16.59 16.43 22.21 13.01 5.52 5.09 15.78 9.10 1.24 1.47 26.76 17.84 15.96 4.21
5.10 1.96 5.91 4.76 1.09 2.1b 2.90 7.81 4.55 2.72 0.88 1.25 3.27 5.92 1.23 5.84 1.48 7.70 0.84 0.76 0.86 5.63 3.77 5.27 6.66
7.58
5.32
22.00
3.48
overall average a
this work
Unless noted otherwise, the data were obtained from ref 25. b Data taken from ref 29.
Figure 1. Sensitivity of the predictions obtained by models to polymer molar volumes.
(PIB) and polystyrene (PS) are estimated by the following empirical expressions, because they cannot be obtained directly.
F ) 0.929297 - (5.46739 × 10-4)t + (3.10118 × 10-7)t2 (for PIB26) (19) F ) 1.0865 - (6.19 × 10-4)t + (1.36 × 10-7)t2 (for PS27) (20) where the density F is given in units of g/cm3. When considering the effect of temperature-dependent density on the activity
coefficient calculations for the PIB(53000)/ethyl acetate and PVAc(141420)/ethyl acetate systems (where PVAc denotes poly(vinyl acetate)), the Tail correlation28 was adopted to estimate the molar volume of the polymers, and then the FV was further calculated. The sensitivity of activity prediction to the density of polymers is elucidated by observing the variation of activity with slight changes in density. It is reflected that UNIFAC-FV is the most sensitive to the density; both the EFV method and the proposed model do not make much difference in the sensitivity, as indicated in Figure 1 for a PVAc(143000)/ benzene mixture. When the density of PVAc approaches a value of 1.21 g/cm3, the average absolute deviations (AADs) for the UNIFAC-FV and EFV methods have minimum values of 7.5% and 7.2%, respectively; the AAD observed using the present method is 7.1%. The AAD by this model approaches a minimum value of 6.9 as the PVAc density approaches a value of 1.22 g/cm3, whereas the AADs obtained using UNIFAC-FV and EFV are 9.5% and 7.6%, respectively. The proposed approach yields acceptable results in the polymer density range of 1.10-1.22 g/cm3. The EFV method produces reasonable predictions, in the range of 1.22-1.30 g/cm3. The UNIFAC-FV method is the last choice, because it brings in the largest deviation in the range of density considered. The solvent activity is observed to increase with the polymer density, and the AAD has a minimum value in the polymer density range of interest. Therefore, one may conclude that the estimation in activity is sensitive to polymer density. Discussion. A total of 34 polymer solutions, such as PS/toluene, PVAc/benzene, and PIB/pentene, were used to
368
Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006
Figure 2. Experimental and calculated activities of cyclohexane in the PIB(100000)/cyclohexane system at 298.15 K.
evaluate the proposed model. The calculated results for solvent activity using the EFV, UNIFAC-FV, and UNIFAC methods are summarized in Table 1 for comparison. It is obvious that the present method yields improved results for most of the systems. For the PIB(100000)/cyclohexane system, the magnitude of the AAD is 1.31% for this work, 3.15% for the EFV method, 3.70% for the UNIFAC-FV method, and 15.51% for the UNIFAC method. For a few systems, the proposed model does not produce good predictions; for example, for a PPO(500000)/benzene mixture, the magnitude of the AAD is 6.66% for this work, 0.92% for the UNIFAC-FV method, and 3.21% for the EFV method. The reason may be that the coordination number setting of z ) 3 is not the optimum value for such system, and improved predictions can be achieved by adjusting the magnitude of z. Similar to other modified models, this model does not bring an obvious improvement for a few systems, such as for a PS(97200)/ethylbenzene mixture. Table 1 shows that the present method generally reproduces the phase behavior for 34 polymer solutions. It yields improved predictions over the other three models, in terms of the average value of AAD. The EFV method primarily fails significantly in PIB/butane systems; here, the molar weight of PIB is significant higher than that of the other polymers. The results are somewhat worse at very high molecular weights, which are consistent with the findings of Kontogeorgis et al.10 The possible reason may be that no adjustable parameters for the FV part are used in the EFV method. On the other hand, the UNIFACFV term contains two empirical constants (b and c1) that have been optimized based on the vapor-liquid equilibrium (VLE) data of ∼30 polymer-solvent solutions. Another reason may be the dependency of FV on the length and flexibility of the polymer chain. Figures 2 and 3 display the calculated activities for PIB(100000)/cyclohexane and PS(900000)/benzene systems, using four models, relative to the experimental observations.25 It is indicated that the present method represents the experiment data over a wide range of solvent weight fractions. Figures 4 and 5 present the calculated solvent activities of PIB(100000)/ cyclohexane and PS (900000)/benzene systems by the methods with and without considering the FV effect. One can conclude that the method that takes the FV effect into consideration produces remarkable improvement. Figures 6-8 show the calculated activities, the combinatorial part, and the residual part for the PIB (40000)/cyclohexane system, using the EFV method, the UNIFAC-FV method, and the proposed method, respectively. The proposed model is observed to yield improved results, as indicated in Figure 6. The shape of the curve for the combinatorial part of the activity
Figure 3. Experimental and calculated activities of benzene in the PS(48000)/benzene system at 303.15 K.
Figure 4. Experimental and calculated activities of benzene in the PS(900000)/benzene system at 318.35 K.
Figure 5. Calculated activities of cyclohexane in the PIB(100000)/ cyclohexane system at 298.15 K with and without considering the freevolume (FV) effect.
is similar to that for activity including both the combinatorial and the residual part, as depicted in Figure 7. The activity contribution to the residual part approaches unity, as indicated in Figure 8. The calculation of an infinite dilution activity coefficient for a PIB(53000)/cyclohexane mixture is performed using the proposed model, to study its applicability. The results are displayed in Figure 9 for the present method and the EFV model, relative to experimental observations.30 Figure 9 shows that the calculated results by the proposed model and the EFV method are almost constant with temperature under constant density. In contrast, the calculated activities by both methods
Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 369
Figure 6. Experimental and calculated activities of cyclohexane in the PIB(40000)/cyclohexane system at 298.15 K.
Figure 7. Activity contribution to the combinatorial-FV part for cyclohexane in the PIB(40000)/cyclohexane system.
Figure 8. Activity contribution to the residual part for cyclohexane in the PIB(40000)/cyclohexane system.
are increased with temperature, under a temperature-dependent density. The former is less sensitive to temperature than the latter. Figures 10 and 11 show the combinatorial and residual parts of the infinite dilution activity coefficient, respectively. In a manner similar to that observed in Figure 8, the magnitude of the residual part approaches unity and is slightly sensitive to temperature. A polar system of PVAc(141420) and ethyl acetate was also studied for comparison. Figure 12 gives the calculated infinite dilution activity coefficients of ethyl acetate, relative to experimental data.30 It is obvious that this model yields remarkable improvements over the EFV method for this polar system. In a manner similar to that observed in the nonpolar
Figure 9. Predicted infinite dilution activity coefficients for cyclohexane in PIB(53000).
Figure 10. Infinite dilution activity coefficients attribute to the combinatorial-FV part of the models for cyclohexane in PIB(53000).
Figure 11. Infinite dilution activity coefficients attribute to the residual part of the models for cyclohexane in PIB(53000).
system of PIB(53000) and cyclohexane, the combinatorial part of the infinite dilution activity coefficient increases with temperature, under a temperature-dependent density and almost remains constant under constant density, as indicated in Figure 13. Figure 14 shows that the value of the residual part approaches unity and is also slightly sensitive to temperature for this polar system. Therefore, one can conclude that the combinatorial part of activity has a vital role in the accuracy of the prediction. Conclusion In the frame of the Gibbs-Helmholtz relation, a new activity model is developed, using lattice theory and taking the free-
370
Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006
that the present method produces improved results over the other three models for most of the systems considered. List of Symbols
Figure 12. Predicted infinite dilution activity coefficients for ethyl acetate in PVAc(141420).
Figure 13. Infinite dilution activity coefficients contribution to the combinatorial-FV part of the models for ethyl acetate in PVAc(141420).
T ) temperature (K) ∆A ) Helmholtz free energy R ) gas constant ∆U ) energy of mixing x ) mole fraction q ) effective segment ) energy t ) temperature (°C) V* ) characteristic volume Na ) Avogadro’s number PIB ) polyisobutylene PVAc ) poly(vinyl acetate) AAD ) (1/N)∑|(Ri,cal - Ri,exp)/Ri,exp| × 100 φ ) volume fraction θ ) area fraction γ ) activity coefficient r ) segments for each molecule z ) coordination number τij ) binary interaction parameter F ) density R ) activity k ) Boltzmann constant VH ) volume of each segment (cm3) PS ) polystyrene PPO ) poly(propylene oxide) Acknowledgment The support of National Nature Science Foundation of China (Grant Nos. 50243014 and 50573063) in preparation of this article is gratefully acknowledged. Literature Cited
Figure 14. Infinite dilution activity coefficients contribution to the residual part of the models for ethyl acetate in PVAc(141420).
volume (FV) effect into account. In a manner similar to the UNIFAC-FV method, the proposed model consists of a combinatorial term, a residual term, and an FV term. The theoretical advantages are that the proposed model is simple in form and accounts for the FV effect. Another advantage is that the interaction energy parameter for unlike species can be directly obtained from that for like species without introducing any adjustable parameter (eq 11). The theoretical disadvantage is that the coordination number is not 10. A total of 34 systems of polymer solutions were used to evaluate the present method. The calculated activities using the EFV, UNIFAC-FV, and UNIFAC methods were also listed for comparison. It suggests
(1) Holten-Andersen, J.; Rasmussen, P.; Fredenslund, Aa. Phase equilibria of polymer solutions by group contribution. 1. Vapor-liquid equilibria. Ind. Eng. Chem. Res. 1987, 26, 1382-1390. (2) Hamedi, M.; Lu¨tzow, N.; Betz, H. S.; Duda, J. L.; Danner, R. P. Thermodynamic Behavior of Ethylene-Styrene Interpolymers. Ind. Eng. Chem. Res. 2001, 40, 3002-3008. (3) Fredenslund, Aa.; Jones, R. L.; Prausnitz, J. M. Group contribution estimation of activity coefficients in nonideal liquid mixtures. AIChE J. 1975, 21, 1086-1099. (4) Vicente, M. S.; Gottifredi, J. C. Estimation of solvent activities in polymers solutions using a group-contribution method. Sep. Purif. Technol. 2001, 22-23, 671-679. (5) Abrams, D. S.; Prausnitz, J. M. Statistical thermodynamics of liquid mixtures: a new expression for the excess Gibbs energy of partly or completely miscible systems. AIChE J. 1975, 21, 116-127. (6) Rolemberg, M. P.; Kra¨henbu¨hl, M. A. Vapor-Liquid Equilibria of Binary and Ternary Mixtures of Benzene, Cyclohexane and Chlorobenzene at 40.0 kPa and 101.3 kPa. J. Chem. Eng. Data 2001, 46, 256-260. (7) Wibawa, G.; Takishima, S.; Sato, Y.; Masuoka, H. Revision of UNIFAC group interaction parameters of group contribution models to improve prediction results of vapor-liquid equilibria for solvent-polymer systems. Fluid Phase Equilib. 2002, 202, 367-383. (8) Pappa, G. D.; Voutsas, E. C.; Tassios, D. P. Prediction of Activity Coefficients in Polymer and Copolymer Solutions Using Simple Activity Coefficient Models. Ind. Eng. Chem. Res. 1999, 38, 4975-4984. (9) Oishi, T.; Prausnitz, J. M. Estimation of solvent activities in polymer solutions using a group-contribution method. Ind. Eng. Chem. Process Des. DeV. 1978, 17, 333-339. (10) Kontogeorgis, G. M.; Fredenslund, Aa. Simple activity coefficient model for the prediction of solvent activities in polymer solutions. Ind. Eng. Chem. Res. 1993, 32, 362-372.
Ind. Eng. Chem. Res., Vol. 45, No. 1, 2006 371 (11) Bogdanic, G.; Vidal, J. A segmental interaction model for liquidliquid equilibrium calculations for polymer solutions. Fluid Phase Equilib. 2000, 173, 241-252. (12) Tritopoulou, E. A.; Pappa, G. D.; Voutsas, E. C.; Economou, I. G.; Tassios, D. P. Modeling of Liquid-Liquid Phase Equilibria in Aqueous Solutions of Poly(ethylene glycol) with a UNIFAC-Based Model. Ind. Eng. Chem. Res. 2003, 42, 5399-5408. (13) Pappa, G. D.; Voutsas, E. C.; Tassios, D. P. Liquid-liquid-phase equilibrium in polymer-solvent system: correlation and prediction of the polymer molecular weight and the temperature effect. Ind. Eng. Chem. Res. 2001, 40, 4654-4663. (14) Jonquieres, A.; Perrin, L.; Arnold, S.; Lochon, P. Comparison of UNIQUAC with related models for modeling vapor sorption in polar materials. J. Membr. Sci. 1998, 150, 125-141. (15) Jonquieres, A.; Perrin, L.; Arnold, S.; Clement, R.; Lochon, P. From binary to ternary systems: general behavior and modeling of membrane sorption in purely organic systems strongly deviating from ideality by UNIQUAC and related models. J. Membr. Sci. 2000, 174, 255-275. (16) Elbro, H. S.; Fredenslund, Aa.; Rasmussen, P. A New Simple Equation for the prediction of Solvent Activities in Polymer Solutions. Macromolecules 1990, 23, 4707-4714. (17) Guggenheim, E. A. Mixtures; Oxford University Press: Oxford, U.K., 1952. (18) Chang, B. H.; Bae, Y. C. Vapor-liquid equilibria and liquid-liquid equilibria calculations of binary polymer solutions. Polymer 2002, 43, 6627-6634. (19) Panayiotou, C.; Vera, J. H. Statistical Thermodynamics of r-mer fluids and their mixtures. Polymer J. 1982, 14 (9), 681-694. (20) Yoo, K.-P.; Shin, M. S.; Yoo, S. J.; You, S.-S.; Lee, C. S. A new equation of state based on nonrandom two-fluid lattice theory for complex mixtures. Fluid Phase Equilib. 1995, 111, 175-201. (21) Hu, Y.; Liu, H.; Shi, Y. Molecular thermodynamic theory for polymer systems. I. A close-packed lattice model. Fluid Phase Equilib. 1996, 117, 100-106.
(22) Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses; Wiley: New York, 1968. (23) Poling, B. E.; Prausnitz, J. M.; O’Connell, J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2000. (24) Lu, H. Z. Handbook for Petrochemicals; Chemical Industry Press: Beijing, 1992. (25) Hao, W.; Elbro, H. S.; Alessi, P. Polymer Solution Data Collection. Part 1. Vapor-Liquid Equilibrium; DECHEMA Chemistry Data Series, Vol. 14; DECHEMA: Frankfurt am Main, Germany, 1992. (26) Kontogeorgis, G. M.; Coutsikos, P.; Tassios, D.; Fredenslund, Aa. Improved models for the prediction of activity coefficients in nearly athermal mixtures. Part 1: Empirical modifications of free-volume models. Fluid Phase Equilib. 1994, 92, 35-66. (27) Romdhane, I. H.; Danner, R. P. Solvent Volatilities from Polymer Solutions by Gas-Liquid Chromatography. J. Chem. Eng. Data 1991, 36, 15-20. (28) Rodges, P. A. Pressure-volume-temperature relationship for polymeric liquid: review of equations of state and their characteristic parameters for 56 polymers. J. Appl. Polym. Sci. 1993, 48, 1061-1080. (29) Tochigi, K.; Futakuchi, H.; Kojima, K. Prediction of vapor-liquid equilibrium in polymer solutions using a Peng-Robinson group contribution model. Fluid Phase Equilib. 1998, 152, 209-217. (30) Kontogeorgis, G. M.; Fredenslund, Aa.; Tassios, D. P. Simple activity coefficient model for the prediction of solvent activities in polymer solutions. Ind. Eng. Chem. Res. 1993, 32, 362-372.
ReceiVed for reView June 27, 2005 ReVised manuscript receiVed October 23, 2005 Accepted November 2, 2005 IE050761G
