Ind. Eng. Chem. Res. 2007, 46, 2191-2197
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Simplified Nonrandom Model for Predicting Solvent Activity in Polymer Solutions Li Yi Le, Qing Lin Liu,* Zhen Feng Cheng, and Xin Bo Zhang Department of Chemical & Biochemical Engineering, College of Chemistry & Chemical Engineering, Xiamen UniVersity, Xiamen, 361005, China
A simplified activity model was developed based on the nonrandom version of statistical thermodynamics. The partition function consists of two parts: one is a random distribution of the empty sites, which corresponds to the ideal solution part, and the other is a correction factor for the actual nonrandom distribution of the empty sites, which corresponds to the excess part. Model parameters are not difficult to obtain in comparison with other model equations. The applicability of this model was demonstrated by calculating the solvent activities in 12 polymer solutions. It is found that the calculated results are consistent with the experimental data. The present model was also applied to describe the phase diagram of polystyrene-cyclohexane and polystyrene-ethyl acetate. It is proved that the method was able to correlate both low critical solution temperature and upper critical solution temperature. Furthermore, extension of this model equation to multicomponent systems is straightforward, so the present model can be taken as an alternative for the description of the phase behaviors in polymer solutions. Introduction Estimation of solvent activities in polymer mixtures is important for processes that are concerned with polymers. A variety of GE models have been used for estimation of the phase behavior for polymer solutions. Of these models, the FloryHuggins (F-H) equations1 are the most widely used expressions because of their simplicity and accuracy. However, these models cannot describe the phase behavior of a system with strong interactive species and of systems only differing in polymer weight. As such, modifications to the F-H theory have been proposed recently. Elbro et al. presented an F-H type equation for the entropy of mixing2 in conjunction with the energy contribution term taken from the UNIQUAC model3 for the energy of mixing to successfully describe the vapor-liquid equilibrium of nearly athermal polymer solutions. Considerable efforts have also been made in recent years toward the development and refinement of statisticomechanical theories of the liquid state to elucidate the F-H interaction parameter χ12. The most widely accepted one was the SanchezLacombe lattice fluid theory for classical fluids4,5 and for polymer solutions.6 Some modifications were made regarding the volume-combining rule used in this theory. Panayiotou has extended this theory to account for gas solubility in polymeric liquids with a more general mixing rule used for the closedpacked volume per segment in the mixture.7 Wen et al. have employed Sanchez-Lacombe lattice fluid theory to elucidate the thermodynamics of PMMA/SAN blends by using a new volume-combining rule to evaluate the closed-packed volume per mer.8 Although the formulation of the existing activity model equation is self-consistent with a sound theoretical background, a less complicated model with fewer parameters without sacrificing accuracy would be convenient for practical use. In this article, a simplified activity model was developed based on the nonrandom version of statistical thermodynamics. The purpose of this work is to develop a simple expression, where the free-volume effects are taken into account and where the parameters of the energy interaction term are independent of * To whom correspondence should be addressed. e-mail: qlliu@ xmu.edu.cn.
composition and can be found from vapor-liquid equilibria of solvent mixtures only. Model Development In this section, we will describe the nonrandom lattice partition function for a mixture of N1 molecules, each consisting of r1 segments (r1-mers) and N2 r2-mer molecules at temperature T and external pressure P. For an approximate description of our system, it is assumed that all molecules are arranged on a quasi-lattice of Nr sites, N0 of which are empty (holes). Molecule segments and holes are assumed in the present work to be a nonrandom mixture. In a general way, we may consider that the partition function of our system can be written as follows:
Q(N,T,P) ) QRQNR
(1)
where QR is the partition function for the hypothetical system where there is a random distribution of the empty sites and QNR is a correction factor for the actual nonrandom distribution of the empty sites. We may then use Flory’s combinatorial expression1 for the number of configurations available to our system. In Sanchez-Lacombe4-6 nomenclature, the pressure ensemble partition function QR in it is maximum term approximation may be written as
QR(T,P) )
()() δ1 σ1
N1
δ2 σ2
N2
( )
Nr! Nq! N0!N1!N2! Nr!
z/2
exp(- E/kT) (2)
E is the potential (attractive) energy, σi is a symmetry number, and δi is a flexibility parameter characteristic of component i. Where
Nr ) N0 + r1N1 + r2N2 ) N0 + rN
(3)
Nq ) N0 + q1N1 + q2N2 ) N0 + qN
(4)
qi ) [ri(z - 2) + 2]/z
(5)
10.1021/ie061198k CCC: $37.00 © 2007 American Chemical Society Published on Web 03/06/2007
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N ) N1 + N 2
(6)
r ) x1r1 + x2r2
(7)
q ) x1q1 + x2q2
(8)
xi is the mole fraction of component i, qi is the average number of external contacts per segment (either molecular or empty sites), and z is the coordination number and is set to 10 in this article. In addition, the total number of external contacts between the molecules is
zNq ) z(N0 + q1N1 + q2N2) ) z(N0 + qN)
(9)
For QNR we adopted the expression9
QNR )
Nrr0!N000![(Nr00/2)!]2 Nrr!N00![(Nr0/2)!]2
(10)
where Nrr is the number of external contacts between the segments of the molecules, N00 is the number of contacts between the empty sites, and Nr0 is the number of contacts between a molecular segment and an empty site. Superscript 0 refer to the case of randomly distributed empty sites. We know from thermodynamic relations
G E ≈ AE )
∆A - RT(x1 ln x1 + x2 ln x2) n1 + n2 GE ) RT
∑ xi ln γi
From statistical thermodynamics, the Helmholtz free energy of our system is obtained from the partition function as follows:
∆A ) - RT ln
(13)
Q Q1Q2
[
(
(14)
()() () () δ1 σ1
N1
δ2 σ2
Q1 ) QR,1QNR,1 )
δ1 σ1
N1
Q2 ) QR,2QNR,2 )
δ2 σ2
N2
N2
QcQNR exp(-E/RT) (15)
φi )
(21)
θij θjj
)
θi θj
τij
(22)
τij ) exp(- θ(ij - jj)/kT)
(23)
θij + θjj ) 1
(24)
ij is the absolute value of the interaction energy between a segment of species i and that of species j, which is assumed as follows for nonspecific interactions
ij ) (iijj)0.5(1 - λij)
(25)
where λij is the binary interaction parameter. ij between holes and molecular species is zero. Where θi and θj are the area fractions on the hole free basis and are defined by
∑ Njqj ) θi/θ
(26)
The holes are supposed to be randomly distributed in the system. The total area fraction occupied by molecules, θ, is defined by
∑ Niqi/Nq
(27)
where the area fraction of component i is given by
θi ) Niqi/Nq
(28)
Qc,2QNR,2 exp(- E2/RT) (17)
QNR,1 ) QNR,2 ) 1
(18)
Integrating eqs 15-18 into eq 14 gives
(
riNi rN
The two-fluid theory is employed for the calculation of ∆ANR in a way similar to that in the derivation of the UNIQUAC model.3 For a binary mixture of components i and j, one considers two central cells. For each central cell, one can write, based on the concept of local area fractions,
θ)
Qc,1QNR,1 exp(- E1/RT) (16)
If the assumption of randomness of pure component segments and holes in the system was made, then
∆A ) - RT ln
(20)
where Na is Avogadro’s number and φi is the close-packed segment fraction for component i, given by
θi ) Niqi/
where
Q ) QRQNR )
) ]
∆AR φ1 Nr φ2 1 1 1 2 ) ln φ1 + ln φ2 + φ1φ2 RT r1 r2 z r1 r2 Na
(11) (12)
A ) - RT ln Q
∆AR is derived from Dudowicz and Freed,10 truncating their expression at its first correction for simplicity, and is given with some modifications for practical purposes by
)
QR QNR + ln ) ∆AR + ∆ANR QR,1QR,2 QNR,1QNR,2 (19)
where ∆AR is the contributions from the combinatorial part in the random array and the energy of random mixing and ∆ANR is the nonrandom part correcting for the effects of nonrandom mixing.
θij can be related to the local area fraction of component i around the segment of component j, θij by
θij ) θij/θ
(29)
τij is the group interaction parameter of i-j pairs being reflective of nonrandomness. The local area fraction is related to the bulk area fraction using eqs 22-24 for components i and j as
θij ) θiτij/
∑ θkτ kj
(30)
Let us suppose that a molecule 1 in the pure liquid has z(0) nearest neighbors; the energy of vaporization per molecule is 1/2 z(0)θ1(0)11(0), where 11(0) characterizes the potential energy of two nearest neighbors in pure liquid 1. In a mixture, the
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2193
central molecule in hypothetical fluid 1 is surrounded by z(1)θ11 molecules of species 1 and z(1)θ21 molecules of species 2, where θ11 is the local area fraction of component 1 about central molecule 1 and θ21 is the local area fraction of component 2 about central molecule 1. We make a transfer for a molecule 1 from pure liquid 1 into the “two-liquid” mixture; the energy released by the condensation process is 1/2 z[θ1111(1) + θ2121(1)] (assume z(1) ) z(0) ) z) and similarly for molecule 2. Then we obtain the energy of mixing ∆E.
1 (1) (0) (0) ∆E ) z[N1q1(θ11(1) 11 + θ2121 - θ1 11 )] + 2 1 (2) (0) (0) z[N2q2(θ22(2) 22 + θ1212 - θ2 22 )] (31) 2 where the superscript 1 stands for mixtures, and 0 for pure liquids. Two assumptions are introduced to simplify the evaluation of eq 31. The first one is that the total area fraction occupied by molecules in pure states is the same as that in mixtures; e.g., relation θi(0) ) θ holds. The second one is that the pair energy in pure states is the same as that in mixtures; e.g., 11(1) ) 11(0) and 22(1) ) 22(0). Dropping the superscript for pair energy and replacing the local surface fractions with those on the hole free basis, eq 31 can be written as
∆E )
zNq θ[θ1θ21(21 - 11) + θ2θ12(12 - 22)] (32) 2
The Helmholtz energy of mixing contributing to the nonrandom part, ∆ANR, is obtained by integrating the Gibbs-Helmholtz equation by using the thermodynamic relation at constant volume and composition
∆ANR ) T
∫01/T ∆Ed(T1)
(33)
Substituting eq 32 into eq 33 gives
zNq ∆ANR zNq θ ln(θ1 + θ2τ21) θ ln(θ2 + θ1τ12) )RT 2Na 1 2Na 2 (34) This formulation goes back to the residual term in the original UNIQUAC when the hole numbers, N0, tend to zero. After inserting eqs 20 and 34 into eq 19, we have
[
(
) ]
φ2 φ1 Nr 1 1 1 2 ∆A ln φ1 + ln φ2 + φ1φ2 ) RT r1 r2 z r1 r2 Na zNq [θ ln(θ1 + θ2τ21) + θ2 ln(θ2 + θ1τ12)] (35) 2Na 1 The activity of solvent in the mixture can be obtained from eq 35 as
[
( ) ( )
ln a1 ) ln φ1 + φ2 1 -
{
]
r1 r1 1 1 2 2 + φ + r2 z r1 r2 2
θ1 θ2τ12 zq1 1 - ln(θ1 + θ2τ21) 2 θ1 + θ2τ21 θ1τ12 + θ2
}
(36)
Equations 35 and 36 are the working equations in this present article. The first part is a return to the Freed Flory-Huggins combinatorial term, and the second part goes back to the residual part of the original UNIQUAC model when the holes in the
systems are negligible. So this method is expected to reproduce improved phase character for polymer solutions since the freevolume effect is being taken into account. The physical significance of the proposed modifications/additions to current practice lies in that the hole effect was introduced into the original UNIQUAC model. While the UNIQUAC uses two adjustable parameters, the present method requires only one binary parameter when the pure molecular parameters ii and V/i are available. Results and Discussion Calculation. The molecular weight and density of polymers used in this article were taken from the database.11-13 The number of sites occupied by a molecule of component i, ri, can be calculated from the relation ri ) V/i /Vh, where the hard core volume, V/i , is considered to be a characteristic parameter of pure component i and is given by following empirical correlation as a function of temperature for reliable and convenient use.14
V/i ) Va + VbT + Vc ln T
(37)
The estimated values of the coefficients Va, Vb, and Vc are summarized in Table 1. The volume for each segment of the molecules and the holes, Vh, was assumed to be a constant value 9.75 × 10-3 m3/kmol as suggested by Panayiotou.9 qi can be obtained from eq 5. The close-packed segment fraction, φi, can be obtained from eq 21. The determination of the group interaction parameter, τij, can be carried out through following three substeps. (a) Determination of the interaction energy between a segment of species i and that of species j, ij. ii can be obtained from eq 38 as suggested by Yoo.14
ii ) (Ea + EbT + Ec ln T)k
(38)
where Ea, Eb, and Ec are the component characteristic parameters, and can be obtained from ref 14, which are given in Table 1. ij can be estimated from eq 25 by with considering the interaction between a segment of species i and that of species j through the binary interaction parameter, λij, which can be obtained by fitting to the experimental liquid-vapor phase equilibrium data.11 (b) Determination of the total surface area fraction occupied by molecules, θ. Nq can be calculated by the relation Nq ) Nh + q, where the number of holes, Nh, can be obtained from r and the reduced volume and q can be obtained from eq 8. So θi can be obtained from eq 28; thus, the surface fraction on the hole free basis, θi, and the θ can be calculated from eqs 26 and 27. (c) Determination of τij. The value of τij can be obtained by substituting the parameters obtained from steps a and b into eq 23. At this stage, the solvent activity can be calculated from eq 36 in this article. Solvent activities for 12 polymer solutions were calculated by the present model equation to evaluate its applicability in the description of phase behavior for polymer/ solvent systems. The deviation between the calculated results and experimental data is reflected by the AAD value listed in Table 2. The agreement between the calculated results and the observations are good. In Table 2, we list a comparison of our model calculations with models such as Entropic-FV (EFV) proposed by Elbro et al.,2 UNIFAC-FV (UFV) presented by Oishi et al.,15 and the standard UNIFAC. Both models are in
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Table 1. Component Characteristic Parameters for Estimation the Hard Core Volume Eq 37 and Energy Eq 38 chemicals
formula
Va
Vb
Vc
Ea
Eb
Ec
range (K)
pentane cyclohexane benzene toluene ethylbenzene chloroform acetic acetate acetone PIB PS PVAc
C5H12 C6H12 C6H6 C7H8 C8H10 CHCl3 C4H8O2 C3H6O naa na na
79.21 41.08 19.08 39.14 28.56 69.98 21.84 67.06
-0.0007 -0.0228 -0.0153 -0.0164 -0.0334 0.0194 -0.0136 0.0375
3.48 11.14 11.90 11.22 16.57 -0.38 12.68 -1.74
100.22 180.65 167.01 191.44 234.54 124.20 270.67 221.51 95.868 84.318 120.287
0.0309 0.0694 0.0292 0.0592 0.0910 -0.0084 0.0714 0.0013 0.0624 0.1117 -0.0035
-2.74 -16.33 -9.84 -16.00 -25.43 -0.29 -31.29 -17.11 0.0 0.0 0.0
273-450 273-513 283-473 313-523 273-473 273-483 273-473 273-473 na na na
a
na, not applicable.
Table 2. Comparison of Calculated Results between Different Methods with Experimental Dataa AADb (%)
a
system
T (K)
N
EFV
UFV
UNIFAC
this work
PIB(40000)-cyclohexane PIB(50000)-cyclohexane PVAc(143000)-benzene PVAc(48000)-benzene PVAc(9000)-acetone PS(63000)-benzene PS(10300)-toluene PIB(1170)-pentane PIB(2250000)-pentane PS(290000)-chloroform PS(97200)-ethylbenzene PVAc(143000)-acetic acetate average
298.15 298.15 303.15 303.15 303.15 288.15 321.65 298.15 298.15 298.15 283.15 303.15
8 11 9 8 5 8 9 5 4 11 7 10
7.49 5.52 8.13 2.78 3.16 2.92 6.05 7.09 4.01 13.39 1.05 3.26 5.40
4.61 1.92 9.98 3.48 1.90 3.87 7.09 4.83 3.81 9.25 0.85 4.29 4.65
24.34 23.94 17.87 11.37 21.92 16.43 20.85 24.39 12.44 15.96 1.24 13.06 16.98
0.58 1.68 7.00 2.76 3.91 1.62 3.25 1.36 0.83 4.46 0.045 3.44 2.58
Literature values from ref 11. b AAD ) (1/N)|(ai,cal - ai,
exp)/ai, exp|
× 100%; N, experimental points; UFV, UNIFAC-FV.
the frame of the UNIFAC method, our proposed model is in the frame of the UNIQUAC method. Effect of Solvent Mass Fraction. Comparison of the estimation of solvent activities by the present model among other methods with the experimental data over various solvent mass fractions was depicted in Figures 1-6. It is shown that the model developed in this study is able to produce improved estimations in solvent activity over EFV2 and UNIFAC-FV15 methods, and the values of the binary interaction parameters between groups for the EFV and UNIFAC-FV theories were obtained from the data bank.16 The trend of the deviation of predicted results from the observations is not the same for different polymer solutions. For an example, Figures 3 and 4 show that the deviation is smaller at low solvent mass fractions but larger at high solvent mass fractions for PVAc(48000)-benzene and PS(10300)-
toluene systems. But it is quite different for PIB(2250000)pentane; the deviation is larger at low solvent mass fractions for this system, as indicated in Figure 2. It is observed that calculation of the surface area fraction of component i, with or without using the hole free basis, does have an effect on the solvent activity prediction. The variation in solvent mass fraction has a slight effect on the calculation of the second term on the right-hand side of the working equation, while it has a remarkable influence on the first-term estimation. Normally, the value for the first term is negative and the second term positive. Effect of Binary Interaction Parameter. It is shown from eq 25 that the binary interaction parameter, λij, is related to ij, while the latter has an impact on the estimation of the randomness factor, τji, which relates to the nonrandom part correcting for the effects of nonrandom mixing through eq 22.
Figure 1. Calculated results against experimental data for solvent activities for polyisobutylene (PIB)(40000)-cyclohexane at 25 °C (λij ) -0.0158).
Figure 2. Calculated results against experimental data for solvent activities for PIB(2250000)-pentane at 25 °C (λij ) -0.0245).
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2195
Figure 3. Calculated results against experimental data for solvent activities for poly(vinyl acetate) (PVAc)(48000)-benzene at 30 °C (λij ) -0.0136).
Figure 6. Calculated results against experimental data for solvent activities for PVAc(143000)-acetic acetate at 30 °C.
Figure 4. Calculated results against experimental data for solvent activities for polystyrene (PS)(10300)-toluene at 48.5 °C (λij ) -0.0110).
Figure 7. Effect of interaction parameter on solvent activities estimation for PVAc(143000)-acetic acetate at 30 °C.
Figure 5. Calculated results against experimental data for solvent activities for PS(290000)-chloroform at 25 °C.
This, consequently, leads to λij having an influence on the solvent activities prediction. We studied in detail how λij can influence the estimation in solvent activities in this section. Using the λij ) 0 calculation in Figures 5 and 6, we are trying to make a point regarding the importance of including interactions. Taking the systems of PVAc(143000)-acetic acetate and PIB(50000)-cyclohexane as examples, the reason for choosing
Figure 8. Effect of interaction parameter on solvent activities estimation for PIB(50000)-cyclohexane at 25 °C.
the system of PVAc(143000)-acetic acetate is that it consists of two components both having hydroxyl moieties, resulting in strong nonideal interaction between the two species in this mixture. The calculation is depicted in Figure 7; the variation for the value of λij is in the range of 0 to -0.2181. It is shown that the calculated results for solvent activities at various λij make a large difference. This is also true for PIB(50000)-
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Table 3. Coefficient of λij ) aij + bijT for Various Polymer-Solvent Systems polymer
polymer Mw
solvent
polystyrene polystyrene
100 000 1 270 000
ethyl acetate cyclohexane
cyclohexane polymer solution, which is different in nature from PVAc(143000)-acetic acetate system. As indicated in Figure 8, the variation for solvent activity with λij ranging from 0 to -0.146 is significant. So one may conclude that λij has appreciable impact on solvent activities prediction. Phase Diagram Correlation. To examine the present model further, we applied the method to the polymer-solvent system exhibiting low critical solution temperature (LCST) and upper critical solution temperature (UCST). The calculation for liquidliquid equilibrium is based on the equality of component chemical potentials in both phases. The condition for the instability of a binary liquid mixture depends on the nonideality of the solution and on the temperature. It may be generally a maximum (upper) or a minimum (lower) temperature on a T-x diagram. The miscibility of polymer and solvent usually increases with temperature until complete miscibility is reached at an UCST. If the temperature is further increased, however, the solution again becomes only partially miscible at a LCST. At the LCST or UCST of a partially miscible mixture,
kind of phase diagram LCST UCST
aij (-)
bij ([1/K] × 103)
-1.201 62 0.162 42
1.762 -0.514
compositions of coexisting phases tends to identical, and the condition for the critical points can be written
∂2∆A ∂3∆A ) )0 ∂φ12 ∂φ13
(39)
In the phase diagram correlation, the adjustable parameter, λij, is calculated as follows suggested by Firouzi17
λij ) aij + bijT
(40)
where aij and bij are assumed to be constant characteristics of the components i, j and are listed in Table 3. The aij and bij values in Table 3 are obtained by fitting to experimental values18 of two systems using the least-squares technique. We find λij values for different temperatures and then curve fit the results. Figure 9 shows the system of polystyrene/cyclohexane. It indicates that the experimental data19 and calculated results for the UCST are in good agreement. Figure 10 displays the cloud point curve for the polystyrene/ethyl acetate system. It suggests that the experiment data20 are well fitted to the calculated data and the correlated LCST is in good agreement with the observation. Conclusion
Figure 9. Cloud point temperature as a function of polymer mass fraction for polystyrene/cyclohexane (Mw ) 1 270 000).
A simplified activity model was developed based on the nonrandom version of statistical thermodynamics. This model is in a form similar to the UNIQAC activity model equation; model parameters ri, φi, θi, etc., are easily calculated. Twelve polymer solution systems including both weak and strong polarity systems, such as PIB(40000)-cyclohexane corresponding to the weak and PVAc(143000)-acetic acetate to the strong, were used to evaluate the model. It is found that predicted results in solvent activity are comparable to the observations. Furthermore, extension of this model to multicomponent is straightforward. One interesting observation is that surface area fraction of component i, which is calculated with or without using the hole free basis, has an effect on the calculated value of the second term on the right-hand side of eq 36, while does not have much effect on the predicted solvent activity. Another one is that the variation of the binary interaction parameter,λij, does cause obvious differences in solvent activity estimation. The present method with an adjustable parameter is able to describe the phase diagram for polymer-solvent systems. This study suggests that the present model correlates the experimental UCST and LCST of various polymer solutions with reasonable accuracy. Acknowledgment The support of the National Nature Science Foundation of China Grant 50573063, the Program for New Century Excellent Talents in University, and the research fund for the Doctoral Program of Higher Education (2005038401) in preparation of this article is gratefully acknowledged. Literature Cited
Figure 10. Cloud point temperature as a function of polymer mass fraction for polystyrene/ethylacetate (Mw ) 100 000).
(1) Flory, P. J. Principles of polymer chemistry; Cornell University Press: New York, 1953.
Ind. Eng. Chem. Res., Vol. 46, No. 7, 2007 2197 (2) Elbro H. S.; Fredenslund, A. a.; Rasmussen P. A New Simple Equation for the Prediction of Solvent Activities in Polymer Solutions. Macromolecules 1990, 23, 4707-4714. (3) 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, 116-128. (4) Sanchez, I. C.; Lacombe, R. H. An elementary molecular theory of classical fluids. Pure fluids. J. Phys. Chem. 1976, 80 (21), 2352-2362. (5) Lacombe R. H.; Sanchez I. C. Statistical thermodynamics of fluid mixtures. J. Phys. Chem. 1976, 80 (23), 2568-2580. (6) Sanchez I. C.; Lacombe R. H. Statistical thermodynamics of polymer solutions. Macromolecules 1978, 11 (6), 1145-1156. (7) Panayiotou C. G. Lattice-fluid theory of polymer solutions. Macromolecules 1987, 20 (4), 861-871. (8) Wen, G. Y.; Sun, Z. Y.; Shi, T. F.; Yang, J.; Jiang, W.; An, L. J.; Li, B. Y. Thermodynamics of PMMA-SAN Blends Application of the Sanchez-Lacombe lattice fluid theory. Macromolecules 2001, 34, 6291-6296. (9) Panayiotou C.; Vera J. H. Statistical thermodynamics of r-mer fluids and their mixtures. Polym. J. 1982, 14, 681-694. (10) Dudowicz, J.; Freed K. F. Role of molecular structure on the thermodynamic properties of melts blends and concentrated polymer solutions comparison of Monte Carlo simulations with the cluster theory for the lattice model. Macromolecules 1990, 23, 4803-4819. (11) Wen H.; Elbro H. S.; Alessi P. Polymer Solution Data Collection. Part 1. Vapor-Liquid Equilibrium; Chemistry Data Series; DECHEMA: Frankfurt am Main, Germany, 1992. (12) 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.
(13) Romdhane I. H.; Danner R. P. Solvent volatilities from polymer solutions by gas liquid chromatography. J. Chem. Eng. Data 1991, 36 (1), 15-20. (14) 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. (15) 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. (16) Poling B. E.; Prausnitz J. M.; O’Connell J. P. The Properties of Gases and Liquids; McGraw-Hill: New York, 2000. (17) Firouzi F.; Modarress H.; Mansoori G. A. Predicting liquid-liquid transition in polymer solutions using the GCLF equation of state. Eur. Polym. J. 1998, 34 (10), 1489-1498. (18) Danner R. P.; High M. S. Handbook of Polymer Solution Thermodynamics; DIPPER, AIChE: New York, 1993. (19) Shultz A. R.; Flory P. J. Phase equilibria in polymer-solvent systems. J. Am. Chem. Soc. 1952, 74, 4760-4767. (20) Bae Y. C.; Lambert S. M.; Soane D. S.; Prausnitz J. M. Cloudpoint curves of polymer solutions from thermooptical measurements. Macromolecules 1991, 24, 4403-4407.
ReceiVed for reView September 14, 2006 ReVised manuscript receiVed January 19, 2007 Accepted February 5, 2007 IE061198K
