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Ind. Eng. Chem. Res. 2001, 40, 4654-4663
Liquid-Liquid Phase Equilibrium in Polymer-Solvent Systems: Correlation and Prediction of the Polymer Molecular Weight and the Pressure Effect Georgia D. Pappa,* Epaminondas C. Voutsas, and Dimitrios P. Tassios Thermodynamics and Transport Phenomena Laboratory, Section II, Department of Chemical Engineering, National Technical University of Athens, 9 Heroon Polytechneiou Str., Zografou Campus, GR-15780 Athens, Greece
A GE model is used for the correlation and prediction of partial miscibility in polymer solutions that exhibit both upper critical solution temperature and lower critical solution temperature as well as hourglass-type phase separation in some cases. The model employs a combinatorial/ free-volume term and the residual term of UNIQUAC, for which interaction parameters between the solvent molecule and the repeating unit (segment) of the polymer are determined. The obtained parameters are then used to predict the effect of the polymer molecular weight on the solvent/polymer partial miscibility. The model is also successfully applied in the prediction of the pressure effect on mutual solubility by incorporating it through the free volumes of the components. Finally the prediction of excess properties using the proposed model is investigated. 1. Introduction Knowledge of phase equilibria in polymer solutions is essential for the design of various processes such as polymerization, devolatization, or drying. In particular, liquid-liquid equilibria (LLE) information is essential for polymerization processes where polymers are produced in one or two liquid phases and unreacted monomers, solvents, and additives must be separated.1 In recent years extensive work is done in the correlation and prediction of LLE in polymer solutions. The proposed models can be classified in the following main categories: (i) Modifications/extensions of the Flory-Huggins model (Koningsveld and Staverman,2 Bae et al.,3 and Enders and de Loos4). (ii) Free-volume theories capable of describing the lower critical solution temperature (Flory5 and Patterson and Delmas6). (iii) Models based on the lattice-fluid theory (Sanchez and Lacombe,7,8 Kleintjens and Koningsveld,9 Panagiotou and Vera,10 and High and Danner11,12). (iv) Local composition models (Heil and Prausnitz,13 Brandani,14 Vera,15 and Chen16). (v) UNIFAC-based models that include the freevolume effects (Iwai et al.17 and Kontogeorgis et al.18). UNIFAC-based models have been shown to be quite successful in vapor-liquid equilibria (VLE) predictions in polymer/solvent systems19-22 and, moreover, to be capable of predicting at least qualitatively all types of phase splits in polymer solutions.18 In this study the free-volume/combinatorial term proposed by Elbro et al.19 is combined with the UNIQUAC residual term23 considering, however, interactions between the solvent molecule and the polymer segment (repeating unit). The proposed model, referred to as the free volume (FV)/ UNIQUAC, can be thus used for prediction purposes. A similar approach was used by Bogdanic and the late * To whom correspondence should be addressed. Tel.: +301 7723137. Fax: +3017723155. E-mail:
[email protected].
Vidal24 for the prediction of the polymer molecular weight (MW) effect on partial miscibility for various polymer solutions, the majority of which exhibit only an upper critical solution temperature (UCST) or a lower critical solution temperature (LCST) phase split. In the present study the FV/UNIQUAC model is applied to systems that exhibit both UCST and LCST as well as an hourglass phase split in some cases. It is also applied in the prediction of the pressure effect on LLE by incorporating pressure-dependent molar volumes for the solvent and the polymer. 2. The Model The model employs the combinatorial term proposed by Elbro et al.19 and the residual term of UNIQUAC.23 The combinatorial expression is similar to the FloryHuggins one, but free-volume fractions are used instead of volume fractions. Thus, both combinatorial and freevolume effects are included in a single combinatorial/ FV term:
ln γcomb-fv ) ln i
φfv φfv i i +1xi xi
(1)
where xi is the mole fraction of component i and φfv i is the fraction of free volumes:
φfv i
)
xiVfv i
∑i
(2)
xiVfv i
the free volume (Vfv i ), which is accessible to other molecules volume, is assumed to be equal to w Vfv i ) Vi - Vi
(3)
where Vi is the molar volume of component i and Vw i is the van der Waals volume as calculated by the method of Bondi.25
10.1021/ie0103658 CCC: $20.00 © 2001 American Chemical Society Published on Web 09/18/2001
Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4655
Note that eq 1 allows for variation of the activity coefficients with pressure by introducing in eq 3 the variation of the molar volumes with pressure. This, of course, implies the existence of an excess volume of mixing and requires relaxation of the assumption of VE ) 0 made by Elbro et al.19 in the theoretical development of eq 1. For the application of the model, every system was assumed to be a mixture of two groups: the solvent molecule and the polymer repeating unit. The systems examined in this study exhibit both UCST and LCST and in some cases hourglass phase split too, and hence the temperature range of the experimental data is very wide, more than 150 K in most cases. For this reason linearly temperature-dependent solvent/segment interaction parameters of the following form were calculated:
Rmn ) R0mn + R1mn(T - 298.15)
(4)
For the estimation of the parameters, the following objective function (OF) was minimized:
OF )
∑ ∑[ln(xIi γIi ) - ln(xIIi
γII i )]
(5)
n exp i
where xIi and xII i are respectively the mole fractions of component i in the liquid phases I and II and γIi and γII i are the corresponding activity coefficients. Logarithms were used because the polymer and solvent activity coefficients differ by several orders of magnitude. 3. Estimation of the Molar Volumes of the Components The combinatorial term of Elbro et al.19 used in this study is very sensitive to the accuracy of the molar volumes used for the components. This was demonstrated in a previous study on VLE prediction in polymer solutions,22 where it was shown that an error introduced to the molar volume of one of the components leads to a double or even triple error in the calculated activity coefficient. It is, thus, necessary to use accurate volumes for both polymer and solvent with respect to temperature and pressure in order to make a reliable evaluation of the performance of the model. In this study, polymer molar volumes were obtained by using the T- and P-dependent Tait correlation with the parameters available in the publication by Rodgers.26 To account for the pressure effect on solvent molar volumes, modifications of the Peng-Robinson (PR) equation of state (EoS) were used. For the nonpolar solvents (e.g., methylcyclohexane), the so-called t-mPR EoS27 was used (see Appendix 1), while for polar solvents, for which t-mPR gives poor saturated liquid volume predictions, the a, b, and m parameters of the PR EoS were fitted to vapor pressures and saturated liquid molar volume experimental data taken from DIPPR as proposed by Kontogeorgis et al.28 The obtained a, b, and m parameters of the EoS are presented in Table 1. 4. Pressure Effect on LLE in Polymer Solutions The effect of pressure on the miscibility in liquid mixtures is associated with the volume change on mixing (or the excess volume of mixing, VE) at fixed
Table 1. PR Parameters for the Polar Solvents solvent
T range (K)
acetone 254-468 diethyl ether 210-350 tert-butyl 240-447 acetate ethyl formate 229-452c
a b (bar‚L2/mol) (L/mol)
m
∆Pa (%)
∆Vb (%)
15.9136 18.6056 29.9782
0.0619 0.0836 0.1148
0.7715 0.39 0.96 0.8625 1.79 0.41 0.9269 0.69 0.77
16.6131
0.1067
0.8036 0.79 0.85
a
∆P % is the average absolute percent deviation in vapor pressure ∆P % ) average|(Ps,calc - Ps,exp)/Ps,exp| × 100. b ∆V % is the average absolute percent deviation in saturated liquid molar volumes ∆V % ) average|(Vcalc - Vexp)/Vexp| × 100. c Parameters for t-PR EoS.
temperature and composition. If VE is positive, then the solubility decreases with increasing pressure, while for negative VE, the solubility increases with increasing pressure.29 The UCST phase separation is mainly connected with differences in intermolecular forces between the solvent and the polymer. Experimentally it has been found that pressure may lower or raise the UCST, corresponding to negative and positive values of VE.30 A parabolic-like pressure dependence of the UCST with a minimum is also reported for a few polymer solutions.31,32 On the other hand, the LCST is mainly attributed to the differences in the free volumes of the polymer and the solvent33 or, in other words, to the fact that the solvent is more expanded than the polymer especially as the critical temperature of the solvent is approached. The polymer is less compressible than the solvent, and hence raising the pressure decreases the free-volume difference and lowers the tendency toward partial miscibility at constant temperature, thus raising the LCST.30,34 As experiments show, the pressure effect is more pronounced on the LCST than on the UCST. There are, however, systems where the LCST phase separation is due to strong interactions between the solvent and the polymer molecules like hydrogen bonding,35 such as, for example, the system poly(ethylene glycol) (PEG)/water for which the LCST is almost constant for 1-50 atm.36 4.1. Results of the Pressure Effect on LLE. The systems examined in this study, the type of phase separation they exhibit, and the temperature range of the available experimental data are listed in Table 2. The polymer polydispersity index (Mw/Mn) is also listed in the same table. As shown, all data used correspond to practically monodispersed polymers, thus avoiding to incorporate the polydispersity effect in the model. The EoS used for the calculation of the solvent molar volume for each system and the obtained solvent/segment interaction parameters are shown in Table 3. For these systems the model yields negative excess volumes, and consequently it predicts increasing miscibility with pressure. System PS(20400)/Acetone (Where PS ) Polystyrene). Though this system is partially miscible for all temperatures at atmospheric pressure (hourglass behavior), raising the pressure results in the appearance of a complete miscibility region at intermediate temperatures (UCST and LCST behavior). Experimental data at atmospheric pressure were used to calculate the necessary interaction parameters, and then predictions were made for higher pressures. Results are presented in Figure 1. A very satisfactory correlation is obtained and, moreover, the pressure effect predictions are remarkably accurate, with deviations of less than 5 K for the UCST and LCST even at the highest pressure of 100 bar.
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Table 2. Experimental LLE Data for Polymer/Solvent Systems at Various Pressures system
Mw/Mn
pressure (bar)
LLE type
T range (K)
data used for
PS(20400)/acetone30
