Ind. Eng. Chem. Res. 2003, 42, 5399-5408
5399
Modeling of Liquid-Liquid Phase Equilibria in Aqueous Solutions of Poly(ethylene glycol) with a UNIFAC-Based Model Efthimia A. Tritopoulou,†,‡ Georgia D. Pappa,† Epaminondas C. Voutsas,† Ioannis G. Economou,‡ and Dimitrios P. Tassios*,† Thermodynamics and Transport Phenomena Laboratory, Section II, School of Chemical Engineering, National Technical University of Athens, 9, Heroon Polytechniou Str., Zografou Campus, GR-15780 Athens, Greece, and Molecular Modeling of Materials Laboratory, Institute of Physical Chemistry, National Research Center for Physical Sciences “Demokritos”, GR-15310, Aghia Paraskevi Attikis, Greece
An activity coefficient model, which is a combination of the entropic-free volume combinatorial term and a UNIFAC residual term, referred to as EFV/UNIFAC, is applied to the correlation and prediction of the molecular weight effect on liquid-liquid equilibria (LLE) of aqueous poly(ethylene glycol) (PEG) solutions. Interaction parameters between the end group of PEG and its repeating unit, as well as between the end group and the water molecule, are estimated by fitting activity coefficient experimental data of mixtures containing low molecular weight compounds. Interaction parameters between the water molecule and the repeating unit of PEG are determined from experimental LLE data for a single PEG molecular weight with water. These interaction parameters are used to predict the effect of the polymer molecular weight on miscibility. The proposed model provides a good compromise between accuracy and simplicity. 1. Introduction Water-soluble polymers are widely used in industrial applications and biotechnology. In particular, solutions of poly(ethylene glycol) (PEG) in water are used in biochemistry and biochemical engineering to separate and purify biological products, biomaterials, proteins, and enzymes from the complex mixtures in which they are produced.1,2 PEG is water-soluble in all proportions at moderate temperatures due to the hydrophilic ethylene oxide groups it contains. For rather short chain lengths, PEG and water are completely miscible at all temperatures. For higher PEG molecular weights (MW), the oriented intermolecular forces, which make the system miscible at moderate temperatures, weaken as temperature increases. Above a certain temperature, the lower critical solution temperature, LCST, the mixture splits into two phases due to the dispersion forces. At even higher temperatures, above the upper critical solution temperature, UCST, the combinatorial entropy of mixing becomes dominant and a single homogeneous phase reappears. Thus, the liquid-liquid equilibria phase envelope for these systems is a closed-loop one. As the PEG MW increases, the region of partial miscibility expands with respect to both temperature and polymer concentration.3,4 Several models based on the Flory-Huggins lattice theory have been used to describe the phase behavior in PEG/water systems. Hu et al.5 developed a doublelattice model based on Freed’s lattice cluster theory, an exact mathematical solution of the Flory-Huggins lattice. Using their model, which takes into account * To whom correspondence should be addressed. Tel.: +302107723232. Fax: +302107723155. E-mail: dtassios@ chemeng.ntua.gr. † National Technical University of Athens. ‡ National Research Center for Physical Sciences “Demokritos”.
specific interactions, Hu et al. obtained excellent fit for systems with LCST but fair fit for systems with a closed miscibility loop. Later, Bae et al.6 returned to the FloryHuggins lattice theory, and using the χ parameter as the product of two functions, one depending on composition and the other on temperature, they obtained very good correlation results with, however, five adjustable parameters per PEG MW. Yu et al.3 developed a model by coupling a cross-association term to deal with hydrogen bonding, the Flory-Huggins theory for the excess entropy of mixing, and the nonrandom two-liquid (NRTL) model for the intermolecular interactions. With five parameters per binary mixture, they obtained very good correlation results for various MWs of PEG in water. Li et al.7 used a modification of the nonrandom two-liquid (NRTL) model to correlate the phase behavior of water-polymer systems with fairly good results for low MW PEG (2270 and 2180) in water. Furthermore, fairly good correlation results have been obtained with more complicated models based on statistical thermodynamics, such as the modification of Flory-Huggins theory that accounts also for the conformation of the PEG chains proposed by Karlstrom8 and the solvation model by Matsuyama and Tanaka.9 All the aforementioned models can be used only to correlate experimental LLE data. Recently, a mean field approximation model was developed by Dormidontova,10 which has predictive capabilities but gives fair results especially for the low PEG MW. In this study a simple UNIFAC-based model for the prediction of the MW effect on LLE of aqueous PEG solutions is presented. The model employs the freevolume/combinatorial term proposed by Elbro et al.,11 which has been shown to perform satisfactorily in polymer solutions.12 The empirical modifications of this term proposed by Kouskoumvekaki et al.13 and Sheng et al.14 are also examined. The energetic interactions are taken into account by the UNIFAC residual term.15 In the proposed approach the PEG/water solution is considered to be a mixture of three groups: the water
10.1021/ie0304154 CCC: $25.00 © 2003 American Chemical Society Published on Web 09/16/2003
5400 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
molecule, the polymer internal chain group (CH2OCH2 or EO in short), and the polymer end group (CH2OH). All pair group interaction parameters, except for the EO/ H2O one, are determined from experimental activity coefficient data of systems containing low MW compounds, while for the EO/H2O pair experimental LLE (binodal curves) data for one PEG MW are used. When the proposed approach is used, the prediction of the PEG MW effect on LLE (binodal curves) in aqueous PEG systems is investigated. Prediction of PEG/water vaporliquid equilibria (VLE) is also considered.
Qk is the area parameter of group k, and Θm is the area fraction of group m, calculated as
Θm )
ln γi ) ln γcomb-FV + ln γres i i
(1)
The combinatorial expression is similar to the FloryHuggins one, but free-volume (FV) fractions are used instead of volume fractions. Thus, both combinatorial and free-volume effects are included in a single combinatorial-FV term,
φFV φFV i i +1xi xi
ln γcomb-FV ) ln i
(2)
where xi is the mole fraction and φFV i is the free-volume fraction of the component i:
φFV i
)
xiVFV i
∑i
(3)
xiVFV i
The free volume (VFV i ) is assumed to be equal to W VFV i ) Vi - Vi
(4)
where Vi is the molar volume and VW i is the van der Waals volume of the component i as calculated by the Bondi method.16 The residual part of the model is that of UNIFAC, which is calculated as the difference between the residual activity coefficient of group k in the solution (Γk) and the corresponding activity coefficient in a reference solution containing only molecules of type i (Γ(i) k ):
ln γres i ) with
(
∑k v(i)k (ln Γk - ln Γ(i)k )
ln Γk ) Qk 1 - ln(
ΘmΨmk
(5)
)
ΘmΨmk) - ∑ ∑ m m ∑n ΘnΨnk
(6)
v(i) k is the number of groups of type k in molecule i,
(7)
∑n QnXn
where the mole fraction of group m in the mixture (Xm) is given by
2. The Model and Required Parameters The model, referred to as EFV/UNIFAC, employs the combinatorial term proposed by Elbro et al.11 and the residual term of UNIFAC.15 In this way, the activity coefficient of the component i (γi) can be expressed as
QmXm
Xm )
∑j v(j)m xj ∑j ∑n
(8)
v(j) n xj
Finally, the group interaction parameter (Ψmn) is given by
(
Ψmn ) exp -
)
Rmn T
(9)
where Rmn is the interaction parameter (IP) between groups m and n. The EFV/UNIFAC model has already been successfully applied in the correlation and prediction of LLE in polymer/solvent solutions17 where, however, nonaqueous systems were involved. Two modifications of the free-volume term (eq 4) were also examined in this work. The first one (EFV-1.2) has been proposed by Kouskoumvekaki et al.11 The authors used the combinatorial term of Elbro et al. to predict the activity coefficients in nearly athermal polymer/ solvent solutions and asymmetric alkane systems. It was shown that the results are improved when a hard core volume larger than the van der Waals one is employed. They concluded that the optimum free volume for this combinatorial term is W VFV i ) Vi - 1.2Vi
(10)
The second modification (EFV-0.9) follows the suggestion of Sheng et al.14 that better representation of the combinatorial effects can be achieved by employing an exponent equal to 0.9 in the free-volume fractions:
φFV i
)
0.9 xi(VFV i )
∑i
(11)
0.9 xi(VFV i )
The molecular formula of PEG is HOCH2-(CH2OCH2)n-CH2OH. Even though the number of the PEG end groups is quite small (two per molecule) compared with the chain groups, it has been pointed out that their presence does affect phase behavior in PEG aqueous solutions, especially for the lower MW PEG,3 due to the hydrogen bonds they form with water molecules as well as with the oxygen atoms present in the EO group. This was further supported by some preliminary results of ours which indicated that not even qualitative correlation results could be obtained when the end groups were neglected. Consequently, in this study aqueous solutions
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5401 Table 1. Groups Involved in PEG/Water Systems and Their vdW Volume and Surface Area Values
H2O one (Table 2):
group
vdW volume (cm3/mol)
dimensionless vdW surface area, Q
Rmn ) Rmn,0 + Rmn,1(T - 298.15)
H2O EO (CH2OCH2) CH2OH
12.37 24.16 18.27
1.4a 1.32 1.74a
The objective function (O.F.) used for the correlation was
a For the H O and CH OH groups the fitted Q values from 2 2 UNIFAC15 were used.
of PEG are treated as a mixture of the two PEG groups (EO and CH2OH) plus the water molecule group. Application of the model requires knowledge of water molar volumes, PEG molar volumes, and the vdW volumes of the three groups involved for the combinatorial-FV term and the vdW surface areas of the groups as well as the interaction parameters (aij) between them for the residual term. Water molar volumes were obtained from DIPPR18 and PEG molar volumes from Rodgers.19 Values for the groups’ vdW volume and surface area (Q), which are calculated from the group contribution method proposed by Bondi,16 are presented in Table 1. To develop a predictive model, and at the same time to provide realistic values to the group interaction parameters, their values, except those of the EO/H2O pair, were estimated independently using experimental activity coefficient data for appropriate mixtures of small molecules. When this approach was used, some additional parameters were also needed: estimation of the CH2OH/H2O parameters required the CH2/H2O and CH2/CH2OH ones while the CH2OH/EO parameters required also those of the CH2/EO pair. The end group containing pairs for which parameters were determined, along with the corresponding database used in the calculations, are presented in Table 2. Finally, the EO/H2O pair interaction parameters were determined using experimental LLE data for a single molecular weight PEG/H2O binary mixture. 3. Evaluation of Interaction Parameters Linear temperature dependency was assumed for the interaction parameters for all pairs except for the EO/
O.F. )
γi,exp - γi,calc
∑| NDi)1 1
ND
γi,exp
|
(12)
× 100
(13)
where γi,exp and γi,calc are the experimental and calculated activity coefficient values respectively and ND is the number of data. Calculations were performed for the combinatorial term of Elbro et al. (EFV) as well as for the two modifications: ΕFV-0.9 and ΕFV-1.2. Results are presented only for EFV, which performs best, and they are compared with predictions of UNIFAC-Dortmund, which is the most recent and updated UNIFAC version. The required molar volumes for all compounds involved were obtained from DIPPR. The interaction parameters were determined as discussed in sections 3.1-3.4 and the obtained values are shown in Table 3 while the correlation results are presented in Table 4. The procedure for the calculation of the EO/H2O pair interaction parameters is presented in section 3.5. 3.1. The CH2/CH2OH Pair. Attempts to determine interaction parameters for the CH2/CH2OH pair with the EFV/UNIFAC model by fitting alcohol/alkane binary infinite dilution activity coefficient data gave poor results, especially in the description of the higher MW alcohols. Similar poor predictions were obtained with the UNIFAC-Dortmund.25 The poor performance of the UNIFAC models is attributed to the lack of an explicit term that accounts for specific hydrogen-bonding interactions. Hydrogen bonding is quite important for low MW alcohols, leading to high γ values but less important for the high MW alcohols whose γ values in alkanes are close to unity. The results for both low and high MW alcohols were improved by introducing the effect of hydrogen bonding in an empirical way through the surface area parameter (Q) of the CH2OH group. The
Table 2. End Group Containing Pairs for Which Interaction Parameters Are Determined and the Corresponding Database NDa
temperature range (K)
γ∞
47
333-463
20
alcohol/water alkane/water
γ∞
44 6
333-383 323-374
20, 21
alkane/ether
γ∞
112
330-390
b
alcohol/ether PEG/alkane
γ ∞,
54 24
313-333 334-363
20, 22, 23, 24
type of data
pairs needed
pairs involved
CH2OH/H2O
CH2/CH2OH
alcohol/alkane
CH2OH/H2O CH2/H2O CH2/EO
EO/CH2OH
EO/CH2OH
systems
γint c
a ND: number of data. b Data reproduced using UNIFAC-Dortmund.25 concentrations.
c
references
γint refers to activity coefficient values at intermediate
Table 3. Calculated Group Interaction Parameters for the EFV/UNIFAC Model CH2 amn,0 (K) CH2 H2O CH2OH CH2OCH2 a
1051.2 52.1 -54.1
H2O amn,1 2.37 -2.27 -0.59
CH2OH
CH2OCH2
amn,0 (K)
amn,1
amn,0 (K)
amn,1
amn,0 (K)
amn,1
1310.5
-3.0
1057.5 739.9
-0.96 -10.15
0.39 a 1.12
377.6
-3.15
147.3 a 25.9
-278.7 a
7.37 a
Values were determined from LLE data of PEG/water systems and are presented in Table 5.
5402 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 Table 4. Results for Binary Systems Containing Low MW Compounds and the Database Used AADa type of data
systems pentane/ethanol decane/ethanol heptadecane/ethanol eicosane/ethanol heptane/propanol hexadecane/propanol octadecane/propanol heptane/butanol hexadecane/butanol octadecane/butanol nonane/pentanol pentane/dodecanol heptane/dodecanol pentane/tetradecanol octane/tetradecanol hexane/hexadecanol heptane/hexadecanol
γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞
1 2 2 4 4 4 3 5 4 3 2 2 1 2 3 3 2
total ethanol/water propanol/water butanol/water pentanol/water octanol/water nonanol/water decanol/water
hexane/water octane/water
diethyl ether/ethanol dipropyl ether/ethanol dibutyl ether/ethanol dipropyl ether/propanol dipropyl ether/hexanol dipropyl ether/octanol
heptane/PEG-4000 nonane/PEG-4000 hexane/PEG-7500 octane/PEG-7500 hexane/PEG-10000 nonane/PEG-10000
47
γ∞ γ∞ γ∞ γ∞ γ∞ γ∞ γ∞
10 4 14 4 4 2 6
total
44
γ∞ γ∞
3 3
total
6
γ∞ γ∞ γ∞ γ∞, γint b γint γint
4 2 6 16 12 14
total
54
γ∞ γ∞ γ∞ γ∞ γ∞ γ∞
4 4 4 4 4 4
total a
ND
24
temperature range (K) Alkane(1)/Alcohol(2) 343 343-363 333-343 333-373 333-353 373-413 333-352 333-373 373-413 333-352 353-373 326-333 333.15 338-348 329-348 333-367 347-367 overall Alcohol(1)/Water(2) 348-371 353-365 333-383 343-363 333-373 333 333-373 overall Alkane(1)/Water(2) 323-373 323-374 overall Ether(1)/Alcohol(2) 313-323 333 323-352 313-323 313-323 313-323 overall Alkane(1)/PEG(2) 334-364 334-364 334-364 334-364 334-364 334-364 overall
Average absolute deviation AAD ) (1/ND)∑|ln γexp - ln γcalc|.
new surface area parameter is a function of the alcohol carbon number, according to the expression
Q′(CH2OH) )
Q(CH2OH) ncR
(14)
where nc is the alcohol carbon number and R an adjustable parameter. In this way, what is achieved is what experimentally expected: the longer the alcohol, the weaker the interaction of the hydroxy group; due to that, the association effect diminishes. A similar approach has been proposed by Anderson and Prausnitz26 for the UNIQUAC model.
b
γint
EFV/UNIFAC (correlation) ln γ1 0.80 0.94 0.32 1.03 0.49 0.10 0.03 0.65 0.05 0.28
ln γ2
0.20 0.06
0.21
UNIFAC-Dortmund (prediction) ln γ1 0.96 1.05 0.11 1.00 0.62 0.16 0.10 0.66 0.12 0.32
0.50 0.09 0.07 0.09 0.05 0.07 0.13
ln γ2
0.34 0.22
0.20
0.32 1.12 0.98 0.07 0.90 0.73 0.59
0.47
0.15
0.51
0.35
0.02 0.06 0.48 0.81 0.07 0.24 0.11
0.05 0.21 0.07 0.34 0.15 0.20 0.13
0.03 0.05 0.00 0.48 1.05 0.53 1.18
0.02 0.14 0.13 0.12 0.24 0.45 0.44
0.26
0.12
0.47
0.22
4.23 3.88
4.45 4.81
4.05
4.63
0.12 0.12 0.08 0.06 0.15 0.16
0.23 0.12 0.19 0.08 0.18 0.19
0.15 0.00 0.11 0.05 0.06 0.06
0.04 0.21 0.11 0.07 0.08 0.07
0.11
0.10
0.07
0.10
1.14 1.23 0.94 1.11 0.83 1.01
1.43 1.60 1.04 1.29 0.86 1.16
1.04
1.23
refers to activity coefficient values at intermediate concentrations.
The exponent, R, and the interaction parameters were simultaneously fitted to γ∞ experimental data for alcohol/ alkane systems. The calculated interaction parameters are listed in Table 3, while R was found equal to 0.33 for alcohols up to nc ) 20. The carbon number-dependent Q values were used for the rest of the n-alcohol containing systems of this work. Satisfactory correlation is obtained for the alcohol/ alkane systems as suggested by the results in Table 4. Compared to UNIFAC-Dortmund, the proposed model performs better especially for the activity coefficients of alcohols in alkanes. In Figure 1, EFV/UNIFAC and UNIFAC-Dortmund are compared for the prediction of
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5403
Figure 1. Prediction of infinite dilution activity coefficients for alcohol/alkane systems with the EFV/UNIFAC. Experimental data taken from DECHEMA.20
activity coefficients for alcohol/alkane systems not included in the correlation database. Experimental data correspond to systems containing alcohols from 2 to 16 carbon atoms and alkanes from 5 to 18 carbon atoms, including cyclohexane, in the temperature range of 333-373 K. Results for the low MW alcohols are similar but a significant improvement is achieved with EFV/ UNIFAC for the higher MW onessfrom 5 to 16 carbon atomsswhich have the lowest activity coefficient values. 3.2. The CH2/H2O and CH2OH/H2O Pairs. Interaction parameters for these two pairs were estimated simultaneously by fitting experimental infinite dilution activity coefficient data for alkane/water and alcohol/ water systems. This approach was chosen because when the CH2/H2O parameters were estimated using data
only for alkane/water systems and then used to determine the CH2OH/H2O parameters from alcohol/water binaries, very poor results were obtained for the latter. Simultaneous fit improves significantly the results for alcohol/water systems as shown in Table 4. In the same table it is also shown that the correlation for the highly nonideal alkane/water binaries is still poor but improved over that of UNIFAC-Dortmund. The poor performance of UNIFAC in alkane/water systems has been pointed out by various researchers.27,28 3.3. The CH2/CH2OCH2 Pair. A very limited number of experimental data for alkane/ether systems containing the CH2OCH2 group are reported in the literature. For this reason the alkane/ether data used for the determination of the EFV/UNIFAC interaction parameters were generated by the UNIFAC-Dortmund. 3.4. The CH2OCH2/CH2OH Pair. Interaction parameters for the CH2OCH2/CH2OH pair were estimated by fitting simultaneously activity coefficient experimental data of alcohol/ether and alkane/PEG systems. The inclusion of experimental γ∞ data of PEG/alkane systems in the correlation database, apart from the apparent reason of using PEG-containing systems such as the one of interest in this study, aimed also to the extension of the temperature range of the database to higher temperatures. Correlation results with the EFV/UNIFAC for both types of systems are presented in Table 4. The results are very satisfactory for the alcohol/ether systems but not so for the alkane/PEG ones where the model underestimates the γ∞ of PEG a little less than the UNIFAC Dortmund. 3.5. The CH2OCH2/H2O Pair. Interaction parameters between the water molecule and the chain group (EO) of the polymer were determined using the LLE experimental data for the system PEG-2290/water of Saeki et al.29 Due to the closed-loop type of phase equilibria involved, quadratic temperature dependency
Figure 2. LLE correlation (dashed line) for PEG-2290/water and prediction (solid lines) for PEG-2180, 3350, 4490, 8000, 100000, 1020000/ water systems with the EFV/UNIFAC model. Experimental data (points) were taken from Saeki et al.,29 Bae et al.,30 and Saraiva et al.31
5404 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 Table 5. Interaction Parameters between H2O and CH2OCH2 Groups Estimated for the EFV/UNIFAC from LLE Data of PEG/Water Systems H2O
CH2OCH2
amn,0 (K) amn,1 amn,2 (K-1) amn,0 (K) amn,1 amn,2 (K-1) H2O CH2OCH2 -226.8
7.75
319.3
-0.0244
2.28
-0.0065
was employed:
Rmn ) Rmn,0 + Rmn,1(T - 298.15) + Rmn,2(T - 298.15)2 (15) The objective function (O.F.) that was minimized is given by the expression
O.F. )
∑i
II [ln(xIi γIi ) - ln(xII i γi )]
[ln(xIi )
-
ln(xII i )]
(16)
Figure 3. Shultz-Flory diagram for the UCST of PEG-2180, 3350, 4490, and 8000/water systems. Experimental data and model predictions.
where xIi and xII i are respectively the mole fractions of component i in the two liquid phases and γIi and γII i are the corresponding activity coefficients. Logarithms were used because the polymer and solvent activity coefficients differ from each other by several orders of magnitude. The obtained values of the EO/H2O interaction parameters are given in Table 5 and the quality of correlation of the LLE data is shown in Figure 2. 4. Prediction of the MW Effect on PEG/Water LLE The determined group interaction parameters were used to predict the effect of the polymer molecular weight on partial miscibility. For this purpose, experimental LLE data for aqueous solutions of PEG of several MWs were used:
PEG MW: 2180 and 1 020 000 from Saeki et al.29 PEG MW: 3350 (1.6), 8000 (1.6), and 100 000 (2.0) from Bae et al.30 PEG MW: 4490 (1.13) from Saraiva et al.31 The numbers in parentheses correspond to the PEG polydispersity (Mw/Mn) where available. The predictions with the proposed model are shown in Figure 2. The results are satisfactory for the UCST and LCST with deviation lower than 15 K, but less so in terms of the width of the phase envelope, where the H2O solubility is underpredicted. The results must be considered quite satisfactory given the simplicity of the model used. It is apparent that for improved results the polymer polydispersity and the specific hydrogen-bonding interactions must be explicitly taken into account. It has been for example
Figure 4. Variation of HE, -TSE, and GE with temperature as predicted with the EFV/UNIFAC model for PEG-2290/water and for PEG weight fraction equal to 0.26.
demonstratedsexperimentally and through molecular simulationsthat in aqueous solutions PEG prefers a helix conformation where at each ethylene oxide unit about three water molecules are associated, leading, thus, to an extended network of hydrogen bonds within the PEG helix.32 This enrichment of water molecules around PEG, which is not taken into account by the model, is probably responsible for the water solubility underprediction in the PEG-rich phase. The effect of polymer molecular weight on the critical solution temperature of polymer-solvent mixtures is typically shown based on the empirical Shultz-Flory relationship33
1 1 1 1 ∝ + Tc(r) Tc(∞) xr 2r
(17)
where r is the number of polymer segments, Tc(r) is the
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5405
Figure 5. LLE correlation (dashed line) for PEG-2290/water and prediction (solid lines) for PEG-2180, 4490, and 8000/water with the EFV-0.9/UNIFAC model. Experimental data (points) were taken from Saeki et al.,29 Bae et al.,30 and Saraiva et al.31
Figure 6. LLE correlation (dashed line) for PEG-2290/water and prediction (solid lines) for PEG-2180, 4490, and 8000/water with the EFV-1.2/UNIFAC model. Experimental data (points) were taken from Saeki et al.,29 Bae et al.,30 and Saraiva et al.31
critical solution temperature for a polymer with r segments, and Tc(∞) is the critical solution temperature for a polymer with infinite molecular weight. In Figure 3, experimental data and EFV/UNIFAC model predictions are shown for the UCST of waterPEG mixtures for variable PEG molecular weight. Interestingly, both experiments and theory satisfy the
Shultz-Flory relationship for a polymer mixture with strong hydrogen-bonding interactions. Similar results were obtained also for the LCST of these mixtures. Figure 4 presents the variation of excess entropy (SE), excess enthalpy (HE), and excess Gibbs free energy (GE) with temperature for the PEG-2290/water mixture, which explains the observed phase behavior of aqueous
5406 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003
PEG solutions. At low temperatures the highly directional hydrogen-bonding forces create local structures, hence the unfavorable to mixing negative SE values, but also very negative HE values, leading to complete miscibility. As the temperatures increases, these interactions become weaker and the dispersion forces dominate, resulting in phase separation. At still higher temperatures the entropy of mixing takes over, leading to complete miscibility. It is also worth noticing that even though the proposed model is basically an empirical one, its predictions are in qualitative agreement with the findings of Smith and Bedrov34 on the basis of molecular simulation calculations for aqueous solutions of PEO oligomers. These authors find that the entropy penalty for dissolving PEO in water decreases significantly with temperature and that the unfavorable interaction of PEO with water at higher temperatures is enthalpy-driven. A similar procedure with that followed for the EFV/ UNIFAC model was also followed for the two other combinatorial expressions: the EFV-0.9 and the EFV1.2. Both empirical modifications of the FV-combinatorial term give less accurate predictions, especially for the high MWs as shown in Figures 5 and 6. The EFV/UNIFAC model with the new estimated interaction parameters was next applied to the prediction of LLE of the water/polypropylene glycol (PPG) and the water/ethylene oxide propylene oxide copolymer (EOPO) systems. The experimental data for the water/ PPG system were taken from Malcolm and Rowlinson35 and for the water/EOPO from Li et al.,7 and they both present LCST phase behavior. In both cases the results are qualitatively correct with maximum LCST deviations of about 30 K but the location of the LCST is shifted to lower polymer concentrations. The performance of the new model was also evaluated for VLE predictions of PEG/water solutions using experimental data for various PEG MWs:
Figure 7. Prediction of VLE of aqueous solutions of PEG-200, 400, 600, 1500, 4000, 6000, 8000, 10000, and 100000 in the temperature range of 293-348 K and in the whole concentration range with the EFV/UNIFAC model and with the set of parameters presented in Table 5. Experimental data were taken from Nimmi et al.,36 Herskowitz and Gottlieb,37 Eliassi et al.,38 and Striolo and Prausnitz.39
PEG MW: 200, 400, 600, 6000, 8000, and 10 000 from Nimmi et al.36 PEG MW: 600, 1500, and 6000 from Herskowitz and Gottlieb37 PEG MW: 4000 from Eliassi et al.38 PEG MW: 100 000 from Striolo and Prausnitz39 These data cover the temperature range of 293-346 K and the whole PEG concentration range. As shown in Figure 7, the results are not satisfactory, especially for the lower water concentrations (low water activity coefficients) where the water activity coefficients are substantially underestimated. This, of course, should be expected from the aforementioned water solubility predictions (Figure 2). Simultaneous regression of VLE and LLE data (the so-obtained parameters are shown in Table 6) gave, as expected, improved water activity coefficient results but poorer LLE prediction as shown in Figures 8 and 9, respectively. The same problem was encountered by Bae et al.,6 who present different sets
Figure 8. Prediction of VLE of aqueous solutions of PEG-200, 400, 600, 1500, 4000, 6000, 8000, 10000, and 100000 in the temperature range 293-348 K and in the whole concentration range with the EFV/UNIFAC model and with the set of parameters presented in Table 6. Experimental data were taken from Nimmi et al.,36 Herskowitz and Gottlieb,37 Eliassi et al.,38 and Striolo and Prausnitz.39 Table 6. Interactions Parameters between H2O and CH2OCH2 Groups Estimated for the EFV/UNIFAC from LLE and VLE Data PEG/Water H2O
CH2OCH2
amn,0 (K) amn,1 amn,2 (K-1) amn,0 (K) amn,1 amn,2 (K-1) H2O CH2OCH2
172.7
2.63
-0.0085
251.4
3.15
-0.0091
of Flory-Huggins, χ, parameters to describe VLE and LLE aqueous PEG data. Conclusions The performance of the EFV combinatorial term coupled with the UNIFAC residual term is examined in the prediction of the MW effect of LLE in aqueous
Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 5407
Figure 9. LLE correlation for PEG-2290/water and prediction for PEG-2180 and 8000/water systems using parameters estimated from VLE and LLE data with the EFV/UNIFAC model. Experimental data (points) taken from Saeki et al.29 and Bae et al.30 and model correlation (dashed line) and predictions (solid lines).
PEG solutions. The required interaction parameters were determined from binary systems of low molecular weight compounds except for the pair CH2OCH2/H2O, which was obtained from the PEG-2290/H2O liquidliquid equilibria data. These interaction parameters were then employed in the prediction of the molecular weight effect of PEG/H2O systems up to PEG-1020000. Upper and lower critical solution temperatures are within 15 K from the experimental data while the obtained concentrations were satisfactory in the low PEG concentration values but not so in the high ones. Use of two modifications of the EFV combinatorial term gave poorer results, indicating no advantage in using them here. Prediction of VLE in PEG/H2O systems with the EFV/ UNIFAC model and the obtained parameters gave poor results, while simultaneous fit of VLE and LLE data gave improved VLE results but, as expected, poorer LLE ones. Simultaneous, thus, representation of VLE and LLE data for PEG/H2O systems is not feasible with this model, which is in agreement with the findings of Bae et al. using a Flory-Huggins model. The obtained results, in conclusion, are satisfactory considering the following: (i) the relative limitations of EFV/UNIFACsand of course all UNIFAC modelsswith systems where hydrogen bonding appears, (ii) the temperature extrapolation involved since the interaction parameterssexcept for the CH2OCH2/H2O pairswere determined in the range 350-390 K and used for LLE calculation in the range 360-560 K; (iii) the PEG polydispersity; and (iv) the simplicity of the model, that is, specific hydrogen-bonding interactions are not explicitly taken into account, combined with the difficulty involved in modeling closed-loop systems.
Literature Cited (1) Albertsson, P. A. Partioning of Cell Particles and Macromolecules, 3rd ed.; John Wiley: New York, 1986. (2) Glass, J. E. Hydrophilic Polymers; American Chemical Society: Washington, DC, 1996. (3) Yu, M.; Nishiumi, H.; Arons, J. D. S. Thermodynamics of Phase Separation in Aqueous Solutions of Polymers. Fluid Phase Equilib. 1993, 83, 357. (4) Prausnitz, J. M.; Lichtenthaler, R. N.; Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 3rd ed.; Prentice Hall: Englewood Cliffs, NJ, 1999. (5) Hu, Y.; Lambert, S. M.; Soane, D. S.; Prausnitz, J. M. Double-Lattice Model for Binary Polymer Solutions. Macromolecules 1991, 24, 4356. (6) Bae, Y. C.; Shim, J. J.; Soane, D. S.; Prausnitz, J. M. Representation of Vapor-Liquid and Liquid-Liquid Equilibria for Binary Systems containing Polymers: Applicability of an Extended Flory-Huggins Equation. J. Appl. Polym. Sci. 1993, 47, 1193. (7) Li, W.; Zhu Z. Q.; Li M. Measurement and calculation of liquid-liquid equilibria of binary aqueous polymer solutions. Chem. Eng. J. 2000, 78, 179. (8) Karlstrom, G. A New Model for Upper and Lower Critical Solution Temperatures in Poly(ethylene oxide) Solutions. J. Phys. Chem. 1985, 89, 4962. (9) Matsuyama, A.; Tanaka, F. Theory of Solvation-Induced Reentrant Phase Separation in Polymer Solutions. Phys. Rev. Lett. 1990, 65, 341. (10) Dormidontova, E. E. Role of Competitive PEO-Water and Water-Water Hydrogen Bonding in Aqueous Solution PEO Behavior. Macromolecules 2002, 35, 987. (11) Elbro, H. S.; Fredenslund, Aa.; Rasmussen, P. A New Simple Equation for the Prediction of Solvent Activities in Polymer Solutions. Macromolecules 1990, 23, 4707. (12) 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. (13) Kouskoumvekaki, I. A.; Michelsen, M. L.; Kontogeorgis, G. M. An improved entropic expression for polymer solutions. Fluid Phase Equilib. 2002, 202, 325.
5408 Ind. Eng. Chem. Res., Vol. 42, No. 21, 2003 (14) Sheng, Y. J.; Panagiotopoulos, A. Z.; Tassios, D. P. Activity Coefficients in Nearly Athermal Model Polymer/Solvent Systems. AIChE J. 1995, 41, 2306. (15) Abrams, D. S.; Prausnitz, J. M. Statistical Thermodynamics of Liquid Mixtures: A New Expression for the Excess Gibbs Energy. AIChE J. 1975, 21, 116. (16) Bondi, A. Physical Properties of Molecular Crystals, Liquids and Glasses; Wiley: New York, 1968. (17) Pappa, G. D.; Voutsas, E. C.; Tassios, D. P. Liquid-LiquidPhase Equilibrium in Polymer-Solvent Systems: Correlation and Prediction of the Polymer Molecular Weigh and the Pressure Effect. Ind. Eng. Chem. Res. 2001, 40, 4654. (18) Daubert, T. E.; Danner, R. P. Physical and Thermodynamic Properties of Pure Compounds: Data Compilation; Hemisphere: New York, 1989. (19) Rodgers, P. A. Pressure-Volume-Temperature Relationships for Polymeric Liquids: A Review of Equations of State and Their Characteristic Parameters for 56 Polymers. J. Appl. Polym. Sci. 1993, 48, 1061. (20) Gmehling J.; Onken U.; Rarey-Nies J. R. Chemistry Data Series; DECHEMA: Frankfurt, 1988. (21) Kojima, K.; Zhang, S.; Hiaki, T. Measuring methods of infinite dilution activity coefficients and a database for systems including water. Fluid Phase Equilib. 1997, 131, 145. (22) Garriga R.; Sa´nchez. F.; Perez P.; Gracia M. Isothermal vapour-liquid equilibrium at eight temperatures and excess functions at 298.15 K of di-n-propylether with 1-propanol or 2-propanol. Fluid Phase Equilib. 1997, 138, 131. (23) Garriga R.; Martinez S.; Perez P.; Gracia M. Vapour pressures at several temperatures between 288.15 and 323.15 K of di-n-propylether with 1-hexanol or 1-octanol. Application of the ERAS model. Fluid Phase Equilib. 1998, 147, 195. (24) Sorensen E. L.; Hao W. Infinite Dilution Activity Coefficients in Poly(ethylene glycol). Fluid Phase Equilib. 1990, 56, 249. (25) Gmehling J.; Li J.; Schiller, M. A Modified UNIFAC Model. 2. Present Parameter Matrix and Results for Different Thermodynamic Properties. Ind. Eng. Chem. Res. 1993, 32, 178. (26) Anderson, T. F.; Prausnitz, J. M. Application of the UNIQUAC Equation to Calculation of Multicomponent Phase Equilibria. 1. Vapor-Liquid Equilibria. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 552. (27) Zhang, S.; Hiaki, T.; Kojima, K. Prediction of infinite dilution activity coefficients for systems including water based on the group contribution model with mixture-type groups. 1. AlkaneH2O and alkanol-H2O mixtures. Fluid Phase Equilib. 1998, 149, 27.
(28) Voutsas, E. C.; Tassios, D. P. Prediction of infinite dilution activity coefficients in binary mixtures with UNIFAC. A critical evaluation. Ind. Eng. Chem. Res. 1996, 35, 1438. (29) Saeki, S.; Kuwahara, N.; Nakata, M.; Kaneko, M. Upper and lower critical solution temperatures in poly(ethylene glycol) solutions. Polymer 1976, 17, 685. (30) Bae, Y. C.; Lambert, S. M.; Soane, D. S.; Prausnitz, J. M. Cloud-Point Curves of Polymer Solutions from Thermooptical Measurements. Macromolecules 1991, 24, 4403. (31) Saraiva, A.; Persson, O.; Fredenslund, A. An experimental investigation of cloud-point curves for the poly(ethylene glycol)/ water system at varying molecular weight distributions. Fluid Phase Equilib. 1993, 91, 291. (32) Tasaki, K. Poly(oxyethylene)-Water Interactions: A Molecular Dynamics Study. J. Am. Chem. Soc. 1996, 118, 8459. (33) Shultz, A. R.; Flory P. J. Phase Equilibria in PolymerSolvent Systems. J. Am. Chem. Soc. 1952, 74, 4760. (34) Smith, G. D.; Bedrov, D. Roles of Enthalpy, Entropy, and Hydrogen Bonding in the Lower Critical Solution Temperature Behavior of Poly(ethylene oxide)/Water Solutions. J. Phys. Chem. B 2003, 107, 3095. (35) Malcolm, G. N.; Rowlinson, J. S. The thermodynamic properties of aqueous solutions of polyethylene glycol, polypropylene glycol and dioxane. Trans. Faraday Soc. 1957, 53, 921. (36) Nimmi, L.; Camargo, M. S.; Meirelles, A. J. A. Water activity in poly(ethylene glycol) aqueous solutions. Thermochim. Acta 1999, 328, 169. (37) Herskowitz, M.; Gottlieb M. Vapor-Liquid Equilibrium in Aqueous Solutions of Various Glycols and Poly(ethylene glycols). 3. Poly(ethylene glycols). J. Chem. Eng. Data 1985, 30, 233. (38) Eliassi, A.; Modarres, H.; Mansoori, G. A. Measurement of Activity of Water in Aqueous Poly(ethylene glycol) Solutions (Effect of excess Volume on the Flory-Huggins χ-Parameter). J. Chem. Eng. Data 1999, 44, 52. (39) Striolo, A.; Prausnitz, J. M. Vapor-liquid equilibria for some concentrated aqueous polymer solutions. Polymer 2000, 41, 1109.
Received for review May 12, 2003 Revised manuscript received July 18, 2003 Accepted July 22, 2003 IE0304154
