2864
Ind. Eng. Chem. Res. 1998, 37, 2864-2872
GENERAL RESEARCH Empirical Method To Correlate and To Predict the Vapor-Liquid Equilibrium and Liquid-Liquid Equilibrium of Binary Amorpous Polymer Solutions Alessandro Vetere Snamprogetti Research Laboratories, San Donato Milanese, Italy
A previous method of the author is modified in order to ensure (i) a better evaluation of the solvent activity coefficients at infinite dilution and (ii) the calculation of liquid-liquid equilibrium (LLE) for athermal solutions also. The first goal is obtained by imposing that the parameter, which corrects the entropic term of the Flory-Huggins equation as required by the previous work, varies not only according to the density of pure components but also according to the molar volume compositions. The second problem, linked to the demixing phenomena occurring at the upper and lower critical solution temperature (UCST and LCST) points for athermal solutions also, suggested a simple modification of the enthalpic term. It is assumed that the χ parameter is not nil for nonpolar mixtures also when the density of pure polymer is remarkably different from the mixture density. While the method is predictive for vapor-liquid equilibrium (VLE) and the weight fraction activity coefficients at infinite dilution, the LLE data can only be correlated by using “ad hoc” empirical parameters. Further, an empirical procedure is proposed to predict the UCST and LCST points of both polar and nonpolar mixtures which requires only the knowledge of the molecular weight of solvents and the density of polymers. A comparison is made with well-known literature methods which predict the equilibrium data for many systems in both the VLE and LLE ranges. Introduction The thermodynamics of polymer solutions of the last 30 years is mainly aimed to face two universal behaviors of polymer containing systems: the phase separations which occur in both the low- and high-temperature range. In fact, polymer mixtures disclose a more complex phase behavior than the usual solutions of components having comparable molar volumes. Beside the well-understood liquid-liquid equilibria obtained by lowering the temperature below the so-called upper critical solution temperature (UCST) point, all polymer solutions are characterized by a limited miscibility at high temperatures, that is, above the lower critical solution temperature (LCST) point. The LCST point is often located close to the solvent critical point but in some instances, as the molecular weight of the polymer increases, it can fall more than 100 °C below Tc. Still more complex phase diagrams with two critical concentrations are predicted by theory and partially verified experimentally (Quian et al., 1991a). However, it was the “unpredictable” lowering of solubility by rising temperature which opened the way to a drastic revision of the paradigmatic FloryHuggins equation (FHE)
ln(γ1) ) ln(φ1) + (1 - V1/V2)φ2 + χφ22
(1)
The most striking feature of demixing in polymer solutions is that this phenomenon is found also in the
so-called athermal solutions such as, for example, butadiene rubbers in saturated and unsaturated hydrocarbons. Since for these mixtures it cannot be invoked as an energetic contribution to the excess Gibbs energy, the inadequacy of eq 1 to treat liquid-liquid equilibrium (LLE) was soon attributed to an approximate evaluation of the entropic term. In fact, the main assumption of the Flory-Huggins statistical theory of polymer solutions is that both solvent molecules and polymer segments occupying a single site in the lattice have the same free volume, so that the mixing process takes place at constant volume. The inability of the Flory-Huggins equation to deal with negative excess volume often found in vapor-liquid equilibrium (VLE) and LLE is surely the weak side of this theory, which is not able to describe, even on a qualitative basis, most of the various phase diagrams related to the polymer/ solvent systems. After the Flory and Huggins era, the literature on polymer solutions grew so impressively that in this paper we quote only very few works among the relevant ones in order to frame problems and to fix some reference points for the method proposed in this work. Several works aimed at saving the simple analytical form of relation (1) have proposed to remove the invariance of the χ parameter assuming that it can vary with both composition and temperature. The last and most refined developments of this approach are represented by the papers of Quian et al. (1991b) and Bae et al. (1993). Following a pionering work by Eichinger
S0888-5885(97)00889-0 CCC: $15.00 © 1998 American Chemical Society Published on Web 05/21/1998
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2865
(1970), these authors assume that
χ ) D(T) B(φ2)
(2)
Both D(T) and B(φ2) are related to temperature and composition, respectively, through simple functions which require five empirical constants to be determined for each binary system. Quian et al. (1991a) showed that, by changing the sign of two constants in the B(φ2) term, the various types of diagrams found in LLE (UCST, LCST, hourglass, and closed loop) can be derived. While not predictive, the method was adequate to correlate with a high accuracy the experimental data of any kind of polymer mixture. Despite these excellent results, the strategy of correcting the inaccuracy of the F-H entropic term by modifying the enthalpic term appears, in principle, questionable. The evaluation of the enthalpic term according to more sophisticated models for the excess Gibbs energy such as the NRTL equation proposed by Heil and Prausnitz (1966) or the predictive UNIFAC method does not remove the above incoherence. A very innovative method which corrects the F-H entropic term was proposed by Elbro et al. (1990) and extensively improved by Kontogeorgis et al. (1993, 1994). These authors take into account the free-volume dissimilarity concept by assuming different co-volumes for the solvent and the polymer molecules to justify the negative volume changes upon mixing, experimentally detected for many polymeric mixtures. The very simple relation obtained for the entropic term is the same as the FHE provided that the free-volume fraction defined as
φfv i )
xi(Vi - Vw)
∑xj(Vj - Vwj)
(3)
substitutes the usual volume fractions. The Elbro relation was elegantly derived by the combinatorial part of the generalized van der Waals partition function by assuming only that molecules have a hard-core volume inaccessible to other molecules. Very recently, Kontogeorgis et al. (1995) have tested the Elbro method coupled with the UNIFAC model for the residual term to calculate the five types of phase diagrams which represent the behavior of polymer mixtures. Their analysis shows that this method, labeled as the Entropic-FV model, is able to correlate and in some instances to predict not only the usual VLE but also the LLE linked to the UCST and LCST points. The most impressive feature of the Entropic-FV model is the use of the same empirical constants predicted by the UNIFAC method for all kinds of equilibrium calculations. For these reasons, the method must be regarded as a touchstone for future developments. The thermodynamics of polymer solutions shares with the usual VLE of monomeric compounds the actual tendency for evaluating through the equation of state (EOS) route the nonidealidy of fluid mixtures. The superiority of the EOS model with respect to the activity coefficient method to treat in a unitary manner both the liquid and the vapor range is well-recognized. Still more important is the dependence on both temperature and density of the interaction forces, as depicted by all EOS. Flory et al. (1964) pioneered also the EOS method in the early 1960s, when the limits of relation (1) became evident. However, the EOS developed by Flory is
inherently very complex and, further, it requires some parameters not always well-known. The EOS method ramified in numberless variations on a theme shares with the Flory method both complexity and the need of unusual parameters. A fair complete account of these works is given by Harismaidis et al. (1994a,b). However, the application of an EOS to solve VLE and LLE in the domain of polymer solutions became a convenient tool for engineering purposes when a particularly simple method of applying the van der Waals EOS to polymers was shown to work well (see Harismaidis et al., 1994a,b). Remarkably, only two density data in the low-pressure range are required to evaluate the van der Waals parameter for the pure polymer. Further, the application of the method to mixtures containing polymers is straighforward since the mixing rules adopted for the a and b constants embodied in the wdW EOS are equally simple. Harismaidis et al. (1994b) have stressed several points of superiority of the EOS method over the Entropic-FV model and more, in general, over the various modifications of the FHE. Among these, two features of the application of the vdWs EOS to polymers deserve mention: (i) The Elbro modification of the FH entropic term follows directly from standard thermodynamic relations when an EOS is applied to calculate the activity coefficient of a compound in a liquid mixture. (ii) The enthalpic term is volume dependent, and it can justify a nonzero value for pseudo athermal polymer solutions also. This latter point is particularly important since it explains the solubility gap occurring at the LCST for several nonpolar mixtures. In this work a recent method proposed by the author (Vetere, 1997) is modified for a 2-fold reason: to improve the prediction of the solvent activity coefficients at infinite dilution and to enlarge its application to the correlation of various types of LLE found in polymer mixtures and not previously considered. This double task will require the modification of both the entropic and enthalpic parts of the FHE according to some criteria which are clearly inspired by the above-quoted literature methods. Proposed Method It is well-known that the inaccuracies of the FHE lead to an overestimation of the entropy of mixing. A simple interpretation of this result is based on the assumption that the solvent molecules can be partly allocated in the free volume of polymer, so that the excess volume of mixing is negative. Obviously, the same process cannot be admitted for polymer molecules, since due to the interconnectivity of the polymer segments only the terminal ones can be inserted in the free volume of the solvent. A rough but effective modification of the FHE which embodies this concept consists of correcting the molar volume of the solvent according to the relation
V1* ) V1/(1 - )
(4)
which ultimately leads to an increase of the volume fraction of the solvent Vetere (1997). The parameter can be calculated through two simple relations
) -0.25 + 0.667(100/V1F2)
(5)
which rules for cyclic solvents and
) -0.05 + 0.600(V1F2)
(6)
2866 Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998
which must be applied for noncyclic solvents. Surprisingly enough, this simple mathematical apparatus enables the prediction of the VLE of a wide class of binary amorphous polymer solutions. Another interesting result of this work is that the equilibrium data can be predicted with fair accuracy by neglecting the enthalpic part of the FHE. There are only very few exceptions, represented by polymers in solution with water, methanol, ethanol, chloroform, and dichloromethane. However, further inquiry revealed that the weight fraction activity coefficient at infinite dilution, calculated according to the relation
Ω1∞ ) (M2/M1)[ln(V1*/V2) + 1 - V1*/V2 + χ] (7) is appreciably higher than the experimental ones. The evaluation of the activity coefficients defined by relation (7) is important from a practical point of view since in industrial practice the solvent must be removed from the polymer up to a few parts per million, that is, in a range of concentration close to infinite dilution. At these conditions, the FHE in its various modified forms inevitably falls in defect, since the correct physical context is that of a liquid adsorbed on a solid instead of a liquid dissolved in another fluid. Empirically, it was found that the prediction of Ω1∞ is considerably improved if V1* is calulated by using the following slightly modified form of the parameter:
* ) [1 - 0.15φ2]
(8)
where the values of are those calculated according to eqs 5 and 6 for cyclic and noncyclic compounds, respectively. Equation 7 clearly shows that the * parameter used to calculate the value of V1* must be composition dependent. The proposed correction is relevant only in the very dilute concentration range, and it is nil for the pure solvent. A second modification devoted to the enthalpic term arises from the inadequacy of the Flory-Huggins parameter, χ, independent of temperature and composition to describe the strong nonidealities which are responsible for phase separations at the UCST and LCST points. In this work we assume that χ varies with both temperature and composition according to simple relations (see eqs 9 and 11-13), since in our model the χ(φi) and χ(T) functions are not introduced to correct the redundance of the FH entropic term. As a result, we propose the following slight modification of the FH enthalpic term
χ(T) ∞ a + bT
(9)
where a is positive at temperatures above the LCST point and negative under the UCST point. A second modification is dictated by less evident reasons. As reminded above, the application of an EOS inevitably led to the conclusion that the enthalpic term depends on the density of the polymeric solution also. However, there are cogent reasons to introduce a χ(F) term also in the frame of the activity coefficient models. The “inescapable” dissimilarity between solvents and polymers is linked to the differences not only in molecular volume but also in density. Admittedly, the polymer is by far denser than a liquid solvent, and this property has important consequences on the excess enthalpy of mixing which are usually overlooked. To illustrate, the ∆He of mixing for regular solutions can be derived from
the following relation:
∆He ) x12g11* + 2x1x2g12* + x22g22* - x1g11 - x2g22 (10) In the usual VLE it is assumed that the asterisked gii* terms, which represent the interaction forces between the ii molecules in solution, equal the corresponding gii terms related to the pure-component interactions. This assumption does not hold for the polymer molecules, since the interactions between the segments of the chain in the solid polymer are appreciably higher than the corresponding interactions in solution. As a consequence, ∆He depends also on the polymeric solution density. We assume that
χ(F) ∞ K(Fp - Fl)
(11)
with
Fl )
∑Fiφi
(12)
As a result, by merging the K constant with the a and b constants, we have
χ ) [(a1 + b1T)(Fp - Fl) + c]/RT
(13)
where c is a term which takes into account specific interactions arising from permanent dipoles or interaction of the acid-base type. Conclusively, the equation used in this work is
ln(γ1) ) ln(φ1*) + (1 - 1/m)φ2* + 1 [(a + b1T)(Fp - Fl) + c]φ22 (14) RT 1 with
φ1* )
x1V1* x1V1* + x2V2
(15)
and φ2* the complement to unity of φ1*. We recall that asterisked volume fractions must be introduced in the entropic term only, for the reasons stated in a previous work (Vetere, 1997). Observations on Demixing in Polymer Mixtures Equation 14 clearly shows that the enthalpic term can be appreciably different from zero also for mixtures of nonpolar polymers in nonpolar solvents when Fp . Fl. On this result rests a simple explanation of the existence of a miscibility gap above the LCST. The interpretation proposed in this work of phase separations in both lowand high-temperature ranges is at variance with the canonical theory. Usually, demixing at the LCST point is attributed to a combinatorial effect: due to the “freevolume dissimilarity” between solvent and polymer molecules, the entropy of dilution is negative and the appearance of two phases is thermodynamically favored. On the contrary, the appearance of an UCST point is attributed to a prevailing enthalpic effect. In fact, the enthalpy of mixing becomes more positive by lowering the temperature, as occurs for mixtures of monomeric compounds. The physical model underlying this work suggests reversing the standard interpretation: the entropic effects largely prevail at low temperatures, while the
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2867 Table 1. Comparison between Old and New Methods To Predict the VLE Data of Binary Polymeric Mixtures no. of systems
type of mixture nonpolar polymers in nonpolar noncyclic solvents nonpolar polymers in nonpolar cyclic solvents polar polymers in polar solvents system containing both polar and nonpolar components
AAD % new old
19
5.60
5.17
17
3.93
4.56
18 19
5.56 7.12
5.05 7.96
enthalpic ones are responsible for the appearence of a LCST point. According to our modification of the FHE, the entropy of mixing decreases as a consequence of volume contraction promoted by the insertion of some solvent molecules in the free volume of polymer molecules. The analytical form of relations (5) and (6) shows that the negative excess volumes are more relevant the lesser is the solvent molar volume, that is, at low temperatures. In other words, an increasing number of solvent molecules can be allocated in the free volumes of polymer cells when V1 decreases with temperature. In any case, it is difficult to account for the limited solubility below the USCT of nonpolar polymers in nonpolar solvents by invoking mainly a positive enthalpy of dilution, since a miscibility gap is not found at any temperature for the usual monomeric mixtures of hydrocarbons. At high temperatures, the molar volume of the solvent expands exponentially, particularly near its critical temperature, while the molar volume of the polymer is still nearly constant. As a consequence, the excess mixing volume is less negative or nil, and the enthalpic contribution to the excess Gibbs energy, which increases according to relation (13), is responsible for the phase separation above the LCST point. This interpretation is by no means new. Soon after the discovery of the LCST by Freeman and Rowlinson (1960), some tentative explanations were reported in the literature which are very close to the above considerations. Allen and Barker (1965) wrote, “the suggestion is that, as the critical temperature is approached, χ increases rapidly to cause phase separation because the solvent becomes expanded. In relation to the solubility parameters of the system, χ increase
with (δ1 - δ2); for a given solvent near its gas-liquid critical temperature... δ1 is sensitive both to temperature and pressure and δ2 is controlled essentially by the chemical nature of the solute”. However, the simpler explanation of the LCST point is, perhaps, that of the discoverers of this phenomenon: “It is presumably the decreasing configurational energy and the increasing molar volume of the pure solvent as it approaches its own gas-liquid critical point that makes it a ‘poorer’ solvent for the polymers” (Freeman and Rowlinson, 1960). Therefore, our interpretation of phase separation in polymer mixtures is more a restoration than a revolution. Experimental Section All the literature data on the binary VLE and the activity coefficient at infinite dilution processed in this work were taken from the monographs by Wen et al. (1991, 1992). The LLE data pertaining to the various phase separations along with the density of pure solvents were taken from the work of Danner and High (1993). The critical data of solvents used to calculate their densities at various temperatures through the Rackett equation were taken from the work of Reid et al. (1987). The results obtained for the various types of equilibria studied in this work will be discussed separately. Binary VLE Data To test the new method with respect to the previous one, we have examined the same binary data reported by Vetere (1997) for three classes of mixtures, namely, nonpolar/nonpolar mixtures, nonpolar/polar mixtures, and mixtures containing both polar and nonpolar components. The polymers in water systems were disregarded since the nonideality of these mixtures can be entirely embodied in the entropic term, as previously shown (Vetere, 1997). The mixtures which disclose negative deviations from Rault’s law were also not considered here because the strong specific interactions between solvent and polymer molecules largely prevail over nonspecific ones represented by the (a1 + b1T)(Fp - Fl) term in eq 13.
Table 2. Experimental and Predicted Weight Fraction Activity Coefficients at Infinite Dilution for Athermal Polymer Solutions Ω1∞ LDPE/nC6
LDPE/nC7 LDPE/3-MeC6 LDPE/nC10 LDPE/nC12 LDPE/2,2,4-TMC5 HDPE/3-MC6 PIB/CC6
PIB/nC5
M2
T (K)
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Elbro-FV
UNIFAC-FV
35 000 35 000 35 000 35 000 35 000 35 000 35 000 82 000 82 000 82 000 82 000 82 000 82 000 105 000 105 000 24 500 53 000 53 000 860 000 260 000 860 000
383.2 398.2 423.2 448.2 473.2 383.2 473.2 393.15 418.25 418.25 393.15 418.25 418.25 418.6 425.8 298.2 323.2 423.2 313.1 313 323.1
5.65 5.73 5.91 5.97 6.10 5.08 5.90 4.91 5.04 4.32 4.02 4.10 4.97 5.48 5.71 4.36 4.56 4.88 4.81 7.33 7.20
5.75 5.80 5.91 6.08 6.37 5.20 5.60 5.04 5.11 4.17 3.79 3.82 4.91 5.15 5.18 4.73 4.77 4.91 4.77 7.05 7.07
5.07 5.19 5.51 5.98 6.79 4.64 5.66 4.56 4.72 4.03 3.73 3.77 4.72 4.78 4.85
4.26 4.27 4.27 4.26 4.19 3.94 3.95
4.33 4.95 3.62 6.15 6.28
4.61 4.45 4.59 7.51 7.54
GK-FV
4.14 3.81 3.86 4.99
2868 Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 Table 3. Experimental and Predicted Weight Fraction Activity Coefficients at Infinite Dilution for Non-athermal Solutions Ω1∞ system
M2
T (K)
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Elbro-FV
UNIFAC-FV
PIB/toluene PIB/toluene PIB/CCl4 LDPE/m-xylene LDPE/m-xylene PVC/acetone PVC/acetone PVC/heptane PVC/MEK PVC/EtOAc PVC/BuOAc PVC/acetaldehyde PVC/cyclohexane PS/benzene PS/benzene PS/benzene PS/EtC6H5 PS/dioxane PS/MEK PIB/benzene PIB/benzene PIB/CCl4 PIB/toluene LDPE/n-heptane PEO/1-butanol PEO/1-propanol PEO/ethanol PEO/hexane PS/CHCl3 PS/acetic acid PS/i-propanol PS/MEK PS/chlorobenzene PS/toluene PS/H2O PBMA/acetone PBMA/chlorobutane PBMA/methanol PMMA/methanol PMMA/1,2-dichloromethane PMMA/methyl acetate PVAC/2-propanol PVAC/2-propanol PVAC/MEK PVAC/ethyl acetate
53 000 53 000 40 000 35 000 35 000 41 000 101 500 41 000 101 500 41 000 40 000 101 500 101 500 3 524 86 700 178 000 97 000 97 000 97 000 53 000 53 000 40 000 40 000 35 000 2 000 10 700 1 000 7 500 2 000 2 000 2 000 96 200 96 200 76 000 20 000 8 716 73 500 8 716 6 107 6 107 85 100 84 300 84 300 83 550 83 550
323.2 423.2 298.2 383.2 473.2 393.2 398.2 383.2 413.2 393.2 413.0 398.2 413.2 400.2 423.2 420.9 423.2 432.2 473.2 323.2 423.2 298.2 298.2 473.2 398.2 393.2 398.2 372.7 502.6 502.6 492.9 420.0 369.5 473.2 502.6 373.0 413.2 393.0 423.0 473.0 398.5 398.2 448.2 398.2 473.2
5.30 4.91 2.53 3.73 3.71 11.8 10.8 45.0 8.69 12.4 12.2 12.0 11.4 4.27 5.69 5.45 4.96 5.17 9.12 5.9 5.6 2.5 6.2 5.4 4.7 5.8 5.7 26.8 3.5 17.9 11.6 7.1 3.5 5.3 76.6 9.3 5.1 21.1 16.2 4.1 6.0.3 7.2 6.5 6.3 6.2
4.28 4.38 3.24 3.83 3.94 8.56 8.63 11.00 7.91 6.39 6.00 10.38 6.32 5.07 5.30 5.30 5.01 5.32 7.86 5.17 5.15 2.53 4.45 5.56 7.25 9.37 11.23 8.71 4.3 9.4 9.3 6.9 3.6 4.9 75.9 8.5 5.3 12.9 15.4 5.3 6.9 8.7 8.8 7.0 6.1
5.28 4.98 2.74 3.59 3.67 9.98 10.13 15.65 8.80 10.04 10.29 10.93 11.01 4.76 5.12 5.10 4.65 4.35 7.87 5.9 5.3 2.7 6.1 5.7 5.0 5.7 5.9 20.7 3.90 14.9 13.2 6.8 3.5 5.3 78.9 8.7 4.8 24.8 17.8 4.4 6.2 7.2 6.1 6.8 6.4
6.19 5.24 2.93 3.62 3.49 10.17 10.66 14.63 7.95 6.48 5.41 10.93 10.46 3.98 4.43 4.44 4.23 4.65 7.95 6.8 5.6 2.9 6.3 3.9 7.5 8.0 9.2 15.2 2.9 13.7 23.7 7.9 3.3 4.3
The following empirical rules were adopted for the constants of eq 13: b1 is nil for all the systems; a1 is equal to 500 cal/mol for nonpolar/nonpolar mixtures and polar/nonpolar mixtures and is nil when both components are polar. The interaction parameter c is equal to 100 when the solvent is polar, with the exception of C1-C3 alcohols for which c ) 200. The value of c is nil for polar polymers in nonpolar solvents. The results reported in Table 1 show the equivalence of the new method with the previous one (Vetere, 1997), but the first can justify the existence of a miscibility gap at the LCST point for nearly athermal solutions also. It must be stressed that for both methods the entropic contribution largely prevails over the enthalpic one. Activity Coefficients at Infinite Dilution Tables 2 and 3 report the results obtaind in predicting Ω1∞ according to eqs 7 and 13 for a wide choice of binary systems, both polar and nonpolar. Since the Fp - Fl term at infinite dilution is nil, χ assumes only two values: zero or c/RT. As a reference point, we have chosen the
8.8 24.2
10.6 10.0 6.4 4.5
so-called literature methods Elbro-FV and UNIFAC-FV reported by Kontogeorgis et al. (1993, 1995), which appear the most reliable among the published ones. On the whole, the proposed method is slightly more accurate than both Elbro-FV and UNIFAC-FV when the nonideality of the mixture is moderate (Ω1∞ < 10), while the literature methods give better results for strongly nonideal mixtures. LLE Data Phase separations in polymeric mixtures occur in the very low concentration range of polymer, so that both the volumetric fraction and the activity of solvent are nearly unity. The thermodynamic condition for the splitting of a liquid mixture in two separate phases is that the excess Gibbs energy of the mixing curve must have two inflection points. The difficulty of revealing these points near to the infinite dilution of a component is discussed in detail by Kontogeorgis et al. (1995) and Harismiadis et al. (1994a). The choice of the isoactivity criterion for both solutes and solvents in both phases to calculate the compositions at which demixing occurs
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2869 Table 4. UCST Predicted According to Various Methods T (K) system HDPE/p-octylphenol HDPE/1-octanola PS/cyclopentanea PS/cyclohexane
PS/methylcyclohexanea PS/acetone PS/ethyl formatea HDPE/n-butyl acetatea PMVPD/n-butyl acetate HDPE/diphenylmethane BR/n-hexanea BR/2,2,3-trimethylbutane BR/2-methylhexanea BR/n-octanea
a
M2
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new UN
Entropic-FV
20 000 77 800 163 000 20 000 77 800 163 000 97 200 200 000 20 400 37 000 43 600 89 000 100 000 10 200 97 200 200 000 4 800 10 300 4 000 10 000 13 600 20 000 61 100 420 000 600 000 20 000 132 000 191 000 376 000 830 000 220 000 355 000 44 500 65 000 104 000
424.35 433.82 436.39 422.81 437.19 441.41 276.20 280.90 279.80 285.60 292.28 296.67 299.83 285.71 321.80 327.40 221.0 270.59 243.6 295.1 415.0 429.52 446.19 286.2 288.0 384.5 271.0 274.95 283.15 287.3 289.6 311.5 307.5 321.7 344.8
409.8 435.2 443.4 420.5 440.8 447.4 276.2 279.0 284.0 290.3 292.0 297.3 298.2 289.4 319.4 324.3 259.9 282.7 243.7 279.1 414.9 423.1 439.3 268.8 271.6 384.5 271.2 275.3 280.2 313.7 307.9 311.7 309.3 318.4 327.6
424.65 453.20 461.65 442.5 426.0 468.5 170.8 177.6 218.0 227.0 229.0 236.5 237.5 245.0 260.4 283.2 242.20 262.85
447.65 478.00 486.75 354.35 392.05 408.2 248.9 269.0 241.5 260.0 268.3 284.5 285.2 279.6 340.7 350.3 305.74 369.04
329.80 332.95 339.20
402.0 407.2 417.6
Systems used to calculate the empirical constants of eqs 18-20.
involves some problems, since the ability of the FHE to describe the activity coefficients of polymer molecules also is not well-tested. Usually, they have extremely low values. Therefore, “LLE in polymer solutions is (almost) totally governed by the polymer molar activity coefficient”, as stated by Saraiva et al. (1995). In this work we adopt the thermodynamic criterion of equating to zero the third derivative of the ∆Ge of mixing with respect to the polymer molar fraction, x2, or, in an equivalent manner, the second derivative of the chemical potential. The differentiation of eq 14, being trivial but very long, is not reported here for the sake of brevity. However, the prediction of both UCST and LCST according to the same empirical constants used to predict the VLE of binary systems is very poor. This limitation is shared also by most literature methods. Among these, the only true prediction is represented of the work by Kontogeorgis et al. (1995), which gives in many cases acceptable results but falls sometimes badly in error. Therefore, in this work we follow an entirely empirical route to predict the UCST and LCST points, while eq 14 is used only to correlate the LLE data by evaluating “ad hoc” empirical values for the three constants embodied in it. Prediction of UCST and LCST Points Although the prediction of the temperatures at which phase separations occur is very difficult by applying a rigorous thermodynamic method, both the UCST and LCST points appear to be linked by very simple rela-
tions to well-known properties of pure compounds, namely, the molecular weight of solvents and the density of polymers. It was empirically found that relations having the form
T (K) ) R + βM1(1 ( r-0.5)
(16)
T (K) ) R + βM1(1 ( r-0.5)/F2
(17)
or
predict with a high accuracy the phase splitting in both low- and high-temperature ranges provided that the binary systems are subdivided into a few classes which take into account the structure or the polarity degree of the considered system. The 1 ( r-0.5 term in eqs 16 and 17 can be considered as the ash of the celebrated Shultz-Flory equation which entails some other quantities in order to predict both the UCST and LCST points. The sign in the 1 ( r-0.5 term is dictated by the critical temperature considered, as it will be shown in the following paragraphs. (a) UCST Points. The following relations are proposed:
T (K) ) 75.0 + 2.53M1(1 - r-0.5)4
(18)
for nonpolar polymers in nonpolar noncyclic solvents
T (K) ) 162 + 1.77M1(1 - r-0.5)3
(19)
2870 Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 Table 5. LCST Predicted According to Various Methods T (K) system PS/acetonea PPO/n-pentanea PP/diethyl ether PIB/benzene HDPE/n-pentanea HDPE/n-heptane HDPE/n-octane BR/n-hexane BR/2-methylhexane BR/2,2,3-trimethylbutane BR/n-octanea PS/cyclopentanea PS/diethyl ether PS/cyclohexanea PS/benzene PS/toluenea
PS/MEK PS/ethyl formatea PS/isopropyl acetatea PS/ethyl acetate PDMA/cyclopentane a
M2
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this work
4 800 10 300 16 000 40 000 18 100 28 700 64 000 72 000 34 900 442 100 93 500 125 000 202 000 76 800 202 000 98 000 191 000 104 000 220 000 830 000 44 500 65 000 104 000 200 000 400 000 2 700 000 4 800 10 000 400 000 670 000 400 000 670 000 2 700 000 37 000 97 200 200 000 400 000 37 000 97 200 200 000 4 000 10 000 37 000 670 000 233 000 10 0000 470 000
466.50 417.40 436.48 432.50 436.10 428.50 420.60 540.5 433.6 414.45 462.0 460.3 456.8 500.77 493.4 394.8 379.7 411.88 395.44 414.7 435.9 416.9 390.5 440.0 435.0 429.5 408 353.1 494.7 491.7 528.3 527.0 527.0 567.2 559.9 557.2 554.9 463.2 448.7 441.1 481.5 451.5 436.7 395.2 425.0 437.8 469.15
466.1 444.0 441.5 433.5 442.6 432.4 419.3 490.0 435.1 410.5 443.8 439.6 434.8 456.4 443.7 405.6 399.2 413.9 406.5 399.3 436.1 429.1 422.1 440.0 435.9 430.6 374.5 353.4 505.9 502.7 475.6 473.3 468.8 576.7 560.3 550.4 545.0 434.1 428.7 421.9 481.8 447.6 437.0 423.8 426.1 429.6 435.4
new UN
Entropic-FV
309.16
481.7 424.11 192.85 187.30 415.90 397.95 376.40 395.65
250.45
279.6
Systems used to calculate the empirical constants of eqs 21-23.
for nonpolar polymers in nonpolar cyclic solvents, and
T (K) ) 20 + 320.6(1 - r-0.5)/(F2)2
(20)
for mixtures containing both polar and nonpolar compounds, regardless of the solvent structure. In relation (20) the polymer density is evaluated at 300 K for all the mixtures to save the predictive character of the method. Table 4 reports the results obtained for several mixtures along with the data predicted by the Kontogeorgis et al. method (1995) for some systems. The predicted UCST are generally very close to the experimental points despite the very few well-known input data required by relations (18)-(20). The only shortcoming of the proposed method lies in its inability to take into account the dependence of UCST on the polymer molecular weight. The variation of UCST with M2 is, however, remarkable only for oligomers. (b) LCST Points. The subdivision into classes and the analytical form of the proposed relations closely follow those reported for the UCST point calculations.
In fact, the following relation holds for mixtures of nonpolar polymers in noncyclic hydrocarbons
T (K) ) 158 + 65.5(M1)0.2(1 + r-0.5)3/(F2)2 (21) while the relation
T (K) ) 93.8 + 4.75M1(1 + r-0.5)2
(22)
can be applied for nonpolar polymers in cyclic hydrocarbons. The last class of the examined mixtures is represented by the polar/nonpolar systems, whose LCST point can be calculated by applying the relation
T (K) ) 153 + 267(1 + r-0.5)
(23)
in which the polymer density is calculated at 400 K, a temperature assumed as roughly representative of the LCST point mean values. The results obtained are reported in Table 5 along with very scattered data given by Kontogeorgis et al. (1995).
Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 2871
Figure 1. UCST and LCST phase diagrams for the acetone/PS (10 300) system: b, calculated data; 0, experimental data.
Correlation of the LLE Data The whole liquid-liquid phase diagram of a binary system is calculated by optimizing the empirical constants embodied in eq 14 according to the usual regression methods in order to fulfill the condition ∂2Gmix/∂2x2 < 0. It is well-known that usually polymer mixtures exhibit both UCST and LCST behavior, and when the molecular weight of the polymer increases, the two phase diagrams can merge in an hourglass-type diagram. As an example, the acetone/polystyrene (PS) systems show both UCST and LCST diagrams at polymer molecular weights equal to 4800 and 10 300, which overlap when the molecular weight increases to 19 800. These systems were chosen to ascertain the capability of the proposed method of dealing with such a complex behavior. It was found that the interaction parameters of eq 14, grouped as A ) (a1 + b1T)(Fp - Fs) + c, can be expressed by the relations
A ) 2910 - 10.0T (K)
(24)
for the UCST envelope and
A ) -7510 + 19.0T (K)
(25)
for the LCST envelope. The results obtained are plotted in Figure 1. The hourglass behavior can be described by the following relation:
A ) 17929 - 102.7T (K) + 0.151T2 (K)
(26)
From the plot of Figure 2 it results that the low polymer concentration branch is that modeled with a minor accuracy. A similar trend is observed also for other literature methods (see, for example, Kontogeorgis et al., 1995). Conclusions This work is inspired by a concept which can be expressed by the words “we must distinguish for better understanding”. Accordingly, the corrective term for the redundance of the FHE is not embodied in the enthalpic χ parameter, as performed in other literature methods. The resulting simplification of the analytical form of the modified FHE proposed in this work is remarkable, since a complex dependence of the χ term on composition is thus avoided. Another objective was the translation in the simple frame of the FHE of some properties of
Figure 2. Hourglass phase diagram for the acetone/PS (19 800) system: b, calculated data; 0, experimental data.
the EOS models, among which the volume-dependent residual term appears of paramount importance. Accordingly, we have assumed a double dependence of χ from both temperature and density without loading the analytical form of the proposed relation with many empirical constants. Conclusively, our model can describe with a fair accuracy the various types of phase equilibria involving polymer and solvent molecules with a number of empirical parameters which range from zero to three, but it is predictive only for VLE measurements. In this respect, the EOS route based on the application of the van der Waals equation appears the most rational and simple way to face the multiform features of polymer solutions. Last, it is asserted that old explanations of demixing at the LCST are more thermodynamically sound than recent ones which have gained wide acceptance. The thesis mainly hinges on a previous work which corrects the redundance of the entropic FH term, according to which the negative excess volume of mixing invoked to justify the phase splitting is less relevant at high than at low temperatures. Nomenclature a ) constant of eq 9 a1 ) constant of eq 13 A ) quantity defined by eq 7 b ) constant of eq 9 b1 ) constant of eq 13 c ) constant in eq 13 B(φ2) ) composition-dependent term of eq 2 d(T) ) temperature-dependent term of eq 2 gii ) enthalpic binary interaction parameters between i pure molecules, cal mol-1 gii* ) enthalpic binary interaction parameter between i molecules in solution, cal mol-1 ∆Ge ) excess Gibbs energy of mixing, cal mol-1 ∆He ) excess enthalpy of mixing, cal mol-1 K ) constant of eq 11, cm3 g-1 m ) parameter defined as V2/V1* m1 ) parameter defined as V1*/V2 M ) molecular weight r ) parameter defined as V2/V1 R ) univeral gas constant, cal mol-1 T-1 T ) absolute temperature, K Vi ) molar volume of the i component, cm3 mol-1 V1* ) corrected molar volume of the solvent, cm3 mol-1 Vw ) van der Waals volume, cm3 mol-1 x ) mole fraction
2872 Ind. Eng. Chem. Res., Vol. 37, No. 7, 1998 Greek Letters R ) constant in eq 18 and 19 β ) constant in eq 18 and 19 γ ) activity ) corrective term for the solvent liquid volume φ ) volumetric fraction φ* ) volumetric corrected fraction χ ) Flory-Huggins interaction parameter F ) density, g cm-3 Ω1∞ ) solvent molecular weight activity coefficient at infinite dilution Superscipt fv ) free volume Subscripts 1 ) solvent 2 ) polymer l ) liquid mixture p ) polymer s ) solvent
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Received for review December 8, 1997 Revised manuscript received March 25, 1998 Accepted March 30, 1998 IE9708891
