Ind. Eng. Chem. Res. 1998, 37, 4463-4469
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An Empirical Method To Predict the Liquid-Liquid Equilibria of Binary Polymer Systems Alessandro Vetere Snamprogetti Research Laboratories, 20097 San Donato Milanese, Italy
An empirical method is described to predict the liquid-liquid equilibria of several binary systems of polymers in both polar and nonpolar solvents according to a simple modification of the FloryHuggins equation. The method hinges on a two-step procedure: (i) first, the UCST or the LCST points are calculated by using a previous method proposed by the author and based on the knowledge of a few pure-component properties only; (ii) second, starting from these points, all the miscibility gap is calculated by applying a thermodynamic constraint and simple empirical rules. The predicted results are in fair accordance with the experimental data in both the lowand high-temperature ranges. The correlation of the hourglass phase diagrams is also considered. Introduction The study of the complex phase diagrams that characterize polymer solutions and blends is important for two main reasons. First, knowledge of the liquid-liquid phase separations often found in both the high- and lowtemperature ranges is required in order to process polymer materials. Second, the demixing phenomena are a touchstone to evaluate current theories aimed at describing the behavior of polymers in solutions. As is well-known, a wide revision of the celebrated FloryHuggins equation (FHE)
ln(a1) ) ln(φ1) + (1 - 1/m)φ2 + χφ22
(1)
was promoted by the discovery that a miscibility gap can be usually obtained simply by raising the temperaure above the so-called lower critical solution temperature, LCST (Freeman and Rowlison, 1960). As a result, the proper balance between the entropic and the enthalpic contributions to the nonideality of polymeric mixtures was more complex than that depicted by eq 1. The first remedy to the inadequacies of the FHE was found in removing the constancy of the Flory term, χ. By assuming that χ varies according to both temperature and composition, all types of liquid-liquid equilibria, such as UCST, LCST, and hourglass shaped and closed-loop diagrams, can be described according to the following simple modification of the enthalpic term of the FHE:
χ ) D(T)B(φ2)
(2)
where D(T) and B(φ2) are both expressed by a threeempirical-constant relationship (Eichinger, 1970; Quian et al., 1991). A further simplification of the B(φ2) term proposed by Bae et al. (1993) enables the correlation of many types of phase diagrams in all the experimental field with an astonishing accuracy. However, all these methods suffer because of their intrinsinc drawback: the χ parameter is seen as a purely empirical term which corrects the FHE for the inadequate treatment of the entropic term. As a result, the prediction of liquid-liquid equilibria with the above methods is not feasible, since the empirical constants embodied in D(T) and B(φ2) cannot be generalized.
A second major improvement over the FHE is represented by the use of an equation of state (EOS) to describe both the entropic and enthalpic contributions to the excess Gibbs energy of mixing of polymeric solutions. This trend originated with Flory and obeys the modern requirement of studying the properties of all the phases at equilibrium with a unifying model. The EOS route has gained a wide acceptance since it was shown that the first and simplest EOS, the van der Waals equation, is able to represent and, in some instances, to predict the phase equilibria of many polymer solutions, including the liquid-liquid equilibria at the UCST and LCST points (Kontogeorgis et al., 1993, 1995; Harismaidis et al., 1994). The algorithm is simple and robust, the need for pure component data is not demanding, and the results are reliable. Further, it must be appreciated from a theoretical point of view that the dependence of the enthalpic term on fluid density directly follows from the analytical form of a cubic EOS without additional hypothesis. The application of the van der Waals equation to the polymeric solution field also justifies the Elbro modification of the FHE (Elbro et al., 1990). In this work, a third procedure was followed, based on the assumption that the thermodynamics of polymer solutions is dominated by the entropic effects, the enthalpic ones representing only a perturbative contribution. As it is shown elsewhere (Vetere, 1997, 1998), the field of application of the FHE can be considerably enlarged thanks to very simple assumptions which save the analytical form of eq 1. The aim of utilizing only very simple tools to solve complex problems is pursued here also for the prediction of the binary liquid-liquid phase equilibria involving oligomers or high molecular weight polymers. Proposed Model In a previous work, it was assumed that the main flaw of the FHE is the redundance of its entropic term (Vetere, 1997). Further, it was shown that a more correct evaluation of the ratio between the entropic and enthalpic contributions can be obtained without changing the analytical form of the FHE by assuming an arbitrary value for the molar volume of pure solvent. To illustrate, if one writes
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4464 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 Table 1. Comparison between Two Methods To Predict the VLE Data of Binary Polymeric Mixtures AAD% type of mixture
no. of systems
Vetere (1998)
this work
nonpolar polymers in nonpolar noncyclic solvents nonpolar polymers in nonpolar cyclic solvents polar polymers in polar solvents system containing both polar and nonpolar components water/polymer polymer/chlorinated solvents
19 18 18 19 7 12
5.17 4.56 5.05 7.96 7.07 10.70
4.72 5.45 5.52 6.93 7.07 8.42
V1* ) V1/(1 - )
(3)
in which is a function of the pure solvent molar volume and polymer density according the relation
) F[1/(V1F2)]
(4)
the backbone of eq 1 is still retained with only the modification of substituting the usual volume fraction φi’s in the entropic term with the following fictitious values:
φ1* )
x1V1* x1V1* + x2V2
(5)
and
φ2* ) 1 - φ1*
(6)
Regarding the enthalpic term, some recent works propose an improvement of the FHE performances by substituting the two-suffix Margules equation with more updated relations, such as the NRTL equation (Vetere, 1997). However, the recognized predominance of the entropic term over the enthalpic one strongly suggests privileging, when possible, of simple analytical relations. Accordingly, the following slightly modified form of the FHE is applied through this work:
ln(a1) ) ln(φ1*) + (1 - 1/m*)φ2* + χφ22
(7)
which is formally equivalent to eq 1. Procedures Although the working equation (7) is very similar to the ones applied in previous works by the author (Vetere, 1997, 1998), the empirical constants embodied in it must be reoptimized in order to have reliable predictive results. Accordingly, the analytical relation which expresses the dependence of the parameter on pure-component properties is
) -0.21 + 0.567[100/(V1F2)]
(8)
when eq 7 is applied to cyclic solvents and
) -0.04 + 0.510[100/(V1F2)]
(9)
when noncyclic solvents are considered. Equations 8 and 9 embody only a slight numerical modification with respect to the corresponding equations reported in Vetere (1997). This adjustment was mainly dictated by the correlation of some infinite dilution weight fraction activity coefficient, Ω1∞, chosen among those reported by Danner and High (1993) for many nonpolar solvents in mixtures with nonpolar polymers. In fact, eq 7
A, J/mol 200 200 850 for C1-C3 alcohols; 0.0 for other compds 400 for ketones; 0.0 for other compds 0.0 6000 for CHCl3 in oxygenated polymers; 400 for other compds
applied with the parameter calculated through eq 8 or 9 is as reliable as the method reported by Vetere (1998) to predict the Ω1∞ data pertaining to several polar and nonpolar mixtures (the complete list of the data processed is reported in Tables 2 and 3 of the quoted reference). Since this work is clearly inspired by the statement “seek simplicity and do not wring its neck”, a further point must be considered: according to the new method, is independent of composition, and the differentiation of eq 7 required by the thermodynamic condition of demixing does not lead to complex relations. Another set of simple rules regards the enthalpic term. Recalling that the χ term can be expressed as A/RT, the constant A is evaluated as shown in Table 1 for each of the five classes of compounds which cover the experimental field according a previous method (Vetere, 1997). The results reported in Table 1 for 92 binary systems justify the use of the simple relation (7). Liquid-Liquid Phase Equilibria The only literature methods to predict the liquidliquid equilibria of polymeric solutions according to a rigorous procedure are those of the Fredenslund and Tassios schools (Kontogeorgis et al., 1993, 1995; Harismaidis et al., 1994). However, the task is inherently very difficult, and the deviations of the calculated data from the experimental ones are in many cases significant. Very recently (Vetere, 1998), it was found that a brutal but very efficient empirical method enables the prediction of both the UCST and LCST points for a wide class of mixtures with a fair accuracy (the overall AAD% is less than 4%). The method relates the UCST and LCST temperatures to a few pure-component properties such as the molecular weight of the solvent and the polymer density through simple relations. Typically, one may write
T(K) ) a ( bM1(1 - m-0.5)/F2
(10)
where the plus and minus signs stand for LCST and UCST, respectively. Six relations of this type represent well the experimental data of six classes of mixtures which cover a wide corpus of literature data with the exception of polar polymers in polar solvents. However, some incongruities found in a fundamental paper by Delmas and Saint-Romain (1974) suggest a revision of the UCST and LCST calculations for nonpolar polymers in nonpolar solvents. Delmas and Saint-Romains studied the critical temperatures in the polybutadiene/ alkane systems. However, the statement that “the branched alkanes are poorer solvents than linear alkanes” since “the substitution of a CH2 group in the normal heptane by a side methyl group greatly reduced its solvent quality” is denied by some of their Table 2 data and their graphical representations of Figures 1
Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4465
Figure 2. UCST and LCST phase diagrams for the BR/n-octane system. (b) Experimental points, (O) calculated points, (s) smoothing experimental curves, (- - -) smoothing calculated curves.
Figure 1. UCST and LCST phase diagrams for the BR/n-hexane system. (b) Experimental points, (O) calculated points, (s) smoothing experimental curves, (- - -) smoothing calculated curves.
and 2. A critical examination of the text leads to the conclusion that the data represented in their Figure 1 refer to 2,2,4-trimethylpentane instead of n-octane and their Figure 2 refer to n-octane instead of 2,2,3-trimethylbutane. The wrong headings of their Tables 1 and 2 are reported also in the monograph by Danner and High (1993). The revision of relation 10 for the class of nonpolar polymers in hydrocarbons promoted by the above criticisms leads to
TU(K) ) 380 - 297(1 - m-0.5)7/M10.2
(11)
for the UCST points of linear hydrocarbons,
TU(K) ) 288.4 + 0.0366M1(1 - m-0.5)2
(12)
for the UCST points of ramified and cyclic hydrocarbons, and
TL(K) ) 59 + 54M10.3(1 + 1.2m-0.5)4/F22
(13)
for the LCST points of all types of hydrocarbons. Equations 11-13, along with the equations reported by Vetere (1998) for the other classes of binary mixtures, are the basis of the liquid-liquid equilibria predictions proposed in this work. The UCST and LCST points represent the temperature at which incipient instability occurs. They are equivalent to the so-called plait point in the usual liquid-liquid equilibria. The thermodynamic condition for this instability is given by a nil value for the first and second derivatives of the FHE with respect to volumetric fractions. Below the UCST point
Figure 3. UCST and LCST phase diagrams for the BR/2,2,5trimethylpentane system. (b) Experimental points, (O) calculated points, (s) smooting experimental curves, (- - -) smoothing calculated curves.
and above the LCST point, the phases at equilibrium have different compositions and the miscibility gap is defined by binodal lines which obey the condition
∂ ln(a1)/∂φ1 < 0
(14)
Looking at eq 7, if φi* is expressed as a function of φi or vice versa, the differentiation becomes very tedious. Therefore, in order to fulfill the condition expressed by
4466 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998
Figure 4. UCST and LCST phase diagrams for the PS/cyclohexane system. (b) Experimental points, (O) calculated points, (s) smoothing experimental curves, (- - -) smoothing calculated curves.
Figure 5. UCST and LCST phase diagrams for the PS/cyclopentane system. (b) Experimental points, (O) calculated points, (s) smoothing experimental curves, (- - -) smoothing calculated curves
eq 14, it is advisable to differentiate the modified FHE with respect to x1. The resulting relation is
A ) AL[1 - K1(TL(K) - T(K))]
2x2χ m* 1 - 1/m* TL(K) in eq 17. The signs in these equations reflect the opposite effect of temperature on A in the low- and high-temperature ranges. Since the nonideality of any system increases with temperature above the LCST point, while the contrary occurs below the UCST point, the same trend must be disclosed by the A values, also. Accordingly, the value of A calculated through eqs 16 and 17 is always greater than AU or AL. K1 is not an universal constant, but it varies in a very narrow range. The following values can be assumed: K1 ) 0.01 for nonpolar polymers in noncyclic hydrocarbons, K1 ) 0.03 for nonpolar polymers in cyclic hydrocarbons, and K1 ) 0.04 for nonpolar polymers in polar solvents. By applying eqs 15-17 along with the above rules for A and K1, the whole miscibility gap can be easily predicted in both the low- and high-temperature fields. Results The literature on liquid-liquid equilibrium data processed in this work were taken from the monograph by Danner and High (1993), except for the two systems quoted above studied by Delmas and Saint-Romain (1974). The polymer density at each temperaure was determined by applying the relations reported by Danner and High (1993), or it used the experimental data
Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4467
Figure 6. UCST and LCST phase diagrams for the PS/ethyl formate system. (b) (Experimental points, (O) calculated points, (s) smoothing experimental curves, (- - -) smoothing calculated curves.
given by Wen et al. (1991). The density of solvents was calculated by applying the Rackett equation using the critical constants reported by Reid et al. (1987). The chosen systems are representative of the following classes of mixtures: nonpolar polymers in linear hydrocarbons, nonpolar polymers in cyclic hydrocarbons, and nonpolar polymers in polar solvents. The results obtained are represented in Figures 1-9, along with the literature experimental points. Some of these systems were studied also by Kontogeorgis et al. (1995), but their data are not reported here to make it easier to read the plots. However, the rough, unpleasant empirical method proposed in this work is always more reliable than the elegant theoretical method of the literature. The comparison with the experimental data reveals that the calculated TU(K) ant TL(K) points are often very close to the experimental ones. On the contrary, appreciable differences can be found in the width of the miscibility gap. As a rule, the calculated values of the polymer concentration are lower then the measured ones in the branch rich in the solvent, and they are higher in the regions rich in the polymer. The variance is partly explained by theoretical reasons. The turbidity methods often used to measure the phase separation in polymer solutions circumscribe the instability region, which is subtended to the liquid-liquid miscibility gap determined by our method. Therefore, the experimental width of the immiscibility curve is inherently narrower than the calculated one. Another source of errors is found when the polymer samples are not of a monodisperse type. Since the polydispersity displaces the critical points to the right (Cowie et al., 1971), in some
Figure 7. UCST , LCST and hourglass phase diagrams for the PS/acetone system. (b) Experimental points, (O) calculated points, (s) smoothing experimental curves, (- - -) smoothing calculated curves.
instance, an opposite phenomenon occurs, and the experimental gap is wider than the calculated one. This case is verified for the mixture of Figure 2. Among the studied systems, of particular interest appear the reliable results obtained for the prediction of the liquidliquid equilibria of polymers in strongly polar solvents such as HDPE/n-octanol and HDPE/octylphenol (see Figures 8 and 9). However, the temperature, TH, at which the UCST and LCST branches merge in the so-called hourglass type equilibria cannot be predicted by the proposed method. These data must be assumed from the experimental literature, from which the whole curve can be derived according the above procedure simply by posing TU(K) ) TL(K) ) TH in eqs 16 and 17. The only example of this type of phase equilibrium given here is represented by the PS/acetone system of Figure 7. Conclusions This work confirms the paramount importance of a proper evaluation of the entropic term of the classical FHE. If this prerequisite is fulfilled, the enthalpic term represented by a two-suffix Margules equation can be mantained without an appreciable loss of accuracy to calculate the phase separations of whatever nature found in polymer solutions. Further, the simple but effective modification of the entropic term assumed in this work preserves the same analytical form of the FHE. As a result, the mathematical apparatus required to correlate or to predict the complex phase equilibria
4468 Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 K1 ) constant of eqs 16 and 17, K-1 m ) parameter defined by V2/V1 m* ) parameter defined by V2/V1* M ) molecular weight R ) universal gas constant, J mol-1 K-1 T(K) ) absolute temperature, K TU(K) ) absolute temperature at the UCST point, K TL(K) ) absolute temperature at the LCST point, K V ) molar volume of the pure liquid, cm3 mol-1 V1* ) corrected molar volume of the solvent, cm3 mol-1 W ) weight fraction x ) molar fraction Greek Letters ) corrective term for the solvent volume φ ) volumetric fraction φ* ) corrected volumetric fraction χ ) Flory-Huggins interaction parameter F ) density, g cm-3 Ω∞ ) molecular weight activity coefficient at infinite dilution Subscripts 1 ) solvent 2 ) polymer Acronyms
Figure 8. UCST and LCST phase diagrams for the HDPE/ n-octanol system. (b) Experimental points, (O) calculated points, (s) smoothing experimental curves, (- - -) smoothing calculated curves .
Figure 9. UCST and LCST phase diagrams for the HDPE/ octylphenol system. (b) Experimental points, (O) calculated points, smoothing experimental curves, (- - -) smoothing calculated curves.
in the field of polymer solution is greatly simplified. In essence, this work proves that the pursuing of empirical solutions to difficult problems is not detrimental to the reliability and accuracy if the inquiry is based on a simplified but correct physical model of reality. Nomenclature a1 ) solvent activity A ) constant of the two-suffix Margules equation, J mol-1 B(φ2) ) composition-dependent term of eq 2 D(T) ) temperature-dependent term of eq 2
BR ) butadiene rubber PS ) polystyrene HDPE ) high-density polyethylene
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Ind. Eng. Chem. Res., Vol. 37, No. 11, 1998 4469 Reid, R. C.; Prausnitz, J. M.; Poling, B. E. The Properties of Gases and Liquids, 4th ed.; McGraw-Hill: New York, 1987. Vetere, A. An Improved Method to Predict the Vapor-Liquid Equilibria of Amorphous Polymer Solutions. Ind. Eng. Chem. Res. 1997, 36, 2466. Vetere, A. Empirical Method to Correlate and to Predict the Vapor-Liquid Equilibrium and the Liquid-Liquid Equilibrium of Binary Amorphous Polymer solutions. Ind. Eng. Chem. Res. 1998, 37, 2864.
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Received for review April 29, 1998 Revised manuscript received July 27, 1998 Accepted August 12, 1998 IE980258M
