Polymer−Solvent Phase Behavior near the Solvent Vapor Pressure

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Ind. Eng. Chem. Res. 1996, 35, 4301-4309

4301

Polymer-Solvent Phase Behavior near the Solvent Vapor Pressure Naveen Koak and Robert A. Heidemann* Chemical and Petroleum Engineering, University of Calgary, 2500 University Drive, N.W., Calgary, Alberta T2N 1N4, Canada

This paper looks at polymer-solvent phase behavior at conditions near the solvent vapor pressure curve where vapor-liquid-liquid equilibria can occur. Data of Kennis (1988) and Kennis et al. (1990) for a high-density polyethylene in n-hexane have been modeled using the SanchezLacombe lattice gas model, the related mean-field lattice gas model of Kleintjens and Koningsfeld (1980), and the perturbed hard-sphere-chain (PHSC) model of Song et al. (1994a,b). Phenomena of interest include LCST behavior and the liquid solvent, vapor solvent, polymer three-phase condition. Some aspects of the computational problems are described in the paper. These include such matters as the relationship between the spinodal and the coexistence curves and the potential for locating metastable liquid-liquid equilibria at conditions near the solvent vapor pressure curve. Convergence of flash calculations proves to be problematic, with the vaporliquid equilibria and liquid-liquid equilibria showing somewhat different character. Some techniques are described for modifying the familiar successive substitution computational algorithm so that convergence can be obtained. Introduction Many polymer processing situations involve mixtures of polymers and solvents under conditions where solubility and phase separations are important issues (Kiran, 1994). The mixtures may be present at high pressures since pressure is an important parameter in optimizing properties of polymeric systems (Kleintjens, 1994; Folie and Radosz, 1995). In this paper, we have focused on the cloud-point data and three-phase conditions reported by Kennis (1988) and Kennis et al. (1990) for solutions of a high-density polyethylene (HDPE) in n-hexane at three different pressures. The polyethylene had a number-average molar mass of 8000 g/mol and a weight-average molar mass of 177 000 g/mol. The temperature and pressure ranges involved cross the n-hexane vapor pressure curve, and, as a consequence, both liquid n-hexane and vapor n-hexane can play a role in the equilibrium and three-phase vapor-liquid-liquid equilibria (VLLE) occur. The three-phase behavior occurs near a lower critical solubility condition, and, together, these phenomena present significant modeling challenges. We have looked at the performance of three equation of state models in correlating and predicting the polyethylene/n-hexane data of Kennis (1988) and Kennis et al. (1990). The three models are (i) the SanchezLacombe equation of state (Lacombe and Sanchez, 1976; Sanchez and Lacombe, 1976, 1978), (ii) the mean-field lattice gas (MFLG) model of Kleintjens and Koningsveld (Kleintjens and Koningsveld, 1980), and (iii) the perturbed hard-sphere-chain (PHSC) equation of state (Song et al., 1994a,b, 1996). The first two of these models have a similar basis in presuming chain molecules and holes situated on a lattice. The third, the PHSC model, employs a reference fluid consisting of chains of hard spheres. The equations for the “chains of hard spheres” reference fluid used in the PHSC model were conceived recently by Chiew (1990) and further developed by Song et al. The SAFT model (Chapman et al., 1989, 1990; * Author to whom correspondence should be addressed. Telephone: (403) 220-8755. FAX: (403) 284-4852. E-mail: [email protected].

S0888-5885(95)00684-1 CCC: $12.00

Huang and Radosz, 1990, 1991; Fu and Sandler, 1995), which has received much recent attention, is a perturbation model with a similar reference fluid based on the work of Wertheim (1987). Song et al. (1994a,b) assert that the Chiew (1990) development is an essentially exact result and that SAFT is based on a “cluster integral” approximation. SAFT is somewhat more complex algebraically than the PHSC equation. In any case, we have not looked at SAFT in this paper. We found it convenient to rewrite the three equation of state models in terms of mole fractions as the composition variable instead of the traditional choice of volume fraction or lattice fraction as the composition variable. In our view, it has been possible to make the equations more transparent and easier to use, particularly for multicomponent mixtures, through this reorganization. The modified forms of the equations are given in the appendices. Computations involving mixtures of high molar mass polymers and relatively low molar mass solvents are hazardous because of the minute mole fractions and activities of polymers, particularly in solvent-rich vapors. A further problem was recently described by Koak and Heidemann (1994) and analyzed by Heidemann and Michelsen (1995). The typical “successive substitution” algorithm for flash calculations can be oscillatory and divergent for mixtures with phase models that show large negative deviations in the excess Gibbs free energy. This was observed for a Flory-Huggins model for two-phase water-soluble polymer systems that was presented by Kang and Sandler (1987). Similar problems that arise in computing with the equation of state models are reported here. Model Parameters The parameters used in the various EOS models are summarized in Table 1. Parameters for n-hexane and high-density polyethylene in the various equations of state are available in Sanchez and Lacombe (1978), Kennis et al. (1990), and Song et al. (1994a,b, 1996). Kennis et al. (1990) report binary interaction parameters for polyethylene/n-hexane for use in the KleintjensKoningsveld MFLG model. There are two binary pa© 1996 American Chemical Society

4302 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 Table 1. Parameters for HDPE/n-Hexane Used in Equilibrium Calculations n-hexane

interaction param

HDPE

ii/kB, K di νii, cm3/mol

Sanchez-Lacombe 476 659 8.356 M/11.477 13.28 12.7

mi γi R0i β0ii β1ii, K β2ii, K2

MFLG, ν0 ) 20 cm3/mol 5.6567 M/17.66 -0.49924 -0.95446 1.0475 0.96874 -1.3177 -0.9711 660.72 342.90 0 247 500

ri σi, Å /kB, K

3.446 4.084 235.6

k12 ) -0.041 44

PHSC 0.049 38M 3.825 324.1

R12 ) -0.25 β012 ) -0.10 β112 ) 0.00

k12 ) -0.10 ζ2 ) 0.895 38

Table 2. n-Hexane Critical Points from Equations of State model

Tc, °C

Pc, bar

Fc, g/cm3

Sanchez-Lacombe mean-field lattice gas PHSC (1994) PHSC (1995) data

253.25 243.61 256.7 263.31 234.7

36.17 41.73 36.0 34.7 30.31

0.1987 0.228 0.200 0.189 0.234

rameters reported in the Kennis et al. paper; i.e., Rij ) -0.10 and β0ij ) -0.25. (See Appendix B for the definitions.) Kennis et al. (1990) computed only the spinodal curves at three pressures and selected the interaction parameters to match the minimum temperatures on the spinodal curves to the minimum temperatures on the three cloud-point isobars. We attempted to reproduce these results. Our calculations indicate that an error was made in reporting the parameters and that they should have been Rij ) -0.25 and β0ij ) -0.10. Binary interaction parameters had to be determined for the Sanchez-Lacombe and PHSC equations of state. In our fitting, we searched for the interaction parameter that gave 127.7 °C as the 6 bar LCST, the temperature we obtained from the MFLG model. The lowest point on the spinodal curve for a binary system is the lower critical solution temperature, and we searched for the interaction parameter that produced the required minimum temperature. For the Sanchez-Lacombe equation of state, the required value was kij ) -0.041 44. (For a definition of kij, see Appendix A.) For the PHSC (1996) equation of state, two binary parameters are available: (i) an interaction parameter in the energy term, kij, and (ii) a “size reduction parameter”, ζ, for the polymer segment number in the attractive perturbation term. (See Appendix C for the definitions.) We used kij ) -0.1 and found ζ ) 0.895 38 to give the desired LCST. Our experience was that the PHSC model could not be fitted to the 6 bar critical point through use of the kij parameter alone. There was no LCST for most values of kij, and the maximum LCST that could be found with any kij was far below the experimental point. A kij ) -0.1, without the size reduction parameter, produces a LCST well below the required value. Solvent-Polymer Phase Behavior n-Hexane. The vapor pressure curve for hexane has a dominant effect on the location of three-phase equilibria in the hexane-polyethylene system. Figure 1 shows the percentage error in the hexane vapor pres-

Figure 1. Percent error in calculated vapor pressures for nhexane.

Figure 2. Calculated VLE, LLE, and spinodals from KleintjensKoningsveld EOS for the system n-hexane + HDPE.

sures produced by the EOS models examined. (The calculations used the parameters for n-hexane supplied by the proposers of the models, as given in Table 1.) For the PHSC EOS both the 1994 and 1996 versions were examined. These two models differ in the temperature dependencies of the attractive term and the temperature-dependent hard-core diameter. Figure 1 shows that all the models fit the experimental vapor pressure data well at the lower temperatures but significant deviations are present at higher temperatures. Furthermore, the parametrization of all the models has been done through least-squares fitting of selected vapor pressure and liquid density data, without concern for the critical point. The EOS critical points for n-hexane are compared with the data in Table 2. In all cases, the vapor pressure curve is extrapolated far beyond the data. Deviations from the experimental vapor pressure curve can be more or less important, depending on the temperature range in which the EOS model is to be used. Cloud-Point Curves. Figures 2-5 show calculated VLE and LLE cloud points (solid lines) and spinodal curves (dashed lines) for the HDPE/n-hexane system at 6, 25, and 50 bar. Figures 2 and 3 are for the Kleintjens-Koningsveld MFLG equation, and Figures 4 and 5 respectively are for the Sanchez-Lacombe and

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4303

Figure 3. Experimental cloud-point curves and calculated VLE, LLE, and spinodals from Kleintjens-Koningsveld EOS for the system n-hexane + HDPE.

Figure 4. Experimental cloud-point curves and calculated VLE, LLE, and spinodals from Sanchez-Lacombe EOS for the system n-hexane + HDPE.

Figure 5. Experimental cloud-point curves and calculated VLE, LLE, and spinodals from PHSC EOS for the system n-hexane + HDPE.

PHSC (1996) equations. The vapor phase composition, which is essentially pure hexane, is not distinguishable from the temperature axis. Figure 2 includes polymer compositions up to 80 mass %, far beyond the range of the Kennis et al. (1990) data, in order to give a more full picture of the phase behavior called for by the models. The 6 and 25 bar isobars show the presence of three-phase lines where the slopes are discontinuous. The lower sections represent LLE equilibria. These curves intersect VLE phase boundaries, and, from the sharp edge onward, the curves correspond

to a liquid in equilibrium with a vapor-like hexane-rich phase. The three-phase condition is at a temperature very close to the boiling temperature of n-hexane (as given by the EOS model) because this is the condition where pure n-hexane liquid and vapor phases can be in equilibrium. The pressure of 50 bars is above the EOS critical pressure of n-hexane, and no three-phase condition is indicated in the calculations. The spinodal curves fall well inside the phase boundaries and do not provide an accurate picture of the equilibria. Furthermore, the LLE curve does not correctly represent the model behavior at temperatures above the three-phase line. A calculated LLE at higher temperature will represent a metastable equilibrium, whereas the correct equilibrium solution will be a VLE condition with significantly different compositions and densities of the phases. At a pressure of 6 bar one may extend the LLE phase boundaries to temperatures far above the three-phase condition, into a region where the correct equilibrium involves a liquid solution and almost pure hexane vapor. Figures 3-5 compare the calculated LLE and VLE cloud points with the Kennis et al. (1990) experimental cloud-point data. The n-hexane/polymer interaction parameters used in the calculations were adjusted to obtain a LCST at 6 bar that was in line with the data. As will be seen in Figure 3, Kennis et al. (1990) succeeded with their parametrization in placing the minimum points on the spinodal curves for the three isobars near the experimental minimum temperatures (i.e., the LCST points in the binary models). The critical temperatures on the 25 and 50 isobars for the SanchezLacombe model (Figure 4) and the PHSC model (Figure 5) appear a bit low, with the Sanchez-Lacombe results farther from the apparent experimental values. (The interaction parameters were fitted to the 6 bar isobar.) The critical solubilities from all three models are displaced to higher mass fractions of polymer than the data indicate. The Sanchez-Lacombe calculations show a flatter coexistence curve with critical compositions not as close to the data as the other two models. Three-Phase Equilibria. The existence of threephase equilibria has implications on the ability of the models to match the data. Metastable LLE curves could be computed at higher temperatures but would be well inside the cloud-point boundaries that are the correct equilibrium conditions according to the models. Parameter estimation procedures must take into account the different kinds of equilibria that can occur. Figure 6 shows some details of the three-phase equilibrium calculated from the Sanchez-Lacombe equation. Several metastable coexistence lines that are shown in the figure were found through two-phase flash procedures. Also indicated is the region of equilibrium between n-hexane-rich vapor and liquid phases that lies below the three-phase temperature. The densities of the n-hexane-rich liquid and vapor phases that coexist on the three-phase line can be quite different, depending on the pressure. For example, the densities of the three phases we calculate from the Sanchez-Lacombe equation at 6 bar are 0.0171 g/cm3 for the vapor, 0.524 g/cm3 for the middle liquid, and 0.572 g/cm3 for the more polymer-rich liquid. The mass fraction of polymer in the middle liquid is 0.004 67, obviously very small but not negligible. The mass fraction of polymer in the second liquid is 0.135. The composition at the LCST (from the model) is 0.04 mass fraction polymer.

4304 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

Figure 6. Calculated VLE, LLE, metastable VLE and LLE, and the three-phase line from Sanchez-Lacombe EOS for the system n-hexane + HDPE.

Figure 8. Liquid phase compositions and temperature along the three-phase lines. Comparison with data of Kennis.

Figure 7. Calculated three-phase and critical lines. Comparison with data of Kennis.

The consequences of the presence of a three-phase line on the experiments to determine the cloud-point curve are interesting. If the polymer mass fraction lies between the LCST composition and 0.135, the new phase formed as the coexistence curve is crossed will be a second liquid phase. However, if the polymer mass fraction exceeds 0.135, the new phase will be a solventrich vapor of a much different density from the polymerrich liquid. The thesis by Kennis (1988) contains data for the three-phase cloud-point temperature and pressure at a series of mass fractions of the polyethylene in n-hexane. Figure 7 shows the temperature-pressure three-phase line data of Kennis (1988) and the computed three-phase lines from the EOS models. Also shown in Figure 7 are the calculated liquid-liquid critical lines for the three models. These lines intersect the three-phase lines at a lower critical end point. The n-hexane/polyethylene system, when characterized as a binary mixture, is apparently type IV or type V in the general classification scheme for critical behavior. The three-phase lines from the EOS models are very close to the n-hexane vapor pressure curves produced by these models and differ from each other because the vapor pressure predictions differ. The Kennis (1988) data lie above the n-hexane vapor pressure curve, the probable cause for the difference being that the polymer

is polydisperse, not monodisperse as is assumed in the EOS calculations. The three-phase lines in Figure 7 locate the boundary between vapor-liquid equilibria (below the line) and liquid-liquid equilibria (above the line). Metastable vapor-liquid equilibria can be calculated at pressures above the line. Similarly, metastable liquid-liquid separations can be calculated at a pressure below the critical end-point pressure. Calculations must be done with caution to avoid accepting these metastable states as the true equilibrium. Figure 8 shows the three-phase cloud-point temperature against the mass fraction of polymer in the liquid phases. The three models and the Kennis (1988) data have very nearly the same critical end-point temperature (where the two liquid phases in equilibrium with the n-hexane-rich vapor are critical), principally because the hexane/polymer interaction parameters were fitted to a low-pressure critical temperature. The compositions of the polymer-rich liquid phases, as given by the models, deviate from the Kennis (1988) data at lower temperatures. The MFLG model and the PHSC model are somewhat better than the Sanchez-Lacombe model in this regard. The polymer mass fraction at the minimum three-phase temperature in the Kennis (1988) data is somewhat uncertain but appears to be at a considerably lower value than is given by any of the models. This character may also be due to the polydisperse nature of the polymer. Overall, the three models represent these complex phenomena reasonably. Equilibrium Calculation Methods We used flash calculations to obtain the various cloudpoint lines and the three-phase conditions. That was possible because the polyethylene was treated as a monodisperse polymer with a number-average molar mass of 8000, following the treatment of Kennis et al. (1990). These calculations were problematic for more than one reason. Problems arise due to the extreme asymmetry in the vapor-liquid equilibria and are compounded by the large differences in the molar masses of the polymer and solvent. Extremely small fugacities and mole fractions for the polymer must be calculated, particularly in a solvent-rich vapor-like phase. In some

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4305

cases, the numbers pass the “underflow” limits of typical computers. Using the Sanchez-Lacombe equation, it was possible to obtain convergence of the flash calculations requiring equality of the chemical potentials, even of the polymer, in all the phases. This was true even though the mole fractions of the polymer fell to the order of 10-200. With the other two equations, however, underflow would occur in the mole fraction of polymer in the vapor. This problem was dealt with by freezing the equilibrium ratio (K ≡ y/x) whenever the polymer mole fraction became lower than 10-75. The difference in chemical potential between the phases was then dropped from the convergence criterion. A different kind of numerical problem is that the successive substitution algorithm itself can become oscillatory and divergent. The analysis of Heidemann and Michelsen (1995) shows that this kind of instability of successive substitution can occur whenever any of the equilibrium phases shows strong negative departures from ideality. In successive substitution procedures, ratios of mole fractions (K factors) are used in an inner loop to find phase amounts and mole fractions. The K factors are updated in an outer loop until conditions of equal chemical potentials in all coexisting phases are reached. Heidemann and Michelsen (1995) suggested a simple “damping” procedure in the K factor updating algorithm that would result in a convergent process. For binary systems around critical points (including the n-hexane/polyethylene systems at temperature just above the LCST), damping proves unnecessary and monotonic convergence to a solution is possible, even if slow. These conclusions also follow from the Heidemann and Michelsen (1995) analysis. A consequence is that VLE and LLE computations can behave differently. Away from critical points the behaviors are dictated by the negative deviations in the liquid phases. Most of the calculations done in the preparation of this paper involved looking only for two-phase VLE or LLE equilibrium. For VLE, the hexane-rich phase was assigned the larger of two volume roots from the equation of state (if two could be found). For LLE, liquid-like volume roots were used for both phases. The K value updating scheme employed is

ln

Kk+1 i

) ln

Kki

(

II µIi 1 µi m RT RT

)

k

(1)

where m is the “damping factor” needed to obtain a convergent process. For multiphase calculations, we employed a slight modification of a scheme presented by Abdel-Ghani (1995) and Abdel-Ghani et al. (1994). There are possibly many phases, each with its own set of K factors defined as Kij ) xij/xˆ i. The reference mole fraction is arbitrary but could be the composition of the mixture being flashed; i.e., xˆ i ) zi. The inner loop calculates phase amounts and mole fractions consistent with the mass balances, using an algorithm based on a proposal by Michelsen (1994). This inner loop algorithm is quite robust and capable of returning the mole fractions of any number of phases, including some that might have zero amounts in the mixture. (The amount of phase j is βj.) Within the subroutine, the normalizing factors for the mole fractions in the phases are calculated, ∑iXij. When the phase amount is zero (βj ) 0), then the normalizing factor is less than 1.0(∑iXij < 1).

Table 3. Iteration Count with Various Damping Factors: VLE Calculations m ) 30

m ) 20

m ) 15

m ) 13

m ) 11

T (°C) IC T (°C) IC T (°C) IC T (°C) IC T (°C) 150.00 149.00 148.00 147.00 146.00 145.00 144.00 143.00 142.00 141.00 140.00 139.00

15 15 15 15 15 14 14 14 16 18 22 37

150.00 149.00 148.00 147.00 146.00 145.00 144.00 143.00 142.00 141.00 140.00 139.00

10 10 10 10 10 10 10 10 10 10 13 24

150.00 149.00 148.00 147.00 146.00 145.00 144.00 143.00 142.00 141.00 140.00 139.00

17 17 17 17 16 15 13 11 8 8 8 17

150.00 149.00 148.00 147.00 146.00 145.00 144.00 143.00 142.00 141.00 140.00 139.00

35 33 26 28 27 25 22 18 14 9 7 14

IC

150.00 149.00 148.00 147.00 146.00 145.00 144.00 143.00 142.00 141.00 140.00 139.00

20001 20001 20001 20001 446 128 68 44 28 17 9 11

Table 4. Iteration Count with Various Damping Factors: LLE Calculations m ) 30 T (°C)

IC

150.00 548 148.00 564 146.00 585 144.00 613 142.00 651 140.00 703 138.00 777 136.00 886 134.00 1062 132.00 1389 130.00 2219 138.00 13100

m ) 20 T (°C)

IC

m ) 10 T (°C)

IC

150.00 364 150.00 180 148.00 374 148.00 185 146.00 389 146.00 192 144.00 407 144.00 202 142.00 433 142.00 214 140.00 467 140.00 232 138.00 516 138.00 256 136.00 589 136.00 293 134.00 707 134.00 352 132.00 925 132.00 461 130.00 1478 130.00 738 128.00 8733 128.00 4366

m)5 T (°C)

m)1

IC

150.00 87 148.00 90 146.00 94 144.00 99 142.00 105 140.00 114 138.00 126 136.00 144 134.00 174 132.00 229 130.00 367 128.00 2182

T (°C)

IC

150.00 148.00 146.00 144.00 142.00 140.00 138.00 136.00 134.00 132.00 130.00 128.00

20001 20001 20001 20001 20001 155 30 25 31 43 71 436

In the outer loop, the equilibrium ratios are updated through

ln Kk+1 ) ln Kkij ij

(

µˆ i 1 µij - ln m RT RT

)

∑iXij

(2)

In this equation, the reference chemical potential is calculated as a weighted average in the phases present; i.e., µˆ i ) ∑jβjµij. The compositions of phases not present, once the outer loop has converged, locate phases that lie above the plane tangent to the Gibbs free energy surface that defines the multiphase equilibrium. The tangent plane distance is, in fact, a minimum at these compositions. Equation 2 may look unfamiliar, but it follows directly from the “tangent plane stability test” proposed by Michelsen (1982). It has been modified here by inclusion of the damping factor, m, which is needed to obtain convergence. Tables 3 and 4 demonstrate the effect of m on the iteration count at different temperatures in VLE and LLE calculations, respectively, using the two-phase algorithm and eq 1. (As noted, when a mole fraction for substance i was less than 10-75, the Ki value was not changed.) These results are for the SanchezLacombe EOS at a pressure of 6 bar with a feed at the LCST composition of 4 mass % polymer. If a specified maximum iteration count was exceeded, then either the calculations were oscillatory nonconvergent or the dampling factor was too large. Table 3, for VLE calculations, shows that as m is decreased from 30 to 20 the iteration count decreases as expected. In this region of the table, the actual number of iterations shown is quite small, mainly because the K value for the polyethylene becomes fixed after a few iterations and the polymer chemical potential difference between the two phases does not enter the convergence criterion. For smaller damping factors

4306 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

the iteration count shows an increase, again suggesting the occurrence of oscillations in computations. For a value of 11 for m, at the higher temperatures shown, the maximum iteration count is exceeded and the calculations fail to converge. The iteration counts for LLE calculations are shown in Table 4. In all these calculations, the polymer mole fractions in the two phases are of similar orders of magnitude and the polymer chemical potentials are driven toward equality. Table 4 shows, as is expected, that damping is not required at temperatures near the LCST but, rather, retards convergence. However, away from the LCST there is an optimum value for m that minimizes the number of iterations required for convergence. Tables 3 and 4 show that the optimum value of m is a function of temperature, pressure, and composition. The optimization of m is a problem that needs to be addressed in order to have an efficient algorithm for these kinds of equilibrium computations. Heidemann and Michelsen (1995) have suggested alternatives to successive substitution for systems where successive substitution can become unstable. Chen, and Duran, and Radosz (1993) propose a Newton-Raphson scheme for LLE equilibria in polymer/solvent systems. The three-phase temperature was located using a modification of the algorithm proposed by Abdel-Ghani et al. (1994) that was described above. Flash calculations were initiated with four phases, i.e., a hexane-rich vapor, a hexane-rich liquid, the feed composition (4.0 mass % polymer) as a liquid, and a polymer-rich liquid. At temperatures near the three-phase condition, a solution was found with two phases in finite amounts and a third (incipient) phase with ∑iXij < 1. The threephase temperature was the unique temperature (at 6 bar) where ∑iXij ) 1 for three phases. Equality of chemical potentials was obtained for both the polymer and the n-hexane in all three phases in these calculations with the Sanchez-Lacombe equation. Figure 9 is a blown up version of Figure 6 with emphasis on the hexane-rich region. In Figure 9 the relation between metastable equilibrium, minimum tangent plane phase composition (Michelsen, 1982), and the spinodal is shown. Our calculations show the expected behavior that, in general, both the metastable VLE and the minimum tangent plane phase compositions can be extended to the spinodal but no further. These characteristics indicate that computations looking for three-phase equilibria have to be performed with care. Discussion All three models examined give reasonable representations of the cloud-point data of Kennis et al. (1990) for a high-density polyethylene in n-hexane and for the conditions along the three-phase line. In our calculations, we used the parameters for hexane and polyethylene that were presented by the model developers. The full parametrization of the MFLG model was given in the Kennis et al. (1990) paper, and we did not adjust their parameter values (except that we detected an interchange between the two binary interaction parameters). The binary parameters in the Sanchez-Lacombe and PHSC models were selected so that the lowest temperature on the 6 bar spinodal curve was the same for all three models. Kennis et al. (1990) had only computed the spinodal curves, so the cloud-point calculations shown in Figures

Figure 9. Calculated VLE, LLE, metastable VLE and LLE, and the three-phase line from the Sanchez-Lacombe EOS. Also shown is the minimum tangent plane phase composition.

2 and 3 are new results. These two figures make very clear that the spinodal is a very poor approximation to the cloud-point curves, especially at temperatures above the three-phase line where the n-hexane-rich phase is a low-density vapor. Even so, the approach to parametrization that focuses on the lowest point on the liquidliquid phase boundary proves to be satisfactory. In this work the polymer has been treated as monodisperse. It is well-known that polydispersity of the polymer can be a very important issue in phase equilibrium behavior and calculations (Koningsveld and Staverman, 1968, Sˇ olc, 1970: Kang and Sandler, 1988; Gaube et al., 1992). We believe that the multicomponent forms of the models examined that are presented in the appendices will assist in such calculations. This will be a subject of further study by us. The computational problems in these polymersolvent systems are quite severe. The successivesubstitution flash calculation procedures are poorly convergent, and alternative methods must be sought. Acknowledgment The authors thank Professor Mark McHugh of Johns Hopkins University for providing listings of computer programs for the Sanchez-Lacombe equation. We are also grateful to Y. Song, T. Hino, S. M. Lambert, and J. M. Prausnitz for prepublication access to their 1996 manuscript. This research was supported by a grant from the Natural Sciences and Engineering Research Council of Canada. Appendix A. The Sanchez-Lacombe Equation of State The extension for use with polymer solutions of Sanchez and Lacombe (1978) has been employed in this work. The model involves three pure-component parameters: di, the number of lattice sites per molecule; νii, the volume occupied per mole of lattice sites; and ii, an attractive energy parameter. There is one temperatureindependent binary parameter, kij, relating the cross energy parameters through ij ) (1 - kij)xiijj.

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4307

The molar Helmholtz free energy (in our notation) is

A ) RT

( ) ∑ ( )

d2 a/RT ν - b/d (ν - b/d) ln + b ν ν nc nc xiRT dib xi ln + xi ln (A.1) ν i)1 i)1 RTd2

( )

∑

where ν is the molar volume. The parameters are calculated from nc

b)

nc

∑ ∑xixjbij i)1 j)1

where

Coefficients g0i and χij are composition dependent, with nc

g0i ) R0i + gii/Q where Q ) 1 -

χij ) Rij + gij(1 - γj)/Q;

and nc

a)

nc

∑ ∑ i)1 j)1

nc

xixjaij )

nc

∑ ∑xixjdidjijνij i)1 j)1

gij ) β0ij + β1ij/T + β2ij/T2 + ...

(A.3)

) RT

RT

( )

) ln

ν

)

ν

1 -

+ ν+d nc nc xiRT Ai° (B.5) xi ln + xi ν i)1 i)1 RT ν

( )

∑

nc

ximi ∑ i)1

(B.6)

(-γi)ximi ∑ i)1

(B.7)

∑ ∑[(R0i + R0j)/2 - fij]xixjmimj i)1 j)1

(B.8a)

b ) ν0

d - 2 [ xibik/b - dk/d](ν b i)1

∑ nc

- d[2

xibik/b - dk/d] ln

2

nc

RTνi)1

xiaik + 1 + ln

nc

ν - b/d

d ) ν0

+

ν

i)1

2(

dkb

nc

+

a1 )

2

d RT

xibik/b - dk/d) (A.5)

i)1

Because of the last term in this expression, the SanchezLacombe lattice gas does not tend to an ideal solution in the low-pressure limit.

) φ0 ln φ0 +

φi

∑ i)1m

i

∑ i)1

g0iφ0φi +

nc-1

Rij/2; j * i 0; j ) i

(B.8b)

and nc

nc

∑ ∑{[gii(1 - γj) + gjj(1 - γi)]/2 - hij}xixjmimj j)1 i)1

with

hij )

{

gij(1 - γj)/2 ) hji; j > i 0; j ) i

Ai° ) mi(R0i + gii) + ln(miν0/RT) RT

nc

ln φi +

{

(B.9a)

Kennis et al. (1990) give the MFLG equations for binary and ternary systems. The Helmholtz free energy of mixing multicomponent systems, in their notation, is nc

nc

fij ≡

with

a2 )

Appendix B. Kleintjens-Koningsfeld Mean-Field Lattice Gas Model

NφRT

ν

-

-

In this expression b, d, a1, and a2 are composition dependent. These coefficients are expressed in terms of parameters given in the Kennis et al. paper as

nc

∆A

(B.4)

a2ν0

a1ν0

ν-b

2 nc

( ) ∑ [( ) ] ∑ [ ( ) ∑ ]

b/d) ln

ν - b/d

ν0

ln

(A.4)

The chemical potential of component k is also found by differentiation:

xkRT

( )( ) ν-b

∆A

∑

(1 - d) d2 P a/RT ν - b/d ) ln - 2 RT ν b ν ν

µk

(B.3)

Coefficients gij are temperature dependent and account for surface contacts. The temperature dependence used by Kennis et al. (1990) is of the form

Pressure is found by differentiation:

(

j>i

Not counting ν0, the parameters for each pure substance may include, mi, R0i, γi, β0ii, β1ii, and β2ii. In terms of molar volume, temperature, and the mole fractions, we obtain

nc

xidi ∑ i)1

(B.2)

and

bij ) didjνij ) didj(νii + νjj)/2 (A.2) d)

γiφi ∑ i)1

(B.9b)

(B.10)

The equation for pressure is found by differentiation: nc

∑ ∑ χijφiφj i)1 j)i+1

(B.1)

In this model, a mole of lattice sites occupies a volume ν0, a substance-independent constant. A molecule of i occupies mi lattice sites. Nφ, the total number of lattice sites, is Nφ ) ν/ν0. The fraction of sites occupied by component i is φi ) ximiν0/ν.

(1 - b/ν0) 1 a1ν0 a2ν0 ν-b P ) - ln - 2 RT ν ν0 ν ν (ν + d)2 (B.11)

(

)

Appendix C. PHSC Equation of State The perturbed hard-sphere-chain (PHSC) equation of state (Song et al., 1994a,b, 1996) can be reduced to a

4308 Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996

relatively simple form by combining terms with like dependence on the mixture molar volume. Our result is

a0 a1 a2 a3 P a ) + + + (C.1) RT ν ν - b (ν - b)2 (ν - b)3 RTν2 where the numerator coefficients and parameter b are functions of the mole fractions in the mixture and ν is the molar volume. The Helmholtz free energy is obtained through integration (Prausnitz et al., 1986). For 1 mol of the PHSC fluid,

A RT

) a1 ln

a2

ν + ν-b

a3

1

a

+ ν-b

+

2 (ν - b)2

RTν

∑i xi ln(xiRT/ν) + ∑i xiAi°/RT

(C.2)

The chemical potential of a component is found by differentiation of the Helmholtz free energy. The Song et al. (1996) versions for the cross-sphere volume and the attractive pressure a parameter, in our notation, are

bij(T) ) NAν(2π/3)σij3Fb(kBT/ij) aij ) NAν

()

2π 3 ij σ RFa(kBT/ij) 3 ij kB

(C.3) (C.4)

Fa and Fb are given as universal functions of the dimensionless temperature. The mixing rules for the cross diameters and energies are

1 σij ) (σi + σj)(1 - λij) 2

and ij ) (iijj)0.5(1 - κij) (C.5)

Two interaction parameters per binary pair are available to fit data, i.e., λij, and κij. Song et al. (1996) provide pure substance and polymer values for σii and ii/kB and for the number of segments in the molecule, ri. For the polymer molecules, r/M values are supplied, where M is the number-average molar mass of the polymer. In our version of the Song et al. (1996) equations, we find

a)

∑i ∑j xixjζiriζjrjaij

and

b)

∑i xiribii/4

(C.6)

The parameter ζi < 1 in the attractive pressure term was suggested by Song et al. (1996) as useful in fitting dilute polymer solution data. Some auxiliary sums are required.

d)

∑i xiribii2/3/4

and

c0 )

r)

∑i xi(ri - 1)

∑i ∑j xixjrirjbij

(C.7) (C.8)

∑i ∑j xixjrirjbij(biibjj/bij)1/3

(C.9)

∑i ∑j xixjrirjbij(biibjj/bij)2/3

(C.10)

c1 ) (3/2) c2 ) (1/2)

The coefficients in the our version of the pressure equation are then given by

a1 ) 1 - a0 ) c0/b - dc1/b2 + d2c2/b3 - r

(C.11)

a2 ) dc1/b - d2c2/b2 - 3br/2

(C.12)

a3 ) d2c2/b - b2r/2

(C.13)

Literature Cited Abdel-Ghani, R. M. EOS Mixing Rules for Multi-Phase Behavior. M.Sc. Thesis, University of Calgary, Calgary, Alberta, Canada, 1995. Abdel-Ghani, R. M.; Phoenix, A. V.; Heidemann, R. A. An Approach to Multiphase Equilibrium Computation. Paper presented at the AIChE Symposium on SCF Science and Technology, 1994. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. SAFT: Equation of State Model for Associating Fluids. Fluid Phase Equilib. 1989, 52, 31-38. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. New Reference Equation of State for Associating Fluids. Ind. Eng. Chem. Res. 1990, 29, 1709-1721. Chen, C.-K.; Duran, M. A.; Radosz, M. Phase Equilibria in Polymer Solutions. Block-Algebra Simultaneous Flash Algorithm Coupled with SAFT Equation of State, Applied to Single-Stage Supercritical Antisolvent Fractionation of Polyethylene. Ind. Eng. Chem. Res. 1993, 32, 3123-3127. Chiew, Y. C. Percus-Yevick Integral Equation Theory for Athermal Hard-Sphere Chains. Part I: Equations of State. Mol. Phys. 1990, 70, 129-143. Folie, B.; Radosz, M. Phase Equilibria in High-Pressure Polyethylene Technology. Ind. Eng. Chem. Res. 1995, 34, 1501-1516. Fu, Y.-H.; Sandler, S. I. A Simplified SAFT Equation of State for Associating Compounds and Mixtures. Ind. Eng. Chem. Res. 1995, 34, 1897-1909. Gaube, J.; Pfenning, A.; Stumpf, M. Thermodynamics of Aqueous Poly(Ethylene Glycol)-Dextran Two-Phase Systems using the Consistent Osmotic Virial Equation. Sixth FPECPD 92-Cortina, July 19-24, 1992; session BIOTECHNOLOGY, 18. Heidemann, R. A. The Classical Theory of Critical Points. Supercritical Fluids, Proceedings of the NATO Advanced Study Institute on Supercritical Fluids, Kemer, Turkey, 1994; Kiran, E., Levelt Sengers, J. M. H., Eds.; Kluwer Academic Publishers: Amsterdam, The Netherlands, 1994. Heidemann, R. A.; Michelsen, M. L. Instability of Successive Substitution. Ind. Eng. Chem. Res. 1995, 34, 958-966. Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284-2294. Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Extension to Fluid Mixtures. Ind. Eng. Chem. Res. 1991, 30, 1994-2005. Kang, C. H.; Sandler, S. I. Phase Behavior of Aqueous TwoPolymer Systems. Fluid Phase Equilib. 1987, 38, 245-272. Kang, C. H.; Sandler, S. I. Effects of Polydispersivity on the Phase Behaviour of Aqueous Two-Phase Polymer Systems. Macromolecules 1988, 21, 3088-3095. Kennis, H. A. J. The Influence of Nitrogen on the Phase Behavior of the System n-Hexane/Linear High Density Polyethylene. Ph.D. Thesis, Technical University of Delft, Delft, The Netherlands, 1988. Kennis, H. A. J.; De Loos, Th. W.; De Swaan Arons, J.; Van Der Haegen, R.; Kleintjens, L. A. The Influence of Nitrogen on the Liquid-Liquid Phase Behavior of the System n-HexanePolyethylene: Experimental Results and Predictions with the Mean-Field Lattice-Gas Model. Chem. Eng. Sci. 1990, 45, 18751884. Kiran, E. Polymer Formation, Modifications and Processing in or with Supercritical Fluids. In Supercritical Fluids, Proceedings of the NATO Advanced Study Institute on Supercritical Fluids, Kemer, Turkey, 1994; Kiran, E., Levelt Sengers, J. M. H., Eds.; Kluwer Academic Publishers: Amsterdam, The Netherlands, 1994. Kleintjens, L. A. Optimizing Properties of Polymer Solutions. In Supercritical Fluids, Proceedings of the NATO Advanced Study Institute on Supercritical Fluids, Kemer, Turkey, 1994; Kiran,

Ind. Eng. Chem. Res., Vol. 35, No. 11, 1996 4309 E., Levelt Sengers, J. M. H., Eds.; Kluwer Academic Publishers: Amsterdam, The Netherlands, 1994. Kleintjens, L. A.; Koningsveld, R. Liquid-Liquid Phase Separation in Multicomponent Polymer Systems. XIX. Mean Field LatticeGas Treatment of the System n-alkane/linear-polyethylene. Colloid Polym. Sci. 1980, 258, 711-718. Koak, N.; Heidemann, R. A. The Correlation of Thermodynamic Properties of Polymer-Water Systems with Flory-Huggins Theory. Paper presented at the 44th Canadian Chemical Engineering Conference, 1994. Koningsveld, R.; Staverman, A. J. Liquid-Liquid Phase Separation in Multicomponent Polymer Solutions. I. Statement of the Problem and Description of Methods of Calculation. J. Polym. Sci., Polym. Chem. Ed. 1968, 6, 305-323. Lacombe, R. H.; Sanchez, I. C. Statistical Thermodynamics of Fluid Mixtures, J. Phys. Chem. 1976, 80, 2568-2580. Michelsen, M. L. The Isothermal Flash Problem. I: Stability. Fluid Phase Equilib. 1982, 9, 1-20. Michelsen, M. L. Calculations of Multi-Phase Equilibria. Comput. Chem. Eng. 1994, 60, 47-58. Peng, D.-Y.; Robinson, D. B. A New Two Constant Equation of State. Ind. Eng. Chem. Fundam. 1976, 15, 59-68. Prausnitz, J. M.; Lichtenthaler, R. N.; de Azevedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria; Prentice-Hall Inc.: Englewood Cliffs, NJ, 1986. Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory

of Classical Fluids. Pure Fluids. J. Phys. Chem. 1976, 80, 23522362. Sanchez, I. C.; Lacombe, R. H. Statistical Thermodynamics of Polymer Solutions. Macromolecules 1978, 11, 1145-1156. Sˇ olc, K. Cloud-Point Curves of Polymer Solutions. Macromolecules 1970, 3, 665-673. Song, Y.; Lambert, S. M.; Prausnitz, J. M. A Perturbed HardSphere Equation of State for Normal Fluids and Polymers. Ind. Eng. Chem. Res. 1994a, 33, 1047-1057. Song, Y.; Lambert, S. M.; Prausnitz, J. M. Liquid-Liquid Phase Diagrams for Binary Polymer Solutions from a Perturbed HardSphere-Chain Equation of State. Chem. Eng. Sci. 1994b, 49, 2765-2775. Song, Y.; Hino, T.; Lambert, S. M.; Prausnitz, J. M. Liquid-Liquid Equilibria for Polymer Solutions and Blends, Including Copolymers. Fluid Phase Equilib. 1996, 117, 69-76. Wertheim, M. S. Thermodynamic Perturbation Theory of Polymerization J. Chem. Phys. 1987, 87, 7323-7331.

Resubmitted for review August 2, 1996 Revised manuscript received August 2, 1996 Accepted August 14, 1996X IE950684X X Abstract published in Advance ACS Abstracts, October 15, 1996.