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Modeling of Liquid-Liquid-Phase Separation in Linear Low-Density Polyethylene-Solvent Systems Using the Statistical Associating Fluid Theory Equation of State Prasanna K. Jog,† Walter G. Chapman,*,† Sumnesh K. Gupta,‡ and Robert D. Swindoll§ Department of Chemical Engineering, Rice University, Houston, Texas 77005, The Dow Chemical Company, Midland, Michigan 48667, The Dow Chemical Company, Freeport, Texas 77541
The statistical associating fluid theory (SAFT) is used to model liquid-liquid-phase equilibria in solutions of linear low-density polyethylene (LLDPE) with hexane, heptane, and octane. The effect of temperature, pressure, polymer concentration, and polymer molecular weight on phase separation is studied. Finally, the effect of polydispersity on cloud point is also considered. SAFT results are compared with experimental data by de Loos et al. (Fluid Phase Equilib. 1996, 117, 40). SAFT can model the phase behavior of the polymer in different solvents at various state conditions with a single adjustable parameter. For a monodisperse polymer in hexane, the critical polymer concentration is linear with respect to the reciprocal of the square root of molecular weight. Calculations with a polydisperse polymer show that the cloud-point curve for a polydisperse polymer differs qualitatively from that of a monodisperse polymer at low polymer concentration. As the polymer concentration is decreased below the critical polymer concentration, the cloud-point pressures decrease for a monodisperse polymer. SAFT predicts that the cloudpoint pressures for a polydisperse polymer continue to increase in qualitative agreement with experimental data. Thus, SAFT can be used to predict the phase behavior of LLDPE in different hydrocarbon solvents. 1. Introduction The phase behavior of polymer solutions is an important industrial and scientific problem. The pressure at which a polymer solution of given composition, polymer molecular weight distribution, and temperature just starts forming a second liquid phase is called the cloudpoint pressure. The locus of cloud-point pressures with varying polymer composition at constant temperature is called the cloud-point curve. The locus of compositions of the incipient liquid phase at the cloud-point pressure is called the shadow curve. The cloud-point and shadow curves are different because the polymer is polydisperse. The polymer in the coexisting (shadow) phase has a molecular weight distribution different from that of the original polymer solution. de Loos et al.1 reported experimental data for the cloud-point curve of linear low-density polyethylene (LLDPE) in three different alkane solvents, viz., hexane, heptane, and octane. In this work we model the polyethylene-solvent phase behavior with hexane, heptane, and octane as solvents using the statistical associating fluid theory (SAFT). The phase behavior of polyolefins with lower hydrocarbon solvents such as propylene, butene, and hexene has been studied by Chen et al.2,3 The cloud-point behavior of polyethylene with higher alkanes such as hexane, heptane, and octane has not been modeled using SAFT, prior to this work. SAFT is a statistical mechanics based equation of state based on the thermodynamic perturbation theory of Wertheim.4-7 Chapman et al.8,9 generalized Wertheim’s theory for associating polyatomic †
Rice University. The Dow Chemical Co., Midland, MI. § The Dow Chemical Co., Freeport, TX. ‡
fluids and derived an engineering equation of state called SAFT. SAFT views a molecule as a chain of tangentially connected spheres. The Helmholtz free energy of a fluid of chainlike molecules is written as a perturbation series:
A ) Aseg + Achain + Aassoc
(1)
where Aseg is the segment free energy, Achain is the chain contribution, and Aassoc is the change in free energy due to association. In the present work we are dealing with nonassociating fluids, so the association contribution will be zero. The segment free energy is written as
Aseg ) Ahs + Adisp
(2)
where Ahs is the hard-sphere contribution and Adisp is the dispersion contribution. Huang and Radosz introduced the Chen-Kreglewski equation of state for the segment term in SAFT. They fitted the parameters for a variety of pure fluids10 and mixtures.11 The phase equilibrium calculations are performed by equating chemical potentials of each component in each phase and by equating the pressures of all of the phases. This allows us to do bubble-point calculations. SAFT has been shown to be a good model for polymer solution phase behavior.3,12-14 2. Polymer Characterization and SAFT Parameters The polymer considered is polyethylene with octene-1 comonomer. The comonomer content was not specified for the sample studied in this work. The molecular
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Table 1. Molecular Characterization of Polyethylene type of polymer comonomer Mn (kg/mol) Mw (kg/mol) density (g/mL) LLDPE
octene-1
33
124
0.920
Table 2. SAFT Parameters for Polyethylene and Solvents polymer
Mn (g/mol)
v00 (mL/mol)
u0/k (K)
M
polyethylene n-hexane n-heptane n-octane
33000 86.178 100.205 114.232
12.0 12.475 12.282 12.234
216.15 202.72 204.61 206.03
0.05096Mn 4.724 5.391 6.045
Figure 2. Cloud-point isobars for the poly(ethylene-octene) (Mn ) 33 000) + hexane system from experiment1 points and SAFT (curves).
Figure 1. Cloud-point isopleths for poly(ethylene-octene) (Mn ) 33 000) in hexane from experiment1 points and SAFT (curves).
characterization of the polymer is given in Table 1. For SAFT calculations the polymer is modeled as a homonuclear chain of segments connected tangentially. The Huang and Radosz correlation for polyethylene is used for SAFT parameters of the polymer independent of the comonomer concentration. Thus, for the cloud-point calculations, we have treated the copolymer as a homopolymer (polyethylene) and the octene-1 comonomer concentration is not required. The polymer is treated initially as a fluid of monodisperse chains of molecular weight equal to the reported number-average molecular weight. The parameters for the polymer and solvents are summarized in Table 2. 3. Polyethylene in an n-Hexane Solvent SAFT can correlate the effect of the polymer concentration, molecular weight, and temperature on the cloud-point pressure of polymer solutions. Figure 1 shows the results of experiment (points) and SAFT (curves) for the cloud-point pressure of polyethylene in n-hexane versus temperature at a constant composition of the polymer. The curves so obtained are called isopleths (constant composition). The same binary interaction parameter k12 is used at all temperatures and compositions. The experimental data were presented by de Loos et al.1 as plots and were extracted using a data digitizer. From Figure 1 we see that the results are best at intermediate compositions (7.7-20.05% of the polymer). At the lowest (3.77%) and highest (25.81%) compositions, SAFT qualitatively predicts the correct trends. Figures 2 and 3 show constant pressure and constant temperature cuts of the cloud-point envelope, respectively. In this case, polymer composition is a variable. SAFT predictions are in very good agreement with the experimental results.
Figure 3. Cloud-point isotherms for the poly(ethylene-octene) (Mn ) 33 000) + hexane system from experiment1 points and SAFT (curves).
4. Effect of Solvent On the basis of the correlation of LLDPE in hexane, SAFT can predict the cloud-point curve in other solvents. Figure 4 shows the isobaric cloud-point curve from experiment (points) and SAFT (curves) for LLDPE in n-hexane, n-heptane, and n-octane. As before, HuangRadosz parameters were used for the solvents. The polymer unary parameters were the same as those in the case of n-hexane. The binary interaction parameter for the LLPDE-n-hexane system was used for nheptane and n-octane solvents. From Figure 4, it can be seen that SAFT accurately predicts the effect of solvent on the cloud-point behavior. It can be seen that all of these systems show lower critical solution temperature (LCST) behavior. In systems showing LCST behavior, the solution splits into two phases at higher temperatures and becomes one phase at lower temperatures.
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Figure 4. Effect of solvent on the cloud-point temperature of poly(ethylene-octene) (Mn ) 33 000 and Mw ) 124 000) at a constant pressure of 3 MPa. Solvents are hexane, heptane, and octane. Experimental data by de Loos et al.1
Figure 6. Effect of molecular weight on the critical polymer composition (weight %) of poly(ethylene-octene) + hexane at 450 K from SAFT (points) and a scaling equation (curve).
root of molecular weight for the upper critical solution temperature. In our case of lower critical solution temperature, the approximate critical weight fraction of polymer was calculated from SAFT for the polyethylene-n-hexane system at T ) 450 K at different molecular weights of polyethylene. The critical weight fraction was determined by extrapolating the phase envelope near the critical point. The critical point can be determined more rigorously by using conditions on derivatives of free energy or pressure, but for the purpose of demonstration of the molecular weight effect on the critical point, the aforementioned approximate method is sufficient. From Figure 6 it can be seen that the critical weight fraction of the polymer follows a scaling law with respect to molecular weight:
wc ) AM-1/2 + B Figure 5. Effect of molecular weight at constant polymer concentration on the cloud-point temperature of the poly(ethyleneoctene) (Mn ) 33 000 and Mw ) 124 000) + heptane system at P ) 5 MPa from experiment1 points and SAFT (curves).
5. Effect of Polymer Molecular Weight on Phase Behavior The effect of polymer molecular weight on cloud-point temperature at fixed pressure and polymer mass fraction is investigated for a solution of polyethylene in n-heptane. The SAFT results in Figure 5 show that the cloud-point temperature decreases with increasing molecular weight until it reaches a plateau at high enough molecular weight. The SAFT results are in qualitative agreement with the experimental data. In SAFT calculations, the polymer was treated as monodisperse and only molecular weight was varied, whereas for experimental data, the samples were polydisperse and the polydispersity index (ratio of weight-average molecular weight Mw to Mn) is different for different data points. As shown in section 6, polydispersity affects the phase behavior of polymer solutions. Thus, polydispersity is partly responsible for the deviation between experimental data and SAFT predictions. The predicted molecular weight dependence can be compared with that predicted by other models. It has been shown from Flory-Huggins theory15 that the critical polymer weight fraction scales with the square
(3)
at constant temperature, where wc is the critical polymer weight fraction in the solution, M is the polymer molecular weight, and A and B are constants dependent on the polymer and solvent. In Figure 6, A ) 555.1 and B ) 0.2326. 6. Effect of Polydispersity The experimental data for an isothermal cloud point shows that the cloud-point pressure continues to increase as the concentration decreases. On the contrary, SAFT shows a critical point, and the pressure at the inception of two-phase equilibrium decreases as the polymer concentration goes below the critical concentration as shown in Figure 7. This is shown by the shadow curve for the monodisperse case (dotted line in Figure 7). As discussed in the previous section, the critical concentration depends on the molecular weight of the polymer. Because the polymer sample used in the experimental study was polydisperse, it is necessary to model the polydisperse polymer in SAFT calculations particularly at low polymer concentration. The polydispersity of the polymer is accounted for by using pseudocomponents to represent different molecular weight fractions. Thus, the polymer is treated as a mixture of chains with the same segment volume parameter (v00) and dispersion energy parameter (u0/ k) differing only in the chain-length parameter (m).
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Figure 7. Cloud-point and shadow curves for poly(ethyleneoctene) + hexane at 450 K from experiment1 points and SAFT (curves). The dotted curve shows the composition of the incipient phase at the cloud point. Results for monodisperse and polydisperse polymers are included. Table 3. Molecular Weight Distribution of Polyethylene on Solvent Free Basis component no.
mol wt
1 2 3 4 5 6 7 8 9 10
63 276 976 3152 9744 29 742 91 957 296 774 1 049 116 4 562 825
weight fraction mole fraction (solvent-free basis) (solvent-free basis) 8.30 × 10-9 6.35 × 10-6 5.66 × 10-4 1.29 × 10-2 1.02 × 10-1 3.11 × 10-1 3.78 × 10-1 1.72 × 10-1 2.41 × 10-2 5.96 × 10-4 Mw ) 12 399.4
4.31 × 10-6 7.58 × 10-4 1.91 × 10-2 1.35 × 10-1 3.45 × 10-1 3.45 × 10-1 1.35 × 10-1 1.91 × 10-2 7.58 × 10-4 4.31 × 10-6 Mn ) 33 000
As suggested by Huang and Radosz,10 the chainlength parameter is assumed to be directly proportional to the molecular weight of the chain. The proportionality constant used here is the same as that used in calculations for monodisperse polymer previously in this work. The binary interaction parameter kij is assumed to be the same between each pseudocomponent and the solvent. Between the pseudocomponents, kij is assumed to be zero. The detailed molecular weight distribution was not reported for the polymer sample.1 The only information given was the number-average molecular weight (Mn) and the weight-average molecular weight (Mw). Because the log-normal distribution is widely used to represent polymer molecular weight distributions, we assumed the log-normal distribution for this study. The parameters of the distribution were calculated from the known Mn and Mw. The continuous molecular weight distribution was discretized using the Hermitian quadrature procedure.16 In this work 10 pseudocomponents were used to represent the molecular weight distribution. The resulting discrete molecular weight distribution of the polymer (on a solvent-free basis) is given in Table 3. Note that this distribution matches the actual Mn and Mw very accurately. Having discretized the polymer into a number of components, the flash problem is formulated as a multicomponent bubble-point problem to generate a P-x diagram at constant temperature. We recently devised a special flash algorithm which takes advantage of the form of the SAFT equation of state so that the computation time is independent of the number of
pseudocomponents used for discretization. The details of the algorithm will be published soon.17 The binary interaction parameter kij between all of the pseudocomponents and the solvent was set to -0.0035, which is the same as that for the monodisperse case. The results of the calculation are shown in Figure 7 as a P-x diagram. As expected the polydispersity of the polymer explains the behavior of the cloud-point curve at low polymer concentrations. Koak et al.14,18 have shown that the experimental data for nearly monodisperse polystyrene in methylcyclohexane show a decrease in the cloud-point temperature at low polymer concentration at constant pressure. It is interesting to note that the diagram shows a three-phase liquid-liquidliquid equilibrium point at a polymer weight fraction of around 3.73%. The existence of a three-phase point was suggested by the discrepancies in the calculated cloud-point pressures with increasing and decreasing polymer concentrations around the three-phase point. The actual three-phase point was determined by solving the phase equilibrium equations for the three phases. The P-x diagram has a first-order discontinuity at that concentration because of the three-phase point. The P-x diagram generated from SAFT is in qualitative agreement with the experimental results especially at low polymer concentrations. Figure 7 shows that, in the region below the threephase point, the polymer concentration in the shadow phase is less than the polymer concentration in the starting phase. The opposite occurs in the region above the three-phase point. One can also expect differences in the molecular weight distribution of the polymer in these two phases. Consider the case of 4.3 MPa pressure. This point is below the three-phase pressure. The polymer in the shadow phase has Mn ) 8314 g/mol and Mw ) 14 305 g/mol (whereas the polymer in the starting phase has Mn ) 33 000 g/mol and Mw ) 124 000 g/mol as shown in Table 1). The opposite happens for the case of 8 MPa pressure, which is above the three-phase pressure. The polymer in the shadow phase has Mn ) 125 979 g/mol and Mw ) 3 147 889 g/mol. In both cases, the molecular weight averages in any phase appear to be related to the concentration of polymer in that phase. Details of these molecular weight distributions are provided in another publication.17 7. Conclusions The SAFT equation of state was used to model the phase behavior of LLDPE in different n-alkane solvents. SAFT successfully modeled the effect of temperature, pressure, and polymer concentration on the cloud point of the polymer solution. Although SAFT calculations are sensitive to the binary interaction parameter, a constant value of the binary interaction parameter was used to model the cloud point at different state conditions (temperature, pressure, and polymer concentration) and also different solvents. The effect of polymer molecular weight was also predicted qualitatively. Thus, the SAFT model is a good predictive tool for this system. The monodisperse polymer model could not account for the phase behavior of the polyethylene-hexane solution at low polymer concentration. The polydisperse polymer model using discrete pseudocomponents qualitatively agrees with the trend in experimental cloud points at low polymer concentration without fitting any additional parameters. SAFT rigorously accounts for chain effects using statistical mechanics. In the case of polymer
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solutions, chain effects are dominant in determining the phase behavior and hence SAFT is a good predictive model for polymer solution phase equilibria. Acknowledgment We thank The Robert A. Welch Foundation and The Dow Chemical Co. for generous financial support. Some of us (P.K.J., S.K.G., and R.D.S.) were clients of Hasan Orbey and will always be grateful for his friendship and support. Literature Cited (1) de Loos, T. W.; de Graaf, L. J.; de Swaan Arons, J. Liquidliquid-Phase Separation in Linear Low-Density PolyethyleneSolvent Systems. Fluid Phase Equilib. 1996, 117, 40. (2) Chen, S.-J.; Radosz, M. Density-tuned polyolefin phase equilibria. 1. Binary solutions of alternating poly(ethylenepropylene) in subcritical and supercritical propylene, 1-butene, and 1-hexene. Experiment and Flory-Patterson model. Macromolecules 1992, 25, 3089. (3) Chen, S.-J.; Economou, I. G.; Radosz, M. Density-Tuned Polyolefin Phase Equilibria. 2. Multicomponent Solutions of Alternating Poly(ethylene-propylene) in Subcritical and Supercritical Olefins. Experimental and SAFT model. Macromolecules 1992, 25, 4987. (4) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. I. Statistical Thermodynamics. J. Stat. Phys. 1984, 35, 19. (5) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. II. Thermodynamic Perturbation Theory and Integral Equations. J. Stat. Phys. 1984, 35, 35. (6) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. III. Multiple Attraction Sites. J. Stat. Phys. 1986, 42, 459. (7) Wertheim, M. S. Fluids with Highly Directional Attractive Forces. IV. Equilibrium Polymerization. J. Stat. Phys. 1986, 42, 477. (8) Chapman, W. G.; Jackson, G.; Gubbins, K. E. Phase equilibria of associating fluids. Chain molecules with multiple bonding sites. Mol. Phys. 1988, 65, 1057.
(9) Chapman, W. G.; Jackson, G.; Gubbins, K. E.; Radosz, M. New reference equation of state for associating liquids. Ind. Eng. Chem. Res. 1990, 29, 1709. (10) Huang, S. H.; Radosz, M. Equation of State for Small, Large, Polydisperse, and Associating Molecules. Ind. Eng. Chem. Res. 1990, 29, 2284. (11) 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. (12) Gregg, C. J.; Stein, F. P.; Radosz, M. Phase behavior of binary ethylene-propylene copolymer solutions in sub and supercritical ethylene and propylene. Fluid Phase Equilib. 1993, 83, 375. (13) Gregg, C. J.; Stein, F. P.; Radosz, M. Phase Behavior of Telechelic Polyisobutylene (PIB) in Subcritical and Supercritical Fluids. 2. PIB Size, Solvent Polarity, and Inter- and IntraAssociation Effects for Blank, Monohydroxy, and Dihydroxy PIB(11K) in Ethane, Propane, Carbon Dioxide, and Dimethyl Ether. Macromolecules 1994, 27, 4981. (14) Koak, N.; Visser, R. M.; de Loos, T. W. High-pressure phase behavior of the systems polyethylene + ethylene and polybutene + 1-butene. Fluid Phase Equilib. 1999, 158-160, 835. (15) Prausnitz, J. M.; Rudiger, N. L.; Azevedo, E. G. Molecular Thermodynamics of Fluid Phase Equlibria; Prentice-Hall: Englewood Cliffs, NJ, 1999. (16) Koak, N. Some Computational and Experimental Aspects of Polymer Solution Phase Behavior; Department of Chemical Engineering, The University of Calgary: Calgary, Alberta, Canada, 1997. (17) Jog, P. K.; Chapman, W. G. An Algorithm for Calculation of Phase Equilibria iIn Polydisperse Polymer Solutions Using the SAFT Equation of State. Macromolecules 2001, submitted. (18) Koak, N.; de Loos, T. W.; Heidemann, R. A. Upper-CriticalSolution-Temperature Behavior of the System Polystyrene + Methylcyclohexane. Influence of CO2 on the Liquid-Liquid Equilibria. Fluid Phase Equilib. 1998, 145, 311.
Received for review June 23, 2000 Revised manuscript received March 27, 2001 Accepted March 27, 2001 IE000604B
