Narrowly Distributed

Apr 22, 2004 - reports on the phase behavior of the system n-hexane/ polydisperse PE.9,10 To our knowledge, information on solutions of nearly ...
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Ind. Eng. Chem. Res. 2004, 43, 2852-2859

Liquid/Liquid Demixing in the System n-Hexane/Narrowly Distributed Linear Polyethylene Matthias Schnell, Sergej Stryuk, and Bernhard A. Wolf* Institut fu¨ r Physikalische Chemie und Materialwissenschaftliches Forschungszentrum, Johannes Gutenberg Universita¨ t Mainz, Jakob Welder-Weg 13, D-55099 Mainz, Germany

Demixing conditions were measured visually for solutions of three narrowly distributed polyethylene samples (M ranging from 6.5 to 380 kg/mol) in n-hexane up to 500 K and 150 bar. This information yields the critical line for infinite molar mass; i.e., it specifies the pT area within which one is safe from phase separation irrespective of the molar mass of the polymer and of the composition of the mixture. The experimental findings are in good qualitative agreement with the predictions of the Sanchez-Lacombe theory applied without any adjustable parameters. The influences of molar masses are, however, underestimated, and the resulting critical compositions are too small. For a more quantitative description, the observed phase behavior of the system was modeled by means of the Flory-Huggins theory, employing a new expression for the concentration dependence of the interaction parameter, which is based on chain connectivity and conformational relaxation. With this procedure, the required systemspecific parameters are obtained from the critical conditions measured for different M values of the polymer. Despite the fact that only the data for the two higher molecular weight samples could be used for that purpose, the calculated binodals and spinodals appear reasonable. Introduction There exist various possibilities for the technical production of polyethylene (PE). One option consists of the polymerization of ethylene in solution using low molecular weight alkanes1 as solvents. Furthermore, PE is sometimes worked up from such solutions.2-4 Because of the fact that these mixtures do not necessarily remain in the homogeneous state under all conditions of interest and that this phenomenon would normally be very adverse, detailed experimental and theoretical knowledge of the phase diagram is of great interest. Considerable research has for this reason already been performed on the phase separation of PE in alkanes of low molecular weight. The published work includes pressure experiments with solutions of PE samples of comparatively narrowly distributed PE in n-butane or n-pentane5,6 and experiments under normal pressure7,8 with solutions in n-hexane. There are also reports on the phase behavior of the system n-hexane/ polydisperse PE.9,10 To our knowledge, information on solutions of nearly monodisperse PE in n-hexane under pressure is still lacking. In this work, we present experimental data for three narrowly distributed PE samples of different molar masses and compare these findings with the predictions of the Sanchez-Lacombe theory.11 Furthermore, we test to which extent a reinterpretation of the Flory-Huggins interaction parameter in terms of chain connectivity and of conformational relaxation of polymer chains is suited for the modeling of these phase diagrams. Experimental Section 2.1. Materials. The polymer samples were synthesized by Polymer Source, Inc. (Dorval, Canada) from purified butadiene in cyclohexane using s-BuLi as the * To whom correspondence should be addressed. Fax: + 49(0)6131-39-24640. E-mail: [email protected].

Table 1. Molecular Weights and Polydispersity Indices D ) Mw/Mn of the Different PE Samples, as determined from GPC Measurements at the NOVA Chemicals Corp., Calgary, Alberta, Canada sample

Mw/(kg mol-1)

D

PE 6.5 PE 67 PE 380

6.5 67.1 382.8

1.035 1.10 1.19

initiator. The obtained polybutadiene was then hydrogenated in a p-xylene solution using the Wilkinson catalyst tris(triphenylphosphine)rhodium(I) chloride and hydrogen. These materials were purified by dissolution in o-xylene at 110 °C (PE 6.5, 5.5 wt %; PE 67, 3.8 wt %; PE 380, 0.7 wt %) and precipitation by means of 2-isopropoxyethanol at 0 °C. The molecular weights and polydispersity indices of the products studied here are collected in Table 1. The solvent n-hexane was obtained from Fluka (Riedelde Hae¨n, Germany) with a purity larger than 98% and degassed in an ultrasonic bath for 20 min. Prior to its use, argon was flushed through the solvent for 10 min. 2.2. Apparatus. Measurements were performed by means of a pressure cell with bellows. This equipment (which is analogous to the one described in detail in the literature12) can be used up to 4000 bar, where the maximum volume amounts to about 20 mL. For temperature control, we use a Eurotherm 2416 system (Eurotherm, Limburg, Germany); T is measured outside the cell with a Ni-Cr-Ni thermocouple and regulated by means of heating tapes (Isopad, Heidelberg, Germany). The accurate temperature inside the cell can be read (approximately to (0.1 °C) from a Pt-100 thermometer placed in its interior. Pressure is applied by pressing oil into the bellows and can be read on a manometer (Keller Druckmesstechnik, Jestetten, Germany) with an accuracy of (0.5%. Cloud points were determined visually through a sapphire window. 2.3. Procedures. In preparation of the measurements, the cell was flushed with argon. Then it was

10.1021/ie034302w CCC: $27.50 © 2004 American Chemical Society Published on Web 04/22/2004

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2853

filled with the required amount of polymer, oxygen-free solvent, and a nut as the stirring tool and sealed. After that, the outer parts of the apparatus were heated to 150 °C, corresponding to approximately 130 °C inside. At this temperature, the cell was brought into the vertical position and turned over by about 180° every 20-30 min for ca. 12 h to guarantee homogeneous mixing of PE and n-hexane. For determination of the cloud points, the cell was placed horizontally and the solution was illuminated through a sapphire window by means of a light guide. Because of the fact that the ranges of the demixing conditions were roughly known for all solutions, T and p could be chosen within the homogeneous region close to the onset of phase separation. Starting with the lowest temperature of interest, the pressure was lowered in small increments and the cloud-point pressure determined by the fact that the reflected light drops drastically as soon as a second phase is segregated. To avoid macroscopic demixing, the pressure is thereafter quickly raised again. The above-described procedure is then repeated at higher temperatures up to 260 °C. 3. Theoretical Background There exist several theories to model the thermodynamics of polymer solutions as a function of pressure and temperature, like the Simha-Somcynsky theory,13 the mean-field-lattice-gas theory,10,14 the Born-GreenYvon lattice theory,15,16 the statistical associating fluid theory,17-19 the perturbed chain statistical associating fluid theory,20,21 or the Sanchez-Lacombe theory.22-25 In this work, we confine the comparison of the experimental results to the predictions of the original SanchezLacombe theory and model them by means of the FloryHuggins theory, using a new expression for the composition dependence of the interaction parameter. In the following, we briefly recall the basic equations of these approaches for the discussion. 3.1. Sanchez-Lacombe. This theory22-25 treats the different pure substances as lattice fluids containing vacancies to account for density changes. The reduced equation of state reads

[

(

˜ ln(1 - F˜ ) + 1 F˜ 2 + p˜ + T

1 F˜ ) 0 r

)]

(1)

where the tildes identify reduced quantities, F stands for the density, and r stands for the number of mers a molecule consists of. Each substance is characterized by the three reduction parameters: T*, p*, and F* (close-packed mass density). The reduced parameters are given by

X ˜ ) X/X*; X ) p, υ, T, F

(2)

where υ* is the volume of a lattice site and υ the volume a mer occupies at given T and p. The following relation holds true for r:

r ) M/F*υ*NL

(3)

where NL is Avogadro’s number, and the total interaction energy per mer is given by

* ) kT* ) p*υ*

(4)

The characteristic parameters X* are usually deter-

mined by fitting eq 1 to the pVT data measured in the p and T ranges of interest. The Gibbs energy of the mixture per mer is according to the Sanchez-Lacombe theory,22,25 as reformulated by Horst,26 given as

〈 〉

{

G ) 〈*〉 -〈F˜ 〉 + 〈P ˜ υ˜ 〉 (1 - 〈F˜ 〉) ln(1 - 〈F˜ 〉) + rN φ2 φ1 F˜ ln 〈F˜ 〉 + 〈T ˜〉 ln φ1 + ln φ2 r r1 r2

[

〈〉

]

[

]}

(5)

The parameters inside the square brackets refer to binary mixtures and were in this work calculated from the values of the pure components by means of the mixing rules formulated in the literature.22,25 3.2. Chain Connectivity and Conformational Relaxation. A realistic modeling of phase diagrams, especially for variable pressure, requires rather sophisticated theories because of the deficiencies of simple approaches, like the Flory-Huggins theory. Two inadequacies of these straightforward descriptions are particularly harmful: (i) the neglect of the fact that the segments of a macromolecule must stay together upon dilution (i.e., ignoring the existence of large-volume elements containing only pure solvent) and (ii) the treatment of the polymer chain as an unchangeable entity (i.e., not accounting for any conformational changes in response to alterations in the environment or solvent quality). For that reason, we have recently rephrased this simple approach such that it accounts for the two phenomena described above.27,28 It allows the quantitative modeling of liquid/vapor29 and liquid/ liquid30 phase equilibria by means of only two physically meaningful adjustable parameters. In terms of the Flory-Huggins theory, the segment molar Gibbs energy of mixing is usually written as

φ2 ∆G C ) (1 - φ2) ln(1 - φ2) + ln φ2 + g(1 - φ2)φ2 RT N (6) where N is the number of segments, defined as its molar volume divided by the molar volume of the solvent, and g the integral interaction parameter. The present approach yields the following expression for g:

g)

R - ζ[1 + (1 - λ)φ] (1 - ν)(1 - νφ)

(7)

The parameter R quantifies the thermodynamic effect of opening a contact between polymer segments by insertion of a solvent molecule at infinite dilution (φ2 f 0) without changing the conformation of the polymer chain. This first step of the mixing process does, in general, not describe the total effect. It is only via a conformational relaxation of the polymer chain, quantified by the parameter ζ, that the equilibria are reached. Such a rearrangement of segments in response to the change in the immediate environment is only absent under Θ conditions. Under these special circumstances, ζ becomes zero and R assumes the value of 0.5. The parameters ν and λ are required to quantify the changes in g associated with an increase of the polymer concentration beyond the infinitely dilute state. The parameter ν accounts primarily for the alterations in the deviation of the entropy of mixing from combinatorial behavior. λ is defined in terms of the Kuhn-Mark-Houwink

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constants K and a (relating the intrinsic viscosity of a given polymer in a certain solvent at specified T and p values to its molar mass) as

λ)

1 + κN-(1-a) 2

(8)

The factor κ is given by

κ ) KF2

( ) F2 M F1 1

a

(9)

where Fi are the densities of the components and M1 is the molar mass of the solvent. As can be seen from the above relations, λ measures the differences in the integral interaction parameter resulting from dissimilar chain lengths. It can be calculated accurately if sufficient information on the system is available. For sufficiently high molecular weight polymers, λ results so close to the limit for infinite M that this parameter can be set equal to 0.5 without noteworthy loss of accuracy. Equations 6 and 7 enable a detailed modeling of phase diagrams if sufficient information on the variation of the critical data (φ2,c plus Tc and pc) with the molecular weight of the polymer (i.e., the number of segments N) is available. By means of the fact that the second and third derivatives of the Gibbs energy must under critical conditions become zero, it is possible to determine the system-specific parameters and their dependence on the variables of state. The procedures employed for the determination of these constants and for the calculation of the phase diagrams by a direct minimization of the Gibbs energy26 have been described in detail.30 4. Results and Discussion Qualitatively, the influences of temperature, pressure, and molar mass of the polymer on the extension of the homogeneous region are naturally the same for all solutions of high molecular weight PE in the alkanes that are liquid at room temperature.1,2,5-10,13,14,16,31-41 Because of the chemical similarity of the high molecular weight compound and the solvents, it is obvious that free-volume effects become particularly apparent. An increase in temperature or a reduction in pressure reduces the density of the solvent drastically. This is equivalent to the production of voids, into which the polymer molecules can “disappear” upon mixing. This exothermal process (formation of additional sites of favorable interaction by hole filling) is naturally associated with negative excess volumes. According to the principle of Le Chatelier Brown, the qualitative consequences of temperature and pressure changes are obvious: Homogeneous solutions will phase separate upon heating (lower critical solution temperatures) or upon a reduction of pressure (upper critical solution pressures). What is of central interest here are the quantitative dependencies of the miscibility of the components on the above variables and the possibilities to predict or at least describe them reliably by means of methods of reasonable complexity. Primary Data. The three diagrams given in Figures 1-3 display for the three PE samples of different molar masses how the cloud-point pressures for solutions of given composition depend on the temperature. In the following sections, we show how the limits of complete miscibility vary with pressure for constant

Figure 1. Cloud-point pressures as a function of temperature for solutions of PE 6.5 in n-hexane. The weight fractions of the polymer and the shift factors X (introduced to emerge the results) are indicated on the graph.

Figure 2. As in Figure 1 but for PE 67.

Figure 3. As in Figure 1 but for PE 380.

temperature and how they depend on the temperature at constant pressure. To this end, the primary data are interpolated or slightly extrapolated according to the linear isopleths shown above. Furthermore, weight fractions are converted into volume fractions (neglecting volume changes upon mixing) by means of the relations for the densities of n-hexane42 and PE43 in order to enable a comparison of the experimental finding with theoretical calculations (Table 2). Two theoretical concepts are studied here: (i) the Sanchez-Lacombe theory, which should be particularly apt to predict the phase diagrams of the present system

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2855 Table 2. Temperature Dependence of the Densities of n-Hexane and of PE at the Indicated Pressures p/bar

n-hexane

polyethylene

cm-3)

cm-3)

Fn-C6/(g ) -0.0015T (K) + 1.1545 Fn-C6/(g cm-3) ) -0.0014T (K) + 1.1146 Fn-C6/(g cm-3) ) -0.0013T (K) + 1.0741

20 40 60

FPE/(g ) -0.0006T (K) + 1.0407 FPE/(g cm-3) ) -0.0006T (K) + 1.0395 FPE/(g cm-3) ) -0.0006T (K) + 1.0383

Table 3. Critical Temperatures and Critical Compositions for the Solutions of the Different PE Samples in n-Hexane at the Indicated Pressures 20 bar

Tc/K φ2,c Tc/K φ2,c Tc/K φ2,c

40 bar 60 bar

6.5 kg mol-1

67 kg mol-1

380 kg mol-1

451 0.138 463 0.138 476 0.138

429.0 0.026 444.0 0.025 461.0 0.024

416.2 0.012 431.4 0.012 446.3 0.013

Table 4. Reduction Parameters of the Components Used for the Calculations According to the Sanchez-Lacombe Theory substance

T*/K

p*/bar

F*/(g cm-3)

PE n-hexane

646.2 516.9

4244 2470

0.905 0.745

because of the chemical similarity of the components, and (ii) the Flory-Huggins theory, using a new approach for the composition dependence of the FloryHuggins interaction parameter, accounting for conformational relaxation and chain connectivity (Table 3). The characteristic data of the pure components required for the Sanchez-Lacombe theory were calculated from published pVT data42,43 and are collected in Table 4. The system-specific parameters used for the modeling of phase diagrams by means of the Flory-Huggins theory are listed in Table 5. Isotherms. In accordance with literature reports,14 the demixing pressures for PE solutions in n-alkanes increase with rising molecular weight of the polymer. The experimental data shown in Figure 4 for 500 K demonstrate that the accuracy of the measurements is considerably higher for the two samples of larger molar mass than for the very short chain polymer. The reason lies in the diminution of the refractive index increment upon a reduction of M. This feature makes the determination of the exact cloud point considerably more uncertain. Figure 4 also shows the binodals predicted by the Sanchez-Lacombe theory; they vary less with molar mass than the measured ones, but by and large the theory matches the experimental results reasonably well. In view of the fact that no adjustable parameter is involved in these calculations, the agreement appears remarkable. Similar calculations for solutions of narrowly distributed PE in n-pentane or n-butane required an adjustable parameter for a reasonable description of the experimental data.5,34 The description of the system n-hexane/PE of broad molecular weight distribution by means of the Sanchez-Lacombe theory naturally requires a modification of the equations.37-40,44 In Figure 5, we compare the phase diagrams measured for the two higher molecular weight samples with

Figure 4. Demixing pressures as a function of composition at 500 K for the three PE samples under investigation. Solid lines are a guide for the eye only. The dashed lines represent the binodal curves predicted by the Sanchez-Lacombe theory for the different systems.

Figure 5. Demixing pressures as a function of composition at 460 K for the two higher molecular weight PE samples. The solid lines are the binodals and the dotted lines the spinodals modeled according to the Flory-Huggins theory by means of the composition dependence formulated in eq 7 and the parameters collected in Table 5.

that modeled by means of the Flory-Huggins interaction parameter accounting for chain connectivity and conformational relaxation. Because of the experimental difficulties with the lowest molecular weight sample mentioned above, it is impossible to fix the critical composition for PE 6.5 with the accuracy required for the determination of the system-specific parameters of that approach. For that reason, we have confined the evaluation to the remaining two samples and compared experiment and theory at 460 K, instead of 500 K, to avoid too much extrapolation. Despite the rather limited

Table 5. Parameters Describing the Composition Dependence of the Flory-Huggins Interaction Parameter by Means of Equation 7 as a Function of Temperature at Different Pressuresa

a

p/bar

ζ

R

ν

20 40 60

7.71578 - 0.01886T (K) 8.42097 - 0.01986T (K) 7.94996 - 0.01815T (K)

2.76758 - 0.00554T (K) 2.97572 - 0.00584T (K) 2.83711 - 0.00534T (K)

26.70095 - 0.06572T (K) 30.95961 - 0.07350T (K) 31.44714 - 0.07228T (K)

These values were obtained by adjusting them to the critical data of the two higher molecular PE samples.

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Figure 6. Demixing pressures as a function of composition for the system n-hexane/PE 380 at the different temperatures indicated on the graph. The solid lines are the binodals and the dotted lines the spinodals modeled according to the Flory-Huggins theory by means of the composition dependence formulated in eq 7 and the parameters collected in Table 5.

input data, the binodals calculated in this manner agree reasonably well. In addition to the changes in the solubility of PE resulting from differences in its molar mass discussed above, it is also of interest to know how the pressure changes the solubility of a given PE sample at different temperatures. Figure 6 shows as an example the experimental findings for the highest molecular PE, together with the modeling by means of the FloryHuggins theory. For validation of the insufficient agreement between experiment and modeling at the highest temperature, it must be kept in mind that the parameters of eq 7 were obtained from the critical points of the two higher molecular weight PE samples only and that the modeling of the pressure range beyond 60 bar requires a rather inaccurate extrapolation of these parameters. An interesting feature of the polymer solutions consists of the well-defined limit the demixing condition reaches as the polymer chains become infinitely long. Despite the deficiencies of the original Flory-Huggins theory, the mathematical relation45 following from it for the dependence of the inverse critical temperature on the molar mass of the polymer normally remains valid. In this section, we replace 1/Tc by pc in this evaluation. Figure 7 shows an example for such an extrapolation to M f ∞. The solubility limits obtained in this manner for different temperatures and pressures are particularly helpful because they mark the maximum possible extension of the two-phase area. Isobars. Figures 8 and 10 show, as an example, the experimental demixing data for 60 bar plus the corresponding predictions of the Sanchez-Lacombe theory and the modeling on the basis of the Flory-Huggins theory. According to the above graph, the quality of the predictions of the Sanchez-Lacombe theory for the temperature influences is comparable to that for the pressure effects. Again the results of changes in the chain length of the polymer are somewhat underestimated. For modeling of the phase diagrams by means of the Flory-Huggins theory and eq 7, it is, above all, the parameter ζ that decides whether a mixture is homogeneous or phase separated under given conditions. For good solvents, the dimensional relaxation is positive,

Figure 7. Generalized Shultz-Flory plot45 for the dependence of the measured critical demixing pressure on the molar mass M of the polymer. Also shown is the corresponding relation calculated by means of the Sanchez-Lacombe theory.

Figure 8. Demixing temperatures as a function of composition at 60 bar for the three PE samples under investigation. Solid lines are a guide for the eye only. The dashed lines represent the binodal curves predicted by the Sanchez-Lacombe theory for the different systems.

Figure 9. Pressure and temperature dependence of the conformational response ζ (cf. eq 7) calculated from the critical conditions measured for the two higher molecular weight PE samples.

and for marginal or bad solvents (where phase separation may but need not take place), it assumes negative values. The critical data measured for the two higher molecular weight PE samples yield the dependence of ζ on p and T given in Figure 9. The information obtained for PE 6.5 was not used for that purpose because of the uncertainty in the determination of the critical composition.

Ind. Eng. Chem. Res., Vol. 43, No. 11, 2004 2857

Figure 10. Demixing temperatures as a function of composition at 60 bar. The solid lines are the binodals, and the dotted line shows the spinodals modeled according to the Flory-Huggins theory by means of the composition dependence formulated in eq 7 and the parameters collected in Table 5 for the lowest molecular PE. The asterisk indicates the critical point of the system n-hexane/PE 6.5 used for the modeling.

Figure 11. Phase diagram of the system n-hexane/PE 380 at the indicated pressures. The solid lines are the binodals and the dotted lines the spinodals modeled according to the Flory-Huggins theory by means of the composition dependence formulated in eq 7 and the parameters collected in Table 4.

The phase diagrams depicted in Figure 10 were calculated by means of the dependencies of ζ, R, and ν on p and T (cf. Figure 9 and Table 4). This set of parameters is unable to model the comparatively low φc values estimated from the maxima of the composition dependence of the demixing pressures or minima in the demixing temperatures, as can be seen from Figure 10. Presently, it remains unclear whether this discrepancy is primarily due to experimental errors or theoretical deficiencies. To enable a direct assessment of the temperature influences on the solubility of PE in n-hexane at different pressures, an example for that dependency is given in Figure 11 for the highest molecular weight polymer. The extrapolation of the critical temperatures obtained for constant pressure to infinite molar mass of the polymer is shown in Figure 12. The limiting values for T∞c stand for the highest temperature at which PE is still completely miscible with n-hexane at 60 bar, irrespective of its chain length. Critical Line for Infinite Chain Length. For practical purposes, it is helpful to quantify the pairs of variates p and T for which one can be sure that a polymer/solvent system remains homogeneous irrespec-

Figure 12. Extrapolation of the critical temperature measured for the present PE samples to the infinite molar mass of PE (Shultz-Flory plot) for 60 bar. The broken line gives the result of the Sanchez-Lacombe theory for this dependence.

Figure 13. Pressure dependence of T∞c , the critical temperature for solutions of infinitely long PE chains in n-hexane. The area within which two liquid phases can coexist, depending on the composition of the system and the molar mass of the polymer is indicated by hatching. Full symbols represent the data extrapolated from the measured critical data. Also shown are the predictions of the Sanchez-Lacombe theory and the results of the modeling by means of the Flory-Huggins theory, where the critical conditions for infinitely long chains are characterized by the absence of conformational relaxation, i.e., by the condition ζ ) 0. The steepest line represents the vapor-pressure curve of the pure solvent. The upper triangle gives the result of Hamada,32 and the lower triangle represents the result of Kodama7 obtained from measurements under the equilibrium vapor pressure of the solvent.

tive of the molar mass of the polymer, its molecular weight distribution, and the composition of the mixture. Figure 13 shows how this dependence looks in the present case. It should be kept in mind that the critical volume fractions φ∞c , corresponding to T∞c , become zero in the present limit. If we contact a pure polymer of infinite molar mass and a pure solvent, under conditions in which two phases coexist, this means that the polymer takes up large amounts of the solvent but the solvent cannot dissolve noticeable amounts of the polymer. 5. Conclusions The critical line for infinitely high molar mass of the polymer provides the most condensed information concerning the possibility of segregation of a second liquid phase from a given solution upon the variation of

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pressure and temperature. For the present system, the experimental information (extrapolation of the critical data according to generalized Shultz-Flory plots) and the predictions of the Sanchez-Lacombe theory and of the modeling by a new approach agree remarkably well. For practical purposes, this means that one can use the information of Figure 13 to make sure that by choosing a proper combination of p and T values mixtures of PE and n-hexane do not demix. Another noteworthy feature consists of the comparatively easy and accurate modeling of binodals and spinodals if a certain minimum of experimental information on the molecular weight dependence of the critical data is available. This possibility should turn out to be particularly helpful in the context of tasks that require exact knowledge on the borderlines between the metastable and the unstable states of a system, like membrane making. Acknowledgment We are very grateful to NOVA Chemicals Corp. (Calgary, Alberta, Canada) for funding the research, donating the polymer samples, and doing some of the characterization. Literature Cited (1) Cheluget, E. L.; Bokis, C. P.; Wardhaugh, L.; Chen, C.-C.; Fisher, J. Modeling Polyethylene Fractionation Using the PerturbedChain Statistical Associating Fluid Theory Equation of State. Ind. Eng. Chem. Res. 2002, 41, 968. (2) Sang-Do, Y.; Kang, I.-S.; Kiran, E. Critical Polymer Concentrations of Polyethylene Solutions in Pentane. J. Chem. Eng. Data 2002, 47, 571. (3) Spahl, R.; Luft, G. Fraktionierungserscheinungen bei der Entmischung von Ethylen-Polyethylen-Gemischen. Angew. Makromol. Chem. 1983, 115, 87. (4) Folie, B.; Radosz, M. Phase Equilibria in High-Pressure Polyethylene Technology. Ind. Eng. Chem. Res. 1995, 34, 1501. (5) Xiong, Y.; Kiran, E. Comparison of Sanchez-Lacombe and SAFT Model in Predicting Solubility of Polyethylene in HighPressure Fluids. J. Appl. Polym. Sci. 1995, 55, 1805. (6) Liu, K.; Kiran, E. Pressure-Induced Phase Separation in Polymer Solutions: Kinetics of Phase Separation and Crossover from Nucleation and Growth to Spinodal Decomposition in Solutions of Polyethylene in n-Pentane. Macromolecules 2001, 34, 3060. (7) Kodama, Y.; Swinton, F. L. Lower Critical Solution Temperatures, Part I Polymethylene in n-Alkanes. Br. Polym. J. 1978, 10, 191. (8) Hamada, F.; Fujisawa, K.; Nakajima, A. Lower Critical Solution Temperature in Linear Polyethylene-n-Alkane Systems. Polym. J. 1973, 4, 316. (9) de Loos, Th. W.; de Graaf, L. J.; de Swaan Arons, J. Liquidliquid-phase separation in linear low-density polyethylene-solvent systems. Fluid Phase Equilib. 1996, 117, 40. (10) 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 behaviour of the system n-hexanepolyethylene: experimental results and predictions with the meanfield lattice-gas model. Chem. Eng. Sci. 1990, 45, 1875. (11) Schnell, M. Phasenverhalten von Polyethylen in Mischsystemen. Ph.D. Dissertation, Johannes Gutenberg Universita¨t, Mainz, Germany, 2003. (12) Horst, R.; Wolf, B. A.; Kinzl, M.; Luft, G.; Folie, B. Shear influences on the solubility of LDPE in ethene. J. Supercrit. Fluids 1998, 14, 49. (13) Nies, E.; Simha, R.; Jain, R. K. LCST phase behavior according to the Simha-Somcynsky theory: application to the n-hexane-polyethylene system. Colloid Polym. Sci. 1990, 268, 731. (14) Kleintjens, L. A.; Koningsveld, R. Liquid-liquid-phase separation in multicomponent polymer systems XIX. Mean-field lattice-gas treatment of the system n-alkane/linear-polyethylene. Colloid Polym. Sci. 1980, 258, 711.

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Received for review December 12, 2003 Revised manuscript received February 18, 2004 Accepted February 26, 2004 IE034302W