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High pressure phase equilibria in the systems linear low density polyethylene + n-hexane and linear low density polyethylene + n-hexane + ethylene: Experimental results and modelling with the Sanchez-Lacombe equation of state
J. of Supercritical Fluids 37 (2006) 115–124
High pressure phase equilibria in the systems linear low density polyethylene + n-hexane and linear low density polyethylene + n-hexane + ethylene: Experimental results and modelling with the Sanchez-Lacombe equation of state I. Nagy a , Th.W. de Loos a,∗ , R.A. Krenz b , R.A. Heidemann b a
Delft University of Technology, Department of Chemical Technology, Physical Chemistry and Molecular Thermodynamics, Julianalaan 136, 2628 BL Delft, The Netherlands b University of Calgary, Department of Chemical and Petroleum Engineering, 2500 University Drive NW, Calgary, Alta., Canada T2N 1N4 Received 30 March 2005; received in revised form 1 August 2005; accepted 24 August 2005
Abstract Cloud point isopleths, bubble-point isopleths and liquid–liquid–vapour bubble point isopleths were measured for a binary system of linear low density polyethylene (LLDPE) and n-hexane and for the ternary system LLDPE + n-hexane + ethylene. The experiments were performed according to the synthetic method in the temperature range 400–500 K and at pressures up to 14 MPa. The LLDPE used was a hydrogenated polybutadiene and was almost monodisperse (Mw /Mn = 1.19). Measured experimental data for the system LLDPE + n-hexane and experimental data for the system LLDPE + ethylene taken from literature [H. Trumpi, Th.W. de Loos, R.A. Krenz, R.A. Heidemann, High pressure phase equilibria in the system linear low density polyethylene + ethylene: experimental results and modeling, J. Supercrit. Fluids 27 (2003) 205–214.] were modelled with the modified Sanchez-Lacombe equation of state. The same LLDPE sample was used in both experiments. The parameters for LLDPE were found by performing a sequence of non-linear regressions on pressure–volume–temperature reference data for molten polyethylene and the experimental cloud point data for the systems LLDPE + n-hexane and LLDPE + ethylene. From this information and a Sanchez-Lacombe fit to nhexane + ethylene data the phase behaviour of the ternary system LLDPE + n-hexane + ethylene can be predicted. Using this procedure the influence of the ethylene concentration on the cloud point pressure is slightly under predicted. Therefore, the LLDPE–ethylene binary interaction parameter was adjusted to ternary LLDPE + hexane + ethylene cloud point data. In this way the modified Sanchez-Lacombe equation gives a very good description of the ternary cloud point curves and an almost quantitative prediction of the ternary bubble point and liquid–liquid–vapour boundary curves. © 2005 Elsevier B.V. All rights reserved. Keywords: Cloud point; Bubble point; Liquid–liquid–vapour phase boundary; Linear low density polyethylene; LLDPE; n-Hexane; Ethylene; Sanchez-Lacombe equation of state
1. Introduction LLDPE is produced by solution copolymerization of ethylene and a 1-alkene in a hydrocarbon solvent. This reaction is commonly performed in the one-phase fluid region where the polymer and the monomers are dissolved in the solvent. For this, the pressure must be high enough to keep the reaction mixture in a single phase at the reaction temperature. The
∗
Corresponding author. Tel.: +31 15 2788478; fax: +31 15 2788713. E-mail address: [email protected] (Th.W. de Loos).
0896-8446/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.supflu.2005.08.004
low-temperature boundary of the one-phase fluid region is determined by the solidification of the LLDPE. The high-temperature boundary, the cloud point curve, is determined by the onset of a liquid–liquid phase split, characterized by a lower critical solution temperature (LCST). For some process variants also the location of the liquid–liquid–vapour boundary is of importance for the separation step. Kennis et al. [1] showed that the addition of nitrogen to a polyethylene + n-hexane system lowers the solvent power for the polymer and shifts the LCST to lower temperatures and higher pressures. This means that the supercritical gas acts as an anti-solvent. Similar effects in other polymer + solvent systems
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with other low molecular weight supercritical fluids have been found [2–6]. de Loos et al. [4] studied the phase behaviour of different LLDPE samples with n-hexane, n-heptane, n-octane, cyclohexane and 2-methyl-pentane. For these systems LCSTtype phase behaviour was also found. These authors showed that the addition of ethylene to a solution of 10 wt.% poly(ethyleneco-1-octene) in n-heptane lowers the lower solution temperature (cloud point) with approximately 14 K/wt.% ethylene added. Various equations of state have been proposed and modified to predict polymer–solvent phase behaviour. Jog et al. [7] used the SAFT equation of state [8] to describe the phase behaviour of the LLDPE + solvent systems investigated by de Loos et al. [4]. The same equation of state was used by ter Horst et al. [6] to model the influence of the addition of supercritical gases on the phase behaviour of systems of polyethylene + cyclohexane and polystyrene + cyclohexane. The Sanchez-Lacombe equation of state [9,10], which is used in this work, was used by Gauter and Heidemann [11] to model the phase behaviour of the systems polyethylene + n-hexane and polyethylene + ethylene. Trumpi et al. [12] used the Sanchez-Lacombe equation of state to describe the phase behaviour of a LLDPE + ethylene system. The Sanchez-Lacombe equation was used in the modeling reported in this manuscript because it is amongst the simplest of the equations that accounts for large differences in molecular chain lengths and because of its successful use in the cited manuscripts. In this paper, experimental cloud point data and bubble point data are presented for mixtures of an LLDPE + n-hexane or n-hexane + ethylene, using the same LLDPE sample that was used by Trumpi et al. [12] to study the phase behaviour of LLDPE + ethylene. The experimental data on these binary LLDPE systems have been fitted using the modified SanchezLacombe equation of state. In the data fitting, the parameters for the polyethylene and a temperature dependent interaction parameter have been adjusted. Additional data on the system n-hexane + ethylene were fitted to obtain the binary interaction parameter of this binary subsystem. The resulting parameters are used to predict the influence of the addition of ethylene on the phase behaviour of LLDPE + n-hexane. 2. Experimental The hexane used had a minimum purity of 99.5% (Fluka puriss p.a.) and was stored over molecular sieve to remove traces of water. The mole fraction purity of the ethylene was greater than 0.9998 (Matheson Gas Products). The polyethylene sample was a hydrogenated polybutadiene (PBD 50000) that was made by DSM and is regarded as a linear low-density polyethylene (LLDPE). The number-average molar mass is Mn = 43,700 g/mol, Mw /Mn = 1.19 and the branch density is 2.05 methyl groups per 100 carbons on the main chain. Slightly different average molar masses were reported for this same polymer in ref. [12]. The SEC chromatogram for the polymer, which is shown in Fig. 1, was digitized and re-analyzed in this study. Since the differences are small, no effort was made to force exact agreement between the two sets of values. Additional
Fig. 1. SEC distribution of the polyethylene sample used: (䊉) pseudocomponents.
molecular characterization of the LLDPE sample appears in Table 1. The experiments were carried out using the so-called Cailletet apparatus according to the synthetic method. A detailed description of the apparatus and the experimental procedure is given elsewhere [13]. A sample of the mixture with known composition is confined over mercury in a narrow glass tube, mounted in a thermostatic bath. At constant temperature cloud points, vapour–liquid bubble points and vapour–liquid–liquid bubble points were measured visually by adjusting the pressure. During the experiments the temperature is maintained constant to within 0.1 K. Critical points for the system LLDPE + n-hexane were measured using a method described in literature [14]. The temperature is measured with a Pt100 resistance thermometer, which was calibrated against a standard thermometer with an accuracy of ±0.01 K. The uncertainty in the measured temperature was approximately 0.02 K. The pressure is applied hydraulically and is measured with a dead weight pressure gauge (De Wit, accuracy ±0.005 MPa). The uncertainty in the experimental bubble-point pressures is 0.01 MPa and in the cloud-point pressure 0.02 MPa. The amounts of LDDPE and n-hexane added to the sample cell are determined by weight. LLDPE and n-hexane are degassed under vacuum using a freeze–thaw technique. Ethylene is added volumetrically. A detailed description of the sample preparation and gas filling apparatus has been given by de Loos [15]. Table 1 Molecular characterization of LLDPE
Mn (g/mol) Mw (g/mol) Mz (g/mol) BD (CH3 /100C) a b
Absolutea
SECb
48,000 52,000 – 2.05
43,700 52,000 59,000 –
Mn from osmometry, Mw from light scattering measurements. Mn , Mw , and Mz , from size exclusion chromatography.
I. Nagy et al. / J. of Supercritical Fluids 37 (2006) 115–124
Fig. 2. Phase diagram for LLDPE + n-hexane. Symbols are experimental data, curves were calculated using modified Sanchez-Lacombe equation.
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Fig. 4. Phase behaviour of LLDPE + n-hexane + ethylene: influence of the addition of ethylene at LLDPE weight fraction of 0.05. Symbols are experimental data, curves were calculated using modified Sanchez-Lacombe equation.
3. Experimental results Fig. 2 shows an experimental isopleth for the system LLDPE + n-hexane at a mass fraction of LLDPE wp = 0.0923 showing a bubble-point curve (L2 + V) → L2 , a cloud point curve L2 → (L1 + L2 ) and a three phase curve L2 L1 V. Similar phase diagrams were determined for mixtures of LLDPE and nhexane with mass fractions LLDPE from 0.0005 to 0.30. These experimental results are reported in Tables 2–4. From the experimental cloud point isopleths for the system LLDPE + n-hexane isothermal cloud point curves at 450, 470 and 490 K were obtained by interpolation. These data are plotted in Fig. 3. The critical weight fraction of LLDPE was found to be (0.0666 ± 0.0012). Within the experimental uncertainty no change of the critical weight fraction with temperature could be found in the temperature range investigated. The critical points do not exactly coincide with the pressure maxima of the cloud
Fig. 3. Isothermal cloud point curves of LLDPE + n-hexane. Symbols: experiments; curves: modified Sanchez-Lacombe fit.
point isotherms as a result of the polydispersity of the polymer. The liquid–liquid critical pressure increases with increasing temperature and the system shows lower critical solution temperatures. Figs. 4–6 show the influence of the addition of ethylene on the phase behaviour of LLDPE + n-hexane. In these figures the LLDPE mass fraction is approximately constant at 0.05, 0.10 and 0.15, respectively, and the ethylene mass fraction varies from 0 to 0.03. The corresponding data are reported in Tables 5–7. The figures show that the addition of ethylene to LLDPE + n-hexane shifts the cloud point temperature to lower temperatures and higher pressures (10 K or 3 MPa/mass% ethylene). With increasing ethylene concentration also the liquid–vapour and the liquid–liquid–vapour bubble-point pressure increases.
Fig. 5. Phase behaviour of LLDPE + n-hexane + ethylene: influence of the addition of ethylene at LLDPE weight fraction of 0.10. Symbols: experiments; curves: modified Sanchez-Lacombe modeling results.
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Table 2 LLDPE + n-hexane: P–T cloud point isopleths at indicated mass fractions wp of LLDPE T (K)
P (MPa)
T (K)
1.706 2.386 3.051 3.746 4.336 4.911 5.611 6.136 6.806 7.281
440.92 445.88 450.92 455.86 460.76 466.02 470.62 475.56 480.46 485.21 490.07 494.94
wp = 0.0024 1.198 1.898 2.668 3.408 4.048 4.703 5.333 5.948 6.498 7.068 7.643 8.223
440.48 445.50 450.36 455.27 460.43 465.40 470.32 475.12 480.08 485.17 490.14 494.91
wp = 0.0049 1.572 2.322 3.017 3.662 4.402 5.052 5.652 6.242 6.867 7.437 8.017 8.547
1.223 1.973 2.738 3.453 4.103 4.798 5.428 6.023 6.648 7.228 7.788 8.353 8.913
435.86 440.94 445.88 450.77 455.64 460.68 465.64 470.62 475.53 480.52 485.32 490.24 495.24
wp = 0.0223 1.563 2.348 3.048 3.748 4.408 5.098 5.763 6.388 6.988 7.588 8.168 8.738 9.298
435.79 440.73 445.66 450.62 455.59 460.51 465.56 470.53 475.45 480.39 485.32 490.13 494.97
wp = 0.0452 1.750 2.490 3.200 3.900 4.590 5.260 5.920 6.560 7.165 7.765 8.345 8.905 9.445
1.964 2.614 3.369 4.099 4.724 5.459 6.094 6.724 7.344 7.944 8.519 9.084 9.619
435.72 440.65 445.61 450.46 455.52 460.47 465.71 470.67 475.56 480.50 485.43 490.46 495.72
wp = 0.06242 1.894 2.624 3.384 4.119 4.789 5.494 6.144 6.734 7.374 7.994 8.559 9.134 9.709
435.89 440.81 445.79 450.61 455.61 460.50 465.30 470.25 475.23 480.14 484.88 489.85 494.89
wp = 0.0654450 1.924 2.749 3.399 4.069 4.774 5.474 6.044 6.724 7.319 7.899 8.469 9.069 9.624
1.909 2.684 3.374 4.084 4.794 5.434 6.029 6.709 7.324 7.899 8.469 9.029 9.629
430.71 435.74 440.62 445.60 450.52 455.46 460.35 465.27 470.15 475.21 480.11 485.29 490.16 495.18
wp = 0.0734 0.973 1.748 2.503 3.223 3.918 4.603 5.263 5.913 6.538 7.188 7.768 8.443 9.018 9.543
436.21 441.16 446.06 450.95 455.85 460.80 465.70 470.83 475.71 480.57 485.49 490.59 495.31
wp = 0.0826 1.814 2.564 3.279 3.989 4.664 5.329 5.969 6.624 7.229 7.834 8.414 8.989 9.529
wp = 0.0005 450.65 455.60 460.52 465.47 470.26 475.12 480.14 485.02 490.13 495.03
wp = 0.0100 435.95 440.86 445.93 450.76 455.68 460.64 465.65 470.57 475.52 480.43 485.28 490.18 495.20 wp = 0.0606 435.90 440.70 445.68 450.67 455.37 460.64 465.45 470.40 475.34 480.24 485.23 490.26 495.15 wp = 0.0706 436.14 441.05 445.86 450.805 455.70 460.81 465.77 470.61 475.57 480.38 485.36 490.26 495.323
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa) wp = 0.0078
435.55 440.60 445.56 450.47 455.40 460.33 465.27 470.19 475.15 480.05 484.91 490.19 495.08
1.099 1.874 2.599 3.274 3.964 4.719 5.269 5.899 6.494 7.099 7.699 8.274 8.809 wp = 0.0585
435.70 440.62 445.52 450.36 460.25 465.20 470.108 475.03 480.03 485.07 490.13 495.02
435.93 440.80 445.73 450.66 455.55 460.52 465.40 470.31 475.21 480.28 485.25 490.00 495.00
2.024 2.719 3.394 4.119 5.449 6.084 6.674 7.334 7.969 8.569 9.124 9.699
wp = 0.06690 1.969 2.594 3.349 4.034 4.684 5.394 5.994 6.644 7.244 7.869 8.454 9.014 9.589 wp = 0.0923
431.30 436.40 441.11 446.17 451.05 456.00 458.55 460.95 463.34 465.91 473.36 475.73 482.45 487.53 491.14 495.73
0.985 1.79 2.475 3.21 3.905 4.59 4.94 5.255 5.575 5.905 6.9 7.175 7.99 8.585 8.98 9.485
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Table 2 (Continued ) T (K)
P (MPa)
T (K)
1.363 2.103 2.798 3.513 4.233 4.898 5.563 6.213 6.843 7.438 8.028 8.598 9.178
wp = 0.19463 435.888 0.989 440.916 1.624 446.062 2.449 451.113 3.149 455.543 3.774 460.414 4.424 465.513 5.144 470.451 5.799 475.400 6.419 480.519 7.074 485.453 7.649 490.548 8.244 495.514 8.849
wp = 0.1534 435.805 440.700 445.574 450.492 455.459 460.303 465.374 470.264 475.266 480.228 485.118 489.966 495.032
P (MPa)
4. Modelling 4.1. Polymer characterization The LLDPE sample used in the experiment is nearly monodisperse (Mw /Mn = 1.19). For the purpose of modelling, 100 pseudo-components, indicated by the symbols in Fig. 1, were generated from the SEC distribution using the anchored quadrature method. 4.2. The modified Sanchez-Lacombe equation The original Sanchez-Lacombe equation had three pure component parameters [9,10]: εi , the attractive energy parameter, v∗i , the volume occupied per mole of lattice sites, and di , the
T (K)
P (MPa)
T (K)
1.644 2.374 3.094 3.849 4.499 5.104 5.774 6.199 6.894 7.499 8.099
451.011 455.705 460.899 465.863 470.829 475.762 480.675 485.650 490.540 495.435
wp = 0.2435 445.479 450.554 455.445 460.460 465.535 470.421 475.462 480.029 484.864 489.887 494.885
P (MPa) wp = 0.3031 1.597 2.297 3.067 3.847 4.547 4.897 5.642 6.247 6.847 7.497
number of lattice sites per molecule. In this work the modified Sanchez-Lacombe equation of Neau [16] is used. The SanchezLacombe parameters for n-hexane and ethylene were calculated from the pure component critical properties and pure component vapour pressures at Tr = 0.7 using the proposal of Gauter and Heidemann [17]. For a polymer, the number of lattice sites per molecule is proportional to the molar mass; i.e., di = Mi dp . Gauter and Heidemann [11] proposed including a volume shift parameter, ci , similar to that suggested by P´eneloux et al. [18] to provide some freedom in matching polymer–solvent equilibrium data while retaining a good fit of PVT reference data for the molten polymer (Table 8). In fitting simultaneously the phase equilibrium data for LLDPE + n-hexane and the literature cloud point data for LLDPE + ethylene [12], the polymer energy parameter εp was
Table 3 LLDPE + n-hexane: P–T bubble point isopleths at indicated mass fractions wp of LLDPE T (K)
P (MPa)
T (K)
0.618 0.703 0.753 0.823 0.903 1.008
410.94 415.91 425.79 430.74 435.63
wp = 0.0049 0.652 0.707 0.827 0.907 0.992
0.614 0.679 0.749 0.819
411.65 416.54 421.46 431.31
wp = 0.0923 0.609 0.674 0.739 0.889
wp = 0.0024 411.396 416.30 421.22 426.08 430.92 436.09 wp = 0.0452 411.199 416.162 421.063 425.985
wp = 0.3031 411.50 416.36 421.27 426.22 431.21 436.10 441.10 446.00
0.667 0.752 0.837 0.907 0.972 1.047 1.147 1.252
P (MPa)
T (K)
P (MPa)
T (K)
0.652 0.707 0.747 0.827 0.992
411.58 416.42 421.41 426.12 431.12
0.613 0.678 0.753 0.828 0.898 0.988
420.924 425.802 430.584 435.601 440.595
wp = 0.0078 410.94 415.92 425.79 430.74 435.64
wp = 0.0223
wp = 0.1946 411.092 415.975 420.932 425.915 430.949 435.888
P (MPa) 0.613 0.673 0.743 0.808 0.888 wp = 0.2435 0.754 0.814 0.899 0.984 1.069
I. Nagy et al. / J. of Supercritical Fluids 37 (2006) 115–124
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Table 4 LLDPE + n-hexane: P–T liquid–liquid–vapour bubble point isopleths at indicated mass fractions wp of LLDPE T (K)
P (MPa)
T (K)
1.251 1.481 1.736 2.026 2.341
440.94 450.82 460.74 470.75 480.45 490.02
1.068 1.273 1.538 1.753 2.128 2.353
431.016 440.959 450.757 460.601 470.530 480.378 490.215
1.164 1.269 1.494
460.793 470.728 480.391
wp = 0.0005 450.58 460.52 470.32 480.3 490.07
1.068 1.278 1.603 1.758 2.053 2.363
440.62 450.62 460.54 470.46 480.31 490.20
0.899 1.079 1.274 1.499 1.759 2.049 2.364
431.31 441.38 451.33 460.96 470.67 480.46 490.32
wp = 0.0452
wp = 0.2435 445.488 450.430 460.276
T (K)
wp = 0.0024
wp = 0.0223 440.970 450.959 460.669 470.571 480.380 490.182
P (MPa)
P (MPa) wp = 0.0049 1.087 1.292 1.517 1.762 2.052 2.367 wp = 0.0923 0.89 1.075 1.275 1.49 1.74 2.03 2.355
T (K)
P (MPa) wp = 0.0078
440.61 450.62 460.54 470.46 480.32 490.20
1.081 1.320 1.519 1.782 2.057 2.371 wp = 0.1946
441.080 450.976 460.856 470.696 480.547 491.112
1.098 1.288 1.513 1.773 2.058 2.418
wp = 0.3031 1.521 1.748 2.087
regarded as variable. For a fixed value of εp , the three remaining polymer parameters, v∗i , dp and cp were found through a non-linear least squares fit of the reference PVT data of Olabisi and Simha [19] for liquid low density polyethylene (LDPE). Linear low-density polyethylene (LLDPE) is assumed to exhibit similar PVT behaviour to low density polyethylene (LDPE). The parameters used for the polymer and for the two solvents are given in Table 8. Note that polymer parameters reported in [12] were obtained by fitting the data in [12] without reference to other cloud-point data for LDPE with solvents. The new low-density polyethylene (LDPE) parameters were regressed to match the liquid–liquid critical points of many LDPE/LLDPE + hydrocarbon mixtures and are expected to be useful regardless of the branch density. Details of the data considered and the fitting method are given in [20].
Fig. 6. Phase behaviour of LLDPE + n-hexane + ethylene: influence of the addition of ethylene at LLDPE weight fraction of 0.15. Symbols: experiments; curves: modified Sanchez-Lacombe modeling results.
The binary interaction parameter between any two species was regarded as temperature dependent, in the form; kij = kija + kijb T
(1)
Six parameters were available to fit the data for a single polymer + solvent system. For the chosen value of εp the pure polymer parameters v∗p , dp , and cp parameters were fit to the LDPE PVT data and the binary parameters, kija and kijb were fit to the cloud point data for a binary polymer + solvent system. The entire procedure is repeated by varying εp until a good fit of both binary systems (LLDPE + ethylene and LLDPE + n-hexane) is achieved. The polymer is nearly monodisperse with a unimodal distribution. Cloud-point curves are not much affected by increasing the number of pseudo-components used to represent the polymer beyond the 7 that were employed in [12]. The critical point is more sensitive. Fig. 7 shows the effect of the number of polymer pseudo-components on the number-average and zaverage molar masses and on the critical polymer mass fraction. Comparison is with the values obtained using 1000 pseudocomponents. The effects of increasing the number of polymer pseudo-components above 40 are small. In calculations with many systems, 100 pseudo-components was adopted as a standard representation. The cost of computation, using methods reported in [20], increases only linearly with the number of pseudo-components. If there is sufficient information available, it is important to represent the original distribution as accurately as possible because different parts of the molar mass distribution affect different parts of the phase diagram. In the modeling of the ternary system LLDPE + nhexane + ethylene two sets of LLDPE–ethylene binary interaction parameters were used. The first set is obtained in the way described above. However, the LLDPE + ethylene phase equilibrium data [12] were obtained at lower temperature and much higher pressure than the experimental results obtained
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Table 5 LLDPE + n-hexane + ethylene: P–T cloud point isopleths at indicated composition T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
wp = 0.0502, we = 0.0118 426.28 1.846 431.16 2.566 436.16 3.276 441.26 4.096 446.14 4.721 451.05 5.421 455.97 6.096 460.90 6.746 465.75 7.396 470.74 7.996 475.64 8.571 480.51 9.176 485.46 9.726 490.27 10.271 495.18 10.841
wp = 0.0501, we = 0.0205 415.92 1.659 420.93 2.459 425.96 3.224 430.76 3.974 435.65 4.734 440.65 5.394 445.51 6.049 450.52 6.769 455.49 7.474 460.43 8.144 465.36 8.724 470.27 9.319 475.28 9.869 480.25 10.394 485.26 10.999 490.18 11.549 495.12 12.074
wp = 0.0499, we = 0.0298 411.24 2.149 416.19 3.021 421.11 3.849 426.01 4.594 430.99 5.324 435.94 6.094 440.90 6.799 445.92 7.394 450.79 8.099 455.84 8.749 460.69 9.369 465.65 9.994 470.62 10.699 475.58 11.299 480.51 11.799 485.42 12.394 490.31 12.894 495.24 13.369
wp = 0.0998, we = 0.0099 426.12 1.774 430.98 2.524 435.95 3.194 440.82 3.969 445.76 4.739 450.70 5.249 455.59 5.944 460.54 6.609 465.50 7.274 470.43 7.844 475.42 8.574 480.45 9.194 485.47 9.849 490.41 10.449 495.30 10.999
wp = 0.1007, we = 0.0196 421.26 2.449 426.08 3.144 431.05 3.949 435.99 4.719 440.73 5.449 445.90 6.199 450.91 6.844 455.88 7.524 460.64 8.224 465.67 8.834 470.72 9.484 475.58 10.094 480.54 10.694 485.56 11.249 490.41 11.819 495.45 12.374
wp = 0.1002, we = 0.0296 410.90 2.199 415.82 3.024 420.77 3.809 425.74 4.629 430.62 5.374 435.61 6.084 440.43 6.824 445.48 7.569 450.46 8.244 455.38 8.934 460.36 9.599 465.45 10.057 470.41 10.734 475.31 11.324 480.19 11.949 485.06 12.349 489.96 12.949
wp = 0.1508, we = 0.0098 430.69 2.074 435.59 2.869 440.64 3.619 445.66 4.349 450.45 5.044 455.37 5.694 460.32 6.349 465.30 6.989 470.23 7.614 475.17 8.229 480.07 8.824 484.95 9.409 489.87 9.964 494.80 10.514
wp = 0.0503, we = 0.0210 420.47 2.094 425.34 2.844 430.32 3.644 435.25 4.369 440.27 5.069 445.20 5.794 450.13 6.519 455.10 7.319 460.07 7.789 464.88 8.509 469.80 9.024 474.77 9.669 479.72 10.229 484.58 10.769 489.52 11.319 494.48 11.884
wp = 0.1506, we = 0.0293 411.03 2.074 415.95 2.874 420.84 3.749 425.82 4.444 431.00 5.209
wp = 0.1506, we = 0.0293 435.98 5.969 440.92 6.689 445.91 7.339 450.89 8.019 455.82 8.639
wp = 0.1506, we = 0.0293 460.76 9.304 465.67 9.959 470.37 10.529 475.46 11.124 480.33 11.669
wp = 0.1506, we = 0.0293 485.26 12.219 490.35 12.759 495.17 13.309
wp is the mass fraction of LLDPE, we is the mass fraction of ethylene.
in this work. So the prediction of the ternary phase behaviour involves an extrapolation of the LLDPE–ethylene binary interaction parameter. To avoid this problem the second set of LLDPE–ethylene binary interaction parameters was fitted to the ternary LLDPE + n-hexane + ethylene cloud point data using the same pure component LLDPE and ethylene parameters as in the first set. The n-hexane–ethylene binary interaction parameters were obtained from a fit to VLE data of this system [21]. 4.3. Modelling results The resulting Sanchez-Lacombe parameters for the pure components and the binary interaction parameters for the different binary systems involved are summarized in Table 8. The
Sanchez-Lacombe model provides a good fit of the two polymer solvent systems over the entire range of compositions, temperatures and pressures. The AAD of the calculated and experimental LDPE density data is 0.35%. The AAD of calculated and experimental cloud point temperatures for LLDPE + nhexane is 1.44 K and for LLDPE + ethylene using the first set of LLDPE–ethylene interaction parameters 1.54 K. The calculated results for LLDPE + n-hexane at 450, 470 and 490 K are plotted in Fig. 3. The LLDPE energy parameter, εp = 4.014 kJ/mol, is of the same magnitude as those reported for LDPE for the original SL equation by Gauter and Heidemann [11] (εp = 4.65 and 4.900 kJ/mol). The liquid–liquid critical polymer mass fractions for LLDPE + n-hexane predicted by the Sanchez-Lacombe model
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Table 6 LLDPE + n-hexane + ethylene: P–T bubble point isopleths at indicated composition T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
wp = 0.0502, we = 0.0118 411.46 1.052 416.37 1.116 421.20 1.181
wp = 0.0501, we = 0.0205 406.00 1.354 410.93 1.434
wp = 0.0499, we = 0.0298 396.49 1.529 401.10 1.604 406.30 1.699
wp = 0.0998, we = 0.0099 406.11 0.969 411.11 1.039 416.06 1.109 421.13 1.189
wp = 0.1007, we = 0.0196 401.65 1.309 406.43 1.374 411.45 1.479 416.30 1.549
wp = 0.1002, we = 0.0296 396.26 1.58 401.23 1.66 406.21 1.75
wp = 0.1508, we = 0.0098 416.15 1.139 421.11 1.219 426.07 1.299
wp = 0.1503, we = 0.0210 400.89 1.354 405.73 1.429 410.70 1.504 415.66 1.589
wp = 0.1506, we = 0.0293 396.32 1.664 401.21 1.744 406.10 1.834 wp is the mass fraction of LLDPE, we is the mass fraction of ethylene.
varies from 0.0547 at 450 K to 0.0748 at 490 K. Experimentally the critical LLDPE mass fractions in this temperature range was found at (0.0666 ± 0.0012). No trend with temperature was observed. In Fig. 8 the results of the modified Sanchez-Lacombe cloud point calculations for the ternary system LLDPE + nhexane + ethylene at 450, 470 and 490 K are plotted. The mass fraction LLDPE is 0.0983, the mass fraction ethylene varies between 0 and 0.03. The dashed curves are the results of the calculations using the LLDPE–ethylene binary interactions parameters obtained from the fit of the phase behaviour of
LLDPE + ethylene. It is clearly seen that at all temperatures the model under-predicts the effect of ethylene on the cloud point pressure. The full curves are the results of the calculations using the LLDPE–ethylene binary interactions parameters obtained from a fit to the ternary cloud point data. The quality of this fit is very good as can be seen from the figure. In Figs. 4–6 the curves are calculated with the modified Sanchez-Lacombe model and the later set of LLDPE–ethylene binary interaction parameters. These figures show that the fit of the experimental ternary cloud point data is also very good at other polymer mass fractions and that the modified Sanchez-Lacombe model gives a very good
Table 7 LLDPE + n-hexane + ethylene: P–T liquid–liquid–vapour bubble point isopleths at indicated composition T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
wp = 0.0502, we = 0.0118 431.1681 1.346 441.13 1.561 450.89 1.746 460.82 1.996 470.58 2.251 480.33 2.551 490.18 2.856
wp = 0.0501, we = 0.0205 420.93 1.588781 430.83 1.784 440.72 1.974 450.70 2.194 460.51 2.444 470.39 2.699 480.24 2.984 490.10 3.269
wp = 0.0499, we = 0.0298 411.32 1.774 421.44 1.964 431.01 2.149 440.90 2.359 450.90 2.589 460.75 2.834 470.56 3.094 480.44 3.369 490.32 3.649
wp = 0.0998, we = 0.0099 431.00 1.364 440.82 1.544 450.72 1.754 460.66 1.989 470.51 2.254 480.43 2.544 490.25 2.869
wp = 0.1007, we = 0.0196 421.34 1.629 431.27 1.804 440.74 2.009 450.63 2.234 460.65 2.474 470.39 2.734 480.38 3.024 490.29 3.319
wp = 0.1002, we = 0.0296 411.27 1.829 421.23 2.009 431.10 2.194 441.04 2.429 450.93 2.664 460.69 2.909 470.55 3.174 480.48 3.449 490.31 3.709
wp = 0.1508, we = 0.0098 430.69 1.384 440.59 1.569 450.49 1.774 460.60 2.029 470.34 2.284 480.41 2.579 490.00 2.889
wp = 0.1503, we = 0.0210 420.578 1.674 430.38 1.859 440.26 2.059 450.09 2.269 459.96 2.519 469.77 2.784 479.68 3.044 489.57 3.354
wp = 0.1506, we = 0.0293 411.03 1.899 420.84 2.084 430.89 2.299
wp = 0.1506, we = 0.0293 440.85 2.519 450.76 2.754 460.61 3.004
wp = 0.1506, we = 0.0293 470.42 3.264 480.19 3.534 490.17 3.799
wp is the mass fraction of LLDPE, we is the mass fraction of ethylene.
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Table 8 Modified Sanchez-Lacombe (MSL) parameters
Energy parameter εp (kJ/mol) Volume per lattice site v∗p (cm3 /mol) Number of lattice sites dp = di /Mi (mol/g) Volume shift parameter cp = ci /Mi (cm3 /g)
LLDPE
n-Hexane
Ethylene
4.014 8.411 0.06986 −0.5059
3.685a
2.288a 7.471a 0.2271a 0
12.550a 0.1122a 0
Binary interaction parameterb for LLDPE + ethylene kij = 2.703 × 10−2 − 7.856 × 10−5 T Binary interaction parameterc for LLDPE + ethylene kij = −5.453 × 10−2 + 2.050 × 10−4 T Binary interaction parameter for LLDPE + n-hexane kij = −1.708 × 10−1 + 3.357 × 10−4 T Binary interaction parameter for n-hexane + ethylene kij = −1.520 × 10−1 + 4.405 × 10−4 T a b c
From Gauter and Heidemann’s parameterization. From fit to LLDPE + ethylene cloud point data. From fit to LLDPE + n-hexane + (1–3 wt.%) ethylene cloud point data.
prediction of the influence of the ethylene concentration on the vapour–liquid and liquid–liquid–vapour bubble point curves. 5. Conclusions
Fig. 7. Effect of the number of pseudo-components on the number-average and z-average molar masses and on the critical polymer mass fraction. Comparison is with results using 1000 pseudo-components.
The system LLDPE + n-hexane shows a liquid–liquid phase split characterised by lower critical solution temperatures and upper critical solution pressures. The addition of ethylene to n-hexane + LLDPE shifts the cloud point curve to lower temperatures and higher pressures with approximately 10 K or 3 MPa per wt.% ethylene. The modified Sanchez-Lacombe equation of state can be used to correlate the phase behaviour of the binary subsystems of ethylene + n-hexane + LLDPE and to predict the phase behaviour of the ternary systems using parameters obtained from the fit of the binary subsystems. The description of the phase behaviour of the ternary system is improved if the binary interaction parameter of the system LLDPE + ethylene is fitted to the ternary data. Acknowledgements The research was sponsored by grants from the Natural Science and Engineering Research Council of Canada and Nova Research and Technology Corporation. The authors are thankful to Mr. W. Poot for his assistance in the phase equilibrium measurements. References
Fig. 8. Cloud point pressure of LLDPE + n-hexane + ethylene as function of the ethylene mass fraction at constant mass fraction of LLDPE of 0.0983. Symbols: experimental data; dashed curves: modified Sanchez-Lacombe predictions using LLDPE–ethylene binary interaction parameter fitted to high pressure LLDPE + ethylene data; full curves: modified Sanchez-Lacombe modeling results using LLDPE–ethylene binary interaction parameter fitted to the ternary LLDPE + n-hexane + ethylene data.
[1] H.A.J. Kennis, Th.W. de Loos, J. de Swaan Arons, R. van der Haegen, L.A. Kleintjens, The influence of nitrogen on the liquid-liquid phase behaviour of the system n-hexane–polyethylene: experimental results and predictions with the mean-field lattice-gas model, Chem. Eng. Sci. 45 (1990) 1875–1884. [2] A.J. Seckner, A.K. McClellan, M.A. McHugh, High-pressure solution behavior of the polystyrene–toluene–ethane system, AIChE J. 34 (1988) 9–16. [3] E. Kiran, W. Zhuang, Y.L. Sen, Solubility and demixing of polyethylene in supercritical binary fluid mixtures: carbon dioxide–cyclohexane, carbon dioxide–toluene, carbon dioxide–pentane, J. Appl. Polym. Sci. 47 (1993) 895–909. [4] Th.W. de Loos, L.J. de Graaf, J. de Swaan Arons, Liquid–liquid phase separation in linear low density polyethylene–solvent systems, Fluid Phase Equilib. 117 (1996) 40–47.
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[5] B. Bungert, G. Sadowski, W. Arlt, Separations and material processing in solutions with dense gases, Ind. Eng. Chem. Res. 37 (1998) 3208– 3220. [6] M.H. ter Horst, S. Behme, G. Sadowski, Th.W. de Loos, The influence of supercritical gases on the phase behavior of polystyrene–cyclohexane and polyethylene–cyclohexane systems, J. Supercrit. Fluids 23 (2002) 181–194. [7] P.K. Jog, W.G. Chapman, S.K. Gupta, R.D. Swindoll, Modeling of liquid–liquid-phase separation in linear low-density polyethylene–solvent systems using the statistical associating fluid theory equation of state, Ind. Eng. Chem. Res. 41 (2002) 887–891. [8] S.H. Huang, M. Radosz, Equation of state for small, large, polydisperse, and associating molecules: extension to fluid mixtures, Ind. Eng. Chem. Res. 30 (1991) 1994–2005. [9] I.C. Sanchez, R.H. Lacombe, An elementary molecular theory of classical fluids. Pure fluids, J. Phys. Chem. 8 (1976) 2352–2362. [10] I.C. Sanchez, R.H. Lacombe, Statistical thermodynamics of polymer solutions, Macromolecules 11 (1978) 1145–1156. [11] K. Gauter, R.A. Heidemann, Modeling polyethylene–solvent mixtures with the Sanchez–Lacombe equation, Fluid Phase Equilib. 183 (2001) 87–97. [12] H. Trumpi, Th.W. de Loos, R.A. Krenz, R.A. Heidemann, High pressure phase equilibria in the system linear low density polyethylene + ethylene: experimental results and modelling, J. Supercrit. Fluids 27 (2003) 205–214.
[13] Th.W. de Loos, H.J. van der Kooi, P.L. Ott, Vapor–liquid critical curve of the system ethane + 2-methylpropane, J. Chem. Eng. Data 31 (1986) 166–168. [14] Th.W. de Loos, W. Poot, G.A.M. Diepen, Fluid phase equilibria in the system polyethylene + ethylene. 1. Systems of linear polyethylene + ethylene at high pressure, Macromolecules 16 (1983) 111–117. [15] Th.W. de Loos, Evenwichten Tussen Flu¨ıde Fasen in Systemen van Lineair Polyetheen en Etheen (Dutch), Ph.D. Thesis, Delft University of Technology, Delft, Netherlands, 1981. [16] E. Neau, A consistent method for phase equilibrium calculation using the Sanchez–Lacombe lattice–fluid equation-of-state, Fluid Phase Equilib. 203 (2002) 133–140. [17] K. Gauter, R.A. Heidemann, A proposal for parametrizing the SanchezLacombe equation of state, Ind. Eng. Chem. Res. 39 (2000) 1115–1117. [18] A. P´eneloux, E. Rauzy, R.A. Fr´eze, A consistent correction for Redlich–Kwong–Soave volumes, Fluid Phase Equilib. 8 (1982) 7–23. [19] O. Olabisi, R. Simha, Pressure–volume–temperature studies of amorphous and crystallizable polymers. I. Experimental, Macromolecules 8 (1975) 206–210. [20] R.A. Krenz, Correlating the fluid phase behaviour of polydisperse polyethylene solutions using the modified Sanchez-Lacombe equation of state, Ph.D. Thesis, University of Calgary, Calgary, Canada, 2005. [21] I. Nagy, R.A. Krenz, R.A. Heidemann, Th.W. de Loos, Vapor–liquid equilibrium data for the ethylene + hexane system, J. Chem. Eng. Data 50 (2005) 1492–1495.
High pressure phase equilibria in the systems linear low density polyethylene + n-hexane and linear low density polyethylene + n-hexane + ethylene: Experimental results and modelling with the Sanchez-Lacombe equation of state I. Nagy a , Th.W. de Loos a,∗ , R.A. Krenz b , R.A. Heidemann b a
Delft University of Technology, Department of Chemical Technology, Physical Chemistry and Molecular Thermodynamics, Julianalaan 136, 2628 BL Delft, The Netherlands b University of Calgary, Department of Chemical and Petroleum Engineering, 2500 University Drive NW, Calgary, Alta., Canada T2N 1N4 Received 30 March 2005; received in revised form 1 August 2005; accepted 24 August 2005
Abstract Cloud point isopleths, bubble-point isopleths and liquid–liquid–vapour bubble point isopleths were measured for a binary system of linear low density polyethylene (LLDPE) and n-hexane and for the ternary system LLDPE + n-hexane + ethylene. The experiments were performed according to the synthetic method in the temperature range 400–500 K and at pressures up to 14 MPa. The LLDPE used was a hydrogenated polybutadiene and was almost monodisperse (Mw /Mn = 1.19). Measured experimental data for the system LLDPE + n-hexane and experimental data for the system LLDPE + ethylene taken from literature [H. Trumpi, Th.W. de Loos, R.A. Krenz, R.A. Heidemann, High pressure phase equilibria in the system linear low density polyethylene + ethylene: experimental results and modeling, J. Supercrit. Fluids 27 (2003) 205–214.] were modelled with the modified Sanchez-Lacombe equation of state. The same LLDPE sample was used in both experiments. The parameters for LLDPE were found by performing a sequence of non-linear regressions on pressure–volume–temperature reference data for molten polyethylene and the experimental cloud point data for the systems LLDPE + n-hexane and LLDPE + ethylene. From this information and a Sanchez-Lacombe fit to nhexane + ethylene data the phase behaviour of the ternary system LLDPE + n-hexane + ethylene can be predicted. Using this procedure the influence of the ethylene concentration on the cloud point pressure is slightly under predicted. Therefore, the LLDPE–ethylene binary interaction parameter was adjusted to ternary LLDPE + hexane + ethylene cloud point data. In this way the modified Sanchez-Lacombe equation gives a very good description of the ternary cloud point curves and an almost quantitative prediction of the ternary bubble point and liquid–liquid–vapour boundary curves. © 2005 Elsevier B.V. All rights reserved. Keywords: Cloud point; Bubble point; Liquid–liquid–vapour phase boundary; Linear low density polyethylene; LLDPE; n-Hexane; Ethylene; Sanchez-Lacombe equation of state
1. Introduction LLDPE is produced by solution copolymerization of ethylene and a 1-alkene in a hydrocarbon solvent. This reaction is commonly performed in the one-phase fluid region where the polymer and the monomers are dissolved in the solvent. For this, the pressure must be high enough to keep the reaction mixture in a single phase at the reaction temperature. The
∗
Corresponding author. Tel.: +31 15 2788478; fax: +31 15 2788713. E-mail address: [email protected] (Th.W. de Loos).
0896-8446/$ – see front matter © 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.supflu.2005.08.004
low-temperature boundary of the one-phase fluid region is determined by the solidification of the LLDPE. The high-temperature boundary, the cloud point curve, is determined by the onset of a liquid–liquid phase split, characterized by a lower critical solution temperature (LCST). For some process variants also the location of the liquid–liquid–vapour boundary is of importance for the separation step. Kennis et al. [1] showed that the addition of nitrogen to a polyethylene + n-hexane system lowers the solvent power for the polymer and shifts the LCST to lower temperatures and higher pressures. This means that the supercritical gas acts as an anti-solvent. Similar effects in other polymer + solvent systems
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with other low molecular weight supercritical fluids have been found [2–6]. de Loos et al. [4] studied the phase behaviour of different LLDPE samples with n-hexane, n-heptane, n-octane, cyclohexane and 2-methyl-pentane. For these systems LCSTtype phase behaviour was also found. These authors showed that the addition of ethylene to a solution of 10 wt.% poly(ethyleneco-1-octene) in n-heptane lowers the lower solution temperature (cloud point) with approximately 14 K/wt.% ethylene added. Various equations of state have been proposed and modified to predict polymer–solvent phase behaviour. Jog et al. [7] used the SAFT equation of state [8] to describe the phase behaviour of the LLDPE + solvent systems investigated by de Loos et al. [4]. The same equation of state was used by ter Horst et al. [6] to model the influence of the addition of supercritical gases on the phase behaviour of systems of polyethylene + cyclohexane and polystyrene + cyclohexane. The Sanchez-Lacombe equation of state [9,10], which is used in this work, was used by Gauter and Heidemann [11] to model the phase behaviour of the systems polyethylene + n-hexane and polyethylene + ethylene. Trumpi et al. [12] used the Sanchez-Lacombe equation of state to describe the phase behaviour of a LLDPE + ethylene system. The Sanchez-Lacombe equation was used in the modeling reported in this manuscript because it is amongst the simplest of the equations that accounts for large differences in molecular chain lengths and because of its successful use in the cited manuscripts. In this paper, experimental cloud point data and bubble point data are presented for mixtures of an LLDPE + n-hexane or n-hexane + ethylene, using the same LLDPE sample that was used by Trumpi et al. [12] to study the phase behaviour of LLDPE + ethylene. The experimental data on these binary LLDPE systems have been fitted using the modified SanchezLacombe equation of state. In the data fitting, the parameters for the polyethylene and a temperature dependent interaction parameter have been adjusted. Additional data on the system n-hexane + ethylene were fitted to obtain the binary interaction parameter of this binary subsystem. The resulting parameters are used to predict the influence of the addition of ethylene on the phase behaviour of LLDPE + n-hexane. 2. Experimental The hexane used had a minimum purity of 99.5% (Fluka puriss p.a.) and was stored over molecular sieve to remove traces of water. The mole fraction purity of the ethylene was greater than 0.9998 (Matheson Gas Products). The polyethylene sample was a hydrogenated polybutadiene (PBD 50000) that was made by DSM and is regarded as a linear low-density polyethylene (LLDPE). The number-average molar mass is Mn = 43,700 g/mol, Mw /Mn = 1.19 and the branch density is 2.05 methyl groups per 100 carbons on the main chain. Slightly different average molar masses were reported for this same polymer in ref. [12]. The SEC chromatogram for the polymer, which is shown in Fig. 1, was digitized and re-analyzed in this study. Since the differences are small, no effort was made to force exact agreement between the two sets of values. Additional
Fig. 1. SEC distribution of the polyethylene sample used: (䊉) pseudocomponents.
molecular characterization of the LLDPE sample appears in Table 1. The experiments were carried out using the so-called Cailletet apparatus according to the synthetic method. A detailed description of the apparatus and the experimental procedure is given elsewhere [13]. A sample of the mixture with known composition is confined over mercury in a narrow glass tube, mounted in a thermostatic bath. At constant temperature cloud points, vapour–liquid bubble points and vapour–liquid–liquid bubble points were measured visually by adjusting the pressure. During the experiments the temperature is maintained constant to within 0.1 K. Critical points for the system LLDPE + n-hexane were measured using a method described in literature [14]. The temperature is measured with a Pt100 resistance thermometer, which was calibrated against a standard thermometer with an accuracy of ±0.01 K. The uncertainty in the measured temperature was approximately 0.02 K. The pressure is applied hydraulically and is measured with a dead weight pressure gauge (De Wit, accuracy ±0.005 MPa). The uncertainty in the experimental bubble-point pressures is 0.01 MPa and in the cloud-point pressure 0.02 MPa. The amounts of LDDPE and n-hexane added to the sample cell are determined by weight. LLDPE and n-hexane are degassed under vacuum using a freeze–thaw technique. Ethylene is added volumetrically. A detailed description of the sample preparation and gas filling apparatus has been given by de Loos [15]. Table 1 Molecular characterization of LLDPE
Mn (g/mol) Mw (g/mol) Mz (g/mol) BD (CH3 /100C) a b
Absolutea
SECb
48,000 52,000 – 2.05
43,700 52,000 59,000 –
Mn from osmometry, Mw from light scattering measurements. Mn , Mw , and Mz , from size exclusion chromatography.
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Fig. 2. Phase diagram for LLDPE + n-hexane. Symbols are experimental data, curves were calculated using modified Sanchez-Lacombe equation.
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Fig. 4. Phase behaviour of LLDPE + n-hexane + ethylene: influence of the addition of ethylene at LLDPE weight fraction of 0.05. Symbols are experimental data, curves were calculated using modified Sanchez-Lacombe equation.
3. Experimental results Fig. 2 shows an experimental isopleth for the system LLDPE + n-hexane at a mass fraction of LLDPE wp = 0.0923 showing a bubble-point curve (L2 + V) → L2 , a cloud point curve L2 → (L1 + L2 ) and a three phase curve L2 L1 V. Similar phase diagrams were determined for mixtures of LLDPE and nhexane with mass fractions LLDPE from 0.0005 to 0.30. These experimental results are reported in Tables 2–4. From the experimental cloud point isopleths for the system LLDPE + n-hexane isothermal cloud point curves at 450, 470 and 490 K were obtained by interpolation. These data are plotted in Fig. 3. The critical weight fraction of LLDPE was found to be (0.0666 ± 0.0012). Within the experimental uncertainty no change of the critical weight fraction with temperature could be found in the temperature range investigated. The critical points do not exactly coincide with the pressure maxima of the cloud
Fig. 3. Isothermal cloud point curves of LLDPE + n-hexane. Symbols: experiments; curves: modified Sanchez-Lacombe fit.
point isotherms as a result of the polydispersity of the polymer. The liquid–liquid critical pressure increases with increasing temperature and the system shows lower critical solution temperatures. Figs. 4–6 show the influence of the addition of ethylene on the phase behaviour of LLDPE + n-hexane. In these figures the LLDPE mass fraction is approximately constant at 0.05, 0.10 and 0.15, respectively, and the ethylene mass fraction varies from 0 to 0.03. The corresponding data are reported in Tables 5–7. The figures show that the addition of ethylene to LLDPE + n-hexane shifts the cloud point temperature to lower temperatures and higher pressures (10 K or 3 MPa/mass% ethylene). With increasing ethylene concentration also the liquid–vapour and the liquid–liquid–vapour bubble-point pressure increases.
Fig. 5. Phase behaviour of LLDPE + n-hexane + ethylene: influence of the addition of ethylene at LLDPE weight fraction of 0.10. Symbols: experiments; curves: modified Sanchez-Lacombe modeling results.
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Table 2 LLDPE + n-hexane: P–T cloud point isopleths at indicated mass fractions wp of LLDPE T (K)
P (MPa)
T (K)
1.706 2.386 3.051 3.746 4.336 4.911 5.611 6.136 6.806 7.281
440.92 445.88 450.92 455.86 460.76 466.02 470.62 475.56 480.46 485.21 490.07 494.94
wp = 0.0024 1.198 1.898 2.668 3.408 4.048 4.703 5.333 5.948 6.498 7.068 7.643 8.223
440.48 445.50 450.36 455.27 460.43 465.40 470.32 475.12 480.08 485.17 490.14 494.91
wp = 0.0049 1.572 2.322 3.017 3.662 4.402 5.052 5.652 6.242 6.867 7.437 8.017 8.547
1.223 1.973 2.738 3.453 4.103 4.798 5.428 6.023 6.648 7.228 7.788 8.353 8.913
435.86 440.94 445.88 450.77 455.64 460.68 465.64 470.62 475.53 480.52 485.32 490.24 495.24
wp = 0.0223 1.563 2.348 3.048 3.748 4.408 5.098 5.763 6.388 6.988 7.588 8.168 8.738 9.298
435.79 440.73 445.66 450.62 455.59 460.51 465.56 470.53 475.45 480.39 485.32 490.13 494.97
wp = 0.0452 1.750 2.490 3.200 3.900 4.590 5.260 5.920 6.560 7.165 7.765 8.345 8.905 9.445
1.964 2.614 3.369 4.099 4.724 5.459 6.094 6.724 7.344 7.944 8.519 9.084 9.619
435.72 440.65 445.61 450.46 455.52 460.47 465.71 470.67 475.56 480.50 485.43 490.46 495.72
wp = 0.06242 1.894 2.624 3.384 4.119 4.789 5.494 6.144 6.734 7.374 7.994 8.559 9.134 9.709
435.89 440.81 445.79 450.61 455.61 460.50 465.30 470.25 475.23 480.14 484.88 489.85 494.89
wp = 0.0654450 1.924 2.749 3.399 4.069 4.774 5.474 6.044 6.724 7.319 7.899 8.469 9.069 9.624
1.909 2.684 3.374 4.084 4.794 5.434 6.029 6.709 7.324 7.899 8.469 9.029 9.629
430.71 435.74 440.62 445.60 450.52 455.46 460.35 465.27 470.15 475.21 480.11 485.29 490.16 495.18
wp = 0.0734 0.973 1.748 2.503 3.223 3.918 4.603 5.263 5.913 6.538 7.188 7.768 8.443 9.018 9.543
436.21 441.16 446.06 450.95 455.85 460.80 465.70 470.83 475.71 480.57 485.49 490.59 495.31
wp = 0.0826 1.814 2.564 3.279 3.989 4.664 5.329 5.969 6.624 7.229 7.834 8.414 8.989 9.529
wp = 0.0005 450.65 455.60 460.52 465.47 470.26 475.12 480.14 485.02 490.13 495.03
wp = 0.0100 435.95 440.86 445.93 450.76 455.68 460.64 465.65 470.57 475.52 480.43 485.28 490.18 495.20 wp = 0.0606 435.90 440.70 445.68 450.67 455.37 460.64 465.45 470.40 475.34 480.24 485.23 490.26 495.15 wp = 0.0706 436.14 441.05 445.86 450.805 455.70 460.81 465.77 470.61 475.57 480.38 485.36 490.26 495.323
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa) wp = 0.0078
435.55 440.60 445.56 450.47 455.40 460.33 465.27 470.19 475.15 480.05 484.91 490.19 495.08
1.099 1.874 2.599 3.274 3.964 4.719 5.269 5.899 6.494 7.099 7.699 8.274 8.809 wp = 0.0585
435.70 440.62 445.52 450.36 460.25 465.20 470.108 475.03 480.03 485.07 490.13 495.02
435.93 440.80 445.73 450.66 455.55 460.52 465.40 470.31 475.21 480.28 485.25 490.00 495.00
2.024 2.719 3.394 4.119 5.449 6.084 6.674 7.334 7.969 8.569 9.124 9.699
wp = 0.06690 1.969 2.594 3.349 4.034 4.684 5.394 5.994 6.644 7.244 7.869 8.454 9.014 9.589 wp = 0.0923
431.30 436.40 441.11 446.17 451.05 456.00 458.55 460.95 463.34 465.91 473.36 475.73 482.45 487.53 491.14 495.73
0.985 1.79 2.475 3.21 3.905 4.59 4.94 5.255 5.575 5.905 6.9 7.175 7.99 8.585 8.98 9.485
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Table 2 (Continued ) T (K)
P (MPa)
T (K)
1.363 2.103 2.798 3.513 4.233 4.898 5.563 6.213 6.843 7.438 8.028 8.598 9.178
wp = 0.19463 435.888 0.989 440.916 1.624 446.062 2.449 451.113 3.149 455.543 3.774 460.414 4.424 465.513 5.144 470.451 5.799 475.400 6.419 480.519 7.074 485.453 7.649 490.548 8.244 495.514 8.849
wp = 0.1534 435.805 440.700 445.574 450.492 455.459 460.303 465.374 470.264 475.266 480.228 485.118 489.966 495.032
P (MPa)
4. Modelling 4.1. Polymer characterization The LLDPE sample used in the experiment is nearly monodisperse (Mw /Mn = 1.19). For the purpose of modelling, 100 pseudo-components, indicated by the symbols in Fig. 1, were generated from the SEC distribution using the anchored quadrature method. 4.2. The modified Sanchez-Lacombe equation The original Sanchez-Lacombe equation had three pure component parameters [9,10]: εi , the attractive energy parameter, v∗i , the volume occupied per mole of lattice sites, and di , the
T (K)
P (MPa)
T (K)
1.644 2.374 3.094 3.849 4.499 5.104 5.774 6.199 6.894 7.499 8.099
451.011 455.705 460.899 465.863 470.829 475.762 480.675 485.650 490.540 495.435
wp = 0.2435 445.479 450.554 455.445 460.460 465.535 470.421 475.462 480.029 484.864 489.887 494.885
P (MPa) wp = 0.3031 1.597 2.297 3.067 3.847 4.547 4.897 5.642 6.247 6.847 7.497
number of lattice sites per molecule. In this work the modified Sanchez-Lacombe equation of Neau [16] is used. The SanchezLacombe parameters for n-hexane and ethylene were calculated from the pure component critical properties and pure component vapour pressures at Tr = 0.7 using the proposal of Gauter and Heidemann [17]. For a polymer, the number of lattice sites per molecule is proportional to the molar mass; i.e., di = Mi dp . Gauter and Heidemann [11] proposed including a volume shift parameter, ci , similar to that suggested by P´eneloux et al. [18] to provide some freedom in matching polymer–solvent equilibrium data while retaining a good fit of PVT reference data for the molten polymer (Table 8). In fitting simultaneously the phase equilibrium data for LLDPE + n-hexane and the literature cloud point data for LLDPE + ethylene [12], the polymer energy parameter εp was
Table 3 LLDPE + n-hexane: P–T bubble point isopleths at indicated mass fractions wp of LLDPE T (K)
P (MPa)
T (K)
0.618 0.703 0.753 0.823 0.903 1.008
410.94 415.91 425.79 430.74 435.63
wp = 0.0049 0.652 0.707 0.827 0.907 0.992
0.614 0.679 0.749 0.819
411.65 416.54 421.46 431.31
wp = 0.0923 0.609 0.674 0.739 0.889
wp = 0.0024 411.396 416.30 421.22 426.08 430.92 436.09 wp = 0.0452 411.199 416.162 421.063 425.985
wp = 0.3031 411.50 416.36 421.27 426.22 431.21 436.10 441.10 446.00
0.667 0.752 0.837 0.907 0.972 1.047 1.147 1.252
P (MPa)
T (K)
P (MPa)
T (K)
0.652 0.707 0.747 0.827 0.992
411.58 416.42 421.41 426.12 431.12
0.613 0.678 0.753 0.828 0.898 0.988
420.924 425.802 430.584 435.601 440.595
wp = 0.0078 410.94 415.92 425.79 430.74 435.64
wp = 0.0223
wp = 0.1946 411.092 415.975 420.932 425.915 430.949 435.888
P (MPa) 0.613 0.673 0.743 0.808 0.888 wp = 0.2435 0.754 0.814 0.899 0.984 1.069
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Table 4 LLDPE + n-hexane: P–T liquid–liquid–vapour bubble point isopleths at indicated mass fractions wp of LLDPE T (K)
P (MPa)
T (K)
1.251 1.481 1.736 2.026 2.341
440.94 450.82 460.74 470.75 480.45 490.02
1.068 1.273 1.538 1.753 2.128 2.353
431.016 440.959 450.757 460.601 470.530 480.378 490.215
1.164 1.269 1.494
460.793 470.728 480.391
wp = 0.0005 450.58 460.52 470.32 480.3 490.07
1.068 1.278 1.603 1.758 2.053 2.363
440.62 450.62 460.54 470.46 480.31 490.20
0.899 1.079 1.274 1.499 1.759 2.049 2.364
431.31 441.38 451.33 460.96 470.67 480.46 490.32
wp = 0.0452
wp = 0.2435 445.488 450.430 460.276
T (K)
wp = 0.0024
wp = 0.0223 440.970 450.959 460.669 470.571 480.380 490.182
P (MPa)
P (MPa) wp = 0.0049 1.087 1.292 1.517 1.762 2.052 2.367 wp = 0.0923 0.89 1.075 1.275 1.49 1.74 2.03 2.355
T (K)
P (MPa) wp = 0.0078
440.61 450.62 460.54 470.46 480.32 490.20
1.081 1.320 1.519 1.782 2.057 2.371 wp = 0.1946
441.080 450.976 460.856 470.696 480.547 491.112
1.098 1.288 1.513 1.773 2.058 2.418
wp = 0.3031 1.521 1.748 2.087
regarded as variable. For a fixed value of εp , the three remaining polymer parameters, v∗i , dp and cp were found through a non-linear least squares fit of the reference PVT data of Olabisi and Simha [19] for liquid low density polyethylene (LDPE). Linear low-density polyethylene (LLDPE) is assumed to exhibit similar PVT behaviour to low density polyethylene (LDPE). The parameters used for the polymer and for the two solvents are given in Table 8. Note that polymer parameters reported in [12] were obtained by fitting the data in [12] without reference to other cloud-point data for LDPE with solvents. The new low-density polyethylene (LDPE) parameters were regressed to match the liquid–liquid critical points of many LDPE/LLDPE + hydrocarbon mixtures and are expected to be useful regardless of the branch density. Details of the data considered and the fitting method are given in [20].
Fig. 6. Phase behaviour of LLDPE + n-hexane + ethylene: influence of the addition of ethylene at LLDPE weight fraction of 0.15. Symbols: experiments; curves: modified Sanchez-Lacombe modeling results.
The binary interaction parameter between any two species was regarded as temperature dependent, in the form; kij = kija + kijb T
(1)
Six parameters were available to fit the data for a single polymer + solvent system. For the chosen value of εp the pure polymer parameters v∗p , dp , and cp parameters were fit to the LDPE PVT data and the binary parameters, kija and kijb were fit to the cloud point data for a binary polymer + solvent system. The entire procedure is repeated by varying εp until a good fit of both binary systems (LLDPE + ethylene and LLDPE + n-hexane) is achieved. The polymer is nearly monodisperse with a unimodal distribution. Cloud-point curves are not much affected by increasing the number of pseudo-components used to represent the polymer beyond the 7 that were employed in [12]. The critical point is more sensitive. Fig. 7 shows the effect of the number of polymer pseudo-components on the number-average and zaverage molar masses and on the critical polymer mass fraction. Comparison is with the values obtained using 1000 pseudocomponents. The effects of increasing the number of polymer pseudo-components above 40 are small. In calculations with many systems, 100 pseudo-components was adopted as a standard representation. The cost of computation, using methods reported in [20], increases only linearly with the number of pseudo-components. If there is sufficient information available, it is important to represent the original distribution as accurately as possible because different parts of the molar mass distribution affect different parts of the phase diagram. In the modeling of the ternary system LLDPE + nhexane + ethylene two sets of LLDPE–ethylene binary interaction parameters were used. The first set is obtained in the way described above. However, the LLDPE + ethylene phase equilibrium data [12] were obtained at lower temperature and much higher pressure than the experimental results obtained
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Table 5 LLDPE + n-hexane + ethylene: P–T cloud point isopleths at indicated composition T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
wp = 0.0502, we = 0.0118 426.28 1.846 431.16 2.566 436.16 3.276 441.26 4.096 446.14 4.721 451.05 5.421 455.97 6.096 460.90 6.746 465.75 7.396 470.74 7.996 475.64 8.571 480.51 9.176 485.46 9.726 490.27 10.271 495.18 10.841
wp = 0.0501, we = 0.0205 415.92 1.659 420.93 2.459 425.96 3.224 430.76 3.974 435.65 4.734 440.65 5.394 445.51 6.049 450.52 6.769 455.49 7.474 460.43 8.144 465.36 8.724 470.27 9.319 475.28 9.869 480.25 10.394 485.26 10.999 490.18 11.549 495.12 12.074
wp = 0.0499, we = 0.0298 411.24 2.149 416.19 3.021 421.11 3.849 426.01 4.594 430.99 5.324 435.94 6.094 440.90 6.799 445.92 7.394 450.79 8.099 455.84 8.749 460.69 9.369 465.65 9.994 470.62 10.699 475.58 11.299 480.51 11.799 485.42 12.394 490.31 12.894 495.24 13.369
wp = 0.0998, we = 0.0099 426.12 1.774 430.98 2.524 435.95 3.194 440.82 3.969 445.76 4.739 450.70 5.249 455.59 5.944 460.54 6.609 465.50 7.274 470.43 7.844 475.42 8.574 480.45 9.194 485.47 9.849 490.41 10.449 495.30 10.999
wp = 0.1007, we = 0.0196 421.26 2.449 426.08 3.144 431.05 3.949 435.99 4.719 440.73 5.449 445.90 6.199 450.91 6.844 455.88 7.524 460.64 8.224 465.67 8.834 470.72 9.484 475.58 10.094 480.54 10.694 485.56 11.249 490.41 11.819 495.45 12.374
wp = 0.1002, we = 0.0296 410.90 2.199 415.82 3.024 420.77 3.809 425.74 4.629 430.62 5.374 435.61 6.084 440.43 6.824 445.48 7.569 450.46 8.244 455.38 8.934 460.36 9.599 465.45 10.057 470.41 10.734 475.31 11.324 480.19 11.949 485.06 12.349 489.96 12.949
wp = 0.1508, we = 0.0098 430.69 2.074 435.59 2.869 440.64 3.619 445.66 4.349 450.45 5.044 455.37 5.694 460.32 6.349 465.30 6.989 470.23 7.614 475.17 8.229 480.07 8.824 484.95 9.409 489.87 9.964 494.80 10.514
wp = 0.0503, we = 0.0210 420.47 2.094 425.34 2.844 430.32 3.644 435.25 4.369 440.27 5.069 445.20 5.794 450.13 6.519 455.10 7.319 460.07 7.789 464.88 8.509 469.80 9.024 474.77 9.669 479.72 10.229 484.58 10.769 489.52 11.319 494.48 11.884
wp = 0.1506, we = 0.0293 411.03 2.074 415.95 2.874 420.84 3.749 425.82 4.444 431.00 5.209
wp = 0.1506, we = 0.0293 435.98 5.969 440.92 6.689 445.91 7.339 450.89 8.019 455.82 8.639
wp = 0.1506, we = 0.0293 460.76 9.304 465.67 9.959 470.37 10.529 475.46 11.124 480.33 11.669
wp = 0.1506, we = 0.0293 485.26 12.219 490.35 12.759 495.17 13.309
wp is the mass fraction of LLDPE, we is the mass fraction of ethylene.
in this work. So the prediction of the ternary phase behaviour involves an extrapolation of the LLDPE–ethylene binary interaction parameter. To avoid this problem the second set of LLDPE–ethylene binary interaction parameters was fitted to the ternary LLDPE + n-hexane + ethylene cloud point data using the same pure component LLDPE and ethylene parameters as in the first set. The n-hexane–ethylene binary interaction parameters were obtained from a fit to VLE data of this system [21]. 4.3. Modelling results The resulting Sanchez-Lacombe parameters for the pure components and the binary interaction parameters for the different binary systems involved are summarized in Table 8. The
Sanchez-Lacombe model provides a good fit of the two polymer solvent systems over the entire range of compositions, temperatures and pressures. The AAD of the calculated and experimental LDPE density data is 0.35%. The AAD of calculated and experimental cloud point temperatures for LLDPE + nhexane is 1.44 K and for LLDPE + ethylene using the first set of LLDPE–ethylene interaction parameters 1.54 K. The calculated results for LLDPE + n-hexane at 450, 470 and 490 K are plotted in Fig. 3. The LLDPE energy parameter, εp = 4.014 kJ/mol, is of the same magnitude as those reported for LDPE for the original SL equation by Gauter and Heidemann [11] (εp = 4.65 and 4.900 kJ/mol). The liquid–liquid critical polymer mass fractions for LLDPE + n-hexane predicted by the Sanchez-Lacombe model
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Table 6 LLDPE + n-hexane + ethylene: P–T bubble point isopleths at indicated composition T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
wp = 0.0502, we = 0.0118 411.46 1.052 416.37 1.116 421.20 1.181
wp = 0.0501, we = 0.0205 406.00 1.354 410.93 1.434
wp = 0.0499, we = 0.0298 396.49 1.529 401.10 1.604 406.30 1.699
wp = 0.0998, we = 0.0099 406.11 0.969 411.11 1.039 416.06 1.109 421.13 1.189
wp = 0.1007, we = 0.0196 401.65 1.309 406.43 1.374 411.45 1.479 416.30 1.549
wp = 0.1002, we = 0.0296 396.26 1.58 401.23 1.66 406.21 1.75
wp = 0.1508, we = 0.0098 416.15 1.139 421.11 1.219 426.07 1.299
wp = 0.1503, we = 0.0210 400.89 1.354 405.73 1.429 410.70 1.504 415.66 1.589
wp = 0.1506, we = 0.0293 396.32 1.664 401.21 1.744 406.10 1.834 wp is the mass fraction of LLDPE, we is the mass fraction of ethylene.
varies from 0.0547 at 450 K to 0.0748 at 490 K. Experimentally the critical LLDPE mass fractions in this temperature range was found at (0.0666 ± 0.0012). No trend with temperature was observed. In Fig. 8 the results of the modified Sanchez-Lacombe cloud point calculations for the ternary system LLDPE + nhexane + ethylene at 450, 470 and 490 K are plotted. The mass fraction LLDPE is 0.0983, the mass fraction ethylene varies between 0 and 0.03. The dashed curves are the results of the calculations using the LLDPE–ethylene binary interactions parameters obtained from the fit of the phase behaviour of
LLDPE + ethylene. It is clearly seen that at all temperatures the model under-predicts the effect of ethylene on the cloud point pressure. The full curves are the results of the calculations using the LLDPE–ethylene binary interactions parameters obtained from a fit to the ternary cloud point data. The quality of this fit is very good as can be seen from the figure. In Figs. 4–6 the curves are calculated with the modified Sanchez-Lacombe model and the later set of LLDPE–ethylene binary interaction parameters. These figures show that the fit of the experimental ternary cloud point data is also very good at other polymer mass fractions and that the modified Sanchez-Lacombe model gives a very good
Table 7 LLDPE + n-hexane + ethylene: P–T liquid–liquid–vapour bubble point isopleths at indicated composition T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
T (K)
P (MPa)
wp = 0.0502, we = 0.0118 431.1681 1.346 441.13 1.561 450.89 1.746 460.82 1.996 470.58 2.251 480.33 2.551 490.18 2.856
wp = 0.0501, we = 0.0205 420.93 1.588781 430.83 1.784 440.72 1.974 450.70 2.194 460.51 2.444 470.39 2.699 480.24 2.984 490.10 3.269
wp = 0.0499, we = 0.0298 411.32 1.774 421.44 1.964 431.01 2.149 440.90 2.359 450.90 2.589 460.75 2.834 470.56 3.094 480.44 3.369 490.32 3.649
wp = 0.0998, we = 0.0099 431.00 1.364 440.82 1.544 450.72 1.754 460.66 1.989 470.51 2.254 480.43 2.544 490.25 2.869
wp = 0.1007, we = 0.0196 421.34 1.629 431.27 1.804 440.74 2.009 450.63 2.234 460.65 2.474 470.39 2.734 480.38 3.024 490.29 3.319
wp = 0.1002, we = 0.0296 411.27 1.829 421.23 2.009 431.10 2.194 441.04 2.429 450.93 2.664 460.69 2.909 470.55 3.174 480.48 3.449 490.31 3.709
wp = 0.1508, we = 0.0098 430.69 1.384 440.59 1.569 450.49 1.774 460.60 2.029 470.34 2.284 480.41 2.579 490.00 2.889
wp = 0.1503, we = 0.0210 420.578 1.674 430.38 1.859 440.26 2.059 450.09 2.269 459.96 2.519 469.77 2.784 479.68 3.044 489.57 3.354
wp = 0.1506, we = 0.0293 411.03 1.899 420.84 2.084 430.89 2.299
wp = 0.1506, we = 0.0293 440.85 2.519 450.76 2.754 460.61 3.004
wp = 0.1506, we = 0.0293 470.42 3.264 480.19 3.534 490.17 3.799
wp is the mass fraction of LLDPE, we is the mass fraction of ethylene.
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Table 8 Modified Sanchez-Lacombe (MSL) parameters
Energy parameter εp (kJ/mol) Volume per lattice site v∗p (cm3 /mol) Number of lattice sites dp = di /Mi (mol/g) Volume shift parameter cp = ci /Mi (cm3 /g)
LLDPE
n-Hexane
Ethylene
4.014 8.411 0.06986 −0.5059
3.685a
2.288a 7.471a 0.2271a 0
12.550a 0.1122a 0
Binary interaction parameterb for LLDPE + ethylene kij = 2.703 × 10−2 − 7.856 × 10−5 T Binary interaction parameterc for LLDPE + ethylene kij = −5.453 × 10−2 + 2.050 × 10−4 T Binary interaction parameter for LLDPE + n-hexane kij = −1.708 × 10−1 + 3.357 × 10−4 T Binary interaction parameter for n-hexane + ethylene kij = −1.520 × 10−1 + 4.405 × 10−4 T a b c
From Gauter and Heidemann’s parameterization. From fit to LLDPE + ethylene cloud point data. From fit to LLDPE + n-hexane + (1–3 wt.%) ethylene cloud point data.
prediction of the influence of the ethylene concentration on the vapour–liquid and liquid–liquid–vapour bubble point curves. 5. Conclusions
Fig. 7. Effect of the number of pseudo-components on the number-average and z-average molar masses and on the critical polymer mass fraction. Comparison is with results using 1000 pseudo-components.
The system LLDPE + n-hexane shows a liquid–liquid phase split characterised by lower critical solution temperatures and upper critical solution pressures. The addition of ethylene to n-hexane + LLDPE shifts the cloud point curve to lower temperatures and higher pressures with approximately 10 K or 3 MPa per wt.% ethylene. The modified Sanchez-Lacombe equation of state can be used to correlate the phase behaviour of the binary subsystems of ethylene + n-hexane + LLDPE and to predict the phase behaviour of the ternary systems using parameters obtained from the fit of the binary subsystems. The description of the phase behaviour of the ternary system is improved if the binary interaction parameter of the system LLDPE + ethylene is fitted to the ternary data. Acknowledgements The research was sponsored by grants from the Natural Science and Engineering Research Council of Canada and Nova Research and Technology Corporation. The authors are thankful to Mr. W. Poot for his assistance in the phase equilibrium measurements. References
Fig. 8. Cloud point pressure of LLDPE + n-hexane + ethylene as function of the ethylene mass fraction at constant mass fraction of LLDPE of 0.0983. Symbols: experimental data; dashed curves: modified Sanchez-Lacombe predictions using LLDPE–ethylene binary interaction parameter fitted to high pressure LLDPE + ethylene data; full curves: modified Sanchez-Lacombe modeling results using LLDPE–ethylene binary interaction parameter fitted to the ternary LLDPE + n-hexane + ethylene data.
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