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Modeling of polymer phase equilibria using Perturbed-Chain SAFT
Fluid Phase Equilibria 194–197 (2002) 541–551
Modeling of polymer phase equilibria using Perturbed-Chain SAFT Feelly Tumakaka a , Joachim Gross b,1 , Gabriele Sadowski a,∗ a
Lehrstuhl für Thermodynamik, Universität Dortmund, Emil-Figge-Strasse 70, 44227 Dortmund, Germany b Fachgebiet für Thermodynamik und Thermische Verfahrenstechnik, Technische Universität Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany Received 20 May 2001; accepted 23 October 2001
Abstract The Perturbed-Chain SAFT equation-of-state is applied to binary and ternary mixtures of polymers, solvents and gases. Using a temperature-independent interaction parameter kij for each binary system, the Perturbed-Chain SAFT equation-of-state gives good correlations of the appropriate phase behavior over wide ranges of conditions. Copolymers or polymers with short-chain branching can be described using the copolymer version of PC-SAFT. In this version, the chain segments are allowed to differ in size and in attractive forces. Using this concept, copolymers like poly(ethylene-co-propylene) and poly(ethylene-co-vinyl acetate) could be modeled using the knowledge of the homopolymer properties and only one additional parameter which describes the attractive interactions between the unlike copolymer segments. Comparisons to the original SAFT model reveal an improvement of the proposed model. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Equation-of-state; Polymer; High pressure; Phase equilibria; Perturbation theory; Chain molecules; Copolymer
1. Introduction The knowledge of polymer phase equilibria is of vital importance for the design and optimization of polymer processes, as polymer production, separation and processing. Polymer systems are in general more complex than systems of low molecular weight substances: due to the large difference in the molecular sizes of polymers and solvents, the phase behavior of polymer systems often exhibit a pronounced density dependence at high temperatures. gE -models fail here, an equation-of-state needs to be used instead. Moreover, considerable experimental effort is generally required for determining the phase equilibria of polymer systems. Therefore, it is important from a practical viewpoint, that an equation-of-state is robust for extrapolations. ∗
Corresponding author. Tel.: +49-231-755-2635; fax: +49-231-755-2572. E-mail addresses: [email protected] (J. Gross), [email protected] (G. Sadowski). 1 Present address: BASF AG, Prozesstechnik GIC/P-Q 920, 67056 Ludwigshafen, Germany. Fax: +49-621-60-73488. 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 7 8 5 - 3
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In previous studies, the authors proposed the Perturbed-Chain SAFT (PC-SAFT) equation-of-state [1,2]. This equation-of-state accounts for the chain-like structure of a molecule in the repulsive as well as in the attractive part. Therefore, it showed to be particularly suitable for the description of polymer systems but also advantageous for any molecules of non-spherical shape [3]. For numerous low molecular weight substances the three pure-component parameters (the segment number m, the segment diameter σ , and the dispersion energy parameter ε/k) were earlier determined [2]. The PC-SAFT model was shown to accurately describe vapor pressures, densities and calorimetric properties of pure components. The equation-of-state was extended to mixtures by applying one-fluid mixing rules. The parameters between a pair of unlike segments are obtained by conventional Berthelot–Lorentz combining rules [2]. Comparisons to the SAFT equation-of-state [4,5] revealed a clear improvement for pure component and mixture properties. In this work, the Perturbed-Chain SAFT equation-of-state is applied to polymer systems, whereas homopolymer systems as well as copolymer systems are considered. Besides binary polymer/solvent systems also ternary systems containing an additional gaseous component are investigated. In order to evaluate the Perturbed-Chain SAFT equation-of-state, comparisons are made with the original SAFT version from Huang and Radosz [4,5] (hereafter SAFT for short). 2. Perturbed-Chain SAFT equation-of-state The model development of the PC-SAFT equation-of-state is in detail described by Gross and Sadowski [2]. PC-SAFT is a theoretical-based equation-of-state which was proposed also for the thermodynamic modeling of systems containing long-chain molecules like polymers. In the framework of PC-SAFT, molecules are assumed to be chains of freely jointed spherical segments. For the compressibility factor z of a non-associating molecule holds: z = zhard chain + zdispersion where zhard chain accounts for the repulsion of the chain-like molecule and is described using the hard-chain expression derived by Chapman et al. [6], which is also used in SAFT. In contrast to the SAFT equation-ofstate, the PC-SAFT model accounts for the non-spherical shape of the molecules also in the dispersion term zdispersion . The perturbation theory of Barker and Henderson [7] was applied using a hard chain fluid as the reference system, whereas a hard sphere reference was considered in earlier SAFT models. The theory was tested for square-well chains versus computer simulations [1] leading to a satisfactory prediction of the simulated behavior. Readjusting of model constants for real substances led to the PC-SAFT equation-of-state [2]. 3. Copolymer concept Recent effort has been devoted to develop equation-of-state for heteronuclear chain fluids. They can serve as simplified models for describing the thermodynamic properties of copolymers which consist of dissimilar monomers [8–10]. In order to characterize the properties of copolymer systems, the Perturbed-Chain SAFT model is extended for heteronuclear hard-chain fluids using the approach suggested by Banaszak et al. [9], and Shukla and Chapman [10]. They presented the extended SAFT version
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to heteronuclear hard chains, referred to as copolymer-SAFT, in which each segment in a chain can have different properties. In the modeling concept of PC-SAFT, a poly(ethylene-co-propylene) for example is comprised of polyethylene segments and of polypropylene segments. This copolymer is on the one hand characterized with pure-component parameters of polyethylene and of polypropylene. On the other hand, segment fractions are required, which describe the amount of different segment types in the copolymer chain, and furthermore bond fractions, which define to a certain extend the arrangement of the segments in chains. The segment fractions and the bond fractions can be estimated on the basis of the molecular structure. Since the dispersion term incorporates one-fluid mixing rules for mixtures, the different segment types in the copolymer chains are also implicated in this theory. In the concept of the copolymer Perturbed-Chain SAFT model, the copolymer chains are build by segments of the considered homopolymers. Consequentially, the modeling of copolymer–solvent mixture requires the appropriate pure-component parameters of the respective homopolymers and of the solvent. Further, three binary parameters have to be declared: two of them are the binary parameters of the homopolymer–solvent pairs. They are determined from the homopolymer–solvent data, hence, independent from copolymer data. Only one additional parameter is needed which accounts for the interactions between the different segment types of different chains. This latter parameter is the only one that has to be determined from copolymer data. For example, the solubility of poly(ethylene-co-vinyl acetate) in cyclopentane is calculated from the pure-component parameters of polyethylene, poly(vinyl acetate) and the solvent cyclopentane. The binary parameters for this system are kij of polyethylene–cyclopentane, kij of poly(vinyl acetate)–cyclopentane, and the third parameter kij for the interactions between ethylene segments and vinyl acetate segments. The advantage of this concept is that it allows calculations over the whole range of copolymer compositions with a minimum of experimental data.
4. Pure-component parameters for polymers The common method for determining pure-component parameters is to fit vapor pressures and liquid density data. This is not possible for polymers, since only the liquid density data are accessible. However, the polymer parameters obtained from density data only, often lead to unsatisfying results in describing the phase behavior of polymer mixtures, as it has also been reported by other authors [11,12]. The reason for this is the low sensitivity of the equation-of-state energy parameter towards liquid densities. A possible approach is to fit simultaneously polymer densities and one single polymer/solvent cloud-point curve to identify the polymer pure-component parameters and the binary parameter kij for this particular polymer/solvent system, which leads to a total of four parameters that need to be optimized. It was shown, that polymer pure-component parameters determined this way are suitable also for other solvent mixtures and can thus be regarded characteristic for a polymer [3]. Table 1 gives correlation results for seven polymers (as earlier published in reference [3]), as well as average deviations for the liquid densities and the appropriate pressure range covered by the density data. Moreover, the binary systems used for the pure-component parameter estimation are listed. The segment number m is here given as a ratio with respect to molecular weight M, since this is a convenient parameter for polymers, where the segment number is a linear function of the molecular weight.
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Table 1 Pure-component parameters of the Perturbed-Chain SAFT equation-of-state for polymers [3]. Polymer
Polyethylene (HDPE) Polyethylene (LDPE) Polypropylene Polybutene Polyisobutene Polystyrene Poly(vinyl acetate) a
m/M (mol/g)
0.0263 0.0263 0.02305 0.014 0.02350 0.0190 0.03211
σ (Å)
4.0217 4.0217 4.1 4.2 4.1 4.1071 3.3972
ε/k (K)
252.0 249.5 217.0 230.0 265.5 267.0 204.65
Density data
Binary data
AAD% ρ
P range (bar)
Solvent
Reference
1.62 1.14 5.55 ∼30a 0.95 1.16 6.14
1–1000 1–1000 1–981 1–1000 100–800 100–1000 100–1000
Ethylene Ethane n-Pentane 1-Butene n-Butane Cyclohexane Cyclopentane
[15] [13] [16] [21] [22] [23,24] [20]
A particular emphasis was put on binary data for the regression of the polybutene pure-component parameters [17].
5. Results This chapter presents modeling results for vapor–liquid and liquid–liquid equilibria of binary a nd ternary homopolymer as well as for binary copolymer systems described with the Perturbed-Chain SAFT equation-of-state. 5.1. Homopolymer systems 5.1.1. Polyethylene systems The pure-component parameters of polyethylene are determined by fitting the solubility data of polyethylene (LDPE) in ethane as reported by Hasch [13] and polyethylene liquid densities. These parameters are subsequently used in the phase equilibrium calculations of other polyethylene mixtures with ethylene, propane, propylene, n-butane and 1-butene. Fig. 1 compares experimental and modeling results. The good agreement of calculation with the experimental data shows that the parameters obtained by this approach are characteristic for polyethylene and thus can be considered as pure-component parameters. It is worth mentioning that only one temperature-independent kij is required for each binary system. The high-pressure equilibrium of polyethylene (HDPE)–ethylene mixtures was extensively investigated by de Loos et al. [15]. Cloud points of this system, along with polyethylene liquid density data, were used for determining the pure-component parameters of HDPE. The molecular mass distribution of the polyethylene sample considered in Fig. 2 was characterized by three weight averages (Mn = 43 kg/mol, Mw = 118 kg/mol, Mz = 231 kg/mol) [15]. Fig. 2 compares experimental and modeling results for three temperatures [3]. The Perturbed-Chain SAFT equation-of-state gives a good representation of the temperature dependence using temperature-independent parameter. However, the region of immiscibility is somewhat too narrow in concentration. The SAFT model suffers from the same deficiency in a more pronounced manner. Fig. 3 gives the solubility of carbon dioxide in HDPE at P = 90 bar. The Perturbed-Chain SAFT model precisely describes this binary data within the temperature range of 125–250 ◦ C using a constant kij parameter [3]. Fig. 4 displays mixtures of HDPE and toluene at T = 120 ◦ C as an example for a vapor–liquid system. The experimental results for two molecular masses of the polymer are well described by the Perturbed-Chain SAFT equation-of-state with kij = 0.
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Fig. 1. High pressure equilibrium of polyethylene (LDPE)–solvents. Symbols are experimental cloud points [13,14], lines are calculation results of Perturbed-Chain SAFT using only one temperature-independent kij for each binary system. The polymer was assumed to be monodisperse.
5.1.2. Polypropylene systems Experimental cloud points for polypropylene–n-pentane mixtures at several temperatures and varying molecular mass of polypropylene are given by Martin et al. [16]. These data were used for the parameter identification of polypropylene as described before. Fig. 5 compares modeling results obtained from
Fig. 2. High pressure equilibrium of HDPE–ethylene at three temperatures. (HDPE: Mn = 43 kg/mol, Mw = 118 kg/mol, Mz = 231 kg/mol). Comparison of experimental cloud points to calculation results of Perturbed-Chain SAFT (kij = 0.0404) and SAFT (kij = 0.0564). The polymer was modeled using three pseudocomponents [3].
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Fig. 3. Solubility of carbon dioxide in polyethylene (HDPE) at P = 90 bar (HDPE: Mn = 87 kg/mol). Comparison of Perturbed-Chain SAFT (kij = 0.181) and SAFT (kij = 0.242) correlation results to experimental data [3].
the Perturbed-Chain SAFT equation-of-state with experimental liquid–liquid data at three temperatures [3]. Since the molecular mass distribution of polypropylene is of moderate width (Mw /Mn = 2.2), the calculations were performed assuming a monodisperse polymer of molecular mass M = Mw = 50.4 kg/mol. The correlation results are in good agreement with the experimental phase behavior. Compressed gases and supercritical gases can be used for separating polymer blends, for separating polymers from solvents, and for fractionating polymers [18]. Thus, it is interesting and a challenge as well to model polymer–solvent–gas mixtures, because deficiencies in an equation-of-state often become particularly apparent when asymmetric multicomponent systems are calculated.
Fig. 4. Vapor–liquid phase equilibrium of polyethylene (HDPE)–toluene at T = 120 ◦ C [25]. Comparison of Perturbed-Chain SAFT (kij = 0) and SAFT (kij = 0) predictions to experimental data [3].
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Fig. 5. Liquid–liquid equilibria of polypropylene (PP)–n-pentane at three temperatures. (PP: Mw = 50.4 kg/mol, Mw /Mn = 2.2). Comparison of experimental cloud points [16] to Perturbed-Chain SAFT calculations ([17], kij = 0.0137). The polymer was assumed to be monodisperse.
Martin et al. [16] presented cloud point measurements for ternary mixtures of polypropylene–npentane–CO2 for various CO2 concentrations (Figs. 6 and 7). The binary kij parameter for polypropylene– n-pentane was already given above (kij = 0.0137, Fig. 5) and the binary parameter for n-pentane–CO2 was given in a previous study. The kij parameter between polypropylene and CO2 was obtained from the ternary mixture [3], adjusted to the data at 42 wt.% CO2 (Fig. 6). This was done because the binary system polypropylene–CO2 shows a moderate sensitivity towards kij , whereas calculations for the ternary
Fig. 6. Cloud point curve of PP–n-pentane–CO2 for various CO2 contents. Initial polymer weight fraction wPP = 0.03 (before the addition of CO2 ). Comparison of experimental cloud points [16] to Perturbed-Chain SAFT calculations (PP–n-pentane: kij = 0.0137, PP–CO2 : kij = 0.177, n-pentane–CO2 : kij = 0.143) [3].
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Fig. 7. Cloud point curve of PP–n-pentane–CO2 for various CO2 contents. Initial polymer weight fraction wPP = 0.097 (before the addition of CO2 ) [3]. The symbols and lines are defined as in Fig. 6.
system react strongly on changes of this binary kij parameter. Figs. 6 and 7 show, that the Perturbed-Chain SAFT is able to describe the pressure shift of the demixing curves with varying CO2 concentrations correctly. 5.2. Copolymer systems The physical properties of polymers can be influenced by varying the comonomer types, their contents as well as their arrangement along the polymer chain. Because of the wide variation in copolymer properties, the description of phase behavior of copolymer–solvent mixtures is very challenging. 5.2.1. Poly(ethylene-co-propylene) PEP system According to the copolymer concept described at the beginning, the calculation of PEP/solvent mixtures requires the pure-component parameters of polyethylene and polypropylene, the kij of polyethylene/solvent, the kij of polypropylene/solvent and the kij of ethylene/propylene segments. The pure-component parameters of the two homopolymers were previously identified independently (see modeling results of polyethylene and polypropylene systems). The kij value of 0.028 for polyethylene/1-butene and of −0.009 for ethylene/propylene segments were also independently determined by Spuhl [17]. Since no polypropylene/1-butene binary data were accessible, the binary parameter for polypropylene/1-butene has to be obtained from the copolymer–solvent mixtures (kij = 0.03) [17]. Fig. 8 shows experimental solubility data of alternating PEP with 50 mol% propylene content in the copolymer in 1-butene (Chen and Radosz [19]) for four different molecular masses and the modeling results with the copolymer version of Perturbed-Chain SAFT. The correlation results are in good agreement with the experimental phase behavior. 5.2.2. Poly(ethylene-co-vinyl acetate) EVA system In the last years, a new class of ethylene-based copolymers has been developed which incorporate polar or hydrogen bonding groups into the backbone of the polymer chain. One typical comonomer
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Fig. 8. Phase behavior of alternating PEP–1-butene (wPEP = 0.15) for different molecular masses. Experimental cloud point data [19] compared to calculations with Perturbed-Chain SAFT [17].
is vinyl acetate. These copolymers present a new challenge to produce and to model due to their unusual structure. They contain both non-polar and polar components. Poly(ethylene-co-vinyl acetate) was investigated in this study because of the availability of experimental data for various comonomer concentrations. Beyer [20] presented the experimental cloud points for the homopolymers polyethylene (LDPE) and poly(vinyl acetate) (PVAc) with cyclopentane as solvent. Furthermore, the phase behavior of poly(ethylene-co-vinyl acetate) at three different copolymer concentrations (11, 25 and 43 mol% of vinyl acetate in EVAs) with different molecular masses in cyclopentane was also investigated. An interesting phase behavior is observed here: starting from polyethylene, an increasing content of the vinyl acetate in the copolymer shifts the demixing curve to lower pressures up to 11 mol% of vinyl acetate in EVA, whereas with further addition of vinyl acetate groups the solubility again decreases. Fig. 9 shows, that the Perturbed-Chain SAFT model is able to capture the observed dependence of temperature, molecular mass as well as the non-monotonic dependence of comonomer concentration. The pure-component parameters for poly(vinyl acetate) and the kij of PVAc/cyclopentane (kij = 0.0233) are determined by fitting the binary data of PVAc/cyclopentane and PVAc pressure–volume–temperature data. The kij of −0.016 for LDPE/cyclopentane is obtained from the experimental cloud points (0 mol% of vinyl acetate in polymer, Fig. 9). Only the kij for ethylene/vinyl acetate segments was fitted to the copolymer data. It is a linear function of vinyl acetate mass fraction in EVA (kij = 0.1431 − 0.1548wVA ). Although this latter binary parameter is a linear function of the vinyl acetate content in the copolymer, the Perturbed-Chain SAFT model gives good representation of the observed non-monotonic dependence of comonomer content. The advantage of this copolymer concept is that the calculation or prediction for any comonomer compositions and any molecular weights is in general possible. As an example, also the prediction of the cloud-point curves for two copolymer compositions of 56 and 74 mol% vinyl acetate content is shown in Fig. 9.
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Fig. 9. Symbols are solubility data of cyclopentane in LDPE, PVAc and EVAs. Molar percentage indicates the vinyl acetate content in EVA. Dashed and solid lines are modeling results with Perturbed-Chain SAFT.
6. Conclusions The Perturbed-Chain SAFT equation-of-state was successfully applied to binary and ternary mixtures of homopolymers, solvents, and gases, as well as to binary copolymer–solvent systems. Pure-component parameters for seven non-associating polymers were identified. In comparison to the original SAFT version, the calculation results with the Perturbed-Chain SAFT model are more accurate using a temperatureindependent binary parameter. Furthermore, copolymer systems could be described from the knowledge of the homopolymer phase behavior using only one additional parameter which describes the interaction of unlike segments. Acknowledgements The authors are grateful to the Deutsche Forschungsgemeinschaft for supporting this work with grants SAD 700/3 and SAD 700/5. We thank Prof. Wolfgang Arlt for his encouragement and support. References [1] J. Gross, G. Sadowski, Application of perturbation theory to a hard-chain reference fluid: an equation-of-state for square-well chains, Fluid Phase Equilib. 168 (2000) 183. [2] J. Gross, G. Sadowski, Perturbed-Chain SAFT: an equation-of-state based on a perturbation theory for chain molecules, Ind. Eng. Chem. Res. 40 (2001) 1244. [3] J. Gross, G. Sadowski, Modeling polymer systems using the Perturbed-Chain statistical associating fluid theory equation-of-state, Ind. Eng. Chem. Res, in press.
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[4] S.H. Huang, M. Radosz, Equation-of-state for small, large, polydisperse and associating molecules, Ind. Eng. Chem. Res. 29 (1990) 2284. [5] S.H. Huang, M. Radosz, Equation-of-state for small, large, polydisperse and associating molecules: extensions to fluid mixtures, Ind. Eng. Chem. Res. 30 (1991) 1994. [6] W.G. Chapman, G. Jackson, K.E. Gubbins, Phase equilibria of associating fluids: chain molecules with multiple bonding sites, Mol. Phys. 65 (1988) 1057. [7] J.A. Barker, D. Henderson, Perturbation theory and equation-of-state for fluids: the square-well potential, J. Chem. Phys. 47 (1967) 2856. [8] M.D. Amos, G. Jackson, BHS theory and computer simulations of linear heteronuclear triatomic hard-sphere molecules, Mol. Phys. 74 (1991) 191. [9] M. Banaszak, C.K. Chen, M. Radosz, Copolymer SAFT equation-of-state: thermodynamic perturbation theory extended to heterobonded chains, Macromolecules 29 (1996) 6481. [10] K.P. Shukla, W.G. Chapman, SAFT equation-of-state for fluid mixtures of hard chain copolymers, Mol. Phys. 91 (1997) 1075. [11] B.M. Hasch, S.-H. Lee, M.A. McHugh, Strengths and limitations of SAFT for calculating copolymer-solvent phase behavior, J. Appl. Polym. Sci. 59 (1996) 1107. [12] S. Behme, Thermodynamik von Polymersystemen bei hohen Drücken, Dissertation, Technische Universität Berlin, 2000. [13] B.M. Hasch, Hydrogen Bonding and Polarity in Ethylene Copolymer–Solvent Mixtures: Experiment and Modeling, Dissertation, Johns Hopkins University Press, Baltimore, MD, 1994. [14] H. Dietzsch, Hochdruck-Copolymerisation von Ethen und (Meth)Acrylsäureestern: Entmischungsverhalten der Systeme Ethen/Cosolvens/poly(Ethen-co-Acrylsäureester)–Kinetik der Ethen-Methylmethacrylat-Copolymerisation, Dissertation, Georg August-Universität zu Göttingen, 1999. [15] 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. [16] T.M. Martin, A.A. Lateef, C.B. Roberts, Measurements and modeling of cloud point behavior for polypropylene/n-pentane and polypropylene/n-pentane/carbon dioxide mixtures at high pressures, Fluid Phase Equilib. 154 (1999) 241. [17] O. Spuhl, Berechnung von Copolymer-Lösungsmittel Phasengleichgewichten mit einer Zustandsgleichung, Diplomarbeit, Technische Universität Berlin, 2000. [18] B. Bungert, G. Sadowski, W. Arlt, Innovative verfahren mit komprimierten gasen, Chem. Ing. Tech. 69 (1997) 298. [19] S.-J. Chen, M. Radosz, Density-tuned polyolefin phase equilibria. 1. Binary solution of alternating poly(ethylene-propylene) in subcritical and supercritical propylene, 1-butene, Macromolecules 25 (1992) 3089. [20] C. Beyer, Untersuchungen zum Phasengleichgewichtsverhalten von Polymer-Lösungsmittelgemischen, Diplomarbeit, Universität Karlsruhe, 1997. [21] N. Koak, R.M. Visser, T.W. de Loos, High pressure phase behavior of the systems polyethylene and ethylene and polybutene + 1-butene, Fluid phase Equilib. 158 (1999) 835. [22] L. Zeman, J. Biros, G. Delmas, D. Patterson, Pressure effects in polymer solution phase equilibria. I. The lower critical solution temperature of polyisobutylene and polydimethylsiloxane in lower alkanes, J. Phys. Chem. 76 (1972) 1206. [23] B. Bungert, G. Sadowski, W. Arlt, Separations and material processing in solutions with dense gases, Ind. Eng. Chem. Res. 37 (1998) 3208. [24] S. Saeki, N. Kuwahara, S. Konno, M. Kaneko, Upper and lower critical solution temperatures in polystyrene solutions, Macromolecules 6 (1973) 246. [25] C. Wohlfarth, Vapour-liquid equilibrium data of binary polymer solutions, Physical Science Data 44, Elsevier, Amsterdom, The Netherlands, 1994.
Modeling of polymer phase equilibria using Perturbed-Chain SAFT Feelly Tumakaka a , Joachim Gross b,1 , Gabriele Sadowski a,∗ a
Lehrstuhl für Thermodynamik, Universität Dortmund, Emil-Figge-Strasse 70, 44227 Dortmund, Germany b Fachgebiet für Thermodynamik und Thermische Verfahrenstechnik, Technische Universität Berlin, Strasse des 17. Juni 135, 10623 Berlin, Germany Received 20 May 2001; accepted 23 October 2001
Abstract The Perturbed-Chain SAFT equation-of-state is applied to binary and ternary mixtures of polymers, solvents and gases. Using a temperature-independent interaction parameter kij for each binary system, the Perturbed-Chain SAFT equation-of-state gives good correlations of the appropriate phase behavior over wide ranges of conditions. Copolymers or polymers with short-chain branching can be described using the copolymer version of PC-SAFT. In this version, the chain segments are allowed to differ in size and in attractive forces. Using this concept, copolymers like poly(ethylene-co-propylene) and poly(ethylene-co-vinyl acetate) could be modeled using the knowledge of the homopolymer properties and only one additional parameter which describes the attractive interactions between the unlike copolymer segments. Comparisons to the original SAFT model reveal an improvement of the proposed model. © 2002 Elsevier Science B.V. All rights reserved. Keywords: Equation-of-state; Polymer; High pressure; Phase equilibria; Perturbation theory; Chain molecules; Copolymer
1. Introduction The knowledge of polymer phase equilibria is of vital importance for the design and optimization of polymer processes, as polymer production, separation and processing. Polymer systems are in general more complex than systems of low molecular weight substances: due to the large difference in the molecular sizes of polymers and solvents, the phase behavior of polymer systems often exhibit a pronounced density dependence at high temperatures. gE -models fail here, an equation-of-state needs to be used instead. Moreover, considerable experimental effort is generally required for determining the phase equilibria of polymer systems. Therefore, it is important from a practical viewpoint, that an equation-of-state is robust for extrapolations. ∗
Corresponding author. Tel.: +49-231-755-2635; fax: +49-231-755-2572. E-mail addresses: [email protected] (J. Gross), [email protected] (G. Sadowski). 1 Present address: BASF AG, Prozesstechnik GIC/P-Q 920, 67056 Ludwigshafen, Germany. Fax: +49-621-60-73488. 0378-3812/02/$ – see front matter © 2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 8 - 3 8 1 2 ( 0 1 ) 0 0 7 8 5 - 3
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F. Tumakaka et al. / Fluid Phase Equilibria 194–197 (2002) 541–551
In previous studies, the authors proposed the Perturbed-Chain SAFT (PC-SAFT) equation-of-state [1,2]. This equation-of-state accounts for the chain-like structure of a molecule in the repulsive as well as in the attractive part. Therefore, it showed to be particularly suitable for the description of polymer systems but also advantageous for any molecules of non-spherical shape [3]. For numerous low molecular weight substances the three pure-component parameters (the segment number m, the segment diameter σ , and the dispersion energy parameter ε/k) were earlier determined [2]. The PC-SAFT model was shown to accurately describe vapor pressures, densities and calorimetric properties of pure components. The equation-of-state was extended to mixtures by applying one-fluid mixing rules. The parameters between a pair of unlike segments are obtained by conventional Berthelot–Lorentz combining rules [2]. Comparisons to the SAFT equation-of-state [4,5] revealed a clear improvement for pure component and mixture properties. In this work, the Perturbed-Chain SAFT equation-of-state is applied to polymer systems, whereas homopolymer systems as well as copolymer systems are considered. Besides binary polymer/solvent systems also ternary systems containing an additional gaseous component are investigated. In order to evaluate the Perturbed-Chain SAFT equation-of-state, comparisons are made with the original SAFT version from Huang and Radosz [4,5] (hereafter SAFT for short). 2. Perturbed-Chain SAFT equation-of-state The model development of the PC-SAFT equation-of-state is in detail described by Gross and Sadowski [2]. PC-SAFT is a theoretical-based equation-of-state which was proposed also for the thermodynamic modeling of systems containing long-chain molecules like polymers. In the framework of PC-SAFT, molecules are assumed to be chains of freely jointed spherical segments. For the compressibility factor z of a non-associating molecule holds: z = zhard chain + zdispersion where zhard chain accounts for the repulsion of the chain-like molecule and is described using the hard-chain expression derived by Chapman et al. [6], which is also used in SAFT. In contrast to the SAFT equation-ofstate, the PC-SAFT model accounts for the non-spherical shape of the molecules also in the dispersion term zdispersion . The perturbation theory of Barker and Henderson [7] was applied using a hard chain fluid as the reference system, whereas a hard sphere reference was considered in earlier SAFT models. The theory was tested for square-well chains versus computer simulations [1] leading to a satisfactory prediction of the simulated behavior. Readjusting of model constants for real substances led to the PC-SAFT equation-of-state [2]. 3. Copolymer concept Recent effort has been devoted to develop equation-of-state for heteronuclear chain fluids. They can serve as simplified models for describing the thermodynamic properties of copolymers which consist of dissimilar monomers [8–10]. In order to characterize the properties of copolymer systems, the Perturbed-Chain SAFT model is extended for heteronuclear hard-chain fluids using the approach suggested by Banaszak et al. [9], and Shukla and Chapman [10]. They presented the extended SAFT version
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to heteronuclear hard chains, referred to as copolymer-SAFT, in which each segment in a chain can have different properties. In the modeling concept of PC-SAFT, a poly(ethylene-co-propylene) for example is comprised of polyethylene segments and of polypropylene segments. This copolymer is on the one hand characterized with pure-component parameters of polyethylene and of polypropylene. On the other hand, segment fractions are required, which describe the amount of different segment types in the copolymer chain, and furthermore bond fractions, which define to a certain extend the arrangement of the segments in chains. The segment fractions and the bond fractions can be estimated on the basis of the molecular structure. Since the dispersion term incorporates one-fluid mixing rules for mixtures, the different segment types in the copolymer chains are also implicated in this theory. In the concept of the copolymer Perturbed-Chain SAFT model, the copolymer chains are build by segments of the considered homopolymers. Consequentially, the modeling of copolymer–solvent mixture requires the appropriate pure-component parameters of the respective homopolymers and of the solvent. Further, three binary parameters have to be declared: two of them are the binary parameters of the homopolymer–solvent pairs. They are determined from the homopolymer–solvent data, hence, independent from copolymer data. Only one additional parameter is needed which accounts for the interactions between the different segment types of different chains. This latter parameter is the only one that has to be determined from copolymer data. For example, the solubility of poly(ethylene-co-vinyl acetate) in cyclopentane is calculated from the pure-component parameters of polyethylene, poly(vinyl acetate) and the solvent cyclopentane. The binary parameters for this system are kij of polyethylene–cyclopentane, kij of poly(vinyl acetate)–cyclopentane, and the third parameter kij for the interactions between ethylene segments and vinyl acetate segments. The advantage of this concept is that it allows calculations over the whole range of copolymer compositions with a minimum of experimental data.
4. Pure-component parameters for polymers The common method for determining pure-component parameters is to fit vapor pressures and liquid density data. This is not possible for polymers, since only the liquid density data are accessible. However, the polymer parameters obtained from density data only, often lead to unsatisfying results in describing the phase behavior of polymer mixtures, as it has also been reported by other authors [11,12]. The reason for this is the low sensitivity of the equation-of-state energy parameter towards liquid densities. A possible approach is to fit simultaneously polymer densities and one single polymer/solvent cloud-point curve to identify the polymer pure-component parameters and the binary parameter kij for this particular polymer/solvent system, which leads to a total of four parameters that need to be optimized. It was shown, that polymer pure-component parameters determined this way are suitable also for other solvent mixtures and can thus be regarded characteristic for a polymer [3]. Table 1 gives correlation results for seven polymers (as earlier published in reference [3]), as well as average deviations for the liquid densities and the appropriate pressure range covered by the density data. Moreover, the binary systems used for the pure-component parameter estimation are listed. The segment number m is here given as a ratio with respect to molecular weight M, since this is a convenient parameter for polymers, where the segment number is a linear function of the molecular weight.
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Table 1 Pure-component parameters of the Perturbed-Chain SAFT equation-of-state for polymers [3]. Polymer
Polyethylene (HDPE) Polyethylene (LDPE) Polypropylene Polybutene Polyisobutene Polystyrene Poly(vinyl acetate) a
m/M (mol/g)
0.0263 0.0263 0.02305 0.014 0.02350 0.0190 0.03211
σ (Å)
4.0217 4.0217 4.1 4.2 4.1 4.1071 3.3972
ε/k (K)
252.0 249.5 217.0 230.0 265.5 267.0 204.65
Density data
Binary data
AAD% ρ
P range (bar)
Solvent
Reference
1.62 1.14 5.55 ∼30a 0.95 1.16 6.14
1–1000 1–1000 1–981 1–1000 100–800 100–1000 100–1000
Ethylene Ethane n-Pentane 1-Butene n-Butane Cyclohexane Cyclopentane
[15] [13] [16] [21] [22] [23,24] [20]
A particular emphasis was put on binary data for the regression of the polybutene pure-component parameters [17].
5. Results This chapter presents modeling results for vapor–liquid and liquid–liquid equilibria of binary a nd ternary homopolymer as well as for binary copolymer systems described with the Perturbed-Chain SAFT equation-of-state. 5.1. Homopolymer systems 5.1.1. Polyethylene systems The pure-component parameters of polyethylene are determined by fitting the solubility data of polyethylene (LDPE) in ethane as reported by Hasch [13] and polyethylene liquid densities. These parameters are subsequently used in the phase equilibrium calculations of other polyethylene mixtures with ethylene, propane, propylene, n-butane and 1-butene. Fig. 1 compares experimental and modeling results. The good agreement of calculation with the experimental data shows that the parameters obtained by this approach are characteristic for polyethylene and thus can be considered as pure-component parameters. It is worth mentioning that only one temperature-independent kij is required for each binary system. The high-pressure equilibrium of polyethylene (HDPE)–ethylene mixtures was extensively investigated by de Loos et al. [15]. Cloud points of this system, along with polyethylene liquid density data, were used for determining the pure-component parameters of HDPE. The molecular mass distribution of the polyethylene sample considered in Fig. 2 was characterized by three weight averages (Mn = 43 kg/mol, Mw = 118 kg/mol, Mz = 231 kg/mol) [15]. Fig. 2 compares experimental and modeling results for three temperatures [3]. The Perturbed-Chain SAFT equation-of-state gives a good representation of the temperature dependence using temperature-independent parameter. However, the region of immiscibility is somewhat too narrow in concentration. The SAFT model suffers from the same deficiency in a more pronounced manner. Fig. 3 gives the solubility of carbon dioxide in HDPE at P = 90 bar. The Perturbed-Chain SAFT model precisely describes this binary data within the temperature range of 125–250 ◦ C using a constant kij parameter [3]. Fig. 4 displays mixtures of HDPE and toluene at T = 120 ◦ C as an example for a vapor–liquid system. The experimental results for two molecular masses of the polymer are well described by the Perturbed-Chain SAFT equation-of-state with kij = 0.
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Fig. 1. High pressure equilibrium of polyethylene (LDPE)–solvents. Symbols are experimental cloud points [13,14], lines are calculation results of Perturbed-Chain SAFT using only one temperature-independent kij for each binary system. The polymer was assumed to be monodisperse.
5.1.2. Polypropylene systems Experimental cloud points for polypropylene–n-pentane mixtures at several temperatures and varying molecular mass of polypropylene are given by Martin et al. [16]. These data were used for the parameter identification of polypropylene as described before. Fig. 5 compares modeling results obtained from
Fig. 2. High pressure equilibrium of HDPE–ethylene at three temperatures. (HDPE: Mn = 43 kg/mol, Mw = 118 kg/mol, Mz = 231 kg/mol). Comparison of experimental cloud points to calculation results of Perturbed-Chain SAFT (kij = 0.0404) and SAFT (kij = 0.0564). The polymer was modeled using three pseudocomponents [3].
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Fig. 3. Solubility of carbon dioxide in polyethylene (HDPE) at P = 90 bar (HDPE: Mn = 87 kg/mol). Comparison of Perturbed-Chain SAFT (kij = 0.181) and SAFT (kij = 0.242) correlation results to experimental data [3].
the Perturbed-Chain SAFT equation-of-state with experimental liquid–liquid data at three temperatures [3]. Since the molecular mass distribution of polypropylene is of moderate width (Mw /Mn = 2.2), the calculations were performed assuming a monodisperse polymer of molecular mass M = Mw = 50.4 kg/mol. The correlation results are in good agreement with the experimental phase behavior. Compressed gases and supercritical gases can be used for separating polymer blends, for separating polymers from solvents, and for fractionating polymers [18]. Thus, it is interesting and a challenge as well to model polymer–solvent–gas mixtures, because deficiencies in an equation-of-state often become particularly apparent when asymmetric multicomponent systems are calculated.
Fig. 4. Vapor–liquid phase equilibrium of polyethylene (HDPE)–toluene at T = 120 ◦ C [25]. Comparison of Perturbed-Chain SAFT (kij = 0) and SAFT (kij = 0) predictions to experimental data [3].
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Fig. 5. Liquid–liquid equilibria of polypropylene (PP)–n-pentane at three temperatures. (PP: Mw = 50.4 kg/mol, Mw /Mn = 2.2). Comparison of experimental cloud points [16] to Perturbed-Chain SAFT calculations ([17], kij = 0.0137). The polymer was assumed to be monodisperse.
Martin et al. [16] presented cloud point measurements for ternary mixtures of polypropylene–npentane–CO2 for various CO2 concentrations (Figs. 6 and 7). The binary kij parameter for polypropylene– n-pentane was already given above (kij = 0.0137, Fig. 5) and the binary parameter for n-pentane–CO2 was given in a previous study. The kij parameter between polypropylene and CO2 was obtained from the ternary mixture [3], adjusted to the data at 42 wt.% CO2 (Fig. 6). This was done because the binary system polypropylene–CO2 shows a moderate sensitivity towards kij , whereas calculations for the ternary
Fig. 6. Cloud point curve of PP–n-pentane–CO2 for various CO2 contents. Initial polymer weight fraction wPP = 0.03 (before the addition of CO2 ). Comparison of experimental cloud points [16] to Perturbed-Chain SAFT calculations (PP–n-pentane: kij = 0.0137, PP–CO2 : kij = 0.177, n-pentane–CO2 : kij = 0.143) [3].
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Fig. 7. Cloud point curve of PP–n-pentane–CO2 for various CO2 contents. Initial polymer weight fraction wPP = 0.097 (before the addition of CO2 ) [3]. The symbols and lines are defined as in Fig. 6.
system react strongly on changes of this binary kij parameter. Figs. 6 and 7 show, that the Perturbed-Chain SAFT is able to describe the pressure shift of the demixing curves with varying CO2 concentrations correctly. 5.2. Copolymer systems The physical properties of polymers can be influenced by varying the comonomer types, their contents as well as their arrangement along the polymer chain. Because of the wide variation in copolymer properties, the description of phase behavior of copolymer–solvent mixtures is very challenging. 5.2.1. Poly(ethylene-co-propylene) PEP system According to the copolymer concept described at the beginning, the calculation of PEP/solvent mixtures requires the pure-component parameters of polyethylene and polypropylene, the kij of polyethylene/solvent, the kij of polypropylene/solvent and the kij of ethylene/propylene segments. The pure-component parameters of the two homopolymers were previously identified independently (see modeling results of polyethylene and polypropylene systems). The kij value of 0.028 for polyethylene/1-butene and of −0.009 for ethylene/propylene segments were also independently determined by Spuhl [17]. Since no polypropylene/1-butene binary data were accessible, the binary parameter for polypropylene/1-butene has to be obtained from the copolymer–solvent mixtures (kij = 0.03) [17]. Fig. 8 shows experimental solubility data of alternating PEP with 50 mol% propylene content in the copolymer in 1-butene (Chen and Radosz [19]) for four different molecular masses and the modeling results with the copolymer version of Perturbed-Chain SAFT. The correlation results are in good agreement with the experimental phase behavior. 5.2.2. Poly(ethylene-co-vinyl acetate) EVA system In the last years, a new class of ethylene-based copolymers has been developed which incorporate polar or hydrogen bonding groups into the backbone of the polymer chain. One typical comonomer
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Fig. 8. Phase behavior of alternating PEP–1-butene (wPEP = 0.15) for different molecular masses. Experimental cloud point data [19] compared to calculations with Perturbed-Chain SAFT [17].
is vinyl acetate. These copolymers present a new challenge to produce and to model due to their unusual structure. They contain both non-polar and polar components. Poly(ethylene-co-vinyl acetate) was investigated in this study because of the availability of experimental data for various comonomer concentrations. Beyer [20] presented the experimental cloud points for the homopolymers polyethylene (LDPE) and poly(vinyl acetate) (PVAc) with cyclopentane as solvent. Furthermore, the phase behavior of poly(ethylene-co-vinyl acetate) at three different copolymer concentrations (11, 25 and 43 mol% of vinyl acetate in EVAs) with different molecular masses in cyclopentane was also investigated. An interesting phase behavior is observed here: starting from polyethylene, an increasing content of the vinyl acetate in the copolymer shifts the demixing curve to lower pressures up to 11 mol% of vinyl acetate in EVA, whereas with further addition of vinyl acetate groups the solubility again decreases. Fig. 9 shows, that the Perturbed-Chain SAFT model is able to capture the observed dependence of temperature, molecular mass as well as the non-monotonic dependence of comonomer concentration. The pure-component parameters for poly(vinyl acetate) and the kij of PVAc/cyclopentane (kij = 0.0233) are determined by fitting the binary data of PVAc/cyclopentane and PVAc pressure–volume–temperature data. The kij of −0.016 for LDPE/cyclopentane is obtained from the experimental cloud points (0 mol% of vinyl acetate in polymer, Fig. 9). Only the kij for ethylene/vinyl acetate segments was fitted to the copolymer data. It is a linear function of vinyl acetate mass fraction in EVA (kij = 0.1431 − 0.1548wVA ). Although this latter binary parameter is a linear function of the vinyl acetate content in the copolymer, the Perturbed-Chain SAFT model gives good representation of the observed non-monotonic dependence of comonomer content. The advantage of this copolymer concept is that the calculation or prediction for any comonomer compositions and any molecular weights is in general possible. As an example, also the prediction of the cloud-point curves for two copolymer compositions of 56 and 74 mol% vinyl acetate content is shown in Fig. 9.
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Fig. 9. Symbols are solubility data of cyclopentane in LDPE, PVAc and EVAs. Molar percentage indicates the vinyl acetate content in EVA. Dashed and solid lines are modeling results with Perturbed-Chain SAFT.
6. Conclusions The Perturbed-Chain SAFT equation-of-state was successfully applied to binary and ternary mixtures of homopolymers, solvents, and gases, as well as to binary copolymer–solvent systems. Pure-component parameters for seven non-associating polymers were identified. In comparison to the original SAFT version, the calculation results with the Perturbed-Chain SAFT model are more accurate using a temperatureindependent binary parameter. Furthermore, copolymer systems could be described from the knowledge of the homopolymer phase behavior using only one additional parameter which describes the interaction of unlike segments. Acknowledgements The authors are grateful to the Deutsche Forschungsgemeinschaft for supporting this work with grants SAD 700/3 and SAD 700/5. We thank Prof. Wolfgang Arlt for his encouragement and support. References [1] J. Gross, G. Sadowski, Application of perturbation theory to a hard-chain reference fluid: an equation-of-state for square-well chains, Fluid Phase Equilib. 168 (2000) 183. [2] J. Gross, G. Sadowski, Perturbed-Chain SAFT: an equation-of-state based on a perturbation theory for chain molecules, Ind. Eng. Chem. Res. 40 (2001) 1244. [3] J. Gross, G. Sadowski, Modeling polymer systems using the Perturbed-Chain statistical associating fluid theory equation-of-state, Ind. Eng. Chem. Res, in press.
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