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Fluid Phase Equilibria 307 (2011) 202–207
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Fluid Phase Equilibria
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S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 203
Table 1
Ternary systems investigated in this study [29]....
204 S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207
Table 2
Energy parameters obtained for the ternary syste...
S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 205
Table 4
Model parameters of the ternary PS solutions (no...
206 S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207
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S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 207
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1 s2.0-s037838121100207 x-main.correlation of thermodynamic modeling and molecular simulations for liquid liquid equilibrium of ternary polymer mixtures based on a phenomenological scaling method

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1 s2.0-s037838121100207 x-main.correlation of thermodynamic modeling and molecular simulations for liquid liquid equilibrium of ternary polymer mixtures based on a phenomenological scaling method

  1. 1. Fluid Phase Equilibria 307 (2011) 202–207 Contents lists available at ScienceDirect Fluid Phase Equilibria journal homepage: www.elsevier.com/locate/fluid Correlation of thermodynamic modeling and molecular simulations for liquid–liquid equilibrium of ternary polymer mixtures based on a phenomenological scaling method Suk Yung Oh, Young Chan Bae∗ Division of Chemical Engineering and Molecular Thermodynamics Laboratory, Hanyang University, Seoul 133-791, Republic of Korea a r t i c l e i n f o Article history: Received 27 September 2010 Received in revised form 27 April 2011 Accepted 30 April 2011 Available online 30 May 2011 Keywords: Molecular thermodynamics Molecular simulation Ternary polymer mixtures a b s t r a c t In our previous study [S.Y. Oh, Y.C. Bae, J. Phys. Chem. B 114 (2010) 8948–8953], we presented a new method to predict liquid–liquid equilibria in ternary simple liquid mixtures by using a combination of a thermodynamic model and molecular simulations. As a continuation of that effort, we extend our previ- ously developed method to ternary polymer systems. In the simulations, we used the dummy atoms to calculate the pair interaction energy values between the polymer segments and the solvent molecules. Furthermore, a thermodynamic model scaling concept is introduced to consider the chain length depen- dence of the energy parameters. This method was applied to ternary mixtures incorporating low to high molecular weight polymers. The method presented here well described the experimental observations using one or no adjustable parameters. © 2011 Elsevier B.V. All rights reserved. 1. Introduction Phase behaviors and thermodynamic properties of polymer mixtures have proven to be of noticeable interest from both exper- imental and theoretical points of view due to their importance for the design and optimization of the polymer production plants. In many cases, most commercial polymeric materials produced are a mixture of multiple components; however, compared to binary systems, experimental and theoretical considerations on multi- component polymer systems have been limited. It is necessary to have a proper model and method able to describe the behaviors of the multi-component mixtures in the widest range of operating conditions. The most widely known theoretical treatment of polymer solu- tions is the lattice model of Flory and Huggins [1,2]. Based on the Flory–Huggins (FH) model, Scott [3] has treated the system sol- vent + non-solvent + polymer with the assumption of “single liquid approximation” and has obtained valuable qualitative results. A similar approach was given by Tompa [4], which has considered the effect of the chain length on the phase diagram. Following these classical works, numerous studies have been done on the applicability and the modification of the FH theory during the last few decades [5–10]. Recently, Hu and co-workers [11–13] reported ∗ Corresponding author. Tel.: +82 2 2298 0529; fax: +82 2 2296 0568. E-mail address: ycbae@hanyang.ac.kr (Y.C. Bae). URL: http://www.labmtl.com (Y.C. Bae). Monte Carlo simulation results of liquid–liquid equilibria (LLE) for ternary chain molecule systems and developed a new lattice model for multi-component systems. Equation of state (EOS) models, based on theories such as the lattice fluid (LF) theory or statistical associating fluid theory (SAFT) have also successfully interpreted the phase equilibrium of the ternary polymer systems at times [14–18]. Almost all of the thermodynamic models are capable of calculat- ing the phase equilibria for ternary systems by using binary model parameters obtained from experimental data for the three sub- stituent binaries. Accurate predictions are often possible for ternary vapor–liquid equilibria (VLE) with binary data alone, but are hardly obtained for ternary LLE [19]. Furthermore, in common cases where constituent binary data are not available, many fitting parameters may have to be used to correlate the phenomena. Due to the lim- ited data on the polymer mixtures, phase equilibrium calculations of ternary polymer mixtures are more difficult than ordinary liquid mixtures. Modern molecular thermodynamic methods combined with molecular simulation have been able to overcome the difficulties originating from the above-mentioned problems [20–22]. In our recent paper [20], we presented a new method to predict LLE in ternary liquid mixtures by using a combination of a thermody- namic model and molecular simulations. A modified double lattice (MDL) theory [23,24] is adopted to express the Helmholtz function of mixing, and the Dreiding force field [25] is used in simulations. In prediction of the phase diagrams, model parameters are simply obtained from the molecular simulations so correlating work is not 0378-3812/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.fluid.2011.04.026
  2. 2. S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 203 Table 1 Ternary systems investigated in this study [29]. System T (K) (1) (2) (3) A 223.15 258.15 298.15 PEG 200 n-Heptane Dichloro methane B 323.15 PS 233,000 2-Butanone 1-Propanol C 323.15 PS 233,000 2-Butanone Methanol D 298 PS 300,000 Ethyl acetate Methanol required. In this study, extending the method to ternary polymer mixtures is presented by revising a previously developed method. Dummy atoms are introduced to represent the polymer segments in simulations, and chain length dependence is accounted for in the energy parameters to scale down that of the monomer–solvent pair. Four real ternary polymer systems are investigated by the method presented here. 2. Thermodynamic model The MDL theory is chosen as a thermodynamic model because this model provided good predictions and more satisfactory results than those given by the FH theory or lattice cluster theory (LCT) [26–28], according to our previous work [20]. The expression of the multi-component lattice model starts with a simple cubic lattice (coordination number, z = 6) containing Nr = K i=1 Niri sites that are composed of K components. Each component i fills the ri lattice sites completely by Ni molecules. Fol- lowing our previous work [20], the Helmholtz function of mixing is given by mixA NrkT = K i=1 i ri ln i + Cˇ 2 K i=1 K j=1 i j 1 ri − 1 rj 2 + K i=1 K j=1 aij i j − ⎛ ⎝ K i=1 K j=1 a∗ ij i j ⎞ ⎠ 2 (1) where kT have their usual meaning, and i(= Niri/Nr) is the volume fraction of species i. aij and a∗ are a generalized function of the reduced interchange energy ˜εij, chain lengths ri and rj as. aij = ˜εij 2 2 − i ri + j rj + 1 ri + 1 rj (2) a∗ ij = ˜εij C 4 (3) where universal constants, Cˇ and C are 0.1415 and 1.7986, respectively [23]. The reduced interchange energy between i and j segments, ˜εij, is given by ˜εij = (εii + εjj − 2εij) kT (4) where εij is the interaction energy between segments i and j. The first two terms of Eq. (1) are the entropy contribution, and the last terms of Eq. (1) are the energy contribution. The chemical potential of component i is necessary for calculat- ing the binodal line. i kT = ∂( mixA/NrkT) ∂Ni T,V,Nj /= i (5) A coexistence curve can be determined from the following con- ditions i = i (i = 1, 2, . . . , K) (6) where i is the change in chemical potential upon isothermally transferring component i from the pure state to the mixture. Super- scripts and denote two phases at equilibrium. For calculation of the phase equilibrium, the critical condition is given by A ≡ A 11 A 12 · · · A 1K A 21 A 22 · · · A 2K . . . A K1 A K2 · · · A KK = 0 (7) and A ≡ ∂ A ∂ 1 ∂ A ∂ 2 · · · ∂ A ∂ K A 21 A 22 · · · A 2K . . . A K1 A K2 · · · A KK = 0 (8) where A ij is defined by A ij = ∂2( mixA/NrkT) ∂ i∂ j T,V (i, j = 1, 2, . . . , K) (9) 3. Molecular simulation Thermodynamic modeling was further applied to the LLEs of real ternary polymer systems chosen from the LLE data handbook [29], Table 1. Pairwise interaction energy parameters were determined by a molecular simulation technique developed by Blanco et al., [30,31] available in the Blends module of the commercial software Materials Studio (version 4.4) from Accelrys. The procedure used in this study involves the following stages. First, the structures of the molecules were constructed using the sketching tools in the materi- als visualizer. The repeat units of polymers, which are polyethylene glycol (PEG) and polystyrene (PS), and solvent molecules, were built in this investigation. Only one molecule of each kind was used in the Blends calculation. After the construction of the molecules, the geometry was optimized by energy minimization, and then the pairwise interaction energy values were calculated by using a Monte Carlo approach that includes constraints arising from the excluded volume constraint method [30,31]. This method allows for the generation of a large number of relative orientations of the two molecules, thereby allowing for a more reliable estimate of the average interaction energy in binary mixtures. The pairwise inter- action is calculated for a specified number of times and a probability distribution, P(εij) is generated. An explicit temperature depen- dence of εij is incorporated by averaging the binding energies using the Boltzmann method: εij T = dεijP(εij)εij exp(−εij/kT) dεijP(εij) exp(−εij/kT) (10) From the simulation results, attractions for pairs were given as a negative pair potential because the interaction energy is defined as the energy required to separate two bodies to an infinite
  3. 3. 204 S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 Table 2 Energy parameters obtained for the ternary systems using molecular simulation. System ε11/k (K) ε22/k (K) ε33/k(K) ε12/k (K) ε13/k (K) ε23/k (K) A 136.36 545.39 424.29 250.65 245.39 482.99 B 747.49 399.79 281.29 521.38 422.89 334.45 C 747.49 399.79 152.76 521.38 292.10 242.45 D 747.49 434.94 152.76 525.06 292.10 254.55 n X X O O ethyl acetate a b polystyrene Fig. 1. Examples of the constructed structures in the simulations, either the whole molecule in the case of solvent or the monomer repeat unit in the case of polymer: (a) ethyl acetate, (b) polystyrene. intermolecular separation. When applying our model, we take an opposite direction of the simulated energy value because the energy definitions of the two methods (our model and Blends) are equal to an amount, but opposite in sign. The results are listed in Table 2. The obtained energy parameters (εii and εij) are further directly used in Eq. (4) or (12) of the thermodynamic model. In the simulations, the average interaction energy was calculated for the generated N configurations. As N is increased, a better average interaction energy is obtained. Here 107 pair configurations were generated during the simulations. Fig. 1 shows examples of the structures in the simulations, either the whole molecule in the case of the solvent (Fig. 1a) or the monomer repeat unit in the case of the polymer (Fig. 1b). Dummy atoms are introduced to consider the effects of polymer connec- tivity, which are indicated by a letter “X” in Fig. 1b. Two dummy atoms were located at the head and tail positions of the polymer segment, making some positions of the polymer segment inacces- sible to solvent molecules. Therefore, any configurations violating this constraint would be rejected. To obtain the optimum struc- ture of the redefined polymer segment, these two dummy atoms represent –CH3 in the united atom approximation. The interaction energies associated with the dummy atoms were not calculated although the dummy atoms were considered to be methyl groups. The repeat units further than one CH3 group away do not hinder the solvent positions, but using a single dummy atom at both ends of the polymer is sufficient to mimic a real polymer segment in these simulations [30]. The Dreiding force field [25] was used in these simulations, which is a simple generic force field based on simple hybridiza- tion rules rather than on specific combinations of atoms. This force field is useful for predicting structures and dynamics of organic, bio- logical, and main-group inorganic molecules. Energetically, the van der Waals interactions are captured by the Lennard–Jones poten- tial, and electrostatic interactions are defined by atomic monopoles and a screened Coulombic term. An explicit Lennard–Jones 12–10 potential represents hydrogen bonding. 1.000.750.500.250.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 (3) (2) 298.15K 253.15K 223.15K (1) Fig. 2. Phase diagram of the ternary liquid–liquid equilibria of a PEG 200 (1) -n- heptane (2) -dichloro methane (3) system at three different temperatures (223.15 K, 253.15 K and 298.15 K). Open symbols are experimental data [29], and the lines are calculated by the model. 4. Results and discussion 4.1. Ternary LLE phase diagrams in polymer systems In ternary systems listed in Table 1, the polymer molecule is regarded as component 1 and the smaller solvent molecule of the two solvents as component 3. We setr3 = 1 for the smallest molecule and the chain length parameters of the components 1 and 2 are given by the classical method as ri = viMWi v3MW3 (i = 1, 2) (11) where vi and MWi are specific volume and molecular weight of an ith component, respectively. In the lattice models, size parameter, ri, is more of volume ratio than weight ratio because the size is related to the occupied sites of the lattice. Therefore, Eq. (11) is more reasonable than weight ratio when using the lattice thermo- dynamics. Application to low molecular weight polymer systems is first considered by using system A of Table 1. Table 3 provides the model parameters used in this system. The predicted results of our model and the experimental cloud point data of PEG 200 solutions at different temperatures (223.15 K, 253.15 K and 298.15 K) [29] are shown in Fig. 2. These systems show Type 1 phase behavior of the Treybal classification [32], i.e. in a ternary system that has only Table 3 Model parameters of the ternary PEG solutions (note: ˜εij = (εii + εjj − 2εij)/kT). System T (K) r1 r2 r3 ˜ε12 ˜ε13 ˜ε23 A 223.15 2.772 2.288 1 0.809 0.313 0.017 258.15 0.699 0.271 0.014 298.15 0.605 0.234 0.012
  4. 4. S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 205 Table 4 Model parameters of the ternary PS solutions (note: ˜εij = (εii + εjj − 2εij − εr )/kT). System r1 r2 r3 εr/k ˜ε12 ˜ε13 ˜ε23 B 2966.4 1 1 61.4 0.323 0.566 0.038 C 5483.9 1 1 87.3 0.323 0.978 0.209 D 7060.8 1 1 104.3 0.444 1.061 0.264 one separate region. Due to the higher values of r1, r2, and ˜ε12, PEG and n-heptane produce a pair of partially miscible liquids. As depicted in Fig. 2, the mixing region of the phase diagram increases slightly as the temperature increases from 223.15 to 298.15 K. The model parameters ri and εij are regarded as temperature indepen- dent in this work, but slightly decreasing ˜εij induces a larger mixing region. Reasonable agreement between the model and experimen- tal data without the use of adjustable parameters confirms the successful application of our proposed method to the ternary poly- mer systems. Accurate predictions were already expected for low molecular weight polymer mixtures because their behaviors are not significantly different from those of single liquid mixtures. For high molecular weight polymer mixtures, prediction is not easy so an additional concept is introduced to interpret unusual phenomena. To model the experimental observations appropri- ately, it turned out to be necessary to introduce a chain length dependence of the energy parameters. For polymer mixtures, local correlations between unlike segments are amplified due to the combined effects of large chain length and self avoidance, leading to a reformulation of the segment-segment interaction parame- ter [33]. For that purpose, Eq. (4) was reformulated by analogy to the scaling concept given by de Gennes [34] for the chain length dependence of the FH interaction parameter, as follows: ˜εij = (εii + εjj − 2εij − εr) kT (12) where εr is the reduced interaction energy of chain length depen- dence and was obtained in the light of the ternary LLE data. The values of εr are expected to increase as r1 increases. Thus, it might be better interpreted by using the functional form of εr, as follows: εr k = ˛(r1 − 1)ˇ (13) Similar functional approaches are given by other literatures [35–37]. Systems B, C, and D of PS in Table 1 [29] were considered to inves- tigate the ternary mixtures of high molecular weight polymers. We set r2 = r3 = 1 for solvent molecules because the chain length of the polymer is very high (r1 r2, r3). Only small differences within 0.1 percent result from r2 being exactly 1 instead of being calculated using Eq. (11). To incorporate the chain length dependence of the polymer, the reduced interchange energies of the pairs of polymer and solvent (1–2 and 1–3) are calculated by Eq. (12), and those of the solvents pair (2–3) are calculated by Eq. (4). For maintaining simplicity of the method, εr for 1–2 and 1–3 are identical for the same system. Table 4 provides the model parameters of the three ternary PS systems and, using these parameters, the calculated results were compared with experimental data as shown in Figs. 3–5. The calcu- lated curve fit fairly well to experimental values using only one scaling parameter (εr). All three PS solutions have one separate region and a relatively high value of the reduced interchange energy (˜ε13) confirmed that the components 1 and 3 (PS and alcohol) pro- duce a partly miscible pair. The values of εr are increased reasonably as r1 increases. Fig. 6 shows the reduced energy parameter versus the polymer chain length parameter. Eq. (13) with ˛ = 0.45 and ˇ = 0.61 gives a good approximation of the chain length depen- dence. As such the present method could be used as a predictive tool in high polymer mixtures using the scaling function with Eq. 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 (3) (2) (1) Fig. 3. Phase diagram of the ternary liquid–liquid equilibria of a PS 233,000 (1) -2- butanone (2) -1-propanol (3) system at 323.15 K. Open symbols are experimental data [29], and the line is calculated by the model. (13) as an universal function of each polymer. However, it is very difficult to accurately predict the phase equilibrium of high polymer mixtures on account of the various limitations in polymer materi- als. 4.2. Significance and limitation The validity of using the molecular simulation to predict the phase equilibrium depends on the theoretical model it is based 1.000.750.500.250.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 (3) (2) (1) Fig. 4. Phase diagram of the ternary liquid–liquid equilibria of a PS 233,000 (1) - 2-butanone (2) -methanol (3) system at 323.15 K. Open symbols are experimental data [29], and the line is calculated by the model.
  5. 5. 206 S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 1.000.750.500.250.00 0.00 0.25 0.50 0.75 1.00 0.00 0.25 0.50 0.75 1.00 (3) (2) (1) Fig. 5. Phase diagram of the ternary liquid–liquid equilibria of a PS 300,000 (1) - ethyl acetate (2) -methanol (3) system at 298 K. Open symbols are experimental data [29], and the line is calculated by the model. upon and the simulation technique applied. MDL theory, of course, has many obstacles for this type of calculation. Our thermodynamic model is based on an incompressible lattice, so the volume change of mixing is not considered. Flexibilities and structural effects of the polymer chains are also neglected; it assumed that all polymers have freely flexible chains. Lastly, polydispersity of the polymer is neglected, and all polymers are regarded as monodisperse. These limitations in dealing with polymers may lead to errors in phase equilibrium calculations. In simulations, choosing a proper force field for a given chemical is important, because calculated values are strongly dependent on the force field applied. The suitability of the Dreiding force field was confirmed again in this work for the method combined with a thermodynamic model. The methodology outlined here can well predict the ternary LLE of mixtures of simple liquids or oligomers, but one scaling parame- ter is introduced for accurate prediction of high polymer mixtures. This shows the possibility of calculating the ternary phase behavior of polymer mixtures where constituent binary data are not avail- able, because the model parameters of any given chemicals can 1000050000 0 50 100 150 εr /Κ r1 (polymer chain length parameter) Fig. 6. Chain length dependence of reduced energy parameter. Open symbols are the reduced energy parameters of the systems B, C, and D. The line is calculated by Eq. (13). be achieved in such a simple manner. Though it has restrictions on prediction, qualitative study or relative comparisons are always possible. For binary polymer mixtures, Fan et al. [30] and Patnaik et al. [22] showed similar studies using molecular simulations with the FH model. They provided a detailed understanding of the mis- cibility of the polymer mixtures qualitatively, but had difficulties in predicting the phase behavior quantitatively. The ultimate goal of applying the method to the real systems was successful from the viewpoint of both quantitative and qualitative prediction. Despite the use of an inevitable scaling parameter, this work provides a new method to predict the ternary polymer mixtures and describes quantitatively the phase behavior of polymer with mixed-solvent systems fairly well in spite of its simplicity. 5. Conclusions A novel approach to calculations of LLE of ternary polymer mix- tures is introduced by combination of a thermodynamic model and molecular simulations. MDL theory and the blend module with a Dreiding force field are adopted as a means to the method. In the calculation of high polymer mixed-solvent systems, a certain degree of scaling on the energy parameters of polymer-solvent pairs is necessary in order to match the experimental observations. Good agreement was obtained between the experimental data and the calculated results with using one or no adjustable parameters. Despite a few limitations, the present method can serve as a basis to develop more efficient methods for practical applications. Acknowledgments This work was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded from the Ministry of Education, Science and Technology (MEST) of Korea for the Center for Next Generation Dye-sensitized Solar Cells (No. 2011-0001055). References [1] P.J. Flory, J. Chem. Phys. 10 (1942) 51–61. [2] M.L. Huggins, J. Phys. Chem. 46 (1942) 151–158. [3] R.L. Scott, J. Chem. Phys. 17 (1949) 268–279. [4] H. Tompa, Trans. Faraday Soc. 45 (1949) 1142–1152. [5] A.R. Schultz, P.J. Flory, J. Am. Chem. Soc. 75 (1953) 5681–5685. [6] C.C. Hsu, J.M. Prausnitz, Macromolecules 7 (1974) 320–324. [7] J. Pouchly, D. Patterson, Macromolecules 9 (1976) 574–579. [8] F.W. Altena, C.A. Smolders, Macromolecules 15 (1982) 1491–1497. [9] E. Favre, Q.T. Mguyen, R. Clement, J. Neel, Eur. Polym. J. 32 (1996) 303–309. [10] T. Hino, J.M. Prausnitz, J. Appl. Polym. Sci. 68 (1998) 2007–2017. [11] J. Jiang, Q. Yan, H. Liu, Y. Hu, Macromolecules 30 (1997) 8459–8462. [12] H. Liu, J. Yang, Q. Xin, Y. Hu, Fluid Phase Equilibr. 261 (2007) 281–285. [13] Q. Xin, C. Peng, H. Liu, Y. Hu, Ind. Eng. Chem. Res. 47 (2008) 9678–9686. [14] G.R. Brannock, D.R. Paul, Macromolecules 23 (1990) 5240–5250. [15] G.D. Pappa, G.M. Kontogeorgis, D.P. Tassios, Ind. Eng. Chem. Res. 36 (1997) 5461–5466. [16] J. Gross, G. Sadowski, Ind. Eng. Chem. Res. 41 (2002) 1084–1093. [17] J.A. Gonzalez-Leon, A.M. Mayes, Macromolecules 36 (2003) 2508–2515. [18] T. Lindvig, M.L. Michelsen, G.M. Kontogeorgis, Ind. Eng. Chem. Res. 43 (2004) 1125–1132. [19] T. Cha, J.M. Prausnitz, Ind. Eng. Chem. Process Des. Dev. 24 (1985) 551–555. [20] S.Y. Oh, Y.C. Bae, J. Phys. Chem. B 114 (2010) 8948–8953. [21] J.H. Yang, Y.C. Bae, J. Chem. Phys. 129 (2008) 064902. [22] S.S. Patnaik, R. Pachter, Polymer 43 (2002) 415–424. [23] J.S. Oh, Y.C. Bae, Polymer 39 (1998) 1149–1154. [24] S.Y. Oh, Y.C. Bae, Polymer 49 (2008) 4469–4474. [25] S.L. Mayo, B.D. Olafson, W.A. Goddard, J. Phys. Chem. 94 (1990) 8897–8909. [26] K.F. Freed, J. Phys. A: Math. Gen. 18 (1985) 871–877. [27] M.G. Bawendi, K.F. Freed, J. Chem. Phys. 88 (1988) 2741–2756. [28] J. Dudowicz, K.F. Freed, Macromolecules 24 (1991) 5076–5095. [29] C. Wohlfarth, CRC Handbook of liquid–Liquid Equilibrium Data of Polymer Solutions, CRC Press, Boca Raton, London and New York, 2007. [30] C.F. Fan, B.D. Olafson, M. Blanco, S.L. Hsu, Macromolecules 25 (1992) 3667–3676. [31] M. Blanco, J. Comp. Chem. 12 (1991) 237–247. [32] R.E. Treybal, Liquid Extraction, 2nd ed., McGraw-Hill, New York, 1963.
  6. 6. S.Y. Oh, Y.C. Bae / Fluid Phase Equilibria 307 (2011) 202–207 207 [33] M.D. Gehlsen, J.H. Rosedale, F.S. Bates, G.D. Wignall, L. Hansen, K. Almadal, Phys. Rev. Lett. 68 (1992) 2452–2455. [34] P.-G. de Gennes, Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca and London, 1979. [35] R. Koningsveld, L.A. Kleintjens, Macromolecules 4 (1971) 637–641. [36] Y. Hu, X. Ying, D.T. Wu, J.M. Prausnitz, Macromolecules 26 (1993) 6817– 6823. [37] L. An, B.A. Wolf, Macromolecules 31 (1998) 4621–4625.

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