Solvent

Article pubs.acs.org/jced

Continuous Thermodynamics of Polydisperse Polymer/Solvent Systems: Polystyrene/Cyclohexane and Polystyrene/Methyl Acetate Mixtures Han Earl Yang and Young Chan Bae* Department of Chemical Engineering and Molecular Thermodynamics Laboratory, Hanyang University, Seoul 133-791, Korea ABSTRACT: Various types of liquid−liquid equilibrium of polydisperse polymer solutions, including upper critical solution temperature (UCST), both UCST and lower critical solution temperature (LCST), and closed loop miscibility, are investigated. Determination of the cloud-point curves for polydisperse polystyrene (PS)/cyclohexane and PS/methyl acetate systems is conducted by thermal optical analysis. In addition, the molecular weight distribution of polydisperse PS is determined by a gel permeation chromatography technique. To describe the phase equilibria of polydisperse polymer solutions, a continuous thermodynamic approach is incorporated into the incompressible lattice model. This appropriately describes the characteristic phase behaviors of polydisperse polymer solutions, and the calculated results show good agreement with the experimental data.

a polymer. Ratzsch and co-workers,13 and Cotterman and Prausnitz14 reported a functional approach that incorporates the chain length distribution of a polymer into the thermodynamic model. Hu et al.15,16 applied the continuous thermodynamic approach to a close-packed lattice model based on Freed’s theory.17−19 Similar works for polydisperse polymer solutions were also reported by Choi and Bae20 and Enders and co-workers.21,22 In addition, applications of continuous thermodynamics to equations of state, including statistical associating fluid theory23,24 and lattice fluid theory,25 have been reported by several researchers. As noted by Hu et al.,15 the primary advantage of the continuous thermodynamic approach is its convenience, which arises from replacing the summation over all components with integration. It should also be noted that the continuous thermodynamic method does not suffer from the arbitrary choice of pseudocomponents because of its usage of the molecular weight distribution function. In this work, we investigate the phase behaviors of a polydisperse polymer solution with the continuous thermodynamic method. The molecular weight distribution of polystyrene (PS) is determined by gel permeation chromatography (GPC). Using a thermal optical analysis (TOA) method, cloud points of the PS/cyclohexane and PS/methyl acetate

1. INTRODUCTION Polymers have many different properties from other materials, not only because of their large molecular weight, but also because most synthetic polymers are polydisperse.1 Here, a “polydisperse polymer” means that the polymer is a mixture of molecules with different chain lengths, which is usually described by a continuous distribution function. It is wellknown that the polydispersity of a polymer has a significant effect on the properties of polymer melts and polymer solutions, including the viscosity,2,3 interfacial tension,4−6 adsorption property,7 and liquid−liquid equilibrium (LLE) of polymer solutions.1 Two main approaches have been used to estimate these properties of polydisperse polymer solutions: the discrete method and the continuous method. The discrete method describes the polydisperse polymer as pseudocomponent systems. In the early stage, the prototype for a polydisperse polymer, which is composed of two monodisperse polymers and a solvent, was investigated theoretically by Tompa.8 Similar works were conducted by Koningsveld and Staverman,9 Flory and Schultz,10 Fujita and co-workers,11 and Yang and Bae.12 However, the discrete method is a crude and arbitrary procedure because the calculated results highly depend on the choice of the pseudocomponents.13 Furthermore, the expense of the phase equilibrium calculation increases with the number of pseudocomponents. To overcome these drawbacks, several researchers developed continuous thermodynamic approaches, in which chemical potentials are defined using the molecular weight distribution of © 2016 American Chemical Society

Special Issue: Proceedings of PPEPPD 2016 Received: June 30, 2016 Accepted: October 13, 2016 Published: October 21, 2016 4104

DOI: 10.1021/acs.jced.6b00555 J. Chem. Eng. Data 2016, 61, 4104−4109

Journal of Chemical & Engineering Data

Article

systems are determined; these exhibit the upper critical solution temperature (UCST)-type and both UCST and lower critical solution temperature (LCST)-type, respectively. Phase equilibrium calculations of these systems are conducted with the continuous thermodynamic approach. In addition, hypothetical descriptions of a polydisperse polyethylene glycol (PEG)/water system, which exhibits closed loop miscibility, are also conducted to investigate the effect of the molecular weight distribution on the various types of LLE. The proposed thermodynamic model appropriately describes the phase behavior of polydisperse polymer solutions and the calculated results show reasonable agreement with the experimental data.

2. EXPERIMENTS 2.1. Chemicals. Polystyrene (PS), cyclohexane, and methyl acetate are obtained from Aldrich and used without further purification. The weight-average molecular weight, numberaverage molecular weight, and molar mass distribution of polydisperse PS were determined by Agilent 1100 S gel permeation chromatography (GPC). 2.2. Measurements for Molecular Weight Distribution of Polystyrene. The molecular weight distribution of polydisperse PS is analyzed by the GPC technique. GPC measurements are made with an Agilent 1100 S liquid chromatography. Tetrahydrofuran is used as the solvent, and the flow rate is 1 mL/min. The normalized experimental GPC data of polydisperse PS is tabulated in Table 1. The determined polydispersity index, weight-average molecular wegith (M̅ w) and number-average molecular weight (M̅ n) are 3.45, 272 320 g/mol and 78 958 g/mol, respectively.

Figure 1. Normalized molecular weight distribution of PS(M̅ w = 272 320 and M̅ n = 78 958). The solid line is a curve calculated with eq 1 and the open squares are the experimental data.

2.3. Measurements of Cloud Point. The cloud points of polydisperse PS solutions were measured via thermal optical analysis (TOA). To accurately measure cloud points, the polymer solution sample was cooled or heated at a scan rate of 0.5 °C/min; the detailed TOA procedure is described elsewhere.26 PS/cyclohexane and PS/methyl acetate systems are investigated and the determined experimental cloud points are listed in Table 2. Table 2. Cloud Point Data of the Cyclohexane/Polystyrene and Methyl Acetate/Polystyrene Systems: T Is Temperature and wi Is the Weight Fraction of Component i,a w2

Table 1. Normalized Experimental GPC Data of Polydisperse PS (M̅ w = 272 320 g/mol, M̅ n = 78 958 g/mol) Mw 1680 2410 3423 4827 6782 9523 13405 18958 26982 38668 55783

−6

W(Mw) × 10 0.15654 0.62790 1.15882 1.60113 1.91874 2.06872 2.13891 2.38594 2.53216 2.54026 2.64712

Mw 80914 117785 171648 249745 361860 521039 744819 1057700 1496530 2122430 3047260

0.002 0.005 0.01 0.03 0.05

−6

W(Mw) × 10 2.64876 2.35331 1.84677 1.29151 0.84110 0.44670 0.19204 0.05751 0.01064 0.00100 0.00000

0.002 0.005 0.01 0.03 0.05 0.002 0.005 0.01 0.03 0.05

Figure 1 shows the GPC data of polydisperse PS and calculated molecular weight distribution with the following Schulz−Flory type distribution function. ⎛ Mw ⎞ ⎛ Mw ⎞ k W (Mw) = ⎜ ⎟ exp⎜ −k ⎟ Mw0Γ(k + 1) ⎝ Mw0 ⎠ ⎝ Mw0 ⎠ k+1

where k is given by 1 k= = 0.4083 (M̅ w /M̅ n ) − 1

a

k

T (K)

w2

Cyclohexane + Polystyrene 296.68 0.08 297.58 0.1 299.11 0.12 300.63 0.15 299.78 0.18 Methyl Acetate + Polystyrene (UCST Region) 290.40 0.08 291.77 0.1 293.92 0.12 294.01 0.15 295.07 0.18 Methyl Acetate + Polystyrene (LCST Region) 406.34 0.08 405.81 0.1 404.55 0.12 401.92 0.15 401.80 0.18

T (K) 299.22 298.41 298.34 297.88 297.02 292.60 291.06 289.45 286.51 284.35 404.35 405.92 407.98 411.3 413.76

Standard uncertainties u are u(T) = 0.05 K, and u(wi) = 0.005.

(1)

3. MODEL DEVELOPMENT Recently, Jung and Bae27 developed a new Flory−Huggins model by incorporating a specific interaction based on a modified double lattice theory.28 This model has a much simple expression for the Helmholtz energy of mixing of polymer solution. In addition, it is applicable to describe the various types of phase equilibrium of polymer solution, such as UCST, LCST, and closed loop miscibility.

(2)

The parameter for the Schulz−Flory type molecular weight distribution, Mw0, is 78 960 which is equal to the numberaverage molecular weight. The calculated result shows good agreement with experimental GPC data. 4105



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where δεSij and δεHij are entropic and enthalpic contributions, respectively. 3.1. Continuous Thermodynamics. In the continuous thermodynamic framework, eq 3 is rewritten as

The Helmholtz energy of mixing from the Flory−Huggins theory is given by ϕ ϕ ΔA mix = 1 ln ϕ1 + 2 ln ϕ2 + χϕ1ϕ2 NrkT r1 r2

(3)

ϕ ΔA mix = 1 ln ϕ1 + NrkT r1

where ϕi and ri are the volume fraction and segment number of component i. Additionally, χ is an energy parameter for the i and j components which is expressed by zεij̃

χi , j =

2

=

zε 2kT

+ ϕ1

Ni , jkT

∫0

zCαδεij̃ (1 − η)η ⎤ ⎥ + 1 + Cαδεij̃ (1 − η)η ⎥⎦

=

εi , j kT εi , j kT +

+2

+

εij̃ = =

kT εi , j kT

−

−

Δμ2 (r ) kT

δεij

(6)

kT

(10)

and

with

Mw0 = m × rn

(11)

= ln(ϕ2W (r )) + 1 −

rϕ ⎛ r⎞ r + 2 ⎜1 − 1 ⎟ r1 r1 ⎝ rn ⎠ (12)

and

Δμ2α (r ) = Δμ2β (r )

(13)

which leads to following two equations. ⎛1 − ϕ β ⎞ ⎛ r1 ⎞ α α 2 2 ⎟ ln⎜⎜ α ⎟ = ϕ2 ⎜1 − α ⎟ + r1(ϕ2 ) χ 1 − ϕ r ⎝ ⎠ ⎝ n 2 ⎠

2 ⎡⎢ η ln η + (1 − η) ln(1 − η) z ⎢⎣

⎞ ⎛ ⎛ r ⎞ − ⎜⎜ϕ2 β ⎜⎜1 − 1β ⎟⎟ + r1(ϕ2 β )2 χ ⎟⎟ rn ⎠ ⎠ ⎝ ⎝

(14)

⎛ϕ α⎛ ⎛ ϕ β W β (r ) ⎞ r ⎞ ln⎜⎜ 2 α α ⎟⎟ = rσ = r ⎜⎜ 2 ⎜1 − 1α ⎟ + (1 − ϕ2 α)2 χ rn ⎠ ⎝ ϕ2 W (r ) ⎠ ⎝ r1 ⎝

(7)

⎛ϕ β⎛ ⎞⎞ r ⎞ − ⎜⎜ 2 ⎜⎜1 − 1β ⎟⎟ + (1 − ϕ2 β )2 χ ⎟⎟⎟⎟ rn ⎠ ⎝ r1 ⎝ ⎠⎠

T ·δεijS

kT

k ⎛ r⎞ kk + 1 ⎛ r ⎞ ⎜ ⎟ exp⎜ −k ⎟ rn Γ(k + 1) ⎝ rn ⎠ ⎝ rn ⎠

Δμ1α = Δμ1β

NijkT

−

ϕ2W (r ) dr = ϕ2

where r1 is the chain length of the solvent equal to unity. The equality of chemical potential between separated phases, denoted as α and β, will be given by

ΔA sec, ij

=

∞

+ r(1 − ϕ2)2 χ

In addition, temperature dependence in δε̃ij is adopted in order to account for the complex specific interaction.29 δεij̃ =

∫0

and

⎛ r⎞ = ln(1 − ϕ2) + ϕ2⎜1 − 1 ⎟ + r1ϕ2 2χ kT rn ⎠ ⎝

2 ⎡⎢ η ln η + (1 − η) ln(1 − η) z ⎢⎣

δεijH

(9)

Δμ1

NijkT

zCαδεij̃ (1 − η)η ⎤ ⎥ + 1 + Cαδεij̃ (1 − η)η ⎥⎦

χϕ2W (r ) dr

where r n is the average chain length defined as ∞ 1/∫ r −1W (r ) dr and the coefficient m is an adjustable 0 parameter. 3.2. Phase Equilibrium Calculation. To calculate the phase equilibrium, the chemical potentials of each component are required. The chemical potentials of the solvent and polydisperse polymer are given by

In the opposite case (i.e., specific interaction occurs between i− i or j−j), the energy parameter is given by εi , j

ln(ϕ2W (r )) dr

∞

W (r ) d r = 1

Mw = m × r

(5)

ΔA sec, ij

zCαδεij̃ (1 − η)η ⎤ ⎥ 1 + Cαδεij̃ (1 − η)η ⎥⎦

r

∞

W (r ) =

where δε̃ij(δεij/kT) is the reduced energy parameter for the specific interactions, Ni,j is the number of i−j pairs, and η is the fraction of specific interactions, which is set to 0.3. The universal constant Cα is determined by fitting with the Monte Carlo simulation data of the Ising lattice; this has a value of 0.4881.28 If a specific interaction occurs between the i and j components, the reduced interaction parameter, ε̃ij, is replaced by εij̃ =

ϕ2W (r )

We assume that the segment number parameter, r, has a linear dependence on the molecular weight and eq 1 is converted into the following equation after normalization.

2 ⎡⎢ η ln η + (1 − η) ln(1 − η) z ⎢⎣

=

∞

where subscripts 1 and 2 denote the solvent and polymer, respectively. W(r) is the distribution function of the polymer segment number and satisfies the following normalization conditions.

(4)

where ε̃ij is the reduced interaction parameter and z is a coordination number that is equal to six for the simple cubic lattice assumption. To incorporate the contribution of a specific interaction, a secondary lattice concept from the modified double lattice theory is adopted. The expression for the Helmholtz energy of mixing of the secondary lattice is28 ΔA sec, i , j

∫0

∫0

(8)

(15) 4106



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From the normalization condition, eq 15 is converted into ϕ2 β =

∫0

∞

ϕ2 αW α(r ) exp(rσ ) dr

Table 3. Adjustable Model Parameters for the Studied Polymer Solutions

(16)

system PS (Mn = 79 000, Mw = 272 320) + cyclohexane

Because the applied thermodynamic model, eq 9, does not contain the moments of the distribution function, the occurring integrals in eq 16 can be solved analytically. After rearrangements, eq 16 results in the following symmetric relations based on the assumption that the distribution function for each separated phase is the Schulz−Flory type.13 ϕ2α α k+1

(rn )

=

PS (Mn = 100 000, Mw = 324 000) + cyclohexane PS (Mn = 79 000, Mw = 272 320) + methyl acetate

ϕ2β

PEG + water

β k+1

(rn )

(17)

ri

energy parameters

r1 = 1 rαn = 245.44 r1 = 1 rαn = 450.00 r1 = 1 rαn = 640.889 r1 = 1 rαn = 25.61

ε/k = 392.016 δε/k = 5828.45 ε/k = 375.36 εH/k = 10119.2 εS/k = 16.14 εS/k = −956.92 δεH/k = 23116.8 δεS/k = 28.02

ref this work 30 this work 12

This symmetric relation permits us to simplify the phase equilibrium calculation because the integrals in eq 16 are converted into analytic equation. Cloud-point curve and shadow curve are obtained from solving eq 14, 15, and 17, simultaneously. The second- and third-order derivatives of the Helmholtz energy of mixing are necessary for the critical point calculation. The second-order derivative, spinodal criterion, gives ∂ 2(ΔA mix /NrkT ) ∂ϕ1

2

=

1 1 + − 2χ = 0 r1(1 − ϕ2) v1ϕ2

(18)

The third-order derivative, critical-point criterion, is expressed as ∂ 3(ΔA mix /NrkT ) ∂ϕ13

=−

v 1 + 22 3 = 0 r1(1 − ϕ2)2 ϕ2 v1

(19)

Figure 2. Cloud and shadow curve for the PS/cyclohexane system. Squares are experimental data. Gray lines are calculated results for PS(M̅ w = 324 000 and M̅ n = 100 000)/cyclohexane system.30 Black lines are calculated results for PS(M̅ w = 272 320 and M̅ n = 78 958)/ cyclohexane system. Solid and dashed lines are calculated cloud points and shadow curve, respectively.

∞ k r W (r ) 0

dr . The critical point calculation is where vk = ∫ conducted by solving eqs 18 and 19, simultaneously. 3.3. Molecular Weight Distribution in Each Separated Phase. The cloud-point curve corresponds to the distribution of the initial polymer which is denoted by the α phase. The conjugated phase (shadow curve), which is denoted by β, always has a different distribution from that of the principal phase except for at the critical point. The molecular weight distribution of the conjugated phase, Wβ(r), is calculated using the weight distribution of the initial polymer, and the following relation is obtained from eq 15. W β (r ) =

ϕ2 α ϕ2 β

W α(r ) exp(rσ ) (20)

4. RESULTS AND DISCUSSION In the continuous thermodynamic framework, the molecular weight distribution of the investigated polymer must be estimated precisely. Using the molecular weight distribution of polydisperse PS (listed in Table 1), the adjustable parameters in the interaction energy parameter are fitted to the experimental cloud-point data. The calculated parameters are listed in Table 3. Figure 2 shows the cloud-point curve and shadow curve of the PS/cyclohexane system which shows UCST-type phase behavior. The phase equilibrium of PS(M̅ w = 324 000 and M̅ n = 100 000)/cyclohexane system30 is also investigated to compare the new experimental data. The calculated molecular weight distributions in the cloud-point curve and shadow curve are illustrated in Figure 3. As shown in Figure 2, the volume

Figure 3. Calculated molecular weight distribution for the PS(M̅ w = 272 320 and M̅ n = 78 958)/cyclohexane system. The red line corresponds to the cloud-point curve. The blue line represent the distribution in shadow curve at T = 299.10 K, ϕα2 = 0.075, and ϕβ2 = 0.031. The green line represents the distribution in shadow curve at T = 300.48 K, ϕα2 = 0.025, and ϕβ2 = 0.054.

fraction of PS in each equilibrated phase has the following relationships. 4107


Journal of Chemical & Engineering Data ϕ2 α < ϕ2 β

when

ϕ2 α < ϕ2 crit

when

ϕ2 α > ϕ2 crit

Article

and ϕ2 α > ϕ2 β

(21)

ϕcrit 2

where is the volume fraction of PS at the critical state. In other words, the cloud-point curve represents the polymer-poor phase in the case of ϕα2 < ϕcrit 2 and vice versa. In terms of the molecular weight distribution, the principal phase has a smaller portion of long chains relative to the conjugated phase when ϕα2 < ϕcrit 2 ; this is the case because the conjugated phase is a polymer-rich phase. The green line in Figure 3 shows the calculated molecular weight distribution in this case and the opposite case is also illustrated as the blue line in Figure 3. It should be noted that the experimental determination of shadow curves using the TOA method is difficult because the conjugated phase has a very small volume.1,20 Therefore, in this work, the shadow curves and the molecular weight distribution in the shadow curve are estimated from the thermodynamic model. Both UCST-type and LCST-type LLEs are also investigated with the polydisperse PS/methyl acetate system. The cloudpoint curve and shadow curve are illustrated in Figure 4. This

Figure 5. Calculated molecular weight distribution for the PS(M̅ w = 272 320 and M̅ n = 78 958)/methyl acetate system. The red line corresponds to the cloud-point curve. The blue line represent the distribution in shadow curve at T = 404.84 K, ϕα2 = 0.075, and ϕβ2 = 0.036. The green line represents the distribution in the shadow curve at T = 401.62, ϕα2 = 0.025, and ϕβ2 = 0.061.

Figure 4. Cloud and shadow curves for the PS(M̅ w = 272 320 and M̅ n = 78 958)/methyl acetate system. The squares are the experimental cloud-point data and the open triangle is the calculated critical point. The solid and dashed lines are the calculated cloud-point curve and shadow curve, respectively.

system also satisfies the relationship given in eq 21. The calculated results for the molecular weight distributions in the cloud point and shadow curves are shown in Figure 5. In the polymer-rich phase, the calculated distribution function is located on the long chain-length region, whereas the polymer poor phase has distribution with short chain length. The calculated results for the PS/cyclohexane and PS/methyl acetate systems show good agreement with the experimental data. In addition, the calculated molecular weight distributions in each equilibrated phase are consistent with the polymer solution LLE behavior. A hypothetical description of the polydisperse PEG/water system is also conducted to investigate the effect of polydispersity on closed-loop type LLE. To determine the energy parameter of this system, LLE data of monodisperse PEG (Mw = 4720)/water are considered.12 Figure 6 illustrates the LLE of the PEG/water system and the molecular weight distribution of each PEG. As indicated in Figure 6a, the chainlength distribution becomes broad and the polydispersity of the

Figure 6. Cloud and shadow curves for the PEG/water system. The squares are experimental LLE data of the monodisperse PEG(Mw = 4720)/water system.12 The red line corresponds to the monodisperse PEG/water system. The green and blue lines are hypothetical polydisperse PEG/water systems with k = 3.33 and 1, respectively. (a) Cloud point and shadow curves for the PEG/water systems and (b) molecular weight distributions of PEG.

polymer increases as the parameter k of the chain-length distribution function decreases. Furthermore, the critical temperature and separation region increase as the polydisper4108



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(11) Tsuyumoto, M.; Einaga, Y.; Fujita, H. Phase Equilibrium of the Ternary System Consisting of Two Monodisperse Polystyrenes and Cyclohexane. Polym. J. 1984, 16, 229−240. (12) Yang, H. E.; Bae, Y. C. Effects of polydispersity on liquid-liquid equilibrium of polymeric fluids. Fluid Phase Equilib. 2016, 417, 220− 228. (13) Ratzsch, M. T.; Kehlen, H. Continuous thermodynamics of polymer systems. Prog. Polym. Sci. 1989, 14, 1−46. (14) Cotterman, R. L.; Prausnitz, J. M. Flash Calculations for Continuous or Semicontinuous Mixtures Using an Equation of State. Ind. Eng. Chem. Process Des. Dev. 1985, 24, 434−443. (15) Hu, Y.; Ying, X.; Wu, D. T.; Prausnitz, J. M. Liquid-Liquid Equilibria for solutions of Polydisperse Polymers. Continuous Thermodynamics for the Close-Packed Lattice Model. Macromolecules 1993, 26, 6817−6823. (16) Hu, Y.; Ying, X.; Wu, D. T.; Prausnitz, J. M. Continuous thermodynamics for polydisperse polymer solutions. Fluid Phase Equilib. 1995, 104, 229−252. (17) Freed, K. F. New lattice model for interacting, avoiding polymers with controlled length distribution. J. Phys. A: Math. Gen. 1985, 18, 871−887. (18) Dudowicz, J.; Freed, K. F. Effect of Monomer Structure and Compressibility on the Properties of Multicomponent Polymer Blends and Solutions: 1. Lattice Cluster Theory of Compressible Systems. Macromolecules 1991, 24, 5076−5095. (19) Madden, W. G.; Pesci, A. I.; Freed, K. F. Phase Equilibria of Lattice Polymer and Solvent: Test of Theories against Simulations. Macromolecules 1990, 23, 1181−1191. (20) Choi, J. J.; Bae, Y. C. Liquid-liquid equilibria of polydisperse polymer systems: applicability of continuous thermodynamics. Fluid Phase Equilib. 1999, 157, 213−228. (21) Enders, S.; Browarzik, D. Modeling of the (liquid + liquid) equilibrium of polydisperse hyperbranched polymer solutions by lattice-cluster theory. J. Chem. Thermodyn. 2014, 79, 124−134. (22) Zeiner, T.; Browarzik, C.; Browarzik, D.; Enders, S. Calculation of the (liquid + liquid) equilibrium of solutions of hyperbranched polymers with the lattice-cluster theory combined with an association model. J. Chem. Thermodyn. 2011, 43, 1969−1976. (23) Chapman, W. G.; Sauer, S. G.; Ting, D.; Ghosh, A. Phase behavior applications of SAFT based equations of statefrom associating fluids to polydisperse, polar copolymers. Fluid Phase Equilib. 2004, 217, 137−143. (24) Paricaud, P.; Galindo, A.; Jackson, G. Examining the effect of chain length polydispersity on the phase behavior of polymer solutions with the statistical associating fluid theory (Wertheim TPT1) using discrete and continuous distributions. J. Chem. Phys. 2007, 127, 154906. (25) Krenz, R. A.; Laursen, T.; Heidemann, R. A. The Modified Sanchez−Lacombe Equation of State Applied to Polydisperse Polyethylene Solutions. Ind. Eng. Chem. Res. 2009, 48, 10664−10681. (26) Bae, Y. C.; Lambert, S. M.; Soane, D. S.; Prausnitz, J. M. CloudPoint Curves of Polymer Solutions from Thermooptical Measurements. Macromolecules 1991, 24, 4403−4407. (27) Jung, J. G.; Bae, Y. C. Liquid-Liquid Equilibria of Polymer Solutions: Flory-Huggins with Specific Interaction. J. Polym. Sci., Part B: Polym. Phys. 2010, 48, 162−167. (28) Oh, J. S.; Bae, Y. C. Liquid-liquid equilibria for binary polymer solutions from modified double lattice model. Polymer 1998, 39, 1149−1154. (29) Choi, J. S.; Yang, H. E.; Lee, C. H.; Bae, Y. C. Comparison of thermodynamic lattice models for multicomponent mixtures. Fluid Phase Equilib. 2014, 380, 100−115. (30) Handbook of Liquid-Liquid Equilibrium Data of Polymer Solutions, Wohlfarth, C., Ed.; CRC Press: Boca Raton, pp131−132, 2008.

sity of polymer increases; this is the case because the distribution function possesses a larger chain-length region compared to that of the monodisperse one.

5. CONCLUSIONS The cloud points of the polydisperse PS/cyclohexane and PS/ methyl acetate systems were measured using a TOA method. The experimental molecular weight distribution of PS was also determined via GPC. In addition, hypothetical descriptions of the polydisperse PEG/water system, which shows closed loop miscibility, were investigated to consider the effect of polymer polydispersity on the various types of LLE. The continuous thermodynamic approach permits phase equilibrium calculations of polydisperse polymer solutions by direct use of the molecular weight distribution, which is obtained experimentally. The cloud-point curve, shadow curve, and molecular weight distribution of each equilibrated phase were estimated from the incompressible lattice model incorporated with a Schulz−Flory-type distribution function. The thermodynamic model that was used showed good agreement with the experimental data and provided a simple calculation procedure to determine the phase equilibrium of the polydisperse polymer solution.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Tel.: +82-2-2298-0529. Fax: +82-2-2296-0568. Funding

This research was respectfully supported by Engineering Development Research Center (EDRC) funded by the Ministry of Trade, Industry & Energy (MOTIE) (No. N0000990). Notes

The authors declare no competing financial interest.

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REFERENCES

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