9678
Ind. Eng. Chem. Res. 2008, 47, 9678–9686
Molecular Thermodynamic Model of Multicomponent Chainlike Fluid Mixtures Based on a Lattice Model Qin Xin,† Changjun Peng,*,† Honglai Liu,*,‡ and Ying Hu‡ Laboratory for AdVanced Materials, State Key Laboratory of Chemical Engineering, and Department of Chemistry, East China UniVersity of Science and Technology, Shanghai 200237, China
The molecular thermodynamic model of polymer solutions based on a close-packed lattice model presented in a previous work has been generally extended to multicomponent chainlike fluid mixtures. The Helmholtz function of mixing contains three terms, i.e., the contribution of athermal mixing of polymer chains, which is calculated by Guggenheim’s theory; the contribution of nearest-neighbor interactions between monomers, which is calculated by Yang et al.’s model of the Helmholtz function of mixing for a multicomponent Ising lattice; and the contribution of the formation of polymer chains from monomers, which is obtained according to the sticky-point theory of Cummings, Zhou, and Stell. The liquid-liquid phase equilibria of ternary chainlike mixtures predicted by this model are in good agreement with Monte Carlo simulation results and superior to the results calculated by Flory-Huggins (FH) theory and revised Freed theory (RFT) obviously. This model not only can describe types 1-3 phase separations of Treybal classification satisfactorily, but can also correlate well the coexistence curves of binary polymer blends systems with an upper critical solution temperature (UCST) or a lower critical solution temperature (LCST). Meanwhile, model parameters correlated from the binary system can be further extended to predict the corresponding liquid-liquid equilibrium of ternary mixtures, including systems of ionic liquids. 1. Introduction Generally, it is very difficult for two kinds of macromolecular polymers to form a homogeneous phase because the chains of different polymers are not miscible at a molecular-level except when there are some strong interaction forces between them. Rather, two or more separate phases usually appear owing to their respective cohesion. Adding some polymers to binary systems can always result in the formation of some classical liquid-liquid equilibria, such as islands and one, two, or three pairs of partially miscible liquids, which correspond to types 0-3, respectively, of the Treybal classification.1 Therefore, polymer blends can be obtained only by special methods. For example, during the course of manufacture of polymer materials, polymer blend solutions are usually obtained by dissolving different polymers in single or mixed solvents first and then eliminating the solvent. Thus, establishing a more advanced model for these multicomponent mixtures is quite necessary for guidance in the preparation and exploitation of polymer materials. The lattice model, as one of the most popular models used in theories of polymer solutions, has marked advantages. After more than fifty years’ development, lattice models of polymer systems, such as binary homopolymer or copolymer solutions, were employed to correlate well with the corresponding Monte Carlo (MC) simulation data and make good predictions for real systems. However, as shown by the simulated results,2-7 the results predicted by all of the existing lattice models8-13 do not match well with MC-simulated results for ternary chainlike molecular systems, which indicates that models of multicomponent mixtures still need to be developed further. Many * To whom correspondence should be addressed. Tel.: 86-2164252921. Fax: 86-21- 64252921. E-mail:
[email protected] (C.P.),
[email protected] (H.L.). † Laboratory for Advanced Materials and Department of Chemistry. ‡ State Key Laboratory of Chemical Engineering and Department of Chemistry.
researchers have carried out useful works based on the lattice model from different aspects and have made great achievements. Kambour et al.14 constructed a model of polymer-copolymer systems based on Flory-Huggins (F-H) theory.8-10 They noticed that the various segmental interactions of copolymers are crucial to the degree of mixing. Based on the mean-field theory of Kambour et al.,14 ten Brinke et al.15 extended the model to random copolymer-copolymer systems and found that miscibility in these systems does not require any specific interaction but rather a “repulsion” between the different covalently bonded monomers of the copolymers. Hino et al.16 developed a lattice theory for liquid-liquid equilibrium of binary systems containing random copolymers by taking into account deviations from random mixing through a nonrandomness factor. To overcome the problems brought by the mean-field theory, Freed11 developed a rigorous analytical solution of the F-H lattice. Lattice cluster theory (LCT), as another landmark in the development of lattice theory, was extended to study more different systems by Freed et al., such as random copolymerhomopolymer systems17 and polymer blend systems.18,19 However, the enormous algebraic complexity brought by LCT is a nonnegligible problem that limits its application to engineering purposes. To reach the precision of LCT, Hu et al.12,13 reported a simple double-lattice model by introducing some necessary parameters to the truncated LCT. This revised Freed theory (RFT) is consistent with MC simulations of coexistence curves of polymer systems.20-22 Subsequently, Chang et al.23 revised RFT by introducing new universal constants to take into account Table 1. Molecular Model Parameters of Six Systems system
r1
r2
r3
ε˜ 12
ε˜ 13
ε˜ 23
a b c d e f
1 1 1 1 1 1
4 4 4 4 4 4
16 24 32 8 8 8
0.250 0.250 0.250 0.333 0.333 0.333
0.500 0.500 0.500 0.556 0.556 0.556
-0.100 -0.100 -0.100 0.100 0.000 -0.200
10.1021/ie800924r CCC: $40.75 2008 American Chemical Society Published on Web 11/01/2008
Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9679
Figure 1. Phase diagrams of ternary liquid-liquid equilibria with type 1 phase separation: (O) simulated results, (- - -) Flory-Huggins model, ( · · · ) RFT, (s) our model, (g) critical point from our model. Table 3. Values of Model Parameters for Systems with a UCST
Table 2. Molecular Model Parameters of Eight Systems system
r1
r2
r3
ε˜ 12
ε˜ 13
ε˜ 23
a b c d e f g h
1 1 1 1 1 1 1 1
4 4 4 4 4 4 4 4
32 24 16 16 16 16 16 16
0.57 0.57 0.57 0.57 0.57 0.57 0.57 0.60
0.42 0.42 0.42 0.42 0.42 0.42 0.43 0.45
0.09 0.09 0.09 0.05 0.00 0.14 0.17 0.25
the chain length dependence of a polymer in a solvent. Later, Chang and Bae24 extended and simplified the modified double-lattice model for polymer solutions to binary polymer blend systems. By comparison with MC simulation data,25,26 their model can describe well the phase behaviors of ordinary and oriented polymer blend systems. Three universal model parameters are included in the model: Cβ and Cγ for the primary lattice and CR for a secondary lattice. Meanwhile,
system a b c d e f g
r1
r2
PBD(1100)-PS(1340) 74.26 83.55 PBD(1200)-PS(2350) 74.82 158.65 PBD(1100)-PS(4370) 74.26 272.47 PaMeS(62100)-PS(58400) 3802.91 3641.18 PaMeS(56100)-PS(49000) 3435.48 3055.10 PEGE(750)-PPG(2000) 43.20 121.79 PEGE(550)-PPG(2000) 31.68 121.79
δε1/k (K) δε2/k2 (K2) 0.65 -0.75 -2.82 -8.09 0.47 -33.25 -28.52
1047.69 1590.60 2345.68 34.99 74.54 15485.05 12166.93
to reduce the mathematical defects, Chang and Bae27 redefined Cγ as a new universal function of interaction energy with Ryu and Gujrati’s numerical results.28 Unfortunately, there is no further report on their model being extended to multicomponent mixtures. It is still a great significance to develop analytically simple, improved theories with a minimal number of adjustable parameters for multicomponent mixtures. Recently, we developed a mixing Helmholtz function model for multicomponent Ising lattice.29 Combined with the
9680 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008
Figure 2. Phase diagrams of ternary liquid-liquid equilibria with types 2 and 3 phase separation: (O) simulated results, (- - -) Flory-Huggins model, ( · · · ) revised Freed theory, (s) our model, (g) critical point from our model.
method of Cummings, Zhou, and Stell,30-32 we constructed a molecular thermodynamic model for mixed-solvent polymer solutions33and random copolymer solutions.34 As a continuation of that effort, we present here a molecular thermodynamic model for multicomponent mixtures based on the
lattice model following the same ideas as we did in ref 33. Section 2 of this article shows the development of the model. Comparisons between the model and simulation results are made in section 3. The final section presents an overall conclusion.
Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9681 Table 4. Values of Model Parameters for PVME-PS Systems with a UCST system a b c d
PVME(95000)-PS(67000) PVME(95000)-PS(10600) PVME(99000)-PS(50000) PVME(99000)-PS(100000)
δε1/k (K) δε2/k2 (K2)
r1
r2
5712.87 5712.87 5953.41 5953.41
4177.38 6608.99 3117.45 6234.90
0.88 2.15 2.40 1.96
-316.96 -816.73 -941.51 -751.25
2. Molecular Thermodynamic Model We consider a multicomponent system that is composed of K components. Ni and ri are the number of molecules and chain length, respectively, of component i. ri ) 1 means that the ith component is solvent. The total number of lattice sites is Nr ) ∑Ki ) 1Niri. Only the nearest-neighbor interactions are considered. To obtain the Helmholtz function of mixing, ∆mixA, of multicomponent chainlike fluid mixtures, we adopt an approach that is analogous to our previous work:33,34 first, decomposing the pure polymers into monomers; second, mixing all of these monomers together; and finally, reconnecting all of the monomers to the final polymer solution. Because of the mutual compensation of the bond energies involved in the dissociation step and the association step, the expression for the Helmholtz function of mixing ∆mixA is written as r K ∆mixS0 ∆mixAIsing Ni ln g(ri) ∆mixA )+ NrkT Nrk NrkT Nr i)1
∑
)-
r ∆mixAIsing
∆mixS0 + Nrk
NrkT
K
-
∑ i)1
˜εij ) ∈ij/kT ) (εii + εjj - 2εij)/kT
∈ij is the exchange interaction energies between segments i and j. εij is the interaction energy between segments i and j. For the last term of eq 1, we followed the same method as Cummings, Zhou, and Stell30-32 and further considered the longrange correlations beyond the nearest-neighbor interactions as Yang et al. did.36 The parameter λi was first introduced by Yang et al.36 for binary polymer solutions, and its expression was established by correlating MC simulation data of critical temperatures and compositions for two systems with chain lengths of 4 and 200 in ref 36 as λi )
(ri - 1)(ri - 2) r2i
(ari + b) with a ) 0.1321 and b ) 0.5918 (7)
(2)
gi is the pair correlation function of component i in the corresponding Ising lattice system calculated by33 g(2) i )
φij )
∑
and θi )
∑Nr
i i
Niqi K
∑Nq
Nij 2Nii +
1/Γij )
ri(z - 2) + 2 (4) z The second term on the right-hand side of eq 1 is the residual Helmoltz function of mixing for a multicomponent Ising lattice and can be directly quoted from the Yang et al.’s model29 qi )
∑∑
M
φiφj˜εij -
j)1
M
˜εik - ˜εjk) +
M
M
j)1 k)1 l)1
)
(11)
and the constraint of normalization of local volume fraction K
φii +
∑φ
ij ) 1
(12)
j*i
we obtain g(2) i ) 1/
K K ∆mixA φi qi θi z K z ) ln φi + φi ln + NrkT r 2 i)1 ri φi 4 i)1 i)1 i
∑
∑
K
+
K
∑ ∑ φ φ ˜ε
i j ij
j)1
K
i j k ij
εik - ˜εjk) ij + ˜
j)1 k)1 K K K
∑ ∑ ∑ ∑ φ φ φ φ ˜ε (ε˜
z 16 i)1
i j k l ij
j)1 k)1 l)1
K
φiφjφkφl˜εij(ε˜ik + ˜εil - ˜εkl) (5)
K
∑ ∑ ∑ φ φ φ ˜ε (ε˜
c z 16 i)1 K
j)1 k)1
(13)
j ij
Finally, substituting eqs 2, 5, and 13 into eq 1 yields an expression for the Helmholtz function of mixing ∆mixA of multicomponent chainlike fluid mixtures, namely
M
φiφjφk˜εij(ε˜ij +
∑φΓ j)1
M
∑∑∑∑
z 16 i)1
M
∑∑∑
c z 16 i)1
Nil
K
i)1
M
(10)
K l*i
(
∑
(3)
qi is the surface area parameter, defined as
r M ∆mixAIsing z ) NrkT 4 i)1
∑
K 2Niiφj ˜εij + ˜εik - ˜εjk ) φk exp Nijφi k)1 2
i i
i)1
il
Combined with the nonrandom factor Γ in the multicomponent Ising lattice system29
K
where z is the coordination number of the lattice. φi and θi are the volume fraction and surface fraction, respectively, of component i and can be calculated by Niri
∑N
and correspondingly
(2)
K
(9)
l*i
φi qi θi ∆mixS0 ∆mixSGuggenheim z ) ) ln φi + φi ln Nrk Nrk r 2 i)1 ri φi i)1 i
φi )
(8)
K
2Nii +
where k is the Boltzmann constant and T is the absolute temperature. The first term on the right-hand side of eq 1 is the athermal entropy of mixing. It can be evaluated directly by using Guggenheim’s model,35 which has been confirmed to be one of the most accurate models for athermal entropy36 on closedpacked lattice model
∑
φii φi
2Nii
φii )
ri - 1 + λi φi ln g(2) i ri
K
(6)
where φii is the local volume fraction
(1)
-
where c ) 1.1 and ε˜ ij is the reduced exchange interaction energies between i and j segments, written as
+
∑ i)1
εil - ˜εkl) ik + ˜
(∑ )
ri - 1 + λi φi ln ri
K
φjΓij (14)
j)1
9682 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008
Figure 3. Phase diagrams of binary liquid-liquid equilibria: (]) experimental data, (*) critical point from our model, (s) calculated results from our model.
Figure 4. Phase diagrams of ternary liquid-liquid equilibria for benzene (1)-heptane (2)-diethylene glycol (3): (0) experimental data, (g) critical point from our model, (s) calculated results from our model.
This expression is a quite generalized lattice model that is suitable for various polymer systems. One can naturally and quickly obtain corresponding molecular thermodynamic models for binary polymer solutions36 and for mixed-solvent polymer solutions.33 For copolymer systems, the contribution of various segmental interactions can be taken into account by revising
34 g(2) i . In our previous work, by introducing a proper expression (2) for gi , we constructed a molecular thermodynamic model of random copolymer solutions that can be extended further to the construction of models for alternative copolymer or block copolymer solutions. When changing the number of components, the chain length, or the kind of polymer, we can obtain models
Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9683
Figure 5. Phase diagrams of binary liquid-liquid equilibria for [C2mim][NTf2] (2)-[P6 6 6 14][NTf2] (3): (]) experimental data, (*) critical point from our model, (s) calculated results from our model. Figure 8. Coexistence curves for the PaMeS-PS system: (0) system d, (∆) system e.
Figure 6. Phase diagrams of ternary liquid-liquid equilibria for [C4mim][NTf2] (1)- [C2mim][NTf2] (2)-[P6 6 6 14][NTf2] (3): (0) experimental data, (g) critical point from our model, (s) calculated results from our model. Figure 9. Coexistence curves for the PEGE-PPG system: (0) system f, (∆) system g.
Figure 7. Coexistence curves for the PBD-PS system: (O) system a, (0) system b, (∆) system c.
applicable to different systems including polymer solutions and polymer mixtures.
(β) (γ) µ(R) (i ) 1, 2,..., K) i ) µi ) ... ) µi
3. Results and Discussion 3.1. Comparisons with Other Theories and Simulation Results. We tried to verify the reliability of our model by making comparisons with some other lattice theories and MC simulation results. Before calculating the liquid-liquid equilibrium curve, the chemical potential of components should be obtained first from the expression
(
∂(∆mixA/kT) µi - µ0i ) kT ∂Ni By solving the equations
Figure 10. Coexistence curves for the PVME-PS system with an LCST: (0) system a, (∆) system b, (O) system c, (]) system d.
)
(15) T,V,Nj*i
(16)
one can obtain the coexistence curve. In addition, the spinodal line and the critical point can be calculated based on the thermodynamic principle. As shown in Figure 1, type 1 phase behavior (Treybal classification1) of a ternary chain mixture that has only one pair of partially miscible liquids is compared for F-H theory, RFT, our model, and MC simulation results. The MC results and the results of F-H theory and RFT are quoted from ref 7. Table 1 presents the corresponding six groups of molecular model parameters. Figure 1a-c reveals the influence of r3 on the phase equilibria when all other model parameters are fixed. The effect of ε˜ 23 on the phase
9684 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008
behavior is also depicted in Figure 1d-f. The comparisons show that there are different deviations among the three models. As exhibited in Figure 1, the Flory-Huggins model always shows a large discrepancy with the simulation results. RFT has clear improvements over the Flory-Huggins model, but still shows significant deviations. Our model is in good agreement with the MC simulation results, especially far from the critical point. However, the deviations of our model near the critical point become larger as the chain length increases. Figure 2 shows a comparison of the results of the three models and MC simulations for type 2 and type 3 phase behaviors.1 All of the results except for those from our model are quoted from ref 22. The first six groups of molecular model parameters in Table 2 can result in type 2 phase separation, which exhibits two separate regions for two-liquid equilibrium mixtures. The parameters in Table 2 for systems g and h exhibit three separate regions for two-liquid equilibrium mixtures as shown in Figure 2g,h, which indicates type 3 phase separation. A comparison of Figure 2h with Figure 2g shows that the areas of incomplete solubility have grown and a small central region of three equilibrium liquids has appeared because of the increase in interchange energy, which is analogous to a decrease in temperature. Because of the absence of MC results, we give only the results of our model in the last three diagrams in Figure 2. Upon comparison with the first five groups, as shown in Figure 2a-e, we found that Flory-Huggins theory cannot describe the phase situation well, whereas RFT provides a great improvement except for some deviations in the region of larger φ1 values. In contrast, our model can reproduce the MC results well. In comparison with Figure 2c and Figure 2f, our model can predict two separate phases when the energy parameter ε˜ 23 reaches 0.14 and three separate phases when ε˜ 23 reaches 0.17 in Figure 2g. The shift from type 2 to type 3 behavior arises solely from the change of energy parameter ε˜ 23. In Figure 2h, the three separate phases of Figure 2g can converge into one large phase by simultaneously increasing three energy parameters while maintaining the same ratio. Increasing the energy parameters in the model has the same effect as decreasing the temperature in experiments. For example, the ternary system of glycol-nitroethane-lauryl alcohol presents a phase diagram of three separate phases at 302.15 K. When the temperature is decreased to 295.15 K, the three separate binodal curves expand and eventually overlap, forming an internal triangle, which would indicate compositions separating into three liquid phases.37 This indicates that our model can qualitatively reproduce the experimental phenomena. Both Figure 1 and Figure 2 reflect that our model can illustrate the effects of the chain length and interchange energy on phase separation behaviors. 3.2. Application to Real Systems. When applying our model to real ternary systems, our attention is focused on rationalizing liquid-liquid equilibria of polymer systems only with the model parameters correlated from the binary systems. The model parameters, namely, chain lengths ri and interchange energy parameters ∈ij/k, must be estimated first. Generally, we regard the component having the smallest molar volume as a solvent and then obtain other parameters by solving eq 16. However, because it is too hard to correlate correctly the parameters in ternary systems with only the group of ternary experimental liquid-liquid equilibrium (LLE) data, we must first correlate the corresponding binary experimental liquid-liquid equilibria to obtain some parameters and then extend the results to calculate the ternary LLE. For binary blend systems, we can calculate the ri values first by one experimental expression. In
the following discussion, we provide applications of our model predicting some real systems, including binary and ternary systems. First, phase diagrams of ternary liquid-liquid equilibria for benzene (1)-heptane (2)-diethylene glycol (3) and the corresponding binary experimental liquid-liquid equilibria would be correlated by our model. All of the experimental data are taken from ref 38. In the binary system of benzene (1)-diethylene glycol(3), benzene is regarded as the solvent with a chain length of r1 ) 1.0. As shown in Figure 3a, we obtained the parameters r3 ) 12.0 and ∈13/k ) 148.8 K by correlating the critical point (wc ) 0.32, Tc ) 361.95 K). The model gives a good prediction of the experimental results. In the system of heptane (2)-diethylene glycol (3), the chain length of heptane (r2 ) 7.5) and the exchange energy (∈23/k ) 109.0 K) are also acquired by solving eq 12 with the LLE data at T ) 323.65 K. A comparison between our model result and the experimental data is depicted in Figure 3b. The remaining exchange energy, ∈12/k, can, in principle, be obtained by correlating the vapor-liquid equilibria or other thermodynamic data. Here, we obtained the parameter ∈12/k ) 39.8 K by correlating the ternary LLE at T ) 361.95 K. Thus far, we have already obatined the needed molecular model parameters through the correlation of the corresponding binary experimental liquid-liquid equilibria. With this information, phase diagrams for benzene (1)-heptane (2)-diethylene glycol (3) ternary system at temperatures of 348.15, 361.95, and 398.15 K and the corresponding critical points were calculated by our model. As shown in Figure 4a-c, the calculated data are consistent over the entire composition range and show few deviations from the experimental data. Meanwhile, the shrinking of the phase separation region with increasing temperature is predicted well, implying a shift of phase separation classification from type 2 to type 1. Because of a lack of experimental LLE data for ternary polymer mixtures, we correlated some other systems instead of polymer mixtures, such as ionic liquid systems. Ionic liquids (ILs) are a type of molten salts at ambient temperature and are used for a novel class of green benign solvents. Meanwhile, there is a strong analogy between the phase diagrams of ionic liquid and polymeric solutions.39 Lou et al.40 extended FH theory to derive the solvent power in the case of different sizes of the IL components based on a “polymerlike” model. Thus, we correlated the liquid-liquid equilibria of a ternary system of three ionic liquids: [C4mim][NTf2] (1)-[C2mim][NTf2] (2)-[P6 41 6 6 14][NTf2] (3). The chain lengths of ionic liquids are all treated as ri > 1 if there is no solvent in the system. Similar to the method mentioned above, the only group of the corresponding binary experimental LLE41 of [C2mim][NTf2] (2)-[P6 6 6 14][NTf2] (3) at 25 °C was also correlated by this model to obtain some model parameters. The values r2 ) 2, r3 ) 20, and ∈23/k ) 85.1 K are the optimized model parameters, and the relative errors between our model and experiments agree with the experimental accuracy as shown in Figure 5. Alternatively, we estimated the remaining parameters as r1 ) 5.0, ∈12/k ) 5.96 K, and ∈13/k ) 26.83K directly by the ternary LLE. Hence, the reduced interchange energies ∈12/kT ) 0.02, ∈13/kT ) 0.09, and ∈23/kT ) 0.29 were obtained. Figure 6 shows experimental phase diagrams and the results of our model for the [C4mim][NTf2] (1)-[C2mim][NTf2] (2)-[P6 6 6 14][NTf2] (3) ternary system. The agreement between the results of experiments and calculations confirms the successful application of our model to the IL solution. More vapor-liquid and liquid-liquid phase behaviors
Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 9685
of such systems containing ionic liquids have also been described well by this model.42 The established model is also suitable for some binary polymer blend systems. Only one parameter, ∈12/k, is needed for binary systems. ri, which is not an adjustable parameter, is calculated by the expression quoted from Chang and Bae′s work24,27 ri )
u Vmi(vdW) Mwi u 10.23(cm3/mol)Mwi
(17)
where Vmi(vdW) and Mwi are the molar van der Waals (vdW) volume and molecular weight, respectively, of component i. The constant 10.23 cm3/mol is the vdW volume of a CH2 group. The superscript u represents a repeat unit of polymers. When applied to oriented systems, Hu et al.′s double-lattice model concept12,13 is used to account for the oriented interaction of the LLE. ∈12/kT is expressed as a function of temperature as ˜ε12 ) ∈12/kT ) δε1/kT + δε2/(kT)2
(18)
where δε1/k and δε2/k2 are two energy parameters. Their values could be calculated by correlating the critical point and coexistence curve. All of the experimental data and the results calculated by our model are listed and compared in the following figures. Figure 7 depicts the model results of the binodal (lines) and the experimental results (symbols) of three coexistence curves for the polybutadiene (PBD)-polystyrene (PS) system with different molecular weights. The experimental LLE results are quoted from ref 43. It is evident that our model obtains a good prediction for the experimental polymer blend. All of the corresponding model parameters are listed in Table 3 for systems a-c. For example, PBD(1100)-PS(1340) denotes that the molecular weight of polybutadiene is Mw ) 1100 and the molecular weight of polystyrene is Mw )1340. The remaining abbreviations have the same meanings. The chain lengths calculated by eq 17 and the energy parameters correlated by the experimental results are all included in Table 3. A plot of experimental and predicted liquid-liquid equilibria for two poly(a-methyl styrene) (PaMeS)-polystyrene (PS) systems against the volume fraction of PS is given in Figure 8. The experimental results are quoted from ref 44, and the model parameters are listed in Table 3 for systems d and e. The agreement between the lines and symbols is quite good. Figure 9 shows a comparison between the model and experimental results for two polyethylene glycol monomethyl ether (PEGE)-polyethylene glycol (PPG) systems.45 Obviously, the fit is quite satisfactory with two adjustable parameters. The chain lengths and energy parameters of Figures 8 and 9 listed in Table 3 for systems f and g were obtained using the same approach as in Figure 7. The binary systems mentioned above present the most common phase diagrams with UCSTs. Subsequently, we correlated four LCST-type phase diagrams as shown in Figure 10, which shows the LLE of poly(vinyl methyl ether) (PVME)-PS systems with different molar weights.46,47 The model parameters were obtained by the method described above and are listed in Table 4. From a thermodynamic viewpoint, LCST-type phase separation can occur under conditions where both enthalpy (∆mixH) and entropy (∆mixS) of mixing become negative and the entropic contribution (∆mixST) to ∆mixG overwhelms ∆mixH. The comparisons show that the correlation is not very good when φ2 is large but is still acceptable. It is much better than Chang and Bae’s correlation. The deviations are mainly due to the high polydispersity of PVME and the neglect of the freevolume effect that leads to the LCST.
4. Conclusion A molecular thermodynamic model of multicomponent chainlike fluid mixtures based on a close-packing lattice was established in this work, which is a quite generalized lattice model for polymer solutions. By changing the number of components, the chain length, or the type of polymer, various of models for different polymer systems can be obtained, such as molecular thermodynamic models for binary polymer solutions36 and for mixed-solvent polymer solutions.33 A molecular thermodynamic model of random copolymer solutions can also be obtained by introducing a proper expression for gi(2) following the same principles.34 It can be concluded that models for other alternative copolymer or block copolymer solutions can be further improved in the same way. Predictions of liquid-liquid phase equilibria for ternary chainlike polymer mixtures can describe types 1-3 phase separations very well, are in nearly perfect agreement with MC simulation results, and substantially outdo Flory-Huggins theory and revised Freed theory. For real systems, the proposed model describes very well phase behaviors of ordinary and oriented polymer blend systems with UCST-type phase diagrams. However, some deviations exist in the correlations of blend systems with LCST-type phase diagrams because of the high polydispersity of the polymers. The lengths of polymers are calculated from the molar vdW volume. The exchange energy, the only adjustable parameter in the model, uses the concept of Hu et al.’s double lattice to describe the dependence on temperature. Meanwhile, the model parameters correlated from real binary systems can be extended to predict the corresponding LLE of ternary mixtures perfectly. The trends of the effect of temperature on the region of phase separation predicted by our model fit the experimental data. A ternary ionic liquid mixture for which the thermodynamic property is quite the same as that of a polymer mixture was also correlated by our model. It shows again that the agreement is satisfactory. All of the results show that the model described herein will be a promising, versatile, and attractive molecular thermodynamic model in the field of practical engineering applications. Acknowledgment Financial support for this work was provided by the National Natural Science Foundation of China (Nos. 20776040, 20736002), the National Basic Research Program of China (973 Program) (No. 2009CB219902), the Program for Changjiang Scholars and Innovative Research Team in University of China (Grant IRT0721), and the 111 Project (Grant B08021) of China. Nomenclature ∆mixA ) Helmholtz energy of mixing, J r ∆mixAIsing ) residual Helmholtz energy of mixing of ternary Ising lattice, J g ) radial distribution function k ) Boltzmann constant ≈ 1.38 × 10- 23 J/Κ Mwi ) molecular weight of component i Nr ) total number of sites of close-packed lattice model Ni ) number of component i q ) surface area parameter ri ) chain length of monomer i ∆mixS0 ) Guggenheim’s athermal entropy of mixing, J/K T ) temperature, K Vmi(vdW) ) molar van der Waals (vdW) volume of a CH2 group z ) coordination number Greek Letters
9686 Ind. Eng. Chem. Res., Vol. 47, No. 23, 2008 Γ ) nonrandom factor ε ) interaction energy, J εij ) interaction energy of i-j pair, J θ ) surface fraction λ ) parameter charactering the long-range correlations µi ) chemical potential of component i φ ) volume fraction or segment fraction Subscripts i, j, k, l ) monomer i, j, k, l, respectively c ) critical value mix ) mixing value Superscripts ˜ ) reduced interchange property u ) repeating unit of polymer
Literature Cited (1) Treybal, R. E. Liquid Extraction, 2nd ed.; McGraw-Hill: New York, 1963. (2) Sariban, A.; Binder, K. Critical properties of the Flory-Huggins lattice model of polymer mixtures. J. Chem. Phys. 1987, 86, 5859–5873. (3) Deutsch, H. P.; Binder, K. Critical Behavior and Crossover Scaling in Symmetric Polymer Mixtures: A Monte Carlo Investigation. Macromolecules 1992, 25, 6214–6230. (4) Mu¨ller, M.; Binder, K. Computer Simulation of Asymmetric Polymer Mixtures. Macromolecules 1995, 28, 1825–1834. (5) Houdayer, J.; Mu¨ller, M. Phase Diagram of Random Copolymer Melts: A Computer Simulation Study. Macromolecules 2004, 37, 4283–4295. (6) Liang, H. J.; He, X. H.; Jiang, W.; Jiang, B. Z. Monte Carlo simulation of phase separation of A/B/A-B ternary mixtures. Macromol. Theory Simul. 1999, 8, 173–178. (7) Jiang, J. W.; Yan, Q. L.; Liu, H. L.; Hu, Y. Monte Carlo Simulations of Liquid-Liquid Equilibria for Ternary Chain Molecule Systems on a Lattice. Macromolecules 1997, 30, 8459–8462. (8) Flory, P. J. Thermodynamics of high polymer solutions. J. Chem. Phys. 1942, 10, 51–61. (9) Flory, P. J. Statistical Thermodynamics of Liquid Mixtures. J. Am. Chem. Soc. 1965, 87, 1833–1838. (10) Huggins, M. L. Some Properties of Solutions of Long-Chain Compounds. J. Phys. Chem. 1942, 46, 151–158. (11) Freed, K. F. New lattice model for interacting, avoiding polymers with controlled length distribution. J. Phys. A: Math. Gen. 1985, 18, 871–887. (12) Hu, Y.; Lambert, S. M.; Soane, D. S.; Prausnitz, J. M. DoubleLattice Model for Binary Polymer Solutions. Macromolecules 1991, 24, 4356–4363. (13) Hu, Y.; Liu, H. L.; Soane, D. S.; Prausnitz, J. M. Binary liquid-liquid equilibria from a double-lattice model. Fluid Phase Equilib. 1991, 67, 65–86. (14) Kambour, R. P.; Bendler, J. T.; Bopp, R. C. Phase Behavior of Polystyrene, Poly(2,6-dimethyl-1,4-phenylene oxide), and Their Brominated Derivatives. Macromolecules 1983, 16, 753–757. (15) ten Brinke, G.; Karasz, F. E.; MacKnight, W. J. Phase Behavior in Copolymer Blends: Poly(2,6-dimethyl-1,4-phenylene oxide) and HalogenSubstituted Styrene Copolymers. Macromolecules 1983, 16, 1827–1832. (16) Hino, T.; Lambert, S. M.; Soane, D. S.; Prausnitz, J. M. Miscibilities in binary copolymer systems. Polymer 1993, 34, 4756–4761. (17) Dudowicz, J.; Freed, K. F. Molecular Influences on Miscibility Patterns in Random Copolymer/Homopolymer Binary Blends. Macromolecules 1998, 31, 5094–5104. (18) Buta, D.; Freed, K. F.; Szleifer, I. Monte Carlo test of the lattice cluster theory: Thermodynamic properties of binary polymer blends. J. Chem. Phys. 2001, 114, 1424–1434. (19) Dudowicz, J.; Freed, K. F.; Douglas, J. F. New patterns of polymer blend miscibility associated with monomer shape and size asymmetry. J. Chem. Phys. 2002, 116, 9983–9996. (20) Yan, Q. L.; Jiang, J. W.; Liu, H. L.; Hu, Y. Phase equilibria for chain molecule systems from Monte Carlo simulations. J. Chem. Ind. Eng. (In Chinese) 1995, 46, 517–523. (21) Jiang, J. W.; Yan, Q. L.; Liu, H. L.; Hu, Y. Computer simulation of liquid-liquid equilibria for ternary chain molecules. J. Chem. Ind. Eng. (In Chinese) 1996, 47, 637–641. (22) Yang, J. Y.; Peng, C. J.; Liu, H. L.; Hu, Y. Monte Carlo Simulations and Theoretical Calculations of Liquid-Liquid Equilibria for Ternary Chain Molecule Systems. J. Chem. Eng. Chin. UniV. 2006, 20, 673–678 (in Chinese).
(23) Chang, B. H.; Ryu, K. O.; Bae, Y. C. Chain length dependence of liquidliquid equilibria of binary polymer solutions. Polymer 1998, 39, 1735–1739. (24) Chang, B. H.; Bae, Y. C. Molecular thermodynamics approach for liquid-liquid equilibria of the symmetric polymer blend systems. Chem. Eng. Sci. 2003, 58, 2931–2936. (25) Mu¨ller, M. Miscibility behavior and single chain properties in polymer blends: A bond fluctuation model study. Macromol. Theory Simul. 1999, 8, 343–374. (26) Panagiotopolous, A. Z.; Quirke, N.; Stapleton, M.; Tildesley, D. J. Phase equilibria by simulation in the Gibbs ensemble Alternative derivation, generalization and application to mixture and membrane equilibria. Mol. Phys. 1988, 63, 527–545. (27) Chang, B. H.; Bae, Y. C. Phase Behaviors of Symmetric Polymer Blend Systems. J. Polym. Sci. B: Polym. Phys. 2004, 42, 1532–1538. (28) Ryu, J. H.; Gujrati, P. D. Lattice theory of a multicomponent mixture of monodisperse polymers of fixed architectures. J. Chem. Phys. 1997, 107, 3954–3966. (29) Yang, J. Y.; Xin, Q.; Sun, L.; Liu, H. L.; Hu, Y.; Jiang, J. W. A new molecular thermodynamic model for multicomponent Ising lattice. J. Chem. Phys. 2006, 125, 164506. (30) Cummings, P. T.; Stell, G. Statistical mechanical models of chemical reactions. II. Analytic solution of the Percus-Yevick approximation for a model of homogeneous association. Mol. Phys. 1985, 55, 33–48. (31) Stell, G.; Zhou, Y. Chemical association in simple models of molecular and ionic fluids. J. Chem. Phys. 1989, 91, 3618–3623. (32) Zhou, Y.; Stell, G. Chemical association in simple models of molecular and ionic fluids. II. Thermodynamic properties. J. Chem. Phys. 1992, 96, 1504–1506. (33) Liu, H. L.; Yang, J. Y.; Xin, Q.; Hu, Y. Molecular thermodynamics of mixed-solvent polymer solutions. Fluid Phase Equilib. 2007, 261, 281–285. (34) Xin, Q.; Peng, C.; Liu, H.; Hu, Y. A Molecular Thermodynamic Model for Binary Lattice Random Copolymer Solutions. Fluid Phase Equilib. 2008, 267, 163–171. (35) Guggenheim, E. A. Mixtures; Oxford University Press: Oxford, U.K., 1952. (36) Yang, J. Y.; Yan, Q. L.; Liu, H. L.; Hu, Y. A molecular thermodynamic model for binary lattice polymer solutions. Polymer 2006, 47, 5187–5195. (37) Francis, A. W. Ternary Systems with Three Separate Binodal Curves. Ind. Eng. Chem. 1956, 60, 20–27. (38) Johnson, G. C.; Francis, A. W. Ternary Liquid System, BenzeneHeptane-Diethylene Glycol. Ind. Eng. Chem. 1954, 46, 1662–1668. (39) Lachwa, J.; Szydlowski, J.; Najdanovic-Visak, V.; Rebelo, L. P. N.; Seddon, K. R.; da Ponte, M. N.; Esperanc¸a, J.M.S.S.; Guedes, H. J. R. Evidence for Lower Critical Solution Behavior in Ionic Liquid Solutions. J. Am. Chem. Soc. 2005, 127, 6542–6543. (40) Lou, P.; Kang, S.; Ko, K. C.; Lee, J. Y. Solubility of Small Molecule in Ionic Liquids: A Model Study on the Ionic Size Effect. J. Phys. Chem. B 2007, 111, 13047–13051. (41) Arce, A.; Earle, M. J.; Katdare, S. P.; Rodrı´guez, H.; Seddon, K. R. Phase equilibria of mixtures of mutually immiscible ionic liquids. Fluid Phase Equilib. 2007, 261, 427–433. (42) Yang, J.; Peng, C.; Liu, H.; Hu, Y. Calculation of Vapor-Liquid and Liquid-Liquid Phase Equilibria for Systems Containing Ionic Liquids Using a Lattice Model. Ind. Eng. Chem. Res. 2006, 45, 6811–6817. (43) Rostami, S.; Walsh, D. J. Simulation of Upper and Lower Critical Phase Diagrams for Polymer Mixtures at Various Pressures. Macromolecules 1985, 18, 1228–1235. (44) Lin, J. L.; Roe, R. J. DSC study of miscibility of polystyrene and poly(a-methylstyrene). Polymer 1988, 29, 1227–1232. (45) Takahashi, H.; Kyu, T.; Tran-Cong, Q.; Yano, O.; Soen, T. Phase separation in mixtures of poly(ethylene glycol monomethylether) and poly(propylene glycol) oligomers. J. Polym. Sci. B: Polym. Phys. 1991, 29, 1419–1425. (46) Walsh, D. J.; Dee, G. T.; Halary, J. L.; Ubiche, J. M.; Millequant, M.; Lesec, J.; Monnerie, L. Application of Equation of State Theories to Narrow Molecular Weight Distribution Mixtures of Polystyrene and Poly(vinylmethyl ether). Macromolecules 1989, 22, 3395–3399. (47) Bae, Y. C.; Shim, J. J.; Soane, D. S.; Prausnitz, J. M. Representation of Vapor-Liquid and Liquid-Liquid Equilibria for Binary Systems Containing Polymers: Applicability of an Extended Flory-Huggins Equation. J. Appl. Polym. Sci. 1993, 47, 1193–1206.
ReceiVed for reView June 11, 2008 ReVised manuscript receiVed September 18, 2008 Accepted September 22, 2008 IE800924R
