Application of Dipolar Chain Theory to the Phase Behavior of Polar

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Ind. Eng. Chem. Res. 2001, 40, 4641-4648

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Application of Dipolar Chain Theory to the Phase Behavior of Polar Fluids and Mixtures Prasanna K. Jog,† Sharon G. Sauer, Jorg Blaesing,‡ and Walter G. Chapman* Rice University, Department of Chemical Engineering, P.O. Box 1982, MS 362, Houston, Texas 77251-1892

Phase behavior is strongly affected by dipolar interactions in a wide range of systems including those containing ketones, aldehydes, ethers, and esters. Multiple polar sites are present in various polar copolymers as well as in polyethers and polyesters. Although theories have been developed for nearly spherical polar molecules and for nonpolar chain molecules, accounting simultaneously for a single multipolar interaction and molecular shape has remained an unsolved problem of statistical-mechanics-based perturbation theory (Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids I; Clarendon Press: Oxford, U.K., 1984. Walsh, J. M.; Gang, J.; Donohue, M. D. Fluid Phase Equilib. 1991, 65, 209). Accurate accounting for the effect of multiple polar sites in nonspherical molecules has been well beyond expectation. In recent work, we solved part of this problem by showing how to accurately predict the properties of chainlike molecules with single or multiple dipolar sites (Jog, P. K.; Chapman, W. G. Mol. Phys. 1999, 97, 307-319). Although we cast this result in terms of the original SAFT equation of state (Chapman, W. G.; Gubbins, K. E.; Jackson, G. Mol. Phys. 1988, 65, 1057-1079. Chapman, W. G. Ph.D. Dissertation, Cornell University, Ithaca, NY, 1988. Chapman, W. G.; Gubbins, K. E.; Jackson, G.; Radosz, M. Ind. Eng. Chem. Res. 1990, 29, 1709-1721), the approach is applicable to other accurate chain fluid equations of state. In this paper, we (a) demonstrate the differences between our approach and previous models, (b) extend our theory to mixtures of polar fluids, and (c) compare our results with experimental data to demonstrate the predictive capabilities of the new theory. Introduction Dipolar interactions are known to have a significant effect on the phase behavior of numerous systems, from those containing esters and ketones to polymeric systems of polar polymers and copolymers. This is evident from the nonideal behavior seen in mixtures in which one of the components is polar and another nonpolar. Many equations of state, including the original SAFT equation of state due to Chapman et al.4-6 and many of its modified forms, do not include a separate long-range polar term. (Variations and applications of the SAFT approach have been the subject of a recent review by Mu¨ller and Gubbins.7) If such an equation of state is used for polar fluids, the polar interaction is effectively included in the van der Waals attraction, resulting in an artificially large pure-fluid attraction energy U11. As we will show, this causes poor prediction of properties for mixtures of polar and nonpolar fluids. In the case of the interaction between a polar component 1 and a nonpolar component 2, the cross van der Waals attraction is usually estimated from the geometric mean of the two pure-fluid attraction energies, U12 ) xU11U22(1 - k12). If U11 is artificially large as a result of the neglect of polar interactions, k12 must then be large and possibly composition-dependent to compensate for the fact that there is no multipolar interaction between components 1 and 2. This reduces the predictive ability of the equation of state. The necessity for this large, * Author to whom correspondence should be addressed. Phone: 713-348-4900. Fax: 713-348-5478. E-mail: [email protected]. † Present address: Dow Chemical, Midland, MI. ‡ Visiting scholar from the Technical University of Berlin, Berlin, Germany.

state-dependent k12 can be removed by correctly including the effect of the multipolar interaction. This is particularly important in polar polymers in which the fraction of polar sites can be varied, as has been shown in the work of Hasch et al.8,9 and Folie et al.10,11 Among the statistical-mechanics-based perturbation theories developed for polar fluids, the u-expansion is the most widely applied.1,12 However, the u-expansion is valid only for mixtures of polar spheres;1 attempts to extend the theory to nonspherical molecules have failed. In previous efforts to develop a theory for nonspherical polar fluids, researchers primarily studied diatomic molecules with the multipole moment along the molecular axis of the molecule. This was considered to be the simplest realistic model to study as it is linear and representative of many small polar molecules. For example, Wojcik et al.13 investigated the thermodynamics and structure of mixtures of quadrupolar hard diatomics and nonpolar molecules, calculating the full angular distribution functions by molecular simulation. Although the convergence of various perturbation theories was tested with molecular simulation, an accurate theory was not found. Although no verifiably rigorous theory has previously been developed, the necessity of modeling nonspherical, polar molecules for engineering applications has resulted in the development and use of approximate models. The model that has been most widely used approximates the dipolar nonspherical molecule as a large polar sphere with the sphere diameter chosen so that the molecular volume is preserved.14-22 Although this model preserves the idea that the contribution to the fluid properties of a single dipole on a molecule becomes weaker as the size of the molecule increases, it is known to be in poor agreement with molecular

10.1021/ie010264+ CCC: $20.00 © 2001 American Chemical Society Published on Web 09/19/2001

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simulation results.1,13 Another crucial limitation of this approach is that it allows only a single polar group per molecule, thus excluding molecules with multiple polar sites such as polar polymers. Other approaches have used site-site perturbation theory and accounted for molecular shape through an interaction volume parameter or diagrammatic expansion techniques. The PACT approach derived by Walsh et al.2 uses an interaction site perturbation theory with polar interactions. This model accounts for the shape of the molecule through an interaction volume parameter that depends on the molecular shape. Sear23 applied a diagrammatic expansion technique similar to that of Wertheim to study the effect of chain formation by dipolar hard spheres at low density, which is energetically favorable. This approach is applicable only at low temperature and low density because it neglects longrange multibody interactions. In a model for water, Mu¨ller and Gubbins applied the u-expansion to a fluid of spheres and allowed these spheres to associate using Wertheim’s associating fluid theory.24 The theory showed reasonably good agreement with molecular simulation results and with experimental data for water. In this work, we propose a method to account for dipolar interactions in chain equations of state for mixtures and demonstrate the differences between our approach and previous models. Our approach is to use physical insight concerning the molecular structure of fluids to suggest reasonable approximations to theories of molecular fluids. Here, we present a statistical mechanical theory for chains with dipolar functional groups based on Wertheim’s thermodynamic perturbation theory of first order.25-28 Because Wertheim’s theory is the basis for the statistical associating fluid theory (SAFT), our approach is similar to that used in developing the SAFT equation of state. In a previous article,3 we showed that, for pure dipolar hard-sphere chains with the dipole oriented perpendicular to the molecular axis, the theoretical results for thermodynamics are in excellent agreement with molecular simulation results. This model is realistic for many real compounds such as ketones, ethers, and polar polymers. We now extend this approach to mixtures of dipolar chains and apply it to model phase equilibria in mixtures of dipolar fluids. In our approach, we account for the fact that the dipoles are located on certain functional groups in a molecule. This allows us to model polymers with multiple dipolar functional groups. In this way, we can predict the effect of changing the concentration of dipolar functional groups in a copolymer. We compare our results with experimental data to demonstrate the predictive capabilities of the new theory. Comparison of the Molecular Sphere and Segment Approaches Most of the prior approaches to account for dipolar interactions in chain equations of state16,17,19 treat the dipolar molecule as a sphere of volume equal to the molecular volume with an ideal dipole at the molecular center. Thus, as the molecule becomes larger, the effect of the dipole weakens because of the larger effective separation of the dipoles. In this treatment of the dipolar interaction, the effect of nonsphericity of the molecule on the polar term is not considered. In addition, when a molecule has multiple polar groups, the molecular dipole moment is used, and as a result, there is no way to account for differing polarity of functional groups

Figure 1. Representative alkanone: (a) segment approach, (b) effective molecular sphere approach.

within a molecule. We call this the “molecular sphere approach”.14-22 In contrast, the approach developed in this work, which we refer to as the “segment approach”, explicitly accounts for multiple dipolar functional groups and the nonspherical shape of the molecule in the polar term. The dipolar contribution to the Helmholtz free energy is obtained by dissolving all of the bonds in a chain and then applying the u-expansion to the resulting mixture of polar and nonpolar spherical segments. We illustrate the difference between the two approaches quantitatively with an example. Consider a pure-fluid system, for example a representative alkanone that can be modeled as a chain of five segments (see Figure 1a). Now, compare this to a molecular sphere with the same volume (Figure 1b). It is obvious that, in both models, the nonpolar segments dilute the dipolar interaction. This is consistent with the idea that polar interactions are less significant in long-chain alkanones than in short-chain alkanones. Nonetheless, this effect is artificially exaggerated in the molecular sphere model because the distance of closest approach of dipoles is greater than in the segment approach. We can demonstrate this by considering the contributions to the Helmholtz free energy of the two methods for pure dipolar fluids. In the u-expansion, the dipolar contribution is an infinite series of terms of second, third, and higher order. The second- and third-order terms are calculated explicitly,29 and the higher-order terms are estimated by the Pade´ approximant of Rushbrooke et al.30 Using the Pade´ approximate,30 the change in free energy due to polar interactions can be written as

Apolar )

A2 1 - A3/A2

(1)

where A2 is the second-order term in the perturbation expansion and A3 is the third-order term.

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For the dipolar molecular sphere approach, A2 is given by1,30,31 4 Amolecule 2 2 1 2 µmolecule )- π F I (F*) 3 2 NkT 9 kT d

( )

(2)

molecule

where µmolecule is the molecular dipole moment, N is the number of molecules, k is the Boltzmann constant, T is the absolute temperature, and dmolecule is the diameter of a sphere with the same volume as the molecule. This diameter is given by

dmolecule ) (mdsegment3)1/3

(3)

where dmolecule is the diameter of the effective spherical molecule, m is the number of segments in the actual molecule, and dsegment is the diameter of a segment. I2 is the integral over the pair correlation function of the reference fluid (here a hard-sphere fluid)

I2(F*) )

3σ3 4π

∫gHS(r,F*)r16 dr

(4)

F is the number density of molecules, and F* ) Fdmolecule3 ) Fmdsegment3 is the reduced density. Using the segment approach,3 we define xp as the fraction of dipolar segments on a molecule. In the case of a single dipolar segment in a molecule, xp ) 1/m. For the segment approach, the second-order contribution A2 in the perturbation expansion for dipolar molecules is given as 4 Asegment 2 2 1 2 µsegment )- π F I (F*) 3 2 NkT 9 kT d

( )

(5)

segment

where F* ) Fmdsegment3 is the reduced segment density, in keeping with the segment basis of the dipolar free energy contribution. For molecules with a single dipolar site, the molecular dipole moment is the same as the segment dipole moment.32 For this example of an alkanone, the segement approach and the molecular sphere approach give different results. From eqs 2-5, it can be seen that

Amolecule 2 Asegment 2

)

1 . m

(6)

As stated above and shown in Figure 1, the contribution to the dipolar free energy based on the molecular sphere approach is smaller than that from the segment approach. By also analyzing the third-order contribution to the free energy and the Pade´ approximate, we obtain a simple relationship between the dipolar contribution to the free energy from the two approaches for pure fluids molecule Apolar segment Apolar

1 ) m

(7)

For molecules with a single polar segment, we see that the contribution of the polar term to the total Helmholtz free energy differs by a factor of the number of segments, and hence, the difference between the two approaches is more pronounced as the chain molecule becomes longer. For mixtures, the relationship between

the two approaches is more complex. Given that our segment approach has been shown to be in excellent agreement with molecular simulation results,3 we conclude that the commonly applied molecular sphere approach underestimates the effect of dipolar interactions for chainlike molecules. In the following, we extend the segment approach to mixtures and demonstrate the predictive capabilities of the resulting equation of state. The Equation of State As we have shown previously for pure polar fluids,3 the dipolar term can be added to the perturbation expansion of the chain equation of state used by SAFT. The total residual Helmholtz free energy is given by

Ares ) Ahs + Adispersion + Apolar + Achain + Aassociation (8) where the Helmholtz free energy is residual to an ideal gas at the same temperature and density as the fluid of interest. The hard-sphere and chain terms in the original SAFT equation of state4-6 are unaffected by the presence of a dipole term, as shown by Jog and Chapman.3 Various approaches to the dispersion term can be used; here, we use the equation of state for spheres due to Chen and Kreglewski.33 The species in this work are nonassociating so that the association contribution to the Helmholtz free energy is zero. SAFT has three pure-component parameters for a pure nonpolar fluid, namely, segment volume (v00), segment dispersion energy (u0/k), and chain length (m).4-6,34 To account for multiple dipolar segments in a molecule, the functionalgroup dipole moment and an additional parameter, xp, which is the fraction of dipolar segments in a chain, are required. Ideally, this parameter should be equal to 1/m for chains with a single dipolar site; however, for real fluids, we make xp an adjustable parameter. A value of xp different from 1/m arises because the SAFT model of the molecule as a homonuclear chain of tangentially connected segments is somewhat oversimplified. As shown previously,3 the dipolar contribution to the Helmholtz free energy is accurately obtained by dissolving all of the bonds in a chain and then applying the u-expansion to the resulting mixture of polar and nonpolar spherical segments. The extension of the mixture A2 and A329 to our case of multiple dipolar segments gives

A2

)-

NkT A3

F

2π

9 (kT)2

∑i ∑j xixjmimjxpixpj

µi2µj2 dij3

I2,ij (9)

)

NkT 5

2

µi2µj2µk2

F2

π 162 (kT)3

∑i ∑j ∑k xixjxkmimjmkxpixpjxpk d d

ij jkdik

I3,ijk (10)

where

I2,ij )

3dij3 4π

∫gHS (r,F*)r-6 dr ) ∞ 3∫1 gHS (r/ij,F*)r/ij-4 dr/ij ij

ij

(11)

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and

Table 1. SAFT and Polar SAFT Parameters for Acetone and Alkane Solvents

I3,ijk )

192 14π 1/2 π 5 5

∫

( )

∫0∞ dr/12 r/12-2∫0∞ dr/13 r/13-2 ×

(σij/σjk)r/12 + (σik/σjk)r/13 |(σij/σjk)r/12 - (σik/σjk)r/13|

dr/23

r/23-2gijk(r/12,r/23,r/13)

v00 component (mL/mol)

×

ψ222(R1,R2,R3) (12) The function ψ222(R1,R2,R3) is a well-known function29 of angles R1, R2, and R3 of the triangle formed by the centers of the three molecules. Also note that, for the integrals I2,ij and I3,ijk, we have defined a reduced radius given by

r/ij )

rij dij

acetone acetone pentane hexane decane dodecane a

13.245 7.765 12.533 12.475 11.723 11.864

u0/k (K)

m

µ (D)

xp

comment

232.47 210.92 200.02 202.72 205.46 205.93

2.954 4.504 4.091 4.724 7.527 8.921

2.72 0 0 0 0 0

0.167 0 0 0 0 0

polar SAFT HRa HR HR HR HR

Huang and Radosz parameters.34

(13)

We assume that the integrals I2,ij and I3,ijk in reduced form are independent of component at a given reduced density. The integrals I2,ij and I3,ijk in eqs 9 and 10 are related to the corresponding pure-fluid integrals by

I2,ij ) I2(Fdx3)

(14)

I3,ijk ) I3(Fdx3)

(15)

where dx3 ) ∑i mixidii3 and I2 and I3 are the corresponding pure-fluid integrals. Another approach that should give similar results would be to use a van der Waals one-fluid approximation for dx. Other thermodynamic quantities such as pressure and chemical potential can be obtained by differentiation of the Helmholtz free energy expression in eq 8. The four pure-component parameters, namely, the segment size (v00), segment dispersion energy (u0/k), number of segments per chain (m), and fraction of polar segments in a chain (xp), are fitted to the saturated liquid density and vapor pressure data of the pure components. For molecules with a single polar site, we use the experimental value for the dipole moment of the molecule. For multiple polar groups, the dipole moment of each functional group calculated from quantum mechanics can also be used. Application of Polar SAFT to Pure Fluids and Mixtures We now apply the polar SAFT model to molecules containing single or multiple dipolar segments. First, we predict the phase behavior of acetone-alkane mixtures. This tests the ability of the theory to model polarnonpolar mixtures and to model differences in molecular size. Second, we calculate the cloud point of poly(ethylene-co-methyl acrylate) in propane and butane solvents as a function of the polar comonomer concentration. This tests the ability to model multiple dipolar functional groups in a complex system. Thus, modeling these systems provides a strong test of the predictive capabilities of SAFT with the new polar term (hereafter referred to as polar SAFT). Phase Equilibria in Acetone-Alkane Binary Systems. To demonstrate the effect of the dipolar term in polar SAFT, we have performed calculations for the acetone-alkane systems both with and without including the polar contribution. Acetone contains a carbonyl group, which has a dipole moment perpendicular to the molecular axis. The equation-of-state parameters for acetone were fitted to the saturated liquid density and

Figure 2. Vapor-liquid equilibrium of acetone-pentane system. Polar SAFT (s) and SAFT (- - -) at k12 ) 0; experimental data36 (b).

vapor pressure data for acetone.35 Huang and Radosz parameters34 are used for the alkane solvents with zero values for the dipole moment and the fraction of polar segments (xp). The solvents we chose are pentane, hexane, decane, and dodecane. The parameters are listed in Table 1. Both (nonpolar) SAFT and polar SAFT have an adjustable binary interaction parameter k12. To test the predictive capability of polar SAFT, this binary interaction parameter is set to zero. Figures 2-5 show the predictions of polar SAFT and (nonpolar) SAFT for the P-x-y diagrams for acetonealkane mixtures. The polar SAFT results show remarkable agreement with the experimental data for each of these systems, where the phase behavior is dominated by the polar effects. However, (nonpolar) SAFT does not give acceptable results at zero k12. In fact, (nonpolar) SAFT predicts slight negative deviations from ideal solution behavior. Each of these systems shows a strong positive deviation from ideality as a result of the polar interactions, or more correctly, the lack of polar interactions between the two components. Note that, in the acetone-pentane system, polar SAFT shows a region of liquid-liquid equilibrium at a low temperature of 258 K, whereas the experimental data at this temperature do not. At low enough temperatures, we would expect a region of liquid-liquid immiscibility for this system. Thus, we conclude that polar SAFT predicts the onset of liquid-liquid equilibrium at a higher temperature for this system than would be observed experimentally. In the second set of calculations, the binary interaction parameter k12 that best fits the P-x-y data at

Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4645

Figure 3. Vapor-liquid equilibrium of acetone-hexane system. Polar SAFT (s) and SAFT (- - -) at k12 ) 0; experimental data36 (b).

Figure 5. Vapor-liquid equilibrium of acetone-dodecane system. Polar SAFT (s) and SAFT (- - -) at k12 ) 0; experimental data36 (b). Table 2. Optimized Binary Interaction Parameters system

polar SAFT k12

SAFT k12

acetone-pentane acetone-hexane acetone-decane acetone-dodecane

0.000 0.004 0.005 0.005

0.090 0.092 0.075 0.078

Figure 4. Vapor-liquid equilibrium of acetone-decane system. Polar SAFT (s) and SAFT (- - -) at k12 ) 0; experimental data36 (b).

different temperatures is selected. The k12 parameter is obtained by minimizing the function N

min f(k12) ) k12

xi - xi,exp

∑1 |min (x

i,exp,1 N

|+ - xi,exp) yi - yi,exp

∑1 |min (y

i,exp,1

| (16)

- yi,exp)

where N is the number of experimental data points, x is the liquid-phase molar composition, and y is the vapor-phase molar composition of a consistently chosen component at the experimental temperature and pressure for a particular data point. The suffix “exp” indicates the experimental composition. In certain cases where the data had very close liquid- and vapor-phase compositions (as in the case of azeotropes), the best k12

Figure 6. Vapor-liquid equilibrium of acetone-hexane system. Polar SAFT (s) and SAFT (- - -) at optimized k12; experimental data36 (b). k12 is correlated to the phase composition data.

was determined manually by repeating calculations at different values of k12. The k12 parameter was fitted both for polar SAFT and (nonpolar) SAFT. Table 2 lists the optimized k12 values for the systems considered here. Polar SAFT needs consistently smaller values of k12 than SAFT. In the case of acetone-pentane, attempts to adjust the binary interaction parameter in the original SAFT model result in phase splitting at the lower temperature (258.15 K), similar to that shown in Figure 2 for polar SAFT with a k12 of zero. Figures 6 and 7 show the results for the acetone-hexane and acetone-dodecane binary systems at optimized k12, respectively. Polar SAFT consistently

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Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 Table 3. Molecular Characterization of Copolymers EMA31 and EMA41 polymer

Mw

mole % MA

EMA10 EMA31 EMA41

17 000 33 000 29 900

10 31 41

Figure 7. Vapor-liquid equilibrium of acetone-dodecane system. Polar SAFT (s) and SAFT (- - -) at optimized k12; experimental data36 (b). k12 is correlated to the phase composition data.

gives better agreement with the experimental results than SAFT. For the acetone-hexane system, polar SAFT gives accurate azeotropic compositions and pressures. By adjusting k12 for the acetone-dodecane, a more accurate fit is obtained for both models. However, (nonpolar) SAFT is unable to capture accurately the shape of the liquid composition curves. Overall, polar SAFT predicts the shape of the pressure-composition curves quite well and shows a significant improvement compared to (nonpolar) SAFT for acetone-alkane mixtures. Phase Equilibria in Polar Copolymer-Solvent Systems. Polar copolymer-solvent systems provide a difficult test of our approach by demonstrating the effect of multiple dipolar functional groups on the properties of polymeric fluids. Hasch et al.8 showed that SAFT cannot predict the phase behavior for polar copolymer solutions and can only be used to correlate the experimental cloud-point data. To illustrate the predictive capability of polar SAFT, we apply the model to the systems studied by Hasch. The polymer considered is poly(ethylene-co-methyl acrylate) (EMA). The goal is to predict the phase behavior of this copolymer in different solvents and for different concentrations of the polar comonomer (methyl acrylate) in the polymer. Hasch et al.9 modeled this system by using the same segment volume and chain length parameters as those for polyethylene. They obtained the dispersion energy parameter by fitting the cloud-point data of EMA with a suitable solvent. Hence, the dispersion energy parameter was a function of methyl acrylate content of the polymer, and the effect of polarity was modeled by adjusting the dispersion energy parameter. The polymers considered here are EMA10, EMA31, and EMA41, where the subscript indicates the molar percentage of methyl acrylate in the polymer. The molecular characterization of these polymers is given in Table 3. Our goal is to demonstrate the predictive ability of our theory by minimizing the number of available adjustable parameters. Therefore, the segment volume (v00), segment energy (u0/k), and chain length (m)

Figure 8. Cloud point of EMA copolymers with 31 and 41 mol % methyl acrylate in butane solvent from polar SAFT (s) and experimental data8,9 (b). k12 ) 0.008 was fit to polyethylenebutane cloud-point data, which is included for comparison. All polymers are at 5 wt % in the solution.

parameters are set to those of polyethylene. The binary interaction parameter k12 is set to the corresponding value for the polyethylene-solvent system, which is based on fitting the experimental data. In addition to these parameters, polar SAFT needs the dipole moment of the functional group (or monomer) and the fraction of polar segments (xp). The experimental value for the dipole moment of methyl acrylate is used as the segment dipole moment. The parameter xp is a function of the segment fraction of methyl acrylate in the polymer. xp is adjusted to fit the cloud-point data for the EMA31-butane system. To predict the phase behavior for other comonomer concentrations, we recognize that xp is proportional to the segment fraction of methyl acrylate, and we assume that the contribution of a functional group to the segment number is proportional to its molecular weight. Thus, xp is given by

yMA xp ) x/p yMA + (Methyl/MMA)(1 - yMA)

(17)

where x/p is the fraction of polar segments for pure methyl acrylate (determined using this equation and xp for the EMA31-butane system), yMA is the mole fraction of methyl acrylate in the polymer, Methyl is the molecular weight of the ethyl group, and MMA is the molecular weight of the methyl acrylate group. With xp fitted to EMA31-butane, we can predict the phase behavior of EMA41-butane. Figure 8 shows the cloud-point pressures calculated by polar SAFT for the EMA41- and EMA31-butane systems as a function of temperature for a solution of 5 wt % of the polymer. The binary interaction parameter is set to that of the

Ind. Eng. Chem. Res., Vol. 40, No. 21, 2001 4647 Table 4. Polar SAFT Parameters for EMA Copolymers and Solvents component

v00 (mL/mol)

u0/k (K)

EMA10 EMA31 EMA41 propane butane

12.0 12.0 12.0 13.457 12.599

216.15 216.15 216.15 193.03 195.11

m

µ (D)

xp

866.3 1681.7 1523.7 2.696 3.458

1.77 1.77 1.77 0 0

0.067 0.15 0.18 0 0

Figure 9. Cloud point of EMA copolymers with 10 and 31 methyl acrylate in propane solvent from polar SAFT (s) and experimental data8,9 (b). k12 ) 0.01. All polymers are at 5 wt % in the solution.

polyethylene-butane system (k12 ) 0.008). The values of the parameter xp, calculated using eq 17, and the other polar SAFT parameters for the polymers and solvents are listed in Table 4. The solvent parameters and the parameters for polyethylene are taken from Huang and Radosz.34 The polar SAFT prediction of the cloud-point curve for EMA41 agrees well with experimental data. Using the xp value fitted to EMA31-butane, we can also predict the cloud-point curves for EMA in propane. We use the same parameters for the polymer except for k12, which we determined to be 0.01. Figure 9 shows the cloud-point curves of EMA31 and EMA10 in propane. The calculations for both EMA31 and EMA10 in propane are completely predictive. Again, the polar SAFT results for EMA31 in propane are in close agreement with the experimental data. Although not shown here, for quantitative agreement, the k12 value for the EMA10 and propane system needs to be adjusted to 0.0125. (Note that Hasch et al.9 used a k12 value of 0.015.) In comparison with the SAFT calculations of Hasch et al.,9 where they adjusted the dispersion energy as a function of methyl acrylate comonomer content and fitted kij for different solvents, the polar SAFT calculations are much more predictive. In our approach, we took the simplistic view that the segment volume, dispersion energy, and chain length of EMA are the same as those of polyethylene. Although in reality these parameters, especially the dispersion energy, are different for the methyl acrylate group than that of the ethyl group, our rather simplistic approach provides good results in comparison with experimental data. This indicates that dipolar effects play an important role in the phase behavior of EMA-alkane solutions.

Conclusion Dipolar interactions have a significant effect on the phase behavior of polar fluids. Predicting the phase behavior of nonspherical molecules with even a single polar group (much less multiple polar groups) has long been an unsolved problem in statistical-mechanicsbased perturbation theory. We have solved this problem, in part, by incorporating the polar term in a chain equation of state (e.g., SAFT) utilizing a segment approach in which the dipoles are assumed to be located on the spherical segments of a chain. This approach is applicable to molecules with multiple dipolar groups; the theory has been validated versus molecular simulation results.3 We have presented a polar SAFT equation of state for mixtures that accurately predicts the effect of multiple dipolar groups and molecular shape on the phase behavior of mixtures of polar and nonpolar components. The effect of nonpolar segments is to dilute the dipole-dipole interaction between the polar segments. Previous approaches considered all molecules to be spherical for the purpose of calculating polar contributions in an equation of state. This molecular sphere approach is limited to the case of a single polar group on a molecule. We have shown that this molecular sphere approach underestimates the effect of the dipolar interaction, thus indicating why the molecular sphere approach is in poor agreement with molecular simulation results. We applied polar SAFT to two significantly different types of systems. For the acetone-alkane mixtures, the polar SAFT results are in much better agreement with the experimental data than are those of (nonpolar) SAFT. Even at a kij value of zero, polar SAFT gives good results for these systems, indicating the predictive capability of the model. For the alkane solutions of the copolymer EMA poly(ethylene-co-methyl acrylate), polar SAFT accurately predicts the effect of the polar methyl acrylate comonomer content and of the solvent on the cloud-point behavior. This is possible only because our approach accounts for multiple polar segments. Finally, the accurate small-chain as well as copolymer results demonstrate the predictive capability of polar SAFT for modeling phase equilibria in a wide range of fluid mixtures from polar monomers to polymers. Acknowledgment We thank the Robert A. Welch Foundation and Dow Chemical Company for their financial support of this work and NSF for a graduate fellowship for one of us (S.G.S.). We also thank John Walsh for encouraging us to look at the problem of polar fluids. Literature Cited (1) Gray, C. G.; Gubbins, K. E. Theory of Molecular Fluids I; Clarendon Press: Oxford, U.K., 1984. (2) Walsh, J. M.; Gang, J.; Donohue, M. D. Thermodynamics of Short-Chain Polar Compounds. Fluid Phase Equilib. 1991, 65, 209. (3) Jog, P. K.; Chapman, W. G. Application of Wertheim’s thermodynamic perturbation theory to dipolar hard sphere chains. Mol. Phys. 1999, 97, 307-319. (4) Chapman, W. G.; Gubbins, K. E.; Jackson, G. Phase equilibria of associating fluids: Chain molecules with multiple bonding sites. Mol. Phys. 1988, 65, 1057-1079. (5) Chapman, W. G. Theory and Simulation of Associating Liquid Mixtures. Ph.D. Dissertation, Cornell University, Ithaca, NY, 1988.

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Received for review March 23, 2001 Revised manuscript received July 26, 2001 Accepted July 30, 2001 IE010264+