Ind. Eng. Chem. Res. 2002, 41, 1043-1050
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Extension of the Elliott-Suresh-Donohue Equation of State to Polymer Solutions J. Richard Elliott, Jr.* and Ramasubramaniam N. Natarajan Chemical Engineering Department, The University of Akron, Akron, Ohio 44325-3906
A methodology is presented for applying the Elliott-Suresh-Donohue (ESD) equation of state to polymer solutions based on knowledge of the polymer’s molecular structure. Group contributions to the ESD shape parameter are presented for 88 functional groups to complement the existing group contribution methods for the solubility parameter and molar volume. Hydrogenbonding contributions are treated explicitly through Wertheim’s theory. The resulting extension provides a framework for solutions that may include any combination of small molecules, polymers, or intermediate compounds such as waxes and asphaltenes. As a demonstration, 20 polymer solutions are studied involving liquid and supercritical solvents exhibiting a range of phase behaviors. The polymers treated are polyethylene, polystyrene, polyisobutylene, poly(ethylene oxide), poly(propylene oxide), poly(vinyl acetate), poly(vinyl chloride), and poly(vinyl alcohol). Solvent partial pressures are generally correlated to high accuracy, but the accuracy for lower and upper critical solution behavior is only qualitative. Introduction Applications involving polymers have steadily gained a more significant role in chemical engineering during recent years.1 These applications compel extensions of the successful engineering methods for small molecules to encompass polymeric compounds in a consistent manner. The most widely applied engineering models currently use group contribution factors such as the UNIFAC model2 or an equation of state like the SoaveRedlich-Kwong (SRK) equation.3 The UNIFAC model can predict phase behavior, even for polymer solutions, where activity coefficients are applicable. The SRK equation is applicable in the presence of high-pressure vapors. However, these methods do not generally include explicit treatment of hydrogen-bonding contributions. For a general treatment including high-pressure vapors, it is possible to combine the UNIFAC formalism with the SRK equation using generalized Huron-Vidal4 methodology for mixing rules. This was the approach of Fischer and Gmehling5 in developing the predictive SRK (PSRK) equation and a similar development by Kalospiros and Tassios.6 Orbey et al.7-9 have adapted the PSRK methodology for polymer solutions to form the polymer SRK (PolySRK) equation. Finally, it is possible to correlate equation of state parameters in terms of linear correlations for various functional groups and extrapolate them to the polymer limit. A recent example of such an approach is given by Benzaghou et al.10 These considerations have motivated the work presented here to develop a predictive model for polymer solutions based on group contribution factors for the pure-component properties. Hydrogen-bonding effects have been incorporated in a manner that has proved successful for solutions containing small molecules, including high-pressure gaseous species, and the predictions can be conveniently tuned to correlate experimental data when available. The basis of the method is the Elliott-Suresh-Donohue (ESD) equation of state.11,12 The ESD equation is generalized to polymers through * To whom correspondence should be addressed. Fax: 330972-5856. E-mail:
[email protected].
a group contribution method to estimate the shape parameter, along with solubility parameters and molar volumes that can also be estimated by group contribution methods. The method presented here could easily be adapted to similar equations of state like the statistical associating fluid theory (SAFT) equation.13 Both the ESD and SAFT equations treat hydrogen bonding by Wertheim’s theory of association.14 Applications of the ESD equation to polymer solutions and blends have not been published previously, but the inclusion of a shape parameter in its initial formulation makes the manner of its extension to polymers straightforward. The ESD equation of state for pure fluids is specified by
Z ) 1 + Zrep + Zatt + Zassoc Zrep ) Zatt )
(1)
4cη 1 - 1.9η
-9.5qYη 1 + 1.7745Yη
Zassoc )
-F2 1 - 1.9η
where c and q ) 1 + 1.90476(c - 1) are factors accounting for the effect of molecular shape on the repulsive and attractive contributions, η ) bF is the packing fraction, b is the molar volume occupied by the molecules themselves, F is the molar density, Y ) exp(β) - 1.0617, β ) 1/kT ) reciprocal of Boltzmann’s constant times absolute temperature, ) disperse attraction energy, F ) 2NdR1/2/[1 + (1 + 4NdR)1/2], Nd ) number of hydrogen-bonding segments per molecule, R ) FKADYHB/(1 - 1.9η), YHB ) exp(βHB) - 1, KAD ) hydrogen-bonding volume, and HB ) energy of hydrogen bonding. Some equations of state, like the SAFT equation, require the user to regress optimal parameters for each pure component every time a new compound is to be considered for application. On the other hand, the ESD
10.1021/ie010346y CCC: $22.00 © 2002 American Chemical Society Published on Web 12/27/2001
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equation, like the SRK equation, is generalized in the sense that the necessary parameters can be estimated by the principle of three-parameter corresponding states. For small molecules, the ESD and SRK equations require only estimates of the critical constants. It would be most convenient if this generalized three-parameter methodology could be adapted to all molecular types, including polymers. The validity of basing parameter estimates for polymers on estimated critical constants is undermined, however, in that the estimates may be unreliable. Orbey et al.7 showed that the estimate of critical pressure had a significant effect on the estimates of the benzene + polystyrene solubility curve. Solubility parameter and molar volume estimates are more common in polymer applications, and the confidence level in these estimates is higher. Therefore, we suggest for polymers a generalized parameter estimation procedure based on the solubility parameter, molar volume, and molecular weight instead of the critical temperature, critical pressure, and acentric factor. The purpose of the present work is twofold: (1) to describe a consistent basis for adapting a generalized form of three-parameter corresponding states to polymers, including hydrogen-bonding polymers, when the critical constants are unknown; (2) to demonstrate this extension for the example of the ESD equation by comparing correlations with experimental data in several solvents. Extending the ESD Equation to Polymers In general, the ESD equation characterizes each pure component in terms of a molecular volume, b, a shape parameter, c, and a van der Waals attractive energy, . For completeness, hydrogen-bonding species require specification of three additional parameters: the number of hydrogen-bonding segments per molecule, Nd, the bonding volume, KAD, and the hydrogen-bonding energy, HB. However, the present work correlates the bonding volume in terms of b and c, and Nd is obvious from the molecular structure. Furthermore, general estimates of the hydrogen-bonding energy have been applied throughout the present work. So, only three parameters, b, c, and , need to be characterized for each component. The relationships for bonding volume and bonding energy are based on recognizing basic trends in the physical interpretations of these parameters. The bonding volume is a quantity that should remain roughly constant on a per segment basis. It represents the volume extending beyond a segment’s core repulsive diameter available for bonding overlaps to occur. Because the molecular volume and shape parameter are both directly proportional to the effective number of segments in the molecule, it is reasonable to assume that KAD ∼ b/c. By correlating the bonding volume for a large database, we obtained the generalized proportionality constant of 0.025, giving KAD ) 0.025 b/c as a workable general relation. For the hydrogen-bonding energy, we assumed 4 kcal/mol for hydroxyl groups and 1.5 kcal/ mol for amine, amide, nitrile, and aldehyde groups. These values were derived from previous studies of hydrogen-bonding energies for a wide range of components.15 Two physical quantities that are common in the study of polymer solutions are the solubility parameter and the liquid molar volume. Broad experience has been compiled over many years in terms of these two quantities, demonstrating their utility and efficacy in describ-
Table 1. Coverage of Compounds and Percent Error in the Correlation for the Shape Parameter, c group hydrocarbons hydroxyls amines, nitriles sulfides, thiols aldehydes, esters, ketones, ethers halocarbons overall
no. in database
% rms error in c
% rms error in v
342 63 68 26 180
5.75 13.64 13.83 2.76 5.16
6.47 7.80 7.86 4.16 5.59
46 725
5.42 7.65
8.57 6.86
ing trends to be expected in the solution and PVT behavior. Furthermore, many correlations of these quantities have been developed, facilitating their application and building confidence in the trends that they indicate as being “tried and true”. In this manner we adapt the parametrization of the equation of state to the best available methods of polymer characterization, decoupling the characterization step from the parametrization step to some extent. Expressions for the solubility parameter and the molar volume in terms of the equation of state are readily derived as shown below, providing two equations for the three unknown parameters. The task remaining is to develop a procedure for specifying the third parameter among b, c, and . One alternative is to apply a vapor-pressure datum, and that would certainly be preferable if any vapor pressure were known. In the absence of vapor-pressure data, however, we recommend a group contribution correlation for the shape parameter, c, as given below. Group contribution factors for the shape parameter, c, were correlated in terms of UNIFAC groups from a database of 725 pure components.16 The database consisted of shape parameters computed by first satisfying the solubility parameter and molar volume constraints from the group contribution method and then solving for the shape parameter that matched the boiling temperature at 10 mmHg. The boiling temperature at 10 mmHg was chosen as a standard vapor pressure because experimental data were available for a much larger number of high molecular weight compounds, especially for hydrocarbons from the API 42 compilation.17 The boiling temperature at 10 mmHg was computed from the standard correlation for vapor pressure when available and from the tabulated data for compounds from the API 42 compilation. The coverage of families of compounds is given in Table 1. The deviations in Table 1 correspond to the deviations of the correlated shape parameters from the values that exactly matched the 10 mmHg boiling temperature. The correlation errors are generally around 10%, a fairly large amount of error. On the other hand, the values of the shape parameters are very sensitive to the estimates of the solubility parameter. In a specific analysis of squalane, for example, changing the estimated solubility parameter by 3% resulted in a 25% change in the shape parameter. Considering that correlations of the solubility parameter may exhibit errors of 3%, 10% accuracy on the shape parameter correlation is difficult to avoid. Nevertheless, we recommend that the shape parameter correlation should only be applied when reliable critical constants or vapor pressures are not available. UNIFAC groups were selected as the basis for the regression because the UNIFAC method is often used in evaluating mixture properties. Note that there is no direct correspondence between the UNIFAC group definitions for a particular molecule and those from, say,
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1045 Table 2. Group Contributions for Estimating the Shape Parameter, c, the Liquid Molar Volume, and the Heat of Vaporization of Polymersa group
∆c
∆V
∆Hvap
group
∆c
∆V
∆Hvap
group
∆c
∆V
∆Hvap
CH3CH2< CH >C< RCH2< >RCH>RC< CH2dCH CHdCH CH2dC CHdC CdC CH2dCdCH ACH ACACCH3 ACCH2 ACCH OH ACOH CH3CO CH2CO CHO CH3COO CH2COO HCOO CH3O CH2O CH-O
0.217 0.187 0.051 -0.159 0.133 0.018 -0.140 0.270 0.272 0.182 0.156 -0.043 0.300 0.129 0.116 0.338 0.219 0.021 0.610 2.044 0.774 0.740 0.709 0.871 0.868 0.700 0.515 0.452 0.333
21.6 15.6 9.6 3.6 15.6 9.6 3.6 32.4 26.4 26.4 20.4 14.4 39.5 13.4 7.4 29.0 23.0 17.0 12.5 19.9 38.9 32.9 23.3 43.0 37.0 43.3 28.0 22.0 16.0
4.116 4.65 2.771 1.284 4.65 2.771 1.284 6.714 7.37 6.797 8.178 9.342 12.9 4.098 12.552 9.776 10.185 8.834 24.529 40.246 18.999 20.041 12.909 22.709 17.759 14.5 10.919 7.478 5.708
FCH2O CH2NH2 CHNH2 CH3NH CH2NH CHNH CH3-RN CH2-RN ACNH2 C5H4N C5H3N CH2CN COOH CH2Cl CHCl CCl CHCl2 CCl2 CCl3 ACCl CH2NO2 CHNO2 ACNO2 CH2SH I Br CHTC CTC Cl(CdC)
0.428 0.269 0.235 0.884 1.194 0.592 0.302 0.398 1.503 0.816 0.816 0.303 0.431 0.421 0.341 0.309 0.447 0.017 0.196 0.274 1.150 1.013 1.078 0.451 0.297 0.327 0.269 0.076 0.148
33.2 32.6 26.6 32.6 26.6 20.6 34.2 28.2 24.4 75.7 69.7 38.7 26.1 35.1 29.1 23.1 48.6 42.6 62.1 26.9 50.2 46.3 31.4 46.7 42.6 25.3 40.2 28.8 19.5
11.227 14.599 11.876 14.452 14.481 14 6.947 6.918 28.453 31.523 31.005 23.34 43.046 13.78 11.985 9.818 19.208 17.574 33.4 11.883 30.644 26.277 19.7 14.931 14.364 11.423 7.751 11.549 7
ACF CF3 CF2 CF COO SiH3 SiH2 SiH Si SiH2O SiHO SiO tert-N CCl2F HCClF CClF2 CONH2 CONHCH3 CONHCH2 CON(CH3)2 CONCH3CH2 CON(CH2)2 C2H5O2 C2H4O2 CH3S CH2S CHS C4H3S C4H2S
0.247 0.363 0.215 0.028 0.675 -0.004 -0.004 -0.004 -0.004 0.086 0.086 0.086 0.077 0.484 0.528 0.364 0.604 0.821 0.792 1.037 1.008 0.979 0.858 0.904 0.436 0.407 0.270 1.065 1.065
18.6 37.2 26.0 14.8 25.7 21.6 58.4 53.7 50.3 33.8 33.8 33.8 12.6 53.8 40.3 45.5 34.3 49.9 43.9 78.9 72.9 66.9 50.0 44.0 39.6 33.6 27.6 65.7 59.7
4.877 8.901 1.86 8.901 13.4 3.4 3.4 3.4 3.4 6.8 6.8 6.8 4.19 13.322 16.6 8.301 41.9 38.5 51.787 38.9 39.1 39.3 36.657 14.956 16.921 17.117 13.265 27.966 28
a
Values in boldface italics indicate supplements to Hoy’s method for volume.
Joback and Reid groups.18 Thus, selecting Joback and Reid groups as the basis would necessitate two steps of group contribution definition before a new component could be used with UNIFAC as well as equations of state.19 By using UNIFAC groups as the basis for all group contributions, all properties for new components can be specified in a single step. There are several alternatives when selecting group contribution methods for estimating solubility parameters, δ [(cal/cm3)1/2], and liquid molar volumes at 298 K, V298 L . We have focused on the method compiled by van Krevelen20 for liquid molar volume and Constantinou and Gani21 for heat of vaporization. Note that van Krevelen recommended that group contributions for polymers be independent of the contributions for solvents. Nevertheless, a comparison of van Krevelen’s molar volume contributions to those of Hoy22 (Table 7.10 of van Krevelen) shows a small discrepancy of only 4%. By comparison, the method of Fedors23 gives much larger discrepancies, especially for polystyrene and poly(vinyl alcohol). Hoy’s contributions have an advantage of being characterized for many more groups than van Krevelen’s. Furthermore, Hoy’s correlation can be improved for small molecules by incorporating a small residual constant that has minimal impact on polymers. ) Hence, we adapt Hoy’s correlation in the form V298 L 12.1 + ∑νi∆Vi. We supplement Hoy’s method in cases of missing groups by applying the database cited in Table 1. With this approach, we obtained 6.9% error by the supplemented Hoy method, compared to 7.8% by Fedors’ method. Note that 3.5% error could be achieved for these databases by regressing new values for all of the group contributions, but we favor Hoy’s method as having been “tried and true” based on years of polymer experience. For completeness, we present in Table 2 the 298 and values of UNIFAC group contributions for Hvap 298 VL along with the group contributions for the shape
parameter, c. Solubility parameters were computed from 298 298 and V298 according to the definition δ ) (Hvap Hvap L 298 1/2 298R)/VL ) . The relationship for the solubility parameter from the ESD equation is derived by expressing the heat of vaporization in terms the internal energy departure function, neglecting the departure function for the vapor. The internal energy departure function is given by the derivative of the free energy departure function. In the present work, we consider the efficient form of Wertheim’s theory treated by Elliott,24 for which the energy and free energy of association can be written as
Aassoc ) 2Nd ln(XA) + Nd(1 - XA) ) RT 2Nd ln(1 - F2/Nd) + F2 (2)
[
]
assoc 2Nd /RT) Uassoc β∂(A ∂F ) ) 1(3) 2Fβ RT ∂β ∂β N - F2
[
d
]
2 ∂F HBxR Nd - F YHB + 1 ) HB ∂β 2 1 + 2FxR Y
(4)
δ2V298 9.5qη(Y + 1.0617) L ) + R/k 1 + 1.7745Yη 2 HB HB FxR(Y + 1) Nd + F (5) YHB 1 + 2FxR where δ is the solubility parameter, V298 is the liquid L molar volume at 298 K, and R is the universal gas constant. Equation 5 is used to match the solubility parameter. The equation of state can be applied at ZL ) 0 to match the molar volume. The value for c was
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assumed to follow a linear relationship with respect to the degree of polymerization.
c ) 1+
∑νi∆ci
(6)
In summary, when no vapor-pressure data are available, we have three unknown quantities (b, c, and /k) and three equations to determine them: ZL ) 0 and eqs 5 and 6. If vapor-pressure data are available, it is extremely valuable to apply the data in the determination of the equation of state parameters. In those cases, eq 6 is replaced by the isofugacity criterion. Typically, the vapor-pressure data for heavy compounds are available at low pressures, because higher saturation pressures lead to temperatures that cause the compounds to degrade thermally. Under these conditions, the vapor phase may be treated as an ideal gas. These observations lead to a simplification of the isofugacity criterion when Mn > 200.
9.5q -4c ln(1 - 1.9ηsat) ln(1 + 1.7745Yηsat) 1.9 1.7745 Psatb FxR (7) - 2 ln(1 + FxR) ) ln sat 1+ η RT 1 + FxR
(
)
Note that the values of ηsat in eq 7 must be computed at the saturation condition. This introduces a new unknown parameter into the set of equations. Fortunately, the availability of the vapor pressure indicates a fourth constraint equation.
ZLsat ) Psatb/ηsatRT
(8)
where ZLsat is calculated by eq 1. Hence, we have four equations and four unknowns in cases where Psat is available. To facilitate understanding of how these equations are applied, examples are given in the appendix. There are a number of detailed considerations that should be clarified for the benefit of readers familiar with previous treatments of polymer solutions by equations of state. First, the reader may recognize that the polymers considered in this study are not liquid at 298 K, and they vary considerably in their crystallinity in the solid state. The group contributions have been regressed from data for compounds that are amorphous at 298 K, however, so application to compounds that are solid at 298 K means that these represent “effective” values for the liquid molar volume. The equation of state treats these as liquid molar volumes as well, so the treatment is consistent. Second, many previous equations of state for polymers have focused on the volumetric behavior of the pure polymer, placing special emphasis on expansivity and compressibility as well as density. In the literature on solutions of small molecules, however, Peneloux et al.25 have established that the precise volumetric behavior of the equation of state has little or no impact on predictions of phase behavior. Extending these observations to polymer solutions and noting that we are primarily interested in the phase behavior of mixtures, we have elected not to become preoccupied with the density properties of pure polymers. Existing methods for estimating density given composition can be applied once the phase compositions have been accurately determined.26
Figure 1. P-x,y diagram for ethane + docosane. Points represent the experimental measurements of Peters et al.33 The curve repesents the result of the ESD equation with kij ) 0.03.
Applications to Polymer Solutions To test the utility of these correlations, it is necessary to compare to experimental measurements for polymers in solution. We consider these comparisons in two sections: first for polyethylene and second for miscellaneous polymers, including polar polymers such as poly(vinyl chloride) and poly(vinyl alcohol). In all cases, the following mixing and combining rules have been applied:
〈4cη〉 9.5〈qηY〉 P F2 )1+ (9) FRT 1 - 1.9η 1 + k1〈Yη〉 1 - 1.9η where η ≡ F∑xibi, 〈4cη〉 ≡ 4F∑∑(cb)ijxixj, 〈qηY〉 ≡ F∑∑xixj(qb)ijYij, (cb)ij ≡ (cibj + cjbi)/2, (qb)ij ≡ (qibj + qjbi)/2, ij ≡ (ij)1/2(1 - kij), and 〈ηY〉 ≡ F∑xibiYi. All polymers were treated as being monodisperse for the present calculations. Applications to Polyethylene Solutions The trends for polyethylene in supercritical solvents are very similar, as illustrated by Figures 1-4. We can organize these results by first considering systems for which the vapor pressure of the heavy component is known (docosane and squalane) and then considering systems for which the vapor pressure is unknown. Alessi et al.27 have compared several equations of state for the ethane + docosane system (Figure 1). The equations considered include the perturbed hard-chain equation,28 the Peng-Robinson equation, and the SRK equation. We reproduce their comparison, along with the results from the method presented here, in Table 3. Figure 2 further illustrates the accuracy when vapor-pressure data are available, as in the case of squalane for which the 10 mmHg temperature is given as 532 K.17 The lines show that even the pure predictions of the ESD equation (kij ) 0) are quite accurate for this system. Squalane and docosane are interesting as examples of heavy hydrocarbons that might be considered as model compounds in studies of waxes and asphaltenes.
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1047
Figure 2. P-x,y diagram for n-hexane + squalane. Points represent the experimental measurements of Joyce and Thies.42 The curves represent the results of the ESD equation with kij ) 0.0.
Figure 4. P-x,y diagram for ethylene + polyethylene (Mn ) 7000). Points represent the experimental measurements of Luft and Lindler.44 The curves represent the results of the ESD equation with kij ) 0.027. Table 3. Percent Bubble Pressure Deviation for an Ethane (1)-Docosane (2) System at 320 Ka x2 0.1079 0.1471 0.21 0.2978 0.3962 0.5023 0.6580 0.7965 %AAD
P (bar) PHCT1 PHCT2 PHCT3 69.21 60.76 51.56 41.08 31.35 23.10 13.48 7.25
SRK
PR
ESD
-64.3 -111.8 15.1 13.0 10.3 -10.3 -36.6 11.8 9.6 4.8 -128.7 3.8 -2.9 6.5 7.2 1.6 -46.3 9.4 3.0 6.0 3.5 0.31 -35.3 12.7 6.0 0.8 1.6 0.09 -30.4 15.4 8.6 -4.3 -6.8 -0.52 -28.6 16 9.7 -13.9 -16.3 0.44 -24.7 19.7 12.7 -18.8 -21.1 -1.5 36.75 18.95 23.91 9.65 9.89 2.45
a Experimental data from Peters et al.33 (k ) 0.032 for the ESD ij equation). PHC, PR, and SRK errors are from Alessi et al.27
Figure 3. P-x,y diagram for ethylene + polyethylene (Mn ) 1940). Points represent the experimental measurements of Wohlfarth et al.43 The curves represent the results of the ESD equation with kij ) 0.03.
The primary effect of increasing the molecular weight of the polymer is to increase the maximum pressure of the phase envelope. All of these calculations were based on a simple bubble-point pressure algorithm assuming a monodisperse polymer with molecular weight equal to the number-average molecular weight. The results in Figure 3 are very similar to those obtained by Sako et al.29 based on a specially developed equation of state. When the number-average molecular weight approaches 7000, the cricondenbar exceeds the range of the experimental study. Figure 4 shows the behavior for 7000 molecular weight. Miscellaneous Polymer Solutions
Figure 5. Cloud-point curves for octadecane + polystyrene (Mn ) 3700). Points represent experimental measurements of van Opstal et al.30 The curves represent the predictions of the ESD equation with kij ) -0.007.
van Opstal et al.30 studied a number of systems containing polystyrene. These consistently exhibited
upper critical solution temperature behavior. Figure 5 illustrates a typical example for the system octadecane
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Figure 6. Calculated and experimental cloud-point pressures for the poly(ethylene-co-vinyl acetate) + ethylene pseudobinary (Mn ) 18 000). Data of Folie et al.31 kij ) 0.051 - 13/T.
Figure 7. Polyisobutylene (PIB) with n-pentane (P), cyclohexane (C), and benzene (B). Average molecular weight of PIB ) 40 000 g/mol. Points are experimental data,36-38 and lines are ESD correlations. (kij values are summarized in Table 4.)
+ polystyrene (Mn ) 3700). The curvature is inaccurately portrayed by the equation of state, but this is fairly complex phase behavior. The upper curve shows that the equation of state predicts lower critical solution temperature behavior for this system at slightly higher temperatures. This inaccurate curvature is typically attributed to the inability of a mean field equation of state like the ESD equation to represent the behavior in the critical region. The phase behavior of this system may be of interest in the context of asphaltene studies because the polymer is fairly low in molecular weight but high in aromatic content. Folie et al.31 have measured phase equilibria for ethylene + poly(ethylene-co-vinyl acetate) (PEVAc)
Figure 8. Benzene-poly(ethylene oxide) (PEO) system. Average molecular weight of PEO ) 5700 g/mol. Points denote experimental results,34 and lines show the ESD correlation.
Figure 9. Benzene-poly(propylene oxide) (PPO) system. Average molecular weight of PPO ) 500 000 g/mol. Points are experimental data,40 and lines show the ESD correlation.
systems. We have applied the methodology presented here to their experimental data for the system with a molecular weight for PEVAc of 18 000. Results are illustrated in Figure 6. Comparing to their results for the SAFT equation, we conclude that their method and the method presented here yield similar accuracy for correlating the phase behavior of this system. The method presented here is relatively simple and general to apply however. Experimental data for other polymer solutions generally take the form of partial pressure data. Trends for systems containing polyisobutylene, poly(ethylene oxide), and poly(propylene oxide) are illustrated in Figures 7-9. Average deviations for these solutions and several additional solutions are presented in Table 4. Included in Table 4 are comparisons for polar polymer solutions
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1049 Table 4. Summary from Analysis of Partial Pressures
polymer
solvent
PS PS PIB PIB PIB PEO PPO PMMA PMMA PVAc PVAc PVAc PVAlc PVC PVC PVC
chloroform methyl ethyl ketone benzene cyclohexane n-pentane benzene benzene methyl ethyl ketone toluene acetone benzene n-propanol water 1,4-dioxane tetrahydrofuran toluene
MW of ref polymer ref (g/mol) 34 35 36 37 38 39 40 41 41 41 41 41 41 41 41 41
290 000 97 200 40 000 40 000 40 000 5 700 500 000 19 700 19 700 170 000 170 000 170 000 67 000 34 000 34 000 34 000
kij
% AAD (PP)
0.028 0.034 0.030 0.019 0.029 0.003 -0.005 0.014 0.010 -0.001 0.003 0.005 0.014 0.020 -0.001 0.004
7.24 3.44 3.04 1.01 6.12 1.01 1.20 3.33 2.52 2.50 2.91 13.31 6.09 5.18 4.89 7.05
such as poly(vinyl chloride) and poly(vinyl alcohol). These correlations show that partial pressure data can be accurately represented by the methods presented here for many systems. Orbey and Sandler32 obtained very similar results using multiparameter mixing rules. No strong conclusions should be drawn from these observations, however, because partial pressure data are fairly insensitive to the polymer characterization.
>C
