4966
Ind. Eng. Chem. Res. 1999, 38, 4966-4974
A Perturbed Hard-Sphere-Chain Equation of State for Polymer Solutions and Blends Based on the Square-Well Coordination Number Model Wei Feng and Wenchuan Wang* College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China
Based on the concept of a perturbed hard-sphere-chain (PHSC), where the square-well fluid is the reference system, a perturbation term is developed from the coordination number model for pure and mixture square-well fluids proposed by this group. Consequently, we derive a new PHSC type equation of state (EOS), which is of much simpler formulation and is even easier to use. The EOS has been extensively tested in terms of a large data bank, consisting of the properties of 37 normal fluids and solvents, 67 polymers. The correlation accuracy for the saturated properties is within the errors in common EOS approach. In particular, the grand average deviation for correlating the liquid densities for 67 polymers is 0.19%, which is of the same accuracy as that for commonly used lattice model EOSs. In all of the calculations, the EOS needs only three temperature- and composition-independent parameters. In addition, the new EOS is used for 40 sets of vapor-liquid equilibrium (VLE) calculations of polymer-solvent systems. Three typical liquid-liquid equilibrium (LLE) systems have been investigated by using the EOS, including both the upper critical solution temperature and the lower critical solution temperature polymer-solvent and polymer blend systems. The calculated binodal and spinodal curves are in good agreement with experimental data. All of the VLE and LLE calculations need conventional mixing rules and only a binary interaction parameter. Introduction In recent years, much attention has been paid to the development of off-lattice equations of state on the perturbation theory for the calculation of thermodynamic properties of polymer systems. Some off-lattice equations of state (EOSs) have been successfully applied to polymer solutions and blends, such as the statistical association fluid theory (SAFT) EOS,1,2 the polymer chain rotators EOS,3 the generalized Flory dimer equation,4 and the EOS for chainlike molecule fluids.5 In particular, the perturbed hard-sphere chain EOS,6,7 consisting of an attractive hard-sphere chain and a van der Waals type perturbation term, has been used to calculate the pressure-volume-temperature (PVT) relationships for pure polymers and solvents and to estimate vapor-liquid equilibria (VLE)8 and liquidliquid equilibria (LLE) for polymer solutions and blends.7,9,10 Recent development of the perturbed hardsphere chain (PHSC) EOS is the work of Hino and Prausnitz.11 After detailed analysis of shortcomings of the model, they attribute some unsatisfactory performances of the PHSC EOS partly to the use of a simple van der Waals perturbation term. As a result, the term is replaced with a theoretical one derived by Chang and Sandler,12 which is based on the second-order theory of Barker and Henderson13 for the square-well (SW) fluid of variable width. By simplifying the analytical solution of Chang and Sandler, Hino and Prausnitz developed a new PHSC EOS and tested it extensively in terms of PVT relationships of pure polymers and solvents as well as VLE and LLE data. The results show that improve* To whom correspondence should be addressed. Tel: +86-10-64433776. Fax: +86-10-64436781. E-mail: wangwc@ mailserv.buct.edu.cn.
ments are obtained through the reformulation of the perturbation term, while maintaining the reference term represented by the modified Chiew’s EOS.6 As discussed by the same authors, there is still some room for further development of the new PHSC EOS. For example, a large coefficient matrix is needed in the EOS for the calculation of the radial distribution functions proposed by Chang and Sandler,12 and all of the coefficients vary with the width of the square well. In addition, the quality of the correlation of saturated properties of pure solvents and polymers is dependent on the width of the square well as well. In the calculation of VLE and LLE for polymer solutions and blends, a semiempirical parameter is introduced to modify the number of segments, s, per polymer molecule. The goal of this work is to develop a practical and user-friendly EOS for polymer systems. Based on the concept of PHSC, where the SW fluid is the reference system, a perturbation term for the SW fluid is introduced from the coordination number model for pure and mixture SW fluids proposed by our group.14 As a result, we derive a new PHSC type equation of state, which is of much simpler formulation and even easier to use. The new EOS is tested extensively in terms of saturated properties of large numbers of polymer and diversified solvents. Using simple mixing rules, the new EOS is extended to VLE and LLE calculations of polymersolvent and polymer blend systems. PHSC EOS from the SW Coordination Number Model A general form of the PHSC type EOS can be expressed as the sum of a reference term and a perturbation term, representing repulsive and attractive interactions, respectively:
10.1021/ie990234v CCC: $18.00 © 1999 American Chemical Society Published on Web 12/06/1999
Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4967
( ) ( ) ( ) P P ) FkBT FkBT
P FkBT
+
ref
(1)
pert
where P is the pressure, F is the number density, kB is the Boltzmann constant, T is the temperature, and subscripts ref and pert signify the reference equation of state and the perturbation term, respectively. Reference EOS. Chiew15 has derived analytical expressions of the EOS for athermal tangent hardsphere-chain models. In this work, we simply use of the EOS proposed for homonuclear hard-sphere chains:
[
]
3η2 1 P 3η + ) + FkBT 1 - η (1 - η)2 (1 - η)3 r - 1 1 + 0.5η (2) r (1 - η)2 η is the hard-sphere site volume fraction, where η ) (π/ 6)Fσ3, F is the number density of hard spheres of the system, σ is the diameter of a hard sphere or a segment, and r is the number of hard spheres per molecule. For hard-sphere-chain mixtures, the compressibility factor is given by15
( ) [ P
FkBT
)
ref
6
ξ0
π 1 - ξ3 1
3
3ξ1ξ2
+
(1 - ξ3)
m
+ 2
(
3ξ2
(1 - ξ3)
]
where ξj is defined as
ξj )
ξ2
Fc(k)rkσkj ∑ 6 k
3
)
(3)
(4)
where Fc(k) is the number density of molecule k, rk is the chain length of the k molecule, and σk is the hard-sphere or segment diameter of component k. Coordination Number Model and EOS of the SW Fluid. The potential energy, φ, between two SW molecules at distance r is given by
{
∞ reσ φ(r) ) - σ < r < λσ 0 r g λσ
(5)
where is the well depth and λ is a parameter characterizing the well width and is set to 1.5 in most cases and in this work. Because the SW molecule has a well-defined coordination shell, several researchers established different coordination number models14,16-18 to describe the structures of pure and mixture SW fluids based on computer simulation data and statistical thermodynamics. Bokis and Donohue19 made a comparison of these models and pointed out that the Guo-Wang-Lu (GWL) model14,20 is in relative good agreement with the internal energy simulation data of SW fluids, and it contains no adjustable parameters. The general form of the GWL coordination number model for pure and mixture SW fluids is
Nji )
4π 3 (λ - 1)F*xjmji exp(jji/kBT) 3
(7)
where
∑i xiσii3
(8)
j3 mji ) σji3/σ
(9)
σ j3 ) and
where σji for a binary SW system containing molecules i and j is simply expressed as
σji ) (σjj + σii)/2
(10)
In eq 6, jji is defined as an effective energy parameter:
(
jji) 1 -
)
F* ∑∑xjximji ji x2
(11)
where ji can be obtained by the combining rule
ji ) (iijj)0.5
3
πm
F* ) Fσ j3
-
∑ Fc(k)(rk - 1) 2 + 3σk(1 - ξ )
2(1 - ξ3)k)1
j about a molecule i, x is the mole fraction, and F* is the reduced density, given by
(6)
where Nji denotes the coordination number of molecule
(12)
On the GWL model, EOSs for the SW pure and mixture fluids were derived.20 The EOSs were tested with a large number of Monte Carlo simulation data and compared with other models. Of notable simplicity and reasonable accuracy, the EOSs provide a basis for the representation of the perturbation term in eq 1, which is discussed in the next section. PHSC EOS Based on the SW Coordination Number Model (PHSC-SWCNM EOS). (1) EOS for Pure Fluids. To establish a PHSC type EOS, the reference term is simply taken from Chiew’s work, and the perturbation term can be obtained from the SW EOS. Considering a molecule consisting of r segments, we assume that the Helmholtz energy is given by
( ) A NkBT
)r
pert
( ) Ar)1 NkBT
(13)
pert
where superscript r ) 1 represents the SW monomer. Obviously, the perturbation term of the compressibility factor can be expressed as
( ) P F c k BT
( )
)r
pert
PSW FkBT
(14)
pert
where Fc is the molecular number density of the system. The superscript SW represents the SW molecule system. As is shown in the paper of Guo, Wang, and Lu,20 the EOS for the SW fluid derived from the GWL model is given by
[ ]
PSW PSW PHS ) + FκBT FκBT FκBT
(15)
pert
where the superscript HS represents hard sphere, and the perturbation term is expressed as
[ ] PSW FκBT
pert
)
(
)
N0 Ω - 1 ΩR′F* + 2R R T*
(16)
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Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999
where R′ is the constant, set to -1/x2, and N0, R, T*, j, and Ω are defined as
N0 )
4π 3 (λ - 1)F* 3
(17)
ξ3 1 - ξ3 N0 2Rm
m
-
(25)
(20)
In eqs 24-26, mij is given by eq 9, and T/ij is given by
j ) (1 - F*/x2)
(21)
T/ij ) kBT/ij
[
3ξ1ξ2
Fπ (1 - ξ )2 3
[
] )]
(
3ξ23
+
(1 - ξ3)3
(
-
ξ3ξ23
]
+
(1 - ξ3)3
)
ξ2 Fc(k)(rk - 1) 2 + 3σk 1 - ξ3 2(1 - ξ3)k)1 m
1
m
m
Ω ) exp(j/kBT)
Note that for polymer systems η here is the site volume fraction, where η ) (π/6)Fσ3, F is the number density of segments, and σ is the diameter of a segment. Obviously, the PHSC-SWCNM EOS contains three parameters, /kB, σ, and r, which can be obtained by fitting saturated properties or PVT data of pure fluids. (2) EOS for Mixture Fluids. The PHSC-SWCNM EOS can be easily extended to mixture fluids, because the reference term for mixtures is given by eq 3 and the perturbation term for mixtures has been derived by Guo, Wang, and Lu.20 Therefore, the EOS for mixtures is given by
M
x2
m
∑i ∑j φiφjmij
Ω′ij ) exp(Rm/T/ij)
[
)1+
1
(24)
(19)
P 1 + 0.5η 4η - 2η )1+r - (r - 1) -1 FckT (1 - η)3 (1 - η)2 N0 Ω - 1 ΩR′F* r + (22) 2R R T*
FkBT
x2
m
T* ) kBT/
( )
( )
m
∑i ∑j φiφjmji
(18)
2
6
R′m ) -
F*
R ) 1 - F*/x2
Substituting eq 16 into eqs 1 and 14, we finally obtain the perturbed hard-sphere-chain EOS based on the SW coordination number (PHSC-SWCNM EOS):
P
Rm ) 1 -
∑
[( )
∑i ∑j φiφjmij
1-
]
R′mF* R′m F* (Ω′ij - 1) + Ω′ij Rm T/ij (23)
where subscript m denotes mixture, φk is the segment fraction of component k, where φk ) xkrk/ m ∑k)1 xkrk, xk is the mole fraction of component k, rk is the number of segments per molecule k, and Fc(k) is the molecular number density of component k. σk is the diameter of a segment of component k, and σ j3 ) m 3 ∑i φiσi . F is the number density of a segment of the system, where F ) Nr/V, Nr is the total number of segments, Fc is the molecular number density of the mixture, where Fc ) N/V, N is the number of molecules, and V is the volume of the mixture, and other variables are expressed as
(26)
(27)
The parameters representing interactions between a pair of unlike segments are obtained from conventional combining rules:
σij ) (σii + σjj)/2
(28)
ij ) (1 - kij)(iijj)0.5
(29)
As is seen in the combining rules, only an adjustable binary interaction parameter kij is needed. Results Pure Component. A databank was used for testing the capability of the PHSC-SWCNM EOS for the description of saturated properties of normal fluids and solvents for polymer solutions and the PVT relationship for polymers. All of the detailed information about the polymer databank, including temperature and pressure ranges, data sources, and abbreviations of polymers, can be referred to our previous work.21 (1) Normal Fluids and Solvents. The PHSCSWCNM EOS is used for the calculation of saturated properties for 37 normal fluids and solvents, including normal alkanes, branched and cyclic alkanes, aromatics, ketones, and halogenated hydrocarbons. These temperature- and composition-independent parameters, /kB, σ, and r, are regressed by fitting simultaneously the saturated vapor pressures and liquid densities. The objective function for regression is the sum of squares of the relative deviations between calculation and experimental values with respect to the experimental values. Table 1 shows the EOS parameters and deviations (see definitions in the footnote of Table 1). For illustration, Figures 1-6 present the results for selected alkanes, ketones, and halogenated hydrocarbons. It is worth noticing that when we plotted the EOS parameters of normal alkanes against the molar mass, Mm, a good linear relationship for parameters r and rσ and a smooth monotonic curve for the energy parameter /kB are obtained, as shown in Figures 7-9. On the basis of our experience, this observation provides a basis for developing a group-contribution EOS for polymer solutions and blends. (2) Polymers. The PHSC-SWCNM EOS has been applied to 67 polymers and copolymers of different structures and bonding forces over a temperature span of 50-150 K and a pressure range of up to 200 MPa. In the regression of EOS parameters for polymers, the objective function is the sum of squares of the relative deviations between calculated and experimental liquid densities alone. The ratio of r/M is a parameter for polymers in the regression, other than r for pure normal
Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4969 Table 1. Calculated Results for Pure Normal Fluids and Solvents AADa
parameters substance
/κ, K
cm3/mol
r
Psat (13)b
Fliq 2.6 (13)b
methane
102-187
171.68
22.476
1.027
3.0
ethane propane butane pentane hexane heptane octane nonane
163-304 274-363 217-425 291-454 252-492 264-523 296-552 244-424
203.246 222.598 232.275 238.451 247.097 246.578 232.078 247.638
18.621 19.497 20.905 19.704 20.586 19.541 16.479 17.912
1.732 2.150 2.535 3.230 3.750 4.421 5.602 6.229
1.1 (16) 1.7 (15) 1.9 (17) 1.9 (23) 3.3 (25) 3.3 (17) 2.6 (17) 0.5 (15)
4.4 (16) 2.0 (15) 4.4 (14) 2.6 (23) 3.6 (25) 3.9 (17) 4.2 (17) 1.3 (15)
cyclopropane cyclobutane cyclopentane cyclohexane cycloheptane cyclooctane
203-386 199-285 274-302 279-541 290-575 288-466
245.506 271.211 347.030 285.986 296.997 375.855
16.806 19.729 21.297 19.956 20.565 23.803
2.184 2.595 2.923 3.384 3.635 4.196
3.7 (20) 0.3 (23) 0.9 (15) 3.6 (15) 1.8 (13)
3.0 (20) 0.1 (23) 0.2 (15) 3.0 (16) 4.1 (15) 0.2 (13)
acetone butanone cyclobutanone 3,3-dimethyl-2-butanone cyclohexanone 2-heptanone 2-octanone
275-493 272-529 285-381 295-403 290-575 274-448 297-446
252.715 237.025 285.507 286.332 369.766 316.036 279.271
12.277 13.666 11.452 18.820 15.868 17.390 16.616
3.586 3.999 4.382 4.451 4.912 5.767 6.324
3.6 (16) 3.0 (21) 1.5 (17) 0.3 (17) 0.1 (21) 1.1 (16) 4.1 (12)
2.8 (16) 0.3 (7) 0.7 (14) 0.3 (11) 0.1 (21) 0.3 (12) 1.3 (11)
benzene toluene ethylbenzene m-xylene
284-550 282-546 277-487 276-582
309.851 311.400 321.679 289.73
20.255 20.739 20.977 17.638
2.748 3.331 3.969 4.455
2.5 (22) 2.7 (16) 1.0 (21) 3.7 (19)
3.0 (22) 2.4 (16) 1.7 (21) 3.2 (19)
chloromethane dichloromethane trichloromethane tetrachloromethane fluoromethane difluoromethane trifluoromethane
193-393 212-510 252-431 253-556 144-288 249-351 191-291
270.105 262.462 286.684 262.997 213.554 190.112 153.273
15.522 13.244 16.118 15.548 12.410 8.564 8.533
2.020 2.909 3.126 3.474 1.995 2.783 3.108
1.5 (20) 3.0 (21) 1.0 (14) 1.8 (23) 2.7 (18) 4.6 (16) 1.5 (15)
1.7 (20) 2.7 (21) 2.7 (14) 6.8 (23) 0.7 (18) 2.2 (16) 2.3 (15)
isobutane 2-methylbutane 2,3-dimethylbutane
219-369 266-447 286-494
223.124 238.630 245.973
20.296 20.538 21.460
2.673 3.025 3.409
0.7 (13) 1.9 (14) 4.3 (20)
4.2 (13) 4.1 (14) 3.5 (20)
a
sat
AAD(P ) ) b
T, K
σ3/mol,
1 N
∑| N
i
|
exp Pcal s (Ti) - Ps (Ti)
Pexp s (Ti)
× 100
liq
AAD(F ) )
1 N
∑|
|
N
exp Fcal l (Ti) - Fl (Ti)
i
Fexp l (Ti)
× 100
Numbers in parentheses indicate the number of data points, and data sources are taken from the literature of Smith and Srivastava.25
fluids and solvents. All of these EOS parameters, /kB, σ, and r/M, and the deviation (see definition in Table 2) are shown in Table 2. Figures 10 and 11 give a graphical description of comparisons between the calculated and experimental results for the polymers polyisobutylene (PIB) and poly(2,6-dimethylphenylene oxide) (PPO). Mixtures. In this work, we focus on applicability of the PHSC-SWCNM EOS to VLE and LLE calculations of polymer solution and blend systems. In all of the calculations, the parameters of pure components are taken directly from the tables of this work, and only a binary interaction parameter kij in eq 29 is needed, which can be fitted by experimental VLE or LLE data. VLE. In this work, polymer solvent solutions are recognized as pseudo binary solutions, which is acceptable for the description of VLE. Therefore, we compare our calculated weight fraction activity values to test the capability of the EOS for VLE calculations. The derivation of an analytical expression of the weight fraction activity coefficient from the EOS is included as Supporting Information for readers’ reference. The VLE data of polymer solutions can be referred to the literature.21 The objective function for regressing the binary interaction parameter k12 is the sum of squares of the
Figure 1. Comparison between calculated and experimental results for vapor pressures of n-alkanes. s: calculated. Symbols: experimental data taken from ref 25.
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Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999
Figure 2. Comparison between calculated and experimental results for liquid volumes of n-alkanes. s: calculated. Symbols: experimental data taken from ref 25.
Figure 3. Comparison between calculated and experimental results for vapor pressures of ketones. s: calculated. Symbols: experimental data taken from ref 25.
relative deviations between calculated and experimental weight fraction activity coefficients of the solvent in a polymer-solvent system. All of the results for 40 data sets are shown in Table 3. LLE. LLE calculations for polymer solutions and blends lie in the determination of the phase separation domain, which is characterized by the binodal and spinodal curves and the critical point. The condition for the binodal compositions in the chemical potentials in the two liquid phases must be equal for all of the components:
(µi - µ0i )′ ) (µi - µ0i )′′
(30)
where superscripts single prime and double prime represent the two phases in LLE and µ0i is the chemical potential of pure component i at the system T, P, and compositions. From the thermodynamic stability crite-
Figure 4. Comparison between calculated and experimental results for liquid volumes of ketones. s: calculated. Symbols: experimental data taken from ref 25.
Figure 5. Comparison between calculated and experimental results for vapor pressures of halogenated alkanes. s: calculated. Symbols: experimental data taken from ref 25.
ria, the spinodal points can be derived by solving the equation
( ) ∂2Gφ ∂φ2
)0
(31)
T,P
where Gφ is the Gibbs energy of the system and φ is the segment fraction of component 1 or 2. The critical solution point is determined by the equation
( ) ∂3Gφ ∂φ3
)0
(32)
T,P
It is worth mentioning that the LLE calculations for polymer solutions and blends are complicated and
Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4971
Figure 6. Comparison between calculated and experimental results for liquid volumes of halogenated alkanes. s: calculated. Symbols: experimental data taken from ref 25.
Figure 8. Close-packed molar volume rσ3 for n-alkanes versus molar mass.
Figure 9. Segment energy /k for n-alkanes versus molar mass. Figure 7. Segment number r for n-alkanes versus molar mass.
rather tedious work. Therefore, analytical expressions for the LLE criteria are essential to the reliable and convenient LLE calculations. The second derivative of the Gibbs energy with respect to the segment fraction in eq 31 has been derived analytically in this work. Because the length of the paper is limited, the expressions are also included as Supporting Information. On the other hand, the third derivative can be obtained from eq 31 by using the numerical method. Similar to our recent work,23 two types of LLE calculations were conducted using the PHSC-SWCNM EOS. (1) Correlations of Spinodal Points. We here correlated the experimental spinodal points for three systems: PS(520000), PS(166000), and PS(51000) with the solvent cyclohexane. Figure 12 shows the correlated results graphically. The EOS parameters of the polymer
and solvent are obtained from Tables 1 and 2, and the binary interaction parameter k12’s are listed in Table 4. (2) Estimations of Binodal and Spinodal Curves from Experimental Cloud Points. In most cases, only experimental cloud-point data are available for polymersolvent and polymer-polymer systems. We calculated the LLE for the systems by using the cloud point information. The procedures for the calculations are referred to the literature.23 Here, we give two examples, shown in Figures 13 and 14. Figure 13 gives the LCST type phase diagram of the system PIB(500000) + benzene. By contrast, Figure 14 presents the UCST type polymer blend system, PS (4370) + PBD (1100). The binary parameters are also listed in Table 4. Discussion and Conclusions The PHSC model provides a reasonable framework for the development of off-lattice EOSs to describe the
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Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999
Table 2. Calculated Results for Polymers Using the PHSC-SWCNM EOS parameters polymer
/κ, K
σ3/mol, cm3/mol
r/M
PDMS PDMS1 PDMS2 PDMS3 PDMS4 PDMS5 PDMS6 PS PPO PIB BPE PVAc PMP HLPE HDPE PA PTFE PVAC PP PoMS PMMA PBMA PECH PCL i-PP PH PMA PEA PEMA HFPC BCPC SMMA20 SMMA60 PEO TMPC PSF PC PB PAR PTHF PVME PA6 PA66 LDPE LDPE-A LDPE-B LDPE-C EP50 EVA18 EVA25 EVA28 EVA40 SAN3 SAN6 SAN18 SAN40 SAN70 POM PEEK i-PMMA PA11 PVDF PET PBD PcHMA LPE LDPE1 grand average
345.08 302.84 320.34 327.62 332.34 337.63 337.21 512.16 447.99 471.93 445.99 4397.08 483.04 418.75 413.68 569.04 277.20 396.87 464.42 532.51 605.98 442.18 483.93 496.98 417.47 505.73 463.98 443.65 491.43 475.52 545.41 515.76 511.13 434.94 452.70 564.58 517.10 482.10 553.87 428.33 462.37 542.07 474.97 405.83 478.06 489.90 484.10 531.59 512.82 514.08 467.66 473.38 535.81 509.23 540.67 564.95 577.52 525.55 533.79 476.08 519.91 431.88 523.01 410.60 512.10 445.56 455.65
21.09 24.54 24.39 23.99 23.55 23.73 23.93 21.09 23.87 21.03 25.71 15.28 37.41 21.11 21.04 22.76 18.31 15.84 32.65 22.80 20.03 19.56 16.26 21.66 24.87 17.29 20.59 20.66 20.73 20.84 20.99 21.93 21.12 20.50 20.58 20.52 20.44 29.89 20.47 20.21 20.90 20.78 21.10 20.98 24.76 25.07 24.94 26.37 21.06 21.06 24.18 24.51 24.66 23.44 23.01 23.40 20.32 20.97 21.70 16.31 22.96 19.07 20.71 16.03 21.74 24.48 24.95
0.0333 0.0340 0.0340 0.0339 0.0341 0.0338 0.0334 0.0375 0.0302 0.0430 0.0380 0.0444 0.0272 0.0457 0.0452 0.0300 0.0177 0.0428 0.0302 0.0355 0.0357 0.0400 0.0375 0.0358 0.0381 0.0413 0.0346 0.0358 0.0362 0.0347 0.0280 0.0350 0.0346 0.036 0.0346 0.0325 0.0336 0.0328 0.0332 0.0364 0.0389 0.0310 0.0310 0.0449 0.0399 0.0391 0.0397 0.0387 0.0347 0.0347 0.0391 0.0410 0.0321 0.0336 0.0337 0.027 0.0363 0.0319 0.0297 0.0417 0.0368 0.0259 0.0304 0.0563 0.0347 0.0398 0.0394
AAD Fla from this work 0.11 (60)b 0.18 (48) 0.15 (40) 0.15 (40) 0.13 (40) 0.13 (40) 0.13 (40) 0.12 (64) 0.16 (56) 0.11 (70) 0.09 (53) 0.06 (100) 0.30 (97) 0.16 (61) 0.14 (67) 0.14 (77) 0.24 (21) 0.08 (62) 0.24 (41) 0.08 (43) 0.29 (48) 0.32 (100) 0.17 (105) 0.14 (126) 0.12 (55) 0.27 (100) 0.33 (100) 0.33 (100) 0.12 (100) 0.22 (100) 0.24 (100) 0.27 (100) 0.28 (100) 0.11 (100) 0.21 (100) 0.19 (100) 0.20 (107) 0.18 (34) 0.13 (100) 0.08 (100) 0.35 (100) 0.36 (100) 0.24 (100) 0.30 (42) 0.22 (100) 0.21 (100) 0.21 (100) 0.22 (100) 0.17 (100) 0.21 (100) 0.21 (100) 0.26 (100) 0.25 (100) 0.24 (100) 0.20 (100) 0.27 (100) 0.25 (100) 0.15 (24) 0.16 (120) 0.31 (93) 0.10 (80) 0.22 103) 0.11 (46) 0.21 (86) 0.15 (89 0.15 (63) 0.16 (78) 0.19
a AAD F ) (1/N)∑N[|Fcal - Fexp|/Fexp] × 100. b Numbers in l i i l l l parentheses indicate the numbers of data points. Data sources are referred to the literature in ref 21.
Figure 10. Calculated and experimental PVT properties of PIB. s: calculated. Symbols: experimental data are referred to ref 21.
Figure 11. Calculated and experimental PVT properties of PPO. s: calculated. Symbols: experimental data are referred to ref 21.
properties of polymer solution systems, in particular. The reference term of a PHSC type EOS can be derived from the contribution to the free energy due to the formation of a chain of monomers, for example, the Chiew’s equation of state15 and its modification6 or recent work of Hu et al.,24 where a sound theoretical hard-sphere-chain term was proposed. As for the perturbation term, the segments are usually expressed as square wells, instead of the oversimplified van der Waals perturbation,7 which partly induces an unsatisfactory performance. Recently, Chang and Sandler12 have presented completely analytical solution for the SW fluids, and it provides a basis for improving the perturbation term. Hino and Prausnitz11 introduced the simplified analytical solution of Chang and Sandler, and some improvements are obtained. In this work, a new perturbation term is developed, based on the GWL coordination number model and the derived EOS14,20 for pure and mixture fluids, while using
Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 4973 Table 3. Calculated Results for Solvent Weight Fraction Activity Coefficients system
T (K)
k12
AAD γa
data source
PS/benzene1 PS/benzene2 PS/benzene3 PS/Benzene4 PS/benzene5 PS/toluene1 PS/toluene2 PS/toluene3 PS/toluene4 PS/toluene5 PS/toluene6 PS/cyclohexane PS/cyclohexane1 PS/cyclohexane2 PS/cyclohexane3 PS/cyclohexane4 PS/acetone PIB/benzene PIB/benzene1 PIB/benzene2 PIB/benzene3 PIB/octane PIB/cyclohexane PIB/cyclohexane1 PIB/cyclohexane2 PMMA/MEKb PMMA/toluene PEO/benzene PEO/benzene1 PEO/benzene2 PEO/benzene3 PEO/benzene4 PEO/benzene5 PDMS/benzene PDMS/benzene1 PDMS/benzene2 PDMS/benzene3 PDMS/benzene4 PDMS/benzene5 PDMS/benzene6
298-333 288-333 298 293 293 293 293 298-353 298-338 298-338 321 297-317 307-317 293 303-338 296 345 298-338 283-312 300 297-323 298 298-338 283-312 298-303 321.65 321.65 318-343 323-343 343.15 343.15 343.85 343.75 303.0 303.0 303.0 303.0 303.0 303 303.0
0.0321 -0.0341 0.1867 0.0230 -0.1004 0.0255 -0.0598 -0.1806 -0.0346 0.5982 -0.0156 0.0391 0.3285 0.2989 0.4539 0.2233 0.4620 0.2330 0.2955 0.2605 0.6006 0.7081 0.0341 -0.0113 0.6676 0.2921 0.0394 -0.1363 -0.2380 -0.2894 -0.2005 -0.2454 -0.4407 -0.3284 -0.3199 -0.3275 -0.2795 -0.2744 -0.2745 -0.2811
9.0(31)c 10.8(31) 7.3(5) 6.5(15) 10.8(10) 5.7(9) 8.0(12) 12.9(18) 8.2(33) 5.7(27) 7.2(9) 7.2(50) 7.1(16) 6.8(10) 8.0(33) 0.8(4) 9.7(7) 9.6(29) 5.8(19) 6.7(22) 5.0(24) 6.7(5) 10.2(30) 7.6(8) 5.3(21) 9.1(7) 10.7(8) 9.7(14) 12.3(13) 11.0(7) 2.9(6) 5.2(4) 5.5(5) 6.4(8) 7.6(8) 6.7(8) 8.5(8) 7.2(8) 8.6(8) 9.8(16)
26 26 26 27 28 27 27 29 30 31 32 33 34 27 30 31 35 36 37 38 39 40 36 41 39 32 32 42 42 43 44 44 44 45 45 45 45 45 45 45
Figure 12. Calculated spinodal curves for the systems PS(520000), PS(166000), and PS(51000) with the solvent cyclohexane. 4, O, and × are experimental spinodal points,46 respectively. 0 is the experimental critical solution point.47
a AAD γ ) 1/N ∑N|γcal - γexp/γexp| × 100. γ is the solvent weight i i i i fraction activity coefficient. b MEK: methyl ethyl ketone. c Numbers in parentheses indicate the number of data points.
Chiew’s hard-sphere-chain EOS as the reference term. The new EOS, PHSC-SWCNM EOS, is of relatively simple formulation and provides easy derivation of analytical expressions for thermodynamic functions, which makes the calculations of VLE and LLE for polymer solution systems reliable and convenient. The EOS has been extensively tested in terms of a large databank, consisting of the properties of 37 normal fluids and solvents, 67 polymers for the saturated properties; the correlation accuracy is within the errors in a common EOS approach. On the other hand, the grand average deviation (see definition in Table 2) for correlating the liquid densities for 67 polymers is 0.19%, which is of the same accuracy as that for commonly used lattice model EOSs.21 In addition, in all of the aforementioned calculations, the EOS needs only three temperature- and composition-independent parameters. The PHSC-SWCNM EOS has been used for VLE calculations of polymer-solvent systems. Compared with 40 sets of experimental data, most relative average deviations for the weight fraction activity coefficients of the solvents are less than 10% (see Table 3). Three typical LLE systems have been investigated using the EOS, including both the UCST and the LCST polymersolvent systems and a polymer blend system. The calculated binodal and spinodal curves are in good
Figure 13. Calculated binodal and spinodal curves for the system PIB(500000) + benzene. 0 represents the experimental cloud point.48 Table 4. Parameters for the PHSC-SWCNM EOS for LLE Calculations system
parameter k12
PS(520000) + cyclohexane PS(166000) + cyclohexane PS(51000) + cyclohexane PIB(500000) + benzene PS(4370) + PBD(1100)
0.3297 0.3309 0.3288 0.3455 0.3767
agreement with experimental data. All of the VLE and LLE have been calculated here by using conventional mixing rules and only a binary parameter. It should be pointed out that the present EOS does not contain a term taking into account the association interactions between molecules. However, similar to the SAFT EOS, the term can be introduced into the EOS, which will be addressed in our future work. In conclusion, many attempts indicate the PHSC type EOSs can be improved by using different reference and
4974
Ind. Eng. Chem. Res., Vol. 38, No. 12, 1999 (12) Chang, J.; Sandler, S. I. Mol. Phys. 1994, 81, 735. (13) Barker, J. A.; Henderson, D. J. Chem. Phys. 1967, 47, 2856. (14) Guo, M.; Wang, W.; Lu, H. Fluid Phase Equilib. 1990a, 60, 37. (15) Chiew, Y. C. Mol. Phy. 1990, 70 (No. 1), 129. (16) Lee, K. H.; Sandler, S. I.; Patel, N. C. Fluid Phase Equilib. 1986, 25,31. (17) Lee, R. J.; Chao, K. C. Mol. Phys. 1987, 61, 1431. (18) Yu, M. L.; Chen, Y. P. Fluid Phase Equilib. 1995, 111, 37. (19) Bokis, C. P.; Donohue, M. D. Ind. Eng. Chem. Res. 1995, 34, 4553. (20) Guo, M.; Wang, W.; Lu, H. Fluid Phase Equilib. 1990b, 60, 221-237. (21) Wang, W.; Liu, X.; Zhong, C.; Twu, C. H.; Coon, J. E. Ind. Eng. Chem. Res. 1997, 36, 2390. (22) Wang, W.; Liu, X.; Zhong, C.; Twu, C. H.; Coon, J. E. Fluid Phase Equilib. 1998, 144, 23. (23) Feng, W.; Wang, W. Ind. Eng. Chem. Res. 1999, 38, 1140. (24) Hu, Y.: Liu, H.; Prausnitz, J. M. J. Chem. Phys. 1996, 104, 396.
Figure 14. Calculated binodal and spinodal curves for polymer blend PS(4370) + PBD(1100). 0 represents the experimental cloud point.49
perturbation terms. This work explores a way for reforming the perturbation term to make the EOS practical and easy to use. Moreover, as is aforementioned, the parameters of the PHSC-SWCNM EOS are of regularity chaining with the structure of molecules. It implies, on our experience,21 that the group-contribution method can be incorporated into the EOS to develop a predictive method for polymer solution systems. Acknowledgment This work was supported by the Petrochemical Corp. of China and The National Science Foundation of China. Supporting Information Available: Derivation of the chemical potential and weight fraction activity coefficient of component k and derivation of the second derivatives of the molar segmental Gibbs with respect to the segment fraction of component k (7 pages). This material is available free of charge via the Internet at http://pubs.acs.org.
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Received for review March 30, 1999 Revised manuscript received September 8, 1999 Accepted September 16, 1999 IE990234V
