1140
Ind. Eng. Chem. Res. 1999, 38, 1140-1148
Simplified Hole Theory Equation of State for Liquid-Liquid Equilibria of Polymer Solutions and Blends Wei Feng and Wenchuan Wang* College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, China
In our previous work (Wang et al., 1997), the simplified hole theory equation of state was successfully used for the estimation of the pressure-volume-temperature relationship and vapor-liquid equilibria of a large number of polymers, solvents, and their solutions. Here, we extend the equation of state to liquid-liquid equilibria (LLE) calculations for a variety of polymer-solvent and polymer-polymer systems. Analytical expressions for the calculation of the LLE are derived, which make the determination of critical solution point, spinodal, and binodal curves convenient and reliable. Several types of LLE calculations are carried out, including predictions and correlations of spinodals and estimations of binodal and spinodal curves. The results indicate that the equation of state is capable of describing fairly the LLE for polymersolvent and polymer-polymer systems including the upper and/or lower critical solution temperature phase behavior. 1. Introduction Liquid-liquid equilibria (LLE) of polymer solutions, including polymer-solvent and polymer-polymer systems, are of theoretical and practical importance in polymerization and polymer processing. Since equations of state (EOS’s) are applicable over a wide range of temperatures, pressures, and molecular sizes, some researchers have recently reported their results for LLE calculations using an EOS method (Lee and Danner, 1996; Wang et al., 1996; Folie and Radosz, 1995). In our previous work, we developed the simplified hole theory (SHT) EOS (Zhong et al., 1993; 1994; Wang et al., 1997) and successfully applied it to the description of the pressure-volume-temperature (PVT) behavior for 67 polymers and 61 solvents and to the description of vapor-liquid equilibria (VLE) for 95 polymer-solvent systems. Moreover, we incorporated the group contribution (GC) method into the SHT EOS. The group parameters for polymers and solvents were obtained for the prediction of PVT relationship for 59 polymers and 59 solvents. A binary group interaction parameters matrix containing 110 parameters was reported for the prediction of weight fraction activity coefficients for polymersolvent systems (Wang et al., 1998). This work is a further development of the SHT EOS. The aim of this project is to extend the SHT EOS to LLE calculations for both polymer-solvent and polymerpolymer systems. First, we derive analytical expressions for the second and third derivatives of the Gibbs free energy with respect to the segmental fraction of a component in the LLE calculations. Then, different types of LLE, including the upper critical solution temperature (UCST) and lower critical solution temperature (LCST) behavior, are estimated for a variety of systems in terms of the experimental critical solution points, spinodal points, and cloud points. The results of extensive calculations, represented by curves and tables, are compared with experimental data to dem* Corresponding author. Fax: +86-10-64436781. E-mail:
[email protected].
onstrate the theory’s capability for estimating LLE of polymer-solvent and polymer-polymer systems. 2. Simplified Hole Theory Equation of State (SHT EOS) SHT EOS for Pure Polymers and Solvents. The simplified hole theory equation of state (SHT EOS) is explained in detail in an article of Wang et al. (1997). For brevity, we here simply present the expression of the SHT EOS as
[
]
(yV ˜ )1/3 P ˜V ˜ 1.1394 2y + ) - 1.5317 1/3 2 T ˜ (yV ˜ ) - 0.9165y T ˜ (yV ˜ ) (yV ˜ )2 (1) where y is the occupied site fraction. By introduction of the Shotttley vacancy (Shottky, 1935) representing the hole in liquids, a simple equation for y was derived (Zhong et al., 1993):
y ) 1 - e-0.52/T˜
(2)
In eqs 1 and 2, the reduced temperature, T ˜ , pressure, P ˜ , and volume, V ˜ , are given by
T ˜ ) T/T*
T* ) qz*/cR
P ˜ ) P/P*
P* ) qz*/sν*
V ˜ ) V/V*
V* ) ν*/M0
(3)
where T, P, and V are temperature, pressure, and specific volume, s is the number of segments per molecule, R is the gas constant. T*, V*, and P* are three characteristic parameters, * and ν* represent the characteristic energy and volume per segment, respectively, and 3c is the number of external degree of freedom per chain. The characteristic flexibility ratio, c/s, for all the molecules is set to a universal constant 1.04 in the SHT EOS. qz is the number of nearest neighbor sites per chain, equal to s(z - 2) + 2, z is the
10.1021/ie9805178 CCC: $18.00 © 1999 American Chemical Society Published on Web 02/10/1999
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1141
coordination number that is set to 8, and M0 is the segmental molecular weight. Equations 1-3 form the so-called SHT EOS. The three substance characteristic parameters T*, V*, and P* (or the molecular parameters *, ν*, and c) can be obtained by fitting PVT data for pure substances, and those for 67 liquid polymers and 61 solvents can be found in our previous article (Wang et al., 1997). SHT EOS for Mixtures. The SHT EOS for a polymer-solvent or polymer-polymer mixture can be expressed (Zhong et al., 1994; Wang et al., 1997) as
2yQ2 P ˜V ˜ ) (1 - W)-1 + (1.1394Q2 - 1.5317) (4) T ˜ T ˜ where the occupied site fraction is given by
y ) 1 - e-〈cs〉/2T˜
(5)
Here 3〈cs〉 represents the compositional average of the number of external freedom per chain. 3〈cs〉 is given by eq 8 for a binary system, and Q ) (yV ˜ )-1, W ) 1/3 0.9165yQ . The reduced properties for a mixture are defined as
P ˜ )
P P*
V ˜ )
P* )
V V*
〈qz〉〈*〉 〈ν*〉
V* )
〈ν*〉 〈M0〉
(6)
where the broken brackets signify compositional average. For a binary mixture, the compositional average variables are defined as
(7)
〈cs〉 ) φ1/cs1 + φ2/cs2
(8)
〈M0〉 ) φ1M01 + φ2M02
(9)
(
(13)
)
ν/111/3 + ν/221/3 2
3
(14)
The two binary interaction parameters can be determined by fitting the properties of the mixture of interest. In summary, given known the substance characteristic parameters, P*, T*, V*, namely the molecular parameters *, ν*, and c (see eq 3), for component 1 and 2 in a binary mixture, one can obtain the parameters in the SHT EOS for a polymer-solvent or polymerpolymer mixture through the combining rules, eqs 7-14, and further derive other thermodynamic properties for the mixture from the EOS, eq 4. LLE Calculations. LLE calculations lie in the determination of the phase separation domain, bounded by the binodal. The binodal compositions can be determined by using an EOS, if pressure and temperature are specified. The condition for the binodal compositions is the chemical potentials in two liquid phases must be equal for all the components (Lee and Danner, 1996):
(15)
Here superscripts prime and double prime represent the two phases in LLE, µ0i is the chemical potential of pure component i at system T and P and µi is the chemical potential of component i at system T, P, and compositions of the mixture. In this work, polymer-solvent solutions and polymer blends are recognized as pseudobinary solutions. Wang et al. (1997) have derived the chemical potential for pure component 1, µ01, at system T and P, in terms of the SHT EOS, expressed as
[
()
s1 - 1 1 1 z-1 ln y ln ln + + s1 s1 s1 e s1
]
(
)
{[
]
ν/11(1 - W)3 1-y ln(1 - y) - RTs1cs1 ln + y Q
[
]
}
2πM01RT yQ2 3 ln (1.1394Q2 - 3.0634) + 2 2 2T ˜ (Nah)
φ22ν/22
(10)
〈*〉 ) θ12/11 + 2θ1θ2/12 + φ22/22
(11)
˜ ν/11 (16) s1PV
〈qz〉 ) φ1q1z + φ2q2z
(12)
where y, T ˜, V ˜ , Q, and W can be calculated by their aforementioned definitions for pure component 1 at system P and T. Na, h, and R are the Avogadro, Planck, and gas constants, respectively. From thermodynamic relations, the chemical potential for component 1 in a binary mixture can be derived as
〈ν*〉 )
+
2φ1φ2ν/12
ν/12 ) k12
µ01 ) RTs1
〈s〉-1 ) φ1/s1 + φ2/s2
φ12ν/11
/12 ) a12(/11/22)1/2
(µi - µ0i )′ ) (µi - µ0i )′′
〈qz〉〈*〉 T* ) 〈cs〉R
T T ˜ ) T*
interaction parameters, a12 and k12, are introduced:
+
in eqs 7-12, qiz is the number of nearest neighbor sites per chain, qiz ) z - 2 + 2/si, si is the number of segments per molecule, θi is the contact fraction, θi ) φiqiz/〈qz〉, for component i, respectively. In the above expressions, subscript i represents component i and i ) 1 and 2 for a binary system. As a result, the segmental fractions φ1 ) N1s1/(N1s1 + N2s2) and φ2 ) 1 - φ1 and N1 and N2 are the number of molecules for components 1 and 2, respectively. 3cs1 and 3cs2 are the number of external degrees of freedom per segment, M01 and M02 are the / / and ν22 are the charsegmental molecular weights, ν11 / / acteristic volumes, and 11 and 22 are the characteristic energies for components 1 and 2, respectively. For the characteristic interaction energy and volume / / and ν12 , two binary between molecules 1 and 2, 12
( )
µ1 ) Gφs1 - s1φ2
∂Gφ ∂φ2
(17)
T,p
where Gφ is the molar segmental Gibbs free energy of the mixture. s1 is the number of segments per molecule 1, and φ2 the segmental fraction of component 2 in the mixture. Since
Gφ ) Aφ + PVφ
(18)
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Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999
Table 1. Parameters in the SHT EOS for Pure Substances, Taken from Wang et al. (1997) params in the SHT EOS substances
T* (K)
P* V* (MPa) (cm3/g)
polystyrene (PS) polyisobutylene (PIB) high-density polyethylene (HDPE) poly(vinyl methyl ether) (PVME) cis-1,4-polybutadine (PBD) poly(cyclohexyl methacrylate) (PCHMA) poly(R-methylstyrene) (PMS) cyclohexane benzene hexane
3587.5 3651.0 2717.9 3095.9 3009.2 3483.1 3871.6 1873.9 1859.9 1748.9
500.3 393.96 575.61 525.21 457.94 502.88 458.81 565.09 712.22 447.39
0.9091 1.0556 1.0602 0.9159 1.0420 0.8528 0.9307 1.0663 0.9332 1.2310
Figure 2. Predicted spinodals for the system PE + hexane at several pressures. 0 represents experimental critical solution points (Nies et al., 1990).
Figure 1. Predicted spinodals for the system PS + cyclohexane. Curves from top to bottom represent the molecular weights of PS: 1 500 000, 527 000, 394 000, 286 000, 166 000, 93 000, 61 500, 51 000, and 35 400, respectively. 0 represents experimental critical solution points (Koningsveld et al., 1970).
where Aφ is the molar segmental Helmhotz free energy of the mixture and Vφ is the segmental volume of the mixture, Vφ ) V/(N1s1 + N2s2), we rearrange the eq 17 as
µ1 ) RTs1
( )]
[
Gφ 1 ∂Gφ - φ2 RT RT ∂φ2
(19)
T,p
By substituting eq 18 into eq 19, we can get the expression of µ1 in terms of Aφ and its derivative with respect to φ2, while Aφ is derived as (Wang et al., 1997) as
φ1 φ1 φ2 φ2 Aφ A ln + ln ) ) RT (N1s1 + N2s2)kT s1 s1 s2 s2
( )
〈s〉 - 1 z-1 ln y 1 - y ln(1 - y) + ln + e y 〈s〉 〈s〉
{[
〈cs〉 ln
]
[
]
2π〈M0〉RT 〈ν*〉(1 - W)3 3 + ln Q 2 (Nah)2 2
Figure 3. Calculated spinodal curves for the systems PS(520000), PS(166000), and PS(51000) with the solvent cyclohexane. 4, O, / are experimental spinodal points (Irvine and Gordon, 1980). 0 is the experimental critical solution point (Koningsveld et al., 1970).
yQ (1.1394Q2 - 3.0634) 2T ˜
}
a result, by combination of eqs 18-20, µ1 can be obtained. Since the derivation of µ1 is very complicated, the detailed expression of µ1, and likewise that of µ2 can be obtained from our previous paper (Wang et al., 1997). In addition to the binodal, the spinodal and critical solution points (CSP) play important roles in LLE calculations. In the unstable region, bounded by the spinodal curve, the system splits into two liquid phases spontaneously. From the thermodynamic stability criteria, the spinodal points can be derived by solving the equation
[ ] ∂2Gφ
(20)
where A is the Helmholtz free energy of the system. As
∂φi2
)0
(21)
T,P
and the critical solution point (CSP) is determined by
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1143 Table 2. Binary Interaction Parameters in SHT EOS for LLE Calculations binary interaction params
binary interaction params
system
a12
k12
system
a12
k12
PS(1500000) + cyclohexane PS(527000) + cyclohexane PS(394000) + cyclohexane PS(286000) + cyclohexane PS(166000) + cyclohexane PS(93000) + cyclohexane PS(615000) + cyclohexane PS(51000) + cyclohexane PS(354000) + cyclohexane PE + hexane (0.1 MPa) PE + hexane (5 MPa) PE + hexane (10 MPa) PE + hexane (20 MPa)
0.9096 0.9006 0.9008 0.9012 0.9105 0.9253 0.9266 0.9278 0.9292 0.9386 0.9166 0.8888 0.7965
1.1489a 1.2887a 1.2885a 1.2950a 1.2820a 1.2607a 1.2655a 1.2671a 1.2733a 0.4121a 0.4440a 0.4847a 0.6195a
PS(51000) + cyclohexane PS(166000) + cyclohexane PS(520000) + cyclohexane PS(200000) + cyclohexane PS(37000) + cyclohexane PIB(50000) + benzene HDPE(82600) + hexane PS(20400) + hexane(LCST) PS(20400) + hexane(UCST) PS(4370) + PBD(1100) PS(58400) + PMS(62100) PS(255300) + PCHMA(101732) PS(100000) + PVME(99000)
0.8415 0.8463 0.9806 0.8841 0.8702 0.7909 0.8911 0.9841 0.9829 0.9619 1.0018 1.0027 1.0159
1.6890b 1.6491b 0.3347b 0.5883c 0.5986c 0.9615c 0.3878c 0.7446c 0.7916c 0.7358c 0.9872c 0.9833c 0.9151c
a Parameters determined by the critical solution point data. b Parameters determined by fitting spinodal data. c Parameters determined by cloud point data.
Figure 4. Calculated binodal and spinodal curves for the system HDPE(82600) + hexane. 0 represents experimental cloud points (Kodama et al., 1978).
Figure 5. Calculated binodal and spinodal curves for the system PS(37000) and PS(200000) + cyclohexane. 0 and / represent experimental cloud points for PS(37000) + cyclohexane and PS(200000) + cyclohexane, respectively (Saeki et al., 1973).
the equation
[ ] ∂3Gφ ∂φi3
)0
(22)
T,P
where i represents component 1 or 2 in a binary system. As a result, eqs 15, 21, and 22 are used in this work for the description of the LLE phase behavior, represented by the binodal and spinodal curves, and the critical solution points for polymer-solvent and polymerpolymer systems. It is noticed that analytical expressions for the first, second, and third derivatives of Gφ with respect to the segmental fraction φi are essential to the implementation of the LLE calculations. Through reasonable simplification and delicate derivations, the analytical expression for the first derivative was obtained by Wang et al. (1997). The second and third derivatives have been derived in this work and are shown in the Appendix. 3. Results and Discussion Several types of LLE calculations are carried out in this work. All the SHT EOS parameters for pure polymers and solvents needed in the LLE calculations
Figure 6. Calculated binodal and spinodal curves forthe system PIB(500000) + benzene. 0 represents experimental cloud points (Lee et al., 1996).
are taken directly from our previous work (Wang et al., 1997) and listed in Table 1. (1) Predictions of Spinodal Curves from the Critical Solution Points. For polystyrene (PS) +
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Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999
Figure 7. Calculated binodal and spinodal curves for the system PS(20400) + hexane. 0 represents experimental cloud points (Zeman et al., 1972).
Figure 9. Calculated binodal and spinodal curves for polymer blend PS(58400) + PMS(62100). 0 represents experimental cloud point (Lin and Roe, 1988).
Figure 8. Calculated binodal and spinodal curves for polymer blend PS(4370) + PBD(1100). 0 represents experimental cloud points (Rostami and Walsh, 1985).
cyclohexane and polyethylene (PE) + hexane systems, the experimental critical solution points were reported by Koningsveld et al. (1970) and Nies et al. (1990), respectively. Because the critical solution point composition of a binary system at the critical temperature and pressure satisfies both the eqs 21 and 22, two unknown binary interaction parameters, a12 and k12, in eqs 13 and 14 were obtained by solving eqs 21 and 22 simultaneously. Then, a temperatures was chosen, which is lower than the CSP’s temperature for the UCST type system or higher than that for the LCST type system. Introducing a given temperature and the two parameters a12 and k12 obtained, we solved eq 21 to get the two spinodal compositions in LLE at the system temperature and pressure. It is noted that the initial guesses of the spinodal compositions can be determined using the method proposed by Lee and Danner (1996). Then by repeating the calculations of the spinodal compositions at various temperatures, we obtained the phase diagram. Figure 1 shows the predicted UCST spinodals for the PS + cyclohexane systems with different molecular
Figure 10. Calculated binodal and spinodal curves for polymer blend PS(255300) + PCHMA(101732). 0 represents experimental cloud points (Rudolf and Cantow, 1995).
weights. The experimental CSP’s for the system PE + hexane were measured at several constant pressures, from 1 to 20 MPa. Since the SHT EOS can reflect the effect of pressure on phase behavior, four LCST type spinodals were predicted, shown in Figure 2. All the binary interaction parameters obtained for the systems above are listed in Table 2. (2) Correlations of Spinodal Points. Experimental spinodal data are very useful for testing the EOS’s capability to describe LLE, because of rigorous thermodynamic definition of the spinodal point, eq 21. Unfortunately, only a few data are available due to difficulties with the measurements. Among them, the spinodal data for the system PS + cyclohexane were thoroughly studied by Stroeks et al. (1990) and are commonly used to test EOS’s. Here, we correlate all the experimental spinodal points for three systems: PS(520000), PS(166000), and PS(51000) with the solvent cyclohexane. The binary interaction parameters a12’s
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1145
4. Conclusions
Figure 11. Calculated binodal and spinodal curves for polymer blend PS(100000) + PVME(99000). 0 represents experimental cloud points (Bae et al., 1993).
and k12’s were obtained by fitting the spinodal points, listed in Table 2. A good agreement between the calculated and experiment results is showed in Figure 3. (3) 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 in the following procedure. First, the critical solution points were determined from the cloud point curves graphically. For example, we can estimate the lowest temperatures for the LCST type systems (or the highest temperatures for the UCST type systems) and their compositions on the cloud point curves and take them as the CSP’s. Then, the binary interaction parameters were obtained as is discussed above. Finally, the binodal and spinodal curves were estimated by solving eqs 15 and 21, respectively. It is worth mentioning that, in order to avoid trivial solutions in the calculation of binodal curves, the method proposed by Lee and Danner (1996) was adopted to solve the binodal compositions. Figures 4-6 show the calculated LCST type binodal and spinodal curves for the polymer-solvent systems HDPE + hexane, PS + cyclohexane, and PIB + benzene. Figure 7 is a good example to show that the SHT EOS is capable of describing the phase behavior of systems exhibiting both UCST and LCST behavior in different temperature ranges. All the binary interaction parameters used for the calculation of phase behavior for these systems are listed in Table 2, too. The SHT EOS was applied to the calculation of the polymer-polymer miscibility behavior for polymer blends. Since all the experimental data are cloud points, the same calculation procedure as that for the polymersolvent systems was used in order to determine the binodal and spinodal curves of polymer blends. Figures 8 and 9 show the UCST type binodal and spinodal curves for the polymer blends PS(4370) + PBD(1100) and PS(58400) + PMS(62100). In addition, Figures 10 and 11 present the LCST type binodal and spinodal curves for the system PS(255300) + PCHMA(101732). All the binary interaction parameters in the SHT EOS for the LLE calculations of the polymer blends are also listed in Table 2.
The simplified hole theory equation of state (SHT EOS) has been extended to LLE calculations for a variety of polymer-solvent and polymer-polymer systems. Analytical expressions for the LLE criteria have been derived. These are essential to the reliable and convenient determination of the critical solution point (CSP), spinodal, and binodal curves. Several types of LLE calculations covering 26 different systems have been carried out in this work: (1) predictions of spinodal curves from the CSP’s (see Figures 1 and 2); (2) correlations of spinodal points (see Figure 3); and (3) estimations of binodal and spinodal curves from experimental cloud points (see Figures 4-11). The results indicate that the SHT EOS is capable of fairly describing the LLE for polymer-solvent and polymer-polymer systems with either the UCST and/or LCST phase behavior. In addition, the SHT EOS can represent the shift from the UCST to the LCST behavior due to the temperature change (see Figure 6). The pressure effect on the LLE phase behavior is also reasonably depicted by using the EOS (see Figure 2). In conclusion, applications of the SHT EOS to the LLE calculations reported here, together with our previous work, indicate that the SHT EOS can be a practical tool for the description of the PVT, VLE, and LLE behavior for polymer-solvent and polymer-polymer solutions. Acknowledgment This work was supported by the Petrochemical Corp. of China and the Natural Science Foundation of China. Appendix: Derivation of the Second and Third Derivatives of the Molar Segmental Gibbs Free Energy with Respect to the Segment Fraction From eq 18, we get
Gφ ) Aφ + PVφ
(A-1)
where, for a binary system, Gφ ) G/(N1s1 + N2s2), Vφ ) V/(N1s1 + N2s2) and Aφ is given by eq 20. The first derivative [∂Gφ/∂φ2]T,P can be found in the Appendix of Wang et al. (1997). Then, we can derive [∂2Gφ/∂φ22]T,P as
[ ] ∂2 G φ
)
2
∂φ2
T,P
3(M02 - M01)(cs2 - cs1) 1 1 + + s2φ2 s1φ1 〈M0〉
3〈cs〉(M02 - M01)2
+
{(
)
1 2 1 + y s2 s1
2〈M0〉 2(cs2 - cs1)(3W - 1) 2
y(1 - W) 2(cs2 - cs1)Q2(1.7091Q2 - 1.5317) ∂y T ˜ ∂φ2
{
}[ ] }
-
T,P
yQ2(4.5576Q - 6.1284) 2 × + T ˜V ˜ (1 - W)V ˜
(cs2 - cs1) 2〈cs〉W
[] { ∂V ˜ ∂φ2
+
T,P
1 - 〈s〉 - 〈cs〉〈s〉 y〈s〉
-
ln(1 - y) y2
}[ ]
〈cs〉Q2(1.7091Q2 - 1.5317) ∂2y T ˜ y(1 - W) ∂φ22
T,P
+ +
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Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999
{
〈cs〉〈s〉 + 〈s〉 - 1
y2〈s〉 2〈cs〉W(3W - 1) 3y2(1 - W)2
[ ] {
2(1 - y) ln(1 - y) + y
+
T,P
y3(1 - y) 〈cs〉Q2(6.8364Q2 - 3.0634) + × yT ˜
}
4〈cs〉W 2
}[ ] [ ] }[ ] [ ] }[ ] [ ] ∂V ˜ ∂φ2
∂y ∂φ2
3yV ˜ (1 - W) T,P 2 〈cs〉Q (3.4182Q2 - 3.0634) ∂y ∂φ2 T ˜2
{ {
〈cs〉yQ2(4.5576Q2 - 6.1268) 2
}[ ] {
∂2V ˜ P 〈ν*〉 RT ∂φ22
+
{
T,P
∂V ˜ ∂φ2
+
T,P
∂T ˜ ∂φ2
〈cs〉(3 - 2W)
3(1 - W)2V ˜2
〈cs〉yQ2(11.3940Q - 9.1902) T ˜ V ˜
2
{ {
2
〈cs〉yQ2(0.5697Q2 - 1.5317) T ˜2
〈cs〉yQ (1.1394Q - 3.0634)
{
2
2
T ˜3
2(cs2 - cs1) ∂〈ν*〉 ∂φ2 〈ν*〉
{
〈cs〉
〈ν*〉
2
∂〈ν*〉 ∂φ2
〈cs〉
-
T,P 2
+
T,P
〈ν*〉
-
2P ∂〈ν*〉 RT ∂φ2
T,P
{
[ ] ∂φ23
+
T,P 2
3〈cs〉(M02 - M01) 〈M0〉3
{
+
T,P
(A-2)
)
]
T,P
∂y ∂φ2
{
-
T,P
}
2〈cs〉W(5W - 1) 3y2V ˜ (1 - W)3
}[ ]
}[ ] [ ]
9V ˜ 3(1 - W)3
∂y ∂φ2
T,P
2
+
T,P
+
2(1 - 〈s〉 - 〈s〉〈cs〉) 1 + 2 y (1 - y) y3〈s〉
+
T,P
+
2
T,P
+
-
}[ ]
〈cs〉Q2(34.1802Q2 - 9.1902)
2
2(cs2 - cs1)(4W - 1) 1 1 1 3 - (cs2 - cs1) + y s2 s1 y(1 - W)
2〈cs〉W
+
y2T ˜
}[ ]
(cs2 - cs1)Q2(5.1273Q2 - 4.5951) ∂2y T ˜ ∂φ22
2
4〈cs〉W(9W2 - 7W + 2)
)
×
+
〈cs〉Q2(82.0368Q2 - 18.3804) ∂V ˜ yT ˜V ˜ ∂φ2
1 3 1 1 + 2 (cs2 - cs1) - 2 + s s y y 2 1
+
T,P
T,P
ln(1 - y)
[ ][ ] {
T,P
2(cs2 - cs1)(4W2 - 3W + 1)
{[(
-
2
3 + 2 + y2〈s〉 y (1 - y) 6ln(1 - y) 2〈cs〉W(3W - 1) + + y3 y2(1 - W)2 〈cs〉Q2(20.5092Q2 - 9.1902) × yT ˜
∂2 y ∂φ22
+ y2(1 - W)2 (cs2 - cs1)Q2(20.5092Q2 - 9.1902) ∂y yT ˜ ∂φ2
T,P
∂T ˜ ∂φ2
3(〈cs〉〈s〉 + 〈s〉 - 1)
T,P 2
(
∂V ˜ ∂φ2
y(1 - W) y 2 2 〈cs〉Q (1.7091Q - 1.5317) ∂3y + T ˜ ∂φ23 T,P y〈s〉
{
-
9(M02 - M01)2(cs2 - cs1) 1 1 ) + s1φ12 s2φ22 2〈M0〉2 T,P 3
[ ][ ]
1 - 〈s〉 - 〈cs〉〈s〉
-
and the third derivative is given by
∂3 G φ
+
T,P
}[ ] }
T ˜ 2V ˜
∂V ˜ ∂φ2
T,P
∂V ˜ ∂φ2
}[ ]
T ˜V ˜2 (cs2 - cs1)yQ2(13.6728Q2 - 18.3804)
{
×
T,P
+
+ (1 - W)2V ˜2 (cs2 - cs1)yQ2(34.1820Q2 - 27.5706)
T,P
PV ˜ ∂ 〈ν*〉 RT ∂φ 2 2
(1 - W)V ˜
(cs2 - cs1)(3 - 2W)
-
∂2T ˜ ∂φ22
T,P
{
+
∂T ˜ ∂φ2
∂T ˜ ∂φ2
-
3(cs2 - cs1)
-
}[ ]
(cs2 - cs1)yQ2(6.8364Q2 - 9.1902) ∂2V ˜ T ˜V ˜ ∂φ22
2
∂V ˜ ∂φ2
(cs2 - cs1)(1.1394Q2 - 3.0634)yQ2 T ˜
[ ] {
T,P
∂T ˜ ∂φ2
}[ ] [ ]
∂T ˜ ∂φ2
}[ ] }[ ] }[ ] }[ ] }[ ] { }[ ] }[ ] { }[ ] [ ] T,P 2
{
{
Q2(41.0184Q2 - 18.3852) T ˜V ˜
∂V ˜ ∂y˜ 4W × (1 - W)2 yV ˜ ∂φ2 T,P ∂φ2 T,P (cs2 - cs1)Q2(10.2546Q2 - 9.1902) ∂y˜ ∂φ2 T ˜2
+
T ˜ V ˜ T,P T,P 〈cs〉 〈cs〉yQ2(2.2788Q2 - 3.0634) + T ˜V ˜ (1 - W)V ˜
{
(cs2 - cs1)
+
〈cs〉Q2(13.6728Q2 - 6.1268) T ˜V ˜
2
∂y ∂φ22
+
}[ ] }[ ] [ ]
61n(1 - y) ∂y 4 3 ∂φ2 y (1 - y) y4
{
〈cs〉Q2(20.5092Q2 - 6.1268) yT ˜
2
∂y ∂φ2
3
-
T,P
2
T,P
∂T ˜ ∂φ2
T,P
+
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999 1147
{
}
} [ ][ ] { [ ][ ] { }[ ] [ ] { }[ ] [ ] ∂2 y ∂φ22
∂2 y ∂φ22
T,P
∂V ˜ ∂φ2
T,P
∂T ˜ ∂φ2
〈cs〉Q2(5.1273Q2 - 4.5951)
+
〈cs〉Q2(20.5092Q2 - 9.1902) T ˜V ˜
+
T,P
∂2 V ˜ ∂φ22
yV ˜ (1 - W)2
T,P
∂y ∂φ2
+
{
3yV ˜ 2(1 - W)3
T,P
〈cs〉Q2(102.5460Q2 - 27.5706) T ˜V ˜
4〈cs〉W(2 - W) 2
∂V ˜ ∂φ2
2
T,P
}
〈cs〉Q (41.0184Q - 18.3804) 2
∂y ∂φ2
2
T ˜ 2V ˜
∂y ∂φ2
∂V ˜ ∂φ2
T,P
T,P
∂T ˜ ∂φ2
〈cs〉Q2(5.1273Q2 - 4.5951) T ˜2
2〈cs〉Q2(5.1273Q2 - 4.5951) T ˜3
〈cs〉yQ2(6.8364Q2 - 9.1902) 2
T ˜ V ˜
〈cs〉yQ2(6.8364Q2 - 9.1902) 2
T ˜ V ˜
〈cs〉yQ2(13.6728Q2 - 18.3804) T ˜ 3V ˜
T,P
∂y ∂φ2
∂V ˜ ∂φ2
∂2T ˜ ∂φ22
{
T,P
∂T ˜ ∂φ2
(1 - W)2V ˜2
+
T,P
∂2 T ˜ ∂φ22
T,P
∂T ˜ ∂φ2
T,P
T,P
∂V ˜ ∂φ2
T,P
T,P
∂V ˜ ∂φ2
T,P
-
T,P
T ˜V ˜
2
3
T ˜
2
×
+
}
T,P
9(1 - W)3V ˜3
}[
T ˜
T ˜
3
∂2 T ˜ ∂φ22 ∂T ˜ ∂φ2
∂3T ˜ ∂φ23
×
+
-
T,P
}[ ] }[ ] }[ ]
〈cs〉yQ2(0.5697Q2 - 1.5317) 2
]
∂V ˜ ∂φ2
2(cs2 - cs1)yQ2(1.7091Q2 - 4.5951) 3
-
T,P
T ˜V ˜2 2 2〈cs〉(5W - 13W + 9)
(cs2 - cs1)yQ2(1.7091Q2 - 4.5951)
{
+
〈cs〉yQ2(34.1820Q2 - 27.5706)
〈cs〉yQ2(68.3640Q2 - 36.7608)
{ {
+
T,P
∂T ˜ ∂φ2
T,P
T ˜4
3(cs2 - cs1) ∂2〈ν*〉 〈ν*〉 ∂φ22
3〈cs〉
〈ν*〉2
∂〈ν*〉 ∂φ2
T,P
T,P
2 3P ∂ 〈ν*〉 RT ∂φ 2 2
∂φ22
T,P
3
-
T,P
3(cs2 - cs1) ∂〈ν*〉 ∂φ2 〈ν*〉2
+
∂2〈ν*〉
∂T ˜ ∂φ2
-
T,P
2
+
T,P
〈cs〉 ∂3〈ν*〉 PV ˜ + RT 〈ν*〉 ∂φ 3 2 T,P 2〈cs〉 ∂〈ν*〉 3 + 〈ν*〉3 ∂φ2
+
T,P
∂V ˜ ∂φ2
T,P
T,P
T,P
∂2 V ˜ ∂φ22
(A-3)
T,P
Literature Cited
〈cs〉 P 〈ν*〉 RT (1 - W)V ˜
[ ][ ] { ∂2 V ˜ ∂φ22
2
T,P
〈cs〉yQ2(2.2788Q2 - 3.0634) ∂3V ˜ T ˜V ˜ ∂φ23
〈cs〉(3 - 2W)
-
T,P
∂T ˜ ∂φ2
∂2 V ˜ ∂φ22
3〈cs〉yQ2(1.1394Q2 - 3.0634)
T,P
∂2 T ˜ ∂φ22
Likewise, the derivatives for component 1, [∂2Gφ/ ∂φ12]T,P and [∂3Gφ/∂φ13]T,P can be obtained.
T,P
∂y ∂φ2
T ˜3
∂T ˜ ∂φ2
2P ∂〈ν*〉 RT ∂φ2
+
T ˜ 2V ˜2
2
-
×
〈cs〉yQ2(34.1820Q2 - 27.5706)
∂V ˜ ∂φ2
-
T,P
[ ][ ][ ] }[ ] [ ] }[ ] [ ] }[ ] [ ] }[ ] [ ] }[ ] [ ] { } [ ] [ ] { }[ ]
}[ ] [ ] { }[ ] { }[ ] { }[ ] { }[ ] [ ] { }[ ] { }[ ] [ ] { }[ ] { }[ ] [ ] 〈cs〉yQ2(3.4182Q2 - 9.1902)
×
T ˜2
T,P
2〈cs〉W
{ { { { {
{
〈cs〉Q2(20.5092Q2 - 9.1902) 2〈cs〉W × T ˜V ˜ yV ˜ (1 - W)2
+
T,P 2
T,P
+
-
Bae, Y. C.; Shim, J. J.; Soane, D. S.; Prausnitz, J. M. Representation of Vapor-Liquid and Liquid-Liquid Equilibria for Binary System Containing Polymers: Applicability of an Extended Flory-Huggins Equation. J. Appl. Polym. Sci. 1993, 47, 1193. Folie, B.; Radosz, M. Phase Equilibria in High-Pressure Polyethylene Technology. Ind. Eng. Chem. Res. 1995, 34, 1501-1516. Hamada, F.; Fujisawa, K.; and Nakajima, A. Lower Critical Solution Temperature in Linear Polyethylene/n-alkaline Systems. Polym. J. 1973, 4, (3), 316. Irvine, P.; Gordon, M. Graph-Like State of Matter. 14. Statistical Thermodynamics of Semidilute Polymer Solution. Macromolecules 1980, 13, 761. Kodama, Y.; Swinton, F. L. Lower Critical Solution Temperatures. Part I. Polyethylene in n-Alkanes. Brit. Polym. J. 1978, 10, 191. Koningsveld, R.; Kleeintijens, L. A.; Shultz, A. R. J. Polym. Sci., Part A2 1970, 8, 1261. Lee, B. C.; Danner, R. P. Group-Contribution Lattice-Fluid EOS: Prediction of LLE in polymer Solutions. AIChE J. 1996, 42, (11), 3223. Lin, J. I.-L.; Roe, R. J. D. Sc. Study of Miscibility of Polystyrene and Poly(methylstyrene). Polymer 1988, 29, 1227. Nies, E.; Stroeks, A.; Simha, R.; Jain, R. K. LCST Phase Behavior according to the Simha-Somcynsky Theory: Application to the n-Hexane Polyethylene System. Colloid Polym. Sci. 1990, 268, 731-743. Rostami, S.; Walsh, D. J. Simulation of Upper and Lower Critical Phase Diagrams for Polymer Mixtures at Various Pressures. Macromolecules 1985, 18, 1228. Rudolf, B.; Cantow, H.-J. Description of Phase Behavior of Polymer Blends by Different Equation of State Theories. 1, Phase Diagrams and Thermodynamic Reasons for Mixing and Demixing. Macromolecules 1995, 28, 6586-6594. Saeki, S.; Kuwahara, N.; Konno, S.; Kaneko, M. Macromolecules 1973, 6, 246. Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory of Classical Fluids, Pure Fluids. J. Phys. Chem. 1976, 80, 2352. Shottky, Z. Z. Phys. Chem., Abt. B 1935, 29, 335. Simha, R.; Somcynsky, T. On the Statistical Thermodynamics of Spherical and Chain Molecule Fluids. Macromolecules 1969, 2, 342. Stroeks, A.; Nies, E. A Modified Hole Theory of Polymeric Fluids. 2. Miscibility Behavior and Pressure Dependence of System Polystyrene/Cyclohexane. Macromolecules 1990, 23, 4092-4098. Wang, W.; Tree, D. A.; High, M. S. A Comparison of Lattice-Fluid Models for the Calculation of the Liquid-Liquid Equilibria of Polymer Solutions. Fluid Phase Equilib. 1996, 114, 4762. Wang, W.; Liu, X.; Zhong, C.; Twu, C. H.; Coon, J. E. Simplified Hole Theory Equation of State for Liquid Polymers and Solvents and Their Solutions. Ind. Eng. Chem. Res. 1997, 36, 2390.
1148
Ind. Eng. Chem. Res., Vol. 38, No. 3, 1999
Wang, W.; Liu, X.; Zhong, C.; Twu, C. H.; Coon, J. E. Group Contribution Simplified Hole Theory Equation of State for Liquid Polymers and Solvents and their Solutions. 1998, 144, 23. Zeman, L.; Patterson, D. Pressure Effects in Polymer Solution Phase Equilibria. 2. Systems Showing Upper and Lower Critical Solution Temperatures. J. Phys. Chem., 1972, 76, 1214. Zhong, C.; Wang, W.; Lu, H. Simplified Hole Theory Equation of State for Polymer Liquids. Fluid Phase Equilib. 1993, 86, 137.
Zhong, C.; Wang, W.; Lu, H. Application of the Simplified Hole Theory Equation of State to Polymer Solutions and Blends. Fluid Phase Equilib. 1994, 102, 173.
Received for review August 7, 1998 Revised manuscript received November 12, 1998 Accepted November 25, 1998 IE9805178
