2390
Ind. Eng. Chem. Res. 1997, 36, 2390-2398
GENERAL RESEARCH Simplified Hole Theory Equation of State for Liquid Polymers and Solvents and Their Solutions Wenchuan Wang,* Xiangling Liu, and Chongli Zhong College of Chemical Engineering, Beijing University of Chemical Technology, Beijing 100029, People’s Republic of China
Chorng H. Twu and John E. Coon Simulation Sciences, Inc., Brea, California 92621
In the simplified hole theory equation of state proposed by our group (Zhong, C.; Wang, W.; Lu, H. Fluid Phase Equilib. 1993, 86, 137), the occupied site fraction for the Simha-Somcynsky equation of state (Simha, R.; Somcynsky, T. Macromolecules 1969, 2, 342) was derived as a simple exponential function of temperature alone, in terms of the theory for an imperfect crystal with thermal defects. In this work, the simplified hole theory equation of state has been applied to an accurate description of the pressure-volume-temperature relationship for 67 liquid polymers and 61 solvents and compared with four commonly used equations of state. An approximate expression for the activity coefficient of a solvent in polymer-solvent solutions has been derived, which enables one to calculate the activity coefficient for the solvent analytically. The grand absolute average deviation between the calculated and experimental solvent activity coefficients for 95 data sets is 2.4%, which indicates that this equation of state is also capable of accurately describing polymer-solvent solutions. Introduction Equations of state (EOSs) are widely used to describe the pressure-volume-temperature (PVT) relationship of liquid polymers and solvents and, furthermore, the thermodynamic properties for polymer-solvent solutions. To establish an EOS, a feasible approach is to figure out a model to represent the physical structure of liquids, in particular for polymers. As is reviewed in Rodgers’ (1993) paper and a handbook of Danner and High (1993), EOS theories for liquid polymers can be roughly classified into three categories: cell models (Flory et al., 1964; Flory, 1965; Prigogine, 1957), latticefluid models (Sanchez and Lacombe, 1976; Panayiotou and Vera, 1982), and hole models. In the hole models, Simha and Somcynsky (1969) presented their EOS (SS EOS) based on the concept that the thermal expansion of liquids is mainly due to the holes, i.e., the vacant cells, while changes of cell volume are also allowed. The SS EOS has been successfully used to represent the PVT behavior for polymer liquids. Dee and Walsh (1988) and Rodgers (1993) made comprehensive comparisons between the EOSs for liquid polymers and have found that the SS EOS provides the best fit to the experimental data for liquid polymers. However, solving the SS EOS is mathematically complex, and it does not lead directly to a principle of corresponding states. This unfortunately hinders the wide application of this model. To remove the drawbacks, we have proposed a so-called simplified hole theory (SHT) EOS (Zhong et al., 1993, 1994). In our previous work, the occupied site fraction for the SS EOS was derived as a simple exponential function of temperature alone, in terms of the theory for an imperfect crystal with thermal defects. Tested by 11 polymers and 3 solvents and their solutions, the S0888-5885(96)00413-7 CCC: $14.00
SHT EOS presents the same quality results as the SS EOS with significantly less computing effort. In this paper we develop our previous work in several aspects. First, the SHT EOS is extended to a large number of polymers and diversified solvents, in particular. The characteristic parameters of the SHT EOS for the polymers and solvents are reported, and the PVT results are compared with four EOSs, which are representatives among the three catogories aforementioned. Next, we adjust the characteristic flexibility ratio (see definition in the subsequent section) for solvents in compliance with that for the polymers, which enables us to describe polymer-solvent systems in a consistent way. Finally, instead of a numerical method used in our original work (Zhong et al., 1994), an analytical expression for the activity coefficient of the solvent in a polymer-solvent solution is derived and tested in terms of the data of the solutions.
Simplified Hole Theory Equation of State (SHT EOS) In the hole theory, allowing for a quasi-lattice with sites either occupied by segments of an s-mer or vacant and changes in cell volume, Simha and Somcynsky (1969) proposed a liquid EOS for polymers, which has been reviewed in detail by Simha (1977). The occupied site fraction, y, in the EOS is determined by the minimization of the configurational Helmholtz free energy and an implicit function of the reduced volume, V ˜ , and the reduced temperature, T ˜, © 1997 American Chemical Society
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2391
s s-1 y + y-1 ln(1 - y) ) [2.409 3c s 6T ˜ (yV ˜ )2 ˜ )-1/3 3.033(yV ˜ )-2] + [2-1/6y(yV 1 /3][1 - 2-1/6y(yV ˜ )-1/3]-1 (1)
[
]
As a result, the SS EOS is
P ˜V ˜ 2y ) [1 - 2-1/6y(yV ˜ )-1/3]-1 + [1.011(yV ˜ )-2 T ˜ T ˜ (yV ˜ )2 1.2045] (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, T*, V*, and P* are three characteristic parameters, * and ν* represent the characteristic energy and volume per segment, respectively, 3c is the number of external degree of freedom per chain, s is the number of segments per molecule, qz is the number of nearest-neighbor sites per chain, equal to s(z - 2) + 2, z is the coordination number that is set to 12, and M0 is the segmental molecular weight. Since the occupied site fraction is an implicit function of V ˜ and T ˜ , shown in eq 1, the SS EOS has to be solved by using a numerical method and does not lead directly to the principle of corresponding states. As noted above, the difficulty in solving the SS EOS is mainly due to the complexity of the expression for the occupied site fraction (eq 1). To overcome this drawback, we resorted to the theory for the thermal defects of an imperfect crystal. By introduction of the Shottky vacancy (Shottky, 1935) representing the hole in liquids, a simple equation for the occupied site fraction, y, was derived (Zhong et al., 1993):
y ) 1 - e-c/2sT
(4)
where 3c is the number of external degrees of freedom per chain. From a previous investigation (Simha and Somcynsky, 1969), the characteristic flexibility ratio (c/s) can be taken as a constant. As is suggested by Zhong et al. (1993), c/s is set to a universal constant 1.04, based on our experience in practical calculations for liquid polymers. Therefore, eq 4 can be further simplified as
y ) 1 - e-0.52/T˜
(5)
By following the approach of Simha and Somcynsky (1969), an EOS can be derived in terms of an explicit occupied site fraction expression (eq 5), except that the coordination number z here is set to 8 for the bodycentered-cubic lattice instead of the face-centered-cubic lattice in the SS EOS:
[
]
(yV ˜ )1/3 P ˜V ˜ 2y 1.1394 ) - 1.5317 + 1/3 T ˜ (yV ˜ ) - 0.9165y T ˜ (yV ˜ )2 (yV ˜ )2
(6) where the reduced and characteristic properties are the same as those in eq 3.
In summary, eqs 5 and 6 form the so-called simplified hole theory (SHT) EOS. Obviously, compared with the SHT EOS, this EOS can be simply solved for the PVT calculations and is easily applied to polymer solutions. PVT Relationship of Liquid Polymers and Solvents A data bank mainly from the compilation of Danner and High (1993) was used for testing the capability of the SHT EOS for the description of the PVT relationship for polymers and solvents. It contains 67 polymers (including 13 copolymers) of different structures, bonding forces, and polarities and 61 solvents including paraffin, aromatics, heavy hydrocarbons, and polar substances over a temperature range of 50-150 K and pressure range from ambient up to 200 MPa. The complete data bank, consisting of abbreviations of polymers and solvents, number of data points, temperature and pressure ranges, and data sources, is included as supporting information. Four Equations of State. In the PVT calculations, the SHT EOS was compared with four commonly used EOSs, which are as follows: (1) Flory, Orwoll, and Vrij (FOV) EOS. Flory et al. (1964) derived their cell model EOS as
V ˜ 1/3 1 P ˜V ˜ ) 1/3 T ˜ V ˜ T ˜ V ˜ -1
(7)
where the reduced properties are defined as
P ˜ ) P/P*, T ˜ ) T/T*, V ˜ ) V/V*,
P* ) s/2ν*2 T* ) s/2ν*ck V* ) Nrν*
(8)
where Nr is the total number of segments, ν* is the reference volume defined by the authors, s is the interaction energy of the molecule per segment, c is a measure of the amount of flexibility and rotation that is present in a molecule per segment, i.e., the vibrational and rotational energy states, and k is Boltzmann’s constant. (2) Prigogine Cell Model (CM) EOS. Using different cell geometry and molecular interaction potentials from Flory, Prigogine (1957) proposed a cell model EOS:
V ˜ 1/3 2 P ˜V ˜ ) 1/3 [1.2045 - 1.011/V ˜ 2] 2 T ˜ V ˜ - 0.8909 T ˜V ˜
(9)
The definitions of the reduced properties are the same as eq 8. (3) Sanchez and Lacombe (SL) EOS. Sanchez and Lacombe (1976) derived an EOS on the lattice-fluid model:
1 - 1/r P ˜ V ˜ 1 ) ln - 2 T ˜ V ˜ -1 V ˜ V ˜ T ˜
(10)
where r is the number of lattice sites occupied by the r-mer. The reduced quantities are given by
T ˜ ) T/T*,
T* ) */k
P ˜ ) P/P*,
P* ) */ν*
V ˜ ) V/V*,
V* ) N(rν*)
(11)
where ν* is defined by the authors as the closed-packed
2392 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Table 1. Calculated Results for Liquid Polymer Densities for Five EOSs and Characteristic Parameters for the SHT EOS AADFa polymer LPE BPE i-PP PIB PDMS PDMS1 PDMS2 PDMS3 PDMS4 PDMS5 PDMS6 PcHMA PMMA PnBMA PET PVAc PTFE PS PoMS PBD PMP PEO PPO PEA PEMA PVC PSF PC PAr PH PTHF PVME PA6 PA66 LDPE LDPE-A LDPE-B LDPE-C HDPE i-PB TMPC HFPC BCPC PECH PCL R-PP EP50 EVA18 EVA25 EVA28 EVA40 SMMA20 SMMA60 SAN3 SAN6 SAN15 SAN18 SAN40 SAN70 PA POM i-PMMA PEEK PMA PA11 PVDF PB
T* (K)
P* (MPa)
V* (cm3/g)
FOV
SL
CM
SS
SHT
2751.7 2914.1 3115.4 3651.0 2233.6 1944.7 2056.8 2104.4 2129.8 2191.2 2204.8 3483.1 6274.2 2967.3 3348.2 2823.1 2872.0 3587.5 3871.6 3009.2 3062.8 2895.9 3150.8 2853.7 2823.1 3973.4 3593.2 3345.6 3617.8 3451.4 3018.1 3095.9 5034.4 3453.1 3046.4 2992.2 3091.5 3038.3 2717.9 3195.2 3072.6 3031.8 3557.4 3536.7 3101.1 2699.8 3764.5 3503.7 3526.4 2922.4 2966.4 3400.8 3348.9 3544.0 3244.7 3509.6 3625.6 4031.9 4370.9 3684.2 2939.9 3429.8 3463.3 3015.3 2996.7 2824.3 3106.8
561.63 472.26 407.78 393.96 345.18 366.26 366.04 371.35 381.78 365.96 356.21 502.88 347.66 537.33 861.12 577.67 369.14 500.30 458.81 457.94 393.59 633.37 682.34 536.31 678.37 476.49 795.90 754.93 687.87 704.79 465.57 525.21 350.36 529.42 395.16 496.98 480.39 490.98 575.61 390.27 638.35 648.65 605.26 535.97 522.18 414.82 385.46 496.68 491.22 518.67 511.20 508.58 536.50 464.45 532.13 510.81 492.33 442.92 500.58 678.51 865.94 624.07 867.77 610.91 756.79 707.08 417.18
1.0646 1.0904 1.1098 1.0556 0.8985 0.9245 0.9198 0.9031 0.8901 0.8929 0.8946 0.8528 0.8791 0.8822 0.6949 0.7740 0.4618 0.9091 0.9307 1.0420 1.1215 0.8267 0.8190 0.8237 0.8109 0.6981 0.7379 0.7631 0.7587 0.8063 0.9541 0.9159 0.7910 0.7580 1.1022 1.0912 1.1014 1.0954 1.0602 1.0981 0.8052 0.8019 0.6593 0.7043 0.8616 1.0579 1.1691 0.8291 0.8308 1.0422 1.1093 0.8647 0.8198 0.8927 0.8763 0.8727 0.8757 0.8779 0.8579 0.7608 0.7111 0.7795 0.7115 0.7949 0.9046 0.5494 1.0931
0.151 0.087 0.398 0.057 0.126 0.134 0.118 0.103 0.103 0.098 0.113 0.158 0.174 0.163 0.071 0.047 0.103 0.153 0.075 0.193 0.261 0.113 0.124 0.268 0.132 0.105 0.156 0.123 0.145 0.140 0.085 0.230 0.222 0.177 0.117 0.233 0.234 0.239 0.151 0.305 0.137 0.205 0.155 0.109 0.139 0.057 0.113 0.185 0.225 0.226 0.204 0.232 0.203 0.229 0.235 0.213 0.217 0.202 0.162 0.117 0.104 0.108 0.113 0.228 0.105 0.058 0.186
0.190 0.121 0.498 0.104 0.172 0.170 0.156 0.142 0.141 0.139 0.153 0.227 0.178 0.292 0.087 0.071 0.143 0.248 0.124 0.292 0.317 0.158 0.112 0.424 0.169 0.157 0.225 0.165 0.215 0.266 0.128 0.391 0.287 0.210 0.154 0.333 0.339 0.341 0.197 0.420 0.166 0.273 0.234 0.205 0.202 0.094 0.081 0.266 0.331 0.326 0.321 0.354 0.317 0.364 0.362 0.323 0.347 0.343 0.300 0.161 0.127 0.188 0.117 0.378 0.138 0.094 0.269
0.065 0.063 0.125 0.031 0.038 0.097 0.076 0.078 0.071 0.075 0.058 0.083 0.247 0.101 0.357 0.018 0.123 0.050 0.050 0.031 0.120 0.109 0.179 0.152 0.038 0.070 0.063 0.123 0.033 0.081 0.061 0.117 0.085 0.061 0.067 0.071 0.070 0.066 0.065 0.083 0.049 0.068 0.096 0.071 0.065 0.073 0.157 0.052 0.056 0.053 0.067 0.057 0.074 0.080 0.055 0.052 0.053 0.117 0.088 0.069 0.050 0.136 0.048 0.140 0.056 0.136 0.078
0.151 0.052 0.084 0.007 0.014 0.093 0.069 0.071 0.064 0.059 0.048 0.066 0.215 0.061 0.047 0.016 0.255 0.030 0.036 0.037 0.119 0.056 0.142 0.109 0.027 0.072 0.035 0.116 0.028 0.037 0.037 0.086 0.048 0.060 0.060 0.049 0.045 0.046 0.068 0.047 0.051 0.054 0.073 0.043 0.045 0.064 0.152 0.037 0.035 0.041 0.042 0.043 0.057 0.036 0.032 0.020 0.017 0.059 0.041 0.059 0.054 0.105 0.083 0.100 0.055 0.144 0.052
0.057 0.046 0.092 0.017 0.114 0.085 0.065 0.067 0.062 0.058 0.049 0.063 0.255 0.063 0.040 0.020 0.155 0.042 0.041 0.021 0.103 0.049 0.159 0.045 0.023 0.064 0.046 0.106 0.045 0.066 0.014 0.024 0.064 0.043 0.057 0.041 0.036 0.046 0.059 0.047 0.045 0.065 0.059 0.059 0.056 0.086 0.161 0.054 0.086 0.069 0.061 0.089 0.093 0.076 0.076 0.042 0.054 0.106 0.068 0.058 0.043 0.079 0.036 0.032 0.059 0.123 0.045
0.159
0.229
0.084
0.065
0.066
grand average a
AADF )
(1/N)∑i|Fcalc i
-
exp Fexp i |/Fi
× 100.
volume of a segment that comprises the molecule, and * is the international energy of the lattice per site.
(4) Simha and Somcynsky (SS) EOS. As is mentioned above, eqs 1 and 2 form the SS EOS.
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2393 Table 2. Calculated Results for Liquid Solvent Densities for Five EOSs and Characteristic Parameters for the SHT EOS AADFa solvent
T* (K)
P* (MPa)
propane n-butane n-pentane n-hexane n-heptane n-octane n-nonane n-decane n-hendecane n-dodecane n-dodecane n-pentadecane n-hexadecane n-octadecane carbon tetrachloride carbon sulfide water ammonia cyclohexane benzene tolune m-xylene o-sylene n-xylene i-butane i-pentane acetonitrile methanol 1-pentanol 1-hexanol 1-heptanol 1-octanol 1-nonanol acetone butan-2-one octan-1-oic acid decan-1-oic acid dodecan-1-oic acid tetradecan-1-oic acid hexadecan-1-oic acid methyl acetate ethyl acetate neopentane oct-1-ene ethylbenzene methylchloride 1-chloropropane 1-chlorobutane tetramethylsilane CCPDb DCCHc NPDd CCHe PPHf OHTg CPPHh HMDSi CHEHj DPDk TMSIl PEHm
1295.5 1509.5 1620.0 1748.9 1830.9 1859.6 1959.8 1976.7 2053.4 2116.7 2003.4 2182.4 2234.8 2251.6 1782.1 1812.1 2379.1 1455.0 1873.9 1859.9 1958.8 2084.9 2117.1 2096.3 1425.5 1531.0 1658.3 1805.1 2065.1 2087.1 2202.5 2155.5 2216.3 1623.9 1715.3 2290.4 2364.4 2392.8 2339.2 2346.8 1618.7 1652.5 1565.2 1812.0 2024.7 1345.3 1816.2 1872.5 1555.2 2587.9 2700.2 2546.8 2644.5 2837.9 2416.5 2446.5 1696.7 2492.5 2503.6 1586.8 2442.9
416.60 374.62 475.25 447.39 440.91 506.68 477.92 528.64 455.01 447.98 523.92 543.30 503.11 546.49 620.04 616.59 1139.0 800.46 565.05 712.22 568.87 549.98 561.50 545.52 395.03 464.52 656.98 500.99 572.10 582.14 621.45 580.92 549.88 627.80 612.47 504.85 505.08 519.29 574.75 589.14 775.99 706.65 351.18 558.10 595.00 695.66 733.63 593.60 378.90 529.30 519.95 626.39 526.57 478.01 504.24 537.41 385.37 530.40 613.79 350.15 573.25
V*
(cm3/g)
1.3572 1.3087 1.2503 1.2310 1.2049 1.1750 1.1717 1.1541 1.1544 1.1526 1.1323 1.1283 1.1288 1.1163 0.5114 0.6508 0.8738 1.2294 1.0663 0.9332 0.9716 0.9912 0.9791 0.9994 1.3100 1.2348 1.0178 1.0358 1.0474 1.0440 1.0572 1.0459 1.0517 0.9960 0.9994 0.9684 0.9873 0.9973 0.9944 0.9988 0.8416 0.8788 1.3253 1.1413 0.9816 0.7693 0.9277 0.9384 1.2079 1.0558 1.0296 0.9836 1.0513 1.0069 1.1113 1.0771 1.0471 1.0781 1.0315 1.2135 1.0425
grand average (1/N)∑i|Fcalc i
FOV
SL
CM
SS
SHT
0.234 0.106 0.041 0.056 0.296 0.176 0.401 0.710 0.649 0.205 0.211 0.249 0.385 0.241 0.146 0.251 0.319 0.140 0.082 0.195 0.066 0.087 0.160 0.099 0.272 0.212 0.055 0.019 0.019 0.087 0.020 0.030 0.547 0.335 0.008 0.024 0.029 0.047 0.055 0.076 0.116 0.078 0.421 0.040 0.230 0.066 0.099 0.156 0.194 0.214 0.117 0.229 0.237 0.271 0.248 0.051 0.248 0.178 0.544 0.232
0.233 0.087 0.055 0.059 0.339 0.282 0.577 0.905 0.276 0.321 0.292 0.372 0.578 0.352 0.233 0.475 0.381 0.166 0.370 0.201 0.154 0.264 0.144 0.503 0.383 0.176 0.020 0.020 0.093 0.020 0.033 0.740 0.517 0.010 0.012 0.009 0.011 0.009 0.138 0.179 0.154 0.561 0.048 0.300 0.129 0.156 0.221 0.322 0.354 0.185 0.368 0.380 0.410 0.382 0.080 0.381 0.280 0.686 0.362
0.273 0.080 0.162 0.132 0.075 0.089 0.162 0.109 0.457 0.083 0.152 0.083 0.101 0.095 0.046 0.051 0.402 0.088 0.198 0.060 0.275 0.152 0.361 0.249 0.092 0.069 0.031 0.017 0.016 0.089 0.017 0.029 0.046 0.052 0.007 0.010 0.006 0.009 0.010 0.054 0.035 0.035 0.234 0.048 0.205 0.205 0.457 0.067 0.148 0.098 0.039 0.098 0.120 0.108 0.118 0.036 0.086 0.076 0.068 0.080
0.240 0.145 0.087 0.085 0.074 0.048 0.167 0.107 0.504 0.059 0.150 0.089 0.075 0.089 0.039 0.054 0.299 0.103 0.070 0.035 0.122 0.070 0.094 0.084 0.076 0.063 0.028 0.024 0.080 0.023 0.031 0.062 0.043 0.009 0.011 0.008 0.010 0.010 0.054 0.027 0.114 0.176 0.038 0.182 0.184 0.091 0.072 0.128 0.078 0.030 0.082 0.090 0.085 0.101 0.065 0.075 0.058 0.102 0.059
0.254 0.073 0.151 0.161 0.060 0.071 0.141 0.109 0.051 0.064 0.172 0.084 0.090 0.085 0.037 0.060 0.765 0.482 0.071 0.077 0.035 0.087 0.053 0.155 0.239 0.057 0.096 0.040 0.024 0.039 0.078 0.045 0.033 0.077 0.056 0.009 0.008 0.010 0.010 0.010 0.046 0.039 0.084 0.229 0.037 0.196 0.644 0.314 0.066 0.131 0.078 0.065 0.084 0.076 0.078 0.090 0.032 0.077 0.089 0.071 0.078
0.203
0.289
0.154
0.097
0.125
exp Fexp i |/Fi
a AADF ) × 100. -, no results available. b CCPD ) 1-cyclopentyl-4-(3-cyclopentylpropyl)dodecane. c DCCH ) 1,7-dicyclopentyl-4-(3-cyclopentylpropyl)heptane. d NPD ) 1-R-naphthylpentadecane. e CCH ) 1-cyclohexyl-3-(2-cyclohexylethyl)hendecane. f PPH ) 1-phenyl-3-(2-phenylethyl)hendecane. g OHT ) 9-n-octylheptadecane. h CPPH ) 9-(3-cyclopentylpropyl)heptadecane. i HDMS ) hesadimethylsiloxane. j CHEH ) 9-(2-cycloxylethyl)heptadecane. k DPD ) 1-R-decalylpentadecane. l TMSI ) tetramethylsilane. m PEH ) 9(2-phenylethyl)heptadecane.
PVT Relationship of Polymers and Solvents. The PVT behavior of 67 polymers and 61 solvents has been fitted to the experimental data by using the SHT EOS. For a comparison, parallel calculations for the four EOSs have been carried out. Table 3 presents detailed results for the characteristic parameters of the SHT EOS and the density deviations, which are defined
in the whole context here as where n represents the
AADF )
1
exp |Fcalc - Fexp × 100 ∑ i i |/Fi N i
(12)
number of data points and F is the liquid density.
2394 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Table 3. Calculated Results for Solvent Weight Fraction Activity Coefficients and Two Binary Interaction Parameters for the SHT EOS by Using Simplified Equations 24-30 system
no. of data
kija
aijb
AADc
system
PIB/n-pentane PIB/n-pentane1 PIB/n-pentane2 PIB/n-octane PIB/neopentane PIB/isopentane PIB/isobutane PIB/cyclohexane PIB/cyclohexane1 PIB/cyclohexane2 PIB/n-butane PIB/benzene PIB/benzene1 PIB/benzene2 PIB/benzene3 PS/m-xylene PS/m-xylene1 PS/toluene1 PS/toluene2 PS/toluene3 PS/toluene4 PS/toluene5 PS/toluene6 PS/toluene7 PS/toluene8 PS/toluene9 PS/n-nonane PS/methyl ethyl ketone PS/methyl ethyl ketone1 PS/methyl ethyl ketone2 PS/methanol PS/ethylbenzene PS/ethylbenzene1 PS/cyclohexane PS/cyclohexane1 PS/cyclohexane2 PS/cyclohexane3 PS/cyclohexane4 PS/cyclohexane5 PS/cyclohexane6 PS/cyclohexane7 PS/carbon tetrachloride PS/carbon tetrachloride1 PS/benzene1 PS/benzene2 PS/benzene3 PS/benzene4 PS/benzene5
9 16 29 5 10 8 7 30 8 21 20 29 19 22 24 13 9 12 11 18 33 27 24 18 9 5 16 17 12 9 6 14 36 50 16 10 33 27 24 18 4 14 4 31 31 5 15 13
1.1060 1.1038 1.1333 1.0935 1.0786 0.9690 1.1062 1.0849 1.0820 1.0900 1.0945 1.0788 1.0692 1.0719 1.1009 1.0959 1.0851 1.0864 1.1077 1.0937 1.0948 1.1079 1.1081 1.0371 1.0853 1.0893 1.0723 1.1159 1.1622 1.1211 1.1483 1.0676 1.0935 1.0738 1.0863 1.1131 1.0940 1.1302 1.1299 1.1303 1.0817 1.1187 1.1113 1.1064 1.0866 1.1225 1.0975 1.0968
1.1876 1.2714 1.1060 1.0221 1.8891 3.2758 1.9627 1.0208 1.0049 0.9538 1.9357 0.8718 0.8605 0.8686 0.8196 0.9149 1.0799 1.0343 0.9731 1.0642 1.0875 1.0372 1.0283 1.1363 1.1131 1.0755 1.1401 1.0290 0.9451 1.0120 -0.7063 0.9434 1.0303 1.0442 0.9986 0.8907 0.9857 0.9581 0.9510 0.9462 1.0166 1.4568 1.5414 0.9070 0.9897 0.8827 0.8903 0.9315
3.3 3.9 1.2 0.1 1.9 3.4 0.9 3.5 0.6 0.1 1.6 5.8 2.5 1.9 0.1 8.2 2.3 8.8 15.3 6.0 1.2 0.1 0.01 0.002 1.7 1.1 6.9 8.7 0.3 3.1 4.1 0.1 3.6 9.8 1.7 5.9 1.4 0.03 0.009 0.002 0.2 2.7 0.1 3.0 8.4 0.2 3.7 10.00
PS/benzene6 PS/acetone PP/n-hexane PMMA/methyl ethyl ketone PMMA/toluene HDPE/n-hexane HDPE/n-decane HDPE/cyclohexane PEO/water PEO/water1 PEO/water2 PEO/water3 PEO/benzene PEO/benzene1 PEO/benzene2 PEO/benzene3 PEO/benzene4 PEO/benzene5 PDMS/n-pentane PDMS/n-hexane PDMS/n-hexane1 PDMS/n-heptane PDMS/n-heptane1 PDMS/n-octane PDMS/n-nonane PDMS/cyclohexane PDMS/benzene PDMS/benzene1 PDMS/benzene2 PDMS/benzene3 PDMS/benzene4 PDMS/benzene5 PDMS/benzene6 PDMS/benzene7 PDMS/benzene8 PDMS/benzene9 PDMS/methyl ethyl ketone PDMS/methyl ethyl ketone1 BR/n-nonane BR/n-hexane BR/ethylbenzene BR/cyclohexane BR/cyclohexane1 BR/cyclohexane2 BR/carbon tetrachloride BR/carbon tetrachloride1 BR/benzene grand average
a
γexp i
no. of data
kija
aijb
AADc
6 7 10 7 8 7 9 6 37 41 20 18 14 13 7 6 4 5 16 15 16 16 10 33 19 16 8 8 8 8 8 8 8 8 7 16 15 16 12 7 22 12 7 6 8 9 8
1.1045 1.1350 1.0945 1.2617 1.2021 1.1048 1.0335 1.0876 0.9854 0.9655 0.9226 0.8731 1.0490 1.0186 1.0477 1.0424 1.0464 1.0457 1.0248 1.0910 1.0149 1.0033 0.4633 1.0096 0.9999 1.0020 0.9889 0.9889 0.9887 0.9894 0.9895 0.9892 0.9892 0.9893 0.9895 0.9901 1.0262 1.0107 1.0098 1.0611 1.0242 1.0537 1.0564 1.0509 1.0749 1.0759 1.0562
0.9297 1.0430 1.0199 0.9551 1.0347 0.8413 1.0470 0.8287 0.9807 0.9549 1.0381 1.1582 0.9868 0.9735 1.0592 1.0498 1.0460 1.0912 0.7767 0.6187 0.8161 0.6447 1.0077 0.6890 0.7488 0.7727 0.7508 0.7504 0.7515 0.6867 0.6851 0.6866 0.7502 0.7074 0.7056 0.6832 0.5927 0.6726 1.1448 1.1174 1.0737 1.0698 0.9943 1.0490 1.2609 1.2509 0.9782
0.7 2.7 0.4 7.1 5.1 0.6 0.8 0.3 3.5 6.4 5.0 6.9 1.6 0.3 0.7 1.3 0.2 0.2 0.2 0.04 0.2 0.3 0.02 0.1 0.1 0.4 0.06 0.1 0.1 0.2 0.1 0.3 0.08 0.08 0.1 0.5 0.3 0.7 8.8 1.1 7.9 4.4 1.0 0.3 0.9 1.4 1.9 2.4
kij is the volume interaction molecular parameter. b aij is the energy interaction molecular parameter. c AAD ) (1/N)∑i|γcalc - γexp i i |/ × 100.
In the regression of the parameters of the SHT EOS, the ambient PVT data were used to obtain initial values for T*, V*, and P*. Then all the PVT data were fitted again to get the parameters needed in the EOS. As is seen in Table 1, with remarkable simplicity, the SHT EOS provides the same accuracy as the SS EOS and is superior to the other three EOSs. It is noted that in the derivation of the SHT EOS the structure of polymer molecules was particularly taken into account. In contrast, we treated the characteristic flexibility ratio c/s for solvents in a rather artificial way, by setting it to either unity or a regressed constant valued about unity (Zhong et al., 1994). This inconsistency will induce inconvenience in the description of the properties for polymer solutions. To remedy this shortcoming, we have attempted the ratio c/s ) 1.04, which is exactly the same as that for polymers, for all the solvents. In this way, we have obtained the characteristic parameters for the SHT and density deviations for the five EOSs, listed in Table 2. It is found that the PVT results of the SHT EOS for solvents are comparable with those for the SS, slightly better than those for the CM, and more accurate than those for FOV, SL EOSs.
Activity Coefficients for Polymer-Solvent Systems SHT EOS for Mixtures. The SHT EOS for a polymer-solvent system can be expressed as (Zhong et al., 1994)
2yQ2 P ˜V ˜ ) (1 - W)-1 + (1.1394Q2 - 1.5317) T ˜ T ˜
(13)
where the occupied site fraction y is given by
y ) 1 - e-〈cs〉/2T
(14)
Q ) (yv)-1 and w ) 0.9165yQ1/3. The reduced properties for a mixture are defined as
T ˜ ) T/T*,
T* ) 〈qz〉〈*〉/〈cs〉R
P ˜ ) P/P*,
P* ) 〈qz〉〈*〉/〈ν*〉
V ˜ ) V/V*,
V* ) 〈ν*〉/〈M0〉
(15)
where the brackets signify compositional average. For
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2395
a binary mixture,
[ () ] [ ]
〈s〉-1 )
Φ1 Φ2 + s1 s2
(16)
〈cs〉 ) Φ1cs1 + Φ2cs2
(17)
〈M0〉 ) Φ1M01 + Φ2M02
(18)
2 〈ν*〉 ) Φ12ν* 11 + 2Φ1Φ2ν* 12 + Φ2 ν* 22
(19)
2 〈*〉 ) θ12* 11 + 2θ1θ2* 12 + θ2 * 22
(20)
〈qz〉 ) Φ1q1z + Φ2q2z
(21)
where qiz ) z - 2 + 2/si, θi ) Φiqiz/〈qz〉 (Φ is the segment fraction and Φ1 ) 1 - Φ2 ) N1s1/(N1s1 + N2s2), N1 and N2 are the numbers of molecules, and S1 and S2 are the numbers of segments per molecule for components 1 and 2, respectively. 3cs1 and 3cs2 are the numbers of external degrees of freedom per segment, M01 and M02 are the segmental molecular weights, ν*11 and ν* 22 are the characteristic volumes, and * 11 and * 22 are the characteristic energies for components 1 and 2, respectively. For the characteristic interaction energy and volume between molecules 1 and 2, *12 and ν*12, two binary interaction parameters, a12 and k12, are introduced: 1/2 * 12 ) a12(* 11 * 22 )
ν* 12 ) k12
(
(22)
)
1/3 1/3 ν* + ν* 11 22 2
3
(23)
The two binary interaction parameters can be determined by fitting the properties of mixtures. Derivation of the Activity Coefficients of Solvents for Polymer Solutions. Since the polymers remain entirely in the condensed phase, the activity coefficients of the solvents in polymer-solvent systems are important thermodynamic properties for polymer processing. The weight fraction activity of a solvent can be derived through the SHT EOS and basic thermodynamic relations. For a binary system,
µ1 ) µ01 + RT ln a1
(24)
where subscript 1 represents the solvent, µ01 and µ1 are the chemical potentials for the pure solvent and the solvent in the binary system, respectively, and a1 is the weight fraction activity of the solvent. The weight fraction activity coefficient of the solvent, γ1, is therefore expressed as
γ1 ) a1/w1
(25)
where w1 is the weight fraction of the solvent in the solution. In this work, polymer-solvent solutions are recognized as pseudo binary solutions, which is acceptable for the description of vapor-liquid equilibria, in particular. Thus, µ01 in eq 24 is
( ) {[
s1 - 1 ln y 1 1 z-1 ln ln + + s1 s1 s1 e s1
µ01 ) RTs1
] }
ν*11 (1 - W)3 1-y ln(1 - y) - RTs1cs1 ln + y Q 2πM01RT yQ2 3 ln (1.1394Q2 - 3.0634) + 2 2T ˜ (Nah)2 s1PV ˜ ν*11 (26) where Na, h, and R are the Avogadro’s, Planck’s, and gas constants, respectively, y, V ˜ , and P ˜ are the occupied site fraction, the reduced volume, and the pressure of the pure solvent at the system T and P, respectively. µ1 in eq 24 is given by
( ) ∂GΦ ∂Φ2
µ1 ) GΦs1 - s1Φ2
(27)
T,P
or rearranged as
( )]
[
µ1 ) RTs1
GΦ 1 ∂GΦ - Φ2 RT RT ∂Φ2
(28)
T,P
and
GΦ ) AΦ + PVΦ
(29)
where GΦ, AΦ, and VΦ are the molar segmental Gibbs and Helmholtz free energies and volume, respectively. Unfortunately, the partial differential on the righthand side of eq 28 can not be solved analytically. Consequently, an elaborate numerical method is needed for the calculation of activities. As is seen in the appendix, for the polymer-solvent systems at ambient pressures, which are usually encountered in practice, the right-hand term of eq 28 can be reasonably approximated to
( ) [( ) ( )( ) {[ [ ] [
GΦ 1 ∂GΦ - Φ2 RT RT ∂Φ2
)
T,P
]
Φ1 1 ln + 1 + ln y + s1 s1
1-y 1 1 z-1 + ln(1 - y) + - 1 ln s1 e y 〈s〉
3〈cs〉Φ2 2〈M0〉
(M02 - M01) - cs1 ln
]
〈ν*〉(1 - W)3 + Q
}
2π〈M0〉RT yQ2 3 ln (1.1394Q2 - 3.0634) + 2 2T ˜ (Nah)2 1 - 〈s〉 - 〈cs〉〈s〉 ln(1 - y) PV ˜ 〈ν*〉 - Φ2 + RT y〈s〉 y2
]
1.833〈cs〉Q1/3 〈cs〉Q2 (1.7091Q2 - 1.5317) × 1-W T ˜
( ) ∂y ∂Φ2
[
- Φ2 T,P
〈cs〉yQ
2
2T ˜2
( ) ∂T ˜ ∂Φ2
( )( )
- Φ2 T,P
]
(1.1394Q2 - 3.0634) × 〈cs〉 ∂〈ν*〉 〈ν*〉 ∂Φ2
(30)
T,P
Substituting eq 30 into eq 28 and combining eqs 2428, the activity or activity coefficient can be solved analytically. However, the simplification imposed will be tested in a subsequent section of this work.
2396 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Table 4. Comparison between the Three Methods for the Calculation of the Activity Coefficients of Solvents method
AADa a
AAD )
regressed s for solvent
numerical method
eqs 24-30
1.9
2.2
2.4
(1/N)∑i|γcalc i
-
exp γexp i |/γi
× 100.
Activity Coefficients for Polymer-Solvent Solutions. A data bank, which is also from the compilation of Danner and High (1993), was used for the test of our model. It contains 95 sets of the activity coefficients of polymer-solvent solutions, covering many polymers and a variety of solvents, e.g., aromatics, long-chain hydrocarbons, and polar substances, and varying from low to high polymer concentrations. The complete data bank, including the systems, number of data points, and data sources is also included as supporting information. All the binary interaction parameters and the deviations for the activity coefficients calculated from the analytical formulas are listed in Table 3. It is found from Table 3 that the ground AAD for the activity - γexp coefficients, defined as AAD ) (1/N)∑i |γcalc i i |/ exp γi × 100, is 2.4%. Most results are of good accuracy, with deviations less than 3-5%. However, somewhat large errors have been observed in the systems whose data were measured in 1940-1950 (PS/tolune 1,2; PS/ methyl ethyl ketone) and in aqueous systems. Besides, for the heavy hydrocarbon, nonane (BR/n-nonane) and the high-concentration polymer solution (PS/cyclohexane), deviations up to 8-10% have been found. We have attempted another two methods for calculations of the activity coefficients for solvents: either the values of the number of segments for solvents, s, are taken as regressed parameters or the activity coefficients are solved numerically without simplification aforementioned. The results in Table 4 indicate that eq 30 is a reasonable approximation for the activity coefficient of the solvent in a polymer solution with a slight accuracy loss, compared with the other two rather complicated methods.
absolute average deviation between the estimated and experimental results for 95 data sets is 2.4%, which indicates that the SHT EOS is also capable of accurately describing polymer-solvent solutions. In summary, through rigorous and extensive tests in this work, the SHT EOS can be recommended for estimations of thermodynamic properties for pure liquid polymers and solvents and their solutions in practice. It is, however, desirable to develop predictive models for EOSs, since one has to deal with a variety of polymers, solvents, and their solutions in chemical processes. In fact, this work provides a basis for this purpose. The group contribution approach has been successfully incorporated into the SHT EOS (Wang et al., 1997). This topic will be addressed in another publication of ours.
Acknowledgment This project was partly supported by the National Natural Science Foundation of China and the Petrochemical Corporation of China.
Supporting Information Available: Data bank consisting of PVT relationship for 67 polymers (including 13 copolymers), 61 solvents, 95 sets of activity coefficients for polymer-solvent solutions, and data references (12 pages). Ordering information is given on any current masthead page.
Appendix: Derivation of the Approximation for the Activity Coefficients of Solvents From eqs 28 and 29 we get
µ1 ) RTs1
GΦ 1 ∂GΦ - Φ2 RT RT ∂Φ2
(A-1)
T,P
GΦ ) AΦ + PVΦ
Discussion and Conclusions In the simplified hole theory equation of state proposed by our group (Zhong et al., 1993), the occupied site fraction for the Simha-Somcynsky equation of state was derived as a simple exponential function of temperature alone, in terms of the theory for an imperfect crystal with thermal defects. In this work, the SHT EOS has been applied to the description of PVT for 67 liquid polymers and compared with four commonly used EOSs. The grand density absolute average deviation for the SHT EOS is 0.066% versus 0.159% for the FOV, 0.229% for the SL, and 0.084% for the CM EOSs. In addition, the SHT EOS is of the same accuracy as the SS EOS but much simpler in computation than it. We have adjusted the characteristic flexibility ratio (c/s) for solvents by setting the value to a constant, 1.04, being consistent with that for polymers. Then, satisfactory results for the PVT behavior for 61 solvents have been obtained, compared with the four EOSs. An approximate expression of the activity coefficient for the solvent in polymer-solvent solutions has been derived, which enables one to calculate the activity coefficient for the solvent analytically. The grand
( )]
[
(A-2)
and
( ) ( ) ∂GΦ ∂Φ2
∂AΦ ∂Φ2
)
T,P
( )
+P
T,P
∂VΦ ∂Φ2
(A-3)
T,P
where the Helmholtz free energy, AΦ, is derived from the SHT EOS for a binary system:
Φ1 Φ1 Φ2 Φ2 AΦ A ) ) ln + ln RT (N1s1 + N2s2)kT s1 s1 s2 s2
( )
〈s〉 - 1 ln y 1 - y z-1 ln + ln(1 - y) + e y 〈s〉 〈s〉
{[
〈cs〉 ln
] [
]
2π〈M0〉RT 〈ν*〉(1 - W)3 3 + ln Q 2 (Nah)2 2
}
yQ (1.1394Q2 - 3.0634) 2T ˜ Therefore, we obtain
(A-4)
[ ] [( ) ] ( )[ ( ) ] {[ ] }
1 ∂GΦ RT ∂Φ2
)
T,P
[( ) ]
( )
Φ2 Φ1 1 1 ln + 1 - ln +1 + s2 s2 s1 s1
∂T ˜ ∂Φ2
1 3 〈cs〉(M02 - M01) 1 z-1 ln + ln y s2 s1 e 2 〈M0〉
[
]
2π〈M0〉RT 〈ν*〉(1 - W)3 3 + ln Q 2 (Nah)2 1 - 〈s〉 - 〈cs〉〈s〉 yQ2 (1.1394Q2 - 3.0634) + 2T ˜ y〈s〉
(cs2 - cs1) ln
ln(1 - y) +
y2
[
]( ) [ ]( ) ( )( )
〈cs〉yQ2 + (1.1394Q2 2 2T ˜ T,P 〈c ∂T ˜ s〉 ∂〈ν*〉 + + 3.0634) ∂Φ2 〈ν*〉 ∂Φ2
[
+
T,P
[ ( ) ( )] +V ˜
T,P
∂〈ν*〉 ∂Φ2
∂〈*〉 ∂Φ2
]
(A-5)
T,P
( ) [( ) ( )( ) {[ [ ] [ T,P
]
2〈M0〉
]
( ) [ ( ) ( )( ) ∂y ∂Φ2
∂T ˜ ∂Φ2
- Φ2 -
2
T,P
P ∂V ˜ 〈ν*〉 Φ2 RT ∂Φ2
+V ˜
]
T,P
-
-
T,P
(A-6)
T,P
where
( ) ∂y ∂Φ2
and
) T,P
[( )
∂〈cs〉 1 T ˜ 2 ∂Φ2 2T ˜
T,P
- 〈cs〉
( )] ∂T ˜ ∂Φ2
T,P
2 2 s2 s1
(A-12)
In our calculations, it has been found that the values of (∂V ˜ /∂Φ2)T,P and (∂〈ν*〉/∂Φ2)T,P are on the order of magnititude from 10-4 to 10-5 for the polymer-solvent systems at ambient pressures. As a result, the last two terms on the right-hand side of eq A-6 can be ignored, for their contributions to the values of the activity coefficients are less than about a thousandth. Finally, we get
)
T,P
]
Φ1 1 ln + 1 + ln y + s1 s1
(M02 - M01) - cs1 ln
( ) ∂y ∂Φ2
T,P
[
- Φ2 -
e-〈cs〉/2T˜ (A-7)
〈cs〉yQ
( ) ∂T ˜ ∂Φ2
(1 - W)V ˜
∂〈ν*〉 ∂Φ2
)
T,P
]
〈ν*〉(1 - W)3 + Q
}
]
〈cs〉W
〈cs〉 〈cs〉yQ ∂V ˜ (2.2788Q2 - 3.0634) V ˜ T ˜V ˜ ∂Φ2 2
2q1zq2z [θ2*22 - θ1* 11 + (θ1 - θ2)* 12] 〈qz〉2 (A-11)
1.833〈cs〉Q1/3 〈cs〉Q2 (1.7091Q2 - 1.5317) × 1-W T ˜
[ ]( ) [ ( ) ( )] - Φ2 -
(A-10)
}
(1.1394Q2 - 3.0634) ×
2T ˜ 〈cs〉 ∂〈ν*〉 - Φ2 〈ν*〉 ∂Φ2
T,P
T,P
〈cs〉yQ
) cs2 - cs1
T,P
2π〈M0〉RT yQ2 3 ln (1.1394Q2 - 3.0634) + 2 2 2T ˜ (Nah) 1 - 〈s〉 - 〈cs〉〈s〉 ln(1 - y) PV ˜ 〈ν*〉 - Φ2 + RT y〈s〉 y2
1.833〈cs〉Q1/3 〈cs〉Q2 (1.7091Q2 - 1.5317) × 1-W T ˜ 2
T,P
2〈M0〉
2π〈M0〉RT yQ 3 ln (1.1394Q2 - 3.0634) + 2 2 2T ˜ (Nah) 1 - 〈s〉 - 〈cs〉〈s〉 ln(1 - y) PV ˜ 〈ν*〉 - Φ2 + RT y〈s〉 y2 2
)
3〈cs〉Φ2
〈ν*〉(1 - W)3 + Q
(M02 - M01) - cs1 ln
(A-9)
1-y 1 1 z-1 + ln(1 - y) + - 1 ln s1 e y 〈s〉
1-y 1 1 z-1 + ln(1 - y) + - 1 ln s1 e y 〈s〉
3〈cs〉Φ2
( )
GΦ 1 ∂GΦ - Φ2 RT RT ∂Φ2
Φ1 1 ln + 1 + ln y + s1 s1
)
( ) ( )]
( ) [( ) ( )( ) {[ [ ] [
and substituting eq A-5 into eq A-1, we get
GΦ 1 ∂GΦ - Φ2 RT RT ∂Φ2
T,P
1 ∂〈qz〉 〈qz〉 ∂Φ2 T,P 1 ∂〈cs〉 (A-8) 〈cs〉 ∂Φ2 T,P
) 2Φ2(ν* 22 - ν* 12 ) - 2Φ1(ν* 11 - ν* 12)
∂〈qz〉 ∂Φ2
T,P
P ∂V ˜ 〈ν*〉 RT ∂Φ2
+
T,P
( )
〈cs〉 〈cs〉yQ (2.2788Q2 - 3.0634) × V ˜ T ˜V ˜ (1 - W)V ˜ ∂V ˜ ∂Φ2
1 ∂〈*〉 〈*〉 ∂Φ2
( )
2
( )
( )
[ ( ) ∂〈cs〉 ∂Φ2
∂y 1.5317) ∂Φ2
〈cs〉W
) -T ˜
T,P
∂〈ν*〉 ∂Φ2
1.833〈cs〉Q1/3 〈cs〉Q2 (1.7091Q2 1-W T ˜
T,P
Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 2397
2T ˜
T,P
2
]
2
]
(1.1394Q2 - 3.0634) ×
( )( )
- Φ2 -
〈cs〉 ∂〈ν*〉 〈ν*〉 ∂Φ2
(A-13)
T,P
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2398 Ind. Eng. Chem. Res., Vol. 36, No. 6, 1997 Characteristic Parameters for 56 Polymers. J. Appl. Polym. Sci. 1993, 48, 1061. Sanchez, I. C.; Lacombe, R. H. An Elementary Molecular Theory of Classical Fluids. Pure Fluids. J. Phys.Chem. 1976, 80, 2352. Shottky, Z. Uber den Mechanismus der Ionenbewegung in Festen Electrolyten. Z. Phys. Chem., Abt. B 1935, 29, 335. Simha, R. Configurational Thermodynamics of the Liquid and Glassy Polymeric States. Macromolecules 1977, 10, 1025. Simha, R.; Somcynsky, T. On the Statistical Thermodynamics of Spherical and Chain Molecule Fluids. Macromolecules 1969, 2, 342. 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, Proceedings of the International Symposium on Molecular Thermodynamics and Molecular Simulation, Tokyo, Jan 12-15, 1997; pp 61-79.
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 Hle Theory Equation of State to Polymer Solutions and Blends. Fluid Phase Equilib. 1994, 102, 173.
Received for review July 18, 1996 Revised manuscript received March 6, 1997 Accepted March 17, 1997X IE9604132
X Abstract published in Advance ACS Abstracts, May 1, 1997.
