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Estimation of PVT for Polymers and Solvent Activities in Polymer Solutions Using Simplified Parameters for the Sako-Wu-Prausnitz Equation of State Katsumi Tochigi,* Takeshi Matsumoto, Kiyofumi Kurihara, and Kenji Ochi Department of Materials and Applied Chemistry, College of Science and Technology, Nihon University, 1-8 Surugadai, Kanda, Chiyoda-ku, Tokyo 101-8308, Japan
For the Sako-Wu-Prausnitz (SWP) cubic equation of state, parameters were determined for 10 polymers [low-density polyethylene, poly(-caprolactone), poly(epichlorohydrin), poly(methyl methacrylate), poly(o-methylstyrene), polypropylene, poly(isobutylene), polystyrene, poly(vinyl acetate), and poly(vinyl chloride)] using literature PVT data ranging from the ambient conditions up to 470 K and 200 MPa. The correlated results of relative volume agreed with average deviations of 0.06% and were better than those of the original SWP equation (Sako et al. J. Appl. Polym. Sci. 1989, 38, 1839-1858). When the SWP equation is combined with the GE mixing rule and the ASOG-FV group contribution method, the solvent activities for polymer solutions composed of 4 polymers and 13 solvents (benzene, toluene, cyclohexane, acetone, butyl acetate, etc.) could be predicted. The prediction accuracy was somewhat better than that of PRASOGFV although the accuracy of the PVT calculation for polymers did not almost influence the prediction accuracy of the vapor-liquid equilibria for binary solutions at normal pressure. Introduction GE
rules1,2
Equation of state (EOS) mixing have been used for calculating phase equilibria, and models have been proposed to calculate the vapor-liquid equilibria (VLE) for polymer solutions.3-8 The EOSs used have been PRSV9 and SRK10 equations. For polymers, however, calculated results of PVT are expected to require modifications, especially for high temperatures and pressures. The three-parameter Sako-Wu-Prausnitz (SWP) cubic EOS proposed by Sako et al.11 has the possibility of calculating PVT for polymers as well as VLE for polymer solutions with good accuracy, because the SWP equation contains a free-volume parameter. Recently, Tork et al.12,13 determined parameters for three polymers and discussed the effect of polydispersity of the polymer on the phase equilibrium calculations of polymer solutions. The calculated accuracy of specific volumes for the polymers in their work was about 0.40.5%. The application of EOS GE mixing rules to the SWP equation has not been considered thus far. This paper deals with the determination of three parameters in the SWP equation for polymers at temperatures up to 470 K and pressures up to 200 MPa. The solvent activities of 138 polymer solutions were predicted by the EOS GE mixing rule using ASOG-FV,14 and accuracy was compared with that of PRASOG-FV.6 SWP EOS The SWP equation can be derived from the following generalized van der Waals partition function Q as given by Prausnitz et al.:15
[ ] [ ] [ ( )]
1 v Q) N! Λ3
N
vf v
N
-E0 exp 2kT
N
[Qr,v]
N
(1)
* Corresponding author. Tel: 81-3-3259-0814. Fax: 81-33293-7572. E-mail:
[email protected].
where N is the number of molecules in total volume v at temperature T, Λ is the de Broglie wavelength depending only on the temperature and molecular mass, Vf is the free volume, k is Boltzmann’s constant, and E0 represents the potential field experienced by one molecule to attractive forces from all others. Here, the free volume and the potential field are represented using the approximation of van der Waals and SRK forms, respectively. Qr,v is factored into an internal part and an external part using Prigogine’s procedure:
Qr,v ) Qr,v(int)Qr,v(ext)
(2)
with Qr,v(ext) depending on density expressed by eq 3:
Qr,v(ext) ) (vf/v)c-1
(3)
where 3c is the total number of external degrees of freedom per molecule. By using the relation P ) kT(∂ ln Q/∂V)T,N, the SWP EOS can be derived as follows:11
P)
RT(v - b + bc) a v(v - b) v(v + b)
(4)
For the case that c equals unity, eq 4 reduces to the SRK equation.10 The pure polymer parameters aii, bi, and ci are given by eq 5a-c as
aii ) aii′r2
(5a)
bi ) bi′r
(5b)
ci ) ci′r
(5c)
where aii′, bi′, and ci′ are pure polymer parameters based on segments and r is the number of segments per molecule. Sako et al.11 and Tork et al.12,13 have cor-
10.1021/ie010463t CCC: $22.00 © 2002 American Chemical Society Published on Web 11/21/2001
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1095
Figure 1. Calculated results of PVT for PIB and PVAc.
Figure 2. Predicted results of solvent activities for cyclohexane (1) + PIB (2) and acetone (1) + PS (2) systems.
related the parameters of solvents as well as polymers with the van der Waals volume, molar refraction, and first ionization potential. Extension to Higher Temperatures and Pressures aii′, bi′, and ci′ in eq 5a-c have been determined using experimental PVT data.16-22 The polymers studied here were low-density polyethylene (LDPE), poly(-caprolactone) (PCL), poly(epichlorohydrine) (PECH), poly(methyl methacrylate) (PMMA), poly(o-methylstyrene) (PoMS), polypropylene (PP), poly(isobutylene) (PIB), polystyrene (PS), poly(vinyl acetate) (PVAc), and poly(vinyl chloride) (PVC). The objective function used was the deviation between the experimental and calculated specific volumes as given by eq 6: NDP
Fobj )
∑ k)1
(
)
vexp - vcal vexp
2
(6)
k
Table 1 shows the values of aii′, bi′, and ci′ for 10 polymers. The values of r have been obtained from the number-average molecular weight divided by the molecular weight of a segment in the polymer. These parameters cover temperatures from 308 to 491 K and pressures up to 200 MPa. Table 2 shows a comparison between experimental and calculated specific volumes using the determined parameters. Although the tem-
Table 1. Determined Parameters aii′, bi′, and ci′ in SWP EOS for Polymers polymera LDPE PCL PECH PMMA PoMS PP PIB PS PVAc PVC
temp [K] 394-444 373-421 333-413 367-432 412-470 353-393 326-376 388-468 308-373 373-423
pressure aii′ bi′ × 104 ci′ [MPa] [J m3 mol-2] [m3 mol-1] [m3 mol-1] ref 0-196 0-200 0-200 1-80 1-180 0-180 1-100 1-200 1-80 1-200
3.793 70.44 3.262 5.474 8.167 1.197 2.032 5.646 4.007 1.358
0.2613 0.8674 0.5707 0.6975 0.9677 0.3933 0.5180 0.8396 0.5836 0.3788
2.126 31.13 1.977 2.670 2.603 1.324 1.322 2.092 2.953 1.151
22 16 16 19 17 16 21 17 18 20
a LDPE ) low-density polyethylene, PCL ) poly(-caprolactone), PECH ) poly(epichlorohydrine), PMMA ) poly(methyl methacrylate), PoMS ) poly(o-methylstyrene), PP ) polypropylene, PIB ) poly(isobutylene), PS ) polystyrene, PVAc ) poly(vinyl acetate), and PVC ) poly(vinyl chloride).
perature dependence of parameters was not considered, the average absolute percent deviation was 0.06%. Calculated results using the original SWP equation11 including molar polarization, first ionization potential, and van der Waals volume as parameters gave much higher errors, as noted in Table 2. It is for this reason that the experimental PVT data were fitted with the only parameter ci′ for the original SWP equations. For some systems, calculation with the original SWP equation was not possible because of the lack of parameters. From Figure 1, the proposed parameters show an improved representation of the data.
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Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002
Table 2. Calculated Results of PVT Using Some Cubic EOSs |∆vs/vs|av. [%]a
a
polymer
this work
original SWP11
LDPE PCL PECH PMMA PoMS PP PIB PS PVAc PVC overall average
0.09 0.06 0.04 0.04 0.04 0.10 0.03 0.08 0.02 0.07 0.06
2.29 n.a. n.a. 3.17 n.a. 1.35 0.59 4.11 2.69 1.64 2.26
Acknowledgment K.T. acknowledges the financial support in part of Grand-in-Aid No. 11650789 (1999) from the Ministry of Education, Science and Culture of Japan. Nomenclature
b)
∑xibi
(7b)
a ) energy parameter in the EOS, J m3 mol-2 ai ) activity of component i b ) size parameter in the EOS, m3 mol-1 c ) parameter in the EOS, m3 mol-1 3c ) total number of external degrees of freedom per molecule GE ) excess Gibbs free energy, J mol-1 NDP ) number of data points Q ) van der Waals partition function R ) gas constant, J mol-1 K-1 T ) absolute temperature, K v ) molar volume, m3 mol-1 vf ) free volume, m3 mol-1 vs ) specific volume, m3 kg-1
c)
∑xici
(7c)
Literature Cited
|∆vs/vs| ) 100/NDP∑|vs,exp - vs,cal|/vs,exp. n.a. ) not available.
Prediction of Solvent Activities for Polymer + Solvent Systems For predictions, the mixing rule was used as follows:
a ) b
aii
∑xi bi +
for 10 polymers. Accuracy of predictions of VLE in polymer solutions using the GE mixing rule and the ASOG-FV model was applied, and it was found that the results were almost the same as those using PRASOGFV.
[
E RT G0 + q RT
∑xi ln
( )] b bi
(7a)
where q ) -0.646 63. Equation 7a-c is the so-called zero-pressure GE mixing rule.1,2,5 VLE in polymer solutions can be evaluated using GE models. The excess Gibbs free energy can be predicted from group solution models such as ASOG-FV,14 UNIFACFV,23 and the entropic-FV24 models. In this study, GE0 was predicted from ASOG-FV. For the pure solvent parameters, ci was set equal to unity and aii and bi were obtained using the generalized equation10 for SRK with critical temperature, critical pressure, and an acentric factor. The ASOG group pair parameters used were those reported by Tochigi et al.25 The number of data sets was 138 composed of 4 polymers (PMMA, PIB, PS, and PVAc) and 13 solvents (benzene, toluene, m-xylene, ethylbenzene, cyclohexane, pentane, hexane, octane, acetone, methyl ethyl ketone, diethyl ketone, propyl acetate, and butyl acetate). Prediction Results The overall average relative deviation was 11.5% for experimental26 and predicted solvent activities. Deviations for Peng-Robinson (PRASOG-FV) equations with the same mixing rule were 11.6%. Figure 2 shows a comparison between predicted and experimental values for cyclohexane + PIB and acetone + PS systems at 298.15 K. The accuracy of the PVT calculations for polymers at high temperatures and pressures was relatively insensitive to the predictions of VLE at low pressures for polymer solutions, as noted by Orbey and Sandler.1 Conclusions Simplified parameters were determined for the SWP equation that provide improved PVT for polymers at higher temperatures and pressures. The new parameters gave correlations of specific volumes to within 0.1%
(1) Orbey, N.; Sandler, S. I. Moleling Vapor-Liquid Equilibria; Cambridge University Press: Cambridge, 1998. (2) Fischer, K.; Gmehling, J. Further Development, Status and Results of the PSRK Method for the Prediction of Vapor-Liquid Equilibria and Gas Solubilities. Fluid Phase Equilib. 1996, 121, 185-206. (3) Orbey, N.; Sandler, S. I. Vapor-Liquid Equilibria of Polymer Solutions Using a Cubic Equation of State. AIChE J. 1994, 40, 1203-1209. (4) Kontogeorgis, G. M.; Harismiadis, V. I.; Fredenslund, Aa.; Tassios, D. P. Application of the van der Waals Equation of State to Polymers I. Correlation. Fluid Phase Equilib. 1994, 96, 65-92. (5) Zhong, C.; Masuoka, H. A New Mixing Rule for Cubic Equations of State and Its Application to Vapor-Liquid Equilibria of Polymer Solutions. Fluid Phase Equilib. 1996, 123, 59-69. (6) Tochigi, K. Prediction of Vapor-Liquid Equilibria in NonPolymer and Polymer Solutions Using an ASOG-Based Equation of State (PRASOG). Fluid Phase Equilib. 1998, 144, 59-68. (7) Tochigi, K.; Kurita, S.; Matumoto, T. Prediction of PVT and VLE in Polymer Solutions Using a Cubic-Perturbed Equation of State. Fluid Phase Equilib. 1999, 158, 313-320. (8) Orbey, H.; Bokis, C. P.; Chen, C.-C. Modeling of Phase Equilibrium in the Low-Density Polyethylene Process: The Sanchez-Lacombe, Statistical Associating Fluid Theory, and Polymer Soave-Redlich-Kwong Equations of State. Ind. Eng. Chem. Res. 1998, 37, 4481-4491. (9) Stryjek, R.; Vera, J. H. An Improved Peng-Robinson Equation of State for Pure Components and Mixtures. Can. J. Chem. Eng. 1996, 64, 323-331. (10) Soave, G. Equilibrium Constants from a Modified RedlichKwong Equation of State. Chem. Eng. Sci. 1972, 27, 1197-1203. (11) Sako, T.; Wu, A. H.; Prausnitz, J. M. A Cubic Equation of State for High-Pressure Phase Equilibria of Mixtures Containing Polymers and Volatile Fluids. J. Appl. Polym. Sci. 1989, 38, 18391858. (12) Tork, T.; Sadowski, G.; Arlt, W.; de Haan, A.; Krooshof, G. Modelling of High-Pressure Phase Equilibria Using the SakoWu-Prausnitz Equation of State. I. Pure-Components and Heavy n-Alkanes Solutions. Fluid Phase Equilib. 1999, 163, 61-77. (13) Tork, T.; Sadowski, G.; Arlt, W.; de Haan, A.; Krooshof, G. Modelling of High-Pressure Phase Equilibria Using the SakoWu-Prausnitz Equation of State. II. Vapor-Liquid Equilibria and Liquid-Liquid Equilibria in Polyolefin Systems. Fluid Phase Equilib. 1999, 163, 79-98.
Ind. Eng. Chem. Res., Vol. 41, No. 5, 2002 1097 (14) Choi, J. S.; Tochigi, K.; Kojima, K. Measurement and Correlation of Vapor-Liquid Equilibria in Polymer Solutions Containing Polystyrene with Polymer ASOG. Fluid Phase Equilib. 1995, 111, 143-156. (15) Prausnitz, J. M.; Lichtenthaler, R. N.; de Azwvedo, E. G. Molecular Thermodynamics of Fluid-Phase Equilibria, 2nd ed.; Prentice-Hall: Englewood Cliffs, NJ, 1986. (16) Rodgers, A. A. Pressure-Volume-Temperature Relationships for Polymeric Liquids: A Review of Equations of State and Their Characteristic Parameters for 56 Polymers. J. Appl. Polym. Sci. 1993, 48, 1061-1080. (17) Quach, A.; Simha, R. Pressure-Volume-Temperature Properties and Transitions of Amorphous Polymers; Polystyrene and Poly(o-methylstyrene). J. Appl. Phys. 1971, 42, 4592-4606. (18) McKinney, J. E.; Goldstein, M. J. Res. Natl. Bur. Stand., Sect. A 1974, 78A, 331. (19) Olabisi, O.; Simha, R. Pressure-Volume-Temperature Studies of Amorphous and Crystalizable Polymers. I. Experimental. Macromolecules 1975, 8, 206. (20) Kim, C. K. Ph.D. Thesis, The University of Texas at Austin, Austin, TX, 1992. (21) Danner, R. P.; High, M. S. Handbook of Polymer Solution Thermodynamics; AIChE: New York, 1993.
(22) Zoller, P.; Walsh, D. Standard Pressure-Volume-Temperature Data for Polymers; Technomic Publishing Co.: Lancaster, PA, 1995. (23) Oishi, T.; Prausnitz, J. M. Estimation of Solvent Activities in Polymer Solutions Using a Group-Contribution Method. Ind. Eng. Chem. Process Des. Dev. 1978, 17, 333-339. (24) Elbro, H. S.; Fredenslund, Aa.; Rasmussen, P. A New Simple Equation for the Prediction of Solvent Activities in Polymer Solutions. Macromolecules 1990, 23, 4707-4714. (25) Tochigi, K.; Tiegs, D.; Gmehling, J.; Kojima, K. Determination of New ASOG Parameters. J. Chem. Eng. Jpn. 1990, 23, 453-463. (26) Hao, W.; Elbro, H. S.; Alessi, P. Polymer Solution Data Collection; DECHEMA Chemistry Data Series: Frankfurt, Germany, 1992; Vol. XIV, Part 1.
Received for review May 21, 2001 Revised manuscript received September 25, 2001 Accepted September 28, 2001 IE010463T
