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(Liquid + liquid) equilibria of binary polymer solutions using a free-volume UNIQUAC-NRF model
J. Chem. Thermodynamics 38 (2006) 923–928 www.elsevier.com/locate/jct
(Liquid + liquid) equilibria of binary polymer solutions using a free-volume UNIQUAC-NRF model H.R. Radfarnia a, C. Ghotbi b
a,*
, V. Taghikhani a, G.M. Kontogeorgis
b
a Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran Centre for Phase Equilibria and Separation Processes (IVC–SEP), Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark
Received 31 May 2005; received in revised form 5 October 2005; accepted 8 October 2005 Available online 17 November 2005
Abstract In this work, a modified free-volume (FV) model based on the UNIQUAC-Nonrandom factor (UNIQUAC-NRF) model developed by Haghtalab and Asadollahi was proposed. While the combinatorial part of the proposed model for activity coefficient takes the same form as that of the entropic free-volume (entropic-FV) model, the residual part is similar to that of the UNIQUAC-NRF model. The proposed model, i.e., the FV-UNIQUAC-NRF model overcomes the main shortcoming of the original UNIQUAC-NRF model in predicting the lower critical solution temperature (LCST) for polymer solutions. The appearance of the LCST is believed to be attributed to the existence of the free volume differences between polymer and solvent molecules. Thus, the models without considering such differences fail to predict the LCST behavior of polymer solutions. The proposed model was applied to correlate the experimental data of (liquid + liquid) equilibria (LLE) for a number of binary polymer solutions at various temperatures. The values for the binary characteristic energy parameters for the proposed model and the FV-UNIQUAC model along with their average relative deviations from the experimental data were reported. It should be stated that the binary polymer solutions studied in this work were considered as monodisperse. The results obtained from the FV-UNIQUAC-NRF model were compared with those obtained from the FV-UNIQUAC model. The results of the proposed model show that the FV-UNIQUAC-NRF model can accurately correlate the experimental data for LLE of polymer solutions studied in this work. Also the error produced from the FV-UNIQUAC-NRF model show the slightly better accuracy in comparison with that from the FV-UNIQUAC model. The clear advantage of the proposed model, contrary to the original UNIQUAC-NRF model, is its capability in predicting the LCST for binary polymer solutions. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Free-volume; Polymer solutions; (Liquid + liquid) equilibria; Phase behavior; UNIQUAC-NRF model
1. Introduction Thermodynamics of polymer solutions is important in many practical applications such as polymerization, devolatilization and the incorporation of plasticizers and other additives. Polymer solutions often exhibit (liquid + liquid) phase separation, which occurs at low polymer concentrations and it depends on temperature, pressure, molecular weight and molecular weight distribution of the polymer in solution. *
Corresponding author. Tel.: +98 21 600 5819; fax: +98 21 602 2853. E-mail address: [email protected] (C. Ghotbi).
0021-9614/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2005.10.005
The upper or lower critical solution temperatures (UCST or LCST) or both of them can be observed in polymer solutions. While existence of the UCST is usually justified by intermolecular forces among molecules, interpretation of the LCST can be more difficult. In generally, the LCST in a polymer solution occurs when an unfavorable entropy effect, negative entropy of mixing, overcomes the negative enthalpies of mixing. The LCST is associated with large differences in thermal expansion of solvent and solute or difference in free-volume percentage between polymer and solvent [1–3]. On the other hand, the appearance of the LCST is attributed to the existence of the free volume differences between polymer and solvent molecules.
924
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928
Thus, the models without considering such differences fail to predict the LCST behavior of polymer solutions. Freevolume (FV) is normally defined as the volume available to the center of mass of a single molecule as it moves about the system while the positions of all other molecules remain fixed. For LLE of polymer solutions, the compressibility or free-volume percentage of polymers in comparison with the low molecular weight compounds must be considered. The most widely used lattice model, the Flory–Huggins theory [4], cannot describe the lower critical solution temperature (LCST) behavior of polymer solutions. Elbro et al. [5] proposed a Flory–Huggins based equation for the entropy of mixing considering the free-volume effects in the combinatorial part. After combining it with the energy contribution taken from the original UNIQUAC model [6] for the free energy of mixing, accurate correlation of (vapor + liquid) equilibria (VLE) of nearly athermal polymer solutions was achieved. Many recently proposed models attempt to improve the description of free-volume effects in polymer solutions [7–9]. Moreover, the entropicFV type model coupled with UNIQUAC residual term was recently used to correlate various types of LLE in polymer systems [10–12]. Recently, using the concept of the NRTL-NRF model [13], Haghtalab and Asadollahi [14] developed the UNIQUAC-NRF model and applied it to (polymer + polymer) systems. The main shortcoming of the model of Haghtalab and Asadollahi is its failure to predict the lower critical solution temperatures (LCST). In this work, the UNIQUAC-NRF model was modified considering the free-volume effects. The proposed model was applied to correlate the experimental data of LLE for binary polymer solutions at various temperatures. The results obtained from the FV-UNIQUAC-NRF model were compared with those obtained from the FV-UNIQUAC model. 2. Theory In the proposed model, the activity coefficient is considered as a sum of two contributions. The expression for the activity coefficient of component i in a polymer solution is given by the following equation: ln ci ¼ ln ccomb þ ln cresid ; i i
ð1Þ
represents the combinatorial contribution to where ln ccomb i the activity coefficient and is directly taken from the entropic-FV model [5]. The second term in equation (1), ln cresid , i is the residual part of the UNIQUAC-NRF model and takes the same form as that in the UNIQUAC-NRF model [14]. In equation (1), the combinatorial part can be expressed according to the following equation: /FV /FV comb i i ln ci ¼ 1 þ ln ; ð2Þ xi xi
and xi are the free-volume and mole fractions of where /FV i component i, respectively. The free-volume fraction is presented as xi V FV ¼ P i FV ; /FV i j xj V j
ð3Þ
is the free-volume for component i and can be where V FV i related to the van der Waals volume as ¼ V i V vdW ; V FV i i
ð4Þ
vdW i
Vi and V are the molar and the van der Waals volumes, respectively. The residual part of the proposed model taken from the UNIQUAC-NRF model can be expressed as [14] 9 8 > > > > < X X Cij Cji = resid þ lnci ¼ qi 1 þ lnCii hj Cij þ ð1 hi Þ hj ln > Cii Cjj > > > j¼1 j¼1 ; : j6¼i 9 8 > > > > < 1X X Ckl Clk = qi ; hk hl ln ð5Þ > 2 k¼1 l¼1 Ckk Cll > > > ; : k6¼i l6¼k;2
where hi and C represent the surface area fraction of component i and the nonrandomness factor, respectively, and expressed using the following relations: xi q hi ¼ P i ; ð6Þ j x j qj sij Cij ¼ P ; ð7Þ k hk skj where q is surface area parameter and sij is the binary interaction parameters of i and j molecules and given by the following equation: a ij sij ¼ exp . ð8Þ RT In equation (8), aij is the binary characteristic energy parameter between i and j molecules. Also, T and R refer to temperature and gas universal constant, respectively. In this work, the binary characteristic energy parameters are assumed to be linearly temperature dependent according to the following relation: aij ¼ aij;1 þ aij;2 ðT T 0 Þ;
ð9Þ
where T0 is a reference temperature equal to 298.15 K. 3. Results and discussion The values for the surface area parameters, the density and the van der Waals volumes of the solvents and polymers are needed in use of the proposed model. The values for the density of solvents and polymers studied in this work were obtained from the values reported in the literature as well as from the Tait equation [15–17], respectively. The data for the van der Waals volumes and surface area parameters are taken from Bondi values [18]. The density
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928
of PEG is estimated using the group contribution method proposed by Elbro et al. [19]. The adjustable characteristic energy parameters of the various FV-models studied were
925
regressed using the binary experimental LLE data available in the literature [17] by minimizing the following objective function using the Nelder–Mead simplex method [20]:
TABLE 1 Comparison of the FV-UNIQUAC and the FV-UNIQUAC-NRF models along with the percent of average relative deviation (AAD) between the calculated and experimental values of LLE for binary polymer solutionsa [17] System
HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-pentane HDPE/n-pentane HDPE/n-pentane HDPE/n-butylacetate HDPE/n-butylacetate HDPE/n-hexane HDPE/n-octane PIB/benzene PIB/diisobutyl ketone PMMA/4-heptanone BR/n-octane BR/n-hexane PS/benzene PS/cyclohexane PS/methylcyclohexane PS/methylcyclohexane PS/methylcyclohexane PEG/tert-butylacetate PEG/water
Mn Æ 103/ (g Æ mol1) 10 36.7
Mw Æ 103/ (g Æ mol1)
Mv Æ 103/ (g Æ mol1)
49.3 76.8 93.5 14.3 97.2 204.9 13.6 64
97.7
135.9 93.5 72 22.7 36.5 44.5 98.9 37 200 17.3 20.2 109
8 2.18
2.29
T/K
456 458 462 461 382 420 416 406 433 406 498 540 287 267 302 262 538 295 289 293 313 317 436
to to to to to to to to to to to to to to to to to to to to to to to
Overall deviation
Np 460 470 464 463 390 426 419 524 503 411 500 544 292 272 449 401 540 297 297 299 323 471 510
5 6 5 5 6 5 4 10 12 5 5 9 10 9 10 10 5 13 8 9 10 10 16
102DAAD FV-UNIQUAC
FV-UNIQUAC-NRF
3.5 5.3 10.4 0.03 2.8 4.5 0.1 9.8 27.8 6.9 14.2 2.9 5.4 18.0 12.8 4.1 1.8 3.0 1.7 2.5 2.9 16.4 21.9
3.6 4.6 11.7 1.4 1.2 0.41 2.9 8.8 26.7 6.1 12.0 1.3 5.3 18.3 10.5 5.8 1.4 2.8 1.4 1.2 5.6 7.5 17.2
7.8
6.9
a
HDPE, high-density polyethylene; PIB, polyisobutylene; BR, polybutadiene; PS, polystyrene; PMMA, poly(methyl methaacrylate); PEG, polyethylene glycol.
TABLE 2 Values for the temperature dependent binary characteristic energy parameters of the FV-UNIQUAC-NRF modela System
a12,1/(J Æ mol1)
a12,2/(J Æ mol1)
a21,1/(J Æ mol1)
a21,2/(J Æ mol1)
HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-pentane HDPE/n-pentane HDPE/n-pentane HDPE/n-butylacetate HDPE/n-butylacetate HDPE/n-hexane HDPE/n-octane PIB/benzene PIB/diisobutyl ketone PMMA/4-heptanone BR/n-octane BR/n-hexane PS/benzene PS/cyclohexane PS/methylcyclohexane PS/methylcyclohexane PS/methylcyclohexane PEG/tert-butylacetate PEG/water
570.5784 252.5359 349.3144 162.0818 248.7292 256.8985 143.2008 187.8951 107.7899 411.1333 5708.687 2024.3144 639.5096 3205.3214 1241.0693 337.3930 144.8773 94.4621 112.4160 99.5582 25.4496 658.0568 242.9925
47.3973 2.9245 1.9188 3.3792 4.3096 7.0348 50.0158 2.3634 0.4095 3.1405 25.6469 3.1538 12.2012 91.6133 11.8028 3.1243 0.6161 68.0785 21.1808 42.0967 16.6354 10.6819 2.0140
3677.3727 5460.0661 400.8595 378.8029 590.4235 631.9411 244.5626 1312.2017 1396.5916 547.5013 6193.2943 2239.9921 1432.7551 3989.4134 2710.2729 1211.8783 1203.6314 1365.6570 1633.9565 1681.7932 1822.1294 2276.8341 15476.3561
41.5085 42.4944 5.2439 6.0590 9.4976 6.7520 4.3363 1.6299 0.0567 5.0233 38.3986 0.4515 13.8663 73.4392 29.3786 0.5133 1.5898 38.5932 23.6655 21.7344 10.5303 28.2787 1.2719
a
1 and 2 refer to polymer and solvent components respectively.
926
OF ¼
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928
X X
2 wexp wcal ; i i
ð10Þ
n exp n comp
and wcal are the experimental and calculated where wexp i i weight fractions, respectively, for all components i, and n exp and n comp are the number of experimental data and components, respectively. Table 1 presents the absolute average deviation DAAD of the results obtained from the LLE experimental data of polymer solutions in the temperature range studied in this work for FV-UNIQUAC-NRF and FV-UNIQUAC models. The absolute average deviation is defined as the following relation: ! 1 X wexp wcal i i DAAD ¼ ð11Þ exp . N wi i As seen from table 1 for most cases the accuracy of the proposed model, i.e., the FV-UNIQUAC-NRF to correlate the experimental data are relatively superior to that of the FV-UNIQUAC model. Tables 2 and 3 give the values for the regressed parameters for both the FV-UNIQUAC-NRF and FV-UNIQUAC models. As previously mentioned the binary characteristic energy parameters are considered to be linearly temperature dependent. Figures 1 and 2 show the comparative results obtained from the FV-UNIQUAC-NRF and FV-UNIQUAC models in correlating the LLE data for HDPE (high-density polyethylene)/n-octane and PEG (polyethylene glycol)/tert-butylacetate solutions. As seen from these figures the proposed model can more accurately correlate the experimental data comparing to the original FV-UNIQUAC model.
FIGURE 1. Comparison of the results obtained from the FVUNIQUAC-NRF and FV-UNIQUAC models with the experimental data of LLE for HDPE (Mw = 93,500 g Æ mol1) in n-octane: (——) FVUNIQUAC-NRF model; (– – –) FV-UNIQUAC model; and (d) experimental data [17].
Figures 3 to 6 show the different phase diagrams obtained from the FV-UNIQUAC-NRF model in comparison with the experimental data of LLE for (polymer + solvent) systems as HDPE/n-pentane, PS (polystyrene)/ methylcyclohexane with mass-average molar mass (Mw) of polymer 17,300 g Æ mol1, PIB (polyisobutylene)/benzene with viscosity-average molar mass (Mv) of polymer 72,000 g Æ mol1 and PEG/water with mean molar mass (Mn) of polymer 2180 g Æ mol1. As can be seen from these figures the results of the proposed model are in good agreement with the experimental data in all cases containing the UCST, LCST and closed-loop behaviors. Closed loop behavior can be usually observed in aqueous polymer solu-
TABLE 3 Values for the temperature dependent binary characteristic energy parameters of the FV-UNIQUAC model System
a12,1/(J Æ mol1)
a12,2/(J Æ mol1)
a21,1/(J Æ mol1)
a21,2/(J Æ mol1)
HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-pentane HDPE/n-pentane HDPE/n-pentane HDPE/n-butylacetate HDPE/n-butylacetate HDPE/n-hexane HDPE/n-octane PIB/benzene PIB/diisobutyl ketone PMMA/4-heptanone BR/n-octane BR/n-hexane PS/benzene PS/cyclohexane PS/methylcyclohexane PS/methylcyclohexane PS/methylcyclohexane PEG/tert-butylacetate PEG/water
261.8223 377.4662 115.6985 34.9917 80.9070 34.9934 380.3499 380.0192 368.8517 380.3761 2025.4991 2025.5025 362.0326 3262.8180 755.7703 385.3235 363.0744 384.9448 65.7785 54.1318 761.1950 727.8382 3314.1271
16.6675 4.9341 6.4221 0.0143 7.5236 0.1243 3.5042 7.5312 6.7169 6.6465 3.9952 0.3909 2.041 45.8375 17.4384 3.2123 0.5295 11.0081 9.5239 8.3283 21.8638 33.1310 33.2290
0.9487 1803.3079 224.8412 109.0072 38.5397 109.0058 544.0734 544.3498 554.3294 544.0565 2239.9679 2239.9977 560.6083 1071.0790 466.937 538.8958 559.6433 539.1224 103.0996 117.4752 458.5633 505.0769 60.2378
20.7064 15.1280 5.6955 0.3001 3.4344 0.2018 9.7247 9.5337 6.3139 7.6903 3.6976 0.4583 0.8466 11.0525 8.9982 3.5194 0.3375 6.0509 5.4613 11.2467 15.9794 9.5741 35.6827
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928 490
927
297
470
296
450
295 294
410
T/K
T/K
430
390 370
293 292
350
291
330
290 310 0
0.05
0.1
0.15
0.2
0.25
289
wp
0
FIGURE 2. Comparison of the results obtained from the FV-UNIQUAC-NRF and FV-UNIQUAC models with the experimental data of LLE for PEG (Mn = 8000 g Æ mol1) in tert-butylacetate system: (——) FV-UNIQUAC-NRF model; (– – –) FV-UNIQUAC model; and (d) experimental data [17].
0.1
0.2
0.3
0.4
0.5
wp FIGURE 4. Comparison of the results obtained from the FV-UNIQUAC-NRF model with the experimental data of LLE for PS (Mw = 17,300 g Æ mol1) in n-methylcyclohexane: (——) FV-UNIQUACNRF model; (d) experimental data [17]. 543.5
430 425
543
420
542.5 415
T/K
T/K
410 405
542 541.5
400 395
541
390
540.5
385
540
380 0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.14
0.01
0.02
0.03
wp FIGURE 3. Comparison of the results obtained from the FV-UNIQUAC-NRF model with the experimental data of LLE for HDPE in npentane with different molecular weights: (——) FV-UNIQUAC-NRF model, Mw of polymer = 97,200 g Æ mol1; (– – –) FV-UNIQUAC-NRF model, Mw = 14,300 g Æ mol1; (r) experimental data, Mw = 97,200 g Æ mol1 [17]; and (d) experimental data, Mw = 14,300 g Æ mol1 [17].
0.04
0.05
0.06
wp FIGURE 5. Comparison of the results obtained from the FV-UNIQUAC-NRF model with the experimental data of LLE for PIB (Mv = 72,000 g Æ mol1) in benzene: (——) FV-UNIQUAC-NRF model; (d) experimental data [17]. 520 510 500 490
T/K
tions. It should be mentioned that for all solutions the polymers studied in this work were considered to be monodispersed and their polydispersities were neglected. To obtain the regressed parameters based on the available experimental data, number average molecular weight, mass average molecular weight, and viscosity average molecular weight have been used, respectively.
480 470 460 450 440
4. Conclusions
430 0
In this work, a free-volume model for polymer solutions was proposed based on the UNIQUAC-NRF model. Using this model, the phase behavior of various binary LLE of polymer solutions was studied. Good agreement between the calculated and experimental data is observed
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
wp
FIGURE 6. Comparison of the results obtained from the FV-UNIQUAC-NRF model with the experimental data of LLE for PEG (Mn = 2180 g Æ mol1) in water: (——) FV-UNIQUAC-NRF model; (d) experimental data [17].
928
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928
with the FV-UNIQUAC-NRF model. Also the results obtained from the proposed model can correlate the experimental data of LLE for binary polymer solutions as accurately as those obtained using the FV-UNIQUAC model. The results show that unlike the original UNIQUACNRF model, the proposed model can also be used to estimate the LCST for binary polymer solutions. This is, in fact, the clearest advantage of this research work. References [1] [2] [3] [4]
B.H. Chang, Y.C. Bae, Polymer 39 (1998) 6449. G. Delmas, D. Patterson, T. Somcynsky, J. Polym. Sci. 57 (1962) 79. B.C. Lee, R.P. Danner, AIChE J. 42 (1996) 3223. P.J. Flory, Principles of Polymer Chemistry, seventh ed., Cornell University Press, Ithaca, NY, 1953. [5] H.S. Elbro, Aa. Fredenslund, P. Rasmussen, Macromolecules 23 (1990) 4707. [6] D.S. Abrams, J.M. Prausnitz, AIChE J. 21 (1975) 116. [7] G.M. Kontogeorgis, Aa. Fredenslund, D.P. Tassios, Ind. Eng. Chem. Res. 32 (1993) 362.
[8] I.A. Kouskoumvekaki, M.L. Michelsen, G.M. Kontogeorgis, Fluid Phase Equilibr. 202 (2002) 323. [9] G.M. Kontogeorgis, G.I. Nikolopoulos, Aa. Fredenslund, D.P. Tassios, Fluid Phase Equilibr. 127 (1997) 103. [10] G. Bogdanic, J. Vidal, Fluid Phase Equilibr. 173 (2000) 241. [11] G.D. Pappa, E.C. Voutsas, D.P. Tassios, Ind. Eng. Chem. Res. 40 (2001) 4654. [12] H.R. Radfarnia, V. Taghikhani, C. Ghotbi, M.K. Khoshkbarchi, J. Chem. Thermodyn. 36 (2004) 409. [13] A. Haghtalab, J.H. Vera, AIChE J. 34 (1988) 803. [14] A. Haghtalab, M.A. Asadollahi, Fluid Phase Equilibr. 171 (2000) 77. [15] C. Yaws, Chemical Properties Handbook, McGraw-Hill, Lamar University, Beaumont, TX, 1999. [16] P. Rodgers, J. Appl. Polym. Sci. 48 (1993) 1061. [17] R.P. Danner, M.S. High, Handbook of Polymer Solution Thermodynamics, DIPPR Project, AIChE, New York, 1993. [18] A. Bondi, Physical Properties of Molecular Crystals, Liquid and Glasses, Wiley, New York, 1968. [19] H.S. Elbro, A. Fredenslund, P. Rasmussen, Ind. Eng. Chem. Res. 30 (1991) 2576. [20] J.A. Nelder, R.A. Mead, Comput. J. 7 (1965) 308.
JCT 05/137
(Liquid + liquid) equilibria of binary polymer solutions using a free-volume UNIQUAC-NRF model H.R. Radfarnia a, C. Ghotbi b
a,*
, V. Taghikhani a, G.M. Kontogeorgis
b
a Department of Chemical and Petroleum Engineering, Sharif University of Technology, Tehran, Iran Centre for Phase Equilibria and Separation Processes (IVC–SEP), Department of Chemical Engineering, Technical University of Denmark, DK-2800 Lyngby, Denmark
Received 31 May 2005; received in revised form 5 October 2005; accepted 8 October 2005 Available online 17 November 2005
Abstract In this work, a modified free-volume (FV) model based on the UNIQUAC-Nonrandom factor (UNIQUAC-NRF) model developed by Haghtalab and Asadollahi was proposed. While the combinatorial part of the proposed model for activity coefficient takes the same form as that of the entropic free-volume (entropic-FV) model, the residual part is similar to that of the UNIQUAC-NRF model. The proposed model, i.e., the FV-UNIQUAC-NRF model overcomes the main shortcoming of the original UNIQUAC-NRF model in predicting the lower critical solution temperature (LCST) for polymer solutions. The appearance of the LCST is believed to be attributed to the existence of the free volume differences between polymer and solvent molecules. Thus, the models without considering such differences fail to predict the LCST behavior of polymer solutions. The proposed model was applied to correlate the experimental data of (liquid + liquid) equilibria (LLE) for a number of binary polymer solutions at various temperatures. The values for the binary characteristic energy parameters for the proposed model and the FV-UNIQUAC model along with their average relative deviations from the experimental data were reported. It should be stated that the binary polymer solutions studied in this work were considered as monodisperse. The results obtained from the FV-UNIQUAC-NRF model were compared with those obtained from the FV-UNIQUAC model. The results of the proposed model show that the FV-UNIQUAC-NRF model can accurately correlate the experimental data for LLE of polymer solutions studied in this work. Also the error produced from the FV-UNIQUAC-NRF model show the slightly better accuracy in comparison with that from the FV-UNIQUAC model. The clear advantage of the proposed model, contrary to the original UNIQUAC-NRF model, is its capability in predicting the LCST for binary polymer solutions. Ó 2005 Elsevier Ltd. All rights reserved. Keywords: Free-volume; Polymer solutions; (Liquid + liquid) equilibria; Phase behavior; UNIQUAC-NRF model
1. Introduction Thermodynamics of polymer solutions is important in many practical applications such as polymerization, devolatilization and the incorporation of plasticizers and other additives. Polymer solutions often exhibit (liquid + liquid) phase separation, which occurs at low polymer concentrations and it depends on temperature, pressure, molecular weight and molecular weight distribution of the polymer in solution. *
Corresponding author. Tel.: +98 21 600 5819; fax: +98 21 602 2853. E-mail address: [email protected] (C. Ghotbi).
0021-9614/$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2005.10.005
The upper or lower critical solution temperatures (UCST or LCST) or both of them can be observed in polymer solutions. While existence of the UCST is usually justified by intermolecular forces among molecules, interpretation of the LCST can be more difficult. In generally, the LCST in a polymer solution occurs when an unfavorable entropy effect, negative entropy of mixing, overcomes the negative enthalpies of mixing. The LCST is associated with large differences in thermal expansion of solvent and solute or difference in free-volume percentage between polymer and solvent [1–3]. On the other hand, the appearance of the LCST is attributed to the existence of the free volume differences between polymer and solvent molecules.
924
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928
Thus, the models without considering such differences fail to predict the LCST behavior of polymer solutions. Freevolume (FV) is normally defined as the volume available to the center of mass of a single molecule as it moves about the system while the positions of all other molecules remain fixed. For LLE of polymer solutions, the compressibility or free-volume percentage of polymers in comparison with the low molecular weight compounds must be considered. The most widely used lattice model, the Flory–Huggins theory [4], cannot describe the lower critical solution temperature (LCST) behavior of polymer solutions. Elbro et al. [5] proposed a Flory–Huggins based equation for the entropy of mixing considering the free-volume effects in the combinatorial part. After combining it with the energy contribution taken from the original UNIQUAC model [6] for the free energy of mixing, accurate correlation of (vapor + liquid) equilibria (VLE) of nearly athermal polymer solutions was achieved. Many recently proposed models attempt to improve the description of free-volume effects in polymer solutions [7–9]. Moreover, the entropicFV type model coupled with UNIQUAC residual term was recently used to correlate various types of LLE in polymer systems [10–12]. Recently, using the concept of the NRTL-NRF model [13], Haghtalab and Asadollahi [14] developed the UNIQUAC-NRF model and applied it to (polymer + polymer) systems. The main shortcoming of the model of Haghtalab and Asadollahi is its failure to predict the lower critical solution temperatures (LCST). In this work, the UNIQUAC-NRF model was modified considering the free-volume effects. The proposed model was applied to correlate the experimental data of LLE for binary polymer solutions at various temperatures. The results obtained from the FV-UNIQUAC-NRF model were compared with those obtained from the FV-UNIQUAC model. 2. Theory In the proposed model, the activity coefficient is considered as a sum of two contributions. The expression for the activity coefficient of component i in a polymer solution is given by the following equation: ln ci ¼ ln ccomb þ ln cresid ; i i
ð1Þ
represents the combinatorial contribution to where ln ccomb i the activity coefficient and is directly taken from the entropic-FV model [5]. The second term in equation (1), ln cresid , i is the residual part of the UNIQUAC-NRF model and takes the same form as that in the UNIQUAC-NRF model [14]. In equation (1), the combinatorial part can be expressed according to the following equation: /FV /FV comb i i ln ci ¼ 1 þ ln ; ð2Þ xi xi
and xi are the free-volume and mole fractions of where /FV i component i, respectively. The free-volume fraction is presented as xi V FV ¼ P i FV ; /FV i j xj V j
ð3Þ
is the free-volume for component i and can be where V FV i related to the van der Waals volume as ¼ V i V vdW ; V FV i i
ð4Þ
vdW i
Vi and V are the molar and the van der Waals volumes, respectively. The residual part of the proposed model taken from the UNIQUAC-NRF model can be expressed as [14] 9 8 > > > > < X X Cij Cji = resid þ lnci ¼ qi 1 þ lnCii hj Cij þ ð1 hi Þ hj ln > Cii Cjj > > > j¼1 j¼1 ; : j6¼i 9 8 > > > > < 1X X Ckl Clk = qi ; hk hl ln ð5Þ > 2 k¼1 l¼1 Ckk Cll > > > ; : k6¼i l6¼k;2
where hi and C represent the surface area fraction of component i and the nonrandomness factor, respectively, and expressed using the following relations: xi q hi ¼ P i ; ð6Þ j x j qj sij Cij ¼ P ; ð7Þ k hk skj where q is surface area parameter and sij is the binary interaction parameters of i and j molecules and given by the following equation: a ij sij ¼ exp . ð8Þ RT In equation (8), aij is the binary characteristic energy parameter between i and j molecules. Also, T and R refer to temperature and gas universal constant, respectively. In this work, the binary characteristic energy parameters are assumed to be linearly temperature dependent according to the following relation: aij ¼ aij;1 þ aij;2 ðT T 0 Þ;
ð9Þ
where T0 is a reference temperature equal to 298.15 K. 3. Results and discussion The values for the surface area parameters, the density and the van der Waals volumes of the solvents and polymers are needed in use of the proposed model. The values for the density of solvents and polymers studied in this work were obtained from the values reported in the literature as well as from the Tait equation [15–17], respectively. The data for the van der Waals volumes and surface area parameters are taken from Bondi values [18]. The density
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928
of PEG is estimated using the group contribution method proposed by Elbro et al. [19]. The adjustable characteristic energy parameters of the various FV-models studied were
925
regressed using the binary experimental LLE data available in the literature [17] by minimizing the following objective function using the Nelder–Mead simplex method [20]:
TABLE 1 Comparison of the FV-UNIQUAC and the FV-UNIQUAC-NRF models along with the percent of average relative deviation (AAD) between the calculated and experimental values of LLE for binary polymer solutionsa [17] System
HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-pentane HDPE/n-pentane HDPE/n-pentane HDPE/n-butylacetate HDPE/n-butylacetate HDPE/n-hexane HDPE/n-octane PIB/benzene PIB/diisobutyl ketone PMMA/4-heptanone BR/n-octane BR/n-hexane PS/benzene PS/cyclohexane PS/methylcyclohexane PS/methylcyclohexane PS/methylcyclohexane PEG/tert-butylacetate PEG/water
Mn Æ 103/ (g Æ mol1) 10 36.7
Mw Æ 103/ (g Æ mol1)
Mv Æ 103/ (g Æ mol1)
49.3 76.8 93.5 14.3 97.2 204.9 13.6 64
97.7
135.9 93.5 72 22.7 36.5 44.5 98.9 37 200 17.3 20.2 109
8 2.18
2.29
T/K
456 458 462 461 382 420 416 406 433 406 498 540 287 267 302 262 538 295 289 293 313 317 436
to to to to to to to to to to to to to to to to to to to to to to to
Overall deviation
Np 460 470 464 463 390 426 419 524 503 411 500 544 292 272 449 401 540 297 297 299 323 471 510
5 6 5 5 6 5 4 10 12 5 5 9 10 9 10 10 5 13 8 9 10 10 16
102DAAD FV-UNIQUAC
FV-UNIQUAC-NRF
3.5 5.3 10.4 0.03 2.8 4.5 0.1 9.8 27.8 6.9 14.2 2.9 5.4 18.0 12.8 4.1 1.8 3.0 1.7 2.5 2.9 16.4 21.9
3.6 4.6 11.7 1.4 1.2 0.41 2.9 8.8 26.7 6.1 12.0 1.3 5.3 18.3 10.5 5.8 1.4 2.8 1.4 1.2 5.6 7.5 17.2
7.8
6.9
a
HDPE, high-density polyethylene; PIB, polyisobutylene; BR, polybutadiene; PS, polystyrene; PMMA, poly(methyl methaacrylate); PEG, polyethylene glycol.
TABLE 2 Values for the temperature dependent binary characteristic energy parameters of the FV-UNIQUAC-NRF modela System
a12,1/(J Æ mol1)
a12,2/(J Æ mol1)
a21,1/(J Æ mol1)
a21,2/(J Æ mol1)
HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-pentane HDPE/n-pentane HDPE/n-pentane HDPE/n-butylacetate HDPE/n-butylacetate HDPE/n-hexane HDPE/n-octane PIB/benzene PIB/diisobutyl ketone PMMA/4-heptanone BR/n-octane BR/n-hexane PS/benzene PS/cyclohexane PS/methylcyclohexane PS/methylcyclohexane PS/methylcyclohexane PEG/tert-butylacetate PEG/water
570.5784 252.5359 349.3144 162.0818 248.7292 256.8985 143.2008 187.8951 107.7899 411.1333 5708.687 2024.3144 639.5096 3205.3214 1241.0693 337.3930 144.8773 94.4621 112.4160 99.5582 25.4496 658.0568 242.9925
47.3973 2.9245 1.9188 3.3792 4.3096 7.0348 50.0158 2.3634 0.4095 3.1405 25.6469 3.1538 12.2012 91.6133 11.8028 3.1243 0.6161 68.0785 21.1808 42.0967 16.6354 10.6819 2.0140
3677.3727 5460.0661 400.8595 378.8029 590.4235 631.9411 244.5626 1312.2017 1396.5916 547.5013 6193.2943 2239.9921 1432.7551 3989.4134 2710.2729 1211.8783 1203.6314 1365.6570 1633.9565 1681.7932 1822.1294 2276.8341 15476.3561
41.5085 42.4944 5.2439 6.0590 9.4976 6.7520 4.3363 1.6299 0.0567 5.0233 38.3986 0.4515 13.8663 73.4392 29.3786 0.5133 1.5898 38.5932 23.6655 21.7344 10.5303 28.2787 1.2719
a
1 and 2 refer to polymer and solvent components respectively.
926
OF ¼
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928
X X
2 wexp wcal ; i i
ð10Þ
n exp n comp
and wcal are the experimental and calculated where wexp i i weight fractions, respectively, for all components i, and n exp and n comp are the number of experimental data and components, respectively. Table 1 presents the absolute average deviation DAAD of the results obtained from the LLE experimental data of polymer solutions in the temperature range studied in this work for FV-UNIQUAC-NRF and FV-UNIQUAC models. The absolute average deviation is defined as the following relation: ! 1 X wexp wcal i i DAAD ¼ ð11Þ exp . N wi i As seen from table 1 for most cases the accuracy of the proposed model, i.e., the FV-UNIQUAC-NRF to correlate the experimental data are relatively superior to that of the FV-UNIQUAC model. Tables 2 and 3 give the values for the regressed parameters for both the FV-UNIQUAC-NRF and FV-UNIQUAC models. As previously mentioned the binary characteristic energy parameters are considered to be linearly temperature dependent. Figures 1 and 2 show the comparative results obtained from the FV-UNIQUAC-NRF and FV-UNIQUAC models in correlating the LLE data for HDPE (high-density polyethylene)/n-octane and PEG (polyethylene glycol)/tert-butylacetate solutions. As seen from these figures the proposed model can more accurately correlate the experimental data comparing to the original FV-UNIQUAC model.
FIGURE 1. Comparison of the results obtained from the FVUNIQUAC-NRF and FV-UNIQUAC models with the experimental data of LLE for HDPE (Mw = 93,500 g Æ mol1) in n-octane: (——) FVUNIQUAC-NRF model; (– – –) FV-UNIQUAC model; and (d) experimental data [17].
Figures 3 to 6 show the different phase diagrams obtained from the FV-UNIQUAC-NRF model in comparison with the experimental data of LLE for (polymer + solvent) systems as HDPE/n-pentane, PS (polystyrene)/ methylcyclohexane with mass-average molar mass (Mw) of polymer 17,300 g Æ mol1, PIB (polyisobutylene)/benzene with viscosity-average molar mass (Mv) of polymer 72,000 g Æ mol1 and PEG/water with mean molar mass (Mn) of polymer 2180 g Æ mol1. As can be seen from these figures the results of the proposed model are in good agreement with the experimental data in all cases containing the UCST, LCST and closed-loop behaviors. Closed loop behavior can be usually observed in aqueous polymer solu-
TABLE 3 Values for the temperature dependent binary characteristic energy parameters of the FV-UNIQUAC model System
a12,1/(J Æ mol1)
a12,2/(J Æ mol1)
a21,1/(J Æ mol1)
a21,2/(J Æ mol1)
HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-heptane HDPE/n-pentane HDPE/n-pentane HDPE/n-pentane HDPE/n-butylacetate HDPE/n-butylacetate HDPE/n-hexane HDPE/n-octane PIB/benzene PIB/diisobutyl ketone PMMA/4-heptanone BR/n-octane BR/n-hexane PS/benzene PS/cyclohexane PS/methylcyclohexane PS/methylcyclohexane PS/methylcyclohexane PEG/tert-butylacetate PEG/water
261.8223 377.4662 115.6985 34.9917 80.9070 34.9934 380.3499 380.0192 368.8517 380.3761 2025.4991 2025.5025 362.0326 3262.8180 755.7703 385.3235 363.0744 384.9448 65.7785 54.1318 761.1950 727.8382 3314.1271
16.6675 4.9341 6.4221 0.0143 7.5236 0.1243 3.5042 7.5312 6.7169 6.6465 3.9952 0.3909 2.041 45.8375 17.4384 3.2123 0.5295 11.0081 9.5239 8.3283 21.8638 33.1310 33.2290
0.9487 1803.3079 224.8412 109.0072 38.5397 109.0058 544.0734 544.3498 554.3294 544.0565 2239.9679 2239.9977 560.6083 1071.0790 466.937 538.8958 559.6433 539.1224 103.0996 117.4752 458.5633 505.0769 60.2378
20.7064 15.1280 5.6955 0.3001 3.4344 0.2018 9.7247 9.5337 6.3139 7.6903 3.6976 0.4583 0.8466 11.0525 8.9982 3.5194 0.3375 6.0509 5.4613 11.2467 15.9794 9.5741 35.6827
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928 490
927
297
470
296
450
295 294
410
T/K
T/K
430
390 370
293 292
350
291
330
290 310 0
0.05
0.1
0.15
0.2
0.25
289
wp
0
FIGURE 2. Comparison of the results obtained from the FV-UNIQUAC-NRF and FV-UNIQUAC models with the experimental data of LLE for PEG (Mn = 8000 g Æ mol1) in tert-butylacetate system: (——) FV-UNIQUAC-NRF model; (– – –) FV-UNIQUAC model; and (d) experimental data [17].
0.1
0.2
0.3
0.4
0.5
wp FIGURE 4. Comparison of the results obtained from the FV-UNIQUAC-NRF model with the experimental data of LLE for PS (Mw = 17,300 g Æ mol1) in n-methylcyclohexane: (——) FV-UNIQUACNRF model; (d) experimental data [17]. 543.5
430 425
543
420
542.5 415
T/K
T/K
410 405
542 541.5
400 395
541
390
540.5
385
540
380 0
0.02
0.04
0.06
0.08
0.1
0.12
0
0.14
0.01
0.02
0.03
wp FIGURE 3. Comparison of the results obtained from the FV-UNIQUAC-NRF model with the experimental data of LLE for HDPE in npentane with different molecular weights: (——) FV-UNIQUAC-NRF model, Mw of polymer = 97,200 g Æ mol1; (– – –) FV-UNIQUAC-NRF model, Mw = 14,300 g Æ mol1; (r) experimental data, Mw = 97,200 g Æ mol1 [17]; and (d) experimental data, Mw = 14,300 g Æ mol1 [17].
0.04
0.05
0.06
wp FIGURE 5. Comparison of the results obtained from the FV-UNIQUAC-NRF model with the experimental data of LLE for PIB (Mv = 72,000 g Æ mol1) in benzene: (——) FV-UNIQUAC-NRF model; (d) experimental data [17]. 520 510 500 490
T/K
tions. It should be mentioned that for all solutions the polymers studied in this work were considered to be monodispersed and their polydispersities were neglected. To obtain the regressed parameters based on the available experimental data, number average molecular weight, mass average molecular weight, and viscosity average molecular weight have been used, respectively.
480 470 460 450 440
4. Conclusions
430 0
In this work, a free-volume model for polymer solutions was proposed based on the UNIQUAC-NRF model. Using this model, the phase behavior of various binary LLE of polymer solutions was studied. Good agreement between the calculated and experimental data is observed
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
0.45
wp
FIGURE 6. Comparison of the results obtained from the FV-UNIQUAC-NRF model with the experimental data of LLE for PEG (Mn = 2180 g Æ mol1) in water: (——) FV-UNIQUAC-NRF model; (d) experimental data [17].
928
H.R. Radfarnia et al. / J. Chem. Thermodynamics 38 (2006) 923–928
with the FV-UNIQUAC-NRF model. Also the results obtained from the proposed model can correlate the experimental data of LLE for binary polymer solutions as accurately as those obtained using the FV-UNIQUAC model. The results show that unlike the original UNIQUACNRF model, the proposed model can also be used to estimate the LCST for binary polymer solutions. This is, in fact, the clearest advantage of this research work. References [1] [2] [3] [4]
B.H. Chang, Y.C. Bae, Polymer 39 (1998) 6449. G. Delmas, D. Patterson, T. Somcynsky, J. Polym. Sci. 57 (1962) 79. B.C. Lee, R.P. Danner, AIChE J. 42 (1996) 3223. P.J. Flory, Principles of Polymer Chemistry, seventh ed., Cornell University Press, Ithaca, NY, 1953. [5] H.S. Elbro, Aa. Fredenslund, P. Rasmussen, Macromolecules 23 (1990) 4707. [6] D.S. Abrams, J.M. Prausnitz, AIChE J. 21 (1975) 116. [7] G.M. Kontogeorgis, Aa. Fredenslund, D.P. Tassios, Ind. Eng. Chem. Res. 32 (1993) 362.
[8] I.A. Kouskoumvekaki, M.L. Michelsen, G.M. Kontogeorgis, Fluid Phase Equilibr. 202 (2002) 323. [9] G.M. Kontogeorgis, G.I. Nikolopoulos, Aa. Fredenslund, D.P. Tassios, Fluid Phase Equilibr. 127 (1997) 103. [10] G. Bogdanic, J. Vidal, Fluid Phase Equilibr. 173 (2000) 241. [11] G.D. Pappa, E.C. Voutsas, D.P. Tassios, Ind. Eng. Chem. Res. 40 (2001) 4654. [12] H.R. Radfarnia, V. Taghikhani, C. Ghotbi, M.K. Khoshkbarchi, J. Chem. Thermodyn. 36 (2004) 409. [13] A. Haghtalab, J.H. Vera, AIChE J. 34 (1988) 803. [14] A. Haghtalab, M.A. Asadollahi, Fluid Phase Equilibr. 171 (2000) 77. [15] C. Yaws, Chemical Properties Handbook, McGraw-Hill, Lamar University, Beaumont, TX, 1999. [16] P. Rodgers, J. Appl. Polym. Sci. 48 (1993) 1061. [17] R.P. Danner, M.S. High, Handbook of Polymer Solution Thermodynamics, DIPPR Project, AIChE, New York, 1993. [18] A. Bondi, Physical Properties of Molecular Crystals, Liquid and Glasses, Wiley, New York, 1968. [19] H.S. Elbro, A. Fredenslund, P. Rasmussen, Ind. Eng. Chem. Res. 30 (1991) 2576. [20] J.A. Nelder, R.A. Mead, Comput. J. 7 (1965) 308.
JCT 05/137