- Email: [email protected]
Segment-based Eyring–Wilson viscosity model for polymer solutions
J. Chem. Thermodynamics 37 (2005) 445–448 www.elsevier.com/locate/jct
Segment-based Eyring–Wilson viscosity model for polymer solutions Rahmat Sadeghi
*
Department of Chemistry, Kurdistan University, Pasdaran Street, Sanandaj, Iran Received 1 October 2004; received in revised form 24 October 2004; accepted 25 October 2004 Available online 8 December 2004
Abstract A theory-based model is presented for correlating viscosity of polymer solutions and is based on the segment-based Eyring mixture viscosity model as well as the segment-based Wilson model for describing deviations from ideality. The model has been applied to several polymer solutions and the results show that it is reliable both for correlation and prediction of the viscosity of polymer solutions at different molar masses and temperature of the polymer. 2004 Elsevier Ltd. All rights reserved. Keywords: Segment-based; Eyring; Wilson; Viscosity; Polymer solutions
1. Introduction Knowledge of the viscosity of polymer solutions is important for practical and theoretical purposes. Viscosity of polymer solutions provides an invaluable type of data in polymer research, development and engineering. Furthermore, the simultaneous investigation of viscosity and volume effects on mixing can be a powerful tool for the characterization of the intermolecular interactions present in these mixtures. Also, knowledge of the dependence of viscosities of polymer solutions on composition is of great interest from a theoretical standpoint since it may lead to better understanding of their fundamental behaviour of polymer solutions. A segment-based model should be more physically realistic for large molecules when diffusion and flow are viewed to occur by a sequence of individual segment jumps into vacancies rather large jumps of the molecule into large vacancies. It has been found that the ratio of heats of activation decreases as molecules of normal paraffins get larger, suggesting that vacancies for jumps are smaller than the size of larger molecules [1]. Re*
Tel./fax: +98 871 6660075. E-mail address: [email protected].
0021-9614/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2004.10.009
cently, Novak et al. [2] developed a segment-based Eyring-NRTL viscosity model for mixtures containing polymers, which uses a combination of the segmentbased Eyring mixture viscosity model and the segment based NRTL model. In continuation of our investigation of the extension of the Wilson model to polymer solutions [3] and (polymer + electrolyte) solutions [4], in this paper, a segmentbased Eyring–Wilson viscosity model is presented for correlating and predicting the viscosity of polymer solutions. The utility of the model is demonstrated with successful representation of viscosity of several polymer solutions at different polymer molar masses and temperatures.
2. Theory The absolute rate theory approach of Eyring provides the following expression for the viscosity of a liquid mixture [1]: gV ¼ Nh expðDg =RT Þ,
ð1Þ
where g is the dynamic viscosity of the mixtures, h is Plancks constant, N is Avogadros number, V is the
446
R. Sadeghi / J. Chem. Thermodynamics 37 (2005) 445–448
molar volume of the mixture, Dg* is the molar Gibbs free energy of activation for the flow process, and T is the absolute temperature. In principal, equation (1) can be used to describe the viscosity of ideal mixtures as well as that of nonideal systems. For an ideal liquid mixture, in the viscosity sense, equation (1) can be written as gideal V ideal ¼ Nh expðDgideal =RT Þ,
ð2Þ
where Videal is the molar volume of the ideal mixture in the thermodynamic sense. The molar Gibbs free energy of activation for flow can be regarded as the sum of an ideal contribution and a correction or excess term. That is, Dg ¼ Dgideal þ gE ,
molar masses with a series of interaction parameters, which provides a predictive capability. Following Novak et al. [2], by application of the segment concept to the Eyring rate-based model and using the segmentbased polymer Wilson model [3] for describing deviations from ideality we have: X lnðgV Þ ¼ X i lnðgi V i Þ þ gE =RT , ð7Þ i
V ¼
X
gi ¼
X
ð8Þ
, ri,I gI
V i ¼
X
,
X
ri,I V i,I
!
Equation (4) clearly shows that the model considers the dynamic viscosity of real solutions as the result the contribution of an ideal term and an excess type. By considering the following relation for ideal mixtures: X lnðgV Þideal ¼ xi lnðgi V i Þ, ð5Þ
ð10Þ
ri,I ,
I
, ri,I xI
XX J
! r j, J x J ,
¼ C
X
X i ln
i
ð11Þ
j
ð4Þ gE =RT
ð9Þ
ri,I ,
X
I
Xi ¼
!
I
I
lnðgV Þ ¼ lnðgV Þideal þ gE =RT :
X
I
ð3Þ
where Dgideal is the ideal mixture contribution, at the same conditions of temperature, pressure, and composition for the real system per mole, and gE is the molar excess free energy of activation for flow. Combining equations (1)–(3) leads to the following expression:
X i V i ,
i
X
! X j H ji ,
ð12Þ
j
H ji ¼ expðEji =CRT Þ,
ð13Þ
Eji ¼ hji hii ,
ð14Þ
i
we have lnðgV Þ ¼
X
xi lnðgi V i Þ þ gE =RT :
ð6Þ
i
where xi, gi and Vi are mole fraction, viscosity and molar volume of component i, respectively. Extension of the concepts of classical thermodynamics to the viscous flow behaviour of liquid mixtures can be made by assuming equivalence between the Gibbs free energy of activation for flow and the equilibrium Gibbs free energy of mixing. Taking this direct relationship between the thermodynamics and the transport property, in this work the polymer Wilson model for excess Gibbs free energy developed in our previous paper [3], is used to account for the viscosity deviation from ideal behaviour. That is to say, the assumed equivalence may be dependent on the flow regime, i.e., the Re number of the flow system used. A segment-based model should be more physically realistic for large molecules when diffusion and flow are viewed to occur by a sequence of individual segment jumps into vacancies rather than large molecule jumps into large vacancies. Advantage of the segment-based models over conventional models for correlation of polymer solution experimental data is that, unlike the classical models, they can cover a wide range of polymer
where ri,I is the number of the segment i in the component I, Vi,I is the molar volume of the segment i in the component I, C is a parameter, which represents the effective coordination number in the system and hji is the enthalpy of interaction between j and i species. In the above relations the species i and j can be solvent molecules or segments and species I and J can be solvent or polymer molecules. The excess term in the Eyring– Wilson viscosity model, like the Wilson excess Gibbs energy expression, has some semiempirical basis, and the model parameters (Eij) carry plausible physical significance resulting from the local composition concept. The segment version of the Wilson model [3] assumes that the liquid has a lattice structure that can be described as cells with central species (segments or solvents) that are surrounded by various species (segments or solvents) in the mixture. The distribution of the species around these central species is determined by enthalpies of interaction for the activated state. The use of a local composition model to model the excess term is a reasonable approach because intermolecular friction and viscosity should be affected by nearest neighbours. Parameters Eij are the composition and temperature-independent parameters referred to here as the segment-based Wilson binary parameters for the activated state. In fact, the Wilson term in the seg-
R. Sadeghi / J. Chem. Thermodynamics 37 (2005) 445–448
447
TABLE 1 The parameters of segment-based Eyring–Wilson viscosity model for some polymer solutions along with the corresponding deviations System
Polymer Mn
T/K
Np
104 Æ E12 1
PEG(1) + 1,2-dimethoxyethane (2) PEG(1) + 1,2-dimethoxyethane (2) PEG(1) + dimethoxymethane (2) PEG(1) + dimethoxymethane (2) PEG(1) + 1,3-dioxolane (2) PEG(1) + 1,3-dioxolane (2) PEG(1) + 1,4-dioxane (2) PEG(1) + 1,4-dioxane (2) PEG(1) + oxolane (2) PEG(1) + oxolane (2) PEG(1) + oxane (2) PEG(1) + oxane (2) PEG(1) + methyl acetate (2) PEG(1) + methyl acetate (2) PEG(1) + methyl acetate (2) PVP (1) + H2O (2) PVP (1) + H2O (2) PVP (1) + H2O (2) PVP (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PPG (1) + H2O (2)
192 408 192 408 192 408 192 408 192 408 192 408 400 400 400 4088 4088 4088 4088 300 400 600 900 1000 1500 2000 3000 4000 6000 404
298.15 298.15 298.15 298.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 298.15 303.15 308.15 298.15 308.15 318.15 328.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15
14 14 14 14 14 14 14 14 14 14 14 14 11 11 11 27 27 27 27 10 10 10 10 10 10 10 10 10 10 22
P Np is the number of experimental points; Dev% is (1/Np) [(gexp gcal)/gexp] · 100.
ment-based viscosity model refers to an energy barrier associated with a segment jump, whereas the Wilson activity coefficient model refers to molecular interactions in the liquid phase. Parameters of the Wilson activity coefficient model are obtained by fitting phase equilibrium data, whereas the parameters of the Wilson viscosity model are obtained by fitting viscosity data.
104 Æ E21
Dev%
Reference
8.38 13.51 6.41 8.90 2.27 3.35 2.32 3.55 3.35 5.47 3.70 5.25 4.63 3.82 6.26 3.17 1.58 2.48 2.94 8.22 3.33 5.43 5.30 2.96 2.83 1.75 4.46 8.90 10.09 2.05
[7] [7] [7] [7] [8] [8] [8] [8] [8] [8] [8] [8] [9] [9] [9] [5] [5] [5] [5] [6] [6] [6] [6] [6] [6] [6] [6] [6] [6] [10]
1
J Æ mol
J Æ mol
2.7918 2.7918 2.6563 2.6563 2.3144 2.3144 2.0995 2.0995 2.4285 2.4285 2.4196 2.4196 2.4770 2.4770 2.4770 0.75309 0.75309 0.75309 0.75309 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 0.45149
2.7918 2.7918 2.6563 2.6563 2.3144 2.3144 2.0995 2.0995 2.4285 2.4285 2.4196 2.4196 2.4770 2.4770 2.4770 8.4383 8.4383 8.4383 8.4383 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 3.7652
ethylene glycol (PEG) + H2O} [6] systems, the pure polymer viscosity was not available. In these cases, the polymer viscosity was treated as an adjustable
3. Results and discussion The applicability of the proposed model has been tested using experimental viscosity data for a variety of polymer solutions. The value of parameter C was set to 10. A value of r = 1 was used for solvents and for polymer the value of r is ratio of the molar mass of polymer to that of segment. The values of molar volume of segments were calculated from the ratio of molar volume of polymers to their degree of polymerization. In the case of polyvinylpyrrolidone (PVP), the molar volume of polymer was calculated from the infinite dilution apparent molal volume of polymer at each temperature calculated from density data of aqueous solutions of PVP [5]. Also, In the case of (PVP + H2O) [5] and {poly-
FIGURE 1. Plot of the experimental viscosity against the viscosity calculated for the model for some {PEG (1) + 1,4- dioxane (2)} systems at T = 303.15 K: (s) PEG200; (n) PEG400; and (–––) calculated.
448
R. Sadeghi / J. Chem. Thermodynamics 37 (2005) 445–448
number-average polymer molar mass. The evaluated parameters along with the corresponding deviation for the systems studied are listed in table 1. As can be seen from table 1, except for the systems (PVP + H2O) and {polypropylene glycol (PPG) + H2O}, in the all systems we have E12 = E21. Also, the model parameters are independent of polymer molar mass and temperature. On the basis of the deviation obtained, given in table 1, we conclude that the proposed model represents the experimental viscosity data of polymer solutions, with good accuracy. To show the reliability of the proposed model, comparison between experimental and correlated viscosity data are shown in figures 1 and 2 for (PEG + 1,4-dioxane) and (PPG + H2O) systems, respectively.
FIGURE 2. Plot of the experimental viscosity against the viscosity calculated for the model for some {PPG (1) + H2O (2)} systems at T = 298.15 K: (s) experimental and (–––) calculated.
parameter. In the case of (PVP + H2O) systems at different temperatures, the following relation was used to consider the temperature dependence of the pure polymer viscosity: g1 ¼ a expðb=T Þ,
ð15Þ
where the parameters a and b have been calculated to be 2.9599 Æ 106 and 9.4633 Æ 103, respectively. Also, in the case of (PEG + H2O) systems at different polymer molar masses, the following Mark–Houwink type relation was used to account for polymer molar mass dependence of the pure polymer viscosity: g1 ¼ aM bn,1 ,
ð16Þ
where the parameters a and b have been calculated to be 4.3804 Æ 106 and 6.6293, respectively, and Mn,1 is the
References [1] W. Kauzmann, H. Eyring, J. Chem. Phys. 62 (1940) 3113– 3117. [2] L.T. Novak, C.C. Chen, Y. Song, Ind. Eng. Chem. Res. 43 (2004) 6231–6237. [3] R. Sadeghi, J. Chem. Thermodyn. 37 (2005) 55–60. [4] R. Sadeghi, J. Chem. Thermodyn. (in press). [5] R. Sadeghi, M.T. Zafarani-Moattar, J. Chem. Thermodyn. 36 (2004) 665–670. [6] S. Kirincic, C. Klofutar, Fluid Phase Equilibr. 155 (1999) 311– 325. [7] F. Comelli, S. Ottani, R. Francesconi, C. Castellari, J. Chem. Eng. Data 47 (2002) 1226–1231. [8] S. Ottani, D. Vitalini, F. Comelli, C. Castellari, J. Chem. Eng. Data 47 (2002) 1197–1204. [9] T.M. Aminabhavi, K. Banerjee, J. Chem. Eng. Data 43 (1998) 852–855. [10] M.T. Zafarani-Moattar, A. Salabat, J. Solution Chem. 27 (1998) 663–673.
JCT 04-200
Segment-based Eyring–Wilson viscosity model for polymer solutions Rahmat Sadeghi
*
Department of Chemistry, Kurdistan University, Pasdaran Street, Sanandaj, Iran Received 1 October 2004; received in revised form 24 October 2004; accepted 25 October 2004 Available online 8 December 2004
Abstract A theory-based model is presented for correlating viscosity of polymer solutions and is based on the segment-based Eyring mixture viscosity model as well as the segment-based Wilson model for describing deviations from ideality. The model has been applied to several polymer solutions and the results show that it is reliable both for correlation and prediction of the viscosity of polymer solutions at different molar masses and temperature of the polymer. 2004 Elsevier Ltd. All rights reserved. Keywords: Segment-based; Eyring; Wilson; Viscosity; Polymer solutions
1. Introduction Knowledge of the viscosity of polymer solutions is important for practical and theoretical purposes. Viscosity of polymer solutions provides an invaluable type of data in polymer research, development and engineering. Furthermore, the simultaneous investigation of viscosity and volume effects on mixing can be a powerful tool for the characterization of the intermolecular interactions present in these mixtures. Also, knowledge of the dependence of viscosities of polymer solutions on composition is of great interest from a theoretical standpoint since it may lead to better understanding of their fundamental behaviour of polymer solutions. A segment-based model should be more physically realistic for large molecules when diffusion and flow are viewed to occur by a sequence of individual segment jumps into vacancies rather large jumps of the molecule into large vacancies. It has been found that the ratio of heats of activation decreases as molecules of normal paraffins get larger, suggesting that vacancies for jumps are smaller than the size of larger molecules [1]. Re*
Tel./fax: +98 871 6660075. E-mail address: [email protected].
0021-9614/$ - see front matter 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.jct.2004.10.009
cently, Novak et al. [2] developed a segment-based Eyring-NRTL viscosity model for mixtures containing polymers, which uses a combination of the segmentbased Eyring mixture viscosity model and the segment based NRTL model. In continuation of our investigation of the extension of the Wilson model to polymer solutions [3] and (polymer + electrolyte) solutions [4], in this paper, a segmentbased Eyring–Wilson viscosity model is presented for correlating and predicting the viscosity of polymer solutions. The utility of the model is demonstrated with successful representation of viscosity of several polymer solutions at different polymer molar masses and temperatures.
2. Theory The absolute rate theory approach of Eyring provides the following expression for the viscosity of a liquid mixture [1]: gV ¼ Nh expðDg =RT Þ,
ð1Þ
where g is the dynamic viscosity of the mixtures, h is Plancks constant, N is Avogadros number, V is the
446
R. Sadeghi / J. Chem. Thermodynamics 37 (2005) 445–448
molar volume of the mixture, Dg* is the molar Gibbs free energy of activation for the flow process, and T is the absolute temperature. In principal, equation (1) can be used to describe the viscosity of ideal mixtures as well as that of nonideal systems. For an ideal liquid mixture, in the viscosity sense, equation (1) can be written as gideal V ideal ¼ Nh expðDgideal =RT Þ,
ð2Þ
where Videal is the molar volume of the ideal mixture in the thermodynamic sense. The molar Gibbs free energy of activation for flow can be regarded as the sum of an ideal contribution and a correction or excess term. That is, Dg ¼ Dgideal þ gE ,
molar masses with a series of interaction parameters, which provides a predictive capability. Following Novak et al. [2], by application of the segment concept to the Eyring rate-based model and using the segmentbased polymer Wilson model [3] for describing deviations from ideality we have: X lnðgV Þ ¼ X i lnðgi V i Þ þ gE =RT , ð7Þ i
V ¼
X
gi ¼
X
ð8Þ
, ri,I gI
V i ¼
X
,
X
ri,I V i,I
!
Equation (4) clearly shows that the model considers the dynamic viscosity of real solutions as the result the contribution of an ideal term and an excess type. By considering the following relation for ideal mixtures: X lnðgV Þideal ¼ xi lnðgi V i Þ, ð5Þ
ð10Þ
ri,I ,
I
, ri,I xI
XX J
! r j, J x J ,
¼ C
X
X i ln
i
ð11Þ
j
ð4Þ gE =RT
ð9Þ
ri,I ,
X
I
Xi ¼
!
I
I
lnðgV Þ ¼ lnðgV Þideal þ gE =RT :
X
I
ð3Þ
where Dgideal is the ideal mixture contribution, at the same conditions of temperature, pressure, and composition for the real system per mole, and gE is the molar excess free energy of activation for flow. Combining equations (1)–(3) leads to the following expression:
X i V i ,
i
X
! X j H ji ,
ð12Þ
j
H ji ¼ expðEji =CRT Þ,
ð13Þ
Eji ¼ hji hii ,
ð14Þ
i
we have lnðgV Þ ¼
X
xi lnðgi V i Þ þ gE =RT :
ð6Þ
i
where xi, gi and Vi are mole fraction, viscosity and molar volume of component i, respectively. Extension of the concepts of classical thermodynamics to the viscous flow behaviour of liquid mixtures can be made by assuming equivalence between the Gibbs free energy of activation for flow and the equilibrium Gibbs free energy of mixing. Taking this direct relationship between the thermodynamics and the transport property, in this work the polymer Wilson model for excess Gibbs free energy developed in our previous paper [3], is used to account for the viscosity deviation from ideal behaviour. That is to say, the assumed equivalence may be dependent on the flow regime, i.e., the Re number of the flow system used. A segment-based model should be more physically realistic for large molecules when diffusion and flow are viewed to occur by a sequence of individual segment jumps into vacancies rather than large molecule jumps into large vacancies. Advantage of the segment-based models over conventional models for correlation of polymer solution experimental data is that, unlike the classical models, they can cover a wide range of polymer
where ri,I is the number of the segment i in the component I, Vi,I is the molar volume of the segment i in the component I, C is a parameter, which represents the effective coordination number in the system and hji is the enthalpy of interaction between j and i species. In the above relations the species i and j can be solvent molecules or segments and species I and J can be solvent or polymer molecules. The excess term in the Eyring– Wilson viscosity model, like the Wilson excess Gibbs energy expression, has some semiempirical basis, and the model parameters (Eij) carry plausible physical significance resulting from the local composition concept. The segment version of the Wilson model [3] assumes that the liquid has a lattice structure that can be described as cells with central species (segments or solvents) that are surrounded by various species (segments or solvents) in the mixture. The distribution of the species around these central species is determined by enthalpies of interaction for the activated state. The use of a local composition model to model the excess term is a reasonable approach because intermolecular friction and viscosity should be affected by nearest neighbours. Parameters Eij are the composition and temperature-independent parameters referred to here as the segment-based Wilson binary parameters for the activated state. In fact, the Wilson term in the seg-
R. Sadeghi / J. Chem. Thermodynamics 37 (2005) 445–448
447
TABLE 1 The parameters of segment-based Eyring–Wilson viscosity model for some polymer solutions along with the corresponding deviations System
Polymer Mn
T/K
Np
104 Æ E12 1
PEG(1) + 1,2-dimethoxyethane (2) PEG(1) + 1,2-dimethoxyethane (2) PEG(1) + dimethoxymethane (2) PEG(1) + dimethoxymethane (2) PEG(1) + 1,3-dioxolane (2) PEG(1) + 1,3-dioxolane (2) PEG(1) + 1,4-dioxane (2) PEG(1) + 1,4-dioxane (2) PEG(1) + oxolane (2) PEG(1) + oxolane (2) PEG(1) + oxane (2) PEG(1) + oxane (2) PEG(1) + methyl acetate (2) PEG(1) + methyl acetate (2) PEG(1) + methyl acetate (2) PVP (1) + H2O (2) PVP (1) + H2O (2) PVP (1) + H2O (2) PVP (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PEG (1) + H2O (2) PPG (1) + H2O (2)
192 408 192 408 192 408 192 408 192 408 192 408 400 400 400 4088 4088 4088 4088 300 400 600 900 1000 1500 2000 3000 4000 6000 404
298.15 298.15 298.15 298.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 303.15 298.15 303.15 308.15 298.15 308.15 318.15 328.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15 298.15
14 14 14 14 14 14 14 14 14 14 14 14 11 11 11 27 27 27 27 10 10 10 10 10 10 10 10 10 10 22
P Np is the number of experimental points; Dev% is (1/Np) [(gexp gcal)/gexp] · 100.
ment-based viscosity model refers to an energy barrier associated with a segment jump, whereas the Wilson activity coefficient model refers to molecular interactions in the liquid phase. Parameters of the Wilson activity coefficient model are obtained by fitting phase equilibrium data, whereas the parameters of the Wilson viscosity model are obtained by fitting viscosity data.
104 Æ E21
Dev%
Reference
8.38 13.51 6.41 8.90 2.27 3.35 2.32 3.55 3.35 5.47 3.70 5.25 4.63 3.82 6.26 3.17 1.58 2.48 2.94 8.22 3.33 5.43 5.30 2.96 2.83 1.75 4.46 8.90 10.09 2.05
[7] [7] [7] [7] [8] [8] [8] [8] [8] [8] [8] [8] [9] [9] [9] [5] [5] [5] [5] [6] [6] [6] [6] [6] [6] [6] [6] [6] [6] [10]
1
J Æ mol
J Æ mol
2.7918 2.7918 2.6563 2.6563 2.3144 2.3144 2.0995 2.0995 2.4285 2.4285 2.4196 2.4196 2.4770 2.4770 2.4770 0.75309 0.75309 0.75309 0.75309 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 0.45149
2.7918 2.7918 2.6563 2.6563 2.3144 2.3144 2.0995 2.0995 2.4285 2.4285 2.4196 2.4196 2.4770 2.4770 2.4770 8.4383 8.4383 8.4383 8.4383 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 6.9222 3.7652
ethylene glycol (PEG) + H2O} [6] systems, the pure polymer viscosity was not available. In these cases, the polymer viscosity was treated as an adjustable
3. Results and discussion The applicability of the proposed model has been tested using experimental viscosity data for a variety of polymer solutions. The value of parameter C was set to 10. A value of r = 1 was used for solvents and for polymer the value of r is ratio of the molar mass of polymer to that of segment. The values of molar volume of segments were calculated from the ratio of molar volume of polymers to their degree of polymerization. In the case of polyvinylpyrrolidone (PVP), the molar volume of polymer was calculated from the infinite dilution apparent molal volume of polymer at each temperature calculated from density data of aqueous solutions of PVP [5]. Also, In the case of (PVP + H2O) [5] and {poly-
FIGURE 1. Plot of the experimental viscosity against the viscosity calculated for the model for some {PEG (1) + 1,4- dioxane (2)} systems at T = 303.15 K: (s) PEG200; (n) PEG400; and (–––) calculated.
448
R. Sadeghi / J. Chem. Thermodynamics 37 (2005) 445–448
number-average polymer molar mass. The evaluated parameters along with the corresponding deviation for the systems studied are listed in table 1. As can be seen from table 1, except for the systems (PVP + H2O) and {polypropylene glycol (PPG) + H2O}, in the all systems we have E12 = E21. Also, the model parameters are independent of polymer molar mass and temperature. On the basis of the deviation obtained, given in table 1, we conclude that the proposed model represents the experimental viscosity data of polymer solutions, with good accuracy. To show the reliability of the proposed model, comparison between experimental and correlated viscosity data are shown in figures 1 and 2 for (PEG + 1,4-dioxane) and (PPG + H2O) systems, respectively.
FIGURE 2. Plot of the experimental viscosity against the viscosity calculated for the model for some {PPG (1) + H2O (2)} systems at T = 298.15 K: (s) experimental and (–––) calculated.
parameter. In the case of (PVP + H2O) systems at different temperatures, the following relation was used to consider the temperature dependence of the pure polymer viscosity: g1 ¼ a expðb=T Þ,
ð15Þ
where the parameters a and b have been calculated to be 2.9599 Æ 106 and 9.4633 Æ 103, respectively. Also, in the case of (PEG + H2O) systems at different polymer molar masses, the following Mark–Houwink type relation was used to account for polymer molar mass dependence of the pure polymer viscosity: g1 ¼ aM bn,1 ,
ð16Þ
where the parameters a and b have been calculated to be 4.3804 Æ 106 and 6.6293, respectively, and Mn,1 is the
References [1] W. Kauzmann, H. Eyring, J. Chem. Phys. 62 (1940) 3113– 3117. [2] L.T. Novak, C.C. Chen, Y. Song, Ind. Eng. Chem. Res. 43 (2004) 6231–6237. [3] R. Sadeghi, J. Chem. Thermodyn. 37 (2005) 55–60. [4] R. Sadeghi, J. Chem. Thermodyn. (in press). [5] R. Sadeghi, M.T. Zafarani-Moattar, J. Chem. Thermodyn. 36 (2004) 665–670. [6] S. Kirincic, C. Klofutar, Fluid Phase Equilibr. 155 (1999) 311– 325. [7] F. Comelli, S. Ottani, R. Francesconi, C. Castellari, J. Chem. Eng. Data 47 (2002) 1226–1231. [8] S. Ottani, D. Vitalini, F. Comelli, C. Castellari, J. Chem. Eng. Data 47 (2002) 1197–1204. [9] T.M. Aminabhavi, K. Banerjee, J. Chem. Eng. Data 43 (1998) 852–855. [10] M.T. Zafarani-Moattar, A. Salabat, J. Solution Chem. 27 (1998) 663–673.
JCT 04-200