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GENERAL RESEARCH Viscosity of Polyelectrolyte Solutions: Experiment and a New Model Jianyong Yang, Nan Liu, Dahong Yu, Changjun Peng, Honglai Liu,* and Ying Hu Department of Chemistry and Lab for Advanced Materials, East China University of Science and Technology, Shanghai 200237, People’s Republic of China
Jianwen Jiang* Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
The viscosity of sodium polyacrylate (NaPAA) and poly(sodium 4-styrenesulfonate) (NaPSS) solutions was experimentally measured with various added salt (NaCl) concentrations at different temperatures. With a low salt concentration, the reduced specific viscosity (ηrs) exhibits a maximum and then a minimum with decreasing polyelectrolyte concentration (Cp) and diverges as Cp approaches zero. A new statistical mechanics model was developed based on the combination of Eyring’s absolute-rate theory and Jiang’s polyelectrolyte model. This model can satisfactorily correlate the viscosity of NaPAA and NaPSS solutions over the entire range of added NaCl concentration and temperature in this study and correctly predict the anomalous concentration-dependent behavior of ηrs. 1. Introduction Polyelectrolytes in aqueous solutions dissociate into polyions and counterions, and the former usually carry considerable charges. Consequently, polyelectrolyte solutions have special dual properties, showing the general characteristics of both neutral polymer and electrolyte solutions. Recently, polyelectrolytes have become increasingly important in chemical industries, biological processes, material sciences, and so forth.1,2 Nevertheless, the understanding of their properties is far from complete, primarily because of the complex long-range electrostatic polyion-polyion and polyion-counterion interactions. For instance, as one of the important transport properties, the viscosity of polyelectrolyte solutions is not fully understood, although toward that end, there have been a large number of studies. Early in the 1940s, Fuoss and co-workers3-5 proposed an empirical relationship for the viscosity of polyelectrolyte solutions from a series of experiments:
ηrs )
a +d 1 + bCp1/2
(1)
where ηrs is the reduced specific viscosity (defined as ηrs ) (η - η0)/(η0Cp), where η and η0 are the viscosity of solution and solvent, respectively), Cp is the polyelectrolyte concentration, and a, b, and d are system-related constants. The value of b is dependent on the polyelectrolyte-solvent interaction, and a + d can be regarded as the intrinsic viscosity in the limit of Cp approaching zero. This relationship has been verified to be accurate * To whom correspondence should be addressed. Tel.: 8621-64252921; fax: 86-21-64252921; E-mail:
[email protected] (for H.L.L.). Tel: 1-302-8316953; fax: 1-302-8311048; Email:
[email protected] (J.W.J.).
in the semidilute regime in which ηrs increases monotonically to a + d with decreasing Cp. However, latter studies6-8 have revealed that the ηrs of the dilute polyelectrolyte solutions (especially with a low salt concentration) shows an intriguing anomalous concentration dependence, in which ηrs has a maximum value at a certain concentration Cp,max. This anomalous behavior cannot be described by the Fuoss relation. Rabin and co-workers8-10 measured the viscosity of dilute solutions of sulfonated sodium neutralized polystyrenes with different molecular weights and found a linear molecular weight dependence of ηrs. Based on the mode-mode coupling approximation of Hess and Klein,11 Rabin and co-workers8-10 further developed a theory in which the specific viscosity, ηsp ) (η - η0)/η0, obeys a scaling law:
ηsp ∝
RHlB2Cp2 κ3
(2)
where RH is the hydrodynamic radius and κ is the inverse Debye screening length (defined as κ2 ) 4πlB(Cp + 2cs), where lB is the Bjerrum length (lB ) e2/(0kT), for which e is the electron charge, 0 the dielectric constant of the solvent, k the Boltzmann constant, and T the absolute temperature) and cs is the monovalent salt concentration). From eq 2, the high-concentration limit of the Fuoss relation, ηsp ∝ Cp1/2, is recovered when cs , Cp; in contrast, one obtains the low-concentration limit, ηsp ∝ Cp2, when cs . Cp. Similar conclusions can be also obtained in the model of Witten and Pincus.12 Sukpisan et al.13 tested Rabin’s scaling relation8-10 under different conditions and proposed that the specific viscosity ηsp at a certain concentration region could be better represented as
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Ind. Eng. Chem. Res., Vol. 44, No. 21, 2005 8121
ηsp ) AX + B
(3)
Table 1. Viscosity of Water at Different Temperaturesa
with 2
X)
Cp
(Cp + 2cs)3/2
(4)
where A and B are constants related to the solution property and molecular weight. They13 also found that, for the polyelectrolyte solutions without added salt, B is negative; in contrast, with the addition of salt, B approaches zero. With this uniform formula, Sukpisan’s model13 can describe the viscosity of both salt-free and salt-added polyelectrolyte solutions and, therefore, is more commonly used than the Rabin’s model. A careful analysis of Sukpisan’s model reveals that, when Cp ) 0 and, consequently, X ) 0, the limiting viscosity ηsp is equal to B, according to eqs 3 and 4. Furthermore, for a polyelectrolyte solution without salt, both ηsp and B should be zero. Apparently, the argument of Sukpisan for B13 fails in this limiting case. On the other hand, the reduced specific viscosity, ηrs ) ηsp/Cp, should approach infinity if ηsp is not equal to zero when Cp ) 0 for a polyelectrolyte solution with added salt. This phenomenon has not, to the best of our knowledge, been reported previously in the literature. To resolve the contradiction to the Sukpisan’s model and to better predict the viscosity behavior, especially in the dilute concentration region, further studies are desirable. In this work, we first perform experimental measurements on the viscosity of sodium polyacrylate (NaPAA) and poly(sodium 4-styrenesulfonate) (NaPSS) aqueous solutions with various added NaCl concentrations at a series of temperatures. We then combine Eyring’s absolute-rate theory14 and a polyelectrolyte thermodynamic model15,16 to develop a new model for the viscosity of polyelectrolyte solutions and further use the model to correlate our experimental data. In section 2, the experimental procedure is briefly described, and then the new model is introduced. The experimental and correlated results are presented and discussed in section 3, first for the polyelectrolyte solutions without salt and then for those with salt, followed by a discussion about the constants in the model. As we shall observe, our model can not only accurately describe the relations of ηsp ∼ X and ηsp ∼ Cp, but also correctly determine the peak values of ηrs and predict the trend of ηrs with Cp approaching zero. Finally, concluding remarks are given in section 4. 2. Experiment and Model 2.1. Experiment. The Ubbelohde viscometer was used in the experiment, and the diameter of the capillary is 0.5 mm. All the experimental processes were performed in a constant temperature tank with precisely controlled temperature. The temperature was measured by a mercury thermometer with an accuracy of (0.1 K. NaPAA and NaPSS were purchased from Aldrich Chemical Company, and their weight-average molecular weights (M h w) are 5100 and 70 000, respectively. Doubly distilled, deionized, and air-removed water was used for the preparation of all solutions. The viscosity of water at temperatures of 303.15-318.15 K is listed in Table 1. NaCl with a purity g99.8% was used for the added salt. We measured the viscosity of NaPAA and NaPSS solutions as a function of polymer concen-
a
temperature, T (K)
viscosity, η0 (mPa s)
303.15 308.15 313.15 318.15
0.8007 0.7225 0.6560 0.5988
Data taken from ref 17.
tration with various NaCl concentrations and at different temperatures. 2.2. Viscosity Model. The solution is assumed to consist of K types of charged hard spheres, and the solvent is represented as a continuum medium with a dielectric constant 0. The polyions are modeled as freely tangent-jointed hard-sphere chains, with a chain length l, monomer segment density Fm, hard-core diameter σm, and segment charge Zme. There also are counterions with a density Fc, diameter σc, and charge Zce. The entire system is electrically neutral: K
Fizi ) 0 ∑ i)1
(5)
According to Eyring’s absolute-rate theory,14 the viscosity of a polyelectrolyte solution at a temperature T is
( ) ( ) hNA f* exp V RT
η)
(6)
where h is Planck’s constant (h ) 6.626 × 10-34 J s), NA the Avogadro constant (NA ) 6.02 × 1023), R the gas constant (R ) 8.314 J mol-1 K-1), V the solution volume, and f* the molar activation Helmholtz energy. Similarly, for the solvent, we have
η0 )
( ) ( ) hNA f* 0 exp V0 RT
(7)
From eqs 6 and 7, the reduced viscosity of the polyelectrolyte solution is
() (
)
f* - f* V 0 η ) exp η0 V0 RT
(8)
For a dilute solution, we suppose V ≈ V0, and denote f ) f* - f* 0 , which is the activation Helmholtz energy of the solution with the pure solvent as a reference. We then have
f η ) exp η0 RT
( )
(9)
where f is a mixing property and can be split into ideal mixing and excess contributions:
f ) fid + fr
(10)
However, one cannot easily evaluate the absolute value of fid; therefore, we use an empirical equation that is similar to the Sukpisan equation13 for the ideal mixing contribution:
( )
fid ηid ) exp ) A′X + B′ + 1 η0 RT
(11)
where A′ and B′ are two constants that are related to
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the nature of polyelectrolyte solutions. For a salt-free system, B′ ) 0. The prime symbols are used to distinguish these parameters from the constants in eq 3. Note that other types of equations can also be used for the ideal mixing contribution to Helmholtz energy. Thus, the reduced viscosity of the polyelectrolyte solution is
( )
η fr ) (A′X + B′ + 1) exp η0 RT
[( )
]
f (el) )
FT
[
-
R02
K
∑
Fi,0zi
4π i)11 + σiΓ
(
ziΓ +
)
πσiPn 2∆
Γ3
+
3π
]
(15)
i
1+ K
2
2
4Γ ) R0
Fiσizi
∑ i)11 + σ Γ
Pn )
∑ i)1
π
(16)
Fiσi3
K
∑
2∆i)11 + σiΓ
(
Fi
(1 + σiΓ)2
zi -
)
πσi2Pn 2∆
2
(17)
The third term (poly) represents the chain connectivity contribution:15
fr(poly) )
[
RT Fm(l - 1) ln ym,m(σm) FT l
]
(18)
where ym,m(σm) is the tangently contact value of the cavity correlation function between two ionic segments, which can also be estimated from the MSA solution:20
ln
y(2) ij (σij)
2 aiajΓ2 1 R0 zizj πσiσjζ2 ) + + -1 ∆ 4πσij 4σ ∆2 πσ R 2 ij
ij 0
Fc(1 - Rc)πσmc3y(eff) mc (σmc) 1 ) (1 - Rm) exp 3τmc
(22)
and
Ac Ac + 1
(23)
with
K where ζk ) ∑i)1 Fiσki , ∆ ) 1 - πζ3/6, β ) 1/(kT), R02 ) βe2/, and Γ and Pn are calculated iteratively from eq 16 and 17.
K
where Rm and Rc are determined by a stick parameter τmc, which measures the strength of the stick interaction and can be calculated using the following equations:
(14)
where FT is the total number density of all species (ions) in the solution, FT ) (Cp + cc + 2cs)NA. For a salt-free solution, cs ) 0 and FT ) (Cp + cc)NA. The term (el) denotes the electrostatic contribution obtained from the mean-spherical approximation (MSA) solution of the Ornstein-Zernike integral equation:20
RT
RT [F ln(1 - Rm) + FcRc + Fc ln(1 - Rc)] FT m (21)
Rc )
3 πζ23 πζ1ζ2 RT ζ2 + fr(hs) ) ζ ln ∆ + 0 FT ζ 2 2∆ 6ζ32∆2 3
(20)
The last term (sticky) is incorporated to account for the specific sticky interaction between the polyion and the counterion:16,18
(13)
where the term (hs) denotes the hard-sphere contribution derived from the Mansoori-Carnahan-StarlingLeland equation of state:19
)
πσi2Pn ai ) z 2∆ 2Γ(1 + σiΓ) i
fr(sticky) )
fr ) fr(hs) + fr(el) + fr(poly) + fr(sticky)
(
R02
(12)
where fr is the excess activation Helmholtz energy, which can be approximately related to the equilibrium mixing residual Helmholtz energy.14 Here, we use Jiang et al.’s polyelectrolyte model15,16,18 to calculate the thermodynamic residual Helmholtz energy, which includes four contributions:
r
with
(19)
Fmπσmc3y(eff) mc (σmc) Ac ) 3τmc
(24)
If the last two terms of eq 13 are equal to zero, this model reduces to a recently developed new viscosity model for electrolyte solutions.21 3. Results and Discussion 3.1. Parameter Determination. There are several parameters in the new viscosity model for polyelectrolyte solutions, including the diameters and charges of the small ions and segments, the sticky parameter, and the chain length, as well as the constants A′ and B′. The ions were considered to be solvated by water molecules, and, hence, the literature hydrated diameters of σNa+ ) 0.352 nm and σCl- ) 0.36 nm were adopted here.22 The polyion chain length was calculated using the formula l ) zmonoM h w/(zmMmono), where zmono and Mmono are the charge and molar mass of the monomer, respectively. To correlate the model with the experimental data more conveniently, we rewrite eq 12 as
Ω)
η ) A′X + B′ + 1 η0 exp[fr/(RT)]
(25)
The plot of parameter Ω on X is linear, and from the slope and intercept, the two constants A′ and B′ can be estimated. The three model parameterssthe segment diameter of the chain σm, the charge of a segment zm, and the sticky parameter τmcswere adjusted until the linear relation of Ω ∼ X from the experimental data was well-represented. In this work, we used the experimental data of the polyelectrolyte solutions with 0.0001 M added NaCl at 308.15 K to correlate these three model parameters. The optimized values are σm ) 0.96 nm, zm ) -0.8, and τmc-1 ) 1.5 for the NaPAA solution, and σm ) 2.0 nm, zm ) -1.25, and τmc-1 ) 1 for the NaPSS solution.
Ind. Eng. Chem. Res., Vol. 44, No. 21, 2005 8123 Table 2. Constant A′ and ARD% Values for the Salt-Free Solutions at Different Temperatures A′ (1/g)1/2
temperature, T (K)
Figure 1. Parameter Ω versus the scaling variable X with cs ) 0.0001 M at 308.15 K. Symbols indicate experimental data ((4) NaPSS and (]) NaPAA), whereas the lines indicate model correlations.
Figure 1 shows the relation of Ω ∼ X for the NaPAA and NaPSS solutions, respectively, with 0.0001 M NaCl at 308.15 K, and a nice linear relation is obtained for each solution. For the NaPAA system, Ω ) 0.3172X + 1.0107, A′ ) 0.3172, B′ ) 0.0107, and the correlative coefficient is 0.9998; for the NaPSS system, Ω ) 0.3959X + 1.0100, A′ ) 0.39592, B′ ) 0.0100, and the correlative coefficient is 0.9995. 3.2. Salt-Free Solutions. For the salt-free polyelectrolyte solutions, B′ ) 0 and eq 25 of the reduced viscosity can be written as
( )
η fr ) (A′X + 1) exp η0 RT
(26)
where only one constant A′ needs to be determined. Table 2 lists the values of A′ of the salt-free solutions at temperatures in the range of 303.15-318.15 K, and the resulting average relative deviations (ARD%) of model correlations from experiments. Figure 2 shows the comparison between the experimental and correlated reduced viscosity (η/η0) of the NaPAA solutions without salt. Good correlations of the experimental data by the model are obtained. As expected, the viscosity predicted by the model is equal to that of pure water when Cp ) 0. The magnitude of ηsp is enhanced with increasing X, and the behavior of ηsp ∼ X is approximately concave at low concentrations, but becomes linear at high concentrations. As a result,
a
ARD%a
303.15 308.15 313.15 318.15
NaPAA System 0.3182 0.3212 0.3240 0.3292
0.22 0.22 0.23 0.21
303.15 308.15 313.15 318.15
NaPSS System 0.4029 0.4069 0.4105 0.4152
0.50 0.48 0.45 0.52
N ARD% ) (100/N)∑i)1 |(η/η0)experiment - (η/η0)theory|/(η/η0)experiwhere N is the number of data points.
ment,
the linear form of eq 3 of the Sukpisan’s model13 is not applicable here for the dilute solutions and can only be used at high concentrations. In remarkable contrast, however, our model works well over the entire range of concentration in this study. As mentioned previously, the relation of ηsp ∼ Cp is similar to ηsp ∝ Cp1/2 when cs ) 0. This is observed in Figure 3, in which the reduced viscosity of the NaPSS solution from the experiment and our model is shown. Perfect agreement is observed between the correlated results and the experimental data. The observation here is consistent with the findings of Fuoss and coworkers,3-5 Rabin and co-workers,8-10 and Pincus.11 Figure 4 shows the reduced specific viscosity ηrs as a function of polyelectrolyte concentration Cp. Again, we obtain good agreement between the model correlations and the experimental data. With decreasing Cp, ηrs increases first slowly at high and intermediate concentrations and then rapidly at dilute concentrations. Whether the value of ηrs reaches a constant or diverges as Cp approaches zero is not obvious from the experimental data. From the model prediction by eq 26, however, it is easy to see the divergence of ηrs. The same concentration-dependent behavior of ηrs can be also obtained from Rabin’s scaling law8-10 and Sukpisan’s model.13 3.3. Salt-Added Solutions. For the salt-added polyelectrolyte solutions, two constants A′ and B′ must be determined, as listed in Table 3. If eq 3 is used to
Figure 2. Reduced viscosity (η/η0) versus the scaling valuable X of the NaPAA solution without salt.
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Figure 3. Reduced viscosity (η/η0) versus the scaling valuable X of the NaPSS solution without salt: (4) experimental data and (s) model correlations.
Figure 4. Reduced specific viscosity (ηrs) vs Cp at 303.15 K without salt: (4) experimental data and (s) model correlations. Table 3. Values of Constants A′ and B′, and ARD, for the Salt-Added Solutions with Different cs at 308.15 K cs (M)
A′ (1/g)1/2
B′
ARD%
1 × 10-5 5 × 10-5 1 × 10-4
NaPAA System 0.3238 0.0039 0.3210 0.0081 0.3172 0.0107
0.03 0.22 0.75
5 × 10-5 1 × 10-4 2.5 × 10-4
NaPSS System 0.4006 0.0071 0.3959 0.0100 0.3802 0.0180
0.34 0.34 0.45
correlate our experimental data, we reach the same conclusion as Sukpisan; that is, B is negative for the salt-free solutions and approaches zero for the saltadded solutions. This may be viewed as additional proof of the reliability of our experimental data. Figure 5 shows the dependence of η/η0 on X of NaPAA solutions with different NaCl concentrations. We can
Figure 5. Reduced viscosity (η/η0) vs Cp of NaPAA at 308.15 K: (4) experimental data and (s) model correlations.
see that our model provides precise correlations for the salt-added solutions, as for the salt-free solutions in Figure 2. However, at a very dilute concentration, η/η0 has a minimum here, which was not observed in Figure 2. The higher the salt concentration cs, the closer to linearity the relationship between ηsp and X. Figure 6 shows ηrs as a function of Cp for the three NaPSS solutions with various added salt concentrations at 308.15 K. When cs ) 5 × 10-5 M, as Cp decreases, ηrs
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Figure 6. Reduced specific viscosity (ηrs) vs Cp of NaPSS at 308.15 K. Symbols denote experimental data ((4) cs ) 5 × 10-5 M, (O) cs ) 1 × 10-4 M), and (×) cs ) 2.5 × 10-4 M)), and lines represent model correlations ((‚ ‚ ‚) cs ) 5 × 10-5 M, (- - -) cs ) 1 × 10-4 M, and (s) cs ) 2.5 × 10-4 M).
first increases, reaching a maximum at Cp,max, then decreases, reaching a minimum at Cp,min, and finally increases rapidly as Cp approaches zero. When cs ) 1 × 10-4 M, ηrs has a similar anomalous concentrationdependent behavior; however, both Cp,max and Cp,min move toward higher concentrations and the peak values of ηrs at the two concentrations simultaneously become lower. If the contributions to ηrs are attributed to two parts (one from the polymer and the other from the salt), the former is then dominant when Cp > Cp,min and vice versa. The occurrence of the maximum in ηrs ∼ Cp was also observed in some earlier studies;6-8 however, the value of Cp,max here is lower than that of Rabin et al.8 under the same conditions. This may imply that the charge density of polyions in our NaPSS sample is greater. However, the existence of the minimum in ηrs ∼ Cp observed in this work was not reported previously in the literature. Further studies are probably required to determine if this is a general phenomenon for other polyelectrolyte solutions. When cs is relatively high (at 2.5 × 10-4 M), the behavior of ηrs changes to a monotonic increase with decreasing Cp, as observed in Figure 4 for the salt-free solutions. Our model gives fairly good correlations for the experimentally observed, complex, anomalous concentration dependence of ηrs when cs ) 5 × 10-5 and 1 × 10-4 M. The predicted values of Cp,max and Cp,min, and the peak values of ηrs are quantitatively consistent with the experimental data. When cs ) 2.5 × 10-4 M, although good correlations are obtained at high and low concentrations, there are deviations at the intermediate concentrations. 3.4. Constants A′ and B′. Figure 7a shows the constant A′ as a function of the solution temperature T-1/2. The constant A′ rises slightly with increasing temperature, because A′ is related to the polyion conformation, and, as expected, there is a more significant stretching of the polyion at a higher temperature. The relation of A′ ∼ T-1/2 is approximately linear, as reported by Sukpisan et al.,13 but with an opposite sign of the slope. Figure 7b shows the effect of salt/ionic strength on A′. We can see that A′ is reduced slightly with increasing cs. The reason is that, at a higher cs value and, hence, a stronger ionic strength, the electrostatic repulsion between the likely charged polyions is screened at a larger degree, which leads to a less-stretched polymer conformation. According to its definition, the constant B′ is only related to cs. The correlated B′ values for the NaPAA and NaPSS solutions at the same cs value are almost
Figure 7. Constant A′ vs (a) T-1/2 and (b) cs. Lines are drawn to guide the eye.
identical (see Table 3). The small deviation is within the experimental statistical error. 4. Conclusions We have measured the viscosity of sodium polyacrylate (NaPAA) and poly(sodium 4-styrenesulfonate) (NaPSS) solutions with various NaCl concentrations at a series of temperatures. The relationship between the specific viscosity (ηsp) and X is observed to be nonlinear, unlike the description of Sukpisan et al.13 For the solutions without salt or with a relatively high salt concentration, the reduced specific viscosity (ηrs) increases monotonically as the polyelectrolyte concentration (Cp) decreases; for the solutions with a low salt concentration, however, ηrs first increases, then decreases, and finally increases again with decreasing Cp. In addition to a maximum, a minimum is observed in ηrs ∼ Cp for the first time. We have also developed a new viscosity model on the basis of the absolute-rate theory and a molecular thermodynamic model, which can satisfactorily correlate the relation of η/η0 ∼ X and η/η0 ∼ Cp for both salt-free and salt-added polyelectrolyte solutions. Our model correctly describes the anomalous concentration-dependent behavior of ηrs. The accuracy of the theoretically based new model makes it particularly well-suited for engineering applications. Nomenclature a, b, d, A, B, A′, B′ ) constants in models C, c ) concentration κ ) inverse of the Debye screening length e ) elementary charge k ) Boltzmann constant X ) scaling variable F ) number density of particles K ) number of different types of particles NA ) Avogadro constant f ) Helmholtz function R ) degree of associations τmc ) sticky parameter η ) dynamic viscosity RH ) hydrodynamic radius lB ) Bjerrum length
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) dielectric constant T ) absolute temperature l ) chain length σ ) diameter of particles h ) Planck constant V ) solution volume R ) gas constant Z ) ion charge magnitude Superscripts and Subscripts 0 ) pure solvent s ) salt rs ) reduced specific * ) activation r ) excess p ) polyelectrolyte sp ) specific m ) segment i ) composition id ) ideal mixing mono ) monomer
Acknowledgment This work was supported by the National Natural Science Foundation of China (Project Nos. 20236010 and 20476025) and Shanghai Municipal Education Commission of China. Literature Cited (1) Hoagland, D. A. In Encyclopedia of Polymer Science and Engineering; Wiley: New York, 2004. (2) The 5th International Symposium on Polyelectrolytes, Amherst, MA, June 2004. (3) Fuoss, R. M.; Strauss, U. P. Electrostatic Interaction of Polyelectrolytes and Simple Electrolytes. J. Polym. Sci. 1948, 3, 602. (4) Fuoss, R. M.; Strauss, U. P. Viscosity Function for Polyelectrolytes. J. Polym. Sci. 1948, 3, 603. (5) Fuoss, R. M.; Cathers, G. I. Polyelectrolytes. III. Viscosities of n-Butyl Bromide Addition Compounds of 4-VinylpyridineStyrene Copolymer in Nitromethane-Dioxane Mixtures. J. Polym. Sci. 1949, 4, 97. (6) Eisenberg, H.; Pouyet, J. Viscosities of Dilute Aqueous Solutions of a Partially Quaternized Poly-4-vinylpyridine at Low Gradients of Flow. J. Polym. Sci. 1954, 13, 85.
(7) Wolf, C. Viscosity of Polyelectrolyte Solutions. J. Phys. (Paris) 1978, 39, 169. (8) Cohen, J.; Priel, Z.; Rabin, Y. Viscosity of Dilute Polyelectrolyte Solutions. J. Chem. Phys. 1988, 88, 7111. (9) Rabin, Y.; Cohon, J.; Priel, Z. The Generalized Fuoss Law. J. Polym. Sci., Part C: Polym. Lett. 1988, 26, 397. (10) Cohen, J.; Priel, Z.; Rabin, Y. Viscosity of Dilute Polyelectrolyte Solutions: Temperature Dependence. J. Chem. Phys. 1990, 93, 9062. (11) Hess, W.; Klein, R. Generalized Hydrodynamics of System of Brownian Particles. Adv. Phys. 1983, 32, 173. (12) Witten, T. A.; Pincus, P. Structure and Viscosity of Interpenetrating Polyelectrolyte Chains. Europhys. Lett. 1987, 3, 315. (13) Sukpisan, J.; Kanaharana, J.; Sirivat, A.; Wang, S. Q. The Specific Viscosity of Partially Hydrolyzed Polyacrylamide Solutions: Effects of Degree of Hydrolysis, Molecular Weight, Solvent Quality and Temperature. J. Polym. Sci. 1998, 36, 743. (14) Glasstone, S.; Laidler, K. J.; Erying, H. The Theory of Rate Processes; McGraw-Hill: New York, 1941. (15) Jiang, J. W.; Liu, H. L.; Hu, Y.; Prausnitz, J. M. A Molecular-Thermodynamic Model for Polyelectrolyte Solutions. J. Chem. Phys. 1998, 108, 780. (16) Cai, J.; Liu, H. L.; Hu, Y. A Explicit Molecular Thermodynamic Model for Polyelectrolyte Solutions. Fluid Phase Equilib. 2000, 170, 255. (17) Tao, K. Russian Chemical Handbook. Part I; Scientific Publishing Company: Beijing, 1958; p 978. (18) Zhang, B.; Yu, D. H.; Liu, H. L.; Hu, Y. Osmotic Coefficients of Polyelectrolyte Solutions, Measurements and Correlation. Polymer 2002, 43, 2975. (19) Mansoori, G. A.; Carnahan, N. F.; Starling, K. E.; Leland, T. W. J. Chem. Phys. 1971, 54, 1523. (20) Blum, L. In Theoretical Chemistry: Advances and Perspective; Henderson, D., Ed.; Academic Press: New York, 1980; Vol. 5, p 1. (21) Jiang, J. W.; Sandler, S. I. A New Model for the Viscosity of Electrolytes Solutions. Ind. Eng. Chem. Res. 2003, 42, 6267. (22) Ball, F.; Planche, H.; Furst, W.; Renon, H. Representation of Deviation From Ideality in Concentrated Aqueous Solutions of Electrolytes Using a Mean Spherical Approximation Molecular Model. AIChE J. 1985, 31, 1233.
Received for review April 26, 2005 Revised manuscript received July 25, 2005 Accepted August 8, 2005 IE0504912
