Ind. Eng. Chem. Res. 2003, 42, 6267-6272
6267
A New Model for the Viscosity of Electrolyte Solutions Jianwen Jiang† and Stanley I. Sandler* Center for Molecular and Engineering Thermodynamics, Department of Chemical Engineering, University of Delaware, Newark, Delaware 19716
A new model for the viscosity of electrolyte solutions has been developed based on the combination of liquid-state theory and absolute-rate theory. Using the McMillan-Mayer framework, the ions are represented as charged hard spheres and the solvent as a continuum. The activation Helmholtz energy in absolute-rate theory is approximated by the equilibrium mixing Helmholtz energy calculated analytically from liquid-state theory with a mean spherical approximation. The new model can satisfactorily correlate all available experimental viscosity data up to 12 mol/L of 20 alkali-halide aqueous solutions at ambient conditions with an overall average relative deviation (ARD) of only 0.29%, while the Kaminsky equation yields an overall ARD of 0.81% with the same number of adjustable parameters. Both monotonic and anomalous concentrationdependent viscosity behavior are well described quantitatively. The adjustable parameters in the model have physical meaning and are related with the degree of ion hydration, in agreement with the Hofmeister series. The simplicity and accuracy of this new model make it particularly well suited for engineering applications. 1. Introduction The viscosity of electrolyte solutions is of importance in engineering applications and of research interest because the long-range electrostatic interactions presented cause difficulty in describing such systems. There has been a long history of investigating the viscosity of electrolyte solutions. In 1905 Gru¨neisen1 observed experimentally that at very low concentrations the viscosity of electrolyte solutions increased nonlinearly with concentration, regardless of the type of electrolyte solution. This effect, named after him, is generally correlated as
η/η0 ) 1 + Axc
(1)
where η and η0 are the viscosities of an electrolyte solution and pure solvent, respectively, A is a positive constant, and c is the electrolyte molarity concentration. Later, Falkenhagen et al.2,3 and Onsager and Fuoss4 established a method of calculating parameter A starting from the Debye-Hu¨ckel theory.5 However, eq 1 is only valid for concentrations up to about 0.01 mol/L. In 1929 Jones and Dole6 proposed an empirical formula
η/η0 ) 1 + Axc + Bc
(2)
for the viscosity of electrolyte solutions, in which A is related to the interactions and the mobilities of solute ions and B is the result of interactions between the solvent and ions. The dominant effect is, in general, the latter one. In eq 2, A is always positive but B may be either positive or negative depending on the degree of solvent structuring introduced by the ions. Usually a positive value of B is associated with structure-making (ordering) ions, whereas a negative value of B is associated with structure-breaking (disordering) ions. †
E-mail:
[email protected]. * E-mail:
[email protected].
Given values of A and B, the Jones-Dole equation can reasonably well describe experimentally observed viscosity behavior, but it is usually valid only for concentrations of less than 0.1 mol/L.7 For more concentrated electrolyte solutions, extended Jones-Dole equations with empirical terms of higher order have been proposed. For example, Kaminsky8 added a quadratic term
η/η0 ) 1 + Axc + Bc + Dc2
(3)
This equation is applicable at higher concentrations than the Jones-Dole equation. Alternatively, Lencka et al.9 developed a speciation-based model for multicomponent, concentrated electrolyte solutions. There are also other empirical methods, such as use of a modified Arrhenius equation, the Einstein equation, etc. For a compilation of the methods, see the handbook by Horvath.10 Currently, empirical equations are the prevailing methods for correlating the viscosity of electrolyte solutions. Methods with a more microscopic basis have also been developed, generally starting with the absolute-rate theory of Eyring.11 This theory is based on the postulate that in viscous flow, similar to chemical reaction, a particle must cross an activation Helmholtz energy barrier in order to move between two adjacent equilibrium states. In some cases the activation Helmholtz energy has been fitted to measured viscosity data of electrolyte solutions, such as in the studies of Good et al.,12,13 Goldsack and Franchetto,14,15 and Abraham and Abraham.16,17 An alternative, more predictive way is to relate the activation Helmholtz energy to the equilibrium mixing Helmholtz energy because a close relationship between the two has been found,11 and the latter can be calculated from existing thermodynamic models. For example, for nonelectrolyte solutions, Wu18 made viscosity predictions using the group contribution UNIFAC model, Cao et al.19,20 correlated viscosity and vapor-liquid equilibrium data using a group contribution thermodynamic viscosity model, and Giro et al.21 calculated viscosity and density based on a modified
10.1021/ie0210659 CCC: $25.00 © 2003 American Chemical Society Published on Web 06/26/2003
6268 Ind. Eng. Chem. Res., Vol. 42, No. 25, 2003
Redlich-Kwong-Soave equation of state. For electrolyte solutions, Esteves et al.22 used the Debye-Hu¨ckel theory with the Guggenheim correction. Though their model is able to describe monotonic behavior in which the viscosity only increases with increasing electrolyte concentration, it fails for the so-called anomalous behavior in which the viscosity first decreases and then increases with increasing electrolyte concentration. Here we present a simple statistical mechanics-based model for the viscosity of electrolyte solutions that exhibit monotonic behavior and solutions with anomalous behavior. As shown in section 2, the model is based on the combination of the absolute-rate and liquid-state theories with the assumption that the Helmholtz energy of activation can be approximated by the equilibrium thermodynamic Helmholtz energy using the analytical mean spherical approximation (MSA) expressions for electrolyte solutions. In section 3, the model is used to correlate all available experimental viscosity data of 20 aqueous alkali-halide solutions at ambient conditions. For comparison, correlations using the Kaminsky equation are also presented. We see that better correlations are obtained with the model presented here and that the values of the model parameters are physically meaningful. 2. Model In this work the electrolyte solution is modeled within the McMillan-Mayer framework, in which the solvent is represented as a continuum of dielectric constant and the ion of type k as a charged hard sphere of number density Fk, diameter σk, and charge magnitude zk. Despite its simplicity, this primitive model captures the essential behavior of electrolyte solutions. Based on the absolute-rate theory, the viscosity of the pure solvent at temperature T is
η0 ) (pNA/V0) exp(f0 /RT) q
The reduced viscosity from the ideal mixing contribution is
ηIM/η0 ) exp(fIM/RT)
One cannot easily evaluate the absolute value of fIM, so instead we use an equation simply similar to the JonesDole equation for the ideal mixing contribution
ηIM/η0 ) 1 + axc + bc
η/η0 ) (1 + axc + bc) exp(fEX/RT)
fEX ) fHS + fEL
[
∆2 fEL )
[
-RT βe2 cNA 4π
(7)
where f can be split into ideal mixing and excess contributions
f ) fIM + fEX
(8)
Fkzk
∑k 1 + Γσ
(
Γzk +
k
) ]
πPnσk 2∆
Γ3
-
3π
(13)
(14)
where ζn ) ∑kFkσkn, ∆ ) 1 - πζ3/6, and Γ is the scaling parameter that can be easily calculated iteratively from
Γ )
From eqs 4 and 5, the reduced viscosity of the electrolyte solution relative to that of the pure solvent is
Assuming that V ≈ V0 and using that f ) f q - f0 q is the activation Helmholtz energy of the solution with the pure solvent as a reference, we have
]
ζ23/ζ32
2
η/η0 ) exp(f/RT)
(12)
πζ1ζ2/2 - ζ23/ζ32 RT 3 2 (ζ /ζ - ζ0) ln ∆ + fHS ) + cNA 2 3 ∆
(5)
(6)
(11)
For dilute electrolyte solutions, the simplest model for fEX is the Debye-Hu¨ckel theory. However, for finite electrolyte concentrations and to account for the excluded volume effect, we use the MSA solution by Blum23 of the Ornstein-Zernike integral equation. This approximation provides analytical expressions for hardsphere and electrostatic contributions
(4)
where p is the Planck constant ()6.626 × J s), NA the Avogadro constant ()6.022 × 1023), R the gas constant ()8.314 J‚mol/K), V0 the solvent volume, and f0q the molar activation Helmholtz energy required to move a solvent particle from a local stable state to an activated state. Similarly, for the electrolyte solution, we have
η/η0 ) (V0/V) exp[(f q - f0 q)/RT]
(10)
where a and b are adjustable parameters. As we will see later, these parameters have physical meaning. Note that this equation is used at all concentrations but only to estimate the ideal mixing contribution, which is different from the Jones-Dole equation that is meant to account for all interactions, but is only useful at low concentrations. When eqs 7-10 are combined, the reduced viscosity of the electrolyte solution is
10-34
η ) (pNA/V) exp(f q/RT)
(9)
βe2
Fk
4
(1 + Γσk)2
∑k
with
Pn )
Fkσkzk
(
(
zk -
π
)
πPnσk2 2∆
Fkσk3
∑k 1 + Γσ / 1 + 2∆∑k 1 + Γσ k
2
(15)
)
(16)
k
Substituting eqs 12-16 into eq 11 yields our model for the reduced viscosity η/η0. 3. Results and Discussion To illustrate its applicability, the new model is used to correlate the viscosity of 20 alkali-halide aqueous solutions at ambient conditions (25 °C and 1 atm). The
Ind. Eng. Chem. Res., Vol. 42, No. 25, 2003 6269 Table 1. Pauling Diameters of Alkali Cations and Halide Anions24
Table 2. Model Parameters, Maximum Concentrations, and ARDsa
ion
Li+
Na+
K+
Rb+
Cs+
F-
Cl-
Br-
I-
salt
σ+ 1
1
b
σp (Å)
1.20
1.90
2.66
2.96
3.38
2.72
3.62
3.90
4.32
LiF NaF KF RbF CsF LiCl NaCl KCl RbCl CsCl LiBr NaBr KBr RbBr CsBr LiI NaI KI RbI CsI
0 0.1252 0.0108 0 0 0.0661 0.0533 0.0719 0.0380 0.0398 0.0296 0.0780 0.0572 0.1237 0.1530 0.0405 0.0953 0.0654 0.1561 0.1488
0 0 0 0.0024 0.0070 0.0065 0.0152 0.0777 0.0760 0.0816 0.0235 0.0366 0.0809 0.0792 0.0839 0.0392 0.0728 0.0903 0.1319 0.1331
0.6680 0.5962 0.2053 0.0725 -0.1392 0.4254 0.0679 -0.2250 -0.3622 -0.5086 0.2788 -0.0492 -0.3717 -0.4988 -0.6337 0.1714 -0.1393 -0.5592 -0.6280 -0.7804
extension to other types of electrolyte solutions and/or at other conditions is straightforward. Taking into account that halide anions are usually less hydrated than alkali cations, we set the diameters of halide anions to σp-, their Pauling (bare crystal) diameters, and consider the diameters of hydrated alkali cations to be concentration-dependent + σ+ ) σ + p + σ1 c
(17)
Table 1 gives the measured Pauling diameters24 respectively for alkali cations and halide anions. Because of the hydration effect, the diameters of hydrated cations are larger than the bare Pauling diameters σ+ p , so the adjustable parameter σ+ should be positive and is 1 found to be so. It has long been recognized that the dielectric constant of an electrolyte solution is less than that of a pure solvent because of two factors. First, the polarization of ions disrupts the solvent structure, and second, ions do not make an orientational contribution to the dielectric constant. To account for these, we assume concentration-dependent dielectric constant
) 0/(1 + 1c)
(18)
where 0 is the dielectric constant of a pure solvent and, based on the discussion above, the adjustable parameter 1 should be positive and indeed is found to be so. Together with a and b in eq 10, there are four parameters in the model. However, we find that for aqueous alkali-halide solutions a is close to 1.6; therefore, we fix the value of a at 1.6. The A parameter in the Jones-Dole equation is also nearly constant at 0.006 for aqueous alkali-halide solutions.10 However, because the physical basis of eq 10 is different from that of the Jones-Dole equation, these two constants are not comparable. All available experimental viscosity data25 for 20 alkali-halide aqueous solutions at ambient conditions have been correlated. Table 2 gives the optimized values for the three adjustable parameters, σ+ 1 , 1, and b, the maximum concentration, and the resulting average relative deviation (ARD) for each solution; in addition, the ARDs of the correlations from the Kaminsky equation, which also has three adjustable parameters, are given. The new model can correlate all of the experimental data very well, with an overall ARD only of 0.29% for all solutions. Upon comparison, the Kaminskry equation yields larger ARDs, with an overall ARD of 0.81%. These results are shown in Figure 1, in which the circles are the experimental relative viscosity data25 and the solid and dashed lines are the correlations of the model and the Kaminsky equation, respectively. The dash-dotted line in the figure separates the solutions that exhibit monotonic behavior (upper left part of the figure) from those that show anomalous behavior (lower right part of the figure). The new model quantitatively correlates both types of behavior well. For the cases in which the viscosity is a monotonic function of concentra-
max c (mol/L) ARD %b ARD %c 0.036 0.725 10.382 8.529 8.771 12.345 5.427 4.174 5.800 7.170 8.243 7.000 3.750 0.344 0.297 4.400 7.845 6.020 2.641 2.000
0.07 0.33 0.56 0.55 0.86 0.64 0.28 0.10 0.17 0.31 0.29 0.30 0.07 0.04 0.03 0.29 0.67 0.20 0.12 0.07
0.00 0.00 1.03 0.95 1.41 6.76 0.30 0.14 0.14 0.63 0.83 1.13 0.01 0.01 0.00 0.31 2.28 0.09 0.11 0.03
a ARD % ) (100/N )∑ND (|ηexp - ηcal|/ηexp), where N D D is the i)1 i i i number of data. b The new model. c The Kaminsky equation.
tion, satisfactory correlations are obtained with only two parameters (setting either σ1 ) 0 or 1 ) 0) as we have done for the alkali-fluoride solutions as examples. In general, the Kaminsky equation also gives good correlations, except at high concentrations, particularly, for LiCl, NaBr, NaI, KF, and CsF solutions. The optimized values of the parameters σ+ 1 and 1 are positive for all solutions, as they should be; σ+ 1 is related to the hydration effect and 1 to the polarization effect. Though with some variation, the parameter σ+ 1 decreases with increasing σ+ p for alkali-chloride solutions but increases for alkali-bromide and alkali-iodide solutions. The parameter 1, as shown in Figure 2, increases with σ+ p up to a constant for all solutions of a common anion and also increases with σp for all solutions of a common cation. The optimized values of the parameter b, in general, are positive for solutions with monotonic behavior but negative for solutions with anomalous behavior. As shown in Figure 3, the b value decreases with increasing σ+ p (or σp ) for all solutions of a common anion (or cation). For better visualization, a three-dimensional plot of the variation of the b value with cation and anion Pauling diameters is shown in Figure 4. The larger the Pauling diameter of an ion, the more negative is the b value and, as shown in Figure 1, the more anomalous is the viscosity behavior. The trend of the b value with ion diameter is in agreement with the Hofmeister (lyotropic) effect26 and may be interpreted in terms of ion hydration. When ions interact with water molecules, either hydrated or unhydrated structures occur. For the structure-making ions (kosmotropes) with small Pauling diameters, water molecules are firmly bound to the surface of these strongly hydrated ions. This enhances the water structure, leading to less water molecule mobility, increasing the viscosity, and resulting in a positive value of b. In contrast, for the structurebreaking ions (chaotropes) with large Pauling diameters, water molecules are weakly bound to the less hydrated or unhydrated ions. This reduces the water intermolecular structure and decreases the viscosity, which gives a negative value of b. As is the case for the
6270 Ind. Eng. Chem. Res., Vol. 42, No. 25, 2003
Figure 1. Relative viscosity of aqueous alkali-halide solutions at ambient conditions. Circles: experiments.25 Solid lines: correlations of the new model. Dashed lines: correlations of the Kaminsky equation. The dash-dotted line separates monotonic (upper left part) and anomalous (lower right part) viscosity behavior.
Figure 2. Plot of parameter 1 versus cation Pauling diameter. Lines are drawn to guide the eye.
Figure 3. Plot of parameter b versus cation Pauling diameter. Lines are drawn to guide the eye.
Ind. Eng. Chem. Res., Vol. 42, No. 25, 2003 6271 Superscripts and Subscripts 0 ) pure solvent EL ) electrostatic EX ) excess HS ) hard sphere IM ) ideal mixing p ) Pauling + ) cation - ) anion q ) activation
Literature Cited
Figure 4. Three-dimensional plot of parameter b versus cation and anion Pauling diameters.
parameter B in the Jones-Dole equation, the b value indicates the degree of order or disorder introduced by ions into the solvent structure. 4. Conclusions On the basis of the absolute-rate and liquid-state theories, we have developed a new statistical mechanicsbased model for the viscosity of electrolyte solutions. The model has successfully been used to correlate all experimental viscosity data of 20 alkali-halide aqueous solutions over a wide range of concentrations at ambient conditions, including solutions exhibiting either monotonic or anomalous concentration-dependent behavior. The physically important hydration effect is captured in the model, the degree of hydration is related to the optimized values of the adjustable parameters, and the trends found are in agreement with the Hofmeister series. The simple, analytical MSA expressions in the new model make it particularly well suited for engineering applications. Acknowledgment The authors thank the U.S. National Science Foundation (Grant CTS-0083709) for financial support of this research. Nomenclature Roman Characters a, b ) parameters from an ideal-mixing contribution A, B ) parameters in the Jones-Dole equation c ) concentration in molarity D ) parameter in the Kaminsky equation f ) Helmholtz energy p ) Planck constant NA ) Avogadro constant R ) gas constant T ) temperature V ) volume z ) ion charge magnitude Greek Characters ) dielectric constant η ) dynamic viscosity F ) ion number density σ ) ion diameter
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Received for review December 30, 2002 Revised manuscript received April 28, 2003 Accepted April 29, 2003 IE0210659
