BARRY R. BRESLAU AND IRVING F. MILLER
1056 polyanion on complex formation has to be understood in terms of the effect it has on the ionization of the carboxyl groups and of the effect related to hydrophobic interactions. The two effects are not, in general, additive, as is evident on comparing series 3,4, and 5.
Acknowledgments. The authors are greatly indebted to J. E. Fields, J. H. Johnson, and F. D. Wharton of the Monsanto Company for preparing the copolymers, for many illuminating discussions and very useful advice.
On the Viscosity of Concentrated Aqueous Electrolyte Solutions1-2 by Barry R. Breslaua and Irving F. Miller Department of Chemical Engineering, Polytechnic Inatitute of Brooklyn, Brooklyn, New York ii801 (Received July 7, 1969)
The viscosities of concentrated aqueous electrolytic solutions have been correlated by use of an equation, developed by Thomas, for the viscosities of concentrated suspensions of macroscopic spheres. The Thomas equation, q/qo = 1 f 2.54 10.0542,where 7/70 is the relative viscosity and 4 is the particle volume fraction, is a truncated form of a seventh-order regression which has been shown effective for 4 < 0.25. For ionic particles, 4 is related to an “effective” rigid volume by I‘,= 4 / c , where c is the salt concentration; Veis obtained directly from viscosity measurements and is shown to be relatively concentration independent. Analysis of viscosity data for 72 salts in aqueous solution results in an additional correlation of 8, with the Jones-Dole B coefficient from which viscosities of concentrated aqueous salt solutio_nscan be estimated. For univalent salts, B = 2.907. - 0.018; for salts involving a multivalent ion, B = 6.06V, - 0.041. The difference between these two correlations is attributed to the differencesin hydrodynamic effectsand ion-solvent interaction.
+
Introduction The properties of aqueous electrolytic solutions are highly specific to the individual ions concerned and generalizations are difficult to find. In the case of viscosity, Jones and Dole4developed an empirical equation for the concentration dependence of viscosity of dilute electrolytic solutions, given by 9/70 =
1
+ A d ; + Bc
(1)
where 7 and qo are the viscosities of the solution and pure solvent, respectively; c is the solute concentration (moles/liter); and A and B are constants specific for the given solute-solvent system. This well-known equation has undergone extensive investigation, especially with respect to the interpretation of the constants A and B . Falkenhagen, et aL16-’ demonstrated that the square root term was due to longrange interionic forces and that the coefficient A could be theoretically calculated from the DebyeHiickel theory. While no quantitative theory exists for the independent determination of the linear B coefficient, important qualitative determinations have been advanced relating it to ion-solvent interaction.8 Since, in general, A / B is 2.5 with the ratio growing as the ions tested became smaller. I n particular, Kurucsev, et uZ.,~’ reported that eq 6 is valid only for ions of radius >5 While very little work has been done in the area of concentrated electrolytic solutions, a considerable effort has been made, both theoretical and empirical, with respect to determining the viscosity of concentrated suspensions. 1 9 ~ 2 0 Equations which result can generally be represented by a polynomial of the form
w.
dm
= 1
+ 2.54 +
+ k2@ + .
,
.
(7)
Vand21demonstrated that the addition of the secondand higher order terms to Einstein’s equation (eq 4) are due to particle interactions of various types. Thomas22made a critical analysis of extensive experimental data collected from 16 different sources, and, using statistical techniques, determined the coefficients
of the power series expressed in eq 7 to the seventh degree. These data were obtained with both rotational and capillary viscometers on systems with a range of particle diameters from 0.99 to 435 p and for such materials as polystyrene, rubber latex, glass, and methyl methacrylate. He further demonstrated that a simple second-degree equation (eq 8) will correlate the experimental data to within 97.5% of the 7/70 value for 4 0.25. q/qo =
1
+ 2.54 + 10.0542
(8)
Since, at 4 = 0.25, the average particle separation is only 0.35 particle diameter, this equation seems to be valid for quite highly concentrated suspensions. To date, even though there is considerable work available from suspension theory, no one has been able to relate suspension to solution theory in such a way that viscosities of concentrated electrolytic solutions can be calculated in a general manner.
Proposed Model In this contribution, eq 8 is taken as the starting point. Performing the transformation presented in eq 5 on eq 8 results in 9/90
= 1
+ 2 . 5 ~ V e+ 1 0 . 0 5 ~ ~ V , ~
(9)
where we have added the subscript e to V to designate it as an “effective” rigid molar volume. Equation 9 may be rearranged to solve for V ,
~ )’ 4(10.05c2)(1- q/qo) v, = - 2 . 5 ~+ d ( 2 . 52(10.05)c2
(10)
If viscosity-concentration data are available for any given salt, its “effective” rigid molar volume, V,, may be obtained from eq 10 as a function of concentration. This computation was performed with the aid of an IBM 360/50 computer for 7 2 different salts; data were obtained from a variety of s o ~ r ~ ecovering s ~ ~ a* ~ ~ ~ ~ (11) R. H. Stokes and R. Mills, “Viscosity of Electrolytes and Related Properties,” Pergamon Press, Oxford, 1965. (12) R. A. Robinson and R. H. Stokes, “Electrolytic Solutions,” 2nd ed, Butterworth, London, 1959. (13) V. E. Asmus, 2. Naturjorsch., 49,589 (1949). (14) A. Einstein, Ann. Phys., 19,289 (1906);34,591 (1911). (15) D.F. T. Tuan and R. M. Fuoss, J.Phys. Chem., 67,1343(1963). (16) J. F. Skinner and R. M.Fuoss, ibid., 68,2998 (1964). (17) T. Kurucsev, A. M. Sargeson, and B. 0. West, ibid., 61, 1567 (1957). (18) 8. P. Moulik, $bid., 72,4682 (1968). (19) F. R. Eirich, Ed., “Rheology, Theory and Applications,” Vol. 1, Academic Press, New York, N.Y., 1956. (20) T. F. Ford, J . Phys. Chem., 64,1168 (1960). (21) V. Vand, J . Phys. Colloid Chem., 5 2 , 277 (1948): 52, 314 (1948). (22) D.G. Thomas, J . Colloid Sci., 20,267 (1966). (23) N. A. Lange, “Handbook of Chemistry,” 10th ed, McGraw-Hill Book Go., Inc., New York, N. Y., 1961. (24) E. W. Washburn, Ed., “International Critical Tables,“ Vol. V, McGraw-Hill Book Co., Ino., New York, N. Y., 1929. Volume 74, Number 6 March 6, 10’70
1058
BARRY R. BRESLAU AND IRVING) F. MILLER
Table I : Effective Rigid Molar Volumes of Salts from Viscosity Data
B , l./mol
0.117 0.045 1.967 0.319 0.206 0.128 0.601 0.271 0.193 0.306 0.228 0.529 0.371 0.292 0.593 -0.052 -0.113 0.371 0.293 0.594 0.740 0.027 0.062 0.043 -0.011 0.023 0.272 0.184 -0.014 0.336 0.113 0.117 0.372 -0.075 -0.058 0.102 0.195 0.567 0.105 0.140 0.126 0.101 0.508 0.371 0.293 0.594 0.420 0.356 0.740 0.333 0,044 0.079 0.062 0,006 0.040 0.195 0.371 -0.014 -0.053 0.102 0.195 0.370 The J O t k T n d of Physical Chemistry
No. of data points
4 7 4 10 4 4 4 4 4 4 4 4 4 4 4
5 5 4 4 4 4 4 4 4 4 5 4 4 7 4 12 4 4 12 4 4 4 3 10 12 15 13 4 4 4 4 4 4 4 4 14 9
4 4 8 4 4 4 4 4 4 4
Concentration range, mol/l.
0.125-1.00 1.00-7.00 0.125-1.00 0.098-3.67 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0,125-1.00 0.125-1.00 0.125-1.00 0.5984.058 0,284-2.00 0.125-1 .OO 0,125-1 00 0.125-1 .oo 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.116-5 .SO6 0.125-1.00 0.125-1.00 1.00-4.00 0.125-1.00 0,525-6.445 0.125-1.00 0.125-1.00 0.500-6.00 0.125-1.00 0.125-1.00 0.125-1.00 0.250-1.00 0.597-4.728 0.441-4.964 0.512-8.047 0.0581-8.726 0,125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.500-7.00 1.00-5.00 0.125-1.00 0.125-1.00 0.972-6.24 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0.125-1.00 0,125-1.00 0.125-1.00
.
P,, Range of V , calcd, l./mol
0.0391-0.0533 0.0247-0.0355 0.1121-0.1180 0.0892-0.1539 0.0401-0.0422 0.0265-0.0331 0.0472-0.1081 0.0518-0.0550 0.0240-0.0409 0.0454-0.0627 0.0536-0.0582 0.0994-0.1122 0.0648-0.0719 0.0485-0.0560 0.1008-0.1212 -0.0138-(+0.001) 0.0463-( - 0.0437) 0.0650-0.0823 0.0557-0.0603 0.1017-0.1161 0.0842-0.0924 0.0122-0.0215 0.0244-0.0299 0.0187-0.0227 0.0026-( 4-0.0046) 0.0081-0.0158 0.0855-0.0957 0.0259-0.0334 -0.0012-(+0,006S) 0.0547-0.0596 0.0395-0.0465 0.0163-0.0566 0.0352-0 0389 0.0289-( 0.0075) - 0.02564 0.0103) 0.0408-0,0478 0.0247-0,0366 0.1918-0.2145 0.0314-0.0398 0.0431-0.0546 0.0420-0.0524 0.0365-0.0484 0.0863-0.0976 0.0639-0 0667 0.0557-0.0615 0.0976-0.1127 0.0660-0.0718 0.0592-0.0726 0.1030-0.1109 0.1089-0.1318 0.0222-0.0413 0.0338-0.0386 0.0317-0.0368 0.0089-0.0173 0.0191-0.0381 0.0729-0.0924 0.0712-0.0775 -0.0047-(-0.0003) -0.0149-( -0.13113) 0.0083-0.0096 0.0386-0.0463 0.0651-0.0681
-
-
-
+ I
-
I
l./mol, av value
0.0457 0.0304 0.1152 0.1048 0.0414 0.0310 0 * 0910 0.0538 0.0348 0.0504 0.0553 0.1044 0.0692 0.0533 0.1095 -0.0069 0,0454 0.0716 0.0579 0.1096 0.0888 0.0154 0.0265 0.0201 0.0011 0.0117 0.0917 0.0309 0.0028 0.0574 0.0416 0.0281 0.0372 -0.0120 -0.0180 0.0448 0.0324 0.2019 0.0331 0.0471 0.0456 0.0397 0.0930 0.0658 0.0596 0.1054 0.0695 0.0644 0.1085 0.1210 0.0307 0.0356 0.0336 0.0139 0.0310 0.0809 0.0738 -0.0021 0.0129 0 * 0091 0.0430 0.0662
-
-
Std dev, %
12.9 12.5 2.08 18.9 2.18 10.0 32.2 2.80 22.4 16.3 3.80 5.25 4.77 6.2 8.03 85.5 2.21 10.5 3.29 6.48 3.95 28.1 9.06 8.95 2.91 30.8 5.24 11.o 104 0 4.01 5.05 68.4 4.57 108.0 37.8 6.92 17.3 7.92 7.86 7.85 7.03 9.06 5.26 2.30 4.07 6.07 37.4 9.04 3.50 8.36 19.9 4.76 7.15 25.9 21 .o 10.8 3.65 90.5 13.2 6.59 9.06 2.11 I
VISCOSITIESOF ELECTROLYTE SOLUTIONS
1059
Table I (Continued) No. of data B , l./mol
0.292 0.593 0.141 -0.076 0.356 0.378 0.579 0.343 0.378 0.136
4 4 4
7 4 4 4 1 1 1
mol/l.
0.125-1.00 0.125-1.00 0.125-1.00 0.188-2.174 0.125-1.00 0.125-1.00 0.125-1 .OO 0.100 0.100 0.100
-
concentration range from 0.1 (approximately the upper 8 fV. limit of the Jones-Dole equation (eq 2)) to Since we were primarily interested in high concentration effects, we have only used salts for which data were available to concentrations of at least 1M. The results of these calculations are presented in Table I. Analysis of this work brings out a number of interesting points. Although for any given salt there is some variation in V , with concentration, this variation did not follow any defined trend. Furthermore, when average “effective” rigid molar volumes, F e , were calculated for each salt over the entire concentration range studied, it was observed that the range variation in V , was most pronounced for those salts having a low value of For example, of the 72 salts investigated, 56 have Ve > 0.03 I./mol; the relative, average standard deviation of V , values corresponding to these 56 salts is only 8.7% as compared to 42.5% Corresponding to the 16 remaining salts having V, 5 0.03 l./mol. To account for this effect, we need only note that salts with low values of V , have only a minor effect on solution viscosity (see eq 9). Consequently, in these cases the (1 - ? / q o ) term in eq 10 is quite small, and V , is obtained by taking the difference of two numbers essentially equal in magnitude. This type of calculation tends to magnify any differences which may result from 1 ~ e. lack of precision in the measured values of ~ 1 / 7 and A reasonable conclukon, therefore, that can be drawn from these results is that there exists a unique value of V,, independent of concentration, and that the observed variation of V , can be attributed to experimental error. With this as a hypothesis, the calculated average “effective” rigid molar volume, V e , is then the best approximation to the true “effective” molar volume for each salt. Having thus obtained unique values for V, based on high concentration viscosity data, we seek an additional correlation with the Jones-Dole viscosity B coefficient, which is also unique, but is based on low concentration data. The arguments for such a correlation are purely qualitative and are based on an understanding of the significance of Ve.
re.
Range of Ve oalod, I./mol
Ve, l./mol, av value
0.0583-0.0635 0.0985-0.1084 0.0209-0.0354 -0.0238-( -0.0101) 0.0608-0.0780 0.0540-0.0611 0.1037-0.1179 0.132 0.132 0.057
0 0606 0.1039 0.0287 -0.0160 0.0700 0.0580 0.1107 0.132 0.132 0.057
Concentration range,
points
I
Std dev, %
3.47 4.25
21.6 38.2 10.7 5.17 5.42
...
... I . .
Embodied in the development of Einstein’s equation (eq 4) are the assumptions that (1) the solution is infinitely dilute, (2) the spherical particles move in a continuum, and (3) there is no slip at the surface of the particles. Under the stipulation of infinite dilution, disturbances of the solvent flow pattern, caused by the presence of particles, do not overlap. As the particle concentration increases, a point is reached where perturbations of solvent flow can no longer be treated as being independent. This results in the power series extension of Einstein’s equation (eq 7) which forms the basis of the Thomas equation (eq 9) from which Ve values are obtained; the restrictions of no slip and a continuous medium (ie., that the radius of the suspended particle is large in comparison to that of a solvent molecule) are thus still in effect. On a microscopic level ionic particles certainly do not satisfy these restrictions, for not only are they essentially of the same dimensions as the solvent molecules but they move with considerable “slip.” The significanceof the V ,value thus obtained, therefore, is that of a “fictitious” or “effective” volume; i.e., it is that volume which a mole of solute particles behave like when considered, for purely hydrodynamic reasons, as rigid macroscopic spheres. A particle which has a major effect on neighboring solvent molecules, from physical considerations alone, would be expected to have a higher Ve than one which has a lesser effect. Since the B coefficient i s an empirical measure of the degree of ion-solvent interaction, a relation should therefore exist, with positive slope, when B is plotted against V e , This plot has been prepared for the 72 salts studied and is presented in Figures 1 and 2. Figure 1 represents a correlation for uni-univalent salts; a least-squares fit of the data presented in Figure 1results in the following correlation. B
= 2.Y0Gce
- 0.018
When values of B are plotted against Ve, for salts involving divalent ions, a second linear relationship, different from eq 11, is obtained. Figure 2 presents such a plot for all salts involving a divalent ion. Again, Volume 7.4, Numher 6 March 6,2970
1060
BARRYR. BRESLAU AND IRVINQ F.MILLER
:::I
range. It is mathematically correct, therefore, to compare coefficients of like terms, resulting in eq 6. I n the present case, however, one compares a secondorder equation with a first-order equation valid over different concentration ranges
c
0.7
0.6
0.5.8
0.3 0.2 -
7/70 =
0.4
17/70
-
0.1
-0.1
[/
-0.2
+ BC 1
(0.002 M
+
< c < 0.1 M )
(2.5Ve)c (1O.05Ve2)c2 (VeC < 0.25)
(2) (9)
Since the coefficient of the square term in eq 9 is always positive
-
0.0
1
1
I
-.06 -.04 -.02
' 0
1
.02
' .04
I
.06
' ' .08 .IO -
B I
.I2
1
.I4
1
.I6
I
.I8
1
.20 .22
ve
Figure 1. Jones-Do1e"B coefficient us. effective rigid molar volume, P,, for 33 uni-univalent salts. Least-squares correlation is given by B = 2.90 P e 0.018; B and are given in liters per mole.
ve
-
> 2.5Ve
(14)
1
a result that is substantiated in eq 11 and 12. Note that one cannot go further than this since it would be incorrect to take the limit of eq 2 and 9 as c + 0 since, in this case, eq 2 is no longer valid and eq 1 must be used.
Results and Discussion 0.9
-
0.80.7
-
0.6
-
0.5
-
LOCI3
-0.041
-
8
0.4
00'.32[
-0.2L"
-.06-.04-.02
'
0
'
'
'
'
.02 .04 .06 .08 .IO
-
'
.I2
'
.I4
I
.I6
'
.18
'
'
.20.22
ve
Figure 2. Jones-Dole B coefficient us. effective rigid molar divalent-univalent; volume Fefor 39 multivalent salts (0, A, univalent-divalent ; 0, divalent-divalent ; V, tri- and tetravalent). Least-squares correlation is given by B = 6.06 0.041 (excluding tri- and tetravalent); B and P, are given in liters per mole.
ve-
a good straight line is obtained with the least-squares fit resulting in the correlation B = 6.06v. - 0.041
(12)
It should be mentioned here for clarity that it is not reasonable to suppose that a relationship such as B = 2.5v.
(13)
should result from these correlations. In the case of comparing Einstein's equation (eq 4) to the modified Jones-Dole equation (eq 2), one compares two linear relationships both valid over the same concentration The Journal of Physical Chemistry
From the correlation presented above, it is now possible to estimate the viscosity of concentrated solutions of electrolyte with confidence from a knowledge of the B coefficient alone. From the salt B coefficient, either obtained from the literature or constructed from the B coefficient of the constituent ions, a value of V. is obtained using eq 11 or 12 depending on which is applicable. This value of Veis then used in eq 9 to predict the viscosity of the salt solution at the concentration of interest. Such an estimate shouId be valid up to concentrations in the 5-6 M region. This approach is demonstrated in Figure 3 in which estimated values for 17/90 are plotted against measured values (all at 25") at a series of concentrations up to 5 M for the 72 salts studied in aqueous solution. The applicability of the technique is obvious. With respect t o explaining the difference in slope between the uni-univalent and multivalent correlations, it should be emphasized that in this approach we are comparing two unique properties of a given salt, both pertaining to viscosity, but both the result of different effects. By going to the second-order Thomas equation (eq 9), one tacitly implies that at high concentrations the greatest contribution to the increase in viscosity of an electrolytic solution is due to a strictly hydrodynamic phenomenon, i.e., the interaction of solvent perturbations. This effect should be quite independent of charge type. On the other hand, when working with the modified Jones-Dole equation (eq 2), one implies that the concentration range is one where solvent flow perturbations can be considered independent and, therefore, the increase in viscosity is solely a result of ionsolvent interaction. This interaction is quite suhstantial for a divalent ion in comparison to a univalent ion of the same size and is reflected in an increased B coefi-
1061
VISCOSITIES OF ELECTROLYTE SOLUTIONS 2.6
2.4
2.2
2 .o
0.0.8
I
I
I
I
I
I
I
I
I .o
I .2
I .4
I .6
1.8
2.0
2.2
2.4
( q+,) Figure 3. Calculated values of relative viscosity
( q / q o ) vs.
exp.
experimental values (0, 1M;
cient. This accounts for the fact that for a given Pe value, divalent salts exhibit a marked increase in B values. Note, in particular, that B = 2.5 Ve is not to be expected in this case and that all that is required is be one of positive slope. that the relationship B = f( re) Merker and Scott26studied the viscosity of solutions of tetrakis (trimethylsilyl) methane in ten different organic solvents and attempted to correlate their results by use of Einstein’s equation (eq 4). The observed viscosities proved to be linear with concentration (at low concentration) but, instead of a slope of 2.5, they found slopes ranging from 0.92 to 3.09. They explain their results by suggesting that, as a result of solute-solvent interaction, small density changes take place at the interface between solute and solvent molecules. These density changes lead to changes in the microscopic viscosity of the solvent at the interface and, thus, appear as increases or decreases
2.6
0, 2
M; A, 3 M ;
0, 4 M ;
., 5 M).
in the Einstein slope, depending on whether the microscopic viscosity is, respectively, higher or lower than the pure solvent value. If these arguments are applied to the aqueous systems studied in our work, the implication is that divalent ions, as a result of their higher charge density, tend to induce much more structuring in the solvent water at the interface than do monovalent ions of the same size and, thus, the microscopic viscosity of the solvent water at the interface is higher than it would be for monovalent ions of the same size. It might be noted that one might expect that the tri- and tetravalent salts would correlate with an equation of still higher slope than eq 12. Unfortunately, there are insufficient data from which to draw a conclusion. The available data are presented in Figure 2. (25) R. L. Merker and M. J. Scott, J. Colloid Sci., 19,245 (1964).
Volume 74, Number 6 March 6,1070
