3600
D. STIGTER
On the Viscoelectric Effect in Colloidal Solutions1
by D. Stigter Western Regional Research Laboratory,z Albany, California (Received August 6 , 1964)
The viscoelectric effect for charged colloidal particles is represented as an apparent shift, A, of the shear surface. The value of A is not very sensitive to the type of kinetic experiment. Comparison of theoretical values with an experimental value A < 1 8. for ionic detergent micelles suggests that the Lyklema-Overbeek estimate of the viscoelectric constant of water is considerably too high.
Introduction In a discussion of electrokinetic theory, Lyklema and Overbeek3 have introduced the viscoelectric effect. This is the change of the viscosity 7 of a liquid under the influence of an electric field E. The treatment is based on the relation4 7 = 7o(l
+ fE2)
=
10.2 X 10-l2 v . - ~cm.2
(2)
The results suggest that inclusion of the viscoelectric effect in electrokinetic theory raises the calculated {-potential substantially in most practical cases. This paper presents evidence to the contrary. The viscoelectric effect is introduced into the theory of intrinsic viscosity and of the rate of self-diffusion of charged colloidal particles with a relatively thin double layer. The treatment is applied to ionic micelles in aqueous detergent solutions and compared with the viscoelectric effect in micellar electrophoresis. The results are compared also with experimental viscosity data on micelles.
Apparent Shift of Shear Surface We wish to develop a convenient representation of the viscoelectric effect. To this end we consider shearing motion in a 1-1-valent salt solution contained between two infinitely large, flat surfaces of shear located a t x = 0 and x = xo in a rectangular coordinate system; see Fig. 1. The liquid is a t rest at x = 0 and has a velocity in the z direction v, = vo The Journal of Physical Chemistry
where 7 may be a function of x. Integration with respect to x yields
(1)
where v0 is the bulk viscosity of the liquid, E is derived from electric double layer theory, and the viscoelectric constantffor water is estimated f
at x = xo. I n the solution, when no pressure gradient is imposed, the equations of viscous flow reduce to
qbv,/bx
=
constant
(4)
Remembering the definition of 7, it is evident that the constant in eq. 4 equals the shearing force F per unit area in the liquid parallel to the flat surfaces. For example, on the stationary surface we have in the case of uniform viscosity 7 = v0
F
=
ro(bvz/ax)x-o= T o v o / ~ o
(5)
We now assume that the surface at x = 0 is charged such that the electrostatic potential at x = 0 is { while the potential in the bulk of the solution vanishes. When the surface charge gives rise to a variable viscosity in the solution near x = 0, we define an effective shear surface at x = A, where A is evaluated from
F = 7lbvZ/bx= qoVo/(xo - A)
(6)
(l! Presented in part at the Kendall Award Symposium of the Division of Colloid and Surface Chemistry at the l 4 i t h National Meeting of the American Chemical Society, Philadelphia, Pa., April, 1964. (2) A laboratory of the Western Utilization Research and Development Division, Agricultural Research Service, U. s. Department of Agriculture. (3) J. Lyklema and J. Th. G . Overbeek, J . Colloid Sei., 16, 501 (1961). (4) E. N. da C. Andrade and C. Dodd, Proe. Roy. Sac. (London), A204, 449 (1951).
VISCOELECTWCEFFECT IN COLLOIDAL SOLUTIONS
t‘
T‘
A
360 1
B
Figure 1. Liquid velocity vz between two flat plates, at x = 0 and 5 = xo,in shear motion: A, with constant liquid viscosity; B, with increased viscosity in liquid near x = 0.
Introducing eq. 1 for q, and assuming x o / A - t m , find from eq. 6
we
Specializing now for E to the Gouy-Chapman theory of thc flat double layer, as in ref. 3 , the result is, for A1
A = A’/’(A - 1)-’’’(2K
)-I
[In {A”’ cosh (e{/2lcT) -
+ (A - l)’/’] In {A1/’cosh ( e { / 2 k T ) + (A - 1)”’) {A”’ (A - l)’/’} {A”’
(A - l)”’)] (8b) where A = 32nrRTf/1000r, c is the concentration of salt in nioles/liter, and K is the reciprocal thickness of the double layer. If an effective shear surface is introduced, eq. 1 for the variable viscosity is replaced by a step function wherein the change from q = qo to 7 = m occurs a t distance A from the true surface of shear. The advantage of an effective shear surface is, of course, that it permits one to apply the usual hydrodynamic relations derived with q = 90, such as the Stokes friction law and the Einstein viscosity theory for spheres. Indeed, it can be shown that in translation as well as in rotation of spheres in viscous liquid the liquid velocity at the surface of the sphere is purely tangential. Hence, for diffusion and intrinsic viscosity of charged spheres with radius a, eq. 8 applies in the limiting case that a / A + m . Incidentally, in a complete analysis of viscosity and self-diffusion one should also take into account the interaction between the charged colloid
particle and thc mobile ions i n its double layer. A t present we assume that such interactions may be separated from the viscoclectric effect and can be described satisfactorily, e.g., by the expressions derived by Booth Lyklcma and Overbeekj report their results as a correction factor to the Helmholtz-Smoluchowski electrophoresis equation. With the help of the potential distance relation in a flat Gouy-Chapman double layer, this correction factor can be translated into an apparent shift A of the shear surface. This representation greatly facilitates the comparison between the viscoelectric effect in, say, viscosity and electrophoresis. It is obvious that, for a given structure of the double layer, the exact value of A depends upon the volume forces on the solution in the double layer region. Hence, we anticipate that A for electrophoresis differs from A for viscosity. This is borne out by the example given in the next section.
Application to Detergent Micelles We have applied eq. 8 t,o micelles of sodium dodecyl sulfate in aqueous sodium chloride solutions, using eq. 2 for f and assuming that one-half of the number of counterions is insidc the true shear surface of the micelles.6 As the theory has been developed for a flat double layer, the present calculations neglect the curvature of the micelle surfacc or, rather, we assunie KU = 03 , a being the radius of the spherical micelle. The results for A, recorded in Table I, range from 7 Table I : Viscoelectric Shift ( A ) of Shear Surface of Micelles of Sodium Ihdecyl Sulfate in Aqueous Sodium Chloride Solutions Calculated for KU = m Ionic strength. rnole/l.
0.0085 0.0153 0.0333
0.0523 0.101
--__ A A
_ I
0.66 0.91
1.4 1.8 2.5
_ _ I _ _ _
Intrinsic viscosity Electrophoresis
I-a
14.8 18 5 11.8 10.7
9.3
9.0 0.2 8.6 7.8 7.0
to 14 1. Introduction of the curvaturcof the iiiicelle surface should decrease the calculated values of A, particularly those a t low ionic strength. This correction for curvature is difficult to evaluate. Ilowver, the theory for the flat surface, Ka = a ,should be ap( 5 ) F. Booth, Proc. IzOy. SOC.(London). A203, 533 (1950); J . Chem. Phya., 2 2 , 1986 (1954). (6) D. Stigter, subniittetl for presentation at the IVth International Congress on Surface-Active Substances, Brussels, September. 1904.
Volume 68, Number 18 December, 1.904
3602
proached reasonably well for KQ = 2.5 fn Table I. I n this case one does not expect a very large correction for curvature. Thus it seems safe to conclude that the theory based on eq. 1 and 2 predicts A values of several
A.
The second o’iservat,ion on Table I is that A is of the same order of magnitude for viscosity as for electrophoresis. At this point we introduce some conclusions derived in a concerted analysis6 of various experiments on micelles of sodium dodecyl sulfate in which the viscoelectric effect was disregarded. The viscosity of micellar solutions, and also the rate of self-diffusion of micelles, showedothat the shear surface of micelles coincides within 1 A. with the surface enveloping the hydrated heads of the micellized ions. This means
The Journal of Physical Chemiatry
D. STIGTEk
that, up to a distance of less than 1 A. froni this envelope, the viscosity of water may be treated as a constant. Conseoyuently, the experimental value of A is less than 1 A. and, for all practical purpose3, the viscoelectric effect may be neglected in the interpretation of intrinsic viscosity data. Returning now to Table I we conclude that (a) the viscoelectric effect may also be neglected in the interpretation of micellar electrophoresis ; (b) the viscoelectric constant of water, as estimated by Lyklenia and Overbeek,3 eq. 2, is probably too high, perhaps wen by one order of magnitude. This author agrees with the final suggestion by Lyklema and Overbeek3 that “a determination of the viscoelectric constant for water and aqueous solutions [is]highly desirable.”
