An electrokinetic study of micellar solutions

An Electrokinetic Study of Micellar Solutions D A V E E. D U N S T A N m AND LEE R. W H I T E Department of Mathematics, University of Melbourne, Parkville, Victoria, 3052, Australia

Received October 17, 1988; accepted February 22, 1989 A theory is developed which enables the conductivity of a micellar suspension to be interpreted in terms of a zeta potential and applied to conductivity measurements on SDS in NaC1 solutions. This conductivity derived zeta potential is comparedwith the zeta potential obtained from mobilitiesmeasured by D. Stigterand K. J. Mysels,(J. Phys. Chem. 45, 59 ( 1955)). Good agreementis obtained over a wide electrolyte concentration range using the hydrodynamic radii obtained from neutron scattering for the theoretical interpretation. © 1990 Academic Press, Inc. 1. INTRODUCTION The electrokinetic equations ( 1 ) which describe colloidal transport p h e n o m e n a are based on a continuum model of hydrodynamics and electrostatics; viz., the water is considered as a continuous m e d i u m of bulk dielectric constant and viscosity up to the "slipping plane," inside which nothing moves. The ions are considered to be noninteracting point charges. We have attempted to experimentally determine the validity of these equations by comparing the electrokinetic (zeta) potentials obtained from conductivity and electrophoretic mobility measurements on polystyrene lattices (2). Experimental difficulties were found with precise preparation of finite volume fractions of latex in a known concentration of supporting electrolyte. The supporting electrolyte concentration appeared to vary with the volume fraction of the latex and consequently the theoretical interpretation of the measured conductance increments in terms of zeta potentials is obscured. For micellar systems, the total ionic concentrations are known and a theory m a y be developed in this case based, not on back-

ground electrolyte concentration, but on total ionic concentrations. We develop such a theory below. Other workers ( 3, 4) have attempted to extract electrokinetic parameters from micellar conductivities without using the exact numerical solutions of the governing equations or this model of the micellar ionic distributions. Conductivity measurements on SDS micelles are reported below and the derived zeta potentials are compared with those obtained from the mobility data of Stigter and Mysels ( 5 ). The theory of low frequency conductance in dilute suspensions of spherical charged partides has been developed using the continuum hypothesis outside the slipping plane by several workers (6, 7, 8). In this treatment the micellar conductivity K can be written as K = Kel(1 + ~ba(~', Ka) + O(t~2)),

[1.1]

where Kel is the conductivity of the neutral supporting electrolyte between the particles (the background electrolyte), q~is the volume fraction of the particles, a is a known (or at least computable) function of the zeta potential ~, and Ka is the ratio of the particle radius Present address: Department of Physical Chemistry, a to the Debye screening length K-t for the background electrolyte. For low volume fracUppsala University, Uppsala, Sweden. tions we neglect the higher order terms in q~. To whom correspondence should be addressed. 147 0021-9797/90 $3.00

Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990

Copyright © 1990 by Academic Press, Inc. All rights of reproduction in any form reserved.

148

DUNSTAN AND WHITE

As discussed above the background electrolyte concentration cannot be assumed constant as ¢ is changed. Indeed, as we shall demonstrate below, Kel = K0(1 +/3(~, ra)¢ + 0(¢2)),

[1.2]

miceUar volume fraction increases the effective excluded volume for the S- 1coion. As its bulk concentration is constant the average solution concentration over the total solution volume will decrease. The total surfactant concentration Cs may be written

where Ko is the conductivity at zero volume fraction of particles. Thus, to first order in the conductivity of the suspension is given by K = Ko(1 + (a + fl)(h).

[1.3]

CaL ee~'(r)/kTdV,

~ S = Cmic + ~

[2.2]

where

Experimental measurement of a + ~ enables the zeta potential to be calculated using the electrokinetic theory (8) for a(g', Ka) and the theory developed below for fl( ~', tea). The electrophoretic mobility is not sensitive to background electrolyte variations in the range caused by the ~ variations in a conductivity experiment, so mobility data can be directly converted to zeta potential using the O'Brien and White treatment ( 1 ) together with the total ionic concentrations given in the experiment.

is the micellar surfactant concentration (Nagg is the micellar aggregation number, Nmicis the number ofmicelles in the total volume V, and NAv is Avogadros number). We are assuming here that ionic species are distributed throughout the water volume Vw according to the usual Boltzman distribution, q ( r ) is the electrostatic potential at position r. For a micellar suspension with added electrolyte, MX, the total salt concentration is

2. THEORY FOR ¢1(~', Ka)

Cx fvw ee~(r)/kTdv' Csa,~ --- --V

Cmic ="

103NaggNmic/(NAvV)

[2.31

[2.4]

The chemical potential of the monomer in the bulk solution, outside the double layer of where Cx is the background concentration of the micelles, is set to be that of the critical X - outside the micellar double layers. Combining [ 2.11, [ 2.2 ], and [ 2.4 ] we have micelle concentration (cmc). Any monomer added above the cmc concentration, including Csal t • c m c cx [2.51 those excluded from the electrical double lay(Cs - cmic) ers, will form micelles. For the general case of a cationic surfactant S- and a counterion M ÷ In a dilute solution, the background electrolyte is overall neutral (i.e., micellar double layers we may write do not overlap) so that we may write Cs = cmc, [2.1] CM = Cs + CX [2.61 where Cs is the surfactant concentration (moles/liter ) outside the micelle double layers. Above the cmc the chemical potential of a monomer in the micelle is constant. While these assumptions may need to be modified at high surfactant concentrations where rodlike micelles are observed, near the cmc this will be a good approximation. This is consistent with the average surfactant concentration in the solution, excluding the micelles, decreasing with increasing total surfactant concentration above the cmc (9). Increasing the Journal of Colloid and Interface Science, Vol. 134,No. 1, January 1990

for the background M + concentration. On the assumption of independent ionic mobilities we may write for the background electrolyte conductivity Kel = CsXs- + CxXx- +

CM)kM +

/ c m c + Cmic -- C s \ ~-£Zgm7 ) , [2.7]

= Kcmc + Ksalt/

where Kcmc is the conductivity of the system (salt plus surfactant) at the cmc (~ = 0) and

MICELLAR SOLUTIONS Ks~t is the conductivity of the salt MX at concentration C~alt. Equation (2.2) may be written as cmc Cs = Cmic+ - V

×(gw--fvw(1--ee~(r)/kT)dV ).

[2.8]

For low qS,the integrand is only nonzero inside the individual micellar double layers so that

fVw( 1 - -

149

electrolyte concentration Csalt ~- cmc. Its dependence on ~ and Ka provides the functionality of fl(~, Ka) via [2.151. The function a(~', Ka) for the particle conductivity should also be calculated at the cmc background electrolyte concentration to this order in ~b. P is evaluated numerically for any given ~" value by first solving the PoissonBoltzmann equation for ~p(r) d2~

2 d~

dr2 + rdr

8a'e =

E

- -

NAy

ee'I'(r)/kr)dV

X 10-3(Csalt + cmc)Sinh ~-~ = Nmic

(1 -

e~(r)/kT)4~rr2dr,

[2.9]

where if(r) can now be considered to be the electrostatic potential at radius r outside an isolated spherical micelle in an infinite sea o f the background electrolyte. Using the result ~b = ---V--

)

[2.101

for spherical micelles of radius a, we may rewrite [2.81 as Cs = Cmi~+ cmc(1 - qS(1 + P)),

using a fourth-order Runge-Kutta technique. Using the above theory for fl( ~', Ka) and the electrokinetic program of O'Brien and White (1) for a(~', Ka) for a given cmc background electrolyte concentration and micelle radius, ~"can be calculated from a theoretical curve of a + fl when experimental values of a +/3 (obtained from the slope of conductivity versus micelle volume fraction) are supplied. F is very sensitive to micelle radius a as is readily seen from its definition. The experimental determination of volume fraction ~ uses the equation

[2.11]

( Cmi~]

where

[2.17]

~b = ~bO\cmc/,

3f[

1~ = ~5

(1 --

e~(r)/kr)r2dr.

[2.121

Substituting [ 2.11 ] into [ 2.7 ] we have Kel = Kcmc +

K~t(1 + r)q~ 1 -~(1

+r)

where =

[2.13]

q~ =

as

[2.14]

where gsalt

fl=K---~ (1 + I ' ) .

[2.15]

To this order F, as given by [ 2.12 ], should be calculated at ~b = 0, i.e., at the background

T47ra3 ,(cmc ~NAv10 -3

].

[2.181

From [2.11], this may be written in terms of experimental variables as

To the required order in ~bwe may write [ 2.13 ] gel = Kcmc(1 +/3q5),

[2.161

~bo((Cs - c m c ) / c m c ) I - ~o(1 + r )

,

[2.19]

which, for small q~, can be well approximated by ( C s - - cmc] = q~0 cmc / "

[2.20]

Again, we note the sensitivity to micellar radius through q~o. Journal of Colloid and Interface Science, Vol. 134, No. 1, January 1990

150

DUNSTAN AND WHITE TABLE I

Calculated ~"Values from Mobility Measurements (5) and Conductivity Measurements Using Micelle Radii Calculated from the Aggregation Number by the Tanford Equation (15) and from the Neutron Scattering Data of Hayter and Penfold (13) Conductivity

(ef/kT)

a (nrn) [NaCI] (M)

Mobility

Surface potential

N~,

Tan ford

Hayter

Tanford

Hayter

(ef/kT)

(e~o/kT)

0.0 0.001 0.005 0.01 0.05 0.1

64 64 65 66 68 75

2.22 2.22 2.23 2.24 2.27 2.32

2.42 2.42 2.44 2.46 2.59 2.68

--6.5 5.9 4.7 3.9

6.4 5.8 5.1 4.5 3.2 3.0

5.6 5.4 5.2 4.8 3.4 2.8

5.7 5.6 5.4 5.2 4.4 3.8

Note. For comparison, the surface potentials obtained by spectroscopic probe measurements by Hartland et al. (12) are also given.

3. MATERIALS AND METHODS Millipore ultrapure water was used in all experiments. The NaC1 was analytical grade. Sigma high purity SDS was used without further purification. The c m c determined by the D u N o u y ring m e t h o d was 7.57 × 10 -3 M . The surface tension showed a 0.5 m N m -~ m i n i m u m after the c m c suggesting a high purity o f sample ( 1 0 ) . The conductivity determ i n e d c m c was 8.3 × 10 -3 M w h i c h is in accord with literature values ( 1 1 ). T h e conductivities were all determined at 1 k H z using a H p 4192a impedance analyzer and YSI3403 cylindrical electrode of cell constant 1.0291 c m -1 at 1 kHz. All measurements were c o n d u c t e d at 25.0 ___0.5°C. Mobility values are taken f r o m Stigter and Mysels w h o used a tracer electrophoresis m e t h o d o f determ i n a t i o n (5). 4. RESULTS AND DISCUSSION T h e calculated zeta potentials o f SDS micelles in varying concentrations o f NaCI electrolyte, obtained from conductiviIy and m o bility measurements are shown in Table I. Table II shows the experimental data f r o m the conductivity measurements. The interpretation o f the conductivity data in terms o f zeta potentials is extremely sensitive to the value o f the micelle h y d r o d y n a m i c radius a. W h e n Journal of Colloid and Interface Science,/Vol. 134, No. 1, January 1990

a is calculated from the aggregation n u m b e r using the T a n f o r d equation (15) and adding a sulphate head group diameter o f 0.46 n m for the SDS micelle (4), the ~'values calculated f r o m conductivity measurements are significantly higher than those obtained from m o bility m e a s u r e m e n t s (5). In fact at low salt concentrations, no ~"value can be obtained to fit the experimental conductivity data using these radii. At high salt concentrations where ~"values were obtained, these zeta values were higher than the value o f surface potential (at the plane o f the sulphate head groups) obtained by Hartland et aL ( 1 2 ) using a spectroscopic probe technique.

TABLE II Experimental Conductivity Data [NaCI] (M)

cmc (X 10-3 M)

Kult (mS cm-I )

gcme (mS cm-~ )

(a + ~)exp

0.0 0.001 0.005 0.01 0.05 0.1

8.31 7.70 6.72 5.42 2.43 1.50

0.0005 0.1320 0.6000 1.1907 5.5350 1.100

0.5555 0.6300 1.0179 1.4800 5.6420 10.30

69.6 62.0 39.6 27.8 7.00 4.02

Note. (a +/3)exp is evaluated using the neutron scattering radius in Eq. [2.20] to calculate micelle volume fractions above the cmc.

MICELLAR SOLUTIONS When a is taken directly from the neutron scattering study of Hayter and Penfold (13), the calculated ~"values from conductivity are in excellent agreement with the ~ values derived from experimental mobilities (Table I). It should be noted that the neutron scattering radii were obtained at 40°C but little difference was found between the values obtained at 25°C and those used here (14). At low electrolyte concentrations, the conductivity zeta potentials are higher than both the mobility values and surface potentials using the neutron scattering radii. This m a y be explained readily by the presence of ionic impurities in the SDS, the effect of which is swamped at higher electrolyte concentrations. At these higher concentrations, the conductivity derived ~"values are below the surface potential values as might be expected if any ion binding in a Stern layer occurs and the slipping plane is located outside the plane of the immobilized ions. At the low electrolyte concentrations, the micellar solutions m a y exhibit nonideality due to double layer overlap. The conductivity concentration plots, from which the data of Table II were obtained, were linear from low volume fractions (where interactions should be minimal) up to volume fractions where some double layer overlap should occur. The possibility therefore exists that at low electrolyte the measured conductivity increment contains a contribution from micellar double layer interactions. The sensitivity to the radii used is problematic. There exists no agreement in the literature as to the exact radii, even for SDS, a well characterized surfactant system. The radii values obtained depend not only on the technique used but on the method of interpretation ( 13, 16-18 ). That agreement between the mobility and conductivity zeta potentials holds over a wide electrolyte concentration range is good evidence that the electrokinetic theory developed here for micellar suspensions is a good

151

description of these systems and that the neutron scattering radii are the best available data. We have therefore shown that simple conductivity measurements m a y be used to establish the zeta potentials of spherical micellar systems. Further measurements of both mobility and conductivity are being conducted on CTAB and D T A B micelles to further elucidate the range of validity of the theory. ACKNOWLEDGMENTS The authors thank Dr. Franz Grieser, University of Melbourne, and Professor Mats Almgren, University of Uppsala, for many helpful discussions. D.D. acknowledges the receipt of a Commonwealth Postgraduate Research Award. REFERENCES 1. O'Brien, R. W., and White, L. R,, J. Chem. Soc. Faraday Trans. 2 74, 1607 (1978). 2. Dunstan, D., Ph.D. thesis, University of Melbourne, 1988. 3. Stigter, D., J. Amer. Chem. Soc. 83, 1663 (1979). 4. Stigter, D., J. Amer. Chem. Soc. 83, 1670 (1979). 5. Stigter, D., and Mysels, K. J., J. Phys. Chem. 45, 59 (1955). 6. O'Brien, R. W., and Hunter, R. J., Canad. J. Chem. 59, 1878 (1981). 7. O'Brien,R. W., J. ColloidlnterfaceSci. 92, 17 (1983). 8. Ohshima,H., Healy,T. W., and White, L. R., J. Chem. Soc. Faraday Trans. 2 79, 1613 (1983). 9. Lindman, B., Puyal, M., Kamenka, N., Rymden, R., and Stilbs, P., J. Phys. Chem. 88, 5048 (1984). 10. Mukerjee, P., and Banerjee, K., J. Phys. Chem. 68, 3567 (1964). 11. Mukerjee, P., and Mysels, K. J., Natl. Bur. Stand. Ref Data Ser. 36, 51 ( 1971). 12. Hartland, G. V., Grieser, F., and White, L. R., J. Chem. Soc. Faraday Trans. 1 83, 591 (1987). 13. Hayter,J. B., and Penfold, J., Colloid Polyrn. Sci. 261, 1022 (1983). 14. Hayter, J., Private communication. 15. Tanford, C., J. Phys. Chem. 76, 3020 (1972). 16. Krahtovil, J. P., J. Colloid Interface Sci. 75, 271 (1980). 17. Bendedouch, D,, Chert, S-H, and Koehler, W. C., J. Phys. Chem. 87, 2621 (1983). 18. Warr, G. G., Grieser, F., and Evans, D. F., J. Chem. Soc. Faraday Trans. 1 82, 1829 (1986).

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