Tracer Electrophoresis. II. The Mobility of the Micelle of S(2)

Interpretation Any interpretation of the experimental results is bound to involve a model of the system. Our considerations are based on a spherical micelle (which is the only one susceptible of exact calcu-

lations) surrounded by an electrical diffuse double layer which neutralizes its charge and whose average thickness depends on the effective ionic strength of the solution, L e ., the ionic strength prevailing outside of these double layers. Both sodium chloride, when present, and any unassociated NaLS contribute to this effective ionic strength. Several theories of electrophoretic behavior of colloidal ions in the .presence of small ions are known and we shall discuss them in some detail later. None of them takes into account the interaction of the colloidal ions with each other. These theories can

therefore be applied only a t“infinite dilution of micelles,” yet as shown by the slope of lines of Fig. 1 this interaction has a marked effect. Infinite dilution of micelles may be assumed a t the critical micelle concentration (or CMC) if properly defined.6 Therefore, mobility values at the CMC are needed and these can be obtained only by extrapolation. Extrapolation.-Extrapolation requires a theory of micellar interaction. An outline of such a theory has been proposed elsewhere,8 and the present data extrapolated according to it. We assume that the repulsion of electric double layers prevents any direct interaction of micelles except where the volume occupied by the double layers is so high that they must overlap. This, is the case in water alone above about 2% concentration and explains the curvature found in this region. Otherwise the true mobility of the micelle, with reference to the solvent and a t constant salt concentration in the bulk of the solution, remains the same. However, C1- ions are mostly excluded from the solution within the double layer so that as the concentration of NaLS increases the salt concentration in the bulk of the solution increases and the mobility of the micelle correspondingly decreases. The slopes thus calculated are shown by the dotted lines in Fig 1. The remaining large part of the slope is attributed to a linear effect of back flow of solution entrained by the double layers aa suggested by Enokssong for sedimentation rate. This frame-of-reference effect is interpolated for intermediate concentrations from values obtained for the well-defined slopes a t 0, 0.01 and 0.1 M NaC1. Finally, these slopes are fitted to the points for each concentration. The average deviation of experimental points from the lines is 0.35% which is only slightly more than the error due to colorimetric dye determinations. The extrapolated mobilities are given closely by the relation loglo u= 3.582 - 0.115 logloC, where C is the total molarity of N a f ions a t the CMC. It may be noted that any reasonable extrapolation gives results within about 1.5% for the mobilities a t the CMC.l0 The 5 Potential.-This potential between the shear surface of the micelle and the bulk of the solution is related to the mobility a t the CMC u, the“thickness” of the double layer 1/~, and(8) D. Stigter, Rec. truu. chim., 73,605 (1954); Thesis, Utreoht 1954. Due t o the use of less accurate values of parameters the f potentials quoted am slightly too low. (9) B. Enoksson, N u t w e, 161, 934 (1948). (10) The CMC values used are from the best line of reference 5.

Jan., 1955

MOBILITY OF MICELLE OF SODIUM LAURYL SULFATE

47

the radius a of the micelle. For the radius a of the micelle we take the results of molecular weight determination by light scattering" assuming a density of 1.14 and a monomolecular hydration layer 1.5 A. thick. This agrees with values obtained from self diffusion measurements12 within the accuracy of the latter. The ion

ic strength determining 1/~ is taken as due t o the total NaLS present a t the CMC plus any NaCl added. Other constants used are ha+= 45, XC.-= 70, XLS-= 18 for ionic mobilities; D= 78.5 for the dielectric constant; and= 0.894 centipoise for the viscosity of the solvent. The relation between mobility and zeta potential can be always expressed as a power series in{u=

with Booth's notation for the coefficients C, and introducing dimensionless units, (2) becomesu!%=x1*x3*

C 1 r

+

c 2 1 2

+ c3rs+

c414

...

(1)

Theories of electrophoretic mobility deal with the values of the coefficients Ci. The first approximation was given by Smoluchowski,13who set C1= D/47rq and omitted the higher terms. This is valid for a double layer which is flat or thin compared to the radius of the particle, Le., KU>> 1. Henry1*took into account the commensurate dimensions of the particle and its double layer and obtained C1 as a function of KU varying between Smoluchowski's value and D/67rq while the higher terms are again omitted. Henry's formula is valid if deformation of the double layer by the electrophoretic motion is negligible. To take this deformation or relaxation effect into account Overbeekl6 calculated the two next terms, C z and C3)for the general case. Later BoothI6 investigated the case of symmetrical electrolytes where CZ= 0 and obtained expressions for Ca and C4. In both these treatments C1has the value calculated by Henry and the higher C's are function of KU and of the mobilities of the ions present. The C3 term is common to both treatments and despite independent derivation and different form, its value is the same within the computational error of about 3%. Normally u is the experimentally accessible quantity and l the calculated one. The application of Overbeek's and especially Booth's expression is quite awkward as it involves the solution of third and fourth degree equations, respectively. A great simplification may be obtained by inverting series (1). For the case a t hand of a symmetrical salt where Cz= 0 we obtain

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