Tracer Electrophoresis. II. The Mobility of the Micelle of S(4)

+

K C t h

(6)

The correction factor/3 can be expanded in even powers of a0p=

1

+ A&$+ A&4+ . . .

(7)

(19)

N. E. Hoskin, Traae.

Faraday Soc., 49, 1471 (1953).

HoskinlO found that this series converges very rapidly by comparing it with results obtained using an electronic computer. The coefficients Az and Ad, given in equations (4,2) of his paper, are complicated functions of Ka.

50

D. STIGTER AND K. J. MYSELS

VOl. 59

Fig.

From the known explicit expression of Verwey and OverbeekZ0for the surface charge of a flat double layer, one obtains for the limit Pf of p as KU becomes infinite

or, after expansion

It is advantageous to introduce the function

For any Qo this function compares the deviation of the Debye-Huckel approximation from the real value with its deviation from the Verwey and Overbeek value for a flat surface. For%<< 1 the expression for DQ reads, in view of equations 7 and 9, DQ= A2/24. I n Fig. 4, D Q is plotted versus log KU. The open circles have been derived from Hoskin's p-values (electronic computer) and equation 3. The squares have been calculated with b= KU and x= Kr, r - a is the distance from with series (7) and equ

ation (8). Although the latter values for a,,= 4 might be the surface of the spherical particle. The expression in brackets is a correction t o the slightly too low because terms higher than Oa4were omitted in series (7), the points indicate that all Debye-Huckel approximation represented by the preceding factor. For large values of b - 2, Le., curves follow quite closely the one for@O<< 1. far from the surface, the correction factor is a con(20) E. J. W. Verwey and J. Th.G . Overbeek,"Theory of Stability stant and equab l/r. of Lyophobia Colloids," Elsevier Publishing Co., New York, N. Y.,1948, pp. 25-26.(21)

As a0increases the effective thickness of the double layer decreases, the system resembles more a flat double layer and the curves shift toward D Q= 1. is finite, KU tends t o zero as u approaches If this limit. Under these conditions, the average potential within the double layer also tends to zero so that the Debye-Huckel approximation becomes valid.21 Therefore, D Q= 0 for KU= 0. On the other hand, according to its definition by (lo), D Q= 1for~u= 03. The curve for@O<< 1 and the points in Fig. 4 permit rather precise estimates of DQ for any values of Qo and KU and the relevant value of p can then be obtained directly with the help of equations 10and 8. Appendix I1 The Potential Distance Relation.-The empirical equation given by equations (4,3) of H~ s k i n '~ reads in our notation

Cf.G . 8 . Hartley, Trans. Faraday Boc., 86, 31 (1935).

Jan., 1955

MOBILITY OF MICELLE OF SODIUM LAURYL SULFATE

51

1 .o

0.90.8 0.7

r;r3

Ij: 0.6

2r3

v

v

+

I 0.5

5 0.40.3 0.2

II

0 . 10

0 . 1Fig. 5.-Interpolation

10 Ka. function for the potential-distance relation, lowest line from equation (13).

1

100

The factor y is a function of 'PO and of Ka and can be estimated by a procedure analogous to that described in Appendix I. In the limiting case of Ka= m, y can be computed by Verwey and Overbeek's explicit potentialdistance relation in a flat double layer.21 Some values are as follows

* oyr

2 3 5 6 8 4 1 1 1.019 1.079 1.178 1.309 1.474 1.656 2.08

It is convenient to define theinterpolation function~ ( 9 0,K a ) - 1 D@(@.o, KU)= (12)yr(W

which again compares the deviations of the Debye-Hiickel approximation for any 'Po from the true value for K a and for the flat double layer. For@O<< 1 the value of D* can be computed by the expressionQ o<<

-1

1

De.= 8be[2e4b&(4b) - ezbE(2b)J with E ( u )= - du (13)

sum:e

which is derived from the series given by Gronwall, LaMer and Sandved22 to express 'P in'powers of the particle charge. The lower curve of Fig. 5 shows the values thus computed. The points for higher 'Po values have been computed with Hoskinl* values of C m (= l/y) and Table IV. For all values of 'PO, Da -+ 0 for Ka -+ 0 and D -+ 1 for Ka --f m . Da is similar to DQ and the dotted lines have been drawn accordingly. In fact, both functions may be regarded as measures of the effect of divergence

of the electric field in the double layer around the spherical particle. This may explain the position of the transition range in the a= 1. neighborhood of K Equation 12 with Fig. 5, and the table of yf values, permits a precise estimate of y for any system of given 'POand Ka. The potential at any point is then given rapidly by equation 11.(22) T. H. Gronwall, V. Mer and R. Sandved, Phyeik. Z., 28, 368 (1928).