MOBILITY OF MICELLE OF SODIUM LAURYL SULFATE
Jan., 1955
45
In Table VI, column 5 represents an arbitrary and empirical solubility parameter for the hydrocarbon-ascribing all the effect of deviation to the alteration of the parameter of the hydrocarbon. It is considered significant that essentially the same TABLE VI numerical value of the parameter is obtained for COMPARISON OF VALUES OF (6, - &)' etc., FROM COMPONENTSeach hydrocarbon, whether it is derived from calculations based on the behavior of mixtures with subA N D SOLUTIONS PROPERTIES stances of higher, or of lower parameter. 1 2 3 4 5 (81 - 62) from GilpnarPnt, Acknowledgments.-The director of this work properties of Hydrocarbon, Temp., Pure cornSoluB. H.) gratefully acknowledges the assistance of System ponents tions" + (14A E-v / u31) o J a(J.grant from the National Science Foundation, a 1.901 1.110 7.89 TiC14-C7Hls 323.2 part of the time made available by which was spent 1.798 1.098 7.54 333.2 on this work; the authors acknowledge the assistCClr-CsH18 323.2 1.271 1,000 8.24 ance of the Atomic Energy Commission (as repre1.000 8.11 333.2 1.218 sented by Carbide and Carbon Chemical's Co.), the 2.65 7.75 C ~ F I ~ C I H I 333.2 ~ 1.56 Minnesota Mining and Manufacturing Co., and C~FigCgH18 333.2 1.73 2.77 8.05 the National Lead Co. in making samples of materials available free or at special prices. a Calculated by equation 3.
empirical parameter) to the non-regular mixtures of a hydrocarbon and a non-hydrocarbon, some derived results of experimental determinations are given in Table VI.
OK.
TRACER ELECTROPHORESIS. 11. THE MOBILITY OF THE MICELLE OF SODIUM LAURYL SULFATE AND ITS INTERPRETATION IN TERMS OF ZETA POTENTIAL AND CHARGE1 BY D. STIGTER~ AND K. J. MYSELS Contribution from the Department of Chemistry, University of Southern California Received July 89, 1864
Measurements of electrophoretic mobility of micelles of sodium lauryl sulfate in water and salt solution a t finite concentration of micelles are extrapolated to infinite dilution of micelles. Corresponding zeta potentials and charges a t the shear surface of the micelle are calculated according to several theories including new modifications of Booth's approach. The great effect of the curvature of the double layer and of relaxation effects is thus established for these systems which are far from ideal even a t the ChlC. A comparison of the zeta potential with that calculated from the charge of the lauryl sulfate ions forming a smooth micelle with a diffuse double layer leads to a large discrepancy. A "roughness" of the surface of the micelle is therefore suggested. The calculations involved required the computation of charge-potential and of potentialdistance relations for double curved layers a t high potential. These general problems are dealt with in appendixes.
The size, charge and shape of a micelle, besides their intrinsic interest, provide experimental tests for any theory of association colloids, yet they are still a matter of uncertainty. In the present paper we shall present accurate measurements of the electrophoretic mobility of micelles of pure sodium lauryl sulfate (NaLS) obtained by the tagging technique suggested by H ~ y e r . This ~ mobility is obviously related to the charge, size and shape of the micelle and to the composition of the solution. These relations are not yet fully elucidated but it is shown that they lead already to a reasonably accurate estimate of the f potential and of the charge of the micelle. The f potential is quite high: 65-100 mv. and the charge corresponds to 25-30% ionization, which is not readily explainable by the conventional picture of a smooth micelle surface. Since the experimental accuracy greatly exceeds that of the interpretation, the present data should be capable of yielding more accurate estimates of charge as the theories are developed further. (1) Presented in part at the J. W. MoBain Memorial Symposium of the Division of Colloid Chemistry at the American Chemical Society meeting, Chicago, September, 1953. (2) Bristol Myers Company Fellow, 1952-1953. Present address, Shell Research Laboratories, Amsterdam, The Netherlands. (3) H. W. Hoyer and K. J. Mysels, THISJOURNAL,64,966 (1950).
Experimental Methods and Materials.-The open tube method of measuring electrophoretic mobilities4 and the sodium lauryl sulfate6 and orange OT tracer6 used have all been described previously. Oil red N-1700 obtained from the American Cyanamid Co. was purified by solution in acetone and precipitation with water repeated twice and followed by double recrystallization from ethanol-benzene. Validity of the Method.-Three questions may be raised concerning the validity of our measurements. Is the mobility of the tracer accurate? Does introduction of the tracer dye modify the micelle? Does the mobility of the tracer equal that of the micelle? The first question has already been discussed in the first paper of this ~ e r i e s . ~The second question, about the disturbing effect of the dye, has been partially answered reviouslya but we can add the following: ( a ) D r . H . W. koyer has found7 that two different dyes, Sudan IV and Nile black, give the same mobilities (3.52 and 3.51 X cm.2 v.-l aec.-l, respectively) for the micelle of 5y0 otassium laurate solutions and we have found that orange 8 T and oil red N-1700 in 3.5% solutions of sodium lauryl sulfate give the same mobilities (3.70, 3.75 and 3.74, 3.73 X cm.2 v.-1 sec.-I, respectively). It would be quite a coincidence if the disturbing effect of different dyes were significant and yet the same. (b) The fact that orange OT in saturated solution does not affect measurably the critical micelle concentration of sodium (4) H. W. Hoyer, K. J. Mysels and D. Stigter, ibid., 68, 385 (19541. (5) R . J. Williams, J. N. Phillips and K. J. Mysels, Trans. Faradny Soc., submitted. (6) K. J. Mysels and D. Stigter, THISJOURNAL, 67, 104 (1953). (7) Ph.D. Dissertation, 1951, University of Southern California.
D. STIGTER AND K. J. MYSELS
46
lauryl sulfate6 suggests strongly that its effect on other micellar roperties should be negligible. &ere remains the question of identity of mobility of dye and micelle or in other words, whether the dye moves only by “riding” in a micelle. This is to be anticipated qualitatively because of the low mobility of the dye and its virtual insolubility in water. Quantitatively, we may assume that in a given soap solution a constant fraction K of the total dye concentration C is dissolved in water and the remainder ( 1 - K ) C is solubilized by the micelles. The total transport of dye J is the sum of the transport j , in water and that j,,, in the micelle with corresponding mobilities U , uw and urn. Hence j, = Ku,C and j, = ( 1 - K)u,C then J = UC = [Ku, ( 1 K)u,]C or
+ -
U
=
um[l
- K(l -
z)]
For a non-ionic dye the ratio uw/umvanishes and for any water insoluble, easily solubilized dye K must be negligible. Colorimetric measurements on orange OT solution6 show that even in our most dilute solutions K does not exceed 1.5 X 10-8 so that any difference between the mobility of this tracer and of the micelle is well within experimental error. Oil red N-1700 is more soluble in water and less easily solubilized by micelles so that K in this case is larger. We found that the mobility of this dye in dilute (0.45%)solution of NaLS was lower by 5 f 2% than that of orange OT while in more concentrated (3.5% NaLS) solution, where K is much smaller, the difference of mobilities was negligible as pointed out above. This confirms qualitatively the above reasoning, although the lack of precision does not permit quantitative proof.
Results.-The results of our measurements are shown in Fig. 1. Each open circle represents an individual measurement. Only measurements in which a source of error, such as a leak, was clearly apparent are omitted.
4.4
a
t L
1.I
\ O\O
I
t
I
I
I
I
,
1 2 3 Concn., @;./lo0cc. Fig. 1.-Electrophoretic mobility at 25 O of sodium lauryl sulfate micelles in water and salt solution: 0, orange OT tracer- 0 , oil red N 1700 tracer; 0,extrapolated values a t the ChC.
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-
VOl. 59
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.
MOBILITY OF MICELLE OF SODIUM LAURYL SULFATE
Jan., 1955
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 ionic 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 =
C1r
+
c212
+ 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, Cz 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
The series in square brackets converge very rapidly for the present data and may be set equal to unity without introducing a significant error. Then, (11) J. N. Phillips and K. J. Mysels, THISJOURNAL,submitted. (12) D. Stigter. R. J. Williams and K. J. Mysels, ibid.. submitted. ( 1 3 ) M. Smoluchowski, Z . ghysilc. Chem., 93, 129 (1918). (14) D. C. Henry, Proc. Roy. Soc. (London), 6133, 106 (1931). (15) J. Th. G. Overbeek, K o l l . Beih., 64, 316 (1943); Thesis, Utrecht, 1941. (16) F. Booth, Proc. Roy. SOC.(London), A205, 514 (1950); on p. 530,line 17,XI*(b) should read 2 / 3 X 1 * ( b ) .
47
with Booth's notation for the coefficients C, and introducing dimensionless units, (2) becomes u! %=x1*-
x3*
+ Ys* + qa*Z3* (3, XI * @zc . XI* (&*)4..
+,,
(3)
where = el/kT ({ = 25.7 mvolts for ap 0 . = 1. a t 25') and u' = (6sqe/DkT)u (u' = 0.752 for u = volt-' see.-' at 25" in water). The factors X " , Y* and Z* depend on KU onlyi4 while p* is related t o the ionic mobilities." Figure 2 shows l values computed from the extrapolated values of Fig. 1 using equation 3 when successive terms are taken into account and also the results obtained with Smoluchowski's expression. Their comparison shows clearly that in our systems, taking into account the thickness of the double layer increases the value by some 50% and introduction of relaxation effects raises it by another 1520%. 120
I
I
I
I
A
\
100
80
--.
60
OCHenry
'a-.-.--$oluchowski
40
20
?
003
OGl
0.05
0;I
0
0.2 0.4 log Ka. Fig. 2.---Zeta otentials of micelles of sodium lauryl sulfate a t the C M 8 in water and salt solutions a~ cidculated by several theories. -0.2
0
We have also used the original Booth series. The results, shown by the broken line in Fig. 2, are some 10% higher than those of the inverted series. This difference is not surprising since only an infinite series is identical with its inverted form. It is very difficult to estimate the error in l when the original Booth equation is to be solved for {. On the other hand, the error introduced by omission of higher terms in the inverted series is expected to be of the order of the last term accounted for, i.e., rBooth - {Overbeek, provided that the (17) Numerical values of Q*/6 are given in reference 15, page 316, table 3.
48
D.
STIO'rER AND
I
