- Email: [email protected]
Diffusion in surfactant solutions
Diffusion in Surfactant Solutions ROBERT M. WEINHEIMER, 1 D. F E N N E L L EVANS, 2,z AND E. L. CUSSLER z Department o f Chemical Engineering, Carnegie-Mellon University, Pittsburgh, Pennsylvania 15213 Received April 14, 1980; accepted September 24, 1980 The diffusion coefficients in water of Triton X-100 and sodium dodecyl sulfate were measured as a function of concentration using the Taylor dispersion technique. For Triton X- 100, a nonionic surfactant, the diffusion coefficient drops from 7.4 x 10-7 ctn2/sec at 0.45 g/liter to 6.45 × 10-7 cm2/sec at 5 g/liter. The diffusion coefficient of methyl yellow solubilized in Triton X-100 is close to that of the surfactant. This behavior is quantitatively consistent with a chemical equilibrium between monomer and micelle. For sodium dodecyl sulfate, an anionic surfactant, the diffusion coefficient increases from 1.76 x 10-6 crn2/sec at 0.01 M to 4.53 × 10-6 cm2/sec at 0.125 M. The increase is less when 0.1 M NaC1 is added. The diffusion coefficient of the methyl yellow solubilized by the SDS is significantly less than that of the surfactant, particularly at low ionic strength. This behavior can be quantitatively explained by including electrostatic coupling between monomer, micelle, and counterion. INTRODUCTION
Diffusion in surfactant solutions is one of the steps in the kinetics of detergency. These chemical kinetics are important in such diverse areas as digestion, drug solubilization, cleaning of fabrics, and degreasing of surfaces (31). Such kinetics includes diffusion and chemical reaction between single molecules and miceHes. Studies of diffusion in these solutions commonly use experimental methods which implicitly emphasize micelle diffusion. In this paper we report the average diffusion coefficients which are those involved in processes like digestion and detergency. These averages include the effect of surfactant monomers as well as surfactant micelles. The dynamic balance between these species produces unusual effects. Current address: Celanese Plastics and Specialties Company, 9800 E. Bluegrass Parkway, Jeffersontown, Ky. 40299. 2 Current address: Department of Chemical Engineering and Material Science, University of Minnesota, Minneapolis, Minn. 55455. 3 To whom correspondence should be addressed.
We also compare the monomer-micelle averages with earlier studies of micelle diffusion. This comparison is instructive because past studies of micelle diffusion have often led to dramatically conflicting results. Two sets of results illustrate these differences for sodium dodecyl sulfate. The older set, measured by Stigter et al. (32), follows micelles tagged with solubilized dye. This set shows that the diffusion coefficientdrops 30% over the concentration range studied. The more recent set, by Corti and Degiorgio (6-9), studies micelle diffusion by laserDoppler light scattering. These measurements indicate that the diffusion coefficient increases by 100% over the same concentration range. This is a major discrepancy. The two data sets do not even suggest the same trend of diffusion coefficients, let alone the same values. In addition, adding sodium chloride decreases the coefficients found by light scattering but increases those found by dye solubilization. The obvious explanation is that the different experimental techniques represent different averages over the various
357
Journal of Colloidand Interface Science, Vol.80, No. 2, April1981
0021-9797/81/040357-12502.00/0 Copyright© 1981by AcademicPress, Inc. All rightsof reproductionin any formreserved.
358
WEINHEIMER, EVANS, AND CUSSLER
1 = Kc'I'c~ [1] forms of monomer and micelle present. In the dye diffusion method the dye is present in which K is an equilibrium constant; cl only in the micelles, and is essentially inand c2 are the concentrations of surfactant soluble in the surrounding solution. This monomer and counterion, respectively; and method thus emphasizes micelles. In the n and q are constants, equal to the average light-scattering technique, the diffusion conumber of surfactant ions and counterions efficient determined is usually thought to be in the micelle. The number one appearing a z-average, higher than a weight average. on the left of this relation represents the Such an average should also emphasize constant activity of the new micelle "phase." micelles. The second picture of micelle formation With the stimulus of these discrepancies, (e.g., Mukerjee (26)) is that of a fast chemiwe will also in this paper explore the concal reaction between micelle and monomer. centration dependence of surfactant diffuThis picture leads to the equilibrium sion. In doing this we will emphasize the diffusion averaged between monomer and cm = K ' c'l'c~, [2] micelle, since this is the average important in areas like digestion and detergency. where Cm is the concentration of micelles; Whenever possible we will compare the re- K ' is a different equilibrium constant; and suits obtained with those found for micelle the other variables are the same as those diffusion. defined above. An important characteristic of this equation results because n and q are often large, of order 50. In this case much THEORY of the surfactant is present as micelles and In this section we develop equations de- adding more surfactant does not change ca scribing micelle diffusion in terms of the dif- and c2 dramatically. This means that Eqs. fusion coefficients of the various species [1] and [2] are often indistinguishable present in solution. In this development, we experimentally. will make many simplifying assumptions and Both these models permit surfactant difthen check the effect of these simplifications fusion by two roughly parallel paths. The with diffusion experiments. In particular, detergent can diffuse as free monomer or as we will assume that micelle reactions like part of a micelle. The same two paths are formation and destruction occur much more also available for counterion diffusion. The rapidly than diffusion does. How valid this models presented here for mutual surfactant assumption is depends on the time of the diffusion take into account these two paralmicelle reactions and the distance over which lel diffusion paths in developing an expresdiffusion occurs. If these reactions have sion relating the overall diffusion coefficients haft-lives around 100 /zsec, then diffusion to the total surfactant concentration. must occur over distances greater than 10 -4 We have developed expressions for both cm. In processes like detergency and diges- ionic and nonionic detergent diffusion for tion, it certainly does. In more exact terms, each of these two models. For nonionic this means that the second Damkrhler num- detergents we give a synopsis for both modber in these systems is very large. els. For ionic detergents the phase separaThere are two limiting pictures of micelle tion model leads to much simpler equations formation in detergent solutions. The first, than the chemical equilibrium model. Acpromulgated by McBain (24), assumes mi- cordingly, we report only the phase separaceUes are a separate phase which separates in tion model for ionic diffusion. Details, inthe solution. For an ionic surfactant this phase cluding equations for the chemical equilibseparation model suggests the equilibrium rium model, are available elsewhere (34). Journal of Colloid andlnterface Science, Vol. 80, No. 2, April 1981
359
DIFFUSION IN SURFACTANT SOLUTIONS
Two Models for Nonionic Diffusion
Diffusion of a nonionic surfactant is described by the two continuity equations (29):
further approximation. Specifically, we assume that Eqs. [7] and [ 10] can be replaced by Cm
0 = DIVZcI - nrm
[3]
0 = Dm~72Cm + rm
[4]
=
K'c'I' = __1 (c - c l ) n
and -
1
(c - CMC).
[11]
n
for which D1 and D m a r e the diffusion coefficients of m o n o m e r and micelle, respectively; ca and cm are the m o n o m e r and micelle concentrations; and rm is the rate of micelle formation. This rate is positive if micelles are in fact formed; it is negative if they are destroyed. If we multiply Eq. [4] by n, add it to Eq. [3], and integrate once, we find the total flux J of the detergent: -J
= O l V c a + nOm•Cm.
[5]
The diffusion coefficients are assumed constant, independent of concentration. For the phase separation of nonionic detergents, Eq. [1] reduces to 1 = Kc~.
[6]
Thus the phase separation model sets c1 equal to a constant, presumably the critical micelle concentration or CMC. F r o m a mass balance, c = cl + ncm, [7] where c is the total detergent concentration. As a result,
1
Vcm = - - Vc n
[8]
and Eq. [5] becomes -J
= OmVc.
[9]
Above the critical micelle concentration, all diffusion occurs by transport of micelles through the surrounding m o n o m e r solution. For the chemical equilibrium model, the situation is slightly more complex. Equation [2] becomes Cm = K'c]'. [10] The mass balance is unchanged from Eq. [7]. Unfortunately, these equations cannot be solved in a simple analytical form without
This assumption will be more accurate well above the CMC, where c far exceeds Cl and most of the surfactant is in the micelles. We can find V c m and VCl from this relation and combine with Eq. [5] to yield
-J
=
[D m + DI( 1 t1'° n \--~--7] x
c - CMC
Vc.
[12]
The quantity in square brackets is the apparent diffusion coefficient D of the surfactant. This apparent coefficient should be proportional to (c - CMC) "/~-~, or since n is commonly large, to (c - CMC)-L Thus the results of the phase separation model in Eq. [9] and of the chemical equilibrium model in Eq. [12] give a diffusion coefficient which varies differently with concentration. The phase separation model leads to a constant value Dm. In contrast, the chemical equilibrium model gives an apparent diffusion coefficient which varies explicitly with concentration. If we plot this apparent coefficient vs (c - CMC) -1, we should obtain a straight line. The intercept of this plot corresponds to the micelle diffusion coefficient, and the slope is related to the monomer diffusion coefficient. Such plots are consistent with experiment, as shown later in this paper. A Phase Separation Model for lonic Diffusion
We now turn to the more complex case of an ionic surfactant. The greater complexity results because all diffusing species are coupled electrostatically, and because the Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
360
WEINHEIMER, EVANS, AND CUSSLER
counterion concentration can be independently altered by adding another electrolyte. However, our basic strategy is the same as in the nonionic case. We first write a flux equation; we then write mass balances and chemical equilibria between the species present; and we finally combine these to find an apparent diffusion coefficient. We begin with the continuity equations for monomer " 1 , " counterion '°2/' and micelle " m " :
O = Dl(V2Cx q- Z1V(c1V-~)) - nrm,
[13]
O = D2(V2c2 -}- z2V(CzV R-R~)) - qrm,
[14]
O =Dm(V2Cm~ZmV(Cm~-~))
static potential. If we multiply Eq. [15] by (n + q), add the result and Eqs. [13]-[14], and integrate once, we again find the total surfactant flux J:
The factor 2 results because we regard the diffusion of 1 mole of monomer and 1 mole of counterion as diffusion of 1 mole of surfactant. This flux equation is subject to the constraint of zero current:
+ rm, [15]
0= in which the Di are the appropriate diffusion coefficients, assumed to be constant; the zi are the charges; and 6/RT is the electro-
-j=
°)
(
- 2 J = D1 Vcl + z~clV-R-~
ziDi VCi q- ZiCi
•
We use this restraint to find Vch/RT; inserting the result into Eq. [16], we find
[D~(D2c2 + q ( q - n ) D m c m ) V c l + De(DlCl + n ( n - q)Dmcm)VC2 + Dm(qDlCl + nD2c2)VCm] DIC 1 A- D2C 2 +
The first term in square brackets is caused by a flux due to the monomer concentration gradient; the second term is due to the counterion concentration gradient; and the third term is due to the micelle concentration gradient. Because the concentrations of the various species are not directly known, they must be written in terms of the total concentration c. Material balances again lead both to Eq. [7] and to the following, C2 = Cs + Cl + (n -- q)Cm
[19]
in which cs is the concentration of counterion present due to sources other than the surfactant, such as added electrolyte. A third constraint is due to the phase separation model, given for this case by Eq. [1]. Journalof Colloidand InterfaceScience,Vol. 80, No. 2, April 1981
[17]
i=1
(n - q)2cmD m
[18]
As in the nonionic case these equations cannot be easily solved without approximation. To do this, we recognize that Eq. [7] can be approximated by 1 1 Cm=--(c-cl)=--(cn n
CMC)
[20]
SO
Vcm = --1 Vc.
[21]
n
Because we are using the phase separation model this may not look like an approximation; however, according to Eq. [1], the addition of added electrolyte will depress the monomer concentration above the CMC. In a similar way, we assume that Eqs. [7] and [19] can be combined as
361
D I F F U S I O N IN S U R F A C T A N T S O L U T I O N S
C2 = Cs + C1 +
(1-
%)(C-
Cl)
unexplored problem in detergency. Finally, we determine Vct by combining Eqs. [1] and [221:
= cs + CMC + (1 - q ) ( c - CMC) [22] VC 1 =
SO
a~ __
Note that this implies ca is a constant, and so implicitly neglects multicomponent effects (11). While we found this neglect was experimentally justified in this set of experiments, we suspect that it can be a major and
-j=
tU
V ( C 2 q/n)
(Lll/n(&t(1--q)C2(l+q/~t')VC \ n 7\
- -\K}
n /C 2
Using Eqs. [21], [23], and [24] we can rewrite Eq. [18] in terms of Vc:
DI(Dzc2 + q(q - n)Dmcm(-q/n)(1 - q/n)(cJcz) + Dz(Dlc~ + n(n - q)Dmcm(1 - q/n)) + Dm(qD,ca + nDzc2)(1/n)
The term in square brackets represents the apparent diffusion coefficient D we measure experimentally. Parallel arguments for the chemical equilibrium model give very similar but more complicated equations (34). These additional complexities do not improve agreement between the theory and the experiments. The development above ignores many chemical complexities of micelle formation (2). It assumes micelles form and are destroyed much faster than the diffusion. It assumes these micelles are of constant size and charge, and that the monomers do not undergo any premicellar association. It assumes that monomer and micelle have different diffusion coefficients, but that these coefficients are constants, independent of concentration and ionic strength. These assumptions are significant, but even with them, the results are complex algebraically. For the diffusion experiments to date, these assumptions are justified. Future experiments may require a more complicated theory. EXPERIMENTAL Sodium dodecyl sulfate (SDS) (BDH Chemicals) was of 99% purity and was used
Vc.
[25]
as received. The surface tension vs concentration curve of this sample showed no minimum in the region of the CMC. Triton X-100 (TX-100) (isooctylphenoxypolyethoxyethanol; Packard Instrument Co., Downers Grove, Ill.); the water-insoluble dye, methyl yellow (p-dimethylaminoazobenzene; Pfaltz and Bauer, Flushing, N. Y.); and reagentgrade sodium chloride (Fisher) were also used as received. Diffusion coefficients were measured with the Taylor dispersion technique. Because this technique is very fully described elsewhere (13, 34) only a synopsis is given here. The equipment basically consists of a long thin tube, through which a solution is slowly flowing. Two tubes were used in this study. Both were made of stainless steel by the Superior Tube Company of Norristown, Pennsylvania. The first tube was 104.55 m long with an inside diameter of 0.0787 cm and the second tube was 36.58 m long with an inside diameter of 0.0508 cm. The longer tube was made into a coil 41 cm in diameter while the shorter tube was made into a coil 25 cm in diameter. The pump was a CMP-2 metering pump (Laboratory Data Control, Riviera Beach, Fla.). As delivered, it proJournal of Colloidand Interface Science, Vol.80. No. 2, April1981
362
WEINHEIMER, EVANS, AND CUSSLER
Samples of slightly different composition were then injected into the flowing stream. This method of measuring the concentration dependence of the diffusion coefficient is that used successfully in the present work. The differences between past studies using the Taylor dispersion method hinge largely on the detector chosen to analyze c as a function of time. In this work, we used two detectors, mounted in series, so we could h a v e two different measurements of c. These detectors were a differential rec - 6 ~t ,,1/2 -- I `R ) e -12D(t-tal/a~t [26] fractometer and a spectrophotometer. The refractometer was Waters Associates Model \tJ Crnax~" R401, capable of measuring refractive index in which c is the surfactant concentration differences of 10-8 RI units. The UV-VIS averaged across the tube's radius; Cmax is absorbance detector, always mounted upthe maximum value of this concentration; stream, was Waters Associates Model 440. is the average surfactant concentration With eight different wavelengths across the without the pulse; t is the time; tR is the ultraviolet-visible spectrum, it was caparesidence time, i.e., the tube's length divided ble of measuring absorbance differences of by the average solution velocity; D is the 0.001 absorbance units. The detector signal surfactant's diffusion coefficient; and a is was recorded by a Bristol 10-in. 10-mV recorder. the tube's radius. To check our technique, we first measThis result, basic to the Taylor dispersion apparatus, has the same general Gaussian ured the diffusion coefficient of urea and of form for the decay by diffusion of a pulse in sucrose, using the differential refractometer concentration. The interesting feature of this to analyze the output. The results were result is that the diffusion coefficient appears analyzed by assuming that the concentration in the numerator of the exponent. As a re- varied linearly with refractive index, and by sult, the measured pulse is broad for slow fitting the data to Eq. [26], using nonlinear diffusion, and narrow for fast diffusion. This least squares and a DEC-20 computer. To is the antithesis of intuition and is the in- our dismay the initial results were routinely teresting feature of the Taylor method. It 10% below the initial value. We finally traced occurs when the experimental conditions are this to an error of 3% in the tube radius such that radial diffusion is fast compared specified by the manufacturer. With this corto convection. This is assured by a very rection, results for urea and sucrose were routinely within 1% of the accepted values. low fluid velocity and a very long thin tube. The measurements of surfactant diffusion This technique has been applied to many liquid systems, including hydrocarbons in also used the differential refractometer. To hydrocarbons; gases in water; and organo- begin these experiments, a solution of known tin compounds in organic solvents. More composition was prepared. This solution importantly, it has been shown that the tech- was used to flush the tube and ultimately nique can be used to measure the concen- became the carrier stream. A small (20/zl) tration dependence of diffusion coefficients sample of a more concentrated solution was in water-ethanol systems (28). This was then injected into the carrier. The measured done by pumping a water-ethanol solution diffusion coefficient was found to be indeof known composition through the tube. pendent of the concentration in the pulse. vided pulse-free flow at six flow rates ranging from 2.4 to 120 ml/hr. The pump motor was later replaced with a Bodine stirring motor with variable speed control for use with the shorter tube. An experiment with this equipment is made as follows. At time zero, a pulse of a different solution is injected through a Rheodyne Model 70-10 valve at the tube's inlet. The shape of this pulse, measured as it flows out of the tube, is given by (3, 33)
Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
DIFFUSION IN SURFACTANT SOLUTIONS The measurements of dye diffusion used the spectrophotometer. The experiments were made in two different ways. In the first, a gradient in dye concentration only is introduced. The injection solution was made by taking a sample from the pump supply and saturating the sample with dye. The next day the supernatant surfactant solution was diluted by a factor of five with more surfactant solution from the pump supply. This procedure assures that the surfactant concentration in the injection and in the flowing stream are equal. In the second type of experiment, gradients in both dye and surfactant concentrations were introduced. The injection solution was made by saturating a concentrated surfactant solution with dye. The supernatant solution was diluted by a factor o f I0 with more of the concentrated surfactant solution. This new solution was that injected. Again, the tube was carefully flushed with the carrier solution before injection. In experiments made with other solutes such as NaC1, we were careful to eliminate concentration differences in all but the desired species. For the experiments with added electrolyte, solvent and solute were both made by dissolving surfactant in salt solution taken from the same vessel to ensure equal salt concentration. However, even when no salt gradient exists, one can develop because of multicomponent diffusion, producing a profile different than that in Eq. [26]. We saw no such profile here but we expect such profiles in experiments which extend our results. Three nuances about this procedure deserve emphasis. First, in all experiments above the CMC, the carrier stream contained surfactant at or above the CMC. This is important. If pure water is used as a carrier and the pulse is above the CMC, the results can be much harder to interpret. Second, even though the pulse is of higher surfactant concentration, it is routinely only a few percent above the carrier surfactant. This small change in concentration is possible because
363
TABLE I Diffusion Coefficients of Triton X-100 in Water as a Function of Concentration at 25°C~ Conch
107 D o TX- 100
107 D ~ Dye
0.02e 0.45 0.51 0.61 0.76 1.01 1.76 5.01
8.25 + 0.45 7.40 _+ 0.05 7.28 ± 0.16 7.32 ± 0.10 7.05 ± 0.28 6.93 ± 0.22 6.93 ± 0.15 6.49 ± 0.12
---8.15 ± 0.77 --7.54 ± 0.14 --
Concentrations are in grams per liter because the molecular weight of this material is not well defined. Diffusion coefficients are in cm2/sec. All measurements were made using the Taylor dispersion apparatus. b These are mutual diffusion coefficients calculated from refractive index. c These are dye diffusioncoefficientscalculated from the absorbance at 436 nm. This concentration is one-tenth the CMC, so the diffusion is dominated by that of the monomer.
the detectors are so sensitive. Third, we occasionally used pulses with lower surfactant concentration than in the carrier. We found the same diffusion coefficients in both cases. RESULTS AND DISCUSSION Both the measured diffusion coefficients o f nonionic and of ionic surfactants show the characteristics predicted by the theory given above. The nonionic and ionic cases are best discussed sequentially.
Nonionic Results The diffusion coefficients of TX-100 decrease slightly with concentration, as shown in Table I. The precision of the measurements above the CMC ranges from + 1 to _+4%. The diffusion coefficients of the waterinsoluble dye, measured at two TX-100 concentrations, are also included in Table I. In both cases, the diffusion coefficient of the dye was slightly higher than the corresponding surfactant diffusion coefficient. We are Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
364
WEINHEIMER, EVANS, AND CUSSLER I
I
I
I
I
9.0
~" B.O E o
~
+ 2%
~o x 7.0
6.(-
i I 0.0 2.0 (Concentration-Critical
I
I i 4.0 Micelle Concentration) -t-~/g)
Fro. 1. Diffusion of Triton X-100 at 25°C. This variation with surfactant concentration is consistent with the chemical equilibrium model for micelle formation, but not with the phase separation model of this process.
not certain whether this difference is significant. Finally, a diffusion coefficient measured well below the CMC is also reported in Table I. This value, which should be dominated by monomer diffusion, is of poor precision due to surfactant adsorption on the tube wall. Such adsorption caused a badly skewed profile when a solution of TX100 below the CMC was injected into pure water. This asymmetry is reduced significantly by having a small amount of TX-100 in the carrier stream. Few diffusion coefficients of TX- 100 have been previously reported in the literature. Giglio and Vendramini (16) found a diffusion coefficient in a Solution of 10 g/liter to be 5.8 x 10-7 cm2/sec. The most concentrated solution studied here is 5 g/liter, at which the TX-100 diffusion coefficient is 6.5 x 10-r cm2/sec. The diffusion coefficient is decreasing with concentration, so a drop to the value found by Giglio and Vendramini seems reasonable. The variation of TX-100 diffusion with concentration is inconsistent with Eq. [9], based on the phase separation model of micelle formation. This equation predicts the diffusion coefficient should be independent of concentration above the CMC. This is not supported by the data. Journal of CoUoid and Interface Science, Vol. 80, N o . 2, April 1981
The diffusion data, shown in Fig. 1, are consistent with Eq. [12], based on the chemical equilibrium model. The intercept in Fig. l, equal to about 6.7 × 10-7 cm2/sec, represents the diffusion of the micelle. If we assume the micelle is a rigid sphere, we can use this result and the Stokes-Einstein equation to calculate the micelle' s diameter. The result is 70/~, consistent with the previous estimate of 80 A (22). However, since these micelles may well be nonspherical, such an estimation is speculative. The slope of the line in Fig. 1, equal to 2 × 10-11 g/cmsec, can in principle be estimated using Eq. [12]. The estimate requires values of the monomer diffusion coefficient D1, the equilibrium constant K , and the aggregation number n. While estimates of all these quantities are available in the literature, they vary widely. As a result, this estimate is not instructive. Plots like that in Fig. 1 are also predicted by other theories (5, 23). These theories assume that the observed diffusion coefficient is an arbitrary average of the diffusion coefficients of the various species present. The average is assumed to be arithmetic, weighted with mole fractions, even though harmonic and geometric averages are often more effective in estimating diffusion (e.g., (12)). This average leads to plots like Fig. 1. The reason for prefering the nonionic theory in this paper is that it substitutes sensible chemistry for arbitrary averaging.
Ionic Diffusion The diffusion coefficient of sodium dodecyl sulfate was also measured as a function of surfactant concentration, both with and without added electrolyte. The results, presented in Table II, show a large increase in SDS diffusion coefficient in the absence of added salt. Over the concentration range studied the coefficient nearly tripled. Most of this increase occurs between 0.01 and 0.06 M. It occurs in the face of a viscosity increase of 30% (21), and so is exactly op-
DIFFUSION
IN SURFACTANT
TABLE
365
SOLUTIONS
II
D i f f u s i o n C o e f f i c i e n t s o f S o d i u m D o d e c y l S u l f a t e a t 25°C ~ Concn
10~ D ~
lip D ¢,'j
1@ D ~'~
Concn in
106 D b
106 D ~'a
106 D e'~
in H 2 0
SDS
Dye
Dye
0.1 M NaCI
SDS
Dye
Dye
-0 . 8 7 + 0.01 ----0 . 7 6 +- 0.01 -----
-1.4 ± 0.11 ----------
0.01 0.02
1.18 ± 0.03 1.20 _ 0 . 0 6
0.91 +_ 0.01
0 . 9 4 ± 0.01
0.03 0.04 0.05
1.28 _+ 0 . 0 2 1.29 ± 0.01 1.35 ± 0 . 0 2
__
m
0 . 8 6 + 0.01
0 . 8 8 -+ 0.01
0.06 0.08 0.10
1.34 ± 0 . 0 5 1.47 ___ 0.11 1.45 ± 0 . 1 2
0.01 0.02 0.0U 0.03 0.04 0.05 0.0525 0.06 0.07 0.125 0.15 s
1.76 2.40 3.0 3.03 3.65 4.03 4.05 4.38 4.53 5.7
± ± ± ± ± -+ -± ± ± ±
0.03 0.17 0.6 0.11 0.08 0.07 0.08 0.07 0.12 1
D
C o n c e n t r a t i o n s a r e in m o l e s p e r l i t e r a n d d i f f u s i o n c o e f f i c i e n t s a r e in c m V s e c . E x c e p t a s n o t e d , all e x p e r i ments were performed using the Taylor dispersion apparatus. b T h e s e d i f f u s i o n c o e f f i c i e n t s w e r e c a l c u l a t e d f r o m t h e r e f r a c t i v e i n d e x profile. c T h e s e d i f f u s i o n c o e f f i c i e n t s w e r e c a l c u l a t e d f r o m t h e a b s o r b a n c e profile a t 4 3 6 n m . a T h e s e e x p e r i m e n t s u s e d a g r a d i e n t in d y e c o n c e n t r a t i o n a l o n e . T h e r e w a s n o g r a d i e n t in s u r f a c t a n t concentration. e T h e s e e x p e r i m e n t s u s e d g r a d i e n t s in b o t h d y e a n d s u r f a c t a n t c o n c e n t r a t i o n . T h e s e e x p e r i m e n t s w e r e p e r f o r m e d u s i n g t h e d i a p h r a g m cell.
posite to that expected. We were initially so surprised by this large increase that we questioned the validity of these results. For this reason we made experiments at the extreme concentrations with the diaphragm cell. The same large increase was seen. Such a large increase has also been seen in water for dodecyl sulfonic acid (25). In the presence of 0.1 M NaC1 as an added electrolyte, the diffusion coefficient of SDS increases much less dramatically with concentration as shown in Table II. The increase in diffusion coefficient across the same range of concentration is only about 25%. The diffusion coefficient of the waterinsoluble dye was measured both with and without added salt. These experiments were made in two ways, with a gradient only in the concentration of the dye, and with a gradient of both dye and surfactant. Results from both types of dye experiments are also reported in Table II. The diffusion of dye is less than that of detergent. The values of detergent diffusion reported in Table II are measured by means of re-
fractive index, and so weigh each molecule roughly equally, be it present as a monomer or in a micelle. This equal weighing means that these diffusion coefficients are those important in mass transfer processes like detergency and digestion. At the same time these averages may differ from those found from other experimental methods which weight micelles more heavily. To date, three types of studies of SDS diffusion have appeared in the literature. Each of the three uses a different experimental method, although all three methods are expected to emphasize transport by micelles. The three are exemplified by the laser-Doppler study of Corti and Degiorgio (6-9), the dye solubilization experiments of Stigter et al. (32), and the radioactive tracer work of Kamenka et al. (20). While these groups of experiments are not always made at exactly the same conditions used here, comparisons are instructive. The laser-Doppler results agree with the refractive index results obtained here, as shown in Fig. 2. For example, our data in Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
366 5q
WEINHEIMER, EVANS, AND CUSSLER ~ # v •
i-
Our Our Our Our
Work, Work, Work, Work,
No Salt, Surfaclant No Salt, Dye O.fM NaCI, Surfactant O.IM N a C
~
g
? c~2
I~-SWM in 0.1M NoCI A-Corti 8¢ Degiorgio O.05M NaCI Irn- SWM in 0.03 M NaCI • - Corti & Degiorgio O.1M NaCI Jo-SWM in Water • - Corti & Degiorgio O.3M NaCI
oL~
I
0.02
I
/
I
I
0,04 0.06 0.08 O.tO SDS Concentration, mol/.~
I
0.12
Fie. 2. Diffusionof sodiumdodecylsulfatemeasured by differentmethods. The increases in surfactantdiffusion result from electrostatic couplingbetween the various speciespresent. The decreasesin dye diffusion reflect the solubilizationof the water-insolubledye by the surfactantmicelles.The abbreviationSWM refers to Stigteret al. (32). water and in 0.1 M NaC1 neatly bracket the light-scattering data at 0.03 M NaCI. Our data at 0.1 M NaC1 are about 10% higher than those found by light scattering, a difference which almost certainly reflects a bias away from monomer diffusion in the lightscattering experiments. At the same time the dye diffusion data of Stigter et al. agree closely with the dye results obtained here, shown in Fig. 2. This good agreement is a consequence of the fact that both our dye data and those of Stigter et al. represent micelle transport. Initially, we were surprised that the light-scattering results agree with our refractive index studies, and not with our dye results. After all, the refractive index measurements represent diffusion averaged over monomers and miJournal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
celles, but biased toward the small, fast monomer. The dye experiments are dominated by micelles, because the dye molecules are in micelles. Light-scattering experiments, we felt, should also be dominated by micelle diffusion, for light-scattering data are averages more strongly influenced by larger particles. Upon reflection, we are not so sure. While light-scattering methods do commonly give weight average or z-average diffusion coefficients, we wonder whether they do so for micelles. After all, micelles are not stable molecules like proteins or synthetic polymers, but are dynamic, constantly adding some monomer molecules and losing others. We suspect this dynamism may give unusual characteristics to laser-Doppler light-scattering experiments. The third set of data for possible comparison are the intradiffusion data of Kamenka et al. (20), called "tracer" diffusion data in some of the literature. These data report the transport of tagged dodecyl sulfate in a ternary system of tagged sodium dodecyl sulfate, untagged sodium dodecyl sulfate, and water. The diffusion coefficients of this tagged anion show a huge drop over the range of concentrations studied here. The interpretation of experiments like these is controversial even for much simpler chemical systems than that used here (e.g., (1, 10)). We believe connecting mutual and intradiffusion in this case will turn out to be very difficult indeed. We now turn to trying to predict the increase in surfactant diffusion using the theory developed earlier in this paper. This theory included micelles which form rapidly from monomers, which are electrically charged, and which include counterion bonding. Unfortunately, the key theoretical result in Eq. [25] suggests no simple plot. Instead, we can predict SDS diffusion by independent measurements of seven quantitie s: the three diffusion coefficients of monomer D1, counterion D2, and miceUe Dr.; the monomer and counterion concentrations c1
367
D I F F U S I O N IN S U R F A C T A N T S O L U T I O N S
and c2; and the aggregation numbers n and q. The values used for counterion and monomer diffusion coefficients result from measurements of radioactive tracers in dilute solutions below the CMC when intradiffusion effects are negligible. For the counterion Clifford and Pethica (5) determined a value of 1.35 x 10-~ cm2/sec in water. For the monomer in water, Kamenka et al. (20) measured a value of 5.7 x 10-~ cm2/sec. The micelle diffusion coefficient over this concentration range is taken as 8.0 x 10-7 cm2/ sec from the dye data (Stigter et al. (32)). The concentrations of monomer and counterion were found from ion-specific electrode experiments (Kale et al. (19)). These data assume that activity coefficients are unity. Counterion and micelle concentrations are then calculated using material balances. The only variables not specified are n and q. The value of n was taken as 50 after that determined by Huisman (18). The ratio of q/n is taken to be 0.84, consistent with the range of values shown in Table III. The exact choice of n and q did not sharply affect the results. The theoretical prediction based on Eq. [25] and these seven independent measurements is compared with our direct experimental results in Fig. 3. Agreement is excellent. Thus the surprising increase observed for mutual diffusion is apparently a result of micelle rearrangement and electrostatics.
o × c3 1
I
0.02
I
I
I
I
0.04 0.06 0.08 0.I0 SDS Concentration, mol/I
I
I
0.12
0.14
FIG. 3. Predictions of surfactant diffusion. The theoretical prediction, shown by the solid line, agrees closely with the actual data points. The prediction is based on independent measurements like monomer diffusion coefficient and micelle aggregation number.
CONCLUSIONS
This work shows how surfactant diffusion occurs by parallel transport of micelles, monomers, and, where present, counterions. For nonionic surfactants the diffusion coefficient drops as the concentration increases and micelle diffusion becomes more important. For ionic surfactants the diffusion coefficient rises as the concentration increases because of electrostatic coupling between the species present. In both cases, the coefficients measured are those important in mass transfer and do not focus on the micelles alone. APPENDIX: N O M E N C L A T U R E a
T A B L E III
c
Ratio of Sodium Ions to Dodecyl Sulfate Ions in SDS Micelles in Water at 25°C
c1 c2
q/n
Method
Reference
0.86 0.85 0.85 0.84 0.82 0.80 0.78
emf light scattering specific ion electrodes emf equilibrium dialysis osmotic coefficients emf
(14) (27) (19) (4) (15) (17) (30)
Cm Cs
CMC D D1 D2 Dm J K
radius of the Taylor dispersion tube total concentration monomer concentration counterion concentration micelle concentration concentration of added electrolyte critical micelle concentration apparent diffusion coefficient monomer diffusion coefficient counterion diffusion coefficient micelle diffusion coefficient total surfactant diffusion flux equilibrium constant for the phase
Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
368
K,
n
q R rm T t tR ,7.1 Z2 Zm
WEINHEIMER, EVANS, AND CUSSLER
separation model for an ionic surfactant (Eq. [1]) equilibrium constant for the mass action model for an ionic surfactant aggregation number of micelle number of counterions bound to a micelle gas constant rate of formation of micelles temperature time residence time in the Taylor dispersion tube charge on the monomer charge on the counterion charge on the micelle electrostatic potential ACKNOWLEDGMENTS
This work was primarily supported by the National Science Foundation, Grant CPE80-14567. Other support came from the National Institute of Arthritis, Metabolic and Digestive Diseases Grant SR01 AM16143-07 and the National Science Foundation Grant CPE80-17376. REFERENCES 1. Albright, J. G., and Mills, R., J. Phys. Chem. 69, 3120 (1%5). 2. Aniansson, E. A. G., Wall, S. N., Almgren, M., Hoffmann, H., Kielmann, J., Ulbricht, W., Zana, R., Lang, J., and Tondre, C., J. Phys. Chem. 80, 905 (1976). 3. Aris, R., Proc. Roy. Soc. London Ser. A. 235, 67 (1956). 4. Botre, C., Crescenzi, V. L., and Mele, A.,J. Phys. Chem. 63, 650 (1963). 5. Clifford, J., and Pethica, B. A., Trans. Faraday Soc. 60, 216 (1964). 6. Corti, M., and Degiorgio, V., Opt. Commun. 14, 358 (1975). 7. Corti, M., and Degiorgio, V., in "Photon Correlation Spectroscopy and Velocimetry" (H. Z. Cummins and E. R. Pikes, Eds.). Plenum, New York, 1977.
Journalof CoUoidand InterfaceScience, Vol.80, No. 2, April1981
8. Corti, M., and Degiorgio, V., Ann. Phys. (Paris) 3, 303 (1978). 9. Corti, M., and Degiorgio, V., Chem. Phys. Lett. 53, 237 (1978). 10. Curran, P. F., Taylor, A. E., and Solomon, A. K., Biophys. J. 7, 879 (1967). 11. Cussler, E. L., "Multicomponent Diffusion," Chap. 6. Elsevier, Amsterdam, 1976. 12. Cussler, E. L., A I C h E J . 26, 43 (1980). 13. Evans, D. F., Chan, C., and Lamartine, B. C., J. Amer. Chem. Soc. 99, 6492 (1977). 14. Feinstein, M. E., and Rosano, H. L., J. Colloid Interface Sci. 24, 73 (1967). 15. Fishman, M. L., and Eirich, E. R., J. Phys. Chem. 75, 3135 (1971). 16. Giglio, M., and Vendramini, A., Phys. Rev. Lett. 38, 26 (1977). 17. Huff, H., McBain, J. W., and Brady, A. P., J. Phys. Colloid Chem. 55, 311 (1951). 18. Huisman, H. F., Proc. K. Ned. Akad. Wet. Ser. B 67, 367 (1964). 19. Kale, K., Cussler, E. L., and Evans, D. F., J. Phys. Chem. 84, 593, (1980). 20. Kamenka, N., Lindman, B., and Brun, B., Colloid Polym. Sci. 252, 144 (1974). 21. Kodama, M., and Minra, M., Bull. Chem. Soc. Japan 45, 2265 (1972). 22. Kushner, L. M., and Hubbard, W. D., J. Phys. Chem. 58, 1163 (1954). 23. Lindman, B., and Brun, B., J. Colloid Interface Sci. 42, 388 (1973). 24. McBain, J. W., Trans. Faraday Soc. 9, 99 (1913). 25. McBain, E., Proc. Roy. Soc. London Ser. A 170, 415 (1939). 26. Mukerjee, P., J. Pharm. Sci. 63, 972 (1974). 27. Phillips, J. N., and Mysels, K. J., J. Phys. Chem. 59, 325 (1955). 28. Pratt, K. C., and Wakeham, W. A., Proc. Roy. Soc. London Ser. A 336, 393 (1974). 29. Robinson, R. G., and Stokes, R. H., "Electrolyte Solutions," Chap. 11. Butterworths, London, 1959. 30. Shedlovsky, L., Jakob, C. W., and Epstein, M. B., J. Phys. Chem. 67, 2075 (1963). 31. Short, B. A., in "Detergency" (W. G. Cutler and R. C. Davis, Eds.). Dekker, New York, 1972. 32. Stigter, D., Williams, R. J., and Mysels, K. J., J. Phys. Chem. 59, 330 (1955). 33. Taylor, G. I., Proc. Roy. Soc. London Ser. A 219, 186 (1953). 34. Weinheimer, R. M., Ph.D. thesis, Carnegie-Mellon University, Pittsburgh, Pa. (1979).
Diffusion in surfactant solutions is one of the steps in the kinetics of detergency. These chemical kinetics are important in such diverse areas as digestion, drug solubilization, cleaning of fabrics, and degreasing of surfaces (31). Such kinetics includes diffusion and chemical reaction between single molecules and miceHes. Studies of diffusion in these solutions commonly use experimental methods which implicitly emphasize micelle diffusion. In this paper we report the average diffusion coefficients which are those involved in processes like digestion and detergency. These averages include the effect of surfactant monomers as well as surfactant micelles. The dynamic balance between these species produces unusual effects. Current address: Celanese Plastics and Specialties Company, 9800 E. Bluegrass Parkway, Jeffersontown, Ky. 40299. 2 Current address: Department of Chemical Engineering and Material Science, University of Minnesota, Minneapolis, Minn. 55455. 3 To whom correspondence should be addressed.
We also compare the monomer-micelle averages with earlier studies of micelle diffusion. This comparison is instructive because past studies of micelle diffusion have often led to dramatically conflicting results. Two sets of results illustrate these differences for sodium dodecyl sulfate. The older set, measured by Stigter et al. (32), follows micelles tagged with solubilized dye. This set shows that the diffusion coefficientdrops 30% over the concentration range studied. The more recent set, by Corti and Degiorgio (6-9), studies micelle diffusion by laserDoppler light scattering. These measurements indicate that the diffusion coefficient increases by 100% over the same concentration range. This is a major discrepancy. The two data sets do not even suggest the same trend of diffusion coefficients, let alone the same values. In addition, adding sodium chloride decreases the coefficients found by light scattering but increases those found by dye solubilization. The obvious explanation is that the different experimental techniques represent different averages over the various
357
Journal of Colloidand Interface Science, Vol.80, No. 2, April1981
0021-9797/81/040357-12502.00/0 Copyright© 1981by AcademicPress, Inc. All rightsof reproductionin any formreserved.
358
WEINHEIMER, EVANS, AND CUSSLER
1 = Kc'I'c~ [1] forms of monomer and micelle present. In the dye diffusion method the dye is present in which K is an equilibrium constant; cl only in the micelles, and is essentially inand c2 are the concentrations of surfactant soluble in the surrounding solution. This monomer and counterion, respectively; and method thus emphasizes micelles. In the n and q are constants, equal to the average light-scattering technique, the diffusion conumber of surfactant ions and counterions efficient determined is usually thought to be in the micelle. The number one appearing a z-average, higher than a weight average. on the left of this relation represents the Such an average should also emphasize constant activity of the new micelle "phase." micelles. The second picture of micelle formation With the stimulus of these discrepancies, (e.g., Mukerjee (26)) is that of a fast chemiwe will also in this paper explore the concal reaction between micelle and monomer. centration dependence of surfactant diffuThis picture leads to the equilibrium sion. In doing this we will emphasize the diffusion averaged between monomer and cm = K ' c'l'c~, [2] micelle, since this is the average important in areas like digestion and detergency. where Cm is the concentration of micelles; Whenever possible we will compare the re- K ' is a different equilibrium constant; and suits obtained with those found for micelle the other variables are the same as those diffusion. defined above. An important characteristic of this equation results because n and q are often large, of order 50. In this case much THEORY of the surfactant is present as micelles and In this section we develop equations de- adding more surfactant does not change ca scribing micelle diffusion in terms of the dif- and c2 dramatically. This means that Eqs. fusion coefficients of the various species [1] and [2] are often indistinguishable present in solution. In this development, we experimentally. will make many simplifying assumptions and Both these models permit surfactant difthen check the effect of these simplifications fusion by two roughly parallel paths. The with diffusion experiments. In particular, detergent can diffuse as free monomer or as we will assume that micelle reactions like part of a micelle. The same two paths are formation and destruction occur much more also available for counterion diffusion. The rapidly than diffusion does. How valid this models presented here for mutual surfactant assumption is depends on the time of the diffusion take into account these two paralmicelle reactions and the distance over which lel diffusion paths in developing an expresdiffusion occurs. If these reactions have sion relating the overall diffusion coefficients haft-lives around 100 /zsec, then diffusion to the total surfactant concentration. must occur over distances greater than 10 -4 We have developed expressions for both cm. In processes like detergency and diges- ionic and nonionic detergent diffusion for tion, it certainly does. In more exact terms, each of these two models. For nonionic this means that the second Damkrhler num- detergents we give a synopsis for both modber in these systems is very large. els. For ionic detergents the phase separaThere are two limiting pictures of micelle tion model leads to much simpler equations formation in detergent solutions. The first, than the chemical equilibrium model. Acpromulgated by McBain (24), assumes mi- cordingly, we report only the phase separaceUes are a separate phase which separates in tion model for ionic diffusion. Details, inthe solution. For an ionic surfactant this phase cluding equations for the chemical equilibseparation model suggests the equilibrium rium model, are available elsewhere (34). Journal of Colloid andlnterface Science, Vol. 80, No. 2, April 1981
359
DIFFUSION IN SURFACTANT SOLUTIONS
Two Models for Nonionic Diffusion
Diffusion of a nonionic surfactant is described by the two continuity equations (29):
further approximation. Specifically, we assume that Eqs. [7] and [ 10] can be replaced by Cm
0 = DIVZcI - nrm
[3]
0 = Dm~72Cm + rm
[4]
=
K'c'I' = __1 (c - c l ) n
and -
1
(c - CMC).
[11]
n
for which D1 and D m a r e the diffusion coefficients of m o n o m e r and micelle, respectively; ca and cm are the m o n o m e r and micelle concentrations; and rm is the rate of micelle formation. This rate is positive if micelles are in fact formed; it is negative if they are destroyed. If we multiply Eq. [4] by n, add it to Eq. [3], and integrate once, we find the total flux J of the detergent: -J
= O l V c a + nOm•Cm.
[5]
The diffusion coefficients are assumed constant, independent of concentration. For the phase separation of nonionic detergents, Eq. [1] reduces to 1 = Kc~.
[6]
Thus the phase separation model sets c1 equal to a constant, presumably the critical micelle concentration or CMC. F r o m a mass balance, c = cl + ncm, [7] where c is the total detergent concentration. As a result,
1
Vcm = - - Vc n
[8]
and Eq. [5] becomes -J
= OmVc.
[9]
Above the critical micelle concentration, all diffusion occurs by transport of micelles through the surrounding m o n o m e r solution. For the chemical equilibrium model, the situation is slightly more complex. Equation [2] becomes Cm = K'c]'. [10] The mass balance is unchanged from Eq. [7]. Unfortunately, these equations cannot be solved in a simple analytical form without
This assumption will be more accurate well above the CMC, where c far exceeds Cl and most of the surfactant is in the micelles. We can find V c m and VCl from this relation and combine with Eq. [5] to yield
-J
=
[D m + DI( 1 t1'° n \--~--7] x
c - CMC
Vc.
[12]
The quantity in square brackets is the apparent diffusion coefficient D of the surfactant. This apparent coefficient should be proportional to (c - CMC) "/~-~, or since n is commonly large, to (c - CMC)-L Thus the results of the phase separation model in Eq. [9] and of the chemical equilibrium model in Eq. [12] give a diffusion coefficient which varies differently with concentration. The phase separation model leads to a constant value Dm. In contrast, the chemical equilibrium model gives an apparent diffusion coefficient which varies explicitly with concentration. If we plot this apparent coefficient vs (c - CMC) -1, we should obtain a straight line. The intercept of this plot corresponds to the micelle diffusion coefficient, and the slope is related to the monomer diffusion coefficient. Such plots are consistent with experiment, as shown later in this paper. A Phase Separation Model for lonic Diffusion
We now turn to the more complex case of an ionic surfactant. The greater complexity results because all diffusing species are coupled electrostatically, and because the Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
360
WEINHEIMER, EVANS, AND CUSSLER
counterion concentration can be independently altered by adding another electrolyte. However, our basic strategy is the same as in the nonionic case. We first write a flux equation; we then write mass balances and chemical equilibria between the species present; and we finally combine these to find an apparent diffusion coefficient. We begin with the continuity equations for monomer " 1 , " counterion '°2/' and micelle " m " :
O = Dl(V2Cx q- Z1V(c1V-~)) - nrm,
[13]
O = D2(V2c2 -}- z2V(CzV R-R~)) - qrm,
[14]
O =Dm(V2Cm~ZmV(Cm~-~))
static potential. If we multiply Eq. [15] by (n + q), add the result and Eqs. [13]-[14], and integrate once, we again find the total surfactant flux J:
The factor 2 results because we regard the diffusion of 1 mole of monomer and 1 mole of counterion as diffusion of 1 mole of surfactant. This flux equation is subject to the constraint of zero current:
+ rm, [15]
0= in which the Di are the appropriate diffusion coefficients, assumed to be constant; the zi are the charges; and 6/RT is the electro-
-j=
°)
(
- 2 J = D1 Vcl + z~clV-R-~
ziDi VCi q- ZiCi
•
We use this restraint to find Vch/RT; inserting the result into Eq. [16], we find
[D~(D2c2 + q ( q - n ) D m c m ) V c l + De(DlCl + n ( n - q)Dmcm)VC2 + Dm(qDlCl + nD2c2)VCm] DIC 1 A- D2C 2 +
The first term in square brackets is caused by a flux due to the monomer concentration gradient; the second term is due to the counterion concentration gradient; and the third term is due to the micelle concentration gradient. Because the concentrations of the various species are not directly known, they must be written in terms of the total concentration c. Material balances again lead both to Eq. [7] and to the following, C2 = Cs + Cl + (n -- q)Cm
[19]
in which cs is the concentration of counterion present due to sources other than the surfactant, such as added electrolyte. A third constraint is due to the phase separation model, given for this case by Eq. [1]. Journalof Colloidand InterfaceScience,Vol. 80, No. 2, April 1981
[17]
i=1
(n - q)2cmD m
[18]
As in the nonionic case these equations cannot be easily solved without approximation. To do this, we recognize that Eq. [7] can be approximated by 1 1 Cm=--(c-cl)=--(cn n
CMC)
[20]
SO
Vcm = --1 Vc.
[21]
n
Because we are using the phase separation model this may not look like an approximation; however, according to Eq. [1], the addition of added electrolyte will depress the monomer concentration above the CMC. In a similar way, we assume that Eqs. [7] and [19] can be combined as
361
D I F F U S I O N IN S U R F A C T A N T S O L U T I O N S
C2 = Cs + C1 +
(1-
%)(C-
Cl)
unexplored problem in detergency. Finally, we determine Vct by combining Eqs. [1] and [221:
= cs + CMC + (1 - q ) ( c - CMC) [22] VC 1 =
SO
a~ __
Note that this implies ca is a constant, and so implicitly neglects multicomponent effects (11). While we found this neglect was experimentally justified in this set of experiments, we suspect that it can be a major and
-j=
tU
V ( C 2 q/n)
(Lll/n(&t(1--q)C2(l+q/~t')VC \ n 7\
- -\K}
n /C 2
Using Eqs. [21], [23], and [24] we can rewrite Eq. [18] in terms of Vc:
DI(Dzc2 + q(q - n)Dmcm(-q/n)(1 - q/n)(cJcz) + Dz(Dlc~ + n(n - q)Dmcm(1 - q/n)) + Dm(qD,ca + nDzc2)(1/n)
The term in square brackets represents the apparent diffusion coefficient D we measure experimentally. Parallel arguments for the chemical equilibrium model give very similar but more complicated equations (34). These additional complexities do not improve agreement between the theory and the experiments. The development above ignores many chemical complexities of micelle formation (2). It assumes micelles form and are destroyed much faster than the diffusion. It assumes these micelles are of constant size and charge, and that the monomers do not undergo any premicellar association. It assumes that monomer and micelle have different diffusion coefficients, but that these coefficients are constants, independent of concentration and ionic strength. These assumptions are significant, but even with them, the results are complex algebraically. For the diffusion experiments to date, these assumptions are justified. Future experiments may require a more complicated theory. EXPERIMENTAL Sodium dodecyl sulfate (SDS) (BDH Chemicals) was of 99% purity and was used
Vc.
[25]
as received. The surface tension vs concentration curve of this sample showed no minimum in the region of the CMC. Triton X-100 (TX-100) (isooctylphenoxypolyethoxyethanol; Packard Instrument Co., Downers Grove, Ill.); the water-insoluble dye, methyl yellow (p-dimethylaminoazobenzene; Pfaltz and Bauer, Flushing, N. Y.); and reagentgrade sodium chloride (Fisher) were also used as received. Diffusion coefficients were measured with the Taylor dispersion technique. Because this technique is very fully described elsewhere (13, 34) only a synopsis is given here. The equipment basically consists of a long thin tube, through which a solution is slowly flowing. Two tubes were used in this study. Both were made of stainless steel by the Superior Tube Company of Norristown, Pennsylvania. The first tube was 104.55 m long with an inside diameter of 0.0787 cm and the second tube was 36.58 m long with an inside diameter of 0.0508 cm. The longer tube was made into a coil 41 cm in diameter while the shorter tube was made into a coil 25 cm in diameter. The pump was a CMP-2 metering pump (Laboratory Data Control, Riviera Beach, Fla.). As delivered, it proJournal of Colloidand Interface Science, Vol.80. No. 2, April1981
362
WEINHEIMER, EVANS, AND CUSSLER
Samples of slightly different composition were then injected into the flowing stream. This method of measuring the concentration dependence of the diffusion coefficient is that used successfully in the present work. The differences between past studies using the Taylor dispersion method hinge largely on the detector chosen to analyze c as a function of time. In this work, we used two detectors, mounted in series, so we could h a v e two different measurements of c. These detectors were a differential rec - 6 ~t ,,1/2 -- I `R ) e -12D(t-tal/a~t [26] fractometer and a spectrophotometer. The refractometer was Waters Associates Model \tJ Crnax~" R401, capable of measuring refractive index in which c is the surfactant concentration differences of 10-8 RI units. The UV-VIS averaged across the tube's radius; Cmax is absorbance detector, always mounted upthe maximum value of this concentration; stream, was Waters Associates Model 440. is the average surfactant concentration With eight different wavelengths across the without the pulse; t is the time; tR is the ultraviolet-visible spectrum, it was caparesidence time, i.e., the tube's length divided ble of measuring absorbance differences of by the average solution velocity; D is the 0.001 absorbance units. The detector signal surfactant's diffusion coefficient; and a is was recorded by a Bristol 10-in. 10-mV recorder. the tube's radius. To check our technique, we first measThis result, basic to the Taylor dispersion apparatus, has the same general Gaussian ured the diffusion coefficient of urea and of form for the decay by diffusion of a pulse in sucrose, using the differential refractometer concentration. The interesting feature of this to analyze the output. The results were result is that the diffusion coefficient appears analyzed by assuming that the concentration in the numerator of the exponent. As a re- varied linearly with refractive index, and by sult, the measured pulse is broad for slow fitting the data to Eq. [26], using nonlinear diffusion, and narrow for fast diffusion. This least squares and a DEC-20 computer. To is the antithesis of intuition and is the in- our dismay the initial results were routinely teresting feature of the Taylor method. It 10% below the initial value. We finally traced occurs when the experimental conditions are this to an error of 3% in the tube radius such that radial diffusion is fast compared specified by the manufacturer. With this corto convection. This is assured by a very rection, results for urea and sucrose were routinely within 1% of the accepted values. low fluid velocity and a very long thin tube. The measurements of surfactant diffusion This technique has been applied to many liquid systems, including hydrocarbons in also used the differential refractometer. To hydrocarbons; gases in water; and organo- begin these experiments, a solution of known tin compounds in organic solvents. More composition was prepared. This solution importantly, it has been shown that the tech- was used to flush the tube and ultimately nique can be used to measure the concen- became the carrier stream. A small (20/zl) tration dependence of diffusion coefficients sample of a more concentrated solution was in water-ethanol systems (28). This was then injected into the carrier. The measured done by pumping a water-ethanol solution diffusion coefficient was found to be indeof known composition through the tube. pendent of the concentration in the pulse. vided pulse-free flow at six flow rates ranging from 2.4 to 120 ml/hr. The pump motor was later replaced with a Bodine stirring motor with variable speed control for use with the shorter tube. An experiment with this equipment is made as follows. At time zero, a pulse of a different solution is injected through a Rheodyne Model 70-10 valve at the tube's inlet. The shape of this pulse, measured as it flows out of the tube, is given by (3, 33)
Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
DIFFUSION IN SURFACTANT SOLUTIONS The measurements of dye diffusion used the spectrophotometer. The experiments were made in two different ways. In the first, a gradient in dye concentration only is introduced. The injection solution was made by taking a sample from the pump supply and saturating the sample with dye. The next day the supernatant surfactant solution was diluted by a factor of five with more surfactant solution from the pump supply. This procedure assures that the surfactant concentration in the injection and in the flowing stream are equal. In the second type of experiment, gradients in both dye and surfactant concentrations were introduced. The injection solution was made by saturating a concentrated surfactant solution with dye. The supernatant solution was diluted by a factor o f I0 with more of the concentrated surfactant solution. This new solution was that injected. Again, the tube was carefully flushed with the carrier solution before injection. In experiments made with other solutes such as NaC1, we were careful to eliminate concentration differences in all but the desired species. For the experiments with added electrolyte, solvent and solute were both made by dissolving surfactant in salt solution taken from the same vessel to ensure equal salt concentration. However, even when no salt gradient exists, one can develop because of multicomponent diffusion, producing a profile different than that in Eq. [26]. We saw no such profile here but we expect such profiles in experiments which extend our results. Three nuances about this procedure deserve emphasis. First, in all experiments above the CMC, the carrier stream contained surfactant at or above the CMC. This is important. If pure water is used as a carrier and the pulse is above the CMC, the results can be much harder to interpret. Second, even though the pulse is of higher surfactant concentration, it is routinely only a few percent above the carrier surfactant. This small change in concentration is possible because
363
TABLE I Diffusion Coefficients of Triton X-100 in Water as a Function of Concentration at 25°C~ Conch
107 D o TX- 100
107 D ~ Dye
0.02e 0.45 0.51 0.61 0.76 1.01 1.76 5.01
8.25 + 0.45 7.40 _+ 0.05 7.28 ± 0.16 7.32 ± 0.10 7.05 ± 0.28 6.93 ± 0.22 6.93 ± 0.15 6.49 ± 0.12
---8.15 ± 0.77 --7.54 ± 0.14 --
Concentrations are in grams per liter because the molecular weight of this material is not well defined. Diffusion coefficients are in cm2/sec. All measurements were made using the Taylor dispersion apparatus. b These are mutual diffusion coefficients calculated from refractive index. c These are dye diffusioncoefficientscalculated from the absorbance at 436 nm. This concentration is one-tenth the CMC, so the diffusion is dominated by that of the monomer.
the detectors are so sensitive. Third, we occasionally used pulses with lower surfactant concentration than in the carrier. We found the same diffusion coefficients in both cases. RESULTS AND DISCUSSION Both the measured diffusion coefficients o f nonionic and of ionic surfactants show the characteristics predicted by the theory given above. The nonionic and ionic cases are best discussed sequentially.
Nonionic Results The diffusion coefficients of TX-100 decrease slightly with concentration, as shown in Table I. The precision of the measurements above the CMC ranges from + 1 to _+4%. The diffusion coefficients of the waterinsoluble dye, measured at two TX-100 concentrations, are also included in Table I. In both cases, the diffusion coefficient of the dye was slightly higher than the corresponding surfactant diffusion coefficient. We are Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
364
WEINHEIMER, EVANS, AND CUSSLER I
I
I
I
I
9.0
~" B.O E o
~
+ 2%
~o x 7.0
6.(-
i I 0.0 2.0 (Concentration-Critical
I
I i 4.0 Micelle Concentration) -t-~/g)
Fro. 1. Diffusion of Triton X-100 at 25°C. This variation with surfactant concentration is consistent with the chemical equilibrium model for micelle formation, but not with the phase separation model of this process.
not certain whether this difference is significant. Finally, a diffusion coefficient measured well below the CMC is also reported in Table I. This value, which should be dominated by monomer diffusion, is of poor precision due to surfactant adsorption on the tube wall. Such adsorption caused a badly skewed profile when a solution of TX100 below the CMC was injected into pure water. This asymmetry is reduced significantly by having a small amount of TX-100 in the carrier stream. Few diffusion coefficients of TX- 100 have been previously reported in the literature. Giglio and Vendramini (16) found a diffusion coefficient in a Solution of 10 g/liter to be 5.8 x 10-7 cm2/sec. The most concentrated solution studied here is 5 g/liter, at which the TX-100 diffusion coefficient is 6.5 x 10-r cm2/sec. The diffusion coefficient is decreasing with concentration, so a drop to the value found by Giglio and Vendramini seems reasonable. The variation of TX-100 diffusion with concentration is inconsistent with Eq. [9], based on the phase separation model of micelle formation. This equation predicts the diffusion coefficient should be independent of concentration above the CMC. This is not supported by the data. Journal of CoUoid and Interface Science, Vol. 80, N o . 2, April 1981
The diffusion data, shown in Fig. 1, are consistent with Eq. [12], based on the chemical equilibrium model. The intercept in Fig. l, equal to about 6.7 × 10-7 cm2/sec, represents the diffusion of the micelle. If we assume the micelle is a rigid sphere, we can use this result and the Stokes-Einstein equation to calculate the micelle' s diameter. The result is 70/~, consistent with the previous estimate of 80 A (22). However, since these micelles may well be nonspherical, such an estimation is speculative. The slope of the line in Fig. 1, equal to 2 × 10-11 g/cmsec, can in principle be estimated using Eq. [12]. The estimate requires values of the monomer diffusion coefficient D1, the equilibrium constant K , and the aggregation number n. While estimates of all these quantities are available in the literature, they vary widely. As a result, this estimate is not instructive. Plots like that in Fig. 1 are also predicted by other theories (5, 23). These theories assume that the observed diffusion coefficient is an arbitrary average of the diffusion coefficients of the various species present. The average is assumed to be arithmetic, weighted with mole fractions, even though harmonic and geometric averages are often more effective in estimating diffusion (e.g., (12)). This average leads to plots like Fig. 1. The reason for prefering the nonionic theory in this paper is that it substitutes sensible chemistry for arbitrary averaging.
Ionic Diffusion The diffusion coefficient of sodium dodecyl sulfate was also measured as a function of surfactant concentration, both with and without added electrolyte. The results, presented in Table II, show a large increase in SDS diffusion coefficient in the absence of added salt. Over the concentration range studied the coefficient nearly tripled. Most of this increase occurs between 0.01 and 0.06 M. It occurs in the face of a viscosity increase of 30% (21), and so is exactly op-
DIFFUSION
IN SURFACTANT
TABLE
365
SOLUTIONS
II
D i f f u s i o n C o e f f i c i e n t s o f S o d i u m D o d e c y l S u l f a t e a t 25°C ~ Concn
10~ D ~
lip D ¢,'j
1@ D ~'~
Concn in
106 D b
106 D ~'a
106 D e'~
in H 2 0
SDS
Dye
Dye
0.1 M NaCI
SDS
Dye
Dye
-0 . 8 7 + 0.01 ----0 . 7 6 +- 0.01 -----
-1.4 ± 0.11 ----------
0.01 0.02
1.18 ± 0.03 1.20 _ 0 . 0 6
0.91 +_ 0.01
0 . 9 4 ± 0.01
0.03 0.04 0.05
1.28 _+ 0 . 0 2 1.29 ± 0.01 1.35 ± 0 . 0 2
__
m
0 . 8 6 + 0.01
0 . 8 8 -+ 0.01
0.06 0.08 0.10
1.34 ± 0 . 0 5 1.47 ___ 0.11 1.45 ± 0 . 1 2
0.01 0.02 0.0U 0.03 0.04 0.05 0.0525 0.06 0.07 0.125 0.15 s
1.76 2.40 3.0 3.03 3.65 4.03 4.05 4.38 4.53 5.7
± ± ± ± ± -+ -± ± ± ±
0.03 0.17 0.6 0.11 0.08 0.07 0.08 0.07 0.12 1
D
C o n c e n t r a t i o n s a r e in m o l e s p e r l i t e r a n d d i f f u s i o n c o e f f i c i e n t s a r e in c m V s e c . E x c e p t a s n o t e d , all e x p e r i ments were performed using the Taylor dispersion apparatus. b T h e s e d i f f u s i o n c o e f f i c i e n t s w e r e c a l c u l a t e d f r o m t h e r e f r a c t i v e i n d e x profile. c T h e s e d i f f u s i o n c o e f f i c i e n t s w e r e c a l c u l a t e d f r o m t h e a b s o r b a n c e profile a t 4 3 6 n m . a T h e s e e x p e r i m e n t s u s e d a g r a d i e n t in d y e c o n c e n t r a t i o n a l o n e . T h e r e w a s n o g r a d i e n t in s u r f a c t a n t concentration. e T h e s e e x p e r i m e n t s u s e d g r a d i e n t s in b o t h d y e a n d s u r f a c t a n t c o n c e n t r a t i o n . T h e s e e x p e r i m e n t s w e r e p e r f o r m e d u s i n g t h e d i a p h r a g m cell.
posite to that expected. We were initially so surprised by this large increase that we questioned the validity of these results. For this reason we made experiments at the extreme concentrations with the diaphragm cell. The same large increase was seen. Such a large increase has also been seen in water for dodecyl sulfonic acid (25). In the presence of 0.1 M NaC1 as an added electrolyte, the diffusion coefficient of SDS increases much less dramatically with concentration as shown in Table II. The increase in diffusion coefficient across the same range of concentration is only about 25%. The diffusion coefficient of the waterinsoluble dye was measured both with and without added salt. These experiments were made in two ways, with a gradient only in the concentration of the dye, and with a gradient of both dye and surfactant. Results from both types of dye experiments are also reported in Table II. The diffusion of dye is less than that of detergent. The values of detergent diffusion reported in Table II are measured by means of re-
fractive index, and so weigh each molecule roughly equally, be it present as a monomer or in a micelle. This equal weighing means that these diffusion coefficients are those important in mass transfer processes like detergency and digestion. At the same time these averages may differ from those found from other experimental methods which weight micelles more heavily. To date, three types of studies of SDS diffusion have appeared in the literature. Each of the three uses a different experimental method, although all three methods are expected to emphasize transport by micelles. The three are exemplified by the laser-Doppler study of Corti and Degiorgio (6-9), the dye solubilization experiments of Stigter et al. (32), and the radioactive tracer work of Kamenka et al. (20). While these groups of experiments are not always made at exactly the same conditions used here, comparisons are instructive. The laser-Doppler results agree with the refractive index results obtained here, as shown in Fig. 2. For example, our data in Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
366 5q
WEINHEIMER, EVANS, AND CUSSLER ~ # v •
i-
Our Our Our Our
Work, Work, Work, Work,
No Salt, Surfaclant No Salt, Dye O.fM NaCI, Surfactant O.IM N a C
~
g
? c~2
I~-SWM in 0.1M NoCI A-Corti 8¢ Degiorgio O.05M NaCI Irn- SWM in 0.03 M NaCI • - Corti & Degiorgio O.1M NaCI Jo-SWM in Water • - Corti & Degiorgio O.3M NaCI
oL~
I
0.02
I
/
I
I
0,04 0.06 0.08 O.tO SDS Concentration, mol/.~
I
0.12
Fie. 2. Diffusionof sodiumdodecylsulfatemeasured by differentmethods. The increases in surfactantdiffusion result from electrostatic couplingbetween the various speciespresent. The decreasesin dye diffusion reflect the solubilizationof the water-insolubledye by the surfactantmicelles.The abbreviationSWM refers to Stigteret al. (32). water and in 0.1 M NaC1 neatly bracket the light-scattering data at 0.03 M NaCI. Our data at 0.1 M NaC1 are about 10% higher than those found by light scattering, a difference which almost certainly reflects a bias away from monomer diffusion in the lightscattering experiments. At the same time the dye diffusion data of Stigter et al. agree closely with the dye results obtained here, shown in Fig. 2. This good agreement is a consequence of the fact that both our dye data and those of Stigter et al. represent micelle transport. Initially, we were surprised that the light-scattering results agree with our refractive index studies, and not with our dye results. After all, the refractive index measurements represent diffusion averaged over monomers and miJournal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
celles, but biased toward the small, fast monomer. The dye experiments are dominated by micelles, because the dye molecules are in micelles. Light-scattering experiments, we felt, should also be dominated by micelle diffusion, for light-scattering data are averages more strongly influenced by larger particles. Upon reflection, we are not so sure. While light-scattering methods do commonly give weight average or z-average diffusion coefficients, we wonder whether they do so for micelles. After all, micelles are not stable molecules like proteins or synthetic polymers, but are dynamic, constantly adding some monomer molecules and losing others. We suspect this dynamism may give unusual characteristics to laser-Doppler light-scattering experiments. The third set of data for possible comparison are the intradiffusion data of Kamenka et al. (20), called "tracer" diffusion data in some of the literature. These data report the transport of tagged dodecyl sulfate in a ternary system of tagged sodium dodecyl sulfate, untagged sodium dodecyl sulfate, and water. The diffusion coefficients of this tagged anion show a huge drop over the range of concentrations studied here. The interpretation of experiments like these is controversial even for much simpler chemical systems than that used here (e.g., (1, 10)). We believe connecting mutual and intradiffusion in this case will turn out to be very difficult indeed. We now turn to trying to predict the increase in surfactant diffusion using the theory developed earlier in this paper. This theory included micelles which form rapidly from monomers, which are electrically charged, and which include counterion bonding. Unfortunately, the key theoretical result in Eq. [25] suggests no simple plot. Instead, we can predict SDS diffusion by independent measurements of seven quantitie s: the three diffusion coefficients of monomer D1, counterion D2, and miceUe Dr.; the monomer and counterion concentrations c1
367
D I F F U S I O N IN S U R F A C T A N T S O L U T I O N S
and c2; and the aggregation numbers n and q. The values used for counterion and monomer diffusion coefficients result from measurements of radioactive tracers in dilute solutions below the CMC when intradiffusion effects are negligible. For the counterion Clifford and Pethica (5) determined a value of 1.35 x 10-~ cm2/sec in water. For the monomer in water, Kamenka et al. (20) measured a value of 5.7 x 10-~ cm2/sec. The micelle diffusion coefficient over this concentration range is taken as 8.0 x 10-7 cm2/ sec from the dye data (Stigter et al. (32)). The concentrations of monomer and counterion were found from ion-specific electrode experiments (Kale et al. (19)). These data assume that activity coefficients are unity. Counterion and micelle concentrations are then calculated using material balances. The only variables not specified are n and q. The value of n was taken as 50 after that determined by Huisman (18). The ratio of q/n is taken to be 0.84, consistent with the range of values shown in Table III. The exact choice of n and q did not sharply affect the results. The theoretical prediction based on Eq. [25] and these seven independent measurements is compared with our direct experimental results in Fig. 3. Agreement is excellent. Thus the surprising increase observed for mutual diffusion is apparently a result of micelle rearrangement and electrostatics.
o × c3 1
I
0.02
I
I
I
I
0.04 0.06 0.08 0.I0 SDS Concentration, mol/I
I
I
0.12
0.14
FIG. 3. Predictions of surfactant diffusion. The theoretical prediction, shown by the solid line, agrees closely with the actual data points. The prediction is based on independent measurements like monomer diffusion coefficient and micelle aggregation number.
CONCLUSIONS
This work shows how surfactant diffusion occurs by parallel transport of micelles, monomers, and, where present, counterions. For nonionic surfactants the diffusion coefficient drops as the concentration increases and micelle diffusion becomes more important. For ionic surfactants the diffusion coefficient rises as the concentration increases because of electrostatic coupling between the species present. In both cases, the coefficients measured are those important in mass transfer and do not focus on the micelles alone. APPENDIX: N O M E N C L A T U R E a
T A B L E III
c
Ratio of Sodium Ions to Dodecyl Sulfate Ions in SDS Micelles in Water at 25°C
c1 c2
q/n
Method
Reference
0.86 0.85 0.85 0.84 0.82 0.80 0.78
emf light scattering specific ion electrodes emf equilibrium dialysis osmotic coefficients emf
(14) (27) (19) (4) (15) (17) (30)
Cm Cs
CMC D D1 D2 Dm J K
radius of the Taylor dispersion tube total concentration monomer concentration counterion concentration micelle concentration concentration of added electrolyte critical micelle concentration apparent diffusion coefficient monomer diffusion coefficient counterion diffusion coefficient micelle diffusion coefficient total surfactant diffusion flux equilibrium constant for the phase
Journal of Colloid and Interface Science, Vol. 80, No. 2, April 1981
368
K,
n
q R rm T t tR ,7.1 Z2 Zm
WEINHEIMER, EVANS, AND CUSSLER
separation model for an ionic surfactant (Eq. [1]) equilibrium constant for the mass action model for an ionic surfactant aggregation number of micelle number of counterions bound to a micelle gas constant rate of formation of micelles temperature time residence time in the Taylor dispersion tube charge on the monomer charge on the counterion charge on the micelle electrostatic potential ACKNOWLEDGMENTS
This work was primarily supported by the National Science Foundation, Grant CPE80-14567. Other support came from the National Institute of Arthritis, Metabolic and Digestive Diseases Grant SR01 AM16143-07 and the National Science Foundation Grant CPE80-17376. REFERENCES 1. Albright, J. G., and Mills, R., J. Phys. Chem. 69, 3120 (1%5). 2. Aniansson, E. A. G., Wall, S. N., Almgren, M., Hoffmann, H., Kielmann, J., Ulbricht, W., Zana, R., Lang, J., and Tondre, C., J. Phys. Chem. 80, 905 (1976). 3. Aris, R., Proc. Roy. Soc. London Ser. A. 235, 67 (1956). 4. Botre, C., Crescenzi, V. L., and Mele, A.,J. Phys. Chem. 63, 650 (1963). 5. Clifford, J., and Pethica, B. A., Trans. Faraday Soc. 60, 216 (1964). 6. Corti, M., and Degiorgio, V., Opt. Commun. 14, 358 (1975). 7. Corti, M., and Degiorgio, V., in "Photon Correlation Spectroscopy and Velocimetry" (H. Z. Cummins and E. R. Pikes, Eds.). Plenum, New York, 1977.
Journalof CoUoidand InterfaceScience, Vol.80, No. 2, April1981
8. Corti, M., and Degiorgio, V., Ann. Phys. (Paris) 3, 303 (1978). 9. Corti, M., and Degiorgio, V., Chem. Phys. Lett. 53, 237 (1978). 10. Curran, P. F., Taylor, A. E., and Solomon, A. K., Biophys. J. 7, 879 (1967). 11. Cussler, E. L., "Multicomponent Diffusion," Chap. 6. Elsevier, Amsterdam, 1976. 12. Cussler, E. L., A I C h E J . 26, 43 (1980). 13. Evans, D. F., Chan, C., and Lamartine, B. C., J. Amer. Chem. Soc. 99, 6492 (1977). 14. Feinstein, M. E., and Rosano, H. L., J. Colloid Interface Sci. 24, 73 (1967). 15. Fishman, M. L., and Eirich, E. R., J. Phys. Chem. 75, 3135 (1971). 16. Giglio, M., and Vendramini, A., Phys. Rev. Lett. 38, 26 (1977). 17. Huff, H., McBain, J. W., and Brady, A. P., J. Phys. Colloid Chem. 55, 311 (1951). 18. Huisman, H. F., Proc. K. Ned. Akad. Wet. Ser. B 67, 367 (1964). 19. Kale, K., Cussler, E. L., and Evans, D. F., J. Phys. Chem. 84, 593, (1980). 20. Kamenka, N., Lindman, B., and Brun, B., Colloid Polym. Sci. 252, 144 (1974). 21. Kodama, M., and Minra, M., Bull. Chem. Soc. Japan 45, 2265 (1972). 22. Kushner, L. M., and Hubbard, W. D., J. Phys. Chem. 58, 1163 (1954). 23. Lindman, B., and Brun, B., J. Colloid Interface Sci. 42, 388 (1973). 24. McBain, J. W., Trans. Faraday Soc. 9, 99 (1913). 25. McBain, E., Proc. Roy. Soc. London Ser. A 170, 415 (1939). 26. Mukerjee, P., J. Pharm. Sci. 63, 972 (1974). 27. Phillips, J. N., and Mysels, K. J., J. Phys. Chem. 59, 325 (1955). 28. Pratt, K. C., and Wakeham, W. A., Proc. Roy. Soc. London Ser. A 336, 393 (1974). 29. Robinson, R. G., and Stokes, R. H., "Electrolyte Solutions," Chap. 11. Butterworths, London, 1959. 30. Shedlovsky, L., Jakob, C. W., and Epstein, M. B., J. Phys. Chem. 67, 2075 (1963). 31. Short, B. A., in "Detergency" (W. G. Cutler and R. C. Davis, Eds.). Dekker, New York, 1972. 32. Stigter, D., Williams, R. J., and Mysels, K. J., J. Phys. Chem. 59, 330 (1955). 33. Taylor, G. I., Proc. Roy. Soc. London Ser. A 219, 186 (1953). 34. Weinheimer, R. M., Ph.D. thesis, Carnegie-Mellon University, Pittsburgh, Pa. (1979).