J. Phys. Chem. 1994, 98, 10243-10248
10243
Aggregation Numbers of Dodecyldimethylamine Oxide Micelles in Salt Solutions Hideki Kaimoto, Kouki Shoho, Shigeo Sasaki, and Hiroshi Maeda' Department of Chemistry, Faculty of Science, Kyushu University, Fukuoka 812, Japan Received: May 30, 1994; In Final Form: July 21, 1994@
Micelle aggregation number m of dodecyldimethylamine oxide was determined as a function of the micelle composition or the degree of ionization of the micelle a M in 0.1 and 0.2 M NaCl at 25 "C with light scattering and the steady state fluorescence probe method. The two methods gave consistent results in most cases. The aggregation numbers at the critical micelle concentration (cmc) were nearly identical for both cationic and nonionic micelles: about 70 and 80 in 0.1 and 0.2 M NaCl, respectively. The aggregation number increased as aM approached around 0.4-0.5.The maximum values were about 100 and 150 for 0.1 and 0.2 M NaC1. At intermediate compositions, the Rayleigh ratio of the scattered light increased with the concentration, and this increase was interpreted as the increase of micelle size with concentration. Apparent aggregation numbers at a finite concentration were evaluated on the assumption of negligible contribution from the second virial coefficient. The idea of the growth of micelles was supported by dynamic light scattering but not by the fluorescence probe method. From the observed variation of the aggregation number at cmc with aM, the width of the aggregation number distribution of nonionic micelles is estimated on the basis of a simple theory.
Introduction Micelle size change with the composition of mixed micelles is an interesting and important issue in the field of surfactant chemistry. The aggregation number of dodecyldimethylamine oxide (DDAO) has been studied over three decades. Herrmann first reported an interesting result of the similar aggregation numbers for both cationic and nonionic micelles.' Later, it was shown2that the aggregation numbers reached the maximum in the region of degree of ionization a M around 0.5. These characteristics have been confirmed by vis~osity,~ the fluorescence probe method: and the dynamic light-scattering measurem e n t ~ .The ~ characteristics of DDAO has been interpreted in terms of the assumed hydrogen bond between nonionic and cationic specie^.^^^^^ It is suggested7from the thermodynamic analysis of the stability of DDAO micelles that the hydrogen bond is formed between two cationic species as well with the same or a weaker strength. The latter case hydrogen bond may explain the results of a calorimetric study? the enthalpy of micellization of DDAO was positive for nonionic but negative or zero for cationics. Potentiometric titrations have been carried out on DDA0,9Jo and valuable information has been derived from them.6$7q11Phase separation at high ionic strength was reported,12 and the data of the aggregation number under a variety of conditions are hoped to elucidate the mechanism of the phase separation. In the present study, we examine the aggregation numbers of DDAO at various micelle compositions in 0.1 and 0.2 M NaCl solutions by means of the static light scattering and the steady-state fluorescence probe m e t h ~ d . ~ JIn~ Jthe ~ range of a M centered at 0.5,possible micelle growth with the surfactant concentration is examined by dynamic light scattering as well as the above two methods. The width of the aggregation number distribution of nonionic micelles is estimated from the observed dependence of m on a M on the basis of a simple theory. Experimental Section Dodecyldimethylamine oxide: About 64 g of dodecyldimethylamine (purity 99.3%), kindly donated by Nippon Oil & ~
@
Abstract published in Advance ACS Abstracts, September 1, 1994.
0022-365419412098-10243$04.50/0
Fat Co. Tokyo, was dissolved in 400 mL of ethanol and aqueous solution of H202 was added three times more than that of the dodecyldimethylamine. Reaction was allowed to proceed first for 3 h, and then 300 mL of distilled water was added and the mixture was kept at 75 "C for 35 h. Then paradium carbon was added and kept at 60 "C for 10 h to decompose remaining H202. The filtrate was concentrated and freeze-dried. The residue was suspended in n-hexane, stirred vigorously, filtered, and washed three times with n-hexane. Then, recrystallization was carried out with acetone three times. Anal. Calcd: H 13.62, C 73.30, N 6.11. Found: H 13.42, C 72.86, N 6.05. Purity was also confirmed by NMR spectra. Dodecylpyridinium chloride (purity '90%: Tokyo Kasei Co.) and hexadecylpyridiniumchloride (HPC, Nakalai Tesque) were recrystallized three times with acetone. We repeated the heat treatment of the recrystallized sample of HPC at 120 "C for 10 min to obtain the anhydrous sample. Doubly distilled water was used throughout in the present study. Light-scattering measurements were carried out at 25.0 f 0.1 "C with a Malvern System 4700 with an Ar ion laser at 488 nm as a light source. The intensity of the incident beam was calibrated with benzene. Solutions were filtered with a Millipore membrane filter (0.025 p)pore size under a N2 gas pressure of ca. 3 atm. This was repeated 5-10 times. The filtrate was gathered in a cell (diameter 10 mm). Scattered light intensities were measured at the scattering angles of 60°, 90", and 120". Little or no angular dependence was observed even at the condition where maximum aggregation numbers were observed. Refractive index increments were taken from the previous study.2 On the basis of the titration data, we estimated the value at a given degree of ionization a M although the original data were given as functions of the degree of neutralization. In 0.2 M NaCl only three data points were reported, and hence interpolations suffered from a significant error except three regions of a M 0,0.45,and 0.95. In the dynamic light-scattering measurements, the time correlation function was well approximated with the sum of two exponential functions. The fast component was assumed to originate from micelle translational movements and used for
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0 1994 American Chemical Society
10244 J. Phys. Chem., Vol. 98, No. 40, 1994
Kaimoto et al.
90
a 50 5.2 4.1
30
1 0
0.5
1
1.5
2
2.5
3
(c - co)I 1 0 ' ~ ~
3.5
50
4
'
I
I
1
0
0.2
0.4
0.6
0.8
1
ah4
Figure 1. Debye plot of the results of light-scattering data at 0.1 M NaC1. For each pH, cmc and the reduced intensity at cmc are denoted as COand Ro, respectively. data analysis. The contribution from the slow component was negligibly small in most cases.
Figure 2. Dependence of aggregation numbers m on a M in 0.1 M NaCl: (0)light-scattering data at cmc; (0)light-scattering data at c g ~ m - (0) ~ ; data from the fluorescence probe method. cmc = 3 x 700
I
1
Results Aggregation Numbers from Static Light Scattering. Examples of the results of the static light scattering (SLS) are shown in Figure 1 in terms of the Rayleigh ratio R. Excess values from that observed below cmc Ro are plotted against micellar concentration (c - CO). The plots give straight lines for pH ranges mostly corresponding to either nonionic (pH 6.2 and 7.8) or cationic micelles (pH 3.0), while nonlinear dependencies are seen at the intermediate range of pH. The slopes of the straight lines provide the second virial coefficient B2 according to
KR(c - c,)/(R - R,) = l/Mw
+ 2B2(c - c,)
(1)
where co represents cmc and
Here n and no are the refractive indices of the solution and the solvent. 1 stands for the wavelength in vacuo. Greater B2 values for cationic micelles than nonionic ones are reasonable since the intermicellar electrostatic interaction is significant for the former, while only physically excluded volume effect contributes to BZ in the latter case. Nonlinear concentration dependence in the intermediate pH range strongly suggests micellar growth with concentration rather than negative Bz values. Aggregation numbers m were evaluated by extrapolation to cmc according to eq 1. The results are given with filled circles in Figures 2 and 3 as functions of a M for 0.1 and 0.2 M NaC1, respectively. At these ionic strengths, both nonionic and cationic micelles have similar size and the sizes scarcely change with ionic strength: about 70 and 80 for 0.1 and 0.2 M NaCl. The cationic micelle in 0.1 M NaCl has significantly greater cmc7 and hence the electrostatic free energy than in 0.2 M NaC1. Nevertheless, micelle sizes are almost identical. This means that the aggregation number of about 70 found for both cationic and nonionic micelles corresponds to the most stable spherical micelle. The spherical compact micelle is also suggested from DLS measurements. Previous results on the size of DDAO micelles are summarized as follows: (a) Aggregation numbers of the nonionic micelle are about SO1 or 9 9 and insensitive to the salt concentration Cs.*s2(b)
37 / 200
0 ' 0
1
1
0.2
0.4
I
4
I
0.6
0.8
1.0
ah4 Figure 3. Dependence of aggregation numbers m on a M in 0.2 M NaC1: (0)light-scattering data at cmc; (0)light-scattering data at cmc at c - cmc = 3 x g ~ m - (0) ~ ; data from the fluorescence probe method. Aggregation numbers of the cationics are slightly greater than those of the nonionics and increase slightly with C,. (c) Growth of micelle size with C, occurs for the range of a M greater than about 0.4 if C, is higher than certain critical value^.^ The present results are consistent with previous results with respect to (a) and (c). Although not much different aggregation numbers were obtained in the previous2 and the present study, a clearly different behavior was observed on the concentration dependence of reduced intensity in the range of a M about 0.5: nonlinear dependence in the present study in contrast with the linear dependence in the previous study.2 This cannot be understood well at present. Light-scattering measurements were also canied out on another sample which was supposed to contain unreacted amine from the elemental analysis. Greater aggregation numbers were found on this sample, especially in the range of a M about 0.5, which was attributed to the probable solubilization of the amine within micelles. We tentatively assess the aggregation number mappat finite concentrations higher than cmc by assuming B2 = 0. The results 3 x lop3 g cm-3 are also shown in obtained at c = cmc Figures 2 and 3 with open circles. The aggregation numbers reach the maxima of about 180 and 700 around a M 0.4-0.5 for 0.1 and 0.2 M NaC1, respectively. An example of the concentration dependence of mappis given in Figure 4 with circles.
+
J. Phys. Chem., Vol. 98, No. 40, 1994 10245
Dodecyldimethylamine Oxide Micelles in Salt Solutions 700 I
I
1
I
1
0.E
zI/
E
3
-
W
o.6
C
0.4
200 0.2
,
0 '
2
4
8
6
1
1
1
12
10
14
0
16
CImM Figure 4. Concentration dependence of the apparent aggregation number mw at pH 5 in 0.2 M NaCl: (0)light-scattering data; (A) data from the fluorescence probe method. The cmc is 0.81 mM. 7 ,
,
,
+pH
0
2
,
I
6
I
I
,
-'
0.5
1
1.5
2
2.5
3
3.5
4
[a]I 105M Figure 6. Dependence of the fluorescence intensity on the quencher concentration [Q] at constant surfactant concentrations at pH = 5.0 in NaC10.2 M: fluorescence probe pyrene; quencher hexadecylpyridinium chloride. Surfactant concentrations (mM): (0)2.47, (D) 5.15, (A) 7.66, (A)10.20, (0)13.40, and (0)15.23.
(A)
2.0
4
,
0
8
10 12
C/mM
14 16
0
2
4
6
8
IO 12 14 16
C/mM
Figure 5. Examples of the dependence of the fluorescence intensity on the surfactant concentration c at a constant quencher concentration: fluorescenceprobe pyrene; quencher hexadecylpyridinium chloride; (A) NaCl 0.1 M, [Q] = 3.33 x M; (B) NaCl 0.2 M, [Q] = 2.97 x 10-5 M. Micelle Aggregation Number by the Steady-State Fluorescence Probe Method.4J3J4We have evaluated the micellar aggregation numbers m by the steady-state fluorescence probe method. We used pyrene as a probe and hexadecylpyridinium chloride (HPC) as a quencher. In Figure 5 examples of the results are shown. The ratio of the fluorescence intensity in the absence (lo) of a quencher to that (I) in the presence is given as follows:
intensities were measured at a fixed value of C as a function of the total quencher concentration [Q]. As shown in Figure 6 , results at pH 5.0 in 0.2 M NaCl give straight lines through the origin except for the case of the lowest C examined. From the slopes of the straight lines we can evaluate the aggregation number m according to eq 3. These values of m are shown in Figure 4 with triangles. They are much smaller and less concentration dependent than the results from SLS under the same conditions. When dodecylpyridinium chloride (DPC) was used as a quencher in place of HPC, the plots according to eq 3 gave negative C* values and further they became nonlinear as a M approached unity. These behaviors were interpreted as DPC repelled from micelles due to charge repulsion. We cannot neglect the solubility of DPC in the aqueous phase under this circumstance and need to evaluate the DPC concentration within micelles which is now different from [Q]. We define the partition coefficient K as (n)l[Qa] where (n)denotes the average number of DPC in a micelle and [Qa] represents the DPC concentration in aqueous phase, i.e., solubility. In terms of K, eq 3 is rewritten as [ln(Z&)]-* = (1/K - cmc/m)/[Q]
[ln(&/I)]-' = (C - cmc)/m[Q]
+ { [QI,}-'C
(4)
(3)
where [Q] denotes the total quencher concentration. According to eq 3, we obtain m from the slope. The intercepts on the abscissa of the linear plots C* are regarded as cmc. The values of m are interpreted as corresponding to those at cmc. The results are given in Figures 2 and 3 with open squares. Agreement between the results from SLS and the fluorescence probe method is generally good except for a range of a M 0.50.6 in 0.2 M NaC1. Under this condition, values of m from SLS were about 1.3 times greater than those from the fluorescence probe method. We can thus conclude that the m values at cmc are reliably evaluated in the present study. The cmc values from the fluorescence probe method or C*, on the other hand, showed good coincidence with cmc values from other methods (such as surface tension) only for those aM near to 0 or 1. In the intermediate values of aM, values of C* were significantly smaller than cmc. To examine the possible micellar growth with surfactant concentration C as suggested from SLS data, fluorescence
According to eq 4, the slopes gave correct values of m, as c o n f i i e d by their coincidence with the results from HPC. From C* values we have evaluated the values of K as (2-6) x lo4 M-' (pH 10, a M 0) and 9.2 x lo3 M-' (pH 5, a M 0.5). The results of the aggregation numbers are summarized in Tables 1 and 2. Dynamic Light Scattering (DLS). To examine possible micellar growth with the surfactant concentration C, we have canied out DLS measurements. We tentatively evaluate the Stokes radius RH from the mutual diffusion constant assuming its identity with self-diffusion constant. This is equivalent to assume that both thermodynamic and hydrodynamic interactions among micelles are negligible. In this paper, RH should be regarded as the apparent equivalent hydrodynamic radius. As shown in Figure 7, the Stokes radii are similar for both nonionic and cationic micelles, and they scarcely depend on the concentration C. These results well correspond to those from SLS. At an intermediate a M (pH 5.0), however, the Stokes radius becomes large and increases with C. These results are also consistent with those from SLS. The hydrodynamic
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10246 J. Phys. Chem., Vol. 98, No. 40, 1994
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I
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Kaimoto et al.
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500 400
3
TABLE 1: Dependence of the Aggregation Number m on pH in 0.1 M NaCl Solutions (A) Light Scattering” pH aM mb maoDC pH aM mb mamc 3.00 0.91 70 4.90 0.47 98 178 3.50 0.82 90 5.20 0.39 90 161 4.00 0.69 113 5.50 0.32 85 117 4.50 0.57 90 156 6.20 0.16 77 81 4.70 0.51 92 170 7.80 0.01 69 (B) Fluorescence Probe Methodo
PH 2.00 3.00 4.00 4.50 5.00
,
aM
m
1.oo 0.91 0.69 0.57
70 80 84 89 94
0.44
PH 5.50 6.00 7.00 8.00 10.00
aM
m
0.32 0.20 0.54
90 81 77 72 65
0.00 0.00
“Estimated range of error. pH f 0.02, a M f 0.02, m f 10%. g~m-~. Evaluated at cmc. Evaluated at c - co = 3 x
TABLE 2: Dependence of the Aggregation Number m on pH in 0.2 M NaCl Solutions (A) Light Scattering“ pH aM mb mamC PH aM mb maaaC 3.00 0.95 69 95 5.30 0.42 117 538 3.50 0.85 81 149 5.50 0.37 87 402 4.00 0.74 101 394 6.20 0.21 81 109 4.80 0.54 148 683 7.80 0.01 66 5.00 0.49 135 646 (B) Fluorescence Probe Method”
PH 2.00 3.00 4.00 4.50 5.00
aM
m
1.00 0.95 0.74 0.61 0.49
82 85 100 109 108
PH 5.50 6.00 7.OO 8.00 10.00
aM
m
0.37 0.25 0.07
112 96 79 78 77
0.00
0.00
Estimated range of error: pH h0.02, a M k0.02, m &lo%. Evaluated at cmc. Evaluated at c - co = 3 x g~m-~.
V, = 0.3233m nm3 Micelles are assumed to be the prolates of major and minor axes a and b. The minor axis b is given as the sum of hydrocarbon chain length b‘ and the polar group contribution lo which is taken to be 0.246 nm:
b = b‘
+ lo
(6)
According to TanfordI5
b‘ = 0.1
+ 0.1265 x 11 = 1.156nm +
(7)
TABLE 3: Comparison of Hydrodynamic Radius of DDAO Micelles at pH 5.0 in 0.2 M NaCl c/10-3 M” 2.42 5.11 7.51 10.08 12.45 14.88 5.04 2.62 7.54 10.04 12.63 14.95 mawb 210 401 513 570 570 649 mc 213 426 576 678 771 862 RH(calcd)/nmd 4.3 8.0 8.6 6.7 9.1 9.5 RH(calcd)/nme 4.3 4.3 6.9 8.7 10.7 11.7 5.6 6.4 9.6 10.6 10.8 RH(obsd)/nm 3.5 Upper and lower values refer to the measurements of SLS and DLS, respectively. Evaluated from SLS assuming B2 = 0. Evaluated from SLS corrected for the contribution from Bz calculated for a prolate containing mappsurfactants. Calculated according to eq 9 using mapp. e Calculated according to eq 9 using m.
is to be noted that calculated RH values refer to unsolvated particles. A good coincidence between them can be concluded from Table 3 if the crudeness of the evaluation of RH above is taken into account. This coincidence supports our interpretation of the concentration dependence in terms of micellar growth. When another estimate for b’ was employed corresponding to the fully extended case,15 Le., b’ = 1.542 nm, smaller calculated RH values were obtained by a factor of 0.7-0.8. f/(6nyRO)= F(p) = (1 - p2)/p2/3In[{ 1 (1 - ~ ~ ) ‘ / ~ > / p ] We roughly examine the contribution from the second virial (8) coefficient term for the case of assumed micelle growth. Judging from the data for fully cationic micelles in 0.2 M NaCl, Finally we obtain shown in Figures 3 and 7, it is pertinent to neglect the electrostatic interaction in evaluating the excluded volume effect (9) of the proposed prolates. We roughly evaluate the second virial Calculated values of RH according to eq 9 are summarized (in coefficient BZ in terms of the excluded volume u of a rod of the third row) and compared with observed ones in Table 3. It the diameter (2b) and the length (2a) as follows:17
The major axis a is also given as a = a’ lo where (I’represents the hydrocarbon contribution. We can detennine a’ as 3V,/ [4~t(b’)~]. Once both a and b of a prolate is evaluated, the ratio of its frictional coefficientfto that of the sphere of the same volume is given as follows16in terms of the axial ratio p (=bl a):
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J. Phys. Chem., Vol. 98, No. 40, 1994 10247
Dodecyldimethylamine Oxide Micelles in Salt Solutions
(10)
u = 4na2b
where M O= 229 is the molecular weight of the monomer and N A denotes the Avogadro number. Values of B2 (lop4cm3 mol g-*) were almost constant over the concentration range: 3.51 at 2.42 mM and 3.45-3.42 for the other concentrations. The contributions from the virial term 2B2M,ynapp(c- cmc) are 0.012 at 2.42 mM but increases up to 0.33 at 14.88 mM. Observed aggregation numbers mapp and RH are accordingly underestimated at high concentrations. However, neglect of the hydrodynamic interaction certainly overestimated the values of RH. We tentatively correct the aggregation number for the contribution from B2 as follows:
m = map,[ 1
+ 2B2Momapp(c- cmc)]
(12)
The results of corrected values of m were given in Table 3. Recalculated values of RH on the basis of the corrected m are obtained and also shown in Table 3. Further iteration would not be significant because of the approximate nature in evaluating both calculated and observed RH values. Simple Theoretical Consideration on the Effect of a M on m at cmc. At a given set of pH, C,, and surfactant concentration c, the average degree of ionization a M and the average aggregation number m (a,) are both determined. However, various micelles having different aggregation numbers m’ and degrees of ionization a M ’ from respective averages are present in the solution; however, their concentrations are negligibly small. For example, the population of uncharged micelles of aggregation number m‘ is generally negligibly small but not zero unless a M is close to zero. We consider the equilibrium between two kinds of the micelles of aggregation number m’ with a, = 0 and a M . The equilibrium is described with the following equation in terms of the chemical potentials of hydrogen ions p~ and micelles of aggregation number m‘, pm: &(a,)
=k(0)
+ aMm’pH
(13)
The right-hand side of eq 17 is given as follows from eqs 18 and 19, if the contribution from the micelle concentration term is neglected: G*(m,aM)/m - G*(mo,O)/mo = RT ln[cmc/cmc(O)] R n a , In a,
+
+ RT ln n,(aM) k ( 0 ) = G*(m’,O) + RT In n,(O) = G*(m’,aM)
+ ln(1 - a,)] (20)
Now the right-hand-side of eq 16 reduces to a,m’pH - [G*(m’,aM) - G*(m’,O)l = aMm’(pH*- pI* pN*) RTm’ ln[cmc(O)/cmc] -
+ +
+
+
m’RnaM{ -In aH In a,/( 1 - a,)} In( 1 - a,)] = -RTm’ ln[(l - a,)cmc/cmc(0)] (21) It is to be noted that several terms cancel out before reaching the final result in eq 21 due to the proton dissociation equilibrium of monomers. To make clear the nature of the approximation inherent in eq 17, a slightly different way of deriving eq 21 is presented here. The standard free energy difference of the two micelles appeared on the left-hand side of eq 17 will be given by the work of charging up the micelle size m’ plus the contribution from the ideal mixing entropy, aside from the chemical part:
+ RTm’[aMIn a, + (1 - a M )h ( 1 - a ~ ) + ] m’aM(pH* + RTln KM) (22)
G*(m’,aM) - G*(m’,O) = &(”,aM)
The last term of the right-hand side of eq 22 refers to the correction for the chemical part in terms of the intrinsic proton dissociation constant of the micelle K M . When the electric free energy Gel(m’,aM)lm’per surfactant molecule is approximated with the experimental value, it is given as follows: G“(m,a,)/m
+
= ~ n [ c m c / c m c ( ~ ) l2.303aM(pKM + -p~,)
- a,>/(1 -
(23)
In eq 23, pK, is the apparent dissociation constant defined as
The chemical potentials consist of the standard part and the concentration dependent part: &(a,)
- pN*)
+ (1 - a,)
pKa = pH
+ lOg[aM/( 1 - a,)]
(24)
(14)
Combining eqs 22 and 23, we have eq 21. Substituting eq 21 into eq 16, we have
(15)
In[n,(aM)/nm(0)] = -m’ In[( 1 - a,)cmc(aM)/cmc(0)]=
-w( aM)m’ (25)
From eqs 13-15: RT h[nm(a,)/nm(0)] = aMm’pH
+ G*(m’,o) - G*(m’,aM) (16)
The difference [G*(m’,aM) - G*(m’,O)] is reasonably approximated with that corresponding to the most probable species:
where
w = In[( 1 - a,)cmc(aM)/cmc(0)]
(26)
Rewriting eq 25 leads to
[G*(m’,aM) - G*(m’,O)]/m’ = G*(m,aM)/m - G*(mo,O)/mo (17) A consideration on the equilibrium between monomers and the micelles with most probable aggregation number leads to
+
+
RT ln[(C - cmc)/m] = m{aMpI* G*(m,a,) (1 - a,)pN*} mRT In cmc mRT{ a, In a,
+
+
( l - aM) G*(mo,O)
The following reasoning to derive the dependence of m on a M was originally developed by Andelman.ls If we can ignore the polydispersity with respect to a,, the average aggregation number m under a condition consistent with a M is given as follows:
+
- a,)> (18)
+ RT ln[{C - cmc(0))/mo] = mopN*+ m&Tln cmc(0) (19)
We assume the Gaussian distribution with respect to the size distribution of nonionic micelles:
Kaimoto et al.
10248 J. Phys. Chem., Vol. 98, No. 40, 1994
where n,(O) and n(0) are the concentrations of the nonionic micelles with aggregation number m' and their sum irrespective of the values of m'. The average aggregation number is m. The standard deviation is u. The integral involved in eq 28 can be carried out when eq 29 is introduced for n,(O) in eq 27:
Jnm(aM) dm' = const x exp[(w2d/4) - wm,] 8/h[ln{Jn,(aM)
d"}] = (wc?/2) - mo
(30) (31)
Finally we have m = m, - w 4 / 2 We calculated w values by eq 26 and evaluated the values of u from eq 32. Constant values of u were obtained, however, only in the region of a M between 0.1 and 0.55. Constant values of u2 were about 50 f 20 and 100 for 0.1 and 0.2 M NaCl, respectively. Clearly u is smaller at 0.1 M NaCl than at 0.2 M. This can be ascribed to the highly polar nature of the nonionic micelles of DDAO. For compact spherical micelles with the average aggregation number m, the width of size distribution is given as f01lows:'~
1 < o/z/mo < 2
or
mo
