301%
KLNETWCS OF MICELI,~ DISSOCIATION model Obviously, our data do not suffice to fix all of t,he required parameters, so that the final choice is not uniquely determined: for that mat)ter, there is no way to exclude more elaborate multiple-equilibrium models involving a,n e x w larger number of parameters. The present model is hesi in the sense that (1) a twoequilibrium model is the simplest one which gives an adequate fit for the data in Figure 3; (2) the curvature or” the dilution shift plots near the first cmc suggests that a = 25 is; about right, and then eq 7 requires n = 12; and (3) the choice Z, --- 60 reflects the fact that
the model cannot give good results unless b is substantially larger than a, but in the systems studied by Ekwall the larger micelles still appeared to be spherical.l* We chose to try to incorporate this feature into the model, tanking note also of the fact that spherical micelles cannot be formed by detergents having twelvecarbon chains when the aggregaliorx xiurxiber is much higher than 60. Once a, n, and b are selected in this way, the remaining parameters are indeed ‘(best” values in the srnse that other choices produced c z ~ r v e s ~ ~ which agree less well with the experimental points.
icelllle Dissociation by Temperature-JumpTechniques.
y Norbert Muller Department of Chemistry, Purdue University, Lafayette, Indiana
.47907
(Received June 1.4, 1QYW)
Pi~hlicalioncosts assisted by the National Science Foundation
Sei-eral authors have reported that when the monomer-micelle equilibrium in an aqueous detergent solution is perturbed by a temperature jump the relaxation time lies between 1 and 300 msec. Rate constants for the dissociation of one molecule from the micelle have been derived from such data and stated to be no larger than 120 sec-I. Since other evidence, especially line widths in nmr spectra, appears to hhow that the rate constants must be several orders of magnitude larger a reexamination of the procedures used to interpret the T-jump data was undertaken. It was found that the mathematical development involves an approximation usually not admissible under the conditions of the T-jump experiments, but even when allowance is made for rhis the T-jump and nmr results cannot be reconciled while retaining a mechanism for micelle dissolution wkmh involves a single slow step. An approximate treatment based on the assumption that dissolution proceeds by a sequence of many steps of nearly equal rate shows that with this mechanism the reported relaxation times are compatible with rate constants in the 106-10e sec-l range when the aggregation numbers are larger than 50.
Introduction The formation and dissolution of micelles in aqueous detergent solutions is so rapid that quantitative measurements of the rdevant rates have become available only relatively recenl ly. Since 1966, rakes of dissocia-
0 ~y~sec.The data mrere interpreted on the ztssurnpl ion that micelle dissociation (1)
is the rate-limiting step in the formation and dissolution processes and used to derive values of i%n,n-l ranging from 0.4 to 120 sec-l. These results stand in flat contradiction to the repeated observationst7that process 1 i s “fast on the nmr
(4) B. C. Bennion and E. M. Eyring, J . Colloid Interfuce Sci., 32, 236 (1970). (5) J. Lang and E. M . Eyring, J . Polgmer Sci., Part A-2, 10, 89 (1972). (6) H. Inoue and T. Nakagawa, J . P h y s . Chem., 70, 108 (1966). (7) N. Muller and I?. E. Platlro, (bid.,75, 547 (1971).
T h e Journal of Phgsical Chemistry, V G ~ 76, . N o . 21, 1572
timescale." 'In every detergent system examined by nmr, the monomeric and micellar species produce a time-averaged spectrum in which individual signals are no broader than signals obtained when monomers are present alone. If for a particular resonance the chemical shift difference between micellar and extramicellar molecules, in frequency units, is vmlc - vaq and the respective mean residence times are 7 m i c and 7 a q the additional broadening expected for an intermediate rate of exhangG is8 Avexcb
(1/~Tz),,Ph
4rfm,c'(1
- fmio)'(Vrnic
Vaq)2(7mio
f
7aq)
(T/2)(Vmic
-
vag)2Tmic
(3)
Typical values of vmle vaq are 15 Hz for phenyl resonances in proton nmr spectra6 and 66 Hz for fluorine signals from trifluoromethyl groups of fluorine-labeled surfactants7 when the spectra are taken in a 14 kG magnetic field Bf the absence of detectable line broadening is taken to imply that A v e x e ~is no larger than 1 Rz, then T ~must , ~be less than 3 X sec on the basis sec for the of the proton data or less than 1.5 X fluorinated species. Since it has been shown7 that in1 roduction of a trifluoromethyl group affects micelle stability only 1o a minor extent, it should have no great effect on 71nlc, and it seems safe to infer that Tmic is less than 1.5 X 10 --4 sec for ail surfactant species with alkyl chains containing up to 13 carbon atoms and moderately compact headgroups. To set a loner limit on the rate constant k,,,...l it is only uecessary to consider that when the aggregation number is n reaction 1 must, on the average, proceed n times in each direction before a particular monomer leaves the mic elk. Thus for sodium trifluorododecyl sulfate, if ?a is about 60 I
k7L8n-i s
v//Tmio
2
4 X IO5 sec-l
(4)
This 1s nearly five orders of magnitude larger than the value deduced3 from T-jump data for unfluorinated sodium dodecyl sulfate. The conclusion that monomer-micelle exchange is very fast is also supported by ultrasonic absorption studiesg of potassium alkyl carboxylates and by an electron spill resonance investigation lo of di-tert-butyl nitroxide in, solutions with sodium dodecyl sulfate, which gave a forward! rate constant of 4.3 X lo5 sec-' for the process micelle. nitroxide
micelle
+ nitroxide
(5)
This situation makes it necessary to reexamine carefully the assumptions and procedures used in treating The Journal of Phfilaical Chemistry, Vol. 76, No. 21, SO72
KHDS Interpretation The accepted mathematical analysis of the T-jump data was presented by Kresheck, et a1.,2 whose work will be designated by the abbreviation 14BDS and whose notation is retained here. The final result of the KHDS development is an equation for the relaxation time
(2)
where f m l o represents the fraction of material in the micellar form taking an appropriate concentrae that f m l C = 1/2. Then rmIC= tion one can ar T ~ and ~ (2) , becomes AYoxch
the T-jump data and to determine whether an alternative interpretation can be developed that would eliminate the disagreement between the results of the various physical methods.
where a is the total detergent concentration and [A], the monomer concentration at equilibrium, taken as equal to the critical micelle concentration, cmc. A desirable feature of this equation is that it implies a linear relation between 1 / and ~ a, in accord with the experimental findings, but it is in fact valid only in the 1 imit when the departure from equilibrium produced by the T jump approaches zero. The following reasoning shows that under the conditions of most of the experiments, the use of eq 6 is not justified. Equation 12 of KHDS contains a term { [Ale x ) " with x = [A] - [A],. The next steps in the derivation essentially involve rewriting this term as [A],"{1 -$- z / [A],] and then replacing { I z/[A],) by 1 nz/[A],, the first two terms of the polynomial expansion, However, as Table I of KRDS shows, a temperature rise of 5" produces an initial value of x = Ill while [Ale is about 6 X ill, so -0.3 X that early in the experiment z/[A]e is about -0.05. Taking the aggregation number as 57
+
+
+
(1
+ z/[A],]'
=
(1
- 0.05)87= 0.0115
(7)
but the first two terms of the expansion are
1
+ nx/[A],
=
1 4- 87(-0.05)
(8)
-3.33
a value not only of the wrong magnitude but having the wrong sign. The range of conditions for which use of eq 6 is appropriate may be determined from
+.
,
.
(9)
Use of the first two terms alone is admissible when b/iAle/ .> k,,,-I and [An-l]e
