E. A. G. Aniansson and S.N . Wall
IO24
n the Kinetics of Step-Wise Micelle Association E. A. G. Aniansson* and S. N. Wali Department of Physical Chemistry, University of Gothenburg, S-402 20 Gothenburg, Sweden Rev~sedManuscript Received October 8 , 1973)
(Received June 29, 1873,
Kinetic equations have been given a form which suggest an analogy with heat conduction and identifies the rate limiting quantities. Relaxation times have been deduced for the net disintegration of and the rearrangement among the micelles. The agreement with existing experimental results is satisfactory.
During the last few years the kinetics of micelle association and dissociation in surfactant solutions have been studied experimentally with various techniques.l-I7 The degree of association is typically of the order of 100, which requires either that drastically simplified assumptions be made in treating the kinetics,lJ2,l8JQor, that a perspicuous method be found for its handling. The following treatment is a m0ve in the latter direction. First we shall put the kinetic equation into a form which suggests an analogy with heat conduction (diffusion, etc.) and identifies the proper rate-limiting quantities. A treatment of the relaxation process at very small deviations from equilibrium will be given and the result compared with existing experimental results. The treatment will then be extended to larger deviations from equilibrium and a plausible explanation given of the apparent linearity still exhibited by the process. Finally, the process of rearrangement among the abundant micelles will be treated by approximating the kinetic equations by a partial differential equation. This treatment is limited to a somewhat idealized case, but an extension to more realistic ones is indicated, heir Interpretation Fortunately, it is very plausible to assume that the association (and dissociation) proceeds in unitary steps giving the set of reactions
. +
A, iAL a
A3 k,-
where A1 represents the surfactant monomer, A2 the dimer, etc. For ionic surfactants, A1 denotes the surfactant ion, and the treatment is limited to cases where the counterions are more easily movable than the surfactant ions, so that they can be assumed to adjust almost immediately to the association of the latter. It is probable that this applies to the vast majority of cases. Letting A, also denote the concentration of the species and A , its equilibrium value, we introduce the relative deviation from the equilibrium
one finds
and particularly for s = 1
s=3
The second term on the right side of (4a) is the net increase in unit time of the number of aggregates of size s and larger. It is thus a reaction flow in aggregation space (SIand we shall therefore denote it by J , Equation 4a is then the equivalent of the continuity equation, as seen more easily when written on the form dA, l d t = J , - JS+*
(4c)
When is so small that it can be neglected in the inner bracket of (5a), which is the case toward the end of jump relaxation experiments, this equation takes the form
There is a close analogy between this equation and the equation for heat conduction in one dimension;20 s corresponds to the space coordinate, [,to the temperature, A , to the mass per unit length (the specific heat capacity = 1), and k,- to the conductivity per unit of mass per unit length. 41 here plays the role of a space-independent driving force acting in the negative s direction in addition to the “derivative” of 4, with respect to s. The rate-limiting quantities in the process are thus the products k,-A, and not k,-, as one might be inclined to think at the outset. Dissociation of Micelles at Small Deviations from Equilibrium In a low-disperse micellar solution the dependence of In A, on s is something similar to that illustrated in Figure 1.23-25Also given in this figure is
obtained from (3), and for comparison, In s. The micelle population is more or less sharply peaked around a most probable s value, in this example 100, and orders of magnitude below and above this region. Although the micelle The Journalof Physical Chemistry, Vol. 78, No. 10, 7974
On the Kinetics of Step-Wise Micelle Association
distribution n for this example is not very broad, about 20% of the average aggregation number (see Further Comments and Conclusions), the ratio k,- fk,+Al changes very little over most of the range. It is then plausible to assume that the variatiions of h,- and k,+ themselves are not very much larger. That large changes of A, can be built up From ratios of h,-/h, + A i only slightly different from unity is d m to the large number of steps. It might then well be that the “conductivity” kB-AS is very much lower in the intermediate micellar region than IBP the region or” abrindant micelles. Returning to the analogy with heat c o ~ ~ d ~ c t ithe o n ,situation is, a t least in part, reminiscent of two large metal blocks connected by a thin wire, one block corresponding to the monomers and the oligomers, the other to the abundant micelles, and the wve corresponding to the region in between. After a n initial, short period of a ustment the main process would be a ~ s e ~ a o ~ ~ t f a1 0 ~ ~from i o ~ one ~ ~ “block” y to the other though the 6‘M71re.f9 To pursue a ~ a ~ ~ u of~ the a ~time ~ odevelopment n of this proes-ss we shall m a k ~the following more explicit assumptions. The s space is split into three parts, 1 _i s 5 s1, SI f 1 5 s 5 s2, and s2 1 5 The conductivities k,-A, are as s l g 2 s(1 f &)A,, is assumed to be negligible compared to the materiai iii the other two parts. We shall also assume that the deviation from equrlibrium i s SO small that (5b) is applicable for all s Then in parts one and three
+
To derive a differential equation for the process we note that, on the average, each micelle added to the third region increase the excess amount of material in that region with the amount a m 3 / a x s> s 2 &As so that
(13)
A short calculation gives
where u2 = Ra2 - ha2,the variance of the distribution. Using (11)one obtains
d m3 1 dt - - l m 3
( i5a)
~
where whereas in part two (6bf where J is the practically s-independent value of J s during the pseudo-stationary phase considered. This s independence follows directly from the analogy with the blocks and wire. Due to the low “mass per unit length” of the wire even small differences between Js and Js+l would lead to large and rapid temperature variations, characteristic of the initial adjustment but not of the pseudo-stationary phase. Summing eq 6a from s = 2 to s = SI, eq 6b from s = s1 t 1to sa, and eq 6b from s =: s 2 -t1to s“, one obtains
t+1 = ~ ” ( 1 - R J
(71
where
R being a resistance formed by series connection of the individual resistances larily that
Ilkq-&. s2 will not change during this initial period, the (amount of material, m3>in this range in general will. Although this process may be difficult to follow experimenically unless the changes in n3 and/or o are large, it may be of some Interest to see how it could be handled theoretically. We shall assume that for s > 53 -b 1 the distribution is Gaussian, i.e
;:his implies with (3) that
This ratio will change with the factor e2/“ when s i s varied from a3 - o to A3 + u , i.e., essentially over the proper micelle range. If u is not too small, e.g., of the order of 10 (Figure I), the ratio of k,- to k,+ will change very little and it would be reasonable to assume that
k,-
(27)
k-
independent of s. We then note that if u i s not too small, A,, and probably also [,, will change slowly with s so that they may be treated as continuous and differentiable variables. Equation 4a will then take the form
Assuming now that l[(s,t)l slsz l / ( k s -A,) or, in the approximation (20), k r - A r , o / n,. In the latter form it gives some information concerning the location, length, and "width" of the narrow pamage. To obtain information on k r - , independent measurements of A, ,o and n, are required. Ira principle, mor^ detailed information on the s dependence of k , - 2 , would be obtainable from (23) and (24) at lii~gerdeviations from equilibrium. Knowing R(&) as a fuirction of $ 3 , one v i ~ d dbe able to obtain I j k S - A , as a furrction of s lasing a suitable deconvolution method. Findly, it i s of ~nterestto compare the reciprocal relaxation times expected from (20) and (35). Assuming 12- and h r bo be of about thr same order of magnitude, and using rieasonable valraes of 0 and the other parameters entering these cquationiR, O W Finds that the rearrangement among the abundant micelles tends to be faster than the overall ~ ~ ~ ~ ~ n t of e ~t rh asame i ~ ~t ~ omicelles. This supports the jnitial assumptions mmde in the present paper. It is not inconceivable, hcwevcr, that in some cases the rate of the tws, processes may he of the same order of magnitude. A t ~ e ~ ~coupling ~ e n the t two will then be necessary.
The expressions sought are, then
and more generally _ ilM, _ _ -- -M,,, ____ - ro?M,., d&"t n12c1i- n32c2
If the micelle distribution is symmetrical, as is the Gaussian distribution, M3 = 0 and u2 i s concentration independent. To get order of magnitude expressions __ we set, as before, N 1, c1 Y Ax c z cmc, and 1232 % @. One then finds that
and
When = O(100) and A,,, jA1 = O(1) A, can be neglected in the denominators and one finds that ( A14 )
~ c ~ ~ ~ w lThe ~ ~authors ~ ~ wish e 1 to2 thank ~ . Dr. Mats Almgren fofor valixebie comments on this manuscript.
Appendix The required quantities are afi8/aAtot and au2/dAtot. To derive expressions for these we shall start from eq 3 from which i;here follows
(All)
and from this again
When u ez n3, one finds from (A14) that E3 changes very little when A,,, changes by an order of magnitude from the order of cmc. A significant relative change in u2 with a similar change in A,,, will occur only for very unsymmetrical and broad distributions. Taking as an example
IJsing these results we shall first obtain the derivatives with respect to In ,141, and then aAtot/d In A1 from which the required derivatives are ~ b t a i n e d . ~ g b . ~ ~ From the definitions
one finds o = a , n 3 = a SO, and M S = Z g 3 . In this case u2 will change by an order of magnitude with a similar relative change of A,,, if u is about as large as SO. For a moderately broad and unsymmetrical distribution such as
A, ?ZS,
-
n3
=
- 2 SAs
+
1
c3
S>S)
one finds that
which again is negligible for u
