Oscillations, Waves, and Interactions - GWDG

386 U. Kaatze and R. Behrends
of the system and, on favourable conditions, can be obtained from experimental data.
At reasonable values of the parameters of Eqs. (41,42) a Gaussian size distribution
of micelles follows also from this procedure. According to
σ 2 = (sf + sr) −1
the variance of the distribution is given by the slope parameters in the dependencies
of the rate constants upon the aggregate size.
The adequacy of the above relations has been demonstrated by considering results
for solutions of proper micelles [75]. As the numerical evaluation of the isodesmic
reaction scheme (Eq. (25)) avoids approximations of the analytical treatment, it is
capable of revealing some special features of short chain surfactant solutions which
remain unnoticed otherwise. For parameters modelling the n-HepACl system the size
distribution of the micellar species at some surfactant concentrations is displayed in
Fig. 19. Some characteristics of these distribution functions attract attention. The
less pronounced relative minimum in the size distribution � Ni(i) � contrasts the deep
minimum presumed by the Teubner-Kahlweit-Aniansson-Wall model for the oligomer
region. Hence there exists a noticeable content of small oligomeric structures in
the short chain surfactant solutions and the separation between the slow and the
fast relaxation process is reduced or absent at all. At C ≈ cmc and C < cmc the
relative maximum in � Ni(i) � , as characteristic for the Gaussian distribution of proper
micelles, is absent. Even at surfactant concentrations distinctly smaller than the cmc,
a noticeable content of oligomeric species is formed.
The eigenrate spectrum of the isodesmic reaction scheme (Eq. (25)) near the cmc
confirms the absence of any slow relaxation process due to the absence of a pronounced
relative minimum in the size distribution � Ni(i) � . As indicated by the
Figure 19. Distribution of the equilibrium concentration ˆ Ni
˜ of aggregates from i
monomers as resulting from the extended Teubner-Kahlweit model [75]. In the numerical
calculations parameters have been chosen to correspond with the n-heptylammonium
chloride/water system at surfactant concentration C.
(43)