Oscillations, Waves, and Interactions - GWDG

380 U. Kaatze and R. Behrends
Figure 12. Distribution of the equilibrium concentration ˆ ˜
Ni ¯ of aggregates from i
monomers for proper micelle systems (full line). Dashed and dotted lines show the slow
and fast response, respectively, of the surfactant system to external disturbances.
as sketched in Fig. 12. Here � �
Ni denotes the equilibirium concentration of species
Ni, kr is the mean reverse rate constant for micelle sizes around the mean ¯m and σ2 is the variance of the size distribution. Using Eq. (27) it is assumed that each step
in the series of reactions is characterized by a well defined free energy change
∆Gi = RT ln (Ki) (28)
as following from the van’t Hoff equation (Eq. (19)). Here ∆Gi = ∆Hi − T ∆Si and
Ki = k f
i /kr i .
On the basis of the Aniansson-Wall model, the Teubner-Kahlweit theory [67,68]
predicts sonic spectra with two Debye-type relaxation terms. These terms can be
identified in a suggestive manner with two modes in the reformation of the equilibrium
distribution of aggregates after a small disturbance. As indicated in Fig. 12, the fast
relaxation process, with relaxation time τf roughly in the range of nanoseconds and
microseconds, is due to the monomer exchange. It is characterized by a change
of the aggregation number m at almost constant number of micelles per volume. In
parallel, a slow process with relaxation time τs on the order of milliseconds or seconds
proceeds by which the final equilibirium between the micelles and the suspending
phase is reached. This slow process has been studied by time domain methods,
predominantly by pressure jump and temperature jump techniques [69]. Here we
discuss the low-frequency relaxation term of experimental spectra (Fig. 13 [70]) in
the light of the fast monomer exchange. The term at even higher frequencies is
assigned to the rotational isomerisation of the alkyl chains within the micellar cores.
For nonionic proper micelle systems with large m, the monomer concentration � �
N1
is usually identified with the cmc. Using the scaled concentration
x = (C − cmc) /cmc (29)