Oscillations, Waves, and Interactions - GWDG

382 U. Kaatze and R. Behrends
Figure 14. Relaxation amplitude Af (•) and relaxation rate τ −1
f
(◦) of the sonic relaxation
term reflecting the fast monomer exchange in sodium dodecylsulfate solutions in water at
25 ◦ C [71] displayed versus concentration exceeding the cmc. Lines indicate the theoretical
relations (Eqs. (30,31)) when according to Eq. (35) the monomer concentration [N1] is used
instead of the cmc in the scaled concentration x (Eq. (29)).
to the concentration [N1] of surfactant monomers and [Ni] of class i aggregates.
Here bi+1 is the equilibrium constant. Assuming the effective degree of dissociation
αi = 1 − Ji/i for proper micelles to be independent of i within the micelle region
(αi = α m), an implicit relation for the monomer concentration as a function of total
surfactant concentration C follows:
where
[N1] = Nγ
�
C
− 1
[N1]
� 1/(m(2−α m) �
Nγ =
1 + α m
�√ 2πσ mbm
� �� (1−α m)/(2−α m)
C
− 1
[N1]
� 1/[m(2−α m)]
(35)
. (36)
Deriving these equations m ≈ m + 1 has been tacitely assumed. Using the monomer
concentration [N1] from Eq. (35) instead of the cmc in the scaled concentration
parameter x (Eq. (29)) and assuming the reasonable degree of dissociation αm =
0.33 [73] the Teubner-Kahlweit-Aniansson-Wall theory nicely represents the experimental
sonic amplitudes and relaxation times. The monomer concentrations following
from the analysis of spectra are shown in Fig. 15 as a function of surfactant concentration.
Also presented in that diagram is the cmc and the [N1]-versus-C dependence
for nonionic surfactant systems, as simulated by α m = 1 in the above relations. With
the ionic surfactant solutions the monomer concentration after reaching the cmc decreases
significantly whereas the [N1] values of the nonionic surfactant system slightly
increase with C. The tendency in the monomer concentration of ionic surfactants to
decrease above the cmc is a well-established fact [5].
Another extension of the Teubner-Kahlweit model is required to properly account
for the ultrasonic attenuation spectra of surfactant systems close to the cmc [74–