Oscillations, Waves, and Interactions - GWDG

Liquids: Formation of complexes and complex dynamics 381
Figure 13. Ultrasonic excess attenuation spectra of aqueous solutions of ndecyltrimethylammonium
bromide at 25 ◦ C at surfactant concentration 0.15 mol/l (•) and
0.5 mol/l (◦). The critical micelle concentration is 0.06 mol/l [70].
with the total amphiphile concentration C = �
i i � �
Ni , the amplitude and relaxation
time of the term reflecting the fast monomer exchange are predicted as [67,68]
Af = π (∆V )2 cmc σ
κS∞R T
2
m x
�
1 + σ2
m x
�−1
(30)
and
τ −1
f
kr
=
σ2 �
1 + σ2
m x
�
, (31)
respectively. In deriving Eq. (30) the same reaction volume ∆V has been assumed
for all reaction steps of the isodesmic scheme. The quantity κS∞ = ϱ −1 c −2 (∞) is
the adiabatic compressibility extrapolated to frequencies well above the relaxation
region. The amplitude of the Teubner-Kahlweit-Aniansson-Wall model increases
monotonously with concentration to asymptotically approach
The relaxation rate depends linearly upon concentration
lim
C→∞ Af = π (∆V )2 cmc
. (32)
κS∞R T
τ −1
f
= aC + b (33)
with slope a = k r /(m cmc) and with b = k r � σ 2 − m −1� . As obvious from Fig. 14,
Af and τ −1
f for sodium dodecylsulfate solutions display quite different concentration
dependencies [71], indicating a significant effect from the ionic nature of the
amphiphile. Obviously, the discrepancy between the experimental findings and the
predictions of the original Teubner-Kahlweit-Aniansson-Wall model reflects considerable
effects from incomplete dissociation of the surfactant [72]. If Ji denotes the
number of undissociated monomers per aggregate of class i, the law of mass action
relates the concentration of free counterions
�
�1/(Ji+1−Ji) [Ni+1]
[Nc] =
(34)
[Ni] [N1] bi+1