Oscillations, Waves, and Interactions - GWDG

Liquids: Formation of complexes and complex dynamics 387
dashed line in Fig. 18 the extended model of stepwise association predicts an increase
of the relaxation rate τ −1
max when, at C below the cmc, the surfactant concentration
decreases. An increase distinctly smaller than the one obtained from sonic attenuation
spectra, however, is predicted by the extended Teubner-Kahlweit model. The
theoretical relaxation-rate-versus-concentration relation also suffers from the inappropriate
assumption of a concentration independent mean aggregation number m,
leading to a wrong slope d(τ −1
max)/dC at C > cmc. Interesting, however the numerical
evaluation of the isodesmic reaction scheme yields the simultaneous presence of two
fast relaxation terms with similar relaxation rates and relaxation amplitudes. These
terms cannot be represented by a single Debye term but result in an unsymmetric
broadening of the relaxation time spectrum as indicated by the Hill relaxation time
distribution. Furtheron the numerical simulation reveals ultrafast relaxation contributions
reflecting the monomer exchange of oligomeric species. This oligomer process
is assigned to the high-frequency Debye-type relaxation term in the sonic attenuation
spectra of short chain surfactant solutions because this term is unlikely due to
the hydrocarbon chain isomerisation in the micelle cores. Structural isomerisations
of such short chains are expected to display relaxation characteristics to the sonic
attenuation spectrum at frequencies well above measurement range.
4 Local fluctuations in concentration
4.1 Noncritical dynamics
Binary liquid mixtures may minimize their free energy by forming a microheterogeneous
structure which fluctuates rapidly in time. The local fluctuations in the
concentration of the constituents relax by diffusion. Based on the dynamic scaling
hypothesis [82–85]
τξ = ξ2
, (44)
2D
the characteristic relaxation time τξ is assumed to be given by a characteristic length
ξ of the system and by the mutual diffusion coefficient D. In critical mixtures the
fluctuation correlation length ξ covers vast ranges of size and follows a power law [6–
11]
ξ = ξ0ɛ −˜ν
(45)
thus tending to mask the individual properties of the system. In Eq. (45) ξ0 is an
individual amplitude, ˜ν a universal exponent and
|T − Tc|
ɛ =
is a scaled (reduced) temperature. In systems displaying noncritical dynamics the
critical temperature is not reached so that the fluctuation correlation length does not
diverge.
Ultrasonic attenuation spectra due to noncritical concentration fluctuations [86–
90] extend over a broader frequency range than Debye type processes with a discrete
relaxation time (e. g. Eq. (20)). An example of a spectrum is shown in Fig. 20. Also
Tc
(46)