Oscillations, Waves, and Interactions - GWDG

Liquids: Formation of complexes and complex dynamics 391
the stronger the tendency to form clusters in order to avoid unfavourable interactions
with water. The effect of cluster formation is noticeably smaller for solutes with
branched hydrophobic parts. Almost no fluctuations of local concentration exist
for the tetramethylurea/water system (ξmax = 2 · 10 −10 m [90]), whereas ξmax =
30 · 10 −10 m for aqueous solutions of n-butylurea (Fig. 22).
4.2 Critical demixing
The anomalies in the thermodynamic and transport properties, which are induced
by long wavelength fluctuations in the order parameter, associated with the phase
transition near a critical point, have attracted much interest from both a theoretical
and an experimental point of view. Considerable attention has been directed towards
ultrasonic attenuation spectrometry as the method allows to verify or disprove the
dynamic scaling hypothesis, particularly if combined with quasielastic light scattering
and shear viscosity measurements. Various theoretical models for the description of
ultrasonic spectra of critical systems exist, the most prominent are the dynamic
scaling theory [97–100], the mode coupling theory [101–104], and the more recent
intuitive theory proceeding from a description of the bulk vicosity near the critical
point [105,106].
The dynamic scaling model predicts the critical contribution
α c λ = αλ − α b λ
to the total attenuation per wavelength, αλ = αλ, to be given by
(57)
α c λ = cA(T )F (Ω) . (58)
In this relation A is an amplitude factor, only weakly depending on frequency, and
F (Ω) is the scaling function with reduced frequency
Ω = 2πν
. (59)
Γ(ɛ)
The relaxation rate of order parameter fluctuations, Γ(ɛ) = τ −1
ξ , is assumed to follow
a power law
−Z0 ˜ν
Γ(ɛ) = Γ0ɛ
with the universal dynamical critical exponent Z0 and the critical exponent ˜ν of the
fluctuation correlation length mentioned before.
In correspondence with an empirical form of the dynamic scaling model [99] the
scaling functions of the Bhattacharjee-Ferrell (BF), Folk-Moser (FM), and Onuki
(On) models can be favourably represented by the relation [107]
Fx(Ω) =
�
1 + 0.414
� x �nx Ω1/2 Ω
�−2 with Ωx 1/2 (x = BF, FM, On) denoting the scaled half-attenuation frequency of Fx
and nx an exponent that controls the slope Sx(Ω = Ωx 1/2 ) = dFx(Ω)/d ln(Ω)|Ω of
1/2
(60)
(61)