2018-(I)-Critical Contribution, Dynamic Scaling and Crossover Theory
Article Index
The scaling function F_x_BF(Ω):
Last decades, various theoretical expressions for the universal scaling function of ultrasonic attenuation spectra have been presented for critically demixing binary fluids. In Fig.(4.2) three prominent examples are shown:
In 1981 Bhattacharjee and Ferrell have presented the scaling function for sonic attenuation in analytical form:
Because of the difficulties to treat this integral, an empirical function has been developed:
with the half attenuation frequency Ω1/2 . The Folk-Moser (FM) scaling function FBF (Ω) as resulting from the renormalization group theory of the mode-coupling model and the function FOn (Ω), which Onuki (On) derived from an intuitive description of the bulk viscosity near a consolute point, are not available from theory in analytical form. Therefore, Behrends at al. developed a quasi-universal empirical form of scaling functions, in correspondence with the empirical function of
Bhattacharjee and Ferrell. These forms are shown in Fig.(4.2), and described by following relation:
where x denotes BF , FM and On, while Ωx and Ωx denote a characteristic frequency and the half-attenuation frequency of the scaling function, respectively. Finally, using the relation:
to evaluate experimental attenuation coefficient data along with relaxation rates Γ(ε) from light scattering and shear viscosity, it is possible to decide about the quality and validity of the scaling functions. The determination of the scaling function F(Ω) is usually based on the fact that A and cs are only weakly dependent upon temperature. Therefore, the scaling function can be derived as the ratio:
The measured total attenuation data contain contribution from critical fluctuation but also from noncritical processes. Assuming that all parts contribute to the ultrasonic attenuation spectrum additively, the ultrasonic spectra can be analytically represented as a sum of N Debye terms, the background contribution and the critical contribution: :
The Debye-terms can be likewise replaced by Hill-terms.