Oscillations, Waves, and Interactions - GWDG

388 U. Kaatze and R. Behrends
Figure 20. Ultrasonic excess attenuation spectrum of a mixture of water and 2-(2butoxyethoxy)ethanol
(C4E2) at 25 ◦ C [88]. The mole fraction of C4E2 is x = 0.04. Dashed
and dotted lines are graphs of a Debye relaxation spectral term with discrete relaxation
time τD [34] and of a Romanov-Solov’ev term [91], respectively. The full curve represents
the unifying model of noncritical concentration fluctuations ([89], Eq. (53)).
given in that diagram is the graph of a relaxation term RRS(ν) as resulting from
the Romanov-Solov’ev theory of noncritical concentration fluctuations [91–93]. This
term also represents the experimental data only insufficiently. Several extensions
to the Romanov-Solov’ev theory have been made [87,89,94,95]. Here we sketch only
the last unifying model [89] because it combines the relevant aspects of all previous
theories.
In the unifying model changes in the local composition of the binary liquids are
assumed to occur along two possible pathways, one of which is an elementary chemical
reaction with discrete relaxation time τ0 and the other one a diffusion process
with mutual diffusion coefficient D. The time behaviour of the fluctuations is then
controlled by the differential equation
∂Φ(r, t)
∂t
=
�
D∇ 2 − 1
τ0
�
Φ(r, t) (47)
with Φ(r, t) denoting the autocorrelation function of the order parameter, namely the
deviation of the local concentration from the mean. Spatial Fourier transformation
yields
�
ˆΦ(q, t) = Φ(r, t) exp(ırq) dr (48)
with wave vector q. In the q space the simpler differential equation
follows with
r
∂ ˆ Φ(q, t)
∂t
τ −1
g
= − 1
ˆΦ(q, t) (49)
τg
= Dq 2 + τ −1
0 . (50)