Oscillations, Waves, and Interactions - GWDG

Liquids: Formation of complexes and complex dynamics 371
Figure 2. Complex plane representation of the frequency dependent shear viscosity of
n-dodecanol at 25 ◦ C [32]. Figure symbols indicate data from different shear impedance
resonator cells. The circular arc is a plot of the spectral function defined by Eq. (14) with
the values for the parameters ηs(0), ηs(∞), and τs found by a nonlinear regression analysis.
with discrete relaxation time. In this function ηs(∞) is the extrapolated highfrequency
shear viscosity and As = ηs(0)−ηs(∞) is the relaxation amplitude. Hence,
in general, a frequency dependent shear viscosity has to be taken into account by
using
α(ν) = 2πν2
c 3 ϱ
as the more general version of Eq. (5).
2.2 Attenuation spectrometry
�
4
3 ηs(ν)
�
+ ηv(ν)
Within the frequency range from 12 kHz to 4.6 GHz, which is currently available for
the acoustical spectrometry of liquids, the wavelength λ of the sonic field within the
sample varies by a factor of about 4·10 5 . Due to the quadratic frequency dependence
of the asymptotic high-frequency term in α (Eq. (7)) the variation in the attenuation
coefficient is even as large as (4 · 10 5 ) 2 = 1.6 · 10 11 . It is thus impossible to cover
the frequency range with only one method of measurement. The spectra discussed
in this review have been obtained applying two different techniques and numerous
specimen cells, each one matched to a frequency range and to the sample properties,
in order to reach a maximum sensitivity and to reduce experimental errors to as
small as possible values.
At low frequencies (ν � 20 MHz), where normally α is small, the resonator principle
is appropriate as it is based on convoluting the acoustical path via multiple reflections,
hereby increasing the effective pathlength of interaction with the sample. Calibration
measurements using a reference liquid with carefully adjusted sound velocity and
density are necessary in order to correct the measured data for intrinsic resonator
loss. Spherical resonator cells [35] as well as cylindrically shaped cavities for quasione-dimensional
wave propagation are in use. Popular are biplanar cavities with both
(15)