Oscillations, Waves, and Interactions - GWDG

370 U. Kaatze and R. Behrends
Figure 1. Ultrasonic attenuation spectrum in the frequency normalized format for ndodecanol
at 25 ◦ C [32]. The curve is the graph of a relaxation spectral function with two
Debye-type relaxation terms. The dashed line shows the α/ν 2 value following from Eq. (12),
assuming a frequency independent shear viscosity.
Constancy of the shear viscosity within the frequency range of measurements is not
fulfilled with all liquids. Examples are polymer melts [26,27], the frequency dependent
shear viscosity of which reflects modes of chain conformational isomerisation [27–
30]. Another example is the chain isomerisation of n-alkanes [31] and alcohols [32].
Figure 1 shows a frequency normalized plot of the ultrasonic attenuation spectrum
of n-dodecanol at 25◦C. Also given by the dashed line is the contribution
�
α
ν2 �
ηs0
= 8π2
3c 3 ϱ ηs(0) (12)
that would result on assumption of a frequency independent shear viscosity ηs(0)
as measured with a capillary viscosimeter or a falling ball viscosimeter. The finding
of experimental α/ν2 data smaller than � α/ν2� is a direct indication of the
ηs0
shear viscosity of n-dodecanol to be subject to a relaxation. Shear wave impedance
spectrometry [33] has confirmed this conclusion by revealing a fequency dependent
complex shear viscosity
ηs(ν) = η ′ s(ν) − ıη ′′
s (ν) . (13)
In Eq. (13) the real part η ′ s(ν) represents the irreversible viscous molecular processes,
whereas the negative imaginary part η ′′
s (ν) considers the reversible elastic mechanisms
of the visco-elastic liquid. Figure 2 presents the shear viscosity of n-dodecanol,
measured between some MHz and about 100 MHz, in a suggestive complex plane
representation. The data evidently fit to the semicircular arc which is given as the
graphical representation of the Debye-type relaxation spectral function [34]
As
Rs(ν) = ηs(∞) +
1 + ı ωτs
(14)