Oscillations, Waves, and Interactions - GWDG

336 R. Pottel, J. Haller and U. Kaatze
propagating within the sample, decreases exponentially along the axis of wave propagation.
The absorption coefficient α in the exponential decay is one of the parameters
that is determined in frequency domain spectrometry.
According to the Kramers-Kronig relations absorption is correlated with dispersion
in the sound velocity and speed of light, respectively, within the sample. The
dispersion in the sound velocity of liquids is notoriously small and thus normally not
considered in the discussion of results. We mention, however, that the complexation
kinetics of electrolyte solutions has been studied in the frequency range 3-200 MHz
solely by high-precision sound velocity dispersion measurements [39,40].
In the dielectric spectrometry of dipolar liquids it is common practice to also
measure dispersion and to verify consistency of the results thereby. Hence dielectric
measurements typically involve complex quantities, preferably complex transfer
functions or complex reflection coefficients of appropriate specimen cells, instead of
only a scalar decay function. The dielectric properties of a sample are expressed in
terms of a frequency dependent complex quantity, the permittivity
ɛ(ν) = ɛ ′ (ν) − iɛ ′′ (ν) = 1
ɛ0
P (t)
+ 1 . (2)
E(t)
Here ɛ ′ (ν) and ɛ ′′ (ν) are the real part and negative imaginary part of the permittivity
at frequency ν, i2 = −1, and ɛ0 is the electrical field constant. ɛ ′ represents the
component of polarization P that is in phase with the electrical field E = Ê · eiωt
and ɛ ′′ represents the contribution with a π/2 phase shift. The simplest relaxation
spectral function is the Debye function [41]
ɛ(ν) = ɛ(∞) +
ɛ(0) − ɛ(∞)
1 + iωτ
with discrete relaxation time τ and angular frequency ω = 2πν. This function
corresponds with an exponential decay in the time domain. In the frequency range
up to 100 GHz the dielectric spectrum of water can be well described by the Debye
relaxation function [42,43]. As an example the water spectrum at 25 ◦ C is shown in
Fig. 2 where also the extrapolated low-frequency (“static”) permittivity ɛ(0) and the
extrapolated high-frequency permittivity ɛ (∞) are indicated.
Due to electrical conductivity ionic liquids, in which we are interested here, display
an additional contribution
ɛ ′′ σ(ν) = σ/(ɛ0ω) (4)
to the total loss
(3)
ɛ ′′
tot(ν) = ɛ ′′ (ν) + ɛ ′′ σ(ν) . (5)
Because of the frequency dependence in ɛ ′′ σ, inversely proportional to ν, the conductivity
contribution masks the dielectric contribution ɛ ′′ at low frequencies and renders
measurements difficult or impossible at all at small ν. In Eq. (4) the specific electric
conductivity σ is assumed independent of ν within the frequency range under consideration.
Figure 3 illustrates the situation by a dielectric spectrum for an aqueous
solution of sodium chloride. The salt concentration of that solution is less than one
tenth of the salinity of the North Sea and is only one third of a physiological solution.