in Fig. 1b, which has been redrawn from Funke et al. [42]. At low frequencies, a dc
plateau is typically observed, but at higher frequencies σ
0
ðωÞ is found to increase
with frequency. According to Fig. 1b, at even higher frequencies the conductivity
should again reach another plateau region. The velocity autocorrelation function
corresponding to the spectrum of Fig. 1b consists of a Dirac Delta function at t ¼ 0,
but also of a negative contribution that approaches zero at longer times. Such a
negative autocorrelation has been termed backward correlation of ionic motion.
It can be understood by using the general concept of the MIGRATION model
developed by Funke and coworkers for commonly studied ion conductors such as
inorganic glasses or crystals, polymeric materials, and ionic liquids [42–44]. The
central idea of this model is that because of their mutual repulsive Coulomb
interactions, equally charged mobile ions tend to stay apart from each other. If a
mobile ion leaves its site by hopping into a vacant neighboring site, mismatch is
created. The system then tends to reduce the mismatch, which can be done either by
a correlated backward hop of the ion itself or by a rearrangement of its neighboring
ions. In the first case, which forms the backward correlation effect on short time
scales, the previous forward jump of the ion under consideration turns out to be
unsuccessful, whereas in the second case the ion successfully moves to a new site.
Successful hops are the basis for long-range ion transport.
With the MIGRATION concept in mind, one can easily understand the frequency
dependence of σ
0
ðωÞ as presented in Fig. 1b. At small frequencies, the corresponding
time window Δt ¼ 1=ω is large enough to count only successful hops contributing to
the long-range ion transport. In the conductivity spectrum, this corresponds to a dc
conductivity plateau probed at low frequencies (see Fig. 1b). With increasing
frequency, however, the time window of observation becomes small enough to also
count forward jumps, which – on a longer time scale – will be cancelled by a
backward hop. The dynamic conductivity therefore registers all jumps not proven
unsuccessful within the given time window and the conductivity increases monotonously with frequency. When the time-window is so small that each ionic jump
contributes to the conductivity, the latter should become constant again (highfrequency plateau) (see Fig. 1b). The existence of such a high-frequency plateaus
has indeed been reported for experimental conductivity spectra of some crystalline
ion-conductors, but these plateaus occur at frequencies much higher (typically in the
gigahertz to terahertz regime); see for example [42] and references given therein.
2.2 Early Dielectric and Conductivity Spectra of PEC
As already mentioned in the Introduction, pioneering work on solid PEC dates back
to the 1960s, when Michaels and coworkers published systematic studies on the
frequency dependence of the complex permittivity [2, 3]. They investigated the
influence of the RH and the concentration of dopant salt like NaBr, respectively, on
the complex permittivity of PEC. The complexes were made of poly (vinyl benzyl
trimethyl ammonium chloride) (PVBTAC) and sodium poly(styrene sulfonate)
(NaPSS).
Ion Conduction in Solid Polyelectrolyte Complex Materials
103
plateau is typically observed, but at higher frequencies σ
0
ðωÞ is found to increase
with frequency. According to Fig. 1b, at even higher frequencies the conductivity
should again reach another plateau region. The velocity autocorrelation function
corresponding to the spectrum of Fig. 1b consists of a Dirac Delta function at t ¼ 0,
but also of a negative contribution that approaches zero at longer times. Such a
negative autocorrelation has been termed backward correlation of ionic motion.
It can be understood by using the general concept of the MIGRATION model
developed by Funke and coworkers for commonly studied ion conductors such as
inorganic glasses or crystals, polymeric materials, and ionic liquids [42–44]. The
central idea of this model is that because of their mutual repulsive Coulomb
interactions, equally charged mobile ions tend to stay apart from each other. If a
mobile ion leaves its site by hopping into a vacant neighboring site, mismatch is
created. The system then tends to reduce the mismatch, which can be done either by
a correlated backward hop of the ion itself or by a rearrangement of its neighboring
ions. In the first case, which forms the backward correlation effect on short time
scales, the previous forward jump of the ion under consideration turns out to be
unsuccessful, whereas in the second case the ion successfully moves to a new site.
Successful hops are the basis for long-range ion transport.
With the MIGRATION concept in mind, one can easily understand the frequency
dependence of σ
0
ðωÞ as presented in Fig. 1b. At small frequencies, the corresponding
time window Δt ¼ 1=ω is large enough to count only successful hops contributing to
the long-range ion transport. In the conductivity spectrum, this corresponds to a dc
conductivity plateau probed at low frequencies (see Fig. 1b). With increasing
frequency, however, the time window of observation becomes small enough to also
count forward jumps, which – on a longer time scale – will be cancelled by a
backward hop. The dynamic conductivity therefore registers all jumps not proven
unsuccessful within the given time window and the conductivity increases monotonously with frequency. When the time-window is so small that each ionic jump
contributes to the conductivity, the latter should become constant again (highfrequency plateau) (see Fig. 1b). The existence of such a high-frequency plateaus
has indeed been reported for experimental conductivity spectra of some crystalline
ion-conductors, but these plateaus occur at frequencies much higher (typically in the
gigahertz to terahertz regime); see for example [42] and references given therein.
2.2 Early Dielectric and Conductivity Spectra of PEC
As already mentioned in the Introduction, pioneering work on solid PEC dates back
to the 1960s, when Michaels and coworkers published systematic studies on the
frequency dependence of the complex permittivity [2, 3]. They investigated the
influence of the RH and the concentration of dopant salt like NaBr, respectively, on
the complex permittivity of PEC. The complexes were made of poly (vinyl benzyl
trimethyl ammonium chloride) (PVBTAC) and sodium poly(styrene sulfonate)
(NaPSS).
Ion Conduction in Solid Polyelectrolyte Complex Materials
103