the deviation from the theoretical stoichiometry will be higher than for all other
compositions with a larger number of small excess ions. In addition, the network of
complexes with x ¼ 0.5 deviates from the networks of all other compositions
where extrinsic compensation is not just a defect, but defined by the excess ions.
In addition to the obvious, differing structural properties of PSS-rich and
PDADMAC-rich PEC explained in Sect. 3.1, the differences in shape of the
conductivity spectra for PSS-rich and PDADAC-rich PEC (see Fig. 6) also imply
structural differences between both types of PEC. As a consequence, the energy
landscape in which ions can move and thus contribute to the long-range transport
probed by the dc conductivity will also differ. If we assume no connectivity of
anion-rich regions in a phase-separated PDADMAC-rich PEC, the anion transport
will be drastically decreased. In addition to the large size of the Cl
À ions, such
structural differences might be a reason for the low Cl
À mobility.
Having discussed the long-range ion transport, as probed by the dc conductivity
in the different types of PEC, the next section discusses ion dynamics in the
different PEC on a more local scale, i.e., at higher frequencies.
3.4 Modeling of Conductivity Spectra
There are many different models describing the overall shape of the frequencydependent conductivity, which cannot all be summarized here. Most of the models
have been developed to describe inorganic disordered ion-conducting materials
such as glasses. PEC, forming polymeric glasses with mobile charge carriers, can in
many respects, be described by the same models. One successful approach uses
computer simulations for treating the ion motions in static energy landscapes within
a random barrier model [52–54]. The random barrier model takes disorder into
account by choosing an energy landscape with randomly distributed barriers of
different heights. In most of these simulations, Coulomb interactions between the
moving ions are neglected. In some works, however, the influence of Coulomb
interactions on the scaling properties of conductivity spectra is discussed [52].
Other successful models describing the overall shape of frequency-dependent
conductivity spectra are the counterion model and the dipolar model of Dieterich
and coworkers [55] Here, Monte Carlo simulations are used to describe the hopping
motions of small ions. All Coulomb interactions between mobile ions and their
immobile counterions are taken into account. Disorder is introduced by randomly
placing fixed counterions into the center of cubes forming a lattice. In the counterion
model, long range transport of the mobile cations is possible; however, the dipolar
model only considers local motions of the mobile ions in the vicinity of their
immobile neighboring counterions (dipoles). Interactions between fluctuating dipoles
are taken into account.
As a further development, the MIGRATION model considers microscopic
ion hopping in an energy landscape that changes with time [42]. A distribution of
barriers in the energy landscape is not taken into account. Instead, the model
Ion Conduction in Solid Polyelectrolyte Complex Materials
115
compositions with a larger number of small excess ions. In addition, the network of
complexes with x ¼ 0.5 deviates from the networks of all other compositions
where extrinsic compensation is not just a defect, but defined by the excess ions.
In addition to the obvious, differing structural properties of PSS-rich and
PDADMAC-rich PEC explained in Sect. 3.1, the differences in shape of the
conductivity spectra for PSS-rich and PDADAC-rich PEC (see Fig. 6) also imply
structural differences between both types of PEC. As a consequence, the energy
landscape in which ions can move and thus contribute to the long-range transport
probed by the dc conductivity will also differ. If we assume no connectivity of
anion-rich regions in a phase-separated PDADMAC-rich PEC, the anion transport
will be drastically decreased. In addition to the large size of the Cl
À ions, such
structural differences might be a reason for the low Cl
À mobility.
Having discussed the long-range ion transport, as probed by the dc conductivity
in the different types of PEC, the next section discusses ion dynamics in the
different PEC on a more local scale, i.e., at higher frequencies.
3.4 Modeling of Conductivity Spectra
There are many different models describing the overall shape of the frequencydependent conductivity, which cannot all be summarized here. Most of the models
have been developed to describe inorganic disordered ion-conducting materials
such as glasses. PEC, forming polymeric glasses with mobile charge carriers, can in
many respects, be described by the same models. One successful approach uses
computer simulations for treating the ion motions in static energy landscapes within
a random barrier model [52–54]. The random barrier model takes disorder into
account by choosing an energy landscape with randomly distributed barriers of
different heights. In most of these simulations, Coulomb interactions between the
moving ions are neglected. In some works, however, the influence of Coulomb
interactions on the scaling properties of conductivity spectra is discussed [52].
Other successful models describing the overall shape of frequency-dependent
conductivity spectra are the counterion model and the dipolar model of Dieterich
and coworkers [55] Here, Monte Carlo simulations are used to describe the hopping
motions of small ions. All Coulomb interactions between mobile ions and their
immobile counterions are taken into account. Disorder is introduced by randomly
placing fixed counterions into the center of cubes forming a lattice. In the counterion
model, long range transport of the mobile cations is possible; however, the dipolar
model only considers local motions of the mobile ions in the vicinity of their
immobile neighboring counterions (dipoles). Interactions between fluctuating dipoles
are taken into account.
As a further development, the MIGRATION model considers microscopic
ion hopping in an energy landscape that changes with time [42]. A distribution of
barriers in the energy landscape is not taken into account. Instead, the model
Ion Conduction in Solid Polyelectrolyte Complex Materials
115