focuses on the time dependence of the energy landscape occurring when mobile
ions leave their site. This model has one big advantage over the other models
because it yields a set of analytic equations with fitting parameters. These equations
allow calculation of conductivity spectra that are in agreement with experimental
spectra, where the exact spectral shape differs slightly from system to system.
For this reason, the MIGRATION model was chosen for performing fits to PEC
spectra and will described in more detail.
The acronym MIGRATION stands for MIsmatch Generated Relaxation for the
Accommodation and Transport of IONs. The model considers in a detailed way
backward correlations between the successive jumps of a mobile ion. As already
outlined in Sect. 2.1, these backward correlations give rise to the negative part in the
velocity autocorrelation function and thereby to dispersion of the conductivity. The
model distinguishes between the “single particle route” and the “many particle
route”. Both can reduce mismatch, which is created when a charged particle leaves
its site. In the first case, the previous forward jump of the ion under consideration is
cancelled out by a correlated backward jump, whereas in the second case the ion has
successfully moved to a new site with a rearrangement of charges in the environment causing relaxation of the mismatch. Successful hops result in long-range ion
transport. The rate of relaxation on the many-particle route is related to singleparticle functions such as the velocity autocorrelation function, from which the
complex conductivity can be derived via Eq. (3). Local mismatch is, however, not
only reduced by the rearrangement of the neighboring ions, but at the same time
also progressively shielded. An empirical parameter K is introduced to quantify the
time dependence of the shielding effect. More details are given in [42] and
references therein. The parameter K turns out to modify the shape of the conductivity spectra, increasing values of K resulting in a more gradual onset of the
dispersion. In most single cation glasses and also in many crystals, the value of K
is found to be close to 2.0. The parameter K appears to be related to the effective
number density of mobile ions. The smaller the number density, the higher is the
value of K. Accordingly, materials with small concentrations of mobile cations are
reproduced by model spectra with larger K values than those with high cation
contents. Large values of K have also been reported for mixed cation glasses and for
materials with a low dimensionality of the pathways for the ion transport [42, 56].
The main focus in reviewing modeling approaches in PEC is here on the NaPSSrich compositions with x > 0.50, where a phase separation can be excluded. The
following discussion will analyze the ion transport mechanism on a microscopic
scale by looking at the shape of the conductivity spectra. The corresponding
conductivity spectra are representative for all compositions with x > 0.50 (see
also Fig. 6). Figure 12 shows one conductivity spectrum of 0.60 NaPSS · 0.40
PDADMAC taken at 393 K. The dashed line was obtained on the basis of the
MIGRATION concept with the model parameter K ¼ 2.4 [57].
Whereas the model curve describes the experimental spectrum very well at low
and at high experimental frequencies, one sees variations in the intermediate
frequency range. These deviations remain, even if the value of the parameter K is
varied.
116
C. Cramer and M. Scho ¨nhoff
ions leave their site. This model has one big advantage over the other models
because it yields a set of analytic equations with fitting parameters. These equations
allow calculation of conductivity spectra that are in agreement with experimental
spectra, where the exact spectral shape differs slightly from system to system.
For this reason, the MIGRATION model was chosen for performing fits to PEC
spectra and will described in more detail.
The acronym MIGRATION stands for MIsmatch Generated Relaxation for the
Accommodation and Transport of IONs. The model considers in a detailed way
backward correlations between the successive jumps of a mobile ion. As already
outlined in Sect. 2.1, these backward correlations give rise to the negative part in the
velocity autocorrelation function and thereby to dispersion of the conductivity. The
model distinguishes between the “single particle route” and the “many particle
route”. Both can reduce mismatch, which is created when a charged particle leaves
its site. In the first case, the previous forward jump of the ion under consideration is
cancelled out by a correlated backward jump, whereas in the second case the ion has
successfully moved to a new site with a rearrangement of charges in the environment causing relaxation of the mismatch. Successful hops result in long-range ion
transport. The rate of relaxation on the many-particle route is related to singleparticle functions such as the velocity autocorrelation function, from which the
complex conductivity can be derived via Eq. (3). Local mismatch is, however, not
only reduced by the rearrangement of the neighboring ions, but at the same time
also progressively shielded. An empirical parameter K is introduced to quantify the
time dependence of the shielding effect. More details are given in [42] and
references therein. The parameter K turns out to modify the shape of the conductivity spectra, increasing values of K resulting in a more gradual onset of the
dispersion. In most single cation glasses and also in many crystals, the value of K
is found to be close to 2.0. The parameter K appears to be related to the effective
number density of mobile ions. The smaller the number density, the higher is the
value of K. Accordingly, materials with small concentrations of mobile cations are
reproduced by model spectra with larger K values than those with high cation
contents. Large values of K have also been reported for mixed cation glasses and for
materials with a low dimensionality of the pathways for the ion transport [42, 56].
The main focus in reviewing modeling approaches in PEC is here on the NaPSSrich compositions with x > 0.50, where a phase separation can be excluded. The
following discussion will analyze the ion transport mechanism on a microscopic
scale by looking at the shape of the conductivity spectra. The corresponding
conductivity spectra are representative for all compositions with x > 0.50 (see
also Fig. 6). Figure 12 shows one conductivity spectrum of 0.60 NaPSS · 0.40
PDADMAC taken at 393 K. The dashed line was obtained on the basis of the
MIGRATION concept with the model parameter K ¼ 2.4 [57].
Whereas the model curve describes the experimental spectrum very well at low
and at high experimental frequencies, one sees variations in the intermediate
frequency range. These deviations remain, even if the value of the parameter K is
varied.
116
C. Cramer and M. Scho ¨nhoff