5.1 Principle of Scaling
The conductivity spectra of many materials possess a temperature-independent
shape and can, therefore, be superimposed to a so-called “mastercurve” [68]. This
is commonly termed “TTSP” (time-temperature superposition principle). Validity
of the TTSP implies that an enhancement of temperature causes a general acceleration of all dynamic processes in the material and thus a shift on the time axis, or,
more relevant for conductivity spectra, on the frequency axis. The validity of the
TTSP for the real part of the complex conductivity σ
0 /σ dc can be expressed by a
function σ
0 /σ dc ¼ F(ν/ν 0 ), where ν 0 is an individual scaling parameter for each
conductivity isotherm. The scaling function, F, is independent of temperature. An
appropriate choice of ν 0 for each curve is necessary to superimpose spectra
measured at different temperatues to a master curve. A straightforward approach
is to assign ν 0 to the onset frequency of the conductivity dispersion, ν*, which can be
defined in various manners. In almost all studies, the definition σ
0 (ν*) ¼ 2σ dc is used
(see the stars in Fig. 5). TTSP then means that all conductivity spectra, for example
from Fig. 5, do exactly overlap to form one master curve when the spectra are shifted
along the log(ν) and the log(σ
0 T) axis in order to make the onset points overlap.
The special case of Summerfield scaling [69, 70] is fulfilled if σ dc T and ν*
are proportional to each other. Connecting the onset frequencies in a plot of log(σ
0 T)
versus log(ν) yields a straight line of slope 1. Summerfield scaling can be expressed by:
σ
0
ðνÞ
σ dc
¼ F
ν
σ dc Á T
(7)
and it describes the straightforward case where the temperature enhancement
causes the acceleration of all dynamic processes without affecting structural
properties such as the number density of charges or pathways of ion transport.
Then, the temperature influence on conductivity spectra can be described by a shift
along a line of slope 1 in the log(σ
0 T) versus log(ν) diagram.
Deviations from this simple form of scaling may imply that the shape of the
spectra is changing with temperature. In this case, TTSP is not valid any more. On
the other hand, deviations from Eq. (7) might imply that the spectral shape remains
identical, but the shift along either axis required to produce the master curve differs,
i.e., the simple proportionality between σ dc T and ν* is not valid anymore. This was
also found for some materials [71, 72]. In the case that the slope of the line
connecting the onset frequencies exceeds one, σ dc T increases more strongly with T
than does ν*. One way to describe this type of deviation from Summerfield scaling is
to include an additional scaling factor for the frequency scale, which itself depends
on temperature:
σ
0
ðνÞ
σ dc
¼ F
ν
σ dc Á T
Á f ðTÞ
. With f(T) included, all spectra taken at
different temperatures can then be scaled on a master curve. The scaling relation:
Ion Conduction in Solid Polyelectrolyte Complex Materials
125
The conductivity spectra of many materials possess a temperature-independent
shape and can, therefore, be superimposed to a so-called “mastercurve” [68]. This
is commonly termed “TTSP” (time-temperature superposition principle). Validity
of the TTSP implies that an enhancement of temperature causes a general acceleration of all dynamic processes in the material and thus a shift on the time axis, or,
more relevant for conductivity spectra, on the frequency axis. The validity of the
TTSP for the real part of the complex conductivity σ
0 /σ dc can be expressed by a
function σ
0 /σ dc ¼ F(ν/ν 0 ), where ν 0 is an individual scaling parameter for each
conductivity isotherm. The scaling function, F, is independent of temperature. An
appropriate choice of ν 0 for each curve is necessary to superimpose spectra
measured at different temperatues to a master curve. A straightforward approach
is to assign ν 0 to the onset frequency of the conductivity dispersion, ν*, which can be
defined in various manners. In almost all studies, the definition σ
0 (ν*) ¼ 2σ dc is used
(see the stars in Fig. 5). TTSP then means that all conductivity spectra, for example
from Fig. 5, do exactly overlap to form one master curve when the spectra are shifted
along the log(ν) and the log(σ
0 T) axis in order to make the onset points overlap.
The special case of Summerfield scaling [69, 70] is fulfilled if σ dc T and ν*
are proportional to each other. Connecting the onset frequencies in a plot of log(σ
0 T)
versus log(ν) yields a straight line of slope 1. Summerfield scaling can be expressed by:
σ
0
ðνÞ
σ dc
¼ F
ν
σ dc Á T
(7)
and it describes the straightforward case where the temperature enhancement
causes the acceleration of all dynamic processes without affecting structural
properties such as the number density of charges or pathways of ion transport.
Then, the temperature influence on conductivity spectra can be described by a shift
along a line of slope 1 in the log(σ
0 T) versus log(ν) diagram.
Deviations from this simple form of scaling may imply that the shape of the
spectra is changing with temperature. In this case, TTSP is not valid any more. On
the other hand, deviations from Eq. (7) might imply that the spectral shape remains
identical, but the shift along either axis required to produce the master curve differs,
i.e., the simple proportionality between σ dc T and ν* is not valid anymore. This was
also found for some materials [71, 72]. In the case that the slope of the line
connecting the onset frequencies exceeds one, σ dc T increases more strongly with T
than does ν*. One way to describe this type of deviation from Summerfield scaling is
to include an additional scaling factor for the frequency scale, which itself depends
on temperature:
σ
0
ðνÞ
σ dc
¼ F
ν
σ dc Á T
Á f ðTÞ
. With f(T) included, all spectra taken at
different temperatures can then be scaled on a master curve. The scaling relation:
Ion Conduction in Solid Polyelectrolyte Complex Materials
125