View
226
Download
0
Category
Preview:
Citation preview
8/2/2019 modelefewx
1/8
Modelling frequency-dependent conductivities and permittivities in the
framework of the MIGRATION concept
K. Funke*, R.D. Banhatti
University of Munster, Institute of Physical Chemistry and Sonderforschungsbereich 458, Schlossplatz 4, D-48149 Munster, Germany
Received 12 December 2002; received in revised form 23 April 2003; accepted 12 June 2003
Abstract
Frequency-dependent conductivities and permittivities of ion-conducting materials reflect the dynamics of the mobile ions.Experimentally, the development of the ion dynamics with time can thus be monitored from less than a picosecond up to minutes and
hours. In this paper, we show that the characteristics of frequency-dependent conductivities and permittivities of solid ion conductors are well
reproduced within the framework of the MIGRATION concept. The meaning of the acronym is MIsmatch Generated Relaxation for the
Accommodation and Transport of IONs. In this model treatment, a simple set of rules is introduced in order to describe the essence of the ion
dynamics in terms of a physical picture of the most relevant elementary processes. The rules are expressed in terms of three coupled rate
equations, which form the basis for deriving frequency-dependent model conductivities and permittivities as well as mean square
displacements.
D 2004 Elsevier B.V. All rights reserved.
PACS: 66.30.Dn; 73.61.Jc; 77.22.dKeywords: Ion dynamics; Frequency-dependent conductivities and permittivities; Mean square displacement; Sodium germanate glass
1. Introduction
The frequency-dependent electric and dielectric proper-
ties of ion-conducting materials with disordered structures,
glassy or crystalline, are determined by the hopping
dynamics of the mobile ions. Therefore, valuable informa-
tion on the elementary hopping processes and, in particu-
lar, on correlations between hops is obtainable from a
study of the frequency-dependent conductivity, r(x), and
the frequency-dependent permittivity, e(x) [1]. These two
functions are the constituent parts of the complex conduc-
tivity, r(x) = r(x) + ixe0e(x). Here, x denotes the angular
frequency, while e0 is the permittivity of free space.
In the following, we consider data below microwave
frequencies, i.e., in a frequency regime where r(x) has to be
attributed to the hopping motion of the ions, while vibra-
tional contributions to r(x) may still be neglected. In this
frequency range, vibrations and fast ionic polarisations as
well as even faster (electronic) processes contribute a
constant value, e(l), to e(x), while the remaining part,
e(x) e(l), results from the hopping motion. Sometimes,we will use the notation rHOP(x) = r(x) ixe0e(l).
A surprising experimental observation is made when
conductivity spectra, r(x), of different fast ion conducting
materials, i.e., of structurally disordered crystals and
glassy electrolytes, are compared with each other. In fact,
many of them are found to be virtually identical in shape,
not only for a given material at different temperatures, but
even for different materials. This is most clearly seen
when a scaled conductivity is plotted versus a scaled
angular frequency. This property of scaling is known as
the time temperature superposition principle [212]. In
our notation, rS(xS) , t he s c al ed c on du ct iv it y i s
rS(x) = r(x)/r(0), and the scaled angular frequency is
xS =x/x0. Here, x0 marks the onset of the conductivity
dispersion on the x scale. A quantitative definition will
be given later, see Eq. (13).
Figs. 1 and 2 are, respectively, plots of unscaled and
scaled ionic conductivities of a particular glassy ion con-
ductor, i.e., of a sodium germanate glass of composition
0167-2738/$ - see front matterD 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.ssi.2003.06.003
* Corresponding author. Tel.: +49-251-8323418; fax: +49-251-
8329138.
E-mail addresses: K.Funke@uni-muenster.de (K. Funke),
banhatt@uni-muenster.de (R.D. Banhatti).
www.elsevier.com/locate/ssi
Solid State Ionics 169 (2004) 18
8/2/2019 modelefewx
2/8
0.2Na2O0.8GeO2 [8,13]. Virtually the same scaled curve,rS(xS), as in Fig. 2 has also been obtained for many other
glassy and crystalline fast ion conductors. Therefore, this
master curve is considered representative.
Recently, a model concept was introduced which repro-
duces and explains experimental data such as those ofFigs.
1 and 2. This model, called the concept of mismatch andrelaxation (CMR) [14], is based on the previous jump
relaxation model [15]. Although the CMR is, indeed, very
successful in reproducing conductivity spectra, it does not
yield the proper low-frequency behaviour of the permittiv-
ity, e(x), i.e., a finite dc plateau value, e(0).
In this contribution, we wish to show that a subtle but
physically meaningful modification of the model is suffi-
cient to produce the correct long-time behaviour, while the
dynamics at shorter times remain unchanged. At its new
level, the model is called the MIGRATION concept, the
acronym standing for MIsmatch Generated Relaxation for
the Accommodation and Transport of IONs. In the MIGRA-
TION concept, the hopping motion of the mobile ions is
described in terms of three coupled rate equations (instead
of two in the CMR[14]). The solid lines included in Figs. 1
and 2 result from this model treatment. As the differences
between the CMR and the MIGRATION concept show up at
long times only, they are not visible in these two figures.
This paper is organised as follows. In Section 2, the time-
dependent correlation factor, W(t) [15], is introduced in
order to provide an effective means of expressing frequen-
cy-dependent conductivities and permittivities as well as
ionic mean square displacements. Section 3 is devoted to the
derivation of the MIGRATION concept and to a comparison
of model functions and experimental data. We will inparticular show that our present model treatment is able to
reproduce not only conductivity spectra, but also frequency-
dependent permittivities and time-dependent mean square
displacements. On the other hand, it is not meant to describe
or predict the dependence of ionic transport properties on
the particular structure and composition of the material
under consideration. A brief conclusion is given in the final
Section 4.
2. Frequency-dependent versus time-dependent
functions
Before considering the MIGRATION concept explicitly,
we wish to review the relationships between some relevant
functions. The transition from the frequency-dependent
functions, which are usually determined experimentally, to
time-dependent functions, is most easily performed, if theHaven ratio is close to unity, its own frequency dependence
being negligible [14,16]. This condition, which appears to
be fulfilled in most structurally disordered ionic materials, is
in the following assumed to be valid. In this case, according
to linear response theory [17], the complex conductivity
caused by the hopping motion, rHOP(x), is in a good
approximation proportional to the Fourier transform of the
autocorrelation function of the velocity of the mobile ions,
hv(0)v(t)iHOP. The latter function is characterised by a sharppeak at t= 0, plus a decaying negative contribution at t>0.
While the sharp peak reflects the self-correlation of the
velocity during hops, the decaying negative component
results from a decaying probability for an ensuing back-
ward hop, if the hop performed at t= 0 defines the
forward direction.
At this point, it is useful to introduce the time-dependent
correlation factor, W(t), which is the normalised integral of
hv(0)v(t)iHOP, with W(0) = 1. At the same time, W(t) is alsothe normalised derivative of the mean square displacement
due to hops, hr2(t)iHOP. At long times, W(t) tends to W(l).This value is the fraction of successful hops. While all
hops contribute to the high-frequency plateau of the con-
ductivity, r(l), only the successful ones contribute to the
low-frequency plateau, r(0), cf. Fig. 3.Fig. 1. Conductivity isotherms of 0.2Na2O0.8GeO2 glass. Fits are madeusing the MIGRATION concept.
Fig. 2. Scaled conductivity plot for 0.2Na2O0.8GeO2 glass. Fit is madeusing the MIGRATION concept.
K. Funke, R.D. Banhatti / Solid State Ionics 169 (2004) 182
8/2/2019 modelefewx
3/8
The relationship between r(x) and W(t) is
rx
rl 1
Zl
0
dWt
dtcosxtdt: 1
The expression of Eq. (1) tends to W(l) at low frequencies
and to unity at high frequencies. When plotted in loglogrepresentations, r(x)/r(l) and W(t) are approximately (but
not exactly) mirror images of each other, see Figs. 3 and 4.
In view of experimental data such as those of Fig. 2, we
also wish to express r(x)/r(0) in terms of W(t):
rx
r0 1 x
Zl
0
Wt
Wl 1
sinxtdt: 2
This equation results from Eq. (1) by dividing it by W(l),
then replacing d(W(t)/W(l))/dt with d[(W(t)/W(l))1]/dtand finally performing an integration by parts. With the
notations rS(x) = r(x)/r(0) and WS(t) = W(t)/W(l) for the
scaled functions, it is evident from Figs. 3 and 4 that again,in log log representations, rS(x) and WS(t) are approxi-
mately mirror images of each other.
Eq. (2) is the real part of Eq. (3):
rHOPx
r0 1 ix
Zl
0
WSt 1expixtdt: 3
Considering the imaginary part of Eq. (3) and dividing it
by the angular frequency, we obtain a reduced permittivity,
eR(x), which is caused by the hopping motion of the mobile
ions:
eRx e0ex el
r0
Zl
0
WSt 1cosxtdt:
4
This function, eR(x),decays from eR(0) = e0De/r(0)=e0(e(0)e(l))/r(0), to zero, cf. Fig. 5. Here, De is sometimes called
dielectric strength of relaxation. The quantity eR(0) has the
dimension of time and was first introduced by Sidebottom.
Itis, therefore, sometimes called the Sidebottom time, tSB[7,8]:
tSB eR0 e0De
r0: 5
In Figs. 5 and 6, we compare the frequency-dependent
reduced permittivity, eR(x), with a corresponding function
of time, hrLOC2
(t)iR. The latter is defined by
hr2LOCtiR hr2ti 6Dt
6D
Zt0
WStV 1dtV: 6
It is a reduced localised mean square displacement describ-
ing that part of the ionic motion which does not contribute
to macroscopic diffusion, but remains localised. Its long-
time limiting value, hrLOC2 (l)iR, is of particular interest.
According to Eqs. (4) and (6), it is related to De= e(0) e(l)via
hr2LOCliR eR0
e0De
r0 tSB: 7
From Figs. 5 and 6 it is evident that a similar kind of
approximate mirror symmetry as already found for r(x)/
r(l) and W(t) also applies foreR(x) and hrLOC2 (t)iR. Again,
the frequency-dependent function is obtained from the time-
dependent function by first forming the time derivative and
then the cosine Fourier transform.
Fig. 3. Model conductivity spectrum in a log log representation indicating
two possibilities of scaling, cf. Eqs. (1) and (2).
Fig. 4. Time-dependent correlation factor as obtained from the MIGRA-
TION concept, in a loglog representation, again indicating the two
possibilities of scaling.
Fig. 5. Reduced-permittivity spectrum as defined in Eq. (4), in a log linear
representation. Note that the low-frequency limit of this function is the
Sidebottom time, tSB.
K. Funke, R.D. Banhatti / Solid State Ionics 169 (2004) 18 3
8/2/2019 modelefewx
4/8
In this section, we have shown that WS(t) 1 acts as aturn-table relating frequency-dependent conductivities and
permittivities as well as mean square displacements to oneanother. Therefore, it may be considered a key function
for modelling and understanding conductivity and permit-
tivity spectra. In the following section, we will show how
W(t), and hence WS(t) 1, are obtained from the MI-GRATION concept and thus become available for deriv-
ing model spectra such as those included in Figs. 13,5
and 6.
3. The MIGRATION concept
The solid lines in Figs. 16 have all been calculatedusing the following equation for WS(t),
dWS=dt x0W2
S lnWSlnWS Nl; 8
as well as the equations of Section 2. The meaning of x0and N(l) will be explained later in this section.
Eq. (8) results from the rate equations which constitute
the MIGRATION concept. These are
Wt
Wt B gt; 9
gt
gt C0WtNt; 10
Nt
Nt C0WtNt Nl: 11
In Eqs. (9) to (11), a dot denotes a differentiation with
respect to time. The equations contain three time-dependent
functions. These are the correlation factor, W(t), the mis-
match function, g(t), and a number function, N(t). The latter
two functions will be explained shortly. Here, it is important
to note that g(t) decays from g(0) = 1 to g(l) = 0 as W(t)
decays from W(0)= 1 to W(l). As a consequence of Eq. (9)
we find
Wl expB; 12
where both W(l) and B depend on temperature.
In Eq. (10),y0 denotes the (temperature-dependent)elementary hopping rate of the mobile ions. Therefore,
x0 C0Wl C0expB 13
is the rate of successful hops, i.e., the random hopping rate.
The equation for dWS/dt which results directly fromthe rate equations still differs from Eq. (8) by featuring
ClnWS instead of lnWS, with C= (N(0) N(l))/B, in theexpression in the brackets. Constructing r(x) with the help
of WS(t) as derived from this equation, we see that x0C
marks the onset of the conductivity dispersion on the
angular frequency scale, indicating the transition from
random to non-random hopping. On the other hand, we
expect this particular angular frequency to be x0 itself, cf.
Eq. (13). Therefore, C must be a temperature-independent
constant of the order of one. For simplicity, we assume C= 1
as a working hypothesis and, consequently, use Eq. (8). Of
course, C can be included in our equations at any time, if
this is necessitated by further experimental results.
At this point, we emphasise thatx0 is the only parameter
that can be determined on the basis of experimental spectra
such as those of Figs. 1 and 2. In addition, experimental
permittivity spectra provide a possibility to determine N(l),
while B is obtained from measurements of high-frequency-
plateau conductivities. It is considered the strength of ourmodel treatment that, irrespective of the particular values of
any parameters, it is able to produce a well-defined scaled
conductivity master spectrum, see Fig. 2. At the same
time, realistic scaled permittivities and mean square dis-
placements are also derived, see below. This is, indeed,
possible on the basis of the above set of simple rate
equations, which are meant to capture the essence of the
ion dynamics in a physical picture.
We now briefly outline the physical concept leading to
Eqs. (9)(11). The central idea of the MIGRATION concept
is the following. After each hop of a mobile ion, mismatch is
created between its own position and the arrangement of its
neighbours. There are two possible ways for the system to
reduce the mismatch. Either the neighbours rearrange or the
central ion hops back into its previous site. This explains
the existence of forwardbackward correlations of succes-
sive hops.
Suppose mismatch is created by a hop of a mobile ion at
time t= 0. Then, at times t> 0, the mismatch function, g(t)
(see Refs. [14,15]), represents a normalised distance be-
tween the actual position of the ion and the position at
which it would be optimally relaxed with respect to the
momentary arrangement of its mobile neighbours. The
function g(t), therefore, varies with time from g(0) = 1 to
Fig. 6. Reduced localised mean square displacement in a loglinear
representation, cf. Eq. (6). Note that the long-time limit of this function is
the Sidebottom time, tSB.
K. Funke, R.D. Banhatti / Solid State Ionics 169 (2004) 184
8/2/2019 modelefewx
5/8
g(l) = 0, describing the way mismatch decays because the
neighbouring ions rearrange. The negative time derivative,
g(t), is thus the rate of mismatch relaxation on the many-particle route. On the other hand, W(t)/W(t) is the rate ofmismatch relaxation on the single-particle route, with the
ion hopping backwards. Here, the factor 1/W(t) is required,
since we consider only cases where the ion is (still or again)at its new site at time t. Eq. (9), therefore, expresses the
assumption that the rates of relaxation on the single- and
many-particle routes are proportional to each other at all
times. In other words, the tendency of the ion to hop
backwards is assumed to be proportional to the tendency
of its neighbours to rearrange.
In Eq. (10), we consider the rate of decay of g(t). Here it
is important to realise thatg(t) plays the role of a normalised
dipole moment. Its dipole field is the driving force felt by
the mobile ions, inducing their rearrangement and, as a
consequence, the concomitant decay of g(t) itself. As the
rearrangement of the surrounding ion cloud proceeds, the
central dipole becomes increasingly shielded. This means
that two effects occur simultaneously. One is the reduction
of g(t) with time. The other is the shrinking of the effective
volume of the dipole field, i.e., the reduction of the effective
number of mobile neighbours, N(t), which are still available
for the relaxation on the many-particle route.
In Eq. (10), we try to grasp the essence of the time-
dependent processes described in the preceding paragraph.
We first assume that g(t) is proportional to the convolu-tion of the driving force, which is g(t) itself, and the velocity
autocorrelation function of the neighbouring ions. The latter
function is assumed to be the same as for the central ion.
Apart from a factor, the product, g(t)W(t), turns out to be anexcellent approximation for this convolution, the reason
being that g(t) varies with time much more slowly than
W(t) does. Secondly, g(t) will be proportional to theelementary hopping rate, y0, and thirdly, to the time-
dependent effective number of neighbours available for
the relaxation, N(t). We thus obtain Eq. (10). Note that
any unknown but constant factors are included in the
definition of N(t).
The time dependence of N(t) is certainly not easy to
derive in a simple model. Eq. (11), which is used in the
MIGRATION concept and is found to reproduce experi-
mental spectra very well, is equivalent to
dNt Nl=dt
Nt Nl C0WtNt
gt
gt: 14
The equation thus implies that the difference, N(t) N(l),and g(t) decay in the same fashion. Physically, the number
N(l) represents those nearest neighbours of the central
ion, which are not shielded by others and, therefore, will
always experience the dipole field. Comparing model con-
ductivity spectra to experimental ones such as those of Fig.
1, we find that N(l) is much smaller than N(0).
The previous concept of mismatch and relaxation (CMR)
differed from our present treatment in two respects. In the
first place, it did not contain any non-zero value of N(l).
This implied that, in the limit of long times, both g(t) and W(t) varied with time as 1/t instead of decaying expo-nentially. As a consequence, the low-frequency plateaux
observed in experimental permittivities could not be repro-duced. The second difference concerned the product,
g(t)N(t), which was not necessarily regarded as proportional
to g2(t) as in our present treatment, but was allowed to vary
as gK(t). It was, however, shown that for a large number of
ionic materials the optimum value of K was close to 2.0
[14]. Nevertheless, examples were found where the expo-
nent K was clearly larger than 2.0, e.g., 2.6 [14,18]. In this
paper, we wish to keep our equations as simple as possible
and have, therefore, chosen K to be 2.0. In a later publica-
tion, we will present model equations allowing for K larger
than 2.0. It will then be shown that slight variations of K
imply corresponding slight variations of the shapes of the
model spectra. Those minor systematic differences between
experimental and model spectra, which are still visible in the
present paper, can, indeed, be removed by varying the value
of K.
In Fig. 7, we present the experimental permittivities,
e(x), of glassy 0.2Na2O0.8GeO2 [8,13] taken at the sametemperatures as the conductivities of Fig. 1. Note that there
are considerable experimental uncertainties in the permit-
tivity spectra, in particular in the limit of low frequencies,
where unavoidable electrode effects are always present, but
also at higher frequencies. The solid l ines are our
corresponding model spectra as obtained via Eqs. (8) and
(4). In all of these spectra, the same value fore(l) has beenadded to the expression for e(x) e(l). Remarkably, it ispossible to reproduce the frequency-dependent permittiv-
ities measured at different temperatures, and in particular
their low-frequency limiting values, with only one temper-
Fig. 7. Permittivity spectra of 0.2Na2O0.8GeO2 glass at differenttemperatures. Note that the low-frequency data of e(x) are affected by
electrode polarisation effects. At low frequencies, the permittivity has to be
constant. Fits have been made using the treatment of the MIGRATION
concept with N(l) = 0.075, cf. Eqs. (8) and (4).
K. Funke, R.D. Banhatti / Solid State Ionics 169 (2004) 18 5
8/2/2019 modelefewx
6/8
ature-independent value of N(l), 0.075. Evidently, N(l)
reflects a temperature-independent structural property, in
accordance with the physical meaning of Eq. (14).
For the purpose of scaling, i.e., for generating scaled
model spectra for both conductivity and permittivity, it is
useful to rewrite Eq. (8) with a scaled time, tS = tx0:
dWS=dtS W2
S lnWSlnWS Nl: 15
Eq. (15) may be regarded as resulting from the following
scaled rate equations. With a suitably scaled mismatch
function, gS(tS) =Bg(tS) = l nWS(tS), and with the dot now
denoting a differentiation with respect to scaled time, these
equations are:
WStS
WStS gStS; 16
gStSgStS
WStSNtS; 17
NtS
NtS WStSNtS Nl: 18
Eqs. (17) and (18) imply that the difference, N(tS) N(l), must be proportional to gS(tS). Here, it is important to
note that any temperature-dependent proportionality factor
between N(tS) N(l) and gS(tS) would destroy the propertyof scaling. However, there is no such factor in the expres-
sion
NtS Nl N0 NlgtS CgStS gStS;
19
since we have put C= 1.
The scaled experimental conductivity spectrum and the
corresponding scaled model spectrum have already been
presented in Fig. 2. Note that the scaled time-dependent
correlation factor, WS(tS), as obtained from Eq. (15), dis-
plays a temperature dependence at short scaled times, since
WS(0) = exp(B) = 1 / W(l). This remains, however, invisible
in Fig. 2, since the sine Fourier transform of Eq. (2) gives
little weight to the short-time values of WS(tS).
In Fig. 8, we present the scaled experimental permittivitiesof glassy 0.2Na2O0.8GeO2, along with the correspondingscaled permittivities derived from the MIGRATION concept.
The scaled dimensionless permittivity function is eS(xS),
with
eSxS eRxSx0
Zl
0
WStS 1cosxStSdtS
20
and xS =x/x0 as in Fig. 2. To obtain the scaled experimental
data, we have used the known values ofe(l), r(0), and x0,
with x0 from the fits ofFig. 1. For the scaled model spectra at
different temperatures, we have calculated temperature-de-pendent functions WS(tS), based on the assumption that B
accounts for the larger part of EDC/kT. Here, EDC is the
activation energy of the dc conductivity. Although there is a
cosine Fourier transform in Eq. (4), it turns out that the model
functions, eS(xS), display only little temperature dependence.
In comparison with the experimental uncertainties involved
in the data, these variations may still be neglected.
In the limit of low frequencies, the scaled permittivity,
eS(xS), tends to tSB,S, which is the scaled Sidebottom time,
tSB,S = tSBx0, cf. Eq. (5). Likewise, the scaled dimensionless
localised mean square displacement,
hr2LOCtSiS hr2LOCtSiRx0
ZtS0
WStSV 1dtSV;
21
cf. Eq. (6), tends to the same dimensionless value, tSB,S, in
the limit of long times. This is seen in Fig. 9. The figure
again includes both experimental data and a model curve.
To obtain the experimental data points in Fig. 9, the data
corresponding to T= 323 K have been interpolated, and a
continuous function has thus been generated. This function
has then been Fourier transformed and integrated, providing
the discrete values plotted in Fig. 9. The model curve in Fig.
9 has been obtained using Eq. (21), with the model function,WS(tS), generated from Eq. (15) for 323 K.
In Fig. 10, both scaled mean square displacements,
hr2tSiS hr2tSix0=6D
ZtS0
WStSVdtSV 22
and hrLOC2 (tS)iS, are plotted versus scaled time in a loglog
representation. In hr2(tS)iS, a transition is observed from asubdiffusive behaviour at short scaled times to ordinary
macroscopic diffusion at long scaled times. The subdiffu-
sive behaviour results from the preference of the ions for
correlated backward hops, while any memory for the
Fig. 8. Scaled permittivity of 0.2Na2O0.8GeO2 glass as defined by Eq.(20). Solid curve obtained from the MIGRATION concept with
N(l) = 0.075.
K. Funke, R.D. Banhatti / Solid State Ionics 169 (2004) 186
8/2/2019 modelefewx
7/8
previous site is lost in the diffusive long-time regime. To
locate the transition on the time scale, two different times
have been introduced. One is t0 = 1/x0, the other is theSidebottom time, tSB. The two times are not identical, since
the scaled Sidebottom time,
tSB;S x0tSB
Zl
0
WStS 1dtS; 23
cf. Eqs. (4) and (5), is not unity. Fig. 10 includes a graphic
visualisation of the scaled Sidebottom time, tSB,S. At tSB,S,
the line with slope one, representing macroscopic diffusion,
intersects the horizontal line, (hrLOC2(l)iS). In an unscaled
notation, this means that
hr2LOCli 6DtSB: 24
It is worth mentioning that hrLOC2 (l)i is a temperature-
independent quantity. This is immediately seen from Eq.
(24), since D is proportional to x0, while tSB is proportional
to 1/x0. We have thus shown that the long-time mean square
value of the displacement caused by the localised, non-
translational hopping motion is independent of temperature.
This is perfectly in line with an observation made earlier,
when introducing scaled frequency, cf. Figs. 1 and 2.
Indeed, apart from the dynamic processes occurring at very
short times, the only effect of temperature is to speed up or
slow down the hopping motion of the ions, while the basicrules and geometrical details of the mechanism remain
unchanged [19].
4. Conclusion
The MIGRATION concept is based on the previous
CMR model. The latter already provided the possibility to
reproduce and explain experimental conductivity spectra
taken from a wide variety of ionic materials with disordered
structures. On the other hand, the CMR did not yield the
proper low-frequency behaviour of the permittivity, i.e., its
finite dc plateau. In this paper, we have shown that a finite
value of the low-frequency permittivity is, indeed, obtained
if the CMR is modified in a physically meaningful, subtle
way. The modification concerns the long-time behaviour of
the number function, N(t). This function describes the time-
dependent shielding of the dipole field, and hence it is a
measure of the number of mobile neighbours still available
for mismatch relaxation on the many-particle route. In our
present MIGRATION concept, N(t) remains positive in the
limit of long times, reflecting the obvious fact that the
nearest neighbours will never be shielded. Not excluding
the nearest neighbours from the possibility of rearranging,
even when the process of shielding is already quite ad-vanced, is decisive for the initial hop of the central ion
eventually to prove successful. As a consequence, the
localised, non-translational mean square displacement
remains finite at long times, and so does the permittivity
at low frequencies.
Fig. 10. Loglog representation of hr2(tS)iS and hr2LOC(tS)iS of 0.2Na2O0.8GeO2 glass. The model fit is obtained by using Eqs. (21) and (22).
Fig. 9. Log linear representation of the scaled localised mean square
displacement of glassy 0.2Na2O0.8GeO2. The solid line shows the fit fromthe MIGRATION concept, cf. Eq. (21).
K. Funke, R.D. Banhatti / Solid State Ionics 169 (2004) 18 7
8/2/2019 modelefewx
8/8
Glasses with different compositions differ in their char-
acteristic low-frequency values, eS(0), requiring different
values of N(l). Correlations between N(l) and the overall
number density of mobile ions will be studied in a forth-
coming paper.
Acknowledgements
It is a pleasure to thank M.D. Ingram and A. Heuer for
many fruitful discussions. M.D. Ingram also helped with a
critical reading of the manuscript. We would like to thank B.
Roling and C. Martiny for the use of their impedance data
on sodium germanate. Financial support from the Fonds der
Chemischen Industrie is hereby acknowledged. Sonderfor-
schungsbereich 458 is funded by the German Science
Foundation, DFG.
References
[1] K. Funke, C. Cramer, Curr. Opin. Solid State Mater. Sci. 2 (1997)
483.
[2] H.E. Taylor, Trans. Faraday Soc. 52 (1956) 873.
[3] J.O. Isard, J. Non-Cryst. Solids 4 (1970) 357.
[4] J.R. Dyre, J. Appl. Phys. 64 (1988) 2456.
[5] H. Kahnt, Ber. Bunsenges. Phys. Chem. 95 (1991) 1021.
[6] B. Roling, K. Funke, A. Happe, M.D. Ingram, Phys. Rev. Lett. 78
(1997) 2160.
[7] D.L. Sidebottom, Phys. Rev. Lett. 82 (1999) 3653.
[8] B. Roling, C. Martiny, K. Funke, J. Non-Cryst. Solids 249 (1999)
201.
[9] B. Roling, C. Martiny, S. Bruckner, Phys. Rev., B 63 (2000) 214203.
[10] K. Funke, S. Bruckner, C. Cramer, D. Wilmer, J. Non-Cryst. Solids
307310 (2002) 921.
[11] C. Cramer, S. Bruckner, Y. Gao, K. Funke, R. Belin, G. Taillades, A.
Pradel, J. Non-Cryst. Solids 307 310 (2002) 905.
[12] M. Cutroni, A. Mandanici, P. Mustarelli, C. Tomasi, M. Federico,
J. Non-Cryst. Solids 307 310 (2002) 963.
[13] C. Martiny, Diploma Thesis, University of Muenster (1998).
[14] K. Funke, R.D. Banhatti, S. Bruckner, C. Cramer, C. Krieger, A.
Mandanici, C. Martiny, I. Ross, Phys. Chem. Chem. Phys. 4 (2002)
3155.
[15] K. Funke, Prog. Solid State Chem. 22 (1993) 111.
[16] M.P. Thomas, N.L. Peterson, Solid State Ionics 14 (1984) 297.
[17] R. Kubo, J. Phys. Soc. Jpn. 12 (1957) 570.
[18] C. Cramer, S. Brueckner, Y. Gao, K. Funke, Phys. Chem. Chem.
Phys. 4 (2002) 3214.
[19] S. Summerfield, Phila. Mag., B 52 (1985) 9.
K. Funke, R.D. Banhatti / Solid State Ionics 169 (2004) 188
Recommended