IOP PUBLISHING REPORTS ON PROGRESS IN PHYSICS
Rep. Prog. Phys. 72 (2009) 046501 (15pp) doi:10.1088/0034-4885/72/4/046501
Fundamental questions relating to
ion conduction in disordered solids
Jeppe C Dyre
1
, Philipp Maass
2,3
, Bernhard Roling
4
and
David L Sidebottom
5
1
DNRF centre ‘Glass and Time,’ IMFUFA, Department of Sciences, Roskilde University, Postbox 260,
DK-4000 Roskilde, Denmark
2
Institut f
¨
ur Physik, Technische Universit
¨
at Ilmenau, D-98684 Ilmenau, Germany
3
Fachbereich Physik, Universit
¨
at Osnabr
¨
uck, Barbarastraße 7, D-49069 Osnabr
¨
uck, Germany
4
Fachbereich Chemie, Philipps-Universit
¨
at Marburg, Hans-Meerwein Str., D-35032 Marburg, Germany
5
Department of Physics, Creighton University, Omaha, NE 68178, USA
Received 9 May 2008, in final form 14 January 2009
Published 16 March 2009
Online at
stacks.iop.org/RoPP/72/046501
Abstract
A number of basic scientific questions relating to ion conduction in homogeneously disordered
solids are discussed. The questions deal with how to define the mobile ion density, what can be
learnt from electrode effects, what the ion transport mechanism is, the role of dimensionality
and what the origins of the mixed-alkali effect, the time-temperature superposition, and the
nearly constant loss are. Answers are suggested to some of these questions, but the main
purpose of the paper is to draw attention to the fact that this field of research still presents
several fundamental challenges.
Contents
1. Introduction 1
2. How to define mobile ion density? 2
3. What can be learnt from electrode polarization? 4
4. What causes the mixed-alkali effect? 5
5. What is the origin of time-temperature
superposition (TTS)? 6
6. What causes the nearly constant loss (NCL)? 9
7. What is the ion transport mechanism? 9
8. What is the role of dimensionality? 10
9. Concluding remarks 11
Acknowledgments 11
Appendix A. Relation between the long-time
mean-square displacement and the low-frequency
behavior of the ac conductivity 11
Appendix B. Accurate representation of the RBM
universal ac conductivity 12
References 12
1. Introduction
Ion conduction in glasses, polymers, nanocomposites, highly
defective crystals and other disordered solids plays an increas-
ingly important role in technology. Considerable progress has
been made recently, for instance with solid-oxide fuel cells,
electrochemical sensors, thin-film solid electrolytes in bat-
teries and supercapacitors, electrochromic windows, oxygen-
separation membranes, functional polymers, etc [1–9]. The
applied perspective is an important catalyst for work in this
field. In this paper, however, the focus is on basic scientific
questions. This is relevant because ion transport in disordered
materials remains poorly understood. There is no simple,
broadly accepted model; it is not even clear whether any gen-
erally applicable, simple model exists. Given the intense cur-
rent interest in the field—with hundreds of papers published
each year—it is striking that there is no general consensus on
several fundamental questions [10]. This is in marked con-
trast to other instances of electrical conduction in condensed
matter where a much better understanding has been achieved,
e.g. for electronic conduction in metals, semiconductors and
superconductors, as well as for ion conduction by defects in
crystals.
This paper summarizes and discusses basic scientific ques-
tions relating to ion conduction in (mainly) homogeneously
disordered solids [11–21]. The main motivation is not to
suggest or provide answers, but to inspire further research
into the fundamentals of ion conduction in disordered solids.
0034-4885/09/046501+15$90.00 1 © 2009 IOP Publishing Ltd Printed in the UK
Rep. Prog. Phys. 72 (2009) 046501 J C Dyre et al
A question that is not addressed below, which has been a point
of controversy particularly during the last decade, is how to
best represent ac data, via the conductivity or the electric mod-
ulus [22–25]. By now this has been thoroughly discussed in
the literature, and we refer the interested reader to the dis-
cussions in [26–28] that present and summarize the differing
viewpoints.
2. How to define mobile ion density?
Ion motion indisordered solids is fundamentally different from
electronic conduction in crystalline solids. Ions are much
heavier than electrons so their motion is far less governed by
quantum mechanics. Below typical vibrational frequencies
(100 GHz) ion motion can be described by activated hopping
between (usually) charge-compensating sites. Moving ions
carry charge, of course, and thus produce an electrical response
which can be detected by a variety of experimental techniques.
Unlike crystals, the potential-energy landscape experienced
by an ion in a glass or otherwise disordered solid is irregular
and contains a distribution of depths and barrier heights, as
sketched in figure 1. The varying energies result from differing
binding energies at residence sites and differing saddle point
energies between residence sites, and they are influenced by
interactions between the ions. With increasing time scale, the
ions explore larger parts of space by overcoming higher energy
barriers.
Following standard arguments, suppose ions with charge
q are subjected to an electric field E. The field exerts the force
qE on each ion, resulting in an average drift velocity v in the
field direction. The ion mobility µ is defined by µ = v/E.
If the number of mobile ions per unit volume is n
mob
, the
current density J is given by J = qn
mob
v. Thus we obtain
the following expression for the dc conductivity, σ
dc
≡ J/E:
σ
dc
= qn
mob
µ. (1)
This equation expresses the simple fact that the conductivity
is proportional to the ion charge, to the number of mobile
ions and to how easily an ion is moved through the solid. As
such, equation (1) is an excellent starting point for discussing
how the conductivity depends on factors like temperature and
chemical composition. Or is it? We shall now argue that
the above conventional splitting of the conductivity into a
product of mobility and mobile ion density involves non-trivial
assumptions.
Except at very high temperatures ion motion in solids
proceeds via jumps between different ion sites. Most of the
time an ion vibrates in a potential-energy minimum defined by
the surrounding matrix. This motion does not contribute to the
conductivity except at frequencies above the gigahertz range;
only ion jumps between different minima matter. The mobility
reflects the long-time average ion displacement after many
jumps. The fact that ions spend most of their time vibrating
in potential-energy minima, however, makes the definition of
mobile ion density less obvious: how to define the number of
mobile ions when all ions are immobile most of the time?
Intuitively, equation (1) still makes sense. Imagine a
situation where some ions are tightly bound (‘trapped’) while
(a)
(b)
(
c
)
Figure 1. Schematic figures illustrating ion jumps in a disordered
landscape, here in one dimension. The arrows indicate attempted
jumps. Most of these are unsuccessful and the ion ends back in the
minimum it tried to leave: if the barrier is denoted by E,ifT is the
temperature and k
B
is Boltzmann’s constant, according to rate
theory the probability of a successful jump is exp(−E/k
B
T). This
implies that on short time scales only the smallest barriers are
surmounted. As time passes, higher and higher barriers are
surmounted, and eventually the highest barriers are overcome. In
more than one dimension the highest barrier to be overcome for dc
conduction is determined by percolation theory; there are even
higher barriers, but these are irrelevant because the ions go around
them.
others are quite mobile. In this situation one would obviously
say that the density of mobile ions is lower than the total ion
density. The problem, however, is that the tightly bound ions
sooner or later become mobile and the mobile ions sooner or
later will be trapped: by ergodicity, in the long run all ions
of a given type must contribute equally to the conductivity.
Thus, on long time scales the ‘mobile’ ion density must be
the total ion concentration. This ‘long run’ may be years or
more, and ions trapped for so long are for all practical purposes
immobile. Nevertheless, unless there are infinite barriers in
the solid, which is unphysical, in the very long run all ions are
equivalent.
The question of how many ions contribute to the
conductivity makes good sense, however, if one specifies a
time scale. Thus, for a given time τ it makes perfect sense to
ask: on average, how many ions move beyond pure vibration
within a time window of length τ ? If the average concentration
of ions moving over time τ is denoted by n
mob
(τ ) and n is the
2
Rep. Prog. Phys. 72 (2009) 046501 J C Dyre et al
total ion concentration, ergodicity is expressed by
n
mob
(τ →∞) = n. (2)
An obvious question is how to determine n
mob
(τ )
experimentally. A popular method of determining the ‘mobile
ion density’—without explicit reference to time scale—is by
application of the Almond–West (AW) formalism [29–31]
that takes advantage of the frequency dependence of the
conductivity. We proceed to discuss this approach. First note
that in ion conductors with structural disorder, the short-time
ion dynamics is characterized by back-and-forth motion over
limited ranges, ‘subdiffusive’ dynamics, whereas the long-time
dynamics is characterized by random walks resulting in long-
range ion transport, ‘diffusive’ dynamics (figure 1)[32–36].
The back-and-forth motion leads to dispersive conductivity at
high frequencies, while the long-range transport leads to the
low-frequency plateau marking the dc conductivity (figure 2).
There is experimental evidence that in materials with high ion
concentration, on short time scales only part of the ions are
actively involved in back-and-forth motion [37, 38].
A widely applied description of conductivity spectra in
the low-frequency regime (i.e. below 100 MHz) is a Jonscher
type power law,
σ
(ν) = σ
dc
1+
ν
ν
∗
n
, (3)
where we have written the equation in a form such that the
crossover frequency marking the onset of ac conduction, ν
∗
,
is given by σ
(ν
∗
) = 2σ
dc
. Equation (3) is sometimes referred
to as the Almond–West (AW) formula, although Almond
and West did not consider Jonscher’s ‘universal dielectric
response’ of disordered systems, but introduced their formula
to describe defective crystals with an activated number of
charge carriers. Nevertheless, when applying equation (3)to
strongly disordered systems, as, for example, ionic glasses,
many authors in the literature follow the physical interpretation
suggested by Almond and West and identify the crossover
frequency with a ‘hopping rate’. Thus combining this ansatz
with the Nernst–Einstein relation gives
n
AW
=
6k
B
T
q
2
a
2
σ
dc
ν
∗
(4)
as an equation to determine the number density of ‘mobile
ions’, n
AW
(after assuming jump lengths a = 2–3 Å).
However, if one accepts that equation (3) provides a
good fit to spectra in the low-frequency regime—it generally
fails at frequencies above 100 MHz— the estimate of an
effective number density of ‘mobile ions’ based on equation (4)
is questionable. Application of the fluctuation–dissipation
theorem implies the following expression, where t
∗
≡ 1/ν
∗
and H and γ are numbers that are roughly of order unity (H
is an in principle time-scale-dependent Haven ratio [39–41]
reflecting ion–ion correlations, and γ ∼ 2 is a numerical
factor reflecting the conductivity spectrum at the onset of ac
conduction, see appendix A):
σ
dc
=
nq
2
6k
B
T
r
2
(t
∗
)
γH
ν
∗
. (5)
(a)
(b)
Figure 2. (a) Schematic figure showing the real part of the ac
conductivity as a function of frequency at three different
temperatures. As temperature is lowered, the dc conductivity
decreases rapidly. At the same time the frequency marking the onset
of ac conduction also increases (actually in proportion to the dc
conductivity, compare with the BNN relation discussed below
(equation (13)). (b) The real part of the ac conductivity at three
different temperatures for a lithium-phosphate glass. The circles
mark the frequency for onset of ac conduction.
Combining equations (4) and (5) yields
n
AW
n
=
1
γH
r
2
(t
∗
)
a
2
. (6)
If the mean-square displacement obeys r
2
(t
∗
)a
2
one has n
AW
n
mob
(τ = 1/ν
∗
), but unfortunately the
quantity r
2
(t
∗
) does not generally have this approximate
value. In simple models where all ions have similar jump
rate, r
2
(t
∗
) is roughly a
2
times the fraction of ions that
have jumped within time t
∗
. It is not possible to model
the universally observed strong frequency dispersion of the
conductivity without assuming a wide spread of jump rates,
however, and in such models such as the random barrier model
(RBM) considered below r
2
(t
∗
) is much larger than a
2
.
Generally, r
2
(t
∗
)/a
2
gives an approximate upper limit for
the fraction of ions that have moved in the time window
t
∗
. Ignoring the less significant factor γH, this implies that
n
mob
(t
∗
)<n
AW
. To summarize, only in models without
3
Rep. Prog. Phys. 72 (2009) 046501 J C Dyre et al
significantly varying jump rates does n
AW
give an estimate
of how many ions on average jump over a time interval of
length t
∗
.
An alternative suggestion for obtaining information about
the ‘number of mobile ions’ is based on analyzing the electrode
polarization regime of conductivity spectra for ion conductors
placed between blocking electrodes [42–45]. However,
theoretical analyses of the spectra are often based on Debye–
H
¨
uckel-type approaches [42–45], the applicability of which
is far from obvious at high ion density. Thus while it is a
potentially useful idea, more theoretical work is needed before
observations of electrode effects may lead to safe conclusions
regarding the number of mobile ions (see the next section that
outlines the simplest approximate description); one still needs
to specify the time scale that the number of mobile ions refers
to.
Solid-state NMR methods such as motional narrowing
experiments [46–48] and the analysis of multi-time correlation
functions of the Larmor frequency[49, 50] provideinformation
about the number of ions moving on the time scale that these
methods monitor (milliseconds to seconds).
The question ‘what is the density of mobile ions?’ is
thus well defined only when it refers to a particular time scale.
According to standard ergodicity arguments, if the time scale
is taken to infinity, all ions contribute equally and the density
of mobile ions is the total ion density n. A natural choice of
time scale is that characterizing the onset of ac conduction,
the t
∗
of the above equations. Choosing this time scale leads
to a classification of ion conductors into two classes: those
for which n
mob
(t
∗
) is comparable to the total ion density n:
n
mob
(t
∗
) n (‘strong electrolyte case’ [13]), and those for
which n
mob
(t
∗
) n (‘weak electrolyte case’ [51, 52]). The
latter class includes solids where ion conduction proceeds by
the vacancy mechanism (section 7).
3. What can be learnt from electrode polarization?
As is well known, the ac conductivity σ(ω) is a complex
function. Thus associated with the real part there is also
an imaginary component; the latter determines the real part
of the frequency-dependent permittivity. For the study of
ion conduction in disordered solids the use of blocking or
partially blocking metal electrodes is convenient. In this case,
the high-frequency parts of ac conductivity and permittivity
spectra are governed by ion movements in the bulk of the
solid electrolyte, while the low-frequency part is governed
by so-called ‘electrode polarization’ effects, as shown in
figure 3. Since the ions are blocked by the metal electrode,
there is accumulation or depletion of ions near the electrodes,
leading to the formation of space-charge layers. The voltage
drops rapidly in these layers, which implies a huge electrical
polarization of the material and a near-absence of electric
field in the bulk sample at low frequencies. The build-up of
electrical polarization and the drop of the electric field in the
bulk are reflected in an increase in the ac permittivity and a
decrease in the ac conductivity with decreasing frequency [53].
For completely blocking electrodes σ(0) = 0, of course.
Whenever both ions and electrons conduct, a number of
(a)
(b)
Figure 3. (a) At high frequencies the nearly constant loss (NCL)
regime appears where the conductivity becomes almost proportional
to frequency (data for a lithium-phosphate glass). (b) The electrode
polarization effects on the real part of the conductivity and the real
part of the dielectric constant at high temperature for a
Na–Ca–phosphosilicate glass.
electrochemical techniques exist for evaluating transference
numbers of ions and electrons, including galvanic cells,
polarization and permeation techniques [4, 54–56].
Systematic experimental and theoretical studies of
electrode polarization effects in electrolytes began in the 1950s
with works carried out by Macdonald [57], Friauf [58], Ilschner
[59], Beaumont [60] and others. Their approaches were
based on differential equations for the motion (diffusion and
drift) of charge carriers under the influence of chemical and
electrical potential gradients. These equations were combined
with the Poisson equation and linearized with respect to the
electric field. Thereby, expressions for the ac conductivity
and permittivity at low electric field strengths were derived.
These are mean-field approaches in the sense that a mobile
charge carrier interacts with the average field produced by the
electrode and the other mobile carriers [61].
When charge carrier formation and recombination can be
neglected and the sample thickness L is much larger than the
space-charge layer thickness, the theoretical expressions can
be approximately mapped onto the simple electrical equivalent
circuit shown in figure 4 if the frequency dependence of the
4
Rep. Prog. Phys. 72 (2009) 046501 J C Dyre et al
Figure 4. Simplified electrical equivalent circuit describing the
low-field ac conductivity and permittivity spectra of solid
electrolytes between blocking (R
ct
=∞) or partially blocking
electrodes. The right element describes the bulk sample properties,
the left describes the space-charge layer capacitance and charge
transfer resistance (the frequency dispersion of the bulk conductivity
may be taken into account by replacing the resistor R
B
by a
frequency-dependent impedance).
bulk conductivity is ignored. Ion transport in the bulk is
described by the R
B
C
B
element. The space-charge layers
are described by a capacitance C
EP
, and, in the case of
discharge of the mobile ions at one electrode, a generally
large parallel charge transfer resistance R
ct
. The R
ct
C
EP
element acts in series with the R
B
C
B
element. In the cases
C
EP
C
B
and R
ct
R
B
that usually apply, the equivalent
circuit leads to the following expressions for the frequency-
dependent conductance G(ω) (real part of the admittance) and
capacitance C(ω) (imaginary part of the admittance):
G(ω) ≡ σ
(ω)
A
L
=
(1/R
ct
+ R
B
/R
2
ct
) + ω
2
R
B
C
2
EP
(1+R
B
/R
ct
)
2
+ (ωR
B
C
EP
)
2
(7)
and
C(ω) ≡
0
(ω)
A
L
= C
EP
1+ω
2
R
2
B
C
B
C
EP
(1+R
B
/R
ct
)
2
+ (ωR
B
C
EP
)
2
(8)
with A denoting the sample area, and C
EP
/C
B
= L/(2L
D
),
where the Debye length L
D
is defined by
L
2
D
≡
0
bulk
k
B
T
˜n
mob
e
2
. (9)
From these expressions one can calculate the number density
of mobile ions ˜n
mob
, which is the density of mobile ions
referring to the time scale for build-upof electrode polarization,
˜n
mob
= n
mob
(τ
ep
) where τ
ep
= R
B
C
EP
.
In the absence of ion discharge, i.e. when R
ct
→∞,
the equivalent circuit reduces to an RC element in series
with a capacitor. The existence of a finite charge transfer
resistance leads to the occurrence of a conductance plateau
at low frequencies with plateau value G
s
given by
G
s
=
1
R
B
+ R
ct
. (10)
In addition, the static capacitance C
s
becomes slightly smaller
than C
EP
:
C
s
= C
EP
R
ct
R
ct
+ R
B
2
. (11)
This mean-field approach should apply to materials with
low ˜n
mob
, such as ionic defect crystals and diluted electrolyte
solutions. Its applicability to disordered solids with high ion
density is far from obvious. Nevertheless, a large number
of ac spectra of ion-conducting glasses and polymers were
traditionally analyzed and interpreted utilizing the above
equations. Thereby, number densities of mobile ions were
calculated and compared with the total ion content of the
samples. For instance, Sch
¨
utt and Gerdes concluded that in
alkali silicate and borosilicate glasses only between 1 and
100 ppm of the alkali ions are mobile [62]. Similar results were
obtained by Tomozawa on silica glass with impurity ions [43]
and by Pitarch et al from voltage-dependent measurements on
a sodium aluminosilicate glass [63]. Klein et al carried out
measurements on ionomers containing alkali ions and found
ratios of mobile alkali ions to the total alkali ion content ranging
from about 10 to 500 ppm [44].
For a critical discussion of such experimental results and
their interpretation, it is important to consider limitations of
both experiment and theory. Regarding the experimental
situation, there are in particular two important points: (i) the
roughness of the electrode/solid electrolyte interface is usually
not taken into account. Especially in a frequency range where
the length scale of the potential drop at the electrodes is
comparable to the roughness of the interface, the roughness
must have a considerable influence on the ac conductivity and
permittivity. (ii) The surface-near regions of ion conductors
often exhibit a chemical composition that is significantly
different from the bulk. For instance, in ionic glasses surface
corrosion is initiated by an alkali-proton exchange. Such
deviations from the bulk composition should have a strong
influence on the ac spectra when the potential drop occurs very
close to the surface, i.e. at high capacitance values close to the
static capacitance plateau and in the static capacitance plateau
regime.
From a theoretical point of view, serious limitations of
the applicability of mean-field approaches to disordered solids
derive from: (i) the interactions between the ions are not taken
into account; (ii) surface space charges in disordered solids
may exist even without the application of an external electric
field, due to ion exchange processes at the surface or due to an
interaction of mobile ions with the metal electrode. Thus more
sophisticated theories should take into account the possibility
of an open-circuit potential difference between electrodes and
solid electrolyte.
In summary, considerable efforts in both experiment and
theory are required in order to carry out measurements on
well-defined electrode/electrolyte interfaces and to obtain a
better theoretical understanding of what kind of information
may be derived from electrode polarization effects. It is
clearly worthwhile to pursue this direction of research, and it
would also be worthwhile to look into what can be learnt from
electrode effects in the strong-field case where the electrode
polarization becomes nonlinear.
4. What causes the mixed-alkali effect?
A prominent phenomenon occurring in ion-conducting glasses
is the mixed-alkali effect (for reviews, see [64–67]). This
effect is the increase in the mobility activation energy of
5
