1. PHYSICAL REVIEW E 87, 042308 (2013)
Examination of methods to determine free-ion diffusivity and number density from analysis
of electrode polarization
Yangyang Wang,1,*
Che-Nan Sun,2
Fei Fan,3
Joshua R. Sangoro,1
Marc B. Berman,4
Steve G. Greenbaum,4
Thomas A. Zawodzinski,2,5
and Alexei P. Sokolov1,3
1
Chemical Sciences Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
2
Materials Science and Technology Division, Oak Ridge National Laboratory, Oak Ridge, Tennessee 37831, USA
3
Department of Chemistry, University of Tennessee, Knoxville, Tennessee 37996, USA
4
Department of Physics and Astronomy, Hunter College of the City University of New York, New York, New York 10065, USA
5
Department of Chemical and Biomolecular Engineering, University of Tennessee, Knoxville, Tennessee 37996, USA
(Received 11 October 2012; published 18 April 2013)
Electrode polarization analysis is frequently used to determine free-ion diffusivity and number density in
ionic conductors. In the present study, this approach is critically examined in a wide variety of electrolytes,
including aqueous and nonaqueous solutions, polymer electrolytes, and ionic liquids. It is shown that the electrode
polarization analysis based on the Macdonald-Trukhan model [J. Chem. Phys. 124, 144903 (2006); J. Non-Cryst.
Solids 357, 3064 (2011)] progressively fails to give reasonable values of free-ion diffusivity and number density
with increasing salt concentration. This should be expected because the original model of electrode polarization
is designed for dilute electrolytes. An empirical correction method which yields ion diffusivities in reasonable
agreement with pulsed-field gradient nuclear magnetic resonance measurements is proposed. However, the
analysis of free-ion diffusivity and number density from electrode polarization should still be exercised with
great caution because there is no solid theoretical justification for the proposed corrections.
DOI: 10.1103/PhysRevE.87.042308 PACS number(s): 66.10.Ed, 77.22.Jp
I. INTRODUCTION
One of the fundamental problems impeding the under-
standing of ionic conductors is the inability to unambiguously
estimate the free-ion diffusivity and number density from the
measured dc conductivity. One possible way to determine
these quantities is to analyze the electrode polarization (EP)
effect [1–20] observed by dielectric spectroscopy. In a typical
dielectric measurement, cations and anions migrate under
the applied electric field and accumulate on the surface of
electrodes. The resulting space-charge polarization gives rise
to a dramatic increase of the dielectric permittivity at low
frequencies. The EP phenomenon is ubiquitous in all ionic
conductors, including aqueous solutions, polymer electrolytes,
ionic liquids, and superionic glasses. Recent years have
evidenced a renewed interest in evaluating the free charge
carrier mobility and number density of ionic conductors from
the EP effect. In particular, various versions of Poisson-Nernst-
Planck model of electrode polarization [1,2,5,7,21] have been
applied to evaluate the ion mobility of polymer electrolytes,
hydrogels, and ceramics [13,22,23]. Despite these efforts, the
diffusivities obtained by EP analysis have not been carefully
compared with the results from other experimental techniques.
One of the current theoretical models of electrode po-
larization, proposed by Macdonald [1] and Trukhan [2], is
based on the Debye-H¨uckel type of approach, where the
Nernst-Planck equation for ionic motion is combined with the
Poisson equation and linearized with respect to the electric
field. Other models [4,9,10,16,19] are typically based on
similar methods. It is not obvious that the EP analysis will
hold at high ion concentrations, when the ionic interaction
strength exceeds the thermal energy. However, almost all ionic
*
Corresponding author: wangy@ornl.gov
conductors of practical interests are concentrated systems. A
closer examination of the EP model, especially in the cases of
high ion concentrations, seems necessary.
In the present study, we report the results of dielectric
measurements on a wide variety of ionic conductors. The free-
ion number density and diffusivity in these materials are ana-
lyzed according to the current theoretical model (Macdonald-
Trukhan model) of electrode polarization. Although the EP
analysis yields reasonable ionic dissociation energy, the ob-
tained ionic diffusivities are not in agreement with pulsed-field
gradient (PFG) NMR measurements. Possible origins of the
failure are discussed and a successful empirical correction
method is proposed. Despite the agreement of the estimated
diffusion with NMR data after corrections, we suggest that the
analysis of ion diffusivity and number density with the current
EP model should be exercised with great caution.
II. MATERIALS AND METHODS
A. Materials
To acquire a general understanding of the electrode po-
larization phenomenon, different types of ionic conductors
were investigated, including polymer electrolytes, aqueous and
nonaqueous solutions, and ionic liquids. The details of these
samples are summarized as follows:
(a) PPG-LiTFSI (polymer electrolyte): Polypropylene gly-
col (PPG) is commonly used as a model system in polymer
electrolytes studies. For the present investigation, a PPG
with an average molecular weight of 4000 g/mol was
purchased from Scientific Polymer Products. The polymer
electrolytes were prepared by dissolving the polymer and
the bis(trifluoromethane) sulfonimide lithium salt (LiTFSI)
from Sigma-Aldrich in methanol and subsequently removing
the solvent in a vacuum oven. The vacuum procedure was
042308-11539-3755/2013/87(4)/042308(9) ©2013 American Physical Society

2. YANGYANG WANG et al. PHYSICAL REVIEW E 87, 042308 (2013)
first carried out at room temperature for 2 weeks and then
at 60 ◦
C for another 2 weeks. Here, we report the results
of three different salt concentrations: 1.6 wt.%, 5.8 wt.%,
and 11 wt.%.
(b) LiCl-H2O (aqueous solution): A lithium chloride (LiCl)
aqueous solution was prepared using deionized water and
anhydrous ultradry lithium chloride powder (99.995%) from
Alfa Aesar. The molar ratio of H2O and LiCl was 7.3. The
details of sample preparation can be found in the earlier
study [24].
(c) LiCF3SO3-ethanol (nonaqueous solution): A lithium
trifluoromethanesulfonate (LiCF3SO3) ethanol solution was
prepared using ethanol and LiCF3SO3 powder from Sigma-
Aldrich. The salt concentration was 0.1 mol/L.
(d) NaCl–(glycerol-H2O mixture) (nonaqueous solution):
Glycerol (Fisher) and ultrapure H2O (Alfa Aesar) were mixed
at a molar ratio of 3:7. Puratronic sodium chloride (NaCl, Alfa
Aesar) was then dissolved in the glycerol-H2O mixture with
the aid of gentle heating. The weight percentage of NaCl was
0.1%.
(e) [OMIM][TFSI] (ionic liquid): We reanalyzed the
previously published data of an ionic liquid, 1-octyl-
3-methylimidazolium (OMIM) bis(trifluoromethylsulfonyl)
imide (TFSI) [18]. The broadband dielectric measurement
of this sample was carried out in F. Kremer’s group
at the University of Leipzig, using a Novocontrol alpha
analyzer (0.1 Hz–10 MHz) and an HP impedance analyzer
(1 MHz–1.8 GHz).
B. Methods
1. Dielectric measurements
Broadband dielectric measurements were carried out at
various temperatures using a Novocontrol Concept 80 system.
Standard parallel plate sample cell (Novocontrol), liquid
parallel plate sample cell (BDS 1308, Novocontrol), and a
home-made parallel plate sample cell were used in this study.
The surfaces of the first two sample cells were gold-plated,
whereas the homemade cell was made of brass. The electrode
surfaces were carefully polished and cleaned before each
experiment. Cooling and heating of the samples were achieved
by using a Quatro N2 cryostat. The dielectric measurements
were all performed in the linear response regime with voltage
no more than 0.1 V. The detailed information about the
electrode material, sample thickness, and applied voltage is
provided in Table I.
TABLE I. Electrode material, sample thickness, and applied
voltage.
Electrode Sample Applied
Sample material thickness (mm) voltage (V)
PPG-LiTFSI (1.6%) Gold 0.054 0.1
PPG-LiTFSI (5.8%) Gold 0.054 0.1
PPG-LiTFSI (11%) Gold 0.054 0.1
LiCl-H2O Gold 0.9 0.1
LiCF3SO3-ethanol Gold 0.5 0.1
NaCl–(glycerol-H2O) Brass 0.5 0.03
[OMIM][TFSI] Gold 0.155 0.05
2. PFG-NMR measurements
Pulsed-field gradient nuclear magnetic resonance (PFG-
NMR) measurements of PPG-LiTFSI were performed at
CUNY-Hunter College. The PPG-LiTFSI samples were slowly
added to 5-mm NMR tubes up to 4 cm in height and then
transferred to a vacuum oven to remove the residual moisture.
After overnight vacuum, the sample tubes were rapidly capped
and sealed with Parafilm. PFG-NMR measurements were car-
ried out on a Varian Direct Drive spectrometer in conjunction
with a JMT superconducting magnet of field strength 7.1 T.
The spectrometer equipped with Doty pulse-field-gradient
gradient probe could yield a maximum gradient strength of
10 T/m for the diffusion measurements. Temperature control
was achieved by a Doty temperature module. The samples
were equilibrated for 30 min before each experiment.
III. RESULTS AND DISCUSSIONS
A. Method of EP analysis
The analysis of the electrode polarization is based on the
simplified Macdonald-Trukhan model [13,21,23,25]. It has
been shown that for a 1:1 electrolyte solution, where cations
and anions carry the same amount of charges, by assuming
(1) all the ions have equal diffusivity, and (2) the sample
thickness L is much larger than the Debye length LD, the
complex dielectric function of electrode polarization in a plane
capacitor can be effectively described by the Debye relaxation
function,
ε∗
(ω) = εB +
εEP
1 + iωτEP
, (1)
where εB is the bulk permittivity, εEP = (L/2LD − 1)εB,
and τEP is the characteristic relaxation time of electrode
polarization. τEP is defined by the bulk resistance RB and the
interfacial capacitance CEP: τEP = RBCEP. In the Macdonald-
Trukhan model, CEP is simply determined by the Debye length
LD and the bulk permittivity εB as
CEP = ε0εB
A
2LD
, (2)
with A being the surface area of the electrode.
The total number density of free ions (n), which is the sum
of the number densities of free cations (n+) and anions (n−),
can be related the Debye length by the following equation:
LD =
ε0εBkBT
(n+ + n−)q2
=
ε0εBkBT
nq2
. (3)
Here q is the amount of charges carried by an ion and T is
the absolute temperature. One can, therefore, obtain the free-
ion number density and diffusivity by analyzing the dielectric
spectra of electrode polarization. It has been shown [7,23,25]
that the ion diffusivity can be calculated as
D =
2πfmaxL2
32(tan δ)3
max
, (4)
where (tanδ)max is the maximum value of ε /ε in the frequency
range of EP and fmax is the frequency at the tanδ maximum.
The free-ion number density can be obtained from the dc
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3. EXAMINATION OF METHODS TO DETERMINE FREE-ION . . . PHYSICAL REVIEW E 87, 042308 (2013)
-1 1 3 5 7
0
20
40
60
80
(c)
tanδ
log10
f [Hz]
(tanδ)max
fmax
1
3
5
7
Full development of EP
ε'
log10
ε',ε" NaCl-(glycerol-H2
O)
ε"
Onset of EP
(a)
-9
-8
-7
-6
-5
σ'
σ"(b)
log10
σ',σ"[S/cm]
FIG. 1. (Color online) Dielectric spectrum of NaCl–(glycerol-
H2O) at -5 ◦
C. (a) Complex permittivity. (b) Complex conductivity.
(c) Tanδ. Here, the main “relaxation” is due to electrode polarization.
Solid lines denote fits by Eq. (1). The onset and full development of
electrode polarization are indicated by the arrows.
conductivity and ion diffusivity according to the definition
of dc conductivity and the Einstein relation,
n =
σkBT
Dq2
. (5)
It should be noted that in our current approach the dissociation
of salts is not assumed to be complete, but the dissociation-
association dynamics are assumed to be much faster than the
macroscopic electrode polarization.
A representative dielectric spectrum of our samples is
shown in Fig. 1, where the ε , ε , and tanδ of NaCl–(glycerol-
H2O) are plotted as a function of frequency. On the high-
frequency side, the real part of permittivity is independent of
frequency while the imaginary part increases with decreasing
frequency as ε ∼ f −1
, exhibiting the normal behavior of
dc conduction. On the low-frequency side, the complex
permittivity shows a Debye-like shape, due to the electrode
polarization effect. The corresponding fmax and (tanδ)max can
be used to analyze the free-ion number density and diffusivity
using the method outlined above. One can define the onset
of EP as the minimum in σ and the “full development”
of EP as the maximum in σ . The solid lines in Fig. 1
represent the fits using Eq. (1). The shape of the spectrum
can be clearly approximated by the Debye function up to
the “full development” of electrode polarization. In reality,
no electrodes can be perfectly blocking, even in the case of
gold. As a result, substantial deviation from Eq. (1) occurs at
low frequencies. In order to describe the spectra in the whole
frequency region, more sophisticated models [22,26] have to
be considered. However, it should be emphasized that the main
goal of this study is to see if one can evaluate free-ion number
density and diffusivity from the interfacial capacitance due to
electrode polarization (CEP), which can be essentially captured
by the Eq. (1). Here, the low-frequency response (below the
frequency of σ maximum) does not affect our analysis.
It is necessary to clarify the relation of our current approach
to the model of electrode polarization with generation and
recombination (dissociation and association) effect (denoted
as EP-GR model) [27–29]. First, we explicitly assume that the
dissociation of salt is generally incomplete. In this sense, the
generation and recombination effect is already considered in
our model. On the other hand, we assume that the dissociation
and association are sufficiently fast and, therefore, they do not
directly contribute to the low-frequency electrical response,
i.e., electrode polarization. In the limit of fast dissociation-
association dynamics (characteristic dissociation and associa-
tion time τEP), the two models would be essentially identical.
This situation is demonstrated using simulated spectra in
Fig. 2(a). When the dissociation and association dynamics
are slow, our model will qualitatively differ from the EP-GR
model. The EP-GR model predicts an additional “relaxation”
due to ion dissociation and association in the low-frequency
end of the spectrum [Fig. 2(b)]. However, this situation
requires the rate of ion dissociation and association to be
-6 -4 -2 0
1
2
3
4
5
1/τEP
ε"
ε'
Slow Dissociation/
Association
(b)
log10
ε',ε"
log10
ωτ
1/τ
1/τA
1
2
3
4
5
ε"
log10
ε',ε"
(a)
Fast Dissociation/
Association
ε'
FIG. 2. (Color online) Demonstration of representative spectra
from the EP-GR model [29] in the limit of (a) fast and (b) slow
dissociation and association dynamics. ε∗
= εB (1 + iωτ)/(iωτ +
tanh Y/Y), where εB is the bulk permittivity, τ is the conductivity
relaxation time, and Y = (ϒ)1/2
M(1 + iωτ)1/2
. M = L/2LD, with
the Debye length LD defined by Eq. (3). ϒ = (iωτ + 2rAτ)/(iωτ +
rAτ), where rA is the rate of association. For simplicity, the term
of dissociation has been neglected, under the assumption that the
rate of association is much larger than the rate of dissociation
(incomplete dissociation). The bulk permittivity is assumed to be
10, and the magnitude of electrode polarization M is assumed to be
104
. Therefore, τEP = Mτ = 104
τ. In the case of “fast” dissociation or
association, the characteristic association time τA = 1/rA is assumed
to be 102
τ. In the slow case, τA = 106
τ.
042308-3

4. YANGYANG WANG et al. PHYSICAL REVIEW E 87, 042308 (2013)
2 3 4 5 6 7
16
17
18
PPG-LiTFSI (11%)
PPG-LiTFSI (1.6%)
PPG-LiTFSI (5.8%)
log10
n[cm
-3
]
1000/T [K
-1
]
LiCl-H2
O
LiCF3
SO3
-ethanol
[OMIM][TFSI]
NaCl-(glycerol-H2
O)
(a)
0 2 4 6
-6
-4
-2
0
PPG-LiTFSI (11%)
NaCl-(glycerol-H2
O)
LiCF3
SO3
-ethanol
[OMIM][TFSI]
LiCl-H2
O
PPG-LiTFSI (5.8%)
PPG-LiTFSI (1.6%)
log10
n/ntot
1000/T [K
-1
]
100% Dissociation
(b)
FIG. 3. (Color online) Temperature dependence of (a) free-ion
number density n and (b) free-ion fraction n/ntot, evaluated from the
electrode polarization effect using the Macdonald-Trukhan model.
Solid lines denote Arrhenius fits. n = n0 exp(−Edis/kBT ). The
horizontal dashed line indicates the limit of 100% dissociation.
smaller than the characteristic relaxation rate of electrode
polarization, which is typically several orders of magnitude
smaller than the conductivity relaxation rate. As a result, such
ultraslow ion dissociation and association dynamics should not
occur in most real physical systems. At least, it is not observed
in any of the samples in the present study. Moreover, our
estimated energy for dissociation (see below) is rather low and
cannot lead to very slow dissociation or association process.
B. Failure of the EP analysis
Figure 3(a) presents the free-ion number density, deter-
mined from the EP analysis, as a function of 1000/T for all
the samples in this study. As a first observation, the free-ion
number density n decreases with decrease of temperature in
all samples except LiCl-H2O, where n is almost a constant.
The temperature dependence of ion number density can be
described by the Arrhenius equation,
n = n0 exp(−Edis/kBT ), (6)
where Edis is the dissociation energy and n0 is the number
density in the high-temperature limit. These features are
in accordance with earlier investigations on polymer elec-
trolytes using similar approaches [13,25,30]. It is important
to mention that the temperature dependence of free-ion
concentration from EP analysis is at odds with the studies
by radiotracer diffusion [31], Raman spectroscopy [32], and
computer simulations [33,34]. As pointed out earlier [25],
this apparent contradiction possibly arises from the fact that
different experimental techniques are sensitive to only certain
populations of ions and different timescales of ion association
[30]. Moreover, ions at high concentration can form aggregates
that will complicate analysis of all the experimental data. These
discussions are out of the scope of the present study.
For a simple chemical equilibrium between free ions and
ion pairs for a weak electrolyte solution, AB A+x
+ B−x
,
Kdis =
[A+x
][B−x
]
[AB]
=
cαcα
c(1 − α)
= c
α2
1 − α
≈ cα2
, (7)
where c is the salt concentration and α is the free-ion fraction.
It follows that α decreases with increase of salt concentration
c as α ∼ c−1/2
, whereas the total free-ion concentration cα in-
creases with c as cα ∼ c1/2
. On the other hand, Fig. 3(a) shows
that in PPG-LiTFSI, the free-ion concentration decreases with
increasing salt content. This result clearly contradicts the
general understanding of electrolyte solutions. As we shall
see later, this abnormal behavior of free-ion number density is
due to the failure of EP analysis.
The temperature dependence of free-ion fraction n/ntot is
shown in Fig. 3(b), where the Arrhenius fits are extrapolated
to infinitely high temperature, i.e., 1000/T = 0. The total ion
number density ntot is calculated from the total amount of salt in
the solution, under the assumption of complete dissociation. If
the electrode polarization analysis could indeed quantitatively
capture the free-ion concentration, one would expect n0 ≈ ntot.
In other words, log10(n/ntot) should be zero at 1000/T = 0.
The physical picture is that the ions should become fully
dissociated at sufficiently high temperature. Strictly speaking,
this na¨ıve picture cannot be true since the thermodynamics of
ion solvation is not considered. Nevertheless, the n/ntot → 1
limit should still serve as a good guide for checking the
validity of the electrode polarization analysis. In PPG-LiTFSI
(1.6%) and NaCl–(glycerol-H2O), the n0 in the Arrhenius
fit is fairly close to the total ion number density ntot in the
sample. However, such agreement is not found in all the other
samples. In particular, the free ions in LiCl-H2O are only
about 30 ppm of the total ions in the solution. Since aqueous
solutions are generally considered as strong electrolytes where
salts fully dissociate into free cations and anions, the result of
LiCl-H2O is obviously at odds with the current understanding
of electrolyte solutions. Furthermore, while recent studies of
room temperature ionic liquids have demonstrated that most
of the ions exist as free ions [15], the EP analysis suggests
that the free-ion fraction is only on the order of 100 ppm. It is,
therefore, quite evident that the EP analysis underestimates the
free-ion concentration in PPG-LiTFSI (5.8%), PPG-LiTFSI
(11%), LiCl-H2O, LiCF3SO3-ethanol, and [OMIM][TFSI].
A problem interconnected with underestimation of free-ion
concentration is the overestimation of ion diffusivity, which is
a natural consequence of the definition of ionic conductivity.
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5. EXAMINATION OF METHODS TO DETERMINE FREE-ION . . . PHYSICAL REVIEW E 87, 042308 (2013)
4.5 5.0 5.5 6.0 6.5 7.0 7.5
-10
-8
-6
-4
EP
NMR
log10
D[cm
2
/s]
1000/T [K
-1
]
0 2 4 6
16
18
20
22
log10
n[cm
-3
]
1000/T [K
-1
]
Complete
Dissociation
FIG. 4. (Color online) Temperature dependence of ion diffusivity
for LiCl-H2O, determined from electrode polarization and PFG-
NMR. Inset: Temperature dependence of free-ion number density
(uncorrected). The horizontal dashed line indicates the level of total
ions in the sample.
This issue is highlighted in Fig. 4, where we compare the
diffusivity from EP analysis with that from PFG-NMR mea-
surement for the LiCl-H2O sample. The diffusion coefficient
from EP analysis is about four orders of magnitudes higher
than the values obtained from NMR experiments. The inset
of Fig. 4 shows that corresponding free-ion number density
in LiCl-H2O is considerably lower than the number density
at complete dissociation. Similar problems are also found in
PPG-LiTFSI (5.8%), PPG-LiTFSI (11%), LiCF3SO3-ethanol,
and [OMIM][TFSI].
Overestimating ion diffusivity seems to be a generic
problem in the electrode polarization analysis of ionic mo-
bility. Macdonald and coworkers recently analyzed a polymer
electrolyte containing polyethylene imine (PEI) and LiTFSI
with a N:Li molar ratio of 400:1 [26]. The estimated ion
mobility is 4.87 × 10−4
cm2
/(Vs) at 20 ◦
C, corresponding to
a diffusion coefficient of 1.23 × 10−5
cm2
/s. This value is
typical for aqueous solutions at room temperature and is too
high for a polymer. The diffusivity of Li+
in infinitely dilute
aqueous solution at 25 ◦
C is 1.029 × 10−5
cm2
/s [35]. In
general, Walden’s rule should apply at high temperature, i.e.,
the ion diffusivity should be controlled by the viscosity of
the surrounding medium. Since the (microscopic) viscosity of
PEI is significantly higher than water, the EP analysis clearly
overestimates the ion diffusivity.
Despite the problem of underestimating free-ion con-
centration and overestimating ion diffusivity, the electrode
polarization analysis does seem to be able to reasonably
describe the temperature dependence of free-ion concentra-
tion. The dissociation energy Edis from the Arrhenius fit is
in the range of 0.002–0.08 eV, which is commonly seen in
these ionic conductors. In addition, Edis seems to correlate
with the dielectric constant of the electrolytes: While near
zero Edis is found in the strong electrolyte LiCl-H2O, the
Edis of weak electrolyte PPG-LiTFSI is considerably higher.
It is important to indicate that the dissociation energy in the
PPG-LiTFSI series does not vary significantly with the salt
concentration in the studied range. This result is in agreement
with the general expectations for polymer electrolytes and
further confirms the EP analysis can qualitatively capture the
change of free-ion concentration with temperature. It is worth
noting that Runt and coworkers [13] proposed a different fitting
procedure by forcing n0 to be the stoichiometric value ntot in
the Arrhenius fit. It is obvious from the Fig. 3(b) that this
approach cannot render reasonable fits for samples other than
PPG-LiTFSI (1.6%) and NaCl–(glycerol-H2O). In particular,
the dissociation energy in the polymer electrolytes would
change significantly with salt concentration—a result that does
not seem to be consistent with our general understanding of
electrolyte solutions.
C. Corrections to the EP analysis
It is well known that electrode polarization can be affected
by the roughness and chemical composition of electrodes
[15,17,36]. These effects are not considered in the Macdonald-
Trukhan model of electrode polarization. As a result, it is
quite likely that the model may only qualitatively describe
the dielectric response but fail to yield quantitative values.
We have seen that the EP analysis can produce reasonable
dissociation energy, but the absolute level of free-ion number
density can be off by orders of magnitude. Based on these
observations, we propose that the free-ion number density
and diffusivity from EP analysis can be simply corrected by
using a proportionality constant. If salts fully dissociate in the
infinitely high temperature limit, then the true free-ion number
density should be described by the Arrhenius equation,
˜n =
ntot
n0
n =
ntot
n0
[n0 exp(−Edis/kBT )]
= ntot exp(−Edis/kBT ), (8)
where ˜n is the free-ion number density after correction, n
is the free-ion number density before correction, ntot is the
total ion number density at complete dissociation, and Edis is
the dissociation energy from the original EP analysis. Here,
we rescale the original free-ion concentration n by a factor
of ntot/n0, assuming complete dissociation at infinitely high
temperature limit. Similarly, we need to correct the diffusivity
by a factor of n0/ntot,
˜D = D
n0
ntot
, (9)
with ˜D being the corrected diffusivity and D the original
diffusivity from the EP analysis. It should be emphasized that
possible entropic contribution to Edis is not considered in our
correction.
In Figs. 5–7, the corrected and uncorrected free-ion
diffusivities are compared with the results of PFG-NMR
measurements, along with the diffusivities Dσ evaluated from
the Nernst-Einstein relation, assuming 100% ion dissociation.
Here, we include the examples of three polymer electrolytes,
one aqueous solution, and one ionic liquid. In all samples, the
uncorrected diffusivities from EP analysis are much higher
than the values obtained from NMR measurements. The
difference is as large as four orders of magnitudes in the
case of LiCl-H2O. It is worth pointing out that PFG-NMR
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6. YANGYANG WANG et al. PHYSICAL REVIEW E 87, 042308 (2013)
4.5 5.0 5.5 6.0 6.5 7.0
-16
-14
-12
-10
-8
-6
-4
After
correction
LiCl-H2
O
DEP
, Corrected
DEP
, Uncorrected
Dσ
D, PFG-NMR, Averaged
log10
D[cm
2
/s]
1000/T [K
-1
]
Before
correction
FIG. 5. (Color online) Temperature dependence of diffusivity
for LiCl-H2O. Filled (black) circles denote diffusivity from the EP
analysis, after correction. Unfilled (black) circles denote diffusivity
before correction. Half-filled (black) circles denote diffusivity from
the Nernst-Einstein relation. (Red) Diamonds denote averaged diffu-
sivity of lithium and chloride from PFG-NMR [24]. Here, the chloride
diffusivity is calculated from the lithium diffusivity according to the
Stokes-Einstein relation.
experiments measure the average diffusivities of both free
ions and ion pairs, whereas the EP phenomenon in theory
only reflects the diffusivity of free ions. However, there is no
physical model to support the idea that the free-ion diffusivity
can differ from the average value by orders of magnitude. In
the case of aqueous LiCl solution the majority of ions are
expected to be dissociated, and even if we have only 10% of
free ions, the average diffusivity cannot be more than 10 times
smaller than the diffusivity of free ions.
After the proposed correction, reasonable agreement be-
tween EP analysis and PFG-NMR measurement is found
in all samples, although there is still a slight difference
between diffusivities determined by these two methods. This
discrepancy might be due to either the over-simplification in
the EP analysis (e.g., by assuming cations and anions have
equal diffusivity) or the real difference in diffusivity of free
ions and ion pairs.
D. Possible origins of the failure
In the preceding discussion, we show that the EP analysis
fails for many ionic conductors but can be corrected by
rescaling free-ion number density and diffusivity by a certain
factor. This correction is somewhat expected, as the effects
of surface roughness and electrode materials have not been
taken in account in the current theoretical model of electrode
polarization.
In addition to the known effects of the roughness and
chemical composition of electrodes, it has been suggested that
nonlinear dielectric response could also lead to the failure
of the EP analysis [37]. However, this is not the case in
our measurements. Figure 8 presents the dielectric spectra
of PPG-LiTFSI (11%) at different voltages. In the frequency
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6
-14
-12
-10
-8
-6
-4
After correction
DEP
, Corrected
DEP
, Uncorrected
Dσ
DF
, PFG-NMR
DLi
, PFG-NMR
log10
D[cm
2
/s]
1000/T [K
-1
]
PPG-LiTFSI (1.6%)
(a)
Before correction
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6
-14
-12
-10
-8
-6
-4
After
correction
PPG-LiTFSI (5.8%)
DEP
, Corrected
DEP
, Uncorrected
Dσ
DF
, PFG-NMR
DLi
, PFG-NMR
log10
D[cm
2
/s]
1000/T [K
-1
]
(b)Before
correction
3.0 3.2 3.4 3.6 3.8 4.0 4.2 4.4 4.6
-14
-12
-10
-8
-6
-4
After
correction
DEP
, corrected
DEP
, uncorrected
Dσ
DF
, PFG-NMR
DLi
, PFG-NMR
log10
D[cm
2
/s]
1000/T [K
-1
]
PPG-LiTFSI (11%)
(c)Before
correction
FIG. 6. (Color online) Temperature dependence of diffusivity
for three PPG-LiTFSI samples: (a) PPG-LiTFSI (1.6%); (b) PPG-
LiTFSI (5.8%); and (c) PPG-LiTFSI (11%). Filled (black) circles
denote diffusivity from the EP analysis, after correction. Unfilled
(black) circles denote diffusivity before correction. Half-filled (black)
circles denote diffusivity from the Nernst-Einstein relation. Up- (red)
triangles denote fluoride diffusivity from PFG-NMR. Down- (blue)
triangles denote lithium diffusivity from PFG-NMR.
042308-6

7. EXAMINATION OF METHODS TO DETERMINE FREE-ION . . . PHYSICAL REVIEW E 87, 042308 (2013)
3.0 3.3 3.6 3.9 4.2 4.5 4.8
-12
-10
-8
-6
-4
After
correction
DEP
, Corrected
DEP
, Uncorrected
Dσ
D, PFG-NMR, Average
log10
D[cm
2
/s]
1000/T [K
-1
]
[OMIM][TFSI]
Before
correction
FIG. 7. (Color online) Temperature dependence of diffusivity for
ionic liquid [OMIM][TFSI]. Filled (black) circles denote diffusivity
from the EP analysis, after correction. Unfilled (black) circles
denote diffusivity before correction. Half-filled (black) circles denote
diffusivity from the Nernst-Einstein relation. (Red) Diamonds denote
diffusivity from PFG-NMR [18].
range of 1 Hz–10 MHz, the dielectric response below 0.3 V
is essentially independent of the applied voltage, i.e., there is
a well-defined linear response regime. Since our dielectric
measurements were all carried out at voltages no larger
than 0.1 V, the failure of the EP analysis is not related to
the nonlinear response. In additional, although the dielectric
behavior becomes nonlinear at 1.0 V, the spectrum between
106
Hz (the onset of electrode polarization) and 103
Hz (the
full development of electrode polarization) is still identical to
0 1 2 3 4 5 6 7
0
2
4
6
(b)
1.0 V
0.3 V
0.1 V
0.03 V
0.01 V
log10
ε'
log10
f [Hz]
0
2
4
6
log10
ε"
PPG-LiTFSI (11%), 59
o
C
(a)
0 2 4 6
-1
0
1
2
log10
tanδ
log10
f [Hz]
FIG. 8. (Color online) Frequency dependence of imaginary
(a) and real (b) permittivity for PPG-LiTFSI (11%) at various applied
voltages. Inset: Corresponding tan δ = ε /ε .
that in the linear regime. Our data analysis protocol, described
in Sec. II, relies only on the high-frequency response. The
low-frequency nonlinear behavior will not play any role in our
EP analysis.
It is perhaps useful to indicate that Colby and coworkers
have proposed a dimensionless parameter [37], defined as
Qλ2
B
Aq
=
qV
kBT
nλ3
B
16π
, (10)
to characterize the degree of nonlinearity in their dielectric
measurement of single-ion conductors. Here Q is the charge
accumulated on the electrode surface, A is the electrode surface
area, q is the charge carried by the ions, V is the applied
voltage, and λB is the Bjerrum length. It has been argued that
the EP analysis should remain valid as long as Qλ2
B/Aq is
19 20 21 22
-6
-4
-2
0
2
LiCl-H2
O
[OMIM][TFSI]
PPG-LiTFSI (11%)
PPG-LiTFSI (5.8%)
LiCF3
SO3
-ethanol
PPG-LiTFSI (1.6%)
log10
(n0
/ntot
)
log10
ntot
[cm
-3
]
NaCl-(glycerol-H2
O)
(a)
-1 0 1 2
-6
-4
-2
0
2
(b)
PPG-LiTFSI (11%)
PPG-LiTFSI (5.8%)
[OMIM][TFSI]
LiCF3
SO3
-ethanol
PPG-LiTFSI (1.6%)NaCl-(glycerol-H2
O)
LiCl-H2
O
log10
(n0
/ntot
)
log10
(λB
/LD
)
FIG. 9. (Color online) Correction factor n0/ntot as a function of
(a) ntot and (b) λB/LD. n0 is the prefactor in the free Arrhenius fit
of free-ion concentration n(T ) : n = n0 exp(−Edis/kBT ). ntot is the
total ion number density in the material. The degree of correction
to the experimental ion number density n from the EP analysis is,
therefore, determined by the ratio of n0 and ntot. λB is the Bjerrum
length and LD is the Debye length. The horizontal dashed lines
indicate the limit of no correction, i.e., log10(n0/ntot) = 0.
042308-7

8. YANGYANG WANG et al. PHYSICAL REVIEW E 87, 042308 (2013)
much less than 1. The physical interpretation is that nonlinear
electrode polarization starts to occur when the ion separation
on the electrode surface is smaller than the Bjerrum length. For
the PPG-LiTFSI (11%) sample, Fig. 8 shows that nonlinear
electrode polarization occurs at approximately 0.3 V. On the
other hand, the above criterion predicts that the critical voltage
is 0.25 V, when Qλ2
B/Aq = 1. The linearity criterion proposed
by Colby and coworkers seems to be in good agreement with
our experimental observation.
The current theoretical model of electrode polarization
[1,13] is based on a Debye-H¨uckel-type approach, where the
ionic interaction strength is assumed to be much smaller than
the thermal energy kBT . Therefore, the model is expected
to fail at high salt concentrations, when strength of ionic
interaction increases. The ratio of Bjerrum length λB and
Debye length LD can be used to roughly determine whether
an electrolyte is in the dilute state. When λB/LD 1, the
system typically can be considered as a dilute solution, and
Debye-H¨uckel-type approaches should apply.
We recall that the free-ion concentration from the EP
analysis (without correction) can be described by the Arrhenius
equation: n = n0 exp(−Edis/kBT ). In general, n0 is not the
same as the total ion number density ntot in the material. The
initial free-ion concentration n from the EP analysis therefore
needs to be corrected by a factor of ntot/n0: ˜n = n(ntot/n0).
In other words, the ratio of n0 and ntot reflects the degree of
correction that needs to be introduced in the EP analysis. In
the case of no correction, n0/ntot = 1.
Figure 9 shows n0/ntot as a function of (a) ntot, and
(b) λB/LD. The goal of this presentation is to explore a
possible correlation between the degree of correction in the
EP analysis and the extent to which the system deviates
from the dilute limit. It is clear from Fig. 9 that the original
Macdonald-Trukhan model fails progressively with increasing
salt concentration ntot and λB/LD. Significant correction has to
be made in concentrated electrolytes. Interestingly, the degree
of correction, n0/ntot, seems to correlate better with the total
ion concentration ntot rather than λB/LD. The linear fit of
log10(n0/ntot) and ntot has a R2
of 0.80, whereas the R2
for log10(n0/ntot) and λB/LD is only 0.31. This observation
suggests that the failure of EP model may also be related
to the structure of the compact double layer, where the bulk
permittivity plays a much less significant role.
IV. CONCLUSIONS
The electrode polarization analysis is used to estimate
the ion diffusivity and number density of a wide variety of
ionic conductors. Although the analysis does yield reasonable
estimates in a few cases, it fails for most of the materials.
For systems with high ion concentration, the EP analysis
overestimates the free-ion diffusivity while underestimates the
free-ion number density. Since the original EP model is based
on the Debye-H¨uckel approach, it is expected to fail at high ion
concentrations. An empirical method is proposed to correct
the results at high ion concentrations, and ion diffusivities
that are in close agreement with PFG-NMR measurements are
obtained. Despite this success, the analysis of ion diffusivity
and number density from electrode polarization should still
be exercised with great caution, because there is no solid
theoretical justification for the proposed corrections.
ACKNOWLEDGMENTS
The authors thank A. L. Agapov and M. Nakanishi for
fruitful discussions. This research was sponsored by the
Laboratory Directed Research and Development Program of
Oak Ridge National Laboratory, managed by UT-Battelle,
LLC, for the US Department of Energy. F.F. thanks the NSF
Polymer Program (DMR-1104824) for funding. J.R.S. and
A.P.S. acknowledge the financial support from the DOE-
BES Materials Science and Engineering Division. The NMR
program at Hunter College is supported by a grant from the
US DOE-BES under Contract No. DE-SC0005029.
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