chapter 5 debye-falkenhagen dynamics of electric double layer...
TRANSCRIPT
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Chapter 5
Debye-Falkenhagen Dynamics of
Electric Double Layer on
Partially Blocking Heterogeneous
Electrode
“God made the bulk; the surface was invented by the devil”.
-Wolfang Pauli
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Abstract
A phenomenological theory of electric double layer (EDL) polarization of an
electrode (in the absence and presence of electroactive species) is obtained us-
ing the Debye-Falkenhagen equation for the potential. The influence of surface
heterogeneities on the compact layer causes the distribution in relaxation time,
resulting in constant phase element (CPE) response. This contribution of the
compact layer is included through the current balance boundary constraint at
the outer Helmholtz plane. The results for the impedance and the capacitance
are obtained in terms Debye screening length and dynamic polarization length -
which is dependent on the surface heterogeneity parameter. At frequencies less
than the characteristic compact layer relaxation frequency, the EDL is controlled
by the compact layer dynamics. The intermediate frequency shows the emergence
of Gerischer and Warburg like behavior for systems with lower concentration and
lower diffusion coefficients of ions. At frequencies larger than the characteristic
diffuse layer relaxation frequency the EDL behaves like a resistor. Theoretical
results capture various observations in the experimental capacitance dispersion
data.
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1 Introduction
The electric double layer (EDL) formed at an electrode/electrolyte interface is
ubiquitous in interfacial science. In recent years, the properties of EDL have been
exploited in nanofluidics, for example to make nanofluidic transistors (electrolytic
gate) [1], in capacitive deionization of water (through electrosoption) [2] and in
energy storage devices (supercapacitors [3], batteries [4], fuel cells [5]) etc. The
EDL region in such a system is very different from the bulk of the electrolyte
or idealized homogeneous (smooth) electrode/electrolyte system. These systems
are strongly influenced by the electrode surface heterogeneity and nanometer-scale
surface morphology. This is because a typical width of EDL is few nanometers
comparable to morphological disorder, and crystallographic or material surface
heterogeneity determines the electric interaction between ions and electrode sur-
face. These complexities influence the static behavior, i.e charge stored [6] as well
the dynamical aspect, i.e constant phase element (CPE) [7], capacitance disper-
sion [8, 9], anomalous diffusion [10], etc. Despite its huge application, the physical
mechanism of electrode polarization is still not understood completely, especially
on the nano-meter scales, where various length and phenomenological time scales
can convolute with each other to provide complex behavior. The question of
surface inhomogeneities, roughness and the nature of electrode material on the
electrode side; and inhomogeneous ion transport on the electrolyte side, inclusion
of effect of adsorption/desorption and charge transfer further brings in additional
complexity, making this problem theoretically challenging.
Broadly, two common phenomena observed in EDL are relaxation and disper-
sion [8, 9, 11, 12, 13]. Relaxation is a general process and may include dispersive
processes with one or more discrete times leading to distribution of relaxation
times [12]. These occurs when a solid electrode is in contact with various elec-
trolytes viz., aqueous, liquid crystals [14], glassy electrolytes [3], ionic liquids [11]
and polyelectrolytes [15]. Usually these systems causes nonuniform change of elec-
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trical behavior of the electrode/electrolyte interface and often assumed to be the
possible cause of ubiquitous electrode polarization [14, 16, 17, 18, 19, 20, 21, 22,
23, 24, 25, 26, 27, 28, 29]. In general, the possible cause of change in the elec-
trical behavior leading to electrode polarization may be attributed mainly to two
factors:
1. Dispersion due to inhomogeneous transport of ions due to diffusion (adsorp-
tion/desorption), convection and migration.
2. Dispersion due to surface factors like roughness, atomic scale inhomogeneities,
polycrystalline surface, grain boundaries or defects of electrode.
The dispersion or dynamics in diffuse layer is addressed in through the Debye-
Falkenhagen equation. The surface heterogeneities leads to distribution in relax-
ation times in adjacent compact layer which causes CPE behavior in the dynamics.
Pajkossy emphasized through his experimental work that CPE behaviour is due
surface atomic scale heterogeneities [33] hence it will be appropriate to assume
that CPE mainly originate from the dynamics of compact layer region of EDL.
Geometrically and atomically uniform electrode, viz, liquid mercury and well
defined single crystals of noble metal, exhibit ideally capacitive impedance in
absence of charge transfer and adsorption processes. However, commonly used
polycrystalline solid electrodes (even the smooth one) exhibit anomalous frequency
dependent capacitance. This capacitance is characterize as a power-law function
of frequency, that is, C(ω) ∝ (iω)α−1. CPE exponent, α ≤ 1, is dependent on
the heterogeneity of electrode surface [33]. One of the most widely used simple
approach to understand the dispersion and relaxation is often approximated by
a constant phase element (CPE) with the empirical representation of impedance
has the following form:
Z(CPE) = Rs
(1 +
1
(iωτdl)α
)(1.0.1)
where i =√−1, a characteristic relaxation time τdl is given by τdl = RsCdl,
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Rs is the bulk (ohmic) resistance of electrolyte and Cdl is the electric double
layer capacitance. The frequency exponent α is usually related to constant phase
element (CPE). For an ideally polarizable electrode, α = 1, the EDL is ideal
and there is no dispersion and any deviation from this RC circuit behavior, the
electrode is said to be “polarized”[16]. The characteristic dispersion observed may
arise from the dispersion of solution resistance Rs or the interfacial capacitance
Cdl due to surface or interfacial heterogeneity.
+ +
Static (Equilibrium Picture)
Non-polarized Dynamic (Non-equilibrium Picture)
Polarized
Ionic cloud
Motion
Relaxation τ� = ��� �⁄
Perturbation
(Potential, Electric Field, Current)
Figure 5.1: Model of ionic relaxation after Debye-Falkenhagen [30].
Various models are developed to interpret electrical double layer relaxation
and dispersion[14, 16, 17, 18, 19, 22, 23, 24, 25, 28, 29]. One of the earliest model
of ion relaxation in the bulk was given by Debye-Falkenhagen [30] (see Figure
5.1) and an extension of Debye-Falkenhagen model for ion relaxation near smooth
electrode/electrolyte interface is developed by Ferry [18]. The characteristic time
of relaxation of the ionic atmosphere is given by τD = 1/κ2D, where κ−1 is the
Debye length and D the diffusion constant of ions. τD also called the Debye
relaxation time is time taken by the ion to diffuse out from ionic cloud of size
κ−1. Some real electrochemical system may involves finite length separation in
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Figure 5.2: Schematic diagram of dynamics of electric double layer (EDL) at aheterogeneous planar electrode in absence of a charge transfer process. The EDLconsists of Helmholtz layer (ZH), diffuse layer (ZDF ), and bulk (ZB) impedance.The interfacial boundary constraint is obtained at OHP using flux balance con-dition between compact and diffuse layers. |σZH |−1 is the dynamic polarizationlength where ZH is the known compact layer impedance. rH , κ
−1 are thickness ofcompact layer and diffuse layer, respectively.
electrodes comparable to Debye length, here one identifies the geometry-dependent
time scale in electrochemical cell [20, 21, 25]. Bazant et al. [25] have shown using
dimensional analysis that the response time is a geometric average of Debye time
τD and diffusion time τL = L2/D as
τc =√τD τL = (L/κD) (1.0.2)
Here, the cell behaves like RC circuit with a single time τc and shows geometric
dependent through the electrode separation L.
In this chapter, we develop a model of electrode polarization in EDL formed
near a heterogeneous planar electrode (Figure 5.2). The objective of the paper is
to understand the interplay between the compact and diffuse layers dynamics in-
terms of various length scales and time scales emerging out of the electric double
layer reorganization when a sinusoidal time varying potential is applied.
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Figure 5.2 shows the schematic diagram of electric double layer (EDL) dynam-
ics at a heterogeneous (polycrystalline) planar electrode. The outer Helmholtz
plane is at a distance rH (OHP) from electrode surface and, separates the compact
and the diffuse part of the EDL. We assume that the surface heterogeneity mainly
influence the compact layer dynamics while the diffuse double layer (with rela-
tively weak electric field region) dynamics is governed by the (linearized) Debye-
Falkenhagen equation for the potential. Usually the geometric separation length
L in between the working and counter electrodes is much larger than Debye length
(κ−1) (there is no overlaps of the two diffuse layers) hence its appropriate to use
bulk boundary condition away from the electrode. Our model essentially view the
interface to be blocking to charge transfer but allows transfer of ions from the
diffuse layer to the Helmholtz layer. The surface boundary condition is obtained
by matching the compact and diffuse layers flux or current at OHP. This allow us
to model compact and diffuse layers independently. Contributions from the diffuse
layer are accounted in details through the (linearized) Debye-Falkenhagen equa-
tion but for simplicity, we assume heterogeneity exponent dependent distribution
of compact layer relaxation time. The model predictions are based on the relative
importance of two phenomenological lengths, viz, dynamic polarization ((σZH)−1)
and Debye lengths, while dynamics is controlled by smaller of two lengths. Mathe-
matically our boundary conditions are classified as the Robin boundary condition
at OHP.
2 Helmholtz Layer Model of Heterogeneous Poly-
crystalline Electrode and Constant Phase El-
ement (CPE)
Real electrode are never ideal surfaces except for liquid mercury and single crystal.
It was first recognized by Frumkin [31] that the charge accumulation at different
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crystal faces are different hence their capacitance differs. The dynamic response of
heterogeneous polycrystalline solid electrodes causes the capacitance dispersion.
The deviation from ideal capacitive behavior is usually represented by
Z(CPE) ∝ 1/(ιω)γ (2.0.1)
where ι is√−1. The exponent γ is depending on the heterogeneity of electrode
surface and usually its magnitude less than one. γ = 1 is for perfectly homoge-
neous electrode like mercury and single crystal. The physical interpretation of
CPE and its exponent γ is not clearly understood[32]. The possible connection
between capacitance dispersion and surface roughness [32], polycrystalline [33],
surface disorder [34, 35, 36] and atomic scale inhomogeneities [37] is reported.
Recent experimental studies by Pajkossy and coworkers have shown that capaci-
tance dispersion on solid electrodes is due to surface heterogeneities on the atomic
scale rather than microscopic roughness in the range size about 10 nm to 100 µm.
Form a geometric point of view, the scale of roughness may be different. Surface
features less than 10 nm are considered as atomic scale roughness or disordered.
Here surface disorder means not only atomic scale non-uniformity but energetic
inhomogeneities along the surface as well.
Here in this section we address this issue of capacitance dispersion at atomic
scale due to surface heterogeneities and disorder. Two cases: blocking electrodes
(when charge transfer resistance Rct → ∞) and partially blocking (when Rct is
finite) electrodes are considered.
2.1 Blocking Electrodes
One of the first analysis of anomalous dispersion of capacitance was done using the
ladder-RC model by Brug et al. [35]. We also apply similar model for the compact
(Helmholtz) layer which consist of a site independent resistive (RH) and a site
dependent capacitive (cH(s)) elements. The dynamic site dependent admittance
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density of the compact (Helmholtz) layer is
yH(s, ω) =1
zH(ω)=
1
RH(1 + 1/ιωτH(s))(2.1.1)
where the site dependent Helmholtz layer relaxation time is given by τH = RHcH(s).
The local Helmholtz layer capacitance density cH(s) is a site-dependent and we
assume having the form: cH(s) = cavH exp(s), where s = ln(τH(s)/τ
0H), and
τ 0H = RHcavH , where c
avH is a site averaged net compact layer capacitance resulting
from the combined contributions of all surface sites.
The site average admittance density of the Helmholtz layer is
yH(ω) =
∫ ∞−∞
yH(s, ω)F (s)ds
=1
RH
∫ ∞−∞
(1− 1
1 + ιωτH(s)
)F (s)ds (2.1.2)
where F (s) is the normalized distribution function. Now since the capacitance
at different crystal faces or sites in a polycrystalline electrode is different, we
assume that the time scale of relaxation is distributed through a distribution
function F (s). As in ref. [35] we assume that the RC elements of a single crystal
face in a polycrystalline electrode consisting of several crystal faces and electrode
equivalent circuit model RC elements are distributed in a ladder like fashion and
the distribution function as [35]
F (s) =1
π
[sin[(1− γ)π]
exp(γs) + exp(−γs)− 2 cos[(1− γ)π]
](2.1.3)
Using the normalized condition
∫ ∞−∞
F (s)ds = 1 (2.1.4)
and integrating Equation 2.1.2 we have the Helmholtz layer admittance of the
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polycrystalline electrode as
yH(ω) =1
ZH(ω)=
1
RH
(1− 1
1 + (ιωτ 0H)γ
)(2.1.5)
2.2 Partially Blocking Electrodes
The admittance yh(ω) of HL consisting a Voigt element is
yH(s, ω) =1
zH(ω)=
1
RH
(1− 1
1 +RH yp
)(2.2.1)
where yp is the admittance of the parallel combination of cH and Rct at the surface
site considered and
yp = (Rct)−1 + ιωcH (2.2.2)
Now if the surface consist of sites in which the charge transfer proceeding at any
location of the surface is shunted by capacitance cH pertaining to this location,
we may assume a ladder network of Voigt element whose distribution is given by
Eq. 2.1.3. The corresponding admittance may be written as
yH(ω) =
∫ +∞−∞
yH(s, ω)F (s)ds
=1
RH
[1−
∫ +∞−∞
1
1 +RHR−1ct + ιωτ
0H
F (s)ds
](2.2.3)
On solving the integral we have
yH(ω) =1
ZH=
1
RH
[1− 1
1 +RHR−1ct + (ιωτ
0H)
γ(1 +RHR−1ct )
1−γ
](2.2.4)
When Rct → ∞ the Eq. 2.2.4 reduces to Eq. 2.1.5.
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3 Diffuse Layer Model and Solution
Let’s consider a plane electrode in contact with an electrolyte solution. The general
description of transport is based on conservation of species in the diffuse layer is
given by∂ci∂t
= −(∇.Ni) +Ri (3.0.1)
where ci is the concentration of species i, Ni is the net flux vector for species i,
and Ri is the rate of generation of species i. For a dilute solution, the flux for
any species can be written in terms of contribution from convection, diffusion and
migration as
Ni = vci −Di∇ci − ziuiFci∇ϕ (3.0.2)
The first term represent the contribution from ion drift velocities v when an os-
cillating voltage is applied to the electrode. The second term represent the con-
tribution from diffusion of particle i with diffusion constant Di. The third term
represent contribution from migration when an ion with ionic charge zi moves
with mobility ui. When there is no generation/recombination reaction Ri = 0.
Neglecting the ionic drift and using Einstein equation relating mobility ui and
the diffusion coefficient Di, ui =DikbT
and substituting Ni in the general transport
equation we have the Nernst-Plank equation,
∂ci∂t
= −∇.(−Di∇ci ∓ziDie
kbTci∇ϕ) (3.0.3)
Now the potential distribution is determined by Poisson equation which is
∇2ϕ = −ρϵ=e∑
i ciziϵ
(3.0.4)
where ϕ is a potential, ϵ is the dielectric constant of medium, ρ is the net charge
density and ci is the local concentration ions in the electrolyte. Now assuming
D+ = D− = D and using ρ = e(z+c+ − z−c−) we have the Debye-Falkenhagen
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equation[30] in terms of charge density as
1
D
∂ρ
∂t= (∇2 − κ2)ρ (3.0.5)
where κ−1 is the Debye screening length
κ−1 =
√ϵrϵ0kBT
2NAe2I(3.0.6)
where I = (1/2)∑n
i=1 ciz2i is the ionic strength, kB the Boltzmann’s constant,
NA is the Avogadro number, T the temperature, ϵ0 the permittivity of free space
and ϵr dielectric constant of the solvent. Now we assume near equilibrium local
concentration ions ci is given by Boltzmann distribution
ci = c∞i exp(±zieϕ/kBT ) (3.0.7)
Nonlinear contributions in concentration is ignored for electrochemical techniques
with small external potential perturbations. Now for a symmetrical z+ = z− = z
electrolytes and assuming c∞+ = c∞− = c0, linearizing Eq. 3.0.7 under condition
low applied potential ϕ≪ kBT/ze and substituting in Eq. 3.0.5, we obtained the
linearized Debye-Falkenhagen (LDF) equation for potential ϕ
∂ϕ
∂t= D(∇2 − κ2)ϕ (3.0.8)
The partial differential Eq. 3.0.8 may be looked upon as a analogue of diffusion
equation in terms of potential with a source term κ2ϕ. For a sinusoidal applied
potential at a planar electrode, ϕ(z, t) = ϕ(z) eiωt the complex potential in LDF
equation is simplified to
[∇2 − κ2 (1 + i ωτD)
]ϕ = 0 (3.0.9)
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where τD = 1/κ2D is the Debye time [38], i.e. time taken by a particle to diffuse
the ionic cloud of thickness κ−1. For example, for an ion with D = 10−6cm2/s, the
Debye time for 10−2 M symmetrical 1 : 1 electrolyte is τD ≈ 9× 10−8s. Limiting
high frequency behavior in EDL relaxation is controlled by τD hence Boltzmann
approximations involved in Eq. 3.0.8 will be applicable for ω < 1/τD.
Equation 3.0.9 represent the dynamic picture of the interface, when a sinusoidal
potential is applied at the interface (Fig. 5.2). The current density (j) crossing
an interface is given by Ohm’s law
j = −σ∇ϕ (3.0.10)
where σ is interfacial conductivity of electrolyte at OHP. We assume that the
current arriving at OHP is given by Eq. 3.0.10. This current is equal to the ratio
of potential drop at the OHP (ϕ̃ − ϕ), to the impedance (ZH) of the Helmholtz
layer, i.e. (ϕ̃− ϕ)/ZH where ϕ̃ is the potential of the electrode and ϕ is potential
at and beyond the OHP. Now equating the current on both sides of the Helmholtz
layer we have the Robin (or impedance) boundary condition [25, 39, 40]
−σ(∂ϕ/∂z)|z=0 = (ϕ̃− ϕ(z = 0))/ZH (3.0.11)
where ZH = 1/yH(ω) the impedance of the Helmholtz layer. This boundary
condition is similar to that used by Bazant et al., [25] in context of diffuse layer
charge dynamics due to potential step applied on finite boundaries. Similar bound-
ary conditions have been used by Istkovich et. al., [41] for electrode/solid elec-
trolyte interface. Eq. 3.0.11 is more general boundary condition than one used in
ref. [39] and is relevant for impedance analysis of electric double layer dynamics.
The above boundary condition introduces a complex phenomenological length :
ΛC(ω) = 1/σZH . The quantity | ΛC(ω) |−1 is the effective thickness of interfacial
layer where the electrode is polarized and decides the crossover frequency scales
from purely capacitive regimes to resistive regimes we called it the “polarization
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cbHaL
2 4 60
0.2
0.4
0.6
0.8
1
1.2
log w Hs-1L
H»s
z H»l D
L-1
DHbL
2 4 60
0.2
0.4
0.6
0.8
1
1.2
log w Hs-1LH»s
z H»l D
L-1
Figure 5.3: Plot of |(σzH)lD|−1 vs log ω. (a) effect of concentration. The plotswere generated for 1 : 1 electrolyte with concentration cb (10
−5 M) 1, 3 and 5 and(b) effect of diffusion coefficient. The diffusion coefficient D (10−5 cm2/s) is variedfrom 1, 5 and 15.
length”. It may be rewritten for simplicity as ΛC(ω) = 1/(σ/i ωCH +W ) where
W = σRH is a characteristic resistive length which may be called as modified
Wagner number [40, 42]. The value of | ΛC(ω) |−1 is decided by the compact layer
capacitance CH , resistance RH and the interfacial conductivity of the electrolyte
σ. At low frequency the polarization length is larger than Debye length and at
high frequency it is smaller than Debye length. When the electrode separation
between two parallel electrodes is much larger that Debye length than the finite
boundary can be approximated as semi-infinite boundary value problem. Under
large electrode separation limit, the other bulk boundary condition is ϕ(∞) = 0.
Figure 5.3 shows the plot of (|σZH |lD)−1 vs the log ω. Figure 5.3 (a) shows the
effect of concentration variation of electrolyte. (|σZH |lD)−1 < 1 in all the frequen-
cies except at very dilute concentration case (5µM) where (|σZH |lD)−1 > 1 (b)
shows the effect of diffusion constant variation. In this case also (|σZH |lD)−1 < 1
in all the frequencies except when diffusion coefficient D(10−5cm2/s) = 15 where
(|σZH |lD)−1 > 1. Hence the dynamic polarization length is smaller than con-
centration dependent Debye screening length in most of the cases. The region
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influence by dynamics is within the Debye screening layer thickness. This reason
brings in the notion of independence of the working and counter electrodes in the
electrolytic cell; hence cell impedance is sum of two impedances. But for a cell
with very large area of counter electrode, the dominant cell impedance is from the
working electrode. Hence single working electrode modeling for understanding the
dynamics of electric double layer is sufficient in most electrochemical systems.
4 Solution- Admittance of EDL
The solution of Eq. 3.0.9 along with its boundary constraints is obtained as follows.
The procedure used to determine the solution of the above boundary value problem
is mathematically isomorphic with the problem of quasireversible charge transfer
at an electrode [43]. Now the linearized Debye-Falkenhagen equation with external
sinusoidal perturbation for potential ϕ(z) is
(∂2
∂z2− κ2eff
)ϕ = 0 (4.0.1)
where κeff = κ(1 + iωτD)1/2. Now the solution of Eq.(2.1.1) is
ϕ = Ae−κeff z +Beκeff z (4.0.2)
where the coefficients A and B are determined by the boundary conditions. Using
the semi-infinite (bulk) boundary condition ϕ(∞) = 0 we have B = 0. Using
the surface boundary condition at Helmholtz plane (z = 0) −σ(∂ϕ/∂z)|z=0 =
(ϕ̃− ϕ)/ZH we have
A =ϕ(z = 0)
1 + σZHκ√1 + iωτD
(4.0.3)
Now the potential is
ϕ =
(ϕ(z = 0)
1 + σZH κ√1 + iωτD
)e−κ
√1+iωτD z (4.0.4)
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The total admittance which is the ratio of total current by the potential applied
is given by
Y (ω) = −(σ/ϕ(z = 0))∫
(dϕ/dz)|z=0dS0 (4.0.5)
where dS0 ≡ dx dy is surface element at the OHP plane. Substituting the value
of Eq. 4.0.4 in 2.1.5 we have the total Debye-Falkenhagen admittance Y (ω) for a
polycrystalline heterogeneous electrode
Y (ω) =A0
ZDF (ω)=
A0ZH + ZG
(4.0.6)
where A0 is the area of electrode. The impedance simply looks like series addition
of compact layer admittance ZH and the diffuse layer impedance
ZG =1
σκ√1 + iωτD
=Z0
D1/2√ωD + iω
(4.0.7)
where Z0 = (D/σ) and ωD = 1/τD = κ2D. Equation 4.0.7 looks as reminiscence
of Gerischer impedance or the finite length Warburg type (FLWT) impedance[44].
At high frequencies, the ratio the Debye screening length and to the ion diffusion
length is large so (κ−1/√D/ω)2 ≫ 1 and Eq. 4.0.7 gives the usual Warburg
impedance [45, 46] and is
ZW =Z0√iω D
(4.0.8)
Thus we see that Eq. 4.0.6 essentially general enough and consists both the
Gerischer and Warburg component. It is worth noting that Eq. 4.0.6 is basically
different from pure Gerischer or finite-length-Warburg type and pure Warburg
impedance in that it consist the Helmholtz layer impedance usually neglected.
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5 Limiting Laws
In order to understand behavior of admittance given by Eq. 5.0.9, we consider the
following frequency regimes.
5.1 Low Frequency Limits
At low frequency κ−1 < | ΛC(ω) |−1, from Eq. 4.0.6 we have
Y (ω) =
(A0ZH
)[1− ΛC(ω)
κ+
(ΛC(ω)
κ
)2+ · · ·
](5.1.1)
The leading contribution in admittance originates from the Helmholtz layer and
its characteristic relaxation frequency is ωH = (CHRH)−1. When the frequency
is ω < ωH , Eq. 5.1.1 shows that the EDL behaves like an ideally polarizable
blocking electrode.
5.2 High Frequency Limits
At high frequency | ΛC(ω) |−1< κ−1, from Eq. 4.0.6 we have
Y (ω) = A0σκ
[1− κ
ΛC(ω)+
(κ
ΛC(ω)
)2+ · · ·
](5.2.1)
When the frequency is higher than the characteristic diffuse layer relaxation fre-
quency, ωD < ω, Eq. 5.2.1 shows that the leading order contribution to the inter-
facial admittance comes from the diffuse layer resistance. This observation will
allow us to see the effects of concentration on the admittance of planar elec-
trode. The admittance is resistive and the characteristic frequency at which
the EDL relaxes is the characteristic frequency at which diffuse layer relaxes,
ωD = (σκ)/CD = σ/(ϵ0ϵD). But σ/(ϵ0ϵD), is also a characteristic conductivity
relaxation frequency [16].
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At the crossover region, the frequency range is confined to ωH < ω < ωD and
| ΛC(ω) |−1 is comparable to κ−1. In this case the crossover frequency is resultant
of both processes- relaxation of Helmholtz layer and the relaxation of diffuse layer;
and we expect the dispersions in electrochemical response of EDL.
Now expanding Eq. 4.0.6 in the ratio of the Debye screening length to ion
diffusion length, we have
Y (ω) = A0κσ
[1
1 + κσZH+
i
2(1 + κσZH)2
( ωκ2D
)2+
1 + 3κσZH8(1 + κσZH)3
( ωκ2D
)4− · · ·
]. (5.2.2)
At high frequencies, κ−1/√D/ω > 1, then the higher terms will contribute. Gun-
ning et al. [23] obtained the impedance depended on zeta potential of the diffuse
double layer. The theory developed here differs in the inclusion of the Stern layer
along with the diffuse double layer. Our theory focuses on the boundary constraint
at the OHP rather than the “slipping plane” [23].
Alternatively we can calculate the impedance of EDL from Eq. 4.0.6 as
Z =1
Y(5.2.3)
Also the real Zre and the imaginary Zim component are obtained as
Zre =Yre
Y 2re + Y2im
(5.2.4)
Zim = −Yim
Y 2re + Y2im
(5.2.5)
Consequently the magnitude of impedance is obtained as
|Z(ω)| =√Z2re + Z
2im (5.2.6)
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The phase of the EDL is obtained as
ϕ(ω) = tan−1(ZimZre
)(5.2.7)
6 Complex Capacitance
It is possible to obtained the dynamic capacitive response through the relation
between capacitance and impedance as
C(ω) =1
ιωZ(ω)(6.0.1)
Now the impedance Z(ω) can be written under its complex form as
Z(ω) = Zre(ω) + Zim(ω) (6.0.2)
From Eq. 6.0.1 and 6.0.2 we have
C(ω) =1
ω(ιZre(ω)− Zim(ω))=
−(Zim(ω) + ιZre(ω))ω|Z(ω)|2
(6.0.3)
Now we define the complex capacitance as
C(ω) = Cre(ω)− ιCim(ω) (6.0.4)
Comparing Eq. 6.0.3 and 6.0.3 we have
Cre(ω) =Zim(ω)
ω|Z(ω)|2(6.0.5)
and
Cim(ω) = −Zre(ω)
ω|Z(ω)|2(6.0.6)
where Cre(ω) is the real part of the complex capacitance of EDL. The low frequency
value of Cre(ω) corresponds to the capacitance of the interface. Cim(ω) is the
203
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imaginary part of the complex capacitance of EDL. It corresponds to the energy
losses due to process like rotation of molecules, solvent resistance and dielectric
losses.
7 Results and Discussions
7.1 Effect of Electrode Heterogeneity on the Debye-Falkenhagen
Dynamics of EDL
Figure 5.4(a) shows the effect of heterogeneity of electrode on the log-log plot
of magnitude of impedance |Z(ω)| and frequency ω. The dotted line in the plot
represent the ideal smooth electrode where the heterogeneity parameter γ = 1. As
we increase γ from 0.94, 0.96, 0.98 to 1 the magnitude of impedance is increase
with change in the slope of the impedance plot. This increase is more in the low
frequencies. The plots clearly indicate the right trends observed in experiments
[33, 37]. At high frequencies all the plots merge to the ideal smooth response and
plateau off indicating the influence of solution resistance.
Figure 5.4(b) shows the effect of heterogeneity of electrode on the phase angle
Φ vs log ω. As we increase γ from 0.94 to 1 there is a gradual lift of phase Φ <
90◦ to Φ = 90◦.
Figure 5.4(c) shows the effect heterogeneity of electrode on the real part of
capacitance. The dotted line represent the capacitance of ideally smooth electrode
where the heterogeneity parameter γ =1. We see that as the heterogeneity of the
electrode is increase by lowering the value of γ the real part of capacitance is
increase. The slope of the capacitance plots is increased from that of the pure
ideal smooth electrode.
204
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HaL
g
-2 0 2 4 60
2
4
6
8
log Hwês-1L
logH»
ZȐW
cm2 L
HbL
g
-2 0 2 4 60
20
40
60
80
100
log Hwês-1L
FHw
Lêde
gree
gHcL
-2 0 2 4 6-1
-0.5
0
0.5
1
1.5
2
log Hwês-1L
log
CreHm
F êcm
2 L
Figure 5.4: Effect of surface heterogeneity on (a) impedance (b) phase and (c)capacitance. The plots were generated for 1 : 1 electrolyte whose concentration is10−1 M, using D = 5× 10−5 cm2/s, ϵ0 = 8.854 ×10−14 (F/cm), ϵH = 3, rH= 0.5nm, RH = 10 Ω-cm
2, Rct = 108 Ω-cm2 and area of electrode A0 = 1 cm
2. Theheterogeneity parameter γ used in the curves are 0.94, 0.96, 0.98, 1.
205
-
7.2 Effect of Charge Transfer Resistance on the Debye-
Falkenhagen Dynamics of EDL formed at Electrode
with Heterogeneity
Figure 5.5(a) shows the effect of charge transfer resistance (Rct) on the log-log
plot of magnitude of impedance |Z(ω)| and frequency ω. At low value of Rct,
the impedance response shows three regimes: (1) low frequency characterized by
constant impedance (plateau) with slope zero, (2) intermediate regime where the
slope of impedance curve is < 1 and (3) high frequency regime with constant
impedance with slope zero indicating purely resistive controlled regime. Increase
in the value of Rct the response of the electrode becomes more and more blocking
and the resistive plateau in the low frequency regime disappears and merges with
the blocking response.
Figure 5.5(b) shows the effect of Rct on the phase angle Φ vs log ω. At low fre-
quency the phase show resistive Rct controlled regime where phase Φ → 0◦ as the
value of Rct is lowered. At intermediate frequency the phase value Φ → 90◦ indi-
cating a capacitive nature but less that 90◦, because of CPE behavior is observed
which decrease to a zero value at high frequencies indicating resistive nature. The
overall behavior creates a phase hump (maximum) in phase vs frequency plot.
7.3 Effect of Concentration and Diffusion Coefficient on
Debye-Falkenhagen Impedance and Phase: Gerischer
and Finite-Length-Warburg Behavior
Figures 5.6 illustrates the effect of concentration (a & b) and diffusion coefficient
variation (c & d) on impedance and phase. The curves were plotted using Eq.
4.0.6. At low concentration we observed the impedance have three regimes. At low
frequencies we observed the CPE behavior with slope of impedance curve < −1,
at intermediate frequencies a step in impedance response, where slope changes
206
-
HaL
Rct
-2 0 2 4 61
2
3
4
5
6
7
8
log Hwês-1L
logH»
ZȐW
cm2 L
Rct
HbL
-2 0 2 4 60
20
40
60
80
100
log Hwês-1L
FHw
LFigure 5.5: Effect of charge transfer resistance on (a) impedance and (b) phase.The plots were generated for 1 : 1 electrolyte whose concentration is 10−3 M, usingD = 5×10−5 cm2/s, ϵ0 = 8.854 ×10−14 (F/cm), ϵH = 3, rH= 0.5 nm, RH = 50 Ω-cm2, area of electrode A0 = 1 cm
2 and heterogeneity parameter γ = 0.98. Thecharge transfer resistance Rct(Ω-cm
2) is varied as 105, 106, 107.
from a value of zero to (−1/2) indicating the Gerischer and the finite-length-
Warburg type as predicted by Eqs. 4.0.6, 4.0.7 and 4.0.8 and at large frequencies,
a purely resistive regime is observed where the slope of impedance curve is zero.
The phase shows a much rich variety of regimes. At low frequencies Φ → 90◦, but
at intermediate frequencies the phase show a hump (maximum at frequency ωD)
where Φ < 45◦. This behavior indicates the Gerischer or finite-length-Warburg
type behavior and a large frequencies a resistive regime where Φ → 0◦.
As we vary the concentration from low to high concentration, the intermediate
impedance step regimes is gradually decreased and finally becomes indistinguish-
able with the resistive regime. This may be understood as the Debye-Hückel
screening length is decrease by increasing the concentration of electrolyte the dy-
namics of diffuse layer relaxation becomes faster and faster and correspondingly
the frequency (ωD) shifts to larger frequencies. This behavior is more clearly seen
in phase plots with a phase hump.
Similarly at low diffusion coefficient (c & d) we also observe three impedance
207
-
cb
HaL
0 1 2 3 4 5 6 70
1
2
3
4
5
logHwês-1L
logH»Z»êW
cm2 L
cb
HbL
0 1 2 3 4 5 6 70
20
40
60
80
logHwês-1L
FHw
êdeg
reeL
D
HcL
0 1 2 3 4 5 6 70
1
2
3
4
5
logHwês-1L
logH»Z»êW
cm2 L
D
HdL
0 1 2 3 4 5 6 70
20
40
60
80
logHwês-1L
FHw
êdeg
reeL
Figure 5.6: Effect of concentration variation is shown in (a) & (b). The con-centration is varied as cb(µ M) = 5, 10, 100. Effect of electrolyte conductivityvariation is shown in (c) and (d). The diffusion coefficient D (cm2/s) is variedfrom 5× 10−5, 5× 10−7, 5× 10−9. The graphs are plotted using ϵ0 = 8.854 ×10−14(F/cm), cb = 10
−6 M, ϵH = 3.5, rH= 0.53 nm, RH = 20Ω-cm2, Rct = 10
8Ω-cm2,γ = 0.98, A = 0.038cm2.
208
-
regimes: CPE regime, Gerischer and finite-length-Warburg regime and resistive
regime as observed in (a & b). But diffusion have a large effect on both the
impedance and phase behavior. The diffusion coefficient (D) is varied to mimic
aqueous to solid electrolytes. The D ∼ 10−8 to 10−10 s, typical of solid elec-
trolytes, show presence of Gerischer or finite-length-Warburg behavior with at
step in impedance and a hump approaching to the Warburg phase in impedance
plots. At low value of (D = 10−9 s), the impedance plot shows that Gerischer
regime (plateau region where slope of impedance curve is zero) is reduced and
switch to Warburg response ( impedance regime where slope is (1/2), occur at
much lower frequency. This is also more clearly seen in phase plot (d) where a
plateau (almost constant value of Φ < 45◦) indicating Warburg behavior is ob-
served which gradually decreases and almost vanish at larger D values. This is
because as we increase the value of D, the kinetics of ion diffusion from a Debye
sphere (with size κ−1) becomes faster and faster leading to smaller τD and corre-
spondingly the diffuse layer relaxes at larger frequencies. Increasing the value of
D we observed that the magnitude of phase in the intermediate (Gerischer and
finite-length-Warburg) regime is gradually decreased and almost vanish at large
D making the response indistinguishable to the resistive behavior.
7.4 Effect of Concentration and Diffusion Coefficient on
Debye-Falkenhagen Capacitance
Figure 5.7(a & b) shows the real Cre(ω) (solid lines) and imaginary Cim(ω) (dashed
lines) part of Debye-Falkenhagen capacitance change vs. log ω with change in
(a) concentration and (b) diffusion coefficient of the electrolyte. The plots were
obtained employing Eqs. 6.0.5 and 6.0.6.
As we decrease the concentration of electrolyte the we observed that the fre-
quency of switchover from capacitive to resistive regime is shifted to progressively
higher frequencies. Cre(ω) sharply increases when the frequency ω < ωD and
tends to becomes frequency independent when ω < ωH . Thus for frequencies
209
-
cb
HaL
1 2 3 4 5 60
0.1
0.2
logHwês-1L
Creêm
F
Cimêm
F
HbL
D
1 2 3 4 5 60
0.1
0.2
logHwês-1L
Creêm
F
Cimêm
F
Figure 5.7: Effect of concentration variation is shown in (a). The concentration cbis varied from 10−4, 10−3 to 10−2 M. Effect of diffusion coefficient is shown in (b).The diffusion coefficient D (10−5 cm2/s) is varied from 1, 2 and 4. The graphsare plotted using ϵ0 = 8.854 ×10−14 (F/cm), ϵH = 3.5, rH= 0.53 nm and area ofelectrode A0 = 0.038 cm
2.
ωH < ω < ωD we observed capacitance dispersion. This is the characteristic
of the electrode/electrolyte interface where capacitance dispersions are observed.
These plots clearly shows that the low frequency side of capacitance curves is
dominated by the properties of compact HL.
Correspondingly the imaginary part of capacitance (shown by dashed lines)
represent the capacitive dispersion related to energy losses as function of concen-
tration. The plots shows the imaginary part of capacitance goes through max-
imum at a frequency ωm. This time constant has been described as dielectric
relaxation time characteristic of the electrolyte [49]. This relaxation time defines
the limit between the predominantly resistive behavior at frequencies higher than
ωD and capacitive behavior at lower frequencies. This is the single relaxation time
corresponds to the value of 45◦ for the phase angle Φ. As the concentration of
electrolyte is diluted, ωm is to higher frequencies.
Figure 5.7(b) shows the effect of change of conductivity through diffusion co-
efficient (D). As the value of D is lowered we observed that switchover frequency
210
-
ò ò ò ò òò
òò
ò
ò
òò ò
á
á
á
á
á
á áá á á ááá
æ æ æ æ æ æææ
æ
æ
æ
æ
æ
æ
ææ ææææææ
HaLæ 3.2 M LiCl Hglycerol+ waterL
HbLá 3.2 M LiCl HglycerolL
HcL ò 0.001 M aq. NH4F
2 3 4 5 6
0.0
0.2
0.4
log Hf êHzL
Creêm
F
Figure 5.8: Comparison of theory with experiment. Data were taken from ref.[8, 9]. Physical parameters used to fit the data are listed in Table 5.1.
ωm is shifted at lower frequency and the time of Debye-Falkenhagen dynamics of
EDL becomes slower and slower, with lesser and lesser capacitive energy losses
(as seen in decreasing magnitude of Cim with decrease in D). The low frequency
of Debye-Falkenhagen dynamic capacitance is affect relatively more compared to
high frequency capacitance.
7.5 Comparison of the Experimental Capacitance Disper-
sion Data with Theory
Figure 5.8 shows the theoretical plots fits with experimental capacitance dispersion
data for Hg/electrolyte interface. The plots were generating using Eq. 6.0.5. The
values of various physical quantities used to generate the plots are listed in Table
5.1. The data set (a) represent the 1.3 M LiCl in glycerol-water and (b) 3.2 M
LiCl in glycerol system. Both the capacitance dispersion data taken from ref. [9].
The data set (c) represent the 0.001 M aq. NH4F system and was taken from ref.
211
-
[8].
As can be seen from the Table 5.1, we employ the hydrated ion size of ion as
the thickness of the HL (rH) [6]. The dielectric constant, ϵH of HL is also listed.
The typical HL values of dielectric constant ϵH are much lower than the bulk
dielectric values ϵD. In case of data set (a) and (b) we found that the diffusion
constant D (cm2/s) are 15 and 0.06 respectively. These orders are in typical range
of glycerol-water [50]. These values may be understood as follows: the diffusion
constant of data set (b) is much higher than data set (a) because set (b) is much
more viscous than compared to set (a) which consist of 5.7 % (vol) water. Thus
the addition of water in fact increase the conductivity of the electrolyte. Since
ions in glycerol has lower mobility compared to water, this factor leads to slowing
down of the ion movement and creates a delay in the EDL response, which can
be seen as a frequency dependent phenomena. This results in the lowering of
frequency dependence when the electrolyte is change from glycerol-water to pure
glycerol and because of delay in ion movement, the capacitive response deviates
from the ideal behavior and leads to capacitance dispersion.
Table 5.1: Results of fitting the capacitance dispersion data of mercury electrode.Here is ϵH dielectric constant and rH (nm) is effective thickness (corresponding tohydrated ion size) of Helmholtz layer. The effective diffusion coefficient D (10−6
cm2/sec) and Helmholtz layer resistance RH (Ω-cm2) used as fitting values are
listed.
Data E S cb ϵH ϵD rH A0 D RHFig.4 (a) LiCl gly+H2O 3.2 6 42.5
a 0.37b 25 15 5.5Fig.4 (b) LiCl gly 3.2 6 42.5a 0.37 b 25 0.06 13.95Fig.4 (c) NH4F H2O 10
−3 3.5 78.6 0.53b 38 18.3 26.65
E- Electrolyte, S- solvent, gly- Glycerol, cb- molar concentration, A0 (10−3)- area of electrode in
cm2 , a- [47], b- [48]
In agreement with the theory, the data shows two crossover frequencies. The
effect of electrolyte conductivity on capacitance is exactly same as in experiment.
Increasing the electrolyte conductivity by increasing diffusion coefficient, increases
212
-
the crossover frequency (ωD) of capacitance dispersion.
8 Conclusions
In this paper we present a phenomenological theory of electrode polarization. We
have accounted the EDL interface consisting of compact HL and diffuse layer. By
using the mixed (Robin) boundary and the Poisson-Nersnt-Planck equation for the
transport of ions we have presented analytical equations for admittance/impedance
and complex capacitance of EDL. The treatment presented here is suited for
the admittance/impedance analysis of electrode polarization. The validity of the
model is restricted to cases where electrode separation length is much greater than
the Debye screening length. In other words the dynamics of the EDL is consid-
ered to be independent of the counter electrode constituting an electrochemical
cell. Further the linearization of the of concentration for low applied potential im-
plies its application to steady state or near equilibrium conditions. Consequently
the model is very much simplified, but able to predict the dynamics of electrode
polarization leading to capacitance dispersion.
The following conclusions can be drawn for Debye-Falkenhagen dynamics of
electrode with heterogeneity from the model developed here:
1. The dynamic response of electrical double layer near an electrode is strongly
dependent on the phenomenological polarization length which is smaller and
comparable to Debye length at low and high frequency, respectively. Hence
working and counter electrode act independently in most experimental cells
where electrodes separation is much larger than the Debye length.
2. Helmholtz layer is most affected by the surface heterogeneity while diffuse
layer remains unaffected. Hence influence of heterogeneity included through
dispersion in relaxation times in Helmholtz layer.
3. The Helmholtz and diffused layers relaxation frequencies divide the whole
213
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frequency range into three regimes. The low frequency regime is character-
ized by the capacitance of EDL is that of Helmholtz layer. Whereas the high
frequency is characterized by the capacitance of the EDL is that of diffuse
layer. For concentrated electrolytes the crossover frequency is indicated by a
transition from capacitance to resistive behavior and an S-shaped crossover
region in the capacitance plot. The capacitance dispersion that arises in the
crossover regime is found to depend on the concentration and conductivity
of the electrolyte.
4. For low concentration (highly dilute) and small diffusion coefficient elec-
trolytes theory predicts Gerischer and finite-length-Warburg type behavior
and a step in impedance plot in the intermediate regime along with CPE
behavior at low frequency and resistive behavior at high frequencies.
5. Decrease in the concentration of electrolyte results in the shift of the crossover
region progressively to lower frequencies. The capacitance dispersion is seen
for frequencies where ωH < ω < ωD.
6. The theory agrees with the capacitance dispersion data which account for the
variation in diffusion coefficient and Debye length in the mercury/electrolyte
for an aqueous, glycerol and glycerol-water system.
Finally, we conclude taking into account the HL and the diffuse layer and
employing the PNP transport equation leads to insightful understanding to the
phenomena of ubiquitous electrode polarization in EDL interface. The problem
discussed here and its boundary constraint can be further generalized to account
separate contributions of inner and outer Helmholtz layers as well effect of the
arbitrary profile electrodes and their details will be discussed elsewhere.
214
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Bibliography
[1] R. Karnik, R. Fan, M. Yue, D. Li, P. Yang, A. Majumdar, Nano. Lett, 5
(2005) 943.
[2] P. M. Biesheuvel, B. van Limpt, A. van der Wal, J. Phys. Chem. C 113 (2009)
5636.
[3] C.R. Mariappan, T.P. Heins, B. Roling, Solid State Ionics, 181 (2010) 859.
[4] M. Park, X. Zhang, M. Chung, G. B. Less, A.M Sastry, J. Power Sources
195(2010) 7904.
[5] P.M. Biesheuvel, A.A. Franco, M.Z. Bazant, J. Electrochem. Soc. 156 (2009)
B225.
[6] D. C. Grahame, Chem. Rev., 41 (1947) 441.
[7] Z. Kerner and T. Pajkossy, Electrochim. Acta 46 (2000) 207.
[8] R. D Amstrong, W. P Race, J. Electroanal. Chem. 33 (1971) 285.
[9] R. D Amstrong, W. P. Race J. Electroanal. Chem. 34 (1972) 244.
[10] J. R. Macdonald, L. R. Evangelista, E. K. Lenzi, G. Barbero, J. Phys. Chem.
C 115 (2011) 7648.
[11] A. Serghei, M. Tress, J. R. Sangoro, F. Kremer, Phys. Rev B 80 (2009)
184301.
215
-
[12] J. R. Macdonald, J. App. Phy 107 (2010) 101101.
[13] H. Sanabria, J. H. Miller Jr., Phy. Rev E 74, (2006) 051505.
[14] G. Barbero, A. L. Alexe-Ionescu, Liquid Crystal, 32 ( 2005) 943.
[15] R. J. Klein, S. Zhang, S. Dou, B. H. Jones, R. H. Colby, J. Runta, J. Chem.
Phys. 124 (2006). 144903.
[16] E. Barsoukov, J. R Macdonald, Impedance Spectroscopy Theory, Experiment,
and Applications; Wiley-Interscience, Second Edition, 2005.
[17] H.P. Schwan, Biophysik, 3 (1966) 181.
[18] J. D. Ferry, J. Chem. Phy. 16 (1948) 737.
[19] J. R. Macdonald, J. Chem. Phy 29 (1958) 1346.
[20] J. R. Macdonald, Phy. Rev 92 (1953) 4.
[21] J. R. Macdonald, J. Chem. Phy 54 (1971) 2026.
[22] R. P. Buck, J. Electroanal. Chem. 23 ( 1969) 219.
[23] J. Gunning, D. Y. C. Chan, L. R. White, J. Colloid. Interface Sci. 170 (1995)
522.
[24] H. Zhou, M. A. Preston, R. D. Tilton, L. R. White, J. Colloid. Interface Sci.
292 (2005) 277.
[25] M. Z. Bazant, K. Thornton, A. Ajdari Phys. Rev. E. 70 (2004) 021506.
[26] F. Beunis, F. Strubbe, M. Marescaux, J. Beeckman, K. Neyts, A. R. M.
Verschueren, Phys. Rev. E. 78 (2008) 011502.
[27] M. Marescaux, F. Beunis, F. Strubbe, B. Verboven, K. Neyts, Phys. Rev. E.
79 (2009) 011502.
216
-
[28] J. R. Macdonald, Trans. Faraday Soc. 66 (1970) 943.
[29] J. R. Macdonald, J. Phys.: Condens. Matter 22 (2010) 495101.
[30] H Falkenhagen, Electrolytes , Oxford University, Oxford 1934.
[31] A N. Frumkin. J. Res. Inst. Catal. Hokkaido Univ., 15 (1967) 61.
[32] T. Pajkossy, J. Electroanal. Chemistry, 364 (1994) 111.
[33] T. Pajkossy, Solid State Ionics 176 (2005) 1997.
[34] Z. Kerner, T. Pajkossy, J Electroanal. Chem. 448 (1998) 139.
[35] G. J. Brug, A. L. G. V. D. Eeden, M. S.-Rehbach, J. H. Sluyters, J Elec-
troanal. Chem. 176 (1984) 275.
[36] A. Eftekhari, M. Kazemzad, M. Keyanpour-Rad, Applied Surf. Science 239
(2005) 311.
[37] Z. Kerner, T. Pajkossy, J Electrochim. Acta 46 (2000) 207.
[38] J. R. Macdonald, Phy. Rev. 91 (1953) 412.
[39] R. Ball, M. Blunt, J. Phys. A : Math. Gen. 21 (1988) 197.
[40] E. Bosco, Engineering Analysis with Boundary Elements 32 ( 2008) 770.
[41] E. M. Itskovich, A. A. Kornyshev, and M. A. Vorotyntsev, Phys. Status Solidi
A 39, (1977) 229.
[42] C. Wagner, J. Elect. Soc. 98 (1951) 116.
[43] R. Kant, S.K. Rangarajan, J. Electroanal. Chem. 396 (1995) 285.
[44] D. R. Franceschetti, J. R. Macdonald, R. P Buck, J. Electrochem. Soc. 138
(1991) 1368.
[45] E. Warburg, Ann. Phys. Chem. (Ser. 3) 67 (1899) 493.
217
-
[46] E. Warburg, Ann. Phys. (Leipzig) 6 (1901) 125.
[47] R. A. Robinson, R. H. Stokes, Electrolyte Solution Butterworths, Sec. Edi-
tion, 1968.
[48] E. R. Nightingale. Jr J. Phy. Chem. 63 (1959) 1381.
[49] K. S. Cole, R. H. Cole, J. Chem. Phys. 9 (1941), 341.
[50] J. E. Evanoff, W. E Harris, Can. J. Chem. 56 (1978), 574.
218