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

183

184

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.

185

186

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

191

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

197

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].

201

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

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

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

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

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