Impedance spectroscopy study of aluminum electrocrystallization from basic molten salt (AlCl3-NaCl-KCl)

Article abstract of DOI:10.1016/j.electacta.2007.02.068

Aluminum electrocrystallization is studied by means of electrochemical impedance spectroscopy (EIS). A model based on random birth and deterministic growth of monolayers is proposed, in which the edges are assumed to follow a propagation law. The high frequency impedance data show charge transfer reaction of AlCl₄^- reduction while the low frequency features signifies the growth mode of deposits. The inductive response observed in the course of polycrystalline deposition reflects the activation of electrode area while a capacitive loop appears in regular growth. Parameters of impedance model in this system can be calculated from the fitting of experimental data to the Faradaic impedance function derived theoretically. The physical parameters of this function are analyzed by means of the dependence of simulated EIS spectra on kinetic parameters.

Full text of DOI:10.1016/j.electacta.2007.02.068

Electrochimica Acta 52 (2007) 5437–5443
Impedance spectroscopy study of aluminum electrocrystallization
from basic molten salt (AlCl3–NaCl–KCl)
M. Jafariana,∗, F. Gobalb, I. Danaeea, M.G. Mahjania
a Departement of Chemistry, K.N. Toosi University of Technology, P.O. Box 15875-4416, Tehran, Iran
b Departement of chemistry, Sharif University of Technology, P.O. Box 11365-9516, Tehran, Iran
Received 24 September 2006; received in revised form 7 February 2007; accepted 27 February 2007
Available online 4 March 2007
Abstract
Aluminum electrocrystallization is studied by means of electrochemical impedance spectroscopy (EIS). A model based on random birth and
deterministic growth of monolayers is proposed, in which the edges are assumed to follow a propagation law. The high frequency impedance data
show charge transfer reaction of AlCl4− reduction while the low frequency features signifies the growth mode of deposits. The inductive response
observed in the course of polycrystalline deposition reflects the activation of electrode area while a capacitive loop appears in regular growth.
Parameters of impedance model in this system can be calculated from the fitting of experimental data to the Faradaic impedance function derived
theoretically. The physical parameters of this function are analyzed by means of the dependence of simulated EIS spectra on kinetic parameters.
© 2007 Elsevier Ltd. All rights reserved.
Keywords: Impedance; Electrocrystallization; Nucleation; Growth; Equivalent circuit
1. Introduction
which has proved to be useful in both applied and fundamental
electrochemical studied [3–5] as well as in other disciplines [6].
The power of the technique is its ability to distinguish among
the interfacial processes with different time constants [7]. EIS
is a common and useful technique for studying metals electro-
crystallization [8–12]. One of its advantages is the possibility
of studying the relaxation of electrode surface from adsorbed
species and growth of layers. The impedance spectra of the
deposition processes of metals usually show one or several
relaxations (capacitive or inductive) [13]. Various processes are
generally involved in metal elecrodeposition: ion transport, ion
adsorption, multi-step charge transfer, nucleation and growth
[14–16]. Hence any of the above processes can limit the deposi-
tion rate and must be correctly incorporated in a model for metal
deposition. The surface diffusion of Adatoms can also be a
determining step in the overall mechanism of electrocrystalliza-
tion (Adatoms model) [17–20]. In a number of metals deposition
studies, EIS has provided convincing evidence for ion diffusion
and charge transfer [21–24]. Two dimensional nucleation and
growth and the interactions between growth centers have been
extensively studied [25–28]. The impedance characteristic of the
response of the deposition successive layers to potential steps of
small amplitude have also been studied [29,30]. It is known that
the formation and development of growth centers during elec-
Aluminum has a wide range of applications, due to a com-
bination of three outstanding properties: it is inexpensive, it has
a low density, and it is resistant towards corrosion because it
is always covered with a passivating native oxide film. On the
other hand, the development of high energy density secondary
batteries continues to be an active research area. Aluminum
covered electrode is an attractive electrode because of its high
electrochemical equivalent weight (2.98 Ah/g) leading to high
theoretical gravimetric and volumetric energy densities [1].
In our earlier work electrocrystallization of aluminum onto
graphite from a molten electrolyte containing AlCl3–NaCl–KCl
was studied by the method of current transient. In the early
stage of the deposition and at low cathodic potentials, two-
dimensional (2D) nucleation and layer-by-layer growth with
some overlapping followed the initial double layer charging.
The processes manifested themselves as peaks on decaying
chronoamperograms [2]. Electrochemical impedance spec-
troscopy (EIS) is an example of frequency response analysis
∗
Corresponding author.
E-mail address: mjafarian@kntu.ac.ir (M. Jafarian).
0013-4686/$ – see front matter © 2007 Elsevier Ltd. All rights reserved.
doi:10.1016/j.electacta.2007.02.068

5438
M. Jafarian et al. / Electrochimica Acta 52 (2007) 5437–5443
trodeposition can give rise to a variation of the active area of the
electrode and the subsequent changes of the process overpoten-
tial. It has been shown that the relaxation of the active area can
generate either an inductive or a capacitive loop characterized
by distributed time constants [31]. From a different approach
based on the assumption of successive monolayers randomly
generated and obeying a given ageing law, an analytical
expression for the electrode impedance has been derived [31].
The impedance model construction is frequently the final
aim of an experimental impedance study [32]. As the aim of
impedance modeling is knowledge enhancement, the applied
models should possess physical significance, i.e., they should
correspond to the structure and the properties of processes taking
place in the system under study [32]. The parameters identifi-
cation can be made by using these impedance models that gives
important information about the influence of the physical param-
eters. The identification procedures are in principle numerical
computation techniques and, therefore, sophisticated software
is necessary for this calculation [33]. Electrodeposition of a
number metals have been studied by impedance measurements
[34–38].
quardt method for optimization of functions and Macdonald
weighting for the real and imaginary parts of the impedance
[42].
3. Result and discussion
Fig. 1 presents a cyclic voltammogram obtained in the
potential range 0.6 to −1.2 V/Al at 140 ◦C (without IR drop cor-
rection) with scan rate of 60 mV s−1. The electroactive entity,
probably AlCl4−, starts to reduce at −0.81 V/Al. In the reverse
scan the cross-over loop signifying nucleation/growth [41] in the
course of a cathodic scan is accompanied by the anodic dissolu-
tion peak of Al at −0.27 V/Al. Subtracting the uncompensated
resistance did not affect the on-set of deposition (stripping)
although surely affected the slopes and peaks positions. The
high negative overpotential for deposition is largely due to the
energy required to initiate nucleation while the negative on-set of
stripping is traced to unstable structure of the deposit compared
to more compact bulk Al.
Fig. 2a and b presents complex plane plots recorded
at various applied cathodic potential steps in the range of
−0.82 to −0.86 V/Al. The complex plane plots exhibit fast
electron-transfer and two relaxation processes characterized by
distributed time constant. At high frequency, the distorted capac-
itive loop is related−to the charge transfer resistance for one step
reduction of AlCl4 in parallel with double-layer capacitance.
At low frequency distorted capacitive and inductive loops are
observed which are characteristics of nucleation and growth of
Al [31]. The processes give rise to change in the active area of
the substrate surface.
The aim of this work is the analysis of impedance character-
istics of the aluminum electrocrystallization process in a basic
molten salt (AlCl3–NaCl–KCl) [2]. The analysis of the theo-
retical impedance function provides important information on
the process parameters. This information allows the EIS spec-
trum simulation and, therefore, predicts the system behavior
with regard to the variation of the experimental conditions.
2. Experimental
Materials used in this work were analytical grade of Merck
origin. AlCl3 was redistilled while NaCl and KCl were dried
at 300 ◦C prior to use. Fused electrolyte with the composition
of 66–20–14 wt.% (AlCl3–NaCl–KCl) which is equivalent to
48.2–33.4–18.4 mol% had the melting point starting at around
115 ◦C and terminating nearly at 120 ◦C as measured in this
laboratory. The molten electrolyte was a one phase clear liq-
uid at 140 ◦C where electrochemical studies were performed.
Handling of the materials and the procedures concerning the
preparation of the fused electrolyte were much the same as
reported elsewhere [39,40,2].
3.1. Faradaic impedance function
On the growth sites Me* provided by nucleation, metal depo-
sition occurs through electron transfer:
Me ∗ + Men+ + ne ↔ Me + Me∗
characterized by a current density i given by
i = i0[e−αnFη/RT − e(α−1)nFη/RT
]
(1)
The experiments were carried out at 140 ◦C in a conven-
tional three-electrode cell with a hand polished graphite rod with
the surface area of 0.50 cm2 forming the working electrode. Its
potential was monitored against an aluminum (99.999% purity)
reference electrode placed inside the melt. A large graphite rod
was used as the counter electrode.
TheelectrochemicalcellwaspoweredbyEG&Gmodel273A
potentiostat/galvanostat coupled with Solartron 1255 frequency
response analyzer run by a PC through M270 and M398 com-
mercial software. The methods of cyclic voltammetry (CV) and
impedance spectroscopy (EIS) were employed. The electrical
impedance is calculated without subtracting the uncompensated
resistance and the double layer capacitance. Fitting of exper-
imental data to the proposed theoretical models was done by
means of home written least square software based on the Mar-
Fig. 1. Cyclic voltammogram of the graphite electrode in basic molten salt
(140 ◦C) with (1) and without (2) IR drop correction. The scan began at 0.6 V/Al
with a scan rate of 60 mV s−1
.

M. Jafarian et al. / Electrochimica Acta 52 (2007) 5437–5443
5439
Then the steady-state current and the Faradaic impedance ZF
have been shown [31] to take the following forms:
I = ihl0mτ
ZF−1 = R−t 1 + ihl0m
(5)
(6)
ꢃ
ꢄ
ꢀm
ꢀv
−
F(w)
m ꢀE vꢀE
where E is the electrode potential. The charge transfer resistance,
Rt, and the Fourier transform of the aging function, F(w), are
given by
ꢃ
ꢄ
∂I
R−t 1
=
(7)
∂E
L
and
ꢀ
∞
F(w) =
f(u, τ) exp(−jwu) du
(8)
0
respectively. Whereas I and E are considered to vary in phase,
ꢀm can be reasonably assumed to be retarded. Finally ZF
becomes the following function of the two time-constant τ and
τn:
ꢅ
ꢃ
ꢄ
ꢆ
Fig. 2. Experimental complex plane of impedance plots measured for deposition
of Al on graphite at different cathodic potential: (a)—(1) −0.82 and (2) −0.83;
(b)—(3) −0.84, (4) −0.85 and (5) −0.86 V/Al.
1
ZF
1
F(w)
1
λ
1
=
+
+ 1 − λ
−
(9)
Rt
τ
RP 1 + jwτn
Rt
The delay time, τn, including both the induction time for nucle-
ation [31] and any retarding event such as the slow desorption
of inhibiting species and λ denotes the fraction of nuclei that
needs the delay time τn before entering into the aging process.
In Eq. (9), the charge transfer resistance is Rt and the polarization
resistance is designated as RP. For a simple ageing function:
where the i0 is the exchange current density, α the transfer coeffi-
cient of the cathodic reaction and η is the cathodic overpotential,
and other symbols have their usual meanings.
The growth of two-dimensional nuclei will take place through
edges advancement with the rate, v, is
ꢁ
ꢂ
u
u
τ
f(u, τ) = exp −
(10)
Mi
τ
v =
(2)
ρF
the Fourier transform is
τ
where M is the atomic weight and ρ is the density of the deposit.
In this model an active edge of height h is considered to
propagate at the rate of v = dr/dt which is proportional to the
currentdensityiwhenthedischargeandincorporationofionsare
assumed to take place simultaneously on the growth sites [31]. r
is the distance of propagation and the edge length l = l0f(u, τ) is
defined from the ageing law f(u, τ), where u = r/v is the mono-
layer age and τ denotes the time elapsed when the monolayer
attains its maximal size lm = l0 max[f(u, τ)] for a propagation
distance r0 such that
F(w) =
(11)
(1 + jwτ)2
This function is similar to the potentiostatic transients corre-
sponding to the instantaneous nucleation and growth of a 2D
layer.
Usually the electrode impedance, Ze = Zreal + jZimag, is
obtained by assuming Z in parallel with the double layer capac-
itance Cd:
1
Ze
1
=
+ jCdw
(12)
Z
ꢀ
∞
r0
τ =
=
f(u, τ) du
(3)
Inthiscasethemultilayerformationhasbeentreatedasacascade
process, in which the (n + 1)th layer is born out of the nth layer
so that the current can be evaluated by a recurrence relationship.
Such an approach to multilayer formation implies both the birth
(by nucleation) and the death (as a consequences of overlap)
of each monolayer. In Fig. 2a and b, two relaxation processes
show up, one (with time constant τn) due to the retardation of
nucleation, the other (roughly speaking with time constant τ) is a
result of the propagation of edges. In this analysis of the birth and
growth of monolayers, it seems that at intermediate frequencies,
the capacitive feature originates from the propagation of edges
over a finite distance whereas, at low frequencies the inductive
v
0
The successive monolayers have been assumed to be generated
at a mean nucleation rate and to follow the same ageing law as
nuclei formed at random in time and space [31].
The current for metal deposition is thus given by
ꢀ
ꢁ
ꢂ
∞
m
v
r
I = ihL = ihl0
f(r, r0)
t −
dr
(4)
v
0
where m is the nucleation rate, m/v the size distribution function
of edges and L denotes the total length of edges.

5440
M. Jafarian et al. / Electrochimica Acta 52 (2007) 5437–5443
response essentially reflects the slow desorption of inhibiting
adsorbates, highly likely Cl− ions resulting from the deposition
of Al which allows the edges to propagate over a longer distance.
Surely, the removal of the adsorbed cations by the growing Al
nuclei can in principle feed a reverse current into the circuit
giving rise to an inductive loop. However, the high chances of
the readsorption of these species on the emerging Al surface
tends to weaken the argument.
Accordingly, with assuming constant phase element [42]
instead of double layer capacitance, the Faradaic impedance is
A
B
Fig. 3. Fitting of Faradaic impedance function (—) on experimental (· · ·) nyquist
measured at cathodic potential −0.83 V/Al.
Ze =
with
A =
−
j
(13)
A2 + B2
A2 + B2
is lower and desorption of inhibiting species is faster and also
that the fraction of nuclei that need τn is diminished. The time
constant τ that decreases with overpotential indicates that the
mean life-time of edges is lower in higher overpotential and a
monolayer promptly attains its maximal size and that a complete
monolayer forms at a faster rate at higher overpotentials [2]. The
ln Rct is found to be linear function of cathodic potential which
is an indication of the prevalence of charge transfer process at
the high frequency side of the impedance spectrum.
The characteristic points are the singular points of the
Faradaic impedance function and their calculation can provide
interesting information about the impedance parameters of the
system. When the frequency tends to infinity (initial point of the
first loop), the Faradaic impedance as given by Eq. (13) shows
the following value for characteristic point:
1
(1 − w2τ2)λ
+
Rt
(2w2τ2 + w4τ4 + 1)RP(1 + w2τn2)
w2τλτn
−
(2w2τ2 + w4τ4 + 1)RP(1 + w2τn2)
(1 − w2τ2) ((1/RP) − (λ/RP) − (1/Rt))
+
2w2τ2 + w4τ4 + 1
nπ
+ Cwn cos
(14)
2
and
(1 − w2τ2)λwτn
B = −
(2w2τ2 + w4τ4 + 1)Rp(1 + w2τn2)
λwτ
(2w2τ2 + w4τ4 + 1)RP(1 + w2τn2)
wτ ((1/RP) − (λ/RP) − (1/Rt))
w→∞
Z
Z
= 0
(16)
(17)
−
−
imag
w→∞
= RS
real
nπ
+ Cwn sin
(15)
(2w2τ2 + w4τ4 + 1)
2
On the other hand when the frequency tends to zero the Faradaic
impedance is independent on charge transfer resistance and
shows the following value:
To corroborate this equation, the experimental data are fitted to
Eq. (13) in the applied potential interval of −0.82 to −0.86 V/Al,
Fig. 3, and taking into account the double layer capacitance.
These fittings agree with the experimental data. The evolution
of the impedance parameters with respect to the applied poten-
tial is consistent with the model, Table 1. It is observed that
all impedance parameters decrease with increasing cathodic
potential. The retardation time τn necessary for the propagation
w→0
Z
= 0
(18)
(19)
imag
w→0
Z
= RP
real
The third case is characterized by the fact that the imaginary
part of the Faradaic impedance is zero. Thus, in this situation,
the frequency value in the low frequency end of the spectrum
depends on all impedance parameters:
√
−ττn(λτRt − τnRt + λτnRt + RPτn)(−λτnRt + τRP − τRt)
imag=0
wZ
=
(20)
λτnRtτ2 − τRtτn2 + τλRtτn2 + τRPτn2
distance to be lengthened decreases with increasing overpoten-
tialshowingthatinhigherpotentialinductiontimefornucleation
A hie−rarchically distributed equivalent circuit for simulating
AlCl4 reduction and Al crystallization is reproduced in Fig. 4
Table 1
Impedance parameters calculated from the fitting of Eq. (13) to the experimental data of Fig. 2
E (V)
Rct (ꢁ)
RP (ꢁ)
τ (s)
τn (s)
λ
C (F)
n
Rs
−0.82
−0.83
−0.84
−0.85
−0.86
19
12
10
7
12
8
7
5.5
4
0.017
0.03
0.02
0.01
0.008
0.9
0.7
0.3
0.09
0.04
0.81
0.79
0.77
0.75
0.765
0.0001
0.0002
0.0003
0.0003
0.0005
0.75
0.71
0.7
0.71
0.71
11
11
10.9
10.9
11
4

M. Jafarian et al. / Electrochimica Acta 52 (2007) 5437–5443
5441
where
ꢁ
ꢂ
n2π
1
K = C2wn2 cos
+
(25)
2
R3 + R4
ꢃ
ꢄ
2
ꢁ
ꢂ
n2π
n2π
1
S = C2wn2 cos
+
ꢂꢂ
2
R3 + R4
ꢁ
ꢁ
2
2
n 2
2
+ C2(w ) sin
(26)
(27)
2
ꢃ
ꢄ
ꢁ
2
2
n
2
2
Fig. 4. Equivalent circuit used for aluminum electrocrystallization on graphite
from molten salt. R1: electrode resistance plus solution resistance; R2: charge
transfer resistance; R3: surface diffusion resistance; R4: lattice formation resis-
tance; R5: inductor’s resistance; Q1: double layer capacitance; Q2: surface
diffusion capacitance; L: inductance.
2
K
S
C2(w ) (sin(n2π/2))
D = R2 +
+
S2
ꢂ
n2π
E = C2wn2 sin
(28)
(29)
(30)
2
F = R25 + L2w2
K
H = R2 +
S
on the basis of Adatom model [17–20]. The Adatom model
assumes that the metal ions are discharged onto the electrode
surface and then after diffusion along the electrode surface the
Adatoms are incorporated into the metal at growth sites located
at the edges of nucleus. The crystallization impedance is there-
fore a function of diffusion in the solution, charge transfer,
surface diffusion and lattice incorporation and the correspond-
ing equivalent circuit consists of surface diffusion element with
lattice formation resistance.
and subsequently, this equivalent circuit under a sinusoidal
potential perturbation should reproduce the experimental mea-
surements.
The values of the circuit elements of the considered equiv-
alent circuit (Fig. 4) derived by fitting experimental data to
Eqs. (22)–(24) at various dc-offset potentials are presented in
Table 2. Appreciable variation of the charge transfer resistance
defined by R2 are detected also in agreement with the previous
On the basis of the proposed equivalent circuit and using
constant phase elements instead of capacitances the frequency
dependent impedance is derived as
n
n
n
n
2
2
2 2
Lwj + (R5 + R3 + R4/C2w j R3 + C2w j R4) + R2
n
1
n
1
n
n
n
n
2
2
2 2
+ C1w j (R3 + R4/C2w j R3 + C2w j R4) + R2(Lwj + R5)
1
Ze
1
R1
=
+
(21)
n
2
n
n
n
2 2
2
(R3 + R4/C2w j R3 + C2w j R4)(Lwj + R5)
Accordingly, the Faradaic impedance of equivalent circuit is
Ze = Zreal + Zimag (22)
model (Fig. 5). The impedance of the CPE element is defined as
ZCPE = 1/C(jω)n and the interpretation of this element depends
on the n values [42].
(H/D) + (R5/F) + C1wn1 cos(n1π/2)
((H/D) + (R5/F) + C1wn1 cos(n1π/2))2
+ (−(Lw/F) + (E/SD) + C1wn1 sin(n1π/2))2
Comparing the theoretical Faradaic crystallization
impedance, Eq. (13), and the Faradaic impedance of the
equivalent circuit, Eq. (22), the relations between the resis-
tances Rt, RP and the equivalent circuit elements are established.
This relations are shown in the following system of equations:
Zreal
=
(23)
Rt = R2
(31)
(32)
(Lw/F) − (E/SD) − C1wn1 sin(n1π/2)
((H/D) + (R5/F) + C1wn1 cos(n1π/2))2
ꢃ
ꢄ
Zimag
=
(R4 + R2 + R3)R5
RP =
R3 + R5 + R4 + R2
+ (−(Lw/F) + (E/SD) + C1wn1 sin(n1π/2))2
Which are in agreement with data presented Tables 1 and 2.
(24)
Table 2
Impedance parameters calculated from the fitting of Eq. (22) to the experimental data of Fig. 2
E (V)
R1 (ꢁ)
R2 (ꢁ)
R3 + R4 (ꢁ)
R5 (ꢁ)
C1 (F)
C2 (F)
L (H)
n1
n2
−0.82
−0.83
−0.84
−0.85
−0.86
11
11
11
11
11
18.6
11.5
9
7
4
68
45
24
15
12
14
9.5
8
6.5
5.7
0.0001
0.0005
0.0003
0.0002
0.0003
0.02
150
120
60
40
20
0.72
0.7
0.71
0.72
0.72
0.7
0.7
0.7
0.71
0.7
0.055
0.065
0.085
0.15

5442
M. Jafarian et al. / Electrochimica Acta 52 (2007) 5437–5443
and in this case the impedance is not controlled by the men-
tioned phenomena. λ has no influence on the high frequency
loop indicating that charge transfer step is independent of λ.
Fig. 7 presents complex plane plots derived at different value
of τ. It can be seen that as this parameter decreases the first loop
remain unaffected but the low frequency features (capacitive and
inductive) diminish in agreement with the changes in λ observed
above (Fig. 6).
4. Conclusion
Fig. 5. Fitting of simulated diagram of nyquist from equivalent circuit (—) on
experimental nyquist (· · ·) at cathodic potential −0.83 V/Al.
Electrodeposition of aluminum onto a graphite electrode
from molten (AlCl3–NaCl–KCl) salts as studied by the methods
of cyclic voltammetery and impedance spectroscopy revealed
that the electrodeposition at low cathodic potentials is not con-
trolled by diffusion. Cyclic voltammetery indicates nucleation
and growth of aluminum electrodeposition. The high frequency
semi-circle of complex plane plot is related to charge transfer
−
resistance for AlCl4 reduction and ln Rct versus potential is
found to be linear. The low frequency features of complex plane
plots appear to be strongly dependent on overpotential. The pres-
ence of low-frequency features can be interpreted in terms of the
relaxations of the electrode area on the basis of an impedance
model for the birth and growth of monolayers. In this analysis the
capacitive feature has been ascribed to the propagation of edges
over finite distances whereas the inductive feature essentially
reflects desorption of inhibiting adsorbates.
The EIS spectrum simulation allows a study of the Faradaic
system through the equivalent circuit element and, therefore,
predicts the behavior of the system with regard to the variation
of the experimental conditions.
Fig. 6. Simulated diagrams for the evolution of the EIS spectrum with regard to
the λ variation. The impedance parameters are: Rct = 12 ꢁ; RP = 8 ꢁ; τ = 0.03 s;
τn = 0.7 s; Cd = 0.0002 F cm−2; n = 0.71; Rs = 11 ꢁ; λ1 = 0.7; λ2 = 0.5; λ3 = 0.3;
λ4 = 0.2; ␭5 = 0.1.
3.2. EIS dependence on parameter λ and τ
The fraction of nuclei which need induction time τn affects
both the capacitive and the inductive loops appearing in the
intermediate and high frequency domains, Fig. 6. So, as this
parameter decreases, the size of both loops decreases. For
λ = 0.34 and lower, the capacitive feature in intermediate fre-
quencies disappears and for λ = 0.22 and lower, the inductive
feature appears as two loops in low frequencies. For low value
of λ the propagation of edges and monolayer formation are fast
References
[1] P.R. Gifford, J.B. Palmisano, J. Electrochem. Soc. 135 (1988) 650.
[2] M. Jafarian, M.G. Mahjani, F. Gobal, I. Danaee, J. Electroanal. Chem. 588
(2006) 190.
[3] O.E. Barcia, O.R. Mattos, Electrochim. Acta 35 (1990) 1601.
[4] G. Philippin, J. Delhalle, Z. Mekhalif, Appl. Surf. Sci. 212/213 (2003) 530.
[5] S.A.M. Refaey, G. Schwitzgebel, Appl. Surf. Sci. 135 (1998) 243.
[6] J.R. Macdonald, Impedance Spectroscopy, Wiley, New York, 1987.
[7] C. Cachet, U. Stroder, R. Wiart, Electrochim. Acta 27 (1982) 903.
[8] C. Cachet, B. Saidani, R. Wiart, Electrochim. Acta 34 (1989) 1249.
[9] C. Cachet, B. Saidani, R. Wiart, Electrochim. Acta 33 (1988) 405.
[10] C. Cachet, B. Saidani, R. Wiart, J. Electrochem. Soc. 138 (1991) 678.
[11] C. Cachet, Z. Chami, R. Wiart, Electrochim. Acta 32 (1987) 465.
[12] X. Cheng, G. Li, E.A. Kneer, B. Vermeire, H.G. Parks, S. Raghavan, J.
Electrochem. Soc. 145 (1998) 352.
[13] C. Cachet, C. Gabrielli, F. Huet, M. Keddam, R. Wiart, Electrochim. Acta
28 (1983) 899.
[14] J.O’M. Bockris, G.A. Razumney, Fundamental Aspect of Electrocrystal-
lization, Plenum Press, New York, 1967.
[15] E. Budevski, B. Conway, J.O’.M. Bockris, in: E. Yeager, S.U. Khan, R.E.
White (Eds.), Comprehensive Treatise of Electrochemistry, vol. 7, Plenum
Press, New York, 1983, p. 399.
[16] A.R. Despic, B. Conway, J.O’.M. Bockris, in: E. Yeager, S.U. Khan, R.E.
White (Eds.), Comprehensive Treatise of Electrochemistry, vol. 7, Plenum
Press, New York, 1983, p. 451.
[17] M. Fleischmann, S.K. Rangarajan, H.R. Thirsk, Trans. Faraday Soc. 63
(1967) 1240.
Fig. 7. Simulated diagrams for the evolution of the EIS spectrum with regard to
the τ variation. The impedance parameters are: Rct = 12 ꢁ; RP = 8 ꢁ; λ = 0.79,
τn = 0.7 s; Cd = 0.0002 F cm−2; n = 0.71; Rs = 11 ꢁ; τ1 = 0.01 s; τ2 = 0.05 s;
τ3 = 0.1 s; τ4 = 0.15 s; τ5 = 1 s.

M. Jafarian et al. / Electrochimica Acta 52 (2007) 5437–5443
5443
[18] M. Fleischmann, S.K. Rangarajan, H.R. Thirsk, Trans. Faraday Soc. 63
[29] R.D. Armstrong, A.A. Metcalfe, J. Electroanal. Chem. 63 (1975) 19.
[30] R.D. Armstrong, A.A. Metcalfe, J. Electroanal. Chem. 71 (1976) 5.
(1967) 1251.
[19] M. Fleischmann, S.K. Rangarajan, H.R. Thirsk, Trans. Faraday Soc. 63
(1967) 1256.
[31] C. Cachet, I. Epelboin, M. Keddam, R. Wiart, J. Electroanal. Chem. 100
(1979) 745.
[20] S.K. Rangarajan, J. Electroanal. Chem. 17 (1968) 61.
[21] J.P. Diard, B. Le Gorrec, Surf. Technol. 13 (1981) 127.
[22] J.A. Harrison, D.R. Sandbach, J. Electroanal. Chem. 85 (1977) 125.
[23] H.B. Sierra Alcazar, J.A. Harrison, Electrochim. Acta 22 (1977) 627.
[24] M. Aslam, J.A. Harrison, C.E. Small, Electrochim. Acta 25 (1980) 657.
[25] M. Fleischmann, H.R. Thirsk, in: P. Delahay (Ed.), Advanced Electrochem-
istry and Electrochemical Engineering, vol. 3, Interscience, New York,
1963, p. 123.
[26] S.J.A. Harrison, H.R. Thirsk, in: A.J. Bard (Ed.), Electroanalytical Chem-
istry, vol. 5, Marcel Dekker, New York, 1971, p. 67.
[27] S.K. Rangarajan, J. Electroanal. Chem. 46 (1973) 125.
[28] R.D. Armstrong, J.A. Harrison, J. Electrochem. Soc. 116 (1969) 328.
[32] Z. Stoynov, Electrochim. Acta 35 (1990) 1493.
[33] J.R. Macdonald, Solid State Ionics 58 (1992) 97.
[34] I. Epelboin, M. Joussellin, R. Wiart, J. Electroanal. 101 (1979) 281.
[35] C. Cachet, M. Froment, R. Wiart, Electrochim. Acta 24 (1979) 713.
[36] I. Epelboin, M. Ksouri, R. Wiart, J. Electrochem. Soc. 122 (1975) 1206.
[37] E. Chassaing, R. Wiart, Electrochim. Acta 29 (1984) 649.
[38] I. Epelboin, R. Wiart, J. Electrochem. Soc. 118 (1971) 1577.
[39] K. Grjotheim, K. Mathiasovsky, Acta Chem. Scand. A34 (1980) 666.
[40] R.W. Berg, H.A. Hjuler, N.J. Bjerrum, Inorg. Chem. 23 (1984) 557.
[41] R. Greef, R. Peat, L.M. Peter, D.J. Pletcher, Robinson Instrumental Meth-
ods in Electrochemistry, Ellis Horwood, Chichester, 1985 (Chapter 9).
[42] J.R. Macdonald, Solid State Ionics 13 (1984) 147.