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Page 15 of 57
Accepted Manuscript
15
indistinguishable, even if oxygen vacancies appeared in the formal analysis. However, in this
particular case, where
δ
=
χ
= 3, even the concentration of the interstitial does not enter the
problem, because of Equation (1) and because rate of Reaction (2), Figure 1, is also considered
to be irreversible and hence does not depend on the interstitial concentration at the m/bl
interface. Thus, the Faradaic impedance does not include a relaxation for the concentration of
any defect, but transport clearly occurs. In order to overcome this artifact, we added a Warburg
impedance for defect motion in accordance with Chao, Lin, and Macdonald [22].
2
/
1
2
/
1
j
Z
W
(45)
where
is the Warburg coefficient and
is the angular frequency. As previously shown [22],
the Warburg coefficient can be written as
ss
I
D
)
1
(
2
/
2
/
1
2
/
1
(46)
where
I
ss
is the steady-state passive current density,
is the electric field strength within the
barrier layer,
D
is the defect diffusivity, and
is the polarizability of the barrier layer/solution
interface. Parenthetically, the Warburg coefficient may be used to obtain the diffusivity of the
defect within the barrier layer, by rearranging Equation (46) to yield:
2
2
2
2
/
1
2
ss
I
D
(47)
as is shown below.
It is important to note that the electronic defect structure of the barrier layer, and hence
the electronic character (whether p-type or n-type) is defined by the distributions of the defects
across the layer. This is because cation vacancies are electron acceptors, which dope the barrier
Page 16 of 57
Accepted Manuscript
16
layer p-type, whereas cation interstitials and oxygen vacancies are electron donors, thereby
doping the barrier layer n-type.
Estimation of defect distributions within the barrier layer in the steady state and in the general
case can be performed via analytical solution of linear balance equations (see below).
Accordingly, by knowing the kinetic parameters obtained from the EIS studies, it would be
possible to calculate the concentrations of the defects in the barrier layer and hence to predict the
electronic character.
3. Impedance Model
The electrochemical reactions that occur at the metal/film and film/solution interfaces of
the barrier layer (Figure 1) contribute to the impedance of a system. Therefore, the total
impedance of the system, which comprises all of the contributory interfacial phenomena,
including those at the metal/film/solution interphase plus the resistance of the solution can be
represented by an electrical equivalent circuit. Figure 2 shows the equivalent circuit for the
metal/film/solution interphasial system that is proposed, in order to analyze the passivity of iron.
Figure 2
(Electrical equivalent circuit)
The total impedance of the system is expressed by Equation (48):
s
w
f
randles
e
cg
R
Z
Z
Z
Z
Z
Z
1
1
1
1
(48)
where
Z
f
is the Faradaic impedance associated with the reactions occurring at the metal/film and
film/solution interfaces.
Z
w
is the Warburg impedance associated with the transport of defects
through the barrier layer, and is expressed by Equations (45) to (47),
Z
e
is the electronic
Page 17 of 57
Accepted Manuscript
17
impedance, due to the transport of electronic defects (electrons and/or holes), and
C
g
is the
geometrical capacitance of the barrier layer. In the present analysis, we represent the
geometrical capacitance with a constant phase element (
CPE
), because, on a polycrystalline
substrate, there is expected to exist a distribution of passive film thickness on the various grain
faces exposed to the solution and hence it is expected that a distribution will exist in the
capacitance of the film.
R
s
is the resistance of the electrolyte between the oxide layer and the tip
of the Luggin probe, and
Z
randles
is the impedance associated with the redox reactions occurring at
the barrier layer/solution interface and expressed by Equation (49)
1
1
1
Cdl
Rct
randles
Z
Z
Z
(49)
Note that
Z
randles
exists in series with
Z
e
, because the redox reaction, which it represents, accepts
the electronic charge that by-passes the Faradaic impedance,
Z
f
(Figure 2).
The electronic impedance,
Z
e
, can be expressed as a complex number, the component
values of which can be calculated from fundamental theory. In most of the previous studies
reported by this group [39-41] an electronic resistance (
R
e,h
) has been used instead of
Z
randles
in
series with
Z
e
. The role of
R
e,h
is to short-circuit the Faradaic and defect Warburg impedances, as
indicated by the electrical analog given in Figure 2. However, since free electrons do not exist in
the solution, the electronic charge moving through the barrier layer must be accepted by some
redox couple in the solution at the bl/solution interface. This redox couple could be
H
2
/H
+
,
O
2
/H
2
O
, or
Fe
3+
/Fe
2+
, for example, depending upon the potential and the species present in the
solution. This reaction possesses an impedance, which we describe here by a Randles equivalent
electrical circuit, and which is included in
R
e,h
. If
R
e,h
is low, the majority of the current will
Page 18 of 57
Accepted Manuscript
18
flow through this parallel path and not through
Z
f
and
Z
w
, with the consequence that the faradaic
and defect transport impedances are “short circuited” and the analysis will become insensitive to
the kinetic and transport processes involving the generation, annihilation, and transport of point
defects. This issue was recognized from the earliest development of the PDM [20-22].
Accordingly, our experiments over the past thirty two years have been carried out to render
R
e,h
to be as large as possible, by excluding to the greatest extent possible all “tramp” redox species
from the solution.
In series with the redox impedance, but in parallel with the Faradaic impedance (
Z
f
) and
the defect Warburg impedance (
Z
W
), Figure 2, is the electronic impedance of the film, due to the
ionization of electrons from donors (metal interstitials and oxygen vacancies) into the conduction
band and the generation of electron holes in the valence band by the acceptance of electrons from
the valence band by cation vacancies. Below we estimate the electronic impedance of the barrier
layer,
Z
e
, on the basis of Bojinov et al.’s model [42-44]. In turn, and in accordance with this
model,
Z
e
is calculated by using the Young relation [45]:
L
e
e
j
x
dx
Z
0
0
ˆ
ˆ
)
(
(50)
where
)
(
x
e
is the electronic conductivity profile in the barrier layer,
ω
is angular frequency,
ˆ
is the dielectric constant of the film, and
0
ˆ
is the dielectric permeability of the vacuum. The
electronic conductivity of the barrier layer is proportional to the concentration of electronic
charge carriers,
C
e
, i.e.
e
e
e
FC
(51)
Page 19 of 57
Accepted Manuscript
19
where
RT
FD
e
e
/
is the mobility of the electronic charge carriers, which is assumed to be
constant, and
D
e
is diffusion coefficient of electrons.
We will adopt the following approximation for
C
e
:
O
i
e
C
2
C
C
(52)
which simply states that the concentration of conducting electrons in the conduction band is
directly determined by the concentration of electron donors in the barrier layer with each oxygen
vacancy and metal interstitial contributing 2 and
electrons, respectively, to the conduction
band.
In References [42-44], the concentrations of defects were calculated in accordance with
the Fromhold and Cook equations [45]. Here, we will estimate
C
i
and
C
O
on the basis of the
PDM. In doing so, it is convenient to present flux densities of the ionic defect,
J
k
, in the film in
following form:
k
k
k
k
k
C
U
dx
dC
D
J
(53)
where
RT
FD
z
U
k
k
k
(54)
has the dimension of velocity. Under steady-state conditions
0
dx
dJ
k
(55)
and Equation (55) has the following solution