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Page 5 of 57
Accepted Manuscript
5
Figure 1.
(PDM Reactions)
The physico-chemical basis of the PDM is shown in Figure 1. Briefly, the model
postulates that defect generation and annihilation reactions occur at the metal/barrier layer (m/bl)
and the barrier layer/outer layer (bl/ol) interfaces and that these reactions essentially establish the
point defect concentrations within the barrier layer. Analytical expressions for the rate constants
for the reactions, as derived by the method of partial charges, are summarized in Table 1.
Table 1.
(Rate constants)
The electron current density,
I
, which is sensed in an external circuit, is given by:
7
0
5
4
3
2
1
)
(
)
(
)
(
k
C
k
k
k
k
C
k
F
I
i
L
v
(1)
where
L
C
is the concentration of cation vacancies at the m/bl interface and
0
i
C
is the
concentration of cation interstitials at the bl/ol interface. Note that Equation 1 does not depend
upon the concentration of oxygen vacancies or upon the rate constant for Reaction (6), Figure 1.
Thus, no relaxations in the impedance response involve oxygen vacancies, but this is essentially
an artifact of considering Reaction (3), Figure 1, to be irreversible. If this reaction was
considered to be reversible, then a relaxation involving oxygen vacancies would be present.
Furthermore, the concentration of
H
+
is considered to be constant, corresponding to a well-
buffered solution, and is included in the definition of
k
7
, as indicated in Equation 4.
Using the method of partial charges, the rate constants for the reactions are found to be of
the form:
L
b
I
R
V
a
k
k
i
ol
i
i
i
)
(
exp
0
,
i
= 1, 2, 3
(2)
Page 6 of 57
Accepted Manuscript
6
)
(
exp
0
I
R
V
a
k
k
ol
i
i
i
,
i
= 4, 5
(3)
and
n
0
H
H
ol
7
0
7
7
C
C
)
I
R
V
(
a
exp
k
k
(4)
where
n
is the kinetic order of barrier layer dissolution with respect to
H
+
. In deriving these
expressions theoretically, it is assumed that a resistive outer layer,
R
ol
, exists on the surface of the
barrier layer and that the passive current flows through the barrier layer to a remote cathode,
which is the normal experimental configuration. Because of this, the potential that exists at the
bl/ol interface must be corrected from that applied at the reference electrode located at the outer
layer/solution interface by the potential drop across the outer layer, where
R
ol
(Ω cm
2
) is the
specific resistance of the outer layer. We recognize, however, that in the experiments reported in
this paper,
EDTA
was added to the solution to suppress the formation of the outer layer, so that
R
ol
= 0
and hence the impact of the outer layer is moot. It is included in the theory reported here
only for the sake of completeness.
Let us assume that the applied potential changes sinusoidally around some mean value
(
V
)
in accordance with Equation (5):
t
j
Ve
V
V
V
V
(5)
where
ω
is an angular frequency and
V
is the amplitude. The bar over a letter refers to the
corresponding value under steady-state conditions. Accordingly, in the linear approximation
Page 7 of 57
Accepted Manuscript
7
have the following response
j t
f
f
fe
, where
f
represents current density,
I
, and values on
which
I
depends, namely,
L
,
0
i
C
,
L
C
, and the various rate constants
.
Our task, then, is to calculate the faradic admittance,
Y
F
, which is defined as:
V
I
V
I
Z
Y
F
F
1
(6)
where
Z
F
is the faradic impedance. Note that
I
, is a function of the potential at the bl/ol interface
(
U
), but the potential that is modulated is that at the outer layer/solution (ol/s) interface (
V
), or
close to it, depending upon the exact placement of the tip of the Luggin probe. The two
potentials are related by
I
R
V
U
ol
(7)
It is evident that,
ol
F
F
R
Y
Y
1
1
1
0
or
0
0
1
F
ol
F
F
Y
R
Y
Y
(8)
where
0
F
Y
is the admittance calculated in the absence of the outer layer, assuming that the
potential at the bl/ol interface is
U
under steady-state conditions. We see that
0
F
F
Y
Y
as
R
ol
→
0 and
ol
F
R
Y
/
1
for
0
F
Y
; that is, the interphasial impedance becomes controlled by the
outer layer in the limit of an infinitely large outer layer specific resistance or an infinitely small
barrier layer admittance.
Page 8 of 57
Accepted Manuscript
8
The values of
U
and other steady state values can be easily calculated. Assuming some
arbitrary value of
U
, we can immediately calculate
i
k
,
i
= 4, 5, 7 from Equations (3) and (4).
From the rate equation for the change in thickness of the barrier layer, which is written as
7
3
k
k
dt
dL
(9)
we have
7
3
k
k
, i.e.
n
H
H
SS
C
C
k
k
b
pH
b
C
C
U
b
a
a
L
0
0
3
0
7
3
3
3
7
3
3
7
ln
1
(10)
After that, the values
i
k
(
i
= 1, 2) can be calculated by using Equations (2).
The values of the steady-state concentrations
0
i
C
and
L
C
(concentrations of metal
interstitials at the bl/ol interface and oxygen vacancies at bl/ol interface) can be found by
equating the rates at each location to yield:
4
1
L
v
k
C
k
(11)
0
2
5
i
k
C
k
(12)
and
0
3
6
O
k
C
k
(13)
Page 9 of 57
Accepted Manuscript
9
respectively. Equations (11) to (13) follows from the conditions that steady state fluxes of cation
vacancies, cation interstitials, and oxygen vacancies, respectively, are constants (do not depend
upon position inside the barrier layer).
Finally, we calculate the values of
7
0
5
4
3
2
1
)
(
)
(
)
(
k
C
k
k
k
k
C
k
F
I
i
L
v
(14)
and
I
R
U
V
ol
(15)
i.e. we calculate the dependence
)
(
V
I
. As the actual value of
U
, we will choose the value at
which
V
equals the prescribed value, because no outer layer is assumed to exist in the
experiment. Practically, the task is reduced to the solution of the single equation
)
(
U
I
R
U
V
ol
relative to the unknown value
U
(the voltage at the bl/ol interface).
We see that, if we have a code for calculating the admittance of the system in the absence
of the outer layer,
0
F
Y
, we can calculate the admittance in the presence of the outer layer,
F
Y
, by
using Equation (8), assuming that
0
F
Y
is calculated at the steady state applied potential that
equals
U
(but not
V
).
2.1. Calculation of
0
F
Y
.
As follows from Equation (1) we have in the linear form:
U
C
I
U
C
I
U
L
I
I
U
I
U
I
Y
i
i
L
v
L
v
L
U
F
0
0
0
(16)