ВУЗ: Не указан
Категория: Не указан
Дисциплина: Не указана
Добавлен: 02.04.2021
Просмотров: 340
Скачиваний: 1
Page 10 of 57
Accepted Manuscript
10
where
7
7
0
5
5
4
4
3
3
2
2
1
1
)
(
)
(
)
(
a
k
C
a
k
a
k
a
k
a
k
C
k
a
F
I
i
L
v
U
(17)
3
3
2
2
1
1
b
k
b
k
C
k
b
F
I
L
e
L
(18)
1
k
F
I
L
v
(19)
0
5
(
)
i
I
F
k
(20)
Here, it is assumed that
t
j
Ue
U
U
U
U
and we identify the four terms on the right
side as arising from relaxations with respect to the applied potential
V
, the thickness of the
barrier layer with respect to the voltage at the bl/ol interface,
U
, cation vacancies,
L
C
, and cation
interstitials,
0
i
C
, respectively. Note the absence of a term for the relaxation of oxygen vacancies,
because, again, the concentration of oxygen vacancies does not appear in the current [Equation
(1)], as a consequence of assuming Reaction (3), Figure 1, to be irreversible.
Let us now calculate
U
C
U
C
U
L
i
L
v
0
and
,
. It is convenient to start with
U
L
. The rate of
change of the thickness of the barrier layer is described by Equation (9). Accordingly, by taking
the total differential, we have
V
a
k
L
b
k
U
a
k
k
k
Le
j
dt
L
d
t
j
7
7
3
3
3
3
7
3
)
(
(21)
or
3
3
7
7
3
3
)
(
b
k
j
a
k
a
k
U
L
L
U
(22)
Page 11 of 57
Accepted Manuscript
11
which is the result that we desire.
2.2. Calculation of
U
C
i
0
The flux density of interstitials is
i
i
i
i
i
KC
D
x
C
D
J
(23)
Here,
D
i
is the diffusion coefficient of the cation interstitials,
K = Fε/RT
, where
ε
is the electric
field strength inside the barrier layer, and
T
is the temperature. The continuity equation is
x
C
K
D
x
C
D
t
C
i
i
i
i
i
2
2
(24)
with the boundary conditions
0
at
5
x
KC
D
x
C
D
C
k
i
i
i
i
i
(25)
Substitution
t
j
i
i
i
e
C
C
C
into Equations (23) to (25) and linearization of boundary
conditions relative to
ΔU
and
ΔL
yields:
x
C
K
D
x
C
C
j
i
i
2
i
2
i
(26)
or
0
at
)
(
0
0
0
5
0
5
x
C
K
D
x
C
D
C
U
a
C
k
i
i
x
i
i
i
i
(27)
Page 12 of 57
Accepted Manuscript
12
L
x
C
K
D
x
C
D
)
L
b
U
a
(
k
L
i
i
L
x
i
i
2
2
2
at
(28)
Analytical solution of the linear boundary problem (26) – (28) can be easily obtained and the
sought value
U
C
i
/
0
can be presented in the following bulky form:
U
L
C
C
U
B
A
U
C
iL
iU
i
0
0
0
(29)
where
21
12
22
11
12
11
2
21
22
1
0
)
(
)
(
a
a
a
a
a
a
b
a
a
b
C
U
U
iU
(30)
21
12
22
11
12
11
2
0
)
(
a
a
a
a
a
a
b
C
L
iL
(31)
2
D
/
j
4
K
K
r
i
2
2
2
,
1
(32)
L
r
i
L
r
i
i
i
e
D
K
r
a
e
D
K
r
a
k
D
K
r
a
k
D
K
r
a
2
1
)
(
,
)
(
,
)
(
,
)
(
2
22
1
21
5
2
12
5
1
11
(33)
2
2
2
2
2
2
0
5
5
1
,
,
b
k
b
a
k
b
C
a
k
b
L
U
i
U
(34)
The reader should note that the expressions given above for cation interstitials are exactly
the same for oxygen vacancies, with the oxidation number,
, being replaced by 2, Subscript 2
being replaced by Subscript 3, and Subscript 5 being replaced by Subscript 6, so as to identify
the correct reactions in Figure 1.
Page 13 of 57
Accepted Manuscript
13
2.3. Calculation of
U
C
L
v
By analogy it can be shown that:
L
U
L
C
C
U
Be
Ae
U
C
L
vL
L
vV
L
r
L
r
L
r
L
2
2
1
(35)
where
21
12
22
11
21
1
11
2
12
2
22
1
2
1
)
(
)
(
a
a
a
a
e
a
b
a
b
e
a
b
a
b
C
L
r
U
U
L
r
U
U
L
vU
(36)
21
12
22
11
12
2
11
2
1
2
a
a
a
a
e
a
b
e
a
b
C
L
r
L
L
r
L
L
vL
(37)
2
/
4
2
2
2
,
1
v
D
j
K
K
r
(38)
where
L
r
v
L
r
v
v
v
e
k
D
K
r
a
e
k
D
K
r
a
D
K
r
a
D
K
r
a
2
1
]
)
[(
,
]
)
[(
,
)
(
,
)
(
1
2
22
1
1
21
2
12
1
11
(39)
1
1
2
1
1
2
4
4
1
,
,
b
C
k
b
C
a
k
b
a
k
b
L
v
L
L
v
U
U
(40)
By substituting Equations (22), (29) and Equation (35) into Equation (16) we have the final
result
)
(
)
(
0
0
0
0
U
iL
iU
i
U
L
vL
L
vU
L
v
U
L
U
F
L
C
C
I
L
C
C
I
L
I
I
Y
(41)
If
k
1
= 0, we have
Page 14 of 57
Accepted Manuscript
14
)
(
0
0
0
0
U
iL
iU
i
U
L
U
F
L
C
C
I
L
I
I
Y
(42)
If, in addition, we can neglect changes in the cation vacancy concentration (for example, at high
frequencies) or χ = δ, we have
U
L
U
F
L
I
I
Y
0
(43)
3
3
7
7
3
3
0
)
(
b
k
j
a
k
a
k
I
I
Y
L
U
F
(44)
where constants
I
U
and
I
L
are described by Equations (17) and (18). In the simplest case, when it
is possible to neglect changes in the thickness of boundary layer (for example when
L
is less
than diameter of the iron atom), the impedance of the barrier layer reduces to the ohmic
resistance
U
F
F
I
Y
Z
0
0
/
1
.
The reader will again note that the expression for the current, Equation (1), does not
contain the concentration of oxygen vacancies. This is, because Reaction (3), Figure 1, for the
generation of oxygen vacancies is considered to be irreversible and because Reaction (6) does
not contribute to the electron current, as previously noted. However, physically, the transport of
oxygen vacancies across the barrier layer still occurs, and hence is expected to impact the film
growth rate at the m/bl interface and hence the current from that process, and may determine the
total passive current density if the oxygen vacancy were found to be the dominant defect in the
barrier layer. In the case of iron, the dominant defect is the metal interstitial. Importantly, it
should be noted, as has already been done, that the equations describing the transport of oxygen
vacancies are identical with those describing the transport of metal interstitials, except that the
charge
χ
is replaced by 2. Thus, the contributions to the impedance of each of these defects are