146
3 – Transport in ionic solids
6. Crystallization is accompanied by
2 the appearance of a structural order, and
2 the formation of microcrystallites separated by grain boundaries.
These two factors manifest themselves in the impedance diagram by an
increase in the bulk resistivity of the material and by the appearance of a
supplementary resistance at the grain boundaries.
Solution 3.10 – Electrochemical coloration
1. Literal expressions for partial ionic conductivity
For the ionic conductivity σ i , we use the following expression derived from
the Nernst-Einstein relation:
z F RT
D i
i
i
2 2
i
σ =
6 @
where [i] denotes the concentration of the ionic species i. The equilibrium
involving the Schottky disorder
0 m V ′
K + V
•
Br
allows us to write
K s = [V ′
K ] [V
•
Br ]
By neglecting the concentrations of the electronic species (n and p), we have
[ ] [
]
V
V
K
k
B r
S
=
=
:
l
or
RT
F
D
K e
e
K
2
K
– kT
1.26 eV – kT
1.1eV
σ =
°
°
RT
F
D
K e
K
2
K
– kT
2.36 eV
σ =
°
°
with
D
c m s
10
K
2
2 1
–
–
=
°
The same reasoning leads to the expression for the ionic conductivity of
bromine:
RT
F
D
K e
Br
2
Br
– kT
3.71eV
σ =
°
°
2. Assume that the electronic transport number is negligible. The ratio of partial
ionic conductivities is
D
D
D e
D e
Br
K
Br
K
kT
2.36 3.71
Br
K
kT
1.35 eV
σ
σ =
=
−
−
+
−
°
°
°
°