Environmental Engineering Reference
In-Depth Information
where h
•
is a positive hole and V
′
Ag
is a negatively charged vacancy on Ag sublat-
tice. This means that iodine on the lattice surface of AgI will create extra acceptor
centers like vacancies on silver (V
Ag
), which becomes negatively charged
(V
′
Ag
) by accepting electrons from the valence band, thus injecting positive holes
in the valence band.
Assuming the validity of simple mass action law, one can write the following
expression for equilibrium at the outer interface of the growing film:
[V
′
Ag
]
⋅
p
K
e
(
T
)
(5.66)
P
1/2
I
2
where
K
e
(
T
) is the equilibrium constant at any temperature
T
,[V
′
Ag
] is the concen-
tration of singly negatively charged metal vacancies in number per m
3
,
p
is the
concentration of positive holes in number per m
3
, and
P
I
2
is the pressure of iodine
vapor in the environment. Since inherent concentration of cation vacancies are
already high (predominantly ionic lattice) with fulfillment of the existence of
virtual ionic current equilibrium across the iodide film, incorporation of extra
iodine into the growing halide lattice will not bring in much change in the concen-
tration of cation vacancies. So the concentration of positive holes (
p
) will be
decided by the pressure of the oxidant (iodine). Accordingly, Eq. 5.66 is modified
to the following form assuming [V
′
Ag
] to remain virtually constant:
K
e
(
T
)
[V
p
P
1/2
I
2
′
Ag
]
K
e
(
T
)
(5.67)
As pointed out earlier, virtual ionic current equilibrium does exist across the
iodide film for such a system, so the rate of iodide layer thickening will be con-
trolled by the transport of positive holes, i.e., by the conductivity of positive
holes (
σ
h
•
). Then one can write from Eq. 5.67:
σ
h
•
p
P
1/2
I
2
(5.68)
The above relation has been experimentally verified for the growth of AgCl,
AgBr, and AgI films on silver [20] for film thicknesses exceeding certain values.
The electrical field strength (
E
volt/m) in such cases is very weak and
can be neglected. The expression (5.68) is obtained from the relation
V
/
ξ
σ
q
i
c
i
v
i
,
where
q
i
charge (in coulomb) on the rate limiting species,
v
i
charge mobility
in m
2
V
1
s
1
which remains virutally constant at any temperature (
T
), and
C
i
concentration of the rate-limiting species expressed in number per m
3
.
One can reexamine the Wagner's theoretical relation (5.63) for Ag-AgI-I
2
(v)
system. In this case,
t
1
1
,
t
2
t
I
t
Ag
,
0 (iodide sublattice remains almost
stationary), and
t
3
t
h
•
. Therefore, one may write,
t
h
•
σ
σ
h
•
σ
h
•
P
1/2
I
2
(at constant
temperature), where,
σ
h
•
hole conductivity at
P
I
2
1 atm (1.01325
10
5
N.
m
2
) and
Z
2
Z
I
1
unity.
