Geology Reference
In-Depth Information
molecules can potentially be readily detached or attached, respectively; that is, we
assume
dn
dt
¼
a
s
A
s
b
k
s
¼
A
s
k
V
m
ð
5
:
3
Þ
V
m
where b is the thickness of a monolayer, and k
s
is the rate coefficient at the
potential site. We can write a
s
bk
s
¼
k since it is only this product that is directly
measured. Tsai and Raj (
1982b
) introduce a rate proportional to the molar con-
centration of the diffusing species instead of to the number of reaction sites but, as
they point out, only the product of this term with the rate coefficient is measurable
and so empirically the two models cannot be distinguished from deformation
experiments alone.
Returning to (
5.3
) and using an argument analogous to that used in deriving
(
3.12
), we can now write
k
s
Dl
s
ð
Þ
k
s
¼
RT
under the assumption that
Dl
s
RT ; here the k
s
are of the form A exp
Q
=
R
ð Þ
where the A are constants with dimensions of frequency and the Q's are activation
energies for the reactions. Then the (
5.3
) become
dt
¼
ba
so
A
so
k
so
dn
ð
Dl
so
Þ
ð
5
:
4a
Þ
V
m
RT
dt
¼
ba
si
A
si
k
si
dn
ð
Dl
si
Þ
ð
5
:
4b
Þ
V
m
RT
at source and sink, respectively. In the case of diffusion, (
3.28
) can be used to give,
in the ideal case,
dn
dt
¼
jA
t
¼
A
t
cD
ð
Dl
D
Þ
RT
l
or, using (
5.2
),
dn
dt
¼
cD
A
t
½
C
2
V
m
r
1
r
3
ð
Þ
Dl
so
ð
Þ
Dl
si
ð
Þ
ð
5
:
5
Þ
l
RT
Two situations must now be considered:
1. If Dl
so
and Dl
si
can be treated as dependent variables, they can be eliminated
from (
5.4
)to(
5.5
) and the resulting value of dn
=
dt substituted in (
5.1
) to give
C
0
V
m
cD
1
þ
V
m
cD
A
t
bl
A
t
lV
V
m
r
1
r
3
ð
Þ
e
¼
ð
5
:
6
Þ
RT
1
1
a
si
A
si
k
si
a
so
A
so
k
so
þ
