Geology Reference
In-Depth Information
We approach this problem by considering the diffusion of a simple ionic sub-
stance of formula A
a
B
b
in a matrix. We assume that the substance is fully ionized
into charged species A
b
þ
and B
a
:
If one species tends to diffuse faster than the
other, a Nernst field will build up, as just mentioned, which will slow down the rate
of diffusion of this species. The effect of the Nernst field can be taken into account
by using the full electrochemical potential l in (
3.23
), where l
¼
l
þ
zF/, l
being the chemical potential defined without taking into account the interaction
between the charge and an electric potential, z the charge number of the species,
F the Faraday constant, and / the electric potential. If, following Howard and
Lidiard (
1964
), we ignore coupling terms, we can use (
3.23
) to write the fluxes of
A and B as
j
A
¼
L
A
dl
A
dl
A
dx
bFE
dx
¼
L
A
ð
3
:
36
Þ
j
B
¼
L
B
dl
B
dl
B
dx
¼
L
B
dx
þ
aFE
ð
3
:
37
Þ
where E
¼
d/
=
dx is the Nernst electric field and L
A
;
L
B
are the phenomeno-
logical coefficients. In order to maintain stoichiometry and electrical neutrality, we
also have
bj
A
¼
aj
B
ð
3
:
38
Þ
The three Eqs. (
3.36
-
3.38
) enable us to eliminate E in the expressions for j
A
and
J
B
and to obtain the total molecular flux j as
j
¼
j
A
a
¼
j
B
L
A
L
B
b
2
L
A
þ
a
2
L
B
dl
dx
¼
L
dl
b
¼
ð
3
:
39
Þ
dx
where l
¼
al
A
þ
bl
B
is the chemical potential of the molecular species and L is
the phenomenological coefficient relating it to the total molecular flux. Using
(
3.23
) and (
3.27
) and restricting consideration to the ideal case so that c
¼
1
;
we
can put L
¼
cD
=
RT
;
L
A
¼
c
A
D
A
=
RT and L
B
¼
c
B
D
B
=
RT
;
where c, c
A
;
c
B
and D,
D
A
;
D
B
are the amount-of-substance concentrations and the diffusion coefficients
of the molecular species and the ions A, B, respectively; also we can put
c
¼
c
A
=
a
¼
c
B
=
b
:
Using these relations in (
3.39
), then leads to the expression
D
A
D
B
bD
A
þ
aD
B
D
¼
ð
3
:
40
Þ
for the diffusion coefficient for the molecular species in terms of the diffusion
coefficients of the constituent ions. This expression applies to the ideal case, which
includes the tracer diffusion and self-diffusion cases, D
A
and D
B
being measured
independently as tracer diffusion coefficients by, say, radioactive tracer methods.
More generally, the values of D
i
for the species i in (
3.40
) have to be divided by
the appropriate factors
ð
1
þ
d ln c
i
=
d ln c
i
Þ
from (
3.27
). From (
3.40
), it follows that
