Geology Reference
In-Depth Information
For nonsteady-state conditions, combining (
3.24
) with an equation of continuity
(
Sect. 3.4
) leads to Fick's second law,
or
2
c
ox
2
o
c
ot
¼
o
D
o
c
ox
o
c
ot
¼
D
o
ð
3
:
25
Þ
ox
where the second form applies only for D independent of concentration. In the
latter case, it is identical with the heat flow equation, for which many solutions are
known (Carslaw and Jaeger
1959
). It is to be noted, however, that, in contrast to
the case of heat flow, the conductivity parameter in (
3.24
) is identical to the
diffusivity parameter in (
3.25
), that is, in atomic diffusion, the diffusion coefficient
and the diffusivity are identical.
Equations (
3.25
) are commonly used in experimental studies. Starting from a
known distribution, the concentration profile is measured after a certain time and
compared with the appropriate solution of (
3.25
) in order to evaluate D (Crank
1975
). In the case of a semi-infinite solid with zero initial concentration, placed in
contact with a reservoir that maintains a constant concentration c
0
of the diffusing
species at the surface, the concentration profile at time t is given by
n
o
p
Dt
cx
; ðÞ¼
c
0
1
erf
x
=
2
where erf z is the error function. In this case, the value of c falls to about 0
:
5 c
0
at
x
p
p
Dt
Dt
and to 0
:
1 c
0
at x
¼
2
. It is, therefore, convenient in approximate
p
Dt
p
Dt
calculations to use
as a ''diffusion distance''.
The relationship between the mobility M and the diffusion coefficient D can be
obtained if we relate the chemical potential to the concentration through the
equation of state
or 2
l
¼
l
þ
RT ln cc
ð
3
:
26
Þ
where l
is the chemical potential in a reference state, c the activity coefficient,
R the gas constant, and T the absolute temperature. Substituting this relation into
(
3.23
) and comparing with (
3.24
) leads to
1
þ
d ln c
d ln c
D
¼
RTM
ð
3
:
27
Þ
and hence to
cD
dl
dx
j
¼
ð
3
:
28
Þ
1
þ
d
ln c
RT
d
ln c
The so-called thermodynamic factor 1
þ
d ln c
=
d ln
ð Þ
is unity for ideal mixing, as
in the case of low concentrations or self-diffusion, in which case, (
3.27
) becomes
the well-known Einstein relation D
¼
RTM
:
