Geology Reference
In-Depth Information
(Manning
1968
,
1974
). In the case of one component chemical diffusion, F is
equal to
kT d ln c
=
d
ð Þ;
thus leading back to (
3.28
). In a multicomponent system,
there will be additional terms relating to the concentration gradients of the other
components; in general, these terms will appear as cross-terms D
ij
i
6¼ ð Þ
in (
3.30
)
but where only one additional species is involved the coupling effect may con-
veniently be expressed as an additional factor in the primary diffusion coefficient,
as is done for the ''vacancy wind'' effect in simple cases of diffusion by a vacancy
mechanism (Flynn
1972
, Chap. 8; Manning
1968
,
1974
).
From (
3.35
), it is seen that the main part of the temperature dependence of D,as
expressed in an experimental activation energy Q,(
3.29
), lies in D
. Through
(
3.32
) and (
3.34
), Q will therefore contain mainly contributions from E but it may
also include a contribution from the temperature dependence of f in more complex
crystals. Often it is possible to distinguish a higher temperature, ''intrinsic''
regime, where diffusion involves thermally generated defects and Q reflects con-
tributions from both defect formation and migration terms in E, from a lower
temperature, ''extrinsic'' regime where defects already present are involved and
Q is lower due to there being no defect formation term in E (this concept of an
intrinsic regime should not be confused with the intrinsic diffusion coefficient
mentioned earlier).
The pre-exponential term D
0
in (
3.30
) is correspondingly seen to be of the order
of d
2
m exp DS
=ð Þ
if a and f are taken to be of the order of unity and TDS is the
entropic part of E.IfDS is small and d * 0.1 nm, D
0
will be roughly of the order
of 10
-6
m
2
s
-1
; however, it is difficult in general to estimate DS and reported
values of D
0
range many orders of magnitude either way from 10
-6
m
2
s
-1
.
When the diffusing species is ionized and an electric field E is present, the drift
force F is equal to zeF where z is the effective charge number and e the elementary
charge so that, in (
3.33
), v
F
¼
uzeF where u is the electric mobility of the charged
species. Through the Einstein relation D
r
¼
ukT
=
ez (Atkins
1986
, p. 675), a
theoretical diffusion coefficient D
r
can be obtained from electrical conductivity
measurements that give u. The ratio D
=
D
r
is known as the Haven ratio (Le Claire
1976
), the determination of which is helpful in identifying the diffusion mecha-
nism (for example, the Haven ratio is equal to the correlation factor f in the case of
a simple vacancy mechanism). Another aspect of ionic diffusion is that the dis-
placement of entities of one sign only sets up an internal electric field, known as a
diffusion or Nernst potential (Manning
1968
, Chap. 7), which acts on the oppo-
sitely charged species. This potential tends to rise to a level sufficient to maintain
equal fluxes in opposite directions for identically charged species or equal fluxes in
the same direction for oppositely charged species, thus preserving stoichiometry
and neutrality. For this reason, it is often more appropriate in a macroscopic
treatment to view the neutral components as the diffusing species. However, the
problem then arises of relating the diffusion coefficient of the molecular species to
the diffusion coefficients of the constituent ionic or atomic species, which may be
more easily measured, for example, by isotopic tracer methods.
