Geology Reference
In-Depth Information
j
1
¼
D
11
dc
1
dx
D
12
dc
2
dx
D
1n
dc
n
dx
j
2
¼
D
21
dc
1
dx
D
22
dc
2
dx
D
2n
dc
n
dx
.
j
n
¼
D
n1
dc
1
dx
D
n2
dc
2
dx
D
nn
dc
n
ð
3
:
31
Þ
dx
where, for each component i (i
¼
1
;
2
;
...n), j
i
is the flux density in terms of the
chosen concentration units with respect to a suitable frame of reference and dc
i
=
dx
is the gradient in concentration with respect to the same frame; the n
2
coefficients
D
ij
then specify the diffusion properties of the system and are sometimes called the
practical diffusion coefficients
. Quantities relating to the nth component can be
omitted in (
3.31
) if the frame of reference is chosen, so that there is no net flux
with respect to it and provided the system is closed, thus making the nth quantities
dependent.
There is, in general, no a priori reason to assume that the matrix
½
of the D
ij
is
symmetric in a given frame of reference or that any of its components will be zero,
although the off-diagonal terms (i
6¼
j) are commonly smaller than the diagonal
terms (i
¼
j). However, it is in general possible to transform the frame of reference
so as to diagonalize
½
, making all D
ij
(i
6¼
j) equal to zero and so relating the
diffusion coefficient for each component uniquely to its concentration gradient in
this frame; this property follows from the Onsager reciprocal relations (Cooper
1974
). It follows immediately that, in a binary system (n
¼
2
;
) only one diffusion
coefficient is needed to describe the interdiffusion of the two components in a
frame of reference for which there is no net flux. However, in a ternary system
there are not only two independent coefficients (D
11
;
D
22
) relating the diffusion of
a given substance to its own concentration gradient, but there are also coupling
coefficients (D
12
;
D
21
) relating this diffusion to the concentration gradient of the
other independent component. It should also be borne in mind that over a sub-
stantial range of compositions of a given chemical system the values of D
ij
can, in
general, be expected to be concentration dependent.
We cannot pursue here the manifold ramifications of multicomponent diffusion;
see, for example, Brady (
1975a
,
1975b
), Anderson and Graf (
1976
), Lasaga
(
1979
), Anderson (
1981
). However, attention may be drawn to the following
points of principle:
(1) A choice of the chemical components considered to constitute the system has
to be made in the light of the physics of the situation and the applications in
mind. From a thermodynamic point of view, the components will in general be
molecular species for which chemical potentials can be defined. In practice,
concentration gradients for ionic or isotopic species may be measured and
analyzed, but constraints such as electrical neutrality and stoichiometric bal-
ance tend eventually to reduce the essence of the situation to the diffusion of
