Geology Reference
In-Depth Information
1963
), Adda and Philibert (
1966
), Manning (
1968
,
1974
), Flynn (
1972
), Christian
(
1975
), Crank (
1975
), Le Claire (
1976
), Anderson (
1981
), Kirkaldy and Young
(
1987
), Philibert (
1991
), Allnatt and Lidiard (
1993
), Wilkinson (
2000
), Mehrer
(
2007
), and Cussler (
2009
).
The region of space through which the diffusion occurs is normally 3-dimen-
sional but in cases of strong anisotropy or of interfacial or pipe diffusion it can be
essentially 2D or even 1D. In the following sections, the diffusion equations will
be written for 1D diffusion but they are readily generalized to 3Ds, in which case
the diffusion coefficient becomes a second rank tensor with symmetry properties
appropriate to the material.
3.5.2 One Mobile Component
We deal first with the elementary case of the diffusion of one mobile component in
a matrix. Since the change in energy in an elementary volume as a result of adding
substances to it while other variables are held constant is determined by the
chemical potential of the substance, it can be expected that under isothermal and
isobaric conditions, the diffusive flux of the component will be determined by the
gradient of its chemical potential and that the relationship will be a simple
supports this view and we therefore write, for isothermal and isobaric conditions,
j
¼
L
d
dl
dx
¼
cM
dl
ð
3
:
23
Þ
dx
where j is the flux density, that is, the amount of substance passing per unit time
t through unit cross-sectional area in a defined frame of reference, l the chemical
potential, x the distance in the same frame of reference, and L
d
a phenomeno-
logical, kinetic or transport coefficient, which can also be written as the product of
the amount-of-substance concentration c and the mobility M (in this section, for
convenience, we use c instead of c
N
for the concentration). So long as volume
changes can be neglected, the immobile matrix serves as a frame of reference. The
rate of diffusion of the substance in the matrix can then be described by the
parameter M.
In practice, however, information about the rate of diffusion is derived from
measurements of concentration profiles and is expressed in terms of a diffusion
coefficient D, defined as the ratio of the flux density of substance j to the negative
of the gradient of the concentration (the SI units of D are m
2
s
-1
). This definition
stems from Fick's first law (Fick
1855
)
j
¼
D
dc
dx
ð
3
:
24
Þ
Note that D may depend on the concentration c.
