Geology Reference
In-Depth Information
molecular components (Brady
1975b
; Lasaga
1979
; Lasaga et al.
1977
).
Crystal defects such as vacancies must also be represented among the com-
ponents when it is implicit that the crystal lattice is conserved in the diffusion.
(2) A choice must also be made of a reference frame for the fluxes and the
concentration gradients, bearing in mind that, with different components dif-
fusing at different rates, volumes may be changing or centers of mass or
volume moving relative to external axes and that the values of the diffusion
coefficients will depend on the choice of the reference frame (Anderson
1981
;
Brady
1975a
; Crank
1975
; De Groot and Mazur
1962
). The following are
some of the possibilities:
(a) Laboratory frame, such as a notional grid fixed to one end of the specimen.
While commonly the initial choice for representing measured concentra-
tion profiles, it is often not the most appropriate for subsequent analysis
where the movement of components relative to each other is of interest.
(b) Velocity-fixed frames which move with the mean displacement of volume,
mass, substance, or a particular component of substance relative to the
laboratory frame—called, respectively, a volume-fixed, mass-fixed,
molar-fixed, or nth component-fixed frame. The volume-fixed frame is
particularly appropriate when the volume of the system remains constant
while the center of volume moves relative to the laboratory frame. Dif-
fusion coefficients measured relative to a volume-fixed frame are some-
times called ''standard diffusion coefficients'' (Hooyman et al.
1953
). The
nth component-fixed frame is often also called the solvent-fixed frame. If
the terms involving the nth component are omitted in (
3.31
), it is implicit
that an appropriate velocity-fixed frame is used.
(c) Inert marker frame based on inert markers embedded in the system or, in the
case of crystals, notionally attached to the lattice or unit cell (lattice-fixed
frame). This frame is used when it is desired to eliminate from consideration
any bulk flow of substance, that is, common displacement of all compo-
nents. Diffusion coefficients referred to an inert marker frame are sometimes
called ''intrinsic diffusion coefficients'' (Hartley and Crank
1949
) since they
relate to the intrinsic mobility of the individual component relative to a
section through which no bulk flow occurs. For a binary system, the inter-
diffusion coefficient D
V
referred to a volume-fixed frame is related to the
''intrinsic'' diffusion coefficients D
1
;
D
2
of the two components, 1, 2 by
D
V
¼
x
1
D
1
þ
x
2
D
2
;
where x
1
;
x
2
are the respective mole fractions.
(3) Since the practical measurement of diffusion coefficients is usually done under
nonsteady-state conditions, a generalization of Fick's second law is also needed,
obtained by combining (
3.31
) with an equation of continuity. In cases involving
chemical diffusion coefficients, where concentration dependence is likely to
arise, methods of solution are more complex than for concentration-independent
cases such as tracer diffusion, as is illustrated in the Matano treatment of inter-
diffusion in a binary system; see Crank (
1975
) for analysis in binary systems and
