Geology Reference
In-Depth Information
is to suppose that, during the climb, material is transported to or from a reservoir
consisting of the surface of a cylinder of radius R, R being the average distance to
neighboring dislocations or other structural discontinuities or surfaces that can act
in such a case is
J
=
l
¼
2pkDl
In
ð
R
=
r
Þ
ð
6
:
20
Þ
(Carslaw and Jaeger
1959
, p. 189), where J is the flux from a length l of dislocation
when the driving potential difference between the cylinder surface and the dislo-
cation is taken to be the difference in chemical potential Dl (assuming r and T as
independent variables), r the effective radius of the dislocation core, and k the
quantity M
=
V
m
where M is the mobility of the diffusing material and V its molar
vbl
=
V
m
;
which substituted in (
6.20
) leads to
2p
In
ð
R
=
r
Þ
MDl
b
v
¼
ð
6
:
21
Þ
or
v
¼
MDl
b
ð
6
:
22
Þ
The mobility M in (
6.22
) is related to the diffusion coefficient D by the Einstein
work term in the chemical potential as Dl
¼
V
m
r if there is no potential barrier at
the source or sink. Substituting these quantities in (
6.22
) leads again to
v
DV
m
r
bRT
ð
6
:
23a
Þ
or, in the approximation that V
m
Lb
3
;
where L is the Avogadro number, to
v
Db
2
r
kT
ð
6
:
23b
Þ
The two results in Eqs. (
6.19
) and (
6.23a
) are, of course, equivalent apart from
minor differences in the approximations due to slight differences implicit in the
posing of the two climb models.
In applying (
6.23a
) to elements, V
m
and D are the molar volume and self-
diffusion coefficient for the atomic species. In the case of compounds, V
m
is the
molar volume of the molecular species constituting the crystal, and D is the
effective self-diffusion coefficient for this species, which can be obtained from
the self-diffusion coefficients of the constituent atomic species.
