Geology Reference
In-Depth Information
Q
SO
SOURCE
−Δμ
SO
−Δμ
D
−Δμ
=
C
2
V
m
( )
σ
1
− σ
2
Q
SI
−Δμ
SI
SINK
TRANSFER PATH
Fig. 5.2 Components of thermodynamic potential difference driving transfer of material from
source to sink
The relative shortening of the body, represented by that of the domain in Fig. 5-1,
is of the order of nV
m
=
V where n is the amount of substance transferred (in moles) and
V is the volume of the domain. The macroscopic strain rate e in the direction of r
1
is
then given by
e
¼
C
1
V
m
V
dn
dt
ð
5
:
1
Þ
where C
1
is a dimensionless geometrical factor, generally[1, that takes into account
the actual orientations of sources and sinks and the actual relative movements of the
domains required to fit them together again after transferring the material.
The overall thermodynamic potential difference
Dl driving the transfer of
material is given by
Dl
¼
C
2
V
m
r
1
r
ð Þ
where l is the chemical potential and
C
2
is another dimensionless geometrical factor, generally \1, taking into account
r
1
r
3
is the applied stress difference. The quantity
Dl is made up of three parts
(Fig.
5.2
), the potential differences
Dl
so
and
Dl
si
required to drive any reac-
tions involved in releasing or attaching material at source and sink, respectively,
and the potential difference
Dl
D
required to drive the diffusion in the transfer
path. Thus, we can write
ð
Dl
so
Þþ
Dl
si
ð
Þþ
Dl
D
ð
Þ
C
2
V
m
r
1
r
3
ð
Þ
ð
5
:
2
Þ
We now relate dn
=
dt separately to the reactions at source and sink and to the
diffusion in the transfer path. In the case of the reactions a formulation analogous
to (
3.5
) can be used under the assumption that the reaction will be first order, with
a rate proportional to the number of sites in source or sink at which atoms or
