Geology Reference
In-Depth Information
where A
0
;
v
0
are empirical constants, Q is an empirical activation energy
(enthalpy), m and DA
are the empirical parameters
ð
olnv
=
olns
Þ
T
and
ð
RT
=
b
Þð
olnv
=
os
Þ
T
;
respectively, and R is the gas constant (note that DA
and m are
formally related by m
sbDA
=
RT ). In Eqs. (
6.15
) and (
6.16
) the activation area
and activation energy have been multiplied by the Avogadro constant, relative to
the values in (
6.14
), and the entropic part of the energy has been subsumed under
v
0
;
A
0
.
Following
Sect. 3.2.4
,
(
6.16
) should be the more appropriate form to use when
sbDA
2RT
:
This situation tends to arise at relatively high stress and low tem-
perature, when there are important viscous drag effects such as the Peierls resis-
tance or the analogous low-temperature drag on screw dislocations in iron, thought
to arise from the extension of core into a zonal structure in three intersecting
planes (Hirsch
1960
; Hirth and Lothe
1982
, p. 369; Louchet
1979
; Mitchell et al.
1963
). Alternatively, if viscous drag effects are still controlling the dislocation
velocity at relatively low stress and high temperature with sbDA
2RT
;
then the
form in (
6.15
) should be the more appropriate and m could be expected to be close
to unity.
Turning to the experimental situation, there have been various direct mea-
surements of mean dislocation velocity in glide as a function of stress and tem-
perature, using etching, X-ray topography or transmission electron microscopy for
tracking the dislocations (for reviews, see Alexander and Haasen
1968
; Gilman
1969
; Haasen
1978
, Chap. 11; Sprackling
1976
, Chap. 9). The stress to which the
velocity is related is commonly, and logically, taken to be the ''effective stress'' s
e
acting locally on the dislocation. The effective stress is estimated as the applied
stress minus the internal stress s
i
arising from other dislocation, s
i
being taken to
be equal to aGbq
1
=
2
(when a is a numerical constant near to unity, G the shear
modulus, b the Burgers vector, and q the dislocation density; a rationalization for
this expression will be given in
Sect. 6.6.2
). When the experimental results are
fitted to the form (
6.15
), the exponent m is found to vary widely, from values of
approximately unity for silicon and germanium to values of 100 or more for
copper. The value of m can also vary markedly with effective stress, so that several
velocity/stress regimes can be distinguished as the stress is increased, commonly
with m * 1 at very low or very high velocities and with higher values of m at
intermediate velocities (Haasen
1978
, p. 263).
Observed values of m
1 are consistent with a predominance of viscous drag
effects, and determinations of activation area may help to identify the controlling
factor. Thus, in the case of germanium and silicon where a value of m
1 is found
at around 700-1,000 K and applied stresses of 10-100 MPa (still a relatively low-
temperature regime for these materials), the activation area, calculated from
DA
¼
mkT
=
sb
;
is of the order of a few times b
2
(Louchet and George
1983
). This
result is suggestive of the dislocation motion being controlled by a local effect in
the core, such as kink nucleation or migration. In contrast, measurements on most
metals and ionic crystals at relatively low temperatures, analyzed in a similar way,
tend to give values of m much greater than unity and values of DA
large compared
