In [99] the following equation describing interstitial
cluster kinetics is given:

(14.1) 
where
is the interstitial diffusivity, is
the average interatomic spacing, is the capture radius
expressed in units of , is the concentration of
interstitials trapped in clusters, is the concentration of
free interstitials and is the annealing temperature.
Here the main formula of the model for the change of the concentration
of clustered interstitials is

(14.2) 
where denotes the concentration of clustered interstitials,
time, the concentration of unclustered interstitials, and
the equilibrium concentration of interstitials (which can
be found by solving
). There is a number of
parameters to be adjusted:
,
,
(the reaction constants); the exponents
,
,
, and
,
, and
; and finally
.
The reaction constants have the form
with
,
, and
.
Here is the temperature (in Kelvin) and
the Boltzmann
constant. Since the coefficients
,
, and
are positive, the first two
terms in (14.2) are responsible
for the formation of clusters and the last term for the dissolution.
The sum of interstitials counted in and remains constant and
the initial value of is
.
The ratio of the concentration of the unclustered
interstitials and its equilibrium concentration is often called the
interstitial supersaturation. Here additional exponents modify the
interstitial supersaturation which appears in the form
and
.
Figure 14.2:
Result for the high parameter set corresponding to
parameter values shown in
Table 14.3. The logarithm
(base ) of the simulated and measured concentration
of interstitial clusters is shown depending on
time
.

Figure 14.3:
Result for the low parameter set corresponding to
parameter values shown in
Table 14.3. The logarithm
(base ) of the simulated and measured concentration
of interstitial clusters is shown depending on
time
.

The first term
describes the joining of two clusters and thus
the expected values for the exponents are
. The second growth term
governs the case when an unclustered interstitials joins an
interstitial cluster. Here we can expect the exponents to be unity.
The second factor is a linear combination of and with an
exponent.
Comparing (14.1) and
(14.2), the growth term of
(14.1), basically being a reaction constant times ,
is split into two parts providing greater flexibility: one depending
on a modified interstitial supersaturation term and one depending on a
modified interstitial supersaturation term times
. In the dissolution term an exponent which was
later found to be is added.
Clemens Heitzinger
20030508