Chemistry Reference
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been observed to appear close to
T
g
for some crystal forms [8
11]. This mode of growth
is thought to be diffusionless, and hence occurs much faster than the diffusion controlled
growth described in Equation 5.5.
-
5.2.3 Overall Kinetics of Crystallization
The overall kinetics of crystallization from the melt at a particular temperature depends
on the rate of nucleation, the induction or lag time for nucleation to occur, and the crystal
growth rate. Based on the schematic shown in Figure 5.2, it would be unlikely that an
Arrhenius-type relationship between the crystallization rate and temperature would hold
for a very large temperature range. Thus, it can be very dif
cult to conduct accelerated
stability testing at higher temperatures and extrapolate results to lower temperatures with
good predictive success.
One of the most popular models to describe crystallization from amorphous systems
is that of Kolmogorov
Avrami (KJMA) [12,13]. This theory is based on
the assumption that changes in the degree of crystallization with time depend on the
volume fraction of noncrystallized material and that rates of both nucleation and growth
are time independent and that nucleation is homogeneous. If preseeding is absent, the lag
time for nucleation is often signi
-
Johnson
-
Mehl
-
cant and can be incorporated yielding the following
equation:
n
x
t
1
exp
K
t
τ
;
(5.6)
where
x
(
t
) is the fraction transformed into the crystalline phase at time
t
,
K
is a constant
determined by the nucleation and growth constants,
is the induction time (time before
any crystalline material can be detected), and
n
is a constant characteristic of the
experimental conditions called the Avrami exponent. The value of
n
can be used to
understand possible mechanisms responsible for transformations. This model has been
widely applied to pharmaceutical systems [14
τ
16]. Zhou et al. [15] studied the
crystallization of amorphous nifedipine. The experimental data obtained were
-
tted
to different models. The KJMA model showed a good
fit in the range of
x
0.05
-
0.8. At
=
the early stages of crystallization (for
x
0.05
-
0.6), experimental values
t the KJMA
=
model with
n
0.8. The change in the exponent was
attributed to the relative contributions of nucleation and crystal growth at different stages
of the crystallization reaction.
The preceding discussion of nucleation and growth shows that crystallization is a
complex phenomenon, in
3 and changed to
n
4 for
x
0.6
-
=
=
=
uenced by both thermodynamic and kinetic factors. At lower
temperatures, the undercooling is large (the system is further from the melting tempera-
ture), and there is a greater thermodynamic driving force for crystallization. At higher
temperatures, mobility of the material will be higher and crystallization may proceed
faster. Therefore, thermodynamic factors favor crystallization at lower temperatures,
while kinetic factors favor crystallization at higher temperatures. Thus, the crystallization
rate will be at a maximum somewhere between
T
g
and
T
m
, as shown in Figure 5.2, and
will depend on the complex interplay of thermodynamic and kinetic factors. In order to
understand the tendency of amorphous formulations to crystallize during storage, the
