Crystallization of Amorphous Pharmaceuticals at Ambient and Elevated …
67
Fig. 7 Example of the Avrami–Avramov plot performed on the data for NIM collected at T =
328 K. The evolution of normalized real permittivity ε
N (orange circles) and its first derivative
versus the natural logarithm of the time (black squares)
It is worth noting that the derivative:
dε
N (t)
d(ln(t − t 0 ))
= n
t − t 0
τ cr
n
exp
−
t − t 0
τ cr
n
(7)
reaches the maximum value of ε
N (d
2
ε
N / d(ln (t − t 0 ))
2
= 0) at t = τ cr + t 0 .
Therefore, assuming that t 0 = 0 s, one can determine the value of characteristic time
of the crystallization process (τ cr ) from the d
2
ε
N / d(ln t)
2 peak maximum ((ε
N )
max ).
In a similarly simple way it is possible to calculate also n i.e. the another Avramov
parameter which is related to the nucleation dimensionality. This parameter can be
calculated by employing the following equation:
n =
ε
N
max
0.368
(8)
There is also an alternative method that can be used to evaluate the parameter
n. This method is based on drawing a tangent to the experimentally determined
sigmoidal curve ε
N (ln t) (see dashed line in Fig. 7). By determining the values of
lnt 1 and lnt 2 , which corresponds to the points of intersection of the tangent line with
the horizontal straight lines, constructed at the limit values of ε
N i.e. at 0 and 1, it is
possible to establish the n parameter from the following formula:
n =
e
ln t 2 − ln t 1
(9)
67
Fig. 7 Example of the Avrami–Avramov plot performed on the data for NIM collected at T =
328 K. The evolution of normalized real permittivity ε
N (orange circles) and its first derivative
versus the natural logarithm of the time (black squares)
It is worth noting that the derivative:
dε
N (t)
d(ln(t − t 0 ))
= n
t − t 0
τ cr
n
exp
−
t − t 0
τ cr
n
(7)
reaches the maximum value of ε
N (d
2
ε
N / d(ln (t − t 0 ))
2
= 0) at t = τ cr + t 0 .
Therefore, assuming that t 0 = 0 s, one can determine the value of characteristic time
of the crystallization process (τ cr ) from the d
2
ε
N / d(ln t)
2 peak maximum ((ε
N )
max ).
In a similarly simple way it is possible to calculate also n i.e. the another Avramov
parameter which is related to the nucleation dimensionality. This parameter can be
calculated by employing the following equation:
n =
ε
N
max
0.368
(8)
There is also an alternative method that can be used to evaluate the parameter
n. This method is based on drawing a tangent to the experimentally determined
sigmoidal curve ε
N (ln t) (see dashed line in Fig. 7). By determining the values of
lnt 1 and lnt 2 , which corresponds to the points of intersection of the tangent line with
the horizontal straight lines, constructed at the limit values of ε
N i.e. at 0 and 1, it is
possible to establish the n parameter from the following formula:
n =
e
ln t 2 − ln t 1
(9)