General Concepts of Crystallization: Some Recent …
3
treatment of experimental data of nucleation and growth kinetics but even a quantitatively correct description, in other cases, it may fail even dramatically. Consequently,
the question arises on how these problems can be explained and overcome, what the
origin is for the success of CNT in some and its failure in other cases. These problems are considered here in detail starting with some recently obtained results and
directing then the attention to problems we consider as highly perspective in the
further development of theoretical and experimental analysis of crystal nucleation
and growth. The general considerations are supplemented by a section dealing with
selected specific features of polymer crystallization.
2 Classical Nucleation Theory
2.1 Thermodynamic Driving Force of Crystallization
in Dependence on Temperature and Pressure
One of the basic ingredients in the application of classical nucleation theory to the
description of experimental data consists of the appropriate specification of the work,
W c , of critical cluster formation in nucleation:
W c =
1
3
σ A, A = 4π R
2
,
p β − p α
+ σ
dA
dV
= 0
( 1 )
where σ is the surface tension, A is the surface area of a critical cluster supposed to
be of spherical shape with a radius R corresponding to the surface of tension, p is the
pressure and V is the volume, the indices α and β specify the parameters of the critical
crystal clusters (α), respectively, the parameters of the ambient phase (β) where the
aggregates of the new phase are formed. W c determines widely the probability of
formation of a supercritical cluster of the newly evolving phase capable to a further
deterministic growth, respectively, the steady-state nucleation rate, J:
J = J 0 exp
−
W c
k B T
(2)
The steady-state nucleation rate is equal to the number of critical clusters formed
per unit time in a unit volume of the ambient phase. Here k B is the Boltzmann
constant, T is the absolute temperature.
Critical crystallites are, in general, not of spherical but of different shapes, anyway,
also crystallization can be described in terms of the above given simplified model
as explained in detail in [7]. In brief, for any state of the ambient phase the pressure
in the critical cluster is determined by the equilibrium conditions. It follows that for
any value of the work of critical cluster formation, it is always possible to determine
3
treatment of experimental data of nucleation and growth kinetics but even a quantitatively correct description, in other cases, it may fail even dramatically. Consequently,
the question arises on how these problems can be explained and overcome, what the
origin is for the success of CNT in some and its failure in other cases. These problems are considered here in detail starting with some recently obtained results and
directing then the attention to problems we consider as highly perspective in the
further development of theoretical and experimental analysis of crystal nucleation
and growth. The general considerations are supplemented by a section dealing with
selected specific features of polymer crystallization.
2 Classical Nucleation Theory
2.1 Thermodynamic Driving Force of Crystallization
in Dependence on Temperature and Pressure
One of the basic ingredients in the application of classical nucleation theory to the
description of experimental data consists of the appropriate specification of the work,
W c , of critical cluster formation in nucleation:
W c =
1
3
σ A, A = 4π R
2
,
p β − p α
+ σ
dA
dV
= 0
( 1 )
where σ is the surface tension, A is the surface area of a critical cluster supposed to
be of spherical shape with a radius R corresponding to the surface of tension, p is the
pressure and V is the volume, the indices α and β specify the parameters of the critical
crystal clusters (α), respectively, the parameters of the ambient phase (β) where the
aggregates of the new phase are formed. W c determines widely the probability of
formation of a supercritical cluster of the newly evolving phase capable to a further
deterministic growth, respectively, the steady-state nucleation rate, J:
J = J 0 exp
−
W c
k B T
(2)
The steady-state nucleation rate is equal to the number of critical clusters formed
per unit time in a unit volume of the ambient phase. Here k B is the Boltzmann
constant, T is the absolute temperature.
Critical crystallites are, in general, not of spherical but of different shapes, anyway,
also crystallization can be described in terms of the above given simplified model
as explained in detail in [7]. In brief, for any state of the ambient phase the pressure
in the critical cluster is determined by the equilibrium conditions. It follows that for
any value of the work of critical cluster formation, it is always possible to determine