General Concepts of Crystallization: Some Recent …
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properties of critical clusters as described in terms of Gibbs theory, Eq. 4, or are
governed by more advanced relations. Such analysis could supply us possibly with
additional suggestions concerning the applicability of CNT, respectively, its limits
in application to crystal nucleation.
4.2 Interplay of Crystallization and Glass Transition
Deviations of the properties of critical clusters as compared with the properties of
the evolving macroscopic phases can be found frequently in different types of phase
transformation processes. Going beyond such general type of behavior, crystallization is characterized by an additional particular feature which may be denoted as
interplay of crystallization and glass transition. Here a variety of problems can be
distinguished [45], we will concentrate on only some of them. As the starting point,
we take an experimental fact observed first around 1980 [46] which turned out in the
course of subsequent studies to be a very general phenomenon, an unexpected type
of dependence of the work of critical cluster formation on temperature [16, 18, 44].
Mentioned result is found based on measurements of both steady-state nucleation
rate and time-lag in nucleation. The time-lag can be described theoretically in terms
of CNT by Eq. 21. Utilizing this relation, one can replace it in the pre-exponential
term in Eq. 3 diffusion coefficient or viscosity by the time-lag. Having at one’s
disposal both parameters, steady-state nucleation rate, and time-lag data, one can
then determine via Eq. 2 how the work of critical cluster formation depends on
temperature (or pressure if the respective measurements will be performed). In line
with CNT, it decreases with decreasing temperature starting at the melting or liquidus
temperature but this decrease is observed only down to temperatures corresponding to
the maximum of the steady-state nucleation rate (or the conventional glass transition
temperature). With a further decrease of temperature, the work of critical cluster
formation increases then again in contradiction to expectations based on CNT.
In [46], such behavior was interpreted originally as a consequence of a similar
temperature dependence of the surface tension. This interpretation is followed by
some authors till now but can be hardly given a foundation in terms of Gibbs’ classical
theory of capillarity [16–18]. In addition, it contradicts a variety of measurements
showing a decrease of the latent heat of melting with the size of the crystallites and
general rules like the principle of le Chatelier-Braun: With an increase of the degree of
metastability, the surface tension is expected to decrease to favor nucleation processes
counteracting the mentioned increase of the level of deviation from equilibrium.
For this reason, other factors have been analyzed with respect to the question
whether they allow one to interpret the described above behavior. In a first such
attempt [16], it was checked whether elastic stresses evolving as the result of critical
cluster formation may be responsible for the observed increase of the work of critical
cluster formation. Utilizing the theoretical concepts derived in terms of CNT sketched
briefly here earlier it turns out that stresses do have an effect but it is not sufficient
for an explanation of the experimental data. In a next study [47], in order to reconcile
15
properties of critical clusters as described in terms of Gibbs theory, Eq. 4, or are
governed by more advanced relations. Such analysis could supply us possibly with
additional suggestions concerning the applicability of CNT, respectively, its limits
in application to crystal nucleation.
4.2 Interplay of Crystallization and Glass Transition
Deviations of the properties of critical clusters as compared with the properties of
the evolving macroscopic phases can be found frequently in different types of phase
transformation processes. Going beyond such general type of behavior, crystallization is characterized by an additional particular feature which may be denoted as
interplay of crystallization and glass transition. Here a variety of problems can be
distinguished [45], we will concentrate on only some of them. As the starting point,
we take an experimental fact observed first around 1980 [46] which turned out in the
course of subsequent studies to be a very general phenomenon, an unexpected type
of dependence of the work of critical cluster formation on temperature [16, 18, 44].
Mentioned result is found based on measurements of both steady-state nucleation
rate and time-lag in nucleation. The time-lag can be described theoretically in terms
of CNT by Eq. 21. Utilizing this relation, one can replace it in the pre-exponential
term in Eq. 3 diffusion coefficient or viscosity by the time-lag. Having at one’s
disposal both parameters, steady-state nucleation rate, and time-lag data, one can
then determine via Eq. 2 how the work of critical cluster formation depends on
temperature (or pressure if the respective measurements will be performed). In line
with CNT, it decreases with decreasing temperature starting at the melting or liquidus
temperature but this decrease is observed only down to temperatures corresponding to
the maximum of the steady-state nucleation rate (or the conventional glass transition
temperature). With a further decrease of temperature, the work of critical cluster
formation increases then again in contradiction to expectations based on CNT.
In [46], such behavior was interpreted originally as a consequence of a similar
temperature dependence of the surface tension. This interpretation is followed by
some authors till now but can be hardly given a foundation in terms of Gibbs’ classical
theory of capillarity [16–18]. In addition, it contradicts a variety of measurements
showing a decrease of the latent heat of melting with the size of the crystallites and
general rules like the principle of le Chatelier-Braun: With an increase of the degree of
metastability, the surface tension is expected to decrease to favor nucleation processes
counteracting the mentioned increase of the level of deviation from equilibrium.
For this reason, other factors have been analyzed with respect to the question
whether they allow one to interpret the described above behavior. In a first such
attempt [16], it was checked whether elastic stresses evolving as the result of critical
cluster formation may be responsible for the observed increase of the work of critical
cluster formation. Utilizing the theoretical concepts derived in terms of CNT sketched
briefly here earlier it turns out that stresses do have an effect but it is not sufficient
for an explanation of the experimental data. In a next study [47], in order to reconcile