16
J. W. P. Schmelzer and C. Schick
experimental data and CNT, we assumed an increase of the size, d 0 , of the structural
units that control nucleation with decreasing temperature for temperatures below the
nucleation rate maximum, T < T max . This hypothesis was tested for several glassforming liquids, where crystal formation proceeds by bulk homogeneous nucleation.
It can explain also the temperature dependence of the nucleation rate in the range
T < T max , where the description of nucleation rate by CNT drastically fails. The
size of the structural units can be correlated either with the size of the cooperatively
rearranging regions (CRR) or connected with an effective size parameter, accounting
for corrections in the theoretical treatment of the kinetics of aggregation in multicomponent systems via a quasi-one-dimensional description.
In a third approach [48], a model for the description of crystal nucleation is
proposed incorporating into classical nucleation theory concepts of spatial heterogeneity of glass-forming liquids. It is assumed that nucleation processes may proceed
with detectable rates only in liquid-like (soft) regions and are suppressed in solidlike (rigid) parts. Determining appropriately the fraction of liquid-like, respectively,
solid-like regions in dependence on temperature, this approach allows one to achieve
a satisfactory agreement between classical nucleation theory and experiment not
only at relatively high temperatures but also at temperatures lower than that of the
nucleation rate maximum. The model was tested successfully on several silicate and
polymer glasses revealing homogeneous volume nucleation. Some other phenomena
in the interplay of crystallization and glass transition are also discussed in this analysis
giving an independent verification of the validity of our basic assumption.
But there exists also another feature, we consider as so far not appropriately incorporated into the theoretical description of crystallization if one would like to account
appropriately for the interplay of crystal phase formation and glass transition. In the
analysis of the theoretical description of stress development and stress relaxation it
has been shown by us that the effect of elastic stresses on crystal nucleation depends
basically on the ratio of the time-lag, the time to establish steady-state conditions
in nucleation, and the Maxwellian relaxation time. For liquids, this ratio has to be
consequently large to prevent the effect of elastic stresses. For glasses as frozen-in
liquids, the opposite situation should be fulfilled, i.e., this ratio should tend to zero in
order to obtain in the theory the limiting cases of a Hookean solid as a special limiting
case. Accounting for a curvature dependence of the surface tension we arrived at the
conclusion that near to the conventional glass transition temperature (corresponding
to a viscosity 10
12 Pa s) this ratio is of the order of one and has to tend to zero below
the glass transition temperature.
However, once this is the case, another problem arises. In CNT, the thermodynamic
driving force is computed as the difference between the bulk states of the system both
in the crystalline states and the metastable liquid. As already mentioned, the critical
crystal cluster may have, however, different properties as compared to the respective
macroscopic crystal phase. But, in addition, once the mentioned ratio tends to zero,
the initial state of the liquid will not refer to the metastable equilibrium state but to
a particular non-equilibrium state realized in the course of cooling. Both the thermodynamic driving force for crystal nucleation and the surface tension will depend
J. W. P. Schmelzer and C. Schick
experimental data and CNT, we assumed an increase of the size, d 0 , of the structural
units that control nucleation with decreasing temperature for temperatures below the
nucleation rate maximum, T < T max . This hypothesis was tested for several glassforming liquids, where crystal formation proceeds by bulk homogeneous nucleation.
It can explain also the temperature dependence of the nucleation rate in the range
T < T max , where the description of nucleation rate by CNT drastically fails. The
size of the structural units can be correlated either with the size of the cooperatively
rearranging regions (CRR) or connected with an effective size parameter, accounting
for corrections in the theoretical treatment of the kinetics of aggregation in multicomponent systems via a quasi-one-dimensional description.
In a third approach [48], a model for the description of crystal nucleation is
proposed incorporating into classical nucleation theory concepts of spatial heterogeneity of glass-forming liquids. It is assumed that nucleation processes may proceed
with detectable rates only in liquid-like (soft) regions and are suppressed in solidlike (rigid) parts. Determining appropriately the fraction of liquid-like, respectively,
solid-like regions in dependence on temperature, this approach allows one to achieve
a satisfactory agreement between classical nucleation theory and experiment not
only at relatively high temperatures but also at temperatures lower than that of the
nucleation rate maximum. The model was tested successfully on several silicate and
polymer glasses revealing homogeneous volume nucleation. Some other phenomena
in the interplay of crystallization and glass transition are also discussed in this analysis
giving an independent verification of the validity of our basic assumption.
But there exists also another feature, we consider as so far not appropriately incorporated into the theoretical description of crystallization if one would like to account
appropriately for the interplay of crystal phase formation and glass transition. In the
analysis of the theoretical description of stress development and stress relaxation it
has been shown by us that the effect of elastic stresses on crystal nucleation depends
basically on the ratio of the time-lag, the time to establish steady-state conditions
in nucleation, and the Maxwellian relaxation time. For liquids, this ratio has to be
consequently large to prevent the effect of elastic stresses. For glasses as frozen-in
liquids, the opposite situation should be fulfilled, i.e., this ratio should tend to zero in
order to obtain in the theory the limiting cases of a Hookean solid as a special limiting
case. Accounting for a curvature dependence of the surface tension we arrived at the
conclusion that near to the conventional glass transition temperature (corresponding
to a viscosity 10
12 Pa s) this ratio is of the order of one and has to tend to zero below
the glass transition temperature.
However, once this is the case, another problem arises. In CNT, the thermodynamic
driving force is computed as the difference between the bulk states of the system both
in the crystalline states and the metastable liquid. As already mentioned, the critical
crystal cluster may have, however, different properties as compared to the respective
macroscopic crystal phase. But, in addition, once the mentioned ratio tends to zero,
the initial state of the liquid will not refer to the metastable equilibrium state but to
a particular non-equilibrium state realized in the course of cooling. Both the thermodynamic driving force for crystal nucleation and the surface tension will depend