32
K. Adrjanowicz
dispersion (ε
) and the loss peak (ε
). When crystallization takes place, the dielectric
response of glass-forming liquids changes with time. As can be seen, in the real part
of the dielectric permittivity we observe a decrease of the static permittivity increment, while in the imaginary part the same situation is reflected by a gradual drop of
the α-peak intensity. The decrease of the dielectric strength with time indicates that
the number of reorientating dipoles is drastically reduced as crystallization proceeds.
In the liquid phase, reorientational movements of the molecules are freed, while in
the crystalline phase restricted. Thus, we can use this very useful feature to follow the
crystallization kinetics of the glass-forming systems with the use of dielectric spectroscopy. The idea relies, however, on the assumption that the total dipole moment is
associated with a molecule as a whole. In such a case, changes recorded in the loss
spectra during crystallization can be assumed to give equivalent information about
the kinetics of the transformation as that obtained from the diffraction measurements.
This might not be essentially true for more complex systems like polymers for which
the dipole moment might be attached to the skeletal bonds or located only in some
flexible side groups.
Changes occurring in the static permittivity can be used to follow crystallization
kinetics after normalization according to the following formula:
ε N (t) =
ε
initial − ε
(t)
ε
initial − ε
final
(7)
where ε
initial and ε
final are the initial and final values of the real part of dielectric
permittivity at selected frequency taken from the low-frequency (static) regime. For
imaginary part of the dielectric permittivity, the ongoing changes as due to crystallization can be followed by analyzing the intensity of the α-relaxation peak. Its
complete disappearance and a flat signal, as demonstrated in Fig. 3b, points out for
complete crystallization of the sample (as reorientational movements in crystals are
not possible anymore). The exemplary of the normalized curve plotted versus time is
shown in the inset of Fig. 3. In such a case, the two limiting values –0 and 1—refers
to 0 and 100% crystallinity. To determine the rate of crystallization the normalized
dielectric data are fitted using Avrami equation [60]:
ε
N (t) = 1 − exp
−kt
n
(8)
where n is the Avrami exponent, and k is the crystallization constant rate related
to nucleation I and growth U rates via the relation k = IU
n−1 . In turn, the Avrami
parameter depends on the growth mechanisms and crystal shape. It typically varies
within 1–4. Solid red lines presented in the insets of Fig. 3b are the best fits of the
experimental data to the Avrami equation. It should be noted that the values of k
calculated using Eq. 8 are given as seconds to the power of (−n). Therefore, to have
the crystallization rates presented in more practically meaningful units, i.e., s
−1 , the
values obtained from Eq. 8 should be corrected by the value of the Avrami parameter.
K. Adrjanowicz
dispersion (ε
) and the loss peak (ε
). When crystallization takes place, the dielectric
response of glass-forming liquids changes with time. As can be seen, in the real part
of the dielectric permittivity we observe a decrease of the static permittivity increment, while in the imaginary part the same situation is reflected by a gradual drop of
the α-peak intensity. The decrease of the dielectric strength with time indicates that
the number of reorientating dipoles is drastically reduced as crystallization proceeds.
In the liquid phase, reorientational movements of the molecules are freed, while in
the crystalline phase restricted. Thus, we can use this very useful feature to follow the
crystallization kinetics of the glass-forming systems with the use of dielectric spectroscopy. The idea relies, however, on the assumption that the total dipole moment is
associated with a molecule as a whole. In such a case, changes recorded in the loss
spectra during crystallization can be assumed to give equivalent information about
the kinetics of the transformation as that obtained from the diffraction measurements.
This might not be essentially true for more complex systems like polymers for which
the dipole moment might be attached to the skeletal bonds or located only in some
flexible side groups.
Changes occurring in the static permittivity can be used to follow crystallization
kinetics after normalization according to the following formula:
ε N (t) =
ε
initial − ε
(t)
ε
initial − ε
final
(7)
where ε
initial and ε
final are the initial and final values of the real part of dielectric
permittivity at selected frequency taken from the low-frequency (static) regime. For
imaginary part of the dielectric permittivity, the ongoing changes as due to crystallization can be followed by analyzing the intensity of the α-relaxation peak. Its
complete disappearance and a flat signal, as demonstrated in Fig. 3b, points out for
complete crystallization of the sample (as reorientational movements in crystals are
not possible anymore). The exemplary of the normalized curve plotted versus time is
shown in the inset of Fig. 3. In such a case, the two limiting values –0 and 1—refers
to 0 and 100% crystallinity. To determine the rate of crystallization the normalized
dielectric data are fitted using Avrami equation [60]:
ε
N (t) = 1 − exp
−kt
n
(8)
where n is the Avrami exponent, and k is the crystallization constant rate related
to nucleation I and growth U rates via the relation k = IU
n−1 . In turn, the Avrami
parameter depends on the growth mechanisms and crystal shape. It typically varies
within 1–4. Solid red lines presented in the insets of Fig. 3b are the best fits of the
experimental data to the Avrami equation. It should be noted that the values of k
calculated using Eq. 8 are given as seconds to the power of (−n). Therefore, to have
the crystallization rates presented in more practically meaningful units, i.e., s
−1 , the
values obtained from Eq. 8 should be corrected by the value of the Avrami parameter.