154
K. Adrjanowicz and R. Richert
induced this polarization. In a polar liquid in which molecules carry a permanent
dipole moment μ, this polarization is comprised of two components. One is electronic
polarization leading to a permittivity ε ∞ which is approximately equal to the square
of the refractive index, ε ∞ ≈ n
2 . For a given compound, ε ∞ is mostly a matter of the
density, and therefore this contribution to polarization is not so different for liquid
and crystalline states. The situation is different for the dipole orientation contribution
to polarization, which rests on the orientational degrees of freedom of dipoles and
adds the amount of ε = ε s − ε ∞ , which can be much larger than ε ∞ itself. The
feature that is exploited in the context of crystallization studies is that, to a very good
approximation, the crystalline state does not contribute to the dielectric relaxation
amplitude ε. With these approximations, ε
crystal
= 0 and ε
crystal
∞
= ε
liquid
∞ , the
dielectric technique is assumed to yield the volume fraction of the liquid via:
υ
liquid
υ total =
ε
mi xture
ε liquid
(3)
For liquids with moderate polarity, say ε < ε ∞ , Eq. 3 may serve as a good
approximation to the liquid volume fraction, υ
liquid /υ
total . For materials with large
dielectric constants, the situation of a partly crystallized sample is that of a mixture
of a volume υ
liquid of liquid with dielectric constant ε s with a volume υ
crystal with
a much lower dielectric constant ε ∞ . The problem inherent in this situation is that
the dielectric contributions of the two distinct phases do not simply add to the total
dielectric constant, ε mix , observed for the mixture. Instead, the dielectric permittivity
of such heterogeneous systems has to be determined by a proper mixing formula
[18, 19]. One simple example is the Maxwell–Wagner–Sillars (MWS) theory [20,
21], which determines the frequency dependent permittivity of the composite, ε
∗
c
(ω), from those of the filler, ε
∗
f (ω), and the matrix, ε
∗
m (ω), for a given filler volume
fraction ϕ and filler shape characterized by its depolarization factor n. For the static
limit and designating the crystal as filler, we have the special case ε
∗
c (ω) = ε mix ,
ε
∗
f (ω) = ε ∞ , and ε
∗
m (ω) = ε s , with ϕ = υ
crystal /υ
total . For this case, the MWS relation
reads:
ε mi x = ε s
[nε ∞ + (1 − n)ε s ] + φ(1 − n)[ε ∞ + ε s ]
[nε ∞ + (1 − n)ε s ] − φn[ε ∞ + ε s ]
(4)
where ε s and ε ∞ refer to the permittivities of the pure liquid. The result is based on
mean-field approximations and valid only for volume fractions ϕ < 0.2.
Examples of this nonlinear mixing effect are shown in Fig. 4, indicating that
the deviations from the often assumed linearity are more severe for high dielectric
constant liquids, and that disk-like spheroids can result in a significant reduction of
the apparent crystal volume fraction relative to the actual value. For instance, for the
ε s = 80 and n = 2/3 case, a reduction of (ε mix − ε ∞ )/(ε s − ε ∞ ) from 1.0 to 0.8 would
be interpreted as crystal volume fraction of ϕ = 20% based on linearity, whereas the
MWS calculation yields ϕ = 8%. Note again that more sophisticated approaches are
required for accurate predictions for higher crystal content, ϕ > 20%. The conclusion
K. Adrjanowicz and R. Richert
induced this polarization. In a polar liquid in which molecules carry a permanent
dipole moment μ, this polarization is comprised of two components. One is electronic
polarization leading to a permittivity ε ∞ which is approximately equal to the square
of the refractive index, ε ∞ ≈ n
2 . For a given compound, ε ∞ is mostly a matter of the
density, and therefore this contribution to polarization is not so different for liquid
and crystalline states. The situation is different for the dipole orientation contribution
to polarization, which rests on the orientational degrees of freedom of dipoles and
adds the amount of ε = ε s − ε ∞ , which can be much larger than ε ∞ itself. The
feature that is exploited in the context of crystallization studies is that, to a very good
approximation, the crystalline state does not contribute to the dielectric relaxation
amplitude ε. With these approximations, ε
crystal
= 0 and ε
crystal
∞
= ε
liquid
∞ , the
dielectric technique is assumed to yield the volume fraction of the liquid via:
υ
liquid
υ total =
ε
mi xture
ε liquid
(3)
For liquids with moderate polarity, say ε < ε ∞ , Eq. 3 may serve as a good
approximation to the liquid volume fraction, υ
liquid /υ
total . For materials with large
dielectric constants, the situation of a partly crystallized sample is that of a mixture
of a volume υ
liquid of liquid with dielectric constant ε s with a volume υ
crystal with
a much lower dielectric constant ε ∞ . The problem inherent in this situation is that
the dielectric contributions of the two distinct phases do not simply add to the total
dielectric constant, ε mix , observed for the mixture. Instead, the dielectric permittivity
of such heterogeneous systems has to be determined by a proper mixing formula
[18, 19]. One simple example is the Maxwell–Wagner–Sillars (MWS) theory [20,
21], which determines the frequency dependent permittivity of the composite, ε
∗
c
(ω), from those of the filler, ε
∗
f (ω), and the matrix, ε
∗
m (ω), for a given filler volume
fraction ϕ and filler shape characterized by its depolarization factor n. For the static
limit and designating the crystal as filler, we have the special case ε
∗
c (ω) = ε mix ,
ε
∗
f (ω) = ε ∞ , and ε
∗
m (ω) = ε s , with ϕ = υ
crystal /υ
total . For this case, the MWS relation
reads:
ε mi x = ε s
[nε ∞ + (1 − n)ε s ] + φ(1 − n)[ε ∞ + ε s ]
[nε ∞ + (1 − n)ε s ] − φn[ε ∞ + ε s ]
(4)
where ε s and ε ∞ refer to the permittivities of the pure liquid. The result is based on
mean-field approximations and valid only for volume fractions ϕ < 0.2.
Examples of this nonlinear mixing effect are shown in Fig. 4, indicating that
the deviations from the often assumed linearity are more severe for high dielectric
constant liquids, and that disk-like spheroids can result in a significant reduction of
the apparent crystal volume fraction relative to the actual value. For instance, for the
ε s = 80 and n = 2/3 case, a reduction of (ε mix − ε ∞ )/(ε s − ε ∞ ) from 1.0 to 0.8 would
be interpreted as crystal volume fraction of ϕ = 20% based on linearity, whereas the
MWS calculation yields ϕ = 8%. Note again that more sophisticated approaches are
required for accurate predictions for higher crystal content, ϕ > 20%. The conclusion