144
7 Liquid Crystals
7.1.2 Various Thermotropics
This section is not intended to cover all liquid-crystalline phases identified in real
materials but to give an idea of liquid crystal structures necessary to understand
the importance of molecular natures described in the preceding sections. However,
it would be fruitful for more profound studies to learn which molecular properties
we retain and ignore to describe respective phases adequately. The readers who
need the information to identify specific phases through experiments should consult
specialized books [1].
7.1.2.1 Nematic Phase
The liquid crystal introduced in Chap. 6 exhibits fluidity due to the dynamically
random location of molecules and the anisotropy in its physical property. This phase
is regarded as the one that is brought about by the positional melting while keeping the
orientational order of molecules. This phase has the highest symmetry among various
liquid crystals and is called a nematic phase. The nematic (N) phase is uniaxial due
to the uniaxial alignment of anisotropic molecules. A unit vector called the director,
usually written as n (Fig. 7.1), specifies the uniaxial direction. The uniaxial nature
of the nematic phase implies that the molecular anisotropy around the uniaxial axis
is completely averaged out by thermal motion. That is, the molecules in nematic
phases can be assumed axially symmetric. Besides, no reports of the polar nematic
phase in the director exist, even if constituting molecules are polar (having the head
and tail). Thus, n and −n are physically equivalent. The equivalence implies that
the molecules in nematic phases have no distinction between its head and tail. It is,
therefore, suitable to imagine a highly anisotropic spheroid, irrespective of prolate
or oblate, as an abstract molecule in nematic phases.
A nematic order parameter expresses the degree of anisotropy. Because of the
lack of the distinction of head and tail described above, a simple average of, say,
a molecular director defined as a vector from the head to the tail, has no meaning.
Instead, the nematic order parameter s is defined as
s = =P 2 (cos θ),
(7.1)
where
P 2 (x) =
1
2
(3x
2
− 1)
(7.2)
is the second-order Legendre polynomial, θ the angle between the molecular axis
and the director n, and f is the ensemble average of f (over the system). If all
molecules orient their axes along n (θ = 0 for all molecules), s = 1 is obtained. On
the other hand, for the random (uniform) distribution of molecular orientations, an
integration over polar coordinates yields s = 0 because of
7 Liquid Crystals
7.1.2 Various Thermotropics
This section is not intended to cover all liquid-crystalline phases identified in real
materials but to give an idea of liquid crystal structures necessary to understand
the importance of molecular natures described in the preceding sections. However,
it would be fruitful for more profound studies to learn which molecular properties
we retain and ignore to describe respective phases adequately. The readers who
need the information to identify specific phases through experiments should consult
specialized books [1].
7.1.2.1 Nematic Phase
The liquid crystal introduced in Chap. 6 exhibits fluidity due to the dynamically
random location of molecules and the anisotropy in its physical property. This phase
is regarded as the one that is brought about by the positional melting while keeping the
orientational order of molecules. This phase has the highest symmetry among various
liquid crystals and is called a nematic phase. The nematic (N) phase is uniaxial due
to the uniaxial alignment of anisotropic molecules. A unit vector called the director,
usually written as n (Fig. 7.1), specifies the uniaxial direction. The uniaxial nature
of the nematic phase implies that the molecular anisotropy around the uniaxial axis
is completely averaged out by thermal motion. That is, the molecules in nematic
phases can be assumed axially symmetric. Besides, no reports of the polar nematic
phase in the director exist, even if constituting molecules are polar (having the head
and tail). Thus, n and −n are physically equivalent. The equivalence implies that
the molecules in nematic phases have no distinction between its head and tail. It is,
therefore, suitable to imagine a highly anisotropic spheroid, irrespective of prolate
or oblate, as an abstract molecule in nematic phases.
A nematic order parameter expresses the degree of anisotropy. Because of the
lack of the distinction of head and tail described above, a simple average of, say,
a molecular director defined as a vector from the head to the tail, has no meaning.
Instead, the nematic order parameter s is defined as
s = =P 2 (cos θ),
(7.1)
where
P 2 (x) =
1
2
(3x
2
− 1)
(7.2)
is the second-order Legendre polynomial, θ the angle between the molecular axis
and the director n, and f is the ensemble average of f (over the system). If all
molecules orient their axes along n (θ = 0 for all molecules), s = 1 is obtained. On
the other hand, for the random (uniform) distribution of molecular orientations, an
integration over polar coordinates yields s = 0 because of