Chemistry Reference
In-Depth Information
been accepted (1958). According to the Frank theory, the elastic energy of
a deformed nematic liquid crystal can be expressed as
F
=
F
0
+
1
2
d
3
r
[
K
11
(
∇·
n
)
2
+
K
22
(
n
·∇×
n
)
2
V
+
K
33
(
n
×∇×
n
)
2
]
,
(1.14)
where
F
0
is the constant term for an undeformed sample.
Because of the spontaneously helical structure in the cholesteric liquid
crystals, the second term in the integral must be revised to
n
·∇×
n
+
2
π
P
0
2
,
1
2
K
22
(1.15)
where
P
0
is the intrinsic helical pitch. The supplement favors the molecular
twist. The form illustrates that without an external field, the director of
liquid crystals is naturally twisted with the period of the helix being the
pitch
P
0
.
The form of free energy for smectic liquid crystals is different. If there are
no defects in the smectic liquid crystals, the curl of
n
,
∇×
n
, must be zero.
Thus, no twist and bend deformations exist in the smectic liquid crystals.
In addition, there is an energy penalty associated with the translational
deformation. For example, the displacement of smectic layer
u
will cause
an additional term of elastic energy
2
B
d
u
2
,
1
(1.16)
dz
where
B
is the elastic modulus, assuming that the z axis is perpendicular
to the layers for the undeformed sample.
The elasticity theory has been used in dealing for example with the
response of liquid crystals to external fields (electric, magnetic, mechanical
force), defects,
etc
.
The above argument is suitable for polymer liquid crystals as well. In
fact, the static properties of polymer liquid crystals are basically the same
as those for the low molecular mass liquid crystals. But their dynamics are
quite different because of their polymeric structure.
1.5.3. Frederiks transitions
We now apply the elasticity theory of liquid crystals to analyze the
deformation of liquid crystals under an external magnetic field.
