Chemistry Reference
In-Depth Information
appears. It is a wedge disclination. If the same amount of liquid crystals
is needed to be moved, as shown in Figure 1.21(d), the final picture of the
director field will look like that shown in Figure 1.21(e). The disclination
line L is then +1
/
2 in strength.
There is a simple process to produce a disclination: rotate the direc-
tors on two slips respectively by
ω
1
ω
2
=
ω
.
Thus the same disclination line is produced. The process is named the de
Gennes-Friedel process. One can prove that the de Gennes-Friedel process
is equivalent to the Volterra process for nematic liquid crystals. The oper-
ation
P
v
of the Volterra process can in fact be divided into the translation
and rotation steps,
i.e.
, first, translate the directors (
T
) and then rotate
them around themselves (
P
g
). The latter is actually the de Gennes-Friedel
process. In other words
and
ω
2
and make
ω
1
−
P
v
=
P
g
+
T.
(1.24)
Because there is no dislocation in nematics which have only translational
symmetry, consequently
P
v
=
P
g
.
(1.25)
1.6.2. Strength of disclination in nematics
The relative rotation of two opposite slips is
ω
=2
mπ
where
m
is the
strength of the disclination, which can be either an integer or semi-integer,
i.e.
,
3/2,
etc.
. The general definition of the strength of the
disclination is the sum of the deformation angles, in radian measure, made
with respect to a laboratory axis when going along a closed contour around
the disclination
±
1/2,
±
1,
±
1
2
π
m
=
dψ,
(1.26)
L
where
ψ
is the angle of the director with respect to a certain direction,
e.g.
,
the
x
axis and L is the closed contour. For convenience, we calculate instead
the sum of the angles
θ
of the directors with respect to the tangents of the
closed contour,
i.e.
m
=1+
1
2
π
dθ.
(1.27)
L
The sign of
m
depends on whether
n
and the closed contour (
i.e.
, the
Burger's circle) have the same direction.
The algebraic sum of the disclinations are not arbitrary, and the sum
of all the strengths in a medium is a topological constant,
i.e.
, the Euler
