Chemistry Reference
In-Depth Information
the dislocation may appear because of the discontinuity of the translational
order.
1.6.1. The Volterra process
Both dislocation and disclination can be produced by the well-known
Volterra process. Take a cylinder of a medium and do the following
operations in sequence on it:
(1) cut off a thin cylinder of diameter
r
c
along the axis of the cylinder;
(2) cut a plane via the axis of cylinder and then move two opposite slips of
the cut plane by a displacement of either translation
b
or rotation
ω
,
or both together. Hence the relative displacement of two slips is
d
(
r
)=
b
+
ω
×
r
;
(1.23)
(3) If
b
or
ω
is consistent with the translational or orientational symmet-
rical element of the matter, the gap resulting from the displacement
d
can be filled by a piece of bulk medium and two slips will fit perfectly
with the filled piece. If the operation (2) makes the medium overlap,
we need to take away an additional piece of medium instead and thus
still keep the perfect fit of the two slips.
(4) The medium will then relax to a new state owing to the internal
interactions. Consequently, a singular line L is left.
The six basic Volterra processes are depicted in Figure 1.20, among
them the three intrinsic processes on
b
produce dislocations while the three
intrinsic operations associated with
ω
produce disclinations. In Figure 1.20,
assuming
σ
is the normal of the cut plane,
(a)
b
⊥
L
, but
b
σ
, the line L is an edge dislocation;
(b)
b
⊥
L
, and
b
⊥
σ
, L is an edge dislocation;
(c)
b
L
, L is a screw dislocation;
(d)
ω
L
, L is a wedge disclination;
(e)
ω
⊥
L
, but
ω
σ
, L is a twist disclination;
(f)
ω
⊥
L
, and
ω
⊥
σ
, L is a twist disclination.
The dislocations have been extensively studied so we limit ourselves
mainly to the disclination. Figure 1.21 sketches a sample of nematic liquid
crystals, the short bars denoting the directors which all lie on the sheet
of paper. Now cut the sample along the normal and via the line L, see
