Rheological Theory and Simulation of Surfactant Nematic Liquid Crystals - Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals - page 45

Figure 2.10 Schematic representation of an NLC confi ned in a capillary, displaying a
periodic array of

1 point defects along its axis. This texture is known as escaped
radial with point defects, ERPD.

+ 1 and −

defect point texture arises. This metastable texture is known as an escape
radial with point defects (ERPD). Hence we would expect that for micellar
nematics under capillary confi nement a similar periodic point defect texture
may arise.
In general, texturing due to confi nement requires that the characteristic
system of size H be greater than the extrapolation length

extra :

K

extra = < H (2.48)

γ 2

and in principle for a given LC this length can be changed by modifying the
interface.
Point defects have another aspect of interest to the understanding of textur-
ing processes. Figure 2.11a shows the connection between point defects and
disclination loops, given in terms of director trajectories. The fi gure shows that
a radial defect point is equivalent to a

1
2 disclination loop and that a hyper-
bolic defect point is equivalent to a 2 disclination loop. Figure 2.11b shows the
corresponding LdG computations of a defect loop at the center of a sphere;
this fi gure and associated computations demonstrate that radial

−

1 points are
disclination loops and that the radius of the loop is approximately four to fi ve
times the molecular length scale.

+

2.3.1.2 Disclination Lines Wedge and twist disclinations are possible
according to the rotation axis: Wedge (twist) lines are parallel (perpendicular)
to the rotation axis (de Gennes and Prost, 1993; Rey, 2009). Wedge disclination
lines are associated with planar orientation given by n

= (cos φ , sin φ , 0),

φ is

the polar angle of a cylindrical coordinate system, M is the strength of the
disclination, and

= M ϕ + ϕ 0 , where φ is the director angle with respect to the x direction, ϕ

ϕ 0 is a constant that determines the overall orientation of the
defect. Figure 2.12 shows director fi elds around typical singular wedge disclina-
tion lines. Since defects have a core and a distortion energy associated, short-
and long-range energy calculations are used to establish their stability (Rey,
2009). It turns out that the defect energy is proportional to the disclination