Rheological Theory and Simulation of Surfactant Nematic Liquid Crystals - Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals - page 56

D

D

>> D

D

D

D

≈ D

D

(2.55)

o splay , o twist , o bend , o twist ,

which is attributed to the fact that for discotics, bend backfl ow is weak
while splay backfl ow is strong: αη

2

>> αηγ

2

, >> η

splay (Santos and Amato, 1 21

1999 ).
Backfl ow plays an important role in startup, cessation, reversal of fl ow, and
magnetic reorientation. For startup shear fl ow, for example, backfl ow or
reorientation - induced fl ow is a rotational viscosity reduction mechanism since
the rotation viscosity γ 1 will be reduced for planar modes:

( )

2

2 2

αθ sin − α θ cos

2 3

(2.56)

=−

γ

1

( )

η cos

2

θ ++ η αθ cos

2

sin

2

θ

1 2 1

is the director angle from the fl ow direction. Pattern formation under
magnetic fi elds is observed in lyotropic rodlike nematic polymers (Berret,
1997; Berret et al., 2001; Roux et al., 1995) and also in potassium laurate/
decanol/water solutions (Kuzma et al., 1989) where the viscosity reduction due
to backfl ow produces a spatially periodic fastest growing mode; for static fi elds,
the patterns in both systems have different symmetries. For lyotropic nematic
polymers, Figure 2.20 shows the initial (left) and fi nal stage (right) of the
response to an imposed magnetic fi eld H along x .
Banded patterns of nematic polymers in a thin fi lm of thickness D of wave-
length L arise for suffi ciently large external magnetic fi elds H normal to an
initially planar director with the growth rate s :

where θ

( )

( ) −

2

( )

2

χ HD π

ηη

/ K + K D L /

a 22 33

s (2.57)

=

( )

2

− LD /

bend twist

where the numerator is the net driving force given by the fi eld minus the
resisting elastic torque, and the numerator is the operating viscosity

bend due
to bending of the director in the plane with a superposed twist viscosity

η

η

twist
due to confi nement in thickness in the y direction. The pattern symmetry in

H

X

y

D H L

H

z

Figure 2.20 Pattern formation of a nematic LC with positive magnetic susceptibility
under an external magnetic fi eld H (left). The fastest growing mode has a periodicity
L in the z direction (right).