Rheological Theory and Simulation of Surfactant Nematic Liquid Crystals - Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals - page 61

2

=

h
h

α

t t (2.63)

α β

β

In terms of shear strain γ , Eq. (2.62) can be written in the following alternative

but equivalent form:

( ) = ( )

2 2

ηγ h , γ ηγ h , γ (2.64)

αα α ββ β

( )

γγ =

i t

i = αβ ,

(2.65)

i

Using Eq. (2.63) the equivalence between γ α and γ β follows, by director

substitution,

==

h
h

α

γ γ t γ t = γ (2.66)

β β β α β α

β

where the original Ericksen's scaling relation is satisfi ed in Eq.

(2.61). It has been directly confi rmed that the computational results obey
the Ericksen scaling properties, but one should keep in mind that this scaling
holds for a given preparation of the system (i.e., initial conditions, boundary
conditions) but breaks down in the presence of stabilities leading to difference
steady - state solutions (Ericksen, 1969 ; Han and Rey, 1994a - c ; Rey, 1993a,b ).

γ h

2

= γ h

2

αα ββ

2.4.2 Wormlike Micellar Nematics

As mentioned above, wormlike micelles are observed when surfactant con-
centration is above 20-30 wt %. Their properties have received considerable
attention in the last years due to their remarkable structural and complex
viscoelastic rheological fl ow behavior (Cates, 1987; Larson and Doi, 1991;
Spenley and Cates, 1994; Spenley et al., 1993, 1996; Yang, 2002). Nematic
phases exhibit long-range orientational and translational order with no posi-
tional order, whereas hexagonal phases show both orientational and transla-
tional long-range order of the centers of mass of the micelles (Chen et al.,
1977; Clawson et al., 2006; De Melo et al., 2007; Hendrix et al., 1983; Itri and
Amaral, 1990, 1993; Larson, 1999; Lawson and Flautt, 1967; Leadbetter and
Wrighton, 1979; Lukascheck et al., 1995; Muller et al., 1997; Onsager, 1949;
Ostwald and Pieransky, 2000; Quist et al., 1992; Richtering et al., 1994; Schmidt
et al., 1988; Semenov and Khokhlov, 1988).

2.4.2.1 Banded Patterns Banded patterns are normal to the fl ow direc-
tion after cessation of fl ow for shear rates up to 50 s − 1 , the pattern formation
time is proportional to the square of the applied shear, and the pattern coars-
ening rate at suffi ciently low preshear is well described by a diffusive process.
The pattern is associated with relaxation of elastic energy. For shear rates
greater than 50 s − 1 , the material fl ow aligns. These features were found to be