Rheological Theory and Simulation of Surfactant Nematic Liquid Crystals - Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals - page 26

exhibiting variable and rich rheological behavior (Cates, 1987; Cates and
Candau 1990; Larson, 1999; Larson and Doi, 1991; Spenley and Cates, 1994;
Spenley et al., 1993, 1996; Yang, 2002). These systems exhibit Maxwell-type
behavior in small-amplitude oscillatory shear fl ow and saturation of the shear
stress in steady simple shear, which leads to shear banding fl ow (Cates, 1987;
Cates and Candau 1990; Larson, 1999; Larson and Doi, 1991; Spenley and
Cates, 1994; Spenley et al., 1993, 1996; Yang, 2002). In the nonlinear viscoelastic
regime, elongated micellar solutions also exhibit remarkable features, such as
the presence of a stress plateau in steady shear fl ow past a critical shear rate
accompanied by slow transients to reach steady state (Cates, 1987; Cates and
Candau 1990; Larson, 1999; Larson and Doi, 1991; Spenley and Cates, 1994;
Spenley et al., 1993, 1996; Yang, 2002). Prediction of the fl ow behavior of vis-
coelastic surfactants by constitutive equations has been a challenging issue
(Acierno et al., 1976; Calderas et al., 2009; De Kee and Fong, 1994; Giesekus,
1966, 1982, 1984, 1985; Herrera et al., 2009, 2010; Marrucci, 1985). In the
nematic phase, the rheology of wormlike micelles has similar features as lyo-
tropic nematic polymers and other complex systems (Calderas et al., 2009).
The rheological classifi cation of uniaxial nematic LCs is to a large extent based
on a single parameter known as the “reactive” or “tumbling” parameter, which
describes the ability of a mesogen to align in a simple shear fl ow. The tumbling
parameter is a ratio of the aligning effect of strain to the rotational effect of

vorticity. For rodlike nematics,

λ

0. The
name reactive refers to the fact that the entropy production is independent of
this parameter and hence its sign is undetermined. Hence we fi nd the following
cases (Larson, 1999; Rey, 2007, 2010):

λ > 0, whereas for disklike nematic, λ <

1. Rodlike nematics: fl ow aligning for λ > 1, nonaligning for 0 < λ < 1 .

2. Disklike nematics: fl ow aligning for λ < − 1, nonaligning for − 1 < λ < 0 .

When | 1, strain overcomes vorticity effects, and the average orientation is

close to the velocity for rods and the velocity gradient for disks, respectively,
while for |

λ | >

1 complex periodic and steady three-dimensional (3D) orienta-
tion modes arise (Tsuji and Rey, 2000). A general expression for the tumbling
parameter is given by the product of a shape function

λ | <

( p ) and a thermorheo-

( )

logical function gT

,,(,,) ϕ ST ϕ γ

(Rey, 2010):

aligning effect of strain
rotation effect of vorticity

( ) ( ) λ

= = β T ,, ϕ gST ( ϕ γ )

(2.2)

where β is a function of the effective aspect ratio, T is the temperature, φ is

γ

the mesogen volume fraction, is the shear rate, and S is the molecular align-

ment along the average orientation. For monomeric rodlike nematics with a
lower temperature smectic A phase, a suffi cient decrease in T affects

)
and triggers a rheological aligning-to-nonaligning transition, as in the case of
8CB. Likewise in monomeric discotic nematics with a lower columnar phase,

β ( T , φ