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Finally, observation probabilities of all fl ow modes for all the fl ow regimes
are also shown in Figure 2.23 (right). In the fi gure, the probabilities are plotted
for each fl ow mode, and thus for any Ericksen number the sum of the prob-
abilities is 1. For example, for region I the probability of the IE mode is 1 and
others are zero, and for region 4 the probability of OEA and OTP fl ow modes
are almost 0.5, and the others are zero. The reason for the transition between
regimes 2 and 3 arises from the loss of freedom in the director rotation direc-
tion. The boundary layer thickness increases with decreasing Er, and thus the
two out-of-plane nucleation points become closer. When the points come
closer than a certain correlation length, their rotation directions are also
restricted by each other, and then the system loses the OEC[2] fl ow mode and
changes from regime 3 to regime 2. Summarizing this section, the complete
model given in Section (2.2) [Eq. (2.35)] predicts extensive multistability
phenomena, involving planar, chiral achiral, steady, and time periodic nodes.
The range and richness of the multistability is due to the presence of the two
compatibilization mechanisms predicted by the complete theory (Tsuji and
Rey, 1998 ).
2.4.2.2 Banded Textures
Banded texture predictions during fl ow and
after cessation of fl ow have been compared with experimental data using LE
and LdG models. The consistency between the two model predictions have
been established (Tsuji and Rey, 1998). In the LE model of banded textures
during fl ow, the pattern formation is driven by the OP mode, which nucleates
a periodic array of elliptical splay-twist-bend inversion wall in the velocity/
velocity gradient plane with a wavelength close to the shear cell thickness. In
the LdG model of banded textures, the nucleation and growth of OP modes
give rise to a heterogeneous nonplanar orientation fi eld that relaxes through
the formation of a periodic texture. Figure 2.24 shows the out-of-plane com-
ponent profi le after cessation of fl ow, for
R
=
0 at the following dimensionless
times: (a)
t
*
2, (b) 4, (c) 6, (d) 8, (e) 12, (f) 16, and (g) 20. The small source
of the out-of-plane component near the surface is connected across the bulk
region, and the banded texture is formed with almost the same director con-
fi guration as that during fl ow. Then, the texture relaxes through the shrinking
of the director out - of - plane region.
Lastly it was observed (Santos and Amato, 1999) that for shear rates greater
than 50 s
− 1
, a monodomain was observed. This is in full agreement with high
shear rate fl ow-aligning region 8 of Figure 2.24 and with the predicted texture
length-scale divergence for De
=
>
1 (Tsuji and Rey, 1998 ).
2.4.2.3 Transient Shear Response
Transient shear rheology of cetyl-
pyridinium chloride/hexanol (CPCL/Hex) indicates that wormlike micelles
respond similarly as nonaligning lyotopric nematic polymers (Berret, 1997;
Berret et al., 2001; Roux et al., 1995). In particular, the scaling properties of
the transient shear stress, the effect of preshear, and the fact that the time to
reach steady state is inversely proportional to shear rate were reported. The
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