Rheological Theory and Simulation of Surfactant Nematic Liquid Crystals - Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals - page 62

similar to those of lyotropic nematic polymers of the nonaligning type (Roux
et al., 1995). Using given fl ow kinematics and a 1D spatial description, a full
rheological phase diagram is based on the LdG for nonaligning nematics in
terms of the length scale ratio R and the Ericksen number.
In Figure 2.23, the left schematic shows the rheological phase diagram
in terms of the eight director modes across the cell thickness. At very small
E , elasticity prevails and tumbling is arrested. The dotted parabola corre-
sponding to small E contains the out-plane modes, where regions 3-5 display
multistability.
The main features of these fl ow modes are summarized in the following
explanation [Fig. 2.23 (left)]:

1 . In - Plane Elastic - Driven Steady State (EE) The steady state of this
planar mode arises due to the long-rate order elasticity stored in the deformed
tensor order parameter fi eld. In this planar mode there is no orientation
boundary layer behavior because there is no fl ow alignment in the bulk
region.
2 . In-Plane Tumbling Wagging Composite State (IT) In this time -
dependent planar mode, the director dynamic in the bulk region is rotational,
and in the boundary layer it is oscillatory. The boundary between the bulk
region and each boundary layer is characterized by the periodic appearance
of the abnormal nematic state, which is characterized by two equal eigenvalues
of the tensor order parameter (i.e., μ l ) and follows a smoothly defect-

free transition from the rotation bulk region to the fi xed director anchoring
at the surfaces by a director resetting mechanism. The insert in Figure 2.23
is discussed in full detail below and shows schematics of the existing stable
solutions of Eq. (2.2).
3 . In - Plane Wagging State (IW) In this plane mode, the director dynamics
over the entire fl ow geometry is periodic oscillatory with an amplitude
decreasing from a maximum at the centerline to zero at the two bounding
surfaces.
4 . In - Plane Viscous - Driven Steady State (IV) In this plane mode, the
director profi le shows a fl ow-aligning bulk region and two boundary layers.
On traversing the boundary, the director rotates from the aligning angle to the
fl ow direction at the walls.
5 . Out - of - Plane Elastic - Driven Steady State with Achiral Structure (OEA)
In this nonplanar mode, the director shows steady twist structures, and the
twist angle profi les are symmetric with respect to the centerline. The steady
state arises due to the long-range order elasticity. Similar solutions are pre-
sented by the Leslie-Ericksen solutions. Following the bottom-to-top bound-
ing surface, the net director twist rotation is nil.
6 . Out-of-Plane Elastic-Driven Steady State with Chiral Structure (OEC[n])
In this nonplanar mode, the director shows steady twist structure, with n

μ k = μ r >

π

( n = 1, 2) radian difference between the anchoring angles at the lower and