Rheological Theory and Simulation of Surfactant Nematic Liquid Crystals - Self-Assembled Supramolecular Architectures: Lyotropic Liquid Crystals - page 64

upper bounding surfaces, but without the presence of defects or disclinations.
The different anchoring conditions are smoothly connected by the chiral direc-
tor structure. A similar OEC[2] solution is predicted by the Leslie-Ericksen
equations.

7 . Out - of - Plane Tumbling - Wagging Composite State with Periodic Chirality
(OTP) In this nonplanar mode, the bulk director dynamics is planar and
rotational, and in the two boundary layers it is nonplanar and rotational, and
in the two boundary layers it is nonplanar oscillatory. The spatial profi les of
the periodic director motion are antisymmetric. The director fi eld exhibits
periodic chirality, that is, after a cycle of 2 rotation of the bulk director, the

system periodically essentially recovers the spatially homogeneous director
confi gurations [i.e., n

π

≅ (1, 0, 0)] for 0 ≤ y * ≤ 1 .

Chiral Structure
(OTC) The director dynamics is in-plane rotational in the bulk region and
out-of-plane oscillatory in the boundary layers, the directors at the upper and
lower bounding surfaces have opposite directions, and the system never recov-
ers to the spatial homogeneous director confi guration. Figure 2.23 is a sche-
matic of the rheological phase diagram given in terms of R and Er, clearly
indicating the parametric regions where the four planar modes and the fi ve
nonplanar modes are predicted. The dashed line containing regions 2-5
denotes the parametric envelope within which the out-of-plane modes exist
and are stable; outside the dashed line envelope (regions 1, 6, 7, and 8) only
stable planar modes exist. If R is suffi ciently low, stable nonplanar modes do
not exist at any Ericksen number. If R is suffi ciently high, the stable nonplanar
modes share a boundary with the planar elastic-driven steady-state mode
(region 1) at lower Er and with a time-dependent tumbling-wagging composite
mode (region 6) at high Er numbers. Thus out-of-plane modes arise when the
R and Er numbers have signifi cant magnitudes. In the nine inserts shown in
Figure 2.23, the director orientation is denoted by thick arrows and the dashed
lines denote the bounding surfaces ( y *

8 . Out-of Plane Tumbling-Wagging Composite State with π

0, 1); the fi xed director surface ori-
entation is denoted by the arrow lying on each dashed line. The symbol

=

⊗
denotes a director orientation along the vorticity axis. The main features of
the spatial distributions are that two models (IE, OEC[2]) display one mono-
tonic region behavior, while the other seven (IT, IW, IV, OEA, OEC[1], OTP,
OTC) display a boundary layer and bulk region behavior. In the latter region
the boundary layers are indicated by two thin lines parallel to the dashed lines.
The dynamics of the director for the transient modes (IT, IW, OTP, OTC) are
indicated as follows: the double arrowheads represents director wagging
(oscillations), and the full circle with an arrow represents director tumbling
(rotation). The director orientation of the lower surface of the inserts corre-
sponding to OEC[1] and OTC shows that n 1, 0, 0 and that the modes

are chiral. The full twist shown in the insert for OEC[2] shows that this mode
is also chiral. In addition, as shown below, the OTP mode exhibits periodic
chirality.

= −