##
Flow dynamics of focal conic domains in smectic-A liquid crystals

Smectic-A liquid crystals are formed of molecules that arrange themselves
into layers. Like crystals, which have rows and columns and layers of atoms
in a regular array, smectic-A materials have layers -- but in each layer
the order is random, like a liquid. These smectic layers have a strong
preference to be equally spaced. (If we distort the smectic, the layers
prefer to bend than to deviate from equal layer spacing.)
This leads to defects that form **focal conics** -- ellipses
and hyperbolas.

In this set of animations, we show simulation results for the dynamical
evolution of smectic-A liquid crystals. Experiments are often performed
by putting the smectic between two microscope slides. The slides can have
different anchoring conditions.
**Planar** boundary conditions are treatments which align
the smectic molecules parallel to the glass slides, meaning the layers
are perpendicular to the slide. **Homeotropic** boundary
conditions align the layers parallel to the slide (and hence flat).
Planar boundary conditions naturally attach one of the two confocal conics
in each domain to the microscope slide, so imaging the top and bottom is
the natural thing to do. For homeotropic boundary conditions, the top
and bottom are flat and boring, so experimentalists (and simulators)
examine instead an intermediate height in the sample.
In each case, the images are taken with light (simulated or real)
transmitted through crossed polarizers on either side: the
dark and light regions show the defect structures.

### Focal conic formation and growth

The first and second simulations show the nucleation and growth of focal
conic defects starting from random initial conditions.
### Periodic shear

In this set of animations, we show simulation and experimental results for
the flow dynamics of smectic-A liquid crystals subject to shear strain,
with one of the glass slides oscillating sideways with respect to the other.
In the first three simulations, we initialize our movie by evolving from a
random initial configuration for a time T where clear focal conics are formed.
We then apply the shear dynamical equations of motion for an amplitude of
oscillation equals twice the gap size, and a frequency of 2 \pi / T.
The animations show three periods of oscillation.
#### Shear, planar boundary conditions

First we compare theory with experiment for sheared planar boundary
conditions,
where the smectic layers are perpendicular to the upper and lower glass
slides. (The long axis of the smectic molecules, which point normal to the
layers, are hence parallel to the glass slides.)

For planar anchoring under external shear, the focal conics in the
experiments and simulations seem qualitatively similar.

#### Shear, homeotropic boundary conditions

Second, we compare theory with experiment for sheared
homeotropic boundary conditions
(where the long axis of the smectic molecules are perpendicular to the
glass slides, and hence the layers are parallel).

Here the simulation differs from the experiment in at least two important
ways. First, the simulation has large 'bands' separating the cross-hatched
circles (representing a vertical focal conic defect representing the dimple
near the stem of the 'apple-shaped' focal conic layers. Second, the
experiment under shear destroys the original focal conic defects, and
develops a new pattern of domains with eccentricity perpendicular to the
direction of shear. Whether this represents differences in the parameters
of the simulation and the experiment (relative elastic or viscous constants
or rates), or important new physics (dislocations omitted from the simulation)
remains to be discovered.

Last modified: January 14, 2015
James P. Sethna,
sethna@lassp.cornell.edu;
This work supported by the Basic Energy Sciences division of the
Department of Energy, through grant DE-FG02-07ER46393, and by the National
Science Foundation CBET-PMP 1232666 and DMR-1056662.

Statistical Mechanics: Entropy, Order Parameters, and Complexity,
now available at
Oxford University Press
(USA,
Europe).