Liquid Crystal Hydrodynamics with Variable Order
When disclinations are present, the LC ordering
becomes biaxial near the core and the order parameter drops (as
represented by the background colour of the figure to the right).
We have developed Finite Element discretisations of the QianSheng
Equations in both 2D and 3D. The QianSheng equations are a generalisation
of EricksenLeslie theory for LC hydrodynamics to include changes
in the order parameter [1]. The Finite Element Method is well suited
to resolve the rapid variations in order parameter about disclination
whilst still being able to model large container sizes.
 Three elastic constants
 Flexoelectric effect
 Anisotropic weak anchoring [2]
Theory
The order and the orientation of the liquid crystal are represented
by a symmetric and traceless tensor, the Qtensor:
where and represent
the degree of order about the vectors
and respectively.
The equilibrium Qtensor field minimizes the Landaude Gennes (LdG)
free energy functional:
where is an open bounded subset of
with boundary . The freeenergy densities
and
are due to elastic and electrostatic contributions respectively and
is the surface freeenergy density. The bulk freeenergy density
constrains the degree of order of the LC:
where , and are thermotropic constants,
to are elastic constants, is the
LeviCivita antisymmetric tensor and is the electric potential. Here, the saddlesplay
modulus has been neglected. The flexoelectric polarization
can be defined as
The surface contribution takes the form [2]
This is an anisotropic anchoring energy; a different energy penalty
can be assigned to azimuthal and zenithal perturbations from the
preferred (easy) alignment direction. The first term is necessary
to maintain the surface order parameter,
and are unit vectors orthogonal to the easy
direction and and
are anchoring strengths. A more complete description of the anchoring
types supported by this expression and a means to calculate the
coefficient can be found in [2].
Implementation [3]
The implementation is well suited to the modelling of defect movement
in realistically sized geometries, via the following features:
 Adaptive meshing scheme based on an empirical error estimate
 CrankNicholson time integration using a variable time step
 Matrices calculated using Gauss integration, allowing curved
edge elements to be used
This implicit scheme leads to a larger memory use than the constant
order program. However, the rapid spatial variations in the order
parameter near disclinations severely limits the time step for explicit
methods (<1ns!). We prefer the implicit method in this case to
achieve reasonable simulation times. Flow of the liquid crystal
is calculated by solving the NavierStokes equations with a stress
tensor that takes into account the anisotropy of the LC.
The following assumptions are made:
 Liquid crystal is incompressible
 LowReynold's number approximation
 Electric potential and velocity fields are pseudosteady with
respect to the changing Qtensor
These assumptions could be dropped if required, at the cost of
an increase in simultion time.
The 2D and 3D implementations differ. In 2D second order shape
functions are used for the Qtensor, velocity and electric potential.
A mixed interpolation is used for the NavierStokes equation to
avoid oscillatory pressure solutions. Therefore first order shape
functions are used for the pressure. In 3D first order shape functions
are used for all variables, and a stabilized form of the NavierStokes
equations.
Results
Fig. 1 shows an example simultion results for a type of bistable
display device.
Fig. 1: Defect movement in a Zenithally Bistable
Nematic (ZBN) style device, with a negative dielectric anisotropy
liquid crystal (director colour represents the order parameter)
[1] T. Qian and P. Sheng, “Generalized hydrodynamic equations
for nematic liquid
crystals,” Physical Review E, vol. 58, no. 6, pp. 7475–7485,
1998.
[2] E. Willman, F. A. Fern´andez, R. James, and S. E. Day,
“Phenomenological
anisotropic anchoring energy of nematic liquid crystals for the
Landaude Gennes
theory,” to appear in IEEE Transactions on Electron Devices.
[3] R. James, E. Willman, F. A. Fernandez, and S. E. Day, “Finiteelement
modeling
of liquidcrystal hydrodynamics with a variable degree of order,”
IEEE Transactions
on Electron Devices, vol. 53, no. 7, pp. 1575–1582, 2006.
This page last modified
8 July, 2007
by r.james
