Eugene C. Gartland

Multidimensional Director Modeling Using the Q Tensor Representation in a Liquid Crystal Cell and Its Application to the π Cell with Patterned Electrodes

To estimate numerically multidimensional director configurations in a liquid crystal cell, it is important to use the Q tensor representation of the strain free energy because it solves the problem of the difference between the directors, n and -n, in the Frank-Oseen free energy representation. In this paper, we discuss the numerical methods for calculating the multidimensional director configurations, using Berremans Q tensor representation. Numerical issues discussed include the relaxation method for the director calculation, the liquid crystal (LC)/glass interface problem, the boundary conditions for the electric potential, and the possible ways to obtain faster convergence. We compare the calculated results obtained from the Frank-Oseen and Q tensor representations. By considering a π cell with patterned electrodes, we show the consistency of the model used with experimental observations. The calculated data explain well the position shift of the defects that appear in the test π cell.

A new configurational transition in inhomogeneous nematics

Abstract At the wall in a hybrid cell with strong anchoring, the nematic director is parallel to one wall and perpendicular to the other. Usually, the free energy is minimized by a configuration where the director orientation changes continuously with position across the cell. The boundary conditions can also be satisfied, however, by a biaxial configuration without such rotation. Under certain conditions, such as under increased curvature strains, a transition can take place between these configurations. The transition typically occurs when the wavelength of the deformation becomes comparable to the coherence length of the material. The hybrid cell considered is a simple illustrative example; in real systems, such a transition may be expected in highly strained thermotropics, or in strained lyotropics which are easily made biaxial.

Finite Element Analysis of the Landau--de Gennes Minimization Problem for Liquid Crystals

This paper describes the Landau--de Gennes free-energy minimization problem for computing equilibrium configurations of the tensor order parameter field that characterizes the molecular orientational properties of liquid crystal materials. Analytical and numerical issues are addressed. Conditions guaranteeing well-posedness (existence, regularity) of the problem are given, as is a nonlinear finite element convergence analysis.

Electric-field-induced nematic-cholesteric transition and three-dimensional director structures in homeotropic cells

We study the phase diagram of director structures in cholesteric liquid crystals of negative dielectric anisotropy in homeotropic cells of thickness d which is smaller than the cholesteric pitch p. The basic control parameters are the frustration ratio d/p and the applied voltage U. Upon increasing U, the direct transition from completely unwound homeotropic structure to the translationally invariant configuration (TIC) with uniform in-plane twist is observed at small d/p < or = 0.5. Cholesteric fingers that can be either isolated or arranged periodically occur at 0.5 < or = d/p<1 and at the intermediate U between the homeotropic unwound and TIC structures. The phase boundaries are also shifted by (1) rubbing of homeotropic substrates that produces small deviations from the vertical alignment; (2) particles that become nucleation centers for cholesteric fingers; (3) voltage driving schemes. A novel reentrant behavior of TIC is observed in the rubbed cells with frustration ratios 0.6 < or = d/p < or = 0.75, which disappears with adding nucleation sites or using modulated voltages. In addition, fluorescence confocal polarizing microscopy (FCPM) allows us to directly and unambiguously determine the three-dimensional director structures. For the cells with strictly vertical alignment, FCPM confirms the director models of the vertical cross sections of four types of fingers previously either obtained by computer simulations or proposed using symmetry considerations. For rubbed homeotropic substrates, only two types of fingers are observed, which tend to align along the rubbing direction. Finally, the new means of control are of importance for potential applications of the cholesteric structures, such as switchable gratings based on periodically arranged fingers and eyewear with tunable transparency based on TIC.

Asymmetric transmissive behavior of liquid-crystal diffraction gratings.

We have used computer simulations to predict that the beam-steering efficiency of a common liquid-crystal diffraction grating will depend on which side is presented to the incident beam. The finite-difference time-domain method and the Helmholtz-Kirchhoff diffraction integral were utilized to simulate the performance of an idealized configuration of the grating.

Comparison of Analytical Calculations to Finite-Difference Time-Domain Simulations of One-Dimensional Spatially Varying Anisotropic Liquid Crystal Structures.

There is a need for an accurate optical simulation tool for liquid crystal structures with director variations in more than one dimension. Finite-difference time-domain (FDTD) computation of Maxwells equations is one approach to consider. The FDTD method has been in use in the electrical engineering community for years, but has only recently been applied to liquid crystal structures. Little is known about the accuracy of FDTD simulations of typical liquid crystal structures, in which the dielectric tensor can undergo large local spatial variations. This paper compares FDTD simulations and analytic solutions of two liquid crystal problems: the twisted-nematic cell and Bragg reflection from a planar cholesteric layer.

A Preconditioned Nullspace Method for Liquid Crystal Director Modeling

We present a preconditioned nullspace method for the numerical solution of large sparse linear systems that arise from discretizations of continuum models for the orientational properties of liquid crystals. The approach effectively deals with pointwise unit-vector constraints, which are prevalent in such models. The indefinite, saddle-point nature of such problems, which can arise from either or both of two sources (pointwise unit-vector constraints, coupled electric fields), is illustrated. Both analytical and numerical results are given for a model problem.

On the uniform convergence of Scharfetter-Gummel discretization in one dimension

A convergence analysis is given for the Scharfetter–Gummel discretization of proto-type one-dimensional continuity equations as arise in the drift-diffusion system modeling semiconductors. These are linear, second-order, boundary-value problems whose coefficient functions are

Strong stability of compact discrete boundary value problems via exact discretizations

O(1)

Newton iteration for partial differential equations and the approximation of the identity

but can have derivatives that are