









Simulating Nematic Liquid Crystal Textures
Perry Ellis and Alberto FernandezNieves
IntroductionNematic liquid crystals are birefringent rodlike molecules that prefer to align with their long axis roughly parallel. We use the director n , an average of the orientation, to describe a nematic configuration. Under confinement in a toroidal geometry, we find that the director field exhibits a doublytwisted structure.
Our director ansatz incorporates this doubletwist via the parameters ω and γ, which correspond to letting the director splay and twist, repsectively. We find these parameters by minimizing the Frank Free Energy including contributions from saddlesplay distortions while leaving ω and γ as free parameters. Once ω and γ are determined, we simulate crossedpolarized textures from our director ansatz and compare the simulated textures agaiunst our experimental textures. This gives us the ability to evaluate our director ansatz against our experminetal results.
Simulation DetailsWe simulate the crossedpolarized textures using Jones Calculus, a method of evolving and propagating polarized states through a birefringent material. Jones Calculus neglects refraction and reection, instead just treating light as a bundle of parallel rays. A polarization state is represented by a 2 x 1 Jones Vector and a birefringent element is represented by a 2 x 2 Jones Matrix. As an example, Figure 3 shows how the Jones Vector for +45 degree linearly polarized light interacts with the Jones Matrix for a quarterwave plate with the fast axis at 0 degrees to produce an output Jones Vector of righthand circularly polarized light.
We take the goemetry we're interested in and fill it with the desired director field. Then we break the configuration up into volume elements of size Δ^{3}. We want the size of each volume element to be small enough such that we can take the enclosed director to be constant. This allows us to treat each cube as a discrete optical element. Now we can use Jones Calculus to determine the net effect of all the optical elements along an incident ray of light. After the light has passed through the analyzer, the final intensity is recorded on the output texture. We then repeat this process for all the rays passing through the bulk. Thus, $$I(\vec{\rho}) = \hat{\Theta}_A \, \hat{\vartheta}(\vec{\rho}) \, \vec{E}_P, $$ where \( \rho \) is the position vector in the output image, \( I(\vec{\rho}) \) is the intensity in the output image, \( \vec{E}_P\) is the initial polarization state of the light ray, \( \hat{\Theta}_A \) is the analyzer, and \( \hat{\vartheta}(\vec{\rho}) \) is the operator for a series of optical elements along the ray passing through \(\rho\). In terms of the individual elements, $$ \hat{\vartheta}(\vec{\rho}) = \hat{\Theta}(\vec{\rho})_N\hat{\Theta}(\vec{\rho})_{N1} \cdots \hat{\Theta}(\vec{\rho})_1 = \prod_{i = 1}^{N(\vec{\rho})}\!\hat{\Theta}(\vec{\rho})_i\, ,$$ where \( \hat{\Theta}(\vec{\rho})_i\) is the operator for the optical element associated with the i^{th} Δ ^{3} volume element along the ray.

Soft Condensed Matter Laboratory, School of Physics, Georgia Institute of Technology
770 State Street NW, Atlanta, GA, 303320430, USA
Phone: 4043853667 Fax: 4048949958
alberto.fernandez [at] physics.gatech.edu