Introduction Nematic liquid crystals are cylindrically rod-like shaped molecules, which can be characterized by the orientations of their directors. Directors are unit vectors that describe the average orientation of a local group of liquid crystals. Liquid crystals play a very important role in liquid crystal displays (LCD). The cylindrically rod-like molecules exhibit a phenomenon known as bistability. Bistability allows nematic liquid crystals to geometrically orientate themselves differently even under the same boundary treatment. For the purposes of our research, we restrict this study to 2 dimensional geometries and cases where the elasticity constants are equal. Bistability affects the directors of nematic liquid crystal systems, which in turn affects their overall free energies as well. By investigating free energies of nematic liquid crystals, we can try to understand which geometric distribution is more preferred from an energy perspective. Our Interests We want to gain a deeper understanding of the bistability phenomena for nematic liquid crystals in 2D. Due to the mathematical complexity required for this research, Mathematica had to be implemented throughout the entirety of this project. Physicist S. Burylov had managed to calculate the free energies of nematic liquid crystals in a circular cavity for both the planar radial and planar polar distributions[1]. For our research, we want to expand upon Burylov’s findings and try to understand the bistability phenomena in more general 2D polygonal cavities. Planar Polar (R) with Planar Radial (L) We are interested in the same two forms of alignment: planar radial and planar polar. In 1997, S. Burylov 1 successfully calculated the total free energies of nematic liquid crystals for planar radial and planar polar distributions in a circular boundary treatment. These calculations will eventually be verified by our calculations with an entirely different and independent approach. Methods A specific type of a conformal mapping known as the Schwarz-Christoffel transform maps a circle to a polygon on the complex plane. With the proper parameters, the Schwarz-Christoffel map can produce images of regular N-sided convex polygons. Regular convex polygons are the simplest of all polygons on the Euclidean plane. Polygonal cavities generated by the Schwarz-Christoffel map are given below. Once a specific regular polygon is selected, the Frank energy density expression can be integrated throughout this polygonal cavity to obtain the total free energy. For the Frank expression given below, n represents the director while K 1,2,3 represent the elasticity constants. Note, when K 1,2,3 are equal, the Schwarz-Christoffel solves Laplace’s equation. We want to obtain a general analytic express for the total free energy of nematic liquid crystals in any N-sided regular convex polygon. The Schwarz- Christoffel map is usually given as a derivative and requires integration in order to acquire a usable mathematical form. Schwarz-Christoffel Map For N-Sided Convex Polygon in Derivative Form Ultimately, only a few polygons can be generated using an analytical mathematical expression. Because most polygons are irregular, they will have to undergo numerical integration. Furthermore, even if the Schwarz-Christoffel map can be written as an useful expression w(z), its Frank free energy density formula can be quite complicated. Even so, much can be achieved in understanding bistability just from just regular convex N-sided polygons. Geometrically simple polygons also allow us to model the director orientations along defects areas. Analytical Schwarz Christoffel Expression for Regular N-Sided Convex Polygon After doing quite a bit of research, we eventually found a much better mathematical strategy and formulation for our research. This approach was entirely new and led to fascinating results. It is outlined in the next section. Conclusions An analytical expression for the total free energy of a N-sided regular convex polygon has finally been found. Analytical integration poses to be a difficult challenge but can be numerically calculated with our free energy integral for difficult geometric boundaries. Analysis has been done for the equilateral triangle, which is the simplest regular convex polygon in 2D. Plots of the Hessian and Jacobian for the triangular cavity have been given below. Hessian(T) and Jacobian(B) for N = 3 The Frank energy density expression is simply written as H/J instead. In other words, the integrand H/J is the new energy density and this is a much faster and simpler method for calculating energy densities for nematic liquid crystals in any cavity generated by a conformal map including the Schwarz-Christoffel transform. The analytical expression for the free energy density of a N-sided regular convex polygon is given below. n denotes the number of sides. The variables u and v denote the position of the liquid crystal. Note, n must greater than or equal to 3. Michael Ding, Timothy Atherton Department of Physics, Case Western Reserve University, Cleveland, Ohio Literature Cited [ 1] S. V. Burylov, Zh. Eksp. Teor. Fiz. 112, (November 1997) [2] S. Chandrasekhar, N. Madhusudana. Liquid Crystals. Annual Review Mater. Sci. 1980, 10, [3] T. Driscoll, L. Trefethen. Schwarz-Christoffel Mapping. Cambridge Monographs on Applied and Computational Mathematics Energies of Nematic Liquid Crystals in 2D Polygonal Cavities Results We derived a completely new mathematical expression for the free energy and referred to it as the “free energy integral.” It turned out that the Frank’s free energy density expression had a strong and important indirect relationship with conformal maps. As analytic functions, conformal maps preserve the angular features during the mapping [3]. This meant that the director orientations transformed as the boundary transformed into a polygon. In other words, the Schwarz-Christoffel transform already contained very important information concerning the orientation of the directors. We were able to derive the following mathematical formula for calculating the total free energy of nematic liquid crystals in any 2D boundary treatment generated but by all conformal maps, not just the Schwarz-Christoffel. Free Energy Integral for Nematic Liquid Crystals This free energy integral simply requires both the Hessian and Jacobian of any conformal map; call it w(z). Due to our restriction of only 2 dimensions, the Hessian and Jacobian need only be 2x2 matrices. Keep in the mind that w(z) maps z(x, y) to w(u, v). Furthermore, we were also able to derive an analytical expression for the director orientations in any boundary generated by the w(z) map, which allows us to explore directors near discontinuities. Directors Formula for Nematic Liquid Crystals in Polygonal Cavities A key component of these formulae is that it utilizes w(z) as a conformal map which maps a circular boundary to any conformal boundary. In fact, if the w(z) mapping was not applied, then the free energy integral would simply calculate the total free energy of a nematic liquid crystal system in a circular boundary. We did this free energy integral calculation and it matched with Burylov’s free energies. This confirmed our theory and mathematical derivations for the free energy integral and both the planar radial and planar polar distributions [1]. Energies of Planar Radial and Planar Polar Interestingly, Burylov’s result for the free energy of the planar polar distribution was an approximation. The ε term denotes the higher terms of the Taylor expansion. Furthermore, with this method, we found an analytical expression for the free energy density of nematic liquid crystals for any N-sided regular convex polygon., Director Orientation of Local Group of Nematic LC