Numerics for Liquid Crystals with Variable Degree of Orientation
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Numerics for Liquid Crystals with Variable Degree of Orientation Ricardo H. Nochetto1, Shawn W. Walker2 and Wujun Zhang1 1
2
Department of Mathematics, University of Maryland Department of Mathematics, Louisiana State University
Abstract We consider the simplest one-constant model, put forward by J. Eriksen, for nematic liquid crystals with variable degree of orientation. The equilibrium state is described by a director field n and its degree of orientation s, where the pair (n, s) minimizes a sum of Frank-like energies and a double well potential. In particular, the Euler-Lagrange equations for the minimizer contain a degenerate elliptic equation for n, which allows for line and plane defects to have finite energy. Using a special discretization of the liquid crystal energy, and a strictly monotone energy decreasing gradient flow scheme, we present a simulation of a plane-defect in three dimensions to illustrate our method.
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INTRODUCTION
Complex fluids are ubiquitous in nature and industrial processes and are critical for modern engineering systems [20, 11]. Liquid crystals [27, 16, 14, 3, 2, 10, 6, 21, 22, 4, 26] are a relatively simple example of a material with microstructure that may or may not be immersed in a fluid with a free interface [30, 29]. Several numerical methods for liquid crystals have been proposed [5]. Some are based on harmonic mappings [7, 1, 15, 19, 23] where a unit vector field (called the director field) is used to represent the orientation of liquid crystal molecules; in these methods, they only model the equilibrium state. Some methods for the dynamics of liquid crystals can be found in [18, 24, 28]. The main result of our work is a finite element method (FEM) with provable stability and convergence properties, which we use to explore equilibrium configurations of liquid crystals via gradient flows. The equilibrium theory, found in [17, 16, 27], deals with a director field n, that represents the orientation of liquid crystal molecules, and a scalar parameter s, −1/2 < s < 1, that represents the degree of alignment that molecules have with respect to n. The main purpose of this model is to represent line and plane defects with finite elastic/free energy. Some related work can be found in [12, 13, 22, 21, 25, 4, 26]. The most relevant work that we know of is in [14, 7] which also considers the variable
degree-of-orientation s parameter. However, in both cases they regularize the model to avoid an inherent degeneracy introduced by the s parameter. The regularization is completely artificial and removes the ability to truly capture line and plane defects. The purpose of the regularization is purely mathematical. Our method builds on [8, 9] and consists of a special discrete energy that does not use any regularization, hence we can compute minimizers that exhibit line and plane defects. In this proceedings paper, we only briefly review the continuous model and show some numerical results.
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ERICKSEN’S ONE CONSTANT MODEL
In Ericksen’s model [17], the configuration of liquid crystals
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