Derivation and Application of an Algorithm for the Numerical Calculation of the Local Orientation of Nematic Liquid Crystals

Abstract

D e r iv a tio n a n d A p p lic a tio n o f a n A lg o rith m fo r th e N u m e ric a l C a lc u la tio n o f th e L o c a l O r ie n ta tio n o f N e m a tic L iq u id C r y s ta ls A. Kilian and S. Hess Institut für Theoretische Physik, Technische Universität Berlin, Berlin Z. Naturforsch. 44 a, 693 -703 (1989); received June 16, 1989 Starting from a relaxation equation for the alignment tensor, an algorithm is derived which allows the numerical calculation of the dynamic and static behavior of the director field n with the correct nematic symmetry property, where n and -n are equivalent. As a first application, a two-dimensio nal problem is treated where the typical nematic defects with half-integer winding numbers only occur when the algorithm with the correct nematic symmetry property is used. Furthermore, the method is applied to the static and dynamic behavior of a Frederiks cell with strong and weak anchoring.

On the simulation of the director field of a nematic liquid crystal

“…The algorithm [6] is based on the equation of change for the director field n, (r, t) in an electric field E, d -y I ~n p , = KAn,n, + E,E,,E,,E, with the constraint n,np = constant. Here, yI is the Leslie coefficient, K the Frank elasticity coefficient, and E, is the difference E,, -E~ between the relative dielectric coefficients parallel and perpendicular to the applied field at the given temperature.…”

“…Therefore, we had to find conditions which favour the occurrence of disclinations. One method used in former simulations [6] are competing boundary conditions, e.g. a capillary tube with a homeotropic easy axis.…”

“…Afterwards, the reverse process was performed (dashed line at bottom). The time corresponding to ten iteration steps is shorter than the relaxation time, especially near the critical field strength ECit (the relaxation time zreI is proportional to IE -E,,I -I [6]). Therefore, this is a non-equilibrium process,…”

Computer simulation study of a nematogenic lattice model based on an elastic energy mapping of the pair potential

“…In Important aspects of liquid crystal science and technology are related to their elastic properties: for example, this approach, special care is needed in order to ensure that the algorithm preserves the underlying nematic they are involved in a number of phenomena where the director is manipulated by external ® elds, as in display symmetry with respect to inversion of the director ® eld (see, e.g. [9] ). Metropolis Monte Carlo approaches have devices, and a knowledge of the elastic behaviour is also important in the study of defects [1] .…”

<title>Influence of the interpixel region in liquid crystal diffraction gratings</title>

“…The electric field is determined from the gradient of the electrostatic potential (Equation 4) and a director distribution is computed using the forward time, central difference method applied to the system of equations typified in Equation 5. The new director distribution is used to solve the inhomogeneous fmite difference representation of Laplace's Equation given in Equation 6. From the electrostatic potential determined by convergence of Equation 6, the electrostatic field is recalculated using Equation 4 and substituted back into the system of equations used to determine the director distribution.…”