A Convergent Finite Element Scheme for the Q-Tensor Model of Liquid Crystals Subjected to an Electric Field

We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which, we show that it is possible to simulate the Fr\'eedericksz transition with this scheme.

[1]  F. Weber,et al.  On the Convergence of an IEQ-based first-order Numerical Scheme for the Beris-Edwards System , 2023, ArXiv.

[2]  Hung-Chang Jau,et al.  Multifunctional Liquid Crystal Smart Glass with Light Field Shaping, Dimming, and Scattering Control , 2022, Advanced Photonics Research.

[3]  Shawn W. Walker,et al.  Numerical method for the equilibrium configurations of a Maier-Saupe bulk potential in a Q-tensor model of an anisotropic nematic liquid crystal , 2021, J. Comput. Phys..

[4]  Franziska Weber,et al.  A convergent numerical scheme for a model of liquid crystal dynamics subjected to an electric field , 2021, ArXiv.

[5]  Varun M. Gudibanda,et al.  Convergence analysis of a fully discrete energy-stable numerical scheme for the Q-tensor flow of liquid crystals , 2020, SIAM J. Numer. Anal..

[6]  John A. Mackenzie,et al.  A Moving Mesh Method for Modelling Defects in Nematic Liquid Crystals , 2019, J. Comput. Phys. X.

[7]  Epifanio G. Virga,et al.  Variational Theories for Liquid Crystals , 2018 .

[8]  Xiang Xu,et al.  An Elementary Proof of Eigenvalue Preservation for the Co-rotational Beris-Edwards System , 2018, J. Nonlinear Sci..

[9]  Jie Shen,et al.  A New Class of Efficient and Robust Energy Stable Schemes for Gradient Flows , 2017, SIAM Rev..

[10]  Xiang Xu,et al.  Dynamics and Flow Effects in the Beris-Edwards System Modeling Nematic Liquid Crystals , 2017, Archive for Rational Mechanics and Analysis.

[11]  Ricardo H. Nochetto,et al.  The Ericksen model of liquid crystals with colloidal and electric effects , 2017, J. Comput. Phys..

[12]  Jie Shen,et al.  A stable scheme and its convergence analysis for a 2D dynamic Q-tensor model of nematic liquid crystals , 2017 .

[13]  N. Walkington,et al.  Q-tensor model for electrokinetics in nematic liquid crystals , 2016, 1612.03446.

[14]  P. Aursand,et al.  On the Dynamics of the Weak Fréedericksz Transition for Nematic Liquid Crystals , 2016 .

[15]  Jia Zhao,et al.  Semi-Discrete Energy-Stable Schemes for a Tensor-Based Hydrodynamic Model of Nematic Liquid Crystal Flows , 2016, J. Sci. Comput..

[16]  Nigel J. Mottram,et al.  Introduction to Q-tensor theory , 2014, 1409.3542.

[17]  Horng-Show Koo,et al.  LCD-based color filter films fabricated by a pigment-based colorant photo resist inks and printing technology , 2006 .

[18]  Joseph A. Castellano,et al.  Liquid Gold: The Story Of Liquid Crystal Displays and the Creation of an Industry , 2005 .

[19]  Katsuhiko Satoh,et al.  A COMPUTER SIMULATION INVESTIGATION OF THE FREEDERICKSZ TRANSITION FOR THE NEMATIC PHASE OF A GAY-BERNE MESOGEN , 2003 .

[20]  P. Bos,et al.  Fast Q-tensor method for modeling liquid crystal director configurations with defects , 2002 .

[21]  A. Sonnet,et al.  Dynamics of dissipative ordered fluids. , 2001, Physical review. E, Statistical, nonlinear, and soft matter physics.

[22]  Philip J. Bos,et al.  Multidimensional Director Modeling Using the Q Tensor Representation in a Liquid Crystal Cell and Its Application to the π Cell with Patterned Electrodes , 1999 .

[23]  Timothy A. Davis,et al.  Finite Element Analysis of the Landau--de Gennes Minimization Problem for Liquid Crystals , 1998 .

[24]  S. Pikin,et al.  Orienting effect of an electric field on nematic liquid crystals , 1973 .

[25]  V. Fréedericksz,et al.  Theoretisches und Experimentelles zur Frage nach der Natur der anisotropen Flüssigkeiten , 1927 .

[26]  Juan Pablo Borthagaraya,et al.  Chapter 5-The Q-tensor model with uniaxial constraint , 2021 .

[27]  Hongyun Wang,et al.  Optical Fredericks Transition in a Nematic Liquid Crystal Layer , 2015 .

[28]  M. Yoneya,et al.  Physics of Liquid Crystals , 2014 .

[29]  W. Helfrich.  Electric Alignment of Liquid Crystal , 1973 .