Pattern Formation for Nematic Liquid Crystals-Modelling, Analysis, and Applications

We summarise some recent results on solution landscapes for two-dimensional (2D) problems in the Landau– de Gennes theory for nematic liquid crystals. We study energy-minimizing and non energy-minimizing solutions of the Euler–Lagrange equations associated with a reduced Landau-de Gennes free energy on 2D domains with Dirichlet tangent boundary conditions. We review results on the multiplicity and regularity of solutions in distinguished asymptotic limits, using variational methods, methods from the theory of nonlinear partial differential equations, combinatorial arguments and scientific computation. The results beautifully canvass the competing effects of geometry (shape, size and symmetry), material anisotropy, and the symmetry of the model itself, illustrating the tremendous possibilities for exotic ordering transitions in 2D frameworks.

[1]  Pingwen Zhang,et al.  High-Index Optimization-Based Shrinking Dimer Method for Finding High-Index Saddle Points , 2019, SIAM J. Sci. Comput..

[2]  Iain W. Stewart,et al.  The Static and Dynamic Continuum Theory of Liquid Crystals , 2001 .

[3]  H. H. Wensink.  Polymeric Nematics of Associating Rods: Phase Behavior, Chiral Propagation, and Elasticity , 2019, Macromolecules.

[4]  F. Lin,et al.  Static and dynamic theories of liquid crystals , 2001 .

[5]  Iam-Choon Khoo,et al.  Introduction to Liquid Crystals , 2006 .

[6]  Anders Logg,et al.  Automated Solution of Differential Equations by the Finite Element Method: The FEniCS Book , 2012 .

[7]  Apala Majumdar,et al.  From molecular to continuum modelling of bistable liquid crystal devices , 2017 .

[8]  G. Ranganath.  Defects in liquid crystals , 1990 .

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

[10]  Solution landscape of a reduced Landau–de Gennes model on a hexagon , 2020, 2003.07643.

[11]  W. Maier,et al.  Eine einfache molekulare Theorie des nematischen kristallinflüssigen Zustandes , 1958 .

[12]  Dmitry Golovaty,et al.  Dimension Reduction for the Landau-de Gennes Model in Planar Nematic Thin Films , 2015, Journal of Nonlinear Science.

[13]  Lei Zhang,et al.  Recent developments in computational modelling of nucleation in phase transformations , 2016 .

[14]  P. Bauman,et al.  Analysis of Nematic Liquid Crystals with Disclination Lines , 2011, 1106.5031.

[15]  Pingwen Zhang,et al.  Transition pathways between defect patterns in confined nematic liquid crystals , 2019, J. Comput. Phys..

[16]  A. Mosley.  Liquid crystal applications and uses —volume 1 , 1991 .

[17]  J. Lagerwall.  An Introduction to the Physics of Liquid Crystals , 2016 .

[18]  A. Majumdar,et al.  Order reconstruction patterns in nematic liquid crystal wells , 2014, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences.

[19]  C Tsakonas,et al.  Multistable alignment states in nematic liquid crystal filled wells , 2007 .

[20]  Jesse Freeman,et al.  in Morse theory, , 1999 .

[21]  Peter Palffy-Muhoray.  Orientationally Ordered Soft Matter: The Diverse World of Liquid Crystals , 2007 .

[22]  André R. Studart,et al.  Three-dimensional printing of hierarchical liquid-crystal-polymer structures , 2018, Nature.

[23]  A. Majumdar,et al.  Tailored nematic and magnetization profiles on two-dimensional polygons. , 2021, Physical review. E.

[24]  Apala Majumdar,et al.  Elastic anisotropy of nematic liquid crystals in the two-dimensional Landau-de Gennes model , 2021, 2105.10253.

[25]  S. Hess,et al.  Alignment tensor versus director: Description of defects in nematic liquid crystals. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[26]  Apala Majumdar,et al.  Multistability in planar liquid crystal wells. , 2012, Physical review. E, Statistical, nonlinear, and soft matter physics.

[27]  A. Majumdar,et al.  Surface, size and topological effects for some nematic equilibria on rectangular domains , 2020 .

[28]  L. Onsager.  THE EFFECTS OF SHAPE ON THE INTERACTION OF COLLOIDAL PARTICLES , 1949 .

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

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

[31]  Alfredo Bautista,et al.  An explicit finite differences scheme over hexagonal tessellation , 2001, Appl. Math. Lett..

[32]  Apala Majumdar,et al.  Equilibrium order parameters of nematic liquid crystals in the Landau-de Gennes theory , 2008, European Journal of Applied Mathematics.

[33]  Xavier Lamy,et al.  Bifurcation Analysis in a Frustrated Nematic Cell , 2013, J. Nonlinear Sci..

[34]  J. C. Jones.  Defects, flexoelectricity and RF communications: the ZBD story , 2017 .

[35]  J. Ericksen.  Liquid crystals with variable degree of orientation , 1991 .

[36]  G. Friedel,et al.  Les états mésomorphes de la matière , 1922 .

[37]  C. Yakacki,et al.  Liquid crystal elastomer shell actuators with negative order parameter , 2019, Science Advances.

[38]  Hiroshi Yokoyama,et al.  Tristable nematic liquid-crystal device using micropatterned surface alignment , 2002, Nature.

[39]  Frédéric Hélein,et al.  Asymptotics for the minimization of a Ginzburg-Landau functional , 1993 .

[40]  A. Isihara,et al.  Theory of Liquid Crystals , 1972 .

[41]  O. Lindenmeyer,et al.  Beiträge zur Kenntniss des Cholesterins , 1863 .

[42]  A. Majumdar,et al.  Free energy pathways of a multistable liquid crystal device. , 2015, Soft matter.

[43]  Defects , 2018, Fundamentals of Solid State Engineering.

[44]  Sven Strauss.  Liquid Crystals Applications And Uses , 2016 .

[45]  A. Majumdar,et al.  The Well Order Reconstruction Solution for three-dimensional wells, in the Landau–de Gennes theory , 2019, International Journal of Non-Linear Mechanics.

[46]  Apala Majumdar,et al.  Error Analysis of Nitsche’s and Discontinuous Galerkin Methods of a Reduced Landau–de Gennes Problem , 2020, Comput. Methods Appl. Math..

[47]  Apala Majumdar,et al.  Discontinuous Galerkin Finite Element Methods for the Landau-de Gennes Minimization Problem of Liquid Crystals , 2019, IMA Journal of Numerical Analysis.

[48]  Apala Majumdar,et al.  Landau–De Gennes Theory of Nematic Liquid Crystals: the Oseen–Frank Limit and Beyond , 2008, 0812.3131.

[49]  Kenneth Stephenson,et al.  Explorations in Complex Analysis , 2012 .

[50]  Apala Majumdar,et al.  Order Reconstruction for Nematics on Squares and Hexagons: A Landau-de Gennes Study , 2017, SIAM J. Appl. Math..

[51]  J. Lagerwall,et al.  A new era for liquid crystal research: Applications of liquid crystals in soft matter nano-, bio- and microtechnology , 2012 .

[52]  J. Alvarado,et al.  Colloidal liquid crystals in rectangular confinement: theory and experiment. , 2014, Soft matter.

[53]  Apala Majumdar,et al.  Order Reconstruction for Nematics on Squares with Isotropic Inclusions: A Landau-De Gennes Study , 2018, SIAM J. Appl. Math..