Article
Automation & Control Systems
Razvan-Dumitru Ceuca
Summary: The study examines the behavior of embedded particle lattice in a nematic host, considering both cubic symmetry and loss of cubic symmetry cases. By adjusting the surface anchoring energy density and corresponding coefficients, the phase transition temperature can be influenced. Additionally, loss of cubic symmetry introduces an additional term in the free energy functional, describing a change in alignment preference of liquid crystal particles.
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
(2021)
Article
Mathematics, Applied
Razvan-Dumitru Ceuca, Jamie M. Taylor, Arghir Zarnescu
Summary: In this study, we investigate the impact of boundary rugosity in nematic liquid crystalline systems. A highly general formulation is employed to handle multiple liquid crystal theories simultaneously. Utilizing Gamma convergence techniques, we demonstrate that the fine-scale surface oscillations can be substituted by an effective homogenized surface energy in a simpler domain. Convergence rates are then quantitatively examined in a simplified setting.
COMMUNICATIONS IN CONTEMPORARY MATHEMATICS
(2023)
Article
Mathematics, Applied
Giacomo Canevari, Jamie M. Taylor
Summary: The study investigates a non-local free energy functional that models the competition between entropy and pairwise interactions, with nematic liquid crystals as a specific case. The research extends previous work on the behavior of these models in the large-domain limit to establish Holder bounds for (almost-)minimizers on bounded domains. The proof techniques bear analogy with recent work on singularly perturbed energy functionals, particularly in the context of the Ginzburg-Landau and Landau-de Gennes models.
NONLINEAR ANALYSIS-THEORY METHODS & APPLICATIONS
(2022)
Review
Chemistry, Physical
Jun-ichi Fukuda
Summary: This article presents numerical studies on the exotic structures of chiral liquid crystals, particularly focusing on those in thin films. The research shows that thin films of chiral liquid crystals can exhibit a diverse variety of ordered structures, including a hexagonal lattice of topological line defects. This provides an interesting platform for investigating exotic structures of orientational order arising from the frustration between bulk ordering and spatial confinement with surface anchoring.
LIQUID CRYSTALS REVIEWS
(2022)
Article
Mathematics
Qiao Liu
Summary: This study focuses on the partial regularity of suitable weak solutions to the 3d co-rotational Beris-Edwards system, presenting an improved version of the Caffarelli-Kohn-Nirenberg criterion in terms of the velocity gradient. The findings highlight the importance of the velocity field over the Q-tensor field in the partial regularity theory of the 3d co-rotational Beris-Edwards system.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Chemistry, Physical
M. A. Aliev
Summary: The nematic ordering of V-shaped molecules with two semiflexible arms connected by one end at an external angle is investigated using the Landau theory of phase transitions. The phase diagram shows stable regions of isotropic, prolate uniaxial, oblate uniaxial, and biaxial nematic phases. The phase transitions from isotropic to uniaxial and from uniaxial to biaxial nematic are of the first and second order, respectively. The stability area of the biaxial phase decreases with decreasing arm stiffness and a re-entrant biaxial phase is possible in a certain range of angles. The tricritical points appear at the curves of the uniaxial to biaxial nematic transition in the phase diagrams within certain intervals of the stiffness parameter.
JOURNAL OF MOLECULAR LIQUIDS
(2023)
Article
Mathematics, Applied
Yucen Han, Apala Majumdar
Summary: We study nematic equilibria in an unbounded domain with a two-dimensional regular polygonal hole of K edges using a reduced Landau-de Gennes framework, which complements our previous work on nematic equilibria confined inside regular polygons. The dimensionless model parameters, lambda and gamma *, represent the ratio of the hole's edge length to the nematic correlation length and the nematic director at infinity, respectively. In the limit of lambda -> 0, the limiting profile exhibits two interior point defects outside a generic polygon hole, except for a triangle and a square. For a square hole, the limiting profile has either no interior defects or two line defects, depending on gamma *, while a triangular hole only has a unique interior point defect outside the hole. In the limit of lambda -> infinity, there are at least ((2)(K)) stable states, and the presence of gamma * enhances bistability compared to the interior problem. Our work provides new insights into manipulating the existence, location, and dimensionality of defects.
JOURNAL OF NONLINEAR SCIENCE
(2023)
Article
Mathematics
Ho-Man Tai, Yong Yu
Summary: We investigate the 3D Landau-de Gennes theory with finite temperature to understand the spherical droplet problem. By rigorously constructing biaxial-ring solutions and split-core-segment solutions, we provide theoretical confirmation for the numerical findings of Gartland-Mkaddem in [15]. The paper also addresses the structures of disclinations.
JOURNAL OF FUNCTIONAL ANALYSIS
(2023)
Article
Computer Science, Interdisciplinary Applications
Cody D. Schimming, Jorge Vinals, Shawn W. Walker
Summary: This study presents a numerical method based on a tensor order parameter description for achieving fully anisotropic elasticity, extending the Landau-de Gennes Q-tensor theory to anisotropic phases. By introducing a microscopic model of the nematogen and a constraint on eigenvalue bounds of Q, physically valid order parameter Q is ensured while allowing for more general gradient energy densities. The method is demonstrated in specific two-dimensional examples and in three dimensions for various defect cases, showing successful results in cases where the Landau-de Gennes model with elastic anisotropy fails.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Baoming Shi, Yucen Han, Lei Zhang
Summary: By studying the solution landscape and bifurcation diagrams of confined nematic liquid crystals, novel solution states are discovered, and the effects of geometrical anisotropy on confined defect patterns are revealed.
SIAM JOURNAL ON APPLIED MATHEMATICS
(2022)
Article
Chemistry, Multidisciplinary
Huanhuan Zhao, Zhidong Zhang, Jiliang Zhu, Xuan Zhou
Summary: In this study, the double-twist director configuration in a cylindrical shell with degenerate planar anchoring is investigated using Landau-de Gennes theory and differential iteration method. The combined effects of saddle-splay elasticity and curvature on the spontaneous chiral structure are analyzed, revealing two domain wall structures and a new 1/2 defect ring structure between opposite-handed domains. The stabilities of these structures are analyzed through an energy comparison.
Article
Mathematics, Applied
Dmitry Golovaty, Michael Novack, Peter Sternberg
Summary: This study introduces a quartic energy based on the Q-tensor within the framework of the Landau-de Gennes theory, offering a wider range of elastic constant choices compared to the commonly considered cubic theory for modeling nematic-to-isotropic phase transitions. A rigorous connection is established between this theory and its Oseen-Frank counterpart using Gamma-convergence arguments, with strong convergence of the associated minimisers also proven.
EUROPEAN JOURNAL OF APPLIED MATHEMATICS
(2021)
Article
Physics, Multidisciplinary
Liang De-Shan, Huang Hou-Bing, Zhao Ya-Nan, Liu Zhu-Hong, Wang Hao-Yu, Ma Xing-Qiao
Summary: Algebraic topology, algebraic geometry, and category theory are new branches of mathematics that have had significant interactions with modern physics in recent decades. Topological phenomena, such as the distribution of viral particles in spatial environments and the formation of nanovesicles and condensates, play important roles in various systems. Research based on Landau-de Gennes theory constructs models to simulate topological charge distribution in liquid crystals, revealing how the size of the disc affects equilibrium positions and the angle between topological charges in the range of 140 to 180 degrees.
ACTA PHYSICA SINICA
(2021)
Article
Mathematics, Applied
Shibin Dai, Joseph Renzi, Steven M. Wise
Summary: This paper investigates the Gamma-limit of the degenerate de Gennes-Cahn-Hilliard equation, revealing that it is proportional to the interface area, determined by the de Gennes coefficient and the double well potential.
COMMUNICATIONS IN MATHEMATICAL SCIENCES
(2023)
Article
Multidisciplinary Sciences
Yucen Han, Joseph Harris, Apala Majumdar, Lei Zhang
Summary: This study investigates the effects of elastic anisotropy on Landau-de Gennes critical points in nematic liquid crystals on a square domain. Various symmetric critical points are discovered and the stabilizing effects of L-2 are proven. Numerical bifurcation diagrams illustrate the interplay of elastic anisotropy and geometry in nematic solution landscapes.
PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES
(2022)
Article
Mathematics
Jiri Dadok, Peter Sternberg
JOURNAL OF GEOMETRIC ANALYSIS
(2018)
Article
Mathematics, Applied
Nam Q. Le, Peter J. Sternberg
ANNALI DI MATEMATICA PURA ED APPLICATA
(2019)
Article
Mathematics, Applied
Dmitry Golovaty, Peter Sternberg, Raghavendra Venkatraman
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2019)
Article
Mathematics, Applied
Yaniv Almog, Leonid Berlyand, Dmitry Golovaty, Itai Shafrir
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2019)
Article
Mathematics, Applied
Dmitry Golovaty, Michael Novack, Peter Sternberg, Raghavendra Venkatraman
ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS
(2020)
Article
Mathematics, Applied
Dmitry Golovaty, Michael Novack, Peter Sternberg
Summary: This study introduces a quartic energy based on the Q-tensor within the framework of the Landau-de Gennes theory, offering a wider range of elastic constant choices compared to the commonly considered cubic theory for modeling nematic-to-isotropic phase transitions. A rigorous connection is established between this theory and its Oseen-Frank counterpart using Gamma-convergence arguments, with strong convergence of the associated minimisers also proven.
EUROPEAN JOURNAL OF APPLIED MATHEMATICS
(2021)
Article
Mathematics, Applied
Stan Alama, Lia Bronsard, Dmitry Golovaty
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
(2020)
Article
Mathematics
Dmitry Golovaty, Michael Novack, Peter Sternberg
Summary: In this study, a one-dimensional variational problem related to a model for cholesteric liquid crystals is considered, with a focus on the higher energy penalty incurred by the twist deformation of the nematic director compared to other modes of deformation. By introducing a small parameter epsilon and an Allen-Cahn-type energy functional augmented by a twist term, the behavior of the energy as epsilon approaches zero is investigated. The existence of local energy minimizers classified by their overall twist, the Gamma-limit of the relaxed energies, and the inclusion of twist and jump terms are demonstrated.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2021)
Article
Mathematical & Computational Biology
Dmitry Golovaty, Young-Ki Kim, Oleg D. Lavrentovich, Michael Novack, Peter Sternberg
MATHEMATICAL MODELLING OF NATURAL PHENOMENA
(2020)
Article
Physics, Fluids & Plasmas
Christopher Conklin, O. M. Tovkach, Jorge Vinals, M. Carme Calderer, Dmitry Golovaty, Oleg D. Lavrentovich, Noel J. Walkington