Geometry and mechanics of disclination lines in 3D nematic liquid crystals.
TL;DR: In this article, the authors analyze the structure of 3D disclination lines in terms of the director deformation modes around the disclination, as well as the nematic order tensor inside the core.
Abstract: In 3D nematic liquid crystals, disclination lines have a range of geometric structures. Locally, they may resemble +1/2 or −1/2 defects in 2D nematic phases, or they may have 3D twist. Here, we analyze the structure in terms of the director deformation modes around the disclination, as well as the nematic order tensor inside the disclination core. Based on this analysis, we construct a vector to represent the orientation of the disclination, as well as tensors to represent higher-order structure. We apply this method to simulations of a 3D disclination arch, and determine how the structure changes along the contour length. We then use this geometric analysis to investigate three types of forces acting on a disclination: Peach–Koehler forces due to external stress, interaction forces between disclination lines, and active forces. These results apply to the motion of disclination lines in both conventional and active liquid crystals.
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TL;DR: In this article, a trigonometric and hyperbolic type traveling wave solutions are produced by using the sub-equation analytical method by taking into account the Kerr Law properties of the equation defining nematic liquid crystals.
Abstract: In this study, trigonometric and hyperbolic type traveling wave solutions are produced by using the sub-equation analytical method by taking into account the Kerr Law properties of the equation defining nematic liquid crystals. The resulting traveling wave solutions of the equation play an important role in the energy transport in soliton molecules in liquid crystals. In addition, the solitary wave behaviors obtained for different values of the constants in the produced traveling wave solutions are discussed. The hyperbolic type traveling wave solutions of the equation defining the nematic liquid crystals incorporating Kerr Law property is represented as dark and singular solitons. Ready-made package programs are used for algebraic operations and graphic drawings. It is emphasized that the analytical method is effective, useful and valid.
18 citations
TL;DR: In this article, the intrinsic geometry of elastic distortions in three-dimensional liquid crystals is described and necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor.
Abstract: We give a description of the intrinsic geometry of elastic distortions in three-dimensional nematic
liquid crystals and establish necessary and sufficient conditions for a set of functions to represent
these distortions by describing how they couple to the curvature tensor. We demonstrate that, in
contrast to the situation in two dimensions, the first-order gradients of the director alone are not
sufficient for full reconstruction of the director field from its intrinsic geometry: it is necessary to
provide additional information about the second-order director gradients. We describe several
different methods by which the director field may be reconstructed from its intrinsic geometry.
Finally, we discuss the coupling between individual distortions and curvature from the perspective
of Lie algebras and groups and describe homogeneous spaces on which pure modes of distortion
can be realised.
10 citations
TL;DR: In this paper, a coarse-grained approach was developed to describe the dynamics of skyrmions in chiral nematic liquid crystals, where a localized excitation was represented by a few macroscopic degrees of freedom, including the position of the excitation and the orientation of the background director.
Abstract: Recent experiments have found that applied electric fields can induce motion of skyrmions in chiral nematic liquid crystals. To understand the magnitude and direction of the induced motion, we develop a coarse-grained approach to describe dynamics of skyrmions, similar to our group's previous work on the dynamics of disclinations. In this approach, we represent a localized excitation in terms of a few macroscopic degrees of freedom, including the position of the excitation and the orientation of the background director. We then derive the Rayleigh dissipation function, and hence the equations of motion, in terms of these macroscopic variables. We demonstrate this theoretical approach for 1D motion of a sine-Gordon soliton, and then extend it to 2D motion of a skyrmion. Our results show that skyrmions move in a direction perpendicular to the induced tilt of the background director. When the applied field is removed, skyrmions move in the opposite direction but not with equal magnitude, and hence the overall motion may be rectified.
9 citations
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TL;DR: In this paper, the concept of nbits was introduced by exploiting a quaternionic mapping from liquid crystals (LCs) to the Poincar\'e-Bloch sphere, which can be used as a computational resource.
Abstract: Liquid crystals (LCs) can host robust topological defect structures that essentially determine their optical and elastic properties. Although recent experimental progress enables precise control over localization and dynamics of nematic LC defects, their practical potential for information storage and processing has yet to be explored. Here, we introduce the concept of nematic bits (nbits) by exploiting a quaternionic mapping from LC defects to the Poincar\'e-Bloch sphere. Through theory and simulations, we demonstrate how single-nbit operations can be implemented using electric fields, in close analogy with Pauli, Hadamard and other common quantum gates. Ensembles of two-nbit states can exhibit strong statistical correlations arising from nematoelastic interactions, which can be used as a computational resource. Utilizing nematoelastic interactions, we show how suitably arranged 4-nbit configurations can realize universal classical NOR and NAND gates. Finally, we demonstrate the implementation of generalized logical functions that take values on the Poincar\'e-Bloch sphere. These results open a new route towards the implementation of classical and non-classical computation strategies in topological soft matter systems.
2 citations
TL;DR: In this paper, the intrinsic geometry of elastic distortions in three-dimensional nematic liquid crystals is described and necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor.
Abstract: We give a description of the intrinsic geometry of elastic distortions in three-dimensional nematic liquid crystals and establish necessary and sufficient conditions for a set of functions to represent these distortions by describing how they couple to the curvature tensor. We demonstrate that, in contrast to the situation in two dimensions, the first-order gradients of the director alone are not sufficient for full reconstruction of the director field from its intrinsic geometry: it is necessary to provide additional information about the second-order director gradients. We describe several different methods by which the director field may be reconstructed from its intrinsic geometry. Finally, we discuss the coupling between individual distortions and curvature from the perspective of Lie algebras and groups and describe homogeneous spaces on which pure modes of distortion can be realised.
References
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TL;DR: This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments, and highlights the experimental relevance of various semimicroscopic derivations of the continuum theory for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material.
Abstract: This review summarizes theoretical progress in the field of active matter, placing it in the context of recent experiments. This approach offers a unified framework for the mechanical and statistical properties of living matter: biofilaments and molecular motors in vitro or in vivo, collections of motile microorganisms, animal flocks, and chemical or mechanical imitations. A major goal of this review is to integrate several approaches proposed in the literature, from semimicroscopic to phenomenological. In particular, first considered are ``dry'' systems, defined as those where momentum is not conserved due to friction with a substrate or an embedding porous medium. The differences and similarities between two types of orientationally ordered states, the nematic and the polar, are clarified. Next, the active hydrodynamics of suspensions or ``wet'' systems is discussed and the relation with and difference from the dry case, as well as various large-scale instabilities of these nonequilibrium states of matter, are highlighted. Further highlighted are various large-scale instabilities of these nonequilibrium states of matter. Various semimicroscopic derivations of the continuum theory are discussed and connected, highlighting the unifying and generic nature of the continuum model. Throughout the review, the experimental relevance of these theories for describing bacterial swarms and suspensions, the cytoskeleton of living cells, and vibrated granular material is discussed. Promising extensions toward greater realism in specific contexts from cell biology to animal behavior are suggested, and remarks are given on some exotic active-matter analogs. Last, the outlook for a quantitative understanding of active matter, through the interplay of detailed theory with controlled experiments on simplified systems, with living or artificial constituents, is summarized.
3,314 citations
TL;DR: Aspects of the theory of homotopy groups are described in a mathematical style closer to that of condensed matter physics than that of topology in this paper, where the focus is on mathematical pedagogy rather than on a systematic review of applications.
Abstract: Aspects of the theory of homotopy groups are described in a mathematical style closer to that of condensed matter physics than that of topology. The aim is to make more readily accessible to physicists the recent applications of homotopy theory to the study of defects in ordered media. Although many physical examples are woven into the development of the subject, the focus is on mathematical pedagogy rather than on a systematic review of applications.
1,612 citations
799 citations
TL;DR: Experiments described here support the simple "one-scale" model for cosmic string evolution, as well as some qualitative predictions of string statistical mechanics.
Abstract: Liquid crystals are remarkably useful for laboratory exploration of the dynamics of cosmologically relevant defects. They are convenient to work with, they allow the direct study of the "scaling solution" for a network of strings, and they provide a model for the evolution of monopoles and texture. Experiments described here support the simple "one-scale" model for cosmic string evolution, as well as some qualitative predictions of string statistical mechanics. The structure of monopoles and their apparent cylindrical but not spherical symmetry is discussed. A particular kind of defect known as texture is described and is shown to have a dynamical instability—it can decay into a monopole-antimonopole pair. This decay process has been observed occurring in the liquid crystal, and studied with numerical simulations.
545 citations
TL;DR: A simple analytical model for the defect dynamics is developed which reproduces the key features of both the numerical solutions and recent experiments on microtubule-kinesin assemblies.
Abstract: Liquid crystals inevitably possess topological defect excitations generated through boundary conditions, through applied fields, or in quenches to the ordered phase. In equilibrium, pairs of defects coarsen and annihilate as the uniform ground state is approached. Here we show that defects in active liquid crystals exhibit profoundly different behavior, depending on the degree of activity and its contractile or extensile character. While contractile systems enhance the annihilation dynamics of passive systems, extensile systems act to drive defects apart so that they swarm around in the manner of topologically well-characterized self-propelled particles. We develop a simple analytical model for the defect dynamics which reproduces the key features of both the numerical solutions and recent experiments on microtubule-kinesin assemblies.
324 citations