1
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Chen Y, Jing B, Chang Z, Gong J. Luminescent Möbius Strip of a Flexible Halogen-Bonded Cocrystal Evolved from Ring and Helix. JACS AU 2022; 2:2686-2692. [PMID: 36590258 PMCID: PMC9795465 DOI: 10.1021/jacsau.2c00469] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 08/25/2022] [Revised: 09/21/2022] [Accepted: 09/27/2022] [Indexed: 06/17/2023]
Abstract
Luminescent Möbius strip microstructures have been created for the first time based on flexible organic single crystals via a template-free solution self-assembly. We herein demonstrated a rationally designed morphological evolution toward Möbius strips from rings and helixes. Our findings lay the foundation for the future construction of complex matters with predetermined morphologies and functions from crystal systems.
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Affiliation(s)
- Yifu Chen
- State
Key Laboratory of Chemical Engineering, School of Chemical Engineering
and Technology, Tianjin University, Tianjin 300072, People’s Republic of China
- Collaborative
Innovation Center of Chemical Science and Engineering, Tianjin 300072, People’s Republic of China
- Haihe
Laboratory of Sustainable Chemical Transformations, Tianjin 300192, People’s Republic of China
| | - Bo Jing
- State
Key Laboratory of Chemical Engineering, School of Chemical Engineering
and Technology, Tianjin University, Tianjin 300072, People’s Republic of China
- Collaborative
Innovation Center of Chemical Science and Engineering, Tianjin 300072, People’s Republic of China
- Haihe
Laboratory of Sustainable Chemical Transformations, Tianjin 300192, People’s Republic of China
| | - Zewei Chang
- State
Key Laboratory of Chemical Engineering, School of Chemical Engineering
and Technology, Tianjin University, Tianjin 300072, People’s Republic of China
- Collaborative
Innovation Center of Chemical Science and Engineering, Tianjin 300072, People’s Republic of China
- Haihe
Laboratory of Sustainable Chemical Transformations, Tianjin 300192, People’s Republic of China
| | - Junbo Gong
- State
Key Laboratory of Chemical Engineering, School of Chemical Engineering
and Technology, Tianjin University, Tianjin 300072, People’s Republic of China
- Collaborative
Innovation Center of Chemical Science and Engineering, Tianjin 300072, People’s Republic of China
- Haihe
Laboratory of Sustainable Chemical Transformations, Tianjin 300192, People’s Republic of China
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2
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Sleiman JL, Burton RH, Caraglio M, Gutierrez Fosado YA, Michieletto D. Geometric Predictors of Knotted and Linked Arcs. ACS POLYMERS AU 2022; 2:341-350. [PMID: 36254317 PMCID: PMC9562465 DOI: 10.1021/acspolymersau.2c00021] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Abstract
![]()
Inspired by how certain proteins “sense”
knots and
entanglements in DNA molecules, here, we ask if local geometric features
that may be used as a readout of the underlying topology of generic
polymers exist. We perform molecular simulations of knotted and linked
semiflexible polymers and study four geometric measures to predict
topological entanglements: local curvature, local density, local 1D
writhe, and nonlocal 3D writhe. We discover that local curvature is
a poor predictor of entanglements. In contrast, segments with maximum
local density or writhe correlate as much as 90% of the time with
the shortest knotted and linked arcs. We find that this accuracy is
preserved across different knot types and also under significant spherical
confinement, which is known to delocalize essential crossings in knotted
polymers. We further discover that nonlocal 3D writhe is the best
geometric readout of the knot location. Finally, we discuss how these
geometric features may be used to computationally analyze entanglements
in generic polymer melts and gels.
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Affiliation(s)
- Joseph L. Sleiman
- School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
| | - Robin H. Burton
- School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
| | - Michele Caraglio
- Institut für Theoretische Physik, Universität Innsbruck, Technikerstraße 21A, A-6020 Innsbruck, Austria
| | - Yair Augusto Gutierrez Fosado
- School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
| | - Davide Michieletto
- School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
- MRC Human Genetics Unit, Institute of Genetics and Cancer, University of Edinburgh, Edinburgh EH4 2XU, United Kingdom
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3
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Orlandini E, Micheletti C. Topological and physical links in soft matter systems. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2021; 34:013002. [PMID: 34547745 DOI: 10.1088/1361-648x/ac28bf] [Citation(s) in RCA: 9] [Impact Index Per Article: 3.0] [Reference Citation Analysis] [Abstract] [Key Words] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/16/2021] [Accepted: 09/21/2021] [Indexed: 06/13/2023]
Abstract
Linking, or multicomponent topological entanglement, is ubiquitous in soft matter systems, from mixtures of polymers and DNA filaments packedin vivoto interlocked line defects in liquid crystals and intertwined synthetic molecules. Yet, it is only relatively recently that theoretical and experimental advancements have made it possible to probe such entanglements and elucidate their impact on the physical properties of the systems. Here, we review the state-of-the-art of this rapidly expanding subject and organize it as follows. First, we present the main concepts and notions, from topological linking to physical linking and then consider the salient manifestations of molecular linking, from synthetic to biological ones. We next cover the main physical models addressing mutual entanglements in mixtures of polymers, both linear and circular. Finally, we consider liquid crystals, fluids and other non-filamentous systems where topological or physical entanglements are observed in defect or flux lines. We conclude with a perspective on open challenges.
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Affiliation(s)
- Enzo Orlandini
- Department of Physics and Astronomy, University of Padova and Sezione INFN, Via Marzolo 8, Padova, Italy
| | - Cristian Micheletti
- SISSA, International School for Advanced Studies, via Bonomea 265, Trieste, Italy
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4
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Senyuk B, Mundoor H, Smalyukh II, Wensink HH. Nematoelasticity of hybrid molecular-colloidal liquid crystals. Phys Rev E 2021; 104:014703. [PMID: 34412251 DOI: 10.1103/physreve.104.014703] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/08/2021] [Accepted: 06/25/2021] [Indexed: 11/07/2022]
Abstract
Colloidal rods immersed in a thermotropic liquid-crystalline solvent are at the basis of so-called hybrid liquid crystals, which are characterized by tunable nematic fluidity with symmetries ranging from conventional uniaxial nematic or antinematic to orthorhombic [Mundoor et al., Science 360, 768 (2018)SCIEAS0036-807510.1126/science.aap9359]. We provide a theoretical analysis of the elastic moduli of such systems by considering interactions between the individual rods with the embedding solvent through surface-anchoring forces, as well as steric and electrostatic interactions between the rods themselves. For uniaxial systems, the presence of colloidal rods generates a marked increase of the splay elasticity, which we found to be in quantitative agreement with experimental measurements. For orthorhombic hybrid liquid crystals, we provide estimates of all 12 elastic moduli and show that only a small subset of those elastic constants play a relevant role in describing the nematoelastic properties. The complexity and possibilities related to identifying the elastic moduli in experiments are briefly discussed.
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Affiliation(s)
- B Senyuk
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
| | - H Mundoor
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
| | - I I Smalyukh
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA.,Department of Electrical, Computer, and Energy Engineering, Materials Science and Engineering Program and Soft Materials Research Center, University of Colorado, Boulder, Colorado 80309, USA.,Chemical Physics Program, Departments of Chemistry and Physics, University of Colorado, Boulder, Colorado 80309, USA.,Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, Colorado 80309, USA
| | - H H Wensink
- Laboratoire de Physique des Solides, Université Paris-Saclay & CNRS, UMR 8502, 91405 Orsay, France
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5
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Eun J, Pollard J, Kim SJ, Machon T, Jeong J. Layering transitions and metastable structures of cholesteric liquid crystals in cylindrical confinement. Proc Natl Acad Sci U S A 2021; 118:e2102926118. [PMID: 34373332 PMCID: PMC8379955 DOI: 10.1073/pnas.2102926118] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Our study of cholesteric lyotropic chromonic liquid crystals in cylindrical confinement reveals the topological aspects of cholesteric liquid crystals. The double-twist configurations we observe exhibit discontinuous layering transitions, domain formation, metastability, and chiral point defects as the concentration of chiral dopant is varied. We demonstrate that these distinct layer states can be distinguished by chiral topological invariants. We show that changes in the layer structure give rise to a chiral soliton similar to a toron, comprising a metastable pair of chiral point defects. Through the applicability of the invariants we describe to general systems, our work has broad relevance to the study of chiral materials.
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Affiliation(s)
- Jonghee Eun
- Department of Physics, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea
| | - Joseph Pollard
- Department of Physics, Durham University, Durham DH1 3LE, United Kingdom
| | - Sung-Jo Kim
- Department of Physics, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea
- Center for Soft and Living Matter, Institute for Basic Science, Ulsan 44919, Republic of Korea
| | - Thomas Machon
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
| | - Joonwoo Jeong
- Department of Physics, Ulsan National Institute of Science and Technology, Ulsan 44919, Republic of Korea;
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6
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Yuan Y, Keller P, Smalyukh II. Elastomeric nematic colloids, colloidal crystals and microstructures with complex topology. SOFT MATTER 2021; 17:3037-3046. [PMID: 33491729 DOI: 10.1039/d0sm02135k] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/12/2023]
Abstract
Control of physical behaviors of nematic colloids and colloidal crystals has been demonstrated by tuning particle shape, topology, chirality and surface charging. However, the capability of altering physical behaviors of such soft matter systems by changing particle shape and the ensuing responses to external stimuli has remained elusive. We fabricated genus-one nematic elastomeric colloidal ring-shaped particles and various microstructures using two-photon photopolymerization. Nematic ordering within both the nano-printed particle and the surrounding medium leads to anisotropic responses and actuation when heated. With the thermal control, elastomeric microstructures are capable of changing from genus-one to genus-zero surface topology. Using these particles as building blocks, we investigated elastomeric colloidal crystals immersed within a liquid crystal fluid, which exhibit crystallographic symmetry transformations. Our findings may lead to colloidal crystals responsive to a large variety of external stimuli, including electric fields and light. Pre-designed response of elastomeric nematic colloids, including changes of colloidal surface topology and lattice symmetry, are of interest for both fundamental research and applications.
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Affiliation(s)
- Ye Yuan
- Department of Physics, University of Colorado, Boulder, CO 80309, USA.
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7
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Smalyukh II. Review: knots and other new topological effects in liquid crystals and colloids. REPORTS ON PROGRESS IN PHYSICS. PHYSICAL SOCIETY (GREAT BRITAIN) 2020; 83:106601. [PMID: 32721944 DOI: 10.1088/1361-6633/abaa39] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
Humankind has been obsessed with knots in religion, culture and daily life for millennia, while physicists like Gauss, Kelvin and Maxwell already involved them in models centuries ago. Nowadays, colloidal particles can be fabricated to have shapes of knots and links with arbitrary complexity. In liquid crystals, closed loops of singular vortex lines can be knotted by using colloidal particles and laser tweezers, as well as by confining nematic fluids into micrometer-sized droplets with complex topology. Knotted and linked colloidal particles induce knots and links of singular defects, which can be interlinked (or not) with colloidal particle knots, revealing the diversity of interactions between topologies of knotted fields and topologically nontrivial surfaces of colloidal objects. Even more diverse knotted structures emerge in nonsingular molecular alignment and magnetization fields in liquid crystals and colloidal ferromagnets. The topological solitons include hopfions, skyrmions, heliknotons, torons and other spatially localized continuous structures, which are classified based on homotopy theory, characterized by integer-valued topological invariants and often contain knotted or linked preimages, nonsingular regions of space corresponding to single points of the order parameter space. A zoo of topological solitons in liquid crystals, colloids and ferromagnets promises new breeds of information displays and a plethora of data storage, electro-optic and photonic applications. Their particle-like collective dynamics echoes coherent motions in active matter, ranging from crowds of people to schools of fish. This review discusses the state of the art in the field, as well as highlights recent developments and open questions in physics of knotted soft matter. We systematically overview knotted field configurations, the allowed transformations between them, their physical stability and how one can use one form of knotted fields to model, create and imprint other forms. The large variety of symmetries accessible to liquid crystals and colloids offer insights into stability, transformation and emergent dynamics of fully nonsingular and singular knotted fields of fundamental and applied importance. The common thread of this review is the ability to experimentally visualize these knots in real space. The review concludes with a discussion of how the studies of knots in liquid crystals and colloids can offer insights into topologically related structures in other branches of physics, with answers to many open questions, as well as how these experimentally observable knots hold a strong potential for providing new inspirations to the mathematical knot theory.
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Affiliation(s)
- Ivan I Smalyukh
- Department of Physics, Department of Electrical, Computer and Energy Engineering, Materials Science and Engineering Program and Soft Materials Research Center, University of Colorado, Boulder, CO 80309, United States of America
- Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, CO 80309, United States of America
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8
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Wand CR, Bates MA. Chiral nematic liquid crystals in torus-shaped and cylindrical cavities. Phys Rev E 2019; 100:052702. [PMID: 31869937 DOI: 10.1103/physreve.100.052702] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2019] [Indexed: 11/07/2022]
Abstract
We present a Monte Carlo simulation study of chiral nematic liquid crystals confined in torus-shaped and cylindrical cavities. For an achiral nematic with planar degenerate anchoring confined to a toroidal or cylindrical cavity, the ground state is defect free, with an untwisted director field. As chirality is introduced, the ground state remains defect free but the director field becomes twisted within the cavity. For homeotropic anchoring, the ground state for an achiral nematic within a toroidal cavity consists of two disclination rings, one large and one small, that follow the major circumference of the torus. As chirality is introduced and increased, this ground state becomes unstable with respect to twisted configurations. The closed nature of the toroidal cavity requires that only a half integer number of twists can be formed and this leads to the ground state being either a single disclination line that encircles the torus twice or a pair of intertwined disclination rings forming stable, knotted defect structures.
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Affiliation(s)
- Charlie R Wand
- Department of Chemistry, University of York, Heslington, York YO10 5DD, United Kingdom
| | - Martin A Bates
- Department of Chemistry, University of York, Heslington, York YO10 5DD, United Kingdom
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9
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Affiliation(s)
- Thomas Machon
- H.H. Wills Physics Laboratory, Tyndall Avenue, United Kingdom
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10
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Najafi S. Topological entanglement of interlocked knotted-unknotted polymer rings. SOFT MATTER 2019; 15:1916-1921. [PMID: 30734820 DOI: 10.1039/c8sm02530d] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
Topological entanglements in biopolymers could drive them to certain internal statics and dynamics with important implications for biological functions. In this study, by means of molecular dynamics simulations, we demonstrate that the minimal crossing pattern of a braid plays a major role in its structural and dynamical properties; the braid consists of a knotted ring and an interlocked entwined unknotted polymer ring. In particular, we show that depending on the bending rigidity of the chains, the conformational energy of the braid can be either lower or higher than the unlocked polymer rings. Additionally, we find that a non-identical crossing pattern in the braid could distinctly enforce concerted internal conformational fluctuations between the interlocked rings.
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Affiliation(s)
- Saeed Najafi
- Department of Chemical and Biomolecular Engineering, University of Pennsylvania, Philadelphia, PA 19104, USA.
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11
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Orientation, elastic interaction and magnetic response of asymmetric colloids in a nematic liquid crystal. Sci Rep 2019; 9:81. [PMID: 30643211 PMCID: PMC6331558 DOI: 10.1038/s41598-018-36467-0] [Citation(s) in RCA: 11] [Impact Index Per Article: 2.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/09/2018] [Accepted: 11/22/2018] [Indexed: 11/09/2022] Open
Abstract
Colloidal particles in nematic liquid crystals create elastic distortion and experience long-range forces. The symmetry of elastic distortion and consequently the complexity of interaction strongly depends largely on the liquid crystal anchoring, topology and shape of the particles. Here, we introduce a new nematic colloidal system made of peanut-shaped hematite particles. We report experimental studies on spontaneous orientation, mutual interaction, laser assisted self-assembly and the effect of external magnetic fields on the colloids. Majority of the colloids spontaneously orient either parallel or perpendicular to the nematic director. The colloids that are oriented perpendicularly exhibit two types of textures due to the out of plane tilting, which is corroborated by the Landau-de Gennes Q-tensor modelling. The transverse magnetic moment of the peanut-shaped colloids is estimated by using a simple analysis based on the competing effects of magnetic and elastic torques.
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12
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Machon T, Alexander GP. Woven Nematic Defects, Skyrmions, and the Abelian Sandpile Model. PHYSICAL REVIEW LETTERS 2018; 121:237801. [PMID: 30576189 DOI: 10.1103/physrevlett.121.237801] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/08/2018] [Indexed: 06/09/2023]
Abstract
We show that a fixed set of woven defect lines in a nematic liquid crystal supports a set of nonsingular topological states which can be mapped on to recurrent stable configurations in the Abelian sandpile model or chip-firing game. The physical correspondence between local skyrmion flux and sandpile height is made between the two models. Using a toy model of the elastic energy, we examine the structure of energy minima as a function of topological class and show that the system admits domain wall skyrmion solitons.
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Affiliation(s)
- Thomas Machon
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, United Kingdom
| | - Gareth P Alexander
- Department of Physics and Centre for Complexity Science, University of Warwick, Coventry, CV4 7AL, United Kingdom
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13
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Everts JC, Ravnik M. Complex electric double layers in charged topological colloids. Sci Rep 2018; 8:14119. [PMID: 30237464 PMCID: PMC6147863 DOI: 10.1038/s41598-018-32550-8] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2018] [Accepted: 09/10/2018] [Indexed: 11/09/2022] Open
Abstract
Charged surfaces in contact with liquids containing ions are accompanied in equilibrium by an electric double layer consisting of a layer of electric charge on the surface that is screened by a diffuse ion cloud in the bulk fluid. This screening cloud determines not only the interactions between charged colloidal particles or polyelectrolytes and their self-assembly into ordered structures, but it is also pivotal in understanding energy storage devices, such as electrochemical cells and supercapacitors. However, little is known to what spatial complexity the electric double layers can be designed. Here, we show that electric double layers of non-trivial topology and geometry -including tori, multi-tori and knots- can be realised in charged topological colloidal particles, using numerical modelling within a mean-field Poisson-Boltzmann theory. We show that the complexity of double layers -including geometry and topology- can be tuned by changing the Debye screening length of the medium, or by changing the shape and topology of the (colloidal) particle. More generally, this work is an attempt to introduce concepts of topology in the field of charged colloids, which could lead to novel exciting material design paradigms.
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Affiliation(s)
- Jeffrey C Everts
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia.
| | - Miha Ravnik
- Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000, Ljubljana, Slovenia.,Department of Condensed Matter Physics, Jozef Stefan Institute, Jamova 39, 1000, Ljubljana, Slovenia
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14
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Hashemi SM, Ravnik M. Nematic colloidal knots in topological environments. SOFT MATTER 2018; 14:4935-4945. [PMID: 29740657 DOI: 10.1039/c8sm00539g] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/08/2023]
Abstract
The role of environment in shaping material properties is of great significance, but less is known about how non-trivial topology of the environment couples to material states, which can be of non-trivial topology themselves. In this paper, we demonstrate the role of the topology of the environment on the formation of complex nematic fields and defect structures, specifically in the system of nematic colloidal knots. The topological environments around knotted colloidal particles are suggested to exist as spherical surface-patterned nematic cavities imposing radial, uniform or hyperbolic nematic profiles. We show that topologically different nematic environments significantly affect and create differences in the colloidal field structure created within the environment, such as the location, profile and number of topological defects. Specifically, we demonstrate that topological environments in combination with knotted colloidal particles of non-trivial topology lead to the formation of diverse nematic knotted and linked fields. These fields are different adaptations of the knotted shape of the colloidal particles, creating knots and links of topological defects as well as escaped-core defect-like solitonic structures. These are observed in chiral nematic media but here are stabilised in achiral nematic media as a result of the distinct shape of the knotted colloidal particle, with a double helix segment and nematic environmental patterns. More generally, this paper is a contribution towards understanding the role of environment, especially its topology, on the response and defect formation in elastic fields, such as in nematic liquid crystal colloids.
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Affiliation(s)
- S Masoomeh Hashemi
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, Ljubljana, 1000, Slovenia.
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15
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16
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Wang Y, Zhang P, Chen JZY. Topological defects in an unconfined nematic fluid induced by single and double spherical colloidal particles. Phys Rev E 2017; 96:042702. [PMID: 29347619 DOI: 10.1103/physreve.96.042702] [Citation(s) in RCA: 7] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/18/2017] [Indexed: 06/07/2023]
Abstract
We present numerical solutions to the Landau-de Gennes free-energy model under the one-constant approximation for systems of single and double spherical colloidal particles immersed in an otherwise uniformly aligned nematic liquid crystal. A perfect homeotropic surface anchoring of liquid-crystal molecules on the spherical surface is considered. A large parameter space is carefully examined, including those in the free-energy model and those describing the dimer configurations and the background liquid-crystal orientation. The stability of the resulting liquid-crystal defects appearing in the neighborhood of the colloidal dimer pair is analyzed in light of the numerical results for their free energies. A number of scenarios are considered: a free dimer pair in a nematic fluid where the free-energy ground states are described in terms of a phase diagram, and a constrained dimer pair where the interparticle distance and the relative orientation of the distance vector to the nematic director can be manipulated. We pay particular attention to the nonsymmetric solutions, which yield several metastable defect states that can be observed in real systems. The high-precision numerical calculations are based on a spectral method, which is an enabling factor that allows us to compare the subtle difference in the free energies of different defect structures.
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Affiliation(s)
- Yiwei Wang
- LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
| | - Pingwen Zhang
- LMAM and School of Mathematical Sciences, Peking University, Beijing 100871, People's Republic of China
| | - Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, Ontario N2L 3G1, Canada
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17
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Michieletto D, Marenduzzo D, Orlandini E, Turner MS. Ring Polymers: Threadings, Knot Electrophoresis and Topological Glasses. Polymers (Basel) 2017; 9:E349. [PMID: 30971026 PMCID: PMC6418951 DOI: 10.3390/polym9080349] [Citation(s) in RCA: 18] [Impact Index Per Article: 2.6] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 07/05/2017] [Revised: 08/04/2017] [Accepted: 08/05/2017] [Indexed: 01/12/2023] Open
Abstract
Elucidating the physics of a concentrated suspension of ring polymers, or of an ensemble of ring polymers in a complex environment, is an important outstanding question in polymer physics. Many of the characteristic features of these systems arise due to topological interactions between polymers, or between the polymers and the environment, and it is often challenging to describe this quantitatively. Here we review recent research which suggests that a key role is played by inter-ring threadings (or penetrations), which become more abundant as the ring size increases. As we discuss, the physical consequences of such threadings are far-reaching: for instance, they lead to a topologically-driven glassy behaviour of ring polymer melts under pinning perturbations, while they can also account for the shape of experimentally observed patterns in two-dimensional gel electrophoresis of DNA knots.
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Affiliation(s)
- Davide Michieletto
- SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK.
| | - Davide Marenduzzo
- SUPA, School of Physics and Astronomy, University of Edinburgh, Peter Guthrie Tait Road, Edinburgh EH9 3FD, UK.
| | - Enzo Orlandini
- Dipartimento di Fisica e Astronomia, Sezione INFN, Università di Padova, Via Marzolo 8, 35131 Padova, Italy.
| | - Matthew S Turner
- Department of Physics and Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, UK.
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18
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Micheletti C, Orland H. Efficient Sampling of Knotting-Unknotting Pathways for Semiflexible Gaussian Chains. Polymers (Basel) 2017; 9:polym9060196. [PMID: 30970873 PMCID: PMC6432015 DOI: 10.3390/polym9060196] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/10/2017] [Revised: 05/17/2017] [Accepted: 05/19/2017] [Indexed: 11/16/2022] Open
Abstract
We propose a stochastic method to generate exactly the overdamped Langevin dynamics of semi-flexible Gaussian chains, conditioned to evolve between given initial and final conformations in a preassigned time. The initial and final conformations have no restrictions, and hence can be in any knotted state. Our method allows the generation of statistically independent paths in a computationally efficient manner. We show that these conditioned paths can be exactly generated by a set of local stochastic differential equations. The method is used to analyze the transition routes between various knots in crossable filamentous structures, thus mimicking topological reconnections occurring in soft matter systems or those introduced in DNA by topoisomerase enzymes. We find that the average number of crossings, writhe and unknotting number are not necessarily monotonic in time and that more complex topologies than the initial and final ones can be visited along the route.
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Affiliation(s)
- Cristian Micheletti
- International School for Advanced Studies (SISSA), Physics Area via Bonomea 265, I-34136 Trieste, Italy.
| | - Henri Orland
- Institut de Physique Théorique, CEA, CNRS, UMR3681, F-91191 Gif-sur-Yvette, France.
- Beijing Computational Science Research Center, No.10 East Xibeiwang Road, 100193 Beijing, China.
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Park S, Liu Q, Smalyukh II. Colloidal Surfaces with Boundaries, Apex Boojums, and Nested Elastic Self-Assembly of Nematic Colloids. PHYSICAL REVIEW LETTERS 2016; 117:277801. [PMID: 28084760 DOI: 10.1103/physrevlett.117.277801] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/20/2016] [Indexed: 06/06/2023]
Abstract
Self-assembly of colloidal particles is poised to become a powerful composite material fabrication technique, but remains challenged by a limited control over the ensuing structures. We develop a new breed of nematic colloids that are physical analogs of a mathematical surface with boundary, interacting with the molecular alignment field without inducing defects when flat. However, made from a thin nanofoil, they can be shaped to prompt formation of self-compensating defects that drive preprogramed elastic interactions mediated by the nematic host. To show this, we wrap the nanofoil on all triangular side faces of a pyramid, except its square base. The ensuing pyramidal cones induce point defects with fractional hedgehog charges of opposite signs, spontaneously align with respect to the far-field director to form elastic dipoles and nested assemblies with tunable spacing. Nanofoils shaped into octahedrons interact as elastic quadrupoles. Our findings may drive realization of low-symmetry colloidal phases.
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Affiliation(s)
- Sungoh Park
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
| | - Qingkun Liu
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
| | - Ivan I Smalyukh
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- Department of Electrical, Computer and Energy Engineering, Soft Materials Research Center and Materials Science and Engineering Program, University of Colorado, Boulder, Colorado 80309, USA
- Renewable and Sustainable Energy Institute, University of Colorado and National Renewable Energy Laboratory, Boulder, Colorado 80309, USA
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20
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Kedia H, Foster D, Dennis MR, Irvine WTM. Weaving Knotted Vector Fields with Tunable Helicity. PHYSICAL REVIEW LETTERS 2016; 117:274501. [PMID: 28084747 DOI: 10.1103/physrevlett.117.274501] [Citation(s) in RCA: 8] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 07/17/2016] [Indexed: 06/06/2023]
Abstract
We present a general construction of divergence-free knotted vector fields from complex scalar fields, whose closed field lines encode many kinds of knots and links, including torus knots, their cables, the figure-8 knot, and its generalizations. As finite-energy physical fields, they represent initial states for fields such as the magnetic field in a plasma, or the vorticity field in a fluid. We give a systematic procedure for calculating the vector potential, starting from complex scalar functions with knotted zero filaments, thus enabling an explicit computation of the helicity of these knotted fields. The construction can be used to generate isolated knotted flux tubes, filled by knots encoded in the lines of the vector field. Lastly, we give examples of manifestly knotted vector fields with vanishing helicity. Our results provide building blocks for analytical models and simulations alike.
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Affiliation(s)
- Hridesh Kedia
- Department of Physics, James Franck Institute, Enrico Fermi Institute, The University of Chicago, 929 E 57th St., Chicago, Illinois 60637, USA
| | - David Foster
- HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK
| | - Mark R Dennis
- HH Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 1TL, UK
| | - William T M Irvine
- Department of Physics, James Franck Institute, Enrico Fermi Institute, The University of Chicago, 929 E 57th St., Chicago, Illinois 60637, USA
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21
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Abstract
We give the global homotopy classification of nematic textures for a general domain with weak anchoring boundary conditions and arbitrary defect set in terms of twisted cohomology, and give an explicit computation for the case of knotted and linked defects in R3, showing that the distinct homotopy classes have a 1–1 correspondence with the first homology group of the branched double cover, branched over the disclination loops. We show further that the subset of those classes corresponding to elements of order 2 in this group has representatives that are planar and characterize the obstruction for other classes in terms of merons. The planar textures are a feature of the global defect topology that is not reflected in any local characterization. Finally, we describe how the global classification relates to recent experiments on nematic droplets and how elements of order 4 relate to the presence of τ lines in cholesterics.
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Affiliation(s)
- Thomas Machon
- Department of Physics and Centre for Complexity Science , University of Warwick , Coventry CV4 7AL, UK
| | - Gareth P Alexander
- Department of Physics and Centre for Complexity Science , University of Warwick , Coventry CV4 7AL, UK
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22
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Potestio R, Tubiana L. Discretized knot motion on a tensioned fiber induced by transverse waves. SOFT MATTER 2016; 12:669-673. [PMID: 26510521 DOI: 10.1039/c5sm01766a] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
Topological entanglement is a ubiquitous feature of many biological as well as artificial polymers and fibers. While the equilibrium properties of entangled chains have been the subject of several studies, little is known about their out-of-equilibrium behavior. Here, we address the problem of a stretched knotted fiber driven by a periodic force applied to one of its termini. We show that the onset of standing waves kinetically traps the knot in spatially localized states where the amplitude of the oscillations is maximal, while the knot normal diffusive dynamics is replaced by a discrete jump dynamics.
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Affiliation(s)
- Raffaello Potestio
- Max Planck Institute for Polymer Research, Ackermannweg 10, 55128 Mainz, Germany.
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23
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Horner KE, Miller MA, Steed JW, Sutcliffe PM. Knot theory in modern chemistry. Chem Soc Rev 2016; 45:6432-6448. [DOI: 10.1039/c6cs00448b] [Citation(s) in RCA: 61] [Impact Index Per Article: 7.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/15/2023]
Abstract
This tutorial review provides an introduction to the mathematics of knots and topological concepts in the context of the chemical sciences.
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Affiliation(s)
- Kate E. Horner
- SPOCK Group
- Department of Mathematical Sciences
- Durham University
- Durham
- UK
| | | | | | - Paul M. Sutcliffe
- SPOCK Group
- Department of Mathematical Sciences
- Durham University
- Durham
- UK
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24
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Ravnik M, Čopar S, Žumer S. Particles with changeable topology in nematic colloids. JOURNAL OF PHYSICS. CONDENSED MATTER : AN INSTITUTE OF PHYSICS JOURNAL 2015; 27:354111. [PMID: 26291540 DOI: 10.1088/0953-8984/27/35/354111] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
We show that nematic colloids can serve as a highly variable and controllable platform for studying inclusions with changeable topology and their effects on the surrounding ordering fields. We explore morphing of toroidal and knotted colloidal particles into effective spheres, distinctively changing their Euler characteristic and affecting the surrounding nematic field, including topological defect structures. With toroidal particles, the inner nematic defect eventually transitions from a wide loop to a point defect (a small loop). Trefoil particles become linked with two knotted defect loops, mutually forming a three component link, that upon tightening transform into a two-component particle-defect loop link. For more detailed topological analysis, Pontryagin-Thom surfaces are calculated and visualised, indicating an interesting cascade of defect rewirings caused by the shape morphing of the knotted particles.
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Affiliation(s)
- Miha Ravnik
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, 1000 Ljubljana, Slovenia
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25
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Orlova T, Aßhoff SJ, Yamaguchi T, Katsonis N, Brasselet E. Creation and manipulation of topological states in chiral nematic microspheres. Nat Commun 2015; 6:7603. [PMID: 26145716 PMCID: PMC4506501 DOI: 10.1038/ncomms8603] [Citation(s) in RCA: 55] [Impact Index Per Article: 6.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/22/2015] [Accepted: 05/25/2015] [Indexed: 12/26/2022] Open
Abstract
Topology is a universal concept that is encountered in daily life and is known to determine many static and dynamical properties of matter. Taming and controlling the topology of materials therefore constitutes a contemporary interdisciplinary challenge. Building on the controllable spatial properties of soft matter appears as a relevant strategy to address the challenge, in particular, because it may lead to paradigmatic model systems that allow checking theories experimentally. Here we report experimentally on a wealth of complex free-standing metastable topological architectures at the micron scale, in frustrated chiral nematic droplets. These results support recent works predicting the formation of free-standing knotted and linked disclination structures in confined chiral nematic fluids. We also demonstrate that various kinds of external fields (thermal, electrical and optical) can be used to achieve topological remote control. All this may foster the development of new devices based on topologically structured soft media.
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Affiliation(s)
- Tetiana Orlova
- Laboratoire Ondes et Matière d'Aquitaine, University of Bordeaux, CNRS, 351 cours de la Libération, Talence F-33400, France
| | - Sarah Jane Aßhoff
- Laboratory for Biomolecular Nanotechnology, MESA+ Institute for Nanotechnology, University of Twente, PO Box 207, Enschede 7500AE, The Netherlands
| | - Tadatsugu Yamaguchi
- Laboratory for Biomolecular Nanotechnology, MESA+ Institute for Nanotechnology, University of Twente, PO Box 207, Enschede 7500AE, The Netherlands
| | - Nathalie Katsonis
- Laboratory for Biomolecular Nanotechnology, MESA+ Institute for Nanotechnology, University of Twente, PO Box 207, Enschede 7500AE, The Netherlands
| | - Etienne Brasselet
- Laboratoire Ondes et Matière d'Aquitaine, University of Bordeaux, CNRS, 351 cours de la Libération, Talence F-33400, France
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26
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Senyuk B, Behabtu N, Martinez A, Lee T, Tsentalovich DE, Ceriotti G, Tour JM, Pasquali M, Smalyukh II. Three-dimensional patterning of solid microstructures through laser reduction of colloidal graphene oxide in liquid-crystalline dispersions. Nat Commun 2015; 6:7157. [DOI: 10.1038/ncomms8157] [Citation(s) in RCA: 51] [Impact Index Per Article: 5.7] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/09/2014] [Accepted: 04/10/2015] [Indexed: 11/09/2022] Open
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27
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Pylypovskyi OV, Kravchuk VP, Sheka DD, Makarov D, Schmidt OG, Gaididei Y. Coupling of chiralities in spin and physical spaces: the Möbius ring as a case study. PHYSICAL REVIEW LETTERS 2015; 114:197204. [PMID: 26024195 DOI: 10.1103/physrevlett.114.197204] [Citation(s) in RCA: 13] [Impact Index Per Article: 1.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/20/2014] [Indexed: 05/23/2023]
Abstract
We show that the interaction of the magnetic subsystem of a curved magnet with the magnet curvature results in the coupling of a topologically nontrivial magnetization pattern and topology of the object. The mechanism of this coupling is explored and illustrated by an example of a ferromagnetic Möbius ring, where a topologically induced domain wall appears as a ground state in the case of strong easy-normal anisotropy. For the Möbius geometry, the curvilinear form of the exchange interaction produces an additional effective Dzyaloshinskii-like term which leads to the coupling of the magnetochirality of the domain wall and chirality of the Möbius ring. Two types of domain walls are found, transversal and longitudinal, which are oriented across and along the Möbius ring, respectively. In both cases, the effect of magnetochirality symmetry breaking is established. The dependence of the ground state of the Möbius ring on its geometrical parameters and on the value of the easy-normal anisotropy is explored numerically.
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Affiliation(s)
| | | | - Denis D Sheka
- Taras Shevchenko National University of Kiev, 01601 Kiev, Ukraine
| | - Denys Makarov
- Institute for Integrative Nanosciences, IFW Dresden, 01069 Dresden, Germany
| | - Oliver G Schmidt
- Institute for Integrative Nanosciences, IFW Dresden, 01069 Dresden, Germany
| | - Yuri Gaididei
- Bogolyubov Institute for Theoretical Physics, 03143 Kiev, Ukraine
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28
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D'Adamo G, Marenduzzo D, Micheletti C, Orlandini E. Electric field controlled columnar and planar patterning of cholesteric colloids. PHYSICAL REVIEW LETTERS 2015; 114:177801. [PMID: 25978263 DOI: 10.1103/physrevlett.114.177801] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/22/2014] [Indexed: 06/04/2023]
Abstract
We study how dispersions of colloidal particles in a cholesteric liquid crystal behave under a time-dependent electric field. By controlling the amplitude and shape of the applied field wave, we show that the system can be reproducibly driven out of equilibrium through different kinetic pathways and navigated through a glassylike free energy landscape encompassing many competing metastable equilibria. Such states range from simple Saturn rings to complex structures featuring amorphous defect networks, or stacks of disclination loops. A nonequilibrium electric field can also trigger the alignment of particles into columnar arrays, through defect-mediated force impulses, or their repositioning within a plane. Our results are promising in terms of providing new avenues towards controlled patterning and self-assembly of soft colloid-liquid crystal composite materials.
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Affiliation(s)
- G D'Adamo
- SISSA, International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy
| | - D Marenduzzo
- SUPA, School of Physics and Astronomy, University of Edinburgh, Mayfield Road, Edinburgh EH9 3JZ, United Kingdom
| | - C Micheletti
- SISSA, International School for Advanced Studies, via Bonomea 265, I-34136 Trieste, Italy
| | - E Orlandini
- Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
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29
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Topological zoo of free-standing knots in confined chiral nematic fluids. Nat Commun 2015; 5:3057. [PMID: 24419153 DOI: 10.1038/ncomms4057] [Citation(s) in RCA: 59] [Impact Index Per Article: 6.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/26/2013] [Accepted: 12/03/2013] [Indexed: 11/08/2022] Open
Abstract
Knotted fields are an emerging research topic relevant to different areas of physics where topology plays a crucial role. Recent realization of knotted nematic disclinations stabilized by colloidal particles raised a challenge of free-standing knots. Here we demonstrate the creation of free-standing knotted and linked disclination loops in the cholesteric ordering fields, which are confined to spherical droplets with homeotropic surface anchoring. Our approach, using free energy minimization and topological theory, leads to the stabilization of knots via the interplay of the geometric frustration and intrinsic chirality. Selected configurations of the lowest complexity are characterized by knot or link types, disclination lengths and self-linking numbers. When cholesteric pitch becomes short on the confinement scale, the knotted structures change to practically unperturbed cholesteric structures with disclinations expelled close to the surface. The drops with knots could be controlled by optical beams and may be used for photonic elements.
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30
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Abstract
Geometric shape and topology of constituent particles can alter many colloidal properties such as Brownian motion, self-assembly, and phase behavior. Thus far, only single-component building blocks of colloids with connected surfaces have been studied, although topological colloids, with constituent particles shaped as freestanding knots and handlebodies of different genus, have been recently introduced. Here we develop a topological class of colloids shaped as multicomponent links. Using two-photon photopolymerization, we fabricate colloidal microparticle analogs of the classic examples of links studied in the field of topology, the Hopf and Solomon links, which we disperse in nematic fluids that possess orientational ordering of anisotropic rod-like molecules. The surfaces of these particles are treated to impose tangential or perpendicular boundary conditions for the alignment of liquid crystal molecules, so that they generate a host of topologically nontrivial field and defect structures in the dispersing nematic medium, resulting in an elastic coupling between the linked constituents. The interplay between the topologies of surfaces of linked colloids and the molecular alignment field of the nematic host reveals that linking of particle rings with perpendicular boundary conditions is commonly accompanied by linking of closed singular defect loops, laying the foundations for fabricating complex composite materials with interlinking-based structural organization.
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31
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Zuhail KP, Sathyanarayana P, Seč D, Čopar S, Škarabot M, Muševič I, Dhara S. Topological defect transformation and structural transition of two-dimensional colloidal crystals across the nematic to smectic-A phase transition. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 91:030501. [PMID: 25871035 DOI: 10.1103/physreve.91.030501] [Citation(s) in RCA: 7] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/22/2014] [Indexed: 06/04/2023]
Abstract
We observe that topological defects in nematic colloids are strongly influenced by the elasticity and onset of smectic layering across the nematic (N) to smectic-A (SmA) phase transition. When approaching the SmA phase from above, the nematic hyperbolic hedgehog defect that accompanies a spherical colloidal inclusion is transformed into a focal conic line in the SmA phase. This phase transformation has a strong influence on the pairwise colloidal interaction and is responsible for a structural transition of two-dimensional colloidal crystals. The pretransitional behavior of the point defect is supported by Landau-de Gennes Q-tensor modeling accounting for the increasing elastic anisotropy.
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Affiliation(s)
- K P Zuhail
- School of Physics, University of Hyderabad, Hyderabad-500046, India
| | - P Sathyanarayana
- School of Physics, University of Hyderabad, Hyderabad-500046, India
| | - D Seč
- J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - S Čopar
- J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - M Škarabot
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - I Muševič
- J. Stefan Institute, Jamova 39, SI-1000 Ljubljana, Slovenia
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000 Ljubljana, Slovenia
| | - S Dhara
- School of Physics, University of Hyderabad, Hyderabad-500046, India
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32
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Beller DA, Gharbi MA, Liu IB. Shape-controlled orientation and assembly of colloids with sharp edges in nematic liquid crystals. SOFT MATTER 2015; 11:1078-1086. [PMID: 25523158 DOI: 10.1039/c4sm01910e] [Citation(s) in RCA: 29] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/04/2023]
Abstract
The assembly of colloids in nematic liquid crystals via topological defects has been extensively studied for spherical particles, and investigations of other colloid shapes have revealed a wide array of new assembly behaviors. We show, using Landau-de Gennes numerical modeling, that nematic defect configurations and colloidal assembly can be strongly influenced by fine details of colloid shape, in particular the presence of sharp edges. For cylinder, microbullet, and cube colloid geometries, we obtain the particles' equilibrium alignment directions and effective pair interaction potentials as a function of simple shape parameters. We find that defects pin at sharp edges, and that the colloid consequently orients at an oblique angle relative to the far-field nematic director that depends on the colloid's shape. This shape-dependent alignment, which we confirm in experimental measurements, raises the possibility of selecting self-assembly outcomes for colloids in liquid crystals by tuning particle geometry.
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Affiliation(s)
- Daniel A Beller
- Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
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33
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Bauer T, Banzer P, Karimi E, Orlov S, Rubano A, Marrucci L, Santamato E, Boyd RW, Leuchs G. Observation of optical polarization Möbius strips. Science 2015; 347:964-6. [DOI: 10.1126/science.1260635] [Citation(s) in RCA: 241] [Impact Index Per Article: 26.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/02/2022]
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34
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Abstract
Nematic braids are reconfigurable knots and links formed by the disclination loops that entangle colloidal particles dispersed in a nematic liquid crystal. We focus on entangled nematic disclinations in thin twisted nematic layers stabilized by 2D arrays of colloidal particles that can be controlled with laser tweezers. We take the experimentally assembled structures and demonstrate the correspondence of the knot invariants, constructed graphs, and surfaces associated with the disclination loop to the physically observable features specific to the geometry at hand. The nematic nature of the medium adds additional topological parameters to the conventional results of knot theory, which couple with the knot topology and introduce order into the phase diagram of possible structures. The crystalline order allows the simplified construction of the Jones polynomial and medial graphs, and the steps in the construction algorithm are mirrored in the physics of liquid crystals.
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35
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Tasinkevych M, Campbell MG, Smalyukh II. Splitting, linking, knotting, and solitonic escape of topological defects in nematic drops with handles. Proc Natl Acad Sci U S A 2014; 111:16268-73. [PMID: 25369931 PMCID: PMC4246280 DOI: 10.1073/pnas.1405928111] [Citation(s) in RCA: 32] [Impact Index Per Article: 3.2] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Topologically nontrivial field excitations, including solitonic, linked, and knotted structures, play important roles in physical systems ranging from classical fluids and liquid crystals, to electromagnetism, classic, and quantum field theories. These excitations can appear spontaneously during symmetry-breaking phase transitions. For example, in cosmological theories, cosmic strings may have formed knotted configurations influencing the Early Universe development, whereas in liquid crystals transient tangled defect lines were observed during isotropic-nematic transitions, eventually relaxing to defect-free states. Knotted and solitonic fields and defects were also obtained using optical manipulation, complex-shaped colloids, and frustrated cholesterics. Here we use confinement of nematic liquid crystal by closed surfaces with varied genus and perpendicular boundary conditions for a robust control of appearance and stability of such field excitations. Theoretical modeling and experiments reveal structure of defect lines as a function of the surface topology and material and geometric parameters, establishing a robust means of controlling solitonic, knotted, linked, and other field excitations.
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Affiliation(s)
- Mykola Tasinkevych
- Max-Planck-Institut für Intelligente Systeme, D-70569 Stuttgart, Germany; Institut für Theoretische Physik IV, Universität Stuttgart, D-70569 Stuttgart, Germany;
| | - Michael G Campbell
- Department of Physics, University of Colorado, Boulder, CO 80309; Liquid Crystal Materials Research Center, University of Colorado, Boulder, CO 80309
| | - Ivan I Smalyukh
- Department of Physics, University of Colorado, Boulder, CO 80309; Liquid Crystal Materials Research Center, University of Colorado, Boulder, CO 80309; Department of Electrical, Computer, and Energy Engineering and Materials Science and Engineering Program, University of Colorado, Boulder, CO 80309; and Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, CO 80309
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36
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Machon T, Alexander GP. Knotted defects in nematic liquid crystals. PHYSICAL REVIEW LETTERS 2014; 113:027801. [PMID: 25062237 DOI: 10.1103/physrevlett.113.027801] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 02/27/2014] [Indexed: 06/03/2023]
Abstract
We show that the number of distinct topological states associated with a given knotted defect, L, in a nematic liquid crystal is equal to the determinant of the link L. We give an interpretation of these states, demonstrate how they may be identified in experiments, and describe the consequences for material behavior and interactions between multiple knots. We show that stable knots can be created in a bulk cholesteric and illustrate the topology by classifying a simulated Hopf link. In addition, we give a topological heuristic for the resolution of strand crossings in defect coarsening processes which allows us to distinguish topological classes of a given link and to make predictions about defect crossings in nematic liquid crystals.
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Affiliation(s)
- Thomas Machon
- Department of Physics and Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, United Kingdom
| | - Gareth P Alexander
- Department of Physics and Centre for Complexity Science, University of Warwick, Coventry CV4 7AL, United Kingdom
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37
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Castles F. Liquid crystal research highlights. LIQUID CRYSTALS TODAY 2013. [DOI: 10.1080/1358314x.2013.861962] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 10/26/2022]
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