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Teixeira AW, Tasinkevych M, Dias CS. Particle-based model of liquid crystal skyrmion dynamics. SOFT MATTER 2024; 20:2088-2099. [PMID: 38348527 DOI: 10.1039/d3sm01422c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 02/29/2024]
Abstract
Motivated by recent experimental results that reveal rich collective dynamics of thousands-to-millions of active liquid crystal skyrmions, we have developed a coarse-grained, particle-based model of the dynamics of skyrmions in a dilute regime. The basic physical mechanism of skyrmion motion is related to squirming undulations of domains with high director twist within the skyrmion cores when the electric field is turned on and off. The motion is not related to mass flow and is caused only by the reorientation dynamics of the director field. Based on the results of the "fine-grained" Frank-Oseen continuum model, we have mapped these squirming director distortions onto an effective force that acts asymmetrically upon switching the electrical field on or off. The resulting model correctly reproduces the skyrmion dynamics, including velocity reversal as a function of the frequency of a pulse width modulated driving voltage. We have also obtained approximate analytical expressions for the phenomenological model parameters encoding their dependence upon the cholesteric pitch and the strength of the electric field. This has been achieved by fitting coarse-grained skyrmion trajectories to those determined in the framework of the Frank-Oseen model.
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Affiliation(s)
- A W Teixeira
- Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal.
- Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, P-1749-016 Lisboa, Portugal
| | - M Tasinkevych
- Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal.
- Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, P-1749-016 Lisboa, Portugal
- SOFT Group, School of Science and Technology, Nottingham Trent University, Clifton Lane, Nottingham NG11 8NS, UK
- International Institute for Sustainability with Knotted Chiral Meta Matter, Hiroshima University, Higashihiroshima 739-8511, Japan
| | - C S Dias
- Centro de Física Teórica e Computacional, Faculdade de Ciências, Universidade de Lisboa, 1749-016 Lisboa, Portugal.
- Departamento de Física, Faculdade de Ciências, Universidade de Lisboa, P-1749-016 Lisboa, Portugal
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De Matteis G, Martina L, Naya C, Turco V. Skyrmion tubes in achiral nematic liquid crystals. Phys Rev E 2022; 105:024701. [PMID: 35291124 DOI: 10.1103/physreve.105.024701] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/31/2021] [Accepted: 01/19/2022] [Indexed: 06/14/2023]
Abstract
We analyze the interaction with uniform external fields of nematic liquid crystals within a recent generalized free energy posited by Virga and falling in the class of quartic functionals in the spatial gradients of the nematic director. We review some known interesting solutions, i.e., uniform heliconical structures, which correspond to the so-called twist-bend nematic phase and we also study the transition between this phase and the standard uniform nematic one. The twist-bend phase is further reproduced by three-dimensional simulations. Moreover, we find liquid crystal configurations, which closely resemble some novel, experimentally detected, structures called Skyrmion tubes. Skyrmion tubes are characterized by a localized cylindrically symmetric pattern surrounded by either twist-bend or uniform nematic phase. We study the equilibrium differential equations and find numerical solutions and analytical approximations.
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Affiliation(s)
- G De Matteis
- Dipartimento di Matematica e Fisica, Università del Salento, Via per Arnesano, C.P. 193 I-73100 Lecce, Italy
- INFN, Sezione di Lecce, Via per Arnesano, C. P. 193 I-73100 Lecce, Italy
- GNFM-INDAM, Città Universitaria - P.le Aldo Moro 5, C. P. 00185 Roma, Italy
| | - L Martina
- Dipartimento di Matematica e Fisica, Università del Salento, Via per Arnesano, C.P. 193 I-73100 Lecce, Italy
- INFN, Sezione di Lecce, Via per Arnesano, C. P. 193 I-73100 Lecce, Italy
| | - C Naya
- INFN, Sezione di Lecce, Via per Arnesano, C. P. 193 I-73100 Lecce, Italy
- Department of Physics, Stockholm University, AlbaNova University Center, 106 91 Stockholm, Sweden
| | - V Turco
- Dipartimento di Matematica e Fisica, Università del Salento, Via per Arnesano, C.P. 193 I-73100 Lecce, Italy
- INFN, Sezione di Lecce, Via per Arnesano, C. P. 193 I-73100 Lecce, Italy
- GNFM-INDAM, Città Universitaria - P.le Aldo Moro 5, C. P. 00185 Roma, Italy
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3
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Long C, Selinger JV. Coarse-grained theory for motion of solitons and skyrmions in liquid crystals. SOFT MATTER 2021; 17:10437-10446. [PMID: 34761790 DOI: 10.1039/d1sm01335a] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/13/2023]
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.
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Affiliation(s)
- Cheng Long
- Department of Physics, Advanced Materials and Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA.
| | - Jonathan V Selinger
- Department of Physics, Advanced Materials and Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA.
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4
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Leonov AO. Surface anchoring as a control parameter for shaping skyrmion or toron properties in thin layers of chiral nematic liquid crystals and noncentrosymmetric magnets. Phys Rev E 2021; 104:044701. [PMID: 34781482 DOI: 10.1103/physreve.104.044701] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/08/2021] [Accepted: 10/01/2021] [Indexed: 11/07/2022]
Abstract
Existence of topological localized states (skyrmions and torons) and the mechanism of their condensation into modulated states are the ruling principles of condensed matter systems, such as chiral nematic liquid crystals (CLCs) and chiral magnets (ChM). In bulk helimagnets, skyrmions are rendered into thermodynamically stable hexagonal skyrmion lattice due to the combined effect of a magnetic field and, e.g., small anisotropic contributions. In thin glass cells of CLCs, skyrmions are formed in response to the geometrical frustration and field coupling effects. By numerical modeling, I undertake a systematic study of skyrmion or toron properties in thin layers of CLCs and ChMs with competing surface-induced and bulk anisotropies. The conical phase with a variable polar angle serves as a suitable background, which shapes skyrmion internal structure, guides the nucleation processes, and substantializes the skyrmion-skyrmion interaction. I show that the hexagonal lattice of torons can be stabilized in a vast region of the constructed phase diagram for both easy-axis bulk and surface anisotropies. A topologically trivial droplet is shown to form as a domain boundary between two cone states with different rotational fashion, which underpins its stability. The findings provide a recipe for controllably creating skyrmions and torons, possessing the features on demand for potential applications.
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Affiliation(s)
- Andrey O Leonov
- Chirality Research Center, Hiroshima University, Higashi-Hiroshima, Hiroshima 739-8526, Japan; Department of Chemistry, Faculty of Science, Hiroshima University Kagamiyama, Higashi Hiroshima, Hiroshima 739-8526, Japan; and IFW Dresden, Postfach 270016, D-01171 Dresden, Germany
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5
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Lavrentovich OD. Design of nematic liquid crystals to control microscale dynamics. LIQUID CRYSTALS REVIEWS 2021; 8:59-129. [PMID: 34956738 PMCID: PMC8698256 DOI: 10.1080/21680396.2021.1919576] [Citation(s) in RCA: 6] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/17/2020] [Accepted: 04/11/2021] [Indexed: 05/25/2023]
Abstract
The dynamics of small particles, both living such as swimming bacteria and inanimate, such as colloidal spheres, has fascinated scientists for centuries. If one could learn how to control and streamline their chaotic motion, that would open technological opportunities in the transformation of stored or environmental energy into systematic motion, with applications in micro-robotics, transport of matter, guided morphogenesis. This review presents an approach to command microscale dynamics by replacing an isotropic medium with a liquid crystal. Orientational order and associated properties, such as elasticity, surface anchoring, and bulk anisotropy, enable new dynamic effects, ranging from the appearance and propagation of particle-like solitary waves to self-locomotion of an active droplet. By using photoalignment, the liquid crystal can be patterned into predesigned structures. In the presence of the electric field, these patterns enable the transport of solid and fluid particles through nonlinear electrokinetics rooted in anisotropy of conductivity and permittivity. Director patterns command the dynamics of swimming bacteria, guiding their trajectories, polarity of swimming, and distribution in space. This guidance is of a higher level of complexity than a simple following of the director by rod-like microorganisms. Namely, the director gradients mediate hydrodynamic interactions of bacteria to produce an active force and collective polar modes of swimming. The patterned director could also be engraved in a liquid crystal elastomer. When an elastomer coating is activated by heat or light, these patterns produce a deterministic surface topography. The director gradients define an activation force that shapes the elastomer in a manner similar to the active stresses triggering flows in active nematics. The patterned elastomer substrates could be used to define the orientation of cells in living tissues. The liquid-crystal guidance holds a major promise in achieving the goal of commanding microscale active flows.
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Affiliation(s)
- Oleg D Lavrentovich
- Advanced Materials and Liquid Crystal Institute, Department of Physics, Materials Science Graduate Program, Kent State University, Kent, OH 44242, USA
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6
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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: 33] [Impact Index Per Article: 8.3] [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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Tai JSB, Smalyukh II. Surface anchoring as a control parameter for stabilizing torons, skyrmions, twisted walls, fingers, and their hybrids in chiral nematics. Phys Rev E 2020; 101:042702. [PMID: 32422774 DOI: 10.1103/physreve.101.042702] [Citation(s) in RCA: 20] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/19/2019] [Accepted: 03/25/2020] [Indexed: 06/11/2023]
Abstract
Chiral condensed matter systems, such as liquid crystals and magnets, exhibit a host of spatially localized topological structures that emerge from the medium's tendency to twist and its competition with confinement and field coupling effects. We show that the strength of perpendicular surface boundary conditions can be used to control the structure and topology of solitonic and other localized field configurations. By combining numerical modeling and three-dimensional imaging of the director field, we reveal structural stability diagrams and intertransformation of twisted walls and fingers, torons, and skyrmions and their crystalline organizations upon changing boundary conditions. Our findings provide a recipe for controllably realizing skyrmions, torons, and hybrid solitonic structures possessing features of both of them, which will aid in fundamental explorations and technological uses of such topological solitons. Moreover, we discuss how other material parameters can be used to determine soliton stability and how similar principles can be systematically applied to other liquid crystal solitons and solitons in other material systems.
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Affiliation(s)
- Jung-Shen B Tai
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
| | - Ivan I Smalyukh
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- Materials Science and Engineering Program, Soft Materials Research Center and Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA
- Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, Colorado 80309, USA
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8
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Duzgun A, Nisoli C, Reichhardt CJO, Reichhardt C. Commensurate states and pattern switching via liquid crystal skyrmions trapped in a square lattice. SOFT MATTER 2020; 16:3338-3343. [PMID: 32196037 DOI: 10.1039/c9sm02312g] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/10/2023]
Abstract
Using continuum based simulations we show that a rich variety of skyrmion liquid crystal states can be realized in the presence of a periodic obstacle array. As a function of the number of skyrmions per obstacle we find hexagonal, square, dimer, trimer and quadrimer ordering, where the n-mer structures are a realization of a molecular crystal state of skyrmions. As a function of external field and obstacle radius we show that there are transitions between the different crystalline states as well as mixed and disordered structures. We discuss how these states are related to commensurate effects seen in other systems, such as vortices in type-II superconductors and colloids interacting with two dimensional substrates.
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Affiliation(s)
- A Duzgun
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
| | - C Nisoli
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
| | - C J O Reichhardt
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
| | - C Reichhardt
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.
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9
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De Matteis G, Martina L, Naya C, Turco V. Helicoids in chiral liquid crystals under external fields. Phys Rev E 2019; 100:052703. [PMID: 31869894 DOI: 10.1103/physreve.100.052703] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 06/11/2019] [Indexed: 11/07/2022]
Abstract
Cholesteric liquid crystals, subject to externally applied magnetic fields and confined between two parallel planar surfaces with strong homeotropic anchoring conditions, are found to undergo transitions to different types of helicoidal configurations with disclinations. Analytical and numerical studies are performed in order to characterize their properties. In particular, we produce a phase diagram for the transitions from the nematic state to the helicoidal phases in terms of the molecular chirality and the strength of the applied magnetic field.
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Affiliation(s)
- G De Matteis
- Dipartimento di Matematica e Fisica, Università del Salento, Via Arnesano, 73100 Lecce, Italy.,GNFM-INDAM, Città Universitaria, Piazzale Aldo Moro 5, Casella Postale 00185 Roma, Italy
| | - L Martina
- Dipartimento di Matematica e Fisica, Università del Salento, Via Arnesano, 73100 Lecce, Italy.,INFN, Sezione di Lecce, Via Arnesano, Casella Postale 193I-73100 Lecce, Italy
| | - C Naya
- INFN, Sezione di Lecce, Via Arnesano, Casella Postale 193I-73100 Lecce, Italy
| | - V Turco
- Dipartimento di Matematica e Fisica, Università del Salento, Via Arnesano, 73100 Lecce, Italy.,INFN, Sezione di Lecce, Via Arnesano, Casella Postale 193I-73100 Lecce, Italy
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10
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Krakhalev MN, Rudyak VY, Prishchepa OO, Gardymova AP, Emelyanenko AV, Liu JH, Zyryanov VY. Orientational structures in cholesteric droplets with homeotropic surface anchoring. SOFT MATTER 2019; 15:5554-5561. [PMID: 31243424 DOI: 10.1039/c9sm00384c] [Citation(s) in RCA: 12] [Impact Index Per Article: 2.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/09/2023]
Abstract
The dependency of orientational structures in cholesteric droplets with homeotropic surface anchoring on the helicity parameter has been studied by experiment and simulations. We have observed a sequence of structures, in which the director configurations and topological defects were identified by comparison of polarized microscopy pictures with simulated textures. A toron-like and low-symmetry intermediate layer-like structures have been revealed and studied in detail. The ranges of stability of the observed structures have been summarized in a general diagram and explained by the helicity parameter dependence of the free energy terms.
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Affiliation(s)
- Mikhail N Krakhalev
- Kirensky Institute of Physics, Federal Research Center - Krasnoyarsk Scientific Center, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk 660036, Russia and Institute of Engineering Physics and Radio Electronics, Siberian Federal University, Krasnoyarsk 660041, Russia
| | - Vladimir Yu Rudyak
- Faculty of Physics, Lomonosov Moscow State University, Moscow, 119991, Russia.
| | - Oxana O Prishchepa
- Kirensky Institute of Physics, Federal Research Center - Krasnoyarsk Scientific Center, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk 660036, Russia and Institute of Engineering Physics and Radio Electronics, Siberian Federal University, Krasnoyarsk 660041, Russia
| | - Anna P Gardymova
- Institute of Engineering Physics and Radio Electronics, Siberian Federal University, Krasnoyarsk 660041, Russia
| | | | | | - Victor Ya Zyryanov
- Kirensky Institute of Physics, Federal Research Center - Krasnoyarsk Scientific Center, Siberian Branch, Russian Academy of Sciences, Krasnoyarsk 660036, Russia
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11
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Eun J, Kim SJ, Jeong J. Effects of chiral dopants on double-twist configurations of lyotropic chromonic liquid crystals in a cylindrical cavity. Phys Rev E 2019; 100:012702. [PMID: 31499771 DOI: 10.1103/physreve.100.012702] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/22/2019] [Indexed: 06/10/2023]
Abstract
We investigate how chiral dopants affect the chiral symmetry breaking of lyotropic chromonic liquid crystals (LCLCs) focusing on the double-twist (DT) director configurations in a cylindrical capillary. LCLCs of unusual elastic properties tend to exhibit chiral director configurations under confinement despite the absence of intrinsic chirality. The DT director configuration in a cylindrical cavity with a degenerate planar anchoring, resulting from the large saddle-splay-to-twist elastic modulus ratio, is a representative example. Here we start by reexamining the DT configuration of nematic disodium cromoglycate in a cylindrical capillary and estimate the ratio of saddle splay to bend modulus K_{24}/K_{3}=0.5±0.1. Additionally, we study the DT configurations of the chiral nematic LCLCs with chiral dopants. The DT configuration becomes homochiral when the dopant concentration surpasses the critical concentration. We characterize these chiral DT configurations and provide a theoretical model on their energetics. Finally, we observe how the enantiomeric excess of chiral dopants determines the director configuration when dopants of two different handednesses are mixed.
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Affiliation(s)
- Jonghee Eun
- Department of Physics, Ulsan National Institute of Science and Technology, Ulsan, Republic of Korea
| | - Sung-Jo Kim
- Center for Soft and Living Matter, Institute for Basic Science, Ulsan, Republic of Korea
| | - Joonwoo Jeong
- Department of Physics, Ulsan National Institute of Science and Technology, Ulsan, Republic of Korea
- Center for Soft and Living Matter, Institute for Basic Science, Ulsan, Republic of Korea
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12
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Diep HT. Phase Transition in Frustrated Magnetic Thin Film-Physics at Phase Boundaries. ENTROPY 2019; 21:e21020175. [PMID: 33266891 PMCID: PMC7514657 DOI: 10.3390/e21020175] [Citation(s) in RCA: 10] [Impact Index Per Article: 2.0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 12/06/2018] [Revised: 02/03/2019] [Accepted: 02/10/2019] [Indexed: 12/25/2022]
Abstract
In this review, we outline some principal theoretical knowledge of the properties of frustrated spin systems and magnetic thin films. The two points we would like to emphasize: (i) the physics in low dimensions where exact solutions can be obtained; (ii) the physics at phase boundaries where interesting phenomena can occur due to competing interactions of the two phases around the boundary. This competition causes a frustration. We will concentrate our attention on magnetic thin films and phenomena occurring near the boundary of two phases of different symmetries. Two-dimensional (2D) systems are in fact the limiting case of thin films with a monolayer. Naturally, we will treat this case at the beginning. We begin by defining the frustration and giving examples of frustrated 2D Ising systems that we can exactly solve by transforming them into vertex models. We will show that these simple systems already contain most of the striking features of frustrated systems such as the high degeneracy of the ground state (GS), many phases in the GS phase diagram in the space of interaction parameters, the reentrance occurring near the boundaries of these phases, the disorder lines in the paramagnetic phase, and the partial disorder coexisting with the order at equilibrium. Thin films are then presented with different aspects: surface elementary excitations (surface spin waves), surface phase transition, and criticality. Several examples are shown and discussed. New results on skyrmions in thin films and superlattices are also displayed. By the examples presented in this review we show that the frustration when combined with the surface effect in low dimensions gives rise to striking phenomena observed in particular near the phase boundaries.
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Affiliation(s)
- Hung T Diep
- Laboratoire de Physique Théorique et Modélisation, Université de Cergy-Pontoise, CNRS, UMR 80892, Avenue Adolphe Chauvin, CEDEX, 95302 Cergy-Pontoise, France
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13
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Fukuda JI, Nych A, Ognysta U, Žumer S, Muševič I. Liquid-crystalline half-Skyrmion lattice spotted by Kossel diagrams. Sci Rep 2018; 8:17234. [PMID: 30467358 PMCID: PMC6250727 DOI: 10.1038/s41598-018-35514-0] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/01/2018] [Accepted: 11/05/2018] [Indexed: 11/09/2022] Open
Abstract
Skyrmions are swirl-like topological entities that have been shown to emerge in various condensed matter systems. Their identification has been carried out in different ways including scattering techniques and real-space observations. Here we show that Kossel diagrams can identify the formation of a hexagonal lattice of half-Skyrmions in a thin film of a chiral liquid crystal, in which case Kossel lines appear as hexagonally arranged circular arcs. Our experimental observations on a hexagonal lattice of half-Skyrmions and other defect structures resembling that of a bulk cholesteric blue phase are perfectly accounted for by numerical calculations and a theoretical argument attributing strong reflections yielding Kossel lines to guided mode resonances in the thin liquid crystal film. Our study demonstrates that a liquid crystal is a model system allowing the investigation of topological entities by various optical means, and also that Kossel techniques are applicable to the investigation of thin systems with non-trivial photonic band structures including topologically protected optical surface states.
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Affiliation(s)
- Jun-Ichi Fukuda
- Department of Physics, Kyushu University, 744, Motooka, Nishi-ku, Fukuoka, 819-0395, Japan.
- National Institute of Advanced Industrial Science and Technology (AIST), 1-1-1 Umezono, Tsukuba, 305-8568, Japan.
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000, Ljubljana, Slovenia.
| | - Andriy Nych
- Department of Molecular Photoelectronics, Institute of Physics, prospect Nauky, 46, Kyiv, 03680, Ukraine.
- Condensed Matter Department, Jožef Stefan Institute, Jamova 39, SI-1000, Ljubljana, Slovenia.
| | - Uliana Ognysta
- Department of Molecular Photoelectronics, Institute of Physics, prospect Nauky, 46, Kyiv, 03680, Ukraine
- Condensed Matter Department, Jožef Stefan Institute, Jamova 39, SI-1000, Ljubljana, Slovenia
| | - Slobodan Žumer
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000, Ljubljana, Slovenia
- Condensed Matter Department, Jožef Stefan Institute, Jamova 39, SI-1000, Ljubljana, Slovenia
| | - Igor Muševič
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska 19, SI-1000, Ljubljana, Slovenia
- Condensed Matter Department, Jožef Stefan Institute, Jamova 39, SI-1000, Ljubljana, Slovenia
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14
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Electrically driven three-dimensional solitary waves as director bullets in nematic liquid crystals. Nat Commun 2018; 9:2912. [PMID: 30046035 PMCID: PMC6060142 DOI: 10.1038/s41467-018-05101-y] [Citation(s) in RCA: 29] [Impact Index Per Article: 4.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/12/2018] [Accepted: 06/08/2018] [Indexed: 11/08/2022] Open
Abstract
Electric field-induced collective reorientation of nematic molecules is of importance for fundamental science and practical applications. This reorientation is either homogeneous over the area of electrodes, as in displays, or periodically modulated, as in electroconvection. The question is whether spatially localized three-dimensional solitary waves of molecular reorientation could be created. Here we demonstrate that the electric field can produce particle-like propagating solitary waves representing self-trapped "bullets" of oscillating molecular director. These director bullets lack fore-aft symmetry and move with very high speed perpendicularly to the electric field and to the initial alignment direction. The bullets are true solitons that preserve spatially confined shapes and survive collisions. The solitons are topologically equivalent to the uniform state and have no static analogs, thus exhibiting a particle-wave duality. Their shape, speed, and interactions depend strongly on the material parameters, which opens the door for a broad range of future studies.
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15
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Duzgun A, Selinger JV, Saxena A. Comparing skyrmions and merons in chiral liquid crystals and magnets. Phys Rev E 2018; 97:062706. [PMID: 30011572 DOI: 10.1103/physreve.97.062706] [Citation(s) in RCA: 21] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/20/2017] [Indexed: 06/08/2023]
Abstract
When chiral liquid crystals or magnets are subjected to applied fields or other anisotropic environments, the competition between favored twist and anisotropy leads to the formation of complex defect structures. In some cases, the defects are skyrmions, which have 180^{∘} double twist going outward from the center, and hence can pack together without singularities in the orientational order. In other cases, the defects are merons, which have 90^{∘} double twist going outward from the center; packing such merons requires singularities in the orientational order. In the liquid crystal context, a lattice of merons is equivalent to a blue phase. Here we perform theoretical and computational studies of skyrmions and merons in chiral liquid crystals and magnets. Through these studies, we calculate the phase diagrams for liquid crystals and magnets in terms of dimensionless ratios of energetic parameters. We also predict the range of metastability for liquid crystal skyrmions and show that these skyrmions can move and interact as effective particles. The results show how the properties of skyrmions and merons depend on the vector or tensor nature of the order parameter.
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Affiliation(s)
- Ayhan Duzgun
- Department of Physics and Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
| | - Jonathan V Selinger
- Department of Physics and Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
| | - Avadh Saxena
- Theoretical Division and Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA
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16
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Schneider S, Wolf D, Stolt MJ, Jin S, Pohl D, Rellinghaus B, Schmidt M, Büchner B, Goennenwein STB, Nielsch K, Lubk A. Induction Mapping of the 3D-Modulated Spin Texture of Skyrmions in Thin Helimagnets. PHYSICAL REVIEW LETTERS 2018; 120:217201. [PMID: 29883134 DOI: 10.1103/physrevlett.120.217201] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 10/24/2017] [Revised: 03/29/2018] [Indexed: 06/08/2023]
Abstract
Envisaged applications of Skyrmions in magnetic memory and logic devices crucially depend on the stability and mobility of these topologically nontrivial magnetic textures in thin films. We present for the first time quantitative maps of the magnetic induction that provide evidence for a 3D modulation of the Skyrmionic spin texture. The projected in-plane magnetic induction maps as determined from in-line and off-axis electron holography carry the clear signature of Bloch Skyrmions. However, the magnitude of this induction is much smaller than the values expected for homogeneous Bloch Skyrmions that extend throughout the thickness of the film. This finding can only be understood if the underlying spin textures are modulated along the out-of-plane z direction. The projection of (the in-plane magnetic induction of) helices is further found to exhibit thickness-dependent lateral shifts, which show that this z modulation is accompanied by an (in-plane) modulation along the x and y directions.
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Affiliation(s)
- S Schneider
- Institute for Metallic Materials, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
- Institut für Festkörper- und Materialphysik, Technische Universität Dresden, 01062 Dresden, Germany
| | - D Wolf
- Institute for Solid State Research, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
| | - M J Stolt
- Department of Chemistry, University of Wisconsin-Madison, 1101 University Avenue, Madison, Wisconsin 53706, USA
| | - S Jin
- Department of Chemistry, University of Wisconsin-Madison, 1101 University Avenue, Madison, Wisconsin 53706, USA
| | - D Pohl
- Institute for Metallic Materials, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
- Dresden Center for Nanoanalysis, Technische Universität Dresden, 01062 Dresden, Germany
| | - B Rellinghaus
- Institute for Metallic Materials, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
- Dresden Center for Nanoanalysis, Technische Universität Dresden, 01062 Dresden, Germany
| | - M Schmidt
- Department Chemical Metal Science, Max Planck Institute for Chemical Physics of Solids, Nöthnitzer Str. 40, 01187 Dresden, Germany
| | - B Büchner
- Institute for Solid State Research, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
| | - S T B Goennenwein
- Institut für Festkörper- und Materialphysik, Technische Universität Dresden, 01062 Dresden, Germany
- Center for Transport and Devices of Emergent Materials, Technische Universität Dresden, 01062 Dresden, Germany
| | - K Nielsch
- Institute for Metallic Materials, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
- Institute of Materials Science, Technische Universität Dresden, Helmholtzstr. 7, 01069 Dresden, Germany
| | - A Lubk
- Institute for Solid State Research, IFW Dresden, Helmholtzstr. 20, 01069 Dresden, Germany
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17
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Fukuda JI, Žumer S. Reflection spectra and near-field images of a liquid crystalline half-Skyrmion lattice. OPTICS EXPRESS 2018; 26:1174-1184. [PMID: 29401994 DOI: 10.1364/oe.26.001174] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/14/2017] [Accepted: 12/21/2017] [Indexed: 06/07/2023]
Abstract
We investigate numerically the optical properties of a hexagonal half-Skyrmion lattice exhibited by a highly chiral liquid crystal confined between two parallel plates. Our study focuses on the near and far-field reflection for normally incident light with different polarizations. We show that, when the wavelength of the incident light is longer than a threshold value, the reflectivity is almost insensitive to the polarization of the incident light, although the intensity profiles of the reflected light, in particular in the near-field regime, depend significantly on the polarization. The former property is attributable to the quasi two-dimensional nature of the half-Skyrmion lattice, that is, almost uniform orientational order along the direction normal to the confining plates. Our results for the intensity of reflected light generated by evanescent as well as propagating contributions suggest that direct evidence of the formation and structure of half-Skyrmions could be provided by near-field optics with resolutions higher than that of conventional optical microscopy.
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18
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19
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Topology-commanded optical properties of bistable electric-field-induced torons in cholesteric bubble domains. Sci Rep 2017; 7:16149. [PMID: 29170409 PMCID: PMC5701030 DOI: 10.1038/s41598-017-16241-4] [Citation(s) in RCA: 23] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/06/2017] [Accepted: 10/20/2017] [Indexed: 11/24/2022] Open
Abstract
Nowadays, complicated topological defects enable many experimental manipulations and configurational simulations of active soft matter for optical and photonic applications. Investigation of topological defects in soft anisotropic materials enables one to better understand three-dimensional orientation fields in cholesteric liquid crystals. Here, we describe optical properties of bistable bubble domain (BD) texture torons in a thin layer of cholesteric liquid crystal (CLC), frustrated by homeotropic anchoring conditions, and reliably switchable by a random process. The control of macroscopic optical density and diffraction efficiency of the BD texture is demonstrated by a selection of a confinement ratio of the CLC. Experimentally reconstructed CLC director profile reveals the topology of BD torons allowing consideration of naturally occurring BD texture for applications in optical and photonic devices, which are bistably switchable between active and transparent optical states.
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20
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Abstract
Skyrmions are topologically protected continuous field configurations that cannot be smoothly transformed to a uniform state. They behave like particles and give origins to the field of skyrmionics that promises racetrack memory and other technological applications. Unraveling the non-equilibrium behavior of such topological solitons is a challenge. We realize skyrmions in a chiral liquid crystal and, using numerical modeling and polarized video microscopy, demonstrate electrically driven squirming motion. We reveal the intricate details of non-equilibrium topology-preserving textural changes driving this behavior. Direction of the skyrmion’s motion is robustly controlled in a plane orthogonal to the applied field and can be reversed by varying frequency. Our findings may spur a paradigm of soliton dynamics in soft matter, with a rich interplay between topology, chirality, and orientational viscoelasticity. A skyrmion is a topological object originally introduced to model elementary particles and a baby skyrmion is its two-dimensional counterpart which can be realized as a defect in liquid crystals. Here the authors show that an electric field can drive uniform motion of baby skyrmions in liquid crystals.
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21
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Langner MC, Roy S, Huang SW, Koralek JD, Chuang YD, Dakovski GL, Turner JJ, Robinson JS, Coffee RN, Minitti MP, Seki S, Tokura Y, Schoenlein RW. Nonlinear Ultrafast Spin Scattering in the Skyrmion Phase of Cu_{2}OSeO_{3}. PHYSICAL REVIEW LETTERS 2017; 119:107204. [PMID: 28949160 DOI: 10.1103/physrevlett.119.107204] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/03/2016] [Indexed: 05/26/2023]
Abstract
Ultrafast x-ray scattering studies of the topological Skyrmion phase in Cu_{2}OSeO_{3} show the dynamics to be strongly dependent on the excitation energy and fluence. At high photon energies, where the electron-spin scattering cross section is relatively high, the excitation of the topological Skyrmion phase shows a nonlinear dependence on the excitation fluence, in contrast to the excitation of the conical phase which is linearly dependent on the excitation fluence. The excitation of the Skyrmion order parameter is nonlinear in the magnetic excitation resulting from scattering during electron-hole recombination, indicating different dominant scattering processes in the conical and Skyrmion phases.
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Affiliation(s)
- M C Langner
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
| | - S Roy
- Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley California 94720, USA
| | - S W Huang
- Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley California 94720, USA
| | - J D Koralek
- Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - Y-D Chuang
- Advanced Light Source, Lawrence Berkeley National Laboratory, Berkeley California 94720, USA
| | - G L Dakovski
- Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - J J Turner
- Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - J S Robinson
- Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - R N Coffee
- Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - M P Minitti
- Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
| | - S Seki
- RIKEN, Center for Emergent Matter Science, Wako 351-0198, Japan
- PRESTO, Japan Science and Technology Agency, Tokyo 102-0075, Japan
| | - Y Tokura
- RIKEN, Center for Emergent Matter Science, Wako 351-0198, Japan
- Department of Applied Physics, University of Tokyo, Tokyo 113-8656, Japan
| | - R W Schoenlein
- Materials Science Division, Lawrence Berkeley National Laboratory, Berkeley, California 94720, USA
- Linac Coherent Light Source, SLAC National Accelerator Laboratory, Menlo Park, California 94025, USA
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22
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Afghah S, Selinger JV. Theory of helicoids and skyrmions in confined cholesteric liquid crystals. Phys Rev E 2017; 96:012708. [PMID: 29347177 DOI: 10.1103/physreve.96.012708] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 02/22/2017] [Indexed: 06/07/2023]
Abstract
Cholesteric liquid crystals experience geometric frustration when they are confined between surfaces with anchoring conditions that are incompatible with the cholesteric twist. Because of this frustration, they develop complex topological defect structures, which may be helicoids or skyrmions. We develop a theory for these structures, which extends previous theoretical research by deriving exact solutions for helicoids with the assumption of constant azimuth, calculating numerical solutions for helicoids and skyrmions with varying azimuth, and interpreting the results in terms of competition between terms in the free energy.
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Affiliation(s)
- Sajedeh Afghah
- Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
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23
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Ivanov AV, Bessarab PF, Aksenova EV, Romanov VP, Uzdin VM. Energy surface and minimum energy paths for Fréedericksz transitions in bistable cholesteric liquid crystals. Phys Rev E 2016; 93:042708. [PMID: 27176367 DOI: 10.1103/physreve.93.042708] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/13/2015] [Indexed: 11/06/2022]
Abstract
The multidimensional energy surface of a cholesteric liquid crystal in a planar cell is investigated as a function of spherical coordinates determining the director orientation. Minima on the energy surface correspond to the stable states with particular director distribution. External electric and magnetic fields deform the energy surface and positions of minima. It can lead to the transitions between states, known as the Fréedericksz effect. Transitions can be continuous or discontinuous depending on parameters of the liquid crystal which determine an energy surface. In a case of discontinuous transition when a barrier between stable states is comparable with the thermal energy, the activation transitions may occur, and it leads to the modification of characteristics of the Fréedericksz effect with temperature without explicit temperature dependencies of liquid crystal parameters. A minimum energy path between stable states on the energy surface for the Fréedericksz transition is found using the geodesic nudged elastic band method. Knowledge of this path, which has maximal statistical weight among all other paths, gives the information about a barrier between stable states and configuration of director orientation during the transition. It also allows one to estimate the stability of states with respect to the thermal fluctuations and their lifetime when the system is close to the Fréedericksz transition.
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Affiliation(s)
- A V Ivanov
- Saint Petersburg State University, 199034, Saint Petersburg, Russia.,Science Institute of the University of Iceland, VR-III, 107 Reykjavik, Iceland
| | - P F Bessarab
- Saint Petersburg State University, 199034, Saint Petersburg, Russia.,Department of Materials and Nanophysics, Royal Institute of Technology (KTH), SE-16440 Kista, Sweden
| | - E V Aksenova
- Saint Petersburg State University, 199034, Saint Petersburg, Russia
| | - V P Romanov
- Saint Petersburg State University, 199034, Saint Petersburg, Russia
| | - V M Uzdin
- Saint Petersburg State University, 199034, Saint Petersburg, Russia.,Saint Petersburg National Research University of Information Technologies, Mechanics and Optics, 197101 Saint Petersburg, Russia
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24
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Varanytsia A, Chien LC. Photoswitchable and dye-doped bubble domain texture of cholesteric liquid crystals. OPTICS LETTERS 2015; 40:4392-4395. [PMID: 26421539 DOI: 10.1364/ol.40.004392] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/05/2023]
Abstract
We demonstrate control of the transmittance of the naturally formed bubble domain (BD) texture of cholesteric liquid crystals (CLC) with negative dielectric anisotropy confined into a cell with homeotropic surface anchoring. By using a photosensitive chiral dopant with variable helical twisting power under light irradiation, control of packing density of bubbles, spatial patterning, and all optical switching between bistable states with different optical densities is achieved. By introducing dichroic dye into the CLC mixture, a bistable and switchable by applied electric field guest-host system is obtained. The light dimming properties of dye-doped BD CLC systems may lead to development of a wide range of applications.
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25
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Zhang Q, Ackerman PJ, Liu Q, Smalyukh II. Ferromagnetic Switching of Knotted Vector Fields in Liquid Crystal Colloids. PHYSICAL REVIEW LETTERS 2015; 115:097802. [PMID: 26371682 DOI: 10.1103/physrevlett.115.097802] [Citation(s) in RCA: 28] [Impact Index Per Article: 3.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/17/2015] [Indexed: 06/05/2023]
Abstract
We experimentally realize polydomain and monodomain chiral ferromagnetic liquid crystal colloids that exhibit solitonic and knotted vector field configurations. Formed by dispersions of ferromagnetic nanoplatelets in chiral nematic liquid crystals, these colloidal ferromagnets exhibit spontaneous long-range alignment of magnetic dipole moments of individual platelets, giving rise to a continuum of the magnetization field M(r). Competing effects of surface confinement and chirality prompt spontaneous formation and enable the optical generation of localized twisted solitonic structures with double-twist tubes and torus knots of M(r), which exhibit a strong sensitivity to the direction of weak magnetic fields ∼1 mT. Numerical modeling, implemented through free energy minimization to arrive at a field-dependent three-dimensional M(r), shows a good agreement with experiments and provides insights into the torus knot topology of observed field configurations and the corresponding physical underpinnings.
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Affiliation(s)
- Qiaoxuan Zhang
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- Materials Science and Engineering Program, University of Colorado, Boulder, Colorado 80309, USA
| | - Paul J Ackerman
- Department of Physics, University of Colorado, Boulder, Colorado 80309, USA
- Department of Electrical, Computer and Energy Engineering, 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
- Materials Science and Engineering Program, University of Colorado, Boulder, Colorado 80309, USA
- Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA
- Liquid Crystal Materials Research Center, University of Colorado, Boulder, Colorado 80309, USA
- Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, Colorado 80309, USA
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26
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Ackerman PJ, van de Lagemaat J, Smalyukh II. Self-assembly and electrostriction of arrays and chains of hopfion particles in chiral liquid crystals. Nat Commun 2015; 6:6012. [PMID: 25607778 PMCID: PMC4354077 DOI: 10.1038/ncomms7012] [Citation(s) in RCA: 45] [Impact Index Per Article: 5.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/15/2014] [Accepted: 12/01/2014] [Indexed: 12/22/2022] Open
Abstract
Some of the most exotic condensed matter phases, such as twist grain boundary and blue phases in liquid crystals and Abrikosov phases in superconductors, contain arrays of topological defects in their ground state. Comprised of a triangular lattice of double-twist tubes of magnetization, the so-called ‘A-phase’ in chiral magnets is an example of a thermodynamically stable phase with topologically nontrivial solitonic field configurations referred to as two-dimensional skyrmions, or baby-skyrmions. Here we report that three-dimensional skyrmions in the form of double-twist tori called ‘hopfions’, or ‘torons’ when accompanied by additional self-compensating defects, self-assemble into periodic arrays and linear chains that exhibit electrostriction. In confined chiral nematic liquid crystals, this self-assembly is similar to that of liquid crystal colloids and originates from long-range elastic interactions between particle-like skyrmionic torus knots of molecular alignment field, which can be tuned from isotropic repulsive to weakly or highly anisotropic attractive by low-voltage electric fields. Topological defects can be spontaneously generated to thermodynamically stabilize a variety of peculiar condensed matter phases for technological applications. Here, Ackerman et al. show electrically controllable self-assembly of knotted defects into periodic arrays in chiral liquid crystals.
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Affiliation(s)
- Paul J Ackerman
- 1] Department of Physics, University of Colorado, Boulder, Colorado 80309, USA [2] Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA
| | - Jao van de Lagemaat
- 1] Department of Physics, University of Colorado, Boulder, Colorado 80309, USA [2] National Renewable Energy Laboratory, Golden, Colorado 80401, USA [3] Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, Colorado 80309, USA
| | - Ivan I Smalyukh
- 1] Department of Physics, University of Colorado, Boulder, Colorado 80309, USA [2] Department of Electrical, Computer and Energy Engineering, University of Colorado, Boulder, Colorado 80309, USA [3] Renewable and Sustainable Energy Institute, National Renewable Energy Laboratory and University of Colorado, Boulder, Colorado 80309, USA [4] Liquid Crystal Materials Research Center and Materials Science and Engineering Program, University of Colorado, Boulder, Colorado 80309, USA
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