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Zid M, Pal K, Harkai S, Abina A, Kralj S, Zidanšek A. Qualitatively and Quantitatively Different Configurations of Nematic-Nanoparticle Mixtures. NANOMATERIALS (BASEL, SWITZERLAND) 2024; 14:436. [PMID: 38470767 DOI: 10.3390/nano14050436] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Grants] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/31/2024] [Revised: 02/20/2024] [Accepted: 02/22/2024] [Indexed: 03/14/2024]
Abstract
We consider the influence of different nanoparticles or micrometre-scale colloidal objects, which we commonly refer to as particles, on liquid crystalline (LC) orientational order in essentially spatially homogeneous particle-LC mixtures. We first illustrate the effects of coupling a single particle with the surrounding nematic molecular field. A particle could either act as a "dilution", i.e., weakly distorting local effective orientational field, or as a source of strong distortions. In the strong anchoring limit, particles could effectively act as topological point defects, whose topological charge q depends on particle topology. The most common particles exhibit spherical topology and consequently act as q = 1 monopoles. Depending on the particle's geometry, these effective monopoles could locally induce either point-like or line-like defects in the surrounding LC host so that the total topological charge of the system equals zero. The resulting system's configuration is topologically equivalent to a crystal-like array of monopole defects with alternating topological charges. Such configurations could be trapped in metastable or stable configurations, where the history of the sample determines a configuration selection.
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Affiliation(s)
- Maha Zid
- Jožef Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia
- Jožef Stefan International Postgraduate School, Jamova cesta 39, 1000 Ljubljana, Slovenia
| | - Kaushik Pal
- University Centre for Research and Development (UCRD), Department of Physics, Chandigarh University, Ghruan, Mohali 140413, Punjab, India
| | - Saša Harkai
- Jožef Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia
- Faculty of Mathematics and Physics, University of Ljubljana, Jadranska cesta 19, 1000 Ljubljana, Slovenia
| | - Andreja Abina
- Jožef Stefan International Postgraduate School, Jamova cesta 39, 1000 Ljubljana, Slovenia
| | - Samo Kralj
- Jožef Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
| | - Aleksander Zidanšek
- Jožef Stefan Institute, Jamova cesta 39, 1000 Ljubljana, Slovenia
- Jožef Stefan International Postgraduate School, Jamova cesta 39, 1000 Ljubljana, Slovenia
- Faculty of Natural Sciences and Mathematics, University of Maribor, Koroška cesta 160, 2000 Maribor, Slovenia
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Hölbl A, Mesarec L, Polanšek J, Iglič A, Kralj S. Stable Assemblies of Topological Defects in Nematic Orientational Order. ACS OMEGA 2023; 8:169-179. [PMID: 36643572 PMCID: PMC9835183 DOI: 10.1021/acsomega.2c07174] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Grants] [Track Full Text] [Download PDF] [Figures] [Subscribe] [Scholar Register] [Received: 11/07/2022] [Accepted: 12/07/2022] [Indexed: 06/17/2023]
Abstract
We considered general mechanisms enabling the stabilization of localized assemblies of topological defects (TDs). There is growing evidence that physical fields represent fundamental natural entities, and therefore these features are of interest to all branches of physics. In general, cores of TDs are energetically costly, and consequently, assemblies of TDs are unfavorable. Owing to the richness of universalities in the physics of TDs, it is of interest to identify systems where they are easily experimentally accessible, enabling detailed and well-controlled analysis of their universal behavior, and cross-fertilizing knowledge in different areas of physics. In this respect, thermotropic nematic liquid crystals (NLCs) represent an ideal experiment testbed for such studies. In addition, TDs in NLCs could be exploited in several applications. We present examples that emphasize the importance of curvature imposed on the phase component of the relevant order parameter field. In NLCs, it is represented by the nematic tensor order parameter. Using a simple Landau-type approach, we show how the coupling between chirality and saddle splay elasticity, which can be expressed as a Gaussian curvature contribution, can stabilize Meron TDs. The latter have numerous analogs in other branches of physics. TDs in 2D curved manifolds reveal that the Gaussian curvature dominantly impacts the assembling and stabilization of TDs. Furthermore, a strong enough curvature that serves as an attractor for TDs is a respective field that could be imposed in a fast enough phase transition. Assemblies of created TDs created in such a disordered environment could be stabilized by appropriate impurities.
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Affiliation(s)
- Arbresha Hölbl
- Faculty of Natural
Sciences and Mathematics, University of
Maribor, Koroška
160, 2000 Maribor, Slovenia
| | - Luka Mesarec
- Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia
| | - Juš Polanšek
- Faculty of Natural
Sciences and Mathematics, University of
Maribor, Koroška
160, 2000 Maribor, Slovenia
| | - Aleš Iglič
- Faculty of Electrical Engineering, University of Ljubljana, Tržaška 25, 1000 Ljubljana, Slovenia
| | - Samo Kralj
- Faculty of Natural
Sciences and Mathematics, University of
Maribor, Koroška
160, 2000 Maribor, Slovenia
- Jožef Stefan Institute, Jamova 39, 1000 Ljubljana, Slovenia
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Further Study on C-Eigenvalue Inclusion Intervals for Piezoelectric Tensors. AXIOMS 2022. [DOI: 10.3390/axioms11060250] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 11/16/2022]
Abstract
The C-eigenpair of piezoelectric tensors finds applications in the area of the piezoelectric effect and converse piezoelectric effect. In this paper, we provide some characterizations of C-eigenvectors by exploring the structure of piezoelectric tensors, and establish sharp C-eigenvalue inclusion intervals via Cauchy–Schwartz inequality. Further, we propose the lower and upper bounds of the largest C-eigenvalue and evaluate the efficiency of the best rank-one approximation of piezoelectric tensors. Numerical examples are proposed to verify the efficiency of the obtained results.
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Rosseto MP, Selinger JV. Modulated phases of nematic liquid crystals induced by tetrahedral order. Phys Rev E 2022; 105:024708. [PMID: 35291139 DOI: 10.1103/physreve.105.024708] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/30/2021] [Accepted: 02/15/2022] [Indexed: 06/14/2023]
Abstract
Recent theoretical research has developed a general framework to understand director deformations and modulated phases in nematic liquid crystals. In this framework, there are four fundamental director deformation modes: twist, bend, splay, and a fourth mode Δ related to saddle-splay. The first three of these modes are known to induce modulated phases. Here, we consider modulated phases induced by the fourth mode. We develop a theory for tetrahedral order in liquid crystals, and show that it couples to the Δ mode of director deformation. Because of geometric frustration, the Δ mode cannot fill space by itself, but rather must be accompanied by twist or splay. Hence, it may induce a spontaneous cholesteric phase, with either handedness, or a splay nematic phase.
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Affiliation(s)
- Michely P Rosseto
- Departamento de Física, Universidade Estadual de Maringá, Maringá, Paraná 5790-87020-900, Brazil
| | - Jonathan V Selinger
- Department of Physics, Advanced Materials and Liquid Crystal Institute, Kent State University, Kent, Ohio 44242, USA
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Brand HR, Pleiner H. Two-fluid model for a fluid with tetrahedral-octupolar order. Phys Rev E 2021; 104:044705. [PMID: 34781489 DOI: 10.1103/physreve.104.044705] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/13/2021] [Accepted: 10/05/2021] [Indexed: 11/07/2022]
Abstract
We investigate the macroscopic dynamics of a two-fluid system with tetrahedral order. As all normal-fluid two-fluid systems one has-compared to a simple fluid-the velocity difference between the two subsystems and the concentration of one component as additional macroscopic variables. Depending on the type of system, the concentration can either be a conserved quantity or relax on a long, but finite timescale. Due to the existence of the tetrahedral order such a system breaks parity symmetry. Here we discuss physical systems without preferred direction in real space, meaning that our description applies to optically isotropic materials. We find a number of reversible as well as dissipative dynamic cross-coupling terms due to the additional octupolar order, when compared to a fluid mixture. As the most interesting cross-coupling term from an experimental point of view, we identify a dissipative cross-coupling between the relative velocity and the usual velocity gradients. Applying a shear flow in a plane, this dissipative cross-coupling leads to a velocity difference perpendicular to the shear plane. As a result one can obtain a spatially homogeneous oscillation of the relative velocity. In addition, this induced relative velocity can couple as a function of time and space to the concentration, which gives rise to an overdamped propagating soundlike mode, where the overdamping arises from the fact that velocity difference is a macroscopic variable and not strictly conserved. We also show that electric field gradients are connected with an analogous reversible cross-coupling and can lead in a planar shear geometry to an overdamped propagating mode as well.
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Affiliation(s)
- Helmut R Brand
- Department of Physics, University of Bayreuth, 95440 Bayreuth, Germany.,Max Planck Institute for Polymer Research, 55021 Mainz, Germany
| | - Harald Pleiner
- Max Planck Institute for Polymer Research, 55021 Mainz, Germany
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Xu J, Chen JZY. General liquid-crystal theory for anisotropically shaped molecules: Symmetry, orientational order parameters, and system free energy. Phys Rev E 2021; 102:062701. [PMID: 33466056 DOI: 10.1103/physreve.102.062701] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 09/14/2020] [Accepted: 11/12/2020] [Indexed: 11/07/2022]
Abstract
A general theory of liquid crystals is presented, starting from the group-theory symmetry analysis of the constituting molecules. A particular attention is paid to the type of elastic free-energies and their relationships with the molecular symmetries. The orientational order-parameter tensors are identified for each molecular symmetry, in a consideration of consistently keeping the leading, characteristic elastic free energies in a model. The order parameters are expressed in terms of symmetric traceless tensors, some of high orders, for all major molecular symmetries, including seven groups of axial symmetries and seven groups of polyhedral symmetries. For spatially inhomogeneous liquid crystals, the couplings of these tensors in the elastic energies are derived by expanding the interaction energies between these molecules. The aim is to provide a general view of the molecular symmetries of individual molecules, orientational order parameters characterizing the orientational distribution functions, and the elastic free energies, all under one single group-theory approach.
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Affiliation(s)
- Jie Xu
- LSEC & NCMIS, Institute of Computational Mathematics and Scientific/Engineering Computing (ICMSEC), Academy of Mathematics and Systems Science (AMSS), Chinese Academy of Sciences, Beijing 100190, China
| | - Jeff Z Y Chen
- Department of Physics and Astronomy, University of Waterloo, Ontario, Canada N2L 3G1
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Pedrini A, Virga EG. Liquid crystal distortions revealed by an octupolar tensor. Phys Rev E 2020; 101:012703. [PMID: 32069528 DOI: 10.1103/physreve.101.012703] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/07/2019] [Indexed: 11/07/2022]
Abstract
The classical theory of liquid crystal elasticity as formulated by Oseen and Frank describes the (orientable) optic axis of these soft materials by a director n. The ground state is attained when n is uniform in space; all other states, which have a nonvanishing gradient ∇n, are distorted. This paper proposes an algebraic (and geometric) way to describe the local distortion of a liquid crystal by constructing from n and ∇n a third-rank, symmetric, and traceless tensor A (the octupolar tensor). The (nonlinear) eigenvectors of A associated with the local maxima of its cubic form Φ on the unit sphere (its octupolar potential) designate the directions of distortion concentration. The octupolar potential is illustrated geometrically and its symmetries are charted in the space of distortion characteristics, so as to educate the eye to capture the dominating elastic modes. Special distortions are studied, which have everywhere either the same octupolar potential or one with the same shape but differently inflated.
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Affiliation(s)
- Andrea Pedrini
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy
| | - Epifanio G Virga
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy
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Mirantsev LV, Sonnet AM, Virga EG. Lifting ordered surfaces: Ellipsoidal nematic shells. Phys Rev E 2018; 98:012701. [PMID: 30110838 DOI: 10.1103/physreve.98.012701] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/26/2018] [Indexed: 11/07/2022]
Abstract
When a material surface is functionalized so as to acquire some type of order, functionalization of which soft condensed matter systems have recently provided many interesting examples, the modeler faces an alternative. Either the order is described on the curved, physical surface where it belongs, or it is described on a flat surface that is unrolled as preimage of the physical surface under a suitable height function. This paper applies a general method that pursues the latter avenue by lifting whatever order tensor is deemed appropriate from a flat to a curved surface. We specialize this method to nematic shells, for which it also provides a simple but perhaps convincing interpretation of the outcomes of some molecular dynamics experiments on ellipsoidal shells.
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Affiliation(s)
- Leonid V Mirantsev
- Institute of the Problems of Mechanical Engineering, Academy of Sciences of Russia, St. Petersburg 199178, Russia
| | - André M Sonnet
- Department of Mathematics and Statistics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow G1 1XH, Scotland, United Kingdom
| | - Epifanio G Virga
- Dipartimento di Matematica, Università di Pavia, Via Ferrata 5, 27100 Pavia, Italy
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