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Ishii Y, Zhou Y, He K, Takanishi Y, Yamamoto J, de Pablo J, Lopez-Leon T. Structural transformations in tetravalent nematic shells induced by a magnetic field. SOFT MATTER 2020; 16:8169-8178. [PMID: 32555908 DOI: 10.1039/d0sm00340a] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/11/2023]
Abstract
The role of applied fields on the structure of liquid crystals confined to shell geometries has been studied in past theoretical work, providing strategies to produce liquid crystal shells with controlled defect structure or valence. However, the predictions of such studies have not been experimentally explored yet. In this work, we study the structural transformations undergone by tetravalent nematic liquid crystal shells under a strong uniform magnetic field, using both experiments and simulations. We consider two different cases in terms of shell geometry and initial defect symmetry: (i) homogeneous shells with four s = +1/2 defects in a tetrahedral arrangement, and (ii) inhomogeneous shells with four s = +1/2 defects localized in their thinner parts. Consistently with previous theoretical results, we observe that the initial defect structure evolves into a bipolar one, in a process where the defects migrate towards the poles. Interestingly, we find that the defect trajectories and dynamics are controlled by curvature walls that connect the defects by pairs. Based on the angle between Bs, the local projection of the magnetic field on the shell surface, and n+½, a vector describing the defect orientations, we are able to predict the nature and shape of those inversion walls, and therefore, the trajectory and dynamics of the defects. This rule, based on symmetry arguments, is consistent with both experiments and simulations and applies for shells that are either homogeneous or inhomogeneous in thickness. By modifying the angle between Bs and n+½, we are able to induce, in controlled way, complex routes towards the final bipolar state. In the case of inhomogeneous shells, the specific symmetry of the shell allowed us to observe a hybrid splay-bend Helfrich wall for the first time.
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Affiliation(s)
- Yoko Ishii
- Department of Physics, Graduate School of Science, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto, 606-8562, Japan
| | - Ye Zhou
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA.
| | - Kunyun He
- UMR No. 7083, CNRS, Gulliver, ESPCI Paris, PSL Research University, 10 Rue Vauquelin, 75005 Paris, France.
| | - Yoichi Takanishi
- Department of Physics, Graduate School of Science, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto, 606-8562, Japan
| | - Jun Yamamoto
- Department of Physics, Graduate School of Science, Kyoto University, Oiwake-cho, Kitashirakawa, Sakyo-ku, Kyoto, 606-8562, Japan
| | - Juan de Pablo
- Pritzker School of Molecular Engineering, University of Chicago, Chicago, Illinois 60637, USA.
| | - Teresa Lopez-Leon
- UMR No. 7083, CNRS, Gulliver, ESPCI Paris, PSL Research University, 10 Rue Vauquelin, 75005 Paris, France.
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Nitschke I, Reuther S, Voigt A. Liquid crystals on deformable surfaces. Proc Math Phys Eng Sci 2020; 476:20200313. [PMID: 33071582 DOI: 10.1098/rspa.2020.0313] [Citation(s) in RCA: 5] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/23/2020] [Accepted: 07/17/2020] [Indexed: 11/12/2022] Open
Abstract
Liquid crystals with molecules constrained to the tangent bundle of a curved surface show interesting phenomena resulting from the tight coupling of the elastic and bulk-free energies of the liquid crystal with geometric properties of the surface. We derive a thermodynamically consistent Landau-de Gennes-Helfrich model which considers the simultaneous relaxation of the Q-tensor field and the surface. The resulting system of tensor-valued surface partial differential equation and geometric evolution laws is numerically solved to tackle the rich dynamics of this system and to compute the resulting equilibrium shape. The results strongly depend on the intrinsic and extrinsic curvature contributions and lead to unexpected asymmetric shapes.
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Affiliation(s)
- Ingo Nitschke
- Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany
| | - Sebastian Reuther
- Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany
| | - Axel Voigt
- Institute of Scientific Computing, TU Dresden, 01062 Dresden, Germany.,Center for Systems Biology Dresden (CSBD), Pfotenhauerstraße 108, 01307 Dresden, Germany.,Cluster of Excellence Physics of Life (PoL), 01062 Dresden, Germany
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Abstract
Knotted fields enrich a variety of physical phenomena, ranging from fluid flows, electromagnetic fields, to textures of ordered media. Maxwell's electrostatic equations, whose vacuum solution is mathematically known as a harmonic field, provide an ideal setting to explore the role of domain topology in determining physical fields in confined space. In this work, we show the uniqueness of a harmonic field in knotted tubes, and reduce the construction of a harmonic field to a Neumann boundary value problem. By analyzing the harmonic field in typical knotted tubes, we identify the torsion driven transition from bipolar to vortex patterns. We also analogously extend our discussion to the organization of liquid crystal textures in knotted tubes. These results further our understanding about the general role of topology in shaping a physical field in confined space, and may find applications in the control of physical fields by manipulation of surface topology.
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Affiliation(s)
- Xiuqing Duan
- School of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
| | - Zhenwei Yao
- School of Physics and Astronomy, and Institute of Natural Sciences, Shanghai Jiao Tong University, Shanghai 200240, China
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Napoli G, Vergori L. Influence of the extrinsic curvature on two-dimensional nematic films. Phys Rev E 2018; 97:052705. [PMID: 29906963 DOI: 10.1103/physreve.97.052705] [Citation(s) in RCA: 4] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2018] [Indexed: 06/08/2023]
Abstract
Nematic films are thin fluid structures, ideally two dimensional, endowed with an in-plane degenerate nematic order. In this paper we examine a generalization of the classical Plateau problem to an axisymmetric nematic film bounded by two coaxial parallel rings. At equilibrium, the shape of the nematic film results from the competition between surface tension, which favors the minimization of the area, and the nematic elasticity, which instead promotes the alignment of the molecules along a common direction. We find two classes of equilibrium solutions in which the molecules are uniformly aligned along the meridians or parallels. Depending on two dimensionless parameters, one related to the geometry of the film and the other to the constitutive moduli, the Gaussian curvature of the equilibrium shape may be everywhere negative, vanishing, or positive. The stability of these equilibrium configurations is investigated.
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Affiliation(s)
- Gaetano Napoli
- Dipartimento di Matematica e Fisica "E. De Giorgi", Università del Salento, 73100 Lecce, Italy
| | - Luigi Vergori
- Dipartimento di Ingegneria, Università degli Studi di Perugia, 06125 Perugia, Italy
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Allahyarov E, Voigt A, Löwen H. Smectic monolayer confined on a sphere: topology at the particle scale. SOFT MATTER 2017; 13:8120-8135. [PMID: 29075732 DOI: 10.1039/c7sm01704a] [Citation(s) in RCA: 12] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/07/2023]
Abstract
The impact of topology on the structure of a smectic monolayer confined to a sphere is explored by particle-resolved computer simulations of hard rods. The orientations of the particles are tangential to the sphere and either free or restricted to a prescribed director field with a latitude or longitude orderings. Depending on the imprinted topology, a wealth of different states are found including equatorial smectic with isotropic poles, equatorial smectic with empty poles, a broken egg-shell like modulated smectic, a capped nematic with equatorial bald patches, equatorial nematic with empty poles, and a situation with 4 or 8 half-strength topological defects. Potentially these states could be verified in experiments with Pickering emulsions of droplets with colloidal rods. The unique nature of dipolar structures consisting of positive and negative half-strength disclinations is revealed. These structures, classified by their density and interaction with other defects in the system, relieve the strain of the poles by separating closely positioned half-strength defects. The proximity of these structures to the half-strength defects might enhance the structural diffusion of the defects across the system.
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Affiliation(s)
- Elshad Allahyarov
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine Universität Düsseldorf, Universitätstrasse 1, 40225 Düsseldorf, Germany
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