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Severino PG, Kamien RD. Escape from the second dimension: A topological distinction between edge and screw dislocations. Phys Rev E 2024; 109:L012701. [PMID: 38366472 DOI: 10.1103/physreve.109.l012701] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/13/2023] [Accepted: 12/13/2023] [Indexed: 02/18/2024]
Abstract
Volterra's definition of dislocations in crystals geometrically distinguishes edge and screw defects according to whether the Burgers vector is perpendicular or parallel to the defect. A homotopy-theoretic analysis of dislocations as topological defects fails to differentiate edge and screw. Here we bridge the gap between the geometric and topological descriptions by demonstrating that there is a topological difference between screw and edge defects. Our construction distinguishes edge and screw based on the disclination-line pairs at the core of smectic dislocations. By exploiting the connection between topology and geometry in the form of Gaussian curvature, this analysis results in an invariant for dislocations in the saddle-splay vector. This construction can be generalized to crystals with triply periodic order.
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Affiliation(s)
- Paul G Severino
- Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104, USA
| | - Randall D Kamien
- Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, Pennsylvania 19104, USA
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Missaoui A, Susser AL, Aharoni H, Rosenblatt C. Energetics of topographically designed Smectic-A oily streaks. SOFT MATTER 2023; 19:3733-3738. [PMID: 37190940 DOI: 10.1039/d3sm00306j] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/17/2023]
Abstract
A thin Smectic-A liquid crystal (LC) film is deposited on a polymer vinyl alcohol-coated substrate that had been scribed with a uniform easy axis pattern over a square of side length L ≤ 85 μm. The small size of the patterned region facilitates material distribution to form either a hill (for a thin film) or divot (for a thick film) above the scribed square and having an oily streak (OS) texture. Optical profilometry measurements vs. film thickness suggest that the OS structure aims to adopt a preferred thickness z0 that depends on the nature of the molecule, the temperature, and the surface tension at the air interface. We present a phenomenological model that estimates the energy cost of the OS layer as its thickness deviates from z0.
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Affiliation(s)
- Amine Missaoui
- Department of Physics, Case Western Reserve University, Cleveland, Ohio, 44122, USA.
| | - Adam L Susser
- Department of Physics, Case Western Reserve University, Cleveland, Ohio, 44122, USA.
| | - Hillel Aharoni
- Department of Complex Systems, Weizmann Institute of Science, Rehovot, Israel
| | - Charles Rosenblatt
- Department of Physics, Case Western Reserve University, Cleveland, Ohio, 44122, USA.
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Wensink HH, Grelet E. Elastic response of colloidal smectic liquid crystals: Insights from microscopic theory. Phys Rev E 2023; 107:054604. [PMID: 37329078 DOI: 10.1103/physreve.107.054604] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/31/2023] [Accepted: 05/05/2023] [Indexed: 06/18/2023]
Abstract
Elongated colloidal rods at sufficient packing conditions are known to form stable lamellar or smectic phases. Using a simplified volume-exclusion model, we propose a generic equation of state for hard-rod smectics that is robust against simulation results and is independent of the rod aspect ratio. We then extend our theory by exploring the elastic properties of a hard-rod smectic, including the layer compressibility (B) and bending modulus (K_{1}). By introducing weak backbone flexibility we are able to compare our predictions with experimental results on smectics of filamentous virus rods (fd) and find quantitative agreement between the smectic layer spacing, the out-of-plane fluctuation strength, as well as the smectic penetration length λ=sqrt[K_{1}/B]. We demonstrate that the layer bending modulus is dominated by director splay and depends sensitively on lamellar out-of-plane fluctuations that we account for on the single-rod level. We find that the ratio between the smectic penetration length and the lamellar spacing is about two orders of magnitude smaller than typical values reported for thermotropic smectics. We attribute this to the fact that colloidal smectics are considerably softer in terms of layer compression than their thermotropic counterparts while the cost of layer bending is of comparable magnitude.
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Affiliation(s)
- H H Wensink
- Laboratoire de Physique des Solides-UMR 8502, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - E Grelet
- Centre de Recherche Paul Pascal-UMR 5031, CNRS, Université de Bordeaux, 33600 Pessac, France
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Abstract
Matter self-assembling into layers generates unique properties, including structures of stacked surfaces, directed transport, and compact area maximization that can be highly functionalized in biology and technology. Smectics represent the paradigm of such lamellar materials - they are a state between fluids and solids, characterized by both orientational and partial positional ordering in one layering direction, making them notoriously difficult to model, particularly in confining geometries. We propose a complex tensor order parameter to describe the local degree of lamellar ordering, layer displacement and orientation of the layers for simple, lamellar smectics. The theory accounts for both dislocations and disclinations, by regularizing singularities within defect cores and so remaining continuous everywhere. The ability to describe disclinations and dislocation allows this theory to simulate arrested configurations and inclusion-induced local ordering. This tensorial theory for simple smectics considerably simplifies numerics, facilitating studies on the mesoscopic structure of topologically complex systems.
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Monderkamp PA, Wittmann R, Te Vrugt M, Voigt A, Wittkowski R, Löwen H. Topological fine structure of smectic grain boundaries and tetratic disclination lines within three-dimensional smectic liquid crystals. Phys Chem Chem Phys 2022; 24:15691-15704. [PMID: 35552573 DOI: 10.1039/d2cp00060a] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/12/2022]
Abstract
Observing and characterizing the complex ordering phenomena of liquid crystals subjected to external constraints constitutes an ongoing challenge for chemists and physicists alike. To elucidate the delicate balance appearing when the intrinsic positional order of smectic liquid crystals comes into play, we perform Monte-Carlo simulations of rod-like particles in a range of cavities with a cylindrical symmetry. Based on recent insights into the topology of smectic orientational grain boundaries in two dimensions, we analyze the emerging three-dimensional defect structures from the perspective of tetratic symmetry. Using an appropriate three-dimensional tetratic order parameter constructed from the Steinhardt order parameters, we show that those grain boundaries can be interpreted as a pair of tetratic disclination lines that are located on the edges of the nematic domain boundary. Thereby, we shed light on the fine structure of grain boundaries in three-dimensional confined smectics.
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Affiliation(s)
- Paul A Monderkamp
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany.
| | - René Wittmann
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany.
| | - Michael Te Vrugt
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany
| | - Axel Voigt
- Institut für Wissenschaftliches Rechnen, Technische Universität Dresden, 01062 Dresden, Germany
| | - Raphael Wittkowski
- Institut für Theoretische Physik, Center for Soft Nanoscience, Westfälische Wilhelms-Universität Münster, 48149 Münster, Germany
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, 40225 Düsseldorf, Germany.
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Hocking BJ, Ansell HS, Kamien RD, Machon T. The topological origin of the Peierls–Nabarro barrier. Proc Math Phys Eng Sci 2022. [DOI: 10.1098/rspa.2021.0725] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/12/2022] Open
Abstract
Crystals and other condensed matter systems described by density waves often exhibit dislocations. Here we show, by considering the topology of the ground state manifolds (GSMs) of such systems, that dislocations in the density phase field always split into disclinations, and that the disclinations themselves are constrained to sit at particular phase values in the GSM. This constraint on the location of the disclinations results in an energy barrier to dislocation glide. Consequently, the topology of the GSM alone gives rise to a Peierls–Nabarro barrier.
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Affiliation(s)
- Brook J. Hocking
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK
| | - Helen S. Ansell
- Department of Physics and Astronomy, Northwestern University, Evanston, IL 60208, USA
- Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA
| | - Randall D. Kamien
- Department of Physics and Astronomy, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA
- Department of Mathematics, University of Pennsylvania, 209 South 33rd Street, Philadelphia, PA 19104, USA
| | - Thomas Machon
- H. H. Wills Physics Laboratory, University of Bristol, Bristol BS8 1TL, UK
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Monderkamp PA, Wittmann R, Cortes LBG, Aarts DGAL, Smallenburg F, Löwen H. Topology of Orientational Defects in Confined Smectic Liquid Crystals. PHYSICAL REVIEW LETTERS 2021; 127:198001. [PMID: 34797147 DOI: 10.1103/physrevlett.127.198001] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/08/2021] [Revised: 06/28/2021] [Accepted: 09/22/2021] [Indexed: 06/13/2023]
Abstract
We propose a general formalism to characterize orientational frustration of smectic liquid crystals in confinement by interpreting the emerging networks of grain boundaries as objects with a topological charge. In a formal idealization, this charge is distributed in pointlike units of quarter-integer magnitude, which we identify with tetratic disclinations located at the end points and nodes. This coexisting nematic and tetratic order is analyzed with the help of extensive Monte Carlo simulations for a broad range of two-dimensional confining geometries as well as colloidal experiments, showing how the observed defect networks can be universally reconstructed from simple building blocks. We further find that the curvature of the confining wall determines the anchoring behavior of grain boundaries, such that the number of nodes in the emerging networks and the location of their end points can be tuned by changing the number and smoothness of corners, respectively.
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Affiliation(s)
- Paul A Monderkamp
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
| | - René Wittmann
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
| | - Louis B G Cortes
- School of Applied and Engineering Physics, Cornell University, Ithaca, New York 14853, USA
| | - Dirk G A L Aarts
- Department of Chemistry, Physical and Theoretical Chemistry Laboratory, University of Oxford, South Parks Road, Oxford OX1 3QZ, United Kingdom
| | - Frank Smallenburg
- Laboratoire de Physique des Solides, CNRS, Université Paris-Saclay, 91405 Orsay, France
| | - Hartmut Löwen
- Institut für Theoretische Physik II: Weiche Materie, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, 40225 Düsseldorf, Germany
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Vitral E, Leo PH, Viñals J. Phase-field model for a weakly compressible soft layered material: morphological transitions on smectic-isotropic interfaces. SOFT MATTER 2021; 17:6140-6159. [PMID: 34080608 DOI: 10.1039/d1sm00488c] [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
A coupled phase-field and hydrodynamic model is introduced to describe a two-phase, weakly compressible smectic (layered phase) in contact with an isotropic fluid of different density. A non-conserved smectic order parameter is coupled to a conserved mass density in order to accommodate non-solenoidal flows near the smectic-isotropic boundary arising from density contrast between the two phases. The model aims to describe morphological transitions in smectic thin films under heat treatment, in which arrays of focal conic defects evolve into conical pyramids and concentric rings through curvature dependent evaporation of smectic layers. The model leads to an extended thermodynamic relation at a curved surface that includes its Gaussian curvature, non-classical stresses at the boundary and flows arising from density gradients. The temporal evolution given by the model conserves the overall mass of the liquid crystal while still allowing for the modulated smectic structure to grow or shrink. A numerical solution of the governing equations reveals that pyramidal domains are sculpted at the center of focal conics upon a temperature increase, which display tangential flows at their surface. Other cases investigated include the possible coalescence of two cylindrical stacks of smectic layers, formation of droplets, and the interactions between focal conic domains through flow.
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Affiliation(s)
- Eduardo Vitral
- Department of Mechanical Engineering, University of Nevada, 1664 N. Virginia St., Reno, NV 89557, USA.
| | - Perry H Leo
- Department of Aerospace Engineering and Mechanics, University of Minnesota, 110 Union St. SE, Minneapolis, MN 55455, USA
| | - Jorge Viñals
- School of Physics and Astronomy, University of Minnesota, 116 Church St. SE, Minneapolis, MN 55455, USA
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