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Sakata N, Mishina R, Ogawa M, Ishihara K, Koda Y, Ozawa M, Shimokawa K. Handlebody decompositions of three-manifolds and polycontinuous patterns. Proc Math Phys Eng Sci 2022; 478:20220073. [PMID: 35510221 PMCID: PMC9053369 DOI: 10.1098/rspa.2022.0073] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Download PDF] [Figures] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/27/2022] [Accepted: 03/10/2022] [Indexed: 11/12/2022] Open
Abstract
We introduce the concept of a handlebody decomposition of a three-manifold, a generalization of a Heegaard splitting, or a trisection. We show that two handlebody decompositions of a closed orientable three-manifold are stably equivalent. As an application to materials science, we consider a mathematical model of polycontinuous patterns and discuss a topological study of microphase separation of a block copolymer melt.
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Affiliation(s)
- N Sakata
- Department of Mathematics, Saitama University, Saitama 338-8570, Japan
| | - R Mishina
- Department of Mathematics, Saitama University, Saitama 338-8570, Japan
| | - M Ogawa
- Department of Mathematics, Saitama University, Saitama 338-8570, Japan
| | - K Ishihara
- Faculty of Education, Yamaguchi University, Yamaguchi 753-8511, Japan
| | - Y Koda
- Department of Mathematics, Hiroshima University, Hiroshima 739-8511, Japan
| | - M Ozawa
- Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, Tokyo 154-8525, Japan
| | - K Shimokawa
- Department of Mathematics, Saitama University, Saitama 338-8570, Japan.,Department of Mathematics, Ochanomizu University, Tokyo 112-8610, Japan
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2
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Reddy A, Feng X, Thomas EL, Grason GM. Block Copolymers beneath the Surface: Measuring and Modeling Complex Morphology at the Subdomain Scale. Macromolecules 2021. [DOI: 10.1021/acs.macromol.1c00958] [Citation(s) in RCA: 12] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/14/2022]
Affiliation(s)
- Abhiram Reddy
- Department of Polymer Science and Engineering, University of Massachusetts Amherst, Amherst, Massachusetts 01003, United States
| | - Xueyan Feng
- Department of Materials Science and Engineering, Texas A&M University, College Station, Texas 77843, United States
| | - Edwin L. Thomas
- Department of Materials Science and Engineering, Texas A&M University, College Station, Texas 77843, United States
| | - Gregory M. Grason
- Department of Polymer Science and Engineering, University of Massachusetts Amherst, Amherst, Massachusetts 01003, United States
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Hain TM, Schröder-Turk GE, Kirkensgaard JJK. Patchy particles by self-assembly of star copolymers on a spherical substrate: Thomson solutions in a geometric problem with a color constraint. SOFT MATTER 2019; 15:9394-9404. [PMID: 31595280 DOI: 10.1039/c9sm01460h] [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
Confinement or geometric frustration is known to alter the structure of soft matter, including copolymeric melts, and can consequently be used to tune structure and properties. Here we investigate the self-assembly of ABC and ABB 3-miktoarm star copolymers confined to a spherical shell using coarse-grained dissipative particle dynamics simulations. In bulk and flat geometries the ABC stars form hexagonal tilings, but this is topologically prohibited in a spherical geometry which normally is alleviated by forming pentagonal tiles. However, the molecular architecture of the ABC stars implies an additional 'color constraint' which only allows even tilings (where all polygons have an even number of edges) and we study the effect of these simultaneous constraints. We find that both ABC and ABB systems form spherical tiling patterns, the type of which depends on the radius of the spherical substrate. For small spherical substrates, all solutions correspond to patterns solving the Thomson problem of placing mobile repulsive electric charges on a sphere. In ABC systems we find three coexisting, possibly different tilings, one in each color, each of them solving the Thomson problem simultaneously. For all except the smallest substrates, we find competing solutions with seemingly degenerate free energies that occur with different probabilities. Statistically, an observer who is blind to the differences between B and C can tell from the structure of the A domains if the system is an ABC or an ABB star copolymer system.
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Affiliation(s)
- Tobias M Hain
- College of Science, Health, Engineering and Education, Mathematics and Statistics, Murdoch University, 90 South Street, 6150 Murdoch, Western Australia, Australia.
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de Campo L, Castle T, Hyde ST. Optimal packings of three-arm star polyphiles: from tricontinuous to quasi-uniformly striped bicontinuous forms. Interface Focus 2017. [PMID: 28630673 DOI: 10.1098/rsfs.2016.0130] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/27/2022] Open
Abstract
Star-shaped molecules with three mutually immiscible arms self-assemble to form a variety of novel structures, with conformations that attempt to minimize interfacial area between the domains composed of the different arms. The geometric frustration caused by the joining of these arms at a common centre limits the size and shape of each domain, encouraging the creation of complex and interesting solutions. Some solutions are tricontinuous, and these solutions (and others) share aspects of bicontinuous structures with amphiphilic assemblies as similar molecular segregation factors are at work. We describe both highly symmetric and balanced structures, as well as unbalanced solutions that take the form of intricately striped amphiphilic membranes. All these patterns can result in chiral assemblies with multiple networks.
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Affiliation(s)
- Liliana de Campo
- Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, Australia.,Australian Nuclear Science and Technology Organisation, ACNS, Lucas Heights, Australia
| | - Toen Castle
- Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, Australia.,Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, PA, USA
| | - Stephen T Hyde
- Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, Australia
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Cumming BP, Schröder-Turk GE, Debbarma S, Gu M. Bragg-mirror-like circular dichroism in bio-inspired quadruple-gyroid 4srs nanostructures. LIGHT, SCIENCE & APPLICATIONS 2017; 6:e16192. [PMID: 30167193 PMCID: PMC6061894 DOI: 10.1038/lsa.2016.192] [Citation(s) in RCA: 5] [Impact Index Per Article: 0.7] [Reference Citation Analysis] [Abstract] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 04/05/2016] [Revised: 07/11/2016] [Accepted: 07/18/2016] [Indexed: 05/25/2023]
Abstract
The smooth and tailorable spectral response of Bragg mirrors has driven their pervasive use in optical systems requiring customizable spectral control of beam propagation. However, the simple nature of Bragg mirror reflection prevents their application to the control of important polarization states such as circular polarization. While helical and gyroid-based nanostructures exhibiting circular dichroism have been developed extensively to address this limitation, they are often restricted by the spectral inconsistency of their optical response. Here we present the fabrication and characterization of quadruple-gyroid 4srs nanostructures exhibiting bio-inspired Bragg-mirror-like circular dichroism: a smooth and uniform band of circular dichroism reminiscent of the spectrum of a simple multilayer Bragg-mirror. Furthermore, we demonstrate that the circular dichroism produced by 4srs nanostructures are robust to changes in incident angle and beam collimation, providing a new platform to create and engineer circular dichroism for functional circular polarization manipulation.
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Affiliation(s)
- Benjamin P Cumming
- Laboratory of Artificial-Intelligence Nanophotonics and CUDOS, School of Science, RMIT University, Melbourne, Victoria 3001, Australia
- Centre for Micro-Photonics, Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
| | - Gerd E Schröder-Turk
- School of Engineering and Information Technology, Mathematics and Statistics, Murdoch University, Murdoch, Western Australia 6150, Australia
| | - Sukanta Debbarma
- Laser Physics Centre and CUDOS, Research School of Physical Sciences and Engineering, Australian National University, Canberra, Australian Capital Territory 2601, Australia
| | - Min Gu
- Laboratory of Artificial-Intelligence Nanophotonics and CUDOS, School of Science, RMIT University, Melbourne, Victoria 3001, Australia
- Centre for Micro-Photonics, Faculty of Science, Engineering and Technology, Swinburne University of Technology, Hawthorn, Victoria 3122, Australia
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6
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Jiang K, Zhang J, Liang Q. Self-Assembly of Asymmetrically Interacting ABC Star Triblock Copolymer Melts. J Phys Chem B 2015; 119:14551-62. [PMID: 26492084 DOI: 10.1021/acs.jpcb.5b08187] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.2] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/28/2022]
Affiliation(s)
- Kai Jiang
- Hunan Key
Laboratory for
Computation and Simulation in Science and Engineering, School of Mathematics
and Computational Science, Xiangtan University, Hunan 411105, China
| | - Juan Zhang
- Hunan Key
Laboratory for
Computation and Simulation in Science and Engineering, School of Mathematics
and Computational Science, Xiangtan University, Hunan 411105, China
| | - Qin Liang
- Hunan Key
Laboratory for
Computation and Simulation in Science and Engineering, School of Mathematics
and Computational Science, Xiangtan University, Hunan 411105, China
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