1
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Pedersen MC, Hyde ST, Ramsden S, Kirkensgaard JJK. Mapping hyperbolic order in curved materials. SOFT MATTER 2023; 19:1586-1595. [PMID: 36749349 DOI: 10.1039/d2sm01403c] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/18/2023]
Abstract
Nature employs an impressive range of topologically complex ordered nanostructures that occur in various forms in both natural and synthetic materials. A particular class of these exhibits negative curvature and forms periodic saddle-shaped surfaces in three dimensions. Unlike pattern formation on flat or positively curved surfaces like spherical systems, the understanding of patterning on such surfaces is highly complicated due to the structures being intrinsically intertwined in three dimensions. We present a new method for visualisation and analysis of patterns on triply periodic negatively curved surfaces by mapping to two-dimensional hyperbolic space analogous to spherical projections in cartography thus effectively creating a more accessible "hyperbolic map" of the pattern. Specifically, we exemplify the method via the simplest triply periodic minimal surfaces: the Primitive, Diamond, and Gyroid in their universal cover along with decorations from a soft materials, whose structures involve decorations of soft matter on negatively curved surfaces, not necessarily minimal.
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Affiliation(s)
| | | | - Stuart Ramsden
- National Computational Infrastructure (NCI) Vizlab, Australian National University, Australia
| | - Jacob J K Kirkensgaard
- Niels Bohr Institute, University of Copenhagen, Denmark.
- Department of Food Science, University of Copenhagen, Denmark
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2
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Li L, Xu Z, Li W. Emergence of Connected Binary Spherical Structures from the Self-assembly of an AB 2C Four-Arm Star Terpolymer. Macromolecules 2022. [DOI: 10.1021/acs.macromol.2c01919] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/30/2022]
Affiliation(s)
- Luyang Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Zhanwen Xu
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
| | - Weihua Li
- State Key Laboratory of Molecular Engineering of Polymers, Key Laboratory of Computational Physical Sciences, Department of Macromolecular Science, Fudan University, Shanghai 200433, China
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3
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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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4
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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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5
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Kim H, Arras MML, Mahalik JP, Wang W, Yu DM, Chernyy S, Goswami M, Kumar R, Sumpter BG, Hong K, Smith GS, Russell TP. Studies on the 3-Lamellar Morphology of Miktoarm Terpolymers. Macromolecules 2018. [DOI: 10.1021/acs.macromol.8b01548] [Citation(s) in RCA: 11] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 01/08/2023]
Affiliation(s)
- Hyeyoung Kim
- Department of Polymer Science and Engineering, University of Massachusetts Amherst, 120 Governors Drive, Amherst, Massachusetts 01003, United States
| | | | - J. P. Mahalik
- Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37916, United States
| | | | - Duk Man Yu
- Department of Polymer Science and Engineering, University of Massachusetts Amherst, 120 Governors Drive, Amherst, Massachusetts 01003, United States
| | - Sergey Chernyy
- Department of Micro- and Nanotechnology, Technical University of Denmark, Produktionstorvet, 2800 Lyngby, Denmark
| | | | - Rajeev Kumar
- Department of Mathematics, University of Tennessee, Knoxville, Tennessee 37916, United States
| | | | | | | | - Thomas P. Russell
- Department of Polymer Science and Engineering, University of Massachusetts Amherst, 120 Governors Drive, Amherst, Massachusetts 01003, United States
- Beijing Advanced Innovation Center for Soft Matter Science and Engineering, Beijing University of Chemical Technology, Bejing 100029, China
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6
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Chernyy S, Kirkensgaard JJK, Mahalik JP, Kim H, Arras MML, Kumar R, Sumpter BG, Smith GS, Mortensen K, Russell TP, Almdal K. Bulk and Surface Morphologies of ABC Miktoarm Star Terpolymers Composed of PDMS, PI, and PMMA Arms. Macromolecules 2018. [DOI: 10.1021/acs.macromol.7b02485] [Citation(s) in RCA: 14] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 12/11/2022]
Affiliation(s)
- Sergey Chernyy
- DTU
Nanotech, Technical University of Denmark, Produktionstorvet, 2800 Lyngby, Denmark
| | | | | | - Hyeyoung Kim
- Department
of Polymer Science and Engineering, University of Massachusetts Amherst, 120 Governors Drive, Amherst, Massachusetts 01003, United States
| | | | | | | | | | - Kell Mortensen
- Niels
Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
| | - Thomas P. Russell
- Department
of Polymer Science and Engineering, University of Massachusetts Amherst, 120 Governors Drive, Amherst, Massachusetts 01003, United States
| | - Kristoffer Almdal
- DTU
Nanotech, Technical University of Denmark, Produktionstorvet, 2800 Lyngby, Denmark
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7
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Miyase H, Asai Y, Takano A, Matsushita Y. Kaleidoscopic Tiling Patterns with Large Unit Cells from ABC Star-Shaped Terpolymer/Diblock Copolymer Blends with Hydrogen Bonding Interaction. Macromolecules 2017. [DOI: 10.1021/acs.macromol.6b02406] [Citation(s) in RCA: 28] [Impact Index Per Article: 4.0] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Haruko Miyase
- Department of Applied
Chemistry,
Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
| | - Yusuke Asai
- Department of Applied
Chemistry,
Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
| | - Atsushi Takano
- Department of Applied
Chemistry,
Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
| | - Yushu Matsushita
- Department of Applied
Chemistry,
Graduate School of Engineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8603, Japan
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8
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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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9
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De Nicola A, Kawakatsu T, Rosano C, Celino M, Rocco M, Milano G. Self-Assembly of Triton X-100 in Water Solutions: A Multiscale Simulation Study Linking Mesoscale to Atomistic Models. J Chem Theory Comput 2015; 11:4959-71. [DOI: 10.1021/acs.jctc.5b00485] [Citation(s) in RCA: 30] [Impact Index Per Article: 3.3] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Antonio De Nicola
- Dipartimento
di Chimica e Biologia, Università di Salerno, I-84084 via
Ponte don Melillo, Salerno, Italy
| | - Toshihiro Kawakatsu
- Department
of Physics, Tohoku University, Aoba, Aramaki, Aoba-ku, Sendai 980-8578, Japan
| | - Camillo Rosano
- Biopolimeri
e Proteomica, IRCCS AOU San Martino-IST, Istituto Nazionale per la Ricerca sul Cancro, Largo R. Benzi 10 I-16132 Genova, Italy
| | - Massimo Celino
- ENEA, C.R. Casaccia, Via Anguillarese 301, I-00123 Roma, Italy
| | - Mattia Rocco
- Biopolimeri
e Proteomica, IRCCS AOU San Martino-IST, Istituto Nazionale per la Ricerca sul Cancro, Largo R. Benzi 10 I-16132 Genova, Italy
| | - Giuseppe Milano
- Dipartimento
di Chimica e Biologia, Università di Salerno, I-84084 via
Ponte don Melillo, Salerno, Italy
- IMAST S.c.a.r.l.
Technological District in Polymer and Composite Engineering, P. leBovio 22, I-80133, Napoli, Italy
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10
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Fischer MG, de Campo L, Kirkensgaard JJK, Hyde ST, Schröder-Turk GE. The Tricontinuous 3ths(5) Phase: A New Morphology in Copolymer Melts. Macromolecules 2014. [DOI: 10.1021/ma5016352] [Citation(s) in RCA: 9] [Impact Index Per Article: 0.9] [Reference Citation Analysis] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/29/2022]
Affiliation(s)
- Michael G. Fischer
- Institut
für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7B, 91058 Erlangen, Germany
- Applied Maths, Research School of Physics & Engineering, The Australian National University, Canberra ACT 0200, Australia
- Adolphe Merkle
Institute, Chemin des Verdiers 4, CH-1700 Fribourg, Switzerland
| | - Liliana de Campo
- Applied Maths, Research School of Physics & Engineering, The Australian National University, Canberra ACT 0200, Australia
- Australian
National Science and Technology Organisation, Bragg Institute, New
Illawarra Road, Lucas Heights NSW 2234, Australia
| | | | - Stephen T. Hyde
- Applied Maths, Research School of Physics & Engineering, The Australian National University, Canberra ACT 0200, Australia
| | - Gerd E. Schröder-Turk
- Institut
für Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstr. 7B, 91058 Erlangen, Germany
- Applied Maths, Research School of Physics & Engineering, The Australian National University, Canberra ACT 0200, Australia
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