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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] [What about the content of this article? (0)] [Affiliation(s)] [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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Abstract
Perpendicular arrangements in hierarchical nanostructures show superior mechanical properties and provide great opportunities for the development of advanced membranes because different channels are connected by the perpendicular blocks. To obtain these perpendicular hierarchical nanostructures, we use a simple ABC-star terpolymer because of the existence of a conjunction point by using the A block as a polymer network template, which guides the BC phase separation accordingly. When χBC is 10, the formed phase and the corresponding phase diagram of ABC-star are similar to those of the AB2 triblock because of the mixture between the B and C blocks. Interestingly, at increased χBC, the B and C blocks phase separate, leading to the formation of a series of perpendicular nanostructures, including perpendicular lamellae-in-lamellae (L⊥), perpendicular lamellae-in-cylinder (C⊥), and even perpendicular three-dimensional polymer networks (G⊥). The corresponding stability regime of each phase is identified through the dedicated comparison of free energy, which can well explain the missing phases in Monte Carlo simulations. Our proposed design route according to the target structures and the calculated phase diagram can provide useful guidance for the experimental observation of these perpendicular nanostructures.
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Affiliation(s)
- Mingshuang Qian
- Faculty of Materials Science and Chemical Engineering, Ningbo University, Ningbo, Zhejiang 315211, China
| | - Yuci Xu
- Faculty of Materials Science and Chemical Engineering, Ningbo University, Ningbo, Zhejiang 315211, China
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Mukhtyar AJ, Escobedo FA. Computing free energy barriers for the nucleation of complex network mesophases. J Chem Phys 2022; 156:034502. [DOI: 10.1063/5.0079396] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/14/2022] Open
Affiliation(s)
- Ankita J. Mukhtyar
- Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14850, USA
| | - Fernando A. Escobedo
- Robert Frederick Smith School of Chemical and Biomolecular Engineering, Cornell University, Ithaca, New York 14850, USA
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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] [What about the content of this article? (0)] [Affiliation(s)] [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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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] [What about the content of this article? (0)] [Affiliation(s)] [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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Affiliation(s)
- 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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Abstract
We present coarse-grained simulations of the self-assembly of 3-armed ABC star polyphiles. In systems of star polyphiles with two arms of equal length the simulations corroborate and expand previous findings from related miktoarm star terpolymer systems on the formation of patterns containing columnar domains whose sections are 2D planar tilings. However, the systematic variation of face topologies as the length of the third (unequal) arm is varied differs from earlier findings regarding the compositional dependence. We explore 2D 3-colored foams to establish the optimal patterns based on interfacial energy alone. A generic construction algorithm is described that accounts for all observed 2D tiling patterns and suggests other patterns likely to be found beyond the range of the simulations reported here. Patterns resulting from this algorithm are relaxed using Surface Evolver calculations to form 2D foams with minimal interfacial length as a function of composition. This allows us to estimate the interfacial enthalpic contributions to the free energy of related star molecular assemblies assuming strong segregation. We compare the resulting phase sequence with a number of theoretical results from particle-based simulations and field theory, allowing us to tease out relative enthalpic and entropic contributions as a function of the chain lengths making up the star molecules. Our results indicate that a richer polymorphism is to be expected in systems not dominated by chain entropy. Further, analysis of corresponding planar tiling patterns suggests that related two-periodic columnar structures are unlikely hypothetical phases in 4-arm star polyphile melts in the absence of sufficient arm configurational freedom for minor domains to form lens-shaped di-gons, which require higher molecular weight polymeric arms. Finally, we discuss the possibility of forming a complex tiling pattern that is a quasi-crystalline approximant for 3-arm star polyphiles with unequal arm lengths.
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Marson RL, Phillips CL, Anderson JA, Glotzer SC. Phase behavior and complex crystal structures of self-assembled tethered nanoparticle telechelics. Nano Lett 2014; 14:2071-2078. [PMID: 24641517 DOI: 10.1021/nl500236b] [Citation(s) in RCA: 23] [Impact Index Per Article: 2.3] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 06/03/2023]
Abstract
Motivated by growing interest in the self-assembly of nanoparticles for applications such as photonics, organic photovoltaics, and DNA-assisted designer crystals, we explore the phase behavior of tethered spherical nanoparticles. Here, a polymer tether is used to geometrically constrain a pair of nanoparticles creating a tethered nanoparticle "telechelic". Using simulation, we examine how varying architectural features, such as the size ratio of the two end-group nanospheres and the length of the flexible tether, affects the self-assembled morphologies. We demonstrate not only that this hybrid building block maintains the same phase diversity as linear triblock copolymers, allowing for a variety of nanoparticle materials to replace polymer blocks, but also that new structures not previously reported are accessible. Our findings imply a robust underlying ordering mechanism is common among these systems, thus allowing flexibility in synthesis approaches to achieve a target morphology.
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Affiliation(s)
- Ryan L Marson
- Materials Science and Engineering, ‡Department of Applied Physics, and §Chemical Engineering, University of Michigan , Ann Arbor, Michigan 48109, United States
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Kirkensgaard JJK, Evans ME, de Campo L, Hyde ST. Hierarchical self-assembly of a striped gyroid formed by threaded chiral mesoscale networks. Proc Natl Acad Sci U S A 2014; 111:1271-6. [PMID: 24474747 PMCID: PMC3910609 DOI: 10.1073/pnas.1316348111] [Citation(s) in RCA: 35] [Impact Index Per Article: 3.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [Key Words] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Indexed: 11/18/2022] Open
Abstract
Numerical simulations reveal a family of hierarchical and chiral multicontinuous network structures self-assembled from a melt blend of Y-shaped ABC and ABD three-miktoarm star terpolymers, constrained to have equal-sized A/B and C/D chains, respectively. The C and D majority domains within these patterns form a pair of chiral enantiomeric gyroid labyrinths (srs nets) over a broad range of compositions. The minority A and B components together define a hyperbolic film whose midsurface follows the gyroid minimal surface. A second level of assembly is found within the film, with the minority components also forming labyrinthine domains whose geometry and topology changes systematically as a function of composition. These smaller labyrinths are well described by a family of patterns that tile the hyperbolic plane by regular degree-three trees mapped onto the gyroid. The labyrinths within the gyroid film are densely packed and contain either graphitic hcb nets (chicken wire) or srs nets, forming convoluted intergrowths of multiple nets. Furthermore, each net is ideally a single chiral enantiomer, induced by the gyroid architecture. However, the numerical simulations result in defect-ridden achiral patterns, containing domains of either hand, due to the achiral terpolymeric starting molecules. These mesostructures are among the most topologically complex morphologies identified to date and represent an example of hierarchical ordering within a hyperbolic pattern, a unique mode of soft-matter self-assembly.
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Affiliation(s)
| | - Myfanwy E. Evans
- Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstrasse 7B, 91058 Erlangen, Germany; and
| | - Liliana de Campo
- Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, ACT 0200, Australia
| | - Stephen T. Hyde
- Niels Bohr Institute, University of Copenhagen, 2100 Copenhagen, Denmark
- Department of Applied Mathematics, Research School of Physical Sciences, Australian National University, Canberra, ACT 0200, Australia
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Abstract
Since the discovery of the liquid-crystalline state of matter 125 years ago, this field has developed into a scientific area with many facets. This Review presents recent developments in the molecular design and self-assembly of liquid crystals. The focus is on new exciting soft-matter structures distinct from the usually observed nematic, smectic, and columnar phases. These new structures have enhanced complexity, including multicompartment and cellular structures, periodic and quasiperiodic arrays of spheres, and new emergent properties, such as ferroelctricity and spontaneous achiral symmetry-breaking. Comparisons are made with developments in related fields, such as self-assembled monolayers, multiblock copolymers, and nanoparticle arrays. Measures of structural complexity used herein are the size of the lattice, the number of distinct compartments, the dimensionality, and the logic depth of the resulting supramolecular structures.
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Affiliation(s)
- Carsten Tschierske
- Institut für Chemie, Organische Chemie, Martin-Luther-Universität Halle-Wittenberg, 06120 Halle Saale, Germany.
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Affiliation(s)
- Stephen T. Hyde
- Department of Applied Mathematics, Research School of Physics and Engineering, The Australian National University, Canberra, Australian Capital Territory 0200, Australia
| | - Gerd E. Schröder-Turk
- Theoretische Physik, Friedrich-Alexander Universität Erlangen-Nürnberg, Staudtstrasse 7B, 91058 Erlangen, Germany
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Kirkensgaard JJK. Striped networks and other hierarchical structures in AmBmCn (2m+n)-miktoarm star terpolymer melts. Phys Rev E Stat Nonlin Soft Matter Phys 2012; 85:031802. [PMID: 22587114 DOI: 10.1103/physreve.85.031802] [Citation(s) in RCA: 6] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [What about the content of this article? (0)] [Affiliation(s)] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 11/12/2011] [Indexed: 05/31/2023]
Abstract
Using dissipative particle dynamics simulations we give numerical evidence of the formation of "striped" (or AB alternating) diamond and gyroid network structures and other hierarchical morphologies in A(m)B(m)C(n) (2m+n)-miktoarm star terpolymers where the main variable is the ratio x=n/m with m,n being the number of equal length polymer arms of A and B and C, respectively. The formed networks are purely a result of the star topology, as clearly shown by direct comparison with parallel ABC miktoarm star terpolymer simulations with matching overall composition. Progressively changing x, the system adopts the following phase sequence: three-colored lamellae, C spheres embedded in AB lamellae, C spheres decorating AB lamellae, three-colored [6.6.6] tiling, AB striped diamond network, AB striped gyroid network, AB striped hexagonally arranged cylinders, and finally AB striped globular aggregates. The striped gyroid is particularly interesting as it constitutes an inherently chiral structure made from achiral building blocks.
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Affiliation(s)
- Jacob Judas Kain Kirkensgaard
- Department of Basic Sciences and Environment, Faculty of Life Sciences, University of Copenhagen, Copenhagen, Denmark.
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