Finite Packings of Spheres

Betke, U.; Henk, Martin

doi:10.1007/pl00009341

Cited by 28 publications

(38 citation statements)

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“…Figure 5 shows an example of what is to be found in the supplementary material. 52 The best packings we found for the 5 Platonic solids and 13 Archimedean solids agreed excellently with literature, 48,63,65,89,90 yielding results within 0.002 of the literature value. For the truncated tetrahedra, we discovered a new crystal structure which improved upon the literature value of the densest packing.…”

Section: Close-packed Crystal Structures For Anisotropic Particlessupporting

confidence: 86%

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Crystal-structure prediction via the Floppy-Box Monte Carlo algorithm: Method and application to hard (non)convex particles

Graaf

Filion

Maréchal

et al. 2012

The Journal of Chemical Physics

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In this paper, we describe the way to set up the floppy-box Monte Carlo (FBMC) method [L. Filion, M. Marechal, B. van Oorschot, D. Pelt, F. Smallenburg, and M. Dijkstra, Phys. Rev. Lett. 103, 188302 (2009)] to predict crystal-structure candidates for colloidal particles. The algorithm is explained in detail to ensure that it can be straightforwardly implemented on the basis of this text. The handling of hard-particle interactions in the FBMC algorithm is given special attention, as (soft) short-range and semi-long-range interactions can be treated in an analogous way. We also discuss two types of algorithms for checking for overlaps between polyhedra, the method of separating axes and a triangular-tessellation based technique. These can be combined with the FBMC method to enable crystal-structure prediction for systems composed of highly shape-anisotropic particles. Moreover, we present the results for the dense crystal structures predicted using the FBMC method for 159 (non)convex faceted particles, on which the findings in [J. de Graaf, R. van Roij, and M. Dijkstra, Phys. Rev. Lett. 107, 155501 (2011)] were based. Finally, we comment on the process of crystalstructure prediction itself and the choices that can be made in these simulations.

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Section: Close-packed Crystal Structures For Anisotropic Particlessupporting

confidence: 86%

“…45 Employing the FBMC technique in this way connects the field of materials science with fields as diverse as discrete geometry, number theory, and computer science. [46][47][48][49][50] We only briefly go into this here and refer the reader to Ref. 45 for a discussion of the recent developments in analysing densest packings.…”

Section: Introductionmentioning

confidence: 99%

Crystal-structure prediction via the Floppy-Box Monte Carlo algorithm: Method and application to hard (non)convex particles

Graaf

Filion

Maréchal

et al. 2012

The Journal of Chemical Physics

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“…Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing [13], the densest packings of the other non-tiling Platonic solids that we obtain are their corresponding densest lattice packings [22,23].…”

Section: Introductionmentioning

confidence: 99%

Dense packings of polyhedra: Platonic and Archimedean solids

2009

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Understanding the nature of dense particle packings is a subject of intense research in the physical, mathematical and biological sciences. The preponderance of previous work has focused on spherical particles, and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of nonoverlapping, non-tiling polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find the dense packings of each of the Platonic solids in three-dimensional Euclidean space R 3 , except for the cube, which is the only Platonic solid that tiles space. We find the densest known packings of tetrahedra, icosahedra, dodecahedra, and octahedra with densities 0.823 . . ., 0.836 . . ., 0.904 . . ., and 0.947 . . ., respectively. It is noteworthy that the densest tetrahedral packing possesses no long-range order. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. We also derive a simple upper bound on the maximal density of packings of congruent nonspherical particles, and apply it to Platonic solids, Archimedean solids, superballs and ellipsoids. Provided that what we term the "asphericity" (ratio of the circumradius to inradius) is sufficiently small, the upper bounds are relatively tight and thus close to the corresponding densities of the optimal lattice packings of the centrally symmetric Platonic and Archimedean solids. Our simulation results, rigorous upper bounds, and other theoretical arguments lead us to the conjecture that the densest packings of Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This can be regarded to be the analog of Kepler's sphere conjecture for these solids. The truncated tetrahedron is the only non-centrally symmetric Archimedean solid, the densest known packing of which is a non-lattice packing with density at least as high as 23/24 = 0.958333 . . .. We discuss the validity of our conjecture to packings of superballs, prisms and anti-prisms as well as to high-dimensional analogs of the Platonic solids. In addition, we conjecture that the optimal packing of any convex, congruent polyhedron without central symmetry generally is not a lattice packing. Finally, we discuss the possible applications and generalizations of the ASC scheme in the predicting the crystal structures of polyhedral nanoparticles and the study of random packings of hard polyhedra.2

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“…Our results illustrate the intricate relation between phase behavior and building-block shape, and can guide future experimental studies on polyhedral-shaped nanoparticles. DOI: 10.1103/PhysRevLett.111.015501 PACS numbers: 61.46.Df, 64.70.MÀ, 64.75.Yz, 82.70.Dd Recent advances in experimental techniques to synthesize polyhedron-shaped particles, such as faceted nanocrystals and colloids [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the ability to perform self-assembly experiments with these particles [15][16][17][18][19][20][21][22], have attracted the interest of physicists, mathematicians, and computer scientists [23][24][25][26][27]. Additionally, predicting the densest packings of hard polyhedra has intrigued mathematicians since the time of the early Greek philosophers, such as Plato and Archimedes [28,29].…”

mentioning

confidence: 99%

“…In their research, the close-packed crystals of these particles were studied using sedimentation experiments and simulations. They created exotic superlattices, and their results also tested several conjectures on the densest packings of hard polyhedra [23,[25][26][27]. However, Henzie et al did not examine the finite-pressure behavior of the system.…”

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confidence: 99%

Phase Diagram and Structural Diversity of a Family of Truncated Cubes: Degenerate Close-Packed Structures and Vacancy-Rich States

et al. 2013

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Using Monte Carlo simulations and free-energy calculations, we determine the phase diagram of a family of truncated hard cubes, where the shape evolves smoothly from a cube via a cuboctahedron to an octahedron. A remarkable diversity in crystal phases and close-packed structures is found, including a fully degenerate crystal structure, several plastic crystals, as well as vacancy-stabilized crystal phases, all depending sensitively on the precise particle shape. Our results illustrate the intricate relation between phase behavior and building-block shape, and can guide future experimental studies on polyhedral-shaped nanoparticles. DOI: 10.1103/PhysRevLett.111.015501 PACS numbers: 61.46.Df, 64.70.MÀ, 64.75.Yz, 82.70.Dd Recent advances in experimental techniques to synthesize polyhedron-shaped particles, such as faceted nanocrystals and colloids [1][2][3][4][5][6][7][8][9][10][11][12][13][14] and the ability to perform self-assembly experiments with these particles [15][16][17][18][19][20][21][22], have attracted the interest of physicists, mathematicians, and computer scientists [23][24][25][26][27]. Additionally, predicting the densest packings of hard polyhedra has intrigued mathematicians since the time of the early Greek philosophers, such as Plato and Archimedes [28,29]. Modern computer platforms have made it possible to perform simulations of these systems, which has resulted not only in an improved understanding of the experimentally observed phenomenology in colloidal suspensions of such particles, but also in improved Ansätze for the morphology of their closepacked configurations [24,[30][31][32][33][34][35].The self-assembly of the basic building blocks at finite pressures may differ substantially from the packings achieved at high (sedimentation and solvent-evaporation) pressures. For instance, liquid-crystal, plastic-crystal, vacancy-rich simple-cubic, and quasicrystalline mesophases are stabilized by entropy alone under non-closepacked conditions of hard anisotropic particle systems [30][31][32][33][34]36,37]. Predicting the phase behavior from the shape of the building blocks alone is therefore a major challenge in materials science and is crucial for the design of new functional materials. It is thus not surprising that numerous studies have been devoted to providing simple guidelines for predicting the self-assembly from the particle shape alone [32][33][34].Recently, Henzie et al.[15] reported the shapecontrolled synthesis of truncated cubes. In their research, the close-packed crystals of these particles were studied using sedimentation experiments and simulations. They created exotic superlattices, and their results also tested several conjectures on the densest packings of hard polyhedra [23,[25][26][27]. However, Henzie et al. did not examine the finite-pressure behavior of the system. Mapping the full phase diagram for the system of truncated cubes is thus important, not only from a fundamental perspective but also to guide future experimental self-assembly studies to fabricate new func...

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Finite Packings of Spheres

Abstract: We show that the sausage conjecture of László Fejes Tóth on finite sphere packings is true in dimension 42 and above.

Cited by 28 publications

References 7 publications

Crystal-structure prediction via the Floppy-Box Monte Carlo algorithm: Method and application to hard (non)convex particles

Crystal-structure prediction via the Floppy-Box Monte Carlo algorithm: Method and application to hard (non)convex particles

Dense packings of polyhedra: Platonic and Archimedean solids

Phase Diagram and Structural Diversity of a Family of Truncated Cubes: Degenerate Close-Packed Structures and Vacancy-Rich States

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