Densest lattice packings of 3-polytopes

Betke, U.; Henk, Martin

doi:10.1016/s0925-7721(00)00007-9

Cited by 70 publications

(55 citation statements)

References 24 publications

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“…Much progress has recently been made in enumerating and characterizing the densest packings of polyhedral shapes [4,5,6]. While polyhedral nanocrystals have been assembled into ordered superstructures [7,8,9], directing their self-assembly into densest packings requires precise control of particle shape [10], polydispersity [11], interactions, and driving forces [12].…”

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confidence: 99%

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Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices

et al. 2011

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confidence: 99%

“…For any convex shape, however, the densest lattice packing can be calculated explicitly [4,13]. Lattice packings are a special class of arrangements in which all particles are oriented identically and positioned at sites of a crystalline lattice.…”

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confidence: 99%

Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices

et al. 2011

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“…In contrast, the bulk densest packing of anisotropic bodies has been thoroughly investigated in 3D Euclidean space (60)(61)(62)(63)(64)(65). This work has revealed insight into the interplay between packing structure, particle shape, and particle environment.…”

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confidence: 99%

Clusters of polyhedra in spherical confinement

Teich

Anders

Klotsa

et al. 2016

Proc. Natl. Acad. Sci. U.S.A.

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Dense particle packing in a confining volume remains a rich, largely unexplored problem, despite applications in blood clotting, plasmonics, industrial packaging and transport, colloidal molecule design, and information storage. Here, we report densest found clusters of the Platonic solids in spherical confinement, for up to N = 60 constituent polyhedral particles. We examine the interplay between anisotropic particle shape and isotropic 3D confinement. Densest clusters exhibit a wide variety of symmetry point groups and form in up to three layers at higher N. For many N values, icosahedra and dodecahedra form clusters that resemble sphere clusters. These common structures are layers of optimal spherical codes in most cases, a surprising fact given the significant faceting of the icosahedron and dodecahedron. We also investigate cluster density as a function of N for each particle shape. We find that, in contrast to what happens in bulk, polyhedra often pack less densely than spheres. We also find especially dense clusters at so-called magic numbers of constituent particles. Our results showcase the structural diversity and experimental utility of families of solutions to the packing in confinement problem.clusters | confinement | packing | colloids | nanoparticles P henomena as diverse as crowding in the cell (1, 2), DNA packaging in cell nuclei and virus capsids (3, 4), the growth of cellular aggregates (5), biological pattern formation (6), blood clotting (7), efficient manufacturing and transport, the planning and design of cellular networks (8), and efficient food and pharmaceutical packaging and transport (9) are related to the optimization problem of packing objects of a specified shape as densely as possible within a confining geometry, or packing in confinement. Packing in confinement is also a laboratory technique used to produce particle aggregates with consistent structure. These aggregates may serve as building blocks (or "colloidal molecules") in hierarchical structures (10, 11), information storage units (12), or drug delivery capsules (13). Experiments concerning cluster formation via spherical droplet confinement (13-20) are of special interest here. Droplets are typically either oil-in-water or water-inoil emulsions, and particle aggregation is induced via the evaporation of the droplet solvent. Clusters may be hollow [in which case they are termed "colloidosomes" (13)] or filled, depending on the formation protocol, and may contain a few (15) to a few billion (14) particles. Clusters of several metallic nanoparticles are especially intriguing given their ability to support surface plasmon modes over a range of frequencies (21). The subwavelength scale of these clusters means that their optical response is highly dependent on their specific geometry (22). Consequently, control over their structure enables control over their optical properties, with implications for cloaking (23), chemical sensing (24), imaging (25), nonlinear optics (26), and the creation of so-called metafluids (27)(28)(29), a...

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“…Let us briefly comment on our choice only to search through those combinations of a, b, c for which holds that everybody of the set {M , M a , M b , M c } is touched byt he three other ones. It is shown that, for some convex bodies, there are other contact situations giving minimal volume [3]. Veryp r o b a b ly this also holds for non-convex bodies, but little is known about that.…”

Section: Boxes Their Related Lattices and Calculating Lattice Packingsmentioning

confidence: 99%

“…Having reformulated the minimal-volume box problem as a densest lattice packing problem we have to look for a method which determines for a given body M the densest lattice packing. For polyhedral convex M such a method exists [3]. However, for most bio-macromolecules m and typical layer widths around m,t h es h a p eo fM is non-convex.…”

Section: Boxes Their Related Lattices and Calculating Lattice Packingsmentioning

confidence: 99%

Constructing a Near-Minimal-Volume Computational Box for Molecular Dynamics Simulations with Periodic Boundary Conditions

Bekker

Berg

Wassenaar

2003

Lecture Notes in Computer Science

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Abstract. In many M.D. simulations the simulated system consists of a single macromolecule in a solvent. Usually, one is not interested in the behaviour of the solvent, so, the CPU time may be minimized by minimizing the amount of solvent. For a given molecule and cut-off radius this may be done by constructing a computational box with near minimal volume. In this article a method is presented to construct such a box, and the method is tested on a significant number of macromolecules. The volume of the resulting boxes proves to be typically 40% of the volume of usual boxes, and as a result the simulation time decreases with typically 60%.

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Densest lattice packings of 3-polytopes

Cited by 70 publications

References 24 publications

Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices

Self-assembly of uniform polyhedral silver nanocrystals into densest packings and exotic superlattices

Clusters of polyhedra in spherical confinement

Constructing a Near-Minimal-Volume Computational Box for Molecular Dynamics Simulations with Periodic Boundary Conditions

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