Clusters of polyhedra in spherical confinement

Abstract

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...

On the Form and Growth of Complex Crystals: The Case of Tsai-Type Clusters

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“…Moreover, attempts to mimic reality more closely by producing dense clusters of spheres of two appropriate radii via the simulation technique described in [20] did not result in a TDI-like cluster structure in 50 attempts. This indicates that the explanation of the TDI cluster as a purely volume-minimizing one under very high isotropic pressure in the melt is incomplete.…”

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“…Such depletion interactions scale with thermal energy (k B T), and thus enable tunable and reversible attractions (14,15). Clever design of shaped or patterned colloids yields "lock-and-key" colloidal interactions (16,17) and so-called "colloidal molecules" (18,19). Grafting ligand-functionalized molecules to colloidal surfaces enables molecular sensing (20,21), and sophisticated design of colloidal self-assembly (22,23).…”

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“…Polyhedra with small asphericity and high rotational symmetry are expected to form such mesophases in between the disordered and crystalline states [17]. Packing small numbers of hard icosahedra in a hard spherical container results in clusters that resemble or match sphere clusters (optimal sphere codes) despite significant faceting of these objects [18].…”