Densest local sphere-packing diversity: General concepts and application to two dimensions

Packing problems have been a source of fascination for millennia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals, and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable, and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the "geometric-structure" approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and "order" maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research.

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“…48 It is noteworthy that some optimal spherical codes 278 are related to the densest local packing of spheres around a central sphere. 279,280…”

Section: F Remarks On Packing Problems In Non-euclidean Spacesmentioning

confidence: 99%

Perspective: Basic understanding of condensed phases of matter via packing models

Torquato

2018

The Journal of Chemical Physics

134

139

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“…The optimal spherical code problem is related to the densest local packing (DLP) problem in R d (Hopkins et al, 2010a), which involves the placement of N nonoverlapping spheres of unit diameter near an additional fixed unit-diameter sphere such that the greatest radius R from the center of the fixed sphere to the centers of any of the N surrounding spheres is minimized. Let us recast the optimal spherical code problem as the placement of the centers of N nonoverlapping spheres of unit diameter onto the surface of a sphere of radius R such that R is minimized.…”

Section: Remarks On Packing Problems In Non-euclidean Spacesmentioning

confidence: 99%

Jammed hard-particle packings: From Kepler to Bernal and beyond

2010

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Understanding the characteristics of jammed particle packings provides basic insights into the structure and bulk properties of crystals, glasses, and granular media and into selected aspects of biological systems. This review describes the diversity of jammed configurations attainable by frictionless convex nonoverlapping ͑hard͒ particles in Euclidean spaces and for that purpose it stresses individual-packing geometric analysis. A fundamental feature of that diversity is the necessity to classify individual jammed configurations according to whether they are locally, collectively, or strictly jammed. Each of these categories contains a multitude of jammed configurations spanning a wide and ͑in the large system limit͒ continuous range of intensive properties, including packing fraction , mean contact number Z, and several scalar order metrics. Application of these analytical tools to spheres in three dimensions ͑an analog to the venerable Ising model͒ covers a myriad of jammed states, including maximally dense packings ͑as Kepler conjectured͒, low-density strictly jammed tunneled crystals, and a substantial family of amorphous packings. With respect to the last of these, the current approach displaces the historically prominent but ambiguous idea of "random close packing" with the precise concept of "maximally random jamming." Both laboratory procedures and numerical simulation protocols can and, frequently, have been used for creation of ensembles of jammed states. But while the resulting distributions of intensive properties may individually approach narrow distributions in the large system limit, the distinguishing varieties of possible operational details in these procedures and protocols lead to substantial variability among the resulting distributions, some examples of which are presented here. This review also covers recent advances in understanding jammed packings of polydisperse sphere mixtures, as well as convex nonspherical particles, e.g., ellipsoids, "superballs," and polyhedra. Because of their relevance to error-correcting codes and information theory, sphere packings in high-dimensional Euclidean spaces have been included as well. Some remarks are also made about packings in ͑curved͒ non-Euclidean spaces. In closing this review, several basic open questions for future research to consider have been identified.

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“…One of the main applications of this problem outside mathematics is material science, in which one considers locally rigid packings of solid bodies and nanoparticles (see, for example, [3,14,20]). Note also that most configurations of physical particles that define the minimum of potential energy are also locally rigid.…”

Section: Introductionmentioning

confidence: 99%

Extremal problems of circle packings on a sphere and irreducible contact graphs

Musin

Tarasov

2015

Proc. Steklov Inst. Math.

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Recently, we have enumerated (up to isometry) all locally rigid packings of congruent circles (spherical caps) on the unit sphere with the number of circles N < 12. This problem is equivalent to the enumeration of irreducible spherical contact graphs. In this paper, we show that using the list of irreducible contact graphs, one can solve various problems on extremal packings such as the Tammes problem for the sphere and projective plane, the problem of the maximum kissing number in spherical packings, Danzer's problems, and other problems on irreducible contact graphs.

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Densest local sphere-packing diversity: General concepts and application to two dimensions

Cited by 16 publications

References 26 publications

Perspective: Basic understanding of condensed phases of matter via packing models

Perspective: Basic understanding of condensed phases of matter via packing models

Jammed hard-particle packings: From Kepler to Bernal and beyond

Extremal problems of circle packings on a sphere and irreducible contact graphs

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