Hard Particle Packings

Densest Local Packings


The Densest Local Packings of Identical Spheres in Three Dimensions

The Densest Local Packings of Identical Disks in Two Dimensions

The Densest Local Packings of Spheres in Any Dimension and the Golden Ratio


Densest Local Packings

A rich but simple model for objects (e.g., molecules) interacting via a strong repulsive radial potential is a packing of identical nonoverlapping spheres. However, there are important knowledge gaps concerning critical spatial arrangements of finite numbers of spheres. These knowledge gaps are due to the complexity of the N-body interactions in a packing spheres, where in maximal density packings, for example, the spatial position of any single sphere depends generally on positions of the other N-1. The densest packings of N d-dimensional spheres with centers within radius Rmin(N) of the center of a fixed same-size sphere were until recently generally not known except for very small numbers of d and N. We call this latter problem the densest local packing (DLP) problem.


The Densest Local Packings of Identical Spheres in Three Dimensions

--- A comparison of Rmin(N) to RBar(N), where RBar(N) is the densest local packing from among subsets of Barlow packings, where the Barlow packings, including the well-known fcc and bcc close-packed sphere packings, are the densest infinite packings of identical spheres in three dimensions.


We have used a novel algorithm combining nonlinear programming methods with a random search of configuration space to find the densest local packings of spheres in three-dimensional Euclidean space. Our results reveal a wealth of information about packings of spheres, including counterintuitive results concerning the physics of dilute mixtures of spherical solute particles in a solvent composed of same-size spheres and about the presence of unjammed spheres (rattlers) in the densest local structures. Comparing the densest local packings to minimal-energy configurations of points interacting with short-range repulsive pair potentials, e.g., 12−6 Lennard-Jones, we find that they are in general completely different, a result that has possible implications for nucleation theory. We also find a variety of unusually dense, perfectly symmetric densest local packings, and employ our knowledge of the Rmin(N) to construct a new realizability condition on pair correlation functions for packings of any number of spheres. Our results are published in Densest Local Packing Diversity. II. Application to Three Dimensions ( Phys. Rev. E 83 , 011304 (2011)) by A. B. Hopkins, F. H. Stillinger, and S. Torquato.

--- Similarity metric comparison between subsets of Barlow packings and densest local packings of spheres in three dimensions.


Comparing the densest local packings to subsets of Barlow packings using a similarity metric (with 1.0 being the most similar to a Barlow packing 0.0 the least similar), we find that many densest local packings are very similar to Barlow packings and many are not. However, almost always, the densest local packings are most similar to subsets of Barlow packings with the highest symmetry and least number of radial coordination shells, i.e., the fcc Barlow packing. This is due to the fact that the spheres in the densest local packings are also distributed into only a very small number of radial shells.

--- The three shells of the N=84 densest local packing, which has perfect tetrahedral symmetry. In the diagrams, the points are the sphere centers and the lines drawn between them indicate contact between spheres within a single shell. The spheres in each shell are all the exact same distance from the central sphere, which is not depicted here.

Shell1 Shell2 Shell3

We also find many cases of perfect symmetry among the densest local packings, including rotational and reflection symmetries. These cases include point group symmetry for the sphere centers corresponding to two-fold, three-fold and four-fold perfect rotational symmetry, as well as mirror reflection symmetry, octahedral symmetry, and tetrahedral symmetry. Notably, we do not find any examples of perfect five-fold (or icosahedral) rotational symmetry, though there are a great many densest local packings that have imperfect icosahedral symmetry, to within a tolerance of a few percent of a sphere’s diameter.


The Densest Local Packings of Identical Disks in Two Dimensions

--- The densest local packings of N disks with centers within a radius Rmin(N) of a fixed central disk

DLP 15 example
N=15, point group D5h


The densest local packings of N identical nonoverlapping spheres within a radius Rmin(N) of a fixed central sphere of the same size are obtained using a nonlinear programming method operating in conjunction with a stochastic search of configuration space. Knowledge of Rmin(N) in d-dimensional Euclidean space allows for the construction both of a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. In this paper, we focus on the two-dimensional circular disk problem. We find and present the putative densest packings and corresponding Rmin(N) for selected values of N up to N = 348 and use this knowledge to construct such a realizability condition and upper bound. We additionally analyze the properties and characteristics of the maximally dense packings, finding significant variability in their symmetries and contact networks, and that the vast majority differ substantially from the triangular lattice even for large N. Our work has implications for packaging problems, nucleation theory, and surface physics.

N=24 N=40 N=162
    N=24, point group D3h     N=40, point group D2h       N=162, point group Ci

See A. B. Hopkins, F. H. Stillinger and S. Torquato, Densest Local Sphere-Packing Diversity: General concepts and application to two dimensions, Phys. Rev. E 81, 041305 (2010) for the paper. The packing details, including images and coordinates of presumed optimal packings, can be found here.


The Densest Local Packings of Spheres in Any Dimension and the Golden Ratio

The optimal spherical code problem involves 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. In Spherical Codes, Maximal Local Packing Density, and the Golden Ratio ( J. Math. Phys. 51 , 043302 (2010)) by A. B. Hopkins, F. H. Stillinger, and S. Torquato, we prove that in any dimension, all solutions between unity and the golden ratio to the optimal spherical code problem for N spheres are also solutions to the corresponding DLP problem. It follows that for any packing of nonoverlapping spheres of unit diameter, a spherical region of radius less than or equal to the golden ratio centered on an arbitrary sphere center cannot enclose a number of sphere centers greater than one more than the number that than can be placed on the region’s surface.


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