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

Adam B. Hopkins, Frank H. Stillinger, and Salvatore Torquato
Phys. Rev. E 81, 041305 – Published 20 April 2010

Abstract

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. The knowledge of Rmin(N) in d-dimensional Euclidean space Rd 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 Rd. 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 an 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.

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  • Received 8 February 2010

DOI:https://doi.org/10.1103/PhysRevE.81.041305

©2010 American Physical Society

Authors & Affiliations

Adam B. Hopkins and Frank H. Stillinger

  • Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA

Salvatore Torquato

  • Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, Program in Applied and Computational Mathematics, Princeton Center for Theoretical Science, Princeton University, Princeton, New Jersey 08544, USA and School of Natural Sciences, Institute for Advanced Study, Princeton, New Jersey 08544, USA

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Vol. 81, Iss. 4 — April 2010

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