### Abstract

The determination of the maximal packing arrangements of two-dimensional, binary hard disks of radii R_{S} and R_{L} (with R_{S} ≤ R_{L}) for sufficiently small R_{S} amounts to finding the optimal arrangement of the small disks within a tricusp: the nonconvex cavity between three close-packed large disks. We present a particle-growth Monte Carlo algorithm for the generation of geometric packings of equi-sized hard disks within such a tricusp. The first 19 members of an infinite sequence of maximal density structures thus produced are reported. In addition, the Monte Carlo algorithm is applied to the geometric packing of disks within a flat-sided equilateral triangle and compared to published results for that packing problem. We perform an analysis of geometric properties of the packings, e.g. packing fractions and symmetries of structures confined to both containers. Interestingly, we find a non-monotonic increase in the packing fraction with increasing number of disks packed within both the flat-sided triangle and tricusp. It is important to note that for disk packings within a flat-sided equilateral triangle, this non-monotonic behavior of the packing fraction had not been reported in previously published works. For the flat-sided equilateral triangle, local maxima occur at the triangular integers N_{S} = 1,3,6,10,15⋯, as well as N_{S} = 12, where N_{S} is the number of disks in each packing. However, local maxima for packings within the tricusp exist at N_{S} = 1,3,6,10,18⋯. Finally, we analyze the asymptotic approach to the upper bound on the packing fraction of the infinite sequence of maximal structures of disks confined to the tricusp.

Original language | English (US) |
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Pages (from-to) | 428-446 |

Number of pages | 19 |

Journal | Physica A: Statistical Mechanics and its Applications |

Volume | 342 |

Issue number | 3-4 |

DOIs | |

State | Published - Nov 1 2004 |

### All Science Journal Classification (ASJC) codes

- Statistics and Probability
- Condensed Matter Physics

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## Cite this

*Physica A: Statistical Mechanics and its Applications*,

*342*(3-4), 428-446. https://doi.org/10.1016/j.physa.2004.05.082