Existence of isostatic, maximally random jammed monodisperse hard-disk packings

Significance Disordered particle packings are ubiquitous in all areas of science. Although disordered jammed packings of hard spheres are readily observed in 3D, the story is quite different for disks. Most 2D packing protocols tend to produce highly ordered disk arrangements, suggesting that truly disordered jammed disk packings might not exist in 2D. The maximally random jammed (MRJ) configurations we observe have a complete lack of crystallinity and are distinct from the typical jammed configuration that is most probable upon rapid compression. A protocol-independent geometric-structure approach allows us to create, identify, and analyze these previously elusive packings. Our results shed new light on the nature of randomness, which is an issue that arises across the physical, mathematical, and biological sciences. We generate jammed packings of monodisperse circular hard-disks in two dimensions using the Torquato–Jiao sequential linear programming algorithm. The packings display a wide diversity of packing fractions, average coordination numbers, and order as measured by standard scalar order metrics. This geometric-structure approach enables us to show the existence of relatively large maximally random jammed (MRJ) packings with exactly isostatic jammed backbones and a packing fraction (including rattlers) of ϕ=0.826. By contrast, the concept of random close packing (RCP) that identifies the most probable packings as the most disordered misleadingly identifies highly ordered disk packings as RCP in 2D. Fundamental structural descriptors such as the pair correlation function, structure factor, and Voronoi statistics show a strong contrast between the MRJ state and the typical hyperstatic, polycrystalline packings with ϕ≈0.88 that are more commonly obtained using standard packing protocols. Establishing that the MRJ state for monodisperse hard disks is isostatic and qualitatively distinct from commonly observed polycrystalline packings contradicts conventional wisdom that such a disordered, isostatic packing does not exist due to a lack of geometrical frustration and sheds light on the nature of disorder. This prompts the question of whether an algorithm may be designed that is strongly biased toward generating the monodisperse disk MRJ state.

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