Maximally random jammed packings of Platonic solids: Hyperuniform long-range correlations and isostaticity

Abstract

We generate maximally random jammed (MRJ) packings of the four nontiling Platonic solids (tetrahedra, octahedra, dodecahedra, and icosahedra) using the adaptive-shrinking-cell method [S. Torquato and Y. Jiao, Phys. Rev. E 80, 041104 (2009)]. Such packings can be viewed as prototypical glasses in that they are maximally disordered while simultaneously being mechanically rigid. The MRJ packing fractions for tetrahedra, octahedra, dodecahedra, and icosahedra are, respectively, 0.763±0.005, 0.697±0.005, 0.716±0.002, and 0.707±0.002. We find that as the number of facets of the particles increases, the translational order in the packings increases while the orientational order decreases. Moreover, we show that the MRJ packings are hyperuniform (i.e., their infinite-wavelength local-number-density fluctuations vanish) and possess quasi-long-range pair correlations that decay asymptotically with scaling r(-4). This provides further evidence that hyperuniform quasi-long-range correlations are a universal feature of MRJ packings of frictionless particles of general shape. However, unlike MRJ packings of ellipsoids, superballs, and superellipsoids, which are hypostatic, MRJ packings of the nontiling Platonic solids are isostatic. We provide a rationale for the organizing principle that the MRJ packing fractions for nonspherical particles with sufficiently small asphericities exceed the corresponding value for spheres (∼0.64). We also discuss how the shape and symmetry of a polyhedron particle affects its MRJ packing fraction.