Athermal jamming of soft frictionless Platonic solids

Abstract

A mechanically based structural optimization method is utilized to explore the phenomena of jamming for assemblies of frictionless Platonic solids. Systems of these regular convex polyhedra exhibit mechanically stable phases with density substantially less than optimal for a given shape, revealing that thermal motion is necessary to access high-density phases. We confirm that the large system jamming threshold of 0.623 ± 0.003 for tetrahedra is consistent with experiments on tetrahedral dice. Also, the extremely short-ranged translational correlations of packed tetrahedra observed in experiments are confirmed here, in contrast with those of thermally simulated glasses. Although highly ordered phases are observed to form for small numbers of cubes and dodecahedra, the short correlation length scale suppresses ordering in large systems, resulting in packings that are mechanically consistent with "orientationally disordered" contacts (point-face and edge-edge contacts). Mild nematic ordering is observed for large systems of cubes, whereas angular correlations for the remaining shapes are ultrashort ranged. In particular the angular correlation function of tetrahedra agrees with that recently observed experimentally for tetrahedral dice. Power-law scaling exponents for energy with respect to distance from the jamming threshold exhibit a clear dependence on the "highest-order" percolating contact topology. These nominal exponents are 6, 4, and 2 for configurations having percolating point-face (or edge-edge), edge-face, and face-face contacts, respectively. Jamming contact number is approximated for small systems of tetrahedra, icosahedra, dodecahedra, and octahedra with order and packing representative of larger systems. These Platonic solids exhibit hypostatic behavior, with average jamming contact number between the isostatic value for spheres and that of asymmetric particles. These shapes violate the isostatic conjecture, displaying contact number that decreases monotonically with sphericity. The common symmetry of dual polyhedra results in local translational structural similarity. Systems of highly spherical particles possessing icosahedral symmetry, such as icosahedra or dodecahedra, exhibit structural behavior similar to spheres, including jamming contact number and radial distribution function. These results suggest that although continuous rotational symmetry is broken by icosahedra and dodecahedra, the structural features of disordered packings of these particles are well replicated by spheres. Octahedra and cubes, which possess octahedral symmetry, exhibit similar local translational ordering, despite exhibiting strong differences in nematic ordering. In general, the structural features of systems with tetrahedra, octahedra, and cubes differ significantly from those of sphere packings.

Variable-cell method for stress-controlled jamming of athermal, frictionless grains

“…7) tetrahedra to density [14]. The extrapolated minimal density (0.625 [14]) is very similiar to the jamming threshold density of athermal, soft tetrahedra (0.62-0.64 [6,7]). …”

“…[21][22][23], enable arbitrary shape deformations that are inherent to the definition of strict jamming [1]. Based on such an approach, Jiao and Torquato [11] asserted that cell-shape variations account for the increased jamming threshold density (0.763) and face-face contact number (> 2 per grain) relative to our previous results on systems jammed in fixed-shape cells (0.62-0.64 jamming threshold [6,7] and 1 face-face contacts per grain [7]). …”

“…Jamming has recently been simulated with similar methods for systems of various grain shapes, including ellipses [5], Platonic solids [6][7][8], arbitrarily-shaped polyhedral grains [9], and irregular, polydispersed grains [10]. While soft and hard systems of spheres have exhibited consistency among their jammed densities [2][3][4], disparities have been observed between simulated soft and hardtetrahedra packings [6,8,11] and those of experiments [12,13]. Attempts have also been made to correlate the fraction of 'face-jointed' (a stricter condition than a face- * Electronic address: kyle.c.smith@gmail.com † Electronic address: tsfisher@purdue.edu face contact, as in Ref.…”

“…In addition, recent experiments suggest that metastable jammed states can form under shear [16]. Shape modulation of the periodic cell was an essential part of the densification protocol in hard-tetrahedra systems [11], while cell shape was fixed in previous investigations of soft tetrahedra [6,7].…”

Internal friction and absence of dilatancy of packings of frictionless polygons

“…Moreover, if assembled under an isotropic state of stress, rigid frictionless grains stabilize with positions realizing a local minimum of volume in the configuration space [13,17], and as a consequence it is a common procedure to set μ s = 0 in numerical simulations in order to produce disordered packings of maximum density, for spheres [17,21], or other particle shapes, in two [22,23] and three dimensions [20,[24][25][26][27]. Such frictionless isotropic packings have recently been studied for their specific, barely rigid structure [15,24], or simply used as convenient reference initial states [21][22][23], as they are relatively well reproducible and exhibit little dependence on assembling procedure [17].…”