Dense crystalline packings of ellipsoids

Abstract

An ellipsoid, the simplest non-spherical shape, has been extensively used as models for elongated building blocks for a wide spectrum of molecular, colloidal and granular systems [1][2][3][4] . Yet the densest packing of congruent hard ellipsoids, which is intimately related to the high-density phase of many condensed matter systems, is still an open problem. We discover a novel dense crystalline packing of ellipsoids containing 24 particles with a quasi-square-triangular (SQ-TR) tiling arrangement, whose packing density ɸ exceeds that of the SM2 crystal 5 for aspect ratios α in (1.365, 1.5625), attaining a maximal ɸ ≈ 0.75806… at α = 93/64. We show that SQ-TR phase is thermodynamically stable at high densities over the aforementioned α range and report a novel phase diagram for self-dual ellipsoids. The discovery of SQ-TR crystal suggests novel organizing principles for non-spherical particles and self-assembly of colloidal systems.Dense packings of hard particles have been widely used as models for a variety of condensed matter systems, including glasses, crystals, heterogeneous materials, and granular media [1][2][3][4] . The optimal (maximally dense) packing arrangement is also intimately related to the crystalline phase of the associated particle system and its high-density phase behavior 3 . The pursuit of the optimal packing for given particle shape has long been an intriguing and challenging problem in discrete geometry 6 . Indeed, it took almost four centuries to prove the famous Kepler's conjecture 6 posed in 1611 that the densest packing of spheres is the face-centered cubic (FCC) lattice with a packing density ɸ = π/√18 ≈ 0.7405 7 . For congruent nonspherical particles that do not tile three-dimensional (3D) Euclidean space R 3 , the optimal packing problem is only solved for infinite cylinders 8 and rhombic dodecahedra with a clipped corner 9 . So far no rigorous mathematical theory has been put forth for constructing or proving optimal packings of nonspherical particles. Recently, the discovery of the candidate optimal packings is significantly advanced via computer simulations. [13][14][15][20][21][22][23][24] , to name but a few. No counterexample of the Ulam's conjecture 25 that all convex shapes in R 3 pack better than spheres has been found yet.An ellipsoid, the simplest non-spherical shape, can be obtained from a sphere through an affine transformation. The ratios of the semiaxes of an ellipsoid can be expressed as a:b:c=α: β :1 (a≥b≥c), where α is the aspect ratio, and β (0≤β≤1) is the skewness. When β = 1/2, the ellipsoid is neither prolate (β=0) nor oblate (β=1), and is referred to as the self-dual ellipsoid. In this work, we discover via Monte Carlo simulations a novel crystalline packing of self-dual ellipsoids (β=1/2) that is denser than the corresponding SM2 packing for aspect ratios α in (1.365, 1.5625), attaining a maximal φ ≈ 0.75806… at α = 93/64. The smallest periodic unit cell contains 24 ellipsoids with a quasi-square-triangle tiling arrangement. Henceforth, we refer to...

Dense Disordered Jammed Packings of Hard Spherocylinders with a Low Aspect Ratio: A Characterization of Their Structure

“…Ellipsoids, both uniaxial and biaxial: − Arguably, they are the most direct generalization of hard spheres since an ellipsoid is an affine transformation of a sphere; yet, the densest-known packings of hard ellipsoids are significantly more complicated and denser ,,, than the packings that result from applying the same affine, ϕ-invariant, transformation to the densest packings of hard spheres.…”

“…Numerous hard-nonspherical particles whose densest-known and dense disordered jammed packings have been investigated are actually familiar since the school years: Ellipsoids, both uniaxial and biaxial: − Arguably, they are the most direct generalization of hard spheres since an ellipsoid is an affine transformation of a sphere; yet, the densest-known packings of hard ellipsoids are significantly more complicated and denser ,,, than the packings that result from applying the same affine, ϕ-invariant, transformation to the densest packings of hard spheres. (Circular right) Cylinders: The only hard-nonspherical particle for which a mathematical proof of the densest packings was released, and an experimental measurement of the probability distribution of the number of contacts per hard particle in dense disordered jammed packings was reported The Platonic polyhedra as well as the Archimedean polyhedra: − Out of all of these hard polyhedra, the hard tetrahedron emerged for the assortment of its candidate densest-known packings − , and seemingly being the hard convex nonspherical particle that disorderedly packs most densely. ,,, …”

Dense packings of geodesic hard ellipses on a sphere

“…21 One of the prototypical cases for studying the effects of anisotropy on particle packing are systems of ellipsoids, the most simple aspherical shapes that are routinely used to model particles in a variety of soft matter, granular, and molecular systems. They are known to improve the density of jammed disordered states [22][23][24][25] and can form unusually dense crystal packings, 26,27 with numerous investigations highlighting the mechanical properties of jammed configurations of hard ellipsoids and ellipses. 19,24,[28][29][30][31][32] Additionally, ellipses were found to maximize the packing fraction of random sequential adsorption in two dimensions among the set of shapes of smoothed n-mers and spherocylinders.…”

Measure of overlap between two arbitrary ellipses on a sphere

“…appears in many different fields of research, both in experimental realizations and in numerical models used to study them. Ellipsoids appear in Gay-Berne (anisotropic Lennard-Jones) models [1] of liquid crystals as a coarse-grained replacement for the full molecular structure [2][3][4], in colloidal dispersions with an anisotropic dispersed phase [5][6][7], and in granular and jammed matter [8][9][10][11], where random and optimal packings are of particular interest [12,13]. All these examples are, however, Euclidean-yet many experimental systems call for a confinement of particles to a curved surface, often that of a sphere.…”