Optimal packings of superballs

Dense hard-particle packings are intimately related to the structure of low-temperature phases of matter and are useful models of heterogeneous materials and granular media. Most studies of the densest packings in three dimensions have considered spherical shapes, and it is only more recently that nonspherical shapes (e.g., ellipsoids) have been investigated. Superballs (whose shapes are defined by ${\left|x_1\right|}^{2p}+{\left|x_2\right|}^{2p}+{\left|x_3\right|}^{2p}\le 1$) provide a versatile family of convex particles $\left(p\ge 0.5\right)$ with both cubic-like and octahedral-like shapes as well as concave particles $\left(0<p<0.5\right)$ with octahedral-like shapes. In this paper, we provide analytical constructions for the densest known superball packings for all convex and concave cases. The candidate maximally dense packings are certain families of Bravais lattice packings (in which each particle has 12 contacting neighbors) possessing the global symmetries that are consistent with certain symmetries of a superball. We also provide strong evidence that our packings for convex superballs $\left(p\ge 0.5\right)$ are most likely the optimal ones. The maximal packing density as a function of $p$ is nonanalytic at the sphere point $\left(p=1\right)$ and increases dramatically as $p$ moves away from unity. Two more nontrivial nonanalytic behaviors occur at $p_c^{\ast }=1.1509\dots$ and $p_o^{\ast }=\text{ln}3/\text{ln}4=0.7924\dots$ for “cubic” and “octahedral” superballs, respectively, where different Bravais lattice packings possess the same densities. The packing characteristics determined by the broken rotational symmetry of superballs are similar to but richer than their two-dimensional “superdisk” counterparts [Y. Jiao et al., Phys. Rev. Lett. 100, 245504 (2008)] and are distinctly different from that of ellipsoid packings. Our candidate optimal superball packings provide a starting point to quantify the equilibrium phase behavior of superball systems, which should deepen our understanding of the statistical thermodynamics of nonspherical-particle systems.

Figure 1

Superdisks with different values of the deformation parameter $p$.

Figure 2

(Color online) Superballs with different values of the deformation parameter $p$.

Figure 3

(Color online) Dense packings of (a) superdisks and (b) superballs with $p=1.5$ generated via the Donev-Torquato-Stillinger algorithm with the growth rate $\gamma ={10}^{-5}$. The packing densities are ${\varphi }_a=0.9123\dots$ and ${\varphi }_b=0.7072\dots$, respectively. One can see that the superdisk packing is nearly completely crystallized, while the superball packing shows no significant signs of crystallization. The white “chord” in each superdisk indicates one of its symmetry axis.

Figure 4

(Color online) Candidate optimal packings of superballs. (a) The ${\mathbb{C}}_0$-lattice packing of superballs with $p=1.8$. (b) The ${\mathbb{C}}_1$-lattice packing of superballs with $p=2.0$. (c) The ${\mathbb{O}}_0$-lattice packing of superballs with $p=0.8$. (d) The ${\mathbb{O}}_1$-lattice packing of superballs with $p=0.55$.

Figure 5

(Color online) Illustrations of certain rotational symmetries of the cube and regular octahedron. (a) Twofold rotational symmetry of a cube (axis shown coincides with a face diagonal). (b) Threefold rotational symmetry of a cube (axis shown coincides with a body diagonal). (c) Fourfold rotational symmetry of a regular octahedron.

Figure 6

(Color online) (a) The stretched square-lattice layer of spheres. (b) The stretched lattice layer of superballs viewed from the [110] direction.

Figure 7

(Color online) The ${\mathbb{C}}_0$ lattice for superballs in the cubic regime viewed from the [110] direction. The structure remains invariant when it is rotated $180°$ about an axis in the [110] direction, showing its twofold rotational symmetry.

Figure 8

(Color online) Density versus deformation parameter $p$ for the packings of convex superballs. Insert: around $p_c^{\ast }=1.1509\dots$, the two curves are almost locally parallel to each other.

Figure 9

(Color online) The ${\mathbb{C}}_1$ lattice for superballs in the cubic regime viewed from the [111] direction. The thin dark lines show the three orthogonal directions, which clearly exhibits the threefold rotational symmetry of the structure.

Figure 10

(Color online) Contacting points of a (a) sphere and (b) a schematic superball (shown as a cube). The view is along the [111] direction. The surface contacting points will continuously move along the surface diagonals as the $p$ value changes from unity to $\infty$. Every contacting point has a symmetric image about the center of the superball and their distributions have threefold rotational symmetry.

Figure 11

(Color online) The ${\mathbb{C}}_1$-lattice packing of superballs obtained by stacking superball chains. (a) The chain as depicted in the text. (b) The layer constructed by stacking the chains. (c) The packing obtained by stacking the layers.

Figure 12

(Color online) (a) Face-centered cubic packing of spheres viewed as a laminate of face-centered square layers from the [001] direction. (b) Similar laminate of face-centered square planar layers of superballs in the octahedral regime viewed from the [001] direction.

Figure 13

(Color online) Projections of a ${\mathbb{O}}_1$-lattice subpacking on three orthogonal coordinate planes. (a) Projection on the (001) plane. (b) Projection on the (010) plane. (c) Projection on the (100) plane.

Figure 14

(Color online) The ${\mathbb{O}}_1$-lattice packing of superballs obtained by stacking superball chains. (a) The superball chain. (b) The layer constructed by stacking the chains. (c) The packing obtained by stacking the layers.

Figure 15

(Color online) Density versus deformation parameter $p$ for the packings of concave superballs. Inset: a concave superball with $p=0.1$, which will become a three-dimensional cross at the limit $p\to 0$.

Figure 16

(Color online) A superdisk circumscribed by a hexagon formed from three pairs of parallel lines.