Densest Local Structures of Uniaxial Ellipsoids

Connecting the collective behavior of disordered systems with local structure on the particle scale is an important challenge, for example, in granular and glassy systems. Compounding complexity, in many scientific and industrial applications, particles are polydisperse, aspherical, or even of varying shape. Here, we investigate a generalization of the classical kissing problem in order to understand the local building blocks of packings of aspherical grains. We numerically determine the densest local structures of uniaxial ellipsoids by minimizing the Set Voronoi cell volume around a given particle. Depending on the particle aspect ratio, different local structures are observed and classified by symmetry and Voronoi coordination number. In extended disordered packings of frictionless particles, knowledge of the densest structures allows us to rescale the Voronoi volume distributions onto the single-parameter family of $k$-Gamma distributions. Moreover, we find that approximate icosahedral clusters are found in random packings, while the optimal local structures for more aspherical particles are not formed.

Popular Summary

Disordered materials such as sand piles pose an important challenge to physicists because their grains pack together in a bewildering variety of local motifs, unlike ordered crystals. Mathematicians have answered the question of how many spherical particles can simultaneously touch a central sphere. The solution to this so-called “kissing problem” is exemplified by the icosahedral cluster with 12 neighbor spheres. However, even this simple packing problem (ignoring gravity) remains unsolved for aspherical grains. Here, we identify the densest-packing motifs formed by ellipsoidal particles and analyze their appearance as building blocks of extended granular packings.

In order to generalize the kissing problem to aspherical particles, we optimize the local packing density, defined via the Voronoi cell. This work can be analogously thought of as finding the minimum in an energy landscape in which the Voronoi cell volume corresponds to the energy. We numerically determine the densest local structures, and we find that asphericity increases both the local packing density and the number of neighbors. Surprisingly, many of the densest structures exhibit a high degree of symmetry. Moreover, in random close-packed ellipsoids, variations on these densest motifs appear more frequently than predicted by current theories. Our formulation of the kissing problem can be applied to arbitrarily shaped particles and even mixtures of particles with different sizes and shapes. Such materials arise naturally in many application areas such as geology and industry, where aspherical particles are the norm rather than an exception.

We expect that our findings can be applied to refine physicists’ theories of granular systems, as well as other disordered systems, such as glasses, random networks, and complex fluids.

Figure 1

Densest local structures of ellipsoids. Top row: The 12-neighbor icosahedral cluster of spheres, colored to preserve a threefold rotational axis and three twofolds (left picture), and colored to preserve a fivefold axis plus five twofolds (right picture). Bottom row: Prolate 15-neighbor and oblate 14-neighbor structures generalizing the above. Black or white belts mark the equators of the particles. The red line is the central particle’s distinguished axis. Particles of the same color are related by symmetry operations. Boxes label the number of particles in the respective rings.

Figure 2

Dense local structures of ellipsoids: Local packing fraction of the central Set Voronoi cell vs the aspect ratio $\alpha$ of the particles. We identify several branches of candidate structures, which are distinguished by the number and symmetry of Voronoi neighbors. The plot symbols mark the number of neighbors ($N=12$: filled circle, $N=14$: filled triangle, $N=15$: filled downward triangle, $N=17$: downward triangle), while different line colors mark a change in symmetry. The solid line at the bottom of the graph represents the densest-known crystal structure, and “ico” is the icosahedral cluster. Usually, in the optimal structures, the contact number $C$ is equal to the number of Voronoi neighbors $N$. One exception is the oblate $N=14$ structure, which loses contacts around $\alpha \approx 0.85$ (see kink). A detailed description of the structures is given in the text.

Figure 3

Variants of the icosahedral cluster for slightly aspherical ellipsoids. Left picture: Neighbors squashed in the normal direction, leading to oblate ellipsoids. Right picture: Neighbors expanded in the tangential direction, leading to prolate ellipsoids. The center plot magnifies Fig. 2.

Figure 4

Evolution of the densest structural motifs with aspect ratio $\alpha$. In addition to the number of particles in each ring (color-coded boxes), we give the point group for each structure. The oblate structures marked $N=15$ and $N=14$ have no nontrivial symmetries. The 1-5-5-1 structures (gray) never exceed the 3-3-3-3.

Figure 5

(a) Distributions of the rescaled variable $({\varphi }_l-{\varphi }_g)/\sigma$. For reference, a Gaussian distribution is shown (black dotted curve). (b) Behavior of the standard deviation of local packing fractions $\sigma$. The gray points are reference data from Ref. [15].

Figure 6

Distribution of Voronoi neighbor count and local packing fraction. The bold vertical bars indicate the densest local structures for each $N$ (see Fig. 2). The vertical dashed line marks the densest known crystalline packing ${\varphi }_{\mathrm{cry}}$ [7], and the triangular tics on the axes mark the global packing fraction and mean Voronoi neighbor count of the dense disordered packings. Bottom row: Marginal distributions of the above for different $N$. Top right plot: Radial distribution of the supracrystalline clusters, ${\varphi }_l>{\varphi }_{\mathrm{cry}}$.

Figure 7

Distribution of rescaled Set Voronoi cell volumina in random ellipsoid packings. (a) Data for different aspect ratios $\alpha$, compared with two $k$-Gamma distributions with $k=13$ and $k=17$. The inset shows the same data on a linear scale. (b) Residual after subtracting a best-fit $k$-Gamma distribution (see text).

Figure 8

Morphometric analysis of the dense random packings of $\alpha =0.8$ ellipsoids. The horizontal axis shows the local packing fraction of the individual Set Voronoi cells. The vertical axis shows the morphometric distance $\mathrm{\Delta }$ of each cell from the cell of two of our optimal (densest) structures. Left panel: Clusters in the packing approach the 3-3-3-3 structure (green bullet). Right panel: There is no trend towards the 1-6-6-1 structure (green bullet).