Dionysian Hard Sphere Packings Are Mechanically Stable at Vanishingly Low Densities

High strength-to-weight ratio materials can be constructed by either maximizing strength or minimizing weight. Tensegrity structures and aerogels take very different paths to achieving high strength-to-weight ratios but both rely on internal tensile forces. In the absence of tensile forces, removing material eventually destabilizes a structure. Attempts to maximize the strength-to-weight ratio with purely repulsive spheres have proceeded by removing spheres from already stable crystalline structures. This results in a modestly low density and a strength-to-weight ratio much worse than can be achieved with tensile materials. Here, we demonstrate the existence of a packing of hard spheres that has asymptotically zero density and yet maintains finite strength, thus achieving an unbounded strength-to-weight ratio. This construction, which we term Dionysian, is the diametric opposite to the Apollonian sphere packing which completely and stably fills space. We create tools to evaluate the stability and strength of compressive sphere packings. Using these we find that our structures have asymptotically finite bulk and shear moduli and are linearly resistant to every applied deformation, both internal and external. By demonstrating that there is no lower bound on the density of stable structures, this work allows for the construction of arbitrarily lightweight high-strength materials.

Figure 1

The construction of a Dionysian packing in two and three dimensions. Left: (I) A row of $n=5$ circles $a$ (purple) lie on a strictly convex curve $\mathcal{C}$ such that each circle kissing its neighbors. (II) A row of $n=5$ circles $b$ (orange and yellow) are placed such that they kiss two circles $a$ from below and a circle $b$ on either side. The rightmost $b$ circle is constrained such that its center lies on the vertical line tangent to the rightmost $a$ circle. (III) A row of $n-1=4$ circles $c$ (blue) lie on a horizontal line and kiss two $b$ circles above. (IV) A bridge is formed by reflecting the circles about the dotted lines of symmetry. Three bridges are combined and their centers are filled as shown (gray). Because of the periodic boundary conditions, each of the bridges wraps around the unit cell to contact the central gray spheres twice such that each unit cell contains six half-bridges or three full bridges. The resulting packing, which is jammed and shear stable, has a very low density and is a Dionysian packing in the limit as $n\to \infty$. Right: A three dimensional mechanically stable packing at arbitrarily low densities. Such a construction contains the same three types of spheres as in the two dimensional analog but with additional symmetries and an entirely unrelated set of spheres filling the void region (gray). The three dimensional Dionysian packing has a much narrower set of convex curves $\mathcal{C}$ for which overlaps do not occur (as detailed in the Supplemental Material [16]). This requires a much more subtle curvature of $\mathcal{C}$ which is not apparent to the naked eye in this figure.

Figure 2

Top right inset: demonstration of the definition of a gap for a circle. The $b$ circles, indexed by $i$, oscillate in size and are separated into two categories labeled by squares and triangles. Bottom left inset: The gap value for both square and triangular marked spheres asymptotes in two and three dimensions. When the asymptotic gap value is subtracted, the gap sizes follow a power law of $N^{-1}$ as they reach their respective asymptotic values.

Figure 3

The dimensionless bulk $K$ and shear $G$ moduli per sphere for Dionysian and amorphous packings in a unit cell as a function of the number of spheres $N$. The green line represents a two dimensional triangular packing, the magenta line represents a three dimensional FCC packing, and red and blue represent two dimensional and three dimensional packings, respectively. The dashed curves with open symbols represent $G_{110}$, the shear modulus in direction (1,1,0), whereas the solid curves with closed symbols represent $G_{100}$. The results are exact for the Dionysian packings and crystals. For the amorphous systems, sufficiently many systems were sampled to make the standard error bars smaller than the plot markers. In the limit of large $N$, the bulk modulus per sphere asymptotes to a positive value in two and three dimensions for all of the systems. The shear modulus for crystals and Dionysian packings plateaus for large $N$ indicating that these remain very stiff. On the other hand, the amorphous packings have a shear modulus that decreases like $1/N$ [22].