# Ordered and Disordered Packings

** Maximally Dense Packings **

A problem that has been a source of fascination to mathematicians and scientists for centuries is the determination of the densest arrangement(s) of particles that do not tile space and the associated maximal density. The preponderance of previous work has focused on spherical particles, but even for this simple shape the problem is notoriously difficult. Indeed, Kepler’s conjecture concerning the densest sphere packing arrangement was only proved by Hales in 2005. Much less is known about the densest packings of nonspherically shaped particles. The papers given below describes research that we have carried out on such topics.

## Dense Ellipsoid Packings

We Ref. [1], we report our discovery of the densest known crystalline packing of congruent ellipsoids (see Figure 1), which are periodic packings with two particles in the fundamental cell. This family of new packings consists of crystalline arrangements of ellipsoids with a wide range of aspect ratios whose densities are all above that of the densest Bravais lattice packing of monodisperse spheres, i.e., \( \phi_2 = 0.7405 \) for the face-centered-cubic lattice (see Figure 2). Notably, we observe a remarkably high maximal density of \( \phi_2 = 0.7707 \) for ellipsoid aspect ratios \( \alpha \gt \sqrt{3} \) and when each ellipsoid is contact with 14 neighboring particles. Our results are directly relevant to understanding the equilibrium behavior of systems of hard ellipsoids, and, in particular, the solid and glassy phases.

Figure 1: Schematic of an ellipsoid with semiaxes \(a,b\) and \(c\) (into page) with aspect ratios \( \alpha=b/a \) and \( \beta=c/a=1\) (ellipsoid of revolution). We refer to the ratio of the largest to smallest semiaxes as the “maximal aspect ratio” and denote it with \(\delta\).

Figure 2: The density of the laminate crystal packing of ellipsoids as a function of the aspect ratio \(\alpha=1\) (\beta=1). The point \( \alpha=1 \) corresponding to the FCC lattice sphere packing is shown, along with the two sharp maxima in the density for prolate ellipsoids with \(\alpha=\sqrt{3}\) and oblate ellipsoids with \(\alpha=1/\sqrt{3}\), as illustrated in the insets. The presently maximal achievable density is highlighted with a thicker line, and is constant for \(\delta\gt\sqrt{3}\). Figure and caption from Ref. [1].

- A. Donev, F. H. Stillinger, P. M. Chaikin, and S. Torquato, Unusually Dense Crystal Packings of Ellipsoid,
**Physical Review Letters**,**92**, 255506 (2004).

## Maximally Dense Superdisk Packings

Almost all studies of the densest particle packings consider convex particles. In Ref. [1], we provide exact constructions for the densest known two-dimensional packings of superdisks whose shapes are defined by \(|x_1|^{2p} + |x_2|^{2p} \lt 1 \) (see Figure 1) and thus contain a large family of both convex (\( p \gt 0.5 \)) and concave ( \(0\lt p\lt 0.5\) ) particles. Our candidate maximal packing arrangements are achieved by certain families of Bravais lattice packings (see Figure 2), and the maximal density is nonanalytic at the “circular-disk” point (\(p=1\) ) and increases dramatically as \(p\) moves away from unity. Moreover, we show that the broken rotational symmetry of superdisks influences the packing characteristics in a nontrivial way.

Figure 1: Superdisks with different deformation parameters \(p\). Figure and caption taken from Ref. [1].

Figure 2: Candidate optimal packings of superdisks with \(p=1.8\) (left), and \(p=2.0\) (right). Figure taken from Ref. [1].

## Maximally Dense Superball Packings

In Ref. [2], we provide analytical constructions for the densest known superball packings for all convex and concave cases, defined by \(|x_1|^{2p}+|x_2|^{2p}+|x_3|^{2p}\le 1\). 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 (\(p\gt 0.5\)) are most likely the optimal ones. The maximal packing density as a function of p is nonanalytic at the sphere point \(p=1\) and increases dramatically as p moves away from unity. Two more nontrivial nonanalytic behaviors occur at \(p_c=1.1509\) and \(p_o =\ln{3}/\ln{4}=0.7924\) for cubic and octahedral superballs, respectively, where different Bravais lattice packings possess the same densities.

Figure 3: Superballs with different deformation parameters \(p\). (a) \(p=0.5\), (b) \(p=0.75\), (c) \(p=1..0\), and (d) \(p=2.0\). Figure taken from Ref. [2].

Figure 4: Candidate optimal packings of superballs with \(p=1.8\) (top left), \(p=2.0\) (top right), \(p=0.8\) (bottom left), and \(p=0.55\) (bottom right). Figure taken from Ref. [2].

- Y. Jiao, F. H. Stillinger and S. Torquato, Optimal Packings of Superdisks and the Role of Symmetry,
**Physical Review Letters**,**100**, 245504 (2008). - Y. Jiao, F. H. Stillinger and S. Torquato, Optimal Packings of Superballs,
**Physical Review E**,**79**, 041309 (2009). Please see*Erratum*.

## Dense Packings of Tetrahedra, Icosahedra, Dodecahedra, and Octahedra

Figure 1: The five Platonic solids are the tetrahedron (P1), icosahedron (P2), dodecahedron (P3), octahedron (P4) and cube (P5). The 13 Archimedean solids are the truncated tetrahedron (A1), truncated icosahedron (A2), snub cube (A3), snub dodecahedron (A4), rhombicosidodecahedron (A5), truncated icosidodecahedron (A6), truncated cuboctahedron (A7), icosidodecahedron (A8), rhombicuboctahedron (A9), truncated dodecahedron (A10), cuboctahedron (A11), truncated cube (A12), and truncated octahedron (A13). The cube (P5) and truncated octahedron (A13) are the only Platonic and Archimedean solids, respectively, that tile space. Figure and caption taken from Ref. [1].

Dense particle packings have served as useful models of the structures of liquid, glassy and crystalline states of matter, granular media, heterogeneous materials and biological systems. Probing the symmetries and other mathematical properties of the densest packings is a problem of interest in discrete geometry and number theory. Prior work in this field has focused mainly on spherical particles, but very little is known about dense polyhedral packings. In Refs. [1] and [2], we formulate the generation of dense packings of polyhedra as an optimization problem by using an adaptive fundamental cell subject to periodic boundary conditions; a method termed the “adaptive shrinking cell” scheme (see Ref. [2] for details).

Figure 2: Portions of the densest packing of (a) tetrahedra obtained from our simulations, and the optimal lattice packings of the (b) icosahedra, (c) dodecahedra, and (d) octahedra to which our simulations converge. Figure and caption taken from Ref. [1].

Using a variety of multi-particle initial configurations, we find the densest known packings of the four non-tiling Platonic solids (the tetrahedron, octahedron, dodecahedron and icosahedron) in three-dimensional Euclidean space (see Figures 1 and 2). The densities of these packings are \(0.782…, 0.947…, 0.904…\) and \(0.836…\), respectively. Unlike the densest tetrahedral packing, which must not be a Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Combining the simulation results with derived rigorous upper bounds and theoretical arguments leads to the conjecture that the densest packings of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings which is the analogue of Kepler’s sphere conjecture for these solids.

## Dense Periodic Packings of Regular Tetrahedra

The determination of the densest packings of regular tetrahedra (one of the five Platonic solids) is attracting great attention as evidenced by the rapid pace at which packing records are being broken and the fascinating packing structures that have emerged. In Ref. [3], we provide the most general analytical formulation to date to construct dense periodic packings of tetrahedra with four particles per fundamental cell. This analysis results in six-parameter family of dense tetrahedron packings that includes as special cases recently discovered “dimer” packings of tetrahedra, including the densest known packings with density \(\phi=4000/4671=0.856347…\) (see Figure 3). This study strongly suggests that the latter set of packings are the densest among all packings with a four-particle basis. Whether they are the densest packings of tetrahedra among all packings is an open question, but we offer remarks about this issue. Moreover, we describe a procedure that provides estimates of upper bounds on the maximal density of tetrahedron packings, which could aid in assessing the packing efficiency of candidate dense packings.

Figure 3: In the top panel [(a) and (b)], we show the two different configurations of the densest packings of four tetrahedra (two dimers) with \( \phi=12250/14319=0.855506… \) within their corresponding rhombohedral fundamental cells of the two-parameter family. In the bottom panel [(c) and (d)], we show the two different configurations of the densest packings of four tetrahedra (two dimers) with \(\phi=4000/4671=0.856347…\) within their corresponding rhombohedral fundamental cells of the three-parameter family.

## Dense Periodic Packings of Truncated Tetrahedra

The Platonic and Archimedean polyhedra possess beautiful symmetries and arise in many natural and synthetic structures. Understanding how nonspherical particles pack is a first step toward a better understanding of how biological cells pack. Probing the symmetries and other mathematical properties of the densest packings is a problem of interest in discrete geometry and number theory. Recently, there has been a large effort devoted to finding dense packings of polyhedra. Although organizing principles for the types of structures associated with the densest polyhedron packings have been put forth, much remains to be done to find the maximally dense packings for specific shapes. In Ref. [4], we analytically construct the densest known packing of truncated tetrahedra with packing fraction \(207/208=0.995 192 …\), which is amazingly close to unity and strongly implies the optimality of the packing (see Figure 4). This construction is based on a generalized organizing principle for polyhedra that lack central symmetry. Moreover, we find that the holes in this putative optimal packing are small regular tetrahedra, leading to a new tiling of space by regular tetrahedra and truncated tetrahedra. We also numerically study the equilibrium melting properties of what apparently is the densest packing of truncated tetrahedra as the system undergoes decompression. Our simulations reveal two different stable crystal phases, one at high densities and the other at intermediate densities, as well as a first-order liquid-crystal phase transition.

Figure 4: Structures along the continuous deformation path that bring the Conway-Torquato packing of truncated tetrahedra with \(\phi=23/24\) to the putative optimal packing with \(\phi=207/208\). The associated deformation parameter \(\gamma\) for each packing is (a) \(\gamma=0\) (the Conway-Torquato packing), (b) \(\gamma=2/27\), (c) \(\gamma=4/27\), and (d) \(\gamma=2/9\) (our current putative optimal packing). Figure and caption taken from Ref. [4].

This research has garnered significant attention: our work in Ref. [1] was the featured cover story of the August 13th, 2009 issue of *Nature* and was the subject of multiple discussions in the popular press; including *Nature News and Views*, *Nature Podcasts*, *Nature: Making the Paper*, *Princeton News*, *American Mathematical Society*, and a *Wolfram Demonstration Project*. Additionally, our work in Ref. [4] was featured on the cover of the October 21st, 2011 issue of *The Journal of Chemical Physics*.

- S. Torquato and Y. Jiao. Dense Packings of the Platonic and Archimedean Solids,
**Nature**,**460**, 876 (2009). Please see*Erratum*and*Corrigendum*. - S. Torquato and Y. Jiao. Dense Packings of Polyhedra: Platonic and Archimedean Solids,
**Physical Review E**,**80**, 041104 (2009). Please see*Erratum*. - S. Torquato and Y. Jiao, Exact Constructions of a Family of Dense Periodic Packings of Tetrahedra,
**Physical Review E**,**81**, 041310 (2010). - Y. Jiao and S. Torquato, Communication: A Packing of Truncated Tetrahedra That Nearly Fills All of Space and its Melting Properties,
**Journal of Chemical Physics**,**135**, 151101 (2011).

## Dense Periodic Packings of Tori

Dense packings of nonoverlapping bodies in three-dimensional Euclidean space \(\mathbb{R}^3\) are useful models of the structure of a variety of many-particle systems that arise in the physical and biological sciences. In Ref. [1], we investigate the packing behavior of congruent ring tori in \(\mathbb{R}^3\), which are multiply connected nonconvex bodies of genus 1, as well as horn and spindle tori. Specifically, we analytically construct a family of dense periodic packings of unlinked tori guided by the organizing principles originally devised for simply connected solid bodies detailed in Ref. [2]. We find that the horn tori as well as certain spindle and ring tori can achieve a packing density not only higher than that of spheres (i.e., \(\pi/\sqrt{18}=0.7404…\)) but also higher than the densest known ellipsoid packings (i.e., \(0.7707…\)). In addition, we study dense packings of clusters of pair-linked ring tori (i.e., Hopf links), which can possess much higher densities than corresponding packings consisting of unlinked tori (see Figure 1). For more information on the aforementioned organizing principles, please see slides from the invited talk “Packing Nonspherical Particles: All Shapes Are Not Created Equal” given by Professor Torquato at the American Physical Society March Meeting in Boston on February 28, 2012. The slides are also available on the APS website. For more information,

Figure 1: (left) A dense packing of unlinked ring tori. (right) A dense packing of linked ring tori.

- R. Gabbrielli, Y. Jiao, and S. Torquato, Dense Periodic Packings of Tori,
**Physical Review E**,**89**, 022133 (2014). - S. Torquato and Y. Jiao, Organizing Principles for Dense Packings of Nonspherical Hard Particles: Not All Shapes are Created Equal,
**Physical Review E**,**86**, 011102 (2012).

** Densest Local Packings **

*d*-dimensional Spheres

## Densest Local Packings of \(d\)-dimensional Spheres

The densest local packings of \(N\) identical nonoverlapping spheres within a radius \(R_{min}(N)\) of a fixed central sphere of the same size are obtained using a nonlinear programming method operating in conjunction with a stochastic search of configuration space in Ref. [1]. The knowledge of \(R_{min}(N)\) \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) allows for the construction both of a realizability condition for pair-correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings in Rd. In this paper, we focus on the two-dimensional circular disk problem. We find and present the putative densest packings and corresponding \(R_{min}(N)\) selected values of \(N\) up to \(N=348\) (see Figure 1) and use this knowledge to construct such a realizability condition and an upper bound. We additionally analyze the properties and characteristics of the maximally dense packings, finding significant variability in their symmetries and contact networks, and that the vast majority differ substantially from the triangular lattice even for large \(N\). Our work has implications for packaging problems, nucleation theory, and surface physics.

Figure 1: A conjectured optimal packing (point group \(D_{5h}\)) for \(N=15\) and \(R_{min}(15)=1.873123…\) with encompassing sphere of radius \(R_{min}(15)+0.5=2.373123…\). Figure and caption taken from Ref. [1].

In Ref. [2], we extend our analysis to three-dimensional spheres and obtain the densest local packings of \(N\) identical nonoverlapping spheres within a radius \(R_{min}(N)\) of a fixed central sphere of the same size for selected values of \(N\) up to \(N=1054\). We analyze the properties and characteristics of the densest local packings in \(\mathbb{R}^d\) and employ knowledge of the \(R_{min}(N)\), using methods applicable in any dimension \(d\) to construct both a realizability condition for pair correlation functions of sphere packings and an upper bound on the maximal density of infinite sphere packings. Additionally, we observe that the densest local packings are dominated by the dense arrangement of spheres with centers at distance \(R_{min}(N)\). In particular, we find two “maracas” packings at \(N = 77\) and \(N = 93\), each consisting of a few unjammed spheres free to rattle within a “husk” composed of the maximal number of spheres that can be packed with centers at respective \(R_{min}(N)\). Our results reveal a wealth of information about packings of spheres, including counterintuitive results concerning the physics of dilute mixtures of spherical solute particles in a solvent composed of same-size spheres and about the presence of unjammed spheres (rattlers) in the densest local structures.

Lastly, in Ref. [3], we consider the connection of our densest local packing problem to the spherical code problem which involves the placement of the centers of \(N\) nonoverlapping spheres of unit diameter onto the surface of a sphere of radius \(R\) such that \(R\) is minimized. We specifically prove that in any dimension, all solutions between unity and the golden ratio to the optimal spherical code problem for N spheres are also solutions to the corresponding DLP problem.

- A. B. Hopkins, F. H. Stillinger, and S. Torquato,
*Densest Local Sphere-Packing Diversity: General concepts and application to two dimensions*,**Physical Review E**,**81**, 041305 (2010). - A. B. Hopkins, F. H. Stillinger, and S. Torquato,
*Densest Local Packing Diversity. II. Application to Three Dimensions*,**Physical Review E**,**83**, 011304 (2011). - A. B. Hopkins, F. H. Stillinger, and S. Torquato,
*Spherical Codes, Maximal Local Packing Density, and the Golden Ratio*,**Journal of Mathematical Physics**,**51**, 043302 (2010).

** High Dimensional Packings **

## Background

There has been resurgent interest in hard-sphere packings in dimensions greater than three in both the physical and mathematical sciences. For example, it is known that the optimal way of sending digital signals over noisy channels corresponds to the densest sphere packing in a high dimensional space. These “error-correcting” codes underlie a variety of systems in digital communications and storage, including compact disks, cell phones and the Internet. Physicists have studied hard-sphere packings in high dimensions to gain insight into ground and glassy states of matter as well as phase behavior in lower dimensions. The determination of the densest packings in arbitrary dimension is a problem of long-standing interest in discrete geometry [1] (Conway and Sloane, 1998).

A collection of congruent spheres in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) is called a sphere packing \(P\) if no two of the spheres have an interior point in common. The *packing density* or simply density \(\phi(P)\) of a sphere packing is the fraction of space \(\mathbb{R}^d\) covered by the spheres. We will call

$$\phi_\mathrm{max} = \sup_{P\subset \mathbb{R}^d} \phi(P)$$

the *maximal density*, where the supremum is taken over all packings in \(\mathbb{R}^d\). The sphere packing problem seeks to answer the following question: Among all packings of congruent spheres, what is the maximal packing density \(\phi_\mathrm{max}\), i.e., largest fraction of \(\mathbb{R}^d\) covered by the spheres, and what are the corresponding arrangements of the spheres.

For arbitrary \(d\), the sphere packing problem is notoriously difficult to solve. In the case of packings of congruent \(d\)-dimensional spheres, the exact solution is known for the first three space dimensions. For \(d=1\), the answer is trivial because the spheres tile the space so that \(\phi_\mathrm{max}=1\). In two dimensions, the optimal solution is the triangular lattice arrangement (also called the hexagonal packing) with \(\phi_\mathrm{max}=\pi/\sqrt{12}\). In three dimensions, the Kepler conjecture that the face-centered cubic lattice arrangement provides the densest packing with \(\phi_\mathrm{max}=\pi/\sqrt{18}\) was only recently proved by Hales in 2005.

For \(d\geq4\), the problem remains unsolved. For \(3<d<10\), the densest known packings are Bravais lattices (one sphere per fundamental periodic cell), but in sufficiently large dimensions the optimal packings are likely to be non-Bravais-lattice packings. Each dimension seems to have its own idiosyncrasies, and it is highly unlikely that a single, simple construction will give the best packing in every dimension. Although certain dimensions allow for amazingly dense and symmetric Bravais lattice packings (e.g., \(E_8\) lattice in \(\mathbb{R}^8\) and Leech lattice in \(\mathbb{R}^{24}\)), such “miraculous” dimensions do not seem to persist in sufficiently high dimensions. The determination of bounds on \(\phi_\mathrm{max}\) are the best means of estimating it for arbitrary \(d\). Upper and lower bounds on the density are known, but they differ by an exponential factor as \(d\to\infty\).

Minkowski proved that the maximal density \(\phi^L_\mathrm{max}\) among all Bravais lattice packings for \(d\geq2\) satisfies the lower bound

where \(\zeta(d)=\sum_{k=1}^\infty k^{-d}\) is the Riemann zeta function. One observes that for large values of \(d\), the asymptotic behavior of the *nonconstructive* Minkowski lower bound is controlled by \(2^{-d}\). For the last century, mathematicians have been trying to exponentially improve Minkowski’s lower bound on the maximal density, but this result has proved to be illusive [2].

Our recent work, described below, suggests that disordered sphere arrangements might be the densest packings in sufficiently high dimensiuons and provide the long-sought exponential improvement of Minkowski’s bound. This would imply that disorder wins over order in sufficiently high dimensions.

- J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Spring-Verlag, New York (1998).
- H. Minkowski, Diskontinuitätsbereich für arithmetische Äquivalenz,
**J. reine angel. Math.**,**129**, 220-274 (1905).

## Exactly Solvable Disordered Hard-Sphere Packing Model in Arbitrary-Dimensional Euclidean Spaces

In Ref. [1], we introduce a generalization of the well-known random sequential addition (RSA) process for hard spheres in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\). We show that all of the \(n\)-particle correlation functions (\(g_2\), \(g_3\), etc.) of this nonequilibrium model, in a certain limit called the “ghost” RSA packing, can be obtained analytically for all allowable densities and in any dimension. This represents the first exactly solvable disordered sphere-packing model in arbitrary dimension. The fact that the maximal density \(\phi(\infty)=1/2^d\) of the ghost RSA packing implies that there may be disordered sphere packings in sufficiently high \(d\) whose density exceeds Minkowski’s lower bound for Bravais lattices, the dominant asymptotic term of which is \(1/2^d\).

## New Conjectural Lower Bounds on the Optimal Density of Sphere Packings

In Ref. [2], we use an optimization procedure previously introduced in Ref. [3] as well as a conjecture concerning the existence of disordered sphere packings in \(\mathbb{R}^d\), we obtain a conjectural lower bound on the density whose asymptotic behavior is controlled by \(2^{-0.77865…d}\), thus providing the putative exponential improvement of Minkowski’s bound. The conjecture states that a hard-core nonnegative tempered distribution is a pair correlation function of a translationally invariant disordered sphere packing in \(\mathbb{R}^d\) for asymptotically large \(d\) if and only if the Fourier transform of the autocovariance function is nonnegative. The conjecture is supported by two explicit analytically characterized disordered packings, numerical packing constructions in low dimensions, known necessary conditions that only have relevance in very low dimensions, the fact that we can recover the forms of known rigorous lower bounds, and the “**decorrelation principle**.” This principle states that **unconstrained** correlations in disordered sphere packings vanish asymptotically in high dimensions and that the n-particle correlation function \(g_n\) for any \(n\geq3\) can be inferred entirely (up to some small error) from a knowledge of the number density \(\rho\) and the pair correlation function \(g_2(r)\). A byproduct of our approach is an asymptotic conjectural lower bound on the average kissing number whose behavior is controlled by \(2^{0.22134…d}\), which is to be compared to the best known asymptotic lower bound on the individual kissing number of \(2^{0.2075…d}\). Interestingly, our optimization procedure is precisely the dual of a primal linear program devised by Cohn and Elkies (2002, 2003) to obtain upper bounds on the density, and hence has implications for linear programming bounds. This connection proves that our density estimate can never exceed the Cohn-Elkies upper bound, regardless of the validity of our conjecture.

## Packing Hyperspheres in High-Dimensional Euclidean Spaces

In Ref. [4], we present the first study of disordered jammed hard-sphere packings in four-, five- and six-dimensional Euclidean spaces. Using a collision-driven packing generation algorithm, we obtain the first estimates for the packing fractions of the maximally random jammed (MRJ) states for space dimensions \(d=4, 5,\) and \(6\) to be \(\phi_{MRJ}\simeq 0.46,~0.31, \mathrm{and}~0.20\), respectively. To a good approximation, the MRJ density obeys the scaling form \(\phi_{MRJ}= c_1/2^d +(c_2 d)/2^d\), where \(c_1=-2.72\) and \(c_2=2.56\) , which appears to be consistent with high-dimensional asymptotic limit, albeit with different coefficients. Calculations of the pair correlation function \(g_2(r)\) and structure factor \(S(k)\) for these states show that short-range ordering appreciably decreases with increasing dimension, consistent with a recently proposed “decorrelation principle,” which, among othe things, states that unconstrained correlations diminish as the dimension increases and vanish entirely in the limit \(d\to\infty\). As in three dimensions (where \(\phi_{MRJ}\simeq 0.64\)) , the packings show no signs of crystallization, are isostatic, and have a power-law divergence in \(g_2(r)\) at contact with power-law exponent \(\simeq 0.4\). Across dimensions, the cumulative number of neighbors equals the kissing number of the conjectured densest packing close to where \(g_2(r)\) has its first minimum. Additionally, we obtain estimates for the freezing and melting packing fractions for the equilibrium hard-sphere fluid-solid transition, \(\phi_F \simeq 0.32\) and \(\phi_M \simeq 0.39\) , respectively, for \(d=4\) , and \(\phi_F\simeq 0.19\) and \(\phi_M \simeq 0.24\) , respectively, for \(d=5\). Although our results indicate the stable phase at high density is a crystalline solid, nucleation appears to be strongly suppressed with increasing dimension.

## Random Sequential Addition of Hard Spheres in High Euclidean Dimensions

Employing numerical and theoretical methods, we investigate the structural characteristics of random sequential addition (RSA) of congruent spheres in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\) in the infinite-time or saturation limit for the first six space dimensions ( \(1\leq d\leq 6\) ) in Ref. [5]. Specifically, we determine the saturation density, pair correlation function, cumulative coordination number and the structure factor in each of these dimensions. We find that for \(2\leq d\leq 6\) , the saturation density \(\phi_s\) scales with dimension as \(\phi_s = c_1/2^d +c_2 d/2^d\) , where \(c_1=0.202048\) and \(c_2=0.973872\). We also show analytically that the same density scaling is expected to persist in the high-dimensional limit, albeit with different coefficients. A byproduct of this high-dimensional analysis is a relatively sharp lower bound on the saturation density for any \(d\) given by \(\phi_s\geq (d+2)(1-S_0)/2^{d+1}\) , where \(S_0\in [0,1]\) is the structure factor at \(k=0\) (i.e., infinite-wavelength number variance) in the high-dimensional limit. We demonstrate that a Palàsti-like conjecture (the saturation density in \(\mathbb{R}^d\) is equal to that of the one-dimensional problem raised to the \(d\) th power) cannot be true for RSA hyperspheres. We show that the structure factor \(S(k)\) must be analytic at \(k=0\) and that RSA packings for \(1\leq d\leq 6\) are nearly “hyperuniform.” Consistent with the recent “decorrelation principle,” we find that pair correlations markedly diminish as the space dimension increases up to six. We also obtain kissing (contact) number statistics for saturated RSA configurations on the surface of a \(d\)-dimensional sphere for dimensions \(2\leq d\leq 5\) and compare to the maximal kissing numbers in these dimensions. We determine the structure factor exactly for the related “ghost” RSA packing in \(\mathbb{R}^d\) and demonstrate that its distance from “hyperuniformity” increases as the space dimension increases, approaching a constant asymptotic value of \(1/2\).

## Estimates of the Optimal Density and Kissing Number of Sphere Packings in High Dimensions

The problem of finding the asymptotic behavior of the maximal density \(\phi_\mathrm{max}\) of sphere packings in high Euclidean dimensions is one of the most fascinating and challenging problems in discrete geometry. One century ago, Minkowski obtained a rigorous lower bound on \(\phi_\mathrm{max}\) that is controlled asymptotically by \(1/2^d\) , where \(d\) is the Euclidean space dimension. An indication of the difficulty of the problem can be garnered from the fact that exponential improvement of Minkowski’s bound has proved to be elusive, even though existing upper bounds suggest that such improvement should be possible. Using a statistical-mechanical procedure to optimize the density associated with a “test” pair correlation function and a conjecture concerning the existence of disordered sphere packings [2], the putative exponential improvement on \(\phi_\mathrm{max}\) was found with an asymptotic behavior controlled by \(1/2^{0.77865…d}\). In Ref. [6], we use the same methods to investigate whether this exponential improvement can be further improved by exploring other test pair correlation functions corresponding to disordered packings. We demonstrate that there are simpler test functions that lead to the same asymptotic result. More importantly, we show that there is a wide class of test functions that lead to precisely the same putative exponential improvement and therefore the asymptotic form \(1/2^{0.77865…d}\) is much more general than previously surmised. This class of test functions leads to an optimized average kissing number that is controlled by the same asymptotic behavior as the one found in the aforementioned paper.

- S. Torquato and F. H. Stillinger, Exactly Solvable Disordered Hard-Sphere Packing Model in Arbitrary-Dimensional Euclidean Spaces,
**Physical Review E**,**73**, 031106 (2006). - S. Torquato and F. H. Stillinger, New Conjectural Lower Bounds on the Optimal Density of Sphere Packings,
**Experimental Mathematics 15**, 307 (2006). - S. Torquato and F. H. Stillinger, Controlling the Short-Range Order and Packing Densities of Many-Particle Systems,
**Journal of Physical Chemistry B**,**106**, 8354 (2002). Please see*Erratum*. - M. Skoge, A. Donev, F. H. Stillinger and S. Torquato, Packing Hyperspheres in High-Dimensional Euclidean Spaces,
**Physical Review E 74**, 041127 (2006). - S. Torquato, O. U. Uche and F. H. Stillinger, Random Sequential Addition of Hard Spheres in High Euclidean Dimensions,
**Physical Review E****74**, 061308 (2006). - A. Scardicchio, F. H. Stillinger and S. Torquato, Estimates of the Optimal Density of Sphere Packings in High Dimensions,
**Journal of Mathematical Physics**,**49**, 043301 (2008).

** Disordered Packings **

## Is Random Close Packing of Spheres Well Defined?

Despite its long history, there are many fundamental issues concerning random packings of spheres that remain elusive, including a precise definition of *random close packing* (RCP). In Ref. [1], we argue that the current picture of RCP cannot be made mathematically precise and support this conclusion via a molecular dynamics study of hard spheres using the Lubachevsky-Stillinger compression algorithm. We suggest that this impasse can be broken by introducing a new concept which can be made precise. Roughly speaking, the *maximally random jammed* (MRJ) state is the most disordered packing subject to the condition that it is jammed (mechanically stable) (see Figure 1). For a less technical description of this work, please see the popular press articles in Science and The BBC.

Figure 1: An MRJ packing of identical hard spheres with packing fraction \(\phi_2\approx0.64\).

## Investigating MRJ Sphere Packings Using the Torquato-Jiao Sequential Linear Programming Algorithm

In Refs. [2] and [3], we formulate the problem of generating dense packings of nonoverlapping, nontiling nonspherical particles within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem called the adaptive-shrinking cell ASC formulation. Because the objective function and impenetrability constraints can be exactly linearized for sphere packings with a size distribution in \(d\)-dimensional Euclidean space, it is most suitable and natural to solve the corresponding ASC optimization problem using sequential-linear-programming (SLP) techniques. We implement an SLP solution to robustly produce a spectrum of jammed sphere packings in dimensions two through six with varying degrees of disorder and densities which go up to their respective maximums. This packing protocol, which we refer to as the Torquato-Jiao (TJ) algorithm, is able to ensure that the final packings are truly jammed, produces disordered jammed packings with anomalously low densities, and is appreciably more robust and computationally faster at generating maximally dense packings, especially as the space dimension increases.

Figure 2: (left) A 2-rattler where both rattlers (opaque red spheres) are in the same cage (translucent blue spheres). (right) A 2-rattler with a bottleneck. The cage spheres (translucent) are colored either bright red or dark blue to match the rattler (opaque spheres) that they enclose. The two bottleneck spheres (translucent green spheres marked with opaque dots in their centers) contribute simultaneously to the cage of both rattlers. Figure and caption taken from Ref. [4].

In Ref. [4], we use the TJ algorithm to generate jammed disordered monodisperse sphere packings in 3D of various sizes whose strictly jammed backbones are demonstrated to be exactly isostatic with unprecedented numerical accuracy. Notably, we observe that these MRJ sphere packings have a have a significantly higher degree of disorder (judging by bond-orientational and translational order metrics) than packings generated with the Lubachevksy-Stillinger (LS) protocol. We also observe that the rattler fraction of the TJ packings is significantly lower than that of those prepared via the LS protocol, converging towards \(0.015\) in the infinite-system limit. We find that the rattler pair correlation statistics exhibit strongly correlated behavior which contrasts the conventional understanding that rattlers are distributed randomly throughout the packing. Additionally, we report the existence of dynamically interacting “polyrattlers” which may be found imprisoned in shared cages as well as interacting through “bottlenecks” in the backbone (see Figure 2). Lastly, we discuss how rattlers in hard-sphere packings share an apparent connection with the low-temperature two-level system anomalies that appear in real amorphous insulators and semiconductors.

Figure 3: A 2000-sphere unit cell of a periodic strictly jammed binary packing with small to large sphere size ratio \(\alpha=0.2\), relative number concentration \(x=0.970\) with packing fraction \(\phi=0.785\). Figure and caption taken from Ref. [5].

In Ref. [5], we use the TJ algorithm to explore the MRJ state for binary sphere packings (see Figure 3). Previous attempts to simulate disordered binary sphere packings have been limited in producing mechanically stable, isostatic packings across a broad spectrum of packing fractions. We find that disordered strictly jammed binary packings can be produced with an anomalously large range of average packing fractions \(\phi\) between \(0.634\) and \(0.829\) for small to large sphere radius ratios \(\alpha\) larger than \(0.100\). The average packing fractions of these packings at certain size and number ratios approach those of the corresponding densest known *ordered* packings. These findings suggest, for entropic reasons, that these high-density disordered packings should be good glass formers and that they may be easy to prepare experimentally. The identification and explicit construction of binary packings with such high packing fractions could have important practical implications for granular composites where density is critical both to material properties and fabrication cost, including for solid propellants, concrete, and ceramics. The densities and structures of jammed binary packings at various size and number ratios are also relevant to the formation of a glass phase in multicomponent metallic systems.

In Ref. [6], we generate and study an ensemble of isostatic jammed hard-sphere lattices. These lattices are obtained by compression of a periodic system with an adaptive unit cell containing a single sphere until the point of mechanical stability. We present detailed numerical data about the densities, pair correlations, force distributions, and structure factors of such lattices. We show that this model retains many of the crucial structural features of the classical hard-sphere model and propose it as a model for the jamming and glass transitions that enables exploration of much higher dimensions than are usually accessible.

Figure 4: (left) Exactly isostatic and collectively jammed monodisperse disk packing with \(N=150\) disks. (right) Exactly isostatic and strictly jammed monodisperse disk packing with \(N=110\) disks. Disks are colored to indicate their (backbone) coordination as follows: black = 3 contacts, green = 4 contacts, orange = 5 contacts, and white = rattler (0 contacts). The fundamental cell is outlined in black. Figure and caption taken from Ref. [7].

In Ref. [7], we use the TJ algorithm to show the existence of relatively large maximally random jammed (MRJ) packings with exactly isostatic jammed backbones and a packing fraction (including rattlers) of \(0.826\) (see Figure 4). By contrast, the concept of random close packing (RCP) that identifies the most probable packings as the most disordered misleadingly identifies highly ordered disk packings as RCP in 2D. Fundamental structural descriptors such as the pair correlation function, structure factor, and Voronoi statistics show a strong contrast between the MRJ state and the typical hyperstatic, polycrystalline packings with packing fraction \(0.88\) that are more commonly obtained using standard packing protocols. Establishing that the MRJ state for monodisperse hard disks is isostatic and qualitatively distinct from commonly observed polycrystalline packings contradicts conventional wisdom that such a disordered, isostatic packing does not exist due to a lack of geometrical frustration and sheds light on the nature of disorder.

## Characterization of the Microstructures and Effective Physical Properties of MRJ Sphere Packings

Figure 5: (a) Poisson (overlapping) spheres. (b) Equilibrium hard spheres. (c) MRJ spheres. Figure taken from Ref. [9].

We have extensively studied the microstructures and certain effective physical properties of three-dimensional MRJ sphere packings (see Figure 5) generated via the Torquato-Jiao SLP algorithm. In Ref. [8], we introduced novel Voronoi correlation functions to characterize the structure of MRJ sphere packings across various length scales (see Figure 6). We found that these descriptors are highly sensitive to microstructural details, as they are able to distinguish the structure of MRJ sphere packings (prototypical glasses) from correlated equilibrium hard-sphere liquids and uncorrelated Poisson spheres.

Figure 6: MRJ spheres and their Voronoi cells (red). Figure taken from Ref. [8].

In Ref [9], we determined various correlation functions, spectral functions, hole probabilities, and local density fluctuations of MRJ sphere packings via certain analytical and numerical techniques. Several of these microstructural descriptors arise in rigorous expressions for the effective physical properties of two-phase media. Given that MRJ packings are disordered hyperuniform, one can expect them to possess remarkable physical properties which we demonstrate in Ref. [10]. Notably, we found that a dispersion of dielectric MRJ spheres embedded within a matrix of another dielectric material effectively forms a dissipationless disordered and isotropic two-phase medium for any phase dielectric contrast ratio (see Figure 7).

Figure 7: A two-phase medium consisting of MRJ spheres with dielectric constant \(\epsilon_2\) embedded in a matrix phase of dielectric constant \(\epsilon_1\). Figure taken from Ref. [10].

## Nonuniversality of Density and Disorder in Jammed Sphere Packings

Figure 8: Jammed hard-sphere packings at various densities.

In Ref. [10], we use the TJ algorithm to show for the first time that collectively jammed disordered packings of three-dimensional monodisperse frictionless hard spheres with packing densities as low as \(0.6\) can be produced (see Figure 8). This packing fraction is well below the value of \(0.64\) associated with the MRJ state and entirely unrelated to the ill-defined RCP density. We specifically generate a series of collectively jammed packings whose densities are very narrowly distributed about any density within the range \([0.6,0.74048…]\). Our packings possess varying degrees of disorder, and our results support the view that there is no universal jamming point that is distinguishable based on the packing density and frequency of occurrence. We map our jammed packings onto a density-order-metric plane, which provides a broader characterization of packings than density alone. Other packing characteristics, such as the pair correlation function, average contact number, and fraction of rattlers are quantified and discussed.

- S. Torquato, T. M. Truskett, and P. G. Debenedetti, Is Random Close Packing of Spheres Well Defined?,
**Physical Review Letters**,**84**, 2064 (2000). - S. Torquato and Y. Jiao. Dense Packings of Polyhedra: Platonic and Archimedean Solids,
**Physical Review E**,**80**, 041104 (2009). Please see*Erratum*. - S. Torquato and Y. Jiao, Robust Algorithm to Generate a Diverse Class of Dense Disordered and Ordered Sphere Packings Via Linear Programming
**Physical Review E**,**82**, 061302 (2010). - S. Atkinson, F. H. Stillinger, and S. Torquato, Detailed Characterization of Rattlers in Exactly Isostatic, Strictly Jammed Sphere Packings,
**Physical Review E**,**88**, 062208 (2013). - A. B. Hopkins, F. H. Stillinger, and S. Torquato, Disordered Strictly Jammed Binary Sphere Packings Attain an Anomalously Large Range of Densities,
**Physical Review E**,**88**, 022205 (2013). - Y. Kallus, É. Marcotte, and S. Torquato, Jammed Lattice Sphere Packings,
**Physical Review E**,**88**, 062151 (2013). - S. Atkinson, F. H. Stillinger, and S. Torquato, Existence of Isostatic, Maximally Random Jammed Monodisperse Hard-Disk Packings,
**Proceedings of the National Academy of Sciences**,**111**, 18436 (2014). - M. A. Klatt and S. Torquato, Characterization of Maximally Random Jammed Sphere Packings: Voronoi Correlation Functions,
**Physical Review E**,**90**, 052120 (2014). - M. A. Klatt and S. Torquato, Characterization of Maximally Random Jammed Sphere Packings: II. Correlation Functions and Density Fluctuations,
**Physical Review E**,**94**, 022152 (2016). - M.A. Klatt, and S. Torquato, Characterization of Maximally Random Jammed Sphere Packings. III. Transport and Electromagnetic Properties via Correlation Functions,
**Physical Review E**,**97**, 012118 (2018). - Y. Jiao, F. H. Stillinger, and S. Torquato, Nonuniversality of Density and Disorder of Jammed Sphere Packings,
**Journal of Applied Physics**,**109**, 013508 (2011).

## Disordered Ellipsoid Packings

Figure 1: (left) An experimental packing of M&M’s Candies. The spherical bowl minimizes finite-size effects caused by the boundary. (right) Computer-generated packing of 1000 oblate ellipsoids. Figure and caption taken from Ref. [1].

In Ref. [1], we show experimentally and with a new simulation algorithm that ellipsoids can randomly pack more densely up to \(\phi = 0.68-0.71\) for spheroids with an aspect ratio close to that of M&M’s Candies-and even approach \(\phi\sim0.74\), the packing fraction of the face-centered cubic lattice, for ellipsoids with other aspect ratios (see Figure 1). We suggest that the higher density is directly related to the higher number of degrees of freedom per particle and thus the larger number of particle contacts required to mechanically stabilize the packing. We measure the number of contacts per particle to be \(Z\sim10\) for our spheroids, as compared to \(Z\sim6\) for spheres. Our results have implications for a broad range of scientific disciplines, including the properties of granular media and ceramics, glass formation, and discrete geometry. For a more in-depth analysis on the densities of these packings, including an MRI scan of a bowl of M&M’s candies, see Ref. [2]. For a thorough analysis on the structures of these packings, see Ref [3]. For more information on our numerical packing protocol, see Refs. [4] and [5]. For less-technical discussions on this research, see the popular press articles on CNN, and in Science News.

## Disordered Superball Packings

Figure 2: Typical configurations of MRJ packings of superballs for two different values of the deformation parameter: (a) \(p=0.85\), (b) \(p=1.5\). Figure and caption taken from Ref. [6].

In Ref. [6], we report a study of the maximally random jammed packings of binary superdisks and monodispersed superballs whose shapes are defined by \(|x_1|^{2p}+…+|x_d|^{2p}\le 1\) with \(d=2\) and \(3\), respectively, where \(p\) is the deformation parameter with values in the interval \((0,\infty)\). We generate and examine packings for a variety of shape parameters and find that their densities increase dramatically and non analytically as one moves away from the circular-disk and sphere point \(p=1\) (see Figure 2). We find that these packings are hypostatic, i.e., the average number of contacting neighbors is less than twice the total number of degrees of freedom per particle, while remaining mechanically stable. As a result, the local arrangements of particles are necessarily nontrivially correlated to achieve jamming. We term such correlated structures “nongeneric.” The degree of “nongenericity” of the packings is quantitatively characterized by determining the fraction of local coordination structures in which the central particles have fewer contacting neighbors than average. We also show that such seemingly “special” packing configurations are counterintuitively not rare. As the anisotropy of the particles increases, the fraction of rattlers decreases while the minimal orientational order as measured by the tetratic and cubatic order parameters increases. These characteristics result from the unique manner in which superballs break their rotational symmetry, which also makes the superdisk and superball packings distinctly different from other known nonspherical hard-particle packings.

## Ordered and Disordered Packings of Lens-Shaped Particles

The lens is defined, in three dimensions, as the overlap volume of two congruent spheres. In a collection of recent papers, we have characterized the phase behavior, densest-known packings, and disordered packings of lens-shaped particles for a wide range of aspect ratios. In particular, we have found that lenses and oblate spheroids have qualitatively similar phase diagrams due to their similarity in shape but have more pronounced differences in their high-density crystal phases. While the densest packing of such spheroids is periodic with a two-particle basis, lenses have two degenerate densest packings: i) a periodic packing with a two-particle basis and ii) a Bravais lattice packing. These two crystal phases can stack at will, leading to an infinite number of densest phases, which are generally non-periodic in the direction of stacking. Additional details can be found in Ref. [7].

Figure 3: (a) Schematic of a lens with aspect ratio \(\alpha=b/a=2/3\). (b) An MRJ packing with \(\alpha=2/3\) with particles colored according to the angle that their \(\mathbf{C_{\infty}}\) axis makes with an axis of the laboratory reference frame: the cooler the color of a particle the smaller the angle that its axis forms with that axis of the laboratory reference frame. Figure taken from Ref. [9].

In Ref. [8], we study the behavior of dense lens packings ranging from ‘flat’ lenses to ‘globular’ lenses. Specifically, we find that ‘flat’ lenses form nematic fluid phases, ‘globular’ lenses form a plastic solid phase, while ‘intermediate’ lenses form neither of these mesophases. In general, we did not observe spontaneous formation of crystalline phases (see Figure 3). The structure factors for the jammed states for ‘flat,’ ‘intermediate,’ and ‘globular’ lenses all indicate that they are effectively hyperuniform. Among all possible lens shapes, those with an aspect ratio of \(2/3\) are special because they have the highest packing fraction at jamming while being positionally and orientationally disordered, indicating they are particularly good glass formers. Such packings have a packing fraction only a few percent lower than the densest-known packing. Disordered jammed packings of lenses with this aspect ratio have been studied more extensively in Ref. [9].

- A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Stillinger, R. Connelly, S. Torquato, and P. M. Chaikin, Improving the Density of Jammed Disordered Packings using Ellipsoids,
**Science**,**303**, 990-993 (2004). - A. Donev, R. Connelly, F. H. Stillinger and S. Torquato, Underconstrained Jammed Packings of Nonspherical Hard Particles: Ellipses and Ellipsoids,
**Physical Review E**,**75**, 051304 (2007). - W. Man, A. Donev , F. H. Stillinger, M. T. Sullivan, W. B. Russel, D. Heeger , S. Inati, S. Torquato and P. M. Chaikin, Experiments on Random Packings of Ellipsoids,
**Physical Review Letters**,**94**, 198001 (2005). - A. Donev, S. Torquato, and F. H. Stillinger, Neighbor List Collision-Driven Molecular Dynamics for Nonspherical Hard Particles: I. Algorithmic Details,
**Journal of Computational Physics**,**202**, 737 (2005). - A. Donev, S. Torquato, and F. H. Stillinger, Neighbor List Collision-Driven Molecular Dynamics for Nonspherical Hard Particles: II. Applications to Ellipses and Ellipsoids,
**Journal of Computational Physics**,**202**, 765 (2005). - Y. Jiao, F. H. Stillinger, and S. Torquato, Distinctive Features Arising in Maximally Random Jammed Packings of Superballs,
**Physical Review E**,**81**, 041304 (2010). - G. Cinacchi and S. Torquato, Hard Convex Lens-shaped Particles: Metastable, Glassy and Jammed States,
**Soft Matter**,**14**8205-8218 (2018). - G. Cinacchi and S. Torquato, Hard Convex Lens-Shaped Particles: Characterization of Dense Disordered Packings,
**Physical Review E**,**100**062902 (2019). - G. Cinacchi and S. Torquato, Hard Convex Lens-shaped Particles: Densest-known Packings and Phase Behavior,
**Journal of Chemical Physics**,**143**, 224506 (2015).

## Hyperuniformity and Maximally Random Jammed Packing

Maximally random strictly jammed hard-particle packings are the most disordered configurations of impenetrable particles that are globally incompressible and nonshearable. Such systems are therefore prototypical glasses, lacking long-range order and possessing diverging elastic moduli. In an initial study, we showed that so-called maximally random jammed (MRJ) packings of identical three-dimensional spheres, which can be viewed as a prototypical glass [1], are hyperuniform such that pair correlations decay asymptotically with scaling \(r^{-4}\), which we call *quasi-long-range* correlations [2,3]. Such correlations are to be contrasted with typical disordered systems in which pair correlaitons decay exponentially fast. More recently, we have shown that quasi-long-range pair correlations that decay asymptotically with scaling \(r^{-(d+1)}\) in \(d\)-dimensional Euclidean space \(\mathbb{R}^d\), trademarks of certain quantum systems and cosmological structures, are a universal signature of a wide class of maximally random jammed (MRJ) hard-particle packings, including nonspherical particles with a polydispersity in size [4-6].

More generally, Torquato and Stillinger [7] suggested that certain defect-free strictly jammed packings of identical spheres are hyperuniform. Specifically, they conjectured that any strictly jammed saturated infinite packing of identical spheres is hyperuniform. A saturated packing of hard spheres is one in which there is no space available to add another sphere. Importantly, the Torquato-Stillinger conjecture excludes MRJ packings that may have a mechanically rigid backbone but possess “rattlers” (particles that are not locally jammed but are free to move about a confining cage) because a strictly jammed packing, by definition, cannot contain rattlers. It has been suggested that the ideal MRJ state is rattler-free, implying that the packing is more disordered without the presence of rattlers [8]. Recently, Torquato presented a refined variant of the conjecture: “Any strictly jammed infinite packing of identical spheres that is defect-free is hyperuniform” [9]. Notably, the Torquato-Stillinger conjecture is supported by various numerical experiments; for example, see a recent letter by Rissone, Corwin, and Parisi [10].

Figure 1: (top left) Jammed bidisperse disks. (top right) Jammed polydisperse disks. (bottom left) Jammed bidisperse ellipses. (bottom right) Jammed bidisperse superdisks.

Previous work on jammed materials composed of particles with size and shape distributions was unable to detect hyperuniformity; making it difficult to study such systems within a unified framework. To overcome this obstacle, Zachary, Jiao, and Torquato utilized a more general notion of hyperuniformity involving local-volume-fraction fluctuations [10-12]. They specifically demonstrated that maximally random jammed hard-particle packings share universal global structural signatures, regardless of the particle shapes or relative sizes. They also analyzed the geometry of the void space external to the particles, which is always uniform over the large-scale structure of a jammed packing, and observed that the constraint of strict jamming competes with the maximal randomness of the packing to induce signature quasi-long-range pair correlations in the two-point probability function. Their findings also addressed the common misconception that hyperuniformity is tied to the homogeneity of particles in a jammed material. For a less technical description of this work, please see the popular press articles in Princeton News and MyScience.

- A. B. Hopkins, F. H. Stillinger, and S. Torquato, Nonequilibrium Static Diverging Length Scales on Approaching a Prototypical Model Glassy State,
**Physical Review E**,**86**, 021505 (2012). - A. Donev , F. H. Stillinger, and S. Torquato, Unexpected Density Fluctuations in Disordered Jammed Hard-Sphere Packings,
**Physical Review Letters**,**95**, 090604 (2005). - C. E. Zachary and S. Torquato, Anomalous Local Coordination, Density Fluctuations, and Void Statistics in Disordered Hyperuniform Many-Particle Ground States,
**Physical Review E**,**83**, 051133 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato, Hyperuniform Long-Range Correlations are a Signature of Disordered Jammed Hard-Particle Packings,
**Physical Review Letters**,**106**, 178001 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato, Hyperuniformity, Quasi-Long-Range Correlations, and Void Space Constraints in Maximally Random Jammed Particle Packings. I. Polydisperse Spheres,
**Physical Review E**,**83**, 051308 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato, Hyperuniformity, Quasi-Long-Range Correlations, and Void Space Constraints in Maximally Random Jammed Particle Packings. II. Anisotropy in Particle Shape,
**Physical Review E**,**83**, 051309 (2011). - S. Torquato and F. H. Stillinger,
*Local Density Fluctuations, Hyperuniform Systems, and Order Metrics*,**Physical Review E**,**68**, 041113 1-25 (2003). - S. Atkinson, G. Zhang, A. B. Hopkins, and S. Torquato, Critical Slowing Down and Hyperuniformity on Approach to Jamming,
**Physical Review E**,**94**, 012902 (2016). - S. Torquato, Structural characterization of many-particle systems on approach to hyperuniform states,
**Physical Review E**,**103**052126 (2021). - P. Rissone, E. I. Corwin, and G. Parisi, Long-Range Anomalous Decay of the Correlation in Jammed Packings,
**Physical Review Letters**,**127**038001 (2021). - C. E. Zachary, Y. Jiao, and S. Torquato, Hyperuniform Long-Range Correlations are a Signature of Disordered Jammed Hard-Particle Packings,
**Physical Review Letters**,**106**, 178001 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato. Hyperuniformity, Quasi-Long-Range Correlations, and Void Space Constraints in Maximally Random Jammed Particle Packings. I. Polydisperse Spheres,
**Physical Review E**,**83**, 051308 (2011). - C. E. Zachary, Y. Jiao, and S. Torquato. Hyperuniformity, Quasi-Long-Range Correlations, and Void Space Constraints in Maximally Random Jammed Particle Packings. II. Anisotropy in Particle Shape,
**Physical Review E**,**83**, 051309 (2011).

## Kinetic Effects in Two-Dimensional Packings

Kinetic effects are inevitable in numerical and experimental packing protocols. Specifically, in recent work, we have examined how such effects cause deviations in packing fraction and order with respect to the densest possible configuration of a two-dimensional monodisperse hard-particle packing as a function of compression rate and the particle shape. To study this phenomenon, we use the stochastic Adaptive Shrinking Cell algorithm (see Figure 1 and Ref. [1]) to prepare two-dimensional packings of obtuse scalene triangle, rhombus curved triangle, lens, and “ice cream cone” (a semicircle grafted onto an isosceles triangle) shaped particles with different compression rates.

Figure 1: In the stochastic Adaptive Shrinking Cell algorithm, a low-density ideal gas-like initial configuration (left) of hard particles is densified by continually compressing and shearing the simulation cell. The order and density of the final configuration (middle) is highly dependent on the compression schedule. A useful measure of order and isotropy in the final packings is its scattering pattern (right). Figure taken from Ref. [1].

In Ref. [2], to quantify the kinetic effects on the packing fraction, we define the kinetic frustration index \(K\) which qualitatively describes how much the actual packing fraction deviates from the maximum possible packing fraction. We additionally characterize the short- and long-range ordering in these packings by computing their spectral densities and characterize their contact networks. We find that kinetic effects are minimized when the packings are compressed more slowly, and when the particle shape has more rotational symmetry, is more curved, and is more circular. These findings can be used to aid in the design of laboratory packing protocols. See Ref. [1] for more information.

- Y. Jiao and S. Torquato, Maximally Random Jammed Packings of Platonic Solids: Hyperuniform Long-Range Correlations and Isostaticity,
**Physical Review E**,**84**, 041309 (2011). - C. E. Maher, F. H. Stillinger, and S. Torquato, Kinetic Frustration Effects on Dense Two-Dimensional Packings of Convex Particles and Their Structural Characteristics,
**Journal of Physical Chemistry B**,**125**, 2450 (2021).

## Strictly Jammed Sphere Packings with Anomalously Low Densities

In Ref. [1], we show a family of three-dimensional crystal sphere packings that are strictly jammed (i.e., mechanically stable) with anomalously low densities. This family contains an unaccountably infinite number of crystal packings that are subpackings of the densest crystal packings and are characterized by a high concentration of self-avoiding tunnels (chains of vacancies) that permeate the structures. These packings are not only candidates for the lowest-density rigid packing, but they may have broader significance in condensed matter physics and materials science.

Figure 1: Photograph of a mechanically stable ball-bearing arrangement in two adjacent layers of three nearest-neighbor trivacancy tunnels. Figure and caption from Ref. [2].

Subsequently, we have demonstrated that packings with much larger tunnels can also be strictly jammed in Ref. [2]. In particular, beginning with a strictly jammed hexagonal close-packed (hcp) crystal of monodisperse spheres, one can remove specific subsets of these spheres to produce mechanically stable vacancy arrangements involving compact (equilateral triangle) trivacancies such that they produce linear trivacancy tunnels. Such tunnels can extend over the entire macroscopic length of the hcp medium. In addition, the width of such tunnels is sufficient to allow contained “test” hard spheres with diameters less than \(\sqrt{5}-1=1.23606… \) to migrate over the entire length without contacting the static tunnel-wall spheres. Such a crystal may also be able approach the greatest possible “rattler” density within a jammed monovalent sphere system subject to periodic boundary conditions. These packings may also have practical implications for engineered separation and catalytic

processes.

- S. Torquato and F. H. Stillinger, Toward the Jamming Threshold of Sphere Packings: Tunneled Crystals,
**Journal of Applied Physics**,**102**, 093511 (2007). Please see*Erratum*. - F. H. Stillinger and S. Torquato, Jammed Hard-Sphere Hcp Crystals Permeated With Trivacancy Tunnels,
**Journal of Applied Physics**,**126(47)**194901 (2019).

## Phase Behavior and Maximally Random Jammed Packings of Truncated Tetrahedra

We have mapped out the equilibrium phase diagram and maximally random jammed state of truncated tetrahedra, one of the Archimedean solids. In particular, we observe that as the density increases there are two first-order phase transitions: first a liquid-to-solid phase transition, followed by a solid-to-solid phase transition. The isotropic liquid phase coexists with the Conway-Torquato (CT) crystal phase at intermediate densities, which verifies a claim from one of our previous qualitative studies (see Ref. [1] for more details). At higher densities the CT crystal undergoes a phase transition to the densest known crystal packing. We also characterize the structural characteristics of the hyperuniform MRJ state of truncated tetrahedra. More details on the phase diagram and MRJ packings can be found in Ref. [2]

- Y. Jiao and S. Torquato,
*Communication: A Packing of Truncated Tetrahedra That Nearly Fills All of Space and its Melting Properties*,**Journal of Chemical Physics**,**135**, 151101 (2011). - D. Chen, Y. Jiao, and S. Torquato,
*Equilibrium Phase Behavior and Maximally Random Jammed State of Truncated Tetrahedra*,**Journal of Physical Chemistry B**,**118**, 7981 (2014).

## Geometric Structure Theory for Maximally Random Jammed Packings

Figure 1: Comparison of the MRJ packing density for monodisperse superdisks estimated with the geometric structure theory (solid line) and obtained from molecular dynamics (MD) simulations (red circles), which fail to represent MRJ states, as well as sequential linear programming (SLP) datum for the circle case (blue square). Figure and caption taken from Ref. [1].

Predicting the density of a maximally random jammed (MRJ) packing \(\phi_{MRJ}\), among other packing properties of frictionless particles, still poses many theoretical challenges, even for packings of spheres or disks. Using a geometrical-structure approach, we produce a highly accurate formula for predicting \(\phi_{MRJ}\) for a wide class of two-dimensional frictionless packings, namely, binary convex superdisks, which have shapes that interpolate between disks and squares (tuned via the shape parameter \(p\)). Using specific attributes of MRJ states and a novel organizing principle, our formula yields predictions of \(\phi_{MRJ}\) that are in excellent agreement with corresponding computer-simulation estimates for a large range of particle size ratios and relative small-particle concentrations (see Figure 1). More details can be found in Ref. [1].

- J. Tian, Y. Xu, Y. Jiao, and S. Torquato, A Geometric-Structure Theory for Maximally Random Jammed Packings,
**Scientific Reports**,**5**, 16722 (2015).

Questions concerning this work should be directed to Professor Torquato.