Basic Understanding of Condensed Phases of Matter Via Packing Models

Basic Understanding of Condensed Phases of Matter via Packing Models

S. Torquato∗ Department of Chemistry, Department of Physics, Princeton Center for Theoretical Science, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton New Jersey, 08544 USA

Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the “geometric-structure” approach, which provides a powerful and uniﬁed means to quantitatively characterize individual packings via jamming categories and “order” maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research.

I. INTRODUCTION packing (defined below). Remarkably, nearly four cen- turies passed before Hales proved the general conjecture We will call a packing a large collection of nonoverlap- that there is no other sphere packing in three-dimensional ping (i.e., hard) particles in either a finite-sized container (3D) Euclidean space whose density can exceed that of the fcc packing.38 Even the proof that the densest pack- or in d-dimensional Euclidean space Rd. Exclusion- volume effects that often arise in dense many-particle ing of congruent (identical) circles in the plane is the systems in physical and biological contexts are naturally triangular-lattice packing appeared only 80 years ago; see captured by simple packing models. They have been Refs. 32 and 34 for the history of this problem. studied to help understand the symmetry, structure and One objective of this perspective is to survey recent de- physical properties of condensed matter phases, includ- velopments concerning the simplest but venerable packing liquids, glasses and crystals, as well as the associated ing model: identical frictionless spheres in the absence phase transitions.1–13 Packings also serve as excellent of gravity subject only to a nonoverlap constraint, i.e., models of the structure of multiphase heterogeneous ma- the spheres do not interact for any nonoverlapping con- terials,14–17 colloids,12,18,19 suspensions,20,21 and granu- figuration. This “stripped-down” hard-sphere model can lar media,22,23 which enables predictions of their effective be viewed as the particle-system analog of the idealized transport, mechanical and electromagnetic properties.15 Ising model for magnetic systems,39 which is regarded as Packing phenomena are also ubiquitous in biological one of the pillars of statistical mechanics.40–42 More com- contexts and occur in systems across a wide spectrum plex packing models can include interactions that extend of length scales. This includes DNA packing within the beyond hard-core distances, but our main concern here nucleus of a cell,24“crowding” of macromolecules within is the aforementioned classic version. This model enables living cells, 25 the packing of cells to form tissue,15,26–28 one to uncover unifying principles that describe a broad the fascinating spiral patterns seen in plant shoots and range of phenomena, including the nature of equilibrium flowers (phyllotaxis),29,30 and the competitive settlement states of matter (e.g., liquids and crystals), metastable of territories.15,31 and nonequilibrium states of matter (e.g., supercooled Understanding the symmetries and other mathemat- liquids and structural glasses), and jamming phenomena. ical properties of the densest sphere packings in vari- Jammed sphere packings represent an important sub-

arXiv:1805.04468v1 [cond-mat.stat-mech] 9 May 2018 ous spaces and dimensions is a challenging area of long- set of hard-sphere configurations and have attracted standing interest in discrete geometry and number the- great attention because they capture salient character- ory32–34 as well as coding theory.32,35,36 Packing prob- istics of crystals, glasses, granular media and biological lems are mathematically easy to pose, but they are no- systems22,27,39,43–46 and naturally arise in pure mathe- toriously difficult to solve rigorously. For example, in matical contexts.32,47,48 Jammed packings are those in 1611, Kepler was asked: What is the densest way to which each particle has a requisite number contacting stack equal-sized cannon balls? His solution, known as particles in order for the packing to achieve a certain “Kepler’s conjecture,” was the face-centered-cubic (fcc) level of mechanical stability. This review focuses on arrangement (the way your green grocer stacks oranges). the so-called “geometric-structure” approach to jammed Gauss37 proved that this is the densest Bravais lattice particle packings, which provides a powerful and unified 2 means to quantitatively understand such many-particle II. DEFINITIONS AND BACKGROUND systems. It incorporates not only the maximally dense packings and disordered jammed packings, but a myriad A packing P is a collection of nonoverlapping solid ob- of other significant jammed states, including maximally jects or particles of general shape in d-dimensional Eu- 39,43 random jammed states, which can be regarded to clidean space Rd. Packings can be defined in other spaces be prototypical structural glasses, as well as the lowest- (e.g., hyperbolic spaces and compact spaces, such as the 49 density jammed packings. surface of a d-dimensional sphere), but our primary focus Importantly, the simplified hard-sphere model embod- in this review is Rd.A saturated packing is one in which ies the primary structural attributes of dense many- there is no space available to add another particle of the particle systems in which steep repulsive interparticle in- same kind to the packing. The packing fraction φ is the teractions are dominant. For example, the densest sphere fraction of space Rd covered by the particles. packings are intimately related to the (zero-temperature) A d-dimensional particle is centrally symmetric if it ground states of such molecular systems50 and high- has a center C that bisects every chord through C con- pressure crystal phases. Indeed, the equilibrium hard- necting any two boundary points of the particle, i.e., the sphere model9 also serves as a natural reference system in center is a point of inversion symmetry. Examples of the thermodynamic perturbation theory of liquids char- centrally symmetric particles in Rd include spheres, el- acterized by steep isotropic repulsive interparticle inter- lipsoids and superballs (defined in Section V). A triangle actions at short distances as well short-range attractive and tetrahedron are examples of non-centrally symmet- interactions.6,51 Moreover, the classic hard-sphere model ric 2D and 3D particles, respectively. A d-dimensional provides a good description of certain classes of colloidal centrally symmetric particle for d ≥ 2 is said to pos- systems.18,52–55 Note that the hard-core constraint does sess d equivalent principal axes (orthogonal directions) not uniquely specify the hard-sphere model; there are an associated with the moment of inertia tensor if those di- infinite number of nonequilibrium hard-sphere ensembles, rections are two-fold rotational symmetry axes. Whereas some of which will be surveyed. a d-dimensional superball has d equivalent directions, a We will also review work that describes generalizations d-dimensional ellipsoid generally does not. The reader of this simplified hard-sphere model to include its nat- is referred to Ref. 39 for further discussion concerning ural extension to packings of hard spheres with a size particle symmetries. distribution. This topic has relevance to understanding A lattice Λ in Rd is a subgroup consisting of the inte- dispersions of technological importance, (e.g., solid pro- ger linear combinations of vectors that constitute a basis pellants combustion56 flow in packed beds,57 and sinter- for Rd. In the physical sciences, this is referred to as a ing of powders58), packings of biological cells, and phase Bravais lattice. The term “lattice” will refer here to a behavior of various molecular systems. For example, Bravais lattice only. Every lattice has a dual (or recipro- the densest packings of hard-sphere mixtures are inti- cal) lattice Λ∗ in which the lattice sites are specified by mately related to high-pressure phases of compounds for the dual (reciprocal) lattice vectors.32 A lattice packing 59,60 a range of temperatures. Another natural extension PL is one in which the centroids of the nonoverlapping of the hard-sphere model that will be surveyed are hard identical particles are located at the points of Λ, and all nonspherical particles in two and three dimensions. As- particles have a common orientation. The set of lattice phericity in particle shape is capable of capturing salient packings is a subset of all possible packings in Rd. In features of phases of molecular systems with anisotropic a lattice packing, the space Rd can be geometrically di- pair interactions (e.g., liquid crystals) and is also a more vided into identical regions F called fundamental cells, realistic characteristic of real granular media. In addi- each of which has volume Vol(F ) and contains the cen- tion, we will review aspects of dense sphere packings in troid of just one particle of volume v1. Thus, the lattice high-dimensional Euclidean spaces, which provide use- packing fraction is ful physical insights and are relevant to error-correcting v1 codes and information theory.32,35 φ = . (1) Vol(F ) We begin this perspective by introducing relevant definitions and background (Sec. II). This is followed by a Common d-dimensional lattices include the hypercubic d survey of work on equilibrium, metastable and nonequi- Z , checkerboard Dd, and root Ad lattices; see Ref. 32. librium packings of identical spheres and polydisperse Following Conway and Sloane32, we say two lattices are spheres in one, two and three dimensions (Sec. III). The equivalent or similar if one becomes identical to the other geometric-structure approach to jammed and unjammed possibly by a rotation, reflection, and change of scale, for packings, including their classification via order maps, which we use the symbol ≡. The Ad and Dd lattices is emphasized. Subsequently, corresponding results for are d-dimensional generalizations of the face-centered- sphere packings in high Euclidean and non-Euclidean cubic (FCC) lattice defined by A3 ≡ D3; however, for space dimensions, (Sec. IV) and packings of nonspher- d ≥ 4, they are no longer equivalent. In two dimensions, ∗ ical particles in low-dimensional Euclidean spaces (Sec. A2 ≡ A2 is the triangular lattice. In three dimensions, ∗ ∗ V) are reviewed. Finally, we describe some challenges A3 ≡ D3 is the body-centered-cubic (BCC) lattice. In and open questions for future research (Sec. VI). four dimensions, the checkerboard lattice and its dual 3

∗ are equivalent, i.e., D4 ≡ D4. The hypercubic lattice As usual, we define the nonnegative structure factor d d Z ≡ Z∗ and its dual lattice are equivalent for all d. S(k) for a statistically homogeneous packing as A periodic packing of congruent particles is obtained by ˜ placing a fixed configuration of N particles (where N ≥ S(k) = 1 + ρh(k), (5) 1) with arbitrary orientations subject to the nonoverlap- where h˜(k) is the Fourier transform of h(r) and k is the ping condition in one fundamental cell of a lattice Λ, wave vector. The nonnegativity of S(k) for all k follows which is then periodically replicated. Thus, the packing physically from the fact that it is proportional to the in- is still periodic under translations by Λ, but the N par- tensity of the scattering of incident radiation on a many- ticles can occur anywhere in the chosen fundamental cell particle system.9 The structure factor S(k) provides a subject to the overall nonoverlap condition. The packing measure of the density fluctuations in particle configura- fraction of a periodic packing is given by tions at a particular wave vector k. For any point config- Nv uration in which the minimal pair distance is some posi- φ = 1 = ρv , (2) Vol(F ) 1 tive number, such as a sphere packing, g2(r = 0) = 0 and h(r = 0) = −1, the structure factor obeys the following where ρ = N/Vol(F ) is the number density, i.e., the num- sum rule:62 ber of particles per unit volume. 1 Z For simplicity, consider a packing of N identical d- d [S(k) − 1] dk = −ρ. (6) (2π) d dimensional spheres of diameter D centered at the po- R N sitions r ≡ {r1, r2,..., rN } in a region of volume V in For a single-component system in thermal equilibrium d-dimensional Euclidean space Rd. Ultimately, we will at number density ρ and absolute temperature T , the pass to the thermodynamic limit, i.e., N → ∞, V → ∞ structure factor at the origin is directly related to the 9 such that the number density ρ = N/V is a fixed positive isothermal compressibility κT via the relation constant and its corresponding packing fraction is given by ρkBT κT = S(0), (7) where k is the Boltzmann constant. φ = ρv (D/2), (3) B 1 It is well known from Fourier transform theory that d where ρ is the number density, if a real-space radial function f(r) in R decreases sufficiently rapidly to zero for large r such that all of its even πd/2 moments exist, then its Fourier transform f˜(k), where v (R) = Rd (4) 1 Γ(1 + d/2) k ≡ |k| is a wavenumber, is an even and analytic function at k = 0. Hence, S(k), defined by (5), has an expansion is the volume of a d-dimensional sphere of radius R, and about k = 0 in any space dimension d of the general form Γ(x) is the gamma function. For an individual sphere, 2 4 the kissing or contact number Z is the number of spheres S(k) = S0 + S2k + O(k ), (8) that may simultaneously touch this sphere. In a sphere where S0 and S2 are the d-dependent constants defined packing, the mean kissing or contact number per particle, by Z, is the average of Z over all spheres. For a lattice pack- Z ∞ ings, Z = Z. For statistically homogeneous sphere pack- d−1 d n n S0 = 1 + dρs1(1) r h(r)dr ≥ 0 (9) ings in R , the quantity ρ gn(r ) is proportional to the 0 probability density associated with simultaneously find- n and ing n sphere centers at locations r ≡ {r1, r2,..., rn} d Z ∞ in R ; see Ref. 9 and references therein. With this ρs1(1) d+1 convention, each n-particle correlation function g ap- S2 = − r h(r)dr, (10) n 2d 0 proaches unity when all particles become widely separated from one another for any system without long- where range order. Statistical homogeneity implies that gn d πd/2Rd−1 s (R) == (11) is translationally invariant and therefore only depends 1 Γ(d/2 + 1) on the relative displacements of the positions with respect to some arbitrarily chosen origin of the system, is the surface area of d-dimensional sphere of radius R. i.e., gn = gn(r12, r13,..., r1n), where rij = rj − ri. This behavior is to be contrasted with that of maxi- 43 The pair correlation function g2(r) is of basic interest mally random jammed sphere packings, which possess in this review. If the system is also rotationally invari- a structure factor that is nonanalytic at k = 0 (Ref. 63), ant (statistically isotropic), then g2 depends on the radial as discussed in greater detail in Sec. III E. 64 d distance r ≡ |r| only, i.e., g2(r) = g2(r). The total cor- A hyperuniform point configuration in R is one in relation function is defined by h(r) ≡ g2(r) − 1. Impor- which the structure factor S(k) tends to zero as the tantly, we focus in this review on disordered packings in wavenumber tends to zero, i.e., which h(r) decays to zero for large |r| sufficiently rapidly 61 lim S(k) = 0, (12) so that its volume integral over all space exists. |k|→0 4 implying that single scattering of incident radiation at states as well as the associated phase transitions in molec- infinite wavelengths is completely suppressed. Equiva- ular and colloidal systems. It is well known that the pres- lently, a hyperuniform system is one in which the number sure p of a stable thermodynamic phase in Rd at packing variance σ2 (R) ≡ hN(R)2i − hN(R)i2 of particles within fraction φ and temperature T is simply related to the N + 15 a spherical observation window of radius R grows more contact value of the pair correlation function, g2(D ): slowly than the window volume, i.e., Rd, in the large-R p d−1 + limit. This class of point configurations includes per- = 1 + 2 φg2(D ). (15) fect crystals, many perfect quasicrystals and special dis- ρkBT ordered many-particle systems.64–66 Note that structure- Away from jammed states, it has been proved that the factor definition (5) and the hyperuniformity requirement mean nearest-neighbor distance between spheres, λ, is (12) dictate that the following sum rule involving h(r) bounded from above by the pressure,73 namely, λ ≤ 1 + that a hyperuniform point process must obey: 1/[2d(p/(ρkBT ) − 1]. Z Figure 1 schematically shows the 3D phase behavior ρ h(r)dr = −1. (13) in the φ-p plane. At sufficiently low densities, an in- d R finitesimally slow compression of the system at constant This sum rules implies that h(r) must exhibit negative temperature defines a thermodynamically stable liquid correlations, i.e., anticorrelations, for some values of r. branch for packing fractions up to the “freezing” point The hyperuniformity concept was generalized to incor- (φ ≈ 0.49). Increasing the density beyond the freezing point putatively results in an entropy-driven first-order porate two-phase heterogeneous media (e.g., composites, 70,74 porous media and dispersions).65,67 Here the phase vol- phase transition to a crystal branch that begins at ume fraction fluctuates within a spherical window of ra- the melting point (φ ≈ 0.55). While there is no rigorous dius R and hence can be characterized by the volume- proof that such a first-order freezing transition occurs fraction variance σ2 (R). For typical disordered two- in three dimensions, there is overwhelming simulational V evidence for its existence, beginning with the pioneer- phase media, the variance σ2 (R) for large R goes to zero V 70 −d ing work of Alder and Wainwright. Slow compression like R . However, for hyperuniform disordered two- of the system along the crystal branch must end at one phase media, σ2 (R) goes to zero asymptotically more V of the optimal (maximally dense) sphere packings with rapidly than the inverse of the window volume, i.e., faster √ φ = π/ 18 = 0.74048 ..., each of which is a jammed than R−d, which is equivalent to the following condition packing in which each particle contacts 12 others (see Sec. on the spectral density χ˜ (k):67 V III D). This equilibrium state has an infinite pressure and is putatively entropically favored by the fcc packing.75 lim χ˜V (k) = 0. (14) |k|→0 However, compressing a hard-sphere liquid rapidly, under the constraint that significant crystal nucleation is The spectral density is proportional to scattered intensity suppressed, can produce a range of metastable branches associated with “mass” (volume) content of the phases.68 whose density end points are disordered “jammed” packings,13,15,76 which can be regarded to be glasses. A rapid compression leads to a lower random jammed III. SPHERE PACKINGS IN LOW density than that for a slow compression. The most DIMENSIONS rapid compression ending in mechanically stable packing is presumably the maximally random jammed (MRJ) The classical statistical mechanics of hard-sphere sys- state43 with φ ≈ 0.64. Accurate approximate formulas tems has generated a huge collection of scientific pub- for the pressure along such metastable extensions up to 69 lications, dating back at least to Boltzmann; see also the jamming points have been obtained.15,77 This ideal Refs. 9, 70–72. Here we focus on packings of frictionless, amorphous state is described in greater detail in Sec. congruent spheres of diameter D in one, two and three III E. Note that rapid compression a hard-sphere system dimensions in the absence of gravity. is analogous to supercooling a molecular liquid. It is important to observe that the impenetrability con- Pair statistics are exactly known only in the case of 1D straint alone of this idealized hard-sphere model does not “hard-rod” systems.78 For d ≥ 2, approximate formulas uniquely specify the statistical ensemble. Hard-sphere 9 for g2(r) are known along liquid branches. Approximate systems can be in thermal equilibrium (as discussed in closures of the Ornstein-Zernike integral equation link- Sec. III A) or derived from one of an infinite number of ing the direct correlation function c(r) to the total cor- 15 nonequilibrium ensembles (see Sec. III B for examples). relation function h(r),79 such as the Percus-Yevick (PY) and hypernetted chain schemes,9,80,81 provide reasonably accurate estimates of g2(r) for hard-spheres liquids. Be- A. Equilibrium and Metastable Phase Behavior cause g2(r) decays to unity exponentially fast for liquid states, we can conclude that it must have a correspond- The phase behavior of hard spheres provides powerful ing structure factor S(k) that is an even function and insights into the nature of liquid, crystal and metastable analytic at k = 0; see Eq. (8). Also, since the leading- 5

FIG. 1. The isothermal phase behavior of the 3D hard-sphere model in the pressure-packing fraction plane, adapted from Ref. 15. Three different isothermal densification paths by which a hard- sphere liquid may jam are shown. An infinitesimal compression rate of the liquid traces out the thermodynamic equilibrium path (shown in green), including a discontinuity resulting from the first- order freezing transition to a crystal branch that ends at a maximally dense, infinite-pressure jammed state. Rapid compressions of the liquid while suppressing some degree of local order (blue curves) can avoid crystal nucleation (on short time scales) and produce a range of amorphous metastable extensions of the liquid branch that jam only at the their density maxima. FIG. 2. Pair statistics for an equilibrium hard-sphere fluid in three dimensions at φ = 0.35 as obtained from the PY approximation.9,80 Top panel: Pair correlation function g2(r) versus r/D, where D is order term S0 must be positive because the isothermal the sphere diameter. Bottom panel: The corresponding structure compressibility is positive [cf. (7)], hard-sphere liquids factor S(k) as a function of the dimensionless wavenumber kD, are not hyperuniform. Figure 2 shows g2(r) and the cor- which clearly shows nonhyperuniformity. responding structure factor S(k) in three dimensions at φ = 0.35 as obtained from the PY approximation. nonoverlapping objects into a large volume in Rd that at some initial time is empty of spheres. If an attempt to B. Nonequilibrium Disordered Packings add a sphere at some time t results in an overlap with an existing sphere in the packing, the attempt is rejected Here we briefly discuss three different nonequilibrium and further attempts are made until it can be added with- sphere packings: random sequential addition, “ghost” out overlapping existing spheres. One can stop the addi- random sequential addition, and random close packing. tion process at any finite time t, obtaining an RSA configuration with a time-dependent packing fraction φ(t) but this value cannot exceed the maximal saturation packing 1. Random Sequential Addition fraction φs = φ(∞) that occurs in the infinite-time and thermodynamic limits. Even at the saturation state, the Perhaps one of the most well-known nonequilibrium spheres are never in contact and hence are not jammed in packing models is the random sequential addition (RSA) the sense described in Sec. III C; moreover, the pair cor- packing procedure, which is a time-dependent process relation function g2(r) decays to unity for large r super- 93 that generates disordered sphere packings in d; see Refs. exponentially fast. The latter property implies that the R d 82–90. The RSA packing process in the first three space structure factor S(k) of RSA packings in R must be an 89 dimensions has been used to model a variety of different even and analytic function at k = 0, as indicated in condensed phases, including protein adsorption,85 poly- Eq. (8). It is notable that saturated RSA packings are 89,90 mer oxidation,91 particles in cell membranes84 and ion not hyperuniform for any finite space dimension. implantation in semiconductors.92 In the one-dimensional case, also known as the “car- In its simplest rendition, a RSA sphere packing is pro- parking” problem, the saturation packing fraction was 82 duced by randomly, irreversibly, and sequentially placing obtained analytically: φs = 0.7475979202.... However, 6 for d ≥ 2, φs in the thermodynamic limit has only been d and all realizable densities, which has implications for estimated via numerical simulations. The most precise high-dimensional packings, as discussed in Sec. IV. 90 numerical study to date has yielded φs = 0.5470735 ± 0.0000028 for d = 2 and φs = 0.3841307 ± 0.0000021 for d = 3. Estimates of φs in higher dimensions are discussed in Sec. IV. RSA packings have also been examined for spheres with a size distribution94 and other particle 85,95,96 97 96 shapes, including squares, rectangles, ellipses, 3. Random Close Packing superdisks,98 and polygons,99,100 in R2 and spheroids,101 spherocylinders96 and cubes102,103 in R3. The “random close packing” (RCP) notion was pio- neered by Bernal2,3 to model the structure of liquids, and has been one of the more persistent themes with a venerable history.105–112 Two decades ago, the prevailing notion of the RCP state was that it is the maximum density that a large, random collection of congruent (identical) spheres can attain and that this density is a well- defined quantity. This traditional view has been summa- rized as follows: “Ball bearings and similar objects have been shaken, settled in oil, stuck with paint, kneaded in- side rubber balloons–and all with no better result than FIG. 3. (Color online) Examples of two nonequilibrium pack- (a packing fraction of) ... 0.636”; see Ref. 106. ing models in two dimensions under periodic boundary conditions. 43 Left panel: A configuration of the standard RSA packing at satu- Torquato, Truskett and Debenedetti have argued ration with φs ≈ 0.5. Right panel: A configuration of a ghost RSA that this RCP-state concept is actually ill-defined be- packing at a packing fraction φ very near its maximal value of 0.25. cause “randomness” and “closed-packed” were never de- Note that the packing is clearly unsaturated (as defined in Sec. II) fined and, even if they were, are at odds with one another. and there are no contacting particles. Using the Lubachevsky-Stillinger (LS)113 molecular- dynamics growth algorithm to generate jammed packings, it was shown43 that fastest particle growth rates generated the most disordered sphere packings with φ ≈ 2. “Ghost” Random Sequential Addition 0.64, but that by slowing the growth rates larger packing fractions could be continuously achieved up to the densest value φ = 0.74048 ... such that the degree There is a generalization of the aforementioned stan- max of order increased monotonically with φ. These results dard RSA process, which can be viewed as a “thinning” demonstrated that once can increase the packing fraction process of a Poisson distribution of sphere and is pa- by an arbitrarily small amount at the expense of corre- rameterized by the positive constant κ that lies in the spondingly small increases in order and thus the notion closed interval [0, 1]; see Ref. 104. When κ = 0, one of RCP is ill-defined as the highest possible density that recovers the standard RSA process and when κ = 1, one a random sphere packing can attain. To remedy these obtains the “ghost” random sequential addition (RSA) flaws, Torquato, Truskett and Debenedetti43 replaced the process that enables one to obtain exactly not only the notion of “close packing” with “jamming” categories (de- time-dependent packing fractions but all of the n-particle fined precisely in Sec. III C) and introduced the notion of correlation functions g for any n and d. The reader is n an “order metric” to quantify the degree of order (or dis- referred to Ref. 104 for details about the general model. order) of a single packing configuration. This led them to In the ghost RSA process, one attempts to place supplant the concept of RCP with the maximally random spheres into an initially empty region of space randomly, jammed (MRJ) state, which is defined, roughly speaking, irreversibly and sequentially. However, here one keeps to be that jammed state with a minimal value of an order track of any rejected sphere, which is called a “ghost” metric (see Sec. III C 4 for details). This work pointed sphere. No additional spheres can be added whenever the way toward a quantitative means of characterizing they overlap an existing sphere or a ghost sphere. The all packings, namely, the geometric-structure approach. packing fraction at time t for spheres of unit diameter d is given by φ(t) = [1 − exp(−v1(1)t)]/2 , where v1(R) is We note that whereas the LS packing protocol with the volume of sphere of radius R. Thus, we see that as a fast growth rate typically leads to disordered jammed t → +∞, φ = 2−d, which is appreciably smaller than states in three dimensions, it invariably produces highly the RSA saturation packing fraction φs in low dimen- crystalline “collectively” jammed packings in two dimensions; see Fig. 3 for 2D examples. Nonetheless, it is no- sions. Figure 4 vividly illustrates the differences between table that the ghost RSA process is the only hard-sphere the textures produced in three and in two dimensions packing model that is exactly solvable for any dimension (see Section VII for further remarks). 7

can be simultaneously displaced so that its mem- bers move out of contact with one another and with the remainder set; and 3. Strict jamming: Any collectively jammed configuration that disallows all uniform volume- nonincreasing strains of the system boundary is strictly jammed. These hierarchical jamming categories are closely related to the concepts of “rigid” and “stable” packings found in the mathematics literature,128 and imply that there FIG. 4. Typical protocols, used to generate disordered sphere can be no “rattlers” (i.e., movable but caged particles) packings in three dimensions, produce highly crystalline packings in in the packing. The jamming category of a given pack- two dimensions. Left panel: A highly ordered collectively jammed ing depends on the boundary conditions employed; see configuration (Sec. III C 1) of 1000 disks with φ ≈ 0.88 produced Refs. 39 and 123 for specific examples. Rigorous and using the Lubachevsky-Stillinger (LS) algorithm113 with a fast expansion rate.114 Right panel: A 3D MRJ-like configuration of 500 efficient linear-programming (LP) algorithms have been spheres with φ ≈ 0.64 produced using the LS algorithm with a fast devised to assess whether a particular sphere packing is expansion rate.43 locally, collectively, or strictly jammed.114,129 These jamming categories can now be ascertained in real-system experiments by applying the LP tests to configurational C. Geometric-Structure Approach to Jammed coordinates of a packing determined via a variety of Packings imaging techniques, including tomography130, confocal microscopy131 and magnetic resonance imaging. 132 A “jammed” packing is one in which each particle is in contact with its nearest neighbors such that mechanical stability of a specific type is con- 2. Polytope Picture and Pressure in Jamming Limit ferred to the packing, as detailed below. Two conceptual approaches for their study have emerged. One A packing of N hard spheres of diameter D in is the “ensemble” approach,2,3,22,44,45,115–122 which for a jammed framework in Rd is specified by a Nd- a given packing protocol aims to understand typi- dimensional configurational position vector R = rN ≡ cal configurations and their frequency of occurrence. {r1,..., rN}. Isostatic jammed packings possess the The other, more recently, is the “geometric-structure” minimal number of contacts for a jamming category and approach,43,49,114,123–127 which emphasizes quantitative boundary conditions.133 The relative differences between characterization of single-packing configurations without isostatic collective and strict jammed packings diminish regard to their occurrence frequency in the protocol used as N becomes large, and since the number of degrees of to produce them. Here we focus on the latter approach, freedom is essentially equal to Nd, an isostatic packing which enables one to enumerate and classify packings has a mean contact number per particle, Z, equal to 2d in with a diversity of order/disorder and packings fractions, the large-N limit. Packings having more and fewer con- including extremal packings, such as the densest sphere tacts than isostatic ones are hyperstatic and hypostatic, packing and MRJ packings. respectively. Sphere packings that are hypostatic cannot be collectively or strictly jammed.127 Consider decreasing the density slightly in a sphere 1. Jamming Categories packing that is at least collectively jammed by reducing the particle diameter by ∆D, so that the packing fraction d Three broad and mathematically precise “jamming” is lowered to φ = φJ (1 − δ) , where δ = ∆D/D 1 is categories of sphere packings can be distinguished de- called the jamming gap or distance to jamming. There is pending on the nature of their mechanical stability.64,123 a sufficiently small δ that does not destroy the jamming In order of increasing stringency (stability), for a finite confinement property. For fixed N and sufficiently small sphere packing, these are the following: δ, it can be shown asymptotically, through first order in δ), that the set of displacements accessible to the packing 1. Local jamming: Each particle in the packing is lo- approaches a convex limiting polytope (high-dimensional cally trapped by its neighbors (at least d + 1 con- polyhedron) P.5,72 This polytope P is determined from tacting particles, not all in the same hemisphere), the linearized impenetrability equations114,129 and is nec- i.e., it cannot be translated while fixing the posi- essarily bounded for a jammed configuration. tions of all other particles; Now consider adding thermal kinetic energy to a nearly jammed sphere packing in the absence of rattlers. While 2. Collective jamming: Any locally jammed configura- the system will not be globally ergodic over the full sys- tion is collectively jammed if no subset of particles tem configuration space and thus not in thermodynamic 8 equilibrium, one can still define a macroscopic pressure p cell as well as the centroids and orientations of the par- for the trapped but locally ergodic system by considering ticles. The so-called Torquato-Jiao (TJ) sphere-packing time averages as the system executes a tightly confined algorithm142 is a sequential linear-programming (SLP) motion around the particular jammed configuration RJ. solution of the ASC optimization problem for the special The probability distribution Pf (f) of the time-averaged case of packings of spheres with a size distribution for interparticle forces f has been rigorously linked to the which linearization of the design variables is exact. The contact value r = D of the pair correlation function deterministic SLP solution in principle always leads to 133 g2(r). Moreover, the available (free) configuration vol- strictly jammed packings up to a high numerical toler- ume scales with the jamming gap δ such that the reduced ance with a wider range of densities and degree of disor- pressure is asymptotically given by the free-volume equa- der than packings produced by the LS algorithm. From tion of state:5,72,133 an initial configuration, the TJ algorithm leads to a mechanically stable local “energy” minimum (local density p 1 d ∼ = , (16) maximum), which in principle is the inherent structure ρkBT δ 1 − φ/φJ associated with the starting initial many-particle configuration; see Fig. 5. A broad range of inherent struc- where T is the absolute temperature and ρ is the number tures can be obtained across space dimensions, including density. Relation (16) is remarkable, since it enables one locally maximally dense and mechanically stable pack- to determine accurately the true jamming density of a ings, such as MRJ states,142–145 disordered hyperstatic given packing, even if the actual jamming point has not packings146 and the globally maximally dense inherent quite yet been reached, just by measuring the pressure structures,142,147–150 with very small computational cost, and extrapolating to p = +∞. This free-volume form provided that the system sizes are not too large. has been used to estimate the equation of state along It is notable the TJ algorithm creates disordered “metastable” extensions of the hard-sphere fluid up to jammed sphere packings that are closer to the ideal MRJ the infinite-pressure endpoint, assumed to be disordered state than previous algorithms.143 It was shown that the 15,77 jammed states (see Fig. 1 and Sec. III A). Kamien rattler concentration of these packings converges towards 134 and Liu assumed the same free-volume form to fit data 1.5% in the infinite-system limit, which is markedly lower for the pressure of “metastable” hard-sphere states. than previous estimates for the MRJ state using the LS protocol (with about 3% rattlers).

3. Hard-Particle Jamming Algorithms

For many years, the Lubachevsky-Stillinger (LS) algo- rithm113 has served as the premier numerical scheme to generate a wide spectrum of dense jammed hard-sphere packings with variable disorder in both two and three dimensions.43,124,135 This is an event-driven or collision- driven molecular-dynamics algorithm: an initial configuration of spheres of a given size within a periodic cell are given initial random velocities and the motion of the spheres are followed as they collide elastically and also grow uniformly until the spheres can no longer ex- pand. The final density is generally inversely related to the particle growth rate. This algorithm has been generalized by Donev, Torquato, and Stillinger136,137 to generate jammed packings of smoothly shaped nonspherical particles, including ellipsoids,127,137 superdisks,138, and superballs.139 Event-driven packing protocols with growing particles, however, do not guarantee jamming of the final packing configuration, since jamming is not explicitly incorporated as a termination criterion. FIG. 5. (Color online). A schematic diagram of inherent The task of generating dense packings of particles of structures (local density maxima) for sphere packings of N general shape within an adaptive periodic fundamental spheres, as taken from Ref. 142. The horizontal axis la- cell has been posed by Torquato and Jiao140,141 as an beled XN stands for the entire set of centroid positions, and optimization problem called the adaptive-shrinking cell −φ (“energy”) decreases downward. The jagged curve is the (ASC) scheme. The negative of the packing fraction, −φ boundary between accessible configurations (yellow) and in- (which can be viewed as an “energy”) is to be minimized accessible ones (blue). The deepest point of the accessible subject to constraints, and the design variable are the configurations corresponds to the maximal density packings shape and size of the deformable periodic fundamental of hard spheres. 9

4. Order Metrics der metrics have both strengths and weaknesses. This raises the question of what are the characteristics of a The enumeration and classification of both ordered and good order metric? It has been suggested that a good disordered sphere packings is an outstanding problem. scalar order metric should have the following additional 39,124 Since the difficulty of the complete enumeration of pack- properties: (1) sensitivity to any type of ordering ing configurations rises exponentially with the number without bias toward any reference system; (2) ability to of particles, it is desirable to devise a small set of in- reflect the hierarchy of ordering between prototypical sys- tensive parameters that can characterize packings well. tems given by common physical intuition (e.g., perfect One obvious property of a sphere packing is the pack- crystals with high symmetry should be highly ordered, ing fraction φ. Another important characteristic of a followed by quasicrystals, correlated disordered packings packing is some measure of its “randomness” or degree without long-range order, and finally spatially uncorre- of disorder. Devising such measures is a highly non- lated or Poisson distributed particles); (3) incorporation trivial challenge, but even the tentative solutions that of both the variety of local coordination patterns as well have been put forth during the last two decades have as the spatial distribution of such patterns should be in- been fruitfully applied to characterize not only sphere cluded; and (4) the capacity to detect translational and packings43,64,124,135 but also simple liquids,13,135,151,152 orientational order at any length scale. Moreover, any glasses,135,153 water,154,155 disordered ground states,156 useful set of order metrics should consistently produce re- 15,43 random media,157 and biological systems.27 sults that are positively correlated with one another. It is quite reasonable to consider “entropic” measures The development of improved order metrics deserves con- to characterize the randomness of packings. However, as tinued research attention. pointed out by Kansal et al.124, a substantial hurdle to Order metrics and maps have been fruitfully extended overcome in implementing such an order metric is the to characterize the degree of structural order in con- necessity to generate all possible jammed states or, at densed phases in which the constituent particles (jammed least, a representative sample of such states in an un- or not) possess both attractive and repulsive interac- biased fashion using a “universal” protocol in the large- tions. This includes the determination of the order maps of models of simple liquids, glasses and crystals with system limit, each of which is an intractable problem. 13,135,151,152 154,155 Even if such a universal protocol could be developed, isotropic interactions, models of water, disordered ground states of long-ranged isotropic pair however, the issue of what weights to assign the result- 156 160 ing configurations remains. Moreover, there are other potentials, and models of amorphous polymers. basic problems with the use of entropic measures as order metrics, as we will discuss in Sec. III E, including its significance for certain 2D hard-disk packings. D. Order Maps and Extremal Packings We know that a many-body system of N particles is completely characterized statistically by its N-body The geometric-structure classification naturally em- probability density function P (R; t) that is associated phasizes that there is a great diversity in the types of with finding the N-particle system with configuration R attainable packings with varying magnitudes of overall at some time t. Such complete information is virtually order, packing fraction φ, mean contact number per par- never available for large N and, in practice, one must set- ticle, Z, among many other intensive parameters. Any tle for reduced information, such as a scalar order met- attainable hard-sphere configuration, jammed or not, can ric ψ. Any order metric ψ conventionally possesses the be mapped to a point in this multidimensional parame- following three properties: (1) it is a well-defined scalar ter space that we call an order map. The use of “order function of a configuration R; (2) it is subject typically to maps” in combination with the mathematically precise the normalization 0 ≤ ψ ≤ 1; and, (3) for any two config- “jamming categories” enable one to view and charac- urations RA and RB, ψ(RA) > ψ(RB) implies that con- terize individual packings, including the densest sphere figuration RA is to be considered as more ordered than packing (e.g., Kepler’s conjecture in 3D) and MRJ pack- configuration RB. The set of order metrics that one se- ings as extremal states in the order map for a given jam- lects is unavoidably subjective, given that there appears ming category. Indeed, this picture encompasses not only to be no single universally applicable scalar metric capa- these special jammed states, but an uncountably infinite ble of describing order across all lengths scales. However, number of other packings, some of which have only re- one can construct order metrics that lead to consistent cently been identified as physically significant, e.g., the results, as we will be discussed below. jamming-threshold states49 (least dense jammed pack- Many relevant order metrics have been devised. While ings) as well as states between these and MRJ. a comprehensive discussion of this topic is beyond The simplest order map is the one that maps any hard- the scope of this article, it is useful to here to note sphere configuration to a point in the φ-ψ plane. This some order metrics that have been identified, includ- two-parameter description is but a very small subset of ing bond-orientational order metrics in two158 and three the relevant parameters that are necessary to fully char- dimensions,159 translational order metrics,43,135,156 and acterize a configuration, but it nonetheless enables one hyperuniformity order metrics.64,65 These specific or- to draw important conclusions. Figure 6 shows such 10

portance of analyzing individual packings, regardless of their frequency of occurrence in the space of jammed configurations, as advocated by the geometric-structure approach. The tunneled crystals are subsets of the densest packings but are permeated with infinitely long chains of particle vacancies.49 Every sphere in a tunneled crystal contacts 7 immediate neighbors. Interestingly, the tunneled crystals exist at the edge of mechanical stability, since removal of any one sphere from the interior would cause the entire packing to collapse. The tunneled crystals are magnetically frustrated structures and Burnell and Sondhi162 showed that their underlying topologies 3 FIG. 6. Schematic order map of sphere packings in R in the greatly simplify the determination of their antiferromag- density-order (φ-ψ) plane. White and blue regions contain the at- netic properties. In R2, the “reinforced” kagomépacking tainable packings, blue regions represent the jammed subspaces, with exactly four contacts per particle (isostatic value) is and dark shaded regions contain no packings. The boundaries of 114 evidently the√ lowest density strictly jammed packing the jammed region delineate extremal structures. The locus of 0 points A-A0 correspond to the lowest-density jammed packings. with φmin = 3π/8 = 0.68017 ... and so the line A-A in 3 2 The locus of points B-B0 correspond to the densest jammed pack- R collapses to a single point A in R . ings. Points MRJ represent the maximally random jammed states, i.e., the most disordered states subject to the jamming constraint. E. Maximally Random Jammed States

Among all strictly jammed sphere packings in Rd, the one that exhibits maximal disorder (minimizes some given order metric ψ) is of special interest. This is called the maximally random jammed (MRJ) state43; see Fig. 6. The MRJ state is automatically compromised by passing either to the maximal packing fraction (fcc and its stacking variants) or the minimal possible density for strict FIG. 7. Three different extremal strictly jammed packings in 3 jamming (tunneled crystals), thereby causing any rea- R identified in Fig. 6, as taken from Ref. 39. Left panel: (A) A tunneled crystal with Z = 7 . Middle panel: MRJ packing sonable order metric to rise on either side. A variety of with Z = 6 (isostatic). Right panel: (B) FCC packing with sensible order metrics produce a 3D MRJ state with a Z = 12. packing fraction φMRJ ≈ 0.64 (see Ref. 124), close to the traditionally advocated density of the RCP state, and with an isostatic mean contact number Z = 6 (see Ref. an order map that delineates the set of strictly jammed 133). This consistency among the different order met- packings43,49,114,124 from non-jammed packings in three rics speaks to their utility, even if a perfect order metric dimensions. The boundaries of the jammed region de- has not yet been identified. However, the density of the lineate extremal structures (see Fig. 7). The densest MRJ state is not sufficient to fully specify it. It is possi- 38 0 sphere packings,√ which lie along the locus B-B with ble to have a rather ordered strictly jammed packing at 114,123 124 φmax = π/ 18 ≈ 0.74, are strictly jammed. Point this very same density, as indicated in Fig. 6. The B corresponds to the face-centered-cubic (fcc) packing, MRJ state refers to a single packing that is maximally i.e., it is the most ordered and symmetric densest packing, disordered subject to the strict jamming constraint, re- implying that their shear moduli are infinitely large.125 gardless of its probability of occurrence in some pack- The most disordered subset of the stacking variants of ing protocol. Thus, the MRJ state is conceptually and the fcc packing is denoted by point B0. In two dimen- quantitatively different from random close packed (RCP) sions, the strictly jammed triangular lattice is the unique packings2,3, which, more recently, have been suggested to densest packing161 and so the line B-B0 in R3 collapses be the most probable jammed configurations within an to a single point B in R2. ensemble.44 The differences between these states are even The least dense strictly jammed packings are conjec- starker in two dimensions, e.g., MRJ packings of identical 3 2 tured to be the “tunneled crystals” in R with φmin = circular disks in R have been shown to be dramatically 2φmax/3 = 0.49365 ..., corresponding to the locus of different from their RCP counterparts, including their re- points A-A0.49 These infinitely-degenerate sparse struc- spective densities, average contact numbers, and degree tures were analytically determined by appropriate stack- of order,144 as detailed below. ings of planar “honeycomb” layers of spheres. These con- The MRJ state under the strict-jamming constraint stitute a set of zero measure among the possible packings is a prototypical glass49 in that it is maximally disor- with the same density and thus are virtually impossible dered (according to a variety of order metrics) without to discover using packing algorithms, illustrating the im- any long-range order (Bragg peaks) and perfectly rigid 11

to the same class as Fermi-sphere point processes164 and superfluid helium in its ground state.165,166 This nonanalytic behavior at |k| = 0 implies that MRJ packings are characterized by negative “quasi-long-range” (QLR) pair correlations in which the total correlation function h(r) decays to zero asymptotically like −1/|r|d+1; see Refs. 62, 63, 145, 167–169. The QLR hyperuniform behavior distinguishes the MRJ state strongly from that of the equilibrium hard-sphere fluid170, which possesses a structure factor that is analytic at k = 0 [cf. (8)], and thus its h(r) decays to zero exponentially fast for large r; see Ref. 39. Interestingly, the hyperuniformity concept enables one to identify a static nonequilibrium growing length scale in overcompressed (rapidly compressed) hard-sphere systems as a function of φ up to the jammed glassy state.168,169 This led to the identification of static nonequilibrium growing length scales in supercooled liquids on approaching the glass transition.171 The following is a general conjecture due to Torquato and Stillinger64 concerning the conditions under which certain strictly jammed packings are hyperuniform: Conjecture: Any strictly jammed saturated infinite packing of identical spheres in Rd is hyperuniform.

To date, there is no rigorously known counterexample to this conjecture. Its justification and rigorously known FIG. 8. Pair statistics for packings of spheres of diameter D in the examples are discussed in Refs. 62 and 145. immediate neighborhood of the 3D MRJ state with φ ≈ 0.64. Top 133 A recent numerical study of necessarily finite disor- panel: The pair correlation function g2(r) versus r/D − 1. The dered packings calls into question the connection be- split second peak, the discontinuity at twice the sphere diameter, 172 and the divergence near contact are clearly visible. Bottom panel: tween jamming and perfect hyperuniformity. It is The corresponding structure factor S(k) as a function of the di- problematic to try to test this conjecture using cur- mensionless wavenumber kD/(2π) for a million-particle packing.63 rent packing protocols to construct supposedly disor- The inset shows that S(k) tends to zero (within numerical error) dered jammed states for a variety of reasons. First, we linearly in k hence an MRJ packing is effectively hyperuniform. stress that the conjecture eliminates packings that may have a rigid backbone but possess “rattlers” - – a sub- (i.e., the elastic moduli are unbounded.39,125) Its pair tle point that has not been fully appreciated in numer- 172–174 correlation function can be decomposed into a Dirac- ical investigations. Current numerically generated delta-function contribution from particle contacts and a disordered packings that are putatively jammed tend to 44,63,142,143,175 continuous-function contribution gcont(r): contain a small concentration of rattlers; 2 because of these movable particles, the entire packing Z cont cannot be deemed to be “jammed.” Moreover, it has been g2(r) = δ(r − D) + g2 (r), (17) ρs1(D) recently shown that various standard packing protocols struggle to reliably create packings that are jammed for where s1(R) is given by (11), δ(r) is a radial Dirac delta even modest system sizes.145 Although these packings ap- function and Z = 2d. The corresponding structure factor pear to be jammed by conventional tests, rigorous linear- in the long-wavenumber limit is programming jamming tests114,129 reveal that they are

d−1 not. Recent evidence has emerged that suggest that de- 2Z 2π 2 S(k) ∼ 1+ cos[kD−(d−1)π/4] (k → ∞). viations from hyperuniformity in putative MRJ packings s1(1) kD also can in part be explained by a shortcoming of the nu- (18) merical protocols to generate exactly-jammed configura- cont 145 For d = 3, g2 (r) possesses the well-known feature of a tions as a result of a type of “critical slowing down” as split second peak,163 with a prominent discontinuity at the packing’s collective rearrangements in configuration twice the sphere diameter, as shown in Fig. 8. Inter- space become locally confined by high-dimensional “bot- estingly, an integrable power-law divergence (1/rα with tlenecks” through which escape is a rare event. Thus, α ≈ 0.4) exists for near contacts.63,133 Moreover, an MRJ a critical slowing down implies that it becomes increas- packing in Rd has a structure factor S(k) that tends to ingly difficult numerically to drive the value of S(0) ex- zero linearly in |k| (within numerical error) in the limit actly down to its minimum value of zero if a true jammed |k| → 0 and hence is hyperuniform (see Fig. 8), belonging critical state could be attained; typically,63 S(0) ≈ 10−4. 12

Moreover, the inevitable presence of even a small frac- tocols, “entropic measures” of disorder would identify tion of rattlers generated by current packing algorithms these as the most disordered, a misleading conclusion. destroys perfect hyperuniformity. In summary, the dif- Recently, Atkinson et al.144 applied the TJ algorithm to ficulty of ensuring jamming in disordered packings as generate relatively large MRJ disk packings with exactly the particle number becomes sufficiently large to access isostatic (Z = 4) jammed backbones and a packing frac- the small-wavenumber regime in the structure factor as tion (including rattlers) of φMRJ ≈ 0.827. Uncovering well as the presence of rattlers that degrade hyperunifor- such disordered jammed packings of identical hard disks mity makes it extremely challenging impossible to test challenges the traditional notion that the most proba- the Torquato-Stillinger jamming-hyperuniformity conjec- ble distribution is necessarily correlated with randomness ture on disordered jammed packings via current numer- and hence the RCP state. An analytical formula was ical protocols. This raises the possibility that the ideal derived for MRJ packing fractions of more general 2D 181 MRJ packings have no rattlers, and provides a challenge packings, yielding the prediction φMRJ = 0.834 in the to develop packing algorithms that produce large disor- monodisperse-disk limit, which is in very good agreement dered strictly jammed packings that are rattler-free. with the aforementioned recent numerical estimate.144 A variety of different attributes of MRJ packings generated via the TJ packing algorithm have been investigated in separate studies. In the first paper of a three- F. Packings of Spheres With a Size Distribution part series, Klatt and Torquato176 ascertained Minkowski correlation functions associated with the Voronoi cells of Sphere packings with a size distribution (polydisper- such MRJ packings and found that they exhibited even sity), sometimes called hard-sphere mixtures, exhibit in- stronger anti-correlations than those shown in the stan- triguing structural features, some of which are only be- dard pair-correlation function.176 In the second paper of ginning to be understood. Our knowledge of sphere pack- this series,177 a variety of different correlation functions ings with a size distribution is very limited due in part to that arise in rigorous expressions for the effective physi- the infinite-dimensional parameter space, i.e., all particle cal properties of MRJ sphere packings were computed. In size ratios and relative concentrations. It is known, for the third paper of this series,178 these structural descrip- example, that a relatively small degree of polydispersity tors were used to estimate effective transport and elec- can suppress the disorder-order phase transition seen in tromagnetic properties of composites composed of MRJ monodisperse hard-sphere systems (see Fig. 1).182 Size sphere packings dispersed throughout a matrix phase and polydispersity constitutes a fundamental feature of the showed, among other things, that electromagnetic waves microstructure of a wide class of dispersions of techno- in the long-wavelength limit can propagate without dissi- logical importance, including those involved in composite pation, as generally predicted in Ref. 179. In a separate solid propellant combustion,56 flow in packed beds,57 sin- study, Ziff and Torquato180 determined the site and bond tering of powders,58 colloids,18 transport and mechanical percolation thresholds of TJ generated MRJ packings to properties of particulate composite materials.15 Packings be pc = 0.3116(3) and pc = 0.2424(3), respectively. of biological cells in tissue are better modeled by assum- 27,28 Not surprisingly, ensemble methods that produce ing that the spheres have a size distribution. “most probable” configurations typically miss interest- Generally, we allow the spheres to possess a continu- ing extremal points in the order map, such as the locus of ous distribution in radius R, which is characterized by points A-A0 and the rest of the jamming-region boundary. a probability density function f(R). The average of any R ∞ However, numerical protocols can be devised to yield un- function w(R) is defined by hw(R)i = 0 w(R)f(R)dR. usual extremal jammed states. For example, disordered The overall packing fraction φ is then defined as jammed packings can be created in the entire non-trivial range of packing fraction 0.6 < φ < 0.74048 ...43,124,146 φ = ρhv1(R)i, (19) Thus, in Fig. 6, the locus of points along the boundary of the jammed set to the left and right of the MRJ state where ρ is the total number density, v1(R) is given by 142 (4). Two continuous probability densities that have been are achievable. The TJ algorithm was applied to yield 183 184 disordered strictly jammed packings146 with φ as low as widely used are the Schulz and log-normal distribu- 0.60, which are overconstrained with Z ≈ 6.4, and hence tions. One can obtain corresponding results for spheres are more ordered than the MRJ state. with M discrete different sizes from Eq. (19) by letting 15,111 It is well known that lack of “frustration” in 2D M X ρi analogs of 3D computational and experimental protocols f(R) = δ(R − R ), (20) ρ i that lead to putative RCP states result in packings of i=1 identical disks that are highly crystalline, forming rather large triangular coordination domains (grains). Such a where ρi and Ri are number density and radius of type-i 1000-particle packing with φ ≈ 0.88 is depicted in the particles, respectively, and ρ is the total number density. right panel of Fig. 4, and is only collectively jammed at Therefore, the overall volume fraction using (19) is given PM (i) (i) this high density. Because such highly ordered packings by φ = i=1 φ , where φ = ρiv1(Ri) is the packing are the most probable outcomes for these typical pro- fraction of the ith component. 13

1. Equilibrium and Metastable Behavior nonadditive hard spheres with Coulombic interactions enable one to generate a very wide class of disordered 27 The problem of determining the equilibrium phase be- hyperuniform as well as “multihyperuniform” systems 199,200 havior of hard sphere mixtures is substantially more chal- at positive temperatures. lenging and richer than that for identical hard spheres, including the possibilities of metastable or stable fluid- fluid and/or solid-solid phase transitions (apart from sta- 2. Maximally Random Jammed States ble fluid or crystal phases). While much research remains to be done, considerable progress has been made over the The study of dense disordered packings of 3D poly- years,54,55,185–197 which we only briefly touch upon here disperse additive spheres has received considerable for both additive and nonadditive cases. In additive hard- attention.201–206 However, these investigations did not sphere mixtures, the distance of closest approach between consider their mechanical stability via our modern un- the centers of any two spheres is the arithmetic mean of derstanding of jamming. Not surprisingly, the determi- their diameters. By contrast, in nonadditive hard-sphere nation of the MRJ state for an arbitrary polydisperse mixtures, the distance of closest approach between any sphere packing is a challenging open question, but some two spheres is generally no longer the arithmetic mean. progress has been made recently, as described below. Additive mixtures have been studied both theoretically Jammed states of polydisperse spheres, whether disor- and computationally. Lebowitz exactly obtained the pair dered or ordered, will generally have higher packing frac- correlation functions of such systems with M compo- tions when “alloyed” than their monodisperse counter- nents within the Percus-Yevick approximation.185 Accu- parts. The first investigation that attempted to generate rate approximate equations of state under liquid condi- 3D amorphous jammed sphere packings with a polydis- tions have been found for both discrete188,195 and con- persity in size was carried out by Kansal et al.207 using tinuous189,198 size distributions with additive hard cores. the LS packing algorithm. It was applied to show that Fundamental-measure theory can provide useful insights disordered binary jammed packings with a small-to-large about the phase behavior of hard-sphere mixtures.191,197 size ratio α and relative concentration x can be obtained This theory predicts the existence of stable fluid-fluid co- whose packing fractions exceeds 0.64 and indeed can at- existence for sufficiently large size ratios in additive bi- tain φ = 0.79 for α = 0.1 (the smallest value considered). nary hard-sphere systems, while numerical simulations54 Chaudhuri et al.208 numerically generated amorphous 50- indicate that such phase separation is metastable with 50 binary packings with packing fractions in the range respect to fluid-crystal coexistence and also shows stable 0.648 ≤ φ ≤ 0.662 for α = 1.4. It is notable Clusel et solid-solid coexistence. Nonetheless, it remains an open al.209 carried out a series of experiments to visualize and theoretical question whether a fluid-fluid demixing tran- characterize 3D random packings of frictionless polydis- sition exists in a binary mixture of additive hard spheres perse emulsion droplets using confocal microscopy. for any size ratio and relative composition. In order to Until recently, packing protocols that have attempted quantify fluid-crystal phase coexistence, one must know to produce disordered binary sphere packings have been the densest crystal structure, which is highly nontrivial, limited in producing mechanically stable, isostatic pack- especially for large size ratios, although recent progress ings across a broad spectrum of packing fractions. Many has been made; see Sec. III F 2. previous simulation studies of disordered binary sphere The Widom-Rowlinson model is an extreme case of packings have produced packings that are not mechan- nonadditivity in which like species do not interact and ically stable202,206,210 and report coordination numbers unlike species interact via a hard-core repulsion.186 As as opposed to contact numbers. Whereas a coordination the density is increased, this model exhibits a fluid-fluid number indicates only proximity of two spheres, a con- demixing transition in low dimensions and possesses a tact number reflects mechanical stability and is derived critical point that is in the Ising universality class.192,193 from a jammed network.39,129 More general nonadditive hard-sphere mixtures in which Using an the TJ linear programming packing all spheres interact have been studied. Mixtures of hard algorithm,142 Hopkins et al.211 were recently able to gen- spheres with positive nonadditivity (unlike-particle diam- erate high-fidelity strictly jammed, isostatic, disordered eters greater than the arithmetic mean of the correspond- binary packings with an anomalously large range of packing like-particle diameters) can exhibit fluid-fluid demixing fractions, 0.634 ≤ φ ≤ 0.829, with the size ratio re- ing transition54,194 that belongs to the Ising universal- striction α ≥ 0.1. These packings are MRJ like due to ity class,196 while those with negative nonadditivity have the nature of the TJ algorithm and the use of RSA ini- tendencies to form alloyed (mixed) fluid phases.190 For tial conditions. Additionally, the packing fractions for certain binary mixtures of hard sphere fluids with non- certain values of α and x approach those of the corre- additive diameters, the pair correlation function has been sponding densest known ordered packings.147,148 These determined in the Percus-Yevick approximation.187,190 findings suggest that these high-density disordered pack- Fluid phases of hard sphere mixtures, like their ings should be good glass formers for entropic reasons monodisperse counterparts, are not hyperuniform. How- and hence may be easy to prepare experimentally. The ever, multicomponent equilibrium plasmas consisting of identification and explicit construction of binary packings 14 with such high packing fractions could have important Even the characterization of the densest jammed bi- practical implications for the design of improved solid nary sphere packings offers great challenges and is very propellants, concrete, and ceramics. In this connection, far from completion. Here we briefly review some work a recent study of MRJ binary sphere packings under con- concerning such packings in two and three dimensions. 212 finement is particularly relevant. Ideally, it is desired to obtain φmax as a function of α The LS algorithm has been used successfully to gen- and x. In practice, we have a sketchy understanding of erate disordered strictly jammed packings of binary the surface defined by φmax(α, x). disks213 with φ ≈ 0.84 and α−1 = 1.4. By explicitly Among the 2D and 3D cases, we know most about the constructing an exponential number of jammed pack- determination of the maximally dense binary packings ings of binary disks with densities spanning the spectrum in R2. Fejes Tóth161 reported a number of candidate from the accepted amorphous glassy state to the phase- maximally dense packings for certain values of the radii separated crystal, it has been argued213,214 that there is ratio in the range α ≥ 0.154701 .... Maximally dense no “ideal glass transition”.215 The existence of an ideal binary disk packings have been also studied to determine glass transition remains a hotly debated topic of research. the stable crystal phase diagram of such alloys.222 The 216–218 219,220 Simulational as well as experimental stud- determination of φmax for sufficiently small α amounts ies of disordered jammed spheres with a size distribution to finding the optimal arrangement of the small disks reveal that they are effectively hyperuniform. In such within a tricusp: the nonconvex cavity between three cases, it has been demonstrated that the proper means close-packed large disks. A particle-growth Monte Carlo of investigating hyperuniformity is through the spectral algorithm was used to generate the densest arrangements 216,217 densityχ ˜V (k), which is defined by condition (14). of a fixed number of small identical disks within such a tricusp.223 All of these results can be compared to a 224 relatively tight upper bound on φmax given by 3. Densest Packings α πα2 + 2(1 − α2) sin−1 1 + α The densest packings of spheres with a size distribu- φmax ≤ φU = . (22) tion is of great interest in crystallography, chemistry and 2α(1 + 2α)1/2 materials science. It is notable that the densest pack- Inequality (22) also applies to general multicomponent ings of hard-sphere mixtures are intimately related to packings, where α is taken to be the ratio of the smallest high-pressure phases of molecular systems, including in- disk radius to the largest one. termetallic compounds59 and solid rare-gas compounds60 There is great interest in finding the densest binary for a range of temperatures. sphere packings in 3 in the physical sciences because of Except for trivial space-filling structures, there are no R their relationship to binary crystal alloys found in molec- provably densest packings when the spheres possess a size ular systems; see Refs. 148 and 225 as well as references distribution, which is a testament to the mathematical therein for details and some history. Past efforts to iden- challenges that they present. We begin by noting some tify such optimal packings have employed simple crys- rigorous bounds on the maximal packing fraction of pack- tallographic techniques (filling the interstices in uniform ings of spheres with M different radii R ,R ,...,R in 1 2 M 3D tilings of space with spheres of different sizes) and Rd. Specifically, the overall maximal packing fraction (M) d algorithmic methods, e.g., Monte Carlo calculations and φmax of such a general mixture in R [where φ is de- a genetic algorithm.226,227 However, these methods have fined by (19) with (20)] is bounded from above and be- achieved only limited success, in part due to the infinite (1) low in terms of the maximal packing fraction φmax for a parameter space that is involved. When using traditional monodisperse packing in the infinite-volume limit:15 algorithms, difficulties result from the enormous num- φ(1) ≤ φ(M) ≤ 1 − (1 − φ(1) )M . (21) ber of steps required to escape from local minima in the max max max “energy” (negative of the packing fraction). Hopkins et The lower bound corresponds to the case when the M al.147,148 have presented the most comprehensive deter- (1) components completely demix, each at the density φmax. mination to date of the “phase diagram” for the densest The upper bound corresponds to an ideal sequential binary sphere packings via the TJ linear-programming packing process for arbitrary M in which one takes the algorithm.142 In Ref. 148, 19 distinct crystal alloys 15 limits R1/R2 → 0, R2/R3 → 0, ··· ,RM−1/RM → 0. (compositions of large and small spheres spatially mixed Specific nonsequential protocols (algorithmic or other- within a fundamental cell) were identified, including 8 wise) that can generate structures that approach the up- that were unknown at the time. Using the TJ algorithm, per bound (21) for arbitrary values of M are currently they were always able to obtain either the densest previ- unknown and thus the development of such protocols ously known alloy or denser ones. These structures may is an open area of research. We see that in the limit correspond to currently unidentified stable phases of cer- M → ∞, the upper bound approaches unity, correspond- tain binary atomic and molecular systems, particularly ing to space-filling polydisperse spheres with an infinitely at high temperatures and pressures.59,60 Reference 148 wide separation in sizes221 or a continuous size distribu- provides details about the structural characteristics of tion with sizes ranging to the infinitesimally small.15 these densest-known binary sphere packings. 15

IV. PACKING SPHERES IN HIGH of the former evaluated at negative density.249 This map- DIMENSIONS ping becomes exact for d = 1 and in the large-d limit.

Sphere packings in four and higher-dimensional Eu- clidean spaces are of great interest in the physi- B. Nonequilibrium Disordered Packings Via cal and mathematical sciences; see Refs. 33, 39, Sequential Addition 45, 61, 104, 149, 150, 167, 228–243. Physicists have studied high-dimensional packings to gain in- In Sec. III B 2, we noted that the “ghost” RSA pack- sight into liquid, crystal and glassy states of matter ing104 is a disordered but unsaturated packing construc- in lower dimensions.45,167,229,230,232,235,236,239,240 Finding tion whose n-particle correlation functions are known ex- the densest packings in arbitrary dimension in Euclidean actly and rigorously achieves the infinite-time packing and compact spaces is a problem of long-standing in- fraction φ = 2−d for any d; see Fig. 3 for a 2D realization terest in discrete geometry.32,33,244,245 Remarkably, the of such a packing. optimal way of sending digital signals over noisy chan- Saturated RSA sphere packings have been numerically nels corresponds to the densest sphere packing in a high- generated and structurally characterized for dimensions dimensional space.32,35 These “error-correcting” codes up through d = 6 (Ref. 89). A more efficient numeri- underlie a variety of systems in digital communications cal procedure was devised to produce such packings for and storage, including compact disks, cell phones and the dimensions up through d = 8 (Ref. 90). The current Internet. best estimates of the maximal saturation packing fraction φs for d = 4, 5, 6, 7 and 8 are 0.2600781±0.0000037, 0.1707761 ± 0.0000046, 0.109302 ± 0.000019, 0.068404 ± 90 A. Equilibrium and Metastable Phase Behavior 0.000016, and 0.04230±0.00021, respectively. These are lower than the corresponding MRJ packing fractions in those dimensions (see Sec. IV C). The quantity φs appar- The properties of equilibrium and metastable states ently scales as d · 2−d or possibly d · ln(d) · 2−d for large d; of hard spheres have been studied both theoretically see Refs. 89 and 90. While saturated RSA packings are and computationally in spatial dimensions greater than nearly but not exactly hyperuniform,89 as d increases, the three.167,229,239–241,246,247 This includes the evaluation of 246,247 degree of hyperuniformity increases and pair correlations various virial coefficients across dimensions, as well markedly diminish,90 consistent with the decorrelation as the pressure along the liquid, metastable and crystal principle,61 which is described in Sec. IV E. branches,167,235,239–241 especially for d = 4, 5, 6 and 7. Using the LS packing algorithm, Skoge et al.167 numerically estimated the freezing and melting packing frac- C. Maximally Random Jammed States tions for the equilibrium hard-sphere fluid-solid transition, φF ' 0.32 and φM ' 0.39, respectively, for d = 4, Using the LS algorithm, Skoge et al.167 generated and and φF ' 0.19 and φM ' 0.24, respectively, for d = 5. These authors showed that nucleation appears to be characterized MRJ packings in four, five and six di- strongly suppressed with increasing dimension. The same mensions. In particular, they estimated the MRJ pack- conclusion was subsequently reached in a study by van ing fractions, finding φMRJ ' 0.46, 0.31 and 0.20 for Meel et al..240 Finken, Schmidt and Löwen248 used a va- d = 4, 5 and 6, respectively. To a good approxima- riety of approximate theoretical methods to show that tion, the MRJ packing fraction obeys the scaling form −d −d equilibrium hard spheres have a first-order freezing tran- φMRJ = c12 + c2d · 2 , where c1 = −2.72 and sition for dimensions as high as d = 50. c2 = 2.56, which appears to be consistent with high- dimensional asymptotic limit, albeit with different coef- Any disordered packing in which the pair correlation −d function at contact, g (D+), is bounded (such as equi- ficients. The dominant large-d density scaling d · 2 2 is supported by theoretical studies.61,242,250,251 Skoge et librium hard spheres) induces a power-law decay in the 167 structure factor S(k) in the limit k → ∞ for any dimen- al. also determined the MRJ pair correlation func- sion d given by tion g2(r) and structure factor S(k) for these states and found that short-range ordering appreciably decreases

3d+1 + with increasing dimension, consistent with the decorre- 2 2 Γ(1 + d/2)φg2(D ) lation principle.61 This implies that, in the limit d → ∞, S(k) ∼ 1− √ d+1 cos[kD−(d+1)π/4] . π(kD) 2 g2(r) tends to unity for all r outside the hard-core, except (23) for a Dirac-delta function at contact due to the jamming 61 There is a remarkable duality between the equilibrium constraint. As for d = 3 (where φMRJ ' 0.64), the hard-hypersphere (hypercube) fluid system in Rd and MRJ packings were found to be isostatic, hyperuniform, the continuum percolation model of overlapping hyper- and have a power-law divergence in g2(r) at contact; ; d α + spheres (hypercubes) in R . In particular, the pair con- g2(r) ∼ 1/(r − D) with α ≈ 0.4 as r tends to D . nectedness function of the latter is to a good approxima- Across dimensions, the cumulative number of neighbors tion equal to the negative of the total correlation function was shown to equal the kissing (contact) number of the 16 conjectured densest packing close to where g (r) has its 2 TABLE I. The densest known or optimal sphere packings in first minimum. Disordered jammed packings were also d R for selected d. For each packing, we provide the packing simulated and studied in dimensions 7-10; see Ref. 252. fraction φ and kissing number Z. Except for the non-lattice 10 packing P10c in R , all of the other densest known packings listed in this table are lattice packings: Z is the integer lattice, D. Maximally Dense Sphere Packings A2 is the triangular lattice, Dd is the checkerboard lattice (a generalization of the fcc lattice), Ed is one the root lattices, The sphere packing problem seeks to answer the follow- and Λd is the laminated lattice. For d = 8 and d = 24, it has recently been proved that E and Λ are optimal among all ing question:32 Among all packings of congruent spheres 8 24 d packings, respectively; see Refs. 244 and 245. The reader is in R , what is the maximal packing fraction φmax and referred to Ref. 32 for additional details. the corresponding arrangements of the spheres? Until 2017, the optimal solutions were known only for the first 38 d Packing Packing Fraction, φ Kissing number, Z three space dimensions. For d = 2 and d = 3,√ these 1 1 2 are the triangular lattice (A2) with φmax = π/ 12 = Z √ 0.906899 ... and checkerboard (fcc) lattice (D ) and its 2 A2 π/ 12 = 0.9068 ... 6 √ 3 √ stacking variants with φmax = π/ 18 = 0.74048 ..., re- 3 D3 π/ 18 = 0.7404 ... 12 spectively. We now know the optimal solutions in two 4 D π2/16 = 0.6168 ... 24 4 √ other space dimensions; namely, the E8 and Λ24 lattices 2 5 D5 2π /(30 2) = 0.4652 ... 40 are the densest packings among all possible packings in √ 6 E 3π2/(144 3) = 0.3729 ... 72 8 and 24, respectively; see Refs. 244 and 245. For 6 R R 3 4 ≤ d ≤ 9, the densest known packings are (Bravais) 7 E7 π /105 = 0.2952 ... 126 32 8 E π4/384 = 0.2536 ... 240 lattice packings. The “checkerboard” lattice Dd is be- 8 √ 4 5 4 lieved to be optimal in R and R . Interestingly, the 9 Λ9 2π /(945 2) = 0.1457 ... 272 5 non-lattice (periodic) packing P10c (with 40 spheres per 10 P10c π /3072 = 0.09961 ... 372 10 fundamental cell) is the densest known packing in , 8 R 16 Λ16 π /645120 = 0.01470 ... 4320 which is the lowest dimension in which the best known 12 packing is not a (Bravais) lattice. 24 Λ24 π /479001600 = 0.001929 ... 196360 Table I lists the densest known or optimal sphere packings in Rd for selected d. For the first three space dimensions, the optimal sphere-packing solutions (or their “dual” solutions) are directly related to the best known solutions of the number-variance problem64,65 as well as of two other well-known problems in discrete geometry: the covering and quantizer problems,32,253 but such rela- tionships may or may not exist for d ≥ 4, depending on the peculiarities of the dimensions involved.254 The TJ linear-programming packing algorithm was 149 FIG. 9. (color online) Lattice packings in sufficiently high adapted by Marcotte and Torquato to determine the dimensions are not dense because the “holes” (space exterior d densest lattice packing (one particle per fundamental to the spheres) eventually dominates the space R and hence cell) in some high dimension. These authors applied it become unsaturated.32 For illustration purposes, we consider d for 2 ≤ d ≤ 19 and showed that it was able to rapidly the hypercubic lattice Z . Left panel: A fundamental cell of d and reliably discover the densest known lattice packings Z represented in two dimensions. The distance between the without a priori knowledge of their existence. It was point of intersection of the longest diagonal in the hypercube found to be appreciably faster than previously known al- with the hypersphere boundary√ and the vertex of the cube 255,256 along this diagonal is given by d − 1 for a sphere of unit gorithms at that time. The TJ algorithm was used d radius. This means that Z already becomes unsaturated at to generate an ensemble of isostatic jammed hard-sphere 4 d = 4. Placing an additional sphere in Z doubles the den- lattices and study the associated pair statistic and force sity of 4 and yields the four-dimensional checkerboard lattice distributions.150 It was shown that this special ensem- Z packing D4, which is believed to be the optimal packing in ble of lattice-sphere packings retain many of the crucial 4 R . Right panel: A schematic “effective” distorted represen- structural features of the classical hard-sphere model. tation of the hypersphere within the hypercubic fundamen- It is noteworthy that for sufficiently large d, lattice tal cell for large d, illustrating that the volume content of packings are most likely not the densest (see Fig. 9), the hypersphere relative to the hypercube rapidly diminishes d but it becomes increasingly difficult to find explicit dense asymptotically. Indeed, the packing fraction of Z is given d/2 d packing constructions as d increases. Indeed, the prob- by φ = π /(Γ(1 + d/2)2 ). Indeed, the checkerboard lattice d/2 (d+2)/2 lem of finding the shortest lattice vector in a particular Dd with packing fraction φ = π /(Γ(1 + d/2)2 ) be- lattice packing grows super-exponentially with d and is comes suboptimal in relatively low dimensions because it also in the class of NP-hard problems.257 becomes dominated by larger and larger holes as d increases. For large d, the best that one can do theoretically is 17

32 to devise upper and lower bounds on φmax. The non- with d. They did so using the so-called g2-invariant op- constructive lower bound of Minkowski258 established the timization procedure that maximizes φ associated with a existence of reasonably dense lattice packings. He found radial “test” pair correlation function g2(r) to provide L that the maximal packing fraction φmax among all lattice the putative exponential improvement on Minkowski’s packings for d ≥ 2 satisfies 100-year-old bound on φmax. Specifically, a g2-invariant process250 is one in which the functional form of a “test” ζ(d) φL ≥ , (24) pair correlation g2(r) function remains invariant as den- max 2d−1 sity varies, for all r, over the range of packing fractions 0 ≤ φ ≤ φ∗ subject to the satisfaction of the nonnegativ- where ζ(d) = P∞ k−d is the Riemann zeta function. k=1 ity of the structure factor S(k) and g2(r). When there Note that for large values of d, the asymptotic behavior exist sphere packings with a g2 satisfying these condi- of the Minkowski lower bound is controlled by 2−d. Since tions in the interval [0, φ∗], then one has the lower bound 1905, many extensions and generalizations of the lower on the maximal packing fraction given by bound (24) have been derived32,259–261, but none of these investigations have been able to improve upon the domi- −d nant exponential term 2 ; they instead only improve on φ ≥ φ . (26) the latter by a factor linear in d. max ∗ It is trivial to prove that the packing fraction of a saturated packing of congruent spheres in d satisfies104 R Torquato and Stillinger61 conjectured that a test func- 1 tion g2(r) is a pair-correlation function of a transla- φ ≥ (25) d 2d tionally invariant disordered sphere packing in R for 0 ≤ φ ≤ φ∗ for sufficiently large d if and only if the for all d. This “greedy” bound (25) has the same domi- nonnegativity conditions on S(k) and g2(r). The decorre- nant exponential term as Minkowski’s bound (24). lation principle,61 among other results,39,237,243 provides d Nontrivial upper bounds on φmax in R for any d have justification for the conjecture. This principle states that been derived.33,34,262–264 The linear-programming (LP) unconstrained correlations in disordered sphere packings upper bounds due to Cohn and Elkies33 provides the ba- vanish asymptotically in high dimensions and that the sic framework for proving the best known upper bounds gn for any n ≥ 3 can be inferred entirely (up to small er- on φmax for dimensions in the range 4 ≤ d ≤ 36. Re- rors) from a knowledge of ρ and g2. This is vividly exhib- cently, these LP bounds have been used to prove that ited by the exactly solvable ghost RSA packing process104 no packings in R8 and R24 have densities that can ex- as well as by computer simulations of high-dimensional 167 234 ceed those of the E8 (Ref. 244) and L24 (Ref. 245) MRJ and RSA packings. Interestingly, this opti- lattices, respectively. Kabatiansky and Levenshtein264 mization problem is the dual of the infinite-dimensional found the best asymptotic upper bound, which in the linear program (LP) devised by Cohn and Elkies233 to −0.5990d limit d → ∞ yields φmax ≤ 2 , proving that obtain upper bounds on φmax. the maximal packing fraction tends to zero in the limit Using a particular test pair correlation corresponding d → ∞. This rather counterintuitive high-dimensional 61 property of sphere packings can be understood by recog- to a disordered sphere packing, Torquato and Stillinger found a conjectural lower bound on φmax that is con- nizing that almost all of the volume of a d-dimensional −(0.77865...)d sphere for large d is concentrated near the sphere surface. trolled by 2 , thus providing the first putative exponential improvement on Minkowski’s lower bound (24). Scardicchio, Stillinger and Torquato237 studied a wider class of test functions (corresponding to disordered E. Are Disordered Packings the Densest in High Dimensions? packings) that lead to precisely the same putative exponential improvement on Minkowski’s lower bound and therefore the asymptotic form 2−(0.77865...)d is much more Since 1905, many extensions and generalizations of general and robust than previously surmised. Minkowski’s bound have been derived32, but none of them have improved upon the dominant exponential term Zachary and Torquato265 studied, among other quanti- 2−d. The existence of the unjammed disordered ghost ties, pair statistics of high-dimensional generalizations of RSA packing104 (Sec. III B 2) that rigorously achieves a the periodic 2D kagoméand 3D diamond crystal struc- packing fraction of 2−d strongly suggests that Bravais- tures. They showed that the decorrelation principle is lattice packings (which are almost surely unsaturated in remarkably already exhibited in these periodic crystals sufficiently high d) are far from optimal for large d. in low dimensions, suggesting that it applies for any Torquato and Stillinger61 employed a plausible con- sphere packing in high dimensions, whether disordered jecture that strongly supports the counterintuitive pos- or not. This conclusion was bolstered in a subsequent sibility that the densest sphere packings for sufficiently study by Andreanov, Scardicchio and Torquato243 who large d may be disordered or at least possess fundamen- showed that strictly jammed lattice sphere packings visi- tal cells whose size and structural complexity increase bly decorrelate as d increases in relatively low dimensions. 18

F. Remarks on Packing Problems in Non-Euclidean Spaces

Particle packing problems in non-Euclidean (curved) spaces arises in a variety of fields, including physics,266–268 biology,29,269–272 communications 32 32,36,48,273,274 theory, and geometry. While a compre- FIG. 10. (color online). Superballs with different values of hensive overview of this topic is beyond the scope of the deformation parameter p. We note that p = 0, 1/2, 1, and this review, it is useful to highlight here some of the ∞ correspond to a 3D cross, regular octahedron, sphere and developments for the interested reader. We will limit the cube, respectively. discussion to sphere packings on the positively curved unit sphere Sd−1 ⊂ Rd. The reader is referred to the review by Torquato and Stillinger39 for some discussion A. Simple Nonspherical Convex Shapes of negatively curved hyperbolic space Hd. Recall that the kissing number Z is the number of 1. Ellipsoids spheres of unit radius that simultaneously touch a unit sphere Sd−1.32 The kissing number problem asks for the One simple generalization of the sphere is an ellip- d maximal kissing number Zmax in . The determination soid, the family of which is a continuous deformation of R d of the maximal kissing number in R3 spurred a famous a sphere. A d-dimensional ellipsoid in R is a centrally debate between Issac Newton and David Gregory in 1694. symmetric body occupying the region The former correctly thought the answer was 12, but the 2 2 2 x x x latter wrongly believed that it was 13. The maximal kiss- 1 + 2 + ··· + d ≤ 1, (27) a1 a2 ad ing number Zmax for d > 3 is only known in dimensions 275 276,277 276,277 four, eight, and twenty four. where xi (i = 1, 2, . . . , d) are Cartesian coordinates and A packing of congruent spherical caps on the unit ai are the semi-axes of the ellipsoid. Thus, we see that an sphere Sd−1 in Rd yields a spherical code consisting of the ellipsoid is an affine (linear) transformation of the sphere. centers of the caps.32 A spherical code is optimal if the minimal distance (smallest angular separation between distinct points in the code) is as large as possible. The 2. Superballs reader is referred to the paper by Cohn and Kumar36 and references therein for some developments in the mathe- A d-dimensional superball in Rd is a centrally symmet- matics literature. Interestingly, the jamming of spherical ric body occupying the region codes in a variety of dimensions has been investigated.48 |x |2p + |x |2p + ··· + |x |2p ≤ 1, (28) It is noteworthy that some optimal spherical codes278 are 1 2 d related to the densest local packing of spheres around a where xi (i = 1, . . . , d) are Cartesian coordinates and p ≥ central sphere.279,280 0 is the deformation parameter (not pressure, as denoted in Sec. III A), which controls the extent to which the particle shape has deformed from that of a d-dimensional sphere (p = 1). Thus, superballs constitute a large fam- V. PACKINGS OF NONSPHERICAL ily of both convex (p ≥ 1/2) and concave (0 < p < 1/2) PARTICLES particles (see Fig. 10). A “superdisk”,which is the desig- nation in the 2D case, possesses square symmetry. As p moves away from unity, two families of superdisks with The packing characteristics of equilibrium phases square symmetry can be obtained depending on whether and jammed states of packings of nonspherical par- p < 1 or p > 1. When p < 1/2, the superdisk is concave; ticle are considerably richer than their spherical see Ref. 138. counterparts.74,126,127,132,136,137,140,141,255,281–319 This is due to the fact that nonsphericity of the particle shape introduces rotational degrees of freedom not present in 3. Spherocylinder sphere packings. Our primary interest is in simple 3D convex shapes, such as ellipsoids, superballs, spherocylin- A d-dimensional spherocylinder in Rd consists of a ders and polyhedra, although we remark on more com- cylinder of length L and radius R capped at both ends plex shapes, such as concave particles as well as congru- by hemispheres of radius R , and therefore is a centrally- ent ring tori, which are multiply connected nonconvex symmetric convex particle. Its volume VSC for d ≥ 2 is bodies of genus 1. Organizing principles to character- given by ize and classify very dense, possibly jammed packings of nonspherical particles in terms of shape symmetry of the π(d−1)/2Rd−1 πd/2Rd V = L + . (29) particles19,140,304 are discussed in Sec.V D 5. SC Γ(1 + (d − 1)/2) Γ(1 + d/2) 19

When L = 0, a d-dimensional spherocylinder reduces to a d-dimensional sphere of radius R. Figure 11 shows 3D examples.

FIG. 11. (Color online) Three-dimensional spherocylinders composed of cylinder with length L, caped at both ends with hemispheres with radius R. Left panel: A spherocylinder with aspect ratio L/R = 1. Right panel: A spherocylinder with aspect ratio L/R = 5.

4. Polyhedra FIG. 13. (color online) The thirteen Archimedean solids: truncated tetrahedron (A1), truncated icosahedron (A2), snub cube (A3), snub dodecahedron (A4), rhombicosidodec- ahdron (A5), truncated icosidodecahedron (A6), truncated cuboctahedron (A7), icosidodecahedron (A8), rhombicuboc- tahedron (A9), truncated dodecahedron (A10), cuboctahedron (A11), truncated cube (A12), and truncated octahedron (A13).

B. Equilibrium and Metastable Phase Behavior FIG. 12. (color online). The five Platonic solids: tetrahedron (P1), icosahedron (P2), dodecahedron (P3), octahedron (P4), and cube (P5). Hard nonspherical particles exhibit a richer phase diagram than that of hard spheres because the former The Platonic solids (mentioned in Plato’s Timaeus) are can possess different degrees of translational and orien- convex polyhedra with faces composed of congruent con- tational order; see Refs. 74, 281, 283, 285, 289, 291, 299, vex regular polygons. There are exactly five such solids: 300, 302, 306, 307, 310–312, 314, and 317. Nonspherical- the tetrahedron (P1), icosahedron (P2), dodecahedron particle systems can form isotropic liquids, a variety of (P3), octahedron (P4), and cube (P5) (see Fig. 12) Note liquid-crystal phases, rotator crystals, and solid crystals. that viral capsids often have icosahedral symmetry; see, Particles in liquid phases have neither translational nor for example, Ref. 272. orientational order. Examples of liquid-crystal states in- An Archimedean solid is a highly symmetric, semi- clude a nematic phase in which the particles are aligned regular convex polyhedron composed of two or more (i.e., with orientational order) while the system lacks any types of regular polygons meeting in identical vertices. long-range translational order and a smectic phase in The thirteen Archimedean solids are depicted in Fig. 13. which the particles have ordered orientations and possess This typical enumeration does not count the chiral forms translational order in one direction. A rotator (or plastic) (not shown) of the snub cube (A3) and snub dodecahe- phase is one in which particles possess translational order dron (A4). The remaining 11 Archimedean solids are but can rotate freely. Solid crystals are characterized by non-chiral (i.e., each solid is superposable on its mirror both translational and orientational order. image) and the only non-centrally symmetric one among Ordering transitions in systems of hard nonspherical these is the truncated tetrahedron. particles are entropically driven, i.e., the stable phase is It is noteworthy that the tetrahedron (P1) and the determined by a competition between translational and truncated tetrahedron (A1) are the only Platonic and orientational entropies. This principle was first estab- non-chiral Archimedean solids, respectively, that are not lished in the pioneering work of Onsager,281 where it was centrally symmetric. We will see that the central sym- shown that needle-like shapes exhibit a liquid-nematic metry of the majority of the Platonic and Archimedean phase transition at low densities because in the nematic solids (P2 – P5, A2 – A13) distinguish their dense packing phase the drop in orientational entropy is offset by the arrangements from those of the non-centrally symmetric increase in translational entropy, i.e., the available space ones (P1 and A1) in a fundamental way. for any needle increases as the needle tends to align. 20

The stable phase formed by systems of hard- nonspherical particles is influenced by its symmetry and surface smoothness, which in turn determines the overall system entropy. Here we briefly highlight 3D numerical studies that use Monte Carlo methods and/or free-energy calculations to determine the phase diagrams. Spheroids exhibit not only fluid, solid crystal, and nematic phases for some aspect ratios, but rotator phases for nearly spherical particle shapes at intermediate densities.311,320 Hard “lenses” (common volume to two overlapping identical spheres) have a qualitatively similar phase diagram to hard oblate spheroids, but differences between them are more pronounced in the high-density crystal phase up to the densest-known packings.317 Spherocylinders FIG. 14. Packing fraction φ versus aspect ratio α for MRJ exhibit five different possible phases, depending on the packings of 10,000 ellipsoids as obtained in Ref. 127. The packing fraction and aspect ratio: isotropic fluid, smec- semi-axes here are 1, αβ , α. The inset shows the mean contact 291 tic, nematic, rotator, and solid crystal phases. Apart number Z as a function of α. Neither the spheroid (oblate or from fluid and crystal phases, superballs can form ro- prolate) or general ellipsoids cases attain their isostatic values tator phases at intermediate densities.299,310 Systems of of Z = 10 or Z = 12, respectively. tetragonal parallelepipeds can exhibit liquid-crystalline and cubatic phases, depending on the aspect ratio and 289 packing fraction. Various convex space-filling polyhe- density maximum. The existence of a cusp at the sphere dra (including truncated octahedron, rhombic dodeca- point runs counter to the conventional expectation that hedron, two types of prisms and cube) possess unusual for “generic” (disordered) jammed frictionless particles, liquid-crystalline and rotator-crystalline phases at inter- the total number of (independent) constraints equals the mediate packing fractions.302 Truncated cubes exhibit total number of degrees of freedom df , which has been a rich diversity in crystal structures that depend sensi- 321 312 referred to as the isostatic conjecture. This conjec- tively on the amount of truncation. A study of a wide ture implies a mean contact number Z = 2d , where class of polyhedra revealed that entropy maximization fa- f 307 df = 2 for disks, df = 3 for ellipses, df = 3 for spheres, vors mutual alignment of particles along their facets. d = 5 for spheroids, and d = 6 for general ellipsoids. By analytically constructing the densest known packings f f Since df increases discontinuously with the introduction of congruent Archimedean truncated tetrahedra (which of rotational degrees of freedom as one makes the par- nearly fill all of space), the melting properties of such ticles nonspherical, the isostatic conjecture predicts that systems were examined by decompressing this densest 300 Z should have a jump increase at aspect ratio α = 1 to packing and equilibrating them. A study of the entire a value of Z = 12 for a general ellipsoid. Such a discon- phase diagram of truncated tetrahedra showed that the tinuity was not originally observed by Donev et al.126, system undergoes two first-order phase transitions as the rather, they found that jammed ellipsoid packings are density increases: first a liquid-solid transition and then 314 hypostatic (or sub-isostatic), Z < 2df , near the sphere a solid-solid transition. point, and only become nearly isostatic for large aspect ratios. In fact, the isostatic conjecture is only rigorously true for amorphous sphere packings after removal of rat- C. Maximally Random Jammed States tlers; generic nonspherical-particle packings should generally be hypostatic.127,322 It has been rigorously shown The fact that M&M candies (spheroidal particles with that packings of nonspherical particles are generally not aspect ratio α ≈ 1.9) were shown experimentally to ran- jammed to first order in the “jamming gap” δ [see Sec. domly pack more densely than spheres (φ ≈ 0.66) moti- III C 2], but are jammed to second order in δ due to cur- vated the development of a modified LS algorithm136,137 vature deviations from the sphere.127 In striking contrast to obtain frictionless MRJ-like packings with even higher with MRJ sphere packings, the rattler concentrations of densities for other aspect ratios. This included nearly- the MRJ ellipsoid packings appear practically to van- spherical ellipsoids with φ ≈ 0.74,126,127 i.e., packing ish outside of some small neighborhood of the sphere fractions approaching those of the densest 3D sphere point.127 The reader is referred to recent work on hy- packings. Note that these other densest MRJ packings postatic jammed 2D packings of noncircular particles.319 are realizable experimentally.132 Jiao, Stillinger and Torquato294 have computed the Figure 14 shows separate plots of φ and mean con- packing fraction φMRJ of MRJ packings of binary su- tact number Z as a function of α as predicted by the perdisks in R2 and identical superballs in R3. They 127 more refined simulations of Donev et al.. Each plot found that φMRJ increases dramatically and nonanalyti- shows a cusp (i.e., non-differentiable) minimum at the cally as one moves away from the circular-disk or sphere sphere point, and φ versus aspect ratio α possesses a point (p = 1). Moreover, these disordered packings were 21 √ 140,141 S shown to be hypostatic. Hence, the local particle arrange- of identical spheres√ (e.g., φmax = π/ 12 for d = 2 S ments are necessarily nontrivially correlated to achieve and φmax = π/ 18 for d = 3). The upper bound strict jamming and hence “nongeneric,” the degree of (30) will be relatively tight provided that the aspheric- which was quantified. MRJ packings of binary superdisks ity γ (equal to the ratio of the circumradius to the in- and of ellipses are effectively hyperuniform.323 Note that radius) of the particle is not large. Since bound (30) the geometric-structure approach was used to derive a cannot generally be sharp (i.e., exact) for a non-tiling highly accurate formula for the packing fraction φMRJ particle, any packing whose density is close to the up- of MRJ binary packings of convex superdisks181 that is per bound (30) is nearly optimal, if not optimal. Inter- valid for almost all size ratios, relative concentrations estingly, a majority of the centrally symmetric Platonic and deformation parameter p ≥ 1/2. For the special and Archimedean solids have relatively small aspherici- limit of monodisperse circular disks, this formula pre- ties, explaining the corresponding small differences be- U L dicts φMRJ = 0.837, which is in very good agreement tween φmax and φmax, the packing fraction of the dens- with the recently numerically discovered MRJ isostatic est lattice packing.140,141,258,286 As discussed below, these 144 state with φMRJ = 0.827. densest lattice packings are conjectured to be the densest 3D MRJ packings of the four non-tiling Platonic solids among all packings.140,141 Upper bound (30) will also be (tetrahedra, octahedra, dodecahedra, and icosahedra) relatively tight for superballs (superdisks) for deforma- were generated using the ASC optimization scheme.301 tion parameters p near the sphere (circle) point (p = 1). 318 The MRJ packing fractions for tetrahedra, octahedra, do- Dostert et al. recently obtained upper bounds on φmax decahedra, and icosahedra are 0.763±0.005, 0.697±0.005, of translative packings of superballs and 3D convex bod- 0.716 ± 0.002, and 0.707 ± 0.002, respectively. It was ies with tetrahedral symmetry. shown that as the number of facets of the particles increases, the translational order in the packings increases while the orientational order decreases. Moreover, such 1. Ellipsoids MRJ packings were found to be hyperuniform with a total correlation function h(r) that decays to zero asymptotically with same power-law as MRJ spheres, i.e, like −1/r4. These results suggest that hyperuniform quasi- long-range correlations are a universal feature of MRJ packings of frictionless particles of general shape. How- ever, unlike MRJ packings of ellipsoids, superballs, and superellipsoids, which are hypostatic, MRJ packings of the non-tiling Platonic solids are isostatic.301 In addition, 3D MRJ packings of truncated tetrahedra with an average packing fraction of 0.770 were also generated.314

D. Maximally Dense Packings

We focus here mainly on exact constructions of the densest known packings of nonspherical particles in which the lattice vectors as well as particle positions and orientations are expressible analytically. Where appropriate, we also cite some work concerning packings obtained via FIG. 15. The packing fraction of the “laminated” non-Bravais computer simulations and laboratory experiments. lattice packing of ellipsoids (with a two-particle basis) as a Rigorous upper bounds on the maximal packing frac- function of the aspect ratio α. The point α = 1 corresponding tion φmax of packings of nonspherical particles of general to the face-centered cubic lattice sphere packing is shown, d shape in R can be used to assess their packing efficiency. along with the two sharp maxima√ in the packing fraction for However, it has been highly challenging to formulate up- prolate√ ellipsoids with α = 3 and oblate ellipsoids√ with α = per bounds for non-tiling particle packings that are non- 1√/ 3, as shown in the insets. For both α < 1/ 3 and α > trivially less than unity. It has recently been shown that 3, the packing fraction drops off precipitously holding the particle orientations fixed (blue lines). The presently maximal φmax of a packing of congruent nonspherical particles of volume v in d is bounded from above according to achievable packing fraction φ = 0.7707 ... is√ highlighted with√ P R a thicker red line, and is constant for α ≤ 1/ 3 and α ≥ 3; v see Ref. 288. φ ≤ φU = min P φS , 1 , (30) max max v max S The fact that MRJ-like packings of nearly-spherical el- where vS is the volume of the largest sphere that can lipsoids exist with φ ≈ 0.74 (see Refs. 126 and 127) sug- S be inscribed in the nonspherical particle and φmax is the gested that there exist ordered ellipsoid packings with maximal packing fraction of a packing of d-dimensional appreciably higher densities. The densest known pack- 22 ings of identical 3D ellipsoids were obtained analytically by Donev et al.;288 see Fig. 15. These are exact constructions and represent a family of non-Bravais lattice packings of ellipsoids with a packing fraction that always exceeds the that of the corresponding densest Bravais lattice packing (φ = 0.74048 ...) with a maximal packing fraction of√φ = 0.7707 ...√, for a wide range of aspect ratios (α ≤ 1/ 3 and α ≥ 3). In these densest known packings, each ellipsoid has 14 contacting neighbors and there are two particles per fundamental cell. While a convex “lens” fits snugly within an oblate spheroid of the same aspect ratio, the densest known lens packings are denser for general aspect ratios (except for a FIG. 16. (color online). Packing fraction versus deformation narrow range of intermediate values) than their spheroid parameter p for the packings of convex superballs.139 Insert: ∗ counterparts, achieving the highest packing fraction of Around pc = 1.1509 ..., the two curves are almost locally φ = π/4 = 0.7853 ... in the “flat-lens” limit.317 parallel to each other.

2. Superballs tice, which are apparently optimal in the vicinity of the sphere point and the octahedron point, respectively. The exact maximal packing fraction φ as a function Exact analytical constructions for candidate maxi- max of deformation parameter p is plotted in Fig. 16. As p mally dense packings of 2D superdisks were recently increases from unity, the initial increase of φ is linear proposed for all convex and concave shapes.138 These max in (p−1) and subsequently φ increases monotonically are achieved by two different families of Bravais lattice max with p until it reaches unity as the particle shape tends packings such that φ is nonanalytic at the “circular- max to the cube. These characteristics stand in contrast to disk” point (p = 1) and increases significantly as p those of the densest known ellipsoid packings (see Fig. moves away from unity. The broken rotational symme- 15) whose packing fraction as a function of aspect ratios try of superdisks influences the packing characteristics has zero initial slope and is bounded from above by a in a non-trivial way that is distinctly different from el- value of 0.7707 ....288 As p decreases from unity, the ini- lipse packings. For ellipse packings, no improvement over √ tial increase of φ is linear in (1 − p). Thus, φ is a the maximal circle packing fraction (φ = π/ 12 = max max max nonanalytic function of p at p = 1.138 However, the be- 0.906899 ...) is possible, since the former is an affine havior of φ as the superball shape moves off the sphere transformation of the latter.39 For superdisks, one can max point is distinctly different from that of optimal spheroid take advantage of the four-fold rotationally symmetric packings, for which φ increases smoothly as the aspect shape of the particle to obtain a substantial improvement max ratios of the semi-axes vary from unity and hence has no on the maximal circle packing fraction. By contrast, one cusp at the sphere point.288 These distinctions between needs to use higher-dimensional counterparts of ellipses superball versus ellipsoid packings result from differences (d ≥ 3) in order to improve on φ for spheres. Even max in which rotational symmetries in these two packings are for 3D ellipsoids, φ increases smoothly as the aspect max broken.139 For the small range 0.79248 < p < 1, Ni et ratios of the semi-axes vary from unity,288 and hence has al. 309 numerically found lattice packings that are very no cusp at the sphere point. In fact, 3D ellipsoid pack- slightly denser than those of the theoretically predicted ings have a cusp-like behavior at the sphere point only lattices of Ref. 139. All of the results, support the when they are randomly jammed.126 O0 139 Torquato-Jiao conjecture that the densest packings of all Jiao, Stillinger and Torquato showed that increas- convex superballs are their densest lattice packings.141 ing the dimensionality of a superball from two to three dimensions imbues the optimal packings with structural characteristics that are richer than their 2D counterparts. They obtained analytical constructions for the densest 3. Spherocylinders known superball packings for all convex and concave cases, which are certain families of Bravais lattice pack- The role of curvature in determining dense packings ings in which each particle has 12 contacting neighbors. of smoothly-shaped particles is still not very well under- For superballs in the cubic regime (p > 1), the candidate stood. While the densest packings of 3D ellipsoids are optimal packings are achieved by two families of Bravais (non-Bravais) lattice packings,288 the densest packings of 139,140 lattice packings (C0 and C1 lattices) possessing two-fold superballs appear to lattice packings. For 3D sphe- and three-fold rotational symmetry, respectively. For su- rocylinders with L > 0, the optimal Bravais-lattice pack- perballs in the octahedral regime (0.5 < p < 1), there are ing is very dense and might be one of the actual densest also two families of Bravais lattices (O0 and O1 lattices) packings. This is because the local principal curvature obtainable from continuous deformations of the fcc lat- of the cylindrical surface is zero along the spherocylinder 23

FIG. 18. (color online) Portions of the densest lattice packings of three of the centrally symmetric Platonic solids found by the ASC optimization scheme.140,141 Left panel: Icosahedron packing with packing fraction φ = 0.8363 .... Middle panel: Dodecahedron packing with packing fraction φ = 0.9045 .... FIG. 17. (color online). Packing fraction versus deformation Right panel: Octahedron packing with packing fraction φ = parameter p for the lattice packings of concave superballs.139 0.9473 .... Insert: a concave superball with p = 0.1, which will becomes a 3D cross at the limit p → 0. non-tiling Platonic solids with central symmetry) with densities 0.947 ..., 0.904 ..., and 0.836 ..., respectively. axis (i.e., a “flat” direction), and thus, spherocylinders Unlike the densest tetrahedron packing, which must be can have very dense lattice packings by an appropriate a non-Bravais lattice packing, the densest packings of alignment of the spherocylinders along their axes. The the other non-tiling Platonic solids found by the algo- packing fraction of the densest Bravais-lattice packing is rithm are their previously known densest (Bravais) lat- given by tice packings258,286; see Fig. 18. These simulation results as well as other theoretical considerations led them 4 π L + R to general organizing principles concerning the densest φ = √ √3 , (31) 12 L + 2 6 R packings of a class of nonspherical particles, which are 3 described in Sec. V D 5. where L is the length of the cylinder and R is the radius Conway and Torquato290 showed that the maximally of the spherical caps. This lattice packing corresponds dense packing of regular tetrahedra cannot be a Bra- to stacking layers of aligned spherocylinders in the same vais lattice. Among other non-Bravais lattice packings, manner as fcc spheres and hence there are a uncountably- they obtained a simple, uniform packing of “dimers” infinite number of non-Bravais-lattice packings of sphe- (two particles per fundamental cell) with packing frac- rocylinders (in correspondence to the Barlow stackings of tion φ = 2/3. A uniform packing has a symmetry (in spheres39) with the same packing densities (31). Thus, this case a point inversion symmetry) that takes one the set of dense nonlattice packings of spherocylinders is tetrahedron to another. A dimer is composed of a pair overwhelmingly larger than that of the lattice packing. of regular tetrahedra that exactly share a common face. We will see that these dense packings are consistent with They also found a non-Bravais lattice (periodic) pack- Conjecture 3 described in Sec. V D 5. ings of regular tetrahedra with φ ≈ 0.72, which doubled that the density of the corresponding densest Bravais- lattice packing (φ = 18/49 = 0.367 ...), which was the 4. Polyhedra record before 2006. This work spurred many studies that improved on this density.140,141,293,295–298,324 Kallus et About a decade ago, very little was known about al.296 found a remarkably simple “uniform” packing of the densest packings of polyhedral particles. The dif- tetrahedra with high symmetry consisting of only four ficulty in obtaining dense packings of polyhedra is re- particles per fundamental cell (two “dimers”) with pack- lated to their complex rotational degrees of freedom ing fraction φ = 100/117 = 0.854700 .... Torquato and and to the non-smooth nature of their shapes. It Jiao324 subsequently presented an analytical formulation was the investigation of Conway and Torquato of dense to construct a three-parameter family of dense uniform packings of tetrahedra290 that spurred the flurry of dimer packings of tetrahedra again with four particles activity over several years to find the densest pack- per fundamental cell. Making an assumption about one ings of tetrahedra,140,141,293,295–298,324 which in turn led of these parameters resulted in a two-parameter fam- to studies of the densest packings of other polyhe- ily, including those with a packing fraction as high as dra and many other nonspherical convex and concave φ = 12250/14319 = 0.855506 .... Chen et al.297 used particles.300,304,306,307,312,317 the full three-parameter family to obtain the densest Torquato and Jiao140,141 employed the ASC optimiza- known dimer packings of tetrahedra with the very slightly tion scheme to determine dense packings of the non-tiling higher packing fraction φ = 4000/4671 = 0.856347 ... Platonic solids and of all of the Archimedean solids. For (see Fig. 19). Whether this is the optimal packing is an example, they were able to find the densest known pack- open question for reasons given by Torquato and Jiao.295 ings of the octahedra, dodecahedra and icosahedra (three The fact that the first nontrivial upper bound on the 24

of superballs and other smoothly-shaped centrally symmetric convex particles having surfaces without “flat” directions are given by their corresponding optimal lattice packings.141 Consistent with Conjecture 2, the densest known packing of the non-centrally symmetric truncated tetrahedron is a non-lattice packing with packing fraction φ = 207/208 = 0.995192 ..., which is amazingly close to unity and strongly implies its optimality.300 We note that non-Bravais-lattice packings of elliptical cylinders (i.e., cylinders with an elliptical basal face) that are denser than the corresponding optimal lattice pack- FIG. 19. (color online) A portion of the densest three-parameter ings for any aspect ratio greater than unity have been family of tetrahedron packings with 4 particles per fundamental cell 284,325 4000 constructed, which is consistent with Conjecture 3. and packing fraction φ = 4671 = 0.856347 ... obtained by Chen et al.; see Ref. 295 for a general treatment. A corollary to Conjecture 3 is that the densest packings of congruent, centrally symmetric particles that possess three equivalent principle axes (e.g., superballs) are gen- 139 maximal packing fraction is infinitesimally smaller that erally Bravais lattices. 255 −25 Subsequently, Torquato and Jiao304 generalized the unity (φmax ≤ 1 − 2.6 × 10 ) is a testament to the difficulty in accounting for the orientations of the tetra- aforementioned three conjectures in order to guide one hedra in a dense packing. to ascertain the densest packings of other convex nonspherical particles as well as concave shapes. These generalized organizing principles are explicitly stated as the 5. Organizing Principles for Convex and Concave Particles following four distinct propositions: Proposition 1: Dense packings of centrally symmetric Torquato and Jiao140,141 showed that substantial face- convex, congruent particles with three equivalent axes to-face contacts between any of the centrally symmetric are given by their corresponding densest lattice packings, Platonic and Archimedean solids allow for a higher pack- providing a tight density lower bound that may be opti- ing fraction. They also demonstrated that central sym- mal. metry enables maximal face-to-face contacts when par- Proposition 2: Dense packings of convex, congruent ticles are aligned, which is consistent with the densest polyhedra without central symmetry are composed of packing being the optimal lattice packing. The afore- centrally symmetric compound units of the polyhedra mentioned simulation results, upper bound, and theo- with the inversion-symmetric points lying on the densest retical considerations led to three conjectures concerning lattice associated with the compound units, providing a the densest packings of polyhedra and other nonspherical tight density lower bound that may be optimal. 3 particles in R ; see Refs. 140, 141, and 295. Proposition 3: Dense packings of centrally symmetric Conjecture 1: The densest packings of the centrally concave, congruent polyhedra are given by their corre- symmetric Platonic and Archimedean solids are given by sponding densest lattice packings, providing a tight den- their corresponding optimal (Bravais) lattice packings. sity lower bound that may be optimal. Conjecture 2: The densest packing of any convex, con- Proposition 4: Dense packings of concave, congruent gruent polyhedron without central symmetry generally polyhedra without central symmetry are composed of is not a (Bravais) lattice packing, i.e., the set of such centrally symmetric compound units of the polyhedra polyhedra whose optimal packing is not a lattice is over- with the inversion-symmetric points lying on the densest whelmingly larger than the set whose optimal packing is lattice associated with the compound units, providing a a lattice. tight density lower bound that may be optimal. Conjecture 3: The densest packings of congruent, Proposition 1 originally concerned polyhedra. It is gen- centrally symmetric particles that do not possess three eralized here to include any centrally symmetric convex, equivalent principle axes (e.g., ellipsoids) are generally congruent particle with three equivalent axes to reflect not (Bravais) lattice packings. the arguments put forth by Torquato and Jiao.140,304 Conjecture 1 is the analog of Kepler’s sphere conjecture All of the aforementioned organizing principles in the for the centrally symmetric Platonic and Archimedean form of conjectures and propositions have been tested solids. On the experimental side, it has been shown305 in Ref. 304 against a comprehensive set of both convex that such silver polyhedral nanoparticles self-assemble and concave particle shapes, including but not limited to into the conjectured densest lattice packings of such the Platonic and Archimedean solids,19,140,300 Catalan shapes. Torquato and Jiao141 have also commented on solids,326 prisms and antiprisms,326 Johnson solids,326 the validity of Conjecture 1 to polytopes in four and cylinders,284,325 lenses317, truncated cubes312 and vari- higher dimensions. The arguments leading to Conjec- ous concave polyhedra.326 These general organizing prin- ture 1 also strongly suggest that the densest packings ciples also enable one to construct analytically the dens- 25 est known packings of certain convex nonspherical par- glimpse into the richness of packing models and their ticles, including spherocylinders, and square pyramids capacity to capture the salient structural and physical and rhombic pyramids.304 Moreover, it was shown how properties of a wide class of equilibrium and nonequilib- to apply these principles to infer the high-density equi- rium condensed phases of matter that arise across the librium crystalline phases of hard convex and concave physical, mathematical and biological sciences as well as particles.304 We note that the densest known 2D pack- technological applications. ings of a large family of 2D convex and concave particles (e.g., crosses, curved triangles, moon-like shapes) fully Not surprisingly, there are many challenges and open adhere to the aforementioned organizing principles.308 questions. Is it possible to prove a first-order freezing Interestingly, the densest known packing of any iden- transition in 3D hard-sphere systems? Is the FCC lat- tical 3D convex particle studied to date has a density tice provably the entropically favored state as the maximal density is approached along the stable crystal branch that exceeds√ that of the optimal sphere packing value S in this same system? Can one devise numerical algo- φmax = π/ 18 = 0.7408 .... These results are consistent with a conjecture of Ulam, who proposed to Martin rithms that produced large disordered strictly jammed Gardner327, without any justification, that the optimal isostatic sphere packings that are rattler-free? Is the packing fraction for congruent sphere packings is smaller true MRJ state rattler-free in the thermodynamic limit? than that for any other convex body. There is currently Can one prove the Torquato-Stillinger conjecture that no proof of Ulam’s conjecture. links strictly jammed sphere packings to hyperuniformity? Can one identify incisive order metrics for packings of nonspherical particles as well as a wide class of many- 6. Dense Packings of Tori particle systems that arise in molecular, biological, cos- mological, and ecological systems? What are the appro- We have seen that the preponderance of studies of priate generalizations of the jamming categories for pack- 3 ings of nonspherical particles? Is it possible to prove that dense packings of nonspherical shapes in R have dealt with convex bodies that are simply connected and thus the densest packings of the centrally symmetric Platonic topologically equivalent to a sphere. However, much less and Archimedean solids are given by their corresponding is known about dense packings of multiply connected optimal lattice packings? Upon extending the geometric- solid bodies. Gabbrielli et al.313 investigated the packing structure approach to Euclidean dimensions greater than 3 three, do periodic packings with arbitrarily large unit behavior of congruent ring tori in R , which are multiply connected non-convex bodies of genus one, as well as horn cells or even disordered jammed packings ever provide the and spindle tori. Guided by the aforementioned organiz- highest attainable densities? Can one formulate a disor- d ing principles, they analytically constructed a family of dered sphere-packing model in R that can be rigorously d dense periodic packings of individual tori and found that shown to have a packing fraction that exceeds φ = 1/2 , the horn tori as well as certain spindle and ring tori can the maximal value achievable by the ghost RSA packing? In view of the wide interest in packing problems achieve a packing√ density not only higher than that of spheres (i.e., π/ 18 = 0.7404 ...) but also higher than across the sciences, it seems reasonable to expect that the densest known ellipsoid packings (i.e., 0.7707 ...). substantial conceptual advances are forthcoming. Moreover, they studied 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 Fig. 20 for a specific example.

ACKNOWLEDGMENTS

I am deeply grateful to Frank Stillinger, Paul Chaikin, Aleksandar Donev, Obioma Uche, Antonello Scardicchio, Weining Man, Yang Jiao, Chase Zachary, Adam Hopkins, Etienne´ Marcotte, Joseph Corbo, Steven Atkinson, Remi FIG. 20. The densest known packing of tori of radii ratio Dreyfus, Arjun Yodh, Ge Zhang, Duyu Chen, Jianxiang 2 with a packing fraction φ ' 0.7445. The images (taken Tian, Michael Klatt, Enrqiue Lomba and Jean-Jacques from Ref. 313) show four periodic units viewed from diﬀerent Weis with whom I have collaborated on topics described angles, each containing four tori. in this review article. I am very thankful to Jaeuk Kim, Duyu Chen, Zheng Ma, Yang Jiao and Ge Zhang for VI. CHALLENGES AND OPEN QUESTIONS comments that greatly improved this article. The au- thor’s work on packing models over the years has been Packing problems are fundamental and their solutions supported by various grants from the National Science are often profound. This perspective provides only a Foundation. 26

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