Jammed hard-particle packings: From Kepler to Bernal and beyond

Understanding the characteristics of jammed particle packings provides basic insights into the structure and bulk properties of crystals, glasses, and granular media and into selected aspects of biological systems. This review describes the diversity of jammed configurations attainable by frictionless convex nonoverlapping (hard) particles in Euclidean spaces and for that purpose it stresses individual-packing geometric analysis. A fundamental feature of that diversity is the necessity to classify individual jammed configurations according to whether they are locally, collectively, or strictly jammed. Each of these categories contains a multitude of jammed configurations spanning a wide and (in the large system limit) continuous range of intensive properties, including packing fraction $\varphi$, mean contact number $Z$, and several scalar order metrics. Application of these analytical tools to spheres in three dimensions (an analog to the venerable Ising model) covers a myriad of jammed states, including maximally dense packings (as Kepler conjectured), low-density strictly jammed tunneled crystals, and a substantial family of amorphous packings. With respect to the last of these, the current approach displaces the historically prominent but ambiguous idea of “random close packing” with the precise concept of “maximally random jamming.” Both laboratory procedures and numerical simulation protocols can and, frequently, have been used for creation of ensembles of jammed states. But while the resulting distributions of intensive properties may individually approach narrow distributions in the large system limit, the distinguishing varieties of possible operational details in these procedures and protocols lead to substantial variability among the resulting distributions, some examples of which are presented here. This review also covers recent advances in understanding jammed packings of polydisperse sphere mixtures, as well as convex nonspherical particles, e.g., ellipsoids, “superballs,” and polyhedra. Because of their relevance to error-correcting codes and information theory, sphere packings in high-dimensional Euclidean spaces have been included as well. Some remarks are also made about packings in (curved) non-Euclidean spaces. In closing this review, several basic open questions for future research to consider have been identified.

Figure 1

(Color online) A portion of the densest packing of identical circles in the plane, with centers lying at the sites of a triangular lattice. The fraction of ${\mathbb{R}}^2$ covered by the interior of the circles is $\varphi =\pi ∕\sqrt{12}=0.906899\dots$. The first claim of a proof was made by Thue in 1892. However, it is generally believed that the first complete error-free proof was produced only in 1940 by Fejes Tóth; see 145 and 40 for the history of this problem.

Figure 2

(Color online) Biological examples of packing geometries. Left panel: Schematic representation of the approximate numbers, shapes, and density of packing of macromolecules inside a cell of Escherichia coli. (Illustration by David S. Goodsell, the Scripps Research Institute.) Molecular crowding occurs because of excluded volume effects due to the mutual impenetrability of all of the macromolecules. This nonspecific steric repulsion is always present, regardless of any other attractive or repulsive interactions that might occur between the macromolecules. Thus, molecular crowding is essentially a packing problem (66). Right panel: Closeup of a daisy capitulum (flower head), as given by 138. The most prominent feature is two sets of spirals, one turning clockwise and the other counterclockwise. There are several models that relate such phyllotactic patterns to packings problems (138).

Figure 3

(Color) Three isobaric (constant pressure) cooling paths by which a typical liquid may solidify, represented in a volume vs temperature diagram. An infinitesimal cooling rate from the high-temperature liquid traces out the thermodynamic equilibrium path (shown in green), including a discontinuity resulting from the first-order freezing transition. This reversible path leads to the ground-state defect-free crystalline structure in the $T\to 0$ limit. Very slow but finite cooling rate (not shown) can involve crystal nucleation but typically creates defective crystals. More rapid cooling of the liquid (blue curves) can avoid crystal nucleation, passing instead through a glass transition temperature range, and resulting in metastable glassy solids at absolute zero. The volumes, energies, and other characteristics of those glasses vary with the specific cooling rate employed in their production.

Figure 4

(Color online) Symmetries associated with three particle shapes. Chords pass through each particle centroid. Left panel: A “superdisk” is centrally symmetric and possesses two equivalent principal axes. Middle panel: An ellipse is centrally symmetric but does not possess two equivalent principal axes. Right panel: A triangle is not centrally symmetric.

Figure 5

(Color online) Examples of lattice packings (i.e., Bravais lattices) depicted in two dimensions. Left panel: A portion of a lattice packing of congruent spheres. Each fundamental cell (depicted as a rhombus here) has exactly one assigned sphere center. Right panel: A portion of a lattice packing of congruent nonspherical particles. Each fundamental cell has exactly one particle centroid. Each particle in the packing must have the same orientation.

Figure 6

(Color online) Examples of periodic nonlattice packings depicted in two dimensions. Left panel: A portion of a periodic nonlattice packing of congruent spheres. The fundamental cell contains multiple spheres located anywhere within the cell subject to the nonoverlap constraint. Right panel: A portion of a periodic nonlattice packing of congruent nonspherical particles. The fundamental cell contains multiple nonspherical particles with arbitrary positions and orientations within the cell subject to the nonoverlap constraint.

Figure 7

(Color) Geometric characterization of packings via bond networks. Left panel: Illustrations of the Voronoi and Delaunay tessellations for a portion of a packing of congruent circular disks. The blue and red lines are edges of the Voronoi and Delaunay cells. Right panel: The corresponding contact network shown as green lines.

Figure 8

(Color online) Typical protocols used to generate disordered sphere packings in three dimensions produce highly crystalline packings in two dimensions. Left panel: A three-dimensional MRJ-like configuration of 500 spheres with $\varphi \approx 0.64$ produced using the Lubachevsky-Stillinger (LS) algorithm with a fast expansion rate (191). Right panel: A crystalline collectively jammed configuration (Sec. 4A) of 1000 disks with $\varphi \approx 0.88$ produced using the LS algorithm with a fast expansion rate (60).

Figure 9

(Color online) Illustrations of jamming categories. Leftmost panel: Honeycomb-lattice packing within a rectangular hard-wall container is locally jammed but is not collectively jammed (e.g., a collective rotation of a hexagonal particle cluster, as shown, will unjam the packing). Middle panel: Square-lattice packing within a square hard-wall container is collectively jammed. Rightmost panel: The square-lattice packing shown in the middle panel can be sheared and hence is not strictly jammed. Thus, we see that the square-lattice packing with square hard-wall boundary conditions can only be collectively jammed even in the infinite-volume limit. Thus, the distinction between collective and strict jamming for such packings remains in the thermodynamic limit.

Figure 10

(Color online) The polytope of allowed displacements ${\mathcal{P}}_{\mathbf{\Delta }\mathbf{R}}$ for a locally jammed disk (light shade) trapped among three (left) or six (right, as in the triangular lattice) fixed disks. The exclusion disks (dashed lines) of diameter twice the disk diameter are drawn around each of the fixed disks, along with their tangents (lines) and the polytope ${\mathcal{P}}_{\mathbf{\Delta }\mathbf{R}}$ they bound (black). The polytope for the isostatic (left) and overconstrained (right) case is a triangle and hexagon, respectively.

Figure 11

(Color) The isothermal phase behavior of three-dimensional hard-sphere model in the pressure-packing fraction plane. Increasing the density plays the same role as decreasing temperature of a molecular liquid; see Fig. 3. 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. 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. Adapted from 174.

Figure 12

(Color online) Two types of order. (a) Bond-orientational order contains information about the orientation of the vectors connecting neighboring particles in the packing of interest (left). If these orientations persist throughout the packing, as they do in a triangular lattice (right), the packing is considered to be perfectly bond-orientationally ordered. (b) Translational order contains information about the relative spacing of particles in the packing of interest at some fixed number density (left) relative to that of the densest-particle configuration at the same number density (right).

Figure 13

(Color) Schematic order maps in the density-order $\left(\varphi \text{−}\psi \right)$ plane for the three different jamming categories in ${\mathbb{R}}^3$ under periodic boundary conditions. White and blue regions contain the attainable packings, blue regions represent the jammed subspaces, and dark shaded regions contain no packings. The locus of points $A\text{−}{A}^{\prime }$ correspond to the lowest-density jammed packings. The locus of points $B\text{−}{B}^{\prime }$ correspond to the densest jammed packings. Points MRJ represent the maximally random jammed states, i.e., the most disordered states subject to the jamming constraint. It should be noted that the packings represented are not subject to rattler exclusion.

Figure 14

(Color online) Three different optimal strictly jammed packings identified in the rightmost graph of Fig. 13. Left panel: (A) $Z=7$. Middle panel: MRJ: $Z=6$ (isostatic). Right panel: (B) $Z=12$.

Figure 15

(Color online) A schematic of the disjoint set of nearly jammed packings (dark shaded regions) that develop in multidimensional configuration space as covering fraction $\varphi$ increases. The two axes represent the collection of configurational coordinates. Note that each individual region approaches a polytope in the jamming limit, as discussed in Sec. 4C.

Figure 16

(Color online) Packing protocols can be devised that lead to strictly jammed states at any specific density with a high probability of occurrence anywhere over a wide density range. Shown are histograms of jammed packings that are centered around four different packing fractions: $\varphi \approx 0.60$, 0.64, 0.68, and 0.72, as obtained by 99. The distributions become narrower as the system size increases.

Figure 17

(Color online) Pair statistics for packings in the immediate neighborhood of the three-dimensional MRJ state (106) with $\varphi \approx 0.64$. Left panel: Computational data on the pair-correlation function $g_2\left(r\right)$ vs $r∕D-1$ averaged over five packings of 10 000 spheres (59) of diameter $D$. It is normalized so that at large distances it tends to unity, indicating no long-range order. The split second peak, the discontinuity at twice the sphere diameter, and the divergence near contact are clearly visible. Right panel: The corresponding structure factor $S\left(k\right)$ as a function of the dimensionless wave number $kD∕\left(2\pi \right)$ for a million-particle packing (54). The inset shows the linear $\left(\left|\mathbf{k}\right|\right)$ nonanalytic behavior at $k=0$.

Figure 18

Frequently used hard-sphere size distributions. Left panel: Schulz size distribution for several values of $m$. Right panel: Logarithmic-normal size distribution for several values of $\beta$.

Figure 19

One large disk and two small disks in mutual contact provide the densest local arrangement of binary disks (74). The intersection of the shaded triangle with the three disks yields the local packing fraction ${\varphi }_U=\{\pi {\alpha }^2+2(1-{\alpha }^2)\arcsin [\alpha ∕(1+\alpha )]\}∕2\alpha {(1+2\alpha )}^{1∕2}$, where $\alpha =R_S∕R_L$.

Figure 20

(Color online) A sphere inscribed within the smallest circular cylinder (i) undergoes a linear stretch and shrinkage of the space along the vertical direction leading to a prolate spheroid (ii) and an oblate spheroid (iii). This linear transformation leaves the fraction of space occupied by the spheroids within the cylinders unchanged from that of the fraction of the cylinder volume occupied by the sphere, equal to $2∕3$.

Figure 21

(Color online) Density $\varphi$ vs aspect ratio $\alpha$ for MRJ packings of 10 000 ellipsoids as obtained by 52. The semiaxes here are $1,{\alpha }^{\beta },\alpha$. The inset shows the mean contact number $Z$ as a function of $\alpha$. Neither the spheroid (oblate or prolate) nor general ellipsoids cases attain their isostatic values of $Z=10$ or 12, respectively.

Figure 22

(Color online) Simple examples of hypoconstrained packings in which all particles are fixed, except the central one. Left panel: Generically, four contacting particles are required to trap the central one. Right panel: Special correlated configurations only require three contacting particles to trap the central one. The normal vectors at the points of contact intersect at a common point, as is necessary to achieve torque balance.

Figure 23

(Color online) The packing fraction of the “laminated” non-Bravais lattice packing of ellipsoids (with a two-particle basis) as a function of the aspect ratio $\alpha$. The point $\alpha =1$ corresponding to the face-centered-cubic lattice sphere packing is shown, along with the two sharp maxima in the packing fraction for prolate ellipsoids with $\alpha =\sqrt{3}$ and oblate ellipsoids with $\alpha =1∕\sqrt{3}$, as illustrated in the insets. For both $\alpha <1∕\sqrt{3}$ and $\alpha >\sqrt{3}$, the packing fractions of the laminated packings drop off precipitously holding the particle orientations fixed (lines). The presently maximal achievable packing fraction $\varphi =0.7707\dots$ is highlighted with a thicker line, and is constant for $\alpha ⩽1∕\sqrt{3}$ and $\alpha ⩾\sqrt{3}$ because there is an affine stretch by an arbitrary factor along a direction in a mirror plane of the particle directions; see 53.

Figure 24

(Color online) A portion of the densest packing of congruent ellipses, which is simply an affine transformation of the densest circle packing, i.e., the densest triangular lattice of circles. This maximally dense ellipse packing is not rotationally jammed for any noncircular shape but is strictly translationally jammed.

Figure 25

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

Figure 26

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

Figure 27

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

Figure 28

(Color online) Density vs deformation parameter $p$ for the packings of convex superballs (97). Inset: Around $p_c^{\ast }=1.1509\dots$, the two curves are almost locally parallel to each other.

Figure 29

(Color online) Density vs deformation parameter $p$ for the lattice packings of concave superballs (97). Inset: A concave superball with $p=0.1$, which will become a three-dimensional cross at the limit $p\to 0$.

Figure 30

(Color) Local packing structures with more contacts than average (shown in blue) and those with less contacts than average (shown in pink) in two-dimensional MRJ binary superdisk packings for different values of the deformation parameter $p$, as obtained from 98. Note that the degree of nongenericity decreases (fraction of space occupied by the pink colored particles decreases) as $p$ moves away from its sphere-point value. Here the size ratio (diameter of large superdisks divided by the diameter of the small superdisks) is 1.4 and the molar ratio (number of large superdisks divided by the number of small superdisks) is $1∕3$.

Figure 31

(Color online) The five Platonic solids: tetrahedron (P1), icosahedron (P2), dodecahedron (P3), octahedron (P4), and cube (P5).

Figure 32

(Color online) The 13 Archimedean solids: truncated tetrahedron (A1), truncated icosahedron (A2), snub cube (A3), snub dodecahedron (A4), rhombicosidodecahdron (A5), truncated icosidodecahedron (A6), truncated cuboctahedron (A7), icosidodecahedron (A8), rhombicuboctahedron (A9), truncated dodecahedron (A10), cuboctahedron (A11), truncated cube (A12), and truncated octahedron (A13). This typical enumeration of the Archimedean solids does not count the chiral forms (not shown) of the snub cube (A3) and snub dodecahedron (A4), which implies that the left-handed and right-handed forms of each of these pairs lack central symmetry. The remaining 11 Archimedean solids are nonchiral (i.e., each solid is superposable on its mirror image) and the only noncentrally symmetric one among these is the truncated tetrahedron.

Figure 33

(Color online) Closely related dense tetrahedron packings. Left panel: A portion of one member of the densest two-parameter family of tetrahedron packings with four particles per fundamental cell and packing fraction $\varphi =\frac{12250}{14319}=\mathrm{0.855\hspace{0.17em}506}\dots$ (181). The lighter shade particles represent two dimers (i.e., four tetrahedra) but from this perspective each dimer appears to be a single tetrahedron. Right panel: A portion of one member of the densest three-parameter family of tetrahedron packings with four particles per fundamental cell and packing fraction $\varphi =\frac{4000}{4671}=\mathrm{0.856\hspace{0.17em}347}\dots$ (28; 182). The lighter shade particles represent two dimers. It is clear that these two packings are configurationally almost identical to one another.

Figure 34

(Color online) Portions of the densest lattice packings of three of the centrally symmetric Platonic solids found by the ASC scheme (179, 180). Left panel: Icosahedron packing with packing fraction $\varphi =0.8363\dots$. Middle panel: Dodecahedron packing with packing fraction $\varphi =0.9045\dots$. Right panel: Octahedron packing with packing fraction $\varphi =0.9473\dots$.

Figure 35

(Color online) A fundamental problem in communications theory is to find the best way to send signals or “code words” (amplitudes at $d$ different frequencies) over a noisy channel. Each code word $\mathbf{s}$ corresponds to a coordinate in ${\mathbb{R}}^d$. The sender and receiver desire to design a “code book” that contains a large number of code words that can be transmitted with maximum reliability given the noise inherent in any communications channel. Due to the noise in the channel, the code word $\mathbf{s}$ will be received as $\mathbf{r}\ne \mathbf{s}$ such that $|\mathbf{s}-\mathbf{r}|<a$, where the distance $a$ is the maximum error associated with the channel. 158 showed that the best way to send code words over a noisy channel is to design a code book that corresponds to the densest arrangement of spheres of radius $2a$ in ${\mathbb{R}}^d$. To understand this result, shown are two choices for the code book in two dimensions. Left panel: Three code words (depicted as points labeled 1, 2, and 3) that are circumscribed by circles of radius $a$ are shown in which the distance between any two words is chosen to be less than $2a$. The received signal falls in the region common to all circles and therefore the receiver cannot determine which of the three code words shown was sent. Right panel: By choosing a code book in which the code words are at least a distance $2a$ apart (which corresponds to a sphere packing), any ambiguity about the transmitted code word is eliminated. Since one desires to send many code words per unit volume, the best code book corresponds to the densest sphere packing in ${\mathbb{R}}^d$ in the limit that the number of code words tends to infinity.

Figure 36

(Color online) Lattice packings in sufficiently high dimensions are almost surely unsaturated because the “holes” or space exterior to the spheres dominates ${\mathbb{R}}^d$ (40). To get an intuitive feeling for this phenomenon, it is instructive to examine the hypercubic lattice ${\mathbb{Z}}^d$ (square lattice for ${\mathbb{R}}^2$ and the simple cubic lattice for ${\mathbb{R}}^3$). Left panel: A fundamental cell of ${\mathbb{Z}}^d$ represented in two dimensions. Note that the distance between the point of intersection of the longest diagonal in the hypercube with the hypersphere boundary and the vertex of the cube along this diagonal is given by $\sqrt{d}-1$ for a sphere of unit radius. This means that ${\mathbb{Z}}^4$ already becomes unsaturated at $d=4$. Placing an additional sphere in ${\mathbb{Z}}^4$ doubles the density of ${\mathbb{Z}}^4$ and, in particular, yields the four-dimensional checkerboard lattice packing $D_4$, which is believed to be the optimal packing in ${\mathbb{R}}^4$. Right panel: A schematic “effective” distorted representation of the hypersphere within the hypercubic fundamental cell for large $d$, illustrating that the volume content of the hypersphere relative to the hypercube rapidly diminishes asymptotically. Indeed, the packing fraction of ${\mathbb{Z}}^d$ is given by $\varphi ={\pi }^{d∕2}∕\Gamma (1+d∕2)2^d$. It is the presence of the Gamma function, which grows as $(d∕2)\text{!}$, in the denominator that makes ${\mathbb{Z}}^d$ far from optimal (except for $d=1$). In fact, the checkerboard lattice $D_d$ with packing fraction $\varphi ={\pi }^{d∕2}∕\Gamma (1+d∕2)2^{(d+2)∕2}$ becomes suboptimal in relatively low dimensions because it too becomes dominated by larger and larger holes as $d$ increases.

Figure 37

(Color online) The ghost RSA packing model in ${\mathbb{R}}^d$ is a subset of the Poisson point process and a generalization of the standard RSA process. The latter is produced by randomly, irreversibly, and sequentially placing nonoverlapping objects into a large volume in ${\mathbb{R}}^d$ that at some initial time is empty of spheres (202; 70; 42; 199). If an attempt to add a sphere at some time results in an overlap with an existing sphere in the packing, the attempt is rejected and further attempts are made until it can be added without overlapping existing spheres. As the process continues in time, it becomes more difficult to find available regions into which the spheres can be added, and eventually in the saturation or infinite-time limit no further additions are possible. In the ghost RSA process, a sphere that is rejected is called a “ghost” sphere. No additional spheres can be added whether they overlap an existing sphere or a ghost sphere. The packing fraction at time $t$ for spheres of unit diameter is given by $\varphi \left(t\right)=\{1-\exp [-v_1\left(1\right)t]\}∕2^d$, where $v_1\left(R\right)$ is the volume of sphere of radius $R$ [cf. Eq. (2)]. Thus, we see that as $t\to +\infty$, $\varphi \to 2^{-d}$. Shown is a configuration of 468 particles of a ghost RSA packing in a fundamental cell in ${\mathbb{R}}^2$ at a packing fraction very near its maximal value of 0.25, as adapted from 151. Note that the packing is clearly unsaturated and there are no contacting particles.