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Basic Understanding of Condensed Phases of Matter Via Packing Models

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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, includ- ing 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 jam- ming 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 pack- ing 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
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