Statistical mechanics of random packings: from Kepler and Bernal to Edwards and Coniglio Hernan A. Makse Levich Institute and Physics Department City College of New York jamlab.org... inspired by a Lecture given by Antonio at Boston University in the mid 90’s on unifying concepts of glasses and grains.

Random packings of hard spheres Mathematics PhysicsEngineering Kepler conjecture Information theory Granular matter Glasses Pharmaceutical industry Mining & construction (III) Polydisperse and non-spherical packings Kepler (1611) One of the twenty-three Hilbert's problems (1900). Solved by Hales using computer- assisted proof (~2000). Shannon (1948) Signals → High dimensional spheres Random close packing (RCP) Bernal experiments (1960) (II) High-dimensional packings (I) Unifying concepts of glasses and grains Coniglio, Fierro, Herrmann, Nicodemi, Unifying concepts in granular media and glasses (2004).

Parisi and Zamponi, Rev. Mod. Phys. (2010) Schematic mean-field phase diagram of hard spheres Theoretical approach I: Theory of hard-sphere glasses (replica theory) Jammed states (infinite pressure limit) Approach jamming from the liquid phase. Predict a range of RCP densities Mean field theory (only exact in infinite dimensions). Replica theory: jammed states are the infinite pressure limit of long-lived metastable hard sphere glasses

Edwards and Oakeshott, Physica A (1989), Ciamarra, Coniglio, Nicodemi, PRL (2006). Theoretical approach II: Statistical mechanics (Edwards’ theory) Statistical mechanics Statistical mechanics of jammed matter Hamiltonian Volume function Energy Volume Microcanonical ensemble Number of states Entropy Canonical partition function TemperatureCompactivity Free energy Assumption: all stable configurations are equally probable for a given volume.

The partition function for hard spheres 1. The Volume Function: W (geometry) 2. Definition of jammed state: force and torque balance Volume Ensemble + Force Ensemble Solution under different degrees of approximations

Song, Wang, and Makse, Nature (2008) Song, Wang, Jin, Makse, Physica A (2010) 1. Full solution: Constraint optimization problem 2. Approximation: Decouple forces from geometry. 3. Edwards for volume ensemble + Isostaticity T=0 and X=0 optimization problem: Computer science 4. Cavity method for force ensemble Bo, Song, Mari, Makse (2012)

The volume function is the Voronoi volume Voronoi particle Important: global minimization. Reduce to to one-dimension

Coarse-grained volume function Excluded volume and surface: No particle can be found in: Similar to a car parking model (Renyi, 1960). Probability to find a spot with in a volume V V

Coarse-grained volume function Particles are in contact and in the bulk: Bulk term: Contact term: z = geometrical coordination number mean free volume density mean free surface density

Prediction: volume fraction vs Z Aste, JSTAT 2006 X-ray tomography 300,000 grains Equation of state agrees well with simulations and experiments Theory

Decreasing compactivity X Isostatic plane Disordered Packings Forbidden zone no disordered jammed packings can exist Phase diagram for hard spheres Song, Wang, and Makse, Nature (2008) 0.634

Jammed packings of high- dimensional spheres

P > (c) in the high-dimensional limit (I) Theoretical conjecture of g 2 in high d (neglect correlations) Torquato and Stillinger, Exp. Math., 2006 Parisi and Zamponi, Rev. Mod. Phys., 2010 (II) Factorization of P > (c) Background termContact term Large d 3d3d (mean-field approximation)

Random first order transition theories (glass transition) (I) Density functional theory (dynamical transition) (II) Mode-coupling theory: (III) Replica theory: Kirkpatrick and Wolynes, PRA (1987). Kirkpatrick and Wolynes, PRA (1987); Ikeda and Miyazaki, PRL (2010) Parisi and Zamponi, Rev. Mod. Phys. (2010) No unified conclusion at the mean-field level (infinite d). Neither dynamics nor jamming. Does RCP in large d have higher-order correlations missed by theory?: Test of replica th. Are the densest packings in large dimensions lattices or disordered packings? Edwards’ theory Jin, Charbonneau, Meyer, Song, Zamponi, PRE (2010) Comparison with other theories Isostatic packings (z = 2d) with unique volume fraction Isostatic packings (z = 2d) with ranging volume fraction increasing with dimensions Agree with Minkowski lower bound

Beyond packings of monodisperse spheres Polydisperse packingsNon-spherical packings Clusel et al, Nature (2009) Donev, et al, Science (2004) Higher density? New phases (jammed nematic phase)? Platonic and Archimedean solids Torquato, Jiao, Nature (2009) Glotzer et al, Nature (2010). Ellipses and ellipsoids A first-order isotropic-to-nematic transition of equilibrium hard rods, Onsager (1949)

17 Spheres Dimers Triangles Tetrahedra Spherocylinders Ellipses and ellipsoids Voronoi of non-spherical particles The Voronoi of any nonspherical shape can be treated as interactions between points and lines

Generalizing the theory of monodisperse sphere packings Theory of monodisperse spheres Polydisperse (binary) spheres (dimers, triangles, tetrahedrons, spherocylinders, ellipses, ellipsoids … ) Non-spherical objects Extra degree of freedom Distribution of radius P(r)Distribution of angles P( )

Result of binary packings Binary packings Danisch, Jin, Makse, PRE (2010) RCP (Z = 6)

Results for packings of spherocylinders Baule, Makse (2012) Spherocylinder = 2 points + 1 line. Interactions reduces to 9 regions of line-points, line-line or point-point interactions. Prediction of volume fraction versus aspect ratio: agrees well with simulations Same technique can be applied to any shape. Theory

Cavity Method for Force Ensemble Edwards volume ensemble predicts: Cavity method predicts Z vs aspect ratio:

Forces

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25 Solutions exist No solution Z=2d

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A phase diagram for hard particles of different shapes Phase diagram for hard spheres generalizes to different shapes: Spheres: disordered branch (theory) Spheres: ordered branch (simulations) Dimers Spherocylinders Ellipsoids FCC RCP RLP

Conclusions 1. We predict a phase diagram of disordered packings 2. We obtain: RCP and RLP Distribution of volumes and coordination number Entropy and equations of state 3. Theory can be extended to any dimension: Volume function in large dimensions: Isostatic condition: Same exponential dependence as Minkowski lower bound for lattices.

Definition of jammed state: isostatic condition on Z z = geometrical coordination number. Determined by the geometry of the packing. Z =mechanical coordination number. Determined by force/torque balance.

Sphere packings in high dimensions Most efficient design of signals (Information theory) Optimal packing (Sphere packing problem) Sampling theorem Question: what’s the density of RCP in high dimensions? Rigorous bounds Minkowsky lower bound: Kabatiansky-Levenshtein upper bound: Signal High-dimensional point Sloane