Statistical mechanics of random packings: from Kepler and Bernal to Edwards and Coniglio

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

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

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

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

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

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

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

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

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

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

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

12. Jammed packings of high-dimensional spheres

13. P>(c) in the high-dimensional limit (I) Theoretical conjecture of g2 in high d(neglect correlations) Torquato and Stillinger, Exp. Math., 2006 Parisi and Zamponi, Rev. Mod. Phys., 2010 3d Large d (II) Factorization of P>(c) Background term Contact term (mean-field approximation)

14. Comparison with other theories Isostatic packings (z = 2d) with unique volume fraction Edwards’ theory Jin, Charbonneau, Meyer, Song, Zamponi, PRE (2010) Agree with Minkowski lower bound 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) Isostatic packings (z = 2d) with ranging volume fraction increasing with dimensions 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?

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

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

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

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

20. 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

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

22. Forces

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25. No solution Z=2d Solutions exist 25

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

28. 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.

29. 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.

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