Statistical mechanics of random packings: from Kepler and Bernal to Edwards and Coniglio Hernan A. Makse Levich Institute and Physics Department City College.

Statistical mechanics of random packings: from Kepler and

Bernal to Edwards and Coniglio

Hernan A. MakseLevich 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

MathematicsPhysics Engineering

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 mechanicsStatistical mechanics of

jammed matter

Hamiltonian Volume functionEnergy Volume

Microcanonical ensemble

Number of states

EntropyCanonical partition function

Temperature Compactivity

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

Voronoiparticle

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 VV

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 2006X-ray tomography

300,000 grains

Equation of state agrees well with simulations and experiments

Theory

Decreasing compactivity X

Isostatic plane

Disordered PackingsForbidden zone no disordered jammed packings can exist

Phase diagram for hard spheresSong, Wang, and Makse, Nature (2008)

0.634

Jammed packings of high-dimensional spheres

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

(II) Factorization of P>(c)

Background term Contact term

Large d

3d

(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 packings Non-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 packingsBinary 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

2323

2424

2525

Solutions exist

No

solutionZ=2d

2626

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

EllipsoidsFCC

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