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
Random packings of hard spheres
Physics
Granular matter
Random close packing (RCP)
Bernal experiments (1960)
Mathematics
Engineering
Kepler conjecture
Kepler (1611)
One of the twenty-three Hilbert's
problems (1900).
Pharmaceutical industry
Solved by Hales using computerassisted proof (~2000).
Information theory
Glasses
Shannon (1948)
Mining & construction
(I) Unifying concepts of
glasses and grains
Coniglio, Fierro, Herrmann,
Nicodemi, Unifying concepts
in granular media and
glasses (2004).
Signals → High dimensional spheres
(II) High-dimensional packings
(III) Polydisperse and non-spherical packings
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).
Theoretical approach II: Statistical mechanics (Edwards’ theory)
Edwards and Oakeshott, Physica A (1989), Ciamarra, Coniglio, Nicodemi, PRL (2006).
Statistical mechanics
Hamiltonian
Energy
Statistical mechanics of
jammed matter
Volume function
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.
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
1. Full solution: Constraint optimization problem
T=0 and X=0 optimization problem: Computer science
2. Approximation: Decouple forces from geometry.
3. Edwards for volume ensemble
+ Isostaticity
Song, Wang, and Makse, Nature (2008)
Song, Wang, Jin, Makse, Physica A (2010)
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:
V
Similar to a car parking
model (Renyi, 1960).
Probability to find a spot
with
in a volume V
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
Prediction: volume fraction vs Z
Equation of state agrees well with simulations
and experiments
Aste, JSTAT 2006
X-ray tomography
300,000 grains
Theory
Phase diagram for hard spheres
Song, Wang, and Makse, Nature (2008)
Isostatic plane
Forbidden zone
no disordered jammed
packings can exist
Disordered Packings
Decreasing compactivity X
0.634
Jammed packings of highdimensional 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
3d
Large d
(II) Factorization of P>(c)
Background term
(mean-field approximation)
Contact term
Comparison with other theories
Edwards’ theory
Isostatic packings (z = 2d) with
unique volume fraction
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)
Kirkpatrick and Wolynes, PRA (1987).
(II)
Mode-coupling theory:
Kirkpatrick and Wolynes, PRA (1987); Ikeda and Miyazaki, PRL (2010)
(III) Replica theory:
Parisi and Zamponi, Rev. Mod. Phys. (2010)
Isostatic packings (z = 2d) with
ranging volume fraction
increasing with dimensions
• 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?
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)
Voronoi of non-spherical particles
Spheres
Dimers
Spherocylinders
Ellipses and ellipsoids
Triangles
The Voronoi of any
nonspherical shape can
Tetrahedra
be treated as interactions
between points and lines
17
Generalizing the theory of monodisperse sphere packings
Theory of monodisperse spheres
Polydisperse (binary) spheres
Non-spherical objects
(dimers, triangles, tetrahedrons,
spherocylinders, ellipses, ellipsoids … )
Distribution of radius P(r)
Extra degree of freedom
Distribution of angles P( )
Result of binary packings
Binary packings
RCP (Z = 6)
Danisch, Jin, Makse, PRE (2010)
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
23
24
No
solution
Z=2d
Solutions exist
25
26
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
RLP
Spheres: disordered branch
(theory)
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
Sloane
Signal
Most efficient design of signals
(Information theory)
Sampling theorem
Optimal packing
(Sphere packing problem)
High-dimensional point
Rigorous bounds
Minkowsky lower bound:
Kabatiansky-Levenshtein upper bound:
Question: what’s the density of RCP in high dimensions?
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