Commonalities between Edwards ensemble and glasses Adrian Baule Lin Bo Max Danisch Hernan Makse Yuliang Jin City College of Romain Mari New York Louis Portal Chaoming Song jamlab.org Workshop: Physics of glassy and granular materials July 16-19, Kyoto Wednesday, July 17, 13
Random packings of hard spheres Physics Mathematics Applications Granular matter Information theory Random close packing (RCP) Shannon (1948) Bernal packings (1960) Signals → High dimensional spheres Glasses 2. High-dimensional packings 3. Non-spherical shapes 1. Edwards ensemble for grains and glass theory Volume fraction Force distribution Wednesday, July 17, 13
Theoretical approach I: Statistical mechanics (Edwards’ ensemble) Edwards and Oakeshott, Physica A (1989) Constraint optimization problem Z x, ~ x D ~ Z ( X, T ) = exp[ − W ( ~ x ) /X ] exp[ − H ( ~ f ) /T ] D ~ f Minimize volume (X=0) with constraint of force balance (T=0) and non-overlaping. OPTIMIZATION STATISTICAL PHYS EDWARDS instance sample packing cost function energy volume optimal configuration ground state RCP at X=0 minimal cost ground state energy minimal volume 3 Wednesday, July 17, 13
Theoretical approach II: Mean field theory of jammed hard-sphere (remnant of RSB solution from replica theory) Parisi and Zamponi (2010) Jammed states: J-line s → 0 liquid state splits max { Σ + s } Replica theory: jammed states are the infinite pressure limit of metastable hard sphere glasses • Approach jamming from the liquid phase. [ φ th , φ GCP ] ≈ [0 . 64 , 0 . 68] • Predict a range of RCP densities • Mean field theory based on RSB solution in the glass phase. Wednesday, July 17, 13
(un)Commonalities between Edwards ensemble and RT: 3d 8 densities in 0 . 64 ± 0 . 04 6% max { S } φ th = 0 . 64 φ GCP = 0 . 68 φ MRJ φ c √ φ fcc = 0 . 74 φ edw = 6 / (6 + 2 3) √ φ RLP = 4 / (4 + 2 3) φ onset φ RCP = φ J max { Σ + s } s → 0 φ K = 0 . 62 φ d = 0 . 58 Wednesday, July 17, 13
Very difficult in practice: very small range for 3d equal-size spheres -3 10 0.74 -4 10 -5 10 0.48 crystal -6 10 0.72 -7 10 φ = φ j Pade EOS coexistence 3d Glass fit 4d 0.7 φ j φ j ~ ∼ Equilibrium -6 0.47 10 0.68 -6 4*10 -6 16*10 -6 fluid 32*10 -6 64*10 0.66 -6 128*10 φ = φ j PY EOS 0.64 0.46 0.5 0.55 0.6 0.65 0.7 0.75 0.34 0.36 0.38 0.4 0.42 0.44 0.46 φ φ FIG. 1 (From (Skoge et al. , 2006)) Evolution of the pressure during compression at rate γ in d = 3 (left) and d = 4 (right). The density ϕ is increased at rate γ and the reduced pressure p ( ϕ ) = β P/ ρ is measured during the process. See (Skoge et al. , p ( ϕ ) ϕ 2006) for details. The quantity � ϕ j ( ϕ ) = p ( ϕ ) − d is plotted as a function of ϕ . If the system jams at density ϕ j , p → ∞ and ϕ j → ϕ j . Thus the final jamming density is the point where � � ϕ j ( ϕ ) intersects the dot-dashed line � ϕ j = ϕ . (Left) The dotted line is the liquid (Percus-Yevick) equation of state. The curves at high γ follow the liquid branch at low density; when they leave it, the pressure increases faster and diverges at ϕ j . The curves for lower γ show first a drop in the pressure, which signals crystallization. (Right) All the curves follow the liquid equation of state (obtained from Eq.(9) of (Bishop and Whitlock, 2005)) and leave it at a density that depends on γ . In this case no crystallization is observed. For γ = 10 − 5 the dot-dashed line is a fit to the high-density part of the pressure (glass branch). The arrow marks the region where the pressure crosses over from the liquid to the glass branch. φ GCP = 0 . 68 √ φ edw = 6 / (6 + 2 3) √ φ RLP = 4 / (4 + 2 3) φ MRJ φ RCP φ c φ onset φ fcc = 0 . 74 φ th = 0 . 64 Wednesday, July 17, 13
1. Full solution: Constraint optimization problem Z x, ~ x D ~ Z ( X, T ) = exp[ − W ( ~ x ) /X ] exp[ − H ( ~ f ) /T ] D ~ f T=0 and X=0 optimization problem 2. Approximation: Decouple forces from geometry. Z Z exp[ − H ( ~ f ) /T ] D ~ Z ( X, T ) = exp[ − W ( ~ x ) /X ] D ~ x × f 4. Cavity method for 3. Edwards for volume ensemble force ensemble + Isostaticity 2 2 3 0 1 N X X f ab ˆ n ab H = 6 7 @ A 4 5 a =1 b, ( ab ) ∈ E Song, Wang, Jin, Makse, Physica A (2010) Bo, Mari, Song, Makse (2013) Wednesday, July 17, 13
The Volume function is the Voronoi volume contact network second coordination shell consist of all points closer to the center of the grain than to any other Wednesday, July 17, 13
“Easily” generalizable to other systems equal size polydisperse spheres system ellipsoids, spherocylinders, non-convex particles, rods, sphere/ellipsoids mixtures, etc. any dimension Wednesday, July 17, 13
“Easily” generalizable to other systems equal size polydisperse spheres system ellipsoids, spherocylinders, non-convex particles, rods, sphere/ellipsoids mixtures, etc. any dimension Wednesday, July 17, 13
Analytical formula for Voronoi boundary Voronoi j particle r ij θ ij i r ij / cos θ ij ˆ s � � 1 W i = 1 r ij � 3 ds 2 R min cos θ ij 3 j Important: global minimization. Reduce to one-dimension Wednesday, July 17, 13
Analytical formula for Voronoi boundary Voronoi j particle r ij θ ij i r ij / cos θ ij ˆ s � � 1 W i = 1 r ij � 3 ds 2 R min cos θ ij 3 j Important: global minimization. Reduce to one-dimension Wednesday, July 17, 13
Analytical formula for Voronoi boundary Voronoi j particle r ij θ ij i r ij / cos θ ij ˆ s � � 1 W i = 1 r ij � 3 ds 2 R min cos θ ij 3 j Important: global minimization. Reduce to one-dimension Wednesday, July 17, 13
Analytical formula for Voronoi boundary Voronoi j particle r ij θ ij i r ij / cos θ ij ˆ s � � 1 W i = 1 r ij � 3 ds 2 R min cos θ ij 3 j Important: global minimization. Reduce to one-dimension Wednesday, July 17, 13
Average free-volume per particle Z 1 Z ∞ ( c 3 − 1) p ( c ) dc = ( c 3 − 1) dP > ( c ) w = 1 0 Voronoi Geometrical boundary interpretation of cumulative dist: P > ( c ) c/ 2 − ∂ P > ( c ) = p ( c ) ∂ c Probability to find all particles outside excluded volume and surface: S ∗ ( c ) V ∗ ( c ) Wednesday, July 17, 13
Mean-field approximation analogous to decorrelation principle particles belong to bulk or in contact: Particle gas Similar to car parking problem (Renyi, 1960). V ∗ ( c ) Probability to find a spot with in a volume V V ∗ ( c ) V Particle P ( V ∗ ) = (1 − V ∗ /V ) N → e − ρ V ∗ gas Wednesday, July 17, 13
Calculation of P > (c) V ∗ ( c ) Particles are in contact and in the bulk: r R θ P > ( c ) = P B ( c ) × P c ( c ) c = r/ cos θ ˆ s 2 R S ∗ ( c ) Bulk term: ρ ( w ) = 1 P B ( c ) = e − ρ V ∗ ( c ) w mean free volume density Contact term: P C ( c ) = e − ρ s S ∗ ( c ) √ 1 3 ρ s ( z ) = � S ∗ � = 4 π z z = geometrical coordination number mean free surface density Wednesday, July 17, 13
Average Voronoi volume Self-consistent equation: equal to zero represent the average free-volume of a single particle Wednesday, July 17, 13
Prediction: volume fraction vs z free volume volume function √ w = 2 3 φ = z √ z +2 3 z Equation of state agrees well with simulations and experiments RCP Aste, JSTAT 2006 z = 6 X-ray tomography 1 300,000 grains w = √ 3 6 φ = √ 6 + 2 3 Theory φ = . 634 Wednesday, July 17, 13
Definition of jammed state: geometric coordination z bounded by mechanical coordination Z positions geometrical constraints effectively excludes the ordered states Wednesday, July 17, 13
Edwards phase diagram for hard spheres Isostatic plane Z φ RLP ( Z ) = √ Z +2 3 Forbidden zone no disordered jammed Disordered Packings packings can exist Decreasing compactivity X 0.634 4 6 φ RCP = φ RLP = 3 = 0 . 536 √ 6+2 3 √ 4 + 2 Wednesday, July 17, 13
Jammed packings in high dimensions Minkowsky lower bound: Rigorous bounds Kabatiansky-Levenshtein upper bound: Question: what’s the density of RCP in high dimensions? Conjecture: are disordered packings more optimal than ordered ones? Wednesday, July 17, 13
Conjecture: P > (c) becomes valid in the high-dimensional limit (I) Theoretical conjecture of g 2 in high d (neglect correlations) Torquato and Stillinger, Exp. Math., 2006 3 d Large d (II) Factorization of P > (c) Wednesday, July 17, 13
Comparison with other theories Isostatic packings ( z = 2 d ) with unique volume fraction Edwards’ theory Jin, Charbonneau, Meyer, Song, Zamponi , PRE (2010) Agree with Minkowski lower bound Glass transition RT Isostatic packings ( z = 2 d ) ranging volume fraction increases 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. φ th • Edwards solution seems to corresponds to . Higher entropy state. Wednesday, July 17, 13
Generalizing the theory of monodisperse sphere packings Non-spherical objects Polydisperse spheres (dimers, triangles, tetrahedra, spherocylinders, ellipsoids … ) Distribution of radius P ( r ) Distribution of angles P ( ) Extra degree of freedom treated as in Onsager 1949 Wednesday, July 17, 13
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