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Jamming. Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Corey S. O’Hern Mechanical Engineering, Yale Univ. Leo E. Silbert Physics, Southern Ill. Univ. Jen M. Schwarz Physics, Syracuse Univ. Lincoln Chayes Mathematics, UCLA

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jamming

Jamming

Andrea J. Liu

Department of Physics & Astronomy

University of Pennsylvania

Corey S. O’Hern Mechanical Engineering, Yale Univ.

Leo E. Silbert Physics,Southern Ill. Univ.

Jen M. SchwarzPhysics, Syracuse Univ.

Lincoln ChayesMathematics, UCLA

Sidney R. Nagel James Franck Inst., U Chicago

Brought to you by NSF-DMR-0087349, DOE DE-FG02-03ER46087

mixed phase transitions

rk

rk*

r

Mixed Phase Transitions
  • Recall random k-SAT
  • Fraction of variables that are constrained obeys
  • Finite-size scaling shows diverging length scale at rk*

Monasson, Zecchina, Kirkpatrick, Selman, Troyansky, Nature 400, 133 (1999).

E=0, no violated clauses

E>0, violated clauses

mixed phase transitions3
Mixed Phase Transitions
  • “infinite-dimensional” models
    • p-spin interaction spin glass Kirkpatrick, Thirumalai, PRL 58, 2091 (1987).
    • k-core (bootstrap) Chalupa, Leath, Reich, J. Phys. C (1979); Pittel, Spencer, Wormald, J.Comb. Th. Ser. B 67, 111 (1996).
    • Random k-SAT Monasson, Zecchina, Kirkpatrick, Selman, Troyansky, Nature 400, 133 (1999).

- etc.

  • But physicists really only care about finite dimensions
    • Jamming transition of spheresO’Hern, Langer, Liu, Nagel, PRL 88, 075507 (2002).
    • Knights models Toninelli, Biroli, Fisher, PRL 96, 035702 (2006).
    • k-core + “force-balance” models Schwarz, Liu, Chayes, Europhys. Lett. 73, 560 (2006).
stress relaxation time
Stress Relaxation Time
  • Behavior of glassforming liquids depends on how long you wait
    • At short time scales, silly putty behaves like a solid
    • At long time scales, silly putty behaves like a liquid

Stress relaxation time t: how long you need to wait for system to behave like liquid

glass transition
Glass Transition

When liquid cools, stress relaxation time increases

When liquid crystallizes

Particles order

Stress relaxation time suddenlyjumps

When liquid is cooled through glass transition

Particles remain disordered

Stress relaxation time increases continuously

“Picture Book of Sir John Mandeville’s Travels,” ca. 1410.

jamming phase diagram
A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21 (1998).Jamming Phase Diagram

Temperature

Glass transition

unjammed

jammed

Shear stress

Granular packings

J

1/Density

unjammed state is in equilibrium

jammed state is out of equilibrium

Problem: Jamming surface isfuzzy

point j
C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 88, 075507 (2002).

C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel, Phys. Rev. E 68, 011306 (2003).

Point J is special

It is a “point”

Isostatic

Mixed first/second order zero T phase transition

Point J

Temperature

unjammed

Shear stress

jammed

J

1/Density

soft, repulsive, finite-range

spherically-symmetric

potentials

Model granular materials

how we study point j
Generate configurations near J

e.g. Start w/ random initial positions

Conjugate gradient energy minimization (Inherent structures, Stillinger & Weber)

Classify resulting configurations

How we study Point J

Ti=∞

non-overlapped

V=0

p=0

overlapped

V>0

p>0

or

Tf=0

Tf=0

onset of jamming is onset of overlap
Onset of Jamming is Onset of Overlap

We focus on ensemble rather than individual configs (c.f. Torquato)

Good ensemble is fixed f-fc, or fixed pressure

-

-

-

-

-

-

-

-4

-3

-2

log(- c)

D=2

D=3

  • Pressures for different states collapse on a single curve
  • Shear modulus and pressure vanish at the same fc
how much does f c vary among states
How Much Does fc Vary Among States?

Distribution of fc values narrows as system size grows

Distribution approaches delta-function as N

Essentially all configurations jam at one packing density

Of course, there is a tail up to close-packed crystal

J is a “POINT”

f0

w

point j is at random close packing
Where do virtually all states jam in infinite system limit?

2d (bidisperse)

3d (monodisperse)

These are values associated with random close-packing!

Point J is at Random Close-Packing

log(f*- f0)

f0

w

point j12
Point J is special

It is a “point”

Isostatic

Mixed first/second order zero T transition

Point J

Temperature

unjammed

Shear stress

jammed

J

1/Density

soft, repulsive,

finite-range

spherically-symmetric

potentials

number of overlaps particle z
Number of Overlaps/Particle Z

(2D)

(3D)

-

-

-

-

-

-

-

log(f- fc)

Just abovefc there are Zc

overlapping neighbors per particle

Just below fc, no particles overlap

isostaticity
Isostaticity

What is the minimum number of interparticle contacts needed for mechanical equilibrium?

No friction, spherical particles, D dimensions

Match unknowns (number of interparticle normal forces) to equations (force balance for mechanical stability)

Number of unknowns per particle=Z/2

Number of equations per particle = D

Point J is purely geometrical!

unusual solid properties near isostaticity
L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05)

Excess low-w modes swamp w2 Debyebehavior: boson peak

D(w) approaches constant as f fc(M. Wyart, S.R. Nagel, T.A. Witten, EPL (05))

Unusual Solid Properties Near Isostaticity

Lowest freq mode atf-fc=10-8

Density of Vibrational Modes

f- fc

point j16
Point J is special

It is a “point”

Isostatic

Mixed first/second order zero T transition

Point J

Temperature

unjammed

Shear stress

jammed

J

1/Density

soft, repulsive,

finite-range

spherically-symmetric

potentials

is there a diverging length scale at j
For each f-fc, extract w* where D(w) begins to drop off

Below w* , modes approach those of ordinary elastic solid

We find power-law scaling

L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (2005)

Is there a Diverging Length Scale at J?

w

frequency scale implies length scale
The frequency w* has a corresponding eigenmode

Decompose eigenmode in plane waves

Dominant wavevector contribution is at peak of fT(k,w*)

extract k*:

We also expect with

Frequency Scale implies Length Scale
summary of jamming transition
Summary of Jamming Transition

Mixed first-order/second-order transition

Number of overlapping neighbors per particle

Static shear modulus

Diverging length scale

And perhaps also

jamming vs k core bootstrap percolation
Jamming vs K-Core (Bootstrap) Percolation

Jammed configs at T=0 are mechanically stable

For particle to be locally stable, it must have at least d+1 overlapping neighbors in d dimensions

Each of its overlapping nbrs must have at least d+1 overlapping nbrs, etc.

At f> fc all particles in load-bearing network have at least d+1 neighbors

  • Consider lattice with coord. # Zmax with sites indpendently occupied with probability p
  • For site to be part of “k-core”, it must be occupied and have at least k=d+1 occupied neighbors
  • Each of its occ. nbrs must have at least k occ. nbrs, etc.
  • Look for percolation of the k-core
k core percolation on the bethe lattice
K-core Percolation on the Bethe Lattice

K-core percolation is exactly solvable on Bethe lattice

This is mean-field solution

Let K=probability of infinite k-connected cluster

For k>2 we find

Chalupa, Leath, Reich, J. Phys. C (1979)

Pittel, et al., J.Comb. Th. Ser. B 67, 111 (1996)

  • Recall simulation results

J. M. Schwarz, A. J. Liu, L. Chayes, EPL (06)

k core percolation in finite dimensions
K-Core Percolation in Finite Dimensions

There appear to be at least 3 different types of k-core percolation transitions in finite dimensions

Continuous percolation (Charybdis)

No percolation until p=1 (Scylla)

Discontinuous percolation?

Yes, for k-core variants

Knights models (Toninelli, Biroli, Fisher)

k-core with pseudo force-balance (Schwarz, Liu, Chayes)

knights model
Knights Model

Rigorous proofs that

pc<1

Transition is discontinuous*

Transition has diverging correlation length*

*based on conjecture of anisotropic critical behavior in directed percolation

  • Toninelli, Biroli, Fisher, PRL 96, 035702 (2006).
a k core variant
A k-Core Variant

We introduce “force-balance” constraint to eliminate self-sustaining clusters

Cull if k<3or if all neighbors are on the same side

k=3

24 possible neighbors per site

Cannot have all neighbors in upper/lower/right/left half

discontinuous transition yes
The discontinuity cincreases with system size L

If transition were continuous, c would decrease with L

Discontinuous Transition? Yes

Fraction of sites in spanning cluster

pc 1 yes
Pc<1? Yes

Finite-size scaling

If pc = 1, expect pc(L) = 1-Ae-BL

Aizenman, Lebowitz, J. Phys. A 21, 3801 (1988)

We find

We actually have a proof now that pc<1 (Jeng, Schwarz)

diverging correlation length yes
Diverging Correlation Length? Yes

This value of collapses the order parameter data with

For ordinary 1st-order transition,

diverging susceptibility yes
Diverging Susceptibility? Yes

How much is removed by the culling process?

slide29
BUT

Exponents for k-core variants in d=2 are different from those in mean-field!

Mean field d=2

Why does Point J show mean-field behavior?

Point J may have critical dimension of dc=2 due to isostaticity (Wyart, Nagel, Witten)

Isostaticity is a global condition not captured by local k-core requirement of k neighbors

Henkes, Chakraborty, PRL 95, 198002 (2005).

similarity to other models
Similarity to Other Models

The discontinuity & exponents we observe are rare but have been found in a few models

Mean-field p-spin interaction spin glass (Kirkpatrick, Thirumalai, Wolynes)

Mean-field dimer model (Chakraborty, et al.)

Mean-field kinetically-constrained models (Fredrickson, Andersen)

Mode-coupling approximation of glasses (Biroli,Bouchaud)

These models all exhibit glassy dynamics!!

First hint of UNIVERSALITY in jamming

to return to beginning

E=0

E>0

rk

rk*

r

To return to beginning….
  • Recall random k-SAT
  • Point J

Hope you like jammin’, too!

-c

0

conclusions
Point J is a special point

Common exponents in

different jamming models

in mean field!

But different in finite dimensions

Hope you like jammin’, too!

Thanks to NSF-DMR-0087349

DOE DE-FG02-03ER46087

Conclusions

T

sxy

1/r

J

continuous k core percolation
Continuous K-Core Percolation

Appears to be associated with self-sustaining clusters

For example, k=3 on triangular lattice

pc=0.6921±0.0005, M. C. Madeiros, C. M. Chaves, Physica A (1997).

Self-sustaining clusters don’t exist in sphere packings

p=0.4, before culling

p=0.4, after culling

p=0.6, after culling

p=0.65, after culling

no transition until p 1
E.g. k=3 on square lattice

There is a positive probability that there is a large empty square whose boundary is not completely occupied

After culling process, the whole lattice will be empty

Straley, van Enter J. Stat. Phys. 48, 943 (1987).

M. Aizenmann, J. L. Lebowitz, J. Phys. A 21, 3801 (1988).

R. H. Schonmann, Ann. Prob. 20, 174 (1992).

C. Toninelli, G. Biroli, D. S. Fisher, Phys. Rev. Lett. 92, 185504 (2004).

No Transition Until p=1

Voids unstable to shrinkage, not growth in sphere packings

point j and the glass transition
Point J and the Glass Transition

Point J only exists for repulsive, finite-range potentials

Real liquids have attractions

Attractions serve to hold system at high enough density that repulsions come into play (WCA)

U

Repulsion vanishes at finite distance

r