Jamming

Andrea J. LiuDepartment 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, UCLASidney R. Nagel James Franck Inst., U Chicago

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

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).

rrk* rk

€

f =0 r < rk

*

fc > 0 r > rk*

⎧ ⎨ ⎩

E=0, no violated clauses

E>0, violated clauses

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 spheres O’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 • 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 : how long you need to wait for system to behave like liquid

QuickTime™ and aSorenson Video 3 decompressorare needed to see this picture.

Glass TransitionWhen liquid cools, stress relaxation time

increases• When liquid crystallizes

– Particles order– Stress relaxation time suddenly jumps

• 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.

A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21 (1998).

Jamming Phase Diagram

jammed

unjammed

Temperature

Shear stress

1/DensityJ

Glass transition

Granular packings

unjammed state is in equilibriumjammed state is out of equilibrium

Problem: Jamming surface is fuzzy

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

unjammed

Temperature

Shear stress

1/DensityJ

jammed

soft, repulsive, finite-rangespherically-symmetricpotentials

Model granular materials

• 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

overlappedV>0p>0or

non-overlappedV=0p=0

Ti=∞

Tf=0 Tf=0

€

V (r) =ε 1− r /σ ij( )

αr ≤ σ ij

0 r >σ ij

⎧ ⎨ ⎪

⎩ ⎪

Onset of Jamming is Onset of Overlap

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

Torquato)• Good ensemble is fixed -c, or fixed pressure

•Pressures for different states collapse on a single curve

•Shear modulus and pressure vanish at the same c

D=2D=3

5 4 3 2log (φ- φc)

3

2

1

0

6

4

2

0

8

6

4

2

α=2

α=5/2

α=2

α=5/2

3D

2D

( )a

( )b

( )c

p≈p0(φ−φc)α−1

G≈G0(φ−φc)α−1.5

-

-

-

-

-

-

--4 -3 -2

log(- c)

How Much Does c Vary Among States?

• Distribution of c values narrows as system size grows

• Distribution approaches delta-function as N• Essentially all configurations jam at one packing

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

• J is a “POINT”

0.58 0.6 0.62 0.64φc

1

1.5

2=16N=32N=64N=256N=1024N=4096N

w

0

1 234log N

3

2

1

w≈N−0.55

€

→ ∞

• Where do virtually all states jam in infinite system limit?

2d (bidisperse)3d (monodisperse)

These are values associated with random close-packing!

1 234log N

3

2

1

log(*

- 0

)

€

φ* −φ0 ≈ N−1/ dν

€

ν ≅0.7

φ* =0.639±0.003φ* =0.842±0.001

Point J is at Random Close-Packing

0.58 0.6 0.62 0.64φc

1

1.5

2=16N=32N=64N=256N=1024N=4096N

w

0

• Point J is special– It is a “point”– Isostatic – Mixed first/second order zero T transition

Point J

unjammed

Temperature

Shear stress

1/DensityJ

jammed

soft, repulsive, finite-rangespherically-symmetricpotentials

Number of Overlaps/Particle Z

(2D) (3D)

€

Zc = 3.99 ± 0.02

Zc =5.97±0.03

5 4 3 2log (φ- φc)

3

2

1

0

6

4

2

0

8

6

4

2

α=2

α=5/2

α=2

α=5/2

3D

2D

( )a

( )b

( )c

log(- c)

Z−Zc ≈Z0(φ−φc)0.5

-

-

-

- - - -

Just below c, no particles overlap

Just above c there are

Zc

overlapping neighbors per particle

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! Z=2D

L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05)

• Excess low- modes swamp 2 Debye behavior: boson

peak• D() approaches constant as c (M. Wyart, S.R. Nagel,

T.A. Witten, EPL (05) )

Unusual Solid Properties Near Isostaticity

Density of Vibrational ModesLowest freq mode at -c=10-8

c

• Point J is special– It is a “point”– Isostatic – Mixed first/second order zero T transition

Point J

unjammed

Temperature

Shear stress

1/DensityJ

jammed

soft, repulsive, finite-rangespherically-symmetricpotentials

• For each -c, extract where D() begins to drop off

• Below , 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?

€

*∝ φ −φc( )0.5

QuickTime™ and aTIFF (Uncompressed) decompressor

are needed to see this picture.

• The frequency has a corresponding eigenmode

• Decompose eigenmode in plane waves• Dominant wavevector contribution is at peak of

fT(k,)

• extract k*:

• We also expect with

Frequency Scale implies Length Scale

€

l =2π /k*

⟩⋅∧⟨= ∑2

)exp(),( ii iT rikPkkf ωω

€

l ≅ φ−φc( )−ν + = 0.26

€

k* =ω * /cT

€

k # =ω * /cL

€

ν # ≅ 0.5

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

€

Z =0 φ < φc

Zc + Z0(φ −φc )β φ > φc

⎧ ⎨ ⎩

€

β =0.49 ± 0.03

€

V (r) =0 if r >σ

ε 1− r /σ( )α

if r ≤ σ

⎧ ⎨ ⎩

€

G ≈G0 φ −φc( )α −2φ −φc( )

γ

€

γ=0.48 ± 0.03

€

l ≈l 0 φ −φc( )−ν

€

ν =0.26 ± 0.05

€

ν # ≅ 0.5

• 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

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 c all particles in load-bearing network have at least d+1 neighbors

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)

€

K =0 p < pc

K c + K 0 p− pc( )β =1/ 2

p > pc

⎧ ⎨ ⎩

€

γ=1/2

€

ν =0.26 ± 0.05

• Recall simulation results

€

Z =0 φ < φc

Zc + Z0(φ −φc )β = 0.49±0.04 φ > φc

⎧ ⎨ ⎩

€

γ=0.48 ± 0.03

€

ν =1/4

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

€

ν # =1/2

€

ν # = 0.5(?)

K-Core Percolation in Finite Dimensions• There appear to be at least 3 different types of k-

core percolation transitions in finite dimensions1. Continuous percolation (Charybdis)2. No percolation until p=1 (Scylla)3. Discontinuous percolation?

– Yes, for k-core variantsKnights models (Toninelli, Biroli, Fisher)

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

Knights ModelToninelli, Biroli, Fisher, PRL 96, 035702 (2006).

• Rigorous proofs that

– pc<1

– Transition is discontinuous*– Transition has diverging correlation length*

*based on conjecture of anisotropic critical behavior in directed

percolation

A k-Core Variant•We introduce “force-balance” constraint to eliminate self-sustaining clusters

•Cull if k<3 or 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

• The discontinuity c increases with system size L

• If transition were continuous, c would decrease with L

Discontinuous Transition? YesFr

act

ion o

f si

tes

in s

pannin

g

clust

er

Pc<1? Yes

Finite-size scalingIf pc = 1, expect pc(L) = 1-Ae-BL

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

We find

€

pc (L→ ∞) = 0.396(1)

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

Diverging Correlation Length? Yes

• This value of collapses the order parameter data with

• For ordinary 1st-order transition,

€

ν1 ≅1.5

€

ν1 ≅1.5

€

β ≅1.0

€

ν =1/d = 0.5

Diverging Susceptibility? Yes

How much is removed by the culling process?

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).

€

β =1/2

ν =1/4

€

β ≅1.0

ν ≅1.5

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….• Recall random k-SAT

• Point J

Hope you like jammin’, too!

rrk* rk

-c0

E=0

E>0

• 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

ConclusionsT

xy

1/ J

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).

p=0.4, before culling p=0.4, after culling

p=0.6, after culling p=0.65, after culling

Self-sustaining clusters don’t exist in sphere packings

• E.g. k=3 on square latticeThere 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 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)

Repulsion vanishes at finite distance

U

r