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
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
Stress relaxation time t: how long you need to wait for system to behave like liquid
When liquid cools, stress relaxation time increases
When liquid crystallizes
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.
unjammed state is in equilibrium
jammed state is out of equilibrium
Problem: Jamming surface isfuzzy
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”
Mixed first/second order zero T phase transitionPoint J
soft, repulsive, finite-range
Model granular materials
We focus on ensemble rather than individual configs (c.f. Torquato)
Good ensemble is fixed f-fc, or fixed pressure
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”
These are values associated with random close-packing!Point J is at Random Close-Packing
Just abovefc there are Zc
overlapping neighbors per particle
Just below fc, no particles overlap
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!
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
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?
Mixed first-order/second-order transition
Number of overlapping neighbors per particle
Static shear modulus
Diverging length scale
And perhaps also
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
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)
J. M. Schwarz, A. J. Liu, L. Chayes, EPL (06)
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)
Yes, for k-core variants
Knights models (Toninelli, Biroli, Fisher)
k-core with pseudo force-balance (Schwarz, Liu, Chayes)
Rigorous proofs that
Transition is discontinuous*
Transition has diverging correlation length*
*based on conjecture of anisotropic critical behavior in directed percolation
We introduce “force-balance” constraint to eliminate self-sustaining clusters
Cull if k<3or if all neighbors are on the same side
24 possible neighbors per site
Cannot have all neighbors in upper/lower/right/left half
If pc = 1, expect pc(L) = 1-Ae-BL
Aizenman, Lebowitz, J. Phys. A 21, 3801 (1988)
We actually have a proof now that pc<1 (Jeng, Schwarz)
This value of collapses the order parameter data with
For ordinary 1st-order transition,
How much is removed by the culling process?
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).
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
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
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 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