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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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 NSFDMR0087349, DOE DEFG0203ER46087
rk*
r
Mixed Phase TransitionsMonasson, Zecchina, Kirkpatrick, Selman, Troyansky, Nature 400, 133 (1999).
E=0, no violated clauses
E>0, violated clauses
 etc.
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
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.
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
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 JTemperature
unjammed
Shear stress
jammed
J
1/Density
soft, repulsive, finiterange
sphericallysymmetric
potentials
Model granular materials
e.g. Start w/ random initial positions
Conjugate gradient energy minimization (Inherent structures, Stillinger & Weber)
Classify resulting configurations
How we study Point JTi=∞
nonoverlapped
V=0
p=0
overlapped
V>0
p>0
or
Tf=0
Tf=0
We focus on ensemble rather than individual configs (c.f. Torquato)
Good ensemble is fixed ffc, or fixed pressure







4
3
2
log( c)
D=2
D=3
Distribution of fc values narrows as system size grows
Distribution approaches deltafunction as N
Essentially all configurations jam at one packing density
Of course, there is a tail up to closepacked crystal
J is a “POINT”
f0
w
2d (bidisperse)
3d (monodisperse)
These are values associated with random closepacking!
Point J is at Random ClosePackinglog(f* f0)
f0
w
It is a “point”
Isostatic
Mixed first/second order zero T transition
Point JTemperature
unjammed
Shear stress
jammed
J
1/Density
soft, repulsive,
finiterange
sphericallysymmetric
potentials
(2D)
(3D)







log(f fc)
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 loww 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 IsostaticityLowest freq mode atffc=108
Density of Vibrational Modes
f fc
It is a “point”
Isostatic
Mixed first/second order zero T transition
Point JTemperature
unjammed
Shear stress
jammed
J
1/Density
soft, repulsive,
finiterange
sphericallysymmetric
potentials
Below w* , modes approach those of ordinary elastic solid
We find powerlaw scaling
L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (2005)
Is there a Diverging Length Scale at J?w
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 ScaleMixed firstorder/secondorder 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 loadbearing network have at least d+1 neighbors
Kcore percolation is exactly solvable on Bethe lattice
This is meanfield solution
Let K=probability of infinite kconnected 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 kcore percolation transitions in finite dimensions
Continuous percolation (Charybdis)
No percolation until p=1 (Scylla)
Discontinuous percolation?
Yes, for kcore variants
Knights models (Toninelli, Biroli, Fisher)
kcore with pseudo forcebalance (Schwarz, Liu, Chayes)
Rigorous proofs that
pc<1
Transition is discontinuous*
Transition has diverging correlation length*
*based on conjecture of anisotropic critical behavior in directed percolation
We introduce “forcebalance” constraint to eliminate selfsustaining 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
If transition were continuous, c would decrease with L
Discontinuous Transition? YesFraction of sites in spanning cluster
Finitesize scaling
If pc = 1, expect pc(L) = 1AeBL
Aizenman, Lebowitz, J. Phys. A 21, 3801 (1988)
We find
We actually have a proof now that pc<1 (Jeng, Schwarz)
This value of collapses the order parameter data with
For ordinary 1storder transition,
How much is removed by the culling process?
Exponents for kcore variants in d=2 are different from those in meanfield!
Mean field d=2
Why does Point J show meanfield 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 kcore 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
Meanfield pspin interaction spin glass (Kirkpatrick, Thirumalai, Wolynes)
Meanfield dimer model (Chakraborty, et al.)
Meanfield kineticallyconstrained models (Fredrickson, Andersen)
Modecoupling approximation of glasses (Biroli,Bouchaud)
These models all exhibit glassy dynamics!!
First hint of UNIVERSALITY in jamming
Common exponents in
different jamming models
in mean field!
But different in finite dimensions
Hope you like jammin’, too!
Thanks to NSFDMR0087349
DOE DEFG0203ER46087
ConclusionsT
sxy
1/r
J
Appears to be associated with selfsustaining clusters
For example, k=3 on triangular lattice
pc=0.6921±0.0005, M. C. Madeiros, C. M. Chaves, Physica A (1997).
Selfsustaining 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=1Voids unstable to shrinkage, not growth in sphere packings
Point J only exists for repulsive, finiterange 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