Transcript Jamming Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Corey S.

Jamming
Andrea J. Liu
Department of Physics & Astronomy
University of Pennsylvania
Corey S. O’Hern
Leo E. Silbert
Jen M. Schwarz
Lincoln Chayes
Sidney R. Nagel
Mechanical Engineering, Yale Univ.
Physics, Southern Ill. Univ.
Physics, Syracuse Univ.
Mathematics, UCLA
James Franck Inst., U Chicago
Brought to you by NSF-DMR-0087349, DOE DE-FG02-03ER46087
Mixed Phase Transitions
• Recall random k-SAT
E=0, no violated clauses
E>0, violated clauses
rk
r k*
r
• Fraction of variables that are constrained obeys
•
*
 0
r  rk
f  
*
f c  0 diverging
r  rk length scale at rk*
Finite-size scalingshows
Monasson, Zecchina, Kirkpatrick, Selman, Troyansky, Nature 400, 133 (1999).

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
(1999).
Monasson, Zecchina, Kirkpatrick, Selman, Troyansky, Nature 400, 133
- 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
QuickTime™ and a
Sorenson Video 3 decompressor
are needed to see this picture.
Stress relaxation time t: how long you need to wait for
system to behave like liquid
Glass Transition
When 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.
Jamming Phase Diagram
A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21 (1998).
unjammed state is in equilibrium
jammed state is out of equilibrium
Temperature
Glass transition
unjammed
jammed
Granular packings
Shear stress
J
1/Density
Problem: Jamming surface is fuzzy
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).
soft, repulsive, finite-range
spherically-symmetric
potentials
Temperature
unjammed
jammed
Model granular materials
1/Density
J
Shear stress
• Point J is special
– It is a “point”
– Isostatic
– Mixed first/second order zero T phase transition
How we study Point J
• Generate configurations near J
– e.g. Start w/ random initial positions


1 r / ij 
V (r)  

0

r   ij
Ti=∞
r   ij
– Conjugate gradient energy minimization (Inherent structures, Stillinger & Weber)
• Classify resulting configurations
non-overlapped
V=0
p=0
overlapped
V>0
p>0
or
Tf=0
Tf=0
Onset of Jamming is Onset of Overlap
-2
D=2
D=3
(a)
-4
=2
-6
=5/2
p  p0(f  f c ) 1
-8
•Pressures for different
states collapse on a single
curve
0
(b)
-2
-4
-6
=2
=5/2
G  G0(f  fc ) 1.5
0
(c)
-4
-3
3D
•Shear modulus and
pressure vanish at the
same fc
-2
• We focus log(fon ensemble
fc) rather than individual configs (c.f. Torquato)
1
2D
• 2Good ensemble is fixed f-fc, or fixed pressure
3
5
4
3
log (f- fc)
2
How Much Does fc Vary Among States?
• Distribution of fc values narrows as system size grows
1
2
1.5
N=16
N=32
N=64
N=256
N=1024
N=4096
2
w  N 0.55
f0
1
3
w
1
234
• Distribution
approaches
as N
0.58
0.6
0.62
0.64delta-function
log N
f
• Essentially all configurations jam at one packing
density

Of course, there is a tail up to close-packed crystal
• J is a “POINT”
c

Point J is at Random Close-Packing
1
f *  f0  N 1/ d
2
1.5
log(f*- f0)
N=16
N=32
N=64
N=256
N=1024
N=4096
f0
2
3
1
w
0.58
0.6
0.62
0.64

  0.7
1
234
N
infinitelogsystem
f
• Where do virtually
all states jam in
limit?
2d (bidisperse)
3d (monodisperse)

These are values
* associated with random close-packing!
f  0.842  0.001
c
f*  0.639  0.003
Point J
Temperature
soft, repulsive,
finite-range
spherically-symmetric
potentials
1/Density
unjammed
jammed
Shear stress
J
• Point J is special
– It is a “point”
– Isostatic
– Mixed first/second order zero T transition
(a)
Number of Overlaps/Particle Z
4
=2
Just below
6
fc, no
=5/2
particles
overlap8
Just above
fc there are
Zc
overlapping
neighbors
per particle
0
(b)
2
=2
=5/2
4
6
0
(c)
3D
-1
-2
-3
2D
-5
Z  Zc  Z0 (f  f c ) 0.5
-3
-4
-2
log(ffcf)c)
log (f-
Z c  3.99  0.02
Zc  5.97  0.03
(2D)
(3D)
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
Unusual Solid Properties Near Isostaticity
L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05)
Density of Vibrational Modes
Lowest freq mode at f-fc=10-8
f fc
• Excess low-w modes swamp w2 Debye behavior: boson peak
• D(w) approaches constant as f fc (M. Wyart, S.R. Nagel, T.A.
Witten, EPL (05) )
Point J
Temperature
soft, repulsive,
finite-range
spherically-symmetric
potentials
1/Density
unjammed
jammed
Shear stress
J
• Point J is special
– It is a “point”
– Isostatic
– Mixed first/second order zero T transition
Is there a Diverging Length Scale at J?
w
• 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)
0.5
w* f  fc 
Frequency Scale implies Length Scale
• The frequency w* has a corresponding eigenmode
   0.26
 f  fc 
QuickTime™ and a
TIFF (Uncompressed) decompressor
are needed to see this picture.

• Decompose eigenmode in plane waves
fT (k , w)   i k  Pi exp(ik
r ) 
* i
w
• Dominant wavevector contribution is at peak of fT(k,w )
• extract k*:
• We also expect
with
*
k*  w * /cT
k  w * /c L
#
 2 / k
 #  0.5
2

Summary of Jamming Transition

0
if r  
V(r)  


1
r
/

 if r  
 
• Mixed first-order/second-order transition
• Number of overlapping neighbors per particle

0
Z  

Z

Z
(
f

f
)

c
0
c
• Static shear
modulus
f  fc
f  fc
 2
• Diverging
length
scale
G  G f  f  f  f
0
c
• And perhaps also

0


f  fc 


c

  0.49  0.03
  0.48  0.03
  0.26 0.05
 #  0.5
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 “kcore”, 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 is exactly
solvable on Bethe lattice
• This is mean-field solution
• Let K=probability of infinite
k-connected cluster
• For k>2 we find
• Recall simulation results

0
p  pc 
0
f  fc
K
 Leath, Reich, J. Phys. C (1979)
Z  
 1/ 2
Chalupa,
  0.490.04
K

K
p

p
p

p
Z

Z
(
f

f
)
f  fc


 c
Pittel, 
et al.,c J.Comb.0 Th. Ser. Bc 67, 111 (1996)
0
c
c
  1/2
# 1/4
  1/2

J. M. Schwarz, A. J. Liu, L. Chayes, EPL (06)
  0.48  0.03
 # 0.26 0.05
  0.5(?)
K-Core Percolation in Finite Dimensions
•
There appear to be at least 3 different types of k-core
percolation transitions in finite dimensions
1. Continuous percolation (Charybdis)
2. No percolation until p=1 (Scylla)
3. Discontinuous percolation?
– Yes, for k-core variants
Knights models (Toninelli, Biroli, Fisher)
k-core with pseudo force-balance (Schwarz, Liu, Chayes)
Knights Model
Toninelli, 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
k=3
24 possible neighbors per site
Cannot have all neighbors in
upper/lower/right/left half
•Cull if k<3 or if all neighbors are on the same side
Fraction of sites
in spanning cluster
Discontinuous Transition? Yes
• The discontinuity c increases with system size L
• If transition were continuous, c would decrease with L
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
pc (L  )  0.396(1)

We actually have a proof now that pc<1
(Jeng, Schwarz)
Diverging Correlation Length? Yes
1  1.5
• This value of
collapses the order parameter
data with
1  1.5 transition,
• For ordinary 1st-order
  1.0
 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
  1/2
  1.0
 does
1/ 4 Point J show
  mean-field
1.5
Why
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
• 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
E=0
E>0
• Point J
rk*
rk
Hope you like jammin’,
too!f-fc
0
r
Conclusions
• Point J is a special point
T
• Common exponents in
different jamming models
in mean field!
• But different in finite dimensions 1/r
xy
J
Hope you like jammin’, too!
• Thanks to NSF-DMR-0087349
DOE DE-FG02-03ER46087
Continuous K-Core Percolation
• Appears to be associated with self-sustaining clusters
• For example, k=3 on triangular lattice
Self-sustaining
clusters don’t
exist in sphere
packings
p=0.4, before culling
•
p=0.4, after culling
pc=0.6921±0.0005, M. C. Madeiros, C. M. Chaves, Physica A (1997).
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
Voids unstable to shrinkage, not
growth in sphere packings
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).
Point J and the Glass Transition
• Point J only exists for repulsive, finite-range potentials
• Real liquids have attractions
U
Repulsion vanishes at
finite distance
r
• Attractions serve to hold system at high enough density that
repulsions come into play (WCA)