Transcript Jamming

Jamming
Andrea J. Liu
Department of Physics & Astronomy
University of Pennsylvania
Corey S. O’Hern
Mechanical Engineering, Yale Univ.
Leo E. Silbert
Ning Xu
Vincenzo Vitelli
Matthieu Wyart
Sidney R. Nagel
Physics, S. Illinois U., Carbondale
Physics, UPenn, JFI, U. Chicago
Physics, UPenn
Janelia Farms; Physics, NYU
James Franck Inst., U. Chicago
Jamming
Umbrella concept that aims to tie together
– two of oldest unsolved problems in condensed-matter
physics
• Glass transition
• Colloidal glass transition
– systems only recently studied by physicists
• Granular materials
• Foams and emulsions
¿Is there common behavior in these systems so that we can
benefit by studying them in a broader context?
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
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Speeded
up by
x80
Stress relaxation time t: how long you need to wait for
system to behave like liquid
Glass Transition
Earliest
glassmaking
3000BC
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are need ed to se e th is p icture.
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Glass vessels
from around
1500BC
When liquid is cooled through glass transition
– Particles remain disordered
– Stress relaxation time increases continuously
– Can get 10 orders of magnitude increase in 1020 K range
Colloidal Glass Transition
Suspensions of small (nm-10mm) particles include
– Ink, paint
– McDonald’s milk shakes, …..
– Blood
Micron-sized plastic spheres suspended in water form
crystals
glasses
Stress relaxation time increases with packing fraction
Granular Materials
Materials made up of many distinct grains include
– Pharmaceutical powders
– Cereal, coffee grounds, ….
– Gravel, landfill, ….
San Francisco
Marina District
after Loma Prieta
earthquake
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Static granular packing
behaves like a solid
Shaken granular packing
behaves like a liquid
Foams and Emulsions
Suspension of gas bubbles or liquid droplets
– Shaving cream
– mayonnaise
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QuickTime™ and a
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Courtesy of D. J. Durian
• Foams flow like liquids when sheared
• Stress relaxation time increases as shear stress decreases
Phenomena look similar in all these systems
• No obvious structural signature of jamming
• Dramatic increase of relaxation time near jamming
• Kinetic heterogeneities
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Supercooled
liquids
Courtesy of
S. C. Glotzer
Colloidal
suspensions
Granular
materials
Courtesy of
E. R. Weeks
and
D. A. Weitz
Courtesy of A. S.
Keys, A. R. Abate,
S. C. Glotzer, and
D. J. Durian
These Transitions Are Not Understood
• We understand crystallization and a lot of other
phase transitions
– Liquid-vapor criticality, liquid crystal transitions
– Superconductivity, superfluidity, Bose-Einstein cond…
– Many exotic quantum transitions, etc.
• But glass transition, etc. remain mysterious
– Are they really phase transitions or are they just
examples of kinetic arrest?
• Why are these systems so difficult?
– They are disordered
– They are not in equilibrium
Jamming
Jam ( jam), v. i.
1 To develop a yield stress in a disordered system
2 To have a stress relaxation time that exceeds 103 s in a
disordered system
E.g.
Supercooled liquids jam as temperature drops
Colloidal suspensions jam as density-1 drops
Granular materials jam as driving force drops
Foams, emulsions jam as shear stress drops
¿Can we unify these systems within one framework?
Jamming Phase Diagram
A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21 (1998).
Glass transition
Granular matter
Foams and emulsions
Exp’tal Jamming Phase Diagram
V. Trappe, V. Prasad, L. Cipelletti, P. N. Segre, D. A. Weitz, Nature, 411(N6839) 772 (2001).
1.0
0.8
0.6
0.4
10
kBT/U 0.2
0.0
20
1
2
Pa
Colloids with depletion attractions
3
1/
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).
Problem: Jamming surface is fuzzy
Temperature
unjammed
soft, repulsive,
finite-range
spherically-symmetric
potentials
jammed
1/Density
J
• Point J is special
– Random close-packing
– Isostatic
– Mixed first/second order zero T transition
– Connections to glasses and glass transition
Shear stress
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
T =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
•Shear modulus and pressure
vanish at the same fc
0
(b)
-2
-4
-6
1
=2
=5/2
G  G0(f  fc ) 1.5
0
(c)
-4
•Pressures for different
states collapse on a single
curve
-3 3D
log(f- fc)
•Good ensemble is fixed f - fc
-2
2D
2
3 Durian, PRL 75, 4780 (1995);
D. J.
5
4
3
2
C. S. O’Hern, S. A. Langer,
A.
J.
Liu,
S. R. Nagel, PRL 88, 075507 (2002).
log (f- f )
c
Dense Sphere Packings
¿What is densest packing of monodisperse hard spheres?
Johannes Kepler (1571-1630)
Conjecture (1611)
Thomas Hales
Fejes Tóth
2D Proof (1953) 3D Proof (1998)
2D
3D
triangular is densest possible packing
f   / 12  0.906
FCC/HCP is densest possible packing
f   / 18  0.740
Disordered Sphere Packings
Stephen Hales (1677-1761)
Vegetable Staticks (1727)
J. D. Bernal (1901-1971)
2D
3D
frcp  0.84
frcp  0.64
<
<
f cp  0.906
f cp  0.740
•Random close-packing is not well-defined mathematically
–One can always make a closer-packed structure that is less random

S. Torquato, T. M. Truskett, P. Debenedetti, PRL 84, 2064 (2000).
–But it is highly reproducible. Why? Kamien, Liu, PRL 99, 155501 (2007).
How Much Does fc Vary Among States?
• Distribution of fc values narrows as system size grows
1
N=16
N=32
N=64
N=256
N=1024
N=4096
2
1.5
2
w  N 0.55
3
1
w
0.58
0.6
f0
0.62
fc
0.64
1
234
log N
• Distribution approaches delta-function as N  
• Essentially all configurations jam at one packing density
• J is a “POINT”
Point J is at Random Close-Packing
1
f *  f 0  N 1/ d
2
1.5
log(f*- f0)
N=16
N=32
N=64
N=256
N=1024
N=4096
2
3
1
w
0.58
0.6
f0
0.62
fc
0.64

  0.7
1
234
log N
•Where do virtually all states jam in infinite system
limit?

f *  0.842  0.001 2d (bidisperse)
RCP!
*
f  0.639  0.003 3d (monodisperse)
Most of phase space belongs to basins of attraction of hard
sphere states that have their jamming thresholds at RCP
Point J
Temperature
soft, repulsive,
finite-range
spherically-symmetric
potentials
1/Density
unjammed
jammed
Shear stress
J
• Point J is special
– Random close-packing
– Isostatic
– Mixed first/second order zero T transition
– Connections to glasses and glass transition
4
=2
Number of Overlaps/Particle Z
6
=5/2
Just below
fc, no
8
particles
overlap
Just above
fc there are
Zc
overlapping
neighbors
per particle
0
(b)
2
=2
=5/2
4
6
0
3D
(c)
-1
-2
-3
2D
-5
-4
Z  Zc  Z0 (f  fc ) 0.5
-3
Zc  3.99  0.01 (2D)
Zc  5.97  0.03 (3D)
-2
log(flog (f-fcf)c)
Durian, PRL 75, 4780 (1995).
O’Hern, Langer, Liu, Nagel, PRL 88, 075507 (2002).
Verified experimentally:
Majmudar, Sperl, Luding, Behringer,
PRL 98, 058001 (2007).
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)
equations (force balance for mechanical
stability)
–Number of unknowns per particle=Z/2
–Number of equations per particle = D
James Clerk Maxwell
Z  2D
• Same for hard spheres at RCP Donev, Torquato, Stillinger, PRE 71, 011105 (‘05)
• Point J is purely geometrical! Doesn’t depend on potential
Marginally Jammed Solid is Unusual
L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05)
Density of Vibrational Modes
f fc
• Excess low-w modes swamp w2 Debye behavior: boson peak
• g(w) approaches constant as f fc
• Result of isostaticity M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)
Isostaticity and Boundary Sensitivity
M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)
•For system at fc, Z=2d
•Removal of one bond makes entire
system unstable by introducing one soft
mode
•This implies diverging length as f-> fc +
For f> fc, cut bonds at boundary of circle of size L
Count number of soft modes within circle
Ns  Ld 1  Z  Zc Ld
Define length scale at which soft modes just appear
1
0.5

 f  fc 
Z  Zc
Diverging Length Scale
Ellenbroek, Somfai, van Hecke, van Saarloos, PRL 97, 258001 (2006)
Look at response to
small particle
displacement
Define
h(r) 
f r (r)
2
 f r (r)
2
  0.5

Diverging Time and Length Scales
 / 20.5
w*  f  f c 
   0.26
 f  f c 
w*
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

w
• For each f-fc, extract w* where g(w) begins to drop off
• Below w* , modes approach those of ordinary elastic solid
• Decompose corresponding eigenmode in plane waves
• Dominant wavevector contribution is k*=w*/cT
• We also expect k #  w * /c L with  #  0.5
 2 /k *
Point J
Temperature
soft, repulsive,
finite-range
spherically-symmetric
potentials
1/Density
unjammed
jammed
Shear stress
J
• Point J is special
– Random close-packing
– Isostatic
– Mixed first/second order zero T transition
– Connection to glasses and glass transition
Summary of Jamming Transition

0
if r  
V (r)  

1 r /  if r  
• Mixed first-order/second-order transition (random first-order
phase transition)

• Number of overlapping
neighbors per particle

0
Z  

Z

Z
(
f

f
)
 c
0
c
• Static shear modulus
 2
G  G0 f  f c 
f  fc
f  fc

f  f c 

• Two diverging length scales


f  f 
0
c
• Vanishing frequency scale
w  w 0 f  fc 
*
 1



  0.49  0.03
  0.48  0.03
  0.26  0.05
#
  0.5
Similarity to Other Models
•
In jamming transition we find
– Jump discontinuity & =1/2 power-law in order parameter
– Divergences in susceptibility/correlation length with =1/2 and =1/4
•
This behavior has only been found in a few models
– Mean-field p-spin interaction spin glass
Kirkpatrick, Wolynes
– Mean-field compressible frustrated Ising antiferromagnet
– Mean-field kinetically-constrained Ising models
– Mean-field k-core percolation and variants
– Mode-coupling approximation of glasses
– Replica solution of hard spheres
•
Yin, Chakraborty
Sellitto, Toninelli, Biroli, Fisher
Schwarz, Liu, Chayes
Biroli, Bouchaud
Zamponi, Parisi
These other models all exhibit glassy dynamics!!
First hint of quantitative connection between sphere packings and glass
transition
Point J
Temperature
soft, repulsive,
finite-range
spherically-symmetric
potentials
1/Density
unjammed
jammed
Shear stress
J
• Point J is special
– Random close-packing
– Isostatic
– Mixed first/second order zero T transition
– Connection to glasses and glass transition
Low Temperature Properties of Glasses
• Distinct from crystals
• Common to all amorphous solids
• Still mysterious
– Excess vibrational modes
compared to Debye (boson peak)
– Cv~T instead of T3 (two-level systems)
 ~T2 instead of T3 at low T (TLS)
– K has plateau
– K increases monotonically


crystal
 / 20.5
w*  f  f c 
w*
amorphous
T
ffc
Energy Transport
1
   Ci (T )di
V i
thermal
heat carried
conductivity by mode i
P. B. Allen and
J. L. Feldman,
PRB 48,12581 (1993).
diffusivity
of mode i
Kubo formulation
2

di  2 2  Sij  w i  w j 
3h w i i  j
Kittel’s 1949 hypothesis: rise in  above plateau
due to regime of freq-independent diffusivity
N. Xu, V. Vitelli, M. Wyart, A. J. Liu, S. R. Nagel (2008).
Ioffe-Regel Crossover
• Crossover from weak to strong scattering at wIR
 wIR ~ w* Ioffe-Regel crossover at boson peak
• Unambiguous evidence of freq-indep diffusivity as
hypothesized for glasses
• Freq-indep diffusivity originates from soft modes at J!
Quasilocalized Modes
QuickTime™ and a
TIFF (LZW) decompressor
are needed to see this picture.
• Modes become
quasilocalized near IoffeRegel crossover
• Quasilocalization due to
disorder in coordination z
• Harmonic precursors of
two-level systems?
Relevance to Glasses
• Point J only exists for repulsive, finite-range potentials
• Real liquids have attractions U
Repulsion vanishes at
Attractions serve to hold
system at high enough
density that repulsions
come into play (WCA)
finite distance
r
• Excess vibrational modes (boson peak) believed
responsible for unusual low temp properties of glasses
• These modes derive from the excess modes near Point J
N. Xu, M. Wyart, A. J. Liu,
S. R. Nagel, PRL 98, 175502
(2007).
Glass Transition
Would expect
 : exp(A / T )
Arrhenius behavior
But most glassforming
liquids obey something like

 : exp A / T  T0 
L.-M. Martinez and C. A. Angell,
Nature 410, 663 (2001).
T0 measures “fragility”
x

T0(p) is Linear
•
•
•
3 different types of
Temperature
trajectories to glass
transition
unjammed
– Decrease T at fixed f
– Decrease T at fixed p
jammed
– Increase p at fixed T
Shear stress
4 different potentials
– Harmonic repulsion
J
1/Density
– Hertzian repulsion
– Repulsive Lennard-Jones
(WCA)
– Lennard-Jones
All results fall on consistent
curve!
QuickTime™ and a
TIFF (LZW) decompressor
• T0 -> 0 at Point J!
are needed to see this picture.
Experimental Data for Glycerol
K. Z. Win and N. Menon
350
Tm
300
5
10 Hz
250
10
3
10
T
200
Pg
P0
Tg
150
T0
0
5
10
15
20
P/kb
25
30
Conclusions
• Point J is a special point
T
• First hint of universality in
jamming transitions
xy
• Tantalizing connections to
glasses and glass transition
1/r
J
• Looking for commonalities can yield insight
• Physics is not just about the exotic; it is all around you
Hope you like jammin’, too!--Bob Marley
Bread for Jam: NSF-DMR-0605044
DOE DE-FG02-03ER46087

Imry-Ma-Type Argument
M. Wyart, Ann. de Phys. 30 (3), 1 (2005).
• Upper critical dimension for jamming transition may be 2
• Recall soft-mode-counting argument
Ns  Ld 1  Z  Zc Ld
• Now include fluctuations in Z
Ns  Ld 1  Z  Zc Ld  ZLd / 2
• 
This would explain
– Observed exponents same in d=2 and d=3
– Similarity to mean-field k-core exponents*
*k-core percolation has different behavior in d=2
J. M. Schwarz, A. J. Liu, L. Chayes, EPL 73, 560 (2006)
C. Toninelli, G. Biroli, D. S. Fisher, PRL 96,035702 (2006)
Nature of Vibrational Modes
Visualize 2D displacement vectors
in mode i at atom
Participation ratio
Nature of Vibrational Modes
localized
Nature of Vibrational Modes
localized disturbances merging
Nature of Vibrational Modes
extended
Nature of Vibrational Modes
wave-like
Nature of Vibrational Modes
resonant
Characterize modes in different portions of spectrum.
Mode Analysis of 3D Jammed packings
Stressed
Unstressed
replace compressed
bonds with relaxed
springs.
coordination number
Low
resonant modes have high displacements on under-coordinated particles.
at low
increases weakly as
How Localized is the Lowest Frequency Mode?
• Mode is more and more localized with increasing f
• Two-level systems and STZ’s
Strong Anharmonicity at Low Frequency
Gruneisen parameters
is O(1) for ordinary solids at low
Compression causes increase in stress consistent with scaling of
Stronger anharmonicity at lower frequencies; two-level systems?
Thermal Diffusivity
Gedanken experiment
energy
diffusivity
Thermal
conductivity
sum over all modes
heat capacity
low T
Kinetic theory of phonons in crystals:
speed of sound
mean free path
eg.waves of frequency
incident on a
random distribution of scatterers
(ignore phonon interactions)
Rayleigh scattering
Calculation of Diffusivity
linear response theory & harmonic approx.
Kubo formula for amorphous solids:
AC conductivity
Extract Diffusivity from double limit: N
0
Energy flux matrix elements
calculated directly from eigenmodes and eigenvalues
Diffusivity an intrinsic property of the modes and their coupling independent of
temperature
J. D. Thouless, Phys. Rep. 13, 3, 93 (1974).
P. B. Allen and J. L. Feldman, PRB 48,12581 (1993).
Diffusivity of Jammed Packings
Stressed
Unstressed
replace compressed
bonds with relaxed springs
• Above wL modes are localized and diffusivity vanishes
• Most modes are extended with low diffusivity
• Plateau extends all the way to low w in DC limit
Scaling Collapse
P. Olsson and S. Teitel, cond-mat/07041806
T
• Measure shear viscosity, length scale vs.
shear stress
• Scaling collapse of all data
xy
• Two branches: one below J, one above
• Power-law scaling at Pt. J
1/r
J
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 transition (Charybdis)
2. No percolation until p=1 (Scylla)
3. Mixed transition?
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=0.6, after culling
•
p=0.4, after culling
p=0.65, after culling
pc=0.6921±0.0005, M. C. Madeiros, C. M. Chaves, Physica A (1997).
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).
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 1 1.5 collapses the order parameter
data with  1.0
• For ordinary 1st-order transition,
 1/d  0.5
BUT
• Exponents for k-core variants in d=2 are
different from those in mean-field!
Mean field
  1/2
  1/4
d=2
  1.0
  1.5
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 kcore requirement of k neighbors
Henkes, Chakraborty, PRL 95, 198002 (2005).