Jamming

1
Jamming

• Andrea J. Liu
• Department of Physics Astronomy
• University of Pennsylvania
• Corey S. OHern Mechanical Engineering,
  Yale Univ.
• Leo E. Silbert Physics, S. Illinois U.,
  Carbondale
• Ning Xu Physics, UPenn, JFI, U. Chicago
• Vincenzo Vitelli Physics, UPenn
• Matthieu Wyart Janelia Farms Physics, NYU
• Sidney R. Nagel James Franck Inst., U.
  Chicago

2
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?

3
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 t how long you need to
  wait for system to behave like liquid

Speeded up by x80
4
Glass Transition

• When liquid is cooled through glass transition
• Particles remain disordered
• Stress relaxation time increases continuously
• Can get 10 orders of magnitude increase in 10-20
  K range

Earliest glassmaking 3000BC
Glass vessels from around 1500BC
5
Colloidal Glass Transition

• Suspensions of small (nm-10mm) particles include
• Ink, paint
• McDonalds milk shakes, ..
• Blood
• Micron-sized plastic spheres suspended in water
  form
• Stress relaxation time increases with packing
  fraction

6
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
7
Foams and Emulsions

• Suspension of gas bubbles or liquid droplets
• Shaving cream
• mayonnaise
• Foams flow like liquids when sheared
• Stress relaxation time increases as shear stress
  decreases

Courtesy of D. J. Durian
8
Phenomena look similar in all these systems

• No obvious structural signature of jamming
• Dramatic increase of relaxation time near jamming
• Kinetic heterogeneities

Supercooled liquids
Colloidalsuspensions
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
Courtesy of S. C. Glotzer
9
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

10
Jamming

• Jam ( ), v. i.
• To develop a yield stress in a disordered system
• 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?

glass transition
colloidal glass transition
elastoplasticity
11
Jamming Phase Diagram

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

12
Exptal Jamming Phase Diagram

• V. Trappe, V. Prasad, L. Cipelletti, P. N.
  Segre, D. A. Weitz, Nature, 411(N6839) 772
  (2001).
• Colloids with depletion attractions

13
Point J

• C. S. OHern, S. A. Langer, A. J. Liu and S. R.
  Nagel, Phys. Rev. Lett. 88, 075507 (2002).
• C. S. OHern, L. E. Silbert, A. J. Liu, S. R.
  Nagel, Phys. Rev. E 68, 011306 (2003).
• Problem Jamming surface is fuzzy
• Point J is special
• Random close-packing
• Isostatic
• Mixed first/second order zero T transition
• Connections to glasses and glass transition

soft, repulsive, finite-range spherically-symmetr
ic potentials
14
How we study Point J

• Generate configurations near J
• e.g. Start w/ random initial positions
• Conjugate gradient energy minimization (inherent
  structures, Stillinger Weber)
• Classify resulting configurations

Ti8
15
Onset of Jamming is Onset of Overlap
D2 D3

• Pressures for different states collapse on a
  single curve
• Shear modulus and pressure vanish at the same fc
• Good ensemble is fixed f - fc

D. J. Durian, PRL 75, 4780 (1995) C. S. OHern,
S. A. Langer, A. J. Liu, S. R. Nagel, PRL 88,
075507 (2002).
16
Dense Sphere Packings

• What is densest packing of monodisperse hard
  spheres?

Johannes Kepler (1571-1630) Conjecture (1611)
Thomas Hales 3D Proof (1998)
Fejes Tóth 2D Proof (1953)
triangular is densest possible packing
2D
FCC/HCP is densest possible packing
3D
17
Disordered Sphere Packings
Stephen Hales (1677-1761) Vegetable Staticks
(1727)
J. D. Bernal (1901-1971)
lt
2D
lt
3D

• 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).

18
How Much Does fc Vary Among States?

• Distribution of fc values narrows as system size
  grows
• Distribution approaches delta-function as N
• Essentially all configurations jam at one packing
  density
• J is a POINT

19
Point J is at Random Close-Packing

• Where do virtually all states jam in infinite
  system limit?
• 2d (bidisperse)
• 3d (monodisperse)
• Most of phase space belongs to basins of
  attraction of hard sphere states that have their
  jamming thresholds at RCP

RCP!
20
Point J

• Point J is special
• Random close-packing
• Isostatic
• Mixed first/second order zero T transition
• Connections to glasses and glass transition

soft, repulsive, finite-range spherically-symmetr
ic potentials
21
Number of Overlaps/Particle Z

• (2D)
• 
  (3D)

Just above fc there are Zc overlapping neighbors
per particle
Just below fc, no particles overlap
Verified experimentally Majmudar, Sperl, Luding,
Behringer, PRL 98, 058001 (2007).
Durian, PRL 75, 4780 (1995). OHern, Langer, Liu,
Nagel, PRL 88, 075507 (2002).
22
Isostaticity

• What is the minimum number of interparticle
  contacts needed for mechanical equilibrium?
• Same for hard spheres at RCP Donev, Torquato,
  Stillinger, PRE 71, 011105 (05)
• Point J is purely geometrical! Doesnt depend on
  potential

• No friction, spherical particles, D dimensions
• Match
• unknowns (number of interparticle normal forces)
• equations (force balance for mechanical
  stability)
• Number of unknowns per particleZ/2
• Number of equations per particle D

James Clerk Maxwell
23
Marginally Jammed Solid is Unusual

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

Density of Vibrational Modes
f- fc
24
Isostaticity and Boundary Sensitivity
M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486
(05)

• For system at ?c, Z2d
• Removal of one bond makes entire system unstable
  by introducing one soft mode
• This implies diverging length as ?-gt ?c

For ??gt ?c, cut bonds at boundary of circle of
size L Count number of soft modes within circle
Define length scale at which soft modes just
appear
25
Diverging Length Scale

• Ellenbroek, Somfai, van Hecke, van Saarloos, PRL
  97, 258001 (2006)

Look at response to small particle
displacement Define
26
Diverging Time and Length Scales

• 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?/cT
• We also expect with

27
Point J

• Point J is special
• Random close-packing
• Isostatic
• Mixed first/second order zero T transition
• Connection to glasses and glass transition

soft, repulsive, finite-range spherically-symmetr
ic potentials
28
Summary of Jamming Transition

• Mixed first-order/second-order transition (random
  first-order phase transition)
• Number of overlapping neighbors per particle
• Static shear modulus
• Two diverging length scales
• Vanishing frequency scale

29
Similarity to Other Models

• In jamming transition we find
• Jump discontinuity b1/2 power-law in order
  parameter
• Divergences in susceptibility/correlation length
  with g1/2 and n1/4
• 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 Yin, Chakraborty
• Mean-field kinetically-constrained Ising models
  Sellitto, Toninelli, Biroli, Fisher
• Mean-field k-core percolation and variants
  Schwarz, Liu, Chayes
• Mode-coupling approximation of glasses Biroli,
  Bouchaud
• Replica solution of hard spheres Zamponi, Parisi
• These other models all exhibit glassy dynamics!!
• First hint of quantitative connection between
  sphere packings and glass transition

30
Point J

• Point J is special
• Random close-packing
• Isostatic
• Mixed first/second order zero T transition
• Connection to glasses and glass transition

soft, repulsive, finite-range spherically-symmetr
ic potentials
31
Low Temperature Properties of Glasses

• Distinct from crystals
• Common to all amorphous solids
• Still mysterious
• Excess vibrational modes compared to Debye
  (boson peak)
• CvT instead of T3 (two-level systems)
• ?T2 instead of T3 at low T (TLS)
• K has plateau
• K increases monotonically

32
Energy Transport
P. B. Allen and J. L. Feldman, PRB 48,12581
(1993).
Kittels 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).
33
Ioffe-Regel Crossover

• Crossover from weak to strong scattering at ?IR
• ?IR ? 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!

34
Quasilocalized Modes

• Modes become quasilocalized near Ioffe-Regel
  crossover
• Quasilocalization due to disorder in coordination
  z
• Harmonic precursors of two-level systems?

35
Relevance to Glasses

• Point J only exists for repulsive, finite-range
  potentials
• Real liquids have attractions
• Excess vibrational modes (boson peak) believed
  responsible for unusual low temp properties of
  glasses
• These modes derive from the excess modes near
  Point J

U
Repulsion vanishes at finite distance

• Attractions serve to hold system at high enough
  density that repulsions come into play (WCA)

r
N. Xu, M. Wyart, A. J. Liu, S. R. Nagel, PRL 98,
175502 (2007).
36
Glass Transition
Would expect Arrhenius behavior But most
glassforming liquids obey something like T0
measures fragility
L.-M. Martinez and C. A. Angell, Nature 410, 663
(2001).
37
T0(p) is Linear

• 3 different types of trajectories to glass
  transition
• Decrease T at fixed ?
• Decrease T at fixed p
• Increase p at fixed T
• 4 different potentials
• Harmonic repulsion
• Hertzian repulsion
• Repulsive Lennard-Jones (WCA)
• Lennard-Jones
• All results fall on consistent curve!
• T0 -gt 0 at Point J!

38
Experimental Data for Glycerol
K. Z. Win and N. Menon
39
Conclusions

• Point J is a special point
• First hint of universality in
• jamming transitions
• Tantalizing connections to
• glasses and glass transition
• 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

40
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
• Now include fluctuations in Z
• This would explain
• Observed exponents same in d2 and d3
• Similarity to mean-field k-core exponents
• k-core percolation has different behavior in d2

J. M. Schwarz, A. J. Liu, L. Chayes, EPL 73, 560
(2006) C. Toninelli, G. Biroli, D. S. Fisher, PRL
96,035702 (2006)
41
Nature of Vibrational Modes
Participation ratio
42
Nature of Vibrational Modes
localized
43
Nature of Vibrational Modes
localized disturbances merging
44
Nature of Vibrational Modes
extended
45
Nature of Vibrational Modes
wave-like
46
Nature of Vibrational Modes
resonant
Characterize modes in different portions of
spectrum.
47
Mode Analysis of 3D Jammed packings
Stressed Unstressed replace compressed bonds
with relaxed springs. 
Low resonant modes have high displacements
on under-coordinated particles.
at low increases weakly as
48
How Localized is the Lowest Frequency Mode?

• Mode is more and more localized with increasing
  ??
• Two-level systems and STZs

49
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?
50
Thermal Diffusivity
energy diffusivity
Thermal conductivity
heat capacity
sum over all modes
Kinetic theory of phonons in crystals
speed of sound
mean free path
waves of frequency incident on a random
distribution of scatterers
eg.
(ignore phonon interactions)
Rayleigh scattering
51
Calculation of Diffusivity
Kubo formula for amorphous solids
linear response theory harmonic approx.
AC conductivity
0
Extract Diffusivity from double limit
Energy flux matrix elements calculated
directly from eigenmodes and eigenvalues
Diffusivity an intrinsic property of the modes
and their coupling independent of temperature
P. B. Allen and J. L. Feldman, PRB 48,12581
(1993).
J. D. Thouless, Phys. Rep. 13, 3, 93 (1974).
52
Diffusivity of Jammed Packings

• Above ?L modes are localized and diffusivity
  vanishes
• Most modes are extended with low diffusivity
• Plateau extends all the way to low ? in DC limit

Stressed Unstressed replace compressed bonds
with relaxed springs  
53
Scaling Collapse

• P. Olsson and S. Teitel, cond-mat/07041806
• Measure shear viscosity, length scale vs. shear
  stress
• Scaling collapse of all data
• Two branches one below J, one above
• Power-law scaling at Pt. J

54
K-Core Percolation in Finite Dimensions

• There appear to be at least 3 different types of
  k-core percolation transitions in finite
  dimensions
• Continuous transition (Charybdis)
• No percolation until p1 (Scylla)
• Mixed transition?

55
Continuous K-Core Percolation

• Appears to be associated with self-sustaining
  clusters
• For example, k3 on triangular lattice
• pc0.69210.0005, M. C. Madeiros, C. M. Chaves,
  Physica A (1997).

Self-sustaining clusters dont exist in sphere
packings
p0.4, before culling
p0.4, after culling
p0.6, after culling
p0.65, after culling
56
No Transition Until p1

• E.g. k3 on square lattice
• 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).

Voids unstable to shrinkage, not growth in sphere
packings
57
A k-Core Variant

• We introduce force-balance constraint to
  eliminate self-sustaining clusters
• Cull if klt3 or if all neighbors are on the same
  side

k3 24 possible neighbors per site Cannot have
all neighbors in upper/lower/right/left half
58
Discontinuous Transition? Yes

• The discontinuity ?c increases with system size L
• If transition were continuous, ?c would decrease
  with L

Fraction of sites in spanning cluster
59
Pclt1? Yes

• Finite-size scaling
• If pc 1, expect pc(L) 1-Ae-BL

Aizenman, Lebowitz, J. Phys. A 21, 3801 (1988)
We find
We actually have a proof now that pclt1 (Jeng,
Schwarz)
60
Diverging Correlation Length? Yes

• This value of collapses the order
  parameter data with
• For ordinary 1st-order transition,

61
BUT

• Exponents for k-core variants in d2 are
  different from those in mean-field!
• Mean field d2
• Why does Point J show mean-field behavior?
• Point J may have critical dimension of dc2 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).