jamming andrea j. liu department of physics & astronomy university of pennsylvania corey s....

JammingAndrea J. Liu

Department of Physics & AstronomyUniversity of Pennsylvania

Corey S. O’Hern Mechanical Engineering, Yale Univ.Leo E. Silbert Physics, S. Illinois U., Carbondale Ning Xu Physics, UPenn, JFI, U. ChicagoVincenzo Vitelli Physics, UPennMatthieu Wyart Janelia Farms; Physics, NYUSidney R. Nagel James Franck Inst., U. Chicago

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?

Jamming

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 : how long you need to wait for system to behave like liquid

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Speeded up by x80

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

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are needed to see this picture.



Earliest glassmaking

3000BC

Glass vessels from around

1500BC

Colloidal Glass Transition

Suspensions of small (nm-10m) particles include– Ink, paint– McDonald’s milk shakes, …..– Blood

Micron-sized plastic spheres suspended in water form

Stress relaxation time increases with packing fraction

crystals glasses

Granular MaterialsMaterials made up of many distinct grains include

– Pharmaceutical powders– Cereal, coffee grounds, ….– Gravel, landfill, ….

San Francisco Marina District after Loma Prieta earthquake

Static granular packing behaves like a solid



Shaken granular packing behaves like a liquid

Suspension of gas bubbles or liquid droplets– Shaving cream– mayonnaise

• Foams flow like liquids when sheared• Stress relaxation time increases as shear stress decreases

Foams and Emulsions





Courtesy of D. J. Durian

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

Phenomena look similar in all these systems

Supercooled liquids

Colloidalsuspensions

Courtesy of E. R. Weeks

and D. A. Weitz

Granular materials

Courtesy of S. C. Glotzer

Courtesy of A. S. Keys, A. R. Abate, S. C.

Glotzer, and D. J. Durian

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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

Jam ( ), v. i. 1 To develop a yield stress in a disordered system2 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 dropsFoams, emulsions jam as shear stress drops

¿Can we unify these systems within one framework?

Jamming

j( a m

glass

transit

ion

collo

idal

glass

transit

ion

elastoplasti

city

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

Jamming Phase Diagram

Glass transition

Collo

idal

gla

ss tr

ans.

Granular matter

Foams and emulsions

Exp’tal Jamming Phase DiagramV. Trappe, V. Prasad, L. Cipelletti, P. N. Segre, D. A. Weitz, Nature, 411(N6839) 772 (2001).

Colloids with depletion attractions

1.0

20

10

[Pa]1 30.0 2

0.6

0.4

0.2

0.8

kBT/U 1/

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

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

Point J

unjammed

Temperature

Shear stress

1/DensityJ

jammed

soft, repulsive, finite-rangespherically-symmetricpotentials

• Generate configurations near J– e.g. Start w/ random initial positions

– Conjugate gradient energy minimization (inherent structures, Stillinger &

Weber)

• Classify resulting configurations

How we study Point J

Ti=∞

€

V (r) =ε 1− r /σ ij( )

αr ≤ σ ij

0 r >σ ij

⎧ ⎨ ⎪

⎩ ⎪

overlappedV>0p>0or

non-overlappedV=0p=0

Tf=0 Tf=0

Onset of Jamming is Onset of Overlap

•Pressures for different states collapse on a single curve

•Shear modulus and pressure vanish at the same c

•Good ensemble is fixed - c

D=2D=3

5 4 3 2log (φ- φc)

3

2

1

0

6

4

2

0

8

6

4

2

α=2

α=5/2

α=2

α=5/2

3D

2D

( )a

( )b

( )c

p≈p0(φ −φc)α −1

G ≈G0(φ−φc)α −1.5

-

-

-

-

-

-

--4 -3 -2

log(- c)

D. J. Durian, PRL 75, 4780 (1995);C. S. O’Hern, S. A. Langer, A. J. Liu, S. R. Nagel, PRL 88, 075507 (2002).

Dense Sphere Packings¿What is densest packing of monodisperse hard

spheres?Johannes Kepler (1571-

1630)Conjecture (1611)

Thomas Hales3D Proof (1998)

FCC/HCP is densest possible packing

φ =π / 18≅0.7403D

2Dtriangular is densest possible packing

φ =π / 12≅0.906

Fejes Tóth2D Proof (1953)

Disordered Sphere Packings

Stephen Hales (1677-1761)Vegetable Staticks (1727)

J. D. Bernal (1901-1971)

φrcp ≅0.64φrcp ≅0.842D

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).€

φcp ≅ 0.740

€

φcp ≅ 0.906<

<

How Much Does c Vary Among States?

• Distribution of c values narrows as system size grows

• Distribution approaches delta-function as N• Essentially all configurations jam at one packing

density• J is a “POINT”

1 234log N

3

2

1

w≈N−0.55

€

→ ∞

0.58 0.6 0.62 0.64φc

1

1.5

2=16N=32N=64N=256N=1024N=4096N

w

0

•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

1 234log N

3

2

1

log(*

- 0

)

€

φ* −φ0 ≈ N−1/ dν

€

ν ≅0.7

φ* =0.639±0.003φ* =0.842±0.001

Point J is at Random Close-Packing

RCP!

0.58 0.6 0.62 0.64φc

1

1.5

2=16N=32N=64N=256N=1024N=4096N

w

0

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

Point J

unjammed

Temperature

Shear stress

1/DensityJ

jammed


Number of Overlaps/Particle Z

(2D)

(3D)

Zc =3.99±0.01

Zc =5.97±0.03

5 4 3 2log (φ- φc)

3

2

1

0

6

4

2

0

8

6

4

2

α=2

α=5/2

α=2

α=5/2

3D

2D

( )a

( )b

( )c

log(- c)

Z−Zc ≈Z0(φ −φc)0.5

-

-

-

- - - -

Just below c, no particles overlap

Just above c there are

Zc

overlapping neighbors per particle

Verified experimentally:Majmudar, Sperl, Luding, Behringer, PRL 98, 058001 (2007).

Durian, PRL 75, 4780 (1995).O’Hern, Langer, Liu, Nagel, PRL 88, 075507 (2002).

•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

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! Doesn’t depend on potential

Z=2D

James Clerk Maxwell

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

• Excess low- modes swamp 2 Debye behavior: boson peak

• g() approaches constant as c

• Result of isostaticity M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)

Marginally Jammed Solid is Unusual

Density of Vibrational Modes

c

Isostaticity and Boundary Sensitivity

•For system at c, Z=2d

•Removal of one bond makes entire system unstable by introducing one soft mode

•This implies diverging length as -> c +

For > 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

€

Ns ≈ Ld−1 − Z − Zc( )Ld

€

l ≈1

Z − Zc≈ φ −φc( )

−0.5

M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)

Diverging Length ScaleEllenbroek, Somfai, van Hecke, van Saarloos, PRL 97, 258001 (2006)

Look at response to small particle displacement

Define

€

h(r) = Δfr (r)( )2

− Δfr (r)2

€

ν =0.5

• For each -c, extract where g() begins to drop off

• Below , modes approach those of ordinary elastic solid

• Decompose corresponding eigenmode in plane waves

• Dominant wavevector contribution is k*=*/cT

• We also expect with

Diverging Time and Length Scales



€

l ≅ φ−φc( )−ν + = 0.26

€

l =2π /k*

€

l

€

*∝ φ −φc( )α / 2−0.5

€

*

€

k # =ω * /cL

€

ν # ≅ 0.5

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

Point J

unjammed

Temperature

Shear stress

1/DensityJ

jammed


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

€

Z =0 φ < φc

Zc + Z0(φ −φc )β φ > φc

⎧ ⎨ ⎩

€

β =0.49 ± 0.03

€

V (r) =0 if r >σ

ε 1− r /σ( )α

if r ≤ σ

⎧ ⎨ ⎩

€

G ≈G0 φ −φc( )α −2φ −φc( )

γ

€

γ=0.48 ± 0.03

€

l ≈l 0 φ −φc( )−ν

€

ν =0.26 ± 0.05

€

ν # ≅ 0.5

* ≈ω0 φ −φc( )α −1

Similarity to Other Models• In jamming transition we find

– Jump discontinuity & β=1/2 power-law in order parameter

– Divergences in susceptibility/correlation length with γ=1/2 and ν=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 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

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

Point J

unjammed

Temperature

Shear stress

1/DensityJ

jammed


Low Temperature Properties of Glasses

crystal

T

amorphous

• 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

c

€

*∝ φ −φc( )α / 2−0.5

€

*

Energy Transport

P. B. Allen and J. L. Feldman, PRB 48,12581 (1993).

thermal conductivity

heat carried by

mode i

diffusivity of mode i

κ =1

VCi

i∑ (T )di

di =

π3h2 i

2 Siji≠j∑

2δ i − j( )

Kubo formulation

N. Xu, V. Vitelli, M. Wyart, A. J. Liu, S. R. Nagel (2008).

Kittel’s 1949 hypothesis: rise in κ above plateau due to regime of freq-independent diffusivity

• 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!

Ioffe-Regel Crossover

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Quasilocalized Modes• Modes become

quasilocalized near Ioffe-Regel 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

• Excess vibrational modes (boson peak) believed responsible for unusual low temp properties of glasses

• These modes derive from the excess modes near Point J

Repulsion vanishes at finite distance

U

r

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

N. Xu, M. Wyart, A. J. Liu, S. R. Nagel, PRL 98, 175502 (2007).

Glass Transition

L.-M. Martinez and C. A. Angell, Nature 410, 663 (2001).

Would expect

Arrhenius behavior

But most glassforming liquids obey something like

T0 measures “fragility”

η : exp(A /T )

η : exp A / T −T0( )

x

( )

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T0(p) is Linear

unjammed

Temperature

Shear stress

1/DensityJ

jammed

• 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 -> 0 at Point J!

Experimental Data for Glycerol

350

300

250

200

150

T

302520151050P/kb

105Hz

103

10

T0

P0

Tg

Pg

Tm

K. Z. Win and N. Menon

• 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

ConclusionsT

xy

1/ J

Imry-Ma-Type ArgumentM. 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 d=2 and d=3– Similarity to mean-field k-core exponents*

*k-core percolation has different behavior in d=2

€

Ns ≈ Ld−1 − Z − Zc( )Ld

€

Ns ≈ Ld−1 − Z − Zc( )Ld + δZLd / 2

J. M. Schwarz, A. J. Liu, L. Chayes, EPL 73, 560 (2006)C. Toninelli, G. Biroli, D. S. Fisher, PRL 96,035702 (2006)

Participation ratio

Visualize 2D displacement vectorsin mode i at

atom

Nature of Vibrational Modes

localized


localized disturbances merging


extended


wave-like


resonant

Characterize modes in different portions of spectrum.


StressedUnstressedreplace compressed bonds with relaxed springs.

coordination number

Low resonant modes have high displacements on under-coordinated particles.

at low increases weakly as

Mode Analysis of 3D Jammed packings

How Localized is the Lowest Frequency Mode?

• Mode is more and more localized with increasing • Two-level systems and STZ’s

is O(1) for ordinary solids at low

Gruneisen parameters

Compression causes increase in stress consistent with scaling ofStronger anharmonicity at lower frequencies; two-level systems?

Strong Anharmonicity at Low Frequency

Kinetic theory of phonons in crystals:

mean free path

speed of sound

heat capacity

energy diffusivity

Thermal conductivity

waves of frequency incident on a random distribution of scatterers

(ignore phonon interactions)

eg.

Rayleigh scattering

Gedanken experiment

low T

sum over all modes

Thermal Diffusivity

Kubo formula for amorphous solids:

linear response theory & harmonic approx.

Energy flux matrix elements calculated directly from eigenmodes and eigenvalues

P. B. Allen and J. L. Feldman, PRB 48,12581 (1993).

Diffusivity an intrinsic property of the modes and their coupling independent of temperature

AC conductivity

Extract Diffusivity from double limit:N 0

J. D. Thouless, Phys. Rep. 13, 3, 93 (1974).

Calculation of Diffusivity

• 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

StressedUnstressed replace compressed bonds with relaxed springs

Diffusivity of Jammed Packings

Scaling CollapseP. 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

T

xy

1/ 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

• pc=0.6921±0.0005, M. C. Madeiros, C. M. Chaves, Physica A (1997).

p=0.4, before culling p=0.4, after culling

p=0.6, after culling p=0.65, after culling

Self-sustaining clusters don’t exist in sphere packings

• 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

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

A k-Core Variant•We introduce “force-balance” constraint to eliminate self-sustaining clusters

•Cull if k<3 or 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

• The discontinuity κc increases with system size L

• If transition were continuous, κc would decrease with L

Discontinuous Transition? YesFr

act

ion o

f si

tes

in s

pannin

g

clust

er

Pc<1? Yes

Finite-size scalingIf 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

• This value of collapses the order parameter data with

• For ordinary 1st-order transition, €

ν1 ≅1.5

€

ν1 ≅1.5

€

β ≅1.0

€

ν =1/d = 0.5

BUT

• 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 neighborsHenkes, Chakraborty, PRL 95, 198002 (2005).

€

β =1/2

ν =1/4

€

β ≅1.0

ν ≅1.5

jamming andrea j. liu department of physics & astronomy university of pennsylvania corey s....

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