jamming as the extreme limit of a solid...carl goodrich upenn sidney nagel u chicago tim still upenn...

Jamming as the Extreme Limit of a Solid

Andrea J. LiuDepartment of Physics & Astronomy

University of Pennsylvania

Carl Goodrich UPennSidney Nagel U ChicagoTim Still UPennArjun Yodh UPenn

Tuesday, July 16, 13

Physics of Perfect Crystals

• Start with T=0 perfect crystal– look at vibrational, electronic, etc. properties– add defects as perturbation (chapter 30)


Perturbing away from the crystal


Perturbing away from the crystalBut what about this?


Perturbing away from the crystal

Is there an opposite pole to the perfect crystal, corresponding to rigid solid with complete disorder?

If so, we could describe ordinary solids as somewhere in between

But what about this?


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).

• Study models with smooth transitions – from G/B=0 (like liquid)– to G/B>0 (like crystal)

Jamming Transition for “Ideal Spheres”

JJJJJJ

jammed

unjammed

stress

temperature

1/density

Bubble model for foams D. J. Durian, PRL 75, 4780 (1995).


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).

• Study models with smooth transitions – from G/B=0 (like liquid)– to G/B>0 (like crystal)

Jamming Transition for “Ideal Spheres”

JJJJJJ

jammed

unjammed

stress

temperature

1/density

V (r) =εα1− r

σ⎛⎝⎜

⎞⎠⎟α

r ≤ σ

0 r > σ

⎧

⎨⎪

⎩⎪

Bubble model for foams D. J. Durian, PRL 75, 4780 (1995).


Onset of Jamming in Repulsive Sphere Packings

– (2D)

(3D)Zc = 5.97 ± 0.03

5 4 3 2log (φ-�φc)

3

2

1

06

4

2

0

8

6

4

2

α=2

α=5/2

α=2

α=5/2

3D

2D

(a)

(b)

(c)

log(φ- φc)

Z − Zc ≈ Z0 (φ −φc)0.5

---

- - - -

Just below φc, no particles overlap

Just above φc there are

Zc

overlapping neighbors per particle

Zc = 3.99 ± 0.01

Verified experimentally:G. Katgert and M. van Hecke, EPL 92, 34002 (2010).Durian, PRL 75, 4780 (1995).

O’Hern, Langer, Liu, Nagel, PRL 88, 075507 (2002).Tuesday, July 16, 13

Isostaticity

• What is the minimum number of interparticle contacts needed for mechanical equilibrium?

• No friction, N repulsive spheres, d dimensions• Match

– number of constraints (number of interparticle normal forces)=NZ/2

– number of degrees of freedom =Nd-d

• For large N, Z ≥ 2d

James Clerk Maxwell


Contact Number of Crystal vs. Marginally Jammed Solidlog(Z�Ziso

)

log p

crystal

⇠ p1/2 (harmonic)

perfect crystal

vs

marginally jammed solid

crystal: Z=12marginally jammed solid: Z=Ziso=6


Constraint Counting and G/B• At onset of overlap, φc, can constrain

– all soft modes– compression of the whole system

• So B>0 but G=0 so G/B=0

• Above φc, G/B >0 so φc also marks onset of jamming

5 4 3 2log (φ-�φc)

3

2

1

06

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

-

-

-

-

-4 -3 -2log(φ- φc)

-6

G ≈G0(φ − φc )α −1.5

---

-4 -3 -2

5 4 3 2log (φ-�φc)

3

2

1

06

4

2

0

8

6

4

2

α=2

α=5/2

α=2

α=5/2

3D

2D

(a)

(b)

(c)log(φ- φc)

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


Constraint Counting and G/B• At onset of overlap, φc, can constrain

– all soft modes– compression of the whole system

• So B>0 but G=0 so G/B=0

• Above φc, G/B >0 so φc also marks onset of jamming

5 4 3 2log (φ-�φc)

3

2

1

06

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

-

-

-

-

-4 -3 -2log(φ- φc)

-6

G/B ~ ΔZ

Ellenbroek, Somfai, van Hecke, van Saarloos, PRL 97, 257801 (2006).

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


G/B -> 0 with (ϕ-ϕc)1/2 or Z-Zc appears unique to jamming

X. Mao, A. Souslov, T. C. Lubensky

G/B

ϕc ϕ

hexagonal/fcc/kagome/....

G/B

Zc Z

twisted kagome

randomly decorated square

jamming

ϕc ϕ

G/B

Z

G/B

Zc

Z

G/B

Zc

jamming

randomly diluted hexagonal/fcc/... randomly decorated kagome/....

G/B

Zc Z


Mechanics of crystal vs. marginally jammed solid

log p

crystal

jamming

⇠ p1/2 (harmonic)

log(G/B

)

perfect crystal

vs


crystal: G/B ~ 1marginally jammed solid: G/B -> 0


Consequence: Diverging Length Scale

•For system at φc, Z=2d

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

•This implies diverging length as φ-> φc +

For φ > φc, cut bonds at boundary of size LCount number of soft modes within cluster

Define length scale at which soft modes just appear

�

Ns ≈ Ld−1 − Z − Zc( )Ld

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

L

1Z − Zc

≡1Δz φ − φc( )−0.5


Define ℓ* as size of smallest macroscopic rigid cluster for

system with a free boundary of any shape or size

• ℓ* diverges at Point J as expected from scaling argument

More precisely


Vibrations in Disordered Sphere Packings• Crystals are all alike at low T or low ω

– density of vibrational states D(ω)~ωd-1 in d dimensions

– heat capacity C(T)~Td

• Why? Low-frequency excitations are sound modes. At long

length scales all solids look elastic


Vibrations in Disordered Sphere Packings• Crystals are all alike at low T or low ω

– density of vibrational states D(ω)~ωd-1 in d dimensions

– heat capacity C(T)~Td

• Why? Low-frequency excitations are sound modes. At long

length scales all solids look elastic

BUT near at Point J, there is a diverging length scale ℓL

So what happens?


Vibrations in Sphere Packings

• New class of excitations originates from soft modes at Point J M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)

• Generic consequence of diverging length scale: ℓL≃cL/ω*

ω * /ω0 Δφ1/2

�

ω *D(ω)

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

ω0 ≡

keffm Δφ

α−2( )/2

ℓT≃cT/ω*


Vibrations in Sphere Packings

• New class of excitations originates from soft modes at Point J M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)

• Generic consequence of diverging length scale: ℓL≃cL/ω*

ω * /ω0 Δφ1/2

�

ω *

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

ω0 ≡

keffm Δφ

α−2( )/2

p=0.01

ω*

ℓT≃cT/ω*


Vibrations of crystal vs. marginally jammed solid

perfect crystal

vs


D(

no plane waves even at ω=0 0

0.4

0.8

1.2

0 0.5 1 1.5 2 2.5 3

D(t

)

t

FCC Crystal

D(ω

)


Vibrations of crystal vs. marginally jammed solid

perfect crystal

vs


no plane waves even at ω=0 0

0.4

0.8

1.2

0 0.5 1 1.5 2 2.5 3

D(t

)

t

FCC Crystal

D(ω

)


Back to extreme limitsHow do we connect physics of jamming and physics of crystals? What happens in between?


1. start with a perfect FCC crystal

2. introduce 1 vacancy-interstitial pair

3. minimize the energy

2d illustration




2. introduce 2 vacancy-interstitial pairs


2d illustration



2d illustration


2. introduce 3 vacancy-interstitial pairs





2. introduce M vacancy-interstitial pairs


2d illustration




2. introduce N vacancy-interstitial pairs


2d illustration



Order Parameter

Bond-orientational order

qlm(i) ⌘X

j

Ylm(r̂ij)

Sl(i, j) �X

m

qlm(i) · q⇤lm(j)

f6(i) = fraction

of highly correlated

neighbors (large S6)

Auer and Frenkel. J. Chem. Phys., 120(6):3015, 2004Russo and Tanaka. arXiv, cond-mat.soft, 2012.

f6 = 1 ! crystal

f6 = 0 ! disordered


“Coexistence” of ordered and disordered regions

0 1 2 3 4 5 6 7 8

0 0.2 0.4 0.6 0.8 1

D(f 6

)

f6(i)

global: F60.10.20.30.40.50.60.70.80.9


log p

log(Z�Ziso

)

crystal

jamming

⇠ p1/2 (harmonic)

Connecting jamming and crystal physics

Observed states


log p

Wyart, et al. PRE 72 051306 (2005)

c0

p1/2 Z � Ziso

6

log(Z�Ziso

)

crystal

jamming

⇠ p1/2 (harmonic)

10-1

100

101

10-5 10-4 10-3 10-2 10-1

Z - Z

iso

p

crystal

jamming

Connecting jamming and crystal physics

Observed states


What about systems with intermediate order?

log(Z�Ziso

)

log p

crystal

jamming

⇠ p1/2 (harmonic)

Contact Number

log p

crystal

jamming

⇠ p1/2 (harmonic)

log(G/B

)

Elasticity

Elasticity


Elasticity

l

o

g

(Z �Zi

s

o

)

log p

log

G

B

jamming

crystalfccfcc+vac/intfcc+vacanciesbcc+vacancies


Elasticityfccfcc+vac/intfcc+vacanciesbcc+vacancies


Elasticity

log (Z � Ziso

)

log p

log

G

B

jamming

crystal

fccfcc+vac/intfcc+vacanciesbcc+vacancies


Exclude crystalline states• Include only states where disordered “phase” percolates

in all 3 directions


Exclude crystalline states• Include only states where disordered “phase” percolates

in all 3 directionsStates with intermediate to low order

fall on “jamming surface”

Jammed state is not

only extreme limit but also very robust


How much does jamming scenario apply to real world?• What have we left out? ALMOST EVERYTHING

– frictionK. Shundyak, et al. PRE 75 010301 (2007); E. Somfai, et al. PRE 75 020301 (2007); S. Henkes, et al. EPL 90 14003 (2010).

– long-ranged interactions/attractionsN. Xu, et al. PRL 98 175502 (2007).

– non-spherical particle shapeZ. Zeravcic, et al, EPL, 87, 26001 (2009); M. Mailman, et al, PRL 102, 255501 (2009)

– temperatureC. Schreck, et al. PRL 107, 078301 (2011); A. Ikeda, et al. J. Chem. Phys. 138, 12A507 (2013); L. Wang and N. Xu, Soft Matt. 9, 2475 (2013); T. Bertrand, et al. arXiv:1307.0440.


Real, Thermal Colloidal GlassesVideo microscopy of 2D jammed packing of colloids• NIPA microgel particles ⇒ tune packing fraction

• Track particles over ~30 000 frames ⇒

Extract instantaneous displacements from average position

and the displacement correlation matrix

Chen et al., PRL 105, 025501 (2010)Ghosh et al., Soft Mat 6, 3082 (2010)

microscopeobjective

Ke Chen, Wouter Ellenbroek, Arjun Yodh


• BUT displacement correlation is an equilibrium property, independent of dynamics

• Can use it to obtain vibrational modes of shadow system with same configuration & interactions but without damping

• In harmonic approximation

• Partition function

• Correlation matrix is inverse of stiffness matrix K

Colloids are damped, atoms/molecules are not

V =12uTKu

Z = du exp(−βV )∫

C = uu = K −1

Ghosh, Chikkadi, Schall, Kurchan, Bonn, Soft Mat 6, 3082 (2010)


Boson Peak

Boson peak frequency

2

0.885

0.879

0.872

0.866

0.859

0.850

0.840

0.830

0.07

0.05

0.04

0.02

0.01

0.03

0.06

ω (105 rad/s)4 6 8

D (ω)

0.82 0.86 0.900

0.6

ϕ

0.4

0.2

ω* (

105 r

ad/s

)

Chen et al., PRL 105, 025501 (2010)


Boson Peak

0

2x10-7

4x10-7

6x10-7

8x10-7

1x10-6

D

(ω

)/ω

(s

2/r

ad

2)

0.82 0.84 0.86 0.88 0.900.0

0.2

0.4

0.6

0.8

1.0

1.2

ω*

(10

5 r

ad/s

)

φ

104

105

106

0.0

0.1

0.2

0.3

0.4 0.885

0.879

0.872

0.866

0.859

0.850

0.840

p (ω

)

ω (rad/s)

(a)

(b)

p (ω)

D (ω)/ω

(s2/rad2)

105 106104

ω (rad/s)

Boson peak frequency

2

0.885

0.879

0.872

0.866

0.859

0.850

0.840

0.830

0.07

0.05

0.04

0.02

0.01

0.03

0.06

ω (105 rad/s)4 6 8

D (ω)

0.82 0.86 0.900

0.6

ϕ

0.4

0.2

ω* (

105 r

ad/s

)

Chen et al., PRL 105, 025501 (2010)


Dispersion relation and elastic constants

• From dispersion relation extract sound velocities• From sound velocities extract elastic constants

B

G


G/B behavior• Recall that G/B does not depend on potential• For frictionless particles,

where

• For frictional particles, E. Somfai, et al. PRE 75, 020301 (2007).

where

�z ⌘ z � z0c = 3.3(�� 0c)

�z ⌘ z � z1c = 3.3(�� 1c )


PNIPAM particles are frictional

• one adjustable parameter

frictional

frictionless

�1c


G, B • Interaction most consistent with Hertzian (K. Nordstrom, et al. PRL

105, 175701 (2010))

B/keff

G/keff

ke↵ =

p3✏

2�2

�� µ

c )1/2

kBT/✏ = 3⇥ 10�6

µ ⇡ 0.6 K. Shundyak, et al. PRE 75, 010301 (2007).


G, B • Interaction most consistent with Hertzian (K. Nordstrom, et al. PRL

105, 175701 (2010))

B/keff

G/kefftwo adjustable parameters

ke↵ =

p3✏

2�2

�� µ

c )1/2

kBT/✏ = 3⇥ 10�6

µ ⇡ 0.6 K. Shundyak, et al. PRE 75, 010301 (2007).


Jamming and temperature

A. Ikeda, L. Berthier and G. Biroli, J. Chem. Phys. 138, 12A507 (2013)


Effect of Temperature

kBT* is temperature at which T=0 description breaks down

Bertrand, et al.

where C(N) ! 0 as N ! 1Ikeda, et al.

Wang and Xu

kBT⇤/✏ ⇡ 0.2(�� c)

5/2

kBT⇤/✏ ⇡ 10�3(�� c)

5/2

kBT⇤/✏ ⇡ C(N)(�� c)

5/2

log(�� c)

log kBT/✏




Bertrand, et al.


Wang and Xu

kBT⇤/✏ ⇡ 0.2(�� c)

5/2

kBT⇤/✏ ⇡ 10�3(�� c)

5/2

kBT⇤/✏ ⇡ C(N)(�� c)

5/2

our expt

log(�� c)

log kBT/✏




Bertrand, et al.


Wang and Xu

kBT⇤/✏ ⇡ 0.2(�� c)

5/2

kBT⇤/✏ ⇡ 10�3(�� c)

5/2

kBT⇤/✏ ⇡ C(N)(�� c)

5/2

our expt

Breaks down for what?

log(�� c)

log kBT/✏


Quasilocalized modes predict rearrangements above Tg

• Color contours: Sum (polarization vector magnitudes)2 for each particle over lowest 30 vibrational modes

• white circles: particles that rearranged in relaxation time interval

LETTERS

–15 –10 –5 0 5 10 15 –15 –10 –5 0 5 10 15

0.0200.0280.0350.0430.0500.0580.0650.0730.080

–15 –10 –5 0 5 10 15

15

10

5

0

–5

–10

–15

15

10

5

0

–5

–10

–15

Figure 3 A comparison of the spatial distribution of IR and low-frequency normal modes. Contour plots of the low-frequency mode participation (as in Fig. 2), overlaidwith the location of particles (white circles) whose iso-configurational probability of losing four initial nearest neighbours within 200� is greater than or equal to 0.01.

Two points are worth emphasizing. The mode participationfractions, whose spatial distributions are mapped in Fig. 2, areproperties of the static initial configurations. Our demonstrationof a strong correlation between the mode maps and theirreversible reorganization maps constitutes a significant successin understanding how structure determines relaxation in anamorphous material. Indeed, as Fig. 2 illustrates, we may providesemiquantitative prediction of IR domains as they emerge atrelatively long times from the initial configuration alone. Thesecond point is that, because we have used quenched modes, wehave used only information about the bottom of the local potentialminima. While it is quite possible that the timescale required for areorganization event will depend on the energy barriers associatedwith the transition, our results indicate that the spatial structureof such events is largely determined by the distribution of softquasilocalized modes in the initial configuration.

Given our conclusion that relaxation originates with softquasilocalized modes, it follows that our capacity to predict thesubsequent spatial distribution of the IR depends on how persistentthe mode distribution is in a configuration. After all, should themode maps evolve rapidly then the structural information in agiven configuration would quickly become irrelevant. The fact thatwe observe strong spatial correlations between the initial modesand relaxation some 200� later indicates that the spatial structure ofthe modes does generally persist over such times. This is remarkablegiven that small variations in the quenched modes, indicative of achange in the local minimum (or inherent structure), occur over�1� intervals. Details of these rapid changes are provided in theSupplementary Information. We conclude that the spatial structureof the quenched soft modes can often persist over many changesin the inherent structure. Preliminary results indicate that this

persistence is also found in three-dimensional mixtures (includingtemperatures below the empirical mode-coupling temperature)(see the Supplementary Information).

We do, however, see configurations where the mode structure isnot so stable. An example of this is shown in Fig. 4. In Fig. 4a,b wecompare the mode participation map for the initial configurationwith the map of the maximum participation fraction observed perparticle over five 10� runs starting from the configuration in Fig. 4a.The diVerence in spatial structure between these maps is a measureof the degree of variability of the mode structure. In Fig. 4c,d weoverlay the particles showing IR within 200� (as defined in Fig. 3)over the maps of Fig. 4a,b, respectively. While the mode structure ofthe initial configuration does not provide a quantitative predictorof the spatial distribution of IR (Fig. 4c), the cumulative modestructure sampled over the multiple short runs does (Fig. 4d). Thisresult demonstrates that, even when the soft-mode structure is notstable, the IR still originates with these soft modes, but now thisIR is not well predicted by any single configuration. It seems thatconfigurations such as that analysed in Fig. 4 represent those caughtin transit between configurations with more stable mode structure.

In this paper we have presented two important results relatingto the slow relaxation in a model supercooled liquid. The first isthat the irreversible reorganization originates at the sites of thelow-frequency quasilocalized quenched modes. The second is thatthese modes typically persist for timescales significantly longer thanthe lifetime of a given inherent structure. These results show thatthe spatial location and extent of IR regions at relatively longtimes may be reasonably predicted by a simple, static propertyof the static initial condition. A number of previous reports havelinked localized dynamics with the presence of soft modes16–20

during or immediately before the appearance of the motion in

nature physics VOL 4 SEPTEMBER 2008 www.nature.com/naturephysics 713

Widmer-Cooper, Perry, Harrowell, Reichman, Nat. Phys. 4,711 (2008)


Quasilocalized modes predict rearrangements above Tg

• Color contours: Sum (polarization vector magnitudes)2 for each particle over lowest 30 vibrational modes

• white circles: particles that rearranged in relaxation time interval

LETTERS

–15 –10 –5 0 5 10 15 –15 –10 –5 0 5 10 15

0.0200.0280.0350.0430.0500.0580.0650.0730.080

–15 –10 –5 0 5 10 15

15

10

5

0

–5

–10

–15

15

10

5

0

–5

–10

–15

Figure 3 A comparison of the spatial distribution of IR and low-frequency normal modes. Contour plots of the low-frequency mode participation (as in Fig. 2), overlaidwith the location of particles (white circles) whose iso-configurational probability of losing four initial nearest neighbours within 200� is greater than or equal to 0.01.

Two points are worth emphasizing. The mode participationfractions, whose spatial distributions are mapped in Fig. 2, areproperties of the static initial configurations. Our demonstrationof a strong correlation between the mode maps and theirreversible reorganization maps constitutes a significant successin understanding how structure determines relaxation in anamorphous material. Indeed, as Fig. 2 illustrates, we may providesemiquantitative prediction of IR domains as they emerge atrelatively long times from the initial configuration alone. Thesecond point is that, because we have used quenched modes, wehave used only information about the bottom of the local potentialminima. While it is quite possible that the timescale required for areorganization event will depend on the energy barriers associatedwith the transition, our results indicate that the spatial structureof such events is largely determined by the distribution of softquasilocalized modes in the initial configuration.

Given our conclusion that relaxation originates with softquasilocalized modes, it follows that our capacity to predict thesubsequent spatial distribution of the IR depends on how persistentthe mode distribution is in a configuration. After all, should themode maps evolve rapidly then the structural information in agiven configuration would quickly become irrelevant. The fact thatwe observe strong spatial correlations between the initial modesand relaxation some 200� later indicates that the spatial structure ofthe modes does generally persist over such times. This is remarkablegiven that small variations in the quenched modes, indicative of achange in the local minimum (or inherent structure), occur over�1� intervals. Details of these rapid changes are provided in theSupplementary Information. We conclude that the spatial structureof the quenched soft modes can often persist over many changesin the inherent structure. Preliminary results indicate that this

persistence is also found in three-dimensional mixtures (includingtemperatures below the empirical mode-coupling temperature)(see the Supplementary Information).

We do, however, see configurations where the mode structure isnot so stable. An example of this is shown in Fig. 4. In Fig. 4a,b wecompare the mode participation map for the initial configurationwith the map of the maximum participation fraction observed perparticle over five 10� runs starting from the configuration in Fig. 4a.The diVerence in spatial structure between these maps is a measureof the degree of variability of the mode structure. In Fig. 4c,d weoverlay the particles showing IR within 200� (as defined in Fig. 3)over the maps of Fig. 4a,b, respectively. While the mode structure ofthe initial configuration does not provide a quantitative predictorof the spatial distribution of IR (Fig. 4c), the cumulative modestructure sampled over the multiple short runs does (Fig. 4d). Thisresult demonstrates that, even when the soft-mode structure is notstable, the IR still originates with these soft modes, but now thisIR is not well predicted by any single configuration. It seems thatconfigurations such as that analysed in Fig. 4 represent those caughtin transit between configurations with more stable mode structure.

In this paper we have presented two important results relatingto the slow relaxation in a model supercooled liquid. The first isthat the irreversible reorganization originates at the sites of thelow-frequency quasilocalized quenched modes. The second is thatthese modes typically persist for timescales significantly longer thanthe lifetime of a given inherent structure. These results show thatthe spatial location and extent of IR regions at relatively longtimes may be reasonably predicted by a simple, static propertyof the static initial condition. A number of previous reports havelinked localized dynamics with the presence of soft modes16–20

during or immediately before the appearance of the motion in

nature physics VOL 4 SEPTEMBER 2008 www.nature.com/naturephysics 713

Widmer-Cooper, Perry, Harrowell, Reichman, Nat. Phys. 4,711 (2008)

Why 30? ω*


• The marginally jammed state represents extreme limit at the opposite pole from the perfect crystal

• The behavior of systems over a wide range of order/disorder follows jamming scaling

• So the marginally jammed is a robust extreme limit--more robust than the perfect crystal

• Jamming scenario provides conceptual basis for commonality of low temperature/frequency properties of disordered solids

• relevance to glass transition is still an open question

Summary


Thanks to

Tim Still UPennWouter Ellenbroek Eindhoven Ke Chen UPennArjun Yodh UPenn

Corey S. O’Hern Yale Leo E. Silbert USIStephen A. Langer NIST

Matthieu Wyart NYUVincenzo Vitelli Leiden

Ning Xu USTC

Bread for Jam:

DOE DE-FG02-03ER46087

Sid NagelCarl Goodrich


jamming as the extreme limit of a solid...carl goodrich upenn sidney nagel u chicago tim still upenn...

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