Jamming as the Extreme Limit of a Solid Andrea J. Liu Department of - - PowerPoint PPT Presentation

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Jamming as the Extreme Limit of a Solid Andrea J. Liu Department of - - PowerPoint PPT Presentation

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Jamming as the Extreme Limit of a Solid Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Carl Goodrich UPenn Sidney Nagel U Chicago Tim Still UPenn Arjun Yodh

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SLIDE 1

Jamming as the Extreme Limit of a Solid

Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Carl Goodrich UPenn Sidney Nagel U Chicago Tim Still UPenn Arjun Yodh UPenn

Tuesday, July 16, 13

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SLIDE 2

Physics of Perfect Crystals

  • Start with T=0 perfect crystal

– look at vibrational, electronic, etc. properties – add defects as perturbation (chapter 30)

Tuesday, July 16, 13

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SLIDE 3

Perturbing away from the crystal

Tuesday, July 16, 13

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SLIDE 4

Perturbing away from the crystal

But what about this?

Tuesday, July 16, 13

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SLIDE 5

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?

Tuesday, July 16, 13

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SLIDE 6
  • 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”

J

jammed unjammed

stress temperature 1/density

Bubble model for foams

  • D. J. Durian, PRL 75, 4780

(1995).

Tuesday, July 16, 13

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SLIDE 7
  • 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”

J

jammed unjammed

stress temperature 1/density

V(r) = ε α 1− r σ ⎛ ⎝ ⎜ ⎞ ⎠ ⎟

α

r ≤ σ r > σ ⎧ ⎨ ⎪ ⎩ ⎪

Bubble model for foams

  • D. J. Durian, PRL 75, 4780

(1995).

Tuesday, July 16, 13

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SLIDE 8

Onset of Jamming in Repulsive Sphere Packings

– (2D) (3D) Zc = 5.97 ± 0.03

5 4 3 2 log (φ- φ ) 3 2 1 6

3D 2D (c)

log(φ- φc)

Z − Zc ≈ Z0(φ −φc)

0.5

  • Just below

φc, no particles

  • verlap

Just above φc there are Zc

  • verlapping

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

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SLIDE 9

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

Tuesday, July 16, 13

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SLIDE 10

Contact Number of Crystal vs. Marginally Jammed Solid

log (Z − Ziso) log p crystal

∼ p1/2 (harmonic)

perfect crystal

vs

marginally jammed solid

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

Tuesday, July 16, 13

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SLIDE 11

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

8 6 4 2

α=2 α=5/2 (a)

p ≈ p0(φ −φc)

α −1

  • 4
  • 3
  • 2

log(φ- φc)

  • 6

G ≈ G0(φ − φc )

α −1.5

  • 4
  • 3
  • 2

6 4 2

α=2 α=5/2 (b)

log(φ- φc)

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

Tuesday, July 16, 13

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SLIDE 12

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

8 6 4 2

α=2 α=5/2 (a)

p ≈ p0(φ −φc)

α −1

  • 4
  • 3
  • 2

log(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).

Tuesday, July 16, 13

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SLIDE 13

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

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SLIDE 14

Mechanics of crystal vs. marginally jammed solid

log p crystal jamming

∼ p1/2 (harmonic)

log(G/B)

perfect crystal

vs

marginally jammed solid

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

Tuesday, July 16, 13

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SLIDE 15

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 L Count 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  1 Z − Zc ≡ 1 Δz  φ − φc

( )

−0.5

Tuesday, July 16, 13

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SLIDE 16

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

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SLIDE 17

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

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SLIDE 18

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

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SLIDE 19

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?

Tuesday, July 16, 13

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SLIDE 20

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 ≡ keff m  Δφ

α −2

( )/2

ℓT≃cT/ω*

Tuesday, July 16, 13

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SLIDE 21

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 ≡ keff m  Δφ

α −2

( )/2

p=0.01 ω*

ℓT≃cT/ω*

Tuesday, July 16, 13

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SLIDE 22

Vibrations of crystal vs. marginally jammed solid

perfect crystal

vs

marginally jammed solid

D(

no plane waves even at ω=0

0.4 0.8 1.2 0.5 1 1.5 2 2.5 3 D()

  • FCC Crystal

D ( ω )

Tuesday, July 16, 13

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SLIDE 23

Vibrations of crystal vs. marginally jammed solid

perfect crystal

vs

marginally jammed solid

no plane waves even at ω=0

0.4 0.8 1.2 0.5 1 1.5 2 2.5 3 D()

  • FCC Crystal

D ( ω )

Tuesday, July 16, 13

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SLIDE 24

Back to extreme limits

How do we connect physics of jamming and physics of crystals? What happens in between?

Tuesday, July 16, 13

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SLIDE 25
  • 1. start with a perfect FCC crystal

2d illustration

Back to extreme limits

How do we connect physics of jamming and physics of crystals? What happens in between?

Tuesday, July 16, 13

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SLIDE 26
  • 1. start with a perfect FCC crystal
  • 2. introduce 1 vacancy-interstitial pair

2d illustration

Back to extreme limits

How do we connect physics of jamming and physics of crystals? What happens in between?

Tuesday, July 16, 13

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SLIDE 27
  • 1. start with a perfect FCC crystal
  • 2. introduce 1 vacancy-interstitial pair
  • 3. minimize the energy

2d illustration

Back to extreme limits

How do we connect physics of jamming and physics of crystals? What happens in between?

Tuesday, July 16, 13

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SLIDE 28
  • 1. start with a perfect FCC crystal
  • 2. introduce 2 vacancy-interstitial pairs
  • 3. minimize the energy

2d illustration

Back to extreme limits

How do we connect physics of jamming and physics of crystals? What happens in between?

Tuesday, July 16, 13

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SLIDE 29

2d illustration

  • 1. start with a perfect FCC crystal
  • 2. introduce 3 vacancy-interstitial pairs
  • 3. minimize the energy

Back to extreme limits

How do we connect physics of jamming and physics of crystals? What happens in between?

Tuesday, July 16, 13

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SLIDE 30
  • 1. start with a perfect FCC crystal
  • 2. introduce M vacancy-interstitial pairs
  • 3. minimize the energy

2d illustration

Back to extreme limits

How do we connect physics of jamming and physics of crystals? What happens in between?

Tuesday, July 16, 13

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SLIDE 31
  • 1. start with a perfect FCC crystal
  • 2. introduce N vacancy-interstitial pairs
  • 3. minimize the energy

2d illustration

Back to extreme limits

How do we connect physics of jamming and physics of crystals? What happens in between?

Tuesday, July 16, 13

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SLIDE 32

Order Parameter

Bond-orientational order

qlm(i) ≡ X

j

Ylm(ˆ rij) Sl(i, j) ≡ X

m

qlm(i) · q∗

lm(j)

f6(i) = fraction

  • f highly correlated

neighbors (large S6)

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

f6 = 1 → crystal f6 = 0 → disordered

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SLIDE 33

“Coexistence” of ordered and disordered regions

1 2 3 4 5 6 7 8 0.2 0.4 0.6 0.8 1 D(f6) f6(i) global: F6 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

Tuesday, July 16, 13

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SLIDE 34

log p log (Z − Ziso) crystal jamming

∼ p1/2 (harmonic)

Connecting jamming and crystal physics Observed states

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SLIDE 35

log p

Wyart, et al. PRE 72 051306 (2005)

c0p1/2 ≤ 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 - Ziso p

crystal jamming

Connecting jamming and crystal physics Observed states

Tuesday, July 16, 13

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SLIDE 36

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

Tuesday, July 16, 13

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SLIDE 37

Elasticity

log (Z − Ziso) log p log G B jamming crystal

fcc fcc+vac/int fcc+vacancies bcc+vacancies

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SLIDE 38

Elasticity

fcc fcc+vac/int fcc+vacancies bcc+vacancies

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SLIDE 39

Elasticity

log (Z − Ziso) log p log G B jamming crystal

fcc fcc+vac/int fcc+vacancies bcc+vacancies

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SLIDE 40

Exclude crystalline states

  • Include only states where disordered “phase” percolates

in all 3 directions

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SLIDE 41

Exclude crystalline states

  • Include only states where disordered “phase” percolates

in all 3 directions

Tuesday, July 16, 13

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SLIDE 42

Exclude crystalline states

  • Include only states where disordered “phase” percolates

in all 3 directions

Tuesday, July 16, 13

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SLIDE 43

Exclude crystalline states

  • Include only states where disordered “phase” percolates

in all 3 directions States with intermediate to low order fall on “jamming surface” Jammed state is not

  • nly extreme

limit but also very robust

Tuesday, July 16, 13

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SLIDE 44

How much does jamming scenario apply to real world?

  • What have we left out? ALMOST EVERYTHING

– friction

  • K. 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/attractions

  • N. Xu, et al. PRL 98 175502 (2007).

– non-spherical particle shape

  • Z. Zeravcic, et al, EPL, 87, 26001 (2009); M. Mailman, et al, PRL 102, 255501 (2009)

– temperature

  • C. 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.

Tuesday, July 16, 13

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SLIDE 45

Real, Thermal Colloidal Glasses

Video 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)

microscope

  • bjective

Ke Chen, Wouter Ellenbroek, Arjun Yodh

Tuesday, July 16, 13

slide-46
SLIDE 46
  • 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 = 1 2 uTKu

Z = duexp(−βV)

∫

C = uu = K −1

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

Tuesday, July 16, 13

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SLIDE 47

Boson Peak

Boson peak frequency

0.885 0.879 0.872 0.866 0.859 0.850 0.840

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

Tuesday, July 16, 13

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SLIDE 48

Boson Peak

2x10

  • 7

4x10

  • 7

6x10

  • 7

8x10

  • 7

1x10

  • 6

D (ω)/ω (s

2/rad 2)

0.82 0.84 0.86 0.88 0.90 0.0 0.2 0.4 0.6 0.8 1.0 1.2

ω* (10

5 rad/s)

φ

0.4

0.885 0.879

(a)

D (ω)/ω (s2/rad2)

105 106 104

ω (rad/s)

Boson peak frequency

0.885 0.879 0.872 0.866 0.859 0.850 0.840

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

Tuesday, July 16, 13

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SLIDE 49

Dispersion relation and elastic constants

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

B G

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SLIDE 50

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 − z0

c = 3.3(φ − φ0 c)

∆z ≡ z − z∞

c

= 3.3(φ − φ∞

c )

Tuesday, July 16, 13

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SLIDE 51

PNIPAM particles are frictional

  • one adjustable parameter

frictional frictionless φ∞

c

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slide-52
SLIDE 52

G, B

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

105, 175701 (2010))

B/keff G/keff

keff = √ 3✏ 22

  • − µ

c )1/2

kBT/✏ = 3 × 10−6 µ ≈ 0.6

  • K. Shundyak, et al. PRE 75,

010301 (2007).

Tuesday, July 16, 13

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SLIDE 53

G, B

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

105, 175701 (2010))

B/keff G/keff two adjustable parameters

keff = √ 3✏ 22

  • − µ

c )1/2

kBT/✏ = 3 × 10−6 µ ≈ 0.6

  • K. Shundyak, et al. PRE 75,

010301 (2007).

Tuesday, July 16, 13

slide-54
SLIDE 54

Jamming and temperature

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

Tuesday, July 16, 13

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SLIDE 55

Effect of Temperature

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

Bertrand, et al.

where C(N) → 0 as N → ∞

Ikeda, et al. Wang and Xu

kBT ∗/✏ ≈ 0.2( − c)5/2 kBT ∗/✏ ≈ 10−3( − c)5/2 kBT ∗/✏ ≈ C(N)( − c)5/2 log(φ − φc) log kBT/✏

Tuesday, July 16, 13

slide-56
SLIDE 56

Effect of Temperature

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

Bertrand, et al.

where C(N) → 0 as N → ∞

Ikeda, et al. Wang and Xu

kBT ∗/✏ ≈ 0.2( − c)5/2 kBT ∗/✏ ≈ 10−3( − c)5/2 kBT ∗/✏ ≈ C(N)( − c)5/2

  • ur expt

log(φ − φc) log kBT/✏

Tuesday, July 16, 13

slide-57
SLIDE 57

Effect of Temperature

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

Bertrand, et al.

where C(N) → 0 as N → ∞

Ikeda, et al. Wang and Xu

kBT ∗/✏ ≈ 0.2( − c)5/2 kBT ∗/✏ ≈ 10−3( − c)5/2 kBT ∗/✏ ≈ C(N)( − c)5/2

  • ur expt

Breaks down for what? log(φ − φc) log kBT/✏

Tuesday, July 16, 13

slide-58
SLIDE 58

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

–15 –10 –5 5 10 15 –15 –10 –5 5 10 15

0.020 0.028 0.035 0.043 0.050 0.058 0.065 0.073 0.080

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

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

Tuesday, July 16, 13

slide-59
SLIDE 59

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

–15 –10 –5 5 10 15 –15 –10 –5 5 10 15

0.020 0.028 0.035 0.043 0.050 0.058 0.065 0.073 0.080

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

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

Why 30? ω*

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slide-60
SLIDE 60
  • 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
  • f low temperature/frequency properties of disordered

solids

  • relevance to glass transition is still an open question

Summary

Tuesday, July 16, 13

slide-61
SLIDE 61

Thanks to

Tim Still UPenn Wouter Ellenbroek Eindhoven Ke Chen UPenn Arjun Yodh UPenn Corey S. O’Hern Yale Leo E. Silbert USI Stephen A. Langer NIST Matthieu Wyart NYU Vincenzo Vitelli Leiden Ning Xu USTC

Bread for Jam: DOE DE-FG02-03ER46087

Sid Nagel

Carl Goodrich

Tuesday, July 16, 13