Jamming and the Glass Transition

Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania

Michael Schmiedeberg UPenn Thomas K. Haxton UPenn Ning Xu USTC Sidney R. Nagel UChicago The Jamming Scenario

C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 88, 075507 (2002). temperature C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel, Phys. Rev. E 68, 011306 (2003).

jammed

stress

unjammed J 1/density • Focus on foams/emulsions or frictionless granular material

– soft, repulsive, finite-range spherically-symmetric potentials

• Such systems have T=0 1st/2nd-order phase transition The Jamming Scenario

C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 88, 075507 (2002). temperature C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel, Phys. Rev. E 68, 011306 (2003).

jammed ⎧ α ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ α ⎝ σ ⎠ stress ⎪ ⎩ 0 r > σ unjammed J 1/density • Focus on foams/emulsions or frictionless granular material

– soft, repulsive, finite-range spherically-symmetric potentials

• Such systems have T=0 1st/2nd-order phase transition How we study Point J

• Generate configurations near J ⎧ α ε ⎛ r ⎞ – Start with some initial config ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ α ⎝ σ ⎠ ⎪ ⎩ 0 r > σ

– Conjugate gradient energy minimization (inherent structures, Stillinger & Weber)

• Classify resulting configurations

non-overlapped overlapped V=0 V>0 p=0 or p>0

Tf=0 Tf=0 Onset of Overlap is Onset of Jamming

- 2 (a) D=2 - 4 α=2 D=3 •Pressures for different - 6 α=5/2 states collapse on a single α −1 8 - p ≈ p0(φ −φc) curve

0 (b) -2 α=2 •Shear & bulk moduli, G & B, =5/2 4 α vanish at same φc - α −1.5 G ≈ G0(φ − φc ) - 6 -4 -3 -2 0 (c) 3D φ φ γ≈1/2 log( - ) •G/B ~ ( - c) 1 φ φc 2D

3 D. J. Durian,5 PRL4 75, 47803 (1995); 2 log ( - ) C. S. O’Hern, S. A. Langer,φ φA.c J. Liu, S. R. Nagel, PRL 88, 075507 (2002). 2 (a)

4 Onsetα=2 of Overlap has 1st-Order Character 6 =5/2 Just belowα Just above φ , no c 8 φc there are particles Zc overlap0 (b) overlapping 2 α=2 neighbors =5/2 4 α per particle

0 (c) 3D – Zc = 3.99 ± 0.01 (2D) -1 2D Zc = 5.97 ± 0.03 (3D) -2 0.5 Z − Zc ≈ Z0 (φ −φc) -3 -5 -4 -3 - 2

log(logφ -(φ φ-cφ)c) Verified experimentally: Durian, PRL 75, 4780 (1995). Majmudar, Sperl, Luding, Behringer, O’Hern, Langer, Liu, Nagel, PRL 88, 075507 (2002). 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 James Clerk Maxwell stability) –Number of unknowns per particle=Z/2 –Number of equations per particle = D Z = 2D

Phillips, Thorpe, Boolchand, Edwards, Ball, Blumenfield Isostaticity and Diverging Length Scale M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)

•For system at φc, Z=2d •Removal of one bond makes entire system unstable by adding one soft mode

+ •This implies diverging length as φ-> φc

For φ > φc, cut bonds at boundary of circle of size L Count number of soft modes within circle d −1 d Ns ≈ L − (Z − Zc )L Define length scale at which soft modes just appear 1 −0.5  ≈ ≈ (φ − φc ) Z Z − c

Isostaticity and Diverging Length Scale M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)

•For system at φc, Z=2d •Removal of one bond makes entire system unstable by adding one soft mode

+ •This implies diverging length as φ-> φc

Consequences of Diverging Length Scale • Solids are all alike at low T or ω:

– density of vibrational states D(ω)~ω2 in d=3

– vibrational heat capacity C(T)~T3 – thermal conductivity κ(T)~Cvℓ~T3

• Why? Low-frequency excitations are sound modes. At long length scales all solids look elastic Consequences of Diverging Length Scale • Solids are all alike at low T or ω:

– density of vibrational states D(ω)~ω2 in d=3

– vibrational heat capacity C(T)~T3 – thermal conductivity κ(T)~Cvℓ~T3

• Why? Low-frequency excitations are sound modes. At long length scales all solids look elastic

BUT at Point J, there is a diverging length scale ℓ*

So what happens? Vibrations in Marginally Jammed Solids L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05)

* / 1/2 D(ω) ω ω 0  Δφ ω *

k ω ≡ eff  Δφ(α −2)/2 0 m

• More and more modes in excess of Debye prediction as φ->φc (boson peak)

• New class of excitations distinct from sound modes originates from soft modes at Point J M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05) • Robust for systems near isostaticity Souslov, Liu, Lubensky, PRL 103, 205503 (2009); Mao, Xu, Lubensky arXiv: 09092616 Vibrational Modes Predict Soft Spots • Adams-Gibbs Cooperatively-Rearranging Regions? • Shear Transformation Zones? M. Lisa Manning’s talk next week Summary of Point J ⎧ α ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ α ⎝ σ ⎠ ⎪ ⎩ 0 r > σ • Mixed first-order/second-order transition (RFOT) • Number of overlapping neighbors per particle ⎧ ⎪ 0 φ < φc Z = ⎨ β ≅1/2 Z + z φ − φ φ ≥ φ ⎩⎪ c 0 ( c ) c • Static shear modulus/bulk modulus γ ≅1/2 G / B  φ − φ ( c ) • Two diverging length scales * −ν ≅−1/2   (φ − φc ) † † −ν ≅−1/4   (φ − φc )

Similarity to Other Models • This behavior has only been found in a few models, all in mean-field limit

– 1-RSB p-spin interaction spin glass Kirkpatrick, Thirumalai, Wolynes

– compressible frustrated Ising antiferromagnet Yin, Chakraborty

– kinetically-constrained Ising models Sellitto, Toninelli, Biroli, Fisher

– k-core percolation and variants Schwarz, Liu, Chayes, Toninelli, Biroli, Fisher,Harris, Jeng

– Mode-coupling approximation of glasses Biroli, Bouchaud

– 1-RSB 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 and the Glass Transition

N. Xu, T. K. Haxton, A. J. Liu and S. R. Nagel, PRL 103, 205503 (2009). temperature

jammed

stress

unjammed J 1/density

• How do ideal spheres behave at nonzero temperature? Dramatic Increase of Relaxation Time

Colloidal glass transition Glass transition

⎛ ⎞ ⎛ A ⎞ A τ = τ 0 exp⎜ x ⎟ T T τ = τ 0 exp⎜ x ⎟ ⎝ ( − 0 ) ⎠ ⎝ (φ − φ0 ) ⎠

Brambilla, et al., arXiv/0809.3401

L.-M. Martinez and C. A. Angell, Similar fitting functions. Nature 410, 663 (2001). WHY?? Very Different Underlying Pictures

Colloidal glass transition Map of Lake O'Hara - YellowMaps TopoGlass Guide transition 8/5/09 4:25 PM

Pressure is most important Temperature is most important It governs amount of free It allows system to overcome

volume energy barriers

http://www.yellowmaps.com/topo/_lakeohara_bc/index2.htm Page 1 of 2 Very Different Underlying Pictures

Colloidal glass transition Map of Lake O'Hara - YellowMaps TopoGlass Guide transition 8/5/09 4:25 PM

Pressure is most important Temperature is most important It governs amount of free It allows system to overcome

volume energy barriers There are systems for which these two transitions are the same phenomenon

http://www.yellowmaps.com/topo/_lakeohara_bc/index2.htm Page 1 of 2 Relaxation Time temperature • Look at relaxation time along different jammed trajectories

– Fix p, lower T stress – Fix T, raise p

unjammed J 1/density Relaxation Time temperature • Look at relaxation time along different jammed trajectories

– Fix p, lower T stress – Fix T, raise p

unjammed J 1/density

6 10 5 (a) 10 4 10 ! 3 10 2 10 1 10 -8 -7 -6 -5 -4 10 10 10 10 10 T 6 10 (b) 5 10

4 ! 10

3 10

2 10 3 4 5 6 7 8 10 10 10 10 10 10 1/p Relaxation Time temperature • Look at relaxation time along different jammed

6 trajectories 10 5 (a) – Fix p, lower T 10 stress 4 10 – Fix T, raise p ! 3 10 2 unjammed J 10 1 10 -8 -7 -6 -5 -4 1/density 10 10 10 10 10 T 6 6 10 10 5 (a) (b) 5 10 10 4 10 4 ! 3 ! 10 10 2 3 10 10 1 2 10 -8 -7 -6 -5 -4 10 3 4 5 6 7 8 10 10 10 10 10 10 10 10 10 10 10 T 6 1/p 10 (b) 5 10

4 ! 10

3 10

2 10 3 4 5 6 7 8 10 10 10 10 10 10 1/p Data Collapse! 3 10 -7 6 p=2x10 (a) 10 -6 (a) ⎛ T ⎞ p=2x10 5 -5 10 2 τ p = f p=2x10 -4 1/2 ⎜ ⎟ 4 10 ⎝ p ⎠ p=2x10 10 τ p -8 ! T=10 3 -7

10 /m) T=10 2 -6 ! 1 T=10 10 -5 10 T=10

1 (p 10 -8 -7 -6 -5 -4 10 10 10 10 10 " T 6 0 10 (b) 10 5 10 0 0.1 0.2 0.3 0.4 0.5 0.6 4 ! 10 3

3 T/pT/p! 10

2 10 3 4 5 6 7 8 (b) 10 10 10 10 10 10 Data100 for different (p, T) collapse 1/p 1/2 on to single scaling curve!

/m) 10 ! (p " 1

-5 -4 -3 -2 -1 10 10 103 10 10 p! /# Dimensional Analysis • Recall interaction potential ⎧ 2 ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ 2 ⎝ σ ⎠ particle mass ⎪ ⎩ 0 r > σ • We have 3 dimensional parameters in model: ε, m, σ

interaction energy ε ⎛ T pσ 3 ⎞ Relaxation h , particle diameter τ 2 = ⎜ ⎟ time mσ ⎝ ε ε ⎠ or equivalently pσ ⎛ T pσ 3 ⎞ τ = g , m ⎝⎜ pσ 3 ε ⎠⎟ Dimensional Analysis • Recall interaction potential ⎧ 2 ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ 2 ⎝ σ ⎠ particle mass ⎪ ⎩ 0 r > σ • We have 3 dimensional parameters in model: ε, m, σ

interaction energy ε ⎛ T pσ 3 ⎞ Relaxation h , particle diameter τ 2 = ⎜ ⎟ time mσ ⎝ ε ε ⎠ or equivalently 0 pσ ⎛ T pσ 3 ⎞ τ = g , m ⎝⎜ pσ 3 ε ⎠⎟ 3 10 -7 p=2x10 (a) -6 p=2x10 -5 2 p=2x10 -4 1/2 10 p=2x10 -8 T=10 -7

/m) T=10 -6

! 1 T=10 -5 10 T=10 (p " 0 10 0 0.1 0.2 0.3 0.4 0.5 0.6 Dimensional Analysis3 T/p! (b) 100 1/2

/m) 10 ! (p " 1

-5 -4 -3 -2 -1 10 10 103 10 10 p! /# T, p equally pσ ⎛ T pσ 3 ⎞ ⎛ T ⎞ important! lim τ = lim g , = f 3 3 ⎜ 3 ⎟ ⎜ 3 ⎟ pσ /ε→0 m pσ /ε→0 ⎝ pσ ε ⎠ ⎝ pσ ⎠ • In low p limit, the relaxation time depends only on T/p • Data collapse for different trajectories Data Collapse for Different Potentials • pσ3/ε->0 corresponds to low p limit for soft spheres AND to the hard sphere limit • Should see collapse for anyα≥ 0 including α=0 (hard spheres) ⎧ α ε ⎛ r ⎞ ⎪ ⎜1− ⎟ r ≤ σ V(r) = ⎨ α ⎝ σ ⎠ ⎪ ⎩ 0 r > σ Data Collapse for Different Potentials • pσ3/ε->0 corresponds to low p limit for soft spheres AND to the hard sphere limit • Should see collapse for anyα≥ 0 including α=0 (hard spheres) YES!

pσ ⎛ T ⎞ lim τ = f ΔV~T/p T /ε→0 m ⎝⎜ pσ 3 ⎠⎟ T does work against pressure Hard spheres to open up free volume σ3 Glass transition equivalent to colloidal glass transition as pσ3/ε-> 0 Relaxation times at high pressure

WCA $!" $!# $!$ $!% $!& $!'" 2 T/pσ3=0.06 10 1/2 /m) " 10 $ (p ! T/pσ3=0.2

$ 1 10-6 10-5 10-4 10-3 10-2 10-1 3 p" /#

• As spheres soften, relaxation time decreases (Weitz/Reichman) • New mechanism of relaxation controlled by pσ3/ε emerges Soft spheres have smaller equivalent hard sphere diameter

smaller effective volume fraction

more free volume

faster relaxation

Soft spheres Hard spheres with with diameter σ reduced diameter σHSσ =eff σ <−σ Δσ

How to choose σeff? Rowlinson, Barker-Henderson, Andersen-Weeks-Chandler Strategy for choosing σeff Andersen, Weeks, Chandler J. Chem. Phys. 54, 5237 (1971) • Taylor expand free energy around hard-sphere potential

• Choose σ ef f so that first-order functional derivative of free energy with respect to exp(-V(r)/T) vanishes  dryeff (r) exp[−V(r) / T ] − exp ⎡−Veff (r) / T ⎤ = 0 ∫ { ⎣ ⎦}

V (r) β eff hard-sphere e geff (r) original soft-sphere σ potential potential with eff

• Can calculate σ e ff from the soft-sphere potential and hard- sphere properties alone! • This approximation reproduces captures static quantities beautifully Relaxation time

• Start with soft spheres at arbitrary pressure

• Use ACW to calculate effective hard sphere diameter σeff

• Obtain new packing fraction for effective hard spheres ϕeff Relaxation time

• Start with soft spheres at arbitrary pressure

• Use ACW to calculate effective hard sphere diameter σeff

• Obtain new packing fraction for effective hard spheres ϕeff (a) 104 25 ACW approx rocks!! 20 τcollapses onto /T

3 15

3 " p

10 p 10 1/2 5 hard-sphere curve p

/m) 0

" p for all finite-ranged 102 0.4 0.5 (p # p ! repulsive potentials p p 10 p and pressures studied 0.35 0.4 0.45 0.5 0.55 #eff

(b) 4 10 104

1/2 103

103 /m) 2 " 10 1/2 p (p !

/m) 10 " 102 p (p

! 0.05 0.1 0.15 0.2 0.25 3 p T/(p" ) 10 p p p p 0.05 0.1 0.15 0.2 0.25 3 T/(peff"eff ) Recall Hard-Sphere Curve

25 pσ ⎛ T ⎞ hard-sphere lim f 20 τ = ⎜ 3 ⎟ T /ε→0 m ⎝ pσ ⎠ equation of

/T 15 3 !

p state 10 Hard spheres 5 fit to Carnahan-Starling eqn 0 0.4 0.5 σ3 " Universal Hard-Sphere Master Curve

104 104

3 3 10 p 10

1/2 1/2 p /m) /m) " 102 " 102 p (p p (p ! ! p p p 10 10 p p p p p p p 0.05 0.1 0.15 0.2 0.25 0.05 0.1 0.15 0.2 0.25 3 3 T/(p" ) T/(peff"eff )

pσ peff σ eff peff σ eff τ = τ τ = τ • This implies that m e ff m or eff pσ • Given – hard-sphere master curve – pair interaction potential of soft-sphere system • can calculate relaxation time of soft-sphere system! • Can also use soft spheres to extend master curve Two Mechanisms of Relaxation in Soft Spheres • Temperature opens up free volume against the pressure T = pΔV

– hard spheres are fragile glassformers (super- Arrhenius increase of relaxation time)

• Temperature allows soft spheres to overlap so they behave as hard spheres with smaller diameter (less super-Arrhenius with increasing overlap)

• In energy landscape, canyons do not become deeper but become narrower and more convoluted as p ↑ or T ↓ Connection to Point J • Point J corresponds to double limit T/pσ3 → 0, pσ3/ε → 0 • What is pσ ⎛ T pσ 3 ⎞ ⎛ T ⎞ lim lim τ = lim lim g , = lim f 3 3 3 3 ⎜ 3 ⎟ 3 ⎜ 3 ⎟ T / pσ →0 pσ /ε→0 m T / pσ →0 pσ /ε→0 ⎝ pσ ε ⎠ T / pσ →0 ⎝ pσ ⎠ • Does τ diverge at Point J? Or does it diverge at T/pσ3 > 0? • Does Pt J control glass transition? Or is there an underlying thermodynamic glass transition? temperature temperature

jammed jammed

vs. stress stress

unjammed unjammed J J 1/density 1/density Fitting Forms ⎛ A ⎞ • Vogel-Fulcher form: τ = exp τ 0 ⎜ ⎟ ⎝ T − T0 ⎠

• Elmatad-Chandler-Garrahan form:

Y. S. Elmatad, D. Chandler, J. P. Garrahan, J. Phys. Chem. B 113, 5565 (2009). Form of Scaling Function

Fits: pσ ⎛ T ⎞ ⎛ 0.15 ⎞ lim τ = f 3 f (x) = 0.5exp⎜ ⎟ T /ε→0 m ⎝⎜ pσ ⎠⎟ ⎝ x − 0.05⎠ Vogel-Fulcher form

2 ⎡ ⎛ 1 1 ⎞ ⎤ f (x) = Aexp ⎢B − ⎥ ⎜ x x ⎟ ⎣⎢ ⎝ 1 ⎠ ⎦⎥ Elmatad-Chandler-Garrahan form

• Hard spheres are very fragile! • Can’t distinguish between V-F form and E-C-G form • Can’t tell if there is a thermodynamic glass transition or not • But if not, then Point J controls dynamical glass transition Conclusions temperature • Point J is a special point

• Hint of connection to glass transition in jammed exponents for jamming transition stress • Similarity in form of slow down in unjammed dynamics due to equivalence of J 1/density – hard sphere glass transition – thermal glass transition of soft spheres at low p • Hard spheres tell us everything about soft spheres •Point J controls dynamical glass transition of hard spheres if thermodynamic glass transition does not exist • Still ahead: attractions Thanks to

Michael Schmiedeberg Vincenzo Vitelli Leiden Matthieu Wyart Princeton Tom Haxton Ning Xu Anton Souslov UPenn Tom Lubensky UPenn Sid Nagel Lynn Daniels UPenn Doug Durian UPenn Corey S. O’Hern Yale Leo E. Silbert USI Zexin Zhang UPenn Stephen Langer NIST Ke Chen UPenn J. M. Schwarz Syracuse Arjun Yodh UPenn Lincoln Chayes UCLA Randy Kamien UPenn

Bread for Jam: DOE DE-FG02-03ER46087 German Academic Exchange Service (DAAD) Effective diameter

WCA $!" $!# $!$ $!% $!& $!'" 2 101 T/pσ3=0.06

0.91/2 T/pσ3=0.2 /m) " 10 $ (p

0.8! ! / eff ! 0.7 $ 1 0.6 10-6 10-5 10-4 10-3 10-2 10-1 3 p" /# Hard spheres with 0.5 10-6 10-5 10-4 10-3 10-2 10-1 reduced diameter 3 σ σ = effσ − Δσ p! /" HS

3 • As pσ /ε increases, σeff decreases ⎧ α ε ⎛ r ⎞ ⎪ 1− r ≤ σ 3 V(r) = ⎜ ⎟ T/pσ σeff ⎨ α ⎝ σ ⎠ • As increases, decreases ⎪ ⎩ 0 r > σ Jamming vs. Glass Transition (a) (b) 1/ϕ 1/ϕ

Line of J-points liquid Line of J-points liquid metastable glassy states metastable glassy states 1 1 /ϕJ /ϕJ

1/ϕ 1/ϕ 0 glass phase K 0

T/p T/p BUT: Real Liquids have Attractions • Point J only exists for repulsive, finite-range potentials • Attractions can lead to vapor-liquid transition which preempts Point J

But there is hope: Attractions serve to hold system U at high enough density that Repulsion vanishes at repulsions come into play (WCA) finite distance Behavior of liquids is controlled by repulsions, and attractions are r perturbation Similar Behavior in Jammed State

Zeller & Pohl, PRB 4, 2029 (1971).

• Behavior of amorphous solids is – very different from that of crystals – same in all amorphous solids – still not understood Marginally Jammed Solid L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05) Density of Vibrational Modes

(α −1)/2 ω* ∝ (φ − φc ) ω *

φ− φc

• Density of states is not Debye-like at low ω

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

• Scaling of ω* is robust to systems near isostaticity Souslov, Liu, Lubensky, PRL 103, 205503 (2009); Mao, Xu, Lubensky arXiv: 09092616 Boson Peak

amorphous C/T3

U. Buchenau, et al. PRB 34, crystal 5665 (1986)

• Excess in density of states is tied to peak in heat capacity C(T)/T3 • This frequency/temperature can be tuned in jammed

packings by varying φ-φc Boson Peak

amorphous C/T3

U. Buchenau, et al. PRB 34, crystal 5665 (1986) Correlated with boson peak frequency at ω* • Excess in density of states is tied to peak in heat capacity C(T)/T3 • This frequency/temperature can be tuned in jammed

packings by varying φ-φc Thermal Conductivity

crystal 1 P. B. Allen and κ = ∑Ci (T )di J. L. Feldman, V i PRB 48,12581 (1993).

thermal heat carried diffusivity T3 conductivity by mode i of mode i

κ Kubo formulation π 2 d S i = 2 2 ij δ (ωi − ω j ) 2 3 ω ∑ T  i i≠ j amorphous Kittel’s 1949 hypothesis: rise in κ above plateau due to regime of freq-independent diffusivity Ioffe-Regel Crossover

•Crossover from weak to strong scattering at ω* •At Point J, the diffusivity is flat down to ω=0 •Freq-indep diffusivity originates from Point J

N. Xu, V. Vitelli, M. Wyart, A. J. Liu, S. R. Nagel, PRL 102, 038001 (2008). V. Vitelli, N. Xu, M. Wyart, A. J. Liu, S. R. Nagel, arXiv:0908.2176 Consequences for Thermal Conductivity

• At J, κis finite in harmonic approx crystal amorphous C/T3 ∞ κ (T ) = ∫ dωC(ω)d(ω)g(ω) κ 0

ω d0 up to ωD g0 up to D ~ T kT • So κ up to D   ω D then saturates • For systems withT ω*>0, onset of this behavior occurs at kT*  ω * amorphous boson peak in C T end of plateau in κ

tied together through ω* Consequences for Thermal Conductivity

crystal amorphous C/T3 κ

T amorphous boson peak in C T end of plateau in κ

tied together through ω* Quasilocalized Modes at ω*

N. Xu, Vi Vitelli, A. J. Liu, S. R. Nagel, arXiv:0909.3710 Quasilocalized Modes at ω*

N. Xu, Vi Vitelli, A. J. Liu, S. R. Nagel, arXiv:0909.3710 Quasilocalized Modes at ω* • Modes become quasilocalized near Ioffe- Regel crossover

N. Xu, Vi Vitelli, A. J. Liu, S. R. Nagel, arXiv:0909.3710 Quasilocalized Modes at ω* • Modes become quasilocalized near Ioffe- Regel crossover • High displacements occur in low-coordination regions

N. Xu, Vi Vitelli, A. J. Liu, S. R. Nagel, arXiv:0909.3710 Anharmonicity

0.6

0.4 ) ! p( 0.2

0 -2 10 -3 10 -4 ) 10 !

( -5 10

max -6

V 10 -7 10 -8 10 -9 10 0 0.5 1 1.5 2 2.5 ! • The low-frequency quasi-localized modes have the lowest energy barriers to rearrangement – two-level systems?

– STZ’s? N. Xu, Vi Vitelli, A. J. Liu, S. R. Nagel, arXiv:0909.3710 K-Core Percolation • Culling process – Occupied sites with fewer than k occupied neighbors become vacant • Repeat culling process until no more can be removed • Remaining occupied sites called the k-core

k=3 K-Core Percolation • Culling process – Occupied sites with fewer than k occupied neighbors become vacant • Repeat culling process until no more can be removed • Remaining occupied sites called the k-core

k=3 Jamming vs k-Core (Bootstrap) Percolation J. M. Schwarz, A. J. Liu, L. Chayes, EPL 73, 560 (2006). • Jammed configs at T=0 are • Consider lattice with coord. #

mechanically stable Zmax with sites indpendently • For particle to be locally occupied with probability p stable, it must have at least d • For site to be part of “k-core”, +1 overlapping neighbors in d it must be occupied and have at dimensions least k=d+1 occupied neighbors • Each of its overlapping nbrs • Each of its occ. nbrs must have must have at least d+1 at least k occ. nbrs,etc overlapping nbrs, etc. • Look for percolation of k-core

• At φ> φc all particles in load- bearing network have at least d+1 neighbors Long-Ranged Interactions/Attractions • Point J only exists for repulsive, finite-range potentials • Real liquids have attractions U Repulsion vanishes at Attractions serve to hold finite distance system at high enough density that repulsions r come into play (WCA) • 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). Effect of Particle Shape • Ellipsoids

Z. Zeravcic, N. Xu, A. J. Liu, S. R. Nagel, W. van Saarloos, EPL, 87, 26001 (2009).

• Introduce new rotational band but onset of translational band still scales as for spheres • Ellipsoids controlled by Point J for spheres What happens at Σ>0? . T Look at rheology: stress Σ vs. T, p, γ unjammed jammed Σ

1/φ J Glasses & colloidal glasses shear thin--is there a connection? T. Haxton

Dimensional Analysis T. Haxton (2D) Σσ 2 ⎛ T mσ 2 pσ 2 ⎞ = H ⎜ ,γ , ⎟ ε ⎝ ε ε ε ⎠

or equivalently,

Σ ⎛ T m pσ 2 ⎞ = G , I = γ , p ⎜ pσ 2 p ε ⎟ or equivalently, ⎝ ⎠ T/ε

• Stress is indep of potential for small T, p

Σ ⎛ T m ⎞ lim = F , I ≡ γ 2 ⎜ 2 ⎟ pσ /ε→0 p pσ p ⎝ ⎠ Data Collapse for Rheology

T / pσ 2 = 0.01 p

σ σ / T / pσ 2 = 0.1

m γ I = γ p

Σ ⎛ T m ⎞ lim = F , I = γ 2 ⎜ 2 ⎟ pσ /ε→0 p pσ p ⎝ ⎠

High T p σ 2 , low I: viscous Low T p σ 2, low I: elastic Consequences of Rheology Collapse

σ η mp

m I = γ p

Equivalence of rheology for colloids and glasses T unjammed Σ ⎛ T m ⎞ lim = F , I = γ 2 ⎜ 2 ⎟ pσ /ε→0 p pσ p jammed ⎝ ⎠ Σ p ⎛ T Σ⎞ lim τ = F , 2 ⎜ 2 ⎟ pσ /ε→0 m ⎝ p p⎠ J σ 1/φ

describes jamming surface: universal near Point J!