Soft motions of amorphous solids

Matthieu Wyart

Amorphous solids• structural glasses, granular matter, colloids, dense emulsions

TRANSPORT: thermal conductivity few molecular sizes phonons strongly scattered FORCE PROPAGATION:

L?

ln (T)

Behringer group

L?

Glass Transition

Heuer et. al. 2001

•e

Angle of Repose

h

RearrangementsNon-local

Pouliquen, Forterre

Rigidity``cage ’’ effect:

Rigidity toward collective motions more demanding

Z=d+1: local

characteristic length ?

Maxwell:not rigid

Vibrational modes in amorphous solids?

• Continuous medium: phonon = plane wave Density of states D(ω) N(ω) V-1 dω-1

• Amorphous solids: - Glass: excess of low-frequency modes. Neutron scattering ``Boson Peak” (1 THz~10 K0)

Transport, …

Disorder cannot be a generic explanationNature of these modes?

D(ω) ∼ ω2 Debye

D(ω)/ω2

ω

Amorphous solid different from a continuous bodyeven at L

Unjammed, c

P=0

Jammed, c

P>0

• Particles with repulsive, finite range interactions at T=0• Jamming transition at packing fraction c≈ 0.63 :

O’hern, Silbert, Liu, Nagel

D(ω) ∼ ω0

Crystal:plane waves :: Jamming:??

Jamming ∼ critical point: scaling properties

z-zc=z~ (c)1/2 Geometry: coordination

Excess of Modes:• same plateau is reached for different • Define D(ω*)=1/2 plateau

ω*~ z B1/2

Relation between geometry and excess of modes ??

zc=2d

Rigidity and soft modes

RigidNot rigid soft mode

Soft modes:

RiRjnij=0 for all contacts <ij>

Maxwell: z rigid? # constraints: Nc

# degrees of freedom: Nd

z=2Nc/N 2d >d+1 global

(Moukarzel, Roux, Witten, Tkachenko,...) jamming: marginally connected zc=2d “isostatic”

, Thorpe, Alexander

Isostatic: D(ω)~ ω 0

lattice: independent lines D(ω)~ ω 0

z>zc

*

* = 1/ z ω*~ B1/2/L*~ z B1/2

• main difference: modes are not one dimensional

* ~ 1/ z

L < L*: continuous elastic description bad approximation

Wyart, Nagel and Witten, EPL 2005Random Packing

Ellenbroeck et.al 2006

Consistent with L* ~ z-1

*

Extended Maxwell criterion

f

dE ~ k/L*2 X2 - f X2 stability k/L*2 > f z > (f/k)1/2~ e1/2 ~ (c)1/2

X

Wyart, Silbert, Nagel and Witten, PRE 2005

S. Alexander

Hard Spheres

c0.640.58 cri0.5

1

V(r)

• contacts, contact forces fij

Ferguson et al. 2004, Donev et al. 2004

• discontinuous potential expand E?• coarse-graining in time: < Ri>

Effective Potential

fij(<rij>)?

hij=rij-1

1 d:

Z=∫πi dhij e- fijhij/kT

fij=kT/<hij>

h

Isostatic:

Z=∫πi dhij e- phij/kT p=kT/<h>

Brito and Wyart, EPL 2006

V( r)= - kT ln(r-1) if contactV( r)=0 else

rij=||<Ri>-<Rj>||

G = ij V( rij)

fij=kT/<hij>

• weak (~ z) relative correction throughout the glass phase

•dynamical matrix dF= M d<R> Vibrational modes

z> C(p/B)1/2~p-1/2

Linear Response and Stability

•Near and after a rapid quench: just enough contactsto be rigid system stuck inthe marginally stable region

vitrification

Ln(z)

Ln(p)

Rigid

UnstableEquilibriumconfiguration

vitrification

Activationc

Point defects?Collective mode?

Activationc

Brito and Wyart, J. phys stat, 2007

Granular matter

:

- Counting changes zc = d+1

-not critical z(p0)≠ zc d+1< z <2d

- z depends on and preparation Somfai et al., PRE 2007 Agnolin et Roux, PRE 2008

starth)

h

Hypothesis:

(i) z > z_c

(ii) Saturated contacts:

zc.c.= f(/p)= f(tan ((staron)

(iii) Avalanche starts as z≈ zc.c(start)

Consistent with numerics (2d,: (somfai, staron)

z≈0.2 zc.c(start) ≈ 0.16

Finite h: z -> z +(a-a')/hz +(a-a')/h = f(tan

h c0/ [ c1 tan z]

wyart, arXiv 0807.5109 Rigidity criterion with a fixed and free boundary

Free boundary : z -> z +a'/h

Fixed boundary : z -> z +a/h

a'<a

: effect > *2

Acknowledgement

Tom WittenSid NagelLeo SilbertCarolina Brito

XiL

L

• generate p~Ld-1 soft modes independent (instead of 1 for a normal solid)•argument: show that these modes gain a frequency ω~L-1

when boundary conditions are restored. Then:

D(ω) ~Ld-1/(LdL-1) ~L0

•``just” rigid: remove m contacts…generate m SOFT MODES: High sensitivity to boundary conditions

Isostatic: D(ω)~ ω 0

Wyart, Nagel and Witten, EPL 2005

• Soft modes: extended, heterogeneous

• Not soft in the original system, cf stretch or compress contacts cut to create them

• Introduce Trial modes

• Frequency harmonic modulation of a translation, i.e plane waves ω L-1

D(ω)~ ω0 (variational) Anomalous Modes

R*isin(xi π/L)Ri

xL

z > (c)1/2

A geometrical property of random close packing

maximum density stable to the compression c

relation density landscape // pair distribution function g(r)

1

1+(c)/d

z ~ g(r) dr stable g(r) ~(r-1)-1/2

Silbert et al., 2005

Glass Transition=G relaxation time

Heuer et. al. 2001

•e

Vitrification as a ``buckling" phenomenum

increases

P increases

L