Soft motions of amorphous solids

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) L? Behringer group

Glass Transition Heuer et. al. 2001 • e • 

Angle of Repose h Rearrangements Non-local Pouliquen, Forterre

Rigidity ``cage ’’ effect: Rigidity toward collective motions more demanding Maxwell: not rigid Z=d+1: local characteristic length ?

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 explanation Nature of these modes? Vibrational modes in amorphous solids? D(ω) ∼ ω2 Debye D(ω)/ω2 ω

Amorphous solid different from a continuous bodyeven at L  O’hern, Silbert, Liu, Nagel D(ω) ∼ ω0 • Particles with repulsive, finite range interactions at T=0 • Jamming transition at packing • fraction c≈ 0.63 : Jammed,  c P>0 Unjammed,  c P=0 Crystal:plane waves :: Jamming:??

Jamming ∼ critical point:scaling properties Geometry: coordination zc=2d z-zc=z~ (c)1/2 • 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??

, Thorpe, Alexander Rigidity and soft modes Not rigid  soft mode Rigid Soft modes: for all contacts <ij> RiRjnij=0 Maxwell: z rigid? # constraints: Nc # degrees of freedom: Nd  z=2Nc/N  2d >d+1 global jamming: marginally connected zc=2d “isostatic” (Moukarzel, Roux, Witten, Tkachenko,...)

Isostatic: D(ω)~ ω0  lattice: independent lines  D(ω)~ ω0

z>zc * * = 1/ z  ω*~ B1/2/L*~z B1/2

Random Packing Wyart, Nagel and Witten, EPL 2005 • main difference: modes are not one dimensional • * ~ 1/ z • L < L*: continuous elastic description bad approximation

Ellenbroeck et.al 2006 Consistent with L* ~ z-1

Extended Maxwell criterion S. Alexander f X * dE ~ k/L*2 X2- f X2 stability  k/L*2 > f  z > (f/k)1/2~ e1/2 ~ (c)1/2 Wyart, Silbert, Nagel and Witten, PRE 2005

Hard Spheres V(r)  cri0.5 c0.64 0.58 1 • contacts,contact forces fij Ferguson et al. 2004, Donev et al. 2004

Effective Potential Brito and Wyart, EPL 2006 • discontinuous potential  expand E? • coarse-graining in time: < Ri> fij(<rij>)? h 1 d: Z=∫πi dhij e- phij/kT p=kT/<h> Isostatic: hij=rij-1 Z=∫πi dhij e- fijhij/kT fij=kT/<hij>

fij=kT/<hij>  V( r)= - kT ln(r-1) if contact V( r)=0 else G = ij V( rij) rij=||<Ri>-<Rj>|| • weak (~ z) relative correction throughout the glass phase

Linear Response and Stability • dynamical matrix dF= M d<R> •  Vibrational modes z> C(p/B)1/2~p-1/2 • Near and after a rapid • quench: just enough contacts • to be rigid  system stuck in • the marginally stable region 

vitrification vitrification Ln(z) Rigid Equilibrium configuration Unstable Ln(p) 

Activation  c Point defects? Collective mode?

Activation  c Brito and Wyart, J. phys stat, 2007

Granular matter • : • Counting changes zc = d+1 • not critical z(p0)≠ zc d+1< z <2d • z depends on and preparation • Somfai et al., PRE 2007 • Agnolin et Roux, PRE 2008

start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  h

Rigidity criterion with a fixed and free boundary wyart, arXiv 0807.5109 Fixed boundary : z -> z +a/h Free boundary : z -> z +a'/h a'<a Finite h: z -> z +(a-a')/h z +(a-a')/h = f(tan  h c0/ [ c1 tan z] : effect > *2

Acknowledgement Tom Witten Sid Nagel Leo Silbert Carolina Brito

Isostatic: D(ω)~ ω0 Wyart, Nagel and Witten, EPL 2005 • ``just” rigid: remove m contacts…generate m • SOFT MODES: •  High sensitivity to boundary conditions L Xi L • generate p~Ld-1soft 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

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*isin(xi π/L) Ri x L 

A geometrical property of random close packing z > (c)1/2 maximum density  stable to the compression c relation density landscape // pair distribution function g(r) 1+(c)/d z ~ g(r) dr stable  g(r) ~(r-1)-1/2 1 Silbert et al., 2005

Glass Transition • =G relaxation time  Heuer et. al. 2001 • e

Vitrification as a ``buckling" phenomenum  increases P increases L

Soft motions of amorphous solids