1. Soft motions of amorphous solids Matthieu Wyart

2. 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?

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

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

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

6. 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 K 0 ) Transport, … Disorder cannot be a generic explanation Nature of these modes? D(ω) ∼ ω 2 Debye D(ω)/ω 2 ω

7. Amorphous solid different from a continuous body even 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) ∼ ω 0 Crystal:plane waves :: Jamming:??

8. Jamming ∼ critical point: scaling properties z-z c =  z~ (  c ) 1/2 Geometry: coordination Excess of Modes : same plateau is reached for different  Define D(ω*)=1/2 plateau ω*~  z B 1/2 Relation between geometry and excess of modes ?? z c =2d

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

10. Isostatic: D( ω )~ ω 0  lattice: independent lines  D(ω)~ ω 0

11. z>z c **  * = 1/  z  ω*~ B 1/2 /L*~  z B 1/2

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

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

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

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

16. discontinuous potential  expand E?  coarse-graining in time: Effective Potential f ij ( )? h ij =r ij -1 1 d: Z=∫π i dh ij e - f ij h ij /kT f ij =kT/ h Isostatic: Z=∫π i dh ij e - ph ij /kT p=kT/ Brito and Wyart, EPL 2006

17. V( r)= -  kT ln(r-1) if contact V( r)=0 else r ij =|| - || G =  ij V( r ij ) f ij =kT/  weak (~  z) relative correction throughout the glass phase

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

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

20. Activation cc  Point defects? Collective mode?

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

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

23.  start h) h  Hypothesis: (i) z > z_c (ii) Saturated contacts:  z c.c. = f(  /p)= f(tan (  (staron) (iii) Avalanche starts as  z≈  z c.c (  start ) Consistent with numerics (2d,  : (somfai, staron)  z≈0.2  z c.c (  start ) ≈ 0.16

24. Finite h:  z ->  z +(a-a')/h  z +(a-a')/h = f(tan  h  c 0 / [ c 1 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

25. Acknowledgement Tom Witten Sid Nagel Leo Silbert Carolina Brito

26. XiXi L L generate p~L d-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) ~L d-1 /(L d L -1 ) ~L 0 ``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

27. 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 )~ ω 0 (variational)  Anomalous Modes  R* i   sin(x i π/L)  R i   x L

28.  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

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

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