1. On the density of soft non-linear excitations in amorphous solids Matthieu Wyart, Center for Soft Matter Research, NYU M.Wyart, PRL (2012) E.Lerner, G.During and M.Wyart, Soft Matter (2013) J. Lin, A. Saade, E. Lerner, A. Rosso, M. Wyart submitted

2. Boson Peak Boson Peak: excess soft vibrational modes in glasses. Important: correlates to liquid fragility Sokolov, Novikov One view: disorder in the structure controls amplitude of the boson peak Agreed: - disorder strongly affects the nature of the modes - There are heterogeneities in elasticity - Might be the correct starting point in some cases But: -Disorder is not the main parameter controlling the amplitude of the boson peak in some important materials: Monaco: silica has perhaps a strong boson peak, but crystobalite (crystal) too! Not isolated example (e.g. packing of particles). - In some cases the length scale  characterizing elastic heterogeneities does not enter in the static structure!  structural length

3. Monaco: (i)Why disorder has such a mild effect on the amplitude of the boson peak in some glasses? (ii) If not disorder, what aspects of the microscopic structure governs The amplitude of the boson peak? (iii) Why in a Lennard-Jones the boson peak is very different between the crystal and the glass, but not in silica? Liu, Nagel, Van Saarloos, Wyart 2010 random close packing Soft repulsive particles: Cubic lattice FCC D(w) w w w Proof: z=z c D(w) does not vanish as w->0, independently of disorder. Wyart, Nagel, Witten 2005

4. Mechanical stability Phase diagram of packings of soft particles: coordination and pressure determine boson peak, not disorder Wyart, Silbert, Nagel and Witten 2005 Brito, Wyart 2006 Modes at the boson peak characterized by a length NOT entering in the static structure unstable stable 1 0.1 Ellenbroeck et al

5. Vibrations SiO 2 Tetrahedral structure Model: rigid tetrahedra linked by flexible joint (motivation: bending energy Si-o-Si weak) Si o o o o For any tetrahedra:3+3=6 degrees of freedom 4  3/2=6 constrains  isostatic Trachenko etal, prl 93 135502 (2004)

6. Neglect weak interactions: corrections to boson peak frequency Cf ccc   ?  1 Thz Crystobalite: same open tetrahedral structure: Same argument applies, similar boson peak expected. Conclusion: crystal and glass have a similar peak if local environment of the particles (coordination, pressure) are the same. Yes silica No Lennard-Jones Limitations of our approach: Lennard Jones at ~0 pressure did not work well, boson peak Presumably dominated by fluctuations in the structure. Xu, Wyart, Liu, Nagel, prl 2008

7. Non-linear stability Potentially interacting shear transformation zone: What is their density? How far is the system from a runaway avalanche? 1/ Random close packing, long range interaction 2/ Generic amorphous solid

8. Microscopic structure at random close packing silbert, Liu, Nagel 2006, Donev et al., 2005 Charbonneau, Corwen, Zamponi, Parisi 2012 Pair distribution function at jamming: Force distribution function at jamming: Lerner, During, Wyart 2012 Charbonneau, Corwen, Zamponi, Parisi 2012 What governs the structure?? h

9. Example of non-linear phenomena: plastic flow Combe and roux, prl 2000 non-linear, plastic events: avalanches of rewiring of the contact network cracking : jump in strain are power-law no scale! Microscopic cause for this effect?

10. Non-linear marginal stability of hard particles p p Packing of frictionless hard particles at Pressure p, in a box E= p dV Decreasing volume by changing the network of contact? z=z c isostatic: just enough contact to be rigid Moukarzel, Roux, Trachenko, Witten One contact opened by s, one soft mode, displacement field. Displacement does not decay with distance from the source. Wyart, Annales 2005

11. Two kinds of contacts at low-forces Extended: Localized: Localized contact more numerous at low-force Argument: can be computed requiring that opening such Contacts do not decrease the volume Lerner,During and Wyart, arxiv 1302.3990

12. Stability criterion S limited by the formation of a new contact Contact with weak forces more likely unstable Small gaps limit s, stabilize packings

13. Stability of “extended” contact cf Stability Consistent with saturation

14. Additional results avalanches of contact opening: -impossible if packing strictly stable -Extensive if packing unstable -Power-law like possible only if marginality Stability of type 2 modes is also marginal role of “STZ” in hard spheres played by contact with low forces their density P(f) is singular, singularity such that runaway avalanches almost possible Lerner,During and Wyart, soft matter 2013 Wyart, PRL 2012

15. Soft interactions Local distance to yield stress ? With independence: - What governs Karmakar, Lerner and Procaccia 2010

16. Elasto-plastic Models Discretize space: proximity to yielding If, probability per unit time to yield When yielding occur in site i, stress propagates See talk by Vandembroucq, Barrat, Bocquet, Ajdari

17. Mean-field Similar to Hebraud Lequeux 1998 Long-range interaction: Random kick leads to diffusive equation: Quasi-static regime: absorbing condition in x=0 Very different from depinning! Cf if interaction monotonic, 1 st Conclusion: non-monotonicity of interaction is key

18. decreases with interaction range Marginal stability: requiring that one event triggers another one in average yields Such models lie close to marginality

19. Quadripolar elasto-plastic models: shear and quench In two dimensions: Yield stress Quench x y 1 initial final

20. Quadri-polar model captures well atomistic models No significant dependence on the protocol. Distribution of weakest site Nearly identical to MD Despite lack of independence, relation  and  true

21. Conclusions 1/ Non-linear marginal stability apparent in P(f) governs plasticity at RCP, affect dense suspension flows 2/ Singular distribution of the shear transformation zone density with soft interactions.  depends on interactions range- likely to enter description of yield stress transition Lerner, During, Wyart EPL 2012

22. Acknowledgement Group Members: Gustavo During Edan Lerner Jie Lin Le Yan Alaa Saade Eric de Giuli Collaborators: Alberto Rosso Funding: NSF DMR NSF PMP NYU MRSEC Sloan Fellowship

23. Marginal stabilities in packings System: Modes: Relate: Consequence of marginality: Soft particles above  c Hard particles below  c Vibrational Z(  Anomalous elasticity: strong Boson peak, weak transport, response heterogeneous on large length scales Random close packing Contact change P(f) to g(r) Anomalous plasticity: Crackling, avalanches

24. Rigidity transition Melting: rigidity stems from continuous symmetry breaking, appearance of long-range order Jamming: fluid jams into amorphous solids. No obvious long-range order. What structural aspects confer rigidity? (i)Mechanical stability (ii)small non-linear perturbation do not lead to extensive rearrangements

25. Problem of ensemble Amorphous solids are glassy. Thermal equilibrium does not apply: Principles governing ensemble of configurations visited? Edwards (granular matter): all mechanically stable states are equally likely Simple systems (colloidal glasses, packings): Marginal stability Coulomb glasses: Efros, Schklovskii Spin glasses: Thouless p-spin: Laloux, Kurchan If marginal stability applies: Governs which configurations visited Soft excitations present, affect response

26. Volume change following contact opening Force balance: : external force on particle i (coming from the wall) : magnitude of force in contact Virtual force theorem: for all displacement field Apply to

27. Soft mode No overlap (pythagoras): 12 s Soft mode equation

28. Gap distribution (rattlers are subjected to gravity): Exponents consistent with saturation Two unstable contacts per packing, Independently of system size Type one contacts are marginal force distribution Of type 1 contact: Lerner,During and Wyart, arxiv 1302.3990