1. Marginally Jammed Solids: density of states, diffusion and thermal transport Leiden -- August 26, 2008 Jamming produces rigid material with particle motion restricted by geometrical constraints. What are properties of such solids?

2. Control parameters for rigidity Yield stress J (finite-range, repulsive, spheres) 1/ density T Shear stress Jammed Glass transition Phase boundaries are fuzzy - depend on waiting time. Point J is sharp. What are its properties? Liu

3. New Physics at Point J Mixed-order phase transition Discontinuous order parameter Scaling in other properties Multiple length scales (w/ different scaling exponents) New class of vibration modes Anomalous shear vs. bulk modulus Diffusive energy transport

4. New Physics at Point J But it is highly idealized Neglects: long-range interactions (Xu, Wyart, Liu,) non-spherical particles finite temperature finite shear stress (Olsen & Teitel, Haxton & Liu, van Hecke…) three-body interactions attractive interactions friction (Somfai, van Saarloos et al.) Our program: how is physics altered by their inclusion? Is point J relevant to real systems (i.e., glasses, colloids,…)?

5. Collaborators Andrea J. Liu U. Pennsylvania Corey S. O’Hern UCLA; U. Chicago (now Yale) Leo E. Silbert U. Chicago; UCLA (now SIU) Vincenzo Vitelli U. Pennsylvania Tom Witten U. Chicago Matthieu Wyart U. Chicago; Saclay (now Harvard) Ning Xu U. Chicago; U. Pennsylvania Zorana Zeravcic University of Leiden ITP - Santa Barbara 1997 DOE; NSF-MRSEC

6. Simulations Choose V(r), dimension, dispersity V(r) = ( V 0 /  )(  - r )   r <   = 0 Hard sphere 0 r ≥    = 2 Harmonic  = 2.5 Hertz Generate configurations at fixed density Random ( i.e., T i = ∞ ) initial positions Quench to T = 0 ( inherent structures, Stillinger & Weber) Classify T = 0 configurations: After each perturbation quench to T = 0   time = ∞ T i =∞ Jammed overlaps: V, P > 0 or Unjammed no-overlaps: V, P= 0

7. Isostatic State Minimum number of overlaps needed for mechanical stability Match unknowns (# interparticle normal forces) to equations Frictionless spheres in D dimensions: Number of unknowns per particle = Z/2 Number of equations per frictionless sphere = D   Z c = 2D (Friction or elliptic shape changes Z c ) - - - - - - - - - - - - - - - - - - - - - - We find:Z c = 3.99 ± 0.01 (2D) Z c = 5.97 ± 0.03 (3D) Maxwell criterion for rigidity: global condition - not local. Physics related to connectivity ( like Thorpe, Phillips, Alexander) O’Hern, Liu

8. Implications for structure g(r) diverges at r =   as       (also split, singular, second peak) Length scale (separations of particles)  0 Silbert, Liu r/    -function with 2D neighbors g(r) Slope = -1.0 Slope = 1.0 Height Width  W L g(r peak ) 10 -8 10 -7 10 -6 10 -5 10 -4 10 -3 10 -2 10 8 10 6 10 4 10 2 10 -3 10 -5 10 -7 10 -9

9.  c onset of jamming V(r) µ (  - r)  D = 2, D = 3 See also: D. Durian PRE (1997) -------- ------------ log (  –  c ) - - - -   ≈  - 1   ≈  - 1.5   ≈ 0.5 Pressure and shear modulus   0 at same  c Expected Anomalous  Z scales, (Z discontinuous) log Shear Modulus, G log Pressure c T ~  G /   0.5 ~    (harmonic) O’Hern, Liu

10. Transition: Discontinuous order parameter: # of overlaps Continuous scaling: pressure, shear modulus, extra contacts grow smoothly from 0 above transition Neither first-order nor critical - a mixed transition Schwartz, Liu, & Chayes; Henkes & Chakraborty; Toninelli, Biroli, & Fisher

11. Properties of marginally jammed solid Normal modes in “normal” solids Low-frequency normal modes are long wavelength plane waves. Long wavelengths “average” over disorder. Density of modes, D(  ), from counting waves. D(  )    d-1 in d-dimensions. All solids should behave this way. D(  )  D(  )   2 in 3-D

12. **  *  (   -  c ) 0.49 - c- c Density of states near jamming At threshold no Debye behavior! Excitations not plane waves.  Silbert, Liu

13. Why constant D   ) at jamming? Isostatic system  just enough contacts to stabilize system. Remove one contact  one mode with  = 0. Mode is extended. Remove n contacts  n modes with  = 0. Isostatic Start with periodic boundary conditions Cut at boundary System has n modes with  = 0 Wyart, Witten

14. Construct low-  modes from soft modes Gaps only at broken contacts Even out gaps:  R* 1,i =  R 1,i sin(2  X i /L) New modes have frequency:  ~ 1/L (like lowest acoustic modes) Wyart, Witten How high can  = 0 mode go? restore boundary n modes with  = 0

15. How many slow modes? N(L) ~ L D-1 floppy modes (from cutting boundaries)  ~ 1/L(from evening out gaps) D(  ) ~ N / (  L D ) ~ L 0 (a large constant) Continue dividing system: L/2, L/4,... until atom scale   D(  ) ~ constant -------------------------------------- Higher density: L*   (z-z c ) - 1  (   -  c ) - 0.5  *   (z-z c ) -------------------------------------- Can include “weak” bonds (similar to Thorpe and Phillips)   applicable to Lennard-Jones glasses (small parameter allows using isostatic state as starting point for high coordination glasses with long-range interactions) Wyart, Witten

16. Ellipsoids Z c = 10 (not 6) because 5 degrees of freedom/particle Nearly spherical particles have Z ~ 6  Many unconstrained modes Zorana Zeravcic, N. Xu, A. Liu See also: Donev et al. PRE 2007; Science 2004 (hard spheres) Corey O’Hern in 2D. 3D Gay-Berne potential N = 512 harmonic interaction

17. Normal mode under compression (2D visualization)  = 0.1 N. Xu, V. Vitelli, M. Wyart, A. Liu               p  = D(  )

18. Thermal transport and diffusivity N. Xu, V. Vitelli, M. Wyart, A. Liu Allen and Feldman PRB 1993 At transition

19. Transport in compressed packs d(  )  c T l /3 = d 0 ( l = mean free path)  = c T k (for transverse modes below  *) K l ~ 1 (Ioffe-Regel limit)  IR = c T 2 /3d 0     0.5 ~  N. Xu, V. Vitelli, M. Wyart, A. Liu   

20. What is left out of physics at Point J? Long range interactions (Xu et al.) Friction (Somfai et al.) Non-spherical shapes (Zeravcic et al.) Temperature (Yodh et al.) Finite shear stress Attractions 3-body interactions Extended studies Boson peak for Lennard-Jones glasses. Small parameter allows using isostatic state as starting point even for high coordination glasses. (N. Xu, M. Wyart, A. Liu) Leiden group (Somfai et al.) has extended results to frictional interactions. Yodh et al. measure how one of length scales (from g(r)) varies with T.

21. So, where are we? Transition is special - mixed character, multiple length scales: universal jump in Z  scaling of P, G,  Z with  Marginally jammed solid - no plane waves, no Debye behavior. new class of modes Low-frequency modes highly resonant - Origin of STZs? Adding ellipticity (or friction, long-range ) leaves structure intact Low-T properties of glasses reflect jamming transition at T = 0. D(  0 ) = constant Singularities in g(r) Localized rearrangements at instabilities Heat transport Different ways of looking at modes are related: Diffusion (  IR ) ~ Anomalous modes (  *)