1. Jamming Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Corey S. O’Hern Mechanical Engineering, Yale Univ. Leo E. Silbert Physics, S. Illinois U., Carbondale Ning Xu Physics, UPenn, JFI, U. Chicago Vincenzo Vitelli Physics, UPenn Matthieu Wyart Janelia Farms; Physics, NYU Sidney R. Nagel James Franck Inst., U. Chicago

2. Umbrella concept that aims to tie together –two of oldest unsolved problems in condensed-matter physics Glass transition Colloidal glass transition –systems only recently studied by physicists Granular materials Foams and emulsions ¿Is there common behavior in these systems so that we can benefit by studying them in a broader context? Jamming

3. Stress Relaxation Time Behavior of glassforming liquids depends on how long you wait –At short time scales, silly putty behaves like a solid –At long time scales, silly putty behaves like a liquid Stress relaxation time  : how long you need to wait for system to behave like liquid Speeded up by x80

4. Glass Transition When liquid is cooled through glass transition –Particles remain disordered –Stress relaxation time increases continuously –Can get 10 orders of magnitude increase in 10- 20 K range Earliest glassmaking 3000BC Glass vessels from around 1500BC

5. Colloidal Glass Transition Suspensions of small (nm-10  m) particles include –Ink, paint –McDonald’s milk shakes, ….. –Blood Micron-sized plastic spheres suspended in water form Stress relaxation time increases with packing fraction crystals glasses

6. Granular Materials Materials made up of many distinct grains include –Pharmaceutical powders –Cereal, coffee grounds, …. –Gravel, landfill, …. San Francisco Marina District after Loma Prieta earthquake Static granular packing behaves like a solid Shaken granular packing behaves like a liquid

7. Suspension of gas bubbles or liquid droplets –Shaving cream –mayonnaise Foams flow like liquids when sheared Stress relaxation time increases as shear stress decreases Foams and Emulsions Courtesy of D. J. Durian

8. No obvious structural signature of jamming Dramatic increase of relaxation time near jamming Kinetic heterogeneities Phenomena look similar in all these systems Supercooled liquids Colloidal suspensions Courtesy of E. R. Weeks and D. A. Weitz Granular materials Courtesy of S. C. Glotzer Courtesy of A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian

9. These Transitions Are Not Understood We understand crystallization and a lot of other phase transitions –Liquid-vapor criticality, liquid crystal transitions –Superconductivity, superfluidity, Bose-Einstein cond… –Many exotic quantum transitions, etc. But glass transition, etc. remain mysterious –Are they really phase transitions or are they just examples of kinetic arrest? Why are these systems so difficult? –They are disordered –They are not in equilibrium

10. Jam ( ), v. i. 1 To develop a yield stress in a disordered system 2 To have a stress relaxation time that exceeds 10 3 s in a disordered system E.g. Supercooled liquids jam as temperature drops Colloidal suspensions jam as density -1 drops Granular materials jam as driving force drops Foams, emulsions jam as shear stress drops ¿Can we unify these systems within one framework? Jamming glass transition colloidal glass transition elastoplasticity

11. A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21 (1998). Jamming Phase Diagram Glass transition Colloidal glass trans. Granular matter Foams and emulsions

12. Exp’tal Jamming Phase Diagram V. Trappe, V. Prasad, L. Cipelletti, P. N. Segre, D. A. Weitz, Nature, 411(N6839) 772 (2001). Colloids with depletion attractions

13. C. S. O’Hern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 88, 075507 (2002). C. S. O’Hern, L. E. Silbert, A. J. Liu, S. R. Nagel, Phys. Rev. E 68, 011306 (2003). Problem: Jamming surface is fuzzy Point J is special –Random close-packing –Isostatic –Mixed first/second order zero T transition –Connections to glasses and glass transition Point J unjammed Temperature Shear stress 1/Density J jammed soft, repulsive, finite-range spherically-symmetric potentials

14. Generate configurations near J –e.g. Start w/ random initial positions –Conjugate gradient energy minimization ( inherent structures, Stillinger & Weber) Classify resulting configurations How we study Point J T i =∞ overlapped V>0 p>0 or non-overlapped V=0 p=0 T f =0

15. Onset of Jamming is Onset of Overlap Pressures for different states collapse on a single curve Shear modulus and pressure vanish at the same  c Good ensemble is fixed  -  c D=2 D=3 - - - - - - - -4-3 -2 log(  -  c ) D. J. Durian, PRL 75, 4780 (1995); C. S. O’Hern, S. A. Langer, A. J. Liu, S. R. Nagel, PRL 88, 075507 (2002).

16. Dense Sphere Packings ¿What is densest packing of monodisperse hard spheres? Johannes Kepler (1571-1630) Conjecture (1611) Thomas Hales 3D Proof (1998) FCC/HCP is densest possible packing 3D 2D triangular is densest possible packing Fejes Tóth 2D Proof (1953)

17. Disordered Sphere Packings Stephen Hales (1677-1761) Vegetable Staticks (1727) J. D. Bernal (1901-1971) 2D 3D Random close-packing is not well-defined mathematically –One can always make a closer-packed structure that is less random S. Torquato, T. M. Truskett, P. Debenedetti, PRL 84, 2064 (2000). –But it is highly reproducible. Why? Kamien, Liu, PRL 99, 155501 (2007). < <

18. How Much Does  c Vary Among States? Distribution of  c values narrows as system size grows Distribution approaches delta-function as N Essentially all configurations jam at one packing density J is a “POINT” w 00

19. Where do virtually all states jam in infinite system limit? 2d (bidisperse) 3d (monodisperse) Most of phase space belongs to basins of attraction of hard sphere states that have their jamming thresholds at RCP log(  *-  0 ) Point J is at Random Close-Packing RCP! w 00

20. Point J is special –Random close-packing –Isostatic –Mixed first/second order zero T transition –Connections to glasses and glass transition Point J unjammed Temperature Shear stress 1/Density J jammed soft, repulsive, finite-range spherically-symmetric potentials

21. Number of Overlaps/Particle Z (2D) (3D) log(  -  c ) - - - - - - - Just below  c, no particles overlap Just above  c there are Z c overlapping neighbors per particle Verified experimentally: Majmudar, Sperl, Luding, Behringer, PRL 98, 058001 (2007). Durian, PRL 75, 4780 (1995). O’Hern, Langer, Liu, Nagel, PRL 88, 075507 (2002).

22. No friction, spherical particles, D dimensions –Match unknowns (number of interparticle normal forces) equations (force balance for mechanical stability) –Number of unknowns per particle=Z/2 –Number of equations per particle = D Isostaticity What is the minimum number of interparticle contacts needed for mechanical equilibrium? Same for hard spheres at RCP Donev, Torquato, Stillinger, PRE 71, 011105 (‘05) Point J is purely geometrical! Doesn’t depend on potential James Clerk Maxwell

23. L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05) Excess low-  modes swamp  2 Debye behavior: boson peak g(  ) approaches constant as   c Result of isostaticity M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05) Marginally Jammed Solid is Unusual Density of Vibrational Modes  c

24. Isostaticity and Boundary Sensitivity For system at  c, Z=2d Removal of one bond makes entire system unstable by introducing one soft mode This implies diverging length as  ->  c + For  >  c, cut bonds at boundary of circle of size L Count number of soft modes within circle Define length scale at which soft modes just appear M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05)

25. Diverging Length Scale Ellenbroek, Somfai, van Hecke, van Saarloos, PRL 97, 258001 (2006) Look at response to small particle displacement Define

26. For each  -  c, extract   where g(  ) begins to drop off Below  , modes approach those of ordinary elastic solid Decompose corresponding eigenmode in plane waves Dominant wavevector contribution is k*=  */c T We also expect with Diverging Time and Length Scales 

27. Point J is special –Random close-packing –Isostatic –Mixed first/second order zero T transition –Connection to glasses and glass transition Point J unjammed Temperature Shear stress 1/Density J jammed soft, repulsive, finite-range spherically-symmetric potentials

28. Summary of Jamming Transition Mixed first-order/second-order transition (random first-order phase transition) Number of overlapping neighbors per particle Static shear modulus Two diverging length scales Vanishing frequency scale

29. Similarity to Other Models In jamming transition we find –Jump discontinuity &  =1/2 power-law in order parameter –Divergences in susceptibility/correlation length with  =1/2 and =1/4 This behavior has only been found in a few models –Mean-field p-spin interaction spin glass Kirkpatrick, Wolynes –Mean-field compressible frustrated Ising antiferromagnet Yin, Chakraborty –Mean-field kinetically-constrained Ising models Sellitto, Toninelli, Biroli, Fisher –Mean-field k-core percolation and variants Schwarz, Liu, Chayes –Mode-coupling approximation of glasses Biroli, Bouchaud –Replica solution of hard spheres Zamponi, Parisi These other models all exhibit glassy dynamics!! First hint of quantitative connection between sphere packings and glass transition

30. Point J is special –Random close-packing –Isostatic –Mixed first/second order zero T transition –Connection to glasses and glass transition Point J unjammed Temperature Shear stress 1/Density J jammed soft, repulsive, finite-range spherically-symmetric potentials

31. Low Temperature Properties of Glasses crystal T amorphous Distinct from crystals Common to all amorphous solids Still mysterious –Excess vibrational modes compared to Debye (boson peak) –C v ~T instead of T 3 (two-level systems) –  ~T 2 instead of T 3 at low T (TLS) –K has plateau –K increases monotonically  c

32. Energy Transport P. B. Allen and J. L. Feldman, PRB 48,12581 (1993). thermal conductivity heat carried by mode i diffusivity of mode i Kubo formulation N. Xu, V. Vitelli, M. Wyart, A. J. Liu, S. R. Nagel (2008). Kittel’s 1949 hypothesis: rise in  above plateau due to regime of freq-independent diffusivity

33. Crossover from weak to strong scattering at  IR  IR ~  * Ioffe-Regel crossover at boson peak Unambiguous evidence of freq-indep diffusivity as hypothesized for glasses Freq-indep diffusivity originates from soft modes at J! Ioffe-Regel Crossover

34. Quasilocalized Modes Modes become quasilocalized near Ioffe- Regel crossover Quasilocalization due to disorder in coordination z Harmonic precursors of two-level systems?

35. Relevance to Glasses Point J only exists for repulsive, finite-range potentials Real liquids have attractions Excess vibrational modes (boson peak) believed responsible for unusual low temp properties of glasses These modes derive from the excess modes near Point J Repulsion vanishes at finite distance U r Attractions serve to hold system at high enough density that repulsions come into play (WCA) N. Xu, M. Wyart, A. J. Liu, S. R. Nagel, PRL 98, 175502 (2007).

36. Glass Transition L.-M. Martinez and C. A. Angell, Nature 410, 663 (2001). Would expect Arrhenius behavior But most glassforming liquids obey something like T 0 measures “fragility”

37. T 0 (p) is Linear unjammed Temperature Shear stress 1/Density J jammed 3 different types of trajectories to glass transition –Decrease T at fixed  –Decrease T at fixed p –Increase p at fixed T 4 different potentials –Harmonic repulsion –Hertzian repulsion –Repulsive Lennard-Jones (WCA) –Lennard-Jones All results fall on consistent curve! T 0 -> 0 at Point J!

38. Experimental Data for Glycerol K. Z. Win and N. Menon

39. Point J is a special point First hint of universality in jamming transitions Tantalizing connections to glasses and glass transition Looking for commonalities can yield insight Physics is not just about the exotic; it is all around you Hope you like jammin’, too! --Bob Marley Bread for Jam: NSF-DMR-0605044 DOE DE-FG02-03ER46087 Conclusions T  xy 1/  J

40. Imry-Ma-Type Argument M. Wyart, Ann. de Phys. 30 (3), 1 (2005). Upper critical dimension for jamming transition may be 2 Recall soft-mode-counting argument Now include fluctuations in Z This would explain –Observed exponents same in d=2 and d=3 –Similarity to mean-field k-core exponents* *k-core percolation has different behavior in d=2 J. M. Schwarz, A. J. Liu, L. Chayes, EPL 73, 560 (2006) C. Toninelli, G. Biroli, D. S. Fisher, PRL 96,035702 (2006)

41. Participation ratio Visualize 2D displacement vectors in mode i at atom Nature of Vibrational Modes

42. localized Nature of Vibrational Modes

43. localized disturbances merging Nature of Vibrational Modes

44. extended Nature of Vibrational Modes

45. wave-like Nature of Vibrational Modes

46. resonant Characterize modes in different portions of spectrum. Nature of Vibrational Modes

47. Stressed Unstressed replace compressed bonds with relaxed springs. coordination number Low resonant modes have high displacements on under-coordinated particles. at low increases weakly as Mode Analysis of 3D Jammed packings

48. How Localized is the Lowest Frequency Mode? Mode is more and more localized with increasing  Two-level systems and STZ’s

49. is O(1) for ordinary solids at low Gruneisen parameters Compression causes increase in stress consistent with scaling of Stronger anharmonicity at lower frequencies; two-level systems? Strong Anharmonicity at Low Frequency

50. Kinetic theory of phonons in crystals: mean free path speed of sound heat capacity energy diffusivity Thermal conductivity waves of frequency incident on a random distribution of scatterers (ignore phonon interactions) eg. Rayleigh scattering Gedanken experiment low T sum over all modes Thermal Diffusivity

51. Kubo formula for amorphous solids: linear response theory & harmonic approx. Energy flux matrix elements calculated directly from eigenmodes and eigenvalues P. B. Allen and J. L. Feldman, PRB 48,12581 (1993). Diffusivity an intrinsic property of the modes and their coupling independent of temperature AC conductivity Extract Diffusivity from double limit: N 0 J. D. Thouless, Phys. Rep. 13, 3, 93 (1974). Calculation of Diffusivity

52. Above  L modes are localized and diffusivity vanishes Most modes are extended with low diffusivity Plateau extends all the way to low  in DC limit Stressed Unstressed replace compressed bonds with relaxed springs Diffusivity of Jammed Packings

53. Scaling Collapse P. Olsson and S. Teitel, cond-mat/07041806 Measure shear viscosity, length scale vs. shear stress Scaling collapse of all data Two branches: one below J, one above Power-law scaling at Pt. J T  xy 1/  J

54. K-Core Percolation in Finite Dimensions There appear to be at least 3 different types of k-core percolation transitions in finite dimensions 1.Continuous transition (Charybdis) 2.No percolation until p=1 (Scylla) 3.Mixed transition?

55. Continuous K-Core Percolation Appears to be associated with self-sustaining clusters For example, k=3 on triangular lattice p c =0.6921±0.0005, M. C. Madeiros, C. M. Chaves, Physica A (1997). p=0.4, before culling p=0.4, after culling p=0.6, after culling p=0.65, after culling Self-sustaining clusters don’t exist in sphere packings

56. E.g. k=3 on square lattice There is a positive probability that there is a large empty square whose boundary is not completely occupied After culling process, the whole lattice will be empty Straley, van Enter J. Stat. Phys. 48, 943 (1987). M. Aizenmann, J. L. Lebowitz, J. Phys. A 21, 3801 (1988). R. H. Schonmann, Ann. Prob. 20, 174 (1992). C. Toninelli, G. Biroli, D. S. Fisher, Phys. Rev. Lett. 92, 185504 (2004). No Transition Until p=1 Voids unstable to shrinkage, not growth in sphere packings

57. A k-Core Variant We introduce “force-balance” constraint to eliminate self-sustaining clusters Cull if k<3 or if all neighbors are on the same side k=3 24 possible neighbors per site Cannot have all neighbors in upper/lower/right/left half

58. The discontinuity  c increases with system size L If transition were continuous,  c would decrease with L Discontinuous Transition? Yes Fraction of sites in spanning cluster

59. Pc<1? Yes Finite-size scaling If p c = 1, expect p c (L) = 1-Ae -BL Aizenman, Lebowitz, J. Phys. A 21, 3801 (1988) We find We actually have a proof now that p c <1 (Jeng, Schwarz)

60. Diverging Correlation Length? Yes This value of collapses the order parameter data with For ordinary 1st-order transition,

61. BUT Exponents for k-core variants in d=2 are different from those in mean-field! Mean fieldd=2 Why does Point J show mean-field behavior? Point J may have critical dimension of d c =2 due to isostaticity (Wyart, Nagel, Witten) Isostaticity is a global condition not captured by local k- core requirement of k neighbors Henkes, Chakraborty, PRL 95, 198002 (2005).