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Jamming

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

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Jamming

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  1. Jamming Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Corey S. O’Hern Mechanical Engineering, Yale Univ. Leo E. SilbertPhysics, S. Illinois U., Carbondale Ning Xu Physics,UPenn, JFI,U.Chicago Vincenzo VitelliPhysics, 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 t: 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-10mm) particles include Ink, paint McDonald’s milk shakes, ….. Blood Micron-sized plastic spheres suspended in water form Stress relaxation time increases with packing fraction glasses crystals

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

  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 Granular materials Supercooled liquids Colloidalsuspensions Courtesy of E. R. Weeks and D. A. Weitz Courtesy of A. S. Keys, A. R. Abate, S. C. Glotzer, and D. J. Durian Courtesy of S. C. Glotzer

  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. To develop a yield stress in a disordered system To have a stress relaxation time that exceeds 103 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 glasstransition colloidal glass transition elastoplasticity

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

  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 isfuzzy Point J is special Random close-packing Isostatic Mixed first/second order zero T transition Connections to glasses and glass transition Point J Temperature unjammed Shear stress jammed J 1/Density 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 non-overlapped V=0 p=0 overlapped V>0 p>0 or Tf=0 Tf=0 Ti=∞

  15. Onset of Jamming is Onset of Overlap - - - - - - - -4 -3 -2 log(- c) D=2 D=3 • Pressures for different states collapse on a single curve • Shear modulus and pressure vanish at the same fc • Good ensemble is fixed f - fc 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) Fejes Tóth 2D Proof (1953) triangular is densest possible packing 2D FCC/HCP is densest possible packing 3D

  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 randomS. 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 fc Vary Among States? Distribution of fc 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 f0

  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 Point J is at Random Close-Packing log(f*- f0) w f0 RCP!

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

  21. Number of Overlaps/Particle Z (2D) (3D) - - - - - - - log(f- fc) Just abovefc there are Zc overlapping neighbors per particle Just below fc, no particles overlap 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. 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 • 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 James Clerk Maxwell

  23. L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (‘05) Excess low-w modes swamp w2 Debyebehavior: boson peak g(w) approaches constant as f fc Result of isostaticityM. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05) Marginally Jammed Solid is Unusual Density of Vibrational Modes f- fc

  24. Isostaticity and Boundary Sensitivity M. Wyart, S.R. Nagel, T.A. Witten, EPL 72, 486 (05) • 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

  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 f-fc, extract w* where g(w) begins to drop off Below w* , modes approach those of ordinary elastic solid Decompose corresponding eigenmode in plane waves Dominant wavevector contribution is k*=*/cT We also expect with Diverging Time and Length Scales w

  27. Point J is special Random close-packing Isostatic Mixed first/second order zero T transition Connection to glasses and glass transition Point J Temperature unjammed Shear stress jammed J 1/Density 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 & b=1/2 power-law in order parameter Divergences in susceptibility/correlation length with g=1/2 and n=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 Temperature unjammed Shear stress jammed J 1/Density soft, repulsive, finite-range spherically-symmetric potentials

  31. Low Temperature Properties of Glasses Distinct from crystals Common to all amorphous solids Still mysterious Excess vibrational modes compared to Debye (boson peak) Cv~T instead of T3 (two-level systems) ~T2 instead of T3 at low T(TLS) K has plateau K increases monotonically crystal c amorphous T

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

  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 U Repulsion vanishes at finite distance • Attractions serve to hold system at high enough density that repulsions come into play (WCA) r N. Xu, M. Wyart, A. J. Liu, S. R. Nagel, PRL 98, 175502 (2007).

  36. Glass Transition Would expect Arrhenius behavior But most glassforming liquids obey something like T0 measures “fragility” L.-M. Martinez and C. A. Angell, Nature 410, 663 (2001).

  37. T0(p) is Linear 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! T0 -> 0 at Point J! Temperature unjammed Shear stress jammed J 1/Density

  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 sxy 1/r 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. Visualize 2D displacement vectors in mode i at atom Nature of Vibrational Modes Participation ratio

  42. Nature of Vibrational Modes localized

  43. Nature of Vibrational Modes localized disturbances merging

  44. Nature of Vibrational Modes extended

  45. Nature of Vibrational Modes wave-like

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

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

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

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

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

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