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Jamming Andrea J. Liu Department of Physics & Astronomy University of Pennsylvania Corey S. OHern Mechanical Engineering, Yale Univ. Leo E. Silbert Physics, Southern Ill. Univ. Jen M. Schwarz Physics, Syracuse Univ. Lincoln Chayes Mathematics, UCLA Sidney R. Nagel James Franck Inst., U Chicago Brought to you by NSF-DMR-0087349, DOE DE-FG02-03ER46087

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Mixed Phase Transitions Recall random k-SAT Fraction of variables that are constrained obeys Finite-size scaling shows diverging length scale at r k * Monasson, Zecchina, Kirkpatrick, Selman, Troyansky, Nature 400, 133 (1999). r rk*rk* rkrk E=0, no violated clauses E>0, violated clauses

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Mixed Phase Transitions infinite-dimensional models –p-spin interaction spin glass Kirkpatrick, Thirumalai, PRL 58, 2091 (1987). –k-core (bootstrap) Chalupa, Leath, Reich, J. Phys. C (1979); Pittel, Spencer, Wormald, J.Comb. Th. Ser. B 67, 111 (1996). –Random k-SAT Monasson, Zecchina, Kirkpatrick, Selman, Troyansky, Nature 400, 133 (1999). - etc. But physicists really only care about finite dimensions –Jamming transition of spheres OHern, Langer, Liu, Nagel, PRL 88, 075507 (2002). –Knights models Toninelli, Biroli, Fisher, PRL 96, 035702 (2006). –k-core + force-balance models Schwarz, Liu, Chayes, Europhys. Lett. 73, 560 (2006).

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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

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Glass Transition When liquid cools, stress relaxation time increases When liquid crystallizes –Particles order –Stress relaxation time suddenly jumps When liquid is cooled through glass transition –Particles remain disordered –Stress relaxation time increases continuously Picture Book of Sir John Mandevilles Travels, ca. 1410.

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A. J. Liu and S. R. Nagel, Nature 396 (N6706) 21 (1998). Jamming Phase Diagram jammed unjammed Temperature Shear stress 1/ Density J Glass transition Granular packings unjammed state is in equilibrium jammed state is out of equilibrium Problem: Jamming surface is fuzzy

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C. S. OHern, S. A. Langer, A. J. Liu and S. R. Nagel, Phys. Rev. Lett. 88, 075507 (2002). C. S. OHern, L. E. Silbert, A. J. Liu, S. R. Nagel, Phys. Rev. E 68, 011306 (2003). Point J is special –It is a point –Isostatic –Mixed first/second order zero T phase transition Point J unjammed Temperature Shear stress 1/Density J jammed soft, repulsive, finite-range spherically-symmetric potentials Model granular materials

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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 overlapped V>0 p>0 or non-overlapped V=0 p=0 Ti=Ti= T f =0

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Onset of Jamming is Onset of Overlap We focus on ensemble rather than individual configs (c.f. Torquato) Good ensemble is fixed - c, or fixed pressure Pressures for different states collapse on a single curve Shear modulus and pressure vanish at the same c D=2 D=3 - - - - - - - -4-3 -2 log( - c )

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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 Of course, there is a tail up to close-packed crystal J is a POINT w 0

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Where do virtually all states jam in infinite system limit? 2d (bidisperse) 3d (monodisperse) These are values associated with random close-packing! log( *- 0 ) Point J is at Random Close-Packing w 0

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Point J is special –It is a point –Isostatic –Mixed first/second order zero T transition Point J unjammed Temperature Shear stress 1/Density J jammed soft, repulsive, finite-range spherically-symmetric potentials

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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

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Isostaticity What is the minimum number of interparticle contacts needed for mechanical equilibrium? No friction, spherical particles, D dimensions –Match unknowns (number of interparticle normal forces) to equations (force balance for mechanical stability) –Number of unknowns per particle=Z/2 –Number of equations per particle = D Point J is purely geometrical!

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L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (05) Excess low- modes swamp 2 Debye behavior: boson peak D( ) approaches constant as c (M. Wyart, S.R. Nagel, T.A. Witten, EPL (05) ) Unusual Solid Properties Near Isostaticity Density of Vibrational Modes Lowest freq mode at - c =10 -8 c

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Point J is special –It is a point –Isostatic –Mixed first/second order zero T transition Point J unjammed Temperature Shear stress 1/Density J jammed soft, repulsive, finite-range spherically-symmetric potentials

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For each - c, extract where D( ) begins to drop off Below, modes approach those of ordinary elastic solid We find power-law scaling L. E. Silbert, A. J. Liu, S. R. Nagel, PRL 95, 098301 (2005) Is there a Diverging Length Scale at J?

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The frequency has a corresponding eigenmode Decompose eigenmode in plane waves Dominant wavevector contribution is at peak of f T (k, ) extract k*: We also expect with Frequency Scale implies Length Scale

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Summary of Jamming Transition Mixed first-order/second-order transition Number of overlapping neighbors per particle Static shear modulus Diverging length scale And perhaps also

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Consider lattice with coord. # Z max with sites indpendently occupied with probability p For site to be part of k- core, it must be occupied and have at least k=d+1 occupied neighbors Each of its occ. nbrs must have at least k occ. nbrs, etc. Look for percolation of the k-core Jamming vs K-Core (Bootstrap) Percolation Jammed configs at T=0 are mechanically stable For particle to be locally stable, it must have at least d+1 overlapping neighbors in d dimensions Each of its overlapping nbrs must have at least d+1 overlapping nbrs, etc. At c all particles in load-bearing network have at least d+1 neighbors

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K-core Percolation on the Bethe Lattice K-core percolation is exactly solvable on Bethe lattice This is mean-field solution Let K =probability of infinite k-connected cluster For k>2 we find Chalupa, Leath, Reich, J. Phys. C (1979) Pittel, et al., J.Comb. Th. Ser. B 67, 111 (1996) Recall simulation results J. M. Schwarz, A. J. Liu, L. Chayes, EPL (06)

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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 percolation (Charybdis) 2.No percolation until p=1 (Scylla) 3.Discontinuous percolation? –Yes, for k-core variants Knights models (Toninelli, Biroli, Fisher) k-core with pseudo force-balance (Schwarz, Liu, Chayes)

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Knights Model Toninelli, Biroli, Fisher, PRL 96, 035702 (2006). Rigorous proofs that –p c <1 –Transition is discontinuous* –Transition has diverging correlation length* *based on conjecture of anisotropic critical behavior in directed percolation

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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

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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

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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)

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

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Diverging Susceptibility? Yes How much is removed by the culling process?

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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).

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Similarity to Other Models The discontinuity & exponents we observe are rare but have been found in a few models –Mean-field p-spin interaction spin glass (Kirkpatrick, Thirumalai, Wolynes) –Mean-field dimer model (Chakraborty, et al.) –Mean-field kinetically-constrained models (Fredrickson, Andersen) –Mode-coupling approximation of glasses (Biroli,Bouchaud) These models all exhibit glassy dynamics!! First hint of UNIVERSALITY in jamming

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To return to beginning…. Recall random k-SAT Point J Hope you like jammin, too! r rk*rk* rkrk - c 0 E=0 E>0

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Point J is a special point Common exponents in different jamming models in mean field! But different in finite dimensions Hope you like jammin, too! Thanks to NSF-DMR-0087349 DOE DE-FG02-03ER46087 Conclusions T xy 1/ J

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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 dont exist in sphere packings

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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

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Point J and the Glass Transition Point J only exists for repulsive, finite-range potentials Real liquids have attractions Attractions serve to hold system at high enough density that repulsions come into play (WCA) Repulsion vanishes at finite distance U r

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