Quantum Jamming in the 0 limit Jamming Peter Olsson, Ume University Stephen Teitel, University of Rochester Supported by: US Department of Energy Swedish High Performance Computing Center North what is jamming? QuickTime and a TIFF (Uncompressed) decompressor are needed to see this picture. transition from flowing to rigid

in condensed matter systems the structural glass transition QuickTime and and aa QuickTime None decompressor decompressor None are needed needed to to see see this this picture. picture. are QuickTime and a

None decompressor are needed to see this picture. cool Tm cool Tg liquid short range correlations solid long range correlations

shear stress solid: shear modulus QuickTime QuickTime and and aa None None decompressor decompressor are are needed needed to to see see this this picture. picture. liquid:

shear viscosity glass ?????? correlations the structural glass transition liquid: glass: shear modulus shear modulus shear viscosity shear viscosity

glass transition viscosity diverges equilibrium transition? (diverging length scale) dynamic transition? (diverging time scale) no transition? (glass is just slow liquid) one of the greatest unresolved problems of condensed matter physics transition from flowing to rigid but disordered structure thermally driven sheared foams polydisperse densely packed gas bubbles thermal fluctuations negligible critical yield stress foam has shear flow like a liquid foam ceases to flow and

behaves like an elastic solid transition from flowing to rigid but disordered structure shear driven granular materials large weakly interacting grains thermal fluctuations negligible critical volume density grains flow like a liquid grains jam, a finite shear modulus develops the jamming transition transition from flowing to rigid but disordered structure volume density driven This false color image is taken from Dan Howell's experiments.

This is a 2D experiment in which a collection of disks undergoes steady shearing. The red regions mean large local force, and the blue regions mean weak local force. The stress chains show in red. The key point is that on at least the scale of this experiment, forces in granular systems are inhomogeneous and itermittent if the system is deformed. We detect the forces by means of photoelasticity: when the grains deform, they rotate the polarization of light passing through them. Howell, Behringer, Veje, PRL 1999 and Veje, Howell, Behringer, PRE 1999 isostatic limit in d dimensions number of contacts: number of force balance equations: Nd (for repulsive frictionless particles)

isostatic stability when these are equal Z is average contacts per particle seems well obeyed at jamming c flowing rigid but disordered conjecture by Liu and Nagel (Nature 1998) jamming, foams, glass, all different aspects of a unified phase diagram with three axes: T

glas s temperature volume density applied shear stress surface below which states are jammed (nonequilibrium axis) yield stre

point J is a critical point the epitome of disorder critical scaling at point J influences behavior at finite T and finite . J ss jamming transition point J understanding = 0 jamming at point J may have implications for understanding the glass transition at finite

here we consider the plane at T = 0 in 2D shear viscosity of a flowing granular material shear stress shear flow in fluid state velocity gradient shear viscosity if jamming is like a critical point we expect below jamming above jamming model granular material (Durian, PRL 1995 (foams); OHern, Silbert, Liu, Nagel, PRE 2003)

bidisperse mixture of soft disks in two dimensions at T = 0 equal numbers of disks with diameters d1 = 1, d2 = 1.4 for N disks in area LxLy the volume density is interaction V(r) (frictionless) non-overlapping non-interacting overlapping harmonic repulsion r overdamped dynamics simulation parameters Lx = Ly N = 1024 for < 0.844

finite size effects negligible (cant get too close to c) N = 2048 for 0.844 t ~ 1/N, integrate with Heuns method total shear displacement ~ 10, ranging from 1 to 200 depending on N and animation at: = 0.830 0.838 < c 0.8415 = 10-5 results for small = 10-5 (represents 0 limit, point J) as N increases, vanishes continuously at c 0.8415 smaller systems jam below c results for finite shear stress

c c scaling about point J for finite shear stress critical point J control parameters c , nd , scaling hypothesis (2 order phase transitions):

J c at a 2nd order critical point, a diverging correlation length determines all critical behavior quantities that vanish at the critical point all scale as some power of b, corresponds to rescaling ~ b , ~ b rescaling the correlation length,

~ b1/ , we thus get the scaling law scaling law choose length rescaling factor crossover scaling variable crossover scaling exponent b crossover scaling function

scaling collapse of viscosity point J is a true 2nd order critical point correlation length transverse velocity correlation function (average shear flow along x) regions separated by are anti-correlated distance to minimum gives correlation length

motion is by rotation of regions of size scaling collapse of correlation length diverges at point J shear stress phase diagram in plane ' flowing '

jammed c 0 c point J cz0 volume density conclusions point J is a true 2nd order critical point critical scaling extends to non-equilibrium driven steady states at finite shear stress

in agreement with proposal by Liu and Nagel correlation length diverges at point J diverging correlation length is more readily observed in driven non-equilibrium steady state than in equilibrium state finite temperature?