Jamming Peter Olsson, Umeå University Stephen Teitel, University of Rochester Supported by: US Department of Energy Swedish High Performance Computing Center North Quantum Jamming in the ħ→ 0 limit

what is jamming? transition from flowing to rigid in condensed matter systems

the structural glass transition shear stress solid: shear modulus liquid: shear viscosity liquid solid glass cool T m cool T g short range correlations long range correlations ?????? correlations

the structural glass transition liquid: shear modulus shear viscosity glass: shear modulus 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 transition from flowing to rigid but disordered structure shear driven thermal fluctuations negligible critical yield stress foam has shear flow like a liquid foam ceases to flow and behaves like an elastic solid

granular materials large weakly interacting grains thermal fluctuations negligible transition from flowing to rigid but disordered structure volume density driven the jamming transition critical volume density grains flow like a liquid grains jam, a finite shear modulus develops

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) Z is average contacts per particle isostatic stability when these are equal seems well obeyed at jamming  c

  T J yield stress glass conjecture by Liu and Nagel (Nature 1998) jamming, foams, glass, all different aspects of a unified phase diagram with three axes:   temperature    volume density    applied shear stress (nonequilibrium axis) “point J ” is a critical point “the epitome of disorder” here we consider the   plane at T = 0 in 2D flowing ➝ rigid but disordered surface below which states are jammed jamming transition “point J ” critical scaling at point J influences behavior at finite T and finite . understanding  = 0  jamming at “point J ” may have implications for understanding the glass transition at finite 

shear stress  shear viscosity of a flowing granular material velocity gradient shear viscosity if jamming is like a critical point we expect above jamming below jamming ⇒ shear flow in fluid state

model granular material bidisperse mixture of soft disks in two dimensions at T = 0 equal numbers of disks with diameters d 1 = 1, d 2 = 1.4 for N disks in area L x L y the volume density is interaction V(r) (frictionless) non-overlapping ⇒ non-interacting overlapping ⇒ harmonic repulsion r (Durian, PRL 1995 (foams); O’Hern, Silbert, Liu, Nagel, PRE 2003) overdamped dynamics

L x = L y N = 1024 for  < N = 2048 for  ≥  t ~ 1/N, integrate with Heun’s method total shear displacement ~ 10, ranging from 1 to 200 depending on N and  simulation parameters finite size effects negligible (can’t get too close to  c ) animation at:  =  <  c   = 10 -5

results for small  = (represents  → 0 limit, “point J ”) as N increases,     vanishes continuously at  c ≃ smaller systems jam below  c

results for finite shear stress   c  c

scaling about “point J ” for finite shear stress  scaling hypothesis (2 nd order phase transitions) : at a 2 nd order critical point, a diverging correlation length   determines all critical behavior quantities that vanish at the critical point all scale as some power of   rescaling the correlation length,  → b , corresponds to rescaling   J cc control parameters    c ,  critical “point J ”    ,      ~ b  1/,  ~ b ,    ~ b  we thus get the scaling law

choose length rescaling factor b    crossover scaling variable crossover scaling exponent  scaling law crossover scaling function

scaling collapse of viscosity  point J is a true 2 nd order critical point

correlation length transverse velocity correlation function (average shear flow along x )  distance to minimum gives correlation length  regions separated by  are anti-correlated  motion is by rotation of regions of size 

scaling collapse of correlation length   diverges at point J

phase diagram in  plane volume density  shear stress  jammed flowing “point J ” 0 cc     c         '             '    c   z 0  

conclusions point J is a true 2 nd 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?