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Critical Scaling at the Jamming Transition Peter Olsson, Umeå University Stephen Teitel, University of Rochester Supported by: US Department of Energy Swedish High Performance Computing Center North

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outline introduction - jamming phase diagram our model for a granular material simulations in 2D at T = 0 scaling collapse for shear viscosity dynamic correlation length conclusions

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granular materials large grains ⇒ T= 0 sheared foams polydisperse densely packed gas bubbles structural glass upon increasing the volume density of particles above a critical value the sudden appearance of a finite shear stiffness signals a transition from a flowing state to a rigid but disordered state - this is the jamming transition “point J ” upon decreasing the applied shear stress below a critical yield stress, the foam ceases to flow and behaves like an elastic solid upon decreasing the temperature, the viscosity of a liquid grows rapidly and the liquid freezes into a disordered rigid solid animations from Leiden granular group website flowing ➝ rigid but disordered

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conjecture by Liu and Nagel (Nature 1998) jamming “point J ” is a special critical point in a larger phase diagram with the three axes: volume density T temperature applied shear stress (nonequilibrium axis) understanding T = 0 jamming at “point J ” in granular materials may have implications for understanding the structural glass transition at finite T here we consider the plane at T = 0 in 2D 1/ T J jamming glass surface below which states are jammed

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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

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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)

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dynamics LxLx LyLy LyLy Lees-Edwards boundary conditions create a uniform shear strain interactionsstrain rate diffusively moving particles (particles in a viscous liquid) strain driven by uniform applied shear stress

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L x = L y N = 1024 for < 0.844 N = 2048 for ≥ 0.844 t ~ 1/N, integrate with Heun’s method (t total ) ~ 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: = 0.830 0.838 c 0.8415 = 10 -5

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results for small = 10 -5 (represents → 0 limit, “point J”) as N increases, -1 ( ) vanishes continuously at c ≃ 0.8415 smaller systems jam below c

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results for finite shear stress c c c c

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scaling about “point J” for finite shear stress scaling variables and J cc choose length rescaling factor crossover scaling function crossover scaling variablecrossover scaling exponent Scaling hypothesis (in analogy to 2 nd order equilibrium phase transitions): if we rescale lengths by a factor b, will scale with and according to

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scaling collapse of viscosity stress is a relevant variable unclear if jamming remains at finite point J is a true 2 nd order critical point

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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

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scaling collapse of correlation length diverges at point J

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phase diagram in plane volume density shear stress jammed flowing “point J ” 0 cc c ' ' c z

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conclusions point J is a true 2 nd order critical point correlation length diverges at point J critical scaling extends to non-equilibrium driven steady states at finite shear stress in agreement with proposal by Liu and Nagel shear stress is a relevant variable that changes the critical behavior at point J jamming transition at finite remains to be clarified finite temperature?

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