Critical Scaling at the Jamming Transition

1. 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

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

3. flowing ➝ rigid but disordered granular materials large grains ⇒ T= 0 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” sheared foamspolydisperse densely packed gas bubbles upon decreasing the applied shear stress below a critical yield stress, the foam ceases to flow and behaves like an elastic solid structural glass 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

4. T glass  jamming J 1/ surface below which states are jammed conjecture by Liu and Nagel (Nature 1998) jamming “point J” is a special critical point in a larger 3D 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

5. ⇒ shear flow in fluid state shear stress  velocity gradient shear viscosity below jamming expect above jamming shear viscosity of a flowing granular material

6. for N disks in area LxLy the volume density is non-overlapping ⇒ non-interacting overlapping ⇒ harmonic repulsion r model granular material (O’Hern, Silbert, Liu, Nagel, PRE 2003) bidisperse mixture of soft disks in two dimensionsat T = 0 equal numbers of disks with diameters d1 = 1, d2 = 1.4 interaction V(r)(frictionless)

7. Lees-Edwards boundary conditions create a uniform shear strain  Ly Ly position particle i particles periodic under transformation Lx diffusively moving particles (particles in a viscous liquid) interactions strain rate strain  driven by uniform applied shear stress  dynamics

8. simulation parameters Lx = Ly N = 1024 for < 0.844 N = 2048 for ≥ 0.844 t ~ 1/N, integrate with Heun’s method (ttotal) ~ 10, ranging from 1 to 200 depending onNand finite size effects negligible (can’t get too close to c) animation at: = 0.830  0.838 c ≃ 0.8415 = 10-5

9. 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

10. c c results for finite shear stress 

11. critical “point J” control parameters  J ≡c ,   ,   c we thus get the scaling law bbb scaling about “point J” for finite shear stress  scaling hypothesis (2nd order phase transitions): 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  rescaling the correlation length,  → b, corresponds to rescaling bbb

12. crossover scaling function crossover scaling variable crossover scaling exponent  scaling law bbb choose length rescaling factor b ||

13. jamming transition at possibilities  0 stress  is irrelevant variable  jamming at finite  in same universality class as point J (like adding a small magnetic field to an antiferromagnet)  0 stress  is relevant variable  jamming at finite  in different universality class from point J i) f(z) vanishes only at z 0 finite  destroys the jamming transition (like adding a small magnetic field to a ferromagnet) ii) f+(z)|z - z0|' vanishes as z →z0 from above 1 vanishes as ' (like adding small anisotropy field at a spin-flop bicritical point)

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

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

16. scaling collapse of correlation length   diverges at point J

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

18. critical exponents ≃  ≃  ≃ ≃  ≃ if scaling is isotropic, then expect ≃dx/dy is dimensionless then d ~ dimensionless ⇒ d⇒ d ddt)/zd = (zd) ⇒ z =  + d = 4.83 where z is dynamic exponent

19. conclusions • point J is a true 2nd 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?