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. 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

4. 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 1/   T J jamming glass surface below which states are jammed

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

6. 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 (O’Hern, Silbert, Liu, Nagel, PRE 2003)

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

8. 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

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

11. 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    b      b  we thus get the scaling law    b     b   b  

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

13. 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)   1 vanishes as     ' jamming transition at ii) f + (z)  |z - z 0 |  ' vanishes as z →  z 0 from above (like adding small anisotropy field at a spin-flop bicritical point)

14. 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

15. 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 

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

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

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 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?