Download presentation

Presentation is loading. Please wait.

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 LyLy 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 cc 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 cc 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?

Similar presentations

© 2020 SlidePlayer.com Inc.

All rights reserved.

To make this website work, we log user data and share it with processors. To use this website, you must agree to our Privacy Policy, including cookie policy.

Ads by Google