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

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

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

conjecture by Liu and Nagel (Nature 1998)

jamming “point J” is a special criticalpoint in a larger 3D phase diagramwith the three axes:

volume densityT 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

Jjamming

glas

s

surface below whichstates are jammed

shear stress

shear viscosity of a flowing granular material

velocity gradient

shear viscosity

expectabove jamming

below jamming

⇒ shear flow in fluid state

model granular material

bidisperse mixture of soft disks in two dimensions at T = 0equal numbers of disks with diameters d1 = 1, d2 = 1.4

for N disks in area LxLy the volume density is

interaction V(r) (frictionless)

non-overlapping ⇒ non-interacting

overlapping ⇒harmonic repulsion

r

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

dynamics

Lx

Ly

Ly

Lees-Edwards boundary conditions

create a uniform shear strain

interactions strain rate

diffusively moving particles(particles in a viscous liquid)

position particle i

particles periodicunder transformation

strain driven by uniformapplied shear stress

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 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

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

results for finite shear stress

c

c

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

J

c

control parameters

≡c ,

critical “point J”

,

bbb

we thus get the scaling law

bbb

choose length rescaling factor b ||

crossover scaling variable

crossover scaling exponent

scaling law

bbb

crossover scaling function

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 - z0|

' vanishes as z →z0 from above

(like adding small anisotropy field at a spin-flop bicritical point)

scaling collapse of viscosity

stress is arelevant variable

unclear if jamming remains at finite

point J is a true 2nd 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 rotationof regions of size

scaling collapse of correlation length

diverges at point J

phase diagram in plane

volume density

shea

r st

ress

jammed

flowing

“point J”

0 c

c

'

'

cz

critical exponents

≃

≃

≃

≃

≃

if scaling is isotropic, then expect ≃ dx/dy is dimensionless

then d ~ dimensionless ⇒ d ⇒ d

ddt)/zd = (zd) ⇒ z = + d = 4.83

where z is dynamic exponent

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?