Post on 17-Jan-2016
description
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?