Critical Scaling at the Jamming Transition
description
Transcript of Critical Scaling at the Jamming Transition
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?