Transcript PowerPoint Presentation - Critical Scaling at the Jamming

Jamming
Quantum Jamming
in the ħ→ 0 limit
Peter Olsson, Umeå University
Stephen Teitel, University of Rochester
Supported by:
US Department of Energy
Swedish High Performance Computing Center North
what is jamming?
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transition from flowing to rigid
in condensed matter systems
the structural glass transition
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cool
Tm
cool
Tg
liquid
short range
correlations
solid
long range
correlations
shear stress
solid:
shear modulus
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liquid:
shear viscosity
glass
??????
correlations
the structural glass transition
liquid:
glass:
shear modulus
shear modulus
shear viscosity
shear viscosity
glass transition
viscosity diverges
equilibrium transition? (diverging length scale)
dynamic transition? (diverging time scale)
no transition? (glass is just slow liquid)
one of the greatest unresolved problems
of condensed matter physics
transition from flowing to rigid but disordered structure
thermally driven
sheared foams
polydisperse densely packed gas bubbles
thermal fluctuations negligible
critical yield stress
foam has shear flow
like a liquid
foam ceases to flow and
behaves like an elastic solid
transition from flowing to rigid but disordered structure
shear driven
granular materials
large weakly interacting grains
thermal fluctuations negligible
critical volume density
grains flow like a liquid
grains jam, a finite shear
modulus develops
the jamming transition
transition from flowing to rigid but disordered structure
volume density driven
This false color image is taken from Dan Howell's experiments.
This is a 2D experiment in which a collection of disks undergoes
steady shearing. The red regions mean large local force, and the
blue regions mean weak local force. The stress chains show in red.
The key point is that on at least the scale of this experiment, forces
in granular systems are inhomogeneous and itermittent if the system
is deformed. We detect the forces by means of photoelasticity: when
the grains deform, they rotate the polarization of light passing through
them.
Howell, Behringer, Veje, PRL 1999 and Veje, Howell, Behringer, PRE 1999
isostatic limit in d dimensions
number of contacts:
number of force balance equations: Nd
(for repulsive frictionless particles)
isostatic stability when these are equal
Z is average contacts
per particle
seems well obeyed at jamming
c
flowing ➝ rigid but disordered
conjecture by Liu and Nagel
(Nature 1998)
jamming, foams, glass, all different
aspects of a unified phase diagram
with three axes:
T

surface below which
states are jammed
temperature
 volume density
 applied shear stress

(nonequilibrium axis)
“point J” is a critical point
“the epitome of disorder”
critical scaling at point J influences

behavior at finite T and finite .
J
jamming transition
“point J”
understanding = 0 jamming at “point J” may have
implications for understanding the glass transition at finite 
here we consider the 
plane at T = 0 in 2D
shear viscosity of a flowing granular material
shear stress 
⇒ shear flow in fluid state
velocity gradient
shear viscosity
if jamming is like a critical point we expect
below jamming
above jamming
model granular material
(Durian, PRL 1995 (foams);
O’Hern, Silbert, Liu, Nagel, PRE 2003)
bidisperse mixture of soft disks in two dimensions at T = 0
equal 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
overdamped dynamics
simulation parameters
Lx = Ly
N = 1024 for  < 0.844
finite size effects negligible
(can’t get too close to c)
N = 2048 for ≥ 0.844
t ~ 1/N, integrate with Heun’s method
total shear displacement ~ 10, ranging from 1 to 200
depending on N and
animation at:  = 0.830  0.838 < c  0.8415
 = 10-5
results for small  = 10-5
(represents → 0 limit, “point J”)
as N increases,  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
control parameters
  c , 
critical “point J”



J
,
scaling hypothesis (2nd order phase transitions):
c

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
 ~ b1/ ,
we thus get the scaling law
 ~ b ,
 ~ b
scaling law
choose length rescaling factor
crossover scaling variable
crossover scaling exponent

b

crossover scaling function
scaling collapse of viscosity 
point J is a true 2nd
order critical point
correlation length
transverse velocity correlation function (average shear
flow along x)
regions separated by
 are anti-correlated


distance to minimum gives
correlation length 
motion is by rotation
of regions of size 
scaling collapse of correlation length 
diverges at point J
shear stress 
phase diagram in  plane
 '
flowing
'



c
0
c
“point J”
jammed
c z0
volume density 
conclusions
• point J is a true 2nd order critical point
• critical scaling extends to non-equilibrium
driven steady states at finite shear stress 
in agreement with proposal by Liu and Nagel
• correlation length diverges at point J
• diverging correlation length is more readily observed
in driven non-equilibrium steady state
than in equilibrium state
• finite temperature?