Transcript Granular Materials: A window to studying the Transition

Granular Materials: A
window to studying the
Transition from a nonNewtonian Granular
Fluid To A "Glassy"
system:
aka "The fluid-glass
transition for hard
spheres"
John Drozd and Dr. Colin Denniston
Savage and Jeffrey
J. Fluid Mech. 130, 187, 1983.
Collision rules for dry granular media as
modelled by inelastic hard spheres
Velocity
 r1'   r1  1 1   v n    m 2
   

 r '   r  2 m  m   m
1
1
2 
 2  2
r1
q
r2
m 2   r1  q 
 
 q
 m1   r2  q 
T he velocitydependentcoefficient of restit ution  v n 
determinestheenergy loss :
0.7

v 
1  1   0  n  , v n  v 0
 v n   
 v0 

0 ,
vn  v0


v2  v1   qˆ
0 
v 2  v1   qˆ
As collisions become weaker
(relative velocity vn small),
they become more elastic.
 1 , v 0  ga , a  radiu sof particle
Energyis dissipated.
C. Bizon et. al., PRL 80, 57, 1997.
300 (free fall region)
0.7
a
0.6
Donev et al PRL96
"Do Binary Hard Disks
Exhibit an Ideal Glass
Transition?"
250 (fluid region)
0.5

0.4
polydisperse
monodisperse
0.3
0.2
0.1
20.
b
free fall
10.
fluid
vy
vy
glass
15.
5.
20.
c
1
dvy dt
15.
dvy/dt
200 (glass region)
10
0.1
0.01
10.
0
5.
100
200
300
0.
0
P
P
0.6
2
0.4
v
P
2
0.8
150
d
y
0.2
4
0.2
6
0.4
1
0.6
c
8
0
100
200
y
300
Donev et al PRE 71 P  D k BT1  / c 1
400
y
x
z
Polydispersity means
Normal distribution
of particle radii
The density in the glassy region is a constant.
In the interface between the fluid and the glass does the
density approach the glass density exponentially?
0 = 0.9
0 = 0.95
Interface width seems to increase as 0  1
0.7
a
0.6
0.5

0.4
0.3
0.2
0.1
20.
10.
fluid
glass
vy
vy
free fall
b
15.
5.
20.
dvy dt
15.
10.
5.
0.
2.5
10
1
y
c
0.1
0.01
0
  c  A exp y
100 200 300
How does  depend on (1  0) ?
d
0 = 0.99
Density vs Height in Fluid-Glass
Transition
  c  A exp y
0
1
c
0
0
Log10
2
0
0
3
0
0
4
0
0
160
180
200
220
y
240 260
280
300
0.9
0.95
0.96
0.97
0.98
0.99
0.995
0.999
y
Length Scale in Transition
1
Log10
in
c
A Exp
0.8
  c  A exp y
  1  0 
0.43
"interface width diverges"
1.2
Slope = 0.428  0.007
1.4
3
2.5
Log10
2
1
1.5
0
1
300 (free fall region)
Y Velocity
Distribution
vy
3
2
Poiseuille flow
1
a y 250
250 (fluid region)
0 5 10 15 20 25 30
x
2
vy
1.5
1
Plug flow
snapshot
200 (glass region)
0.5 b y 200
0 5 1015202530
x
2
1.5
vy
Mono-disperse
(crystallized)
only
4
1
Mono
kink
fracture
150
y
0.5 c y 150
0 5 1015202530
x
x
z
300 (free fall region)
y 250
35
vx2
vy
20
15
vy
25
2
30
10
fluid
5
250 (fluid region)
0
5
10 15 20 25 30
x
y 235
235 (At Equilibrium
Temperature)
vx2
1.2
1
vy
0.6
vy
2
0.8
0.4
200 (glass region)
equilibrium
0.2
0
5
vx2
0.2
vy
0.3
2
0.4
vy
Granular
Temperature
10 15 20 25 30
x
y 200
glass
x
0.1
0
5
10 15 20 25 30
x
150
y
z
Fluctuating and Flow Velocity
Experiment by
N. Menon and
D. J. Durian,
Science, 275, 1997.
5
 v v
2
16 x 16
vy
1
v
2/3
flow
0.5
32 x 32
0.2
In Glassy Region !
Simulation results
0.1
0.5
1
5
10
J.J. Drozd and
C. Denniston
vf
Europhysics Letters, 76 (3), 360, 2006
"questionable" averaging over nonuniform regions gives 2/3
vy
 v  v  Tg  v y  v c 
   1 in fluid glass transition
2
i
For 0 = 0.9,0.95,0.96,0.97,0.98,0.99
Subtracting of Tg and vc and not averaging over regions of different vx2
x 16,
0
0.99
x 16 ,
10
Down centre
5
Slope  = 0.9-1.0
Tg
0
1
vx2
Tg
vx2
0.99
20
1
log10
0
2
2
1
0.5
3
0.2
4
0
100
200
y
300
400
0.001
0.01
vy
0.1
vc
1
"Particle Dynamics in
Sheared Granular Matter"
Physical Review Letters 85,
Number 7, p. 1428
(2000)
Experiment:
(W. Losert, L. Bocquet,
T.C. Lubensky and
J.P. Gollub)
vy
h
Velocity Fluctuations
vs. Shear Rate
240
1.7
1.6
1.5
1.4
1.3
1.2
U
5
10
h
15 20 25
x
240 , y g 185 ,
30
0
0.9
v2x
Tg 1 2
1.6
1.8
2
2.2
2.4
3.5
3
log 10
2.5
xv y
U
2
log 10
1.5
1
U
Slope = 0.406  0.018
From simulation
Must Subtract Tg !
Experiment Slope = 0.4
Physical Review Letters 85, Number 7, (2000)
Conclusions
• A gravity-driven hard sphere simulation was used to study
the glass transition from a granular hard sphere fluid to a
jammed glass.
• We get the same 2/3 power law for velocity fluctuations vs.
flow velocity as found in experiment, when each data point
is averaged over a nonuniform region.
• When we look at data points averaged from a uniform
region we find a power law of 1 as expected.
• We found a diverging length scale at this jamming (glass)
to unjamming (granular fluid) transition. Silbert, Liu and
Nagel (PRL 95, 098301 (2005)) also found diverging
length scales near the unjamming transition for vibrations
with jammed packings.
• Finally, we compared our simulation to experiment on the
connection between local velocity fluctuations and shear
rate and found quantitative agreement.