Quasi-two-dimensional granular gas

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Unusual diffusion in quasi-two-dimensional
granular gas
Wennan Chen ??? and Kiwing To ???Institute of
Physics Academia Sinica????????
OUTLINE Introduction granular physics Velocity
distribution of granular gas Diffusion of
quasi-2D granular gas Phenomenological two-state
model Summary Discussions
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Give, and it shall be given unto you a good
measure, pressed down, shaken together, and
running over, they will give into your bosom.
For with what measure you measure, it shall be
measured to you in return.

New Testament Bible, Luke, 638
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Compaction demonstration

• Compaction demonstration video

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Granular Solid

• Force distribution at granular pile bottom
• Vanel et. al., PRE 60, R5040, 1999

pile of sand 1.2 mm diameter 8 cm high 33o repose
angle
localized deposition
uniform deposition
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Granular Solid

• apparent weight
• Vanel, Duran, 97

cylinder radius (mm) 20 bead diameter (mm) 1.5,
2, 3
Where has the missing weight gone?
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Granular Solid

• physical properties are history dependent

density increases due to rearrangement of beads
in packing
Potential energy of a grain
mgd with m 10-2 g, g10m/s2, d1mm 10-7 J
Thermodynamic temperature kT with
k1.38?10-21 J/K, T300K 4?10-21 J
Temperature 0 as compared to granular
energy Thermodynamically non-equilibrium,
meta-stable state
does not relaxes spontaneously relaxes by
external agitation (tapping or vibration)
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Static and dynamicsdead and alive
gas fluid solid
dense, slow flow
quasi-static meta-stable glassy region
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Motivation
Statistical properties of quasi-two-dimensional ga
s with inelastic inter-molecular interaction
gas a collection of small molecules moving
around
inter-molecular interaction exchange of energy
among molecules
inelastic interaction does not conserve energy
statistical properties velocity
distribution temperature diffusion
constant
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hard sphere collision
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elastic collision
from equilibrium statistic mechanics
Maxwell distribution
change in velocity distribution due to collision
Boltzmann Equation
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inelastic collision
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inelastic hard spheres
velocity distribution changes due to collision
Boltzmann-Enskog Equation
collision integral
van Noije and Ernst, Gran. Matter 1, 57, 1998.
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inelastic hard spheres
ltlt
exponential decay at high energy tail
confirmed by direct Monte Carlo simulation Brey,
et. al., PRE 59, 1256, 1999.
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velocity distribution and temperature
molecular gas
elastic collisions
in equilibrium state
temperature
inelastic collisions
granular gas
non-equilibrium steady state
for c gtgt 1
granular temperature
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Unusual diffusion in quasi-two-dimensional
granular gas
MSD 6 D t
diffusion constant
in 2 dimension MSD 4 D t
in 1 dimension MSD 2 D t
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Unusual diffusion in quasi-two-dimensional
granular gas
diffusion constant m2/s
D inversely proportional to N
usual diffusion
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Unusual diffusion in quasi-two-dimensional
granular gas
granular gas a collection of molecules
interact with each other with inelastic
collision
quasi-two-dimensional two dimensional
projection from three dimensional motion
plastic ball diameter 6 mm mass 0.12
gm vibration frequency 20 Hz amplitude
1.84 mm camera resolution 1024x1024 frame
rate 1000 fps
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Unusual diffusion in quasi-two-dimensional
granular gas
get trajectory each particles
measure MSD
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MSD 4D t
diffusive motion
MSD lt(Dx)2gt lt(vt)2gt ltv2gt
t2
ballistic motion
N 1000
slope2
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N 500
D300 mm/s2
N 1000
D100 mm/s2
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4D mm2/s
diffusion increases with N for N lt 1000
Tg 10-9 J
N
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ball dynamics
motion in vertical direction
motion in horizontal direction
collision with top or bottom (type-1)
collision with other balls (type-2)
small velocity fluctuation in horizontal
direction
large velocity fluctuation in horizontal
direction
balls may gain kinetic energy in the horizontal
direction due to inelastic collision
top and bottom act on the balls like a viscous
fluid
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particle trajectory and speed
high speed state
low speed state
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Two state model
ball excited to HSS after colliding with another
ball
ball in HSS relaxes to LSS in t2 due to collision
with top and bottom
1800
500
200
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total area swept
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fraction of ball in HSS
argument break down when ft 1, this happens
when N Nc 945
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Two effect temperature baths
collision with other ball
collision with top or bottom
temperature bath-2, T2
temperature bath-1, T1
effective granular temperature
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Summary and discussions
We studied diffusion in a quasi-two-dimensional
granular gas (Q2DGS) composed of plastic balls
confined in a vertically vibrating thin box.
The motion of the particles in the Q2DGS was
found to follow the Langevin equation with the
top and bottom of the box acting on the balls
like a viscous fluid.
We found that both the granular temperature and
the diffusion constant increased with the number
of balls N in the box for small N.
Based on the velocity distributions and the two
different kinds of horizontal motions observed in
the experiments, we proposed a simple two-state
model to explain the unusual diffusion behavior.
Thank you for your attention !
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Motivation
Can we understand the statistical properties of
granular gas using simple kinetic theory ?
kinetic theory each molecule move according
to Newtons law
statistical properties velocity
distribution temperature diffusion
constant