Granular Clock & Temperature Oscillations in a bidisperse Granular Gas

Pik-Yin Lai ( 黎璧賢 )Dept. of Physics & Center for Complex Systems,

National Central University, Chung-Li,Taiwan

Collaborators:C. K. Chan( 陳志強 ), Institute of Physics Academia Sinica

May Hou ( 厚美英 ), IOP, Chinese Academy of Sciences

Granular materials(顆粒體 )

refer to collections of a large number of discrete solid components.日常生活中所易見的穀物、土石、砂、乃至公路上的車流、輸送帶上的物流等

Granular materials have properties betwixt-and -between solids and fluids (flow).

Basic physics is NOT understood

Complex and non-linear medium

Grains EverywhereGrains Everywhere

Food: almost everything we eat,: rice, cereal, peas...

Engineering: Powder mechanics, soil mechanics

Construction: Rocks, bricks, sand..

Agriculture: transport, storage & manipulation of seeds, grains & foodstuffs

Pharmaceutical: pills & powder processing

Transportation: shock absorption packing

Industries: Mixing & segregation of grains & powder by shaking or rotation

Geological: desert, landslides, earthquake dynamics

1 trillion US$/year

•Discrete & Macroscopic•Hardcore interactions & Dissipative•Zero temperature: mgd/kT ~ 10•Breakdown of hydrodynamics•Friction is important•Dynamically Driven•Inhomogeneous static stresses•Complex Many-body systems

•Mixing & Segregation•Pattern formation•Complex Flow

Physical aspects of Grains

12

Rev. Mod. Phys. 68, 1259 (1996) 71, S374 (1999) 71, 435 (1999)物理雙月刊 23, 503 (2001)

•Far from equilibrium•Dissipation: inelastic collisions•Energy input: vibrating bed (bottom collision)•Dissipation rate ~ input rate steady state

•Heap formation•Granular gas

Grains under excitations: vibrating bed

Properties of Granular Gases

• Particles in “random” motion and collisions• “similar” to molecular gases

But …

• Inelastic Collisions / Highly dissipative• Energy input from vibration table

• Far from thermal equilibrium Brazil Nut Effect, Clustering, Maxwell’s demon

Molecular gases

monodisperse granular gas in compartments: Maxwell’s Demon

Eggers, PRL, 83 5322 (1999)

v

Clustering

• Granular gas in Compartmentalized chamber under vertical vibration

D. Lohse’s group

Maxwell’s Demon is possible in granular systemSteady state: input energy rate = kinetic energy loss rate due to inelastic collisions

N

v

kinetic temp

Evaporation-condensationUnstable !

Bottom plate velocity (input)

Dissipation (output)

Tu

N

VT

grain ~

~2

uRL TT

Evaporation condensation

characteristic

Flux model

kT

mgz

ekT

mgNzn

)(

22 )1(22 )1( naan enendt

dn TV

ha

1~

2

n h 1-n

large V stable; as V decrease bifurcation !

uniform cluster to 1 side

2

1n

2

1n

2

1n is always a fixed point

Eggers, PRL, 83 5322 (1999)

)(hnuareadt

dN

What happens for a binary mixture?

Granular Oscillationsin compartmentalized bidisperse granular gas

NA grain ANB grain B

=NA/NB

Phase Diagram

B

Ao N

N

Objectives

• Quantitative description

• A model to understand the quantitative data

Effects of compartments + bidispersity: Granular Clock

Markus et al, Phys. Rev. E, 74, 04301 (2006)

Big and small grains. Explained by Reverse Brazil Nuts effects

Binary mixture in a single compartment

A B inelastic collision is asymmetric:

A can get K.E. from B (B heats up A & A slows down B)TB is lowered by the presence of A grains ABAB mme

Change of K.E. of A grain due to A-B inelastic collision:BuAu

Dissipation rate of A grain due to A-B inelastic collision:

Binary mixture in a single compartment

)()(

~

)()(

~

2

2

2

2

BB

AA

vq

VT

vp

VT

A B inelastic collision is asymmetric: suppose A gets K.E. from B (B heats up A & A slows down B)TB is lowered by the presence of A grains

ABAB mme

0;0

AB N

q

N

p

AB TT B

A

N

N

Balancing input energy rate from vibrating plate with total dissipation due to collision:

binary mixture of A & B grains in 2 compartments

• Very large V, A & B are uniform in L & R,

• As V is lowered, at some point only

A is free to exchange:

clustering instability of A• TBR gets higher, then B evaporates to L

• Enough B jumped to L to heat up As,

TAL increases A evaporates from L to R

A oscillates !

ABBRBLARAL TTTTTT ;;

(B heats up A & A slows down B)

Flux Model for binary mixture of A & B grains in 2 compartments

L RBL

ALL N

N

BR

ARR N

N

PRL, 100, 068001 (2008)J. Phys. Soc. Jpn. 78, 041001 (2009)

)()(

~

)()(

~

2

2

2

2

BB

AA

vq

VT

vp

VT

Theoretical result for p & q• Balancing input energy and dissipation

due to inelastic collisions:

p() & q() can be calculated theoretically

• is always a fixed point, • stable for V>Vc

• For V<Vc, Hopf bifurcation oscillation

2;

2B

BLA

AL

NN

NN

L R

BL

ALL N

N

BR

ARR N

N

V>Vc

V<Vc

V<Vc

V<Vf

Numerical solution

Model Results• V>Vc, A & B evenly distributed in 2 chambers

• Supercritical Hopf bifurcation near Vc

• V<Vc, limit cycle. Granular clock for A & B.

• Amplitude(v-vc)0.5 [Hopf]

• Period ~ (v- vf)- (numerical solution of Flux model)

• V < Vf , clustering into one chamber

• Saddle-node bifurcation at Vf (to be proved rigorously)

Vc-V (cm/s)

Oscillation amplitude: exptal data

Numerical soln. ofFlux model

Oscillation period

Phase diagram

Analytic results

• Fix point (0,0) loses stability at vc

Supercritical Hopf bifurcation at vc

• Theorem: supercritical Hopf if

verified

Expanding near (0,0):

Analytic result for phase boundary• Fix point (NA/2,NB/2) loses stability at vc

• Vc calculated from

Analytic result for emergent frequency at vc

• Hopf bifurcation at vc :

Larger (more A), longer time to heat up for evaporation smaller freq.

Saddle-node bifurcation at vf

• Phase boundary of vf:

New stable fixed point emerges:

V < Vf , clustering

Other interesting cases:• Tri-dispersed grains : A, B ,C

3-dim nonlinear dynamical system complex dynamics, Chaos…

Other interesting cases:• Bi-dispersed grains in M-compartments:

2(M-1)-dim nonlinear dynamical system complex dynamics,……

3

1 2

Summary• Evaporation /Condensation in granular compartmentalized gas is

unstable when dissipations become important “Maxwell demon”

• Temperature difference is generated spontaneously.

• Each grain type has difference temperature in a bi-disperse vibrating grain mixture because of asymmetric properties of collisions (mass, size,…) [even for single compartment]

• Binary mixtures can generate oscillatory temperature differences in the two compartments

• Oscillations: Hopf bifurcation at vc• Clustering: saddle-node bifurcation at vf• Our model is confirmed by experiments.• Systems with rich and complex dynamics