Temperature Oscillations in a Compartmetalized Bidisperse Granular Gas C. K. Chan 陳志強 Institute of Physics, Academia Sinica, Dept of Physics,National Central University, Taiwan

Collaborators May Hou, Institute of Physics, CAS 厚美英 P. Y. Lai, National Central University 黎璧賢

Content What is a clock? What is special about a granular clock? Unstable Evaporation/Condensation Two temperature in a bi-disperse system Model for bidisperse oscillation Summary

What is a clock ? Periodic motion sun, moon, pendulum etc … Periodic Reaction BZ reaction, enzyme circadian rhythm Periodic Collective behavior suprachiasmatic nuclei, sinoatrial node, comparmentalized granular gases, etc…

BZ reaction From S. Mueller

Granular Oscillation

Second Law  no clock? Belousov-Zhabotinsky reaction A  B  A  B; Why not: A   B Two-compartment granular Clock

Molecular gases

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

monodisperse granular gas in compartments: Maxwell’s Demon Eggers, PRL, (1999) v

Clustering Granular gas in Compartmentalized chamber under vertical vibration D. Lohse’s group

Maxwell’s Demon is possible in granular system Steady state: input energy rate = kinetic energy loss rate due to inelastic collisions N v kinetic temp Evaporation-condensation Unstable ! Bottom plate velocity (input) Dissipation (output) u Evaporation condensation characteristic

Heaping

Flux model n h 1-n large V stable; as V decrease  bifurcation  ! uniform  cluster to 1 side is always a fixed point Eggers, PRL, (1999)

What happens for a binary mixture? What are the steady state? How many granular temperatures ?

Oscillation of millet ( 小米, N=4000) and mung beans ( 绿豆, N=400) F = 20Hz. Amp = 2mm

soda lime glass 138 small spheres diameter : 2 mm 27 large spheres diameter 4 mm box height:7.7 cmx0.73cmx5 cm

Effects of compartments + bidispersity: Granular Clock Markus et al, Phys. Rev. E, 74, (2006) Big and small grains. Explained by Reverse Brazil Nuts effects a=6 mm, f =20 Hz. Times: a=0, b=3.1, c=58.3, d=66.2, e=103.2 s.

Granular Oscillations in compartmentalized bidisperse granular gas 2.6cmx5.4cmx13.3cm barrier at1.5 cm Steel glass balls Radius = 0.5 mm N = 960 f = 60 Hz

Phase Diagram

Model of two temperatures 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 T BR gets higher, then B evaporates to L Enough B jumped to L to heat up As, T AL increases  A evaporates from L to R A oscillates ! (B heats up A & A slows down B)

Model Objectives Quantitative description A model to understand the quantitative data

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) T B is lowered by the presence of A grains Change of K.E. of A grain due to A-B inelastic collision: Dissipation rate of A grain due to A-B inelastic collision:

Binary mixture in a single compartment 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 Balancing input energy rate from vibrating plate with total dissipation due to collision:

Flux Model for binary mixture of A & B grains in 2 compartments L R PRL, 100, (2008) J. Phys. Soc. Jpn. 78, (2009)

is always a fixed point, stable for V>Vc For V<Vc, Hopf bifurcation  oscillation L R

V>Vc V<Vc V<V f Numerical solution

Model Results V>Vc, A & B evenly distributed in 2 chambers Supercritical Hopf bifurcation near V c V<Vc, limit cycle. Granular clock for A & B. Amplitude  (v-v c ) 0.5 [Hopf] Period  ~ (v- v f ) -  (numerical solution of Flux model) V < V f, clustering into one chamber Saddle-node bifurcation at V f (??? to be proved rigorously???)

Vc-V (cm/s) Oscillation amplitude: exptal data Numerical soln. of Flux model

Oscillation period

Phase diagram

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 12

Summary Dissipation is density dependent  “Maxwell demon” Different collision dissipations in binary system  existence of two “granular temperatures” Non-homogeneous temperature with homogenous energy input both spatially and temporally Granular steady state + compartment  oscillations

Thermophoresis or Janus ?

A worm in a temperature bath