Microfluidic crystals: Impossible order

Transcription

1 Microfluidic crystals: Impossible order Tsevi Beatus, Roy Bar-Ziv, T. T. Weizmann Institute International Symposium on Non-Equilibrium Soft Matter Kyoto

2 Outline Micro-fluidic droplets: micron sized bio-reactors. Unexpected order in a 1D droplet array. Many-body effects: phonons & instabilities. From hydrodynamics to electrostatics, solid state and back. The 1D crystal under confinement: Screening. Nature Physics, 2006 Phys Rev Lett,

3 µfluidics: lab-on-a-chip soft polymer glass channel 100µm Quake, Weitz, Whitesides, Stone, Ismagilov 3

4 The onset of disorder First observation ( pairing ) Hint: Interactions? Zigzag instability Collective modes? 4

5 What is the physics of 1D droplet arrays? water oil + surf. u oil 100μm h = 10 μm R = 10 μm a = μm u d = μm/s u drop Geometry: 1D array of discs in 2D. Two forces: friction and drag. Driven far from equilibrium. Broken Symmetry (drag direction). Highly dissipative (~ overdamped) ρ u Re ~ 10 η 4 5

6 Vibrations in linear motion Moving with the crystal Propagating phonons with almost no detectable damping. Interactions? Collective modes? 6

7 Normal modes of a 1D harmonic crystal a m K Dispersion Relation ω ( k) = 2 K / m sin( ka/2) 2 K 2 m ω (k) [rad/s] 0 π a 0 k [cm -1 ] π + a Symmetric bidirectional modes Group velocity vanishes at the end of Brillouin zone v g k ω = = 0 7

8 Phonon spectra of the 1D droplet crystal 2 K 2 m ω (k) [rad/s] xn () t X ( k, ω ) 2 0 a π N = 60 T = 20s L = 1cm 0 k [cm -1 ] π + a Predicted and measured spectrum implies symmetry breaking 8

9 Dispersion relations hint symmetry breaking ω = u d k Cs 250 μm/ s C s /2 ω( k) ω( k) ω ( k) = ω ( k) x y a= 27 μm R= 10 μm u = 360 μm/ sec u = 1730 μm/ sec d oil 9

10 Can waves persist in viscous media? Harmonic crystal 2 2 A A m = κ t 2 x 2 Friction A + μ t ω ~ κ q m m 0, no waves ω ~ iκq 2 can get waves for massive beads μ < κm/ a 2 2 No mass; symmetry-breaking field 0 A A = ξ + μ x t Waves! ω ~( ξ / μ)q Symmetry-breaking Waves 10

11 What is the symmetry-breaking force? Effective potential flow in 2D η v 2 = P ( 2 2 z h ) vr () = 1 4 φ Stokes Lubrication φ 2 = 0 Laplace eq Dipole field φ( r) = uniform + φ ( r) 2 ( ˆ () ) r x φd r = R uoil ud 2 r long-range dipole field F = ξ φ drag d d d 11

12 Dipolar flow mediates interactions in the crystal Long range interaction F r r 2 ij = ξd φ ( i j )~ r Crystal potential φ() r φ ( r r ) j d j Collective modes! 12

13 Droplet interaction: Peloton effect 350 Interactions slow down the crystal. Crystal moves faster w.r.t. oil or slower w.r.t. channel. Similar to sedimenting particles and cyclists the Peloton effect U d [μm/s] Experiment Theory a [μm] 13

14 Peloton leads to longitudinal waves Long longitudinal waves travel against flow 14

15 Dipolar interaction leads to transversal waves Long transversal waves travel with the flow 15

16 Derive wave equations and ω(k) Superposition of dipole fields on a lattice + small vibrations Fdrag = ξ uoil ud Ffriction = μud Equate drag and friction: ( ) Equate forces, get eq. of motion: r = 1 +μ/ ξ u ( r ) ( ) 1 n oil n x, y << a 6C sin s ( jka) ω ( k) = ω x y ( k) π a = ω ( k) 2 3 j= 1 x j Sound velocity 2 2 2π R u d s = 2 oil d 3 a uoil ( ) C u u 16

17 Crystal melting Instabilities Surface melting Zig-zag 17

18 Confined crystal from 2D to 1D Confinement parameter γ 2R W γ = µm γ = 0.40 γ = 0.63 γ = 0.80 What is the effect on phonons? 18

19 Confinement induces screening +q -q +q -q +q -q +q -q F ~ φ ~ x 2 x / W e π unconfined droplet +q -q +q -q potential is screened expect lower ω(k) and Cs confined droplets carries an array of mirror droplets +q -q a dipole array as a plate capacitor 19

20 Anomaly: Phonons move faster unconfined confined 20

21 Screening against incompressibility 1D limit small gaps high resistance Crystal becomes incompressible Expect Cs F ( ) ( x W) ~ φ ~ γ tan πγ /2 exp 2 π / x incompressibility screening 21

22 Phonons change under confinement Breaking of x-y anti-symmetry in x : C s,x decreases in y : C s,y increases Phonon amplitude decays The reason an interplay between Screening of interactions Incompressibility 22

23 Summary Crystalline order in 1D droplet array. Long range forces - Many-body collective modes. Forces induced by symmetry breaking flow. Non-equilibrium driven dissipative system. Which can be described by simple theory. Crystal instabilities Screening under confinement Outlook: Disordered 2D motion Nature Physics, 2006 Physical Review Letters,

24 Extra movies (more peculiarities) Inner flow Kangaroo 1 2D Chain Jaws Kangaroo 2 The return of Kangaroo 24

25 Waves in viscous media induced by symmetry-breaking field Dusty plasma crystals Active membranes Sedimentation of particles Flux lines in type II superconductors under external field Semi-dilute polymer solutions 25

26 Breaking of x-y anti symmetry The reason: breaking of translational invariance in y. With no confinement: x x x n j a 2 n = xφd( rn rj) ( n j) xφd(( ) ) j n j n y y y n j a 2 n = yφd( rn rj) ( n j) yφd(( ) ) j n j n 2 2 Laplace eq: φ = φ ω ( k) = ω ( k) In confinement: ω ( k) ω ( k) x y x d y d φ d x depends on distance from walls y 27

27 Second force: Friction Flow inside droplets Æ Energy dissipation Æ Friction F friction = μ u d (argument) 32