Bridging by Light. Robert Schroll Wendy Zhang. University of Chicago. 9/17/2004 Brown Bag Talk p. 1

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1 Bridging by Light Robert Schroll Wendy Zhang University of Chicago 9/17/2004 Brown Bag Talk p. 1

2 Experiment Experiment done by Alexis Casner and Jean-Pierre Delville, Université Bordeaux I Prepare a microemulsion, by weight 70% toluene, 4% sodium dodecyl sulfate, 17% n-butanol, and 9% water Above T c = 35 C, separates into two micellar phases of differing concentrations Phases have different densities and indices of refraction 9/17/2004 Brown Bag Talk p. 2

3 Experiment n 1 < n 2 n 2 ρ 2 ρ 1 > ρ 2 n 1 ρ 1 Surface tension is 10 6 that of water/air interface Surface is easily deformed by a laser beam 9/17/2004 Brown Bag Talk p. 3

4 Casner and Delville, private communications Experiment 9/17/2004 Brown Bag Talk p. 4

5 Casner and Delville, private communications Experiment 9/17/2004 Brown Bag Talk p. 5

6 Casner and Delville, private communications Experiment Jet formation displays hysteresis 9/17/2004 Brown Bag Talk p. 6

7 Experiment Further increasing the power causes the jet to form a bridge to the bottom of the container Casner and Delville, private communications 9/17/2004 Brown Bag Talk p. 7

8 Overview Can we explain the stability of these shapes? Can we explain the sizes of these shapes? 9/17/2004 Brown Bag Talk p. 8

9 Forces on Surface Surface tension: P σ = 2σH [ H = 1 r (z) 2 (1 + r (z) 2 ) 3/2 ] 1 r(z) (1 + r (z) 2 ) 1/2 r Gravity: P g = ρgz z Radiation pressure: P rad 9/17/2004 Brown Bag Talk p. 9

10 Radiation Pressure Radiation pressure comes from momentum discontinuity at surface Momentum of photon: p γ = E γ n/c For a single photon: p = E γ c n 2 cos θ [ R tan θ 2 tan θ 1 T ] θ θ 2 2 θ 1 n 2 n 1 T = 1 2 ( ) 4n 1 n 2 cos θ 1 θ 2 + 4n 1n 2 cos θ 1 θ 2, R = 1 T (n 2 cos θ 1 +n 1 cos θ 2 ) 2 (n 1 cos θ 1 +n 2 cos θ 2 ) 2 9/17/2004 Brown Bag Talk p. 10

11 Radiation Pressure Illumination of surface [ by intensity ] I(r): P rad = n 2 c cos 2 θ R tan θ 2 tan θ 1 T I(r) 9/17/2004 Brown Bag Talk p. 11

12 Low Power Dimples Previous work has explained low-power dimples as a static balance of these three forces. Casner and Delville, Phys. Rev. Lett. 87, /17/2004 Brown Bag Talk p. 12

Varicose Instability Under the influence of only surface tension, a cylinder will be unstable if its length is greater than its circumference At long

13 Varicose Instability Under the influence of only surface tension, a cylinder will be unstable if its length is greater than its circumference At long lengths, breaking into a series of bubbles is energetically more favorable Casner and Delville, private communications 9/17/2004 Brown Bag Talk p. 13

14 Radiation Pressure Radiation pressure must be holding bridge open Assume a flux density of photons Φ 0 trapped by TIR (total internal reflection) Gives pressure P rad = p γ 2 If radius varies, P rad = p γ cos φ 2 tan φ Φ(z) = p γ 2 cos φ tan φ cos φ tan φ Φ 0 r 0 2 r(z) 2 Φ 0 φ Φ 0 9/17/2004 Brown Bag Talk p. 14

15 Bridge Stability Energy Analysis: Change in surface energy moving from straight to sinusoidal walls, preserving volume Work done against radiation pressure to reach that shape Deformed surface always costs more energy, so varicose instability is overcome. Pressure analysis is more clear (in long wavelength case) 9/17/2004 Brown Bag Talk p. 15

16 Bridge Stability Assume a static solution exists: P 0 = σ r 0 + ρgz Create a disturbance of characteristic sizes r, z ( r z 1) Calculate lowest order change in pressure r 0 z r 9/17/2004 Brown Bag Talk p. 16

17 Bridge Stability P = σ r 2P 0 r 0 = r ( ( 1 z P 0 r 0 r 0 2 ) + σ z 2 σ r r ) r 0 P 0 Static solution means: P 0 = σ r 0 + ρgz > σ r 0 + σ z σ 2 r 2 0 > σ r σ z > 0 2 Thus, P and r will always have opposite signs. The change in pressure will oppose the change in r. 9/17/2004 Brown Bag Talk p. 17

18 Bridge Stability Bends will be unstable: The side bending in will feel a large increase in radiation pressure The side bending out will experience a decrease in radiation pressure 9/17/2004 Brown Bag Talk p. 18

19 Bridge Formation: My Idea Bridge is always stable Difference between dimple and jet is energy to deform surface Explains hysteresis Jet becomes bridge when it hits the bottom Upper fluid wets container Bridge should be stable as power is decreased, until upper surface can no longer support it Bridge should collapse to dimple at same power as jet collapse 9/17/2004 Brown Bag Talk p. 19

20 Bridge Formation: Reality Bridge collapses to a jet Collapse occurs at the same power at which bridge formation occurs 9/17/2004 Brown Bag Talk p. 20

21 Bridge Shape Introduce attenuation of light: Φ(z) = r 0 2 r(z) 2 α 0 e kz Solve for r(z) assuming: Stable, (nearly) vertical solution at z 0 r (z) 1, r (z) 1/r(z) z r 0 0 P 0 Reflections do not change φ or lose light No affect from central column of light 9/17/2004 Brown Bag Talk p. 21

22 Bridge Shape P 0 r 0 2 σ r(z) α 2 0 e kz = r(z) ( + ρgz ) r(z) = σ 2 ρgz 4 ρp 0r 02 α 0 σ ze kz ( ) = B C ζ B ζe ζ ζ kz B σk 2 ρg C 2α 0 P 0 σ/r 0 r 0 2r 0 Self-consistent: for z r 0, we have r (z) 1, r (z) 1/r(z) 9/17/2004 Brown Bag Talk p. 22

23 Bridge Shape r(z) = B ζ ( ) C B ζe ζ C B ζe ζ 1 r(z) C 2 e ζ Corresponds to ignoring buoyancy term C B ζe ζ 1 r(z) BC ζ e ζ/2 Corresponds to ignoring surface tension With experimental values, C B ζ z 10µm 9/17/2004 Brown Bag Talk p. 23

24 End Cap Assume an end cap at depth z 1, with radius r 1. Assume all light r 1 r 2 trapped by TIR by bottom Find where net force on end cap is 0 r2 r 0 r 1 z z 1 0 z 2 9/17/2004 Brown Bag Talk p. 24

25 End Cap Curvature: Treat encap as hemisphere with radius r 1 : F curv = 2πσr 1 Buoyancy: F buoy = ρgz 1 πr 1 2 Radiation pressure: Assume [ a gaussian ] beam profile I(r) = 2P πω 0 exp 2r2 ω 2 0 Two components of radiation pressure: Light from central column Light trapped by TIR 9/17/2004 Brown Bag Talk p. 25

26 End Cap Treat bottom as flat Pretend reflected light has normal incidence 9/17/2004 Brown Bag Talk p. 26

27 End Cap Assume all light attenuates like α 0 e kz F rad = P c ( 2n 2 n n 1 +n 2 α 0 e kz 1 e 2r /ω 0 Balance forces and solve for z 1 : z1 ( ) P c n 1 e 2r /ω 0 P 0 πr 2 0 = πσe k(z 1 z 0 ) r(z 1 ) ) r2 r 0 r 1 z 0 z 2 9/17/2004 Brown Bag Talk p. 27

28 End Cap More mangling gives F 1 = 1 x ( ) F 1 e 2r /ω 0 P n πcp 0 r 0 2 ( 2x ) x 2P 0r 0 2 ρg σ 2 z 1 e k(z 0 z 1 ) Under either approximation (ignoring gravity or surface tension), z 1 drops out completely The solution seems to be stable: F z z 1 < 0 9/17/2004 Brown Bag Talk p. 28

29 Bridge Radius Need to know how much light gets funneled into bridge, and how Assume Φ(φ) is flat Get r 2 by: Assume surface outside of r 0 is minimal Assume r 2 is largest radius that totally reflects incident light r 2 = r 0 n 2 n 1 9/17/2004 Brown Bag Talk p. 29

30 Bridge Radius σr 0 + ρgz 0 r 0 2 = P c n 2 2π ln(cot θ c /2) cos θ c π/2 θ c (exp [ ] 2r 0 2 ω 2 0 exp [ 2r 0 2 ω 0 2 Ignore gravity; work [ to lowest ] order in n: σ = 4 P n 2 r 0 3 πc n 1 ω 2 0 exp 2r 0 2 ω 2 0 Numeric solution with common values gives nonsense Ignoring exponential means r 0 ω 0 2 P ]) 2 n 2 n 2 1 9/17/2004 Brown Bag Talk p. 30

Bridging dielectric fluids by light: A ray optics approach

Eur. Phys. J. E 6, 405 409 (008) DOI 10.1140/epje/i008-10336-1 THE EUROPEAN PHYSICAL JOURNAL E Bridging dielectric fluids by light: A ray optics approach R.D. Schroll 1, E. Brasselet,a, W.W. Zhang 1, and