Holographic interferometry for the study of liquids

Transcription

1 Holographic interferometry for the study of liquids J. Colombani Prof. J. Bert - Laboratoire PMCN Université Claude Bernard Lyon 1 - France 1

2 Etymology latin inter : between greek ολοs : all latin metrum : a measure Holographic interferometry greek γραφειν : to write latin ferire : to hit a measure of the hits inbetween by writing all 2

3 History 1947 : Dennis Gabor invents holography at that time called "wavefront reconstruction" (1900 in Budapest ) - Imperial College, London Nobel prize in physics 1971 " for his invention and development of the holographic method " (nobelprize.org) motivation : electronic microscopy 1962 : Emmett Leith and Juris Upatnieks use the recenty developped laser technology to make 3D photography University of Michigan 3

4 Light = information Wave optics complex amplitude U = A exp(-iφ) φ = 2πδ/λ pathlength difference δ = d(ne) refraction index 3D But : optical detectors see intensity I ~ U 2 = A 2 n and 3D lost 4

5 The equation Thomas Young 1801 : in one point I = Σ n I n but U = Σ n U n (light + light = darkness) Intensity in one point resulting from two waves : I = U 2 = U 1 + U 2 2 = A 1 exp(-iφ 1 ) + A 2 exp(-iφ 2 ) 2 I = A 12 + A A 1 A 2 cos(φ 1 -φ 2 ) interference pattern contains phase difference 5

6 Holography : principle Recording of the hologram reference wave = plane wave : R = r photographic plate θ initial wave object object wave : O = o exp(-iφ) Transmittance of the photographic plate t = t 0 - β τ(r 2 + o o r cos φ) phase of the object recorded on the plate 6

7 Holography : principle Restitution of the image - object removed + reference beam switched on - reference beam diffracted by the hologram reference wave hologram θ θ θ reconstructed object wave : - β τ r 2 o exp(-iφ) direct wave : r (t 0 - βτ r 2 - βτ o 2 ) «virtual image» of the object «real image» of the object reconstructed conjugate of the object wave : - β τ r 2 o exp(iφ) 7

8 Holographic interferometry Superposition of : - reconstructed object beam (recorded at time t 0 ) - real object beam at time t reference wave hologram reference wave hologram reconstruction of the object recorded at time t 0 object wave object wave hologram object at time t 8

9 Experimental setup 9

10 Digital holographic interferometry Hologram recorded by a CCD detector camera instead of photographic plate Diffraction of the reference beam by the interference pattern : numerically instead of physically performed 10

11 Cost without local taxes shutter PC optomechanical elements optical table 1000 Total : (minimum) 11

12 Summary Classical interferometry : interference between objects separated in space Holographic interferometry : interference between objects separated in time 12

13 Possible applications diffusion Soret effect profilometry convection dissolution 13

14 Interdiffusion How to measure a diffusion coefficient with holographic interferometry? c + c/2 c reference hologram c + c time t = 0 t = t 0 14

15 Interdiffusion Diffusion coefficient of a bovine serumalbumine aqueous solution reference hologram λ concentration fringe c = e n/ c between two fringes h time t = 0 t = t 0 15 min 71 min Resolution of Fick s law with this geometry theoretical law of diffusion for the concentration c 0 c ( x, t) = c (erf h-x + erf h+ x ) 2 2 Dt 2 Dt 15

16 Interdiffusion x 25 min. 45 min. 90 min. interface experimental point fit with theoretical law D for each time interface D = 1.3 x 10-9 m 2 s -1 for the mean concentration c+ c/2 = 1.25 x 10-2 g cm -3 16

17 Diffusion through a meniscus Immiscible binary liquid mixture δt water isobutyric acid final equilibrium concentrations initial equilibrium concentrations 17

18 Diffusion through a meniscus concentration fringe c = λ e n/ c between two fringes reference hologram temperature increase 24.7 C 24.9 C acid-rich phase acid water water-rich phase t = 0 t = t 0 30 min 2 days time 18

19 Diffusion through a meniscus initial mass fraction final mass fraction mass fraction of acid time 2 c x 2 meniscus initial mass fraction final mass fraction mass fraction of acid position c t 19

20 Diffusion through a meniscus c D = / t 2 c x 2 D (cm 2 s -1 ) D (cm 2 s -1 ) mass fraction of acid mass fraction of acid 20

21 Soret effect Nonequilibrium thermodynamics Force Flux Fourier thermal T J q heat (1812) Ohm electrical V I current (1827) Fick solutal c J i matter (1855) chemical A T reaction progress 21

22 Soret effect Nonequilibrium thermodynamics Force Flux thermal T J q heat Dufour (1872) Soret (1879) solutal c J i matter Thomson (1892) electrical V I current chemical A T reaction progress 22

23 Soret effect Nonequilibrium thermodynamics Unified by Onsager in 1931 with its reciprocal relations 37 pages in Physical Review Nobel prize in chemistry 1971 (same year as Gabor) «Judged from the number of pages this work is thus one of the smallest ever to be rewarded with a Nobel Prize» (Nobelprize.org) 23

24 Soret effect liquid binary mixture Thermodiffusion T cold c hot Interdiffusion cold T c hot S T ~ c T 24

25 Soret effect thermal gradient imposition logra reference ho min LiCl, H20 concentration fringe C salt-poor T c C salt-rich t = 0 8 days time 0.3 c c = Nλ e n/ c between top and bottom ²c (weight %) S T = 1 c (1-c) c T Time (hours) 25

26 Soret effect Interdiffusion coefficient : D = h 2 π 2 τ 0.3 c characteristic time of the Soret experiment ²c (weight %) τ Time (hours) In one experiment : S T and D 26

27 Convection cold thermosolutal convection salt-poor in LiCl, H 2 0 hot T c salt-rich unstabilizing effect stabilizing effect thermosolutal Rayleigh-Bénard instability stability diagram T liquid at rest / convection? c 27

28 Convection T Ra t (10 5 ) stationary convective regime oscillatory convective regime diffusive regime stability diagram temperature fringe c = ,5 1 1,5 2 2,5 3 3,5 4 cold Ra s (10 8 ) c hot T 28 0 T crit

29 Convection C T Ra t (10 5 ) stationary convective regime oscillatory convective regime diffusive regime 2 c 0 stability diagram 0 0 0,5 1 1,5 2 2,5 3 3,5 4 Ra s (10 8 ) c salt-poor -4.9 C salt-rich 0 T crit 29 T

30 Convection T T 5 ) Ra t ( stationary convective regime oscillatory convective regime diffusive regime ~ min 0 0 0,5 1 1,5 2 2,5 3 3,5 4 Ra s (10 8 ) c 30

31 Convection Convection in extended geometry : E. Piquer, B. Lartigue and M.C. Charrier-Mojtabi Laboratoire d énergétique, Université Paul Sabatier, Toulouse Thermal convection in pure water at 23 C Diffusive regime T = 3.0 C 3 mm Onset of convection T = 5.6 C 143 mm Stationary convective regime T = 5.6 C 31

32 Dissolution of gypsum in water water gypsum CaSO 4,2H 2 0 eference hologram r concentration fringe time t = 0 t = t 0 12 min 60 min 120 min 240 min c = Nλ e n/ c 32

33 Dissolution Resolution of Fick equation in a semi-infinite cell with a chemical reaction at one end molality h = kβ / Dρm sat molality fit equation / experiment m sat = 0,021 mol.kg -1 D = 0,6 x 10-9 m 2.s -1 k = 2 x 10-5 mol.m -2.s -1 33

34 Dissolution Resolution of holographic interferometry? NaCl water 400 µm 34

35 Profilometry 5 mm bovine serumalbumine aqueous solution evaporation cracking crystallisation of a deposited drop 35

36 Profilometry laser expansion - filtration collimation object beam rotating mirror expansion - filtration collimation object reference beam camera 1) Hologram 2) Slight tilt α of the rotating mirror 3) New hologram on the same photographic plate auxiliary fringes 36

37 Profilometry auxiliary fringe parallel to the tilt axis fringe shift p height of the deposit d = p / tan(α+ α) 37

38 Conclusion Holographic interferometry : interference between objects separated in time access to the real time evolution of the field of refraction index in liquids Numerous applications : diffusion dissolution Soret effect convection 38