Diffusing Wave Spectroscopy and µ-rheology : when photons probe mechanical properties

1. Diffusing Wave Spectroscopy and µ-rheology:when photons probe mechanical properties Luca Cipelletti LCVN UMR 5587, Université Montpellier 2 and CNRS Institut Universitaire de France lucacip@lcvn.univ-montp2.fr DWS and µ-rheology

2. Outline • Mechanical rheology and µ-rheology • µ-rheology : a few examples • Mesuring displacements at a microscopic level: DWS • The multispeckle « trick » • Conclusions DWS and µ-rheology

3. Rheology and ... Mechanical rheology: measure relation between stress and deformation (strain) In-phase response  elastic modulusG’(w) Out-of-phase response  loss modulusG"(w) DWS and µ-rheology

4. ... µ-rheology Active µ - Rheology : seed the sample with micron-sized beads, impose microscopic displacements with optical tweezers, magnetic fields etc., measure the stress-strain relation. Passive µ - Rheology : let thermal energy do the job, measure deformation (« weak » materials, small quantities, high frequencies…) DWS and µ-rheology

5. Passive µ-rheology WaterConcentrated solution of DNA (simple fluid) (viscoelastic fluid) Source: D. Weitz's webpage Bead size: 2 mm Key step : measure displacement on microscopic length scales DWS and µ-rheology

6. Outline • Mechanical rheology and µ-rheology • µ-rheology : a few examples • Mesuring displacements at a microscopic level: DWS • The multispeckle « trick » • Conclusions DWS and µ-rheology

7. A simple example: a Newtonian fluid Water: G'(w) = 0, G"(w) =hw D. Weitz's webpage 0.5 mm Mean Square Displacement T. Savin's webpage DWS and µ-rheology

8. Generalization to a viscoelastic fluid Intuitive approach for a Newtonian fluid: taking w = 1/t Rigorous, general approach: or Fourier transform Laplace transform G*(w) = G'(w) + iG"(w) DWS and µ-rheology

9. A Maxwellian fluid(from A. Cardinaux et al., Europhys. Lett. 57, 738 (2002)) Plateau modulus: G0 Relaxation time : tr Viscosity: h = G0tr Rough idea: solid on a time scale << tr, with modulus G0 Liquid on a time scale >> tr, with viscosity h = G0tr solvent viscosity get G0 G0/2 tr 1/tr solvent viscosity DWS and µ-rheology

10. Passive µ-rheology: the key step Seed the sample with probe particles, then : Obtain G’(w), G"(w) Measure mean squared displacement<Dr2(t)> 0.1 µm <Dr2> has to be measured on length scales < 1 nm to 1µm ! 1 nm DWS and µ-rheology

11. Outline • Mechanical rheology and µ-rheology • µ-rheology : a few examples • Mesuring displacements at a microscopic level: DWS • The multispeckle « trick » • Conclusions DWS and µ-rheology

12. q Light scattering: the concept A light scattering experiment Speckle image DWS and µ-rheology

13. From particle motion to speckle fluctuations Dr(t) Dr(t+t) DWS and µ-rheology

14. From particle motion to speckle fluctuations Weakly scattering media (single scattering) Speckles fluctuate if Dr(t) ~ l ~0.5 µm (Dynamic Light Scattering) Dr(t) Dr(t+t) DWS and µ-rheology

15. Diffusing Wave Spectroscopy (DWS): DLS for turbid samples Photon propagation: Random walk Detector DWS and µ-rheology

16. Diffusing Wave Spectroscopy (DWS): DLS for turbid samples L l * Photon propagation: Random walk Detector Speckles fully fluctuate for <Dr2>~l2/ Nsteps = l2/ (L/ l*)2 << l2 Typically: L ~ 0.1-1 cm, l* ~ 10-100 µm <Dr2> as small as a few Å2! DWS and µ-rheology

17. How to quantify intensity fluctuations Photomultiplier (PMT)signal I PMT tc t Intensity autocorrelation function g2-1 tc t (other functions may be used, see L. Brunel's talk) DWS and µ-rheology

18. From g2(t)-1 to <Dr2(t)> • Well established formalism exists since ~1988 • Depends on the geometry of the experiment • A good choice: the backscattering geometry Note: no dependence on l* (corrections are necessary for finite sample thickness, curvature, see L. Brunel's talk) DWS and µ-rheology

19. Outline • Mechanical rheology and µ-rheology • µ-rheology : a few examples • Mesuring displacements at a microscopic level: DWS • The multispeckle « trick » • Conclusions DWS and µ-rheology

20. The problem: time averages! tmax= 20 s Texp ~ 1 day! • I(t) PMT signal • Average over ~ Texp = 103-104tmax • Could be too long! • Time-varying samples? (aging, aggregation...) • Sample should explore all possible • configurations over time (ergodicity). Gels? Pastes? DWS and µ-rheology

21. The Multispeckletechnique Averageg2(t)-1 measured in parallel for many speckles I1(t)I1(t+t) I2(t)I2(t+t) I3(t)I3(t+t) I4(t)I4(t+t) … CCD or CMOS camera DWS and µ-rheology

22. The Multispeckletechnique (MS) tmax= 20 s Texp ~ 20 s! • slow relaxations, • non-stationary dynamics • non-ergodic samples (gels, pastes, foams, concentrated emulsions...) Smart algorithms needed to cope with the large amount of data to be processed, see L. Brunel's talk DWS and µ-rheology

23. Outline • Mechanical rheology and µ-rheology • µ-rheology : a few examples • Mesuring displacements at a microscopic level: DWS • The multispeckle « trick » • Conclusions DWS and µ-rheology

24. µ-rheology and DWS: a well established field, but in its commercial infancy! µ-rheology First paper: Mason & Weitz, 1995 (306 citations) Since then: > 680 papers DWS First papers: 1988 Since then: > 1470 papers DWS and µ-rheology

25. Linear response probed • No inhomogeneous response • Full spectrum at once • No need to load/unload rheometer • Cheaper • Reduced Texp • Time-varying dynamics • Non-ergodic samples • Sensitive to nanoscale motion • Good average over probes • Optically simple & robust • No stringent requirements • on optical properties (turbidity...) MSDWS µ-rheology g2(t)-1 Dr2(t) G'(w), G"(w) Multispeckle DWS µ-rheology DWS and µ-rheology

26. Useful references Useful references: [1] D. Weitz and D. Pine, Diffusing Wave Spectroscopy in Dynamic Light Scattering, Edited by W. Brown, Clarendon Press, Oxford, 1993 [2] M.L. Gardel, M.T. Valentine, D. A. Weitz, Microrheology, Microscale Diagnostic Techniques K. Breuer (Ed.) Springer Verlag (2005) or at http://www.deas.harvard.edu/projects/weitzlab/papers/urheo_chapter.pdf DWS and µ-rheology

27. Additional material DWS and µ-rheology

28. µ-rheology: from <Dr2> to G’, G" or General formulas: Simpler approach (T. Mason, see [2]) assume that locally <Dr2> be a power law: then, with and DWS and µ-rheology

29. l l* Weitz & Pine DWS: qualitative aproach l = 1/rs scattering mean free path l* transport mean free path l* = l /<1-cosq> Number of scattering events along a path across a cell of thickness L: N ~ (L/ l * )2 (l * / l ) [L/ l * 10-100, typically] Change in photon phase due to a particle displacement Dr (over a single random walk step): df ~ <q2><Dr2> ~ k02<Dr2> Total change in photon phase for a path (uncorrelated particle motion): Df ~ k02<Dr2> (L/ l * )2 Complete decorrelation of DWS signal for Df ~ 2p: <Dr2>~l2/ (L/ l *)2 << l2 [<Dr2> as small as a few Å2!!] DWS and µ-rheology

30. DWS: quantitative approach Intensity correlation function g2(t)-1 = b [g1(t)]2 (incoherent) sum over photon paths with t/t = k02< Dr2(t)>/ 6, k0 = 2p/l, and P(s) path length distribution (example: for brownian particles, <Dr2(t)> = 6Dt and t/t = t k02D Note: P(s) (and hence g1) depend on the experimental geometry! for analytical expression of g1 in various geometries (transmission, backscattering) see Weitz & Pine [1] DWS and µ-rheology

31. Backscattering geometry • independent of l*: don’t need to know/measure l*! g1(t) ~ t = (k02D)-1 DWS and µ-rheology

32. Transmission geometry t = (k02D)-1 g1(t) Note: l* has to be determined. Measure transmission Calibrate against reference sample DWS and µ-rheology