1. DWS and µ-rheology 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

2. 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

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

4. 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…)

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

6. 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

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

8. DWS and µ-rheology 8 Generalization to a viscoelastic fluid or taking  = 1/  Intuitive approach for a Newtonian fluid: Rigorous, general approach: Fourier transformLaplace transform G*(  ) = G'(  ) + iG"(  )

9. DWS and µ-rheology 9 A Maxwellian fluid (from A. Cardinaux et al., Europhys. Lett. 57, 738 (2002)) Plateau modulus: G 0 Relaxation time :  r Viscosity:  = G 0  r Rough idea: solid on a time scale <<  r, with modulus G 0 Liquid on a time scale >>  r, with viscosity  = G 0  r get G 0 rr  r G 0 /2 solvent viscosity solvent viscosity

10. DWS and µ-rheology 10 Passive µ-rheology: the key step Measure mean squared displacement Obtain G’(  ), G"(  ) Seed the sample with probe particles, then : has to be measured on length scales < 1 nm to 1µm ! 0.1 µm 1 nm

11. 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

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

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

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

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

16. DWS and µ-rheology 16 Diffusing Wave Spectroscopy (DWS): DLS for turbid samples Photon propagation: Random walk Detector L l * Speckles fully fluctuate for  r 2 >    N steps =   (L/ l* ) 2 <<  Typically: L ~ 0.1-1 cm, l* ~ 10-100 µm  r 2 > as small as a few Å 2 !

17. DWS and µ-rheology 17 How to quantify intensity fluctuations I t Photomultiplier (PMT)signal Intensity autocorrelation function g 2 -1 cc cc (other functions may be used, see L. Brunel's talk) PMT

18. DWS and µ-rheology 18 From g 2 (  )-1 to 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)

19. 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

20. DWS and µ-rheology 20 The problem: time averages! I(t) PMT signal Average over ~ T exp = 10 3 -10 4  max Could be too long! Time-varying samples? (aging, aggregation...) Sample should explore all possible configurations over time (ergodicity). Gels? Pastes?  max = 20 s T exp ~ 1 day!

21. DWS and µ-rheology 21 The Multispeckle technique Average g 2 (  )-1 measured in parallel for many speckles I1(t)I1(t+)I1(t)I1(t+) I2(t)I2(t+)I2(t)I2(t+) I3(t)I3(t+)I3(t)I3(t+) I4(t)I4(t+)I4(t)I4(t+) … CCD or CMOS camera

22. DWS and µ-rheology 22 The Multispeckle technique (MS)  max = 20 s T exp ~ 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

23. 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

24. 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

25. DWS and µ-rheology 25 MSDWS µ-rheology g 2 (  )-1  r 2 (  )G'(  ), G"(  ) Multispeckle DWS µ-rheology Reduced T exp Time-varying dynamics Non-ergodic samples Sensitive to nanoscale motion Good average over probes Optically simple & robust No stringent requirements on optical properties (turbidity...) Linear response probed No inhomogeneous response Full spectrum at once No need to load/unload rheometer Cheaper

26. 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.pdfMicroscale Diagnostic Techniques

27. DWS and µ-rheology 27 Additional material

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

29. DWS and µ-rheology 29 DWS: qualitative aproach Weitz & Pine l l*l* l = 1/  scattering mean free path l* transport mean free path l* = l / 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  r (over a single random walk step):  ~ ~ k 0 2 Total change in photon phase for a path (uncorrelated particle motion):  ~ k 0 2 (L/ l * ) 2 Complete decorrelation of DWS signal for  ~ 2   r 2 >    (L/ l * ) 2 <<   [  r 2 > as small as a few Å 2 !!]

30. DWS and µ-rheology 30 DWS: quantitative approach Intensity correlation function g 2 (t)-1 =  [g 1 (t)] 2 with t/  = k 0 2 / 6, k 0 = 2  /, and P(s) path length distribution (example: for brownian particles, = 6Dt and t/  = t k 0 2 D (incoherent) sum over photon paths Note: P(s) (and hence g 1 ) depend on the experimental geometry! for analytical expression of g 1 in various geometries (transmission, backscattering) see Weitz & Pine [1]

31. DWS and µ-rheology 31 Backscattering geometry g 1 (t) ~  independent of l*: don’t need to know/measure l*!  = (k 0 2 D) -1

32. DWS and µ-rheology 32 Transmission geometry g1(t)g1(t)  = (k 0 2 D) -1 Note: l* has to be determined. Measure transmission Calibrate against reference sample