DWS and µ-rheology 1

Diffusing Wave Spectroscopy and µ-rheology:

when photons probe mechanical properties

Luca CipellettiLCVN UMR 5587, Université Montpellier 2 and CNRS

Institut Universitaire de Francelucacip@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 modulus G’()Out-of-phase response loss modulus G"()

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

Key step : measure displacement on microscopic length scales

Bead size: 2 m

Water Concentrated solution of DNA(simple fluid) (viscoelastic fluid)

Source: D. Weitz's webpage

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

Mean Square Displacement

Water: G'() = 0, G"() =

D. Weitz's webpage

0.5 m

T. Savin's webpage

Dr 6)(2 a

TkD B

6

)(2r

a

TkB

DWS and µ-rheology 8

Generalization to a viscoelastic fluid

or

)(2r

a

TkB

)/1()("

2

ra

TkG B

taking = 1/

Intuitive approach for a Newtonian fluid:

Rigorous, general approach:

Fourier transform Laplace transform

G*() = G'() + iG"()

DWS and µ-rheology 9

A Maxwellian fluid(from A. Cardinaux et al., Europhys. Lett. 57, 738 (2002))

Plateau modulus: G0

Relaxation time : r

Viscosity: = G0r

Rough idea: solid on a time scale << r, with modulus G0

Liquid on a time scale >> r, with viscosity = G0r

get G0

r r

G0/2

solventviscosity

solventviscosity

DWS and µ-rheology 10

Passive µ-rheology: the key step

Measure mean squared displacement <r2(t)>

Obtain G’(), G"()

Seed the sample with probe particles, then :

<r2> has to be measured on length scales < 1 nm to 1µm !

0.1 µm

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

Light scattering: the concept

A light scattering experiment

Speckle image

DWS and µ-rheology 13

From particle motion to speckle fluctuations

r(t)

r(t+)

DWS and µ-rheology 14

From particle motion to speckle fluctuations

r(t)

r(t+)

Weakly scattering media(single scattering)

Speckles fluctuate ifr() ~ ~0.5 µm

(Dynamic Light Scattering)

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

Photon propagation:Random walk

Detector

Ll *

Speckles fully fluctuate forr2> Nsteps = (L/ l* )2 <<

Typically: L ~ 0.1-1 cm, l* ~ 10-100 µm

r2> as small as a few Å2!

DWS and µ-rheology 17

How to quantify intensity fluctuations

I

t

Photomultiplier (PMT)signal

1)(

)()(1)( 22

t

t

tI

tItIg

Intensity autocorrelation function

g2-1

c

c

(other functions may be used, see L. Brunel's talk)

PMT

DWS and µ-rheology 18

From g2()-1 to <r2()>

• Well established formalism exists since ~1988• Depends on the geometry of the experiment

A good choice: the backscattering geometry

2

02

2 )(2exp1)( krg

22

220

n

k

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!

1)(

)()(1)( 22

t

t

tI

tItIg

I(t) PMT signal

• Average over ~ Texp = 103-104 max

Could be too long!

• Time-varying samples? (aging, aggregation...)

• Sample should explore all possibleconfigurations over time (ergodicity). Gels? Pastes?

max= 20 sTexp ~ 1 day!

DWS and µ-rheology 21

The Multispeckle techniqueAverage g2()-1 measured in parallel for many speckles

I1(t)I1(t+)I2(t)I2(t+)

I3(t)I3(t+)I4(t)I4(t+)

…

CCD or CMOS camera

1)(

)()(1)( 2

,

,2

tpp

tppp

tI

tItIg

DWS and µ-rheology 22

The Multispeckle technique (MS)

1)(

)()(1)( 2

,

,2

tpp

tppp

tI

tItIg

max= 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-rheologyFirst paper: Mason & Weitz, 1995 (306 citations)Since then: > 680 papers

DWSFirst papers: 1988Since then: > 1470 papers

19961997199819992000200120022003200420052006200720080

20

40

60

80

100

120

Num

ber

of µ

-Rhe

olog

y pa

pers

Publication year

DWS and µ-rheology 25

MSDWS µ-rheology

g2()-1 r2() G'(), G"()Multispeckle DWS µ-rheology

• 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...)

• Linear response probed• No inhomogeneous response• Full spectrum at once• No need to load/unload rheometer• Cheaper

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 <r2> to G’, G"

General formulas: or

Simpler approach (T. Mason, see [2])assume that locally <r2> be a power law:then,

with

and

DWS and µ-rheology 29

DWS: qualitative aproach

Weitz & Pine

l l*

l = 1/ scattering mean free pathl* transport mean free pathl* = l /<1-cos>

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):

~ <q2><r2> ~ k02<r2>

Total change in photon phase for a path (uncorrelated particle motion): ~ k0

2<r2> (L/ l * )2 Complete decorrelation of DWS signal for ~ 2r2> (L/ l * )2 << [r2> as small as a few Å2!!]

DWS and µ-rheology 30

DWS: quantitative approach

Intensity correlation function g2(t)-1 = [g1(t)]2

with t/ = k02< r2(t)>/ 6, k0 = 2/, and P(s) path length distribution

(example: for brownian particles, <r2(t)> = 6Dt and t/ = t k02D

(incoherent) sum over photon paths

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

g1(t) ~ /6exp~ t independent of l*: don’t needto know/measure l*!

= (k02D)-1

DWS and µ-rheology 32

Transmission geometry

g1(t) = (k02D)-1

Note: l* has to be determined.

Measure transmission

Calibrate against reference sample

LlLl

LlT 3/*5

3/*41

3/*5