diffusing wave spectroscopy and µ-rheology : when photons probe mechanical properties
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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 [email protected]. Outline. Mechanical rheology and µ-rheology µ-rheology : a few examples - PowerPoint PPT PresentationTRANSCRIPT
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 [email protected]
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