Transcript Diffusing Wave Spectroscopy and micro

Diffusing Wave Spectroscopy and µrheology:
when photons probe mechanical properties
Luca Cipelletti
LCVN UMR 5587, Université Montpellier 2 and CNRS
Institut Universitaire de France
[email protected]
DWS and µ-rheology
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Outline
• Mechanical rheology and µ-rheology
• µ-rheology : a few examples
• Mesuring displacements at a microscopic level: DWS
• The multispeckle « trick »
• Conclusions
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Rheology and ...
Mechanical rheology:
measure relation between stress and
deformation (strain)
In-phase response  elastic modulus G’(w)
Out-of-phase response  loss modulus G"(w)
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... 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…)
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Passive µ-rheology
Water
Concentrated solution of DNA
(simple fluid)
(viscoelastic fluid)
Source:
D. Weitz's webpage
Bead size: 2 mm
Key step : measure displacement on microscopic length scales
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Outline
• Mechanical rheology and µ-rheology
• µ-rheology : a few examples
• Mesuring displacements at a microscopic level: DWS
• The multispeckle « trick »
• Conclusions
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A simple example: a Newtonian fluid
Water: G'(w) = 0, G"(w) = hw
D. Weitz's webpage
0.5 mm
T. Savin's webpage
Mean Square Displacement
k BT
2
 r ( )   6D
D
6ha
k BT
h
 r 2 ( ) 
a

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Generalization to a viscoelastic fluid
Intuitive approach for a Newtonian fluid:
h
a
k BT
taking w = 1/
2
 r ( ) 
G" (w )  wh 
k BT
a  r 2 (1 / w ) 

Rigorous, general approach:
or
Fourier transform
Laplace transform
G*(w) = G'(w) + iG"(w)
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A Maxwellian fluid
(from A. Cardinaux et al., Europhys. Lett. 57, 738 (2002))
Plateau modulus: G0
Relaxation time : r
Viscosity: h = G0r
Rough idea: solid on a time scale << r, with modulus G0
Liquid on a time scale >> r, with viscosity h = G0r
solvent
viscosity
get G0
G0/2
r
1/r
solvent
viscosity
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Passive µ-rheology: the key step
Seed the sample with probe particles, then :
Obtain G’(w),
G"(w)
Measure mean squared
displacement <r2(t)>
0.1 µm
<r2> has to be measured on
length scales < 1 nm to 1µm !
1 nm
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Outline
• Mechanical rheology and µ-rheology
• µ-rheology : a few examples
• Mesuring displacements at a microscopic level: DWS
• The multispeckle « trick »
• Conclusions
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Light scattering: the concept
A light scattering experiment
Speckle image
q
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From particle motion to speckle
fluctuations
r(t)
r(t+)
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From particle motion to speckle
fluctuations
Weakly scattering media
(single scattering)
Speckles fluctuate if
r() ~ l ~0.5 µm
(Dynamic Light Scattering)
r(t)
r(t+)
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Diffusing Wave Spectroscopy (DWS):
DLS for turbid samples
Detector
DWS and µ-rheology
Photon propagation:
Random walk
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Diffusing Wave Spectroscopy (DWS):
DLS for turbid samples
L
l*
Detector
Photon propagation:
Random walk
Speckles fully fluctuate for
r2> ~ l2 / Nsteps =
l2 / (L/ l* )2 << l2
Typically: L ~ 0.1-1 cm, l* ~ 10-100 µm
r2> as small as a few Å2!
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How to quantify intensity fluctuations
Photomultiplier (PMT)signal
I
PMT
c
t
Intensity autocorrelation function
g 2 ( )  1 
I (t ) I (t   )
I (t )
2
g2-1
t
1
t
c

(other functions may be used, see L. Brunel's talk)
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From g2()-1 to <r2()>
• Well established formalism exists since ~1988
• Depends on the geometry of the experiment
A good choice: the backscattering geometry
g 2 ( )  1  exp  2

 2n 
k02  

 l 
r 2 ( ) k02 

2
 2
Note: no dependence on l*
(corrections are necessary for finite sample thickness, curvature, see L. Brunel's talk)
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Outline
• Mechanical rheology and µ-rheology
• µ-rheology : a few examples
• Mesuring displacements at a microscopic level: DWS
• The multispeckle « trick »
• Conclusions
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The problem: time averages!
g 2 ( )  1 
I (t ) I (t   )
I (t )
2
t
1
t
max= 20 s
Texp ~ 1 day!
I(t) PMT signal
• Average over ~ Texp = 103-104 max
Could be too long!
• Time-varying samples? (aging, aggregation...)
• Sample should explore all possible
configurations over time (ergodicity). Gels? Pastes?
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The Multispeckle technique
Average 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
g 2 ( )  1 
I p (t ) I p (t   )
I p (t )
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p ,t
1
p ,t
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The Multispeckle technique (MS)
g 2 ( )  1 
I p (t ) I p (t   )
I p (t )
2
p ,t
1
max= 20 s
Texp ~ 20 s!
p ,t
• 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
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Outline
• Mechanical rheology and µ-rheology
• µ-rheology : a few examples
• Mesuring displacements at a microscopic level: DWS
• The multispeckle « trick »
• Conclusions
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µ-rheology
First paper: Mason & Weitz,
1995 (306 citations)
Since then: > 680 papers
DWS
First papers: 1988
Since then: > 1470 papers
Number of µ-Rheology papers
µ-rheology and DWS: a well
established field, but in its commercial
infancy!
120
100
80
60
40
20
0
1996 1997 1998 1999 2000 2001 2002 2003 2004 2005 2006 2007 2008
Publication year
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MSDWS µ-rheology
g2()-1
Multispeckle DWS
• Reduced Texp
• Time-varying dynamics
• Non-ergodic samples
r2()
G'(w), G"(w)
µ-rheology
• Linear response probed
• No inhomogeneous response
• Full spectrum at once
• No need to load/unload rheometer
• Cheaper
• Sensitive to nanoscale motion
• Good average over probes
• Optically simple & robust
• No stringent requirements
on optical properties (turbidity...)
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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
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Additional material
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µ-rheology: from <r2> to G’, G"
or
General formulas:
Simpler approach (T. Mason, see [2])
assume that locally <r2> be a power law:
then,
with
and
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DWS: qualitative aproach
l = 1/rs scattering mean free path
l* transport mean free path
l* = l /<1-cosq>
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):
df ~ <q2><r2> ~ k02<r2>
Total change in photon phase for a path
(uncorrelated particle motion):
f ~ k02<r2> (L/ l * )2
Complete decorrelation of DWS signal for f ~ 2:
r2> ~ l2 / (L/ l * )2 << l2
[r2> as small as a few Å2!!]
Weitz & Pine
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DWS: quantitative approach
Intensity correlation function g2(t)-1 = b [g1(t)]2
(incoherent) sum over photon paths
with t/ = k02< r2(t)>/ 6, k0 = 2/l, and P(s) path length distribution
(example: for brownian particles, <r2(t)> = 6Dt and 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]
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Backscattering geometry
g1(t) ~

~ exp   6t / 

 independent of l*: don’t need
to know/measure l*!
 = (k02D)-1
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Transmission geometry
 = (k02D)-1
g1(t)
Note: l* has to be determined.
Measure transmission T  5l * / 3L  5l * / 3L
1  4l * / 3L
Calibrate against reference sample
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