1. Unexpected drop of dynamical heterogeneities in colloidal suspensions approaching the jamming transition
Luca Cipelletti1,2, Pierre Ballesta1,3, Agnès Duri1,4
1LCVN Université Montpellier 2 and CNRS, France
2Institut Universitaire de France
3SUPA, University of Edinburgh
4Desy, Hamburg
P. Ballesta, A. Duri, and L. Cipelletti, Nature Physics 4, 550 (2008).

2. Soft glassy materials
Eric Weeks

3. Soft glassy materials
Eric Weeks

4. Outline What are dynamical heterogeneities ?
Why should we care about DH ?
How can we measure DH ?
Shaving cream: a model system for DH
Colloids: DH (very) close to jamming

5. What quantities should we measure?
Space and time-resolved correlation functions f(t,t+t,r) or particle displacement
Simulations (« far »  from Tg!)
Granular systems (2D, athermal, see Dauchot’s talk)
(Confocal) microscopy on colloidal systems

6. Simulations (LJ)
L. Berthier, PRE 2002

7. Dynamical length scale in 2D granular media
Keys et al., Nat. Phys. 2007
Lechenault et al., EPL 2008

8. Confocal microscopy on colloidal HS
Weeks et al. Science 00
Weeks et al., J. Phys. Cond. Mat 07
«  »

9. What quantities should we measure?
Space- and time-resolved correlation functions f(t,t+t,r) or particle displacement
Simulations (far  from Tg!)
Granular systems (2D, athermal)
(Confocal) microscopy on colloidal systems
( stringent requirements on particles (size, optical mismatch…), difficult close to jamming)
Time-resolved correlation functions f(t,t+t) (no space resolution)

10. Temporally heterogeneous dynamics
homogeneous

11. Temporally heterogeneous dynamics
homogeneous
heterogeneous

12. Temporally heterogeneous dynamics
homogeneous
heterogeneous

13. Dynamical susceptibility in glassy systems
Supercooled liquid (Lennard-Jones)
<Q(t)>
Lacevic et al., PRE 2002
c4 = N var[Q(t)]

14. Dynamical susceptibility in glassy systems
Nblob regions
c4 = N var[Q(t)] ~ N (1/Nblob) = N/Nblob
c4 (t) ~

15. How can we measure c4?
Time-resolved light scattering experiments (TRC)

16. Experimental setup CCD-based (multispeckle)
Diffusing Wave Spectroscopy
CCD
Camera
Laser beam
Random walk w/ step l*
Change in speckle field mirrors change in sample configuration

17. Time Resolved Correlation
time tw
lag t
2-time correlation function
Cipelletti et. Al JPCM 03, Duri et al. PRE 2005

18. intensity correlation function g2(t) - 1
Average over tw
fixed t, vs. tw
fluctuations of the dynamics
g2(tw,t)
tw (sec)
Average
dynamics
var(g2)(t)
‘dynamical
susceptibility’ c4 (t )
g2(t) - 1

19. Outline What are dynamical heterogeneities ?
Why should we care about DH ?
How can we measure DH ?
Shaving cream: a model system for DH
Colloids: DH (very) close to jamming

20. A « model system »: shaving cream
D.J. Durian, D.A. Weitz, D.J. Pine (1991) Science 252, 686
g2-1 = fraction of paths not rearranged x

21. A « model system »: shaving cream
3D foam (DWS)
Mayer et al. PRL 2004

22. age dependence of c tw
c (tw ,t) 4 t
Coarsening of the foam tw

23. Scaling of c during coarsening4
c (tw ,t)/l*3 (cm-3)
Mayer et al. PRL 2004 4
c Nblob
2<G> (tw)t
Less bubbles more fluctuations!

24. Outline What are dynamical heterogeneities ?
Why should we care about DH ?
How can we measure DH ?
Shaving cream: a model system for DH
Colloids: DH (very) close to jamming

25. Experimental system PVC xenospheres in DOP radius R ~ 5 mm
Polydisperse (~ 33%)
Brownian
Excluded volume interactions
j = 64% – 75% (close to jamming)
L = 2 mm
l* = 200 mm

26. « Diluted » samples
Brownian behavior

27. DWS probes dynamics on a length scale
« Diluted » samples
R/100 !!
DWS probes dynamics on a length scale
ll*/L ~ 10 – 35 nm << R L

28. Concentrated samples: slow dynamics
Fast dynamics
(phototube)
Slow dynamics
(CCD)

29. 2-time intensity correlation function
t0 (sec)
Fit: g2(tw,tw+t) - 1 = aexp[-(t/t0)b]
Initial regime: « simple aging » (t0 ~ tw1.1 ± 0.1)
Crossover to stationary dynamics, large fluctuations of ts

30. Average dynamics
Relaxation time t0 ~
jc = 0.752

31. Average dynamics
Stretching exponent b

32. Fluctuations of the dynamics: c
j = 0.738

33. c vs c4: different normalization
~ correlation volume
In our experiments:
No N factor
N is not known precisely
Need model to extract correlation volume x3 from c

34. Fluctuations of the dynamics: c* vs j

35. Measurement time issue?
Merolle et al., PNAS 2005

36. Measurement time issue?
tseg
g2(t,t)-1
Does c*(tseg,j) depend on tseg ?

37. Not a measurement time issue !

38. Proposed physical mechanism
Competition between :
Growth of x on approaching jc
Smaller displacement associated
with each rearrangement event
(tigther packing)
Nblob c*
More events c*
required to
relax system

39. DWS and intermittent dynamics
Inspired by Durian, Weitz & Pine (Science, 1991)

40. DWS and intermittent dynamics
Inspired by Durian, Weitz & Pine (Science, 1991)
Light is decorrelated x

41. DWS and intermittent dynamics
Inspired by Durian, Weitz & Pine (Science, 1991)
Light is decorrelated x

42. DWS and intermittent dynamics
Inspired by Durian, Weitz & Pine (Science, 1991)
Light is decorrelated x
Number of events between
t and t +t
Mean squared change of phase for
1 event

43. DWS and intermittent dynamics
Inspired by Durian, Weitz & Pine (Science, 1991)
Light is decorrelated x
p = « brownian » rearrangements
p = 2 « ballistic » rearrangements

44. Simulations x Photon paths as random walks on a 3D cubic lattice
Lattice parameter = l*, match cell dimensions
Random rearrangement events of size x3
Calculate with x
Parameters :
p (use one single p for all j) x3
s2f (we expect s2f as j jc )

45. Simulations vs. experiments

46. Simulation parameters
p = supradiffusive motion
x3 - grows continuously with j
- very large!!
Cell thickness!

47. Conclusions
Dynamics heterogeneous
Non-monotonic behavior of c*

48. Conclusions Dynamics heterogeneous Non-monotonic behavior of c*
Competition between
- increasing size of dynamically
correlated regions

49. Conclusions Dynamics heterogeneous Non-monotonic behavior of c*
Competition between
- increasing size of dynamically
correlated regions
- decreasing effectiveness of
rearrangements
Dynamical heterogeneity dictated by the number of rearrangements
needed to relax the system on the probed length scale

50. Thanks to… V. Trappe D. Weitz L. Berthier G. Biroli M. Cloître CNES
Softcomp
ACI
IUF

51. Scaling of c* (revisited)
c* ~ 1 / (# rearrangements in the scattering volume needed
to decorrelate the scattered light)
c* ~ 1/(Nblob Nev)
Nblob , Nev depend on j, q, tw, …

52. Length scale dependence of c
Strongly attractive colloidal gel (Nblob = 1)
Increasing q
Duri & LC, EPL 76, 972 (2006)

53. Strongly attractive gels: scaling of c*
c* ~ var(Nev)/<Nev>2 ~ <Nev>-1
< Nev > ~ tf ~ q-1
c* ~ q
Duri & LC, EPL 76, 972 (2006)

54. Jump size
s d 2
~[x/l*]2
~1/R2
~1/10
d ~ R
d ~ 10-3R

55. Colloidal gel buoyancy-matched polystyrene colloids
low volume fraction ÷ 10-3
screen charges “fast” aggregation (DLCA)
21 nm diam suspended in H2O/D2O
MgCl2 16 mM

56. Time-averaged dynamics
g2(q,t) - 1 ~ [f(q,t)]2
Fast dynamics: overdamped vibrations
(~ 500 nm) Krall & Weitz PRL 1998
Slow dynamics: rearrangements

57. q dependence of tf and p « ballistic » motion
« compressed » exponential

58. A surprising but quite general behavior!
Onion gel
Micellar polycrystal
Conc. Emulsion
Ramos & Cipelletti PRL 2001
Cipelletti et al Faraday Discuss 2003
Laponite
Depletion gels, …
Bandyopadhyay et al. PRL 2004
Chung et al. PRL 2006
f(q,t) µ exp[-(t/tf) p], tf µ q-1, p > 1

59. Compressed exponential
f(q,t) µ exp[-(t/tf) 1.5]

60. c4 increases when decreasing T
Glotzer et al.
c4 increases when decreasing T