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Unexpected drop of dynamical heterogeneities in colloidal suspensions approaching the jamming transition. Luca Cipelletti 1,2 , Pierre Ballesta 1,3 , Agnès Duri 1,4 1 LCVN Université Montpellier 2 and CNRS, France 2 Institut Universitaire de France 3 SUPA, University of Edinburgh

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slide1

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).

outline
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
what quantities should we measure
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
simulations lj
Simulations (LJ)

L. Berthier, PRE 2002

dynamical length scale in 2d granular media
Dynamical length scale in 2D granular media

Lechenault et al., EPL 2008

Keys et al., Nat. Phys. 2007

confocal microscopy on colloidal hs
Confocal microscopy on colloidal HS

Weeks et al. Science 00

Weeks et al., J. Phys. Cond. Mat 07

«  »

what quantities should we measure1
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)

temporally heterogeneous dynamics1
Temporally heterogeneous dynamics

heterogeneous

homogeneous

temporally heterogeneous dynamics2
Temporally heterogeneous dynamics

heterogeneous

homogeneous

dynamical susceptibility in glassy systems
Dynamical susceptibility in glassy systems

Supercooled liquid (Lennard-Jones)

<Q(t)>

Lacevic et al., PRE 2002

c4=N var[Q(t)]

dynamical susceptibility in glassy systems1

c4 (t) ~

Dynamical susceptibility in glassy systems

Nblob regions

c4=N var[Q(t)] ~ N (1/Nblob) = N/Nblob

how can we measure c 4
How can we measure c4?

Time-resolved light scattering experiments (TRC)

experimental setup
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

time resolved correlation

lag t

time tw

Time Resolved Correlation

2-time correlation function

Cipelletti et. Al JPCM 03, Duri et al. PRE 2005

slide18

Average over tw

‘dynamical

susceptibility’ c4(t )

intensity correlation function g2(t) - 1

fixed t, vs.tw

fluctuations of the dynamics

g2(tw,t)

tw (sec)

Average

dynamics

var(g2)(t)

g2(t) - 1

outline1
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
slide20

A « model system »: shaving cream

D.J. Durian, D.A. Weitz, D.J. Pine (1991) Science 252, 686

g2-1 = fraction of pathsnot rearranged

x

slide21

A « model system »: shaving cream

3D foam (DWS)

Mayer et al. PRL 2004

slide22

age dependence of c

tw

c (tw,t)

4

t

Coarsening of the foam

tw

slide23

Scaling of c during coarsening

4

c (tw,t)/l*3(cm-3)

Mayer et al. PRL 2004

4

cNblob

2<G>(tw)t

Less bubbles more fluctuations!

outline2
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
experimental system
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
diluted samples
« Diluted » samples

Brownian behavior

diluted samples1
« Diluted » samples

R/100 !!

DWS probes dynamics on a length scale

ll*/L ~ 10 – 35 nm << R

L

concentrated samples slow dynamics
Concentrated samples: slow dynamics

Fast dynamics

(phototube)

Slow dynamics

(CCD)

2 time intensity correlation function
2-time intensity correlation function

f = 66.4%

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
average dynamics

Average dynamics

Relaxation timet0 ~

jc = 0.752

average dynamics1

Average dynamics

Stretching exponent b

c vs c 4 different normalization

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
slide35

Measurement time issue?

Merolle et al., PNAS 2005

measurement time issue

tseg

tseg

tseg

tseg

tseg

tseg

Measurement time issue?

g2(t,t)-1

Does c*(tseg,j) depend on tseg ?

proposed physical mechanism

Proposed physical mechanism

Competition between :

Growth of x on approaching jc

Smaller displacement associated

with each rearrangement event

(tigther packing)

Nblobc*

More events c*

required to

relax system

dws and intermittent dynamics
DWS and intermittent dynamics

Inspired by Durian, Weitz & Pine (Science, 1991)

dws and intermittent dynamics1
DWS and intermittent dynamics

Inspired by Durian, Weitz & Pine (Science, 1991)

Light is decorrelated

x

dws and intermittent dynamics2
DWS and intermittent dynamics

Inspired by Durian, Weitz & Pine (Science, 1991)

Light is decorrelated

x

dws and intermittent dynamics3
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

dws and intermittent dynamics4
DWS and intermittent dynamics

Inspired by Durian, Weitz & Pine (Science, 1991)

Light is decorrelated

x

p = 1 « brownian » rearrangements

p = 2 « ballistic » rearrangements

simulations
Simulations
  • 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 )
simulations vs experiments
Simulations vs. experiments

experiments

simulations

simulation parameters
Simulation parameters

p = 1.65 supradiffusive motion

x3 - grows continuously with j

-very large!!

Cell thickness!

conclusions
Conclusions

Dynamics heterogeneous

Non-monotonic behavior of c*

conclusions1
Conclusions

Dynamics heterogeneous

Non-monotonic behavior of c*

Competition between

- increasing size of dynamically

correlated regions

conclusions2
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

thanks to
Thanks to…

V. Trappe

D. Weitz

L. Berthier

G. Biroli

M. Cloître

CNES

Softcomp

ACI

IUF

scaling of c revisited
Scaling of c* (revisited)

c* ~ 1 / (# rearrangements in the scattering volume needed

to decorrelate the scattered light)

c* ~ 1/(Nblob Nev)

Nblob,Nevdepend on j, q, tw, …

length scale dependence of c
Length scale dependence of c

Strongly attractive colloidal gel (Nblob = 1)

Increasing q

Duri & LC, EPL 76, 972 (2006)

strongly attractive gels scaling of c
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)

jump size
Jump size

s 2 d 2

~[x/l*]2

~1/R2

~1/10

d ~ R

d ~ 10-3R

colloidal gel

buoyancy-matched polystyrene colloids

  • low volume fraction 10-4 ÷ 10-3
  • screen charges “fast” aggregation (DLCA)

21 nm diam suspended in H2O/D2O

MgCl2 16 mM

Colloidal gel
time averaged dynamics
Time-averaged dynamics

g2(q,t) - 1 ~ [f(q,t)]2

  • Fast dynamics: overdamped vibrations
  • (~ 500 nm) Krall & Weitz PRL 1998
  • Slow dynamics: rearrangements
q dependence of t f and p
q dependence of tf and p

« ballistic » motion

« compressed » exponential

a surprising but quite general behavior
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], tfµq-1,p > 1

compressed exponential
Compressed exponential

f(q,t) µ exp[-(t/tf) 1.5]

slide60

Decreasing T

Glotzer et al.

c4 increases when decreasing T