unexpected drop of dynamical heterogeneities in colloidal suspensions approaching the jamming...

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, France2Institut Universitaire de France3SUPA, University of Edinburgh

4Desy, Hamburg

P. Ballesta, A. Duri, and L. Cipelletti, Nature Physics 4, 550 (2008).

Soft glassy materials

Eric Weeks

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?

Space and time-resolved correlation functions f(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)

L. Berthier, PRE 2002

Dynamical length scale in 2D granular media

Keys et al., Nat. Phys. 2007 Lechenault et al., EPL 2008

Confocal microscopy on colloidal HS

Weeks et al. Science 00 Weeks et al., J. Phys. Cond. Mat 07

« »

What quantities should we measure?

Space- and time-resolved correlation functions f(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+) (no space resolution)

Temporally heterogeneous dynamics

homogeneous

Temporally heterogeneous dynamics

homogeneous heterogeneous

Dynamical susceptibility in glassy systems

Supercooled liquid (Lennard-Jones)

Lacevic et al., PRE 2002

4 N var[Q(t)]

<Q

(t)>

Dynamical susceptibility in glassy systems

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

Nblob regions

4 () ~ '

3 ',',',',0t

tttftttfd rr

How can we measure 4?

Time-resolved light scattering experiments (TRC)

Experimental setup

CCD-based (multispeckle)Diffusing Wave Spectroscopy

CCDCamera

Las

er b

eam

Change in speckle field mirrors change in sample configuration

Random walk w/ step l*

Time Resolved Correlation

time twlag

2-time correlation function

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

intensity correlation function g2(1

Average over tw

Average dynamicsg2(1

fixed , vs. tw

fluctuations of the dynamics

var(g2)() ‘dynamical susceptibility’

tw (sec)g 2(

t w,

)

Outline






A « model system »: shaving creamD.J. Durian, D.A. Weitz, D.J. Pine (1991) Science 252, 686

g2-1 = fraction of paths not rearranged

A « model system »: shaving cream

3D foam (DWS)

Mayer et al. PRL 2004

age dependence of

10-2 10-1 100

10-8

10-7

10-6

10-5

10-4

10-3

tw (sec)

3348 5250 6760 1769610295 2180114265 26118

(t w

,t)

t (sec)

Coarsening of the foam

t

w

tw

tw

4

Scaling of during coarsening

10-1 100 101

10-4

10-3

10-2

10-1

100

101

tw (sec)

3348 5250 6760 17696 10295 21801 14265 26118

(t w

,t)

/l*3 (

cm-3)

(tw)t

Less bubbles more fluctuations!

t

wl

(c

m-3)

tw

10-2 10-1 100

10-8

10-7

10-6

10-5

10-4

10-3

tw (sec)

3348 5250 6760 1769610295 2180114265 26118

(t w

,t)

t (sec)

4

4

Mayer et al. PRL 2004

Nblob

Outline






Experimental system

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

« Diluted » samples

10-6 10-5 10-4 10-3 10-2 10-1 10010-20

10-19

10-18

10-17

10-16

10-15

= 28% = 46%

<r

2 ()>

(m

2 )

(sec)

1Brownian behavior

« Diluted » samples

10-6 10-5 10-4 10-3 10-2 10-1 10010-20

10-19

10-18

10-17

10-16

10-15

= 28% = 46%

<r

2 ()>

(m

2 )

(sec)

1

R/100 !!

DWS probes dynamics on a length scale

l*/L ~ 10 – 35 nm << R

L

Concentrated samples: slow dynamics

1E-4 1E-3 0.01 0.1 1 100.0

0.2

0.4

0.6

0.8

1.0

C:\lucacip\ParisToCopy\FluctuationsTheoryg 2

- 1

(arb .un.)

B ###

Fast dynamics(phototube)

Slow dynamics(CCD)

2-time intensity correlation function

• Initial regime: « simple aging » (0 ~ tw1.1 0.1)

• Crossover to stationary dynamics, large fluctuations of s

101 102 103 104 105

0,00

0,02

0,04

0,06

C:\lucacip\doc\papers\WorkInProgress\2004JapanMeeting2003\Figures\Pierre40pcr030422

tw (sec)

1194 4400 7900 14900 21900 44083 54800

n42 n500 n1000 n2000 n3000 n6169 n7700

g 2(t

w,t w

)-1

(sec)

0 40000 800000

500

1000

C:\lucacip\doc\papers\WorkInProgress\2004JapanMeeting2003\Figures\Pierre40pcr030422

s (se

c)

tw (sec)

TODO: check tw = 0

Fit: g2(tw,tw+1 = aexp[-(/)]

= 66.4%

(

sec)

Average dynamics

Relaxation time 0 ~ 04.001.1

1 c

c = 0.752

Average dynamics

Stretching exponent

Fluctuations of the dynamics:

= 0.738

vs 4: different normalization

In our experiments:

No N factor

~ correlation volume*4

• N is not known precisely• Need model to extract correlation volume 3 from

Fluctuations of the dynamics: vs

2.05.1

1 c

Measurement time issue?

Merolle et al., PNAS 2005

Measurement time issue?

tseg tseg tseg tseg tseg tseg

g2(t,)-1

Does *(tseg,) depend on tseg ?

Not a measurement time issue !

10-3 10-2 10-1 100 101 102 103

10-2

10-1

100

eff

0.637 0.6638 0.693 0.7152 0.7247 0.7377 0.7383 0.7442 0.7455

F F F F F F F F F

*(t

seg)

/*(t

exp)

tseg/0

Proposed physical mechanism

Competition between :

Growth of on approaching c

Smaller displacement associated with each rearrangement event (tigther packing)

Nblob *

More events *required torelax system

DWS and intermittent dynamicsInspired by Durian, Weitz & Pine (Science, 1991)

Light is decorrelated




Number of events betweent and t +

Mean squared change of phase for1 event

2



p = 1 « brownian » rearrangements

p = 2 « ballistic » rearrangements

Simulations

• Photon paths as random walks on a 3D cubic lattice

• Lattice parameter = l*, match cell dimensions

• Random rearrangement events of size 3

• Calculate with

Parameters :

• p (use one single p for all )3

2 (we expect 2

as

c )

2)(12 ),(1),(

s

s tgtg

Simulations vs. experiments

simulations

experiments

Simulation parameters

p = 1.65 supradiffusive motion

3 - grows continuously with -very large!!

Cell thickness!

Conclusions

Dynamics heterogeneous

Non-monotonic behavior of *

Conclusions



Competition between- increasing size of dynamically correlated regions

Conclusions



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…

V. TrappeD. WeitzL. BerthierG. BiroliM. Cloître

CNESSoftcompACIIUF

Scaling of * (revisited)

* ~ 1 / (# rearrangements in the scattering volume needed to decorrelate the scattered light)

* ~ 1/(Nblob Nev)

Nblob , Nev depend on , q, tw, …

Length scale dependence of

Increasing q

104 105

0.01

0.1

*

q (cm-1)

slope 1.14 +/- 0.11

Duri & LC, EPL 76, 972 (2006)

Strongly attractive colloidal gel (Nblob = 1)

Strongly attractive gels: scaling of *

* ~ var(Nev)/<Nev>2 ~ <Nev>-1

< Nev > ~ f ~ q-1

* ~ q

Duri & LC, EPL 76, 972 (2006)

104 105103

104 1.05 ± 0.04

f (se

c)

q (cm-1)

Jump size

2 2

~[/l*]2~1/R2

~1/10

~ R ~ 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

Time-averaged dynamics

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

,)0()(exp),( rrq

• Fast dynamics: overdamped vibrations(~ 500 nm) Krall & Weitz PRL 1998

• Slow dynamics: rearrangements

pfqg exp~1),(2

q dependence of f and p

« compressed » exponential

104 105103

104 1.05 ± 0.04

f (se

c)

q (cm-1)

« ballistic » motion

pfqg exp~1),(2

A surprising but quite general behavior!

Onion gel Micellar polycrystal Conc. Emulsion

f(q,) exp[-(t/f) p], f q-1, p > 1

Laponite Depletion gels, …

Ramos & Cipelletti PRL 2001 Cipelletti et al Faraday Discuss 2003

Bandyopadhyay et al. PRL 2004 Chung et al. PRL 2006

Compressed exponential

f(q,) exp[-(t/f) 1.5]

4 increases when decreasing T

Glotzer et al.

Decreasing T

unexpected drop of dynamical heterogeneities in colloidal suspensions approaching the jamming...

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