dynamical heterogeneity and relaxation time very close to dynamical arrest

Dynamical heterogeneity and relaxation time very close to

dynamical arrestL. Cipelletti

LCVN, Université Montpellier 2 and CNRS

Collaborators: L. Berthier (LCVN), V. Trappe (Fribourg)Students: P. Ballesta, G. Brambilla, A. Duri, D. El MasriPostdocs: S. Maccarrone, M. Pierno

Soft glassy/jammed materials

E. Weeks

Dynamical heterogeneity is ubiquitous!

Granular matter Colloidal Hard Spheres Repulsive disks

Weeks et al. Science 2000Keys et al. Nat. Phys. 2007 A. Widmer-Cooper Nat. Phys. 2008

Outline

• Average dynamics and dynamical hetrogeneity of supercooled colloidal HS

• Temporal fluctuations of the dynamics in

jammed/glassy materials

• Real-space measurements of correlations: ultra-long

range spatial correlations and elasticity

Equilibrium dynamics of supercooled HSEquilibrium dynamics of supercooled HS

10-6 10-4 10-2 1 102 104 106

0.0

0.2

0.4

0.6

0.8

1.0 0.0480 0.5555 0.3096 0.5772 0.4967 0.5818

0.5852 0.5916 0.5970

E23 Dj3 Dj7 Dj11 Dj17 E14 E13 E11 Dj22f(

qR=

3.5,

t)

t (sec)

N

jjjs i

Nqf

1

()0(exp1

),( rrq

Brambilla et al., PRL 2009

Multispeckle DLS• PMMA in decalin/tetralin• a ~ 110 nm• polidispertsity 12.2% (TEM)

(sec)

f s(qa

=2.

75,

)

0.0 0.2 0.4 0.6-4

-2

0

2

4

0.55 0.60

0

2

4

1.0 1.5 2.0 2.5 3.00.12

0.13

0.14

0.15

0 2 4 60

2

4

6

log( / 0)

log( /

0)

log -

log

/2

dependence of the relaxation time

Brambilla et al., PRL 2009

MCT : ~ (c-)-

c= 0.59, = 2.6

VFT: ~ exp[(0-)-]0= 0.614, = 10= 0.637, = 2

Glass/jamming transition in colloidal HS

c ~ .57-.59

Fluid Glass (out of equilibrium)

rcp ~ .64

c ~ .57-.59

Fluid

j = rcp ~ .64

c ~ .57-.59

Fluid Glass

rcp ~ .640

,

p

p ,

MCT

Jamming

Thermodynamic

,

p

Dynamical heterogeneity

Weeks et al. Science 2000

Dynamical heterogeneity

Weeks et al. Science 2000var[f()] 4()

dynamical susceptibility

Dynamical susceptibility in glassy systems

Supercooled liquid (Lennard-Jones)

Lacevic et al., Phys. Rev. E 2002

4 N var[Q(t)]

Dynamical susceptibility in glassy systems

Spatial correlation of the dynamics

Weeks et al. Science 2000

Dynamical heterogeneity: the theoreticians’ trick

222244 )0()()()( StT

c

ktt T

V

BNVENPT

T

tftT

)()(

)()(

tft

For colloidal HS at high

Berthier et al., J. Phys. Chem. 2007

Goal: calculate 4-point dynamical susceptibility 4 ~ size of rearranged region

DH in colloidal HS

Berthier et al., Science 2005

~ (

a)3

dynamical susceptibility 4()

A useful relationship

)()0()( 224 S

)()(

f•

• equilibrium• OK at high

Berthier et al., J. Phys. Chem.2007

~ 0.20 – 0.58

(sec)

dependence of the max of 4

0.35 0.40 0.45 0.50 0.55 0.6010-2

10-1

1

10

10-3 10-2 10-1 1 10 102 103 10410-3

10-1

10

S(0)2

~ (c)-2

(sec)

2/

*

High :deviation from MCT

Brambilla et al., PRL 2009

MCT ~ (c-)-2

Outline

• Average dynamics and dynamical hetrogeneity of supercooled colloidal HS

• Temporal fluctuations of the dynamics in

jammed/glassy materials

• Real-space measurements of correlations: ultra-long

range spatial correlations and elasticity

Time Resolved Correlation (TRC)

time tlag

degree of correlation cI(t,) = - 1< Ip(t) Ip(t +)>p

< Ip(t)>p<Ip(t +)>p

Cipelletti et al. J. Phys:Condens. Matter 2003,Duri et al. Phys. Rev. E 2006

10 102 103 104 105

0.0

0.2

0.4

g 2(t)-

1

(sec)

intensity correlation function g2(1

Average over t

degree of correlation cI(t,) = - 1< Ip(t) Ip(t +)>p

< Ip(t)>p<Ip(t +)>p

Average dynamicsg2(1 ~ f()2

intensity correlation function g2(1

Average over t

degree of correlation cI(t,) = - 1< Ip(t) Ip(t +)>p

< Ip(t)>p<Ip(t +)>p

fixed , vs. t

fluctuations of the dynamics

c I(t,)

-1

t (sec)0 10

4 2x10

410 102 103 104 1050.0

0.2

0.4

g 2(t)-

1

(sec)

Average dynamicsg2(1 ~ f()2 var[cI()] ~ ()

(Polydisperse) colloids near rcp

Ballesta et al., Nature Physics 2008

• PVC in DOP• a ~ 5 µm• polidispertsity ~33%• slightly soft• close to rcp • multiple scattering (DWS) ~ 10 nm << a

Non-monotonic behavior of *()

Fit: g2()-1 ~ exp[-(/)]

relaxation time

stretching exponent

peak of dynamical susceptibilitynon monotonic!

103

104

0 (

s)

0.5

1.0

1.5

2.0

0.60 0.65 0.70 0.750.00

0.02

0.04

eff

Non-monotonic behavior of *()

Competition between:increasing ( as )

increasingly restrained displacementjump size decreases(many events over 0, as )

Model:• random events of size • Poissonian statistics• Quasi-ballistic motion( <r2> ~np, p = 1.65)

103

104

0 (

s)

0.5

1.0

1.5

2.0

0.60 0.65 0.70 0.750.00

0.02

0.04

eff

Simulating the model

0.5

1.0

1.5

2.0

c

b

a

0.00

0.02

0.04

*

10-1 10 103 10510-1

10

103

Figure 3Manuscript NPHYS-2007-05-00481DP. Ballesta, A. Duri, and L. Cipelletti

3

~

volume fraction

cell thickness!

inverse jump size

Outline

• Average dynamics and dynamical hetrogeneity of supercooled colloidal HS

• Temporal fluctuations of the dynamics in

jammed/glassy materials

• Real-space measurements of correlations: ultra-long

range spatial correlations and elasticity

2.3 mm

Measuring by Photon Correlation Imaging

diaphragm

lens

object plane

image plane

focal plane

CC

D

sample

= 90° 1/q ~ 45 nm

Duri et al., Phys. Rev. Lett. 2009

2.3 mm

Local, instantaneous dynamics: cI( t, ,r)

< Ip(t) Ip(t +)>p(r)

< Ip(t)>p<Ip(t +)>p(r)

cI(t, , r) = - 1

Note: << cI(t, , r) >t>r = g2()-1

[g2()-1] ~ f()2

Dynamic Activity Maps

Brownian particlesg2()-1~ exp[-/r], r = 40 s

Colloidal gelg2()-1~ exp[-(/r)1.5], r = 5000 s

cI (t0,r/200 , r)

cI (t0,r/10 , r)

Movie accelerated 10x

Movie accelerated 500x

2 mm

2 mm

0 2 4 60.0

0.5

1.0 Onion gel Colloidal gel "Artificial skin", RH = 12% "Artificial skin" AS, RH = 62% Soft spheres, T = 24.5°C, ~0.69

Soft spheres, T = 28°C, ~0.57 Hard Spheres, ~0.5468 Hard Spheres ~0.5957

r (mm)

G4(

r), G

4(r)

~Spatial correlation of the dynamics:

~ system size in jammed soft matter!

Maccarrone et al., Soft Matter 2010

When does infinity?

• G' > G'' : strain propagation in a predominantly solid-like material

• Magnitude of G' unimportant (G'onions/G'colloidal gel ~ 6x104 !!)

• Origin of elasticity:• entropic (e.g. hard spheres) very small• enthalpic (attractive gels, squeezed particles...) system-size

Evidence of internal stress relaxation

J. Kaiser, O. Lielig,, G. Brambilla, LC, A. Bausch, submitted

Dynamics of actin/fascin networks

0 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300

0

50

100

150

200

250

Average displacement : 0.75 µmStandard deviation: 0.48 µm

cos, where is the angle w/ respect to x axis

ActinFascinJonaR0.1_night.avi: data taken fromF:\4CAMS\Jona\090220_sample4\CCD3\NightDisplacement map calculated for images 3935 and 4100 (MI00001.dat), corresponding to 10510 sec after preparing the sample.boxsize 40 pixels, average over 4 frames, time lag is 330 sec. 1 pixel = 2.2 µm

Y c

oord

inat

e (µ

m)

X coordinate (µm)5 µm

-1.000

-0.8000

-0.6000

-0.4000

-0.2000

0

0.2000

0.4000

0.6000

0.8000

1.000

During polymerization(displacement map over 330 sec)strain field @ late stages of network formation

104 105

0.1

1

<r

( =

330

s)>

time after preparing sample (sec)

ageaverage strain field microcopic dynamics

Conclusions

• DH a general feature of glassy/jammed dynamics

• Supercooled hard spheres: • equilibrium dynamics above the (apparent) MCT divergence• limited to 5-10a

• Jammed materials:• and may decouple!• ~ system size• role of the origin of elasticity

Thanks to...

L. Berthier (LCVN), V. Trappe (Fribourg)

Hard spheresG. Brambilla, M. Pierno, D. El Masri (LCVN), A. Schofield (Edinburgh),G. Petekidis (FORTH)

TRCA. Duri, P. Ballesta (LCVN), D. Sessoms, H. Bissig (Firbourg)

PCIA. Duri, S. Maccarrone (LCVN), D. Sessoms (Fribourg), E. Pashkowski

(Unilever)

FundingCNES, ACI, ANR, PICS, Unilever

Open questions...

DH and aging?

Origin of the dynamics in (undriven) soft materials

Determining the volume fraction dependence of the short-time diffusing coefficient

0.000 0.001 0.002 0.003

1

f(q,

t)

t (sec)

f(q,t)PresentationShortTimeDvsPhi.opj

= 0.2

= 1/Dssq2

Absolute uncertainty: ~4% Relative uncertainty: ~ 10-4

0.00 0.05 0.10 0.15 0.200.6

0.8

1.0 Tokuyama & Oppenheim (T&O) Beenakker (B)

Fitted to T&O Fitted to (B) Scaled to simulation

data

Ds s/D

0

PS gel: 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

PS gels: 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

PS gel: temporally heterogeneous dynamics

<cI(tw,)>tw : average dynamicscI(tw,) @ fixed

temporal fluctuations

Length scale dependence of var(cI)

Increasing q

104 105

0.01

0.1

*

q (cm-1)

slope 1.14 +/- 0.11

Duri & LC, EPL 76, 972 (2006)

PS gel

1.1 +/- 0.1

Scaling of *

* ~ var(n)/<n> ~ 1/<n>

<n> ~ f ~ 1/q

* ~ q

Dynamical susceptibility in glassy systems

Supercooled liquid (Lennard-Jones)

Lacevic et al., Phys. Rev. E 2002

4 N var[Q(t)]

Dynamical susceptibility in glassy systems

4 N

var[Q(t)]

4 dynamics spatially correlated

N regions

Dynamical susceptibility in glassy systems

Spatial correlation of the dynamics

Weeks et al. Science 2000

The smart trick applied to colloidal HS

t

10-3

(t

)

F(t

)

Define

Dynamical heterogeneity: the theoreticians’ trick

222244 )0()()()( StT

c

ktt T

V

BNVENPT

T

tftT

)()(

)()(

tft

For colloidal HS at high

Berthier et al., J. Phys. Chem. 2007

Goal: calculate 4-point dynamical susceptibility 4 ~ size of rearranged region

Evidence of a growing dynamic length scale

~ 0.20 – 0.58Berthier et al., Science 2005

dependence of the max of 4

0.35 0.40 0.45 0.50 0.55 0.6010-2

10-1

1

10

10-3 10-2 10-1 1 10 102 103 10410-3

10-1

10

S(0)2

~ (c)-2

(sec)

2/

*

MCT

High :deviation from MCT

Brambilla et al., PRL 2009

2.3 mm

Photon Correlation Imaging (PCIm)

sample

annular slitlens

object plane

image plane

focal plane

CCD

= 6.4° q = 1 m-1.Duri et al., Phys. Rev. Lett. 2009

Spatial correlation of the dynamics

Other PCIm geometries

diaphragm

lens

object plane

image plane

focal plane

CC

D

sample

sample

diaphragm

lens

object plane

image plane

focal plane

CC

D

beam splitter

a) b)

Additional stuff on diverging