dynamical heterogeneity and relaxation time very close to dynamical arrest
DESCRIPTION
Dynamical heterogeneity and relaxation time very close to dynamical arrest. L. Cipelletti LCVN, Université Montpellier 2 and CNRS Collaborators: L. Berthier (LCVN), V. Trappe (Fribourg) Students: P. Ballesta, G. Brambilla, A. Duri, D. El Masri Postdocs: S. Maccarrone, M. Pierno. - PowerPoint PPT PresentationTRANSCRIPT
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