Dynamical heterogeneity at the
jamming transition of concentrated
colloids
P. Ballesta1, A. Duri1, Luca Cipelletti1,2
1LCVN
UMR 5587
Université Montpellier 2 and CNRS, France
2Institut Universitaire de France
[email protected]
Heterogeneous dynamics
correlation
1.00
0.75
0.50
0.25
0.00
-4
-3
-2
-1
0
1
2
3
10 10 10 10 10 10 10 10
t (arb. un.)
homogeneous
1.00
1.00
0.75
0.75
correlation
correlation
Heterogeneous dynamics
0.50
0.25
0.00
0.50
0.25
0.00
-4
-3
-2
-1
0
1
2
3
-4
-3
-2
-1
0
1
2
3
10 10 10 10 10 10 10 10
10 10 10 10 10 10 10 10
t (arb. un.)
t (arb. un.)
homogeneous
heterogeneous
1.00
1.00
0.75
0.75
correlation
correlation
Heterogeneous dynamics
0.50
0.25
0.50
0.25
0.00
0.00
-4
-3
-2
-1
0
1
2
3
-4
-3
-2
-1
0
1
2
3
10 10 10 10 10 10 10 10
10 10 10 10 10 10 10 10
t (arb. un.)
t (arb. un.)
homogeneous
heterogeneous
Dynamical susceptibility in glassy systems
Supercooled liquid (Lennard-Jones)
Lacevic et al., PRE 2002
c4 ~ var[Q(t)]
Dynamical susceptibility in glassy systems
N regions
c4 ~ var[Q(t)]
c4
dynamics spatially correlated
Decreasing T
Glotzer et al.
c4 increases when decreasing T
Outline
• Measuring average dynamics and c4 in colloidal suspensions
• c4 at very high j : surprising results!
• A simple model of heterogeneous dynamics
Experimental system & setup
PVC xenospheres in DOP
radius ~ 10 mm, polydisperse
j = 64% – 75%
Excluded volume interactions
Laser beam
Experimental system & setup
CCD
Camera
CCD-based (multispeckle)
Diffusing Wave Spectroscopy
Change in speckle field mirrors
change in sample configuration
Probe d << Rparticle
Time Resolved Correlation
lag t
time tw
< Ip(tw) Ip(tw +t)>p
degree of correlation cI(tw,t) =
-1
< Ip(tw)>p<Ip(tw +t)>p
fixed tw, vs. t
2-time intensity correlation
function g2(tw,t) - 1
2-time intensity correlation function
f = 66.4%
tw (sec)
n42
1194
4400
7900
14900
21900
44083
54800
0,04
0,02
1000
n500
ts (sec)
g2(tw,twt )-1
0,06
n1000
n2000
n3000
n6169
n7700
500
0,00
1
10
2
10
3
10
t (sec)
4
10
5
10
0
0
40000
80000
tw (sec)
Fit: g2(tw,t) ~ exp[-(C:\lucacip\doc\papers\WorkInProgress\2004Ja
t /ts (tw))p(tw)]
C:\lucacip\doc\papers\WorkInProgress\2004JapanMeeting2003\Figures\Pierre40pcr03042
• Initial regime: « simple aging » (ts ~ tw1.1  0.1)
• Crossover to stationary dynamics, large fluctuations of ts
2-time intensity correlation function
f = 66.4%
tw (sec)
n42
1194
4400
7900
14900
21900
44083
54800
0,04
0,02
1000
n500
ts (sec)
g2(tw,twt )-1
0,06
n1000
n2000
n3000
n6169
n7700
500
0,00
1
10
2
10
3
10
4
10
5
0
10
0
40000
80000
tw (sec)
t (sec)
Fit: g2(tw,t) ~ exp[-(C:\lucacip\doc\papers\WorkInProgress\2004Ja
t/ts(tw))p(tw)]
C:\lucacip\doc\papers\WorkInProgress\2004JapanMeeting2003\Figures\Pierre40pcr03042
Average dynamics :
< ts >tw , < p >tw
Average dynamics vs j
10
4
10
3
fresh
old reinitialized
ts (sec)
ts ~ |jeff- jc|
-1.01
jc= 0.752
0.64
0.68
0.72
jeff
0.76
Average relaxation time
Average dynamics vs j
1.5
4
ts ~ |jeff- jc|
-1.01
p
ts (sec)
10
fresh
old reinitialized
10
1.0
3
jc= 0.752
0.64
0.68
0.72
jeff
0.76
Average relaxation time
0.5
0.64
0.68
0.72
jeff
0.76
Average stretching exponent
Fluctuations from TRC data
lag t
time tw
< Ip(tw) Ip(tw +t)>p
degree of correlation cI(tw,t) =
-1
< Ip(tw)>p<Ip(tw +t)>p
fixed t, vs. tw
fluctuations of the dynamics
var(cI)(t)
c (t )
Fluctuations of the dynamics vs j
j = 0.74
0.4
c*
0.02
0.2
0.01
0.1
0.00
0.0
2
10
var(cI)
3
4
5
10
10
t (sec)
10
c4
(dynamical susceptibility)
g2(t)-1
var(cI)
0.3
Fluctuations of the dynamics vs j
j = 0.74
c*
0.02
0.06
0.4
0.2
0.01
c*
0.04
g2(t)-1
var(cI)
0.3
0.02
0.1
0.00
0.0
2
10
var(cI)
3
4
5
10
10
t (sec)
10
c4
(dynamical susceptibility)
0.00
0.64
0.68
0.72
jeff
Max of var (cI)
0.76
A simple model of intermittent
dynamics…
A simple model of intermittent
dynamics…
r
fully decorrelated
Durian, Weitz & Pine (Science, 1991)
Fluctuations in the DWP model
Random number of rearrangements
g2(t,t) – 1 fluctuates
r
Fluctuations in the DWP model
Random number of rearrangements
r
g2(t,t) – 1 fluctuates
r increases
fluctuations increase
Fluctuations in the DWP model
1
-1
r
10
4.6r0
-2
2.2r0
var(cI)
10
-3
10
r0
-4
10
-5
r increases
10
-6
10
-7
10
fluctuations increase
-5
-4
-3
-2
10 10 10 10 10
-1
1
t (arb. un.)
2
10 10
Approaching jamming…
partially decorrelated
r
partially decorrelated
Approaching jamming…
Correlation after n events
r
Probability of n events during t
Approaching jamming…
r
Poisson distribution:
Approaching jamming…
r
Poisson distribution:
Random change of phase
Correlated change of phase
Approaching jamming…
r
Poisson distribution:
Random change of phase
Correlated change of phase
Approaching jamming…
r
Poisson distribution:
b  1.5
Average dynamics
increasing j
decreasing sf2
1.0
2.0
increasing j
0.01
1.5
1
10
0.5
p
(s)
g1 (t)
0.1
1.0
0.0
-2
10
10
-1
1
10
t (arb. un.)
2
10
0.5
-3
10
10
-2
10
-1

sf
1
10
10
2
Fluctuations
r
Moderate j : large sf2
Near jamming : small sf2
few events
many events
large flucutations
small flucutations
Fluctuations
increasing j
decreasing sf2
10
1
-1
var(cI)
10
0.1
-2
0.01
10
-3
10
-2
10
-1
10
1
10
t (arb. un.)
2
10
Conclusions
0.06
Dynamics heterogeneous
Competition between
increasing size of dynamically
correlated regions ...
c*
Non-monotonic behavior of c*
0.04
0.02
0.00
0.64
0.68
0.72
jeff
0.76
Conclusions
0.06
Dynamics heterogeneous
Competition between
increasing size of dynamically
correlated regions
and
decreasing effectiveness of
rearrangements
c*
Non-monotonic behavior of c*
0.04
0.02
0.00
0.64
0.68
0.72
jeff
0.76
Conclusions
0.06
Dynamics heterogeneous
Competition between
increasing size of dynamically
correlated regions
and
decreasing effectiveness of
rearrangements
c*
Non-monotonic behavior of c*
0.04
0.02
0.00
0.64
0.68
0.72
jeff
0.76
Dynamical heterogeneity dictated by the number of rearrangements
needed to decorrelate
A further test…
Single scattering, colloidal fractal gel (Agnès Duri)
0.7
0.6
0.5
q=7355 cm
-1
q=9538 cm
-1
q=12393 cm
-1
q=16004 cm
-1
q=20656 cm
-1
-1
g2 (q,t) - 1
q=26506cm
0.4
0.3
q=37771 cm
-1
q=42396 cm
-1
q=52177 cm
-1
0.2
0.1
0.0
-0.1
10
100
1000
t (sec)
10000 100000
A further test…
sf2 ~ q2d 2
look at different q!
A further test…
sf2 ~ q2d 2
2.0
p
1.5
1.0
sf ~ (qd)
2
0.5
10
4
-1
q (cm )
2
look at different q!
A further test…
sf2 ~ q2d 2
look at different q!
2.0
0.1
p
c*
1.5
1.0
sf ~ (qd)
2
0.5
10
4
-1
q (cm )
2
0.01
sf ~ (qd)
2
10
4
-1
q (cm )
2
Fluctuations of the dynamics vs j
0.5
standard deviation(p)/p
standard deviation(ts)/ts
0.5
0.4
0.3
0.2
0.1
0.0
(1/)
0.64
0.68
0.72
jeff
0.76
St. dev. of relaxation time
0.4
0.3
0.2
0.1
0.0
0.64
0.68
0.72
jeff
St. dev. of stretching
exponent
0.76
Average dynamics vs j
10
4
10
3
ts (sec)
ts ~ |jeff- jc|
-1.01
jc= 0.752
0.64
0.68
0.72
jeff
0.76
Average relaxation time
Dynamical hetereogeneity in glassy
systems
Supercooled liquid (Lennard-Jones)
Glotzer et al.,
J. Chem. Phys. 2000
c4 increases when approaching Tg
Conclusions
Non-monotonic behavior of c*
2.0x10
-3
1.0x10
10
-2
10
-3
10
-4
10
-5
10
-6
-3
c*
Dynamics heterogeneous
3.0x10
-3
0.64 0.68 0.72 0.76
0.0
0.64
0.68
0.72
jeff
0.76
Conclusions
Non-monotonic behavior of c*
Many localized, highly effective
rearrangements
2.0x10
-3
1.0x10
10
-2
10
-3
10
-4
10
-5
10
-6
-3
c*
Dynamics heterogeneous
3.0x10
-3
0.64 0.68 0.72 0.76
0.0
0.64
0.68
0.72
jeff
0.76
Conclusions
Non-monotonic behavior of c*
Many localized, highly effective
rearrangements
2.0x10
-3
1.0x10
10
-2
10
-3
10
-4
10
-5
10
-6
-3
c*
Dynamics heterogeneous
3.0x10
-3
0.64 0.68 0.72 0.76
0.0
0.64
Many extended, poorly effective
rearrangements
0.68
0.72
jeff
0.76
Conclusions
0.06
Dynamics heterogeneous
Many localized, highly effective
rearrangements
c*
Non-monotonic behavior of c*
0.04
0.02
0.00
Many extended, poorly effective
rearrangements
Few extended, quite effective
rearrangements
General behavior
0.64
0.68
0.72
jeff
0.76
Time Resolved Correlation
lag t
time tw
< Ip(tw) Ip(tw +t)>p
degree of correlation cI(tw,t) =
-1
< Ip(tw)>p<Ip(tw +t)>p