Dynamical heterogeneity at the jamming transition of a colloidal suspension 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
Soft glassy materials φg ∼0.58
Cheng et al. PRE 02
viscosity
φ
Confocal microscopy image by Eric Weeks
Van Megen et al. PRE 98
Intermediate scattering function
Soft glassy materials
Outline • What are dynamical heterogeneities ? • Why should we care about DH ? • How can we measure DH ? • DH (very) close to jamming
Dynamical Heterogeneity PDF of particle displacements in a dense colloidal suspension
Optical microscopy Kasper et al. Langmuir 98
Confocal microscopy Kegel et al. Science 00
Dynamical Heterogeneity
String-like motion in a LJ supercooled fluid
Donati et al. PRL 98
Dynamical Heterogeneity Relaxation time in a colloidal suspension near close packing
τs (sec)
1000
500
0
0
40000
tw (sec)
80000
Why are DHs important? Crucial role in the slowing down of the dynamics close to the glass transition
• Adam-Gibbs: relaxation through cooperatively rearranging regions. Their size increases approaching the glass transition. • Glass transition as a (dynamical) critical phenomenon ? •
DHs may allow one to discriminate between competing theories
Size of DH: simulations
T=2
T = 0.6 Less mobile particles in a LJ supercooled fluid (Tc ~ 0.435)
Berthier PRE 04
T = 0.45
What quantities should we measure? Space and time-resolved correlation functions f(t,t+τ,r) or particle displacement
• Simulations (far from Tg!) • (Confocal) microscopy on colloidal systems • Granular systems (2D, athermal, see Dauchot’s talk)
Confocal microscopy on colloidal HS From E. Weeks web page
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+τ,r) or particle displacement • Simulations (far from Tg!) • (Confocal) microscopy on colloidal systems (limited statistics, stringent requirements on particles (size, optical mismatch…)) • Granular systems (2D, athermal, see Dauchot’s talk) Time-resolved correlation functions f(t,t+τ) (no space resolution)
Temporally heterogeneous dynamics
correlation
1.00 0.75 0.50 0.25 0.00 -4
-3
-2
-1
0
1
2
10 10 10 10 10 10 10 10
t (arb. un.)
homogeneous
3
1.00
1.00
0.75
0.75
correlation
correlation
Temporally heterogeneous dynamics
0.50 0.25 0.00
0.50 0.25 0.00
-4
-3
-2
-1
0
1
2
10 10 10 10 10 10 10 10
t (arb. un.)
homogeneous
3
-4
-3
-2
-1
0
1
2
3
10 10 10 10 10 10 10 10
t (arb. un.)
heterogeneous
1.00
1.00
0.75
0.75
correlation
correlation
Temporally heterogeneous dynamics
0.50 0.25
0.50 0.25 0.00
0.00 -4
-3
-2
-1
0
1
2
10 10 10 10 10 10 10 10
t (arb. un.)
homogeneous
3
-4
-3
-2
-1
0
1
2
3
10 10 10 10 10 10 10 10
t (arb. un.)
heterogeneous
Dynamical susceptibility in glassy systems
Supercooled liquid (Lennard-Jones)
Lacevic et al., PRE 2002
χ4 = N var[Q(t)]
Dynamical susceptibility in glassy systems
Nblob regions
χ4 = N var[Q(t)] ~ N (1/Nblob) = N/Nblob 3 d χ4 (τ) ~ ∫ r f (0, t ' , t '+t ) f (r, t ' , t '+t )
t'
How can we measure χ4?
Time-resolved light scattering experiments (TRC)
Laser beam
Experimental setup
CCD Camera
Random walk w/ step l*
Change in speckle field mirrors change in sample configuration
CCD-based (multispeckle) Diffusing Wave Spectroscopy
Time Resolved Correlation
lag τ
time tw < Ip(tw) Ip(tw +τ)>p
degree of correlation cI(tw,τ) = -1 < Ip(tw)>pp
< Ip(tw) Ip(tw +τ)>p Time Resolved Correlation degree of correlation cI(tw,τ) = -1 < Ip(tw)>pp Average over tw
intensity correlation function g2(τ) − 1
g2(τ)-1
0.2
0.1
0.0 -2
10
-1
10
1
τ (sec)
g2(τ) − 1
Average dynamics
< Ip(tw) Ip(tw +τ)>p Time Resolved Correlation degree of correlation cI(tw,τ) = -1 < Ip(tw)>pp Average over tw
fixed τ, vs. tw
intensity correlation function g2(τ) − 1
fluctuations of the dynamics 0.31
cI(t,τ)
g2(τ)-1
0.2
0.1
0.30 0.29 0.28
0.0 5
-2
10
-1
10
1.2x10
5
2.0x10
1
t (sec)
τ (sec)
g2(τ) − 1
5
1.6x10
Average dynamics
var(cI)(τ)
‘dynamical susceptibility’ χ4 (τ )
Experimental system
PVC xenospheres in DOP • radius R ~ 10 µm • Polydisperse • Brownian • Excluded volume interactions • ϕ = 64% – 75% ( Note: ϕg ~58%)
« Diluted » samples
-15
10
-16
2
(m )
10
ϕ = 28% ϕ = 46%
-17
10
Brownian behavior
1
-18
10
-19
10
-20
10
-6
10
-5
10
-4
10
-3
10
τ (sec)
-2
10
-1
10
0
10
« Diluted » samples
-15
10
R/100 !!
-16
2
(m )
10
ϕ = 28% ϕ = 46%
-17
10
1
-18
DWS probes dynamics on a length scale
-19
λl*/L ~ 10 – 35 nm
