Spatial correlations and temporal heterogeneity of the slow dynamics of a colloidal fractal gel

Time (N° 1) and Space (N° 2) Resolved Degree of Correlation :

“Multispeckle” Light Scattering experiments (Single Scattering)

• N°1 Far field – Several q ( 4337 cm-1 _{< q < 52177 cm}-1_{) – Time Resolved Dynamics – V}⇔ All the scattering volume

⇔ All the speckled image

• N°2 Imaging geometry – Single q (q = 104_cm-1_{) – Time and Space Resolved Dynamics – V}⇔ Portion of the

scattering volume ⇔ Portion of the speckled image

2 . 4 x 1 05 2 .8 x 1 05 3 .2 x 1 05 3 .6 x 1 05 4 .0 x 1 05 0 . 0 0 . 2 0 . 4 0 . 6 cI ( t , τ ) t ( s e c ) τ = 1 0 s e c τ = 1 0 0 s e c τ = 1 0 0 0 s e c τ = 5 0 0 0 s e c τ = 1 0 0 0 0 s e c τ = 2 0 0 0 0 s e c τ = 4 0 0 0 0 s e c τ = 6 0 0 0 0 s e c τ = 8 0 0 0 0 s e c

Agnès Duri and Luca Cipelletti

Laboratoire des Colloïdes, Verres et Nanomatériaux (LCVN), UMR n°5587, Université Montpellier II, 34095 Montpellier, France

time t _I

p(t): intensity p-th pixel at time t

Spatial Correlations and Temporal Heterogeneity

Spatial Correlations and Temporal Heterogeneity

of the Slow Dynamics of a Colloidal fractal gel

of the Slow Dynamics of a Colloidal fractal gel

lag τ

( )

_{( )} ( )

( )

( ) 1 ) , - τ t I (t) I τ t (t) I I r (t,τ c r V p p r V p p r V p p p I r r r r ∈ ∈ ∈ + + =

Average Dynamics :

Spatially Resolved Intensity Correlation Function (N°1): Spatial Correlation of the Dynamics (N° 2):

( )

τ,

r

-

c

(t,τ

r

)

_t

g

2

r

1

=

I

,

r

Experimental

Experimental

System

System

Polystyrene colloids (d = 20 nm ) suspended in a buoyancy-matching H₂O/D₂O (45/55 vol/vol) ⇔ φ_PS= 6 10-4 + MgCl2⇔ CMgCl2=10 mM

⇒

Fractal gel

Experimental

Experimental

Set Up

Set Up

101 102 103 104 105 -0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 g2 ( q ,τ ) - 1 τ (sec) q=7355 cm-1 q=9538 cm-1 q=12393 cm-1 q=16004 cm-1 q=20656 cm-1 q=26506cm-1 q=37771 cm-1 q=42396 cm-1 q=52177 cm-1

Fit by a “compressed” exponential g₂(q,τ)-1 ∝ exp[-(τ/ τ_f) 3/2_] 104 103 104 slope= - 0.98 τf (s ec ) q (cm-1 ) τ_f∝ q-1⇒ Drift motion τ f t⇒ Aging behaviour

Time Resolved Dynamics – Fluctuations Study:

2 .5 x1 05 3 .0 x1 05 1 03 1 04 τf (s ec ) t (s e c) 1 04 _{1 0}5 _{1 0}6 1 01 1 02 1 03 1 04 1 05 τf (s ec ) t ( s e c ) Large fluctuations ofτ_f Heterogeneous dynamics⇓ 102 104 1E-5 1E-4 1E-3 0.01 τf σ2 cImax τmax σ 2 c I (τ ) τ (sec) q=20656 cm-1 ⇔ 3 µm 0 .0 5 .0 x1 03 1 .0 x1 04 1 .5 x104 2 .0 x1 04 0 .0 5 .0x1 03 1 .0x1 04 s lop e = 0 .8 τ ma x (s e c ) τf (sec) 0 .0 2 .0 x 1 04 4 .0 x 1 04 6 .0 x 1 04 0 .0 0 0 .0 5 0 .1 0 0 .1 5 σ 2 cmaI x q (c m- 1 ) s lo p e = 1 .1 σ2 c Imax ∝ q 4337 cm-1 _{< q < 52177 cm}-1 τ_max∝ τ_f Statistics treatment

Time and Space Resolved Dynamics – Local dynamics Study:

τ

< τ

_f

:

Very long-ranged correlation

⇒ « solid-like » behavior

0 1 2 3 4 5 6 7 0 .0 0 .2 0 .4 0 .6 0 .8 1 .0 Xc I ∆r (m m ) τ = 500 sec τ = 2500 sec τ = 5000 sec τ = 10000 sec τ_f = 1000 sec

Experimental

Experimental

Results

Results

[

] [

]

r c c t t I I t I I r r t r t r r t c r r t c r t c r t c I I r r r r r r r r r r ) , , ( ) , , ( ) , , ( ) , , ( ) , , ( ) , , ( 2 2 +∆ ∆ + − ∆ + − τ σ τ σ τ τ τ τ Speckled Images ∆r

( )

= ΧcIτ ,rr

τ

>

τ

_f

:

Spatial correlation decay

⇒ « fluid-like » behavior

⇓

Fluctuations max at large length scale

[1] L. Cipelletti, H. Manley, R.C. Ball, D. Weitz, J. Phys. Rev. Lett., 2000, 84, 10

[2] L. Cipelletti, H. Bissig, Trappe V, P. Ballesta, S. Mazoyer, J. Phys.: Condens. Matt., 2003, 15, S257- S262 [3] A. Duri, H. Bissig, V. Trappe, P. Ballesta, L. Cipelletti, J. Fluctations and Noise Letters, 2005, 5, 1

References :