Colloidal Gels and Glasses
Dave Weitz
Harvard
Jaci Conrad
Suliana Manley
Hans Wyss
Eric Weeks
Veronique Trappe
Harvard
Harvard
Harvard
Emory
Fribourg
NASA , NSF, Infineum
•Relaxation in colloidal glasses
•Stress bearing chains for solid-like behavior of glass
•Attractive colloidal glasses
•Scaling of the Viscoelasticity of Colloidal Glasses
•Possible models for attractive glass transition
http://www.deas.harvard.edu/projects/weitzlab
EMU 6/8/03
Glass Transition
•Widely studied but poorly understood
•No structural difference between liquid and glass
•Difference defined by time scales
Æ divergent structural relaxation time
New state of matter, or very slow liquid?
What happens microscopically?
Characterized by structural relaxation
Repulsive Colloidal Glasses
liquid
liquid - crystal
coexistence
49%
crystal
54% 58%
63%
74%
φ
“supercooled” glass
φliquid≈0.48
φxtal≈0.54
Maximum packing
φRCP≈0.63
Maximum packing
φHCP=0.74
Viscoelasticity of Glasses
γ
τ
Solid:
Fluid:
τ = Gγ
τ = ηγ
γ = γ 0e
iω t
τ = ⎡⎣G ′ (ω ) + iG ′′ (ω )⎤⎦ γ
Elastic
Viscous
Rheology of Hard Spheres
Colloidal Glasses:
Attractive and Repulsive
U
k BT
Solid:
glass or gel
Fluid
0
Depletion
0.7
Repulsive colloidal glass transition
φ
Gelation: Glass Transition of Clusters
“Jamming” of Clusters to form gel
φ = 0.06, U = 6.0 kBT
Fluid-Clusters
Gel
Brownian Motion
(2 µm particles, dilute sample)
y(t)
0.8
0.6
0.4
0.2
0
0.5 µm
0
5
10
15
20
25
time (s)
Leads to
2⟩ = 2Dt
⟨∆x
normal diffusion:
k BT
D=
6πη a
Particle
size a
viscosity η
Diffusion: dilute samples
Mean square displacement:
⟨∆x2⟩ (µm2)
Displacement distribution function:
P(∆x)
10-1
=1
e
p
Slo
10-2
10-3
10-4
10-5
∆t (s)
-2
0
∆x (µm)
∆t=0.5s
2
Gaussian
Mean square displacement
3D data
⟨x2⟩ µm2
Volume fraction
φ=0.53
“supercooled fluid”
2D data
lag time ∆t (s)
Mean-squared displacement
φ=0.53 -- “supercooled fluid”
φ=0.56, 100 min
(supercooled fluid)
Cage trapping:
•Short times: particles stuck in “cages”
•Long times: cages rearrange
Trajectories of “fast” particles, φ=0.56
shading indicates depth
1 micron
Time Scale and Length Scale
top 5% = tails
of ∆x distribution
95%
Time scale:
∆t* when nongaussian parameter α2
largest
Length scale:
∆r* on average, 5% of particles have
∆r(∆t*) > ∆r*
≈ cage rearrangements
φ=0.53, supercooled fluid
Displacement distribution function
3D
⟨x2⟩ µm2
∆t = 1000 s
2D
lag time ∆t (s)
φ = 0.53: “supercooled fluid”
Nongaussian Parameter
α2 =
x
3x
4
2 2
−1
How to pick ∆t* for glasses?
glasses
Structural Relaxations in a Supercooled Fluid
time
Number of
relaxing
particles
Relaxing particles are highly correlated spatially
Structural Relaxations in a Glass
time
Number of
relaxing
particles
Relaxing particles are NOT correlated spatially
Fluctuations of fast particles
Supercooled fluid φ = 0.56
Glass φ = 0.61
Cluster Properties
Number Nf of fast neighbors to a
fast particle:
Fractal dimension:
φ = 0.56
supercooled fluid
Dependence on Step Size
Bigger step Æ
less likely
Bigger step Æ
cage breaking
Bigger step Æ more V
Bigger step Æ
less crystalline
Cluster size grows as glass
transition is approached
average
cluster
size
volume fraction
Dynamical Heterogeneity:
possible dynamic length scale
Adam & Gibbs: “cooperatively rearranging regions”
(1965)
Simulations:
•Glotzer, Kob, Donati, et al (1997, Lennard-Jones)
Photobleaching:
•Cicerone & Ediger (1995, o-terphenyl)
NMR experiments:
•Schmidt-Rohr & Spiess (1991, polymers)
What is a Glass?
•Glass must have a low frequency shear modulus
•Must have force chains to transmit stress
∆r(∆t) gives no obvious definition of slow
φ = 0.56
5%
10%
20%
30%
35%
40%
(first percolating cluster)
E.R. Weeks et. al, Science 287, 627 (2000)
Topological Change: ∆nn (∆t)
Identify nearest neighbors, calculate ∆nn(∆t)
t0
t0+∆t
∆nn = 1
t0
t0+∆t
∆nn = 2
B. Doliwa and A. Heuer, J. Non-Cryst. Solids 307, 32 (2002).
Percolation clusters break up in supercooled fluids
∆nn = 0
φ=0.52: ∆tbreakup~∆t*
φ=0.56: ∆tbreakup~4∆t*
Glasses Have Connected Cluster for Entire Time
™ Look for connectivity
among ∆nn(∆t)=0 particles
™ Even at ∆t~35,000 s all
∆nn=0 particles form a
connected network
φ=0.60
Total run length is roughly 35,000 s
Number of Edge Particles in Connected Cluster
∆nn=0
φ=0.60
φ=0.56
φ=0.52
Weak Attractive Interaction
Colloid-polymer mixtures
Depletion attraction
fluid
fluid +
polymer
attractive
fluid
gel
Polystyrene polymer, Rg=37 nm + PMMA spheres, rc=350 nm
Attractive Colloidal Glasses
U
k BT
Solid:
glass or gel
Fluid
0
Depletion
0.7
Repulsive colloidal glass transition
φ
Phase diagram: Depends on Range
Short –range
Mw = 96,000 g mol-1
▲ - Gel
∆ - Fluid cluster
Long-range
Mw = 2,000,000 g mol-1
● - Gel
○ - Fluid cluster
Gelation: Glass Transition of Clusters
“Jamming” of Clusters to form gel
φ = 0.06, U = 6.0 kBT
Fluid-Clusters
Gel
Dynamic Light Scattering from
Attractive Colloids
Gelation Transition for Attractive Colloids
Fluid-solid transition at well-defined φ
Colloidal Gel
Confocal microscope image: Slice through gel
PMMA particles, a = 0.35 µm
Colloidal Gel: Chains Connect Particles
Rαβ
20 µm
Confocal microscope: Cut through gel
PMMA particles
Rendered image showing chains
Colloidal Gel: Chains Connect Particles
Probability of 2nd Loop for Length L
100
(a)
Long range
Rg = 38 nm
10-1
f(2)
10-2
Rg = 8.4 nm
10-3
Short range
10-4
1
10
100
L (diameters)
Rendered image showing chains
Short-range interaction – Fewer loops
Long-range interaction – More loops
Fractal scaling
24
Log(g-1)
1.5
g(r/R)
20
16
12
1.0
slope = -1.24
(Df = 1.76)
0.5
L
0.0
0.0
8
0.4
0.8
1.2
Log(r/R)
4
0
0
2
4
6
8
10
12
14
r/R
Rc
M ∝ Rdf
φ(R) ∝ RDf - 3
Rc is set by φ(R=Rc) ≡ φ
Rc = φ(1/df-3)
Viscoelastic behavior
(Udep / kBT = 7.1, ξ = 0.168 )
● φ = 0.15
■ φ = 0.11
♦ φ = 0.08
Scaling behavior
Same Universal Master Curve
Udep/kBT = 7.1, ξ = 0.168
Same Universal Master Curve
Udep/kBT = 7.1, ξ = 0.168
Udep/kBT = 13.7, ξ = 0.04
Udep/kBT = 11.9, ξ = 0.04
Udep/kBT = 5.4, ξ = 0.18
Two Component Model
G’(ω) ∗ b, G’’(ω) ∗ b
G’(ω) = constant = Gp’
G’’(ω) = η ω
φ 1 > φ2 > φ 3
φ1
Scale along the background viscosity
106
φ2
105
b
104
103
1
102
φ3
101
100
Accessible ω
ω*a
102
103
104
105
106
a / µ
107
108
109
Scaled Critical Onset of Plateau Moduli for Colloidal Gels
Depends on Range of Interaction
Exponents Depend on Interaction
Rigidity Percolation
Map φ Æ p
Non-central forces:
G ~ (φ – φ0)3.8
Central Forces:
G ~ (φ – φ0)2
Spring Constant Determined from Thermal Fluctuations
P ( ∆R ) ∝ exp {− U ( ∆R ) k BT }
Rαβ, µm
9.0
(b)
8.8
8.6
8.4
0
40
80
120
time, s
U/kBT
(c)
2.0
α
β
1.0
0.0
-0.2
Movie of Fluctuations
-0.1
0
0.1
R - R (µm)
Harmonic Spring
Curvature is κ
0.2
Length dependence of Spring
Constant
R⊥ ∼ R
R
NO Bending Resistance
Bending Resistance
γ
κ ~ R-1
No dependence on width of chain
γ
κ ∼ R⊥−2 R − d ~ R −3
B
Depends on width of chain
Scaling of Spring Constant with Chain Length
Long-range interaction
Short-range interaction
3
3.5
(a)
(b)
Log(1/Lr 2)
Log(k, kBT/µm2)
2.5
2.0
Log(k Nch, kBT/µm2)
Log(k, kBT/µm2)
3.0
1.5
4
1.0
2
Log(1/Nch)
1
3
0.5
0.0
0.0
-1
0
0.5
1
Log(r 2)
1.0
1.5
Log(Nch)
2
2.0
Centro-symmetric
2.5
0
0.0
0.5
1.0
1.5
Log(R, µm)
Non centro-symmetric
Supports bond bending
2.0
Jamming Transition – Arrest of Motion
Temperature
Vibration, Shaking
Andrea Liu, Sidney Nagel
Nature 386 (1998) 21
Load
Incompatible Stress
Stress
Density
Compatible Stress
Pressure
Osmotic Pressure
Attractive Potential
Jamming Transitions for Colloidal Systems
with Attractive Interactions
kT/U
σ
1/Φ
Jamming Phase Diagram for
Attractive Systems
Proposed by:
Andrea Liu, Sid Nagel
Nature 386, 21 (1998)
U
σ
φ
Spinodal Decomposition of Colloid Polymer
Longer-range Interaction
~3 cm
~16 hrs
A. Bailey, L. Cipelletti, U. Gasser, S. Manely, P. Segre, ISS
~3 cm
Time Evolution of Phase Separation
Short-Time Evolution of Small-Angle Light Scattering after Mix
Scaling of Scattering at Small Angles
Follows Theory – Furukawa
Comparison with Theory: Furukawa
•Like Binary Fluid very close to critical point
• ξ is much larger Æ colloidal particle
Conclusions
• Repulsive glasses have percolation clusters of
slow particles
• Attractive colloidal systems are similar to glasses
• Viscoelastic behavior exhibits scaling
– Defines critical gelation for transition
• Phase behavior depends on range of interaction
• Different rheology for different φ
• Microscopic motion of particles provides insight
into rheology