glasses and gels.04.print - nanoparticles.org · colloidal gels and glasses dave weitz harvard emu...
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Colloidal Gels and GlassesDave Weitz Harvard
EMU 6/8/03
NASA , NSF, Infineum
http://www.deas.harvard.edu/projects/weitzlab
•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
Jaci Conrad HarvardSuliana Manley HarvardHans Wyss HarvardEric Weeks EmoryVeronique Trappe Fribourg
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Glass Transition•Widely studied but poorly understood•No structural difference between liquid and glass•Difference defined by time scales
divergent structural relaxation time
New state of matter, or very slow liquid?What happens microscopically?
Characterized by structural relaxation
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liquidliquid - crystal
coexistence crystal
glass
49% 54% 63% 74%58%
“supercooled”
φ
Maximum packing φHCP=0.74
Maximum packingφRCP≈0.63
φxtal≈0.54φliquid≈0.48
Repulsive Colloidal Glasses
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γ
τ
Solid: Gτ γ=τ ηγ=Fluid: ( ) ( )G iGτ γωω ′′+′= ⎡ ⎤⎣ ⎦
Elastic Viscous
Viscoelasticity of Glasses
0i te ωγ γ=
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Rheology of Hard Spheres
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DepletionDepletion
Colloidal Glasses: Attractive and Repulsive
φRepulsive colloidal glass transition
Solid:glass or gel
0 0.7
Fluid
B
Uk T
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φ = 0.06, U = 6.0 kBT
Gelation: Glass Transition of Clusters
Fluid-Clusters Gel
“Jamming” of Clusters to form gel
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0 5 10 15 20 25
0
0.2
0.4
0.6
0.8
0.5 µmtime (s)
y(t)
Brownian Motion(2 µm particles, dilute sample)
Leads to normal diffusion: ⟨∆x2⟩ = 2Dt
6Bk TD
aπη=
Particlesize a
viscosity η
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Mean square displacement:
⟨∆x2⟩ (µm2)
∆t (s)
Slope=1
Displacement distribution function:
P(∆x)
∆t=0.5s
∆x (µm)-2 0 2
10-1
10-2
10-3
10-4
10-5
Gaussian
Diffusion: dilute samples
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⟨x2⟩ µm2
lag time ∆t (s)
2D data
3D data
Mean square displacement
Volume fraction φ=0.53
“supercooled fluid”
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Cage trapping:
•Short times: particles stuck in “cages”•Long times: cages rearrange
φ=0.56, 100 min(supercooled fluid)
Mean-squared displacementφ=0.53 -- “supercooled fluid”
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1 micronshading indicates depth
Trajectories of “fast” particles, φ=0.56
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95%
top 5% = tailsof ∆x distribution
φ=0.53, supercooled fluid
Time scale:∆t* when nongaussian parameter α2
largest
Length scale:∆r* on average, 5% of particles have
∆r(∆t*) > ∆r*
≈ cage rearrangements
Time Scale and Length Scale
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lag time ∆t (s)
2D
3D⟨x2⟩ µm2
Displacement distribution function
∆t = 1000 s
φ = 0.53: “supercooled fluid”
α2
4
2 23
1= −x
xNongaussian Parameter
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How to pick ∆t* for glasses?
glasses
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Structural Relaxations in a Supercooled Fluid
timeNumber of
relaxing particles
Relaxing particles are highly correlated spatially
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Structural Relaxations in a Glass
timeNumber of
relaxing particles
Relaxing particles are NOT correlated spatially
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Supercooled fluid φ = 0.56 Glass φ = 0.61
Fluctuations of fast particles
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Number Nf of fast neighbors to afast particle:
Fractal dimension:
φ = 0.56supercooled fluid
Cluster Properties
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Dependence on Step Size
Bigger step more V
Bigger step less crystalline
Bigger step cage breaking
Bigger step less likely
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averagecluster
size
volume fraction
Cluster size grows as glass transition is approached
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Adam & Gibbs: “cooperatively rearranging regions”(1965)
Dynamical Heterogeneity:possible dynamic length scale
Simulations: •Glotzer, Kob, Donati, et al (1997, Lennard-Jones)
Photobleaching: •Cicerone & Ediger (1995, o-terphenyl)
NMR experiments: •Schmidt-Rohr & Spiess (1991, polymers)
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What is a Glass?
•Glass must have a low frequency shear modulus•Must have force chains to transmit stress
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E.R. Weeks et. al, Science 287, 627 (2000)
∆r(∆t) gives no obvious definition of slow
5% 10% 20% 30%
φ = 0.56
35%(first percolating cluster)
40%
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Topological Change: ∆nn (∆t)
t0 t0+∆t
Identify nearest neighbors, calculate ∆nn(∆t)
t0 t0+∆t
∆nn = 2∆nn = 1
B. Doliwa and A. Heuer, J. Non-Cryst. Solids 307, 32 (2002).
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Percolation clusters break up in supercooled fluids∆nn = 0
φ=0.52: ∆tbreakup~∆t*
φ=0.56: ∆tbreakup~4∆t*
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Glasses Have Connected Cluster for Entire Time
Total run length is roughly 35,000 s
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
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Number of Edge Particles in Connected Cluster∆nn=0
φ=0.52
φ=0.60
φ=0.56
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Weak Attractive InteractionColloid-polymer mixtures
Polystyrene polymer, Rg=37 nm + PMMA spheres, rc=350 nm
fluid gelattractivefluid
fluid +polymer
Depletion attraction
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DepletionDepletion
Attractive Colloidal Glasses
φRepulsive colloidal glass transition
Solid:glass or gel
0 0.7
Fluid
B
Uk T
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Phase diagram: Depends on Range
Short –rangeMw = 96,000 g mol-1▲ - Gel∆ - Fluid cluster
Long-rangeMw = 2,000,000 g mol-1● - Gel○ - Fluid cluster
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φ = 0.06, U = 6.0 kBT
Gelation: Glass Transition of Clusters
Fluid-Clusters Gel
“Jamming” of Clusters to form gel
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Dynamic Light Scattering from Attractive Colloids
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Gelation Transition for Attractive Colloids
Fluid-solid transition at well-defined φ
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Colloidal Gel
Confocal microscope image: Slice through gelPMMA particles, a = 0.35 µm
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20 µm
Rαβ
Colloidal Gel: Chains Connect Particles
Confocal microscope: Cut through gelPMMA particles
Rendered image showing chains
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Colloidal Gel: Chains Connect Particles
Short-range interaction – Fewer loopsLong-range interaction – More loops
Rendered image showing chains
f(2)
L (diameters)1 10 100
10-4
10-2
10-3
10-1
100
Rg = 38 nm
Rg = 8.4 nm
(a)
Probability of 2nd Loop for Length L
Long range
Short range
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r/R0 2 4 6 8 10 12 14
g(r/R
)
0
4
8
12
16
20
24
Log(r/R)0.0 0.4 0.8 1.2
Log(
g-1)
0.0
0.5
1.0
1.5
slope = -1.24(Df = 1.76)
Rc
L
M ∝ Rdf
φ(R) ∝ RDf - 3
Rc is set by φ(R=Rc) ≡ φRc = φ(1/df-3)
Fractal scaling
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● φ = 0.15
■ φ = 0.11
♦ φ = 0.08
Viscoelastic behavior (Udep / kBT = 7.1, ξ = 0.168 )
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Scaling behavior
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Udep/kBT = 7.1, ξ = 0.168
Same Universal Master Curve
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Udep/kBT = 7.1, ξ = 0.168
Same Universal Master Curve
Udep/kBT = 13.7, ξ = 0.04
Udep/kBT = 11.9, ξ = 0.04
Udep/kBT = 5.4, ξ = 0.18
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Two Component Model
ω * a
G’(ω
) ∗ b
, G’’ (
ω) ∗
b
a / µ102 103 104 105 106 107 108 109
b
100
101
102
103
104
105
106
1
Scale along the background viscosityφ1
φ2
φ3
Accessible ω
G’(ω) = constant = Gp’
G’’(ω) = η ω
φ1 > φ2 > φ3
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Scaled Critical Onset of Plateau Moduli for Colloidal GelsDepends on Range of Interaction
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Exponents Depend on Interaction
Rigidity Percolation
Map φ p
Non-central forces:G ~ (φ – φ0)3.8
Central Forces:G ~ (φ – φ0)2
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9.0
8.8
8.6
8.4
Rα
β , µ
m
time, s0 8040 120
0 0.1 0.2-0.1-0.2
2.0
0.0
1.0U
/kB
T
R - R (µm)
(b)
(c)
α β
Spring Constant Determined from Thermal Fluctuations
Harmonic SpringCurvature is κ
Movie of Fluctuations
( ) ( ){ }exp BP R U R k T∆ ∝ − ∆
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Length dependence of Spring Constant
Bending Resistance
κ ~ R-1
Depends on width of chainNo dependence on width of chain
R
R R⊥ ∼
γ γ2 3~BdR R Rκ −− −
⊥∼
NO Bending Resistance
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3.5
3.0
2.5
2.0
1.5
1.0
Lo
g(k
, k B
T/µ
m2)
(a)
0.0 0.5 1.0 1.5 2.0 2.5Log(Nch)
0.5
0.0
3
-1 0 1 2Log(r 2)
Lo
g(k
Nch
, k B
T/µ
m2)
4
3
2
1
00.5 1.0 1.5 2.00.0
Log(R, µm)
Log(1/Nch)
Log(1/Lr 2)
Lo
g(k
, k B
T/µ
m2)
(b)
Scaling of Spring Constant with Chain Length
Centro-symmetricNon centro-symmetricSupports bond bending
Long-range interaction Short-range interaction
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LoadIncompatible StressStress
Andrea Liu, Sidney NagelNature 386 (1998) 21
DensityCompatible StressPressureOsmotic PressureAttractive Potential
TemperatureVibration, Shaking
Jamming Transition Jamming Transition –– Arrest of MotionArrest of Motion
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kT/U
1/Φ
σ
Jamming Transitions for Colloidal SystemsJamming Transitions for Colloidal Systemswith Attractive Interactionswith Attractive Interactions
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Jamming Phase Diagram forJamming Phase Diagram forAttractive SystemsAttractive Systems
U
σ
φ
Proposed by:Andrea Liu, Sid NagelNature 386, 21 (1998)
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Spinodal Decomposition of Colloid PolymerLonger-range Interaction~3
cm
A. Bailey, L. Cipelletti, U. Gasser, S. Manely, P. Segre, ISS
~16 hrs
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Time Evolution of Phase Separation~3
cm
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Short-Time Evolution of Small-Angle Light Scattering after Mix
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Scaling of Scattering at Small Angles
Follows Theory – Furukawa
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•Like Binary Fluid very close to critical point• ξ is much larger colloidal particle
Comparison with Theory: Furukawa
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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
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