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

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

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

γ

τ

Solid: Gτ γ=τ ηγ=Fluid: ( ) ( )G iGτ γωω ′′+′= ⎡ ⎤⎣ ⎦

Elastic Viscous

Viscoelasticity of Glasses

0i te ωγ γ=

Rheology of Hard Spheres

DepletionDepletion

Colloidal Glasses: Attractive and Repulsive

φRepulsive colloidal glass transition

Solid:glass or gel

0 0.7

Fluid

B

Uk T

φ = 0.06, U = 6.0 kBT

Gelation: Glass Transition of Clusters

Fluid-Clusters Gel

“Jamming” of Clusters to form gel

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

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

⟨x2⟩ µm2

lag time ∆t (s)

2D data

3D data

Mean square displacement

Volume fraction φ=0.53

“supercooled fluid”

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”

1 micronshading indicates depth

Trajectories of “fast” particles, φ=0.56

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

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

How to pick ∆t* for glasses?

glasses

Structural Relaxations in a Supercooled Fluid

timeNumber of

relaxing particles

Relaxing particles are highly correlated spatially

Structural Relaxations in a Glass

timeNumber of

relaxing particles

Relaxing particles are NOT correlated spatially

Supercooled fluid φ = 0.56 Glass φ = 0.61

Fluctuations of fast particles

Number Nf of fast neighbors to afast particle:

Fractal dimension:

φ = 0.56supercooled fluid

Cluster Properties

Dependence on Step Size

Bigger step more V

Bigger step less crystalline

Bigger step cage breaking

Bigger step less likely

averagecluster

size

volume fraction

Cluster size grows as glass transition is approached

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)

What is a Glass?

•Glass must have a low frequency shear modulus•Must have force chains to transmit stress

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%

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).

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

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

Number of Edge Particles in Connected Cluster∆nn=0

φ=0.52

φ=0.60

φ=0.56

Weak Attractive InteractionColloid-polymer mixtures

Polystyrene polymer, Rg=37 nm + PMMA spheres, rc=350 nm

fluid gelattractivefluid

fluid +polymer

Depletion attraction

DepletionDepletion

Attractive Colloidal Glasses

φRepulsive colloidal glass transition

Solid:glass or gel

0 0.7

Fluid

B

Uk T

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

φ = 0.06, U = 6.0 kBT

Gelation: Glass Transition of Clusters

Fluid-Clusters Gel

“Jamming” of Clusters to form 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 gelPMMA particles, a = 0.35 µm

20 µm

Rαβ

Colloidal Gel: Chains Connect Particles

Confocal microscope: Cut through gelPMMA particles

Rendered image showing chains

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

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

● φ = 0.15

■ φ = 0.11

♦ φ = 0.08

Viscoelastic behavior (Udep / kBT = 7.1, ξ = 0.168 )

Scaling behavior

Udep/kBT = 7.1, ξ = 0.168

Same Universal Master Curve

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

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

Scaled Critical Onset of Plateau Moduli for Colloidal GelsDepends on Range of Interaction

Exponents Depend on Interaction

Rigidity Percolation

Map φ p

Non-central forces:G ~ (φ – φ0)3.8

Central Forces:G ~ (φ – φ0)2

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∆ ∝ − ∆

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

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

LoadIncompatible StressStress

Andrea Liu, Sidney NagelNature 386 (1998) 21

DensityCompatible StressPressureOsmotic PressureAttractive Potential

TemperatureVibration, Shaking

Jamming Transition Jamming Transition –– Arrest of MotionArrest of Motion

kT/U

1/Φ

σ

Jamming Transitions for Colloidal SystemsJamming Transitions for Colloidal Systemswith Attractive Interactionswith Attractive Interactions

Jamming Phase Diagram forJamming Phase Diagram forAttractive SystemsAttractive Systems

U

σ

φ

Proposed by:Andrea Liu, Sid NagelNature 386, 21 (1998)

Spinodal Decomposition of Colloid PolymerLonger-range Interaction~3

cm

A. Bailey, L. Cipelletti, U. Gasser, S. Manely, P. Segre, ISS

~16 hrs

Time Evolution of Phase Separation~3

cm

Short-Time Evolution of Small-Angle Light Scattering after Mix

Scaling of Scattering at Small Angles

Follows Theory – Furukawa

•Like Binary Fluid very close to critical point• ξ is much larger colloidal particle

Comparison with Theory: Furukawa

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