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3. Insight from Colloidal Glasses

Peter SchallUniversity of Amsterdam

JMBC WorkshopJamming and glassy behavior in colloids

P. Schall, University of Amsterdam

Colloidal suspensions

10-9 10-6 10-5 10-4 10-3 m10-8 10-7

Atoms GranularDNAPolymers

aColloids

Why important ?

P. Schall, University of Amsterdam

Hard-sphere Phase Diagram

Volume Fraction

Fluid

0.740.540.49

Fluid +Cryst

Crystal

Colloidal Hard Spheres

(Alder, Wainwright 1957)

P. Schall, University of Amsterdam

Hard-sphere Phase Diagram

Volume Fraction

Quench

Colloidal Hard Spheres

0.640.58

Glass

(Alder, Wainwright 1957)

P. Schall, University of Amsterdam

Elastic Modulus ∼ Energy / a3

1eV / 1Å3

=100 GPakT / 1µm3

~1 Pa

Atoms Colloids

Scaling of the Moduli

Colloidal Hard Spheres

P. Schall, University of Amsterdam

3. Dense systemElastic modulus

1. Thermal EnergykT

2. Structure: Glassno long-range order

εkk

σkk

compression shear

Colloidal Glasses

1 µm

εij

σij

Colloidal suspensions

P. Schall, University of Amsterdam

Observation of Colloids

Colloidal Systems

1cm3

1013 particles 10 mm3

1011 particles

Light scattering Microscopy

1mm3

109 particles

106 µm3

2x105 particles

Length Scale

P. Schall, University of Amsterdam

Observation of Colloids

Colloidal Systems

1cm3

1013 particles 10 mm3

1011 particles

Light scattering Microscopy

1mm3

109 particles

106 µm3

2x105 particles

Length Scale

Diameter 1.3 µµµµm

φφφφ ~ 0.59

P. Schall, University of Amsterdam

Collision timeτ = (1/100) s

Total “simulation“ time : 10 h ≈ 3million τTime increment : 1 min ≈ 6000 τ

Colloidal SystemsTime Scale

Colloids: 3D Analogue Computers

P. Schall, University of Amsterdam

Light Scattering

Laser

Detectort

I(t)

τ

Time Auto Correlation Function

� � � � � � � � �

� � ∝ � � ���

P. Schall, University of Amsterdam

Light Scattering

van Megen et al. (PRE 1998)

• Light scattering

Colloidal glass transition

φg ~ 0.57

P. Schall, University of Amsterdam

Weeks, Weitz et al. (Science, 2000)

Colloidal Glasses

• Microscopy

Dynamic heterogeneityCaging

P. Schall, University of Amsterdam

Free volume distribution

Heterogeneous!

P. Schall, University of Amsterdam

Application of stress

Confocal microscopy

γγγγ

50d

60

40

20

0- 0.05 0 0.05 0.1

z / µ

m

∆x / µm

Particledisplacements

P. Schall, University of Amsterdam

Strain and non-affine displacements

x

y

z

di di´

Time t0 t1 Affine transformation : γγγγ

d i´aff = d i + γγγγ d i

neighbors

D2min = ∑ (d i´ - d i) -γγγγ d i)

2

actualchange

affinechange

Symmetric part of γ

Strain tensor εεεεij = εxzεxyεxx

εyzεyyεyx

εzzεzyεzx

P. Schall, University of Amsterdam

0 20 40 60 80 100

60

40

20

0

X(µm)

Z(µ

m)

γ ~ 1 x 10-5 s-1.

-0.1 0 0.1

STZ in colloidal glasses

Affine part

γ τ ~ 0.1.

εεεεxz

x

εεεεxz

P. Schall, University of Amsterdam

--

- -x

z

Incremental strain

Continuum elasticity

(PS, Weitz, Spaepen, Science 2007)

r / µµµµm

εεεεyz

εεεεyz εyz~1r3

slope- 3

x

STZ in colloidal glasses

P. Schall, University of Amsterdam

0 20 40 60 80 100

60

40

20

0

X(µm)

Z(µ

m)

γ ~ 1 x 10-5 s-1.

-0.1 0 0.1 0 0.2 0.4 0.6

0 20 40 60 80 100 X(µm)

Dmin2

Affine part Non-affine part

γ τ ~ 0.1.

εεεεxz

STZ in colloidal glasses

P. Schall, University of Amsterdam

( )( ) ( )

averagespatial

vectordifference

AA

ArArACA

:

: ∆

−∆+=∆ 22

2)()(

)(

Strain correlation : A = εxz

Non-affine correlation : A = D2min

∆∆∆∆ΑΑΑΑ(r)

ΑΑΑΑ(r+∆∆∆∆)

Spatial Correlations

∆∆∆∆∆∆∆∆

∆∆∆∆

P. Schall, University of Amsterdam

0.02

0.01

0

-0.01

-0.02

60

40

20

00 20 40 60 80 100

Z(µ

m)

X(µm)

x

y

z

Affine part: Shear Strain εxz

-30 -15 0 15 30

30

15

0

-15

-30

z / d

Strain Correlation

x / d

0.2

0.1

0

-0.1

-0.2

Quadrupolar Symmetry: Signature of Elasticity

Elastic interactions� Self organization of STZ

P. Schall, University of Amsterdam

1

-0.75

-0.5

-0.25

0

60

40

20

00 20 40 60 80 100

Z(µ

m)

X(µm)

Non-affine part

x

y

z

P. Schall, University of Amsterdam

1

-0.75

-0.5

-0.25

0

60

40

20

00 20 40 60 80 100

Z(µ

m)

X(µm)

Non-affine part

-30 -15 0 15 30

30

15

0

-15

-30

z / d

D2min Correlation

x / d

-1

0.5

0

-0.5

-1

x

y

z

r/σ100 101 102

100

10-1

10-2

10-3

CD

2 min(r

)

α ∼1.3

r

System SizePower-law scaling

up to system size

P. Schall, University of Amsterdam

60

40

20

0- 0.05 0 0.05 0.1

z / µ

m

∆x / µm

Homogeneous

40

20

00 1

Inhomogeneous

γc γ (s-1)

∆x / µm

z / µ

m

Solid-Liquid transition: Shear banding

P. Schall, University of Amsterdam

60

40

20

0- 0.05 0 0.05

z / µ

m

∆x / µm

Homogeneous

γc γ (s-1)

Inhomogeneous

40

20

00 1

∆x / µm

z / µ

m

0.1

0

-0.1

0 20 40 60 80 100

40

20

0

x / µm

εxz

Shear banding transition

0 20 40 60 80 100

60

40

20

0

z / µ

m

x / µm

εxz

P. Schall, University of Amsterdam

0 20 40 60 80 100

40

20

0

X(µm)

Z(µ

m)

εεεεxz

0 20 40 60 80 100 X(µm)

γ ~ 1 x 10-4 s-1.

D2min

-0.1 0 0.1 0 0.2 0.4 0.6

Shear banding

P. Schall, University of Amsterdam

Spatial distribution of Flow ?

1

-0.75

-0.5

-0.25

0

40

20

00 20 40 60 80 100

X(µm)

100

10-1

10-2

10-3

10-4

100 101

r/σ

CD

2 min(r

)

P. Schall, University of Amsterdam

0 20 40 60 80 100 X(µm)

40

20

0

Z(µ

m)

10

´

0

-10

Z

-10 0 10

X

10

´

0

-10

Z-10 0 10 Quadrupolar symmetrysolid like

Strain Correlation function – Shear banded flow

Fundamental Solid � Liquid transitionOrigin ?

No quadrupolar symmetryliquid like

P. Schall, University of Amsterdam

0 2 4 6 8

g(r)

r

3.5

2.5

1.5

0.5

Structural transition ?

Pair correlation function

z

0.008

0

-0.008

6000

4000

0 10 20 30 40

Dila

tion

Den

sity

P. Schall, University of Amsterdam

Shear banding

Shear

Dilation

Very weak correlation betweendilation and shear, Cr~0.01!

P. Schall, University of Amsterdam

Increasing Strain rate60

40

20

0

30

15

0

-15

-30

z/ d

60

40

20

060

40

20

0 0 20 40 60 80 100

30

15

0

-15

-3030

15

0

-15

-30-30 -15 0 15 30

z/µm

z/µm

z/µm

z/ d

z/ d

x/µm x/d

γ.

P. Schall, University of Amsterdam

Anisotropy of Strain correlations

weakly driven:Thermal regime

strongly driven:Stress regime

π2

π2

αααα=-1.5

αααα=-0.8

New correlations withanisotropic, stress-dependent scaling

P. Schall, University of Amsterdam

60

40

20

0

30

15

0

-15

-30

z/ d

60

40

20

060

40

20

0 0 20 40 60 80 100

30

15

0

-15

-3030

15

0

-15

-30-30 -15 0 15 30

z/µm

z/µm

z/µm

z/ d

z/ d

x/µm x/d

γ.Nature of this transition ?

no structural, but

DYNAMICAL transition

Shear banding transition

P. Schall, University of Amsterdam

ρ

P(ρ)

ζ � Density ρ

ρgas ρliquid

Gas – Liquid transition:

First order transition ?

What is the right order parameter ?

P. Schall, University of Amsterdam

First order transition ?

||||δδδδr i |2ζ = = = = ∑i=1

N

∑t=0

T

Dynamic Order Parameter (Garrahan, Chandler 2009)

δ δ δ δ r i

extensive in Space AND Time

ζ /< ζ > ζ /< ζ > ζ /< ζ >ζ /< ζ >

f(ζ)

ζ 2ζ 1

First order transition in 4D space-time

Soft Spheres: beyond the glass transition

... compress beyond close packing

P. Schall, University of Amsterdam

Soft Sphere Glasses

Temperature-sensitive colloids

pNIPAm(Poly-N-isopropoylacrylamide)

TroomThigh

Quench beyond glass transition

Volume Fraction

Fluid

Quench 0.640.58

Glass

P. Schall, University of Amsterdam

Rhology: Hard and Soft Sphere Glasses

Soft Spheres: High elastic component!

P. Schall, University of Amsterdam

Rhology: Hard and Soft Sphere Glasses

Soft SpheresWeaker volume fraction dependence

Soft spheres � Strong GlassesHard Spheres � Fragile Glasses

Mattsson, Weitz (Nature 2010)

P. Schall, University of Amsterdam

Hertzian Interaction

δ

Soft Spheres : Hertzian Interaction

Suspension Modulus

Cloitre, Bonnecaze (J. Rheol. 2006)

P. Schall, University of Amsterdam

Soft Spheres : Hertzian Interaction

(Atomic Force Microscopy)

1. Particle Modulus 2. Pair Distribution Function

(Confocal Microscopy)

P. Schall, University of Amsterdam

Soft Spheres : Hertzian Interaction

Shear Modulus: Model & Measurement

How does elasticity affectmicroscopic relaxation?

P. Schall, University of Amsterdam

Soft Sphere Glasses

Coherentdisplacement

field

Internal Elasticity

P. Schall, University of Amsterdam

Soft Sphere Glasses

Correlations of Displacements

P. Schall, University of Amsterdam

Soft Sphere Glasses

Large Dynamic susceptibility!

φ/φmax

Dyn

. sus

cept

ibili

ty

(φc-φ)

χ4

hard

soft

P. Schall, University of Amsterdam

Soft Sphere Glasses

Long-range correlations ...

... ubiquitous in Soft Matter!

P. Schall, University of Amsterdam

Conclusions

• � Dynamic first order transition in 4D space-time

• Strain correlations :central to material arrest, flow and failure

• New anisotropic correlations:Stress-dependent anisotropic scaling � Strain localization

• Colloidal GlassesInsight into flow of amorphous materials

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