Peter Schall University of Amsterdam - TFE1 | Home · 2013. 2. 28. · Insight from Colloidal...

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 ?


Hard-sphere Phase Diagram

Volume Fraction

Fluid

0.740.540.49

Fluid +Cryst

Crystal

Colloidal Hard Spheres

(Alder, Wainwright 1957)


Hard-sphere Phase Diagram

Volume Fraction

Quench


0.640.58

Glass

(Alder, Wainwright 1957)


Elastic Modulus ∼ Energy / a3

1eV / 1Å3

=100 GPakT / 1µm3

~1 Pa

Atoms Colloids

Scaling of the Moduli



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


Observation of Colloids

Colloidal Systems

1cm3

1013 particles 10 mm3

1011 particles

Light scattering Microscopy

1mm3

109 particles

106 µm3

2x105 particles

Length Scale


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


Collision timeτ = (1/100) s

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

Colloidal SystemsTime Scale

Colloids: 3D Analogue Computers


Light Scattering

Laser

Detectort

I(t)

τ

Time Auto Correlation Function

� � � � � � � � �

� � ∝ � � ��


Light Scattering

van Megen et al. (PRE 1998)

• Light scattering

Colloidal glass transition

φg ~ 0.57


Weeks, Weitz et al. (Science, 2000)

Colloidal Glasses

• Microscopy

Dynamic heterogeneityCaging


Free volume distribution

Heterogeneous!


Application of stress

Confocal microscopy

γγγγ

50d

60

40

20

0- 0.05 0 0.05 0.1

z / µ

m

∆x / µm

Particledisplacements


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


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


--

- -x

z

Incremental strain

Continuum elasticity

(PS, Weitz, Spaepen, Science 2007)

r / µµµµm

εεεεyz

εεεεyz εyz~1r3

slope- 3

x



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



( )( ) ( )

averagespatial

vectordifference

AA

ArArACA

:

: ∆

−

−∆+=∆ 22

2)()(

)(

Strain correlation : A = εxz

Non-affine correlation : A = D2min

∆∆∆∆ΑΑΑΑ(r)

ΑΑΑΑ(r+∆∆∆∆)

Spatial Correlations

∆∆∆∆∆∆∆∆

∆∆∆∆


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


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


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


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


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


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


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

)


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


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


Shear banding

Shear

Dilation

Very weak correlation betweendilation and shear, Cr~0.01!


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.


Anisotropy of Strain correlations

weakly driven:Thermal regime

strongly driven:Stress regime

0°

π2

0°

π2

αααα=-1.5

αααα=-0.8

New correlations withanisotropic, stress-dependent scaling


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


ρ

P(ρ)

ζ � Density ρ

ρgas ρliquid

Gas – Liquid transition:

First order transition ?

What is the right order parameter ?


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


Soft Sphere Glasses

Temperature-sensitive colloids

pNIPAm(Poly-N-isopropoylacrylamide)

TroomThigh

Quench beyond glass transition

Volume Fraction

Fluid

Quench 0.640.58

Glass


Rhology: Hard and Soft Sphere Glasses

Soft Spheres: High elastic component!


Rhology: Hard and Soft Sphere Glasses

Soft SpheresWeaker volume fraction dependence

Soft spheres � Strong GlassesHard Spheres � Fragile Glasses

Mattsson, Weitz (Nature 2010)


Hertzian Interaction

δ

Soft Spheres : Hertzian Interaction

Suspension Modulus

Cloitre, Bonnecaze (J. Rheol. 2006)



(Atomic Force Microscopy)

1. Particle Modulus 2. Pair Distribution Function

(Confocal Microscopy)



Shear Modulus: Model & Measurement

How does elasticity affectmicroscopic relaxation?


Soft Sphere Glasses

Coherentdisplacement

field

Internal Elasticity


Soft Sphere Glasses

Correlations of Displacements


Soft Sphere Glasses

Large Dynamic susceptibility!

φ/φmax

Dyn

. sus

cept

ibili

ty

(φc-φ)

χ4

hard

soft


Soft Sphere Glasses

Long-range correlations ...

... ubiquitous in Soft Matter!


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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