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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)
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Hard-sphere Phase Diagram
Volume Fraction
Quench
Colloidal Hard Spheres
0.640.58
Glass
(Alder, Wainwright 1957)
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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
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Collision timeτ = (1/100) s
Total “simulation“ time : 10 h ≈ 3million τTime increment : 1 min ≈ 6000 τ
Colloidal SystemsTime Scale
Colloids: 3D Analogue Computers
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Light Scattering
Laser
Detectort
I(t)
τ
Time Auto Correlation Function
� � � � � � � � �
� � ∝ � � ���
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Light Scattering
van Megen et al. (PRE 1998)
• Light scattering
Colloidal glass transition
φg ~ 0.57
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Weeks, Weitz et al. (Science, 2000)
Colloidal Glasses
• Microscopy
Dynamic heterogeneityCaging
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Free volume distribution
Heterogeneous!
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Application of stress
Confocal microscopy
γγγγ
50d
60
40
20
0- 0.05 0 0.05 0.1
z / µ
m
∆x / µm
Particledisplacements
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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
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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
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--
- -x
z
Incremental strain
Continuum elasticity
(PS, Weitz, Spaepen, Science 2007)
r / µµµµm
εεεεyz
εεεεyz εyz~1r3
slope- 3
x
STZ in colloidal glasses
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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
∆∆∆∆∆∆∆∆
∆∆∆∆
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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
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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
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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
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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
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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
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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
)
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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
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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
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Shear banding
Shear
Dilation
Very weak correlation betweendilation and shear, Cr~0.01!
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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.
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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
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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
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ρ
P(ρ)
ζ � Density ρ
ρgas ρliquid
Gas – Liquid transition:
First order transition ?
What is the right order parameter ?
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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
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Rhology: Hard and Soft Sphere Glasses
Soft Spheres: High elastic component!
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Rhology: Hard and Soft Sphere Glasses
Soft SpheresWeaker volume fraction dependence
Soft spheres � Strong GlassesHard Spheres � Fragile Glasses
Mattsson, Weitz (Nature 2010)
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Hertzian Interaction
δ
Soft Spheres : Hertzian Interaction
Suspension Modulus
Cloitre, Bonnecaze (J. Rheol. 2006)
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Soft Spheres : Hertzian Interaction
(Atomic Force Microscopy)
1. Particle Modulus 2. Pair Distribution Function
(Confocal Microscopy)
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Soft Spheres : Hertzian Interaction
Shear Modulus: Model & Measurement
How does elasticity affectmicroscopic relaxation?
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Soft Sphere Glasses
Coherentdisplacement
field
Internal Elasticity
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Soft Sphere Glasses
Correlations of Displacements
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Soft Sphere Glasses
Large Dynamic susceptibility!
φ/φmax
Dyn
. sus
cept
ibili
ty
(φc-φ)
χ4
hard
soft
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Soft Sphere Glasses
Long-range correlations ...
... ubiquitous in Soft Matter!
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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