dave weitz harvard - yale university...dynamic light scattering f qt e e t,~ 0 2 0 0 iq r r t mn mn...
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Experimental Colloids I (and I)Dave Weitz
Harvard
http://www.seas.harvard.edu/projects/weitzlab Boulder Summer School 7/24/17
Experimental Colloids I (and I)Dave Weitz
Harvard
http://www.seas.harvard.edu/projects/weitzlab Boulder Summer School 7/24/17
Colloidal particles• Colloidal particles are ubiquitous• Biology
– Viruses, macromolecules, organelles– Probe particles for bioassays– Quantum dots for fluorescent assays– Spores, bacteria
• Processing– Paints, coatings, materials control– Ceramics
Colloidal particles• Key control rheology• Solid particles behave like continuous fluid• Process solids, while flow like fluids• eg Paints and coatings
– Spread paint like a fluid– Solidify into a solid coating
Colloidal particles• Properties set by particle density• Concentration of particles low compared to
normal material• Typical solid: ~1027 atoms/ m3 (1 / nm3)• Colloids: ~1018 particles/ m3 (1 / m3)• Latent heat of phase transitions too small to
measure• Very low pressure: = nkBT• ~ 10-18 x 1.4x10-23 x 300 = 4x10-3 Pa• Gas: 3x1025 molecules/m3 105 Pa = 1 atm
Soft SolidsEasily deformable Low Elastic Constant: V
Eonergylume
Hard Materials
Soft Materials
3
eV
3mBk T
1 Pa
GPa
Soft materials invariably have a larger length scale
Continuous phase fluid
• Thermalization with fluid• Equilibrates particles• Brownian motion
Continuous phase fluid
Hydrodynamic interactions
x x
Colloidal particles
• Ignore hydrodynamic interactions– Thermalize system– Important only for dynamics– No effect on static properties
• Consider just two-body interactions between particles
Colloidal Interactions
Only excluded volume
U
r2a
Hard-sphere interactions
U = 0 r 2aU = r 2a
Colloidal Interactions• van der Waals interactions• Dispersion interactions
– Dipole-induced dipole interactions• Depend on polarizability of material
– Require different materials– Always present for particles in a fluid
van der Waals interactions
•Repulsive•Short-ranged•Dipole-dipole 1/r6
Stabilizing interactions
+++++++
+++++++
--
-
-
-
---
- -- -
-
-
Particle ParticleFluid
Colloidal Interactions - Stabilization
• Screened Coulomb interaction
Inverse screening length
Surface potential
Ion density
Stabilizing interactions
+++++++
+++++++
--
-
-
-
---
- -- -
-
-
Disjoining pressure: Can’t compress ions
Eb
V
S
•Short range attraction•Long range repulsion•Sum: Stabilizing barrier
Eb > kBT Colloid stable against aggregation
Colloidal interactions – stabilizing
Repulsive Spheres
Repulsive interactions
Screen chargesNo Salt Salt
Experimental Techniques
• Light scattering– Static light scattering – Dynamic light scattering– Ultra-small angle dynamic light scattering– Diffusing-wave spectroscopy
• Microscopy• Rheology
– microrheology
Dynamic Light Scattering
= q-1 q =4n/ sin/2
probe structure
• DLS: < I(q,t)I(q,t+) > probe dynamicsf(q,)
Structure and Dyanmics:Light Scattering
• SLS: < I > vs. q
Light Scattering
DYNAMIC LIGHT SCATTERING:Single speckle
STATIC LIGHT SCATTERING:Many speckles
Speckle:Coherence area
Probes characteristic sizes of colloidal particles
4sin
2
nq
Dynamic Light Scattering
2 5 0 3 0 0 3 5 0 4 0 0 4 5 0 5 0 0 5 5 0
7 0
8 0
Inte
nsity
[kH
z]
T im e [ s ]
Measure temporal intensity fluctuations
10 -6 10 -5 10 -4 10 -3 10 -2 10 -1
1
2
Lag tim e () [s ]
Obtain an intensity autocorrelation function
2
( ) ( )I t I tI
Dynamic Light Scattering
, ~ 0f q t E E t
02
,
0 m niq r r t
m n
E E t A e
Measure temporal correlation function of scattered light: Intermediate structure factor
2 2
~q r t
e
2~ q Dte
Time average over all particles
Correlations only between the same particlesCumulant expansion: r2(t) ~ Dt
Physics: How to change the phase of the field by Each particle must move by ~
Ultra Small Angle Light ScatteringProbe Structure
2 deg.
L. Cipelletti
0.07 deg to 5.0 deg100 cm-1 < q < 7000 cm-1
• non-ergodic samples• avoid excessive time averaging
Multispeckle Detection
Average over constant q:
MOST LIGHT IS SCATTERED BACKMILK IS WHITE!!
TRANSMISSION
L
Diffusing Wave Spectroscopy:Very strong scattering
D. Pine, P. Chaikin, E. Herbolzheimer
P(s): DIFFUSION EQUATION
0 *z
t = 0
I (t)
PRL 60, 1134 (88) I(t) # PATHS OF LENGTH s = ct
P(s)
t=0
SINGLE PATH
[MARET & WOLFE]
n SCATTERING EVENTS
s = n PATH LENGTH
t=
EXTENDED SOURCE
* 0
* *0 0 0 0
643
18 6 6 64
3 31 sinh cosh
tL
t t t tL Lg t
CHARACTERISTIC TIME SCALE: * 2
0 L
0.6 m = 2%
d
TRANSMISSION
DWS PROBES MOTION ON SHORTLENGTH SCALES
~ 5000 Å
3
*
2 2 41010Estimate: 10
L
PHASE OF PATH CHANGES WHEN PATH LENGTH CHANGES BY ~1 WAVELENGTH
BUT: LIGHT IS SCATTERED FROM MANY PARTICLES
MOTION OF EACH INDIVIDUAL PARTICLE CAN BE MUCH LESS
CAN MEASURE PARTICLE MOTION ON SCALE OF ~ 5 Å
Confocal Microscopy
detector (PMT)
microscope
laserscreen with
pinhole
sample
rotatingmirrors
screen
Scan many slices,reconstruct 3D image
0.2 m
Confocal microscopy for 3D pictures
sin ( )o t
sin( )o t
G’ G”( sin coscos n )sio t t
E
Elasticity Viscosity
Rheology
Solid: G Fluid: G iG
Elastic Viscous
Mechanical Properties of Soft Materials:Viscoelasticity
0i te
* a 10-2 10-1 100 101 102 103 104 105 106 107 108 109
G'(
)* b
, G
"(
)* b
100
101
102
103
104
105
106
107
Dispersant Series
4.0% CB = 0.1493.0% CB = 0.1112.5% CB = 0.0972.0% CB = 0.0781.8% CB = 0.0731.6% CB = 0.0641.4% CB = 0.056 G'()
G"()
Rheology of soft materials
Scaling plot
liquidliquid - crystal
coexistence crystal
49% 54% 63% 74%
maximum packingHCP=0.74
maximum packingRCP0.63
Hard Sphere Phase Diagram
Increase Decrease Temperature F = U - TS0
Volume Fraction Controls Phase Behavior
Entropy Drives CrystallizationEntropy => Free Volume
Ordered:•Lower configurational
entropy•Higher local entropy•Lower Energy
Disordered:•Higher configurational
entropy•Lower local entropy•Higher Energy
maximum packingHCP=0.74
maximum packingRCP0.63
F = U - TS0
liquidliquid - crystal
coexistence crystal
glass
49% 54% 63% 74%58%
“supercooled”
Maximum packing HCP=0.74
Maximum packingRCP0.63
xtal0.54liquid0.48
Metastable Hard Sphere Phases
State diagram for colloidal particles
Equilibrium
Jammed States:Nonequilibrium solid states
Hard spheres
Polystyrene polymer, Rg=37 nm + PMMA spheres, rc=350 nm
fluid gelattractivefluid
fluid +polymer
Depletion attraction
Controlled Attraction of Colloidal Particles
T. Dinsmore
Weak attractionshigh
Local caging
Glasses
State diagram for colloidal particles
Equilibrium
Jammed States:Nonequilibrium solid states
Hard spheres
Weakly attractive “gels” at intermediate
volume fractions
Strong attractionslow
Network formation
Φ ~ 10-5Fractal gels
Quasi-equilibrium
Non-equilibrium
Weak attractionshigh
Local caging
Glasses
State diagram for colloidal particles
Equilibrium
Jammed States:Nonequilibrium solid states
Hard spheres
Weakly attractive “gels” at intermediate
volume fractions
Strong attractionslow
Network formation
Φ ~ 10-5Fractal gels
Quasi-equilibrium
Non-equilibrium
Hard spheres: -dependent structure factor
-dependent relaxation
Inverse-relaxation follows structure factor
Short-time and long time relaxation processes
-dependent relaxation
Slopes give relaxation rates effective diffusion coefficients
-dependence of viscosityComparison of frequency dependent data
-dependence of short-time diffusion coefficient
Correlates with viscosity
-dependence of long-time diffusion coefficient
Correlates with viscosity
Increasing :Approach to the glass
transition:-dependent relaxationq-dependent relaxation
What is the nature of the relaxation?
a
b
Viscoelasticity of Hard Spheres~0.45-0.56
Glassy plateau
Low frequency relaxation
High frequency response
Fluid contribution
x2 m2
lag time t (s)
2D data
3D data
Mean square displacement – Confocal Microscopy
Volume fraction =0.53
“supercooled fluid”
-relaxation
-relaxation
Cage trapping:
•Short times: particles stuck in “cages”•Long times: cages rearrange
=0.56, 100 min(supercooled fluid)
Mean-squared displacement=0.53 -- “supercooled fluid”
lag time t (s)
2D
3Dx2 m2
1 micronshading indicates depth
Trajectories of “fast” particles, =0.56
lag time t (s)
2D
3Dx2 m2
Displacement distribution function
t = 1000 s
= 0.53: “supercooled fluid”
2
4
2 23
1 x
xNongaussian Parameter
Choose Time with Maximum Non-Gaussian Parameter
95%
top 5% = tailsof x distribution
=0.53, supercooled fluid
Time scale:t* when nongaussian parameter 2
largest
Length scale:r* on average, 5% of particles have
r(t*) > r*
cage rearrangements
Time Scale and Length Scale
Structural Relaxations in a Supercooled Fluid
timeNumber of
relaxing particles
Relaxing particles are highly correlated spatially
Non-Gaussian parameter for glasses
glasses
No well-defined peak
Structural Relaxations in a Glass
timeNumber of
relaxing particles
Relaxing particles are NOT correlated spatially
Supercooled fluid = 0.56 Glass = 0.61
Fluctuations of fast particles
Number Nf of fast neighbors to afast particle:
Fractal dimension:
= 0.56supercooled fluid
Cluster Properties
averagecluster
size
volume fraction
Relaxation events are spatially correlated
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)
Boulder summer school experiments
10-4
10-3
10-2
10-1
100
10-8 10-6 10-4 10-2 100 102 104
(a)
t (sec)
DWS from hard spheres
10-16
10-15
10-14
10-13
10-12
10-11
10-8 10-6 10-4 10-2 100 102 104
10-10
(b)
t (sec)
Mean-squared displacement
MicrorheologyMeasure mean square displacement of probe particles:
Light scattering:Dynamic light scattering
Motion over larger lengths - lower frequenciesDiffusing Wave Spectroscopy
Motion over smaller lengths - higher frequencies
Calculate ModulusGeneralized Stokes-Einstein equation
2
max
Bk Tas
sr
Gs
Transform to storage and loss moduliAnalytic continuation: s = i G G
103
104
105
106
107
10-2 100 102 104 106 108
s (sec1)
G̃(s)(
dyne/cm
2)
“Complex” modulus
100
101
102
103
10-3 10-2 10-1 100 101 102 103
[rad/s]
G'
G''
Light scattering rheology
Weak attractionshigh
Local caging
Glasses
State diagram for colloidal particles
Equilibrium
Jammed States:Nonequilibrium solid states
Hard spheres
Strong attractionslow
Network formation
Φ ~ 10-5Fractal gels
Quasi-equilibrium
Non-equilibrium
Colloidal Stability
Partial StabilityMany collisions to stick
No Stability Sticks every collision
Dilute, stable suspensionC
C
C
CC
C
C
C
6
RkTD
R
Irreversible aggregation
C
C
C
C
C
C
Destablize
1 3
2
2 3
~1~ ~
s csD DC
WRONG!
s
df ~ 1.70
FRACTAL:
• SELF-SIMILARNO CHARACTERISTIC LENGTH SCALE
~ fdM R
df: FRACTAL DIMENSIONNON-INTEGRAL
log M 3df
df < d
log L
DENSITY: DECREASES WITH SIZE
ff
dd d
d
M L LV L
log
log L
FRACTAL
MASS CORRELATIONS:
33-
2-
13d: ( ) ~
12d: ( ) ~
f
f
d
d
c rr
c rr
d = .25 df ~ 1.75
Colloidal Aggregation
Colloidal Gold
DIFFUSION – REACTION –LIMITED AGGREGATION
DIFFUSE MOTION
DIFFUSION-LIMITEDSTICKS WHERE IT FIRST TOUCHES
REACTION-LIMITEDMUST COLLIDE MANY TIMESDIFFUSIVE MOTION NOT IMPORTANT.
d: Euclidean dimension of spaced = 3 real spaced = 2 surface
df: Fractal dimensionAmount of volume occupied by a space filling objectis
dt: Trajectory dimensionFractal dimension of trajectoryRandom walk: dt = 2Ballistic motion: dt = 1No motion dt = 0
DIMENSIONS:
fdM ~ R
DIFFUSION-LIMITED CLUSTER AGGREGATION
1 2 1.75 1.75 2 3.f f td d d
1 2 1.75 0 2 3.f f td d d
NO INTERPENETRATIONBUT CLUSTERS STICK WITH OTHER CLUSTERS
~ 1.8 in 3-d.fd
Brown bag calculation
Gelation of fractal clusters
Gelation of fractal clusters
Gelation of fractal clusters
Gelation of fractal clusters
Rc ~ a-1
Weak attractionshigh
Local caging
Glasses
State diagram for colloidal particles
Equilibrium
Jammed States:Nonequilibrium solid states
Hard spheres
Weakly attractive “gels” at intermediate
volume fractions
Strong attractionslow
Network formation
Φ ~ 10-5Fractal gels
Quasi-equilibrium
Non-equilibrium
Gelation of Attractive ParticlesCarbon Black in Oil
U
0.1 0.8 1.6 2.5 4.0
100 m
Effect of Volume FractionCarbon Black in S150N
U ~ 10 + 2 kT 25oCspacer 23 m
w %
Determination of Volume Fraction
Concentration Series 25oC
' [s-1]102 103
[P
a s]
10-2
10-1
100
4% CB 3% CB 2% CB1.6% CB
Concentration Series 25oC
' [s-1]10-2 10-1 100 101 102 103
[P
a s]
10-2
10-1
100
101
102
4% CB 3% CB 2% CB1.6% CB
Shear breaks aggregates into 0.5 m
units
Volume fraction determined from -dependence of
0.3 3.3 6.4 9.7 14.9
0.1 0.8 1.6 2.5 4.0
100 m
Effect of Volume FractionCarbon Black in S150N
U ~ 10 + 2 kT 25oCspacer 23 m
%
w %
[rad s-1]
10-1 100 101 102
G' (
), G
"(
) [P
a]
10-3
10-2
10-1
100
101
[Pa s]
10-2
10-1
100
101
[rad s-1]
10-1 100 101 102
G' (
), G
"(
) [P
a]
10-3
10-2
10-1
100
101
[Pa s]
10-2
10-1
100
101
[rad s-1]
10-1 100 101 102
G' (
), G
"(
) [P
a]
10-3
10-2
10-1
100
101
[Pa s]
10-2
10-1
100
101
1.2% CB = 0.048 0.8% CB = 0.033 0.4% CB = 0.013
Fluid-Like BehaviorCarbon Black in S150N T=25oC
[rad s-1]
10-1 100 101 102
G' (
), G
"(
) [P
a]
10-2
10-1
100
101
102
[Pa s]
10-1
100
101
102
[rad s-1]
10-1 100 101 102
G' (
), G
"(
) [P
a]
10-2
10-1
100
101
102
[Pa s]
10-1
100
101
102
[rad s-1]
10-1 100 101 102
G' (
), G
"(
) [P
a]10-2
10-1
100
101
102
[Pa s]
10-1
100
101
102
4% CB = 0.149 2.5% CB = 0.097 1.6% CB = 0.064
Solid-Like BehaviorCB in S150N T=25oC
* a 10-2 10-1 100 101 102 103 104 105 106 107 108 109
G'(
)* b
, G
"(
)* b
100
101
102
103
104
105
106
107
Dispersant Series
4.0% CB = 0.1493.0% CB = 0.1112.5% CB = 0.0972.0% CB = 0.0781.8% CB = 0.0731.6% CB = 0.0641.4% CB = 0.056 G'()
G"()
Scaling-Dependence
Carbon Black in S150N U ~ 10 + 2 kT 25oC
a / 102 103 104 105 106 107 108 109
b 100
101
102
103
104
105
106
1
Scaling along the background-viscosity
10-2 10-1 100 101 102 103 104
G'(
) , G
"(
)
10-410-310-210-1100101102103104
experimentally accessible range
0.00 0.05 0.10 0.15 0.20
G'p [Pa]
10-5
10-4
10-3
10-2
10-1
100
101
102
* /
100
101
c
| - c|
10-4 10-3 10-2 10-1
* /
100
101
- 0.13
| - c|
10-3 10-2 10-1 100
G'p [Pa]
10-510-410-310-210-1100101102
4.1
0.00 0.05 0.10 0.15 0.20
G'p [Pa]
10-5
10-4
10-3
10-2
10-1
100
101
102
* /
100
101
c
-Dependence of Fluid-Solid Transition
Effect of DispersantDecrease Interaction Energy
Decrease Aggregation
Amount of Dispersant controls Interaction Energy
Effect of Dispersant ConcentrationCarbon Black in Oil = 0.14 100oC
spacer 6 m
[Disp] w %
100 m
4 3 2 1 0
~ U [kT]3.1 5.1 8.0 12.8
* a 10-2 10-1 100 101 102 103 104 105 106 107 108 109
G'(
)* b
, G
"(
)* b
100
101
102
103
104
105
106
107
G'()
G"()
1.0% Disp U ~ 12.8kT1.5% Disp U ~ 10.0kT2.0% Disp U ~ 8.0kT2.5% Disp U ~ 6.4kT3.0% Disp U ~ 5.1kTconcentration series
Scaling: U-DependenceCarbon Black in S150N ~ 0.14 T=25oC
U0 5 10
* /
100
101
G'p [Pa]
10-5
10-4
10-3
10-2
10-1
100
101
102
|U-Uc|10-2 10-1 100 101
* /
100
101
- 0.14
|U-Uc|10-1 100 101 102
G'p [Pa]
10-510-410-310-210-1100101102
3.9
Uc
Fluid-Solid Transition U-Dependence
Fluid-Solid TransitionWeakly Attractive Systems
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
U
[kT
]
0
5
10
15
20
Solid
Fluid
U
Fluid-Solid TransitionsCarbon Black in Oil
U
[s-1]
10-2 10-1 100 101 102 103 104
[P
a s]
10-2
10-1
100
101
102
4% 3% 2%1.4%0.8%0.4%
[s-1]
10-2 10-1 100 101 102 103 104
[
Pa]
10-3
10-2
10-1
100
101
102
103
o o
Fluidization through Shear
CB in S150N 25oC
Dispersant-Shear Equivalence
x a(disp) [s-1]
10-2 10-1 100 101 102 103 104 105 106
x a
(dis
p) [
Pa]
100
101
102
103
104
o
10-2 10-1 100 101 102 103 104 105 106 [
Pa s]
10-2
10-1
100
101 0% disp.1% disp.2% disp.3% disp.4% disp.
( ) seca disp ( ) seca disp
0.00 0.05 0.10 0.15 0.20
y
[Pa]
10-6
10-5
10-4
10-3
10-2
10-1
100
U [kT]
0 2 4 6 8 10 12 14
y
[Pa]
10-6
10-5
10-4
10-3
10-2
10-1
100
y = (-c)3.4fluid
fluid
solid
solid y = u (U-Uc)2.4
Shear Induced Fluid-Solid-Transition
Phase Boundary in U- Plane
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7
U
[kT
]
0
5
10
15
20
Solid
Fluid
U
Yield Stress as Phase Boundary
0.0 0.1 0.2
yie
ld
[Pa]
10-610-510-410-310-210-1100101102103
8.7 kT 5 kT
16.5 kT
~ 10 + 2 kT
> 20 kT
PMMA - PS
Charged Stabilized PS - Salt
Carbon Black in Oil
y = o ( – c)
0.0
0.2
0.4
0.6
0.8
1.0
10
2012
3
kT/U
1/
[Pa]
Phase Boundary in U – Plane = const.
U/kT0 5 10 15 20
yie
ld
[Pa]
10-4
10-3
10-2
10-1
100
101
= 0.11 = 0.15
= 0.20
= 0.71
= 0.14
PMMA - PSData
Extrapolation
Carbon Black
0.0
0.2
0.4
0.6
0.8
1.0
10
2012
3
kT/U
1/
[Pa]
y = o (U – Uc)
Jamming Phase Diagram for Attractive Systems
U
Proposed by:Andrea Liu, Sid NagelNature 386, 21 (1998)
Completely new way to look at viscosity of soot in oil
Dependence on range of interaction
Attractive colloidal particles
Short Range
Attractive colloidal particles
IntermediateRange
Attractive colloidal particles
Long Range
Long-range attraction
Long Range
Reduce gravity
Spinodal Decomposition of Colloid Polymer
35 hours
Time Evolution of Phase Separation~3
cm
Comparison with Furukawa Theory
Long-time evolution of spinodal decomposition
Short-range attraction
Short Range
How does gelation occur?
What are interparticle interactions?
Structure at gelation Fluid
Structure at gelation Gel
Weak attractive interaction
Cluster distribution: Determine interaction
Cluster distribution: Independent of potential
Interaction energy at gelation
Interaction energy at gelation
Same behavior for all , all
Gelation is proceeded by spinodal decomposition
Implications?
S(q) spinodal decomposition
IncreasingU
Increasingt
Gelation – spinodal decomposition
Schematic Phase Behavior for Colloidal Gels
Attractive glass
U
Attractive interaction Spinodal decomposition attractive glass gelation
Gelation phase diagram
Dynamic arrest of phase separating system
How do crystals melt?
Colloidal crystals melt at grain boundaries
Yodh, Science
What if there are no interfaces?
Born melting:• Elastic catastophe• Elastic modulus goes to zero, and crystal meltsBut how does this actually occur??
High
Volume fraction controls melting
Lower Lowest
Phase diagram of Wigner Crystals
Mean squared displacement
Mean squared displacement
Lindemann parameterFraction of lattice parameter
0.1200.0056
‘Hot’ particles are highly spatially correlated = 0.120
= 0.101‘Hot’ particles are highly spatially correlated
= 0.066‘Hot’ particles are highly spatially correlated
= 0.050‘Hot’ particles are highly spatially correlated
Melted
‘Hot’ particles are strongly correlated in
space
‘Hot’ clusters are fractal
df = 1.70
Cluster-mass distribution
Power-law with exponential cut off
2
Scaling behavior of cluster size
Scaling behavior of cluster volume fraction
Scaling behavior of elasticity
Born (1939)
Scaling behavior
1. Cluster size
2. Cluster volume fraction
3. Elasticity
Scaling behavior
1. Cluster size
2. Cluster volume fraction
3. Elasticity
Second-order character of 3D melting
Hot particles lead to non-affine motion
Breaks force balance weakens lattice
Non-affine motion increases as decreases
Calculate elastic modulus including non-affine motion
Behavior of elasticity
• C44 remains approximately constant with • Non-affine modulus does vanish
2nd order behavior for melting• 1D melting is always 2nd order• 2D melting is through hexatic 2nd order• 3D melting has 2nd order character if the
lattice is perfect• Non-affine shear modulus:
– Provides weakened regions– Mechanical stability is lost– Generalizes Born melting
• 3D melting of Wigner lattice is weakly 1st
order
The End