Colloidal Crystallization:
Nucleation and Growth
Dave Weitz
Urs Gasser
Rebecca Christianson
Peter Schall
Eric Weeks
Konstanz
Harvard
Harvard
Emory
Harvard
Andy Schofield
Peter Pusey
Frans Spaepen
John Crocker
Edinburgh
Edinburgh
Harvard
UPenn
•Colloids as model systems – soft materials
•Nucleation and growth of colloidal crystals
•Defect growth in crystals
•Binary Alloy Colloidal Crystals
NSF, NASA
http://www.deas.harvard.edu/projects/weitzlab
EMU 6/7//04
Colloidal Crystals
Colloids
1 nm - 10 µm solid particles in a solvent
Ubiquitous
ink, paint, milk, butter, mayonnaise,
toothpaste, blood
Suspensions can act like both liquid and solid
Modify flow properties
Control: Size, uniformity, interactions
Colloidal Particles
•Solid particles in fluid
•Hard Spheres
•Volume exclusion
•Stability:
•Short range repulsion
•Sometimes a slight charge
Colloid Particles are:
•Big
Slow
•~ a ~ 1 micron
•Can “see” them
• τ ~ a2/D ~ ms to sec
•Follow individual particle dynamics
Model: Colloid Æ Atom
Phase Behavior
Colloids
Atomic Liquids
U
U
s
Hard Spheres
Osmotic Pressure
Centro-symmetric
Thermalized: kBT
Gas
Liquid
Density
Solid
s
Leonard-Jones
Attraction
Centro-symmetric
Thermalized: kBT
Gas
Liquid
Solid
Phase behavior is similar
Hard Sphere Phase Diagram
Volume Fraction Controls Phase Behavior
liquid
liquid - crystal
coexistence
49%
54%
maximum packing
φRCP≈0.63
crystal
63%
74%
φ
maximum packing
φHCP=0.74
Increase φ => Decrease Temperature
0
F = U - TS
Entropy Drives Crystallization
Entropy => Free Volume
0
F = U - TS
Disordered:
•Higher configurational
entropy
•Lower local entropy
•Higher Energy
Ordered:
•Lower configurational
entropy
•Higher local entropy
•Lower Energy
maximum packing
φRCP≈0.63
maximum packing
φHCP=0.74
Soft Solids
Easily deformable Æ Low Elastic Constant:
Atoms:
eV
3
Α
~GPa
Colloids:
kBT
µ m3
~Pa
Energy
Volume
Easily deformed Æ Shear melt to randomize
Colloidal Particle Æ Atom
Watch each atom!
Confocal Microscopy
Regular
Confocal
detector
lens
objective
in focus
out of focus
pinhole
Confocal microscopy for 3D pictures
0.2 µm
Scan many slices,
reconstruct 3D image
Microscopy and Tracking
Confocal
microscopy:
Particle
tracking:
•30 images/s (512×480 pixels, 2D)
•one 3D “cube” per 6 s
•67 × 63 ×10 µm3
•100× oil / 1.4 N.A. objective
•Identify particles within 0.03 µm (xy)
0.05 µm (z)
•Follow 3000-5000 particles, in 3D
•200-1000 time steps = hours to days
•≈ 4 GB of images per experiment
First direct 3D observation of dynamics
Brownian Motion in Real Time
Colloidal Crystals
1 cm
25 µm
Bragg scattering of visible light
Hexagonal close-packed layers (FCC/HCP)
Nucleation and Growth of Colloidal Crystals
2.3 µm diameter PMMA spheres
Questions:
•How do crystals nucleate?
•What is structure of pre- and post-critical nuclei?
•How does structure evolve with time?
•What is free energy barrier?
Crystallization
∆G = γ ( 4π r
2
) − ∆µ (
Surface energy
4
3
πr
3
)
Chemical potential
∆G
r
How to Identify Crystals
2.3 µm diameter PMMA spheres
Must identify incipient crystal nuclei
Voronoi polyhedra -Delaunay triangulation
(“Wigner-Seitz cell”)
defines nearest neighbor particles
Local Crystallization Order Parameter
P. R. ten Wolde, M. J. Ruiz-Montero,
D. Frenkel: J. Chem. Phys. 104, 9932 (1996)
•Find nearest neighbor connections rij
•Resolve connections in spherical
harmonics:
qlm(i)=⟨Ylm(rij)⟩j
•Examine l=6
•Define Order Parameter: q6m(i) · q6m(j)
•If q6m(i) · q6m(j) > 0.5, then bond (ij) is “crystal-like”
•if particle has ≥ 8 “crystal-like” bonds, it is a crystal-like particle
Colloidal Crystallization
Determination of Size of Critical Nucleus
Crystal Nucleus Structure
R ~ Rc φ = 0.47
Structure of Crystal Nucleus
φ
bcc
fcc
hcp
liquid
0.49
0.00(3)
0.10(3)
0.32(6)
0.58(1)
0.45
0.00(1)
0.58(7)
0.20(3)
0.22(6)
0.43
0.00(3)
0.34(5)
0.25(4)
0.41(1)
RHCP: Random Hexagonally Close Packed
A B C
A
fcc
hcp
Nucleation Rate
Comparable to light scattering measurements
Faster than simulations
Finding Surface Tension
∆G = γ ( 4π r
2
) − ∆µ (
Surface energy
4
3
πr
3
)
Chemical potential
⎛ − ∆G ⎞
2
⎟⎟ ≈ exp − γr
P(r ) ≈ exp⎜⎜
⎝ k BT ⎠
(
(for small r)
)
Measurement of Surface Tension
1000
γ (kT/a2)
0.02
100
0.016
N
0.012
0.4
0.45
0.5
0.55
0.6
Φ
10
1
0
100
200
300
400
500
600
700
800
2
A (µm )
Surface tension is very low
900
1000
Depletion Zone near Surface
φ ( shell )-<φ > / < φ >
0.04
φ=0.373
φ=0.428
0.03
φ=0.492
crystal liquid
0.02
0.01
0.00
-0.01
1
2
3
4
shell
5
6
Crystal Structure through Interface
0.8
(crystal)
cx
(fluid)
0.4
bcc
fcc
hcp
fluid
0.0
1
3
5
7
shell
No bcc structure at all
Random stacking of hexagonal planes
Fractal Structure of Growing Crystallites
103
3.0
df
2.5
2.0
1.5
M
102
1.0
0.4
Φ
0.5
0.6
101
100
100
101
rg (µm)
Single Crystal Growth - Principle
Peter Schall and Frans Spaepen
Laser mask writer
A
B
A
(a)
Fixed stacking sequence
(b)
fcc single crystal
Templated Growth of Single Crystals
SEM micrograph
Confocal micrograph
Thin Film of Single Crystal
Colloidal
crystals
diffracted
beams
transmitted
beam
Imaging Dislocations : Laser Microscope
Colloidal
Crystal
Screen
Lens
Objective
Imaging Dislocations: Excitation Error
Exact two beam
condition
Transmitted
beam
Diffracted
beam
Two beam with
excitation error
Transmitted
beam
Diffracted
beam
Imaging Dislocations: Nearly perfect crystal
0
Exact two beam
condition
Two beam with
excitation error
0
100 µm
“good“ lattice constant
Imaging Dislocations
Exact two beam
condition
Two beam with
excitation error
100 µm
Template stretched by 1.5 %
Nucleation and Growth of Dislocations
• template lattice constant off by 1.5 %
(film recorded
immediately after settling
of ~ 8 additional crystalline
layers
recording time 3.5 hours
Stacking Faults induced by Lattice Mismatch
Growth of Defects
A
B
t=t0
t=t0+16 min
C
t=t0+260 min
11
44
22
33
100 µm
100
60
2
1
3
40
4
FPK
Ffr
Fl
screw
edge
20
80
L [µm]
L [µm]
80
1
60
40
20
0
0
20
40
60
80
100 120 140 160
t [min]
0
0
•Balance forces for growth rate
•Elastic force = viscous drag + line tension
•Exponential functional form
500 1000 1500
t [sec]
Indentation experiment
sewing needle tip
(Crystal at the end of the movie;
there is still a couple of layers
above the layers shown)
Dislocation Dynamics and Interactions
• Indentation
100 µm
Binary Colloidal Crystals
Maximum Volume Fraction
0.8
(Schofield, et al)
AB6 BCC
AB (CsCl)
AB2
AB6 SC
0.6
AB13
AB (NaCl)
0.0
0.2
0.4
0.6
RB/RA
0.8
1.0
AB6
• AB6 occurs in three different structures: simple cubic,
BCC and FCC, which occur at different size ratios from
0.3 to 0.4
• BCC is the fastest to crystallize, with crystals forming
overnight at the optimal size ratio of 0.4
AB6 Binary Alloy Crystal
• Very intense Bragg scattering
• Very large crystals
AB6 Binary Alloy Crystal
•Very well ordered FCC structure
•Small particles induce effective long-range intereaction
•Creates highly ordered large particle lattice
Nucleation and Growth of AB6
Look only at large particles Æ perfect BCC lattice
AB13
• Icosahedra of small
particles, at center of
simple cube of big
particles
• Found naturally in
bimetallic alloys such
as NaZn13
• Stable at size ratios
from 0.5-0.7
10000
0.000
0.005
0.010
(6 0 0)
(6 2 2)
(6 4 2)
(7 3 1)
(8 2 0)
(8 2 2)
(7 5 3)
(5 3 1)
AB13 SLS Spectrum
Red indicates superlattice
(6 2 0)
(4 4 4)
(6 4 0)
(8 0 0)
(7 5 1)
100000
(4 4 0) ?
Intensity
(2 2 2)
(4 0 0)
1000000
(4 2 2)
(4 2 0)
(2 2 0)
(2 0 0)
AB13 Space Sample
RB/RA = 0.57, NB/NA = 19
0.015
0.020
Q (1/nm)
0.025
0.030
0.035
0.040
Mixed Sizes:
AB13 Binary Alloy Crystal
•Very well ordered FCC structure
•Small particles induce effective long-range intereaction
•Creates highly ordered large particle lattice
Conclusions
Real space imaging of colloidal structure and dynamics
•3D observation of crystallization
•Defect growth in crystal films
•Binary Alloy Colloidal Crystals
http://www.deas.harvard.edu/projects/weitzlab