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Colloidal Crystallization: Nucleation and Growth
Dave Weitz Harvard
•Colloids as model systems – soft materials•Nucleation and growth of colloidal crystals•Defect growth in crystals•Binary Alloy Colloidal Crystals
EMU 6/7//04
NSF, NASA
http://www.deas.harvard.edu/projects/weitzlab
Urs Gasser KonstanzRebecca Christianson HarvardPeter Schall HarvardEric Weeks Emory
Andy Schofield EdinburghPeter Pusey EdinburghFrans Spaepen HarvardJohn Crocker UPenn
Colloids
Ubiquitousink, paint, milk, butter, mayonnaise,toothpaste, blood
1 nm - 10 µm solid particles in a solvent
Suspensions can act like both liquid and solidModify flow properties
Control: Size, uniformity, interactions
Colloidal Particles
•Solid particles in fluid•Hard Spheres
•Volume exclusion•Stability:
•Short range repulsion•Sometimes a slight charge
•Big•~ a ~ 1 micron•Can “see” them
Model: Colloid Atom
Slow• τ ~ a2/D ~ ms to sec•Follow individual particle dynamics
Colloid Particles are:
Phase BehaviorColloids Atomic Liquids
Hard SpheresOsmotic PressureCentro-symmetricThermalized: kBT
GasLiquidSolid
Leonard-JonesAttractionCentro-symmetricThermalized: kBT
GasLiquidSolid
Density
Phase behavior is similar
U
s s
U
liquidliquid - crystal
coexistence crystal
49% 54% 63% 74%
φ
maximum packingφHCP=0.74
maximum packingφRCP≈0.63
Hard Sphere Phase Diagram
Increase φ => 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 packingφHCP=0.74
maximum packingφRCP≈0.63
F = U - TS0
Soft SolidsEasily deformable Low Elastic Constant: V
Eonergylume
Colloids: 3Bk Tmµ
~Pa
Atoms: 3eVΑ
~GPa
Colloidal Particle AtomWatch each atom!
Easily deformed Shear melt to randomize
Microscopy and 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
Confocalmicroscopy:
Particletracking:
First direct 3D observation of dynamics
1 cm
25 µmBragg scattering of visible lightHexagonal close-packed layers (FCC/HCP)
Colloidal Crystals
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?
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
Structure of Crystal Nucleus
0.41(1)0.25(4)0.34(5)0.00(3)0.430.22(6)0.20(3)0.58(7)0.00(1)0.450.58(1)0.32(6)0.10(3)0.00(3)0.49
liquidhcpfccbccφ
( ) ( )2 3434γ π µ π∆ = − ∆r rG
Chemical potentialSurface energy
( )2γexpexp)( rTkGrP
B
−≈⎟⎟⎠
⎞⎜⎜⎝
⎛ ∆−≈
(for small r)
Finding Surface Tension
1
10
100
1000
0 100 200 300 400 500 600 700 800 900 1000
N
A (µm2)
0.012
0.016
0.02
0.4 0.45 0.5 0.55 0.6
γ (k
T/a2 )
Φ
Measurement of Surface Tension
Surface tension is very low
1 2 3 4 5 6-0.01
0.00
0.01
0.02
0.03
0.04
shell
φ(sh
ell)
-<φ>
/ <φ>
φ=0.373
φ=0.428
φ=0.492liquidcrystal
Depletion Zone near Surface
1 3 5 7
0.0
0.4
0.8
shell
c x
bcc
fcc
hcp
fluid
(fluid) (crystal)
Crystal Structure through Interface
No bcc structure at allRandom stacking of hexagonal planes
100 101100
101
102
103
rg (µm)
M
0.4 0.5 0.61.0
1.5
2.0
2.5
3.0
Φ
d f
Fractal Structure of Growing Crystallites
(a) (b)
ABA
Single Crystal Growth - Principle
Fixed stacking sequence fcc single crystal
Peter Schall and Frans Spaepen
Laser mask writer
Imaging Dislocations: Excitation Error
Diffractedbeam
Exact two beamcondition
Transmittedbeam
Diffractedbeam
Two beam withexcitation error
Transmittedbeam
100 µm
Imaging Dislocations: Nearly perfect crystal
0 0
“good“ lattice constant
Exact two beamcondition
Two beam withexcitation error
100 µm
Imaging Dislocations
Template stretched by 1.5 %
Exact two beamcondition
Two beam withexcitation error
• template lattice constant off by 1.5 %
(film recordedimmediately after settlingof ~ 8 additional crystallinelayersrecording time 3.5 hours
Nucleation and Growth of Dislocations
t=t0+260 min
0 20 40 60 80 100 120 140 1600
20
40
60
80
L [µ
m]
t [min] 0 500 1000 1500
0
20
40
60
80
100
L [µ
m]
t [sec]
t=t0
100 µm
t=t0+16 min
2233
11 44
A B C
FPK Fl
Ffr
edge
screw12 3 4 1
Growth of Defects
•Balance forces for growth rate•Elastic force = viscous drag + line tension•Exponential functional form
Indentation experiment
(Crystal at the end of the movie; there is still a couple of layersabove the layers shown)
sewing needle tip
Binary Colloidal Crystals
0.0 0.2 0.4 0.6 0.8 1.0
0.6
0.8
AB6 SC
AB6 BCC
AB (NaCl)
AB13
AB2
AB (CsCl)
Max
imum
Vol
ume
Frac
tion
RB/RA
(Schofield, et al)
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 well ordered FCC structure•Small particles induce effective long-range intereaction
•Creates highly ordered large particle 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
AB13 Space SampleRB/RA = 0.57, NB/NA = 19
0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040
10000
100000
1000000
AB13 SLS SpectrumRed indicates superlattice
(7 5
3)
(7 5
1)
(8 2
2)
(8 2
0)
(8 0
0)
(6 4
0)
(4 4
4)
(6 2
2)
(6 2
0)
(4 4
0) ?
(7 3
1)
(6 4
2)
(5 3
1)
(6 0
0)
(4 0
0) (4 2
2)
(4 2
0)
(2 2
2)
(2 2
0)(2
0 0
)
Inte
nsity
Q (1/nm)
Mixed Sizes:AB13 Binary Alloy Crystal
•Very well ordered FCC structure•Small particles induce effective long-range intereaction
•Creates highly ordered large particle lattice
ConclusionsReal 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