Colloidal Crystallization: Nucleation and Growth Dave Weitz Harvard Urs Gasser Konstanz Andy Schofield Edinburgh Rebecca Christianson Harvard Peter Pusey Edinburgh Peter Schall Harvard Frans Spaepen Harvard Eric Weeks Emory John Crocker 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 • τ ~ a2/D ~ ms to sec •Can “see” them •Follow individual particle dynamics Model: Colloid Æ Atom Phase Behavior Colloids Atomic Liquids U U

s s Hard Spheres Leonard-Jones Osmotic Pressure Attraction Centro-symmetric Centro-symmetric Thermalized: kBT Thermalized: kBT Gas Gas Liquid Density Liquid Solid Solid Phase behavior is similar Hard Sphere Phase Diagram Volume Fraction Controls Phase Behavior liquid - crystal liquid coexistence crystal

49% 54% 63% 74% φ

maximum packing maximum packing

φRCP≈0.63 φHCP=0.74 0 Increase φ => Decrease Temperature F = U - TS Entropy Drives Crystallization 0 Entropy => Free Volume F = U - TS

Disordered: Ordered: •Higher configurational •Lower configurational entropy entropy •Lower local entropy •Higher local entropy •Higher Energy •Lower Energy maximum packing maximum packing

φRCP≈0.63 φHCP=0.74 Soft Solids

Æ Energy Easily deformable Low Elastic Constant: Volume

eV ~GPa Atoms: Α 3 kT Colloids: B ~Pa µm3

Easily deformed Æ Shear melt to randomize Colloidal Particle Æ Atom Watch each atom! Confocal Microscopy

Regular Confocal

detector lens pinhole

objective

in focus

out of focus Confocal microscopy for 3D pictures

0.2 µm

Scan many slices, reconstruct 3D image Microscopy and Tracking

•30 images/s (512×480 pixels, 2D) Confocal •one 3D “cube” per 6 s microscopy: •67 × 63 ×10 µm3 •100× oil / 1.4 N.A. objective •Identify particles within 0.03 µm (xy) 0.05 µm (z)

Particle •Follow 3000-5000 particles, in 3D tracking: •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

2 4 3 ∆=G γπ(4 r ) −∆ (3 r ) µπ Surface energy 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 CA fcchcp Nucleation Rate

Comparable to light scattering measurements Faster than simulations Finding Surface Tension

2 4 3 ∆=G γπ(4 r ) −∆ (3 r ) µπ Surface energy Chemical potential

⎛ − ∆G ⎞ 2 P(r) ≈ exp⎜ ⎟ ≈ exp()− γr ⎝ kBT ⎠ (for small r) Measurement of Surface Tension

1000

0.02 ) 2 0.016 (kT/a γ 100 0.012

N 0.4 0.45 0.5 0.55 0.6 Φ

10

1 0 100 200 300 400 500 600 700 800 900 1000 A (µm2)

Surface tension is very low Depletion Zone near Surface

0.04 φ=0.373 φ=0.428

> 0.03 φ crystal liquid φ=0.492

> / < > / 0.02 φ < )- ll e 0.01 h s ( φ 0.00

-0.01 123456 shell Crystal Structure through Interface

0.8 (fluid) (crystal) x c 0.4 bcc fcc hcp fluid

0.0

1357 shell No bcc structure at all Random stacking of hexagonal planes Fractal Structure of Growing Crystallites

103 3.0 2.5 f

d 2.0 1.5 102 1.0 0.4 0.5 0.6 Φ M

101

100 100 101 rg (µm) Single Crystal Growth - Principle

Peter Schall and Frans Spaepen

Laser mask writer A B A

(a) (b)

Fixed stacking sequence fcc single crystal Templated Growth of Single Crystals

SEM micrograph Confocal micrograph Thin Film of Single Crystal

Colloidal crystals

transmitted diffracted beam beams Imaging Dislocations : Laser Microscope

Colloidal Crystal Screen Lens

Objective Imaging Dislocations: Excitation Error

Exact two beam Two beam with condition excitation error

Transmitted Diffracted Transmitted Diffracted beam beam beam beam Imaging Dislocations: Nearly perfect crystal

Exact two beam Two beam with condition excitation error 0 0

100 µm

“good“ lattice constant Imaging Dislocations

Exact two beam Two beam with condition 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 t=t0 B t=t0+16 min C t=t0+260 min 11 44

22 33 100 µm

80 100

60 1 Ffr screw 80

m] 2 F PK Fl µ 3 4 1 40 60 L [ edge m]

µ 40

20 L [ 20 0 0 20 40 60 80 100 120 140 160 0 t [min] 0 500 1000 1500 t [sec] •Balance forces for growth rate •Elastic force = viscous drag + line tension •Exponential functional form 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

0.8

AB6 BCC

AB (CsCl)

AB2 0.6 AB6 SC

AB13

Maximum Volume Fraction Maximum AB (NaCl)

(Schofield, et al) 0.0 0.2 0.4 0.6 0.8 1.0 R /R B A 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 AB13 Space Sample RB/RA = 0.57, NB/NA = 19

(2 0 0) AB13 SLS Spectrum

(2 2 0) (2 2 Red indicates superlattice

1000000 1) (5 3 (4 2 0) (4 2 2) (2 2 2) (6 0 0) (4 0 0) (8 2 2) (8 2 (7 3 1) (7 3 100000 (6 2 2) (6 2 Intensity (8 2 0) (8 2 (6 4 2) (6 4 (7 5 3)

10000 (4 4 0) ? 4 0) (4 (6 2 0) (6 4 0) (6 4 (4 4 4) (8 0 0) (7 5 1) (7 5

0.000 0.005 0.010 0.015 0.020 0.025 0.030 0.035 0.040 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 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