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

Colloidal Crystals

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

Confocal MicroscopyRegular Confocal

lens

objective

detector

pinhole

in focus

out of focus

Scan many slices,reconstruct 3D image

0.2 µm

Confocal microscopy for 3D pictures

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

Brownian Motion in Real Time

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?

Crystallization

( ) ( )2 3434γ π µ π∆ = − ∆r rG

Chemical potentialSurface energy

r

∆G

How to Identify Crystals

2.3 µm diameter PMMA spheres

Must identify incipient crystal nuclei

defines nearest neighbor particles

Voronoi polyhedra --Delaunay triangulation

(“Wigner-Seitz cell”)

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 StructureR ~ Rc φ = 0.47

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

A hcp

RHCP: Random Hexagonally Close Packed

A B C fcc

Nucleation Rate

Comparable to light scattering measurementsFaster than simulations

( ) ( )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

SEM micrograph Confocal micrograph

Templated Growth of Single Crystals

transmittedbeamdiffracted

beams

Colloidalcrystals

Thin Film of Single Crystal

Imaging Dislocations : Laser Microscope

LensScreen

Objective

ColloidalCrystal

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

Stacking Faults induced by Lattice Mismatch

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

100 µm

Dislocation Dynamics and Interactions• Indentation

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 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 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 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