colloidal fluids, glasses, and crystalsphase diagram for charged spheres order-disorder transition...
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Colloidal Fluids, Glasses, and Crystals
Pierre WiltziusBeckman Institute for Advanced Science and Technology
University of Illinois, Urbana-Champaign
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Thermodynamics of Hard Spheres
Hard-sphere interaction potential:
U(r) =
No exact theory available to calculate g(r).
Equation of state for the fluid using Percus-Yevick approximation (Carnahan &Starling, 1969):
Compressibility factor:
Model hard-sphere system:Silica spheres stabilized with a thin organophilic layer and dispersed in cyclohexane (Vrij et al., 1983)
Osmotic compressibility obtained by light scattering
3
32
)1(1)(
φφφφφ
−−++
=Π
=nkT
Z
drfordrfor
≥<∞
0
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Radial Distribution Function of Hard SpheresFluid State
Smith and Henderson (1970)
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Thermodynamics of Hard Spheres (cont.)
Compressibility factor for the ordered state (Hall, 1972):
With:
Coexistence of fluid and liquid for
ββββββββφ /)4(3118.1922.2819.23053.1176.0125.0558.2)( 6542 −++−+−++=Π
=nkT
Z
)74.0/1(4 φβ −=
55.05.0 ≤≤ φ
Alder and Wainwright (1962)
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Radial Distribution Function of Hard SpheresSolid State
Kincaid and Weis (1977)
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Phase Diagram for Charged Spheres
Order-disorder transition for charged spheres in an electrolyte solution. Data of Hachisu, Kobayashi, and Kose (1973) for polystyrene latices with a = 0.085 μm: open circles, disordered; half-filled circles, two-phase; filled circles, ordered; curves predictions of phase boundaries from perturbation theory for a =0.1 μm and 4πa2q=5000e (Russel, 1987)
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Metastability and Crystallization in Hard-Sphere Systems
Large-scale molecular dynamics simulations. Contrary to previous studies, no evidence of a thermodynamic glass transition and after long times the system crystallizes for all φ above the melting point. M. D. Rintoul and S. Torquato (1996)
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References
• N. F. Carnahan and K. E. Starling, J. Chem Phys. 51, 635 (1969)
• A. Vrij, J. W. Jansen, J. K. G. Dhont, C. Pathmamanoharan, M. M. Kops-Werkhoven, and H. M. Fijnaut, Far. Dis. 76, 19 (1983)
• W. R. Smith and D. Henderson, Mol. Phys. 19, 411 (1970)
• K. R. Hall, J. Chem. Phys. 57, 2252 (1972)
• J. M. Kincaid and J. J. Weis, Mol. Phys. 34, 931 (1977)
• W. B. Russel, Dynamics of Colloidal Systems. University of Wisconsin Press (1987)
• M. D. Rintoul and S. Torquato, Phys Rev. Lett. 77, 4201 (1996)
• B. J. Alder and T. E. Wainwright, Phys. Rev. 127, 359 (1962)
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Colloidal glass of 1μm silica spheres
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Interaction potential: Hard-Sphere0.01M LiCl :
decreases double-layer to a few nm
water/glycerol (16 wt% glyc.): decreases van der Waals forces
Pure SiO2 (n = 1.45)
SiO2 withchemically incorporated dye(fluorescein-isothiocyanate)Exc.: 500 nm Emm.: 520 nm
400 nm
1000 nmPolydispersity: 2%
Monodisperse Silica Sphereswith a Fluorescent Core
Langmuir, 8, 2921 (1992)
Alfons’ SPHERES SHOP
Colloidal Model System
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photomultiplier tube
objective lens, e.g. 100x
laser
illuminating aperture
dichroic beamsplitter
confocal aperture
focal planesample
in focus
out of focus
Fluorescence Confocal Scanning Light Microscope
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0 1 2 3 4 5 6 7-0.5
0.0
0.5
1.0
1.5
2.0Radial Distribution Function
φ=61.2 experiment computer simulation
g(r)
r (sphere diameter)
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0 1 2 3 4 5 6 7-0.5
0.0
0.5
1.0
1.5
2.0
φ=61.2 experiment computer simulation
g(r)
r (sphere diameter)
Correlation Functions
Φ = 61.2
experiment
computer simulation
r (sphere diameter) r (sphere diameter)
g 6(r
)
A. van Blaaderen and P. Wiltzius, Science, 270, 1177 (1995)
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10 11 12 13 14 15 16 17 18 190
5
10
15
20
25
30
Perc
enta
ge o
f Nei
ghbo
rs
Voronoi Coordination Numbers
Experiment Computer Simulationvolume fraction = 63.7%
Voronoi Coordination
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volume fraction = 63.7%
2 3 4 5 6 7 8 90
5
10
15
20
25
30
35
40
45P
erce
ntag
e of
Edg
es
Edges/Voronoi Face
Experiment Computer Simulation
Voronoi Coordination
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-0.2 -0.1 0.0 0.10
2
4
6
8
10
12
Experiment Simulation
Perc
enta
ge o
f Bon
ds
Local Bond-Order Parameter W6
Geometry W6
icosahedral -0.170fcc -0.013hcp -0.012bcc 0.013sc 0.013liquid 0.000
Steinhardt, Nelson, Ronchetti (1983)
Local Bond Order Parameters
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Colloidal “Crystal” of 1 μm Silica Spheres
PreparationPreparation• Sediment particles from
dilute suspensions • Form hexagonally close-
packed planes
Problems Problems • Random stacking in
gravity direction• Polycrystalline domains
Rendering of an experimental sediment characterized with confocal scanning optical microscopy.
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φ = 1%
0.01M LiCl in Glycerol/Water
Spin coated PMMA (dye doped):500 nm
Gold: ~5 nm
Cover glass: 170 μm
Silica sphere radii: Fluorescent core 200 nm
Total 1050 nm
1 μma
Colloidal Epitaxy
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A. von Blaaderen and P. Wiltzius, Nature, 385, 321 (1997)
AchievementAchievement• made 400 x 400 x 70 μm3
single crystal of 1 μm diameter silica spheres settled onto a template with [100] pattern
• Face Centered Cubic (FCC) structure
• well oriented
Large Single Crystal of Colloidal Silica
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a = 1.351st layer
a = 1.3510th layer
a = 1.31st layer
Epitaxy Issues
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100 no template 100
100 no template 100
Epitaxy Issues (cont.)
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K. Busch and S. John, PRE, 58, 3896 (1998)
Close-packed FCC Lattice of Silica Spheres in Air
Density of Optical States
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Density of Optical States
Close-packed FCC Lattice of Air Spheres in Silicon
Band structure
K. Busch and S. John, PRE, 58, 3896 (1998)
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Photonic Bandgap Materials
FCC lattice of air spheres surrounded by high dielectric matrix
Requirements to obtain gapn2/n1>3FCC structure
Potential MaterialsTiO2 n=2.5-2.8CdS n=2.5Se n=2.5-3.2GaP n=3.4Si n=3.5
A. van Blaaderen and P. WiltziusNature, 385, 321 (1997)
R. Biswas, et al. Phys. Rev. B 57, 3701, (1998)
FCC crystal of1μm silica spheres settled on template
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TiO2 replica of colloidal assembly
CdSe
Paul Braun
Electrodeposition
Selenium replica of silica colloid
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Charge-Stabilized Colloidal Crystals
10 μm
M. A. Bevan et al. To be submitted.
cover slip
1st layer of crystal
• sediment into a crystal• hexagonal close packing• highly ordered in wet state
u(r)
rvan der Waals
attraction
electrostaticrepulsion
DLVO potential
fluid
confocal
Fourier transform
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Wet crystal does NOT have…• surface-to-surface packing• mechanical stability• order retention when dried• ability to be furtherprocessed
Drying Stresses:• removal of supporting fluid• capillary forces• convection currents
Defects and Disorder
cover slip
air
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add salt
Charge Stabilized
u(r)
rvan der Waals
attraction
electrostaticrepulsion
Screened
u(r)
r
screening
no salt
salt
Concept: Controlled Salt Addition• retain order• gain stability
electrostaticsdominate
screenedelectrostatics
VdW’s attraction“guide” adhesion
Debye length: controls range of coulumbic repulsion
2122
0
1 1−
−⎟⎟⎠
⎞⎜⎜⎝
⎛= ∑
iii
B
ezTK
K ρεε
ρ = # densityε = dielectric constantε0 = permitivity of free spacei = index of ionic species
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Measuring 2D Orientational OrderOrientation:
N = # particlesn = nearest neighborsj = particle indexk = neighbor index
Confocal Image
Voronoi Plot
∑ ∑=ΨN
j
n
k
i jkenN
θ6
6
11
6ψ
( )∑ +=n
kjkjk i
nθθψ 6sin6cos1
6
Ψ6 → 1 = perfect orderΨ6 → 0 = non 6-fold
All points in the polygonare closest to this point
Nearest neighbors sharesides of polygon
θjk1
23
4
65
j
k’s
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( ) ( )ρ
ρ rrg =
r = radial distance from a particle<ρ(r)> = bin averaged # density between r, r+drρ = bulk # densitya = nearest neighbor separation http://www.ccr.buffalo.edu/etomica/app/modules/sites/Ljmd/Background1.html
Measuring 2D Translational OrderRadial Distribution Function:
0 1 2 3 4 5 60
2
4
6
8
1 0
r (μm)
g(r)
a
1st shell
10 μm
confocal image w/ fluorescence
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Early Attempts: Salt Injection
0.020.37gel1000 mM
--gel100 mM
0.080.70polycrystal10 mM
0.320.67polycrystal1 mM
0.600.61polycrystal0.1 mM
0.930.40crystal0 mM
ψ6φAstructure[NaCl]
Adapted from Bevan et al.
gel
polycrystal
Issues:• rate of contraction,
• Brownian equilibration,
• concentration gradients
• shear flow
Sedimentation cell• 1.18 μm SiO2 colloids• H2O with pH ~ 7• Φ ~ 0.01
Equilibrium
0.1 mM10 μm 1000 mM10 μm
⎟⎠⎞
⎜⎝⎛
dtda
R1
⎟⎠⎞⎜
⎝⎛
2RD
shear
confocal confocal
[ ]NaCl∇
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Centrifuge filterwith salt solution• 5000 NMWL cutoff• NaCl added in steps
Sedimentation cell• 1.18 μm SiO2 colloids• H2O with pH ~ 7• Φ ~ 0.01• NO SALT
Confocal Microscope• 3D reconstructions• fluorophore needed• IDL; image processing
10 mM NaCl added
No salt
g(r)
g(r)
Controlled Addition
M. A. Bevan et al. In preparation for submission.
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0 20 40 60 80 100 120 1400 .0
0 .1
0 .2
0 .3
0 .4
0 .5
0 .6
0 .7
0 .8
0 .9
1 .0
1 .00
1 .05
1 .10
1 .15
1 .20
Tracking 2D Order
What about the rest of the crystal?What’s happening in 3D?
Results:• lattice contracts• order retained••
disorder[salt] →∇disordershear →
t = 0 min10 μm t = 132 min10 μm
Ψ6 a / 2R
Time (min)
2 mM 20 mM 200 mM 2000 mM [NaCl] added to filter tube
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Imaging in 3D
10 μm
18.50 μm
~1:1 glycerol to water~0.2 mM Rhodamine 6G
index matching:• decreases scattering• increases observation range• decreases initial order
fluorescent dye:• increases contrast• feature identification• increases initial ionic strength• decreases initial order
glycerol:ngly = 1.47ηgly = 934 mPas
water:nwat = 1.33ηwat = 0.89 mPas
silica:nsilica ~ 1.4
Rhodamine 6G:disassociates in waterneed ~0.1 mM for contrast
14.25 μm
0:1 glycerol to water~0.3 mM Rhodamine 6G
10 μm 10 μm ~2:1 glycerol to water~0.3 mM Rhodamine 6G
http://omlc.ogi.edu/spectra/PhotochemCAD/html/rhodamine6G.html
Rhodamine 6G
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Rhodamine 6G
YEScrystal/gel2 mM
YEScrystal/gel0.2 mM
THESHOLDcrystal0.02 mM
NOcrystal0.002 mM
contraststructure[Rhodamine]• water soluble• dissociates
[R6G] = 0.2 mM
Ψ6 = 0.17, a / 2R = 1.0810 μm
[R6G] = 0.02 mM
Ψ6 = 0.41, a / 2R = 1.1410 μm
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2:1 glycerol waterΨ6 = 0.83, a / 2R = 1.09
10 μm
25 Scan Average: ~25 seconds
NOcrystal2:1saturated*
NOcrystal0:1saturated*
contraststructureglycerol:water
by volume[Prodan]
* concentration was unable to be determined
http://www.probes.com/servlets/structure?item=248
Prodan• non-ionic• water solubility?
Single Scan: ~1 second
2:1 glycerol:water10 μm
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Controlled Dye Addition:• infill 0.2 mM Rhodamine• reduce debris• retain order
Centrifuge filter• 5000 NMWL cutoff• 400 μL of R6G• [R6G] = 0.45 mM
Sedimentation cell• 1.18 μm SiO2 colloids• H2O with pH ~ 7• Φ ~ 0.01
Equilibrated cell• [R6G] ~ 0.2 mM
Initial: reflectance
Ψ6 = 0.93, a / 2R = 1.3010 μm
Final: fluorescence
Ψ6 = 0.88, a / 2R = 1.0810 μm