characterization of the aggregate structure: light scattering...3 physics of light scattering ! in...
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1
Multi-angle static light scattering
Detector
Detector
DLS set-ups
90 degrees
~Back scattering
Characterization of the Aggregate Structure: Light Scattering
2
Physics of Light Scattering
§ The exact determination of the amount and profile of scattered light can be done by Solving Maxwell equation
§ Exact solutions are available only for objects with simple geometries (spheres, cylinders, spheroids…).
§ The solution for spherical particles is usually known as Lorentz-Mie theory
2 2k∇ =E E 2 2k∇ =H H2 2k ω εµ=
Monochromatic radiation with frequency ω
Dielectric permittivity
Magnetic permeability E=Electric Field
H=Magnetic Field
Helmholtz Vector Equations
3
Physics of Light Scattering
§ In the case of a dilute suspension of identical particles, the intensity of the scattered light is given by:
64( ) ( )NI q R P q
λ∝ ⋅ ⋅
N = Number concentration of particles R = Particle Size λ = wavelength of light P(q) = scattering form factor q = scattering wave vector
For R < λ (colloidal size range)
24( ) ( )NI q R P q
λ∝ ⋅ ⋅
For R >> λ
P(q) depends upon the size and shape of particles
4q= sin2
nπ θλ
⎛ ⎞⎜ ⎟⎝ ⎠
4
Physics of Light Scattering
§ The scattering angle and the wavelength of the laser determine the length scale which is probed during the scattering experiment
§ By increasing the angle or decreasing the wavelength, one can change the range of sizes probed during the measurement
§ With light scattering it is possible to span almost 6 orders of magnitude in size (1nm-1mm) (using different instruments)
4q= sin2
nπ θλ
⎛ ⎞⎜ ⎟⎝ ⎠ 1/q = probed lengthscale
5
Physics of Light Scattering V
Blue Sky: blue light scattered by airborne dust particles and droplets Flour in water: Tyndall effect (blue
light scattered)
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Multi-angle static light scattering
Static Light Scattering
7
§ Light scattering is very sensitive to the formation of clusters of particles
§ Non-invasive technique: no manipulation of the sample required (only dilution)
2( ) ( ) ( )I q m P q S q∝ ⋅
Form Factor (depends only on particle size and
shape)
Structure factor (depends only on particles relative
positions)
S(q) = Scattering Structure Factor m = cluster mass (number of particles)
Characterization of the Aggregate Structure: Light Scattering
4q= sin2
nπ θλ
⎛ ⎞⎜ ⎟⎝ ⎠
1/q = probed length-scale
8
2( ) ( ) ( )I q m P q S q∝ ⋅
Characterization of the Aggregate Structure: Light Scattering
Form Factor (depends only on particle size and
shape)
0.0 1.0x10-2 2.0x10-2 3.0x10-210-4
10-3
10-2
10-1
100
101
<P(q
)>
q [nm-1]
(b)
9
Static Light Scattering
0.005 0.01 0.02 0.0410-0.8
10-0.6
10-0.4
10-0.2
100
q (1/nm)
<S(q
)>
3h5h
6h
8h
9hCenter of mass
ri mi
2
2i i
ig
r mR
M=∑
Characterization of Aggregate Structure 2( ) ( ) ( )I q m P q S q∝ ⋅
Structure factor (depends only on particles relative
positions)
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Static Light Scattering: Clusters of particles
-Df
Guinier Regime: Average Radius of Gyration (Rg)
Fractal Regime: Cluster Fractal
Dimension S(q)~q-Df
Rg
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( )I t Intensity fluctuations
0
1( ) lim ( ) ( )T
Tg I t I t dt
Tτ τ
→∞= + ⋅∫
22( ) q Dg A Be ττ −= +
g(τ) = autocorrelation function D = Diffusion Coefficient η = viscosity Rh = hydrodynamic radius
6 h
kTDRπη
=
Dynamic Light Scattering
12
Detector ~Back scattering
Dynamic Light Scattering
13
∑
∑
=
=>=< n
ii
n
iigi
g
Ni
RNi
R
1
2
1
2,
2
2
2
12
1 ,
( )
( )
n
i ii
h ni i
i h i
i N S qR
i N S qR
=
=
< >=∑
∑
§ In the presence of broad populations of clusters, the sizes measured by SLS and DLS are average sizes
• The average quantities show how large clusters dominate (due to the square of the mass weighting)
• The two averages represent different moments of the cluster mass distribution, which evolve in a different manner in time
Characterization of the Aggregate Structure: Light Scattering
14
Population Balance Equation
Smoluchowski Equations solved numerically for each cluster of size i (tricks for numerical solutions)
* 1* * * *
2 2, ,
12
ii
j i j i jj
j ij
i j jdN k N N N k Ndt
− ∞
− −= =
= −∑ ∑
Calculation of complete aggregate population
Calculation of average quantities (moments): (e.g., size, Rh, Rg)
Comparison with experimental data
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Aggregation kernel in Brownian conditions
( )( )
1/ 1/1/ 1/
,
1 1813 4
f f
f f
D DD D
Bi j
i ji jk Tk ij
Wλ
η
⎛ ⎞+ +⎜ ⎟
⎝ ⎠= ⋅ ⋅
810W :
0 2 4 6 8 100
50
100
150
Time (h)
<Rh>
and
<Rg>
[nm
]
WTOT WTOT= WA+WR
repulsion
attraction
Interaction energy
r
Population Balance Equation
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Model predictions using the kernel for partially destabilezed particles including fractal behavior
Rg
Rh
Rg
Rh
MFA® latex, destabilized with NaCl
1.5% volume fraction 0.6% volume fraction
Reaction limited aggregation – Validation with experimental data
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RLCA, df ≈ 2.1 DLCA, df ≈ 1.8
Ri lo
g i
log Ri
df
Fractal concept fd
i
p
Ri kR
⎛ ⎞= ⎜ ⎟⎜ ⎟
⎝ ⎠
10-5 10-4 10-3 10-210-4
10-3
10-2
10-1
100
S(q)
q (1/nm)
Slope = 1.77
Static Light Scattering Image analysis Monte-Carlo simulation DLCA RLCA
Universal behavior – Aggregate structure
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RLCA DLCA Universal behavior – Aggregate structure
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Population balance equations Dimensionless variables
temperature
Initial particles concentration
Dimensionless time
* 1* * * *
2 2, ,
12
ii
j i j i jj
j ij
i j jdN k N N N k Ndt
− ∞
− −= =
= −∑ ∑
Bi , jPi , j =i1/Df + j1/Df( ) 1
i1/Df+1j1/Df
!
"##
$
%&&
4ij( )
λ
Fuchs stability ratio (Activation energy)
τ = t ⋅ 1W⋅8kBT3η
⋅N0viscosity
WTOT WTOT= WA+WR
repulsion
attraction
Interaction energy
r
Population Balance Equation
20
Lin et al., Phys. Review A,41,1 1990 Sandkuehler et al, J. Phys. Chem. B, 2004
Master Curve