Characterization of the Aggregate Structure: Light Scattering...3 Physics of Light Scattering ! In the case of a dilute suspension of identical particles, the intensity of the scattered

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)

6  

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)

10

Static Light Scattering: Clusters of particles

-Df

Guinier Regime: Average Radius of Gyration (Rg)

Fractal Regime: Cluster Fractal

Dimension S(q)~q-Df

Rg

11

( )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

15

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

16  

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

17  

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

18  

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