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### The scattering wave vector, q.

### Inverse q

Light Scattering

It’s how we see the world. Every non-luminous thing we see, we see via light scattering. Reflection is a special form of light scattering.

Non Invasive Probe of

Aerosol Systems

Colloid Science

Biophysics

Condensed Matter Physics

Aerosol Science

Atmospheric Visibility

Earth’s Radiation Budget

Problems Becoming More Complex

Laser Light Scattering

Typically “VU” scattering

incident polarization vertical,

no polarizer on detector, i.e.,

unpolarized.

Rayleigh ScatteringScattering from Small Particles

What do we mean by “small”?

---small compared to .

(Only two length scales, R and ).

Rayleigh Scattering (2)

Since R<<, the point sub volumes within particle

see the same incident phase

are all essentially the same distance

from the detector

Thus they scatter in phase to the detector

(regardless of angle).

Total scattering amplitude

Total scattered intensity

Rayleigh Scattering (3)Unit Analysis

Cross section units: area = (length)2.

But so far we have

σ ~ V2 = (length)6

Rayleigh Scattering (3b)Unit Analysis

Cross section units: area = (length)2 . But so far we have

So there must be another length scale involved—the only other length scale is . So

Consequences of Rayleigh Scattering (2)

- Isotropic scattering

Consequences of Rayleigh Scattering (3)The Tyndall Effect

The increased scattering as a system coarsens, e.g., precipitates.

Is N N V2/4

Hence Is Vparticle.

Single Slit Diffraction Patterns

a = slit width, = light wavelength

Theory leads to

q = (4π/λ)sin(θ/2)

Much more useful than the scattering angle, θ.

Rayleigh-Debye-Gans Theory

[3u-3 (sin u-u cosu)]2

Where u = qR

(a nice dimensionless variable!)

Simply the square of the Fourier Transform of a sphere. Good when

= 2kR|m-1|<1

where m is the particle refractive index and

is the phase shift parameter.

RDG

- RDG is equal to Rayleigh when scattering angle is small in the “Forward Lobe”
- RDG contains Porod’s Law when qR>1

I q-4

Note that since

q = (4π/λ)sin(θ/2),

inverse q has units of length.

Inverse q is the “length scale of the scattering experiment”.

Crossovers and Length Scales. On log-log plots one finds in general that I(q) vs. q crosses over from one slope to another (i.e., from one power law to another) when q passes through a length scale of the scatterer as demonstrate here.

How big are the water drops in the fog?

Why is the edge of the halo red?

Patterns in Mie Scattering(normalized to I(0))Sorensen and Fischbach, Opt. Commun. 173, 145 (2000).

Fractal Aggregate

Rg = Radius of Gyration a root-mean-square radius

N=monomers/aggregate

N ~ RgD

D = Fractal dimension

Guinier Analysis

When

Thus

Regardless of shape

Regardless of refractive index

Plot I(0)/I(q) vs. q2

Slope = Rg2/3

(Recall the Zimm plot of biophysics)

Classic Zimm Plot

cellulose nitrate fraction in acetone (Benoit, Holtzer, and Doty, JPC58, 635 (1954).

Premixed CH4/O2 Flame Soot

Gangopadhyay et al. Appl. Optics 30, 4859 (1991).

Shearing 1 min after the onset of aggregation. Shear rate: 2.3 sec-1. Gel time: 50 ± 10 min.

Bibliography

Light Scattering by Small Particles, H.C. van de Hulst, Wiley, New York (1957).

The Scattering of Light and Other Electromagnetic Radiation, M. Kerker, Academic, New York (1969).

Absorption and Scattering of Light by Small Particles, C.E. Bohren and D.R. Huffman, Wiley, New York (1983).

"Optical Structure Factor Measurements of Soot Particles in a Premixed Flame," Appl. Optics 30, 4859 (1991) S. Gangopadhyay, I. Elminyawi and C.M. Sorensen.

"Light Scattering Measurements of Monomer Size, Monomers per Aggregate and Fractal Dimension for Soot Aggregates in Flames," Appl. Optics 31, 6547 (1992) C.M. Sorensen, J. Cai and N. Lu.

"Test of Static Structure Factors for Describing Light Scattering from Fractal Soot Aggregates," Langmuir 8, 2064 (1992) C.M. Sorensen, J. Cai and N. Lu.

"Comparison of Size and Morphology of Soot Aggregates as Determined by Light Scattering and Electron Microscope Analysis," Langmuir 9, 2861 (1993) J. Cai, N. Lu and C.M. Sorensen.

"Depolarized Light Scattering from Fractal Soot Aggregates," N. Lu and C.M. Sorensen, Phys. Rev. E50, 3109 (1994).

"Scattering and Absorption of Light by Particles and Aggregates," C.M. Sorensen, in Handbook of Surface and Colloidal Chemistry, Ed. K.S. Birdi, CRC Press, Boca Raton, 1997; p. 533-558.

"Light Scattering Study of Fractal Cluster Aggregation Near the Free Molecular Regime," C. Oh and C.M. Sorensen, J. Aerosol Sci. 28, 937 (1997).

"Structure Factor Scaling in Aggregating Systems," H. Huang, C. Oh, and C.M. Sorensen, Phys. Rev. E57, 875 (1998).

"Aerogelation in a Flame Soot Aerosol," C.M. Sorensen, W.B. Hagemann, T.J. Rush, H. Huang, and C. Oh, Phys. Rev. Lett. 80, 1782 (1998).

"Scaling Description of the Structure Factor of Fractal Soot Composites," C.M. Sorensen, C. Oh, P.W. Schmidt and T. Rieker, Phys. Rev. E58, 4666 (1998).

"Scaling Approach for the Structure Factor of a Generalized System of Scatterers," C. Oh and C.M. Sorensen, J. Nanopart. Res. 1, 369 (1999).

"Size Distribution Effect on the Power Law Regime of the Structure Factor of Fractal Aggregates," C.M. Sorensen and G.M. Wang, Phys. Rev. E60, 7143 (1999).

"Patterns in Mie Scattering," C.M. Sorensen and D.F. Fischbach, Opt. Commun. 173, 145 (2000).

"Guinier Analysis for Homogeneous Dielectric Spheres of Arbitrary Size," C.M. Sorensen and D. Shi, Optics Commun. 178, 31 (2000).

"Light Scattering from Fractal Aggregates. A Review," C.M. Sorensen, Aerosol Sci. Tech. 35, 648 (2001).

"Patterns in the Ripple Structure in Mie Scattering," C.M. Sorensen and D. Shi, J. Opt. Soc. Am. 19, 122 (2002).

“Experimental Test of the Rayleigh-Debye-Gans Theory for Light Scattering by Fractal Aggregates,” G.M. Wang and C.M. Sorensen, Applied Optics 41, 4645 (2002).

“Scattering and Adsorption of Light by Particles and Aggregates,” in Handbook of Surface and Colloidal Chemistry, ed. by K.S. Birdi, CRC Press, Boca Raton, 2003, p. 623.

"Observation of Soot Superaggregates with a Fractal Dimension of 2.6 in Laminar Acetylene/Air Diffusion Flames," C.M. Sorensen, W. Kim, D. Fry, A. Chakrabarti, Langmuir 19, 7560-7563 (2003).

"Universal Occurrence of Soot Aggregates with a Fractal Dimension of 2.6 in Heavily Sooting Laminar Diffusion Flames," W. Kim, C.M. Sorensen, A. Chakrabarti, Langmuir 20, 3969-3973 (2004).

"Structure Factor Scaling in Colloidal Phase Separation," J.J. Cerda, T. Sintes, C.M. Sorensen and A. Chakrabarti, Phys. Rev. E 70, 051405 (2004).

"Aggregates, Superaggregates and Gel-Like Networks in Laminar Diffusion Flames," W.G. Kim, C.M. Sorensen, D. Fry and Amit Chakrabarti, J. Aerosol Science (accepted).

“Patterns in Mie Scattering: Evolution when Normalized by the Rayleigh Cross Section,” M.J. Berg, C.M. Sorensen, and A. Chakrabarti, Applied Optics, accepted.

“Multiple Scattering Effects on Optical Structure Factor Measurements,” T. Mokhtari, C.M. Sorensen and A. Chakrabarti, Applied Optics, accepted.

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