Scattering and Absorption of Light by Paticles and Aggregates

19 Scattering and Absorption of Lightby Particles and Aggregates

C.M. Sorensen

CONTENTS

19.1 Introduction ................................................................................................................................................................. 71919.2 Small Particles ............................................................................................................................................................. 720

19.2.1 Polarization Considerations ........................................................................................................................... 72019.2.2 Rayleigh Differential Scattering Cross Section ............................................................................................. 72119.2.3 Rayleigh Total Scattering Cross Section ....................................................................................................... 72319.2.4 Rayleigh Absorption Cross Section............................................................................................................... 723

19.2.4.1 Scattering, Absorption, and Extinction .......................................................................................... 72319.2.5 Rayleigh–Debye–Gans Scattering ................................................................................................................. 724

19.3 Spheres of Arbitrary Size: The Mie Theory ............................................................................................................... 72519.3.1 Mie Differential Scattering Cross Sections ................................................................................................... 725

19.3.1.1 Forward Scattering ......................................................................................................................... 72819.3.1.2 Total Cross Section ........................................................................................................................ 72819.3.1.3 Mie Guinier Regime....................................................................................................................... 72919.3.1.4 Mie Ripples .................................................................................................................................... 730

19.4 Multiple Scattering ...................................................................................................................................................... 73019.5 Fractal Aggregates....................................................................................................................................................... 731

19.5.1 Scattering and Absorption by Fractal Aggregates......................................................................................... 73419.5.2 Structure Factor.............................................................................................................................................. 73519.5.3 Structure Factor of an Ensemble of Polydisperse Aggregates ...................................................................... 73619.5.4 Aggregating Ensembles of Fractal Aggregates ............................................................................................. 73819.5.5 Examples........................................................................................................................................................ 739

Appendix 19A: Cross Sections ........................................................................................................................................... 741Appendix 19B: Scattering Wave Vector ............................................................................................................................. 742Acknowledgments.................................................................................................................................................................... 744References ................................................................................................................................................................................ 744

19.1 INTRODUCTION

The goal of this chapter is to give a brief yet comprehensive description of scattering and absorption of light by sphericalparticles of arbitrary size and aggregates of these particles that have the fractal morphology. This chapter has been updatedfrom its previous appearance in 2003. Changes include a better vision of various patterns in Mie scattering from spheresdiscovered in 2001. I also include a discussion of multiple scattering and recent work on hybrid aggregate structures. Asbefore I have refrained from derivation but have tried to give a physical interpretation of the results I quote. Exact equationsfor scattering and absorption are given, but the emphasis is placed not on the equations themselves but instead on the features,i.e., important functional dependencies, relation to other forms, cardinal points, etc., and on their graphical representation.Hence, I hope I have given the reader a user’s guide to light scattering and absorption that will provide a broad yetwell-connected perspective.

719

Birdi/Handbook of Surface and Colloid Chemistry 7327_C019 Final Proof page 719 14.10.2008 10:39pm Compositor Name: MSubramanian

© 2009 by Taylor & Francis Group, LLC

19.2 SMALL PARTICLES

Small is a relative term. The length scale of light is its wavelength l; hence, a small particle has all its dimensions smallcompared to l. The simplest case, which is considered here, is a spherical particle of radius a. We may then define the sizeparameter as a¼ 2pa=l which is a dimensionless ratio of the two length scales involved.

19.2.1 POLARIZATION CONSIDERATIONS

Since its advent, the light source of choice for light scattering studies of colloids and aerosols has been the laser. This source is almostalways polarized in the vertical plane. Thus, we do not directly discuss the case of unpolarized or natural light, such as that obtainedfrom an ordinary lightbulb or the sun, incident upon the scatterers. This can be an important situation, especially in nature, but itsrules can be inferred from the incident light polarized case, which we discuss here, and it can also be found in earlier work [1–3].

Consider a light wave traveling in the positive z direction incident upon a particle at the origin as drawn in Figure 19.1. Thedirection of propagation is described by the incident wave vector~ki with magnitude

~ki�� ¼ 2p=l (19:1)

where l is the wavelength of the light in the medium. There are two independent linear polarizations, and we initially considerlight polarized along the x-axis. To relate this to typical laboratory situations we will call the x-axis vertical, and the y–z planethe horizontal, scattering plane. Light can be scattered by this particle in any direction described by the scattered wave vector~ksin Figure 19.1. We consider elastic scattering; hence,

~ks�� ¼ ~ki

�� (19:2)

Because these magnitudes are equal, we represent each simply by k. Note that a¼ ka.For particles small compared to the wavelength we can now describe the polarization of the scattered light to be in the

direction of the projection of the incident polarization onto a plane perpendicular to~ks. In vector notation when the incidentpolarization is in the x direction, the polarization direction of the scattered light is the double cross product, k̂s � x̂

� �� k̂s,where k̂s is the unit vector in the scattering direction, i.e., k̂s ¼~ks=

��~ks��. Either of these yield the scattered intensity to be

Iscat / 1� cos2 f sin2 u (19:3)

Most experiments are confined to the scattering plane; hence, f¼ 908, as drawn in Figure 19.2 Then the polarization projectscompletely onto plane p of Figure 19.1 with no angular dependencies. The angle u is the scattering angle; u¼ 0 is forwardscattering.

We now consider incident light with either vertical, V, or horizontal, H, polarizations. The scattered light can be detectedthrough a polarizer set in either the V or H directions. Thus, four scattering arrangements can be obtained described by thescattered intensities as

IVV, IVH, IHV, and IHH (19:4)

wherethe first subscript describes the incident polarizationthe second the detected polarization of the scattered light

yz

Polarization

ki

ks

x

f

q

P

→

→

FIGURE 19.1 Geometry of scattering for an incident wave from the negative z-axis with propagation direction defined by the incident wavevector~ki and with polarization (pol.) parallel to x. Light is scattered to the detector in the direction of the scattering wave vector~ks. Scatteredpolarization will be in the direction of the projection of the incident polarization onto the plane p, which is perpendicular to the detector direction.


720 Handbook of Surface and Colloid Chemistry


IVV is the most common scattering arrangement described in the rest of this chapter.For particles small compared to the wavelength, these four intensities are dependent on u in simple ways. Larger particles

yield more complex functionalities but bear the imprint of the simple functionalities of their smaller brethren. The polarizationrule above and the concept of projection embodied in Equation 19.3 can be used to infer the four intensities. The most commonscattering arrangement measures IVV, which is independent of u, hence isotropic in the scattering plane. The projection ruleimplies IVH¼ IVH¼ 0, and IHH / cos2 u. Figure 19.3 shows both a polar and Cartesian plot of these functionalities.

Unpolarized light is the incoherent sum of two equal components with polarization that differ by 908. To determine thepolarization state of light scattered from an unpolarized incident beam one need only apply the rules above for polarized light tothe two polarized components in linear combination. Figure 19.4 is an example.

19.2.2 RAYLEIGH DIFFERENTIAL SCATTERING CROSS SECTION

Rayleigh first presented a description of light scattering and adsorption for small particles. The conditions for Rayleighscattering are [1–3]

a � 1 (19:5a)

ma � 1 (19:5b)

Then the differential scattering cross section is

ds

dV¼ k4a6

m2 � 1m2 þ 2

��2

(19:6a)

¼ 16p4a6

l4m2 � 1m2 þ 2

��2

(19:6b)

For some elementary elaboration on the meaning of the differential scattering cross section, the reader is referred to Appendix19.A at the end of this chapter. Simply put, the scattered intensity is proportional to the differential scattering cross section.Thus, by Equation A.19.4

yz

H

k i

V

ks

x

f = 90�

q

→

→

FIGURE 19.2 Typical scattering arrangement in which light is incident from the negative z-axis with either (or both) horizontal or verticalpolarization. Scattered light is detected in the horizontal, y–z, scattering plane at a scattering angle of u.

Polar plotsCartesian plots

IVVIIvv

IVH = IHV = 0

IHH ~ cos2 q

IHH

0 90� 180�

= Independent of q,that is isotropic

FIGURE 19.3 Polar and Cartesian plots of Rayleigh scattering for both IVV and IHH.


Scattering and Absorption of Light by Particles and Aggregates 721


IVV ¼ k4a6

r2m2 � 1m2 þ 2

��2

Io (19:7)

Often, the term involving the refractive index, the Lorentz term, is abbreviated as F(m) ¼ (m2 � 1)=(m2 þ 2)�� 2. This makes

Equation 19.6a simply

ds

dV¼ k4a6F(m) (19:8)

A simple dimensionality argument can be made for the length functionalities in Rayleigh scattering. Since the particle is verysmall compared to the wavelength of light, the phase of the incident light is uniform across the volume of the particle; i.e., allsubvolumes of the particle see the same phase. Furthermore, again since the particle is so small, the light scattered to thedetector from all subvolumes of the particle has the same phase at the detector. Thus, the total scattered field at the detector isdirectly proportional to the particle volume, Vpart. Intensity is field amplitude squared, thus I / V2

part. To determine the crosssection realize that it has units of length squared. The argument above implies the cross section is proportional to V2

part, which islength to the sixth power. Thus, a factor of length to the inverse fourth power is missing. There are only two length scales in theproblem, particle size and optical wavelength, l. Particle size has already been used with V2

part. Thus, to achieve the proper unitsfor cross section, a factor of l�4 must be included to yield a cross section proportional to l�4V2

part a result consistent withEquation 19.8.

There are a number of important features to make note of for Rayleigh scattering:

1. IVV scattering is independent of u; i.e., it is isotropic in the scattering plane. IVH¼ IHV¼ 0 and IHH¼ IVV cos2 u asdrawn in Figure 19.3.

2. l�4 dependence: Blue light scatters more than red. This is often associated with the blue of the sky and the red of thesunset [4], but other factors are involved here including the fact that in perfectly clean air (no particles) molecularscattering occurs with a nonzero sum due to small, thermodynamic fluctuations in the air density. Since thefluctuations are small compared to the wavelength, the Rayleigh l�4 dependence ensues.

3. Strong size dependence of a6, which is proportional to the particle volume squared, V2part.

Feature 3 leads to the Tyndall effect [2], which describes the increased scattering from an aggregating colloid or aerosol ofconstant mass. Consider that the total scattering from a particulate system of Rayleigh scatterers of n particles per unit volumehas the proportionality

Iscat / nV2part (19:9)

If the only growth process in the system is aggregation, the mass is neither created nor destroyed. Thus, the mass density isconstant; hence, nVpart is constant. On the other hand Vpart increases during aggregation. Rewriting Equation 19.9 as

Iscat / nVpart � Vpart (19:10)

shows that the scattered intensity increases proportional to as Vpart the system aggregates. This is the Tyndall effect.

Unpolarizedincident light

Colloid

FIGURE 19.4 Unpolarized incident light scatters from colloidal particles to yield polarized scattered light as indicated.




19.2.3 RAYLEIGH TOTAL SCATTERING CROSS SECTION

Integration of the differential cross section over the complete solid angle of 4p yields the total cross section, Equations A.19.5and A.19.6. We consider the scattering arrangement in Figure 19.1 with incident light polarized in the vertical direction to find

s ¼ðds

dVdV (19:11a)

¼ ds

dV

ð2p0

ð1�1

(1� cos2 w sin2 u)d( cos u)dw (19:11b)

¼ 8p3

ds

dV(19:11c)

We can say that the factor 8p=3 comes from integration of the polarization. Equations 19.8 and 19.11c yield

sscat ¼ 8p3

k4a6F(m) (19:12)

Thus, the scattering efficiency, Q¼s = pa2 (Equation A.19.7), is

Qscat ¼ 83a4F(m) (19:13)

Note that the condition for Rayleigh scattering to hold is a � 1; thus Equation 19.13 implies that Rayleigh scatterers are notvery efficient; i.e., it scatters a lot less than its geometric cross section would imply.

19.2.4 RAYLEIGH ABSORPTION CROSS SECTION

The Rayleigh absorption cross section is

sabs ¼ 8p2a3

lIm

m2 � 1m2 þ 2

� �(19:14)

where Im means imaginary part. We use the notation m¼ nþ ik and Im m2 � 1ð Þ= m2 þ 2ð Þ½ � ¼ E(m). Then the absorptionefficiency is simply

Qabs ¼ 4aE(m) (19:15)

As for scattering, a simple dimensionality argument can be made for the absorption cross section. Since the particle is verysmall, the wave completely penetrates the volume of the particle. Hence, all subvolumes of the particle absorb equally to implysabs / Vpart. To make the units match we divide by the only other length scale of the system, the wavelength, to yieldsabs / l�1Vpart

Features of Rayleigh absorption are as follows:

1. Absorption has different dependencies with l and a than scattering

sscat / l�4a6

sabs / l�1a3

a factor of (a=l)3� �

2. If m is real, there is no absorption

19.2.4.1 Scattering, Absorption, and Extinction

When light passes through a medium containing particles, it is attenuated. This attenuation is called extinction and is describedby an exponential decrease of the intensity as it passes through the medium:

Itrans ¼ Ioe�tx (19:16)




whereItrans is the intensity of the light transmitted after passing a distance x through the mediumt is the turbidity of the medium

The turbidity is related to the number density of particles n and their individual extinction cross section sext by

t ¼ nsext (19:17)

Extinction is due to both scattering, which deviates light from the incident path direction, and absorption, which converts thelight into other forms of energy (e.g., heat),

sext ¼ sabs þ sscat (19:18)

These facts are true for particles of all sizes, not just Rayleigh scatterers. From the discussion above two notable features arise:

1. If m is real, extinction is solely due to scattering

sext ¼ sscat (19:19)

2. If m is complex and if the size parameter is small, a< 1, e.g., Rayleigh scatterers, then Equations 19.13 and 19.15imply sabs � sscat. Hence

sext ’ sabs (19:20)

19.2.5 RAYLEIGH–DEBYE–GANS SCATTERING

The equations for Rayleigh scattering are derived under the assumption that the phase of the incident electromagnetic wavedoes not change across the particle. This is achieved by assuming the size of the particle to be small compared to l=2p, hencethe condition a � 1 for spheres. This condition can be relaxed somewhat if the phase across the particle changes onlynegligibly relative to the phase change in the surrounding medium. Thus, a factor of m� 1 is involved as well as the lengthscale l and the size of the particle.

The conditions for Rayleigh–Debye–Gans (RDG) scattering are as follows:

m� 1j j � 1 (19:21a)

r ¼ 2a m� 1j j � 1 (19:21b)

The parameter r in Equation 19.21b is called the phase-shift parameter and represents the difference in phase between a wavethat travels through a particle directly across its diameter and one that travels the same distance through the medium.

Note that condition 19.21b allows for very large particles, i.e., a> 1, so long as m is close enough to unity to satisfy thecondition.

The Rayleigh–Debye–Gans differential scattering cross section in the scattering plane for vertically polarized light(Figure 19.2) is

ds

dVRDG

¼ ds

dVR

3u3

(sin u� u cos u)

� �2(19:22)

where

u ¼ 2a sin u=2 (19:23a)

or

u ¼ qa (19:23b)

In Equation 19.22 the subscripts RDG and R denote Rayleigh–Debye–Gans and Rayleigh. In Equation 19.23b q is thescattering wave vector to be discussed below.




RDG scattering is the no optics, diffraction limit of scattering. By ‘‘no optics’’ I mean the refractive index of the particle isnot important, Equation 19.21a. The term inside the brackets of Equation 19.22 is the Fourier transform of a sphere, hence itrepresents diffraction from a sphere. Thus RDG is simply the Fourier transform of a sphere, squared, and is the threedimensional analogue of single slit Fraunhofer diffraction. In quantum mechanics the analogue is the first Born approximation.

A plot of RDG scattering is shown in Figure 19.5. Note the constant with q, hence constant with scattering angle, ‘‘forwardscattering lobe’’ for qa< 1. In this region the scattering magnitude is equal to the Rayleigh result, Equation 19.8. For qa> 1 aseries of interference ripples are seen with spacing Du¼p which corresponds to Dq¼p=a and Du¼ l=2a and small u.The envelope of this regime has a slope of �4 in this log–log plot to imply the power law (qa)�4. This is the so-called Porodregime. In fact the �4 is equal to �(dþ 1) where d is the dimensionality of the sphere which is d¼ 3. The single slit diffractionpattern plotted in the same manner would look similar, but the slope would be �2 because for a slit d¼ 1.

Notable features of RDG scattering are as follows:

1. At u¼ 0 the cross section equals the Rayleigh cross section.2. The scattering is larger in the forward direction and this anisotropy increases with increasing size.

The RDG absorption cross section is equal to the Rayleigh absorption cross section.

19.3 SPHERES OF ARBITRARY SIZE: THE MIE THEORY

The Rayleigh and RDG theories of scattering and absorption represent solutions to Maxwell’s equations in which approxima-tions could be made due to small size and small index of refraction. For an arbitrary particle these approximations cannot bemade; hence, Maxwell’s equations must be solved exactly. Various symmetries dependent on the shape of the particle, makethis task more tractable, and the most symmetric, hence simplest, is that of a homogeneous sphere, to which this discussion islimited. Mie first presented these solutions and the term Mie scattering is often applied to the general description of scattering ufrom a homogeneous sphere of arbitrary size [1–3].

Mie’s solutions are complex enough to keep us from quoting them here. Moreover, their complexity does not allow for easyphysical interpretation. However, we have used a graphical analysis to discern patterns in the Mie results that allow for astraightforward description of the angular scattering dependence [5–8].

19.3.1 MIE DIFFERENTIAL SCATTERING CROSS SECTIONS

Figure 19.6a shows an example of Mie scattering, Ivv, for a sphere with an index of refraction m¼ 1.05 and a variety of sizesexpressed as the size parameter a. The normalized intensity I(u)=I(0) versus u is plotted. A series of bumps and wiggles are seenwith some periodicities, but with no particularly coherent pattern. Curves for the larger m¼ 1.50 in Figure 19.7a are evenmore complex. The scattering angle u, although conveniently measured in the laboratory, is not the best parameter for plottingthe data. This complexity can be reduced if plots are made with the scattering wave vector q rather than the scattering angle u.The scattering wave vector, derived in Appendix 19.8, is given by

q ¼ 4pl�1 sin u=2: (19:24)

10–7

1 10

qa

Nor

mal

ized

inte

nsity Slope = –4

(Porod’s law)

102

10–6

10–5

10–4

10–3

10–2

10–1

1

FIGURE 19.5 Rayleigh–Debye–Gans scattered intensity versus the dimensionless u¼ qa.




Its physical significance is that the inverse, q�1, is the length scale of the scattering experiment. This means that the scattering issensitive to structures greater than q�1, but it cannot see structures smaller than q�1. Any universal character in the Mie curvesshould be revealed if the differential scattering cross section is plotted versus the ratio of the two length scales involved: theradius of the particle a and the length scale of the scattering experiment q�1. This ratio is the dimensionless product qa and bothFigures 19.4a and 19.5a are replotted in Figures 19.4b and 19.5b as a function of qa.

At small qa a nearly universal ‘‘forward scattering lobe’’ is seen. Near qa � 1, the falloff is approximately described by theGuinier equation, I(q)=I(0) � 1� q2a2=5 (see below). The enhanced backscattering, the glory visible in plots with m¼ 1.50,shows no particular pattern but is compressed into spikes in the large qa part for each size parameter a. The key features are theenvelopes of these plots. Figures 19.6b and 19.7b include lines that roughly describe these envelopes. For small size parametera the envelope is described by a�4 slope, hence (qa)�4 for larger a two slopes are seen to imply (qa)�2 crossing over to (qa)�4

and for m¼ 1.50 the envelope is dominated by a �2 slope although a �4 slope is visible at large qa.Figure 19.8 shows the Rayleigh-normalized Mie scattering intensity i.e., Ivv divided by the differential Rayleigh cross

section, Equation 19.6 [8]. Many of the features described above are contained in this figure. One sees a pattern of power lawswith slope 0, �2, and �4. Figure 19.8 also illustrates the quasiuniversality of Mie scattering on the phase-shift parameter r,described by Sorensen and Fischbach. For each phase-shift parameter value shown, the values of m and a¼ ka vary widely.Despite this variation, the curves for the same r lie with each other, hence the curves are universal with r. This universality isnot perfect, however, with variations of approximately a factor of three for the same r. Hence we use the term ‘‘quasiuniversal’’to describe the r functionality.

1

0.1

0.01

Inte

nsity

1E–3

1E–4

1E–5

0

(a)

30 60 90

q (�)

120 150 180

3612244896

(b)

0.1 1 10

qa

100

–2

–4

1

0.1

3

6

12

24

48

96

0.01

Inte

nsity

1E–3

1E–4

1E–5

m = 1.05ka =

m = 1.05ka =

FIGURE 19.6 (a) Normalized Mie scattering curves as a function of scattering angle for spheres of refractive index m¼ 1.05 and a varietyof size parameters; and (b) same as (a) but plotted versus qa. Lines with slope�2 and�4 are shown. (From Sorensen, C.M. and Fischbach, D.F., Opt. Commun., 173, 195, 2000. With permission.)

18015012090

q (�)

60300 0.1 1 10

qa

100

1

0.1

0.01

Inte

nsity

1E–3

1E–4

(a) (b)

1

0.1

0.01

Inte

nsity 1E–3

1E–4

1E–5

n = 1.50ka = 3

–2

–4

61224

48

96

m = 1.50ka = 3

612244896

FIGURE 19.7 (a) Normalized Mie scattering curves as a function of scattering angle for spheres of refractive index, m¼ 1.50 and a varietyof size parameters; (b) same as (a) but plotted versus qa. Line with slope �2 is shown. (From Sorensen, C.M. and Fischbach, D.F., Opt.Commun., 173, 195, 2000. With permission.)




Figure 19.9 gives a general description of light scattering from a sphere of arbitrary size and real refractive index; it includesthe Rayleigh, RDG, and Mie limits. We see in Figure 19.8 that the Mie scattering patterns start at r ¼ 0 with the RDG limit. Forqa< 1 the RDG curve is flat, i.e., (qa)0 and equivalent to Rayleigh scattering. For qa> 1 it falls off with a negative four powerlaw, the Porod limit, with magnitude (9=2)(qa)�4 times the Rayleigh scattering cross section. Note that this includes Rayleighscattering in the limit l � a because then qa � 1 and only the flat part of the scattering function is obtained.

When r> 1, scattering in the Rayleigh regime decreases relative to true Rayleigh scattering. The relative decrease(remember, the nonnormalized scattering increases with r) is proportional to r2 as depicted in Figure 19.9. For qa> 1 thescattering falls off as (qa)�2 until this functionality crosses the RDG curve at qa¼ r. For qa> r the scattering is identical toRDG scattering, falling off as (qa)�4 for all a and m as long as qa> r. The (qa)�2 functionality between qa¼ 1 and r for r> 1is exact only at those limits. The average Mie curves dip below the (qa)�2 line. This dip is the first interference minimumpresent in all Mie curves and is the strongest of all the minima. Its position is near qa� 3.5.

101

qa

Phase-shift parameter =10

Slope –4

Slope –2

m = 1.05, ka = 100m = 1.20, ka = 25m = 1.50, ka = 10

Phase-shift parameter =100m = 1.05, ka = 1000m = 1.20, ka = 250m = 1.50, ka = 100

102 10310–110–13

10–11

10–9

10–7

Ray

leig

h-no

rmal

ized

sca

ttere

d in

tens

ity

10–5

10–3

10–1

100

FIGURE 19.8 Shown are two sets of Rayleigh-normalized Mie intensity curves. The three curves in each set have the same value of r butdifferent values of ka and m. The bounding envelopes are shown for both sets of curves and illustrate how the envelopes only depend on r.

(9/2) � (qa)–4

~3r(qa)–2

1

Ray

leig

h-no

rmal

ized

mie

sca

ttere

d in

tens

ity

qa~r

~3r–2

~3r–4

1

r = 0 RDG limit

Mie scattering

FIGURE 19.9 Diagram of the averaged Rayleigh-normalized Mie scattering patterns for nonmagnetic uniform spheres of arbitrarysize and real refractive index parameterized by r¼ 2kajm�1j> 1 (thick solid line). In this example, r� 55. The dashed line is the RDGlimit, r ! 0.




We remark that past evaluation of Mie scattering has emphasized the importance of the size parameter a, we now see thatthe phase-shift parameter is also quite significant. In summary, if we ignore the ripples and the glory, Mie scattering displaysthree power law regimes with approximate boundaries (features) as

I / (qa)0 when qa < 1 (19:24a)

/ (qa)�2 when 1 < qa < r (19:24b)

/ (qa)�4 when qa < r (19:24c)

The behavior of Equations 19.24 is illustrated in Figure 19.9.Recognize that the ripple structure that we have ignored by considering only the envelopes will in an experiment by washed

out by any modest particle size polydispersity of geometric width of 20% or more [8].

19.3.1.1 Forward Scattering

Figures 19.8 and 19.9 show that the intensity in the forward scattering Rayleigh regime, qa< 1, becomes less than the Rayleighvalue when r> 1. To investigate this further we plot in Figure 19.10 the Mie forward scattered intensity in the limit u ! 0normalized to the Rayleigh differential cross section as a function of r for spheres of different m. Figure 19.10 shows that theRayleigh-normalized forward scattering falls off roughly as r�2 for r> 1. We summarize the patterns in Figure 19.10 with thefollowing features:

1. All spheres with r

i

1 scatter in the forward direction with the same intensity as the Rayleigh differential crosssection.

2. For r> 1, the curves for different m begin to separate and oscillate. Near r� 2–3, the forward scattering exceedsRayleigh scattering by nearly a factor of three for large m.

3. For r> 5 the oscillations decay and a cr�2 functionality develops, where c is a constant on the order offfiffiffiffiffi10

p.

19.3.1.2 Total Cross Section

It is known that as ka increases beyond 1, spheres scatter the majority of incident light in the forward direction. If forwardscattering dominates, then we expect a connection between the patterns in forward scattering and the total Mie scattered crosssection sMie. Figure 19.11 shows sMie as a function of ka for spheres with different m. We find that

10–1

10–1

10–2

10–3

10–4

10–5

100

101

100 101

r

Relative refractiveindex m Slope –2

~101/2 r–2

Ray

leig

h-no

rmal

ized

forw

ard

inte

nsity

1.801.501.301.101.05

102 103

FIGURE 19.10 Rayleigh-normalized forward scattering (u¼ 0) intensity versus phase-shift parameter, r.




1. Total cross section sMie for spheres with r< 1 is the same as the total Rayleigh cross section sRay¼ (8p=3)(ds=dVRay). Curves of different m in the ka< 1 range show a clear m dependence which is explained by the mfunctionality in Equation 19.7.

2. As a grows to values for which r> 2, the a6 dependence crosses over, through a ripple structure, to a geometric a2

(i.e., the geometrical cross section of the sphere) dependence at large r. The separation between curves of different min the Rayleigh regime begins to vanish.

3. As a continues to increase, all curves converge to a value of twice the geometrical cross section of the sphere, 2pa2,with no m dependence and the ripple structure decays away.

4. For ka � 1, sMie is approximately equal to sRay

� �= c2r4ð Þ, where c2� 10.

19.3.1.3 Mie Guinier Regime

The region of crossover between the functionalities of Equation 19.24a and b is called the Guinier regime and for r ! 0 isdescribed by

ds

dV¼ ds

dV

� �R

1þ 13q2R2

g

� �(19:25)

In Equation 19.25 Rg is the radius of gyration of the scattering object, which can have any shape. If spherical, Rg ¼ffiffiffiffiffiffiffiffi3=5

pa.

We have found [6] for spheres that Equation 19.25 must be modified when r> 1

ds

dV� 1þ 1

3q2R2

g,G

� �(19:26)

In Equation 19.26 the Rayleigh cross section is lost; i.e., at q¼ 0 (u¼ 0) the sphere no longer has its Rayleigh cross section.Also in Equation 19.26 is Rg,G, which we call the ‘‘Guinier regime determined radius of gyration’’ [6]. Only when r¼ 0 doesRg,G¼Rg. Otherwise, it follows a quasiuniversal behavior with r depicted in Figure 19.12. Note that as r!1, Rg,G¼ 1.12Rg,the Fraunhofer diffraction limit. Importantly, use of a Guinier analysis with Equation 19.26 and Figure 19.12 allows for asimple yet accurate experimental measurement of the particle Rg, hence geometric radius a.

10–1

10–5

10–3

10–1

101

103

Tot

al s

catte

ring

cros

s se

ctio

n (n

m2 ) 105

107

100 101

ka

S-polarization Rayleighcross sectionfor m =1.05

Rayleigh cross section

Relative refractiveindex m

Twice geometric cross section

2πa2

1/(10r4 )

1.801.501.301.101.05

r =3for m =1.8

r =3for m =1.05

102 103

FIGURE 19.11 Total Mie cross section sMie for various m and incident light with V-polarization and l¼ 200p nm. The large r behavior ofsMie� 2pa2 is shown as the dotted curve sRay is shown as the dot dashed line for m¼ 1.05. The vertical arrow shows the factor 1=(c2 r4),where c � ffiffiffiffiffi

10p

, relating sRay to sMie, for m¼ 1.05, in the r> 1 range. For clarity, the legend shows the data sets in the same order as theyappear in the figure.




19.3.1.4 Mie Ripples

The ripple structure that we have ignored by considering only the envelopes displays r-dependent structure as well. We will usethe symbol Du to designate the spacing between consecutive ripples when intensity is plotted versus u¼ qa, and Du whenplotted versus u. We find [7]

Du ¼ p for r

i

5 (19:27a)

which is the same as RDG, and

Du ¼ p cos u for r

u

5 (19:27b)

Equation 19.27b is equivalent to

Du ¼ p=a ¼ l=2a for r

u

5 (19:27c)

Equation 19.27c is identical to the angular fringe spacing for single slit Fraunhofer diffraction for a slit of width 2a. A completephysical explanation for this ripple structure and functionality has not yet been given. Equation 19.27c suggests that for highlymonodisperse systems a measurement of the ripple spacing would yield the particle size [9–11].

In summary, features of Mie scattering may be listed as follows:

1. The functionality of the envelopes as depicted in Figure 19.9 and described by Equation 19.24.2. A modified Guinier regime exists as described by Equation 19.26 and Figure 19.12.3. Ripples visible in systems with very narrow size distributions are described in Equation 19.27 and may afford a size

measurement through Equation 19.27c.

19.4 MULTIPLE SCATTERING

The light scattered from an aerosol or colloid will in all practical situations involve scattering from numerous particles. If thelight scatters from any of the particles only once, then the total scattering from the ensemble is a simple sum of all scattering

Change r by a

Rg,

G/R

g

r

m = 1.5

m = 1.33

m = 1.2

1.5

1.0

1.5

1.0

1.5

1.0

0 10 20 30 40 50

FIGURE 19.12 Ratio of the Guinier inferred to real radius of gyration, Rg,G=Rg, versus phase-shift parameter r for spheres with threedifferent refractive indices. The size parameter a¼ kR was varied to vary r. (From Sorensen, C.M. and Shi, D., Opt. Commun., 178, 31, 2000.With permission.)




events. Then the interpretation of the scattering involves direct application of the results described above for scattering fromsingle objects. This single scattering is the ideal situation. However, there is always a finite chance that the scattered lightreaching the detector has scattered sequentially from more than one particle. This is multiple scattering, and its interpretationdoes not involve a straightforward application of the optical scattering cross sections for a single scatterer. Thus it is a situationto be avoided, and to avoid it we must be able to detect or predict it. This I describe below [11].

Consider a system of scatterers with scattering cross section s randomly placed in a right cylindrical scattering volume oflength x with number density n. Imagine light incident upon this volume in a direction parallel to the length x. Conceive of thelight as an ensemble of photons. The rate of photons incident upon the scattering volume is proportional to the incidentintensity I(0), and the rate of photons that pass through the volume the entire distance x is similarly proportional to I(x). What isthe rate I(x)?

We will make the assumption that the photons act like classical particles. We will also assume that the encounter of thephotons with particles in the volume is a Gaussian random process and scattering is not in the exact forward direction. For sucha process we can envision an ensemble of photons encountering a scattering particle but then continuing on along the same pathto possible encounters with other particles. Then the probability that a given photon has s encounters with extincting particlesduring its passage along the entire length x is given by the Poisson distribution:

P(s) ¼ exp �hsið Þ hsi�s

s!(19:28)

In Equation 19.28 hsi is the average number of photon–particle encounters for an ensemble of photons. The average distancetraveled between photon–particle encounters, i.e., the photon mean free path, is

l ¼ x=hsi (19:29)

In the real turbid medium situation, only photons that have no encounters, i.e., s¼ 0, pass out of the far end of the volume. Thusby Equation 19.28

P(0) ¼ exp �hsið Þ (19:30)

Then the light intensity passing through the volume, I(x), is equal to this probability times the incident intensity, I(0), to yield

I(x) ¼ I(0) exp �hsið Þ (19:31)

This is the Lambert–Beer law. Comparison connects the scattering variables to the statistical variables as

hsi ¼ nsx, (19:32)

l ¼ (ns)�1 (19:33)

This theory suggests that the fundamental parameter to describe the extent of multiple scattering is the average number ofscattering events hsi¼ x=l. If hsi is small, multiple scattering is unlikely, if hsi is big, most photons encounter many scatteringevents.

Fortunately the value of hsi is simply obtained either by calculation, with Equation 19.32, or measurement. Measurement isparticularly simple because by Equation 19.31 one needs to measure only the relative amount of incident and transmittedlight, a ratio often called the transmittivity, to obtain hsi¼ ln [I(0)=I(s)]. The veracity of this theory is demonstrated inFigure 19.13 where the light-scattering data with similar hsi but different volume fractions and optical path lengths are plottedtogether (�0.7 0.06, 2.8 0.4, and 12.4 0.3, respectively). The graphs are normalized to 1 on the intensity scale atq¼ 2.1� 103 cm�1. We see that the graphs with similar hsi overlap and show the same behavior. This supports our contentionthat the average number of scattering events is the universal parameter to describe the extent of multiple scattering.

19.5 FRACTAL AGGREGATES

The general problem of scattering and absorption by an aggregate of particles can be very complex. Much of the complexityis due to the difficulty in formulating the electromagnetic wave solution for a system that lacks symmetry and contains many




scattering centers. An additional problem lies in that aggregates can potentially represent an infinite number of differentarrangements of constituent particles. Because we have little use for an infinite number of electromagnetic solutions, weneed the ability to describe classes of aggregates and quantify their features in a manner related to the scattering andabsorptive behavior. The past few decades have seen the development of the fractal concept for quantitative description ofmany aggregates that form in nature [12–16]. The scattering and absorption by fractal aggregates is now fairly well known,has an extensive literature [17–24] and is describable in terms of the quantified fractal parameters. It is the optical propertiesof these fractal aggregates that is the subject of the rest of this chapter. Figure 19.19 shows a picture of a soot fractalaggregate.

A fractal is an object that displays scale invariant symmetry; that is, it looks the same when viewed at different scales. Anyreal fractal object will have this scale invariance over only a finite range of scales. One important consequence of this symmetryis that the density autocorrelation function will have a power law dependence, which can be written as

C(r) ¼ r ~roð Þr ~ro þ~rð Þh i (19:34)

¼ ArD�dh r=jð Þ; r > a (19:35)

Equation 19.34 defines the density correlation function C(r), where r(~r) is the density of material at position~r, and the bracketsrepresent an ensemble average. In Equation 19.35, A is a normalization constant, D is the fractal dimension of the object, andd is the spatial dimension. Also in Equation 19.35 are the limits of scale invariance, a at the smaller scale defined by the primaryor monomeric particle size, and at the larger end of the scale h(r=j) is the cutoff function that governs how the densityautocorrelation function (not the density itself) is terminated at the perimeter of the aggregate near the length scale j. As thestructure factor of scattered radiation is the Fourier transform of the density autocorrelation function, Equation 19.35 isimportant in the development below.

0.1

0.01

1E–3

1

(c)

fv = 8�10–3, x = 4.8 mm, <s> = 12.2fv = 4�10–3, x = 10 mm, <s> = 12.7

(b)

0.1

0.01

I (q

) (a

rbitr

ary

units

)

I (q

) (a

rbitr

ary

units

)

I (q

) (a

rbitr

ary

units

)

1

fv = 4�10–3, x = 2 mm, <s> = 2.5

fv = 2�10–3, x = 4.8 mm, <s> = 3.0

fv = 10–3, x = 10 mm, <s> = 3.2

(a)

0.1

0.01

1E–3 1E–310,000

fv = 10–3, x = 2 mm, <s> = 0.63fv = 5�10–4, x = 4.8 mm, <s> = 0.76

q (cm–1)

1

FIGURE 19.13 Normalized light scattering intensity (arbitrary units) plotted versus q for different average number of scattering events hsiof the photons with 9.6 mm polystyrene particles in a colloidal suspension. The particle volume fraction for fv, x is the optical path length ofthe cells. hsi: (a) 0.7 0.06, (b) 2.8 0.4, (c) 12.4 0.3.




Another important property of fractal aggregates is that their mass scales with their size raised to the fractal dimensionpower. For a naturally occurring aggregate, apparently random to the eye, size is difficult to define in terms of the typicallyragged border and anisotropic shape. Thus, a convenient measure of size that can be precisely quantified is the radius ofgyration, Rg, defined as

R2g ¼

ðr ~rð Þ ~r �~rcmð Þ2d~r

ðr ~rð Þd~r (19:36)

In terms of the correlation function we may show that Rg is given by

R2g ¼

12

ðr2C(r)d~r (19:37)

In Equation 19.36,~rcm is the position of the aggregate center of mass. The radius of gyration is a root-mean-square radius,which is often a useful point of view. Given Rg, a fractal of N monomers or primary particles obeys

N ¼ ko Rg=a� �D

(19:38)

Perhaps the most common aggregation process is diffusion-limited cluster aggregation (DLCA) for which D¼ 1.75 to 1.80.With experimentation on soot aggregates we have found ko¼ 1.23 0.07 [25] and 1.66 0.4 [26]. With simulations, we havefound ko¼ 1.19 0.1 [27] and 1.30 0.07 [28]. Figure 19.15 shows a cartoon of a fractal aggregate.

1 µm

FIGURE 19.14 TEM micrograph of soot collected from an C2H2=air diffusion flame.

Monomerradius = a

N monomersi.e., primary

particles

Rg

FIGURE 19.15 Cartoon of a fractal aggregate.




19.5.1 SCATTERING AND ABSORPTION BY FRACTAL AGGREGATES

To lowest order the scattering and absorption cross sections for a fractal aggregate of N monomers with radius a are simplyrelated to the monomer cross sections as follows [24]

scabs ¼ Nsm

abs (19:39)

dsc

dV¼ N2 ds

m

dVS(q) (19:40)

The superscripts c and m denote cluster and monomer, respectively. S(q) is the static structure factor of the cluster which is theFourier transform of the cluster density autocorrelation function, and hence it contains information regarding the clustermorphology. The structure factor has the asymptotic forms S(0)¼ 1 and S(q) � q�D for q � R�1

g .The simple forms of Equations 19.39 and 19.40 have physical interpretation. Equation 19.39 implies that the absorption is

independent of the state of aggregation; the monomers absorb independently. Equation 19.40 implies that the scattering atsmall q from N monomers is also independent of the state of aggregation; the N scattered fields add constructively to yield theN2 dependence of the intensity. Equations 19.39 and 19.40 are under the assumption that the effects of intracluster multiplescattering within a cluster can be neglected. This assumption is particularly liable for D> 2 because such clusters are notgeometrically transparent; that is, their projection onto a two-dimensional plane would fill the plane. Other factors that can leadto multiple scattering effects are large a (or, better, its size parameter a), N, and m [22]. Current knowledge (see below)indicates that Equations 19.39 and 19.40 are quite good for monomer a

i

0.3, D< 2, and most values of m.Recently, based on an analysis by Farias et al. [29] and our own experimental test of Equation 19.40 [30], we have

concluded that a phase-shift parameter for the cluster aggregate can be defined as

rc ¼ 2kRgjm� 1j (19:41)

When rc

i

3, Equation 19.40 should hold quite well (�10%).The general behavior of S(q) is shown in Figure 19.16. Important features include the following:

1. A scattering angle-independent Rayleigh regime where the cross section is N2 times the monomer Rayleigh crosssection.

2. A Guinier regime near q � R�1g , which is expressed as

dsc

dV¼ N2 ds

m

dV1� 1

3q2R2

g

� �(19:42)

This dependency on q2R2g holds for any morphology, and Equation 19.42 can be used to measure the cluster Rg.

Rayleighregime

I(q)

Guinierregime

Monomerregime

Increasingpolydispersity

~nN2 = NmN

~nN = Nm

Power lawregime

Slope = –D

Slope = –4

a–1Rg–1 q

FIGURE 19.16 Schematic of the general behavior of the structure factor for a fractal aggregate with fractal dimension D, radius of gyrationRg, and monomer radius a. (From Sorensen, C.M., Aerosol Sci. Tech., 35, 241, 2001. With permission.)




3. A power law regime at q > R�1g , where S(q) � q�D. This can be used to measure D.

4. The regime for which q> a�1, where the length scale of the scattering experiment can resolve below the fractal scalingregime and see the individual monomers. In this regime S(q)� q�4, which is known as Porod’s law. This feature is notincluded in Equation 19.40 but may be included by multiplying Equation 19.40 by the form factor, i.e., the normalizeddifferential cross section, for the monomer.

It is worthwhile to compare these features with those listed above for scattering from a homogeneous sphere of arbitrarysize, i.e., Mie scattering. Both fractal aggregates and Mie scatterers display a Rayleigh regime in which no angle dependenceoccurs. In each there follows, with increasing angle, a Guinier regime, which can yield an experimental measurement of Rg.After that follows a power law regime or regimes for which considerable differences appear (e.g., Mie ripples) but which aresimilar because of the decreasing power law with q. Finally, the possibility of enhanced backscattering for fractal aggregates, inanalogy to that seen for spheres, has not been explored to my knowledge.

19.5.2 STRUCTURE FACTOR

The structure factor and the density autocorrelation function are Fourier transform pairs; thus,

S(q) /ðC(r)ei~q�~rd3r (19:43)

Knowledge of one implies knowledge of the other. As sure as Bragg scattering measures the structure of a crystal in q space, sodoes the structure factor of a cluster measured optically represents the structure of the cluster in q space.

Some useful general notions of how the structure factor depends on the structure of the object from which it scatters can beobtained by considering the density profile, not the correlation function, of the object as a function of radial distance r. This isdepicted in Figure 19.17. A particle or cluster with uniform density and a sharp boundary (Figure 19.17a) is a homogeneoussphere and hence its scattering profile is its structure factor when r< 1. It is the sharp boundary that causes the ripples and theuniform density the causes the q�4 envelope. If the boundary is smoothed to a finite width, the ripples become less severe andwith increasing boundary width eventually disappear. If the density is not uniform but instead decreases with increasing r(Figure 19.17b), the power law describing the qRg> 1 q-dependence will decrease from q�4 to, q�D, D< 3, if fractal.

To gain more than a qualitative notion on how the structure factor is related to the structure of the object, an exact form forthe correlation function must be given. A fractal object (cluster) will have C (r) � r D�d as in Equation 19.35. Beyond that, it iscrucial to know the form of the correlation function cutoff function h(r=j) [31–33]. This function must decrease faster than anypower law to effectively cut off the power law of C(r)

The early work [17–19] in this area assumed that h(r=j) was an exponential:

h(r=j) ¼ Ae�(r=j) (19:44)

Smooth edge eliminates ripples

Sharp edge causes Mie ripples

Power law density yields S ~ q–D.

Uniform density causes q –4

(b)

(a)

S (q)

aq

q

r

r

S (q)

r

r

FIGURE 19.17 Examples of qualitative relations between particle or aggregate morphology, as described by the radial density profile,r versus r, and the structure factor.




The advantages of this function are (1) its simplicity, (2) an analytical form for S(q) can be calculated, and it (3) is consistentwith scattering from a fluid near the critical point. Given Equation 19.43 a relation between Rg and j can be determined and thestructure factor can be calculated.

An important quality of the structure factor for the exponential cutoff is that for D¼ 2 it reduces to the so-called Fisher–Burford formula for light scattering from a critical fluid [34]:

S(q) ¼ 1þ 13q2R2

g

� ��1

(19:45)

which is often generalized for D near 2 to

S(q) ¼ 1þ 23D

q2R2g

� ��D=2

(19:46)

The beauty of these equations is their simplicity, but care must be taken because there is evidence [25,31] that the exponentialcutoff and these equations are incorrect for DLCA fractal aggregates.

Another reasonable form for the cutoff is the Gaussian

h(r=j) ¼ Ae�(r=j)2 (19:47)

A cutoff based on physical reasoning is the overlapping spheres cutoff, so named because calculation of C(r) depends on howthe spherical densities, r(r), overlap [35]. We remark that Equations 19.44 and 19.47 can be generalized to

h(r=j) ¼ Ae(�r=j)b (19:48)

with b having any positive, nonzero value. Mountain and Mulholland [21] found b¼ 2.5 0.5 consistent with their computer-generated clusters with D¼ 1.8.

In our laboratory we have attempted to discern which structure factor, hence which cutoff function, describes reality thebest. In one study [31] we fit structure factor measurements from soot aggregates in a premixed CH4=O2 flame to theexponential, Gaussian, Mountain, and Mulholland forms, i.e., Equation 19.48 with b¼ 1, 2, and 2.5. If no polydispersitywas included in the fit, b¼ 1 worked the best. However, with any reasonable polydispersity, b¼ 1, the exponential, failedcompletely. Both b¼ 2 and 2.5 worked well, with b¼ 2, the Gaussian, yielding the best result.

In another study, soot from the same flame was thermophoretically captured and examined via transmission electronmicroscopy (TEM) [25]. The correlation function of soot clusters was determined and the cutoff function extracted from that.Once again, the b¼ 1 exponential failed completely. The best cutoff was the overlapping spheres, which is edifying given itsphysical origin. However, the b¼ 2 Gaussian worked very well failing only when h(r=j) had decreased by a factor of 100.

Thus, a sharp cutoff is highly recommended over the exponential cutoff. Recently, we have reconsidered these issues withregard to the single cluster structure factor and concluded that it is best described by [24,36]

S(q) ¼ 1� 13q2R2

g, qRg

i

1 (19:49a)

¼ C qRg

� ��D, qRg > 1 (19:49b)

In Ref. [36] we have found the sharp cutoffs all imply C¼ 1.0 0.1. Equation 19.49 is consistent with the form first proposedby Dobbins and Megaridis [37]. However, Equation 19.49 is not to be used to fit real data because they neglect the effects ofpolydispersity, which we consider below.

19.5.3 STRUCTURE FACTOR OF AN ENSEMBLE OF POLYDISPERSE AGGREGATES

Any real experiment detecting scattered radiation from an ensemble of fractal aggregates will involve a polydisperse (in clustersize) ensemble. Aggregates are a result of aggregation, which always gives a finite width to the cluster size distribution. Thispolydispersity causes the shape of the observed structure factor to be different from that of the structure factor of any single




cluster in the distribution [36–38]. The single cluster structure factor, dependent on the cutoff and D, was described above. Nowwe follow Ref. [36] to consider how the shape is modified by a distribution in cluster sizes. The modifications will occur in boththe Guinier and power law regimes.

In general, the effective structure factor for an ensemble of aggregates can be written as

Seff (q) ¼ðN2n(N)S qRg(N)

�dN

ðN2n(N)dN (19:50)

where n(N) is the size distribution, i.e., the number of clusters per unit volume with N monomers per cluster. The numberof monomers per cluster and the cluster radius of gyration are related by Equation 19.38.

To compute the results of Equation 19.50 applied to Equation 19.49, we define the ith moment of the size distribution

Mi ¼ðNin(N)dN (19:51)

Note that the moment is an average and could also be written as Ni or hNii. Equation 19.50 yields

Seff(q) ¼ 1� 13q2R2

g,z, qRg,z

i

1 (19:52a)

¼ CCp(qRg,z)�D, qRg,z

u

1 (19:52b)

In Equations 19.52 we have the z-averaged radius of gyration:

R2g,z ¼ a2k�2=D

o

M2þ2=D

M2(19:53)

Obviously, Rg,z can be determined from the slope of S(q) or S(q)�1 versus q2 graph of the data, as we illustrate below.In Equation 19.52b we have what we have called the polydispersity factor Cp,

Cp ¼ M1

M2

M2þ2=D

M2

� �D=2

(19:54)

To calculate Cp we need the analytical form of the size distribution.It is well-established that an aggregating system develops a self-preserving, scaling distribution given by [39,40]

n(N) ¼ Ax�teax (19:55)

where x is the relative size

x ¼ N=s (19:56)

wheres is the mean sizeA and a are the constants

Note that size in this context is the aggregation number, N. The exponent t is a measure of the width of the distribution withlarge t implying a broad distribution. This scaling form is valid when x> 1, with the small x form different. As scatteringstrongly weights the large end, i.e., x> 1, of the distribution, the small x has little effect on the properties of scattering from anensemble of aggregates and hence can be ignored.

Other forms for the size distribution of aggregates exist, but caution must be exercised in their use. For example, theintuitive lognormal distributions are frequently used in the literature. However, we have shown [41] that these distributionsyield erroneous values for distribution moments higher than the second when compared with the exact scaling distribution.For the secondmoment and lower, the distributions agree well. Because scattering involves higher moments, such asM2þ 2=D’M3

for D ’ 2, it is erroneous to use the lognormal distribution for light scattering analysis.




We find the polydispersity factor to be

Cp ¼ 12� t

G(3� t þ 2=D)G(3� t)

� �D=2(19:57)

where G(x) is the gamma function. In Figure 19.18 we graph Cp as a function of the width parameter t for a variety of fractaldimensions D. We find Cp in general to be significantly greater than unity.

In summary, return to Figure 19.16, which schematically illustrates all the quantitative aspects of the structure factor. Theseresults can be expressed as

I(0) ¼ nN2; qRg;z � 1 (19:58)

I(0)=I(q) ¼ 1þ 13q2R2

g,z

� �; qRg;z

i

1 (19:59)

I(q)=I(0) ¼ CCpR�Dg,z q

�D; qRg;z > 1 (19:60)

In fitting data from a real system it is best to analyze the Guinier regime first. Equation 19.58 implies that the inverse normalized(to q¼ 0) scattered intensity is a linear function of q2. Such a plot has a slope of R2

g,z=3. With Rg,z determined by this Guinieranalysis. At large q, ideally qRg,z

u

5, the normalized scattered intensity will obey Equation 19.60.Thus, a log–log plot will have a slope of �D and an intercept related to CCpR�D

g,z , or, as shown below, the product I(q)=I(0)(qRg,z)

D will be constant at large qRg,z and equal to CCp ’ Cp (note, S(q)¼ I(q)=I(0)). Using C¼ 1.0 0.1 and the measured Rg,

z and D, Cp can be extracted to yield, through Figure 19.18 or Equation 19.57, a measure of the cluster size polydispersity.One question that might arise is how the q�D behavior for an aggregate transforms to q�4 as D! 3 from below. Jullien [42]

has studied this question for both the exponential and Gaussian cutoffs. For the exponential cutoff there is an inflection in S(q)after the Guinier regime. For q less than this inflection the negative slope on a log S(q) versus log q plot is greater than D andapproaches 4 as D ! 3. For q greater than the inflection q the negative slope is equal to D. As D ! 3 the inflection q goes toinfinity. For the Gaussian cutoff a noticeable hump appears in log S versus q as D! 3. This hump has an exponential form: exp�q2R2

g=3h i

. At larger q, the curve inflects and begins q�D behavior; i.e., the slope is �D. As D! 3 the inflection point goes toinfinity leaving the exponential.

19.5.4 AGGREGATING ENSEMBLES OF FRACTAL AGGREGATES

Equation 19.58 gives I(0)¼ nN2 where n is the cluster number density and N is the number of monomers per cluster. This canbe rewritten as I(0)¼ nN � N, the first term of which equals the total number of monomers in the entire system. If we assume thisis conserved, then I(0)�N. Thus I(0) increases as the aggregates grow; a Tyndall effect for qRg< 1. Note however byEquations 19.58 and 19.60, I(q) � nN2q�DR�D

g for qRg> 1. Since N � RDg , I(q) � nNq�D which is independent of the state of

aggregation. These results are depicted in Figure 19.19. There we see that an aggregating system of fractal aggregates willincrease in forward scattering intensity, where forward means qRg< 1, but not in backward scattering intensity. This is theTyndall effect for fractal aggregates.

1.51.00.5

D = 1.52.03.0

t0.0−0.5

1

2

3Cp

4

5

FIGURE 19.18 Polydispersity factor Cp, for the large qRg, power law regime of the structure factor S(q) ¼ cCp qRg,z� ��D

for an ensemble ofclusters as a function of the width parameter t of the scaling size distribution for three values of the fractal dimensionD). (From Sorensen, C.M.and Wang, G.M., Phys. Rev. E, 60, 7193, 1999. With permission.)




19.5.5 EXAMPLES

Examples of optical structure factor measurements from aerosol soot fractal aggregates are shown in Figures 19.20 and 19.21.Figure 19.20 is for scattering from soot in a uniform, premixed flame of methane and oxygen [43]. As the height aboveburner increases from 8 to 20 mm, the scattering goes from isotropic to angle dependent. Figure 19.22 shows the inversescattered intensity, normalized to the intensity scattered as zero angle, versus wave vector squared for the data inFigure 19.20. The linearity is in accord with the Guinier formula (Equation 19.59), and the slope is R2

g=3. This is a veryconvenient way to determine the average Rg for any system of scatterers and is the essence of the Zimin plot of biophysics[44]. We have also shown that the curvature of these plots in the regime of qRg

u

1 can be used to measure size distributionwidth [45].

Figure 19.23 shows experimental data [36] for an aerosol with D¼ 1.7 (DLCA) and a colloid with D¼ 2.15, where we cansee different levels at large qRg,z hence different Cp values, hence different degrees of polydispersity as parameterized by t.

Figure 19.24 shows scattering from an aggregating colloid. One sees the bend in the scattering curve progressing to smallerq, hence larger aggregate size as time progresses. Figure 19.24 also displays a good example of the Tyndall effect for fractalaggregation, (cf. to Figure 19.19).

Figure 19.25 shows scattering from an acetylene diffusion flame which is very heavily sooting [46]. The apparatus usedincluded a small angle set-up that allowed scattering from approximately 0.18 to 158. This gives small q, hence large sizes can

Increasing time

log qlo

g l

FIGURE 19.19 Evolution of the scattered intensity for a system of fractal aggregates as the system aggregates with total mass conserved.

30

101

102

103C/O = 0.69h (mm) =

20

17

15

1 10

I (q

) (a

u)

10

8

q (µm–1)

FIGURE 19.20 Optical structure factors for soot aggregates in a premixed CH4=O2 flame at various heights above the burner.




10010

Slope of 1.8

8I (h =18.5)

4I (h =14)

2I (h = 8.5)

I (h = 3)

q (µm–1)10.1

0.1

I (q

)

1

10

0.01

FIGURE 19.21 Optical structure factors for a C2H2=air diffusion flame at various heights above the burner.

300200

q2 (µm–2)

1000

1.0

1.5

2.0

I (0)

/I (q

)

2.5 h = 20 mmC/0 = 0.69

17

Rg =117 nm

97

82

3815

15108

FIGURE 19.22 Guinier analysis of structure factor data in Figure 19.20. By Equation 19.59, I�I versus q2 yields straight lines with slopeR2g=3.

1816141210qRgz

86420

0.0

0.5

1.0

1.5

S(q

)(qR

gz)D

2.0

2.5

3.0

FIGURE 19.23 (qRg,z)DS(qRg,z) plotted versus qRg,z for an aerosol (squares) and a colloid (circles). The constant level at large qRg,z is

(Equation 19.60) equal to Cp. (From Sorensen, C.M. and Wang, G.M., Phys. Rev. E, 60, 7193, 1999. With permission.)




be measured. Figure 19.25 shows at large heights above the burner, e.g., 15 cm, two morphologies for the soot: a fractaldimension of 2.6 for the size range of ca. 1 to 10 m, and a fractal dimension of 1.8 for the size range 0.3 m.

APPENDIX 19A: CROSS SECTIONS

19.A.1 DIFFERENTIAL CROSS SECTIONS

The differential scattering cross section, ds=dV describes the power scattered, Pscat, per unit solid angle, V, (watts=steradian)for an incident intensity (watts=square meter) Io:

Pscat

V¼ ds

dVIo (A:19:1)

Thus, the units of ds=dV are square meter per steradian.

10,000

1.80 ± 0.04

1000

After mixing1 min1.5 min2 min3.5 min5.5 min7.5 min

13.5 min

0.01

0.1

Inte

nsity

(au

)

1

1E–3

23.5 min29 min39 min59 min69 min

q (cm–1)

FIGURE 19.24 Scattered light structure factor for an aggregating colloid at various times after destabilization.

105

–1.8

–2.6

h =15 cm1210754321

104

q (cm–1)

103102

10–3

10–2

10–1

S (q

)

100

101

C2H2

FIGURE 19.25 Scattered light structure factor for soot in an acetylene=air diffusion flame for various heights above the burner, h.




The scattered intensity is the scattered power per unit area of detection:

Iscat ¼ Pscat=A (A:19:2)

The solid angle subtended by the detector at a distance r from the scatterer is

V ¼ A=r2 (A:19:3)

Thus, from Equations A.19.1 through A.19.3, we obtain

Iscat ¼ Iods

dV

1r2

(A:19:4)

We obtain the well known 1=r2 dependence due to the geometry of space as described by Equation A.19.3.

19.A.2 TOTAL CROSS SECTION

The total scattering cross section is found by integration of the differential cross section over the complete solid angle:

s ¼ð4p

ds

dVdV (A:19:5)

This integral must include polarization effects (see main text). The differential element dV in three dimensional Euclideanspace is

dV ¼ dud(cosw) (A:19:6)

(see Figure 19.1 for the polar coordinates u and f).

19.A.3 EFFICIENCIES

The scattering or absorption efficiency is a dimensionless ratio of the either total cross to the projected, onto a plane, area of thescatterer:

Qscat or abs ¼ sscat or abs

Aproj

(A:19:7)

For a sphere of radius a

Aproj ¼ pa2

These quantifies are physically intuitive because they compare the optical cross section to the geometric cross section. If lightwere not a wave, e.g., solely a particle, the sum of the scattering and absorption efficiencies would be unity.

APPENDIX 19B: SCATTERING WAVE VECTOR

Consider a scalar electromagnetic field with incident wave vector~ki incident upon a scattering element at~r as in Figure 19.18.The incident field at~r is

Ei / ei~ki�~r (B:19:1)

where we keep track of phase information only. The field scatters toward the detector in the direction~ks where~ks is the scatteredwave vector. We assume elastic scattering, i.e.,

~ki�� ¼ ~ks

�� ¼ 2pl

(B:19:2)

the field at the detector, which is at ~R is

E(~R) / E(~r)ei~ks�(~R�~r) (B:19:3)




R to detectorkski

rq

→ → →

FIGURE 19.26 Light with incident wave vector~ki scatters from an element at~r into a direction~ks directed toward the detector at scatteringangle u and great relative distance.

Substitution of Equation B.19.1 into B.19.3 yields

E(~R) / eiks�Rei(~ki�~ks)�~r (B:19:4)

The second term of Equation B.19.4 shows that the phase at the detector is a function of the position of the scattering elementand the vector

~q ¼~ki �~ks (B:19:5)

This vector~q is called the scattering wave vector. Its direction is in the scattering plane from~ks to~ki as shown in Figure 19.27.From Figure 19.27 and the elasticity condition, Equation B.19.2, the magnitude of~q is

q ¼ 2ki sin u=2 (B:19:6a)

¼ 4pl�1 sin u=2 (B:19:6b)

The importance of q is that its inverse, q�1 represents the length scale of the scattering experiment. This follows from thesecond term in Equation B.19.4, which cannot be written as

ei~q�~r (B:19:7)

As we have seen, Expression B.19.7 gives the phase at the detector due to a scattering element at~r. The scattering due to anobject will be the sum of such terms, weighted by the scattering density, over the extent of the object. The key point is that inthis sum, the phase expression Equation B.19.7 will not vary significantly if the range of~r is small compared to q�1. Thus, thescattered phase is not sensitive to the overall extent of the object. We might say that the scattering cannot see the object.

Application of this concept to scattering for a particle of extent a implies that if a� q�1, i.e., qa� 1, the scattering cannotsee the morphology of the particle. Furthermore, if q varies but qa remains less than unity there will be no dependence on q, i.e.,no scattering angle dependence. This is the Rayleigh regime for any size particle.

Another consequence of the phase term (Equation B.19.7) is that if qa � 1, the fields from the scattering elements addcoherently. For an object in three dimensions this coherent addition occurs for all three dimensions, and the three dimensionsadd coherently. Thus, the scattered field will be proportional to the amount of matter in the object, which is proportional to thevolume V of a dense particle or the number of monomers per cluster, N, of a ramified object. Since intensity is the field squared,we obtain V2 or N2 dependencies.

ks

qq

→

kI

→

→

FIGURE 19.27 Relationship between the incident~ki and scattered~ks wave vectors, the scattering angle u and the scattering wave vector~q where u is the scattering angle.




In summary, q�1 is the length scale or resolution limit of a scattering experiment. Therefore, the dimensionless product qa,where a is a measure of the extent of an object, determines the boundary of the Rayleigh regime:

qa � 1, I � V2 or N2 not a function of u, the Rayleigh regime

qa > 1, I ¼ I(u)

ACKNOWLEDGMENTS

This work has been supported by grants from the National Science Foundation, the National Institute of Standards andTechnology, and NASA.

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Scattering and Absorption of Light by Paticles and Aggregates

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