Light Scattering and Absorption by Particles C.M. Sorensen Department of Physics, Kansas State University Manhattan, KS

Transcription

1 Light Scattering and Absorption by Particles C.M. Sorensen Department of Physics, Kansas State University Manhattan, KS INTRODUCTION We see the physical universe around us via light scattering. This is true except for the few luminous objects in our world, such as the sun or various light sources, which provide their own light. Everything else, the blue of the sky, the white of clouds, the faces of our friends and this page of this book, scatters light to our detecting eyes. The scientist takes advantage of light scattering by systematic and quantifiable observation to use light scattering as a probe of material systems. Our concern here is largely aerosol systems and one quickly appreciates the importance and potential usefulness of light scattering when the word aerosol engenders in the mind s eye a cloud of particles. Light scattering in aerosol science provides a way to unobtrusively probe such systems to determine size and morphology of the constituent particles and the dynamics of their motion, aggregation or dissipation. Moreover, particles in our environment scatter and absorb light and thereby affect visibility and the Earth s global environment. In this chapter I will describe light scattering and absorption due to particles relevant to aerosol science. For more extensive treatments of light scattering the reader can look to some classic books on the subject (van de Hulst, 1954; Kerker, 1969, Bohren and Huffman, 1983; Mishchencko, Travis and Lacis, 00). PRELIMINARIES The Differential Cross Section. The differential scattering cross section, dcsca / dω,describes the power scattered, P scat, per unit solid angle, Ω, (watts/steradian) for an incident intensity (watts/meter ) I o P Ω dc dω sca = sca I o (1) Thus, the units of dcsca / dω are meter /steradian. The scattered intensity is the scattered power per unit area of detection I sca = P sca / A () The solid angle subtended by the detector a distance r from the scatterer is Ω= A/ r (3) Thus, from Eqs. and 3, we obtain I dc dω 1 sca = I sca o r (4) This equation contains the well-known 1/r dependence due to the 3d geometry of space. Note that the scattered intensity is directly proportional to the differential cross section. Because of this I will often refer to them in a synonymous manner.

2 The Total Scattering Cross Section. The total scattering cross section is found by integration of the differential cross section over the complete solid angle dc C sca sca = dω (5) 4π dω This integral must include polarization effects (see below). The differential element dω in three dimensional Euclidean space is dω= dθd cosϕ (6) ( ) (see Fig. 1, below, for the polar coordinates θ and φ ). Efficiencies. The scattering or absorption efficiency Q is a dimensionless ratio of either total cross to the projected, along the incident direction onto a plane, area of the scatterer: C Q sca or abs sca or abs = (7) A proj For a sphere of radius a Aproj = π a (8) Efficiencies are physically intuitive because they compare the optical cross section to the geometric cross section. If light were not a wave, i.e., solely a particle, the sum of the scattering and absorption efficiencies would be unity independent of size. Extinction and the Albedo. Attenuation of light when it passes through a system of particles is called extinction. It is described by an exponential decrease of the intensity as it passes through the medium known as the Lambert-Beer Law I trans x = I e τ. (9) Here τ is the turbidity of the medium. The turbidity is related to the number density of particles n and their individual extinction cross section C by ext o τ = nc ext (10) Extinction is due to both scattering, which deviates light from the incident path direction, and absorption, which converts the light into other forms of energy (e.g. heat), Cext = Cabs + Csca (11) The albedo, ω, is the ratio of scattering to extinction ω = C sca /C ext. (1) A related parameter is the Rayleigh Ratio which for a particulate system is equal to the scattering cross section, either differential or total, times the number density. Hence Rayleigh Ratios have units of either (meters.steradian) -1 or meters -1. Rayleigh Ratios are usually used in the

3 context of describing light scattering and attenuation in gases and liquids (Kerker, 1969). Polarization The polarization of light is the direction of its electric field vector. This is always perpendicular to the direction of propagation and hence there are two independent polarizations. Natural light that is emitted by the sun and typical home and industrial sources such as light bulbs and fluorescent lights has equal amounts of each of these two independent polarizations. Such light is sometimes called unpolarized, but this is a bit of a misnomer and randomly polarized might be a better term. Most modern scientific light sources are usually lasers, and these are very often, but not always, polarized. A typical laboratory set up positions a laser shining its beam in the horizontal direction with the polarization in the vertical direction. I shall use this configuration to describe scattering. From this the rules for scattering in other arrangements and for natural, unpolarized light are easily perceived. Consider a light wave traveling in the positive z direction incident upon a particle at the origin as drawn in Fig. 1. FIGURE 1. Geometry of scattering for an incident light wave traveling along the positive z-axis with propagation direction defined by the incident wave vector k i and with polarization (pol.) in the vertical x direction. Light is scattered from the small, spherical particle to the detector in the direction of the scattering wave vector k s. The scattered polarization will be in the direction of the projection of the incident polarization at the scatterer onto the plane p, which is perpendicular to the detector direction ks. The direction of propagation is described by the incident wave vector k i with magnitude k i = π / λ where λ is the wavelength of the light in the medium. The incident light is polarized along the vertical x axis. The y-z plane is horizontal and is called the scattering plane. Light is scattered in the direction of the scattered wave vector k s. We consider only elastic scattering; hence, ks = ki (13) Because these magnitudes are equal, we represent each simply by k. For particles small compared to the wavelength, the polarization of the scattered light is in the direction of the projection of the incident polarization at the particle onto a plane perpendicular to k s. In vector notation this is equivalent to the double cross product,

4 ( ˆ s ˆ) k x kˆ s, where k ˆs is the unit vector in the scattering direction, i.e., k ˆ s = k s / k s Either of these yield the scattered intensity to obey the proportionality Isca 1 cos φ sin θ. (15) Most experiments are confined to the scattering plane; hence, φ = 90, as drawn in Fig.. Then the polarization projects completely onto plane p of Fig. 1 with no angular dependence. The angle θ is the scattering angle; θ = 0 is forward scattering. We now consider incident light with either vertical, V, or horizontal, H, polarization. The scattered light can be detected through a polarizer set in either the V or H directions. Thus, four scattering arrangements can be obtained described by the scattered intensities as IVV, IVH, IHV, and I HH where the first subscript describes the incident polarization, the second the detected polarization of the scattered light. I VV is the most common scattering arrangement. FIGURE. A typical scattering arrangement in which light is incident along the positive z axis with either (or both) horizontal or vertical polarization. Scattered light is detected in the horizontal, yz, scattering plane at a scattering angle of θ. For particles small compared to the wavelength, these four intensities are dependent on θ in simple ways. Larger particles yield more complex functionalities but bear the imprint of these simple functionalities. The polarization rule above and the concept of projection embodied in Eq. 15 can be used to infer the four intensities. The most common scattering arrangement measures I VV, which is independent of θ, hence isotropic in the scattering plane. The projection rule or eq. 15 imply IVH = IVH = 0, and Cartesian plot of these functionalities. IHH cos θ. Figure 3 shows both a polar and

5 FIGURE 3 Polar and Cartesian plots of Rayleigh I VV and I HH scattering. Unpolarized light is the incoherent sum of two equal components with polarization that differ in direction by 90 o. To determine the polarization state of light scattered from an unpolarized incident beam one need only apply the rules above for polarized light to the two polarized components in linear combination. Fig. 4 is an example. FIGURE 4. Unpolarized incident light scatters from an aerosol to yield polarized scattered light as indicated. SMALL PARTICLES RAYLEIGH SCATTERING Introduction. A small particle has all its dimensions small compared to λ the wavelength of light. Scattering from small particles is called Rayleigh scattering. A simple dimensionality argument can be made for the length functionalities, of which there are two, in Rayleigh scattering. Since the particle is very small compared to the wavelength of light, the phase of the incident light is uniform across the volume of the particle. If one subdivides the particle into infinitesimal subvolumes, all subvolumes see the same phase. Furthermore, again since the particle is relatively small, the light scattered to the detector from all subvolumes of the particle has the same phase at the detector. Then, the total scattered field at the detector is directly proportional to the number of subvolumes which is in turn proportional to the particle volume, V part. Intensity is field amplitude squared, thus I V part. (16) Cross section is an effective area hence has units of length squared. So far in Eq. 16 we have

6 part V, which is length to the sixth power. Thus, a factor of length to the inverse fourth power is missing. There are only two length scales in the problem, particle size and optical wavelength, λ. Particle size has already been used with V. Thus, to achieve the proper units for cross section, a factor of 4 λ part must be included to yield a cross section with the proper units. Thus, C sca ~ 4 λ V part. (17) This simple derivation did not depend on particle shape; thus the result is independent of shape. Nor did it depend on scattering angle; thus the scattering is independent of scattering angle. i.e. the scattering is isotropic (if we ignore possible polarization effects, see above). Rayleigh Differential Cross Section. Electromagnetic theory can obtain an exact theoretical expression for Rayleigh scattering for the simplest case, a spherical particle of radius a. We may then define the size parameter as α = πa/ λ (18) which is a dimensionless ratio of the two length scales involved. The conditions for Rayleigh scattering are α 1 (19a) mα 1. (19b) Where m is the relative index of refraction of the particle The differential scattering cross section is m = n particle /n medium (0) dcsca 4 6 m 1 = k a dω m π a m 1 = 4 λ m +. (1) () By Eq 4. I VV k a = r 4 6 m m 1 + I o. (3) Often the term involving the refractive index, the Lorentz term, is abbreviated as Fm ( ) = ( m 1)/( m + ). This makes Eq.(1) simply dc sca dω 4 = kafm 6 ( ). (4) Rayleigh scattering has a number of important characteristics:

7 1. Isotropy. The I VV scattering is independent of θ in the scattering plane. IVH = IHV = 0, and IHH = I VV cos θ as drawn in Fig The λ dependence. Blue light scatters more than red. This is associated with the blue of the sky and the red of the sunset (Minneart, 1993), but other (lesser) factors are involved. In perfectly clean air (no particles) molecular scattering occurs due to small, thermodynamic fluctuations in the air density. Since the fluctuations are small compared to the wavelength, the 4 Rayleigh λ dependence ensues. 3. The Tyndall Effect. The strong size dependence of V part ~ a 6 leads to an increase in scattering as a system of particles coarsens. To see this consider that the total scattering from a particulate system of Rayleigh scatterers of n particles per unit volume has the proportionality I sca nv part. (5) If the only growth process in the system is aggregation, the total particulate mass is conserved. Hence, nv part is constant. On the other hand V part increases during aggregation. Rewriting Eq. 5 as Isca nvpart Vpart (6) shows that the scattered intensity increases proportional to as V part the system aggregates. This is the Tyndall effect. Rayleigh Total Cross Section. Integration of the differential cross section over the complete solid angle of 4π yields the total cross section. We consider the scattering arrangement in Fig. 1 with incident light polarized in the vertical direction to find Csca = dcsca dω dω (7) dc sca π 1 = (1 cos ϕ sin θ) d(cos θ) dϕ dω 0 1 (8) 8π dc = sca. 3 dω (9) Since the differential cross section is independent of angle, the factor 8 π /3 comes from integration of the polarization. Equations 1 and 9 yield 8π 4 6 Csca = k a F( m). (30) 3 Thus, the scattering efficiency, Q = C / πa is sca Q sca 8 4 = α F( m). (31) 3 For Rayleigh scatteringα << 1, thus Eq. 31 implies that Rayleigh scatterers are not very efficient; i.e., they scatter a lot less than the geometric cross section would imply. Rayleigh Absorption Cross Section. The Rayleigh absorption cross section is

8 3 π a m abs Im 8 1 C = (3) λ m + where Im means imaginary part. We use the notation m= n+ iκ and Im ( m 1 )/ ( m + ) = E( m). The absorption efficiency is simply Qabs = 4 α E( m) (33) As for scattering, a simple dimensionality argument can be made for the absorption cross section. Since the particle is very small, the wave completely penetrates the volume of the particle. Hence, all subvolumes of the particle absorb equally to imply Cabs Vpart. To make the units match we divide by the only other length scale of the system, the wavelength, to yield C abs 1 λ V part. This is significantly different than scattering. Rayleigh Extinction Cross Section. Extinction is the sum of scattering plus absorption. If the particle s refractive index is purely real, there is no absorption so extinction equals scattering. If there is an imaginary part to the particle s refractive index, the absorption will dominate the scattering in the Rayleigh regime because scattering has an extra factor of α 3 (compare Eqs. 30 and 3), which is much less than unity for a Rayleigh particle. These facts open the possibility for light scattering measurement of the particle size and number density for small particles. The scattered intensity at an arbitrary angle is proportional to the differential cross section and the number density of particles, n. Thus, from Eq. 4, I sca ~ na 6 F. Such a measurement is usually made by calibrating against a known scatterer such as a gas or liquid with known Rayleigh Ratio. A simultaneous turbidity measurement can be made using Eq. 9. The turbidity is related via Eq. 10 to the number density and the extinction cross section which, in this limit, equals the absorption cross section. Then use of Eq. 3 implies τ ~ na 3 E. We now have two equations and two unknowns, n and a, which can be determined. SOFT PARTICLES -- RAYLEIGH-DEBYE-GANS SCATTERING If the refractive index contrast between the scattering particle and the medium is small, i.e, if m is very nearly unity, then the interesting case of Rayleigh-Debye-Gans scattering can occur. The conditions for Rayleigh-Debye-Gans scattering are m 1 1 (34a) ρ = α m 1 1 (34b) Note that Eqs. 34 allow for particles of arbitrary size so long as they are soft, m 1. The parameter ρ in Eq. 34b is called the phase shift parameter and represents the difference in phase between a wave that travels through a particle directly across its diameter and one that travels the same distance through the medium. The Rayleigh-Debye-Gans differential scattering cross section in the scattering plane for vertically polarized light (Figure ) is dc 3 ( ) sca dcsca = 3 sin u u cos u d d (35) Ω Ω u RDG R where ( ) u = αsin θ / (36)

9 or u = qa (37) In Equation 35 the subscripts RDG and R denote Rayleigh-Debye-Gans and Rayleigh, respectively. In Eq. 37 q is the scattering wave vector given by. ( ) ( ) q = 4 π / λ sin θ /. (38) Rayleigh-Debye-Gans scattering is the diffraction limit of scattering. The term inside the brackets of Eq. 35 is the Fourier transform of a sphere; hence it represents diffraction from a sphere. It is completely analogous to single slit Fraunhofer diffraction of sophomore physics fame. Thus RDG is simply the Fourier transform of a sphere, squared. In quantum mechanics the analogue is the first Born approximation. RDG is not limited to spheres. The scattering for an arbitrary shaped particle is, in analogy to Eq. 35, the Fourier transform of the shape squared times the Rayleigh scattering cross section. Plots of RDG scattering are shown in Figs. 5 and 6, linear versus scattering angle theta and double log versus qa, respectively. Note how the profusion of complexity of the former graphs reduces to universal behavior seen in the latter graph. Ah, the joys of unitless variables! Note also in Fig. 6 for qa<1 the scattered intensity is constant with q, hence constant with scattering angle. This is called the forward scattering lobe. The forward scattering lobe starts near qa = 1 which corresponds to θ = λ/πa for small angle (i.e.,sin θ / ~ θ / ). In this region the scattering magnitude is equal to the Rayleigh result, Eq. 1. Interference ripples are seen for qa>1 with spacing Δ u = π which corresponds to Δ q= π / a and Δ θ = λ /a and small θ ; recall single slit diffraction. The envelope of this regime has a slope of -4 to imply the power law (qa) - 4. This is the so-called Porod regime. In fact, the -4 is equal to -(d+1) where d is the dimensionality of the sphere which is d=3. FIGURE 5. Normalized Rayleigh-Debye-Gans scattered intensity versus scattering angle θ for different size parameters.

10 FIGURE 6. Normalized Rayleigh-Debye-Gans scattered intensity versus the dimensionless u=qa. THE SCATTERING WAVE VECTOR In this section we take a bit of a diversion from the discourse of describing scattering from arbitrary particles to describe the physical origins of the scattering wave vector q that appeared, perhaps somewhat mysteriously, above. This section can be skipped; it is enough to know that q exists and to see its utility developed below. However, if the reader desires a firm physical understanding of scattering, this section will take the reader a long way down that path. FIGURE 7. Light with incident wave vector k i scatters from a scattering element at r into a direction k s directed toward the detector at scattering angle θ and great relative distance. Consider a scalar electromagnetic field (polarization effects are not important here) incident upon a scattering element at r with wave vector k i as in Fig. 7. The incident field at r is iki r E e (39) where we keep track of phase information only. We have used a complex representation of the oscillation wave in space. The field scatters toward the detector in the direction k s where k s is the scattered wave vector. i The field at the detector, which is positioned at R, is ik ER ( ) Ere ( ) s ( R r ) (40)

11 Substitution of Eq. 39 into 40 yields ik R i( k k ) r ER ( ) e s e i s (41) The second term of Eq. 41 shows that the phase at the detector is a function of the position of the scattering element r and the vector q = k k (4) This vector q is called the scattering wave vector, and was seen first above in Eq.(38). Its direction is in the scattering plane from k s to k i as shown in Fig. 8. From Fig. 8 and the elasticity condition, Eq. 13, the magnitude of q is i s Which is identical to Eq.(38). q = k sin θ / (43a) 1 = 4πλ sin θ/ (43b) FIGURE 8. Relationship between the incident k i and scattered k s wave vectors, the scattering angle θ and the scattering wave vector q where θ is the scattering angle. The physical significance of q is that its inverse, q -1 represents the length scale of the scattering experiment. This follows from the second term in Equation 41, which can be written as iq r e (44) As we have seen, expression 41 gives the phase at the detector due to a scattering element at r. The scattering due to an object will be the sum of such terms, typically performed with an integral weighted by the scattering density of the object at r, over the extent of the object. This integral is the Fourier transform of the object. A key point is that during this integral, the phase expression. Eq. (44), will not vary significantly if the range of r during the integral is small compared to q -1. Thus, the scattered phase would not be sensitive to the overall extent of the object. We might say that in this case the scattering cannot resolve the object. Application of this concept to scattering for a particle of any extent a implies that 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 isotropic forward scattering regime for any size particle. Only when q > a will scattered intensity show a q, hence an angular, dependence. Moreover, the functionality starts near qa ~1, and this fact can be used to determine the size of the particle SPHERES OF ARBITRARY SIZE AND REFRACTIVE INDEX: THE MIE THEORY The Rayleigh and Rayleigh-Debye-Gans theories of scattering and absorption represent

12 solutions to Maxwell s equations in which approximations have been made due to small size and small index of refraction. For an arbitrary particle Maxwell s equations must be solved exactly. Mie first presented these solutions for the simplest case of a homogeneous sphere and the term Mie scattering is often applied to this case. The equations are, however, not particularly simple to use or to gain physical insight. Below we follow the lesson learned from RDG scattering and use the scattering wave vector q as the independent variable (even better the dimensionless qa) for plotting the Mie scattering differential cross section. Then physical patterns appear (Sorensen and Fischbach, 000; Sorensen and Shi, 000, 00; Berg, Sorensen and Chakrabarti, 006) which lend a great deal of order to the description of scattering by an arbitrary sphere. The Mie Differential Cross Section. Figure 9a shows an example of Mie scattering, I vv, for a sphere with an index of refraction m = 1.50 and a variety of sizes expressed as the size parameterα. The normalized intensity I( θ ) / I(0) vs. θ is plotted. A series of bumps and wiggles are seen with some periodicities, but with no particularly coherent pattern. The scattering angleθ, although conveniently measured in the laboratory, is not the best parameter for plotting the scattered intensity. Figure 9b shows the same intensity plotted versus the dimensionless product qa. Now the data fall into patterns described by power laws. FIGURE 9. (a) Normalized Mie scattering curves as a function of scattering angle for spheres of refractive index, m = 1.50 and a variety of size parameters; (b) same as (a) but plotted vs. qa. Lines with slopes - and -4 are shown. At small qa a nearly universal forward scattering lobe is seen. Near qa ~ 1, the falloff is approximately described by the Guinier equation, Iq ( )/ I(0)~1 qa /5 (see below). The enhanced backscattering, the glory, visible in plots with m = 1.50, shows no particular pattern but is compressed into spikes in the large qa part for each size parameterα. The key features are the envelopes of these plots with slopes - and -4. Figure 7b includes lines that roughly describe these envelopes. Figure 10 shows the Rayleigh-normalized Mie scattering intensity i.e., I vv divided by the differential Rayleigh cross section, Eq. 1. The calculations have been averaged over a small but finite size distribution to eliminate the ripples. Many of the features described above are contained in this figure. One sees a pattern of power laws with slope 0, -, and -4. An important feature is that Fig. 8 demonstrates a quasi-universality of Mie scattering on the phase-shift

13 parameter ρ. For each phase-shift parameter value shown the values of m and α = ka vary widely. Despite this variation, the curves for the same ρ lie with each other, hence the curves are universal with ρ. This universality is not perfect, however, with variations of approximately a factor of 3 for the same ρ. Hence we use the term quasi-universal to describe the ρ functionality. Figure 11 gives a general description of light scattering by a sphere of arbitrary size and real refractive index, Mie scattering. The Mie scattering patterns start at ρ =0 with the RDG limit. For qa<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 power law, the Porod limit, with magnitude (9/)(qa) -4 times the Rayleigh scattering cross section. This includes Rayleigh scattering in the limit λ >> a because then qa 1 and only the flat part of the scattering function is obtained. FIGURE 10. Shown are two sets of Rayleigh-normalized Mie intensity curves. The three curves in each set have the same value of ρ but different values of ka and m. The bounding envelopes are shown for both sets of curves and illustrate how the envelopes only depend on ρ. FIGURE 11. Diagram of the averaged Rayleighnormalized Mie scattering patterns for uniform dielectric spheres of arbitrary size and real refractive index parameterized by ρ = ka m 1, thick solid line. In this example ρ 55. The dashed line is the RDG limit, ρ 0. When ρ >1, scattering in the Rayleigh regime decreases relative to true Rayleigh scattering. The relative decrease (remember, the unnormalized scattering increases with ρ ) is proportional to ρ as depicted in Fig. 11. For qa >1 the scattering now falls off as (qa) - until this functionality crosses the RDG curve at qa ~ ρ qa = ρ. For qa > ρ the scattering is identical to RDG scattering, falling off as (qa) -4 for all a and m as long as qa > ρ. The (qa) - functionality between qa = 1 and ρ for ρ >1 is exact only at those limits. The average Mie curves dip below the (qa) - line. This dip is the first interference minimum near qa 3.5 present in all Mie curves and is the strongest of all the minima. In summary, if we ignore the ripples and the glory, Mie scattering displays three power law regimes I ( qa) 0 when qa< 1 (45a)

14 ( ) when 1< ( ) 4 when qa qa < ρ (45b) qa qa < ρ (45c) The Mie Guinier Regime. The end of the isotropic forward scattering lobe, near qa ~ 1, in both the RGD and Mie scattering regimes is called the Guinier Regime. This regime is very important because it provides a simple and convenient way to size particles with little or no need to know the refractive index. This is also the region of crossover between the functionalities of Eqs. 45a and b. For ρ 0 it is described by 1 Iq ( ) IO ( ) = 1 qrg (46) 3 In Eq. 46 R g is the radius of gyration of the scattering object, which can have any shape. If spherical, Rg = 3/5 a. We have found (Sorensen and Shi, 00) for spheres that Eq. 46 must be modified when ρ > 1 1 Iq ( )~ 1 qr gg,. 3 In Eq. 47 is R g,g, which we call the Guinier regime determined radius of gyration (Sorensen an d Shi, 000). When ρ = 0 Rg, G = Rg. Otherwise, it follows a quasi-universal behavior with ρ depicted in Fig. 1. Note that as ρ R g, G= 1.1Rg, the Fraunhofer diffraction limit. Importantly, use of a Guinier analysis with Eq. 47 and Fig. 1 allows for a simple yet accurate experimental measurement of the particle R g, hence geometric radius a. (47) Change ρ by α 1.5 m = R g,g /R g m = m = ρ FIGURE 1. The ratio of the Guinier inferred to real radius of gyration, Rg, G / R g, vs. phase shift parameter ρ for spheres with three different refractive indices. The size parameter α = kr was varied to vary ρ. The Mie Ripples. Recognize that the ripple structure that we have ignored above by considering only the envelopes will in an experiment by washed out by any modest particle size polydispersity of geometric width of 0% or more (Rieker, et al. 1999). However, the ripple

15 structure contains useful information. Let the symbol Δu designate the spacing between consecutive ripples when the intensity is plotted vs. u=qa, and Δ θ when plotted vs. θ. Then (Sorensen and Shi, 000) Δ u = π for ρ < 5 (48a) Δ u = π cos θ for ρ > 5. (48b) Equation 48b is equivalent to Δ θ = π / α = λ/ a for ρ > 5. (48c) Equation 48c is identical to the angular fringe spacing for Fraunhofer diffraction from a single slit of width a. Equation 48c suggests that for highly monodisperse systems, which would yield good ripple visibility, a measurement of the ripple spacing would yield the particle size (Maron and Elder, 1963; Pierce and Maron, 1964; Kerker et al. 1964). The Mie Total Cross Section. Figure 13 shows C sca as a function of the size parameter α = ka for spheres with different refractive indices m. Curves of different m in the ka < 1 range show a clear m dependence which is explained by the m functionality in Eq. 1. As the size a grows to values for which ρ > 3, the a 6 dependence crosses over, through a ripple structure, to a geometric a (i.e., the geometrical cross section of the sphere) dependence at large ρ, large ka. The separation between curves of different m in the Rayleigh regime begins to vanish. As a continues to increase, all curves converge onto a value of twice the geometrical cross section of the sphere, π a, with no m dependence and the ripple structure decays away. This is a remarkable transformation from an optical entity dependent on the refractive index m, to an apparently nonoptical entity with no m dependence. Moreover, the remaining functionality is remarkable too for it claims that in the geometric limit the scattering cross section is twice the geometric cross section. How is it that the scattering from a large object, a rugby ball for example, is twice its shadow? This fact is known as the Extinction Paradox and has seen a number of explanations the best of which is given by Berg, Sorensen and Chakrabarti, 008a and 008b.

16 FIGURE 13. The total Mie cross section C sca for various m and incident light with V- polarization and λ = 00π nm. The large ρ behavior of Csca π a is shown as the dotted curve; the Rayleigh cross section is shown as the dot dashed line for m = The vertical arrow 4 shows the factor 1/( c ρ ), where c 10, relating the Rayleigh and Mie cross section for m = 1.05, in the ρ > 1 range. For clarity, the legend shows the data sets in the same order as they appear in the figure. EXAMPLE Estimate the total scattering cross section for a spherical water droplet with a diameter of 3.0 microns and index of refraction of The wavelength of the light is micron. Answer. The ρ value for these particles is ρ = 4π(3.0/)(1.33-1)/(0.488) = 1.7 This is large to imply the total cross section is about equal to C sca = πa = 6.8(1.5x10-4 ) = 14. x 10-8 cm. RAYLEIGH-DEBYE-GANS FRACTAL AGGREGATE SCATTERING Fractal Aggregates. The past 5 years have seen the development of the fractal concept for quantitative description of many aggregates that form in Nature (Mandelbrot, 1983; Forrest and Witten, 1979; Jullien and Botet, 1987). A fractal is an object that displays scale invariant symmetry; that is, it looks the same when viewed at different scales. Any real fractal object will have this scale invariance over only a finite range of scales. A consequence of this scale invariance is that the mass scales with it linear size with a power law the exponent of which is call the fractal dimension, D. For an aggregate with an indefinite border the linear size can be well described by its radius of gyration, R g. Then a fractal aggregate of N monomers or primary

17 particles obeys o ( g / ) D N = k R a (49) In Eq.(49) k 0 is a prefactor of order unity and a is the primary particle radius. Perhaps the most common aggregation process is diffusion-limited cluster aggregation (DLCA) which yields clusters for which D = 1.75 to With experimentation on soot aggregates we have found k o = 1.3 ± 0.07 (Cai, Lu and Sorensen, 1995) and 1.66 ± 0.4 (Sorensen and Feke, 1996). Some other workers have found larger values on the order of.4 for soot (Koylu and Faeth, 1994a, 1994b). With simulations, we have found k o = 1.19 ± 0.1 (Sorensen and Roberts, 1997) and 1.30 ± 0.07 (Oh and Sorensen, 1997). Figure 14 shows a fractal aggregate and Fig. 15 shows a cartoon of a fractal aggregate. FIGURE 14. TEM micrograph of soot collected from an C H /air diffusion flame. FIGURE 15. Cartoon of a fractal aggregate.

18 Scattering and Absorption by Fractal Aggregates. The scattering and absorption by fractal aggregates under the Rayleigh-Debye-Gans approximation, so-called RDGFA theory, is now fairly well known (Sorensen, 001). RDGFA assumes that the effects of intracluster multiple scattering can be neglected. This assumption has been shown to be pretty good for D <. It is more suspect for D > because such clusters are not geometrically transparent; that is, their projection onto a two-dimensional plane would fill the plane. Other factors that can lead to multiple scattering effects are large monomer a (or, better, its size parameter α ) N, and m. Based on an analysis by Farias et al and our own experimental tests (Wang and Sorensen, 00), we have concluded (Sorensen, 001) that a phase shift parameter for the cluster aggregate can be defined as ρ c = krg m 1 (50) When this is less than ca. 3, RDGFA is valid. An argument can be made that for D <, the RDGFA gets better with larger N. The matter is not yet settled, and this is an area of research interest. Under RDGFA the scattering and absorption cross sections for a fractal aggregate of N monomers with radius a are simply related to the monomer cross sections as follows (Sorensen, 001). c m C = NC (51) abs dc dω sca = N abs m sca dc dω Sq ( ) The superscripts c and m designate cluster and monomer, respectively. S(q) is the static structure factor of the cluster which is the Fourier transform of the cluster density function, squared, and hence it contains information regarding the cluster structure. The structure factor has the asymptotic forms S(0) = 1 and S(q) ~ q -D 1 for q Rg. The simple forms of eqs. 51 and 5 have physical interpretation. Equation 53 implies that the absorption is independent of the state of aggregation; the monomers absorb independently. Equation 5 implies that at small q the scattering from N monomers is also independent of the state of aggregation; the N scattered fields add constructively to yield the N dependence. There are some variants on the form of the structure factor (Sorensen, 001). The best description is simply (5) S(q) = 1, qrg < 1 (53a) ( ) 1 g = 1 q R / 3 qrg ~ 1 (53b) ( ) D g = C qr qrg > 5 (53c) Where the coefficient C = 1.0 ±0.1 The general behavior of I(q) is shown in Fig. 16. It has many features in common with the descriptions of non-cluster scattering above. At small q there is a scattering angle independent forward scattering lobe, the Rayleigh regime, where the cross section is N times the monomer Rayleigh cross section (recall Eq. 16). Next follows a Guinier regime near qr g ~ 1. This can be D used to measure R g. At yet larger q lies a power law regime, where Sq ( )~ q. This can be used to measure D. Finally, at very large q often inaccessible for light scattering but often observed with small angle X-ray scattering is the regime for which q> a 1, (not drawn) where a is the length scale of the primary particles. In this regime the so-called form factor of the primaries

19 would be seen This is essentially the scattering from single dense particles, e.g. Mie scattering, described above. Figure 17 shows an example of scattering from a titania aerosol. FIGURE 16. Schematic of the general behavior of the structure factor for a single fractal aggregate of N primary particles with aggregate fractal dimension D, radius of gyration R g, and monomer radius a. FIGURE 17. Light scattering structure factor obtained from a titania aerosol. EXAMPLE For the aerosol in Fig. 17, a) what is the approximate mean size of the aggregates? b) What is the mean fractal dimension of the aggregates? a) Comparing Fig. 17 to Fig. 16 we see that the Guinier regime is near 0.1 μm -1. This equals R g -1 so R g = 10 μm. b) Comparing Fig. 17 to Fig. 16 we see that the power law regime has a slope of -1.75; thus D = MULTIPLE SCATTERING

20 Multiple scattering is most easily envisioned with the photon concept of light. We can think of these little bullets as first scattering here and then there before they reach the detector. Warning should be given, however, that the wave/particle duality of light can confound this simple picture (Mishchenko, Travis and Lacis, 00). Regardless, the formulation below is effective in quantitatively describing multiple scattering. In many applications light scattering experiments involve a system of particles. If the light scatters from any of the particles only once, then the total scattering from the ensemble is easily interpreted as an average of single scattering events which are, given the discussions above, well understood. However, there is always a finite chance that the scattered light reaching the detector has scattered sequentially from more than one particle. This is multiple scattering, and its interpretation is difficult. Thus it is a situation to be avoided, and to avoid it we must be able to detect or predict it. Under the assumption that the photons act like classical particles and that their scattering is a Gaussian random process described by the Poisson distribution, one can show that the extinction of the light through a medium containing non-absorbing particles obeys the classic Lambert-Beer law, Eq. 9 (Mokhtari, Sorensen and Chakrabarti, 005). Moreover, the turbidity is the inverse of the photon mean free path l between scattering events, viz. 1 1 τ sca l = = ( C n). (54) The average number of scattering events for the photons s is the ratio of the photon mean free path to the length of the scattering volume < s>= x/. (55) Since the multiple scattering is a Gaussian random process, the probability for s scattering events for a given photon is given by the Poisson distribution as s s Ps () = exp ( s). (56) s! As an experimenter we want s to be small so that the probability of more than one scattering event, i.e. multiple scattering, is small. Fortunately the value of s is simply obtained either by calculation, with Eq. (54), or measurement. Measurement is particularly simple because by Eqs. 54 and 55, s = xτ and τ is easily measured via Eq. 9. A common strategy to avoid multiple scattering is to simply dilute the sample. Sometimes this can t be done without changing the system. Another strategy is to make the scattering length x smaller than the photon mean free path. This can be done by reducing the sample size. Then by Eq. 55, the average number of scattering event <s> can be made far less than one. Scattering from Ensembles of Particles. For particles situated randomly in an aerosol, the most common situation, the total scattered intensity is simply the sum of the scattered intensities from each particle in the illuminated scattering volume. Most often the ensemble of particles is polydisperse with a size distribution n(a) numbers per unit volume. Then the total scattering from the ensemble is I(q) = I(a, q) n(a) da. (57)

21 An important aspect of light scattering is that big particles scatter more than smaller ones. Thus the total scattering is weighted in favor of the bigger particles in the distribution. This can be quite extreme in the Rayleigh regime where the scattering goes as a 6. For example, if the distribution was bidisperse with equal amounts of 40 and 80 nm particles, the 80 nm particles would scatter 64 times more light and hence dominate the scattering; the smaller ones would effectively not be seen. Nonspherical Particles. The problem of how an arbitrarily shaped particle scatters light is quite difficult but major advances have been made in the past two decades. One tack is to divide the particle up into a great many subvolumes which act as dipoles but interact with each other through their scattered fields. An iterative solution for the total scattering can be calculated numerically. This method is called the Dipole-Dipole Approximation, DDA (Draine, 1988; Draine and Flatau, 1994). Another successful tack is the T-Matrix formalism which is also quite complex (Mackowski, 1991, 1994; Mackowski and Mischenko, 1996). Both these methods are beyond the scope of this review. There is some indication that the patterns in Mie scattering for spheres, discussed above and portrayed in Eqs. 45, persist for dense, irregular particles (Hubbard, Eckles and Sorensen, 008). Certainly, there is a Guinier regime which can yield semiquantitative size information. DYNAMIC LIGHT SCATTERING Dynamic Light Scattering (DLS), also known as Quasi-elastic Light Scattering (QELS), Photon Correlation Spectroscopy (PCS) and Light Beating Spectroscopy, is a technique that relies upon temporal fluctuations in the light scattered from an ensemble of particles to determine their motion. Usually the motion is random Brownian diffusion which is quantified by a size dependent diffusion coefficient. The DLS method measures the decay of the temporal fluctuations in the scattered light, which is related to their diffusion which, in turn, is related to their size. Thus a size measurement can be made. Application of DLS to aerosols dates back to Hinds and Reist, 197; application to combustion aerosols dates back to King et al., 198 and Flower, My treatment here will give the salient highlights of DLS; those desiring and more extensive treatment are referred to Berne and Pecora, 1976 and Dahneke, Theory of DLS. The field correlation function. We start with the phase of the scattered light at the detector as given above in Eq. 44, generalized to include the position of the scattering element, which we now take as an entire particle, as a function of time, τ. E( τ )~ e iq r( τ ) (58) This function E(τ) will fluctuate randomly with time if the particle is moving via random Brownian motion, the most common situation. To quantify a fluctuating variable one can compare its value at two different times and average over all these comparisons with a so-called correlation function, viz. g (1) (t) = <E(τ+t)E*(τ)> (59) In Eq. 59 the star means complex conjugate and the brackets < > mean an ensemble or time average (assuming ergodicity) and we have made the reasonable assumption that the average only depends on the time difference, t (an assumption known as stationarity ). Thus from Eqs. 58 and

22 59 we have ( ) ( 0 g(1)( t) ~ e iq r t r ) (60) The average is made by assuming the particles in the ensemble are undergoing random Brownian motion. Then the probability distribution is a Gaussian r /Dt P r, t ~ e Δ Δ (61) ( ) In Eq. 61 Δ r = r () t r (0). The diffusion coefficient D (not to be confused with the fractal dimension) is given by the Stokes-Einstein relation D = kt/6πηa, (6) where k is Boltzmann s constant, T is the absolute temperature, η is the suspending medium shear viscosity, and a is the particle s radius, assumed spherical. Performing the average of Eq. 60 with the probability distribution of Eq. 61, we have (1) ()~ (,) iq Δ g t P Δr t e r dδr (63) (1) ()~ Dq t g t e (64) Equation 64 is our desired result. It says the scattered optical field correlation function decays with a decay time, better known as the correlation time, of 1/Dq. Make note that the field correlation function is the Fourier transform of the probability distribution of the motion. That is a general result. Aerosols present a further complication compared to colloids in that the Stokes-Einstein relation only holds in the limit of large particles. For any particles size the Cunningham corrected diffusion coefficient can be used. The intensity correlation function. With standard detectors such as photomultiplier tubes and photodiodes, we cannot detect the optical field directly; instead we detect its square, the scattered intensity. Thus in DLS we will not measure the correlation function of the scattered field, we will measure the correlation function of the scattered intensity. Fortunately, the two are simply related if the scattering process is random so that the scattered field is Gaussianly distributed. This is achieved whenever there are numerous particles in the scattering volume, the usual case. The relation is called the Siegert relation and is Thus () (1) g () t = I() t I(0) = 1 + g () t. (65) () Dq t g () t = 1+ e. (66) The intensity correlation time, the primary parameter of DLS measurement, is thus

23 τ = 1/Dq. (67) It is worth noting the q dependence. This can be used to adjust the correlation time into an easily measured regime. It is also a good consistency check; any experiment should show this q dependence or else something is wrong. EXAMPLE Consider 1.0 micron diameter particles in air at STP. The viscosity of air is 180 micropoise. What correlation time would you expect using an Ar+ laser operating at 514.5nm with a scattering angle of 90 o? Use cgs units. D = (1.38x10-16 )73/6(3.14)1.8x10-4 )(5x10-5 ) =.x10-7 cm /sec q = [4(3.14)/5.15x10-5 ]sin45 = 1.7x10 5 cm -1 q =.97x10 10 cm - τ = 1/Dq = 1/(.x10-7 cm /sec)(.97x10 10 cm - ) = 7.6x10-5 sec. Flowing Systems. Many aerosol situations involve motion due to flow as well as Brownian motion, e.g., a sooty flame. This flow will have its own characteristic time scale and cause additional beam transit terms. One can show for a flow velocity ν (Chowdury et al., 1984; Taylor and Sorensen, 1986) () Dq t ν 3/ 1 g () t = N 1+ e e + N e t / w1 ν t / w (68) = 1 + (diffusion term)(1 st beam transit term) + ( nd beam transit term) (69) In Eq. 68 <N> is the average number of particles in the scattering volume observed by the detector, w 1 is the focused beam waist of the incident light on the aerosol, w is the beam waist on the sample at the point of observation, not necessarily at the focus, see Fig. 18. Equation 69 attempts to explain Eq. 68. We see the usual diffusion term is multiplied by the first beam transit term. There is also a second beam transit term that is added. This second term has an inverse <N> dependence and hence is only seen in very dilute situations. Figure 18. Manipulation of a Gaussian profile laser beam by a positive lens showing beam waists, w.

24 The potential problem lies with the first beam transit term. If it decays quicker than the diffusional term, i.e. if the beam transit is fast compared to the diffusion, the beam transit will cut off the diffusion term so that it can t be measured to determine particle size. Slowing the flow can alleviate this problem, but sometimes that can t be done. Making the beam wider will also lengthen the beam transit time but this has a subtlety. Simply moving the observation scattering volume to a wider portion of the beam will not affect the first beam transit term since it is dependent only on the focused beam waist regardless of the point of observation. This focused waist is proportional to the focusing lens focal length. One strategy is to not focus the beam very much but that will affect coherence on the cathode necessary for good signal to noise, see below. Thus the experimenter is confronted with a potentially complex give-and-take to optimize the experiment. Chowdury et al have presented a useful graphical method of data analysis when the beam transit time is comparable to the diffusion time. DLS Experimentation. DLS requires a coherent light source. The advent of the laser, as such a source, enabled the technique to be developed in the 1960 s. Most common laboratory lasers such as HeNe, argon ion, Nd:YAG, etc have enough coherence to be useful for DLS. Equation 65 is for perfect coherence and an experimentally realistic formula is () Dq t g () t = B+ Ae (70) We call A the signal strength and B the background. As coherence declines, the signal to noise, A/B, declines. Coherence is a complex topic and will not be discussed here. Standard texts such as Hecht, 1987, or Born and Wolf, 1975 can be consulted. Here it is useful to know that both longitudinal coherence, related to the spectral band width of the light, and transverse coherence are necessary. The former is fixed by the laser you use an typically pretty good. The latter is also a strong function of the laser but can be improved by spatial filtering transverse to the direction of the beam. Thus good transverse coherence can be gained if the laser is operating in the TEM00 mode (transverse electromagnetic). This mode is characterized by a Gaussian beam profile. The donut profile TEM01* will work too but with some loss of signal to noise. Once the light is scattered by the medium, the experimenter has considerable control of transverse coherence and is encouraged to improve it. The concept is based on the van Cittert- Zernike theorem (Born and Wolf, 1975) which states that transverse coherence of light from an incoherent source has an angular size of approximately θ ~ λ/d where λ is the light wave length and d is the spatial extent of the source. This is the diffraction angle. For DLS this means we want a source, an illuminated scattering volume, which is small so that θ is large enough to cover our detector s photosensitive area at some reasonable distance. Notice it also implies we want a small detector. As an example, suppose we have a photomultiplier tube with a.5 mm diameter cathode. Let s place it 500 mm away from the scattering volume. Then it subtends an angle of θ det ~.5/500 = 5 mrad. If we use an unfocused laser beam with diameter of 1 mm, then according to van Cittert-Zernike, the coherence angle is θ coh ~ 0.5 micron/ 1 mm = 0.5 mrad. This is much smaller than the angular size of the detector so the signal to noise will be poor. If, however, we focus our incident beam to a diameter of 0.1 mm then θ coh ~5 mrad and good signal to noise will be achieved. Note this simple example considered diffraction in only the vertical direction since the horizontally propagating beam was restricted by its diameter in the vertical direction. Horizontal is at work too, so we would want to spatially filter, i.e. block, the light from the scattering volume in the horizontal direction as well. Imagine a little square of scattering volume. Ultimately the desire is to get about one coherence area on the photosensitive area for optimal