Scattering of electromagnetic radiation in particulate laden fluids

Profi Eiwrql (ombu~r S(i, 197t) Vo] s PD 73 96

Pergamon Press L'td

Printed in Great Brtlalr~

SCATTERING OF ELECTROMAGNETIC RADIATION IN PARTICULATE LADEN FLUIDS A. R. JONES Imperial College o( Science and Technolooy, Prince Consort Road, London, SW7 2BY, U K

Summary--The main purposes of this article are to introduce the reader to the theoretical treatment of light scattering, and to build on previous reviews by incorporating recent literature. The paper begins with the very basic concepts of wave propagation and radiation, and then reviews the fundamental terminology and theories of scattering by spheres and non-spherical particles, including the integral formulation of the problem• A number of approximate methods of solution are discussed, and in the final section the principles behind the diagnostic applications of scattering are reviewed with emphasis on recent developments. particle size distribution, refractive index, anisotropy and velocity. Here emphasis is placed on principles rather than specific instruments or techniques unless they are of considerable novelty.

I. INTRODUCTION

M a n y practical systems are heterogeneous, and in order to understand the propagation and emission of radiation in them it is necessary to study the interaction of waves with solid particles and liquid droplets• This is the area known as electromagnetic scattering, which generally refers to interaction with any medium, and may include deflection of waves. absorption, changes in frequency and mixing in nonlinear materials• This article is concerned with those processes which are relevant to radiative transfer, combustion studies and pollution monitoring, although the applications are much wider. (The optical properties of paints provide an example.) With this in mind, and considering the question of length, the discussion will be restricted to linear elastic scattering with no frequency change. Information on non-linear and inelastic scattering processes can be found in Refs 1-4• Prior to 1969 light scattering has been very adequately reviewed by Kerker 5 and van de Hulst. 6 These authors do not discuss radiative transfer in particulate media or integral formulations of the scattering problem, though these subjects are reviewed elsewhere. "-1: In the theoretical sections of this article these three strands are brought together in a unified treatment and more recent developments are included. However, length considerations restrict the scope of the paper, and the last section (Section 5) concentrates on the principles behind the diagnostic applications of scattering by particles. While some of the older methods are briefly discussed for completeness, emphasis is placed on the considerable developments since 1969. After defining the basic terminology, a review is given of the way in which scattering problems are sol~ed theoretically, together with a n u m b e r of approximate methods which can be more useful than rigorous theor3 and are the only practical way of obtaining answers in some cases. O n the practical side. methods are described for the measurement of J P [ ~ S \ o [ 5. N~, "

~

2. F U N D A M E N T A L S

A propagating wave can be written in the form = q%cos(kz-~t)

(i)

where q~0 is the amplitude and the bracketed term is the phase, k = 2n/2 is the wave n u m b e r for wavelength 2 and ~o = 2 n f is the angular frequency for frequency f z and t are position and time. The form of this wave is shown in Fig. i. It is an infinite plane wave in the sense that the amplitude and phase are constant everywhere in the x - y plane. A surface of constant phase is a wavefront.

_

,,y

-'I<, FIG. 1. Representation of a simple sinusoidal wave. Generally, the amplitude and phase do vary in three dimensions. The simplest form of realistic wave is that due to a point source which is written v? 0 q~ = - - cos(kr - cot ).

(2)

r

This is a spherical wave with r the distance from the source. A plane wave, which cannot exist in nature, is an approximation of a spherical wave at very large radius of curvature. The energy transported by a wave is proportional to the square of the amplitude. However. the 73

74

A.R. JONES

frequencies of most interest here are those of the visible and near i.r. 0c> 1013 Hz). Real detectors will not respond to such high frequencies and average over m a n y cycles. Thus they detect W--T= lim 1 ~"A2dt"

xA

(3)

,,.~t Jo In this case it can be shown 13 that an equivalent description of the plane wave in eqn. (1) is tp = ~ o e x p ( i { k z - c o t } )

(4)

where i = x / - ~ . * This has the benefit of simplifying the algebra, and is valid provided that the end result is to be the time averaged energy as in eqn. (3). This is found from V 2 = q" q J * =

IVI 2

where W* is the complex conjugate of V. The frequency and wavelength are interrelated through v=f2 where v is the phase velocity. In a vacuum v = c = 3 x 108 m s - 1 and is constant ("the speed of light"). In a material medium the phase velocity differs from c and the ratio C

. . . . .

bly

FIG. 2. Representation of an electromagnetic wave. The polarization is defined to be along the electric vector. More usually the electric and magnetic fields change with time. If this variation is random the wave is unpolarized. The vectors may rotate in a regular fashion. If the rotation is accompanied by a change i n m a g n i t u d e this is elliptical po!arization; if not it is circular polarization. Accepted standard treatises on the subjects of optics and electromagnetic radiation are books by Born and Wolf 13 and Stratton. 14 2.1. Radiation Emitted by Sources

t~

is defined to be the refractive index. The waveldagth in a medium (2) is related to that in a vacuum (20) through

The total rate at which energy is radiated by a source is the total radiation flux ¢ which is measured in watts. Figure 3 shows a small area dA emitting radiation into the element of solid angle dO at the angle 0 to the normal h. The flux contained within

2 = o ; k = vk o. V

Most generally the refractive index is complex. As we shall see in Section 3, the imaginary component arises from conduction in the medium. Writing v = vl + iv z and substituting in eqn. (4) yields

i2

q~ = W0 exp{ - v2koz } • exp{i(v I k o z - ~ o t ) } and ~*

= ~:o exp{ -- 2v2koz } .

Thus we see that v2, or conduction, is responsible for absorption. The waves described by eqns (1), (2) and (4) are scalar. Typically, they may represent the pressure variations associated with sound waves. However, visible and i.r. radiation are not of this type. They form part of the electromagnetic spectrum, energy being transported by oscillating electric and magnetic fields. These are vectors which are orthogonal to each other and to the direction of propagation. This is shown in Fig. 2 where the fields maintain a constant direction relative to the direction of propagation. Such a wave is said to be polarized. * It should be borne in mind that only the real part of eqn. (4) has physical significance.

FIG. 3. Direction and solid angle for the definition of intensity. this cone is simply proportional both to projected area dA cos 0 and the solid angle, i.e.

the

d ~ = l dA cosOdf2 where the constant of proportionality I is the intensity of the source having dimensions of power per unit area per unit solid angle ( W m -2 ster-1). The total flux is evidently ¢'=lfA~2,,

cosodAdf~

The intensity at a wavelength ~. due to a source in thermal equilibrium at a temperature T is given by

Scattering of electromagnetic radiation in particulate laden fluids

75

(a)

"

Par,icle (b)

[ Focal plane

Lens

FIG. 4. Illustration of two modes of diffraction: (a) Fresnel diffraction: (b) Fraunhofer diffraction.

the Planck function I~. .

dl d2

.

v2 e~.c12- 5 . . . ~ exp(c:/2T)- 1

v2 n

e;.Es,a(T)

where e;. is the monochromatic emissivity at the wavelength ,;., and c 1 = 3.74 x 1 0 - ~ 6 W m 2 and c2 = 1.44 x 1 0 - 2 i n k are the Planck constants. v is the refractive index of the medium in which the wave propagates. Although we are concerned with scattering by particles, the incident and scattered waves travel through gases for which v -~ 1. The m a x i m u m value of the emissivity is unity. If this is true for all wavelengths the source is a black body. F o r such a source emitting into a gas we can now write for the total intensity Is

= __T4 I ~ c l ( 2 T ) - S d ( 2 T )

= _a T4"

n ,Io e x p ( c 2 / 2 T ) - 1

n

This is Stefan's law and a = 5.67 x 1 0 - S W m - : K is the Stefan-Boltzmann constant. For real sources it is necessary to write

-4

I = - erT'* 7Z

where, of course, ~: = a T ~ ,o e';EB'~(T)'d£"

The terms diffraction and scattering are often used synonymously and it is worth clarifying the distinction, where it exists. There are two classes of diffraction. In the first, as in Fig. 4(a), light scattered by the particle interferes with unscattered light. This is near field, or Fresnel, diffraction. In the second case the diffraction pattern is observed in the far field; ideally at infinity. This is Fraunhofer diffraction which can be achieved in practice by the use of a lens as in Fig. 4(b). Unscattered light is brought to a focus at the point F, whereas light scattered through an angle 0 is focused at the point D with distance FD ~- Of where f is the lens focal length. Here the diffraction pattern is the same as the scattering pattern, there being no interference with the incident light. There is a further distinction in the theoretical treatment. Diffraction theory is usually an approximation in which scalar waves interact with the crosssection, or aperture, of the particle. A rigorous treatment, however, involves vector fields and the body of the particle and is identical to scattering theory. If the intensity scattered into the spherical polar angles (0, qS) is l~ca (0, ~b) the total power scattered is

(51

For further study of this subject the reader is referred to the book by Hottel and Sarofim. 8 2.2. Interaction of Radiation with Single Particles If electromagnetic radiation of irradiance Io t W m - " l falls upon a particle two processes can occur: scattering and absorption. Scattering is a cause of deflection, which for particles very much larger than the wavelength {strictly £ ~ O) can be described in the geometrical optics terms of reflection and refraction plus a component for diffraction.

V~ca-- C I sca(O, ch) " rZ df~ d4 n where r is distance to the detector. At large distances from the scatterer the wave is quasi-spherical and can be written eikr

E~¢~(0,4)) = - - r i o r

c~t

whence P~¢~ = Jo

[f(O,~)]:sinOdOd~.

For linear scattering the scattered power is simply

76

A.R. JONES

proportional to the irradiance, so that P~ca = Io' C~ca • The constant of proportionality has the dimensions of area and is the scattering cross-section Csca. For absorption we may similarly write

Pabs -/0"Cabs =

where C ~ is the absorption cross-section. Both processes have the effect of removing power from the incident radiation. The total power extinguished is Pext - Psea + Pabs or

Cc,t = Csca+ C~b,

where Ce~t is the extinction cross-section. The cross-sections are not in general equal to the geometrical cross-sectional area Cv of the particle. The ratio Pext

Cext

Po

Cp

Qext ~---

is the extinction efficiency, where Po = Io Cv is the power impinging on the particle. Evidently {~eXt = Qsca'~Qabs •

Figure 5 shows qualitatively how Q . t varies with particle size. Common features are that Q . , is originally zero and rises rapidly to a high value, then falls and oscillates going asymptotically to a value of two. The oscillation is less marked for absorbing than non-absorbing particles. The limit of two for Qe~t is at first sight very surprising because it suggests that the particle blocks off twice as much light as falls upon it. This is the socalled extinction anomaly. The explanation lies in the fact that two processes are occurring, namely the geometrical optics effects reflection, refraction and

absorption for which Q,eo,, = 1. and also the wave phenomenon of diffraction which occurs on the aperture of the particle and for which Qd~tf = 1. Altogether Q¢~, = Qg~om + Qdiff = 2.

However, as we shall see later, the great bulk of diffracted light for large particles lies at very small angles close to 0 = 0 °. Any detector which collec:s light over a finite range of angles, such as the eye or a pyrometer, will see the diffracted light together with the incident light so that the only effective loss is Qgeom = 1. This leads to some difficulty when trying to measure a true extinction cross-section for large particles, as discussed by van de Huls.t. 6

2.3. Radiation in Particle Clouds Scattering by single particles can readily be extended to particulate clouds provided three conditions hold: (a) Each particle scatters as a separate entity, This requires the particles to be sufficiently far apart for there to be no significant electrical interaction between them. A separation greater than about three radii is sufficient, 5 (b) There is no multiple scattering. This is true if the cloud is sufficienfl3' tenuous that on average light once scattered will leave the system without further scattering. (c) There is no optical interference between waves scattered by different particles. Condition (c) is almost certainly satisfied by large numbers of particles suspended in fluids. If there are N identical particles per unit volume and the wave amplitude scattered by each one is ~ j = ~o exp{i~bj}

c~

II

Particle size Wavelength N o n - absorbing .....

Absorbing

FIG. 5. Typical variation of extinction efficiency with particle size.

Scattering of electromagnetic radiation in particulate laden fluids where 0r is phase, then, in the absence of multiple scattering, I"PI2 = 2W

N + ~

The term NC**, has the dimensions of reciprocal length and is called the extinction coefficient K¢,,. Evidently

~ cos(+i-~bi,)

K~t = Ksca+Kab~.

j=|j'=l

is the total scattered intensity per unit volume. For a large n u m b e r of randomly positioned particles the cosine term would sum to zero. Then energies can be added directly. To satisfy condition (a) we imagine a sphere of radius 3a surrounding a particle within ~he volume of this sphere. This leads to an estimate of the m a x i m u m permitted concentration, as shown in Table 1. TABLE 1. Maximum permitted concentration of particulates to avoid electrical interaction effects Particle radius ( l a m )

Concentration (m- 3)

0,1 1.0 10.0 100.0 1000.0

1019 1016 10is 10t° 10~

77

Equation (6) was derived using two further assumptions. The first is that the medium is uniform everywhere. If this is not so the expression must be modified to read I=loexp

-

K~,.dz

.

(7)

It was also assumed that all the particles were identical. At the very least, there will be a range of particle sizes. If we define a particle size distribution such that there are N(a)da particles per unit volume of radius a, then N =

N(a) da 0

and Ke,, t =

N(a)Cext(a)da. 0

Condition (b) is the most difficult to satisfy. The essential requirement is that the optical mean free path be larger than the physical dimensions of the system. Before stating a criterion for this, we examine transmission through a system on the assumption of no multiple scattering. cl

°

"'°

A

.------5 '000 O G 0 D • O~ O0

0 O@

'o o o o

,.o

Q

J

~

FIG. 6. Definition of variables for loss of light by transmission through a scattering medium. In Fig. 6 a beam radiation of intensity ! enters a suspension of particles in a non-absorbing medium contained between two parallel plates of area A. The distance between the plates is dz and there are N particles per unit volume. While crossing the gap the beam encounters N A dz particles each of which has an extinction cross-section C~x,. Consequently, the loss of power is -dP = -AdI

K'x' = Es fo N~(alC~4(a)da" The product K,xtL, or the integral in eqn. (7), is defined to be the turbidity z. The optical mean free path is of the order 1/K** t so that no multiple scattering should occur provided z <~ 1. F o r large spherical particles C,,, ~-2ha 2 so that over a distance of 1 m and letting a = 100Bm we find N < 2 × 107 m -s. Soot particles can be treated by the Rayleigh approximation (p. 82) and over the same length with a = 0 . 1 p m we find N ~<2× 1013m -3 at visible wavelengths. Thus, condition (b) is by far the most restrictive. Further, there is experimental evidence to suggest that multiple scattering effects can be observed for z > 0.1. 5 which would decrease the above concentrations by a factor of 10. We shall see later, that for soot particles K,x, K,b,. This is also true for large coal particles as seen in Fig. 7 where Q,~ ",-0.9. If we recall that a pyrometer would not measure diffraction as a loss, we see that coal particles may be treated approximately as black bodies. In both cases, the losses are entirely due to absorption and the absorptivity becomes Io-I 7~. = - = 1 - exp(-K~b~L). Io

= NAdz'IC~,

or

- d I = NC¢.,I dz and over a length L I = Ioexp(-NCe~t'L).

More generally there may be a mixture of particle types. In this event if there are N~ particles per unit volume o f t y p e j we have

(6)

Since it is generally true that ~t~ = E~ we can determine the monochromatic emissivity for this case. Thus, the absence of multiple scattering requires ~ ~< 0.63 for z < 1 or ~ < 0.1 for r < 0.1.

78

A.R. JONES

v=

I. 86.,'.(~

0.70

0~

]0

20

30

40

2fro X

(iii) Scattering into the beam of radiation of intensity 1';. previously scattered from other volumes. For simplicity consider a cloud of monodisperse particles of radius a. The power incident on any one particle within dO' is l ) ' n a Z . d O '. We define the fraction of this scattered through the angle ® into dO to be p(®)df~Q ..... 4n. The function p~O) is called the phase function or gain, and represents the ratio of scattered power into the direction (9 to the total scattered power. If there are N particles per unit volume the increase in power in the beam becomes 2 1 dP;. = I';."rta •df~' ~ p(®) dOQsea - N d V.

Fio. 7. Variation of absorption efficiency with particle size for a refractive index typical for coal3 ~ We can see that if the more restrictive criterion applies, multiple scattering would have to be accounted for in nearly all radiating particle clouds. 2.4. Inclusion o f Multiple Scattering We note from the above discussion that multiple scattering can be avoided by making z < 1. Whatever the value of Kext, a sufliciently small distance can be chosen to satisfy this criterion. This fact is used to develop the general equation of radiative transfer in which a balance is performed on an elemental volume as in Fig. 8. Contributions to the change in power come from three sources: (i) Scattering losses dP~ = - I a K , , t d z . d A d O (dA = dx dy). (ii) Emission of thermal radiation. Since we may write with small dz e~ = I - e x p [ - K ~ t n . d z ] "-, K ~ . d z the contribution from thermal radiation is dP, = In. , (T)K,,h~dzdA dO.

= I~' p(®) d f l ' - ~ --~dz dA dO. However, this is the contribution from one direction only. Integrating over all directions, adding the three contributions and noting that dP~ = d l ; d A d ~ results in dl~

-

-

dz

=

Ks~a |

- I : ~ K , , , t + I B ~ ( T ) K ~ b , + - ~ - ~ j , l';.p(O)dO'.

'

This expression is often modified by use of the turbidity and the albedo, the latter being defined by ooo = Ks~a/Ke, , •

Then dla = -I:.+(I-ogo)IB.:..(T)+-j-Z_ ~-r -,,,a~°f , l~-p(O) d~'. (8) The multiple scattering problem revolves around solution of the integral term. Usually numerical methods are employed, as reviewed, for example, by Hottel and Sarofim. s In most realistic situations, such as combustion systems, the surrounding fluid may also contain absorbing and emitting species. These have to be allowed for, and are discussed in detail by Hottel and Sarofim. 8

~

x+dIx

},--j

FIG. 8. Definition of variables for derivation of equation of radiative transfer. (eqn. (18)).

Scattering of electromagnetic radiation in particulate laden fluids 3. SINGLE PARTICLE SCA'VI'ERING THEORY Studies of emission by, and transmission through, particulate clouds depend upon the ability to calculate the scattering properties of individual particles. These determine both the phase function p(®) and the albedo to o in eqn. (8). The same is true if scattering is to used as a diagnostic tool where calculation may be needed of the extinction or scattered intensity as a function of angle. This section is concerned with methods, of producing accurate calculations of scattering properties. If the particles are spherical, or of some other simple shape within a limited set, rigorous theories are available. However. in many practical cases the particles do not have shapes for which solutions are known, and, indeed, may be irregular or highly anisotropic.* In such cases it is necessary to turn to numerical methods.

F o r simplicity we restrict the discussion at this stage to homogeneous isotropic media. In this case and ~ are constants and V. D = e.~oV./~ and V-/~ = ##C'eoH. By taking the curl of the first of Maxwell's equations 9 x 9 x/~ = - V 2 / ~ = ito##o V x H = 602/~#0~:'%/~ or

V:/~ + to2#e'~t0eo/~ = 0.

¢×i:=

- a~;

d2E + ~o2lalaogo~ = 0 dz 2 is

E = E0 e x p [ i t o , ~ ~ ] .

ct

This is equivalent to U l o e x p i k z as on p. 74. Thus the phase velocity of the wave is (W%e'%)-i/2 and in Oacuo

1

v/5=P"

a6 VxH=~--

+ J;

V.B=O

(9)

This is the wave equation. The solution of the onedimensional form of eqn. (9) in Cartesian coordinates, i.e.

3.1. Wave Propaoation In an electromagnetic wave the electric (/~) and magnetic (H) field vectors are related through Maxwell's equations

79

In consequence, the refractive index is*

with We see now that we can write eqn. (9) in the form /#~o is the permeability of the medium and /~o = 4n × 1 0 - 7 H m - 1 is the permeability of free space. This definition retains # = 1 in vacuo. B is the magnetic induction. Similarly ee0 is the permittivity with % -,- (1/36zr) x 1 0 - 9 F m -1 and /5 is the electric induction. J is the current density and (r the conductivity, p is the free charge density, which is zero in dielectrics and very short lived in cond u c t o r s ) 4 F o r applications of interest here we may set p = 0 in practice. Using the complex exponential form of the wave, and allowing for a time variation of the form exp ( - i ( o t k Maxwell's equations become x E = ito#l%fl : x f l = - itoe'%E ;

V . D = O. V "B = O

where e' is the complex permittivity given by e' = e + itr/to~,o.

* It is worth clarifying the term anisotropic as used here. It is common in radiative transfer to refer to isotropic scattering by which it is meant that the scattered intensity is almost independent of angle, as for a very small particle. Anisotropic scattering would then show a strong angular dependence, as for a large particle. However. this is not the usual terminolog~ in wave propagation. As used in this paper, anisotropic implies that the properties of the medium or scatterer are dependent upon direction and, consequently, vary with orientation to the plane of polarization of the wave.

V2ff, + k 2 v 2 g = O.

By a similar process the same equation can be derived for H. The energy per unit volume of an electromagnetic wave is

io~[ £ . ~ , + / ~ . ~ ] where D' = e'eo/~. O n integration over the volume it can be shown 14 that the time mean flux per unit area is I

S= ~Re(E×H*)

where S is the Poynting vector and R e implies taking the real part of the argument. This is commonly interpreted as the intensity of the wave. However, care must be exercised in the use of this vector. This is illustrated by the behaviour of an interference pattern. When the angle between the two light beams is 180 ~ it turns out that S = 0. This is correct because in this situation there is a standing wave and there is * The refractive index and, hence, the phase velocity are complex. However, this is an artefact of choosing the complex exPonential wave description, which in itself is unrealistic but was chosen for mathematical simplicity (p. 74). For a further discussion see the book b? StrattonJ 4

80

A.R. JONES

no flux. But, we know that a standing wave can be detected, for example by a probe in a microwave system or an inclined photographic plate in an optical system. This is because real detectors operate by the absorption of photons, and the rate at which these are absorbed depends upon their density. In this case, it is more appropriate to describe the intensity by 16

I oc IEIz

in which the equation is separable. In order to use the boundary conditions the solution must be available in the form of tangential waves which are expressed in orthogonal functions so that all the coefficients a. and b, can be found by the use of just two equations. These criteria limit the number of possible solutions to six co-ordinate systems: cartesian; circular, elliptic and parabolic cylindrical: spherical; conical. A good discussion of this problem is given by Bowman et al. is

and this is the form which will be used throughout this paper.*

El,in e

x.

3.2. Formulation of the Scattering Problem and Boundary" Conditions The most widely used method of obtaining rigorous solutions to scattering problems is to divide space into homogeneous regions, i.e. the interior and exterior of the scatterer. The wave equation is solved in a co-ordinate system relevant to the particle shape, and the interior and exterior waves are related through the electromagnetic boundary conditions. Analytic solutions of the vector wave equation for homogeneous non-magnetic media are constructed from the scalar Helmoltz equation V2tIj + k2etlj = 0 using the vector relationships

L=VEE;

/~=VxdqJ;

1/V=~VxA~I

where fi is any constant vector of unit length. The field vectors are found from

= - Y (a~

+

bd~.),

n

t/=

-

i O-)~O

Y~ ( a . ~ + b ~ ) ,

I

FIG. 9. Definition of scattering angle and electric field components.

In constructing a solution, account has to be taken of the polarization of the waves. This may be defined relative to the plane of measurement. For example, Fig. 9 shows a detector which moves in the ),z-plane about x-axis. The incident electric field may be resolved into two components parallel and perpendicular to the plane of measurement. In this example, ELi, ¢ = Ex,~.¢ and Eii.ine = Ey.ine. Similarly for the scattered field E~ .... = Ex.~¢~ and Ell,sea lies in the yz-plane and is given by Ell.... = Ey,seaCos 0E ..... sin 0. The incident and scattered fields can then be related via the scattering matrix

n

a. and b. being coetlicients of the expansion. These are solved for with the aid of the electromagnetic boundary conditions which state that the components of the electric and magnetic fields which are tangential to the surface are continuous, that is f i x ( E , - E , ) = O;

measurement

Ell.... ) _ e i k r ( s 2

S3)(EI, inc)

E±.,ca/- ~ - \$4

S1/ \E_iinc / "

We note that for isotropic particles there is no coupling between orthogonal vectors and S 3 = $4 -----0.

h x ( / ~ e - H , ) = O.

where n is a unit vector normal to the surface. The subscripts e and t refer to the external and transmitted (internal) waves respectively. Solutions to the scalar wave equation are obtained by the method of separation of variables. This limits the number of solutions to those co-ordinate systems * It can I~e shown that for a single plane wave S oc [Ela, and the two descriptions are synonymous. However, this is not generally true. Further evidence that iEIz is the correct description comes from the calculations of C h o u 17 which show that this form correctly describes the behaviour of an interference pattern in terms of light scattered by small particles.

3.3.

The Mie Solution for Spheres

The most widely used solution is that for the sphere which was developed independently by Lorenz ~9 and Mie, 2° but is commonly called "Mie Theory". Solutions are sought as expansions in spherical harmonies which must satisfy the conditions (a) The field inside the sphere must be finite at r=0.

(b) The field outside the sphere must satisfy the radiation condition at infinity lim {#xVx/~+ikrff.] r~oc,

=0

Scattering of electromagnetic radiation in particulate laden fluids The appropriate solution is h,(l)(kr), a spherical Bessel function of the third kind, which has the property eikr

lim h,°~(kr) ~ r

r ~

or a simple spherical wave.

The complete forms of M, and N, are given by Stratton, ~4 for example, and from these Eo, E¢, Ho and H , may be constructed for both the internal and scattered waves. The boundary conditions match the internal wave with the total external wave, which is the sum of the incident and scattered waves. There are four equations, one for each of the components Eo, E,, H0 and H,~. There are two coefficients each for the scattered and internal waves, and if the series is terminated after N terms there are a total of 4N unknowns. This difficulty is overcome with the aid of the orthogonality relations for the Legendre and trigonometric functions, which decouple the series leaving four equations and four unknowns for each value of n. The solutions are ,/~(v~)rt.(~)- vn,(v~}~(~) r/'~(v~)~.(~) - vr/.(v~)~'~ (~)

a.

b, = v~'.(v~t)q.(~t)-~/.(wt)~/~(~t) where ~/. = ~tj. and ~. = ~h~ and the prime denotes differentiation w i t h respect to the argument, ct - 2hal2 is the particle size parameter. The elements of the scattering matrix are

S,(O) = ~

2n+__.___~l[a,,n.(cosO)+b.r.(cosO)] .=, n(n+ 1) ~.

$2 (0) =

n=l

2n+ 1

[b.~.(cos O)+,:,;.(cos 0)]

n(n + 1------)

with z,(cos 0) = ~

1

P2 (cos 0)

d l r,(cos 0) = ~ P, (cos 0). The efficiency factors are given by 2 = Q~a = ~ . ~ x (2n+ 1){[a.I 2 + ]b.I2} 2 Q~, = ~ - ~ , (2n+ 1)Re{a.+b.}. The Bessel functions and Legendre polynomials are calculated by recurrence relations? ' The series converge rapidly for small particles, but for large particles convergence is slow and computation can be costly. A recent paper by Verner z2 has developed recurrence relations for the Mie coefficients themselves, and holds out the promise of speeding up these calculations. Discussions of scattering by spheres can be found. for example, in the books of Born and Wolf) 3

81

Kerker ~ and Stratton. '4 Kerker also discusses concentric spheres and spheres with shells of inhomogeneous material, The latter is relevant, for example, to oxidizing metals, and a recent discussion of this problem is given by Prishivalko et al. 23 Other interesting solutions include that for two spheres. 24.25 3,4. Non-Spherical Particles The majority of particles are, in reality, not spherical. Examples include agglomerated soot, ground coal, ash, solid metal oxides and so on. In these cases either an equivalent sphere must be assumed, or some other solution relevant to the true shape must be found. There are some coordinate systems in which solutions are possible for certain limited configurations, For example, Schultz 26 gives a solution for a prolate spheroid at nose-on incidence (where there is circular symmetry). However, generally rigorous solutions cannot be found and approximations or numerical methods must be used. Many numerical techniques are available, some based on the integral formulation discussed in the next section. Others assume that the field solutions can be written down as a convergent series such that summation can be terminated at some term giving a finite number of unknown coefficients. By choosing a sufficient number of points on the surface a set of equations can be produced via the boundary conditions to obtain a solution. This is the so-called -. "point matching method". However, problems arise because generally the series do not converge, For example, a series of spherical functions does not converge if the scattering object is far from spherical in shape. The problems can be overcome to some extent by allowing the series to be finite but only finding a "best fit" as, for example, by the method of least squares. Another approach is the extended boundary condition method in which integration is carried out over the surface of the scatterer. These methods have recently been surveyed by Bates, '1 A popular simple shape, because of its flexibility, is the ellipsoid. For example, an oblate spheroid can approximate a disc (certain metal oxides) and a prolate spheroid can represent a long cylinder (say long chain soot agglomerate). A numerical method for spheroidal particles has been developed by Asano and Yamamoto. zv They produce solutions in the form of spheroidal wave functions and "solve" the boundary conditions by expanding in terms of Legendre functions, which are orthogonal. In this way a series of coupled equations are derived which can be solved for the expansion coefficients, 3.5. Integral Formulation of the

Scattering Problem An alternative approach to the scattering problem is to consider the whole of space and allow it to be inhomogeneous, so that the particle is represented by a change in the refractive index of the medium.

82

A.R. JONES

Assuming only non-magnetic media, if the derivation of the wave equation is carried through from Maxwelrs equations one obtains

unknown coefficients.3°-3a These methods are also reviewed by Bates. ~t 4. APPROXIMATE SCA'ITERING THEORIES

V2/~ + k2e ,/~,+ ~ ( 1 Vg •/~) = 0 We have seen that the number of scattering geometries for which rigorous solutions are available are severely limited, and even then they are complicated and time consuming. Thus approximations are needed. The methods available occur at the extremities of size and refractive index.

which may be rearranged into the form (V2 + k~)/~ = f where

4.1. Rayleigh Scattering (~ ~ 1, ~(v- 1[ ~ 1) We note that a scalar equation of the type

Since the particle is very small compared to the wavelength (ct ~ 1) and the phase change is small (~lv- II '~ 1), it is illuminated effectively by a uniform field and oscillates as a dipole. The polarization is proportional to the incident field, or

(V2 +k2)qj = f has the solution

. = v,- f/ dV

p = yEi, ~

where W~ is a solution of

where ~, is the polarizability. The scattered field (see p. 80) is given by

(V2 +k2)~e~ = 0 and G is the scalar Green's function which is a solution of (V2 + k2)G = 6, being a Dirac delta function. (See Morse & Feshbach. 28) The solution of the vector wave equation was derived by Saxon ~2 in terms of the tensor Green's function, and is

E±.~<~/= - ~ - " k o Y t o

1/\Ej_.o j

provided the particle is isotropic. Then, for the intensities we have lit .... = / ~ r ~ k~1712COS2 0,

Elf) = g,.o (~) I1 .... = ~ - ~ k~lTI2

+

and for unpolarized light where P is the Green's function. ~: and ~:' are exterior and interior points, and integration is performed over the volume of the scatterer. The form of the Green's function is essentially a spherical wave emanating from a point source. Consequently this equation simply states that E is the incident field plus a superposition of the fields created at each point r' by sources of strength

kl[~ ' (~')- 1]g(~'). Ultimately, amplitude

Saxon

derives

for

the

scattered

1o i Isc, = ~ . ~ ( 1 +cos20)k4JT[2. The form of these curves is shown in Fig. 10. In Section 4.2., we see that the Rayleigh approximation is valid to within 10% for ~lvl ~< 0.6. A recent paper by Kerker et al. 3s has examined the limits in detail by comparison with Mie theory. These vary with conditions, but roughly ~lvJ ~< 0.2 for validity within o/ 1/o'

The cross-sections are e ikr

g~o,(e) = [ D - (,~'. O ) n ' ] ' - r

where ri' is a unit vector along the direction of scattering, and -

k2 I .

D(n') = -~n

e - a " e [ g (r')-- 1]/~(~')d V'. dv

As they stand, these equations are rigorous. Unfortunately, the internal field E(r') is not known in advance. Thus, again, solution can only be obtained through approximation or numerical methods. The latter can be iterative but there are strict convergence requirements, z9 Other numerical techniques involve expressing the fields in terms of currents in the surface, or over the surface of elemental volumes, ultimately leading to a series of linear equations with

8n 4 2 Cs¢, = -~-koh, I ;

C, bs = 4nkoRe(i7).

(10)

The most common shape assumed is that of the sphere, for which

3V v z - 1 7--- 4re v2+2 where V is the volume of the particle. Examination of eqn, (10) suggests that since ~z <~ 1 if absorption is present it will tend to dominate. This is shown, for example, in Table 2 where calculations were made using rigorous Mie theory for particles of refractive index v = 2 + 1, which is typical for SOOt. 36 Since even large agglomerates of soot rarely exceed 0.1 tam in size, we expect that in the near i.r. with 20 > l Iam that :t < 0.3 and Q~,,~> 35Qs~.,. In fact,

Scattering of electromagnetic radiation in particulate laden fluids

,~3

I±

o H

•.

0°

I

90 °

180 °

~ngle

FIG. 10. Typical polar diagrams of scattered intensity versus angle for a Rayleigh particle.

soot particles are usually much smaller, so that for thermal radiation it is valid to write ~--- 1 - e x p ( - ~ K ~ b . , d z ) with 3c

/v 2 -- 1 \

where c is the mass concentration, p is the density of one particle, and I m implies taking the imaginary component of the argument. A good approximation to the total emissivity can be found by replacing 20 in eqn. (11) with radiation mean wavelength defined by 2~T = 4107 lain K. This leads to 3~ e = 1 - exp(--.fB'Tcdz) where B' is a constant. Alternatively, eqn. (5) can be used. For 2oT ,~ c 2 the PIanck function can be approximated to the Wien equation EB., ( T

~- c l Z o S e x p ( - c 2 / 2 o T )

In practical systems, the presence of radiating gases must be allowed for. Perhaps the simplest means of achieving this is by fitting measured data to a series, as, for example, in the work of Taylor and Foster) 9 If the particles are non-spherical but still fall in the Rayleigh region then they may be treated simply as ellipsoids. Here the polarizability is anisotropic and is given relative to the axes V v2 - 1 ~J = 4zr 1 + ( v 2 - 1 ) L j

j = 1,2,3.

Each Lj is an elliptic integral and they are related by L1+L2+L

3 = 1.

The specific form of L~ is giveri by Kerker. 5 In certain cases there is considerable simplification, as given in Table 3. For a cloud summation should be performed taking into account the orientation of each particle. However, if they are randomly aligned the arithmetic mean may be used, i.e. 1

and use of this leads to

7 = ~(71--Y2+)'3) •

= 1 - 1/( + ~cL,,, TIc 2 )a

where L~ is an appropriate beam length, r is constant suitable to the application3 s TABLE

2. Comparison of Qab, and Q~a for soot using Mie theory.

0.1 0.2 0.3 0.4 0.5

905 115 35 16 9

Using this approximation, Jones 4° has estimated the possible effects of chain-like agglomeration on the emissivity of soot particles, by comparing spheres and prolate spheroids of the same volume. For a typical distribution of elongations taken from the work of Medalia and Heckman 41 and refractive indices from Foster and Howarth 36 the results in Table 4 were determined. In some cases very significant increases in emissivity could occur. Further, on average these particles were only one and a half times as long as wide. Increasing this to only a ratio of 2:1 would increase these percentages by a factor of three-four.

84

A.R. JONES TABLE3. Value of L parameters for Rayleigh scattering by ellipsoids. The axes of the ellipsoid are a. b. c Type

Relationship of Axes

L~

L:

L3

Prolate spheroid

a > b, b = c

l--e'-~½1n[q+e~--I t e~[ \ 1- e /

~" ( I - L , )

½(1-L~)

Oblate spheroid

a < b, b = c

½(1-L1)

½(I-L~)

Flat elliptical disc

b ,> a, c >>a

1

0

0

Long elliptical cylinder

b ~ a, c .~ a

0

Long circular cylinder

b = c, c ~ a

0

--

1-

arctanJ

c

b

b+c

b+c

e = eccentricity; e2 = _f2. Ellipsoids will not necessarily be randomly aligned. Orientation may occur either due to flow gradients or applied fields. There will then be a distribution of alignments and integration must be performed over this. '~2-4.5 4. Percentage relative increase in mean absorption coefficient for agglomeration into chains, approximated as prolate spheroids, as compared to spherical particles

and obtained solutions for ellipsoids in the form of a power series in the ratio of size to wavelength. This solution appears to be limited to kb <%0.95 where b is half the largest dimension. 4.4. Rayleigh-Gans-Debye (RGD) Approximation (Iv-11 ~ 1, ~ l v - I I ,~ 1)

TABLE

2o(~tm)

v

1

Per cent increase

1.6+i0.8

4

Soot

5 10

2.0 + i 1.0 2.2 + i 2.6

13 47

Graphite

--

3.0 + i 2.0

35

4.2.

Approximation of the Mie Series

By expansion of the Mie series in polynomial form, trucation for small particles yields ~ 8 4. //V2

1\//

3v2-2

2

) z,

( Ell.... t \E,.~¢~/

( / v 2 - 1"~

Q~b, = -4~'m~v~--~+ 2) x[1

•

/'v2--1\/v4.+27vZ+38\ 2

This method is not generally applicable to particles suspended in gas streams because of the restriction I v - I I ~ 1. It is most usually applied to polymers in solution, or particles suspended in refractive index matching liquids. However, since it is a very popular method, and has some relevance as it has been used to study carbon blacks suspended in oils, 47 it is included here for completeness. Since the scatterers are weak, it is assumed that there is no reflection ( I v - II ,~ 1) and no added phase shift or absorption (air-11
....]}.

= e(' '°-"i k ~ 7

r

|e'OdV "(;°sO 0~(El,,,c~,_ & lJ \EL.I,¢J

where 6 is the phase at the detector of light scattered by dV. (Note that since I v - 11 ,~ 1,

3dV/v2-,

The formulae suggest that Rayleigh theory is valid to within 10% provided ~lvl < 0.6.

v-,

7 = q-~- \~w~+2 / - Z~-~ dr.) It is customary to write

4.3.

Stevenson's Approximation

The Rayleigh approximation assumes a uniform field, or essentially 2o-*oc(ko---,0), and can be thought of as a solution of Laplace's equation

R(0,4)) = l

i

v.k

e'OdV

whence III..... (

lok2V ]v-I i2 ,.,,,~

:/cosO

V2/~ = O.

Stevenson 46 extended this concept by developing the electric field in the form

= ~ ff~(ik)" m

4.5.

Very Large Particles (~t >>1, :tlv-

11 ~ 1)

This is the limit 2 o --.0(k 0---, zc). Here the interaction of radiation with particles can be described

Scattering of electromagnetic radiation in particulate laden fluids in terms of the reflection and refraction of rays; geometrical optics. A wave effect also takes place on the cross-sectional area of an obstacle, which is diffraction. Thus a full light scattering description for large particles can be obtained by adding together the results of geometrical optics and diffraction theory. This approximation holds well generally for :t > 20, except for rays near grazing incidence or at angles associated with the rainbow where much larger values of ~t would be required. A detailed discussion can be found in van de Hulst. 6 Shipley and Weinman "8 have recently compared various approximations with rigorous Mie theory for values of ~ in the range 200 ~< a ~< 4520. Since diffraction forms the basis of important methods of particle sizing we shall discuss it in some detail. Scattering measurements are invariably made in the far field, and we are concerned here with Fraunhofer diffraction. We have seen (p. 75) that this is synonymous with scattering, and, indeed, rigorous diffraction theory is just scattering theory. In this, the full vector nature of the wave is taken into account. However, it can be shown t3 that for unpolarized waves and a sufficiently large obstacle that a scalar theory is adequate. The simplest approach to scalar diffraction is through the Huygen's principle, in which at any point in space secondary wavelets are produced which are proportional to the incident wave. Thus, if we consider the aperture in Fig. 11 the spherical wave from the point Po at the point Q has the form elkOr

Wi,~ = Wo"

r

and the element of the wave arriving at the point P from a small area dS around Q is eikor

dWp = t P o - - . r

eikos

K(z )

S

"dS

where K(7~) is an inclination factor allowing for variation of amplitude of the secondary wavelet with direction. The total amplitude at P is just the integral over the aperture, i.e. ~PP = qJo

s

K ( Z ) - sr

dS

In light scattering, the incident wave is taken to be plane and the diffraction problem is as illustrated in Fig. 12. The phase of the incident wave is constant over the aperture and is taken to be zero. As s and s' in Fig. 11 become very large compared to the aperture (Fraunhofer diffraction) it is found that s can be approximated as simply s' plus a phase difference relative to s'. Also the only deflection is 0 so that the diffraction integral becomes tfl p ... Ut~0 -eik°s' i i s K lOjei~.,v, sino . . . . . s'

// //

,o// FKs. 11. Definition of variables for derivation of basic diffraction equation. The result of this integration yields an intensity variation with angle of the form [ 2J l (koa sin O) I2

i = iv.i 2

[ ;o S .O

The question arises as to the correct form of K(O). Normally, this is neglected in the far field for small angles where K(O) = 1. However, for small apertures 0 may be large and this correction term would be important. To derive a form for K(O) we return to Huygen's principle and recall that each point on the incident wavefront acts as a point source of a spherical wavelet. To this extent the point source may be considered as a Rayleigh scatterer, for which we already know that (p. 82) Isca ~c ½(I +cos 2 0) in unpolarized light. The diffraction integral may then be thought of as the sum of the amplitudes from all such sources, and is akin to the RayleighG a n s - D e b y e approach. The presence of actual scatterers is unnecessary to this argument. The angular variation for the two incident polarizations arises purely from vector analysis if a wave changes direction. We therefore write l w ' L~~2d'{k°asin0] 2.½(l +cos., 0) It will be seen later that incorporation of the inclination factor makes diffraction theory quite accurate for remarkably small particles, although we note that the strict requirements are 2 ,~ a ,¢ s. A more detailed approach to diffraction is via the scalar wave equation and the Fresnel-Kirchhoff integral. 13 However, we note that purely scalar considerations will not result in the correct form of Ig(0)l 2 which arises from the incoherent addition of the two vector polarizations. 4.6. Anomalous Diffraction (a ,> 1, I v - 11 '~ 1)

dS.

The diffraction integral, eqn. (12), may be written eikos' ff2~ [%c

]

{JK O)I-

.

For a circular aperture dS = p dp dO and we have Wv = W o

~5

K(Olelk . . . . . . . ,. . . . . pdpd4).

(12)

%= *o~, Jo Jo K(O)T(P'4')e'~°°"""°o-'*pdpdrh

86

A.R. JONES

where T(p,~b) is the aperture or pupil function. Normally, the reflected and refracted rays are strongly deflected and, since the main diffraction occurs over a very small angular range close to 0 = 0 °, they do not interfere with the diffraction pattern. Then, for the simple circular aperture T(p,~b) = 0 for p ~< a and T(p,~b) = 1 for p > a. To avoid integration to infinity, we make use of Babinet's principle which states that complementary screens have identical diffraction patterns apart from a phase change. A complementary screen has an aperture function Tc = 1 - T, so that for the circular aperture Tc(p,~b)=l for p<~a and To(o,40=0 for p>a. If the refractive index is close to unity, refraction and reflection are weak and a wave can be almost directly transmitted by the particle. The aperture function then becomes

Tc =

T = eik°(v-l)t;

;

T=I

I - - e ik°(~'-l)L

p

<~a

p>a

To=0

where L is the path length through the sphere. The range of validity of the anomalous diffraction approximation has been explored by Farone and Robinson. .9 4.7. Integral Equation Methodn We saw on p. 82 that the scattered wave can be represented rigorously in terms of an integral which contains the unknown internal field. For refractive indices close to unity, it is reasonable to suppose that the incident wave propagates almost undisturbed and E ~ Ei, ~ =/~o ea°~. This is the Born approximation, but it will be recognized as the RGD approximation (Section 4.4.). The modified Born approximation is similar but puts E = ff,oe~°'. Higher order approximations are written in series form as: = Z J

v2JEj

Born approximation.

E = Eo e x p { ~ v2J~bit

Rytov approximation.

The range of validity of these approximations is said to be Born

I v - l t ,~{ I;

~ l v - l l ,~i 1

Rytov

I v - l l ' ~ 1;

~ l v - l l 2,~ 1.

The internal field can be estimated from geometrical optics. The wave is taken to propagate rectilinearly but the phase is not uniform and is governed by the shape of the object. This is the Wentzel-Kramers-Brillouin WKB approximation. A sophisticated version of this is to use the internal field predicted by the Mie theory for a sphere of similar size to the particle of interest. These methods are reviewed by Saxon ~2 and Bates. at Kerker et al. 5° have used the integral approach to tackle the problem of an inhomogeneous sphere. The internal field was taken to be that of a homogeneous

sphere with permittivity defined by ~-1 ~+2

=' (*')-'- dv. V 3v~(r')+2

Comparisons were made with rigorous theory for two concentric spheres, and good agreement was found provided the two permittivities were not too different and the sphere was not too large. 5. PRINCIPLES O F DIAGNOSTIC A P P L I C A T I O N S O F SCATTERING

The properties which govern scattering, and may in principle be measured by its use, are particle size distribution, concentration, refractive index, shape and orientation. The techniques chosen depend upon the size range of the particle, but we note that the governing factor is 19/2 so that in principle changing the wavelength can always bring a particle within any regime. This may not always be practicable, and visible wavelengths are usually the most convenient. Some of the methods given here are discussed in greater detail by van de Hulst 6 and Kerker 5. They are mentioned briefly in this paper to bring them to the reader's attention, especially where no other methods are given for a certain size range. However, this section is mainly concerned with developments since 1969. 5.1. Rayleigh Scatterers (0 <<,~lvt <~ 0.6) The Rayleigh scattering pattern is independent of size, so its shape is not useful. We can, however, measure the absolute scattered intensity and the extinction of the particle cloud. We saw on p. 82 that

K~a~NV 2 ; Kabs~NV where V is the volume of the particle, and that for absorbing particles Kext = K ~ . In that case measurement of extinction yields N V and the absolute scattered intensity gives N V2. Both N and V are determined. If the particles are non-absorbing the situation is

A+s I0(o,o,o) l

I I

FtG. 12. Definition of variables for Fraunhofer diffraction by an aperture.

Scattering of electromagnetic radiation in particulate laden fluids

87

I0 I

"g ~Z

05

o tn aO

Ol0o

i 30 °

I 6 O*

"~'.-a ~ 9 O*

,

I 120 °

1 t50*

I 1800

Angle

(2ha~2=

FIG. 13. Scattering polar diagram for a particle slightly larger than a Rayleigh scatterer.

more difficult. Here K,,, = KscaaNV 2. In order to measure NV, van de Hulst recommends measuring the composite refractive index of the medium given by ,,'=

5.2.

Small Particles

0.5, v = 1.33).

(0.6 < ~lvl < 5)

As the particle size increases the scattering polar diagram changes, becoming asymmetric with more forward than backward scattering. This is illustrated in Fig. 13. If the particles are sufficiently small, there is a unique relationship between the extent of this asymmetry and the size. Commonly, the ratio of the light intensities is measured at two symmetrical

3v2-1

1 + ~Nv.

The methods are discussed in detail by van de Hulst. 6

133

200

175 /~; ÷ ~ 0.5

~///5

ii'[g/~',20 + ~I0 123

/

~

i

J

~

2

"WO,

k FIG. 14. Variation of disymmetry factor (D = I , , ~45:),l,c~(1351 with particle size and refractive index,

88

A . R . JONES

angles. Usually these are 0 = 45 ° and 135 :. The ratio is termed the disymmetry 5 D = I~(45°)/I~(135

°) >~ I.

Other ratio methods include the polarization ratio 111/1± at 90 °, which is zero for very small particles and rises with size, and the dispersion quotient which is the ratio of the extinction coefficients at two wavelengths.5 These methods are simple to use, but only apply over limited size ranges. Below some minimum, the ratios are independent of size and above the maximum they are no longer monotonic functions and do not, therefore, yield unique values of size. They also depend upon refractive index (see Fig. 14 for disymmetry, for example) and particle shape, both of which are often unknown. Concentrations of particles can be determined by measurement of the absolute scattered intensity.

d A

-2

-3

tl II

II

II Ii If

-4

0•

1 30 °

60 °

90"

1200

150"

I 180"

Angle

FIG.15(b).

5.3. Intermediate Particles (1 < =lvl < 50) As the particle size increases further a lobe structure begins to appear in the polar diagram, the number of lobes being of the order ~lvl. This is seen in Fig. 15. For particles in this size range it is possible to measure size by comparison of theoretical and experimental polar diagrams. This can only be done if the refractive index is known and the scattering polar diagram does not change during the time of measurement. The latter requirement implies that the particle size distribution must remain constant so that the particles do not vary in size and are either stationary or present in sufficient concentration that there is always a representative sample in the test space. These methods are reviewed by Kerker s up to 1969.

Two recent interesting light scattering photometers are based on the optical properties of ellipses 52 and ellipsoids: 3 The latter is illustrated in Fig. 16. Part of the surface of the ellipsoid is a mirror in the form of a circular ribbon. Light scattered by particles at one focus is reflected to the second focus at which is located the detector. The scattering angle is selected by a rotating disc with a small aperture. In this way a complete polar diagram can be displayed

I [L,t,=I -i i

3 3 4"~. 0 10

. . . . .

l

0~

I~ -

~bL

j t!,I i!',tl i! ~V

§

lll~

+ \\

v I,",,

-4.

8'

-5 0°

-4

i 0°

30 °

I

J

6© °

90 °

Anqle FIG. 15(a).

I t20 =

] {50 °

i 180 °

I 30 °

L 69 °

910°

20 °

150 °

180 °

Angle FXG.15(C). FIG. 15. Scattering polar diagrams for larger particles: (a) 2hal2--5, v = 1.33; (b) 2~a//I = I0, v = 1.33; (c) 2~a/~. = 20, v = 1.33 and v = 1.33+/0.10.

Scattering of electromagnetic radiation in particulate laden fluids

89

practice covers a finite region and one can write Qca(Oj} = V Z WpNpl~c~.p(Oj) p

where Wp is a weight factor, Np and ]~:a,p are the number of particles per unit volume and the intensity scattered by particles of size ap. Measurements are made at a number of angles 0j producing a set of simultaneous equations to solve for the coefficients Np.

One of the simplest approaches to handling the data is the method of least squares 55 in which

~t i/ F12~

/x

\\\

//

\

/ \

/ /

/

/

FIG. 16. The Gucker photometer. L--Laser beam: M mirror which is section of ellipsoid: F1, F2--foci of ellipsoid: S I, S2--scattered rays: D--rotating disc: A aperture.

is minimized. Phillips 56 and TwomeyJ" and Backus and Gilbert 5a have developed sophisticated versions of this technique. Both allow for the fact that each Isc,(O~) has an error ej associated with it. The Phillips-Twomey method is a matrix inversion in which restraints are placed on

E ~j2. J

on an oscilloscope in 20ms. 54 This enables measurements of aerosols which are changing in time, provided their time constant is greater than about lOOms.

The Backus-Gilbert technique makes use of the property of the Dirac delta function that

N(a') = f o

N(a)~(a-a')da

arld constructs coefficients such that 5.4.

Particle Size Distribution ~ apWpl~,.p ~ (~(a--a').

Essentially there are three approaches to the P determination of particle size distribution: Chow and Tien 59 have recently looked at these {a} Counting. In this case the particles are measured one at a time so that eventually a methods and concluded that the Phillips-Twomey representative size distribution is built. One such approach is the more powerful. Post 6° also concludes that the Backus-Gilbert method has severe method will be discussed later. (Section 5.5.) (b) Assumption of size distribution. Here a repre- limitations. sentative distribution function is supposed; typically 5.5. Large Particles (a ~ 10} the log-normal or Rosin-Rammler distributions. A search is then enacted to compare measured scatterWe have seen that the number of lobes in ing parameters with theoretical curves to find a best scattering polar diagrams is of the order ctlvt. As the fit to establish the characteristics of the distribution size increases, the pattern becomes more difficult to (e.g. mean size and standard deviation). Possible record accurately and one begins to search for measurements include scattering polar diagrams, 5"~2 and polarization ratio versus angle and wavelength simpler methods. For very large particles (7 >~ 1OO) photography variation of extinction.~ can be applied without too much difficulty, but again A disadvantage of this method is that some prior • increasing sophistication is required as the size is knowledge of the form of the distribution is required. reduced. Some of the problems associated with I~ would not readily reproduce a bimodal distriphotography are due to lenses and these can be bution, for example. avoided with the use of holography. 61 With this {c) Direct inversion of scattering data. Here the size distribution is recovered without prior assump- technique, stationary particles down to 2~m and moving particles down to 5pm have been recontions. We recall that structed. Several particle sizing methods have been reviewed by Jones 62 with special reference to high I~{0)= I " 0 N{a)l~,~{O.a)da concentrations. He concludes that in such circumstances photography is preferable to holography. Holography and photography have one feature in where I is the test volume and is constant. The exercise is to find NIa}. Usually. the distribution in common: at some stage the image must be analysed. JPE{", \ o l 5 Xo -" B

90

A.R. JONES

There are sophisticated techniques to do this but they are not necessarily convenient or economic. Also they are frozen images and we may wish real time information. Lastly, they cannot yield optical properties. Thus other optical techniques are still required in this area. A promising method is the use of the Fraunhofer diffraction pattern (Section 4.5.), which is measured in the focal plane of a lens as in Fig. 4(b). It is found from Mie theory for spheres that for a wide range of refractive indices the intensity may change but the shape of the forward scattered lobe is approximately

However, for large real refractive indices large variations in the ratio arise for ~ > 3. Other size ranges can be covered by changing the angles. There are also methods based on integration of the central maximum over the angular range 0 to 0. Allowing for interference from unscattered light, the result can be expressed in the form R = 1-

t'"~2J~l~sinOl[".sinOdO ~0

|

~sin0

= ½[1 + J02(:~ sin 0 ) + J l -'(z~sin (9)].

io

08

°o

",,o ~ x \o ~\

06

o

• u=

~s ro

133

xu=150

o

Vo=

,~ 0 . 4

\

I 75

/3 t,= 2 0 0

,, ~ ~ . n ~ \ ~0 ×°

v

~,

o

OO= 1 5 0 +

~ 050

[30= 200+

L. I 0 0

&

×

2-

0

I

I

i

I

2

3

N 4

I 5

2~a X

FIG. 17. Variation of ratio of forward scattered intensities (l~a(30°)/l~,~(lY)withparticlesizeandrefractive index.

only dependent upon size. Further the shape can be expressed in terms of the diffraction theory down to quite small sizes of particle. Jones 63 has calculated error charts from a comparison of the Mie and diffraction theories. For non-absorbing particles discrepancies are less than 20% for ~ > 20 and v t > 1.3. At visible wavelengths this corresponds to D > 3 lam. Errors are large at refractive indices close to unity even for large particles as a result of anomalous diffraction (Section 4.6.). For absorbing particles errors greater than 20% are only seen if < 8 (D < 1 lam in the visible) and v 1 > 1.5. Here anomalous diffraction is absent because the internal wave is absorbed. A number of techniques have been explored. 5 The simplest of these are methods which measure the ratio of the scattered intensities at two angles within the forward scattered lobe. As an example, Fig. 17 shows the ratio of the scattered intensities at the angle 30 ° and 15° . For low refractive index and absorbing particles the ratio can be used up to ~ ~ 6.

Desai and Vaidya 64 point to truncation errors which arise with this method if the maximum value of 0 is too small. These lead to broadening of the resulting size distribution. They also conclude that the integration technique is not satisfactory for narrow distributions. Fitzgerald 65 has calculated correction factors to allow for this error for spheres of water in the size range 3 < ~ < 200. Recently Swithenbank et al.66 have developed an instrument ~based on the integration technique. The scattered light is measured over concentric rings of equal area and plots are made of the received energy against ring number. A maximum occurs at a location characteristic of the mean size in the particle cloud. The rings are made of photodetector material so that the distribution can be recovered directly. The result is fed to a computer which inverts the data to fit a R o s i n - R a m m l e r distribution, and prints the result. By changes of focal length of the Fourier transform lens, sizes in the range 5-5001am can be measured.

Scattering of electromagnetic radiation in particulate laden fluids Apart from its dynamic range, this method has a number of advantages. It is insensitive both to motion and refractive index. It will measure fairly dense systems provided a light beam will penetrate, but it is not clear what effects multiple scattering may have. Among its disadvantages is that it will not give a point measurement but integrates along the light beam. Further, there must always be a representative sample within the beam, implying a minimum in the product of concentration and beam area. One either sacrifices resolution or minimum concentration. Swithenback et al. also make the point that particle velocity and, hence, residence time, may be functions of velocity. This would bias the measured distribution, and knowledge of the velocities would be advantageous. Particle velocities can be measured optically by making use of the Doppler effect.6~-73 A particularly simple device is the "fringe anemometer" in which two laser beams cross at an angle 27 as in Fig. 18. This produces an interference pattern having the fringe spacing 2:

~,~.l[

~1

nclde n t lose r

•

beam

space

Scattered wave

FIG. 18. Principle of the fringe anemometer. Thus, the frequency of the scattered light in a fringe anemometer yields the velocity of the scatterer. However, the amplitude depends upon the particle size. The general form of the scattered signal may be written ls~, = A + B cos 2rrfi and several authors have been exploring either the measurement of the mean intensity (A) or the visibility (B/A), the lattcr having the advantage of being a relative value. It can be shown that for large spherical particles the visibility in the extinction should have the form~4. 7s

= 2/2 sin 7

As a particle crosses this fringe system it scatters light with an oscillatory component of intensity, the frequency of which is governed by the time taken to traverse one fringe, i.e.

V = 2 Jl(k:a)

kya

f = u,}.:. iO Aperture

size

a

0.4 °

b c d

4.0 ° 8.0 ° 12.0 °

e

Vext

----

2d,

(k~ a)

kf a

>

¢ c

2

4

6 x

FIc; 19. Comparison of scattered signal visibilit3 at 0 = 0 with I = 2J~Ik raJ kt-a collection aperture.

as

a function of size of

92

A.R. JONES

~0

(a)

TABLE 5. Variation of dynamic range of light scattering in interference pattern for particle sizing Fringe spacing (2s)

Particle size range (p.tmt

12 48 120

1-I0 8-48 20-120

o o

>~ o 5

O0

I

2.0

30

40

50

60

7.0

X 25~

'(b)

{c)

20 %

aperture. 7s's2 For 2 = 0.488rtm the following estimates have been given 79 for variation of fringe spacing as in Table 5. Using the alternative of mean scattered intensity. Yule et al. s3 have claimed a linear relationship with size over a wide range for transparent particles, as indicated in Fig. 21. However. the discrepancy for absorbing particles suggests a strong dependence on refractive index, whereas the visibility method at small angles and apertures is not.

~5 N o n - absorbing

I0 ........

Absorbing

5 0

0.2 0 4

06

Vsca

08

I0

I0

2.0

30

40

5.0

60

6

-

2o(M,m)

FIG. 20. (a) Calculated visibility as a function of particle size. 2/2 s = 0.036, 0 = 3.6°. The two lines encompass the extremes which arise due to varying the refractive index. (b) Visibility histogram for glass ballotini. (c) Measured size distribution; bars from light scattering, histogram from optical microscope.

where k s = 27za/2 I. For small particles a wave theory is needed and several authors have tackled the problem. 1%r6-Ts Results yield the visibility in the scattered light at any required angle. Since forward scattering for large particles is very strong, it would be expected that a detector looking in this direction with a very large aperture would collect nearly all the light, and the result would approximate to that for total scattering. This, indeed, is found to be the case from both diffraction 79 and full wave theory, s° This is illustrated in Fig. 19. Hong and Jones s~ have devised a method of obtaining particle size distributions directly by counting. Particles on the size range 1-101am were measured both in cold gas streams and in flame. The technique can be understood with reference to Fig. 20. Using rigorous wave theory, computer calculations are made of the scattered visibility V~¢,versus particle size at the required angle of measurement over a wide range of refractive indices (Fig. 20(a)). The fringe spacing was selected so that the entire size distribution was encompassed before the first minimum in K¢,. As the particles traversed the test space the scattered signals were recorded and a histogram of K,~ against n u m b e r was developed (Fig. 20(b)). Using Fig. 20(a), this was converted directly into a size distribution. The dynamic range of this method is adjusted via the fringe spacing, angle of collection and detector

IJ°-2 -

l

I

i oo

20o 2o(~.m )

FIG. 21. Variation of mean scattered intensity from a particle in a laser fringe anemometer with size• (After Yule eta/. s3 )

Contrary to the diffraction method, the fringe anemometer techniques give almost point resolution and can measure velocity and size simultaneously. However, only one particle at a time can be tolerated in the test space and this limits the maximum concentration to about 10 ~z m -3. There is, in theory, no lower limit. 5.6. Refractive I n d e x The light scattering properties of particles generally depend on size and refractive index. If the latter is known, then size can be determined unambiguously. If it is not known, then size can only be found with confidence if the method used is insensitive to refractive index. However, it should be possible to use scattering to measure both these properties. This would be particularly valuable in hostile or remote environments. For example, Willis s4 measured the refractive indices of coal by a light scattering method. When cold, he obtained v = 1.81+i0.996 which is in

Scattering of electromagnetic radiation in particulate laden fluids reasonable agreement with other methods, is but in a flame at 170 K he measured v = 2.16 +.i 0.605. Another example is provided by flame soot. The refractive index is known to vary with carbon-hydrogen ratio of the material. 36 Although there is evidence that the refractive index is not strongly temperature dependent 85'86 the composition in a flame is uncertain owing to the presence of additional hydrocarbon compounds which are associated with the soot to an unknown degree. 8v The effective refractive index of soot in a flame may also be determined by the form of agglomeration. Graham ss has suggested that if voids are present v2 - 1

~c-'2 1

Vs+2

O v2c+2

provided )~/2nlv~l > void size. Here v, is the refractive index of the sample, vc that for individual particles and 0 is the volume fraction of particles. The application to sensing atmospheric particulates and pollutants is also apparent. Faxvog 89 has shown that distinctions can be made between absorbing and non-absorbing particles. At visible wavelengths, the former have maximum scattered intensity per unit mass at a size of about 0.2btm, whereas the latter have this maximum between 0.4 and I B m Airborne light scattering photometers have been used by Gibson 9° and by Grams et al. 9' In the latter case the refractive indices of atmospheric soil particles were measured. Equations and calculations for hemi-spherical backscatter have been given by Ch~,lek e t al., 92 and reviews are given by Byer, 93 Cadle and Grams, 94 and Hinkley. 9~ • ,,Eiden has suggested two methods for obtaining s~ze and refractive index. The first of these 96 was the measurement of the degree of polarization, the ratio of the axes of the polarization ellipse and the angle between the plane of this ellipse and the plane of measurement• All these values have the advantage of being independent of concentration. Hunt and Huffman °" have described an instrument capable of measuring all the polarization properties of the scattered light which uses an electrically controlled birefringent plate to vary the incident beam polarization. In his second paper. 9s Eiden expands the Mie coefficients in the form of Fourier series. These can readily be inverted to yield size and refractive index. Crystalline materials are anisotropic, and they must either be treated as such (Section 5.8.) or ways must be found of averaging to obtain an equivalent isotropic refractive index. Barnett and Simon, 99 for example, found that using the arithmetic mean over the three axes can yield good agreement between experiment and theory. Recently. Wang and Greenberg ~°° have applied microwave scattering to this problem and constructed spheres of layers of different materials• They suggest that where there are two characteristic permittivities, as in this example.

93

the permittivity for a direction of polarization ~ to the axis eK is 1

cos2 Z

sin2 Z

/;/

'¢;K

/;I

5.7. I r r e g u l a r P a r t i c l e s By irregular particles we mean those which have a random nature in addition to being non-spherical, such as pulverized coal. It may be expected that such shapes could be treated statistically, or that some averaging or smoothing process may be present. This is found, for example, in the extinction curve where oscillations and resonances are damped out.~°l This feature has been used by ChS'lek et al. 1°2 The resonances in the extinction curve can he associated with maxima in the Mie expansion coefficients. 1°3 These maxima are in turn related to surface waves on the particle and these are damped by irregularities. Thus, for a cloud of randomly aligned irregular particles smoothing occurs, and these authors recommend a means of averaging the Mie coefficients which yields good agreement with experiment. Emslie and Aronson ~°4 dealt with large irregular particles by using geometrical optics for spheres, but letting the edge irregularities behave like small ellipsoids obeying Rayleigh theory. Using a microwave analogue, Zerull et a l ) °5 compared scattering by various non-spherical objects with spheres. They found that diffraction patterns were almost the same, which implies that methods based on this phenomenon are very useful for sizing irregular particles. 5.8. A n i s o t r o p y Anisotropy can be due to refractive index or shape (form anisotropy), or both. Form anisotropy can be readily demonstrated by producing alignment of the particulates and measuring the extinction as a function of polarization. Isotropic particles show no variation, whereas anisotropy results in birefringence. Alignment can be achieved using electric or magnetic fields for appropriate particles. 1°6-1°s A further method is to subject the particle to a flow gradient which produces a torque. Cerf and Scheraga 42 have shown that Brownian motion collisions can balance this force so that the particles tend to align themselves in the flow, provided that the maximum dimension is less than 0.21am. This produces the phenomenon of streaming birefringence. Soot sampled from flames is commonly seen to have the form of long chain agglomerates. Dalzell et al. 1°9 and d'Alessio et al. ~ ° studied sooting flames using light scattering and concluded that agglomerates must be present. However, the3 did not consider anisotropy. Jones and Wong ~'2 used streaming birefringence to demonstrate the presence

94

A.R. JONES

of anisotropic particles in a flame. The shearing force was developed by burning propane at an interface with air on a Wolfhard-Parker burner, with the flow-rate of fuel being much less than that of the air. This leads to the criticism that the agglomerates may have resulted from the peculiar conditions, and that experiments should be conducted on more usual configurations. The optical properties of aggregates of carbon black have been the subject of many papers by Ravey et al. 4"7"112'11a These were generally suspended in fluids and approximations similar to R G D theory were applied. One of the conclusions was that chain agglomerates could be likened to ellipsoids. Anisotropy generally increases the scattering and absorption by particles. 5,~'*'~s The possible significance of this to the emission of radiation by sooting flames was discussed in Section 4.1. To use light scattering generally for the study of anisotropy we retain the complete scattering matrix (p. 80). This shows that if the incident light is polarized, say E.L, inc = 0, then the scattered light contains both parallel and perpendicular components. For a cloud of randomly aligned anisotropic particles there results partial depolarization of the scattered light. The depolarization ratio is defined by p~,(O) = H~(O)fV~,(O) where V~ is the vertical component and H~ is the horizontal component in the scattered light for vertically polarized incident radiation. A ratio Pn for horizontal incident polarization may be similarly defined. Ravey 113 predicts p~ ~ 1% for carbon black agglomerates. An alternative way of studying anisotropy is via the Stokes' matrix.

,,,,\ I,,o,\ sle,o, 1

where l = I n + I ± ; Q = I I I - I ± ; u=2/x/~ll/~cos(~bll - ~ b i ) ; V = 2 ~ 1 1 I ± sin (q~ll-~bl) and ~ represents phase. For a cloud of randomly aligned particles, S=

bl

a2 0 0

0 a 3 b2 " -b 2 a

For isotropic particles a t = a 2 and a 3 a 4. If either of these inequalities is unsatisfied then anisotropy is present. Unfortunately, the converse is not true. A more detailed discussion of the use of the Stoke's matrix is given by Turner. 116 =

6. CONCLUSIONS It can be seen from the foregoing that scattering is a wide subject which has been intensively studied for

many years. Experimental techniques for the determination of spherical particle sizes are numerous. and cover any size range from practically zero to several hundred p.m. the upper limit being above the points at which holography and photography can be readily applied. Developments are still required for means of studying non-spherical particles in more detail and measurement of refractive index: both isotropic and anisotropic. An important area is the influence of scattering by particulates on radiative heat transfer. The effects of irregularity and anisotropy, as for coal or soot particles, need to be studied further, and in dense clouds methods for dealing with multiple scattering when the scattering polar diagram has a convoluted structure are required. The effects of varying refractive index, whether due to composition or temperature, also need further investigation. Rapid numerical techniques are needed to calculate the scattering properties of non-spherical and irregular particles. For those of interest to combustion, where many are present, some statistical averaging could be applied. Soot agglomerate chains have a certain randomness of structure, for example, and a cloud would generally be randomly oriented. We may conclude that there are still many avenues ripe for exploration in this ever expanding field.

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Scattering of electromagnetic radiation in particulate laden fluids

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(Manuscript

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r e c e i v e d 23 A u g u s t

1978)