electrostatics analysis of radiative absorption by sphere clusters in the rayleigh limit:...

Electrostatics analysis of radiativeabsorption by sphere clusters in theRayleigh limit: application to soot particles

Daniel W. Mackowski

An analysis of radiative absorption and scattering by clusters of spheres in the Rayleigh limit is developedwith an electrostatics analysis. This approach assumes that the largest dimension of the cluster issignificantly smaller than the wavelength of the radiation. The electric field that is incident upon andscattered by the cluster can then be represented by the gradient of a potential that in turn satisfiesLaplace’s equation. An analytical solution for the potential that exactly satisfies the boundaryconditions at the surfaces of the spheres is obtained with a coupled spherical harmonics method. Thecomponents of the polarizability tensor and the absorption, scattering, and depolarization factors areobtained from the solution. Calculations are performed on fractallike clusters of spheres, withrefractive-index values that are characteristic of carbonaceous soot in the visible and the IR wavelengths.Results indicate that the absorption cross sections of fractal soot clusters can be significantly larger in themid-IR wavelengths than what is predicted for Rayleigh-limit spheres that have the same totalvolume. The absorption cross section 1relative to a sphere of the same volume2 is dependent on thenumber of spheres in the aggregate for aggregates with up to approximately 100 primary spheres, and forlarger aggregates the relative absorption becomes constant. The predicted spectral variation of sootabsorption in the visible and the mid-IR wavelengths is shown to agree well with experimentalmeasurements.

1. Introduction

The combustion of hydrocarbon fuels invariably re-sults in the formation of small soot particles. Be-cause overall heat-transfer rates from flames areoften dominated by radiative emission from the par-ticles,1 an accurate understanding of the absorptionproperties of soot particles is an essential ingredientin the radiative modeling of combustors. This prop-erty is also important in the interpretation of laser-based diagnostic methods in flames2 and in the predic-tion of the effects of combustion particulates on theatmospheric radiative balance.3The actual task of modeling the absorption of soot

is made difficult by the complicated nature of theparticles. Almost invariably, soot forms as aggre-gates of small, nearly spherical primary particles.4Because Brownian diffusion is the dominant mecha-

The author is with the Department of Mechanical Engineering,Auburn University, Auburn, Alabama 36849.Received 20 June 1994; revisedmanuscript received 16 December

1994.0003-6935@95@183535-11$06.00@0.

r 1995 Optical Society of America.

nism in bringing the primary spheres together as anaggregate, the overall structure of the aggregate canbe describedwell by fractal relationships.5,6 However,soot particles as a whole are profoundly nonsphericalin shape and thus do not readily lend themselves torelatively simple 1e.g,. spherical particle2 radiativemodels.Because the diameters of the primary particles,

which are typically of the order of 10–50 nm, aresignificantly smaller than the visible and the IRwavelengths, it would certainly appear reasonable toassume that the primary spheres can be modeled asradiative dipoles. Beginning with this assumption,Jones7 was the first to develop a model of sootradiative absorption and scattering that explicitlyaccounted for the multiple-sphere nature of the clus-ter. Berry and Percival8 expanded on the analysis ofJones and presented a detailed examination of theradiative behavior of soot clusters that possess afractal structure. One of the most important conclu-sions of this paper was that, for typical refractiveindices of primary soot particles and a fractal dimen-sion less than 2, the absorption of the spheres is notsignificantly perturbed by multiple scattering amongneighboring spheres. Because of this, the absorp-

20 June 1995 @ Vol. 34, No. 18 @ APPLIED OPTICS 3535

tion cross section of the cluster will be well predictedby the sum of the absorption cross sections of theprimary spheres. The lack of significant multiplescattering among the spheres has also formed thebasis of radiative models based on Rayleigh–Gans–Debye theory.2,5,6Recently, I used an exact solution of Maxwell’s

equations for neighboring spheres to investigate thevalidity of the dipole approximation for soot clusters.9Details of the formulation are given in Refs. 9 and 10;only a brief description of the theory is presentedhere. In this formulation the scattered electric fieldfrom the entire cluster is taken to be the superposi-tion of fields that are scattered from each of thespheres. Each of the individual fields, in turn, isrepresented as an expansion of vector spherical har-monics with the origin at the sphere center. Toformulate the boundary conditions at the spheresurfaces, addition theorems are used to translate aharmonic from one sphere origin to another. Afterthe expansions are truncated after an adequate num-ber of multipole orders, a system of linear equations isobtained for the scattered field expansion coefficientsfor each sphere. The cross sections of the cluster arethen obtained directly from the expansion coeffi-cients.Calculations were performed for spheres of size

parameters x 15 2pa@l, where a and l are the sphereradius and the radiation wavelength, respectively2and refractive indices m 15 n 1 ik2 that are typical ofcarbonaceous soot primary particles. An interestingoutcome of the calculations was that in the Rayleighlimit 1x = 02, several multipole orders could be re-quired to resolve the cross sections for highly conduct-ing 1i.e., large nk2 spheres. For example, a two-sphere cluster with each sphere having a sizeparameter x 5 0.01 and a refractive indexm 5 2 1 1irequired 5 multipole orders for three-digit accuracy tobe obtained in the calculated random-orientationcross sections. Using a dipole approximation 1i.e.,the first electric order2 resulted in an absorption crosssection that was ,20% lower than that obtained fromthe exact solution. When the refractive index in-creased to m 5 3 1 2i 1which would correspond tocarbonaceous soot in the mid IR112, the number ofrequired expansion orders increased to 11, and theexact absorption cross section was ,50% greater thanthe dipole solution. Physically, as the spheres be-come more conducting, the electric field about thecontact point betweeen the spheres becomes increas-ingly nonuniform. Because of this, several multipoleorders are thus required for a description of the fieldabout the spheres.These results obviously throw into question the

accuracy of dipole-based models for soot clusters inthe Rayleigh limit. At the same time, the exactmultipole method runs into its own problems for thissituation. Application of this formulation for small,highly conducting spheres will require calculation ofspherical Bessel functions of considerable order 1up to22 for the m 5 3 1 2i case discussed above2 for

3536 APPLIED OPTICS @ Vol. 34, No. 18 @ 20 June 1995

arguments that are significantly less than unity.Unless it is done carefully and with high precision,this can lead to overflow and loss-of-precision numeri-cal errors. Also the exact multipole formulation issomewhat of an overkill in this situation, because themagnetic 1TE2modes become negligible in comparisonwith the electric 1TE2modes.9The objective of this paper is to use an alternative

analytical approach to examine the Rayleigh-limitabsorption and the scattering behavior of fractal sootaggregates. Specifically, an electrostatics analysis isapplied to the sphere cluster. Here it is assumedthat the overall size of the cluster is significantlysmaller than the radiation wavelength. The electricfield can then be represented by the gradient of apotential that satisfies Laplace’s equation. In amethod similar to the exact multipole solution ofMaxwell’s equations, the solution of Laplace’s equa-tion for neighboring spheres can be obtained by asuperposition of solutions about each of the spheres inthe cluster. Gerardy andAusloos used this approachto formulate the general solution for the absorptionscattering properties of neighboring spheres in thelong-wavelength limit.12 The objective of their paperwas primarily to examine the absorption spectra ofmetallic and dielectric powders. Because of this,their paper has not received appreciable attentionamong researchers of soot absorption. However, thisparticular method of solving Laplace’s equation forneighboring sphere geometries has been used ratherextensively in other analogous problems, such asconduction heat transfer in inhomogeneous media13and low-Reynolds-number fluid flow over sphericalaggregates.14,15

2. Analysis

In this section the general solution for radiativeabsorption and scattering of sphere clusters in theelectrostatics limit is presented. The analysis fol-lows, to some extent, that developed by Gerardy andAusloos,12 although the notation complies with thatappearing in Bohren and Huffman.16 In Fig. 1, eachof the NS spheres in the cluster is characterized by aradius ai, a refractive indexmi, and a relative positionXi, Yi, and Zi. The medium in which the spheresreside is assumed to be nonabsorbing with a refrac-tive index of unity. Let E2 denote the electric fieldexternal to the spheres and E1,i denote the fieldwithin the ith sphere. It is assumed that the dimen-sions of the sphere cluster are significantly smaller

Fig. 1. Sphere cluster coordinate system.

than the incident radiation wavelength. Under thiscondition the electric field can be represented by thegradient of a potential, i.e.,

E2 5 =F2, E1,i 5 =F1,i. 112

The potentials F, in turn, will satisfy

=2F 5 0. 122

Because the above equation is linear and homoge-neous, the potential external to the spheres can beconstructed from the superposition of an incidentpotential F0 plus potentials scattered from each of thespheres in the cluster, Fs,i. That is,

F2 5 F0 1 oi51

NS

Fs,i. 132

The individual scattered potentials, in turn, can berepresented by expansions of solid spherical harmon-ics through

Fs,i 5 on51

`

om52n

n

amniumn21ri, ui, fi2, 142

in which amni represent undetermined expansion coef-ficients and umn2 is a solid harmonic defined by

umn2 5 ri21n112Pnm1cos ui2exp1imfi2. 152

In Eq. 152 Pnm is the associated Legendre function and

ri, ui, and fi denote the spherical coordinates centeredabout the origin of sphere i 1Fig. 12. The particularform of the harmonics in Eq. 142 is chosen so that Fs,i1ri = `2 = 0. Likewise, a solution for the potentialwithin each of the spheres can be expressed by

F1,i 5 d0i 1 on51

`

om52n

n

dmniumn11ri, ui, fi2, 162

where

umn1 5 rinPnm1cos ui2exp1imfi2. 172

In the most general form, the incident electric-fieldvector is represented in Cartesian components by

E0 5 exE0x 1 eyE0y 1 ezE0z. 182

Consequently, the incident potential is

F0 5 xE0x 1 yE0y 1 zE0z, 192

where x, y, and z denote the coordinates with respectto an arbitrary origin of the cluster. It is assumedthat the incident vector has unit amplitude, so that

0E0x 02 1 0E0y 0

2 1 0E0z 02 5 1. 1102

When the above constraint is used, the incident

potential can be expressed in terms of solid harmonicsby the following expansion:

F0 5 om521

1

pmum111r, u, f2, 1112

in which the coefficients p depend on the direction ofthe incident vector, as defined by u 5 b and f 5 g, by

p21 5 E0x 1 iE0y 5 sin1b2exp1ig2,

p0 5 E0z 5 cos1b2,

p1 5 21⁄21E0x 2 iE0y2 5 21⁄2 sin1b2exp12ig2. 1122

At the surface of each sphere i in the cluster, theboundary conditions for the potential are16

F21r 5 ai2 5 F1,i1r 5 ai2, 1132

r ? =F21r 5 ai2 5 eir ? =F1,i1r 5 ai2, 1142

in which ei 5 mi2 is the complex permittivity of

sphere i. To apply the boundary conditions, it is firstnecessary to transform the expansions for the scat-tered potentials that originate from all other spheresinto an expansion of harmonics centered about spherei. This can be accomplished with an addition theo-rem for solid spherical harmonics,17 which can bestated in the form

umn21rj, uj, fj2 5 ol51

`

ok52l

l

Cmnkli jukl11ri, ui, fi2,

Ri j . ri. 1152

The addition coefficients C depend solely on thedistance and the direction of travel of origin i fromorigin j as defined by Rij, Ui j, and Fi j. An explicitrelation for these quantities 1for n and l . 12 is

Cmnkli j 5 1212n1m

1n 1 l 2 m 1 k2!

1n 2 m2!1l 1 k2!

3 u1m2k21n1l 221Ri j, Ui j, Fi j2. 1162

Combination of the above addition theorem with theexpansions in Eqs. 142, 162, and 1112 and the boundaryconditions in Eqs. 1132 and 1142 leads to a relationshipbetween the coefficients for the potential from spherei and the coefficients for the potentials from all otherspheres and the incident field. By truncation of theseries expansions after n 5 NO terms a system oflinear equations is obtained for the potential expan-sion coefficients. These equations are

amni 1 fni oj51jfii

NS

ol51

NO

ok52l

l

Cklmni jakl j 5 2f1ipmdn1,

m 5 2n, 2n 1 1, . . . n; n 1 1, 2, . . . NO. 1172


In Eqs. 1172 dn1 is the Kronecker delta function and fniis defined as

fni 5 ai2n11n1ei 2 12

n1ei 1 12 1 1. 1182

The equations for the scattering coefficients, Eqs.1172, are similar to those derived by Gerardy andAusloos.12 A key difference is that the addition theo-rem coefficients in Eq. 1162 are given here in a muchmore concise and simplified form than in the previousstudy.

3. Cross Sections

Following the analysis presented by Bohren andHuffman,16 the cross sections can be obtained byconsideration of the far-field behavior 1i.e., r : RC,where RC is the largest distance between the spheres2of the scattered potential from the cluster. In thislimit the cluster will behave as a dipole, so that

Fs 0 r:RC5p ? r

4pr3, 1192

in which p is the dipole moment of the cluster. Thisquantity, in turn, is related to the incident electricfield through

p 5 aE0. 1202

Equation 1202 defines the polarizability tensor a of thecluster, which will have the structure

a 5 3axx axy axz

axy ayy ayz

axz ayz azz4 . 1212

Combining Eqs. 1192 and 1202 yields

Fs 51aE02 ? r

4pr3. 1222

At large distances from the cluster the potentialexpansions for the individual spheres in Eq. 142 can allbe written about a single origin of the cluster. Inaddition, all but the first 1n 5 12 terms in the expan-sions can be neglected. With the fact that

P1211cos u2 5 21⁄2 sin u, P1

01cos u2 5 cos u,

P111cos u2 5 sin u, 1232

the expansion for the far-field potential becomes

Fs1r : Rc2 5 32 1

2a211

T sin u exp12if2 1 a01T cos u

1 a11T sin u exp1if24 1r2 , 1242


in which the coefficients for the total potential aregiven as

amnT 5 oi51

Ns

amni. 1252

Now consider an incident field that is given by Ex 5

1 and Ey 5 Ez 5 0. The coefficients for the totalpotential corresponding to this state 3obtained thoughsolution of Eqs. 1172, with p’s defined in Eq. 11224 aredenoted amnT,x. When the vector r is expanded into aCartesian frame, Eq. 1222 becomes

Fs,x 5 1axx sin u cos f 1 axy sin u sin f

1 axz cos u21

4pr2. 1262

The components of the polarizability that correspondto x-polarized radiation can then be identified whenEq. 1262 is combined with Eq. 1242, leading to

axx 5 2p12a11T,x 2 a211T,x2,

axy 5 2pi12a11T,x 1 a211T,x2,

axz 5 4pa01T,x. 1272

The above procedure can be repeated for y- andz-polarized radiation and leads to the general formulafor the polarizability tensor,

axn 5 2p12a11T,n 2 a211T,n2,

ayn 5 2pi12a11T,n 1 a211T,n2,

azn 5 4pa01T,n, 1282

where n 5 x, y, or z.From the optical theorem, the absorption cross

sections of the cluster for the three cases of incidentpolarization 1x, y, and z2 are obtained from the diago-nal elements of the polarizability tensor as

Cabs,n 5 k Im1ann2, 1292

where k 5 2p@l is the wave number of the incidentfield. The scattering cross section is obtained from

Csca,n 51

6pk41 0axn 0

2 1 0ayn 02 1 0azn 0

22. 1302

The average cross sections for a randomly orientedcluster are given simply as the average of the x, y, andz cross sections,16 i.e.,

7Cabs8 5 1⁄31Cabs,x 1 Cabs,y 1 Cabs,z2, 1312

7Csca8 5 1⁄31Csca,x 1 Csca,y 1 Csca,z2. 1322

Because the clusters will, in general, be asymmet-ric, and the off-diagonal terms in the polarizabilitytensor will be nonzero, the scattered radiation from

the cluster will be depolarized. Bohren and Huff-man present an analysis of the average scatteringmatrix for an anisotropic dipole,16 and their resultscan be applied directly to the problem at hand. Forunpolarized incident radiation, the differential scatter-ing cross section of the particle is given by

G1u2 537Csca8

80p36 2 M 1 12 1 3M2cos2 u4. 1332

With regard to laser diagnostic measurements of sootin flames, the differential scattering cross sections forvertically and horizontally polarized incident andscattered radiation are of particular interest. Thesequantities are given by

GHH 537Csca8

40p31 2 M 1 12 1 3M2cos2 u4,

GVH 5 GHV 537Csca8

40p11 2 M2,

GVV 537Csca8

40p13 1 2M2. 1342

The first subscript in the above quantities denotes thepolarization state of the scattered radiation, and thesecond denotes the state of the incident radiation,with respect to the scattering plane. For horizontal1H2 the radiation is polarized parallel to the plane, andfor vertical 1V2 the polarization is perpendicular to theplane.The quantity M in the above equations depends on

the principle components 1or eigenvalues2 of the polar-izability tensor through

M 5Re1a1*a2 1 a1*a3 1 a2*a32

0a1 02 1 0a2 0

2 1 0a3 02

. 1352

Rather than finding the eigenvalues, one can deter-mineM directly from a by the use of the relationships

0a1 02 1 0a2 0

2 1 0a3 02 5 o

µo

n

0aµn 02, 1362

where µ and n refer to x, y, and z and

a1 1 a2 1 a3 5 axx 1 ayy 1 azz. 1372

Combining Eqs. 1362 and 1372 into the equation for Myields

M 5

0axx 1 ayy 1 azz 02 2 o

µo

n

0aµn 02

2 oµ

on

0aµn 02

. 1382

For isotropic particlesMwill be unity, and the depolar-ized components of the scattered radiation will bezero.To simplify the subsequent analysis, the spheres in

the cluster are taken to have identical radii and

refractive indices—this is quite reasonable for sootparticles. For the purposes of calculating the scatter-ing coefficients in Eqs. 1172, the sphere radii can bearbitrarily set to unity, with the understanding thatthe radial separation distance Ri j that appears in Eq.1162 represents the relative distance between thesphere centers 1i.e., Ri j 5 2 for touching spheres2 andthat the polarizability of the cluster will be propor-tional to the actual total volume of the cluster. Thepolarizability obtained for this unit-radius, identical-sphere case is denoted a. For a cluster of spheres,each having an actual radius a, Eqs. 1312 and 1322 canbe reexpressed as

7Cabs8 5 4pkaV3E,

7Csca8 58p

3k4aV6F, 1392

where aV 5 NS1@3a is the volume-equivalent radius of

the cluster and E and F are given by

E 51

12pNSIm1axx 1 ayy 1 azz2, 1402

F 51

48p2NS23 0axx 0

2 1 0ayy 02 1 0azz 0

2

1 21 0axy 02 1 0axz 0

2 1 0ayz0224. 1412

Note that for a single sphere E and F will reduce tothe familiar expressions

E 5 Ime 2 1

e 1 2,

F 5 0e 2 1

e 1 202. 1422

It is also useful to define the absorption and theextinction efficiencies of the cluster with respect to aVthrough

7Qabs8 57Cabs8

paV25 4xVE, 1432

7Qsca8 57Csca8

paV258xV4F

3, 1442

in which xV 5 kaV is the volume-equivalent sizeparameter of the cluster. The quantitiesE andFwilldepend only on the refractive index of the spheres andthe structure of the cluster as reflected in the numberof spheres and their relative positions. These quan-tities are therefore useful in comparing the effects ofaggregation in the radiative properties of the sphereswith those obtained for independent, noninteractingspheres.

4. Results

A FORTRAN code with a conjugate-gradient solutionmethod was developed to calculate the potential


coefficients from Eqs. 1172. As expected, the numberof expansion orders required for convergence to beobtained in the results for the cross sections dependedentirely on the proximity of the spheres to each other,their orientations relative to the incident polariza-tion, and the magnitude of the refractive index.Regardless of the refractive index, whenever thespheres were separated by a distance larger thanapproximately 1 radius, only 1 order was required inthe expansions. For contacting spheres, on the otherhand, several orders for convergence could be required.In addition, the absorption cross sections of contact-ing spheres could be appreciably larger than the sumof the cross sections for the independent, isolatedspheres. This behavior is listed in Table 1, whichshows the values for the ratio of Cabs@Cabs,indep andCsca@Csca,indep for a two-sphere cluster for three valuesof refractive index 1m 5 1.6 1 0.6i, 2 1 1i, and 3 1 2i,roughly corresponding to carbonaceous soot in thevisible, near-IR, and mid-IR wavelengths,11 respec-tively2 and the two cases of incident polarization1perpendicular and parallel to the cluster axis2. Theindependent cross sections correspond to Rayleigh-limit values for single spheres with the same refrac-tive index and total volume of the cluster. That is,

Cabs,indep 5 4pkaV3 Ime 2 1

e 1 2, 1452

Csca,indep 58p

3k4aV6 0e 2 1

e 1 202. 1462

Also given in Table 1 are the number of expansionorders required for one to attain three-digit conver-gence in the cross sections. Note that the cluster,when oriented parallel to the field, can have crosssections that are considerably larger than those ob-tained for volume-equivalent spheres. As the spheresbecome more conducting 1i.e., larger nk2, both thedegree of departure from isolated spheres’ behaviorand the number of orders required for convergence tobe obtained increase.The cross-section ratios and truncation limits given

in Table 1 are entirely in keeping with results ob-tained from the exact wave-equation solutions for twospheres with vanishingly small size parameters.9As mentioned above, the need to retain several multi-pole orders for spheres that are, individually, wellwithin the Rayleigh limit is somewhat counterintui-tive. These extra orders result from the fact that the

Table 1. Results for a Two-Sphere Cluster

m PolarizationCabs@

Cext,indep

Csca

@Csca,indep NO

1.6 1 0.6i parallel 1.267 1.246 31.6 1 0.6i perpendicular 0.909 0.907 22.0 1 1.0i parallel 1.558 1.471 52.0 1 1.0i perpendicular 0.872 0.867 33.0 1 2.0i parallel 2.484 2.058 93.0 1 2.0i perpendicular 0.833 0.826 3


electric-field intensity can be highly nonuniform inthe vicinity of the contact point between the spheres—regardless of the uniformity of the incident field.To illustrate, contour plots that show lines of con-stant potential are presented in Figs. 2 and 3 fortouching spheres that have refractive indices of 1.6 10.6i and 3 1 2i, respectively. In Figs. 21a2 and 31a2 theelectric field is polarized parallel to the clusteraxis, and in Figs. 21b2 and 31b2 the polarization isperpendicular to the axis. As can be seen when theplots are compared, a largermagnitude ofm results ina more uniform potential within each of the spheres.

Fig. 2. Isopotential lines, m 5 1.6 1 0.6i, with 1a2 parallel and 1b2perpendicular incident polarization.

Fig. 3. Same as Fig. 2, but withm 5 3 1 2i.

Because the incident potential for parallel polariza-tion varies linearly in the z direction, the mean valuesof the internal potentials for each sphere in this casewill be different. Consequently, a steep jump in thepotential occurs at the contact point, and the magni-tude of this jump increases with increasing 0m 0. Tocompensate for this jump, the gradient of the externalpotential becomes large near the contact point, andmathematically resolving the nonuniformity of theexternal potential requires a relatively large numberof harmonic orders. When the incident field is di-rected perpendicular to the cluster axis 1as shown inFig. 32, the mean values of the internal potential areidentical for both spheres. Accordingly, the varia-tion in external potential at the contact point is notnearly as severe as for parallel polarization.To illustrate the validity of the electrostatics formu-

lation further and to gauge its range of applicability,comparisons of the electrostatics and the wave-equation results are presented in Figs. 4 and 5. Inthese figures the quantities 7Qabs8@xV and 7Qsca8@xV4

Fig. 4. Comparison of electrostatics and wave-equation predic-tions of random-orientation absorption.

Fig. 5. Comparison of electrostatics and wave-equation predic-tions of random-orientation scattering.

are plotted versus xV for a binary cluster. From Eqs.1402 and 1412, 7Qabs8@xV and 7Qsca8@xV4 are equivalent to4E and 8F@3, respectively, and are thus independentof the cluster size. Results indicate that for xV lessthan approximately 0.1, the electrostatics result agreesto within 1% of the wave-equation predictions. Theelectrostatics model is obviously incapable of predict-ing the resonance absorption of the cluster that occursfor xV around unity or the limiting behavior of Q =constant for larger xV. Nevertheless, the electrostat-ics model yields the exact Rayleigh-limit behavior ofthe cluster.Although the computational overhead required in

the electrostatics formulation is considerably smallerthan that for the wave formulation 1primarily becausethe addition theorem in the latter makes it consider-ably more complicated2, both methods still requiresolution of linear equations for the radiative proper-ties of the cluster to be obtained. Because of this,approximate and relatively simple radiative modelsof sphere clusters have been sought. In particular,the chain-aggregate nature of the soot particles hasled previous investigators to suggest that their absorp-tion properties would be similar to those obtained forelongated solids such as prolate spheroids or infinite-length cylinders.18,19The radiative equivalence 1or lack of it2 between

prolate spheroids and chains of spheres can be exam-ined in detail with the formulation developed here.In the Rayleigh limit the absorption cross section of arandomly oriented prolate spheroid is given by16

7Cabs8ps 5kV

3Im3 e 2 1

1 1 L11e 2 121 2

e 2 1

1 1 L21e 2 124 ,1472

in which

L1 51 2 e2

e2 1 12e ln1 1 e

1 2 e2 12 ,

L2 51

211 2 L12. 1482

In Eqs. 1472 and 1482 V is the volume and e is theeccentricity of the spheroid, which is related to theaspect ratio L@a by e 5 1 2 a2@L2. The aspect ratio ofa straight chain of equal-radius spheres, on the otherhand, would correspond simply toNS.Presented in Fig. 6 are the values of 7Cabs8@8Cabs8indep

for a straight chain and a prolate spheroid versusaspect ratio for the three previously used values of therefractive index. In some respects the curves forboth types of particles follow the same trends.Increasing elongation results in larger relative absorp-tion values, and the degree to which absorption isincreased depends strongly on m. For aspect ratiosgreater than 100 the absorption is within a fewpercent of the infinite-length particle limit. On theother hand, the increase in relative absorption for theprolate spheroid is significantly larger than for thechain of spheres. And the difference in the absorp-


tion increase between the two types of particles isdependent on the refractive index. It is interestingto note, however, that the relative absorption for aninfinite-length chain, for all three refractive-indexvalues, is roughly equivalent to that obtained for aprolate spheroid with an aspect ratio of 4. Neverthe-less, it appears from this comparison that the absorp-tion properties of Rayleigh-limit chains of spheresand prolate spheroids, while sharing the same gen-eral trends, do not have a correspondence that isindependent of refractive index.Modeling a soot particle as a straight chain of

spheres may be an improvement over modeling sootas a prolate spheroid, yet this still neglects thepseudorandom structure of actual aggregates. Toinvestigate the absorption characteristics of morerealistic particles, calculations were performed withfractallike clusters of spheres. When formed fromdiffusion-limited aggregation 1DLA2, soot aggregateshave a structure that can be represented by5,6

Ns 5 kf 1Rg

dp2Df, 1492

in which dp is the primary sphere diameter, Rg is theradius of gyration of the cluster, Df is the fractaldimension, and kf is a prefactor constant. For aspecified number of equal-sized spheres in the cluster,Rg is given by

Rg2 5

1

Nsoi51

Ns

ri2, 1502

where ri is the distance from sphere i to the center ofmass of the cluster.Rather than performing numerical DLA simula-

tions to obtain the particle positions in a cluster, asequential algorithm that mimics Eq. 1492 was devel-oped. Beginning with a pair of contacting spheresand specified values of kf and Df, a third sphere was

Fig. 6. Absorption efficiency of straight chains of spheres andprolate spheroids 1normalized by equal-volume sphere absorptionefficiency2 versus the chain length or the spheroid aspect ratio.


randomly attached to the surface of one of the twospheres, with the constraint that the radius of gyra-tion calculated for the three-sphere cluster mustexactly satisfy Eq. 1492. The process was then re-peated for the fourth, fifth, and additional spheres.Values of kf 5 5.8 and Df 5 1.9 were used in thecalculations, which correspond to values obtainedfrom direct DLA simulations.6 The properties of theaggregates generated with this algorithm obeyed theknown statistical relationships of DLA aggregates,5yet the positions could be calculated in a fraction ofthe time required for DLA simulations. An illustra-tion of a 40-sphere cluster generated from the sequen-tial algorithm is given in Fig. 7.Presented in Fig. 8 are the average calculation

Fig. 7. 40-sphere fractal cluster generated with the sequentialalgorithm.

Fig. 8. Absorption, scattering, and 90° polarization behavior offractal clusters and straight chains versus the number of spheres.

results for 10 fractal cluster configurations, in whichthe absorption and the scattering cross sections 1nor-malized by the independent, equivalent sphere val-ues2 and the polarization at u 5 90° are plotted versusthe number of spheres 1NS2 in the cluster. The polar-ization represents the degree to which the scatteredradiation at a scattering angle 90° is polarized forunpolarized incident radiation. This quantity is re-lated toM in Eq. 1382 through16

P190°2 57S12190°28

7S11190°2853 1 2M

6 2 M. 1512

The refractive index of the spheres ism 5 2 1 1i, andfive expansion orders were used in the calculations.In Fig. 8 the error bars denote the standard deviationin the mean among the 10 randomly generated clus-ters. Also given are results calculated for a straightchain. Similar to the trends observed for the chainand the spheroid, the relative absorption of the clus-ter initially increases with the number of spheres.The same general behavior can be seen for thescattering by the cluster, although the departure fromequivalent-sphere behavior is not as severe. For NSgreater than approximately 100 the absorption andthe scattering of the cluster, relative to an equal-volume sphere, become roughly constant. For thegiven refractive index, the absorption and the scatter-ing cross sections of the cluster top out at approxi-mately 125% and 111% of the values obtained for theequivalent sphere, respectively. The effect of theparticular cluster structure 1as reflected in the stan-dard deviation in the results2 also becomes smaller asNS increases.The results for polarization in Fig. 8 show clearly

distinct trends between the fractal cluster and thestraight chain. Realize that for an isotropic particlethe scattered radiation at 90° will be completelypolarized, yielding P190°2 of unity. Results for thecluster show that depolarization is essentially insig-nificant for all but the smallest clusters. The chain,on the other hand, displays a significant depolariza-tion that increases with chain length. The resultsfor the two forms of particles are not unexpected, inthat the polarization reflects the degree of anisotropyof the particle. A straight chain becomes increas-ingly anisotropic with NS, whereas the fractal clusterwould have an increasingly uniform structure. Inaddition, the results suggest that measurement ofdepolarization by Rayleigh-limit clusters will notserve as an effective means of characterizing aggre-gate structure, which is consistent with recent find-ings by Singham and Bohren.20Calculations performed on clusters with fractal

dimensions ranging from 1.7 to 2 indicate that theselarge-sphere limits are essentially independent offractal dimension. On the other hand, the limitsstrongly depend on refractive index. As can be seenin Fig. 6, the relative increase in absorption of thecluster 1relative to independent spheres2 becomeslarger as 0m 0 increases. Because the refractive indexof carbonaceous soot changes considerably thoughout

the visible and the IR wavelengths, this implies thatabsorption properties of flame-generated soot willhave spectral behavior that does not follow sphericalparticle Rayleigh-limit predictions. To examine thisfurther, calculationswere performedwithwavelength-dependent refractive-index values obtained from thedispersion model of Lee and Tien.11 Predicted realand imaginary parts of the refractive index at tempera-tures of 1450 and 300 K are plotted versus wave-length in Fig. 9. The corresponding absorption crosssections of soot clusters 1again relative to independentspheres2 obtained for the refractive-index distributionare plotted in Fig. 10 forNS 5 2, 10, 25, and 100.Several important conclusions can be drawn from

the results presented in Fig. 10. In the visible andthe near-IR wavelengths 1i.e., 0.5 µm , l , 2 µm2the absorption of soot aggregates will be of the orderof 10–20% greater than that predicted from indepen-

Fig. 9. Spectral refractive index used in calculations, from Leeand Tien.11

Fig. 10. Absorption cross section of fractal cluster 1relative to asphere of equal volume2 versus the radiation wavelength.


dent spheres. Because the bulk of radiative heattransfer falls into this wavelength interval, the aggre-gation of soot into clusters will have only a higher-order effect on the rate of thermal emission fromflames or the absorption of solar radiation. On theother hand, aggregation has a dramatic effect on sootabsorption in the mid-IR wavelengths 15–10 µm2.For example, at the 10-µm wavelength the absorptioncross section of a relatively large soot cluster 1i.e.,NS . 1002will be of the order of 200% greater than thecross sections of the primary spheres. Even thoughthis behavior would not be expected to affect radiativeheat transfer significantly, it does have importantimplications on the use of long-wavelength opticaldiagnostic methods for measurement of soot proper-ties in flames.The long-wavelength trends displayed in Fig. 10

are consistent with the conclusions made in previousexperimental measurements of visible and IR absorp-tion of flame-generated soot, conducted by Roesslerand Faxvog21 and Mackowski et al.19 It was ob-served in these investigations that the absorptioncross sections for soot at themid-IR wavelengths wereconsistently larger than what would be expected fromextrapolation, based on spherical models, of visibleand near-IR measurements. Both groups demon-strated that one could qualitatively explain theirmeasurements by modeling the soot particles asprolate spheroids or infinite-length cylinders. Suchmodels, however, did not provide a quantitative de-scription of the structure of the soot aggregate. Thatis, the connection between the aspect ratio of a prolatespheroid 1obtained from curve fitting of the measure-ments2 and the number of spheres in the aggregate isambiguous.A comparison of the electrostatics model predic-

tions for soot aggregate spectral absorption can bemade directly with the data reported in Ref. 21. Inthis paper, measurements were made at wavelengthsof 0.5145 and 10.6 µm of the absorption coefficient ofsoot produced in acetylene–air diffusion flames.When Rayleigh behavior andmonodisperse aggregatesizes are assumed, the ratio of the spectral absorptioncoefficients at the two wavelengths would correspondto kl1@kl2 5 l2El1@l2El2. Table 2 presents the experi-mentally measured ratio along with electrostatics

Table 2. Ratios of Spectral Absorption Coefficients at the 0.5145- and10.6-mmWavelengths for Soot Aggregates as Predicted by the

Electrostatics AESBModel for Fractal Clusters, Rayleigh-Limit SphericalParticles and Infinite-Length Cylinders, and Experimental Measurement

Results from Roessler and Faxvog ARef. 21B

Model k10.5145 µ[email protected] µm2

ESNS 5 2 14.48NS 5 10 11.35Ns 5 50 9.484NS 5 250 8.856

Sphere 21.52Cylinder 1.541

Exp. 10.43


predictions for fractal clusters of 2, 10, 50, and 250spheres. As above, the refractive indices at the twowavelengths were calculated with the dispersionmodel of Lee and Tien11 at a temperature of 300 K1which corresponds to the temperature at which themeasurements were made2. Also presented in Table2 are results obtained for spherical and infinite-length particles with the same refractive indices.The electrostatics predictions can be seen to agreeverywell with themeasurements, with the experimen-tally measured ratio corresponding to that predictedfor an aggregate of between 10 and 50 spheres.Spherical and cylindrical particle predictions, on theother hand, yield ratios that are considerably largerand smaller than the experimental value, respec-tively.Although the agreement between measurements

and theory that is demonstrated in Table 2 is encour-aging, this simple comparison is certainly not meantto be a conclusive validation of the electrostaticsmodel of soot absorption. The most uncertain ele-ment in applying the theory to spectral absorption isthe assumed refractive index of the soot particles.For instance, the use of the refractive-index valuesthat were predicted at 1450 K yielded an absorptioncoefficient ratio of 13.72 for a 250-sphere aggregate,as compared with the value of 8.856 as given in Table2 for the refractive index at 300 K. In addition,the validity of the electrostatics approximation at the0.5145-µm wavelength is questionable, because thesize of the aggregate could be comparable with theincident wavelength. Nevertheless, the results dodemonstrate that the electrostatics model offers thepotential of a quantifiable relationship between thenumber of spheres in a soot aggregate and the IRspectral absorption of the soot. A detailed compari-son of the electrostatics predictions with the five-wavelength soot-absorption measurements presentedin Ref. 19 is planned for a forthcoming paper.

5. Conclusions

This paper offers a means of accurately predicting theabsorption and the scattering properties of aggre-gated soot particles in the Rayleigh limit. The keyconclusions of the paper can be summarized as fol-lows:

1a2 Even though a cluster of spheres will, in theRayleigh limit, behave radiatively as a dipole, severalharmonic orders may be necessary to describe theelectric field about each of the spheres. The requirednumber of orders increases with the magnitude of therefractive indices of the spheres.

1b2 For spheres that have refractive indices typi-cal of soot particles in the mid-IR wavelengths, theabsorption cross section of the cluster can be signifi-cantly greater than that predicted from the sum ofthe absorption cross sections of the individual, nonin-teracting spheres. At these wavelengths, the degreeto which the cluster absorption exceeds the indepen-dent sphere absorption increases with increasingwavelength. On the other hand, absorption crosssections for soot clusters in the visible-to-near-IR-

wavelength region are only slightly larger than thoseobtained from the sum of the individual spheres.

1c2 The absorption cross section of a DLA fractalcluster, relative to the independent spheres, does notappear to depend strongly on the fractal dimension ofthe cluster. It does, however, depend on the numberof spheres in the cluster up toNS < 100 spheres. Forlarger NS, the relative absorption of the cluster isindependent ofNS.

1d2 Provided that accurate estimates of the refrac-tive index of the soot particles can be obtained, theelectrostatics model offers a means to determine thenumber of spheres in soot aggregates from measure-ment of near- and mid-IR spectral absorption coeffi-cients.

This work was partially supported by NationalScience Foundation grant CTS-9108734. The au-thor has benefitted from helpful discussions withRichard Dobbins, Piotr Flatau, and Kirk Fuller.

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