characterization of size and structure of agglomerates and

Characterization of size and structure of agglomeratesand inhomogeneous particles via polarized light

M.P. MenguÈ c° *, S. Manickavasagam

Department of Mechanical Engineering, University of Kentucky, Lexington, KY 40506-0108, USA

Abstract

Characterization of size and structure of small particles is required in many ®elds, includingenvironmental and process control and monitoring, biological and pharmaceutical research, atmosphericremote sensing, as well as combustion systems. In this paper, the use of polarized light and the conceptof Mueller matrix elements for possible particle characterization studies is discussed. A summary of theresearch carried out in our laboratory for application to agglomerates and inhomogeneous spherical andcylindrical particles is presented. Sensitivity of the technique on a number of di�erent physicalparameters is outlined. Finally, an inverse solution methodology is discussed to identify particle/agglomerate characteristics. # 1998 Elsevier Science Ltd. All rights reserved.

1. Introduction

Characterization of small particles, such as those from 10 nm to 100 mm in size, is very

important in many diverse disciplines, including pharmaceutical and biological systems,

environmental and process control and monitoring, atmospheric and oceanographic remote

sensing, as well as combustion studies. The number of papers which appear each year in the

literature attests the importance of the problem. For example, Barth and Flippen in Ref. [1]

have reviewed the particle size analysis literature from 1991 to 1994, and cited more than 450

references. Still, they predicted that the need for more reliable and extensive particle

characterization methodologies will increase, particularly because of environmental concerns,

process control and monitoring demands, and the development of new materials that require

more complete and accurate characterization.

International Journal of Engineering Science 36 (1998) 1569±1593

0020-7225/98/$19.00 # 1998 Elsevier Science Ltd. All rights reserved.PII: S0020-7225(98)00049-4

PERGAMON

* Corresponding author. Tel.: 001606 257 2673; fax: 001 606 257 3304; e-mail:[email protected].

Light, or, in general, any kind of electromagnetic wave, is the most convenient tool for thecharacterization of particles. Everything we notice around ourselves either visually or via adetector is a direct result of interaction between light and matter. Matter absorbs the incidentlight selectively over the wavelength spectrum, which, in e�ect, yields the colors of the objectsas we see them. Emission of light is a direct result of (thermal) excitation of molecules whichmake up matter. And all inhomogeneities, whether they are individual gas molecules, particles,droplets, or liquid/solid surfaces, scatter (i.e. re¯ect, refract and di�ract) the light incident onthem.

The interaction between the light and matter is expressed mathematically via Maxwell'sequations. The solution of these equations requires information about the geometry of theinhomogeneity (e.g. a particle) and its electric and magnetic properties (optical properties) withrespect to its surroundings. For homogeneous spheres, for example, particle diameter, complexindex of refraction and the wavelength of the incident light are needed. Once these equationsare solved, it will be possible to determine how a speci®c particle absorbs and scatters theincident electromagnetic wave on it. The reverse of this idea also holds. If the exact relation forthe interaction between light and particle is known, then particle physical and opticalproperties can be determined from an inverse analysis using this relation, i.e. the Maxwellequations. This inverse problem is the backbone of all non-intrusive particle characterizationstudies.

Analytical solutions of the Maxwell equations are available for only a few speci®c cases.Among those, the Lorenz±Mie (LM) theory which describes the absorption and scattering oflight by homogeneous spheres, is the most well known [2]. If particles are much smaller thanthe wavelength of the light incident on them, the LM theory is further simpli®ed to theRayleigh approximation. The so called ``Rayleigh particles'' absorb light much stronger thanthey scatter it. Therefore, for radiative transfer calculations these particles are considered asabsorbing only. Their scattering characteristics can be used for diagnostic purposes,particularly to determine if they satisfy the ``small'' particle assumption. Very small particlesyield the same scattering pattern (widely known as Rayleigh scattering) regardless of theirshape.

Because of its relative simplicity and availability, the LM theory is widely usedto characterize particles in di�erent physical systems, even if the particles are nothomogeneous spheres. As one may expect, this mathematical simpli®cation is likely toyield physically inaccurate predictions. For many applications, this ``error'' may besmall, unimportant, or comparable with the signal-to-noise ratio of the detection system.However, there are applications where more precise monitoring of particle properties may beneeded.

Characterization of the size and structure of agglomerates is also needed for manyapplications, from the preparation of colloidal suspensions in biological sciences to thedetermination of soot volume fraction in combustion systems. If the individual particles(monomers) which make up an agglomerate are small, then relatively simple numericalapproaches can be adapted to determine the structure of these particles. These techniques havebeen reviewed by Charalampopoulos in Ref. [3], and more recently by Manickavasagam andMenguÈ c° in Ref. [4].

M.P. MenguÈc° , S. Manickavasagam / International Journal of Engineering Science 36 (1998) 1569±15931570

Angular distribution of light scattered by the particles has been used extensively forparticle characterization in the past [1, 2]. This approach is limited in nature as the numberof parameters which can be recovered cannot exceed the number of independentmeasurements. Although absorption and scattering cross sections, and scattering phasefunction can be determined, precise characterization of most particles based on these propertiesis not always possible. The spectral properties of light used in the experiments may help inimproving the accuracy in calculations. Unfortunately, the optical properties of materials are,in general, wavelength dependent, and for most materials this dependency is not readilyknown.

Light, however, has a unique property, called polarization, which can be employed todetermine more intricate details of particle structure. The polarization state of electromagneticwaves was ®rst described by Stokes in 1852, who formulated the electromagnetic wavepropagation using ``measurable quantities'' or ``observables'' rather than amplitudes of electricand magnetic ®elds. Stokes' approach was way ahead of its time. Indeed, the idea ofobservables did not appear in physics until 1925 when Werner Heisenberg used it forapplication to quantum mechanics, and later, in 1954, when Emil Wolf applied it to optics [5].Stokes' paper was forgotten until 1940, when Chandrasekhar used Stoke's formulation toinclude the e�ects of polarized light in the radiative transfer equation [6].

When light in some arbitrary polarization state is incident on a medium, the polarizationstate of the scattered or transmitted light can be related to the polarization state of the incidentlight by a 4 � 4 matrix. This matrix, which is called the Mueller or scattering matrix, is acharacteristic of the medium. It depends on the wavelength of incident light and, in the case oflight scattering, is a function of the scattering angle. It yields invaluable information about thephysical properties of the scattering particulates, and is a necessary and major part ofthe complete optical description of a medium. Yet, this idea was not widely used foridenti®cation of particle properties. Among the ®rst studies on the subject was that of Huntand Hu�man in Ref. [7]. More detailed literature reviews have been given by Kuik et al. inRef. [8], and Govindan et al. in Refs. [9, 10]. Only recently, this approach was applied to morecomplicated problems, including bacterial cells [11, 12], soot agglomerates [4, 9, 10] andphytoplanktons [13, 14].

In this paper, we will concentrate only on light scattering techniques which utilizethe polarization state of electromagnetic waves. After introducing the details of thetechnique, we will discuss its application for agglomerates and radially inhomogeneousspherical and cylindrical particles. We will comment on how the measurements of theseelements yield the necessary tools to identify the optical and physical characteristics of variousscatterers.

As mentioned before, the body of literature on the subject is vast and it is impractical to citeall the relevant papers here. We refer the reader to the following papers for relatively completeand up to date literature reviews: Barth and Flippen in Ref. [1] for the review on particle sizeanalyses; Manickavasagam and MenguÈ c° in Ref. [4] on characterization of agglomerates; Bhantiet al. in Ref. [15] for application to multi-layer spheres; Manickavasagam and MenguÈ c° inRef. [16] for multilayer cylinders; and Govindan et al. in Ref. [9, 10] for experimentaltechniques.

M.P. MenguÈc° , S. Manickavasagam / International Journal of Engineering Science 36 (1998) 1569±1593 1571

2. Theoretical background

If an electromagnetic wave is incident on a particle, it is absorbed and scattered. Theincident and scattered ®elds are related by [2]:

EksE?s

� �� eik�rÿz�

ÿikrs2 s3s4 s1

� �EkiE?I

� ��1�

where s1, s2, s3 and s4 form the amplitude scattering matrix, which depend on the scatteringangle and the orientation of the particle with respect to the incident ®eld vector. Ek and E_

represent the parallel and perpendicular components of the electric ®eld respectively.In the experiments, the intensity, or the time averaged value of the amplitude square is

measured. Therefore, it is more desirable to have a relation between the incident and scatteredintensities than the amplitudes. The intensity and state of polarization of a beam of light canbe completely speci®ed by the 4-element Stokes vector:

�K� �IQUV

0BB@1CCA �2�

Here, I represents the total intensity, Q the di�erence between the horizontally and verticallypolarized intensities, U the di�erence between the +458 and ÿ458 intensities, and V thedi�erence between the right-handed and left-handed circularly polarized intensities [2].The interaction of an inhomogeneity or an optical device with a beam of light can

be described as a transformation of an incident Stokes vector Ki into an emerging Stokesvector Ks:

�Ks� � �S� � �Ki�; �3�or

IsQs

Us

Vs

0BB@1CCA � 1

k2r2

S11 S12 S13 S14

S21 S22 S23 S24

S31 S32 S33 S34

S41 S42 S43 S44

0BB@1CCA

IiQi

Ui

Vi

0BB@1CCA; �4�

where the 4 � 4 matrix [S] is known as the Mueller or scattering matrix. Here, k = 2p/l is thewave number, and r is the distance between the scatterer and the detector.The scattering characteristics of a particle are described exactly by the Mueller matrix

elements. Therefore, if these matrix elements can be measured, one can determine the particlecharacteristics accurately. Note that each of the matrix elements is a function of the scatteringangle y. The Mueller matrix for a cloud of particles is the sum of the individual matrices foreach of the particles.


S11 is the di�erential scattering cross section of the scatterer and its magnitude changes withthe scattering angle. If normalized by the area under the S11±y curve, it represents the phasefunction of the scatterer (which is essential in estimating the distribution of radiation intensityin a medium containing scatterers). S12 is the measure of linearly polarized scattered light foran incident unpolarized light. In general, the angular pattern of S12 shows how much theagglomerate deviates from the Rayleigh scattering regime. S34 indicates how much of anobliquely polarized light (458) is transformed to circularly polarized light because of thescatterer. S33 and S44 have similar angular patterns.

The Sij parameters for a cloud of particles, which are symmetric and randomly oriented, canbe simpli®ed considerably as they are either related to each other or they are zero: S12=S21,S34=S43, S33=S44, S13=S31=S23=S32=S14=S41=S24=S42=0.

In addition to characterizing particles, the Mueller matrix also describes the characteristics ofoptical elements, such as polarizers and retarders. The Mueller matrices for an ideal linearpolarizer [P] and a retarder [R] are given as [2]:

�P� � 1

2

1 cos 2x sin 2x 0cos 2x cos2 2x cos 2x sin 2x 0sin 2x sin 2x cos 2x sin2 2x 00 0 0 0

0BB@1CCA; �5�

�R� � 1

2

1 0 0 00 C2 � S2 cos d SC�1ÿ cos d� ÿS sin d0 SC�1ÿ cos d� S2 � C2 cos d C sin d0 S sin d ÿC sin d cos d

0BB@1CCA; �6�

where C = cos 2b, S= sin 2b. Here x is the angle between the transmission axis of thepolarizer and the parallel axis of the incident beam, b is the angle between the parallel axis ofthe retarder and the horizontal, and d is the retardance of the retarder.

For example, assume that polarization characteristics of a beam is modi®ed by a polarizer[P1] and two retarders [R1] and [R2] before it is incident on a cloud of particles (see Fig. 1).After the light is scattered, it propagates through another retarder [R3] and a polarizer [P2].Then, the Stokes vector of scattered light can be related to that of incident light by:

�Ks� � f�P2� � �R3� � �S�yyy�� R2� � �R1� � �P1�g � �Ki�: �7�

For a cloud of particles that have a plane of symmetry and are randomly oriented, the Muellermatrix has six independent parameters and reduces to:

�S�yyy�� 1

k2r2

S11 S12 0 0S12 S22 0 00 0 S33 S34

0 0 ÿS34 S44

0BB@1CCA �8�


For small Rayleigh spheres, the Mueller matrix is given as:

�S�yyy��Rayleigh �A

k2r2

12 �1� cos2 y� 1

2 �cos2 yÿ 1� 0 012 �cos2 yÿ 1� 1

2 �1� cos2 y� 0 00 0 cos y 00 0 0 cos y

0BB@1CCA; �9�

where

A � x6��m2 ÿ 1

m2 � 2

��2: �10�

Here, x= kD/2 = pD/l is the size parameter, D is the diameter of the spherical particle, l is

the wavelength of the incident beam, and m is the complex index of refraction of the particle

with respect to the medium.

A general expression for the output intensity I {see [K] matrix, Eq. (2) is obtained in terms

of the variables x and b of polarizers and retarders. Since our objective is to identify the

structure of the particles from the measurement of six Mueller matrix elements, six di�erent

expressions are needed. This leads to a system of equations of the form:

�C��Z� � �B�; �11�

�Z� �

S11

S12

S22

S33

S34

S44

0BBBBBB@

1CCCCCCA: �12�

Fig. 1. Schematic of the experimental system used to measure Sij parameters.


Here [C] is the matrix consisting of the coe�cients of the various Mueller matrix elementsobtained using di�erent combinations of the polarizer angles x and the retarder angles b, and[B] is the vector whose elements are the intensities obtained using several polarizer and retarderangle combinations. [C] matrix elements are obtained using a Matlab symbolic manipulator forthe di�erent optical components used [10, 13].Govindan et al. in Ref. [9] have described a procedure to determine an optimum

combination of x and b angles, which allows the measurement or the Mueller matrix elementsaccurately. The criterion used in selecting these angles is based on the condition number.Below, we brie¯y discuss this procedure for the sake of completeness.For a given system of linear equations, the e�ect of the right hand side of the system (in our

case, [B]) on the relative error of solution vector ([Z]) is a function of the condition number ofthe coe�cient matrix [C]. The condition number for matrix [C] is de®ned as CN=k [C]k k [C]ÿ1k and provides a measure of how the relative residual k [B]ÿ [B1] k / k [B] k re¯ects the relativeerror k [Z1]ÿ [Z] k / k [Z] k of the perturbed system of linear equations. If CN is small, then therelative residual and relative errors are proportional to each other. Therefore, our goal is tohave a set of polarizer and retarder angles which yield least error in the Mueller matrixelements of the particles considered.Various random sets of retarder and polarizer angles are chosen to form several di�erent [C].

The relative error is evaluated for di�erent x and b angles. The coe�cient matrix [C] isgenerated by using six sets of polarizer and retarder angles, thus generating a 6 � 6 matrix.The product of [C] and [Z] is evaluated, and [B] is obtained.For a condition number CN of 10, the maximum relative error on S11 was about 0.1%, and

climbed up to 50% for CN=10000 [9]. The relative error on recovered S12, on the other hand,was more sensitive to the condition number of [C]. For CN=10, the maximum relative errorwas about 10%, and for all the other cases it was more than 100%. They concluded that it waspossible to recover the required Mueller matrix elements with acceptable accuracy from theexperiments, if the polarizer and retarder angles were chosen properly.

3. Applications to di�erent particles

In the previous section, we discussed the theory behind the Mueller matrix approach, andconcluded that the scattering matrix elements for particles can be determined fromexperiments. The next question we can ask is: ``How sensitive are the Mueller matrix elementsto the structure of agglomerates and to the inhomogeneity in spherical and cylindricalparticles?`` In this section, we will present the scattering matrix elements determined for threedi�erent particles and investigate their sensitivity to various physical parameters.

3.1. Scattering characteristics of agglomerates

In ¯ames, soot particles are usually present in the agglomerated form. If the soot volumefraction is to be determined in a ¯ame non-intrusively using laser diagnostics, then we shouldhave a good feeling about the structure of these agglomerates.


Here, a few sets of comparisons will be presented to highlight how the the scattering matrixelements may change because of variations in the morphology and the refractive index of sootagglomerates. The e�ect of wavelength of incident radiation on soot agglomerate scatteringcharacteristics is also investigated.For the models discussed below, soot agglomerates with a fractal dimension of Df=1.8 and

cofactor of Kf=5.8 are considered. Fractal structure is described as:

N � KfRg

dp

� �Df

; �13�

where N and dp are the number and diameter of primary spheres, respectively, and Rg is theradius of gyration of the agglomerate [4]. Fig. 2 depicts two di�erent views of a typicalagglomerate used in this study (N= 75). In the calculations, we have considered N values of25, 50, 75, 100, 125 and 150, and monomer diameters (dp) of 20, 40 and 60 nm. Wavelength ofthe incident radiation (l) was taken as 266 or 1000 nm (roughly corresponding to thoseavailable from a Nd:YAG laser).The orientational averaging was performed by rotating the incident ®eld vector with respect

to the particle. The number of orientations considered is 490 for these calculations; thisnumber is based on the accuracy of the most severe cases, which is for the largest agglomerate(150 monomers, with dp=60 nm) at the shortest wavelength (l= 266 nm).

Fig. 2. Two views of a typical soot agglomerate model considered in the calculations; N= 75.


Fig. 3 depicts the angular variation of S11 as a function of N and dp at wavelengths of 266and 1000 nm. For each wavelength, the value of S11 near the forward direction increases by anorder of magnitude as dp goes up from 20 to 40 nm. Numerical results show that it is possibleto recover the S11 pro®le within 1% even if there is 7% error in the experimental data. Thissensitivity suggests that an inverse technique can be developed for obtaining the values of Nand dp, by measuring angular variation of S11, provided that the complex index of refraction isknown.The angular variation of S12 for various N, dp and l values are shown in Fig. 4. The

maxima of S12 gradually shifts from near 608 to forward angles as N is increased. The samebehavior is observed at a given wavelength for increasing dp. For example, in Fig. 4, themaxima occurs near 458 for N= 75 and dp = 20 nm. As dp is changed to 40 nm (for N= 75),the maxima shifts to angles slightly smaller than 308.Functional dependence of the matrix element S34 with respect to N, dp, l and scattering

angle is given in Fig. 5. The most striking feature seen in this ®gure is that with increasingparticle diameter dp, the dip in the pro®le shifts towards the forward angles, and is independent

Fig. 3. Angular pro®les of S11 for di�erent size agglomerates. For two di�erent monomers (dp of 20 and 40 nm),

and at two di�erent wavelengths (266 and 1000 nm).


of the value of N. This also implies that the polydispersity in agglomerate structure may nota�ect this dependency. Another important observation is that with the decreasing sizeparameter (i.e. increasing wavelength), the value of S34 decreases dramatically, suggesting thatultraviolet wavelengths may be more useful for measuring this component.The pro®le of the S12 shown in Fig. 4 suggests the possibility of determining N, if we know

the primary sphere diameter (dp) a priori. The dp value can be obtained from the S34 pro®leindependently, allowing the simultaneous determination of both N and dp. Also, the dp can bemeasured from electron microscope pictures, as the intrusive particle collection techniques arenot likely to alter the individual monomer size. Additionally, measurements performed atlonger wavelengths allow a crude estimation of the primary monomer diameter. It can beobserved from Figs. 4 and 5 that at the wavelength of 1000 nm, the pro®le becomesasymmetric about 908 as dp is increased. For 20 nm monomers, the symmetry is observedalmost independent of N. However, for larger size agglomerates, predictions based on thisbehavior may not be reliable.

Fig. 4. Angular pro®les of S12 for di�erent size agglomerates. For two di�erent monomers (dp of 20 and 40 nm),and at two di�erent wavelengths (266 and 1000 nm).


The e�ect of fractal dimension Df on angular variations and magnitudes of S11 and S12 isshown in Figs. 6 and 7. It is observed that the change in the fractal dimension does not a�ectS11 signi®cantly. On the other hand, S12 varies as much as 100% as a function of the fractaldimension.Note that here we did not present any results for the angular behavior of S44, even

though this information can also be used as a diagnostic tool. In general, the S33 and

Fig. 5. Angular pro®les of S34 for di�erent size agglomerates. For two di�erent monomers (dp of 20 and 40 nm),and at two di�erent wavelengths (266 and 1000 nm).


S44 pro®les are similar, and there is no need to display the additional data here. However,a thorough investigation of the behavior of all Sij elements can provide a diagnostic toolto determine the fractal dimension of large agglomerates. More details on the behaviorof Mueller matrix elements for di�erent size agglomerates are available in the literature inRef. [4].

Fig. 6. E�ect of monomer size and fractal dimension of agglomerates on the angular pro®le of S11. Lower set ofthree curves for l = 1000 nm, and upper set for l = 266 nm.


3.2. Scattering characteristics of multilayer spheres

In this section, we will present the scattering matrix elements determined for a burning coalparticle surrounded by a soot cloud and a water droplet covered by a soot±air mixture. Therefractive index for coal was taken as 1.8 + 0.08 i; for water it was assumed as 1.33 + 0i. For

Fig. 7. E�ect of monomer size and fractal dimension of agglomerates on the angular pro®le of S12. Lower set ofthree curves for l = 1000 nm, and upper set for l = 266 nm.


a soot and air mixture forming the outer layer, the Maxwell±Garnett relation was applied to

®nd an e�ective index of refraction. The refractive indices of soot and air were taken as

1.75 + 0.75i and 1.0 + 0.0i, respectively.

In the ®rst case, the volume fraction of soot was assumed to vary radially in the outside

layer according to the relation:

fv�r� � 1=4�1ÿ r=aL�fv; �14�where fv is either 0.1 or 0.01, r is the radial distance, and aL is the maximum radius of

the multilayered sphere. To account for the radial inhomogeneity, the outer layer was

divided into 100 sub-layers, in each of which the refractive index was assumed to be

uniform. The S11 and S12 elements were then obtained using a computer algorithm developed

by Bhanti et al. in Ref. [15], which was based on the earlier work of Mackowski et al. in

Ref. [17].

In the numerical calculations, the size parameter of the core (xcore) was taken as 100. Soot

and air mixture, which surrounds the coal (core), was assumed to be present in such an extent

as to yield a total size parameter (xc + s + a) of between 100 and 200 (subscripts c, s and a

correspond to coal, soot and air, respectively). Only the Mueller matrix elements, S11, S12, S33

and S34 were obtained because of the symmetric nature of the particle. (For spherically

symmetric particles, S33=S44.)

Fig. 8 depicts the angular variations of S11, S12 and S33 as a function of xc + s + a, for a

given xcore (= 100). Beyond 408, the magnitude of S11 remains almost constant. As xc + s + a

is increased from 120 to 200, which corresponds to an increase in the amount of soot which

surrounds the coal, the value of S11 decreases by 50%. This shows that if scattering

measurements were made carefully for a single coal particle, the rate of soot growth around

the core can be predicted. However, for a cloud of particles, this may not be possible.

The magnitude of S12 decreases as the amount of soot surrounding the coal is increased. At

angles smaller than 608, there is su�cient variation in the magnitude of S12, which can be

measured in experiments. On the other hand, S33 is also sensitive to soot layer, and increasing

xc + s + a yields a decrease in the magnitude of S33. Note that the magnitude of S33 remains

constant beyond 1008.Fig. 9 depicts the e�ect of soot layer deposited on a water droplet, which has a size

parameter of 100. It is assumed that soot accumulates on the droplet and causes an increase in

the e�ective size parameter (up to 140). The outer layer on the droplet is assumed to be a

mixture of soot and air, with ®xed soot volume fraction (see the inset in Fig. 9). The results

presented correspond to a single scattering angle of 458. S11 and S33 pro®les in Fig. 9 show

that the accumulation of soot on the droplet can be traced, particularly if the volume fraction

of soot is larger than 1%.

These results suggest the possibility of predicting the growth of the soot layer around

a burning coal particle or a water droplet from the angular pro®les of the Mueller

matrix elements. The formulation of this problem and the additional details are given in in

Ref. [15].


3.3. Scattering characteristics of cylindrical ®bers

In this section, the scattering matrix elements obtained for radially inhomogeneous in®nite

cylinders are discussed. It is assumed that a planar wave is incident on the ®ber normal to its

axis. The Maxwell equations for the light propagation through the cylinder is solved following

a methodology discussed by Barabas in Ref. [18] and Swathi et al. in Ref. [19]. In this

Fig. 8. E�ect of the size of soot cloud about a coal particle on the angular pro®les of S11, S12 and S33. Sootconcentration distribution in the outer layer varies radially (see Eq. (13).


approach, the inhomogeneous cylinder is discretized into multiple layers and a closed formsolution is obtained for multilayer coated cylinders. From the intensity and polarization stateof scattered light, the Mueller matrix elements are obtained [16].Here, only the results for cylinders with two layers (inner core and an outer coat) are

reported. The refractive indices of inner and outer layers are 1.1±0.01i and 1.4±0.1i,respectively. These values of refractive indices characterize a typical cotton ®ber, which is

Fig. 9. E�ect of the thickness of soot layer on a water droplet on the angular pro®les of S11, S12 and S33. Di�erent,®xed soot volume fractions are considered (see the inset); scattering angle is 458.


modeled as a two layer in®nite cylinder. The size parameters of the cylinder (based on theouter diameter) is allowed to vary from 0.1 to 5.0. For each size parameter considered, theratio of the inner to outer radius (f = ri/ro) is varied from 0.999 to 0.900. Note that an increasein f corresponds to a decrease in the thickness of the cylinder coating. Hence, the resultsobtained for the coated cylinder of a given size parameter, will indicate if the scattering matrixelements are sensitive to a change in the thickness of the coating.Figs. 10±13 depict the angular variations of S11, S12, S33 and S34 elements, respectively, for

size parameters of 0.1, 0.5, 1.0 and 5.0. For the ®rst three size parameters, the f value has asigni®cant measurable e�ect on all the elements. For the smallest size parameter, however, themagnitude of the Sij parameters is very low, making it di�cult to measure. On the other hand,for x= 5, it is not possible to have any meaningful correlation between the angular pro®les ofthe Sij elements and the f parameter.These results show that if the index of refraction and the outer diameter of the ®bers are

available, the Mueller matrix elements may yield a reasonable clue about the coating thickness,particularly for size parameters up to one. This information can be correlated to the maturityof cotton, which is a desirable quantity to assess its quality. Additional details of the e�ect ofdi�erent parameters on the Mueller matrix elements are available in Ref. [16].

4. Inverse analysis

So far, we have discussed a procedure to determine the Sij elements of di�erent shapeparticles. In the inverse problem, the scattering matrix elements are assumed to be measuredfrom experiments and the parameters which would yield the measured values for theseelements are recovered. This is achieved by following an optimization algorithm, which solvesthe forward problem repeatedly to arrive at optimal values of the physical parameters. In thiscase, solution to the forward problem requires intensive computer time. For this reason, adatabase is created which contains Sij elements for several di�erent agglomerates with di�erentvalues of N and dp. The database thus developed along with experimentally measured Sij serveas input to the inverse problem. Note that although we will only discuss the inverse problemfor agglomerates, the same ideas can be extended to other particles easily.In order to solve the inverse problem, a modi®ed Levenberg±Marquardt optimization

algorithm is employed. Since the number of measurements is always greater than the numberof unknown parameters, the system of equations to be solved yields an overdetermined system.Hence, the inverse solution is obtained by minimizing the least squares norm given as:

R �XMi�1�Si ÿ Si�N; dp��2 � FT F ; �15�

where i = 1..M represents the locations (angles) at which the measurements were made, and Tdenotes the transpose. F is de®ned as:

F � Si ÿ Si�N; dp�; �16�


where SÃi are the measured scattering matrix elements, and Si are the values obtained from thedatabase. In order to minimize R, the equation is di�erentiated with respect to each of theunknown parameters, and search directions for unknown parameters are determined. Theresulting expression, which computes the search direction for the variables, is given as:

Fig. 10. Angular pro®les of S11 for di�erent size coated cylinders. f is the ratio of the inner cylinder radius to outer

cylinder radius. The size parameter is based on outer cylinder diameter.


d � �J T J � mI�ÿ1J T F �17�where J is the Jacobian. The algorithm begins with a set of initial values for the

unknown parameters and the iterative procedure is continued until the desired convergence is

achieved.

Fig. 11. Same as Fig. 10, for S12.


For the present case, we obtain Sij elements for a cloud of agglomerates with a known

distribution of N from the database. Then, these data are considered as the experimental data

and are employed as input to the inverse problem. This approach is acceptable as long as the

forward problem models the true physics of the physical system considered. In order to



simulate the experimental conditions, a random error was introduced to the Sij distribution

before it was given as input to the inverse problem. Then, from the inverse analysis, the

distribution function for N is recovered.



4.1. Agglomerate size distribution function

Agglomerates generated in a typical di�usion ¯ame have a broad size distribution. Hence, itis important to measure the dispersion of the distribution (i.e. the standard deviation).Furthermore, for calculations, a suitable mathematical function is needed to describe thedistribution of N.The two-parameter log normal function is often used to represent broad size distributions.

For this case, it is preferable to use a modi®ed form of such a function, truncated outside themeasurement limits Na and Nb. The di�erential number (or count) distribution is the fractionof agglomerates with N within d(ln N), and is given by:

dF�Na;Nb��N�d�lnN� �

Z�N�� lnNb

lnNaZ�x�d�ln x�

; NaRNRNb; �18�

where

Z�x� � 1

�2p�1=2 ln sgexp ÿ 1

2

ln x=Nn

ln sg

� �2" #

; �19�

with x being a dummy variable used for N. In the limit Na40 and Nb41, the denominatorbecomes unity and the distribution shown in Eq. (18) approaches the standard form of thecommon (untruncated) distribution de®ned by:

dF�N�d�lnN� � Z�N�: �20�

This size distribution suggests that half of the agglomerates in a cloud has the number ofprimary spheres N less than the median, Nn. The geometric standard deviation, sg, is ameasure of the breadth of the distribution. About 68% of the particles have N values betweenNn/sg and sgNn. A value of s>2 indicates a very broad distribution, while a closelymonodisperse cloud has sg11.

4.2. Numerical procedure for inverse analysis

A typical algorithm which can be used in inverse analysis is outlined below:

. Sij (y) elements are calculated for di�erent physical cases and used as input to the inverseproblem. In our case, S11, S12, S33 and S34 elements obtained between 10 and 1708 areconsidered. Required parameters can be recovered either by considering only one of theseelements as input, or any combinations of them.

. The number of particles an agglomerate is comprised of is considered to follow a sizedistribution de®ned by a log normal function. Two of the four parameters, Na and Nb, ofsuch a truncated distribution are treated as known. Only Nn and sg are unknown parametersto be determined.


. The primary sphere diameter dp is assumed to be known as it can be obtained from theelectron-microscope pictures.

. Any numerical optimization method can be used to determine the parameters which yieldthe measured quantities as closely as possible.

. A more thorough inverse analysis should assume all parameters, including Df, Kf, dp, Na andNb, as unknown, which is a formidable task.

Following this approach, the size distribution of the agglomerates are calculated. When there isno error in the input data, the inverse algorithm returns the exact values. However, when thereis any error in the input data, then because of the ill-conditioned nature of inverse problemsthe accuracy of the results decreases. A more detailed analysis of this inverse algorithm will beoutlined in a forthcoming paper.

5. Concluding remarks

In this paper, we discussed the possible use of the Mueller matrix elements for thecharacterization of size and structure of small particles. First, we outlined a methodology todetermine scattering patterns of three di�erent types of particles. Results presented foragglomerates, radially inhomogeneous spheres and cylinders showed that Mueller matrixelements reveal much more information about particle characteristics than those based onclassical light scattering techniques. Following this, an inverse algorithm was outlined todetermine the size distribution of agglomerates in ¯ames.Based on the the discussions given here and in our parallel publications, we can draw the

following general conclusions:

. The use of polarized light and the concept of Mueller matrix elements allows us tocharacterize particles with di�erent shapes, sizes and structures. This approach requires aseries of angular measurements and a robust inversion algorithm.

. The experiments that need to be conducted for this purpose are not di�cult, although theyare time consuming. As shown in Fig. 1, the polarization of both the incident and scatteredbeams needs to be modulated, and the measurements should be made at a number ofangular orientations. The use of CCD arrays as detectors may decrease the time required tosweep the entire angular domain.

. The inverse analysis involves the mapping of experimental results onto the theoretical dataobtained for angular Sij pro®les. As in any inverse analysis, the problem is ill-conditioned,and may lead to more than one converged solution. The calculations showed that if therewere no error in the experimental data, then the inverse procedure described in this paperreturns the exact results. However, if there is any random error imposed on the ``inputdata,`` the convergence to exact results may slow down.

. One of the main di�culties we faced in the inversion analysis was because of the order-of-magnitude di�erences between some of the Sij parameters. For example, the S34 pro®le has awealth of information; however, the magnitude of S34 at a given angle is much smaller thanother Sij values. In the inversion, as well as in data collection, this discrepancy a�ects the


reliability of the S34 data signi®cantly. Accuracy of inversion procedure can be increased ifan alternative algorithm is developed where di�erent objective functions are used fordi�erent Sij parameters.

. For agglomerates, the critical parameters needed are N (number of primary spheres in anagglomerate), dp (primary sphere diameter), Df (fractal dimension) and Kf (fractal prefactor).In this study, we assumed ®xed values of Df=1.8 and Kf=5.8, and recovered the N and dpparameters. Note that the knowledge of N is directly related to the volume fraction of sootparticles/agglomerates in a ¯ame and a�ects the radiative transfer calculations signi®cantly.

. In order to expedite the inversion procedure, it is preferable to obtain the information aboutthe soot monomer size dp via ex-situ techniques. An intrusive method is likely to alter thesize and structure of the soot agglomerates (i.e. N, Df and Kf); however, it may not a�ect theprimary particle diameter. Having this information from another measurement will help theexperimentalists to evaluate the distribution of agglomerate size (i.e. N distribution)accurately.

. Agglomerate calculations show that a careful measurement of S34 may allow the independentdetermination of dp. However, the magnitude of S34 is quite small compared with othermatrix elements, and because of this the corresponding experimental data are more prone toerror.

Acknowledgements

This research project has been supported by the DOE-PETC Advanced University CoalResearch Program Grants Nos. DE-FG22-PC92533 and DE-FG22-PC93210.

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characterization of size and structure of agglomerates and

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