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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326

http://d0022-40

n CorrE-mURL

journal homepage: www.elsevier.com/locate/jqsrt

Absorption and scattering by fractal aggregates and by theirequivalent coated spheres

Razmig Kandilian, Ri-Liang Heng, Laurent Pilon n

Mechanical and Aerospace Engineering Department, Henry Samueli School of Engineering and Applied Science, University of California,Los Angeles, Los Angeles, CA 90095, USA

a r t i c l e i n f o

Article history:Received 3 September 2014Received in revised form14 October 2014Accepted 23 October 2014Available online 1 November 2014

Keywords:ScatteringFractal aggregatesSuperposition T-matrixEquivalent particleCoated sphere

x.doi.org/10.1016/j.jqsrt.2014.10.01873/& 2014 Elsevier Ltd. All rights reserved.

esponding author. Tel.: þ1 310 206 5598; faail address: pilon@seas.ucla.edu (L. Pilon).: http://www.seas.ucla.edu/�pilon/ (L. Pilon

a b s t r a c t

This paper demonstrates that the absorption and scattering cross-sections and theasymmetry factor of randomly oriented fractal aggregates of spherical monomers can berapidly estimated as those of coated spheres with equivalent volume and averageprojected area. This was established for fractal aggregates with fractal dimensionranging from 2.0 to 3.0 and composed of up to 1000 monodisperse or polydispersemonomers with a wide range of size parameter and relative complex index of refraction.This equivalent coated sphere approximation was able to capture the effects of bothmultiple scattering and shading among constituent monomers on the integral radiationcharacteristics of the aggregates. It was shown to be superior to the Rayleigh–Debye–Gans approximation and to the equivalent coated sphere approximation proposed byLatimer. However, the scattering matrix element ratios of equivalent coated spheresfeatured large angular oscillations caused by internal reflection in the coating whichwere not observed in those of the corresponding fractal aggregates. Finally, thescattering phase function and the scattering matrix elements of aggregates with largemonomer size parameter were found to have unique features that could be used inremote sensing applications.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Particle aggregation and coagulation is a frequent occur-rence in numerous applications such as combustion systems[1], atmospheric science [2,3], astronomy and astrophysics[4], chemistry [5], and biotechnology [6–8]. Small particlesaggregate to form fractal-like structures changing the lightabsorption and scattering properties of the suspension [2].For example, Fig. 1a, b, c, d, e, and f shows micrographs ofsoot [9], snow [10], cosmic dust [11], gold nanoparticles [12],bacteria [13], and microalgae aggregates, respectively. In allthese systems, knowledge of the radiation characteristics of

x: þ1 310 206 4830.

).

the fractal aggregates are of prime importance for radiationtransfer analysis and remote sensing applications. Theradiation characteristics of soot and aerosol aggregates havebeen studied extensively as reviewed by Sorensen [2].However, to the best of our knowledge, those pertaining toaggregates composed of larger monomers such as micro-algae colonies have not been studied.

Microalgae are single cell photosynthetic microorganismsgrowing in freshwater or seawater. They can be grown inphotobioreactors (PBRs) exposed to solar radiation to producebiofuels as well as various pharmaceuticals and biochemicals[14]. For example, Botryococcus braunii (Fig. 1f) can be used forproducing biofuels for powering jet engines [14]. This speciessecretes exopolysaccharides (EPS), a viscous substance coatingthe cell surface and causing their aggregation into colonies.EPS production is part of a protection mechanism activated inresponse to environmental conditions such as limited

50 nm

50 µm2 µm

1 mm

50 nm

Fig. 1. Micrographs of fractal aggregates of (a) soot [9], (b) snow [10], (c) cosmic dust [11], (d) gold nanoparticles [12], (e) the bacteriaM. luteus [13], and (f)colonie of the microalgae B. braunii.

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326 311

illumination [15], non-optimal temperature [16], high salinity[16], and limited nutrient availability [17]. In addition, a recentstudy demonstrated reversible cell aggregation in concen-trated Chlorella vulgaris cultures used for protein, starch, andlipid production [18]. The authors hypothesized that aggrega-tion occurred at large cell concentration due to the proximityof the cells to one another. Since larger microalgae cellspossess a larger surface charge density compared to smallercells, the electrostatic repulsion between larger cells is muchstronger than that between a large and a small cell [18]. Thisleads to aggregation of the smaller cells in the space createdby the electrostatic repulsion between the larger cells.

To achieve maximum biomass and biofuel productivities,light transfer in the PBRs must be optimized [19–21]. Forexample, a flat-plate PBR should be designed and operatedsuch that the fluence rate at the backwall corresponds tothe photosynthetic compensation point, i.e., the minimumamount of energy required to maintain cell metabolism [21].The latter was reported to be 10 μmolhν=m2 s for the micro-algae Chlamydomonas reinhardtii [22] and 2 μmolhν=m2 s forthe cyanobacteria Arthrospira platensis [19]. Optimizing PBRsfor maximum biomass productivity requires the solution tothe radiative transfer equation and the knowledge of theabsorption and scattering cross-sections as well as the scatter-ing phase function of the microalgal suspension [20]. More-over, the scattering matrix elements of the microalgal sus-pension may be measured for remote sensing of the PBR [23].The radiation characteristics of suspensions composed ofsingle cells can be predicted theoretically [24–27] ormeasuredexperimentally [20,28]. However, theoretical or experimentalcharacterization of the radiation characteristics of suspensionsconsisting of microalgae colonies have received less attention.

Several numerical methods exist to estimate the radiationcharacteristics of aggregates consisting of spherical mono-mers. They include the superposition T-matrix method [29],the generalizedmultiparticle-Mie theory [30], and the volumeintegral method [31]. However, depending on the size of theaggregate, calculations can be time consuming and requiresignificant computational resources [4]. Thus, it would becomputationally far more efficient to approximate the radia-tion characteristics of aggregates with complex morphologyby those of particles with simple shapes such as spheres,coated spheres, or cylinders whose radiation characteristicscan be computed relatively rapidly [32]. For example, Drolenand Tien [33] approximated the absorption and scatteringcross-sections of soot particle aggregates as those of a volumeequivalent solid sphere. More recently, Lee and Pilon [34]demonstrated that the absorption and scattering cross-sections per unit length of randomly oriented linear chainsof spheres can be approximated as those of randomlyoriented infinitely long cylinders with equivalent volumeper unit length. Alternatively, the Rayleigh–Debye–Gans(RDG) approximation provides an analytical expression forthe absorption and scattering cross-sections and the scatter-ing phase function of aggregates based on the assumptionthat the size of the monomers is much smaller than theincident radiation wavelength [35].

This study aims to identify approximations and equiva-lent particles for rapidly and accurately predicting theabsorption and scattering cross-sections and, if possible,the scattering matrix elements of fractal aggregates com-posed of relatively large monodisperse or polydisperseoptically soft spherical monomers. The goal is to facilitatethe predictions of radiation characteristics of fractal

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326312

aggregates with large monomers compared with the wave-length of the incident radiation where numerical methodsare too time consuming and resource intensive for practicalpurposes.

2. Background

2.1. Modeling fractal aggregates

An ensemble of Ns monodisperse spherical monomersof radius a aggregated in a fractal structure satisfies thestatistical rule given by [2]

Ns ¼ kfRg

a

� �Df

ð1Þ

where Rg is the aggregate radius of gyration defined as themean-squared of the distances between the aggregatecenter of mass and the geometric center of each particle.The constants Df and kf are the so-called fractal dimensionand prefactor, respectively. For example, ordered linearchains of spheres (1D), square sheets (2D), and simplecubic (3D) aggregates have fractal dimensions Df of 1.0, 2.0,and 3.0, respectively. On the other hand, fractal dimensionDf of random aggregates depends on the aggregationmechanism [2]. Aggregation by collisions due to Brownianmotion can result in diffusion limited aggregation or inreaction limited aggregation [2]. In diffusion limitedaggregation, particles collide and immediately stick tothe surface of the aggregate [7]. These aggregates typicallyhave fractal dimension Df of 1.75–1.8 [7]. In reactionlimited aggregation, particles may penetrate the aggregatewhen they collide without immediately sticking to thesurface. These aggregates feature fractal dimension Df

equal to 2.2 [7]. In addition, Jackson et al. [8] experimen-tally found that the fractal dimension of phytoplanktonicsuspensions ranged from 2.25 to 2.36. The prefactor kf istypically treated as a fitting constant also known as thepacking factor [7]. It has been reported to range between1.2 and 3.0 for soot particles [35–38]. Sorensen [2] andLapuerta [39] assumed a value of kf for aggregates of sootparticles to be 1.593 corresponding to the most compactpacking of monodisperse monomers.

Fractal aggregate formation can be numerically simulatedusing particle cluster aggregation algorithm [2,40,41]. Thismethod launches a spherical particle of radius a on a randomwalk. A collision between the marching particle and theaggregate leads to adherence of the two if the new aggregatestructure satisfies the prescribed prefactor kf and fractaldimension Df [40]. This procedure is repeated until theaggregate contains the desired number of monomers Ns.

2.2. Scattering matrix

The Stokes vector is composed of the four Stokesparameters I, Q, U, and V describing the intensity anddegree of polarization of an electromagnetic wave [42]. Fora given aggregate whose center of mass is located atthe origin of a spherical coordinate system, the far-fieldscattered intensity at location r in direction s is denoted byIscaðr; sÞ ¼ ðIsca;Qsca;Usca;VscaÞT can be related to theincident irradiance IincðsiÞ ¼ ðIinc;Qinc;Uinc;VincÞT by the

Mueller matrix ½ZðΘÞ� according to [29]

Isca r; s� �¼ 1

r2Z � �� �

Iinc si� � ð2Þ

Here, r is the norm of the location vector r correspondingto the distance between the particle and the observationpoint. The scattering angle Θ is defined as the anglebetween the incident si and the scattered s directions.For randomly oriented aggregates, it is more convenient touse the normalized scattering, or Stokes, matrix ½FðΘÞ�given by [43]

F Θ� �� �¼ 4π

⟨Casca⟩

Z Θ� �� � ð3Þ

where ⟨Casca⟩ is the orientation-averaged scattering cross-

section of the aggregate. It is defined as the fraction of theunpolarized radiant energy incident on the surface of therandomly oriented aggregate that is scattered in anydirection [44]. Similarly, the orientation-averaged absorp-tion cross-section ⟨Ca

abs⟩ represents the fraction of theunpolarized radiant energy incident on the surface of theaggregate that is absorbed [44]. The orientation-averagedextinction cross-section ⟨Ca

ext⟩ is defined as the sum of theabsorption and scattering cross-sections, i.e., ⟨Ca

ext⟩¼⟨Ca

abs⟩þ ⟨Casca⟩ [44]. For randomly oriented aggregates with

a plane of symmetry, the scattering matrix ½FðΘÞ� can beexpressed as [44]

FðΘÞ� �¼F11ðΘÞ F21ðΘÞ 0 0F21ðΘÞ F22ðΘÞ 0 0

0 0 F33ðΘÞ F34ðΘÞ0 0 �F43ðΘÞ F44ðΘÞ

266664

377775 ð4Þ

The asymmetry factor of an aggregate is defined as [42]

g ¼ 14π

Z4πF11 Θ� �

cosΘ dΩ ð5Þ

It is equal to 0, 1, and �1 for isotropic, purely forward, andpurely backward scattering, respectively [44]. In addition,the Henyey–Greenstein (HG) approximate phase function isoften used in radiative transfer analysis for its simplicity asit predicts F11ðΘÞ only as a function of the asymmetry factorg according to [45]

F11;HG Θ� �¼ 1�g2

½1þg2�2g cosΘ�3=2 ð6Þ

Several analytical and numerical methods exist for predict-ing the absorption and scattering cross-sections of aggre-gates as well as their scattering matrix elements asdiscussed in the following sections.

2.3. The Rayleigh–Debye–Gans (RDG) approximation

Monomers of radius a in a given aggregate are char-acterized by (i) their size parameter defined asχs ¼ 2πaλ�1 where λ is the wavelength of radiation invacuum and (ii) their relative complex index of refractionm¼ nþ ik defined as the ratio of the complex indices ofrefraction of the monomers ms ¼ nsþ iks and of the refrac-tive index nm of the non-absorbing surrounding medium[35]. The Rayleigh–Debye–Gans (RDG) approximation pro-vides a closed-form analytical expression for absorption

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326 313

and scattering cross-sections as well as for the scatteringphase function of randomly oriented fractal aggregates[2,35,46]. It is valid for aggregates composed of opticallysoft monomers such that jm�1j51 with small size para-meters, i.e., χs51. Under these conditions, the aggregateabsorption cross-section ⟨Ca

abs;RDG⟩ is the sum of the absorp-tion cross-sections of all constituent monomers and isexpressed as [2,35]

⟨Caabs;RDG⟩¼Ns⟨Cabs;R⟩ ð7Þ

Here, ⟨Cabs;R⟩ is the absorption cross-section of a singlespherical monomer of size parameter χs given, in theRayleigh scattering regime, by [35]

⟨Cabs;R⟩¼λ2χ3

s

πIm

m2�1m2þ2

� �: ð8Þ

Similarly, the aggregate scattering cross-section ⟨Casca;RDG⟩

can be estimated by [2,35]

⟨Casca;RDG⟩¼ 2πN2

s ⟨Csca;vv;R⟩Z π

0

12S qRg

� 1þ cos 2Θ� �

sinΘ dΘ

ð9Þwhere ⟨Csca;vv;R⟩ represents the vertically polarized scatter-ing cross-section for vertically polarized incident radiation[2,35]. For a monomer of size parameter χs in the Rayleighscattering regime, ⟨Csca;vv;R⟩ can be written as

⟨Csca;vv;R⟩¼λ2χ6

s

4π2

m2�1m2þ2

2

ð10Þ

Here, SðqRgÞ is the scattering structure factor of theaggregates describing the intensity of scattered radiationas a function of the scattering wavevector q¼ 4πλ�1

sin ðΘ=2Þ [2]. For an aggregate of fractal dimension Df

and radius of gyration Rg, SðqRgÞ can be expressed as [46]

S qRg

� ¼ 1þ8

3ðqRgÞ2Df

þðqRgÞ8" #�Df =8

ð11Þ

Note that, for very small and very large aggregates com-pared with the radiation wavelength, i.e., for qRg51 andqRgb1, the structure factor simplifies to SðqRgÞ ¼ 1 andSðqRgÞ ¼ ðqRgÞ�Df , respectively [2]. In these two limitingcases, the aggregate scattering cross-section ⟨Ca

sca;RDG⟩ isproportional to Ns

2and Ns, respectively. In the first case, the

scattered waves are in phase and their amplitudes addconstructively [2]. In the second case, the phases arerandom and the waves add randomly [2].

Finally, the unpolarized scattering phase functionF11;RDGðΘÞ of the aggregate predicted by the RDG approx-imation is expressed as [35]

F11;RDG Θ� �¼ 1

2⟨Ca

sca;vv;RDG⟩

⟨Casca;RDG⟩

1þ cos 2Θ� � ð12Þ

where the vertically polarized scattering cross-section ofthe aggregate for vertically polarized incident radiation,denoted by ⟨Ca

sca;vv;RDG⟩, is defined as [35]

⟨Casca;vv;RDG⟩¼Ns⟨Csca;vv;R⟩SðqRgÞ: ð13Þ

The asymmetry factor gRDG of the aggregate can also beestimated from the scattering phase function F11;RDGðΘÞusing Eq. (5).

The validity of the RDG approximation has been inves-tigated in numerous studies. Farias et al. [35] comparedabsorption and scattering cross-sections predicted by theRDG approximation with those estimated by volume inte-gral formulation of Maxwell's equations [31]. The authorsexamined randomly oriented aggregates of fractal dimen-sions Df between 1.0 and 3.0 with monomer size parameterχs ranging from 0.01 to 1.0, and jm�1j between 0.1 and 2.0.The RDG approximation predicted the scattering cross-section of aggregates consisting of 16–256 monodispersemonomers within 10% of those predicted by the volumeintegral method for size parameter χso0:3. However, theaccuracy of predictions of both absorption and scatteringcross-sections by the RDG approximation deteriorated withincreasing size parameter χs [35]. Wang and Sorensen [47]experimentally validated the RDG approximation by com-paring the measured scattering cross-sections at 488 nm ofaggregates with fractal dimension Df of 1.75, composed ofmonodisperse monomers 20 nm in diameter made of SiO2

(m¼1.46) or TiO2 (m¼2.61). Note that at this wavelength,absorption by the aggregates could be ignored. On the otherhand, Bushell [48] measured the scattering intensity ofaggregates of latex particles 4.9 μm in diameter using aforward light scattering photometer. The author found pooragreement between experimental measurements and pre-dictions by the RDG approximation and instead recom-mended using the T-matrix method.

2.4. Numerical predictions of aggregate radiationcharacteristics

The superposition T-matrix method [29], the general-ized multiparticle-Mie theory [30], and the volume inte-gral method [31] provide numerical solutions to Maxwell'sequations for aggregates with arbitrary fractal dimensionand number of monomers. For example, the superpositionT-matrix method estimates the total scattered electromag-netic field at any given location by summing the contribu-tion from each monomer [29].

Liu et al. [49] used the generalized multiparticle-Mietheory to predict the absorption and scattering cross-sections of soot aggregates featuring fractal dimension Df

of 1.4, 1.78, or 2.1 and a fractal prefactor kf of 2.3. Theaggregates were composed of up to 800 monodispersemonomers with size parameter χs¼0.18 and relativecomplex index of refraction m¼ 1:6þ i0:6. They demon-strated that both aggregate absorption and scatteringcross-sections normalized, respectively, by the product ofthe number of monomers in the aggregate and theabsorption or scattering cross-sections of a single mono-mer, i.e., ⟨Ca

abs⟩=Ns⟨Cabs⟩ or ⟨Casca⟩=Ns⟨Csca⟩, increased as a

function of Ns for aggregates of all fractal dimensions andnumber of monomers considered. The authors attributedboth of these observations to multiple scattering. In asimilar study, Liu et al. [50] demonstrated that the normal-ized absorption cross-section per monomer ⟨Ca

abs⟩=Ns⟨Cabs⟩

decreased as a function of Ns for soot aggregates featuringa fractal dimension Df of 1.78 and a fractal prefactor kf of1.3 or 2.3. The aggregates were composed of 20 or moremonomers of size parameter χs of 0.354 and relativecomplex index of refraction m¼ 1:6þ i0:6. The decrease

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326314

in ⟨Caabs⟩=Ns⟨Cabs⟩ was attributed to the fact that outer

particles of the aggregates shielded the inner ones fromthe incident radiation. Moreover, the normalized scatter-ing cross-section per monomer ⟨Ca

sca⟩=Ns⟨Csca⟩ increased asa function of Ns due to multiple scattering [50].

Iskander et al. [31] developed a method for predicting theabsorption and scattering cross-sections and the scatteringmatrix elements of aggregates by solving the volume integralformulation of Maxwell's equations [44]. This method is alsoknown as the discrete dipole approximation and has beenreviewed by Yukin and Hoekstra [51]. Manickavasagam andMengüç [52] used this method to predict the scatteringmatrix elements of randomly oriented soot aggregates withfractal dimension Df and prefactor kf equal to 1.7 and 5.8,respectively. The numerically generated aggregates con-tained up to 150 monodisperse monomers of radius 20, 40,or 60 nm and relative complex index of refraction equal to1:8þ i0:5. The authors concluded that the scattering phasefunction F11ðΘÞ could not be used to identify either thenumber Ns of monomers in an aggregate or the monomerradius a. However, they demonstrated that spectral andangular variations in the scattering matrix element ratioF21ðΘÞ=F11ðΘÞ could be used to determine both Ns and a. Bycontrast, the angular peaks in the scattering matrix elementratio F34ðΘÞ=F11ðΘÞ depended on the monomer radius a butnot on the number of monomers Ns [52]. The authorshypothesized that these peaks could be used to identifythe monomer radius.

These different methods have been used to predict theradiation characteristics of soot aggregates [1,3,40,49,52–55],snow [56], comets [4], and cosmic dust aggregates [4]. Inthese applications, the monomers are relatively small com-pared with the radiation wavelength such that χs51.However, as the size and number of monomers in theaggregate increase, the computational time and resourcesnecessary to predict the radiation characteristics of theaggregates, using any of these numerical methods, increasessharply [4,34].

2.5. Equivalent particle approximations

Latimer [57] approximated fractal aggregates as coatedspheres with equivalent volume and collision diameter.The coating had a relative complex index of refractionidentical to that of the monomers constituting the aggre-gate while the core had the same index of refraction as thesurrounding medium. The ratio of the total volume ofmonomers and the volume of the smallest sphere enclos-ing the aggregate was denoted by F and derived fromfractal theory as [57]

F ¼N1�3=Dfð Þs : ð14Þ

This expression assumed that the aggregate formed a solidsphere when Df was equal to 3.0. The resulting equivalentcoated sphere had an outer diameter equal to the collisiondiameter of the aggregates. The inner ai;L and outer ao;Lradii of the coated sphere were given by [57]

ai;L ¼N1=3s a 1� 1

F1=3

� �and ao;L ¼

a3Ns

F

� �1=3

ð15Þ

Latimer [57] rationalized his approach by hypothesizingthat the morphological features of aggregates composed oflarge monomers ðχsb1Þ did not have any effect on theirradiation characteristics due to their random orientation.The author experimentally measured the scattered inten-sity of a 474 nm laser beam incident by aqueous suspen-sions of aggregates consisting of latex microspheres, withdiameter ranging from 0.26 to 2.05 μm ð1:7rχsr13:6Þ.Theoretical predictions of the scattered laser intensity fellwithin 15% of experimental measurements for scatteringangles 0–101 for all aggregate suspensions. However, therelative error reached up to 80% for scattering anglesgreater than 901 [57]. Therefore, a more rigorous validationof Latimer's method of approximating radiation character-istics of aggregates as volume and collision diameterequivalent coated spheres must be performed.

Finally, Morel and co-workers [58,59] demonstrated thatthe radiation characteristics of spheroidal microorganismswith an aspect ratio smaller than 1.5 can be treated as thoseof spheres. In addition, Lee et al. [60] also established thatrandomly oriented spheroidal microalgae cells, with anaverage aspect ratio of 1.333, could be treated as spheresover the photosynthetically active radiation (PAR) region.

The present study aims to find a simplified method forpredicting the radiation characteristics of fractal aggregates,in particular, those with a large number of monomers oflarge size parameter χs, beyond the range of validity of theRDG approximation. Absorption and scattering cross-sections as well as scattering matrix element ratios of fractalaggregates with fractal dimension ranging from 2.0 to 3.0were computed using the superposition T-matrix method.These results were compared with predictions made by (1)the RDG approximation, (2) Latimer's coated sphere approx-imation, and (3) the volume and average projected areaequivalent coated sphere approximation. Fractal aggregatescomposed of up to 1000 monodisperse or polydispersespherical monomers featuring size parameter ranging from0.01 to 20 were considered.

3. Methods

3.1. Fractal aggregate generation

First, fractal aggregates were generated using the parti-cle cluster aggregation program validated and released byMroczka and co-workers [40,41,61]. All monomers in theaggregate were in contact with each other but did notoverlap. The fractal dimension was taken as Df¼2.25corresponding approximately to that of phytoplankton, aspreviously mentioned in Section 2.1 [8]. The fractal pre-factor kf was taken as 1.59 as prescribed by Sorensen [2] andMroczka et al. [40]. The aggregates generated consisted of2–1000 monomers featuring size parameter χs rangingfrom 0.01 to 20. In addition, ordered aggregates featuringinteger fractal dimension Df equal to 1.0, 2.0, and 3.0 weregenerated and corresponded to linear chain, square pack-ing, and simple cubic packing of sphere, respectively.

The total volume VT of an arbitrary aggregate withpolydisperse spherical monomers of radius ðajÞ1r jrNs

can

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326 315

be written as

VT ¼ ∑Ns

j ¼ 1

4π3a3j ð16Þ

The radius aeq;V of the volume equivalent sphere can beexpressed as

aeq;V ¼ 34π

VT

� �1=3

: ð17Þ

The radius ⟨a⟩ of the volume-averaged monomer is givenby

⟨a⟩¼ 34π

VT

Ns

� �1=3

: ð18Þ

The corresponding volume-averaged size parameter canbe defined as ⟨χs⟩¼ 2π⟨a⟩λ�1. Alternatively, the averagemonomer size parameter χ s for aggregates composed ofpolydisperse monomers can be estimated as

χ s ¼1Ns

∑Ns

j ¼ 1

2πajλ

: ð19Þ

For aggregates consisting of monodisperse monomers, thevolume averaged monomer radius ⟨a⟩ is equal to themonomer radius a and ⟨χs⟩¼ χ s ¼ χs.

The average projected area Ap of the aggregates wasestimated numerically using the method discussed indetail by Heng et al. [62]. In brief, the aggregate's centerof mass was fixed with respect to the observer and theaggregate was rotated through a large number of discreteorientations. Then, the orientation-averaged area pro-jected onto a plane normal to the line of sight wascalculated. Finally, the outer ao;V þAp

and inner ai;V þApradii

of the volume and average projected area equivalentcoated sphere can be expressed as

ao;V þAp¼ Ap

π

!1=2

and ai;V þAp¼ a3

o;V þAp� 34π

VT

� �1=3

ð20ÞThis ensures that the volume of the coating and theprojected area of the equivalent coated sphere were thesame as the total volume VT and the average projected areaAp of the aggregate.

3.2. Radiation characteristic predictions

First, the orientation averaged absorption ⟨Caabs⟩ and

scattering ⟨Casca⟩ cross-sections and scattering matrix ele-

ments of aggregates consisting of monodisperse and poly-disperse spherical monomers were predicted using thesuperposition T-matrix method using the program devel-oped by Mackowski and Mishchenko [29]. The mediumsurrounding the aggregates was non-absorbing and had anindex of refraction equal to that of water in the visible partof the spectrum, i.e., nm¼1.33. Unless stated otherwise,the monomers featured a complex index of refractionmp¼1.355þ i0.004. This resulted in a relative complexindex of refraction m¼mp=nm ¼ 1:0165þ i0:003 represen-tative of various microalgae species [23,60]. The position ofeach monomer and their individual size parameter χs wereprescribed while they were all assumed to have the same

relative complex index of refraction m. Then, theorientation-averaged aggregate absorption ⟨Qa

abs⟩ and scat-tering ⟨Qa

sca⟩ efficiency factors were computed. Finally, theabsorption ⟨Ca

abs⟩ and scattering ⟨Casca⟩ cross-sections of the

randomly oriented aggregates were estimated according to[29]

⟨Caabs=sca⟩ðNs; χs;m;Df ; kf Þ ¼ ⟨Qa

abs=sca⟩ðNs;χs;m;Df ; kf Þπa2eq;Vð21Þ

where the volume equivalent sphere radius aeq;V is givenby Eq. (17).

Moreover, the orientation-averaged absorption ⟨Caabs;RDG⟩

and scattering ⟨Casca;RDG⟩ cross-sections and the scattering

phase function F11;RDGðΘÞ of the aggregates were predictedby the RDG approximation using Eqs. (7), (9), and (12),respectively. For aggregates composed of polydisperse mono-mers, the monomer size parameter χs in Eqs. (8) and (10) wasreplaced by the volume-averaged size parameter ⟨χs⟩.

The absorption and scattering cross-sections and thescattering matrix elements of the volume and averageprojected area equivalent coated spheres were predictedbased on Lorenz–Mie theory using the program developedby Matzler [63]. First, their absorption and scatteringefficiency factors were computed based on (i) the sizeparameters χ i;Ap þV and χo;Ap þV associated with the innerai;V þAp

and outer ao;V þApradii given by Eq. (20) and (ii) the

relative complex index of refraction m. Then, the absorp-tion ⟨Cabs;V þAp

⟩ and scattering ⟨Csca;V þAp⟩ cross-sections

were estimated according to

⟨Cabs=sca;V þAp⟩ðχi;Ap þV ; χo;Ap þV ;mÞ

¼ ⟨Qabs=sca;V þAp⟩ðχ i;Ap þV ;χo;Ap þV ;mÞπa2

o;V þApð22Þ

Similarly, the absorption ⟨Cabs;L⟩ and scattering ⟨Csca;L⟩

cross-sections corresponding to Latimer's equivalentcoated sphere were estimated using Eq. (22) by replacingthe size parameters χ i;V þAp

and χo;V þApwith the size

parameters χi;L and χo;L corresponding to the inner ai;Land outer ao;L radii given by Eq. (15).

Finally, the relative errors in the absorption and scatter-ing cross-sections of the aggregates between predictions bythe superposition T-matrix method and the RDG and theequivalent coated sphere approximations were estimated inorder to identify the best approximation method.

4. Results

4.1. Average projected area

First, the average projected area Ap of aggregates com-posed of monodisperse monomers was computed using thecode developed by Heng et al. [62]. The monomers hadradius a equal to 1, 5, or 10 μm. The number of monomersper aggregates Ns ranged from 2 to 1000 while the fractaldimension Df was taken as 1.0, 1.75, 2.0, 2.25, 2.5, or 3.0. Inall cases, for aggregates of a given fractal dimension Df andmonomers number Ns, the ratio Ap=a2 was found to beconstant. Fig. 2 plots the dimensionless ratio Ap=a2 as afunction of the number of monomers Ns in the aggregate fordifferent values of Df. It indicates that Ap=a2 increased withthe number of monomers Ns for all fractal dimensions

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326316

considered. Moreover, for aggregates with identical mono-mer number Ns and radius a, the average projected areaincreased with decreasing fractal dimension Df. This wasconsistent with fractal theory which dictates that aggre-gates with larger fractal dimension feature a more compactstructure [2]. Finally, the ratio Ap=a2 was fitted by the leastsquares method to a power-law in terms of Ns to yield

Ap

a2¼ πNα

s ð23Þ

where the exponent α was a function of fractal dimensionDf. The constant πwas used to ensure the validity of Eq. (23)in the limiting case of a single sphere when Ns¼1 andAp ¼ πa2. The power α was found to decrease monoto-nously from αmax of 0.92 for linear chains of spheres withDf¼1.0 to αmin of 0.73 for Df¼3.0. The inset of Fig. 2 showsthe reduced variables αn ¼ ðα�αminÞ=ðαmax�αminÞ plottedversus Dn

f ¼ ðDf �1Þ=2 whose least-squares fitting yieldedthe following correlation:

αn ¼ ð1þDn1:8f Þ1=1:8 ð24Þ

For both power-law fits of Eqs. (23) and (24), the coefficientof determination R2 was larger than 0.99. The averageprojected area estimated by Eqs. (23) and (24) was usedto predict the outer radius ao;V þAp

of the equivalent coatedsphere according to Eq. (20).

4.2. Absorption and scattering cross-sections

4.2.1. Effect of aggregate fractal dimensionFig. 3a, b, and c shows the absorption ⟨Ca

abs⟩ andscattering ⟨Ca

sca⟩ cross-sections of randomly oriented

Df =1.0 , a =1 µm Df =2.0 , a =1 µm Df =2.25, a =1 µm Df =2.25, a =5 µm Df =2.25, a =10 µm Df =3.0, a =1 µm Df =3.0, a =5 µm Df =3.0, a =10 µm

Df =2.25 Df =2.0 Df =3.0 Df =1.0

aRg

1

10

100

1000

10000

1 10 100 1000

Ap/a

2

Number of monomers, Ns

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

α*

Df*

Equation (24)

Fig. 2. The ratio Ap/a2 as a function of number Ns of monodispersemonomers in an aggregate for fractal dimension Df equal to 1.0, 1.75, 2.0,2.25, 2.5, and 3.0 and monomer radii a equal to 1, 5, and 10 μm.

aggregates as functions of the number of monodispersemonomers Ns ranging from 2 to 1000 for fractal dimensionDf equal to 2.0, 2.25, and 3.0, respectively. Each monomerfeatured radius a equal to 1 μm and size parameter χsequal to 1, while its relative complex index of refraction mwas equal to 1:0165þ i0:003. Note that the monomer sizeparameter χs of 1 was chosen because it falls outside theRayleigh scattering regime and it was the largest sizeparameter for which computation of aggregates contain-ing as many as 1000 monomers was possible. Fig. 3a–c alsocompares the absorption and scattering cross-sectionspredicted by the superposition T-matrix method withthose estimated by (i) the RDG approximation, (ii) Lati-mer's coated sphere approximation, and (iii) the equiva-lent volume and average projected area coated sphereapproximation. In all cases, the absorption cross-section⟨Ca

abs⟩ was proportional to the number of monomers Ns. Infact, ⟨Ca

abs⟩ was independent of the fractal prefactor anddimension kf and Df. In addition, for all values of Ns and Df

investigated, the three different approximations consid-ered predicted aggregate absorption cross-section ⟨Ca

abs⟩

within less than 1% of predictions by the superposition T-matrix method. This can be attributed to the fact that, foraggregates composed of monomers with relatively smallabsorption index ks (i.e., optically soft), absorption is avolumetric phenomenon. For all approximations, thevolume of material interacting with the incident electro-magnetic wave was identical to the total volume VT of theaggregates. Thus, for aggregates composed of monodis-perse monomers, Eq. (7) can also be expressed as⟨Ca

abs⟩¼ VT=Vs⟨Cabs⟩, where Vs corresponds to the volumeof a single monomer. In other words, the absorption cross-section ⟨Ca

abs⟩ of any randomly oriented fractal aggregateconsidered was proportional to its total volume VT. Thesame conclusion was reached for linear chains of spheres[34] and for bispheres, quadspheres, and rings of spheres[62] made of optically soft monomers.

Similarly, the scattering cross-section ⟨Casca⟩ increased as

a function of number of monomers Ns present in theaggregates. Here, ⟨Ca

sca⟩ was proportional to Nspwhere the

power pwas equal to 1.27, 1.30, and 1.40 for aggregates withfractal dimension Df of 2.0, 2.25, and 3.0, respectively. Thus,unlike for ⟨Ca

abs⟩, the scattering cross-section of the aggre-gate was larger than the sum of the scattering cross-sections of each monomer (i.e., p41) due to multiplescattering. In fact, for a given number of monomers Ns

and a given total volume VT, the aggregates with largerfractal dimension Df featured larger scattering cross-sections ⟨Ca

sca⟩. This was due to the fact that increasing thefractal dimension Df resulted in a smaller average projectedarea (Fig. 2) and a more compact structure more prone tomultiple scattering [53]. The relative error between scatter-ing cross-section ⟨Ca

sca⟩ predicted by the T-matrix methodand that estimated by the RDG approximation was smallerthan 15% for aggregates containing fewer than 100 mono-mers for any fractal dimension Df considered. However, itreached up to 40% for aggregates containing 100 or moremonomers. On the other hand, the relative error in thescattering cross-section predicted by Latimer's coatedsphere approximation was smaller than 18% for fractaldimension Df equal to 2.25 corresponding to the value of

10-4

10-3

10-2

10-1

100

101

102

10-4

10-3

10-2

10-1

100

101

102

1 10 100 1000

Scattering cross-section,(μ m

2)Abs

orpt

ion

cros

s-se

ctio

n,

(μ

m2 )

Number of monomers, Ns

10-4

10-3

10-2

10-1

100

101

102

10-4

10-3

10-2

10-1

100

101

102

1 10 100 1000

Scattering cross-section, (μm

2)Abs

orpt

ion

cros

s-se

ctio

n,

(μm

2 )

Number of monomers, Ns

10-3

10-2

10-1

100

101

102

10-4

10-3

10-2

10-1

100

101

102

1 10 100 1000

Scattering cross-section, (μm

2)Abs

orpt

ion

cros

s-se

ctio

n,

(μm

2 )

Number of monomers, Ns

Df=2.0

Df =3.0

Df=2.25

T-matrix methodRDG approximationEquivalent coated sphere (Latimer)Equivalent coated sphere (V+Ap)

1

1.27 1.30

1.40

1

1

Fig. 3. Absorption ⟨Caabs⟩ and scattering ⟨Ca

sca⟩ cross sections as functions of monomer number Ns in randomly oriented aggregates with fractal dimension Df

of (a) 2.0, (b) 2.25, and (c) 3.0 predicted using the superposition T-matrix method, the RDG approximation, Latimer's [57] coated sphere approximation, andthe volume and average projected area equivalent coated sphere approximation. The aggregates were composed of monodisperse monomers featuring sizeparameter χs¼1 and m¼ 1:0165þ i0:003.

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326 317

Df validated by Latimer [57]. Note that Latimer's choice toreduce the monomer number distribution of a large numberof aggregates to only five discrete bins was arbitrary andmay have introduced larger errors in the theoretical pre-dictions of the scattering cross-section known to be verysensitive to the number of monomers Ns [2,44]. Indeed, therelative error in the scattering cross-section predicted byLatimer's coated sphere approximation reached up to 47%for fractal dimensions Df of 2.0 and 3.0. Due to the relativelylarge discrepancies between predictions by the T-matrixmethod and by Latimer's approximation [57], the latter wasomitted in the remainder of this study. By contrast, thescattering cross-section of the volume and average pro-jected area equivalent coated sphere fell within 7.5% of thatpredicted by the T-matrix method for all values of Ns and Df

considered. Note that predictions of the integral radiationcharacteristics for the volume-equivalent or surface-area-equivalent homogeneous spheres were not as accurate as

those for the volume and average projected area equivalentcoated sphere.

4.2.2. Effect of size parameterFig. 4a–d shows the absorption ⟨Ca

abs⟩ and scattering ⟨Casca⟩

cross-sections of randomly oriented aggregates of mono-disperse monomers with fractal dimension Df¼2.25 as afunction of monomer number Ns for size parameter χs equalto 0.01, 0.1, 0.5, 5, 10, and 20. They compare the super-position T-matrix predictions with those of the RDG approx-imation and for the volume and average projected areaequivalent coated sphere. They indicate that both absorption⟨Ca

abs⟩ and scattering ⟨Casca⟩ cross-sections increased with

increasing number of monomers Ns and size parameter χs.More specifically, Fig. 4a and c establishes that the aggregateabsorption cross-section ⟨Ca

abs⟩ was linearly proportional tothe number of monomers Ns for all size parameters con-sidered. On the other hand, Fig. 4b reveals that the aggregate

10-2

10-1

100

101

102

001011

Scat

teri

ng c

ross

-sec

tion,

(μ

m2 )

Number of monomers, Ns

10-12

10-10

10-8

10-6

10-4

10-2

100

1 10 100 1000

Scat

teri

ng c

ross

-sec

tion,

(μ

m2 )

Number of monomers, Ns

10-1

100

101

102

001011

Abs

orpt

ion

cros

s-se

ctio

n,

(μm

2 )

Number of monomers, Ns

10-4

10-3

10-2

10-1

100

101

1 10 100 1000

Abs

orpt

ion

cros

s-se

ctio

n,

(μm

2 )

Number of monomers, Ns

T-matrix methodRDG approximationEquivalent coated sphere (V+ Ap)

=5

=10

=20

1

1

1

1.17

1.2

1.23

1

1

1

1.46

1.88

2

=20

=10

=5

Fig. 4. (a, c) Absorption ⟨Caabs⟩ and (b, d) scattering ⟨Ca

sca⟩ cross-sections as functions of monomer number Ns in randomly oriented aggregates with fractaldimension Df¼2.25 and composed of monomers with m¼ 1:0165þ i0:003 and size parameter χs ranging from 0.01 to 20 predicted using (i) thesuperposition T-matrix method, (ii) the RDG approximation, and (iii) the volume and average projected area equivalent coated sphere approximation.

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326318

scattering cross-section ⟨Casca⟩ was proportional to Ns

2for

aggregates composed of monomers of size parameterχs ¼ 0:01. It is interesting to note that, in the limiting caseof for small size parameter χs and small aggregates such thatχs51 and qRg51, the RDG approximation predicts that⟨Ca

sca⟩ is proportional to Ns2. Fig. 4b and d indicates that ⟨Ca

sca⟩

was proportional to Nspwith a power-law exponent p that

monotonously decreased from 2 to 1.17 as the monomer sizeparameter χs increased from 0.01 to 20. In fact, in thelimiting case of large size parameter χs and large aggregates,i.e., χs51 and qRg51, the RDG approximation predicts that⟨Ca

sca⟩ is linearly proportional to Ns (i.e., p¼1), as previouslymentioned in Section 2.3.

The absorption cross-section predicted by the RDGapproximation and the volume and average projected areaequivalent coated sphere fell within 5% of the T-matrixpredictions for all values of Ns and χs considered. Inaddition, the relative error in scattering cross-section pre-dictions by the RDG approximation was smaller than 8% forsize parameter χs smaller than 0.5. This confirms andexpands on the findings by Farias et al. [35] and by Wangand Sorensen [47] as previously discussed in Section 2.3.However, it reached 29%, 56%, and 117% for size parametersχs of 5, 10, and 20, respectively. This excessively largerelative error in scattering cross-section renders theRDG approximation unsuitable for predicting the radiation

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326 319

characteristics of aggregates composed of monomers of sizeparameter larger than 1, as previously reported in theliterature [35,38,47,48]. By contrast, the scattering cross-section of the volume and average projected area equivalentcoated sphere fell within 10% of the predictions by theT-matrix method for all values of Ns and χs considered. Inother words, the volume and average projected areaequivalent coated sphere was able to capture the multiplescattering effects on the integral radiation characteristics.

Moreover, note that the maximum number of mono-mers per aggregate that could be simulated decreasedwith increasing monomer size parameter χs. For example,a converged solution for aggregates of 25 monomers ofsize parameter χs ¼ 20 could not be obtained using thesuperposition T-matrix program despite the use of arelatively large computer cluster. Moreover, for a fractalaggregate containing 25 monomers of size parameterχs ¼ 10, the superposition T-matrix code yielded a con-verged solution after 25 h running in parallel on 135 CPUs.Alternatively, predictions of the radiation characteristicsand scattering matrix elements of the correspondingvolume and average projected area equivalent coatedspheres were obtained in 3.6 ms using a computer witha single core CPU. Thus, this approximation could providean invaluable tool for estimating the radiation character-istics of randomly oriented aggregates with reasonableaccuracy, particularly for aggregates with a large numberof large monomers. For example, it could be used ininverse problems aiming to infer the aggregate morphol-ogy and/or the monomer complex index of refraction fromexperimental measurements [36,60,64,65]. It could also beused when the superposition T-matrix method fails toconverge or if the necessary computing resources are notavailable. This is particularly interesting for microalgaecolonies consisting of cells 4–12 μm in diameter (Fig. 1eand f) and featuring size parameter χs ranging between 18and 95 over the PAR region (λ¼400–700 nm). For suchsuspensions, the volume and average projected areaequivalent coated sphere approximation can predict,rapidly and sufficiently accurately, the absorption andscattering cross-sections as well as the asymmetry factorg or the backward scattering fraction b needed to performradiation transfer analysis in PBRs [25,66,67].

Table 1Absorption ⟨Ca

abs⟩ and scattering ⟨Casca⟩ cross-sections of randomly oriented ag

Gaussian radius distribution and standard deviations of 0%, 10%, and 25%. The aparameter χ s¼1 and relative complex index of refraction m¼ 1:0165þ i0:003.

T-matrix Rayleigh–D

Ns χ s stdev χ s ⟨χ s⟩ VT Ap ⟨Caabs⟩ ⟨Ca

sca⟩ ⟨Caabs;RDG⟩

(%) ðμm3Þ ðμm2Þ ðμm2Þ ðμm2Þ ðμm2Þ

256 0 1.0 1.00 1072 386 6.44 0.96 6.4256 10 1.0 1.01 1093 389 6.63 1.01 6.6256 25 1.0 1.08 1335 424 8.03 1.42 8.0512 0 1.0 1.00 2145 707 12.9 2.18 12.8512 10 1.0 1.00 2145 711 12.8 2.14 12.7512 25 1.0 1.06 2567 768 15.4 2.89 15.3

1000 0 1.0 1.00 4189 1341 24.1 4.39 25.01000 10 1.0 1.02 4376 1365 26.0 4.78 25.81000 25 1.0 1.07 5059 1408 30.3 6.2 30.2

4.2.3. Effect of polydispersityTo investigate the effects of monomer polydispersity on

the aggregates' absorption and scattering cross-sections,aggregates composed of 256, 512, and 1000 polydispersemonomers were generated with a Gaussian radius dis-tribution with the same mean radius of 1 μm and standarddeviation of 10% or 25%. Table 1 reports the mean χ s andvolume-averaged ⟨χs⟩ size parameters and the averageprojected area of the aggregates generated. It also com-pares predictions of the corresponding absorption ⟨Ca

abs⟩

and scattering ⟨Casca⟩ cross-sections by the superposition T-

matrix method with those made by the RDG and thevolume and average projected area equivalent coatedsphere approximations. In all cases, the monomer meansize parameter χ s was equal to 1 and the fractal dimensionDf was equal to 2.25. However, the total volume VT and thevolume-averaged size parameter ⟨χs⟩ of the aggregateswith polydisperse monomers were larger than those withmonodisperse monomers for the same number of mono-mers Ns. On the other hand, the average projected area Ap

of the aggregates increased only slightly with polydisper-sity of the monomers. Overall, the ratio Ap=VT of theaverage projected area to the total volume was smallerfor aggregates with polydisperse monomers than withmonodisperse monomers having the same monomernumber and mean radius. In other words, aggregates withpolydisperse monomers were more compact that thosewith monodisperse monomers.

Moreover, for all cases considered, aggregates composedof polydisperse monomers featured larger absorption andscattering cross-sections than those composed of the samenumber of monodisperse monomers Ns. The larger absorp-tion cross-section was due to the fact that the total volumeof the aggregates VT increased with increasing monomerpolydispersity, as reported in Table 1. On the other hand, theincrease in aggregate scattering cross-section ⟨Ca

sca⟩ could beattributed to the fact that the aggregates were morecompact and therefore more prone to multiple scattering.These results were consistent with the conclusion reachedfor aggregates with monodisperse monomers and differentfractal dimensions (Fig. 3).

Finally, the absorption cross-sections of the randomlyoriented aggregates with polydisperse monomers predicted

gregates of fractal dimension Df of 2.25 composed of monomers with aggregates were composed of 256, 512, and 1000 monomers with mean size

ebye–Gans approximation Equivalent coated sphere

Error ⟨Casca;RDG⟩ Error ⟨Ca

abs;VþAp⟩ Error ⟨Ca

sca;VþAp⟩ Error

(%) ðμm2Þ (%) ðμm2Þ (%) ðμm2Þ (%)

0.62 1.14 �19 6.4 0.01 0.83 140.60 1.18 �17 6.6 0.3 0.91 100.75 1.55 �9.2 8.0 0 1.2 140.78 2.64 �21 12.9 0.2 2.0 100.78 2.61 �22 12.8 0 2.0 8.40.65 3.4 �18 15.4 0 2.7 8.0

�3.4 5.9 �34 25.1 4.0 4.4 �0.50.77 6.2 �29 26.0 0 4.5 5.70.33 7.7 �24 30.3 0 5.9 5.7

0

0.2

0.4

0.6

0.8

1

1.2

1 10 100 1000Number of monomers, Ns

0

1

2

3

4

5

6

7

8

1 10 100 1000Number of monomers, Ns

T-matrix methodRDG approximationEquivalent coated sphere (V+ Ap)

k=0.003k=0.03k=0.07

k=0.5

k=0.003k=0.03k=0.07

k=0.5

Fig. 5. Normalized (a) absorption ⟨Caabs⟩=Ns⟨Cabs⟩ and (b) scattering

⟨Casca⟩=Ns⟨Csca⟩ cross-sections as a function of Ns predicted by the super-

position T-matrix method, the RDG approximation, and the volume andaverage projected area equivalent coated sphere approximation fordifferent values of relative absorption index k. All aggregates had fractaldimension Df¼2.25, monomer size parameter χs¼1, and relative refrac-tion index n¼1.0165.

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326320

by both the RDG and the volume and average projected areaequivalent coated sphere approximations fell within 4% ofthe predictions by the superposition T-matrix method.However, the associated scattering cross-section predictedby the RDG approximation suffered from a relative error ofup to 29% for aggregates containing 1000 monomers. Bycontrast, the relative error in the scattering cross-sectionpredicted for the volume and average projected area equiva-lent coated sphere was less than 14% for all aggregatesconsidered. This confirms the validity of volume and averageprojected area equivalent coated sphere approximation inpredicting the absorption and scattering cross-sections ofrandomly oriented fractal aggregates consisting of eithermonodisperse or polydisperse optically soft monomers.

4.2.4. Effect of the relative complex index of refractionFig. 5a and b plots the absorption and scattering cross-

sections of randomly oriented aggregates normalized bythe product of the monomer number Ns and the absorp-tion and scattering cross-sections of a single sphericalmonomer ⟨Ca

abs⟩=Ns⟨Cabs⟩ and ⟨Casca⟩=Ns⟨Csca⟩ as a function

of Ns, respectively. All aggregates featured a fractal dimen-sion Df of 2.25, monodisperse monomers with size para-meter χs ¼ 1, and relative refractive index n¼1.0165. Therelative absorption index k was taken as 0.003, 0.03, 0.07,or 0.5. A ratio ⟨Ca

abs⟩=Ns⟨Cabs⟩ or ⟨Casca⟩=Ns⟨Csca⟩ independent

of Ns and equal to unity would indicate that the absorptionor scattering cross-section of the aggregate is the sum ofabsorption or scattering cross-sections of its constituentmonomers. In fact, the normalized absorption cross-section per monomer ⟨Ca

abs⟩=Ns⟨Cabs⟩ remained constantand equal to unity for aggregates with relative absorptionindex k¼0.003. This was consistent with the RDG approx-imation expression for ⟨Ca

abs;RDG⟩ given by Eq. (7) inSection 2.3. However, for relative absorption index largerthan 0.003, the normalized absorption cross-section wassmaller than 1.0 and decreased with increasing values ofNs and k. Similar observations were made by Liu et al. [50]for soot aggregates composed of monomers with absorp-tion index k of 0.6 and size parameter χs of 0.354, aspreviously discussed in Section 2.4. This can be attributedto the shading of the monomers located inside the aggre-gates by those located on the outside. This phenomenonwas particularly important for aggregates composed oflarge, numerous, and/or strongly absorbing monomers. Inaddition, the scattering cross-section of aggregates com-posed of strongly absorbing monomers was much largerthan for those composed of weakly absorbing monomers.These relatively large absorption and scattering cross-sections caused the incident electromagnetic wave to befully attenuated before it can reach the inner monomers.For such aggregates, the absorption cross-section did notdepend linearly on the material volume interacting with theincident radiation, unlike what has been observed foroptically soft particles aggregates [49,50]. Instead, whenthe penetration depth was smaller than the monomer size,absorption became a surface phenomenon.

Moreover, the normalized scattering cross-section permonomer represented by ⟨Ca

sca⟩=Ns⟨Csca⟩, increased withincreasing Ns and was larger than 1.0 for all four complexindices of refraction considered. In other words, the

scattering cross-section of an aggregate was larger thanthe sum of the scattering cross-sections of its constitutivemonomers. This can be attributed to multiple scattering aspreviously discussed. However, the latter was less signifi-cant for aggregates composed of strongly absorbing mono-mers due to the attenuation of the electromagnetic wavewhich could not emerge from the aggregate. Note thatMishchenko [68] presented the equality ⟨Ca

ext⟩¼Ns⟨Cext⟩ asa necessary condition for single scattering to prevail inmulti-particle aggregates. Moreover, the RDG approxima-tion predictions of ⟨Ca

abs⟩=Ns⟨Cabs⟩ and ⟨Casca⟩=Ns⟨Csca⟩ as a

function of Ns were independent of the relative absorptionindex k. This resulted in very large discrepancies withpredictions by the superposition T-matrix method. On the

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326 321

other hand, the volume and average projected areaequivalent coated sphere predictions featured a relativeerror in the absorption and scattering cross-sections of lessthan 8% and 29%, respectively, compared with the T-matrixmethod for all values of Ns and k considered. Moreover, therelative error between predictions by the coated sphereapproximation and by the T-matrix method decreasedwith increasing monomer absorption index. Indeed, themaximum relative error in the scattering cross-section was29% and 11% for aggregates composed of monomerswith relative complex index of refraction m equal to1:0165þ i0:003 and 1:0165þ i0:5, respectively. This wasin contrast to the relative error between the RDG approx-imation and the superposition T-matrix method predic-tions which increased with monomer absorption index. Infact, the scattering cross-sections estimated by the RDGapproximation differed by more than 50% from those bythe T-matrix method for aggregates composed of morethan Ns¼100 monomers with size parameter χs ¼ 1 andrelative absorption index k larger than 0.03. These resultsestablish that the volume and average projected areaequivalent coated sphere approximation could not onlycapture the effects of multiple scattering but also ofshading among monomers.

Finally, these results indicate that the formation ofcolonies of microalgae in PBRs results in reduced absorp-tion cross-section per cell and increased scattering cross-section per cell. Therefore, light transfer in microalgaesuspensions will be strongly affected by colony formation.

4.3. Scattering phase function

Fig. 6a–f shows the scattering phase function F11ðΘÞ ofrandomly oriented aggregates of fractal dimension Df of 2.25consisting of monomers of size parameter χs of 1 and 5 for anumber of monomers Ns ranging from 9 to 100. Theycompare predictions by the superposition T-matrix methodwith those of (i) the RDG approximation, (ii) the volume andaverage projected area equivalent coated sphere approxima-tion, and (iii) the Henyey–Greenstein phase function givenby Eq. (6) using the asymmetry factor g corresponding to thephase function of the equivalent coated sphere.

First, scattering by randomly oriented aggregates wasincreasingly in the forward direction as the size parameterχs and/or the number of monomers Ns in the aggregateincreased. Fig. 6a–c, corresponding to χs ¼ 1, confirms theconclusions reached by Manickavasagam and Mengüç [52](Section 2.4) that measurements of aggregate scatteringphase function alone could not be used to identify themonomer size parameter χs or their number Ns for sizeparameter χs between 0.5 and 1.5. Indeed, these threeaggregates did not feature any distinguishing characteris-tics that could be used to determine either χs or Ns. On theother hand, aggregates composed of monomers of sizeparameter χs ¼ 5 featured scattering phase functionsF11ðΘÞ with two distinct resonance peaks at scatteringangles Θ of 551 and 1001. These angles depended only onthe monomer size parameter χs. However, the magnitudeof these resonance peaks correlated to the number ofmonomer in the aggregate Ns. Therefore, these two uniquefeatures of the scattering phase function could be used to

determine the monomer's size parameter χs and numberNs in aggregates composed of relatively large monomers.

Moreover, the approximate methods predicted similarphase functions for the aggregates composed of mono-mers of size parameter χs of 1. However, the equivalentcoated sphere scattering phase function featured severalresonance peaks at various scattering angles Θ. Thesepeaks did not correspond to those observed in the phasefunction predicted by the T-matrix method. The numberand magnitude of these resonance peaks increased withincreasing monomer number Ns and size parameter χs.These resonance peaks are characteristic of coated spheres[69] and were due to internal reflection within the coatingwhich acted as a waveguide.

Another indicator of multiple scattering is the aggregatescattering phase function atΘ¼ 01. Indeed, single scatteringby the aggregates requires that F11ð01Þ ¼NsF11;sð01Þ, whereF11;s is the scattering phase function of a single sphere [68].Here, the values of F11ð01Þ of the aggregates composed of 9and 36 monomers of size parameter χs ¼ 1 and relativecomplex index of refraction m¼ 1:0165þ i0:003 was equalto 19.9 and 79.6, respectively. On the other hand, F11;sð01Þ fora single monomer of the same size parameter was equal to2.21. Similarly, F11ð01Þ was 244 and 309 for aggregatescomposed of 16 and 25 monomers of size parameter 5,respectively. The corresponding F11;sð01Þ was equal to 24.7.This further establishes the presence of multiple scatteringin the aggregates considered in Fig. 6.

The inset tables in Fig. 6a–f report the asymmetry factorg corresponding to the scattering phase function predictedby the T-matrix method, the RDG approximation, and thevolume and average projected area equivalent coated sphereapproximation. They indicate that the asymmetry factor gincreased from 0.62 for χs ¼ 1 and Ns¼9 to 0.96 for χs ¼ 5and Ns¼25. The relative error in the asymmetry factor gpredicted by the RDG approximation reached 10% comparedwith the T-matrix method predictions for χs ¼ 5 and Ns¼25.However, it was smaller than 5% for the volume and averageprojected area equivalent coated sphere for all values of χsand Ns considered. The asymmetry factor can be used invarious approximate expressions of the scattering phasefunction including the transport approximation [70] and theHG approximate phase function. In fact, the latter gavereasonable predictions of F11ðΘÞ of the aggregates forχs ¼ 1 using the asymmetry factor corresponding to theequivalent coated sphere (Fig. 6a–c). Note that large errorswere observed for aggregates with larger monomers ðχsb1Þwhich tend to scatter strongly in the forward direction(Fig. 6d–f). However, the use of the HG phase function hasbeen shown to be sufficiently accurate for radiation transferanalysis through strongly forward scattering media such asmicroalgae suspensions containing gas bubbles [66], glasscontaining bubbles [71], red blood cells [72,73], and also inthe field of ocean optics [74,75].

Note that microalgae cells are strongly forward scatter-ing due to their large size compared with the wavelengthof photons in the PAR region. In addition, the solution ofthe radiative transfer equation derived by Pottier et al. [25]based on the two-flux approximation has been shown tooffer a relatively accurate method for predicting thefluence rate in open pond and flat-plate PBRs [25,67].

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326322

Their analytical expression requires only the absorptionand scattering cross-sections, the microorganism concen-tration, as well as the backward scattering ratio b of themicroalgae suspension. The insets to Fig. 6a–e show thevalues of b for each aggregate estimated using the scatter-ing phase function predicted using (i) theT-matrix method, (ii) the coated sphere approximation,(iii) the RDG approximation, and (iv) the HG phase func-tion. They indicate that the backward scattering ratiopredicted using the coated sphere approximation waswithin 30% of that predicted by the T-matrix method foraggregates composed of monomers of size parameterχs ¼ 1. In addition, b was negligibly small for aggregatescomposed of monomers with χs ¼ 5.

Moreover, Berberoğlu et al. [66] used the discreteordinates method with a combination of two Gauss quad-rature having 24 discrete directions per hemisphere topredict the fluence rate in PBRs containing microorgan-isms. Results obtained using the HG approximate phasefunction were in good agreement with those obtainedusing the phase function predicted by the Lorenz–Mietheory of the microorganism suspension. This demon-strates that for the purposes of unpolarized radiationtransfer analysis through microalgae cultures, or any otherstrongly forward scattering media, knowledge of the

Fig. 6. Scattering phase function F11ðΘÞ of randomly oriented aggregates of fractaequal to (a–c) 1.0 or (d–f) 5, m¼ 1:0165þ i0:003 and Ns ranging from 9 to 100 estand for the volume and average projected area equivalent coated sphere, and thefactor g computed using Eq. (5).

integral radiation characteristics ⟨Caabs⟩; ⟨C

asca⟩, g, and b are

sufficient.Finally, the inset to Fig. 6a–e shows the scattering phase

function F11ðΘ¼ 01Þ values obtained by (i) the T-matrixmethod, (ii) for the equivalent coated sphere, (iii) the RDGapproximation, and (iv) the HG phase function. This is ofparticular interest for large aggregates and/or large mono-mers since most of the scattered radiation energy isconcentrated around the forward direction Θ¼ 01. Rela-tively good agreement was found between the values ofF11ð01Þ predicted by the T-matrix approximation andthose predicted for the equivalent coated spheres. In fact,the relative error between the predictions of the twomethods was less than 13% for all size parameters χs andmonomer numbers Ns considered. In addition, the value ofF11ðΘ¼ 01Þ predicted by the HG approximation was accu-rate within 15% for aggregates composed of 36 or 100monomers with size parameter χs ¼ 1. However, it over-estimated F11ð01Þ for aggregates composed of monomerswith larger size parameter.

4.4. Scattering matrix element ratios

Polarized incident radiation and the scattering matrixelements can be used in remote sensing applications to

l dimension Df of 2.25 with monodisperse monomers of size parameters χsimated using the superposition T-matrix method, the RDG approximation,HG phase function. The inset table reports the corresponding asymmetry

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326 323

characterize the morphology of the aggregates defined bya, Ns, kf, Df, VT, and/or Ap.

Fig. 7 plots the normalized scattering matrixelement ratios (a) F21ðΘÞ=F11ðΘÞ, (b) F22ðΘÞ=F11ðΘÞ, (c)F33ðΘÞ=F11ðΘÞ, (d) F34ðΘÞ=F11ðΘÞ, and (e) F44ðΘÞ=F11ðΘÞpredicted by the superposition T-matrix method as func-tions of scattering angle Θ for randomly oriented aggre-gates of fractal dimension Df¼2.25 and consisting of 9, 36,and 100 monomers with size parameter χs¼1 andm¼ 1:0165þ i0:003. They also show the same scatteringmatrix element ratios predicted for the volume andaverage projected area equivalent coated sphere. Thedegree of linear polarization of the aggregates F21ðΘÞ=F11ðΘÞ was identical for all values of Ns considered. Itreached 100% at scattering angle Θ¼ 90○ and was equal to0% at scattering angles Θ of 01 and 1801. The scatteringmatrix element ratio F22ðΘÞ=F11ðΘÞ was equal to 100% forall scattering angles Θ. The scattering matrix elementratios F33ðΘÞ=F11ðΘÞ and F44ðΘÞ=F11ðΘÞ were equal for allaggregates and decreased from 100% at Θ¼ 0○ to �100%at Θ¼ 1801. Finally, the scattering matrix element ratioF34ðΘÞ=F11ðΘÞ was equal to zero for all angles Θ. Resultsfor the different scattering matrix element ratios pre-sented in Fig. 7 for χs ¼ 1 were identical to those for asingle sphere. This indicates the dominant role of singlescattering by the constituent monomers [53,76]. Theseresults confirm the findings by Liu and Mishchenko [53]who demonstrated that increasing the aggregate number

-100

-80

-60

-40

-20

0

20

40

60

80

100

F 21/F

11 (%

)

90

95

100

105

F 22/F

11 (%

)

-100

-80

-60

-40

-20

0

20

40

60

80

100

F 33/F

11 (%

)

Scattering angle, Θ (deg)

-80

-60

-40

-20

0

20

40

60

F 34/F

11 (%

)

Scattering

0 30 60 90 120 150 180 0 30 60 9

0 30 60 90 120 150 180 0 30 60 9

Fig. 7. Scattering matrix element ratios (a) F21ðΘÞ=F11ðΘÞ, (b) F33ðΘÞ=F11ðΘÞ, (c) Ffractal dimension Df¼2.25 containing 9, 36, and 100 monodisperse monomem¼ 1:0165þ i0:003 predicted using the superposition T-matrix method anapproximation.

of monomers Ns up to 400, with size parameter χs ¼ 0:2,did not modify the scattering matrix element ratios.

The equivalent coated sphere featured scattering ele-ment ratios with overall trends similar to those predictedby the T-matrix method. However, they also featuredresonance peaks at scattering angles Θ corresponding tothose observed in the scattering phase function shown inFig. 6a–c and attributed to internal reflectance in thecoating. These results indicate that the volume and aver-age projected area equivalent coated sphere approxima-tion cannot be used for predicting the scattering matrixelements of the actual fractal aggregates. Thus, it will notbe considered further in this section.

Fig. 8 shows the scattering matrix element ratios(a) F21ðΘÞ=F11ðΘÞ, (b) F22ðΘÞ=F11ðΘÞ, (c) F33ðΘÞ=F11ðΘÞ, (d)F34ðΘÞ=F11ðΘÞ, and (e) F44ðΘÞ=F11ðΘÞ predicted by thesuperposition T-matrix method as a function of scatteringangle Θ for randomly oriented aggregates of fractal dimen-sion Df of 2.25 and consisting of 9, 36, and 100 monomers ofsize parameter χs¼5 and m¼ 1:0165þ i0:003. All scatteringmatrix element ratios featured resonance peaks at scatteringangles Θ of 551 and 1001 also observed in the aggregatescattering phase function F11ðΘÞ (Fig. 6d–f). These resonancepeaks appeared in the scattering matrix elements for largeenough monomer size parameter. Similar resonance peakswere observed in the scattering matrix element ratios ofaggregates composed of linear chain of spheres with sizeparameter of 10 [34] and for fractal soot aggregates with

Ns=9Ns=36Ns=100Ns=9Ns=36Ns=100

T-matrix method

Equivalent coated sphere (V+Ap)

=1

angle, Θ (deg)

-100

-80

-60

-40

-20

0

20

40

60

80

100

0 120 150 180

0 120 150 180 0 30 60 90 120 150 180

F 44/F

11 (%

)

Scattering angle, Θ (deg)

34ðΘÞ=F11ðΘÞ, and (d) F44ðΘÞ=F11ðΘÞ of randomly oriented aggregates withrs with size parameter χs ¼ 1 and relative complex index of refractiond the volume and average projected area equivalent coated sphere

Fig. 8. Scattering matrix element ratios (a) F21ðΘÞ=F11ðΘÞ, (b) F22ðΘÞ=F11ðΘÞ, (c) F33ðΘÞ=F11ðΘÞ, (d) F34ðΘÞ=F11ðΘÞ, and (e) F44ðΘÞ=F11ðΘÞ of randomly orientedaggregates with fractal dimension Df¼2.25 containing 9, 16, and 25 monodisperse monomers with size parameter χs ¼ 5 and m¼ 1:0165þ i0:003 predictedusing the superposition T-matrix method.

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326324

Df¼1.82 and kf¼1.19 composed of 200 monomers with sizeparameter χs ¼ 0:6 [53]. Here, the number and angles of theresonance peaks depended on the monomer size parameterχs while their magnitude depended on the number ofmonomers Ns in the aggregates. Indeed, the scattering matrixelement ratios had 1, 2, or 6 resonance peaks for aggregatescomposed of Ns¼9 monomers with size parameter 2.5, 5, or10, respectively (see Supplementary Materials). These con-firm and expand on previous results reported by Mackowskiand Mishchenko [77] illustrating that the resonance anglesof aggregates of up to 5 spherical monomers of size para-meter χs ¼ 5 were equal to those for a single sphere of thesame size parameter. The authors also reported that increas-ing monomer number in an aggregate causes “dampingof the oscillation in the matrix elements” [77]. However,here no such effect could be observed (see SupplementaryMaterials).

Furthermore, the ratio F21ðΘÞ=F11ðΘÞ deviated from unitywhile the scattering element ratios F33ðΘÞ=F11ðΘÞ andF44ðΘÞ=F11ðΘÞ featured similar trends but were not identical.Divergence of the ratio F22=F11 from unity as well as theinequality between the ratios F33=F11 and F44=F11 was alsoobserved by Mishchenko et al. [78] and used as indicators fornonsphericity of randomly oriented bispheres of size para-meter χs ¼ 5 and relative index of refraction m¼ 1:5þ0:005.Such features of the scattering matrix elements can be usedfor remote sensing applications.

5. Conclusion

This study demonstrated that the absorption ⟨Caabs⟩ and

scattering ⟨Casca⟩ cross-sections and the asymmetry factor g

of randomly oriented fractal aggregates consisting ofspherical monomers can be rapidly estimated as those ofcoated spheres with equivalent volume and average pro-jected area. Predictions for ⟨Ca

abs⟩ and ⟨Casca⟩, and g fell within

8%, 29%, and 15%, respectively, for aggregates composedof monodisperse and polydisperse monomers with (i)size parameter χs between 0.01 and 20, (ii) numberNs ranging from 1 to 1000, (iii) relative refractive index of1.0165 and absorption index varying from 0.003 to 0.5, and(iv) for aggregates of fractal dimension ranging from 2.0 to3.0. First, a convenient correlation was derived for theaverage projected area of fractal aggregates with variousfractal dimensions. The proposed equivalent coated sphereapproximation was able to capture multiple scattering in theaggregates and shading among constituent monomers onthe integral radiation characteristics of the aggregate.It was also superior to that proposed by Latimer [57] andto the Rayleigh–Debye–Gans approximation, particularly forlarge values of χs and Ns. In addition, the use of Henyey–Greenstein approximate phase function estimated using theasymmetry factor for the equivalent coated sphere yieldedacceptable predictions of the actual aggregate scatteringphase function for all values of χs and Ns considered.

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326 325

However, the equivalent coated spheres featured scatteringmatrix element ratios significantly different from those ofthe aggregates due to internal reflection in the coating.Finally, the scattering phase function and the scatteringmatrix elements were found to have unique features forlarge monomer size parameter χs. These could be used inremote sensing applications to measure the morphology ofsuch aggregates.

Acknowledgments

This research was supported in part by the U.S. NSF-IGERT program Clean Energy for Green Industry at UCLA(DGE-0903720). Computation was performed on the Hoff-man 2 cluster hosted by the Academic Technology Servicesat the University of California, Los Angeles.

Appendix A. Supplementary data

Supplementary data associated with this paper can befound in the online version at http://dx.doi.org/10.1016/j.jqsrt.2014.10.018.

References

[1] Liou KN, Takano Y, Yang P. Light absorption and scattering byaggregates: application to black carbon and snow grains. J QuantSpectrosc Radiat Transf 2011;112(10):1581–94.

[2] Sorensen CM. Light scattering by fractal aggregates: a review.Aerosol Sci Technol 2001;35(2):648–87.

[3] Takano Y, Liou KN, Kahnert M, Yang P. The single-scattering proper-ties of black carbon aggregates determined from the geometric-optics surface-wave approach and the T-matrix method. J QuantSpectrosc Radiat Transf 2013;125:51–6.

[4] Kimura H, Kolokolova L, Mann I. Optical properties of cometary dust.Astron Astrophys 2003;407(1):L5–8.

[5] Norman TJ, Grant CD, Maganaand D, Zhang JZ, Liu J, et al. Nearinfrared optical absorption of gold nanoparticle aggregates. J PhysChem B 2002;106(28):7005–12.

[6] Guan J, Waite TD, Amal R, Bustamante H, Wukasch R. Rapiddetermination of fractal structure of bacterial assemblages in waste-water treatment: implications to process optimisation. Water SciTechnol 1998;38(2):9–15.

[7] Jiang Q, Logan BE. Fractal dimensions of aggregates determined fromstead-state size distribution. Environ Sci Technol 1991;25(12):2031–8.

[8] Jackson GA, Maffione R, Costello DK, Alldredge AL, Logan BE, DamHG. Particle size spectra between 1 μm and 1 cm at Monterey Baydetermined using multiple instruments. Deep Sea Res Part I:Oceanogr Res Pap 1997;44(11):1739–67.

[9] Friedlander S, Sioutas C. Final report: measurement of the effectivesurface area of ultrafine and accumulation mode PM (pilot project).Technical report R827352, United States Environmental ProtectionAgency, Los Angeles, CA; 2006.

[10] Electron and Agricultural Research Service Confocal MicroscopyLaboratory, Rime and Graupel, United States Department of Agri-culture, Beltsville, MD ⟨http://emu.arsusda.gov/snowsite/rimegraupel/rg.html⟩, 2006.

[11] Jet Propulsion Laboratory, A 10 micron interplanetary dust collected in thestratosphere, National Aeronautic and Space Administration, Pasadena, CA⟨http://stardust.jpl.nasa.gov/science/sd-particle.html#idp-s⟩, 2003.

[12] F.D.A. Keene, Clusters of roughly 30-nanometer gold nanoparticlesimaged by transmission electron microscopy, National Institute ofStandards and Technology, Gaithersburg, MD ⟨http://www.nist.gov/mml/biochemical/cluster-102511.cfm⟩, 2011.

[13] Carr J. Micrococcus luteus 9757, centers for disease control andprevention: public health image library, Atlanta, GA; 2007

[14] Brennan L, Owende P. Biofuels from microalgae: a review oftechnologies for production, processing, and extractions of biofuelsand co-products. Renew Sustain Energy Rev 2010;14(2):557–77.

[15] Dayananda C, Sarada R, Usha Rani M, Shamala TR, Ravishankar GA.Autotrophic cultivation of Botryococcus braunii for the production ofhydrocarbons and exopolysaccharides in various media. BiomassBioenergy 2007;31(1):87–93.

[16] Demura M, Ioki M, Kawachi M, Nakajima N, Watanabe M. Desicca-tion tolerance of Botryococcus braunii (trebouxiophyceae, chloro-phyta) and extreme temperature tolerance of dehydrated cells.J Appl Phycol 2014;26(1):49–53.

[17] Díaz Bayona KC, Atehortúa Garcés L. Effect of different media onexopolysaccharide and biomass production by the green microalgaeBotryococcus braunii. J Appl Phycol 2014;26(5):2087–95. http://dx.doi.org/10.1007/s10811-014-0242-5.

[18] Souliés A, Pruvost J, Legrand J, Castelain C, Burghelea TI. Rheologicalproperties of suspensions of the green microalga Chlorella vulgaris atvarious volume fractions. Rheol Acta 2013;52(6):589–605.

[19] Cornet J-F, Dussap C-G. A simple and reliable formula for assessmentof maximum volumetric productivities in photobioreactors. Biotech-nol Prog 2009;25(2):424–35.

[20] Pilon L, Berberoğlu H, Kandilian R. Radiation transfer in photobio-logical carbon dioxide fixation and fuel production by microalgae.J Quant Spectrosc Radiat Transf 2011;112(17):2639–60.

[21] Pruvost J, Cornet J-F. Knowledge models for the engineering andoptimization of photobioreactors. In: Posten C, Walter C, editors.Microalgal biotechnology. Berlin, Germany: De Gruyter; 2012.p. 181–224.

[22] Takache H, Pruvost J, Cornet J-F. Kinetic modeling of the photosyn-thetic growth of Chlamydomonas reinhardtii in a photobioreactor.Biotechnol Progr 2012;28(3):681–92.

[23] Jonasz M, Fournier G. Light scattering by particles in water:theoretical and experimental foundations. San Diego, CA: AcademicPress; 2007.

[24] Stramski D, Morel A, Bricaud A. Modeling the light attenuation andscattering by spherical phytoplanktonic cells: a retrieval of the bulkrefractive index. Appl Opt 1988;27(19):3954–6.

[25] Pottier L, Pruvost J, Deremetz J, Cornet JF, Legrand J, Dussap CG. Afully predictive model for one-dimensional light attenuation byChlamydomonas reinhardtii in a torus photobioreactor. BiotechnolBioeng 2005;91(5):569–82.

[26] Quirantes A, Bernard S. Light-scattering methods for modeling algalparticles as a collection of coated and/or nonspherical scatterers.J Quant Spectrosc Radiat Transf 2006;100(1–3):315–24.

[27] Berberoğlu H, Yin J, Pilon L. Light transfer in bubble spargedphotobioreactors for H2 production and CO2 mitigation. Int J HydrogEnergy 2007;32(13):2273–85.

[28] Davies-Colley RJ. Optical properties and reflectance spectra of 3shallow lakes obtained from a spectrophotometric study. N Z J MarFreshw Res 1983;17(4):445–59.

[29] Mackowski DW, Mishchenko MI. A multiple sphere T-matrix Fortrancode for use on parallel computer clusters. J Quant Spectrosc RadiatTransf 2011;112(13):2182–92.

[30] Xu Y-L. Electromagnetic scattering by an aggregate of spheres: farfield. Appl Opt 1997;36(36):9496–508.

[31] Iskander MF, Chen HY, Penner JE. Optical scattering and absorptionby branched chains of aerosols. Appl Opt 1989;28(15):3083–91.

[32] Kahnert M, Nousiainen T, Lindqvist H. Review: model particles inatmospheric optics. J Quant Spectrosc Radiat Transf 2014;146:41–58.

[33] Drolen BL, Tien CL. Absorption and scattering of agglomerated sootparticulate. J Quant Spectrosc Radiat Transf 1987;37(5):433–48.

[34] Lee E, Pilon L. Absorption and scattering by long and randomlyoriented linear chains of spheres. J Opt Soc AmA 2013;30(9):1892–900.

[35] Farias TL, Köylü ÜÖ, Carvalho MG. Effects of polydispersity ofaggregates and primary particles on radiative properties of simu-lated soot. J Quant Spectrosc Radiat Transf 1996;55(3):357–71.

[36] Sorensen CM, Cai J, Lu N. Light-scattering measurements of mono-mer size, monomers per aggregate, and fractal dimension for sootaggregates in flames. Appl Opt 1992;31(30):6547–57.

[37] Köylü ÜÖ, Faeth GM, Farias TL, Carvalho MG. Fractal and projectedstructure properties of soot aggregates. Combust Flame 1995;100(4):621–33.

[38] Farias TL, Köylü ÜÖ, Carvalho MG. Range of validity of the Rayleigh–Debye–Gans theory for optics of fractal aggregates. Appl Opt1996;35(33):6560–7.

[39] Lapuerta M, Martos FJ, Martín-González G. Geometrical determina-tion of the lacunarity of agglomerates with integer fractal dimen-sion. J Colloid Interface Sci 2010;346(1):23–31.

[40] Mroczka J, Woźniak M, Onofri FRA. Algorithms and methods foranalysis of the optical structure factor of fractal aggregates. MetrolMeas Syst 2012;19(3):459–70.

R. Kandilian et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 151 (2015) 310–326326

[41] Woźniak M, Onofri FRA, Barbosa S, Yon J, Mroczka J. Comparisonof methods to derive morphological parameters of multi-fractalsamples of particle aggregates from TEM images. J Aerosol Sci2012;47:12–26.

[42] Mishchenko MI, Travis LD, Lacis AA. Scattering, absorption, andemission of light by small particles. Cambridge, UK: CambridgeUniversity Press; 2002.

[43] Mackowski DW. A simplified model to predict the effects ofaggregation on the absorption properties of soot particles. J QuantSpectrosc Radiat Transf 2006;100(1–3):237–49.

[44] Bohren CF, Huffman DR. Absorption and scattering of light by smallparticles. New York, NY: Wiley & Sons; 1983.

[45] Modest MF. Radiative heat transfer. San Diego, CA: Academic Press;2003.

[46] Yang B, Köylü ÜÖ. Soot processes in a strongly radiating turbulentflame from laser scattering/extinction experiments. J Quant Spec-trosc Radiat Transf 2005;93(1–3):289–99.

[47] Wang G, Sorensen CM. Experimental test of the Rayleigh–Debye–Gans theory for light scattering by fractal aggregates. Appl Opt2002;41(22):4645–51.

[48] Bushell G. Forward light scattering to characterize structure of flocscomposed of large particles. Chem Eng J 2005;111(23):145–9.

[49] Liu F, Wong C, Snelling DR, Smallwood GJ. Investigation of absorp-tion and scattering properties of soot aggregates of different fractaldimension at 532 nm using RDG and GMM. Aerosol Sci Technol2013;47(12):1393–405.

[50] Liu F, Snelling DR, Smallwood GJ. Effects of the fractal prefactor onthe optical properties of fractal soot aggregates. In: ASME 2009 2ndMicro/Nanoscale Heat & Mass Transfer International Conference,Shanghai, China; 2009, MNHMT2009-18473.

[51] Yurkin MA, Hoekstra AG. The discrete dipole approximation: anoverview and recent developments. J Quant Spectrosc Radiat Transf2007;106:558–89.

[52] Manickavasagam S, Mengüç MP. Scattering matrix elements offractal-like soot agglomerates. Appl Opt 1997;36(6):1337–51.

[53] Liu L, Mishchenko MI. Effects of aggregation on scattering andradiative properties of soot aerosols. J Geophys Res Atmos2005;110(D11211).

[54] Liu L, Mishchenko MI. Scattering and radiative properties of com-plex soot and soot-containing aggregate particles. J Quant SpectroscRadiat Transf 2007;106(1–3):262–73.

[55] Liu L, Mishchenko MI, Arnott WP. A study of radiative properties offractal soot aggregates using the superposition T-matrix method.J Quant Spectrosc Radiat Transf 2008;109(15):2656–63.

[56] Dlugach JM, Mishchenko MI, Mackowski DW. Numerical simulationsof single and multiple scattering by fractal ice clusters. J QuantSpectrosc Radiat Transf 2011;112(11):1864–70.

[57] Latimer P. Experimental tests of a theoretical method for predictinglight scattering by aggregates. Appl Opt 1985;24(19):3231–9.

[58] Bricaud A, Bedhomme AL, Morel A. Optical properties of diversephytoplanktonic species: experimental results and theoretical inter-pretation. J Plankton Res 2008;10(8):51–73.

[59] Morel A, Ahn YH. Optics of heterotrophic nanoflagellates andciliates: a tentative assessment of their scattering role in oceanicwaters compared to those of bacterial and algal cells. J Mar Res1991;49:177–202.

[60] Lee E, Heng R-L, Pilon L. Spectral optical properties of selectedphotosynthetic microalgae producing biofuels. J Quant SpectroscRadiat Transf 2013;114:122–35.

[61] Skorupski K, Mroczka J, Wriedt T, Riefler N. A fast and accurateimplementation of tunable algorithms used for generation of fractal-like aggregate models. Physica A: Stat Mech Appl 2014;404:106–17.

[62] Heng R-L, Sy KC, Pilon L. Absorption and scattering by bispheres,quadspheres, and circular rings of spheres and their equivalentcoated spheres. J Opt Soc Am A 2015;32(1), accepted.

[63] Mätzler C. MATLAB functions for Mie scattering and absorption,version 2, Institut für Angewandte Physik, Bern, Switzerland, reportno. 2002-11, 2002.

[64] Kandilian R, Lee E, Pilon L. Radiation and optical properties ofNannochloropsis oculata grown under different irradiances andspectra. Bioresour Technol 2013;137:63–73.

[65] Heng R-L, Lee E, Pilon L. Radiation characteristics and opticalproperties of filamentous cyanobacterium Anabaena cylindrica.J Opt Soc Am A 2014;31(4):836–45.

[66] Berberoğlu H, Yin J, Pilon L. Light transfer in bubble spargedphotobioreactors for H2 production and CO2 mitigation. Int J HydrogEnergy 2007;32(13):2273–85.

[67] Lee E, Pruvost J, He X, Munipalli R, Pilon L. Design tool andguidelines for outdoor photobioreactors. Chem Eng Sci 2014;116:18–29.

[68] Mishchenko MI. Electromagnetic scattering by particles and particlegroups: an introduction. New York, NY: Cambridge University Press.

[69] Kerker M. The scattering of light, and other electromagnetic radia-tion. New York, NY: Academic Press; 1969.

[70] Mc Kellar BHJ, Box MA. The scaling group of the radiative transferequation. J Atmos Sci 1981;38:1063–8.

[71] Dombrovski LA, Randrianalisoa J, Baillis D, Pilon L. Use of Mie theoryto analyze experimental data to identify infrared properties of fusedquartz containing bubbles. Appl Opt 2005;44(33):7021–31.

[72] Hammer M, Yaroslavsky AN, Schweitzer D. A scattering phasefunction for blood with physiological haematocrit. Phys Med Biol2001;46(3):65–9.

[73] Friebel M, Roggan A, Mueller G, Meinke M. Determination of opticalproperties of human blood in the spectral range 250 to 1100 nmusing Monte-Carlo simulations with hematocrit-dependent effectivescattering phase functions. J Biomed Opt 2006;11(3):034021.

[74] Haltrin VI. One-parameter two-term Henyey–Greenstein phasefunction for light scattering in seawater. Appl Opt 2002;41(6):1022–8.

[75] Mobley CD, Sundman LK, Boss E. Phase function effects on oceaniclight fields. Appl Opt 2002;41(6):1035–50.

[76] West RA. Optical properties of aggregate particles whose outerdiameter is comparable to the wavelength. Appl Opt 1991;30(36):5316–24.

[77] Mackowski DW, Mishchenko MI. Calculation of the T-matrix and thescattering matrix for ensembles of spheres. J Opt Soc Am A 1996;13(11):2266–78.

[78] Mishchenko MI, Mackowski DW, Travis LD. Scattering of light bybispheres with touching and separated components. Appl Opt1995;34(21):4589–99.