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Journal of Quantitative Spectroscopy &Radiative Transfer

Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348

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Inhomogeneity structure and the applicability of effectivemedium approximations in calculating light scatteringby inhomogeneous particles

Chao Liu, R. Lee Panetta n, Ping YangDepartment of Atmospheric Sciences, Texas A&M University, College Station, TX 77843, USA

a r t i c l e i n f o

Article history:Received 11 November 2013Received in revised form17 March 2014Accepted 18 March 2014Available online 28 March 2014

Keywords:Inhomogeneous particlesEffective medium approximationPSTD

x.doi.org/10.1016/j.jqsrt.2014.03.01873/& 2014 Elsevier Ltd. All rights reserved.

esponding author.ail address: r-panetta@geos.tamu.edu (R. Lee

a b s t r a c t

Atmospheric aerosols commonly occur as inhomogeneous particles with significantinternal variations of refractive indices. For the purpose of light-scattering calculations,effective medium approximations (EMAs) have been developed that seek to model aninhomogeneous particle with an internal variation of refractive index using a homo-geneous particle with a single “effective” refractive index, one whose value is expected toyield approximately the same scattering properties. Here the applicability of four EMAs isinvestigated in the relatively simple case that the particle is a mixture of materials havingjust two different indices of refraction, and consideration is given to the effects of thespatial arrangement of these on the reliability of results obtained using the EMAs. TheEMAs considered are based on the Bruggeman theory, the Maxwell–Garnett theory, andtwo different Wiener bounds. The mixing states considered are relatively regular forms of“layering,” as well as states that involve varying degrees of more irregular “mixing.” Justtwo overall particle shapes are considered: spheres and spheroids. For each inhomoge-neous particle two sets of scattering calculations are performed: calculations treating theinhomogeneous particle itself exactly, and calculations treating an effective homogeneousparticle with the same shape and size but a single refractive index determined using oneof the EMAs. The first kind is done using the core–mantle Mie theory in the case ofstratified composition and the pseudo-spectral time-domain method in the more irregularmixing cases. The second kind of calculation is done using the Lorenz–Mie or T-matrixmethod.

Calculated values of extinction efficiencies, asymmetry factors, and phase matrices of both asingle particle and an ensemble of particles are compared. In comparisons of the EMAs witheach other, the Bruggeman model and Maxwell–Garnett model give similar results, resultswhich differ by small but noticeable amounts from the results obtained from the Wienerbounds. This suggests that the particular choice of one of the EMAs over another is not criticalin calculating the scattering properties of atmospheric particles, especially the bulk-scatteringproperties. Comparison with the numerically exact results shows that the applicability of theEMAs is largely independent of the particle shape, size, or volume fraction of the components,but is significantly affected by different mixing states. It is found that the Bruggeman and

Panetta).

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348332

Maxwell–Garnett theories can only give accurate approximations for the optical properties ofwell-mixed particles, ones for which the scale characterizing the index variation is no largerthan about 0:4=π times the wavelength of the incident light, and that applicability of EMAs forstratified and weakly mixed particles is very limited. These results must of course be viewedwith some caution, given the great number of combinations of particle geometries, mixingstates, and refractive indices that are possible compared to the few idealized ones we examine.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Atmospheric aerosols (e.g. mineral dust, black carbon,and sea salt) affect the climate directly by scattering andabsorption of solar radiation and terrestrial infrared emis-sion and indirectly by aerosol–cloud interaction [1–3].However, quantification of the impact of atmosphericaerosols remains questionable because of the varia-tions in aerosol particle shape, size, and composition.The scattering and absorption properties of complicatedparticles are vital to our understanding of the direct andindirect effects of aerosols [4,5].

Quantitative information about the shape, size, numberof components, and mixing state of natural aerosols isusually unknown [6–8], and, in general, the particles occuras mixtures of various components whose optical proper-ties may vary significantly. However, in radiative transferand remote sensing applications, these inhomogeneousparticles are represented using “equivalent homogeneous”particles having the same shapes and sizes but uniformrefractive indices, making the optical properties mucheasier to estimate [9,10]. This is done by using some formof effective medium approximation (EMA) to calculate aneffective refractive index for the mixture, a value whichwhen given to homogeneous particle is expected to resultin similar optical properties to those of the inhomoge-neous particle [11,12]. A variety of EMAs have beendeveloped and used for atmospheric aerosols.

The first EMA, based on the Maxwell–Garnett (MG)theory, was developed over a century ago [13,14], and wasused on dust aerosols by Longtin et al. [15]. A model basedon the Bruggeman (BR) theory [16,17] is another EMA thathas also been widely used to study the radiative propertiesof mineral aerosols [18,19]. In addition to the MG and BRtheories, Sokolik and Toon [18] chose a simple volumeaverage approximation to calculate the effective refractiveindices of aerosols. Additional EMAs have been proposed:the performances of nine EMAs, developed for a variety ofdifferent shapes, size distributions, physical properties,and mixing structures, have been reviewed by Kolokolovaand Gustafsonm [12].

Although the EMAs and the equivalent homogeneousapproximations are widely used [9,10,12,18,19], theirapplicability and accuracy in atmospheric light-scatteringstudies are still uncertain. Chylek et al. [20] and Kolokolovaand Gustafsonm [12] compared the EMA results withexperiments for spheres with small inclusion amountsand sizes, and both found acceptable agreement betweenthe calculated and measured values. From numericalinvestigations, Perrin and Lamy [21] found the application

of the EMAs to the study of the interaction of light withinhomogeneous dust particles to be limited. However,Voshchinnikov et al. [22,23] indicated that the EMAs couldbe used to simplify the computation of the optical proper-ties of aggregates containing only Rayleigh inclusions. TheEMAs have also been applied to some specific cases, i.e.water coated soot aggregates by Yin and Liu [24] and Liuet al. [25], and the MG was found to provide more accurateapproximations for the single-scattering properties ofcoated aggregates than the BR. More detailed discussionof effective medium theories can be found in [11] andits references. However most of the foregoing studiesused the EMAs with significant limitations on particlesize, shape, mixing structure, or volume fraction of thecomponents, and, in fact, drew substantially differentconclusions.

This study focuses on the case of just two constituentswith different refractive indices, and considers how theinternal arrangement of the constituents affects the abilityof the EMAs to reliably treat problems of light scatteringby inhomogeneous atmospheric aerosols. Four EMAs,three mixing states, and two particle shapes will beinvestigated, and we consider cases of the two compo-nents having volume fractions that vary from 0 to 1. Foreach inhomogeneous particle, approximate and exactmethods are used to calculate its scattering properties.The approximate method treats the inhomogeneous par-ticle as its equivalent homogeneous counterpart, and usesthe EMAs to find the effective refractive indices, withwhich its scattering properties are calculated by theLorenz–Mie theory [26] and the T-matrix theory [27,28].It should be noticed that the Lorenz–Mie and the T-matrixresults are understood as the approximations in this study,because they are used for the equivalent homogeneousparticles, not the original inhomogeneous ones. Represent-ing standards of “truth” for the scattering properties ofthe inhomogeneous particle itself, the core–mantle Mie[29,30] theory and the pseudo-spectral time domainmethod (PSTD) [31,32] are applied. Then the applicabilityand accuracy of the EMAs can be evaluated by comparingthe results from the PSTD/core–mantle Mie methods withthe results from a combination of the EMAs and Lorenz–Mie/T-matrix theories. The scattering properties examinedin making these comparisons are the extinction efficiency(Qext), asymmetry factor (g), and phase matrix elements P11and P12 of both a single and an ensemble of inhomoge-neous particles. Definitions of these scattering propertiesare given in [33].

Section 2 defines the four EMAs that will be used in thisstudy. The particle shapes and mixing states of the

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348 333

inhomogeneous model are introduced in Section 3. Wedescribe the numerical methods for the light scatteringsimulations and the validation of them for inhomogeneousparticles in Section 4. The results are discussed in Section5, and Section 6 concludes the study.

2. The effective medium approximations

Models using the EMAs try to replace a dielectricinhomogeneous particle, i.e. a particle having internalvariations of refractive indices, with a homogeneous onethat has a single “effective” refractive index and approxi-mately the same scattering properties as the inhomoge-neous counterpart. The MG [13,14], whose history datesback to 1904, was one of the first attempts to approximatethe optical properties of inhomogeneous materials. Itconsiders a mixture of two components, with permittiv-ities ϵ1 and ϵ2, and defines the effective permittivity ϵMG tosatisfy the relationϵMG�ϵ2ϵMGþ2ϵ2

¼ f 1ϵ1�ϵ2ϵ1þ2ϵ2

; ð1Þ

where f1 is the volume fraction of the first component. Theasymmetry evident in the formula reflects the intent of theoriginal MG theory to treat the case that the first compo-nent is the “medium,” the second component is the“inclusion,” and the inclusion has a volume fraction thatis assumed to be “small” [34]. We will not make thisrestriction in our examination of the MG-based EMAs. TheBR, on the other hand, treats the two components of aninhomogeneous particle symmetrically [16,17], defining aneffective permittivity ϵBR to satisfy

f 1ϵ1�ϵBR

ϵBRþsðϵ1�ϵBRÞþ f 2

ϵ2�ϵBRϵBRþsðϵ2�ϵBRÞ

¼ 1; ð2Þ

where ϵ1 and ϵ2 are the permittivities of the two compo-nents, and f1 and f2 are their volume fractions. In Eq. (2), sis a geometric factor related to the shape of the inclusion,and, for three-dimensional spherical inclusions, s is 1/3.

In addition to these two popular EMAs, we consider thelower and upper limits of the effective permittivities givenby the Wiener bounds [35]. The absolute bounds ϵmin andϵmax are given by

ϵmin ¼f 1ϵ1þ f 2ϵ2

� ��1

; ð3Þ

ϵmax ¼ f 1ϵ1þ f 2ϵ2: ð4ÞAll four of these EMAs are formulated in terms of the

permittivities of constituents. In studies of atmosphericaerosols, the complex refractive index m is also commonlyused, and is related to the complex permittivity ϵ bym2 ¼ ϵ. Thus

m¼ Re½m�þ Im½m�i¼ ffiffiffiϵ

p: ð5Þ

When we wish to refer to the effective permittivity with-out specifying the particular version, we will use thesymbol ϵeff. We define the corresponding refractive indexto be meff.

In addition to these four EMAs, there are more compli-cated ways to specify effective permittivity, and an exten-sive review of their homogenization strategies can be

found in [12]. While differences in results obtained usingone or another may occur in special circumstances, most ofthe approaches give values for effective permittivity thatgenerally differ from each other by relatively smallamounts. Here we just consider two popular ones andthe two Wiener bounds, together given by Eqs. (1)–(4).

Before comparing the optical properties of the inhomo-geneous and equivalent homogeneous particles, we directlycompare the effective refractive indices given by the fourEMAs. In our comparisons we will be extending someearlier work [36] that considered just volume fractionvariations. Figs. 1 and 2 show two different ways of makingthe comparison, each with the same assumed value of therefractive index of the component whose properties aredenoted by the subscript “1.” Specifically, both figuresassume m1¼1.5þ0.0001i (a typical value for mineral dustat visible wavelength [37]). Variations of the real andimaginary parts of the index of refraction of the othercomponent are then considered in separate figures. In Fig. 1Im½m2� ¼ 0 and variations of the volume fraction f1 of thefirst component (horizontal axis in each subplot) and realpart Re½m2� (vertical axis in each subplot) of the index ofrefraction of the second component are considered. InFig. 2, instead Re½m2� is held at the value Re½m2� ¼ 1:3(water or ice at visible wavelength [38]). Im½m2� (now onthe vertical axis in each subplot) is allowed to vary, whileagain f1 varies on the horizontal axes.

In each of the two figures the top row of panels showsvalues for the real parts of the effective refractive indicesas predicted by the different EMAs, and the bottom rowshows values for the corresponding imaginary parts.Rather than showing the actual values for these quantitiesin the case of each EMA, which would result in rows ofplots that would be visually indistinguishable, we show inthe first panel of each row the result provided by theBruggeman approach, and in the remaining panels of thesame row show the value given by the corresponding EMArelative to the Bruggeman value, i.e. as a ratio to it. Thus itis important to note from the scaling indicated on the colorbars that what by color contrast appears to be a largevariation in relative values reflects in fact quite smalldifferences in absolute values: this is an example of thefeature mentioned above, that different EMAs in commonuse generally give only small differences in effectiverefractive index.

As just explained, the left-most panels of Fig. 1 showRe½mBR� and Im½mBR�, where the subscript “BR” indicatesthat values are calculated by the BR. We see immediatelythat Im½mBR� is sensitive primarily to volume fraction,while Re½mBR� is sensitive to both volume fraction andRe½m2�. In considering the sensitivity of Re½mBR�, it isimportant to recall that Re½m1� ¼ 1:5; which is the valueassigned to the color green in the color bar. This explainsthe value and weak sensitivity of Re½mBR� to variations ofRe½m2� at volume fractions f 1 � 1, in which case theparticle is almost entirely composed of the first compo-nent, as well as the extension of the green region all theway to f 1 ¼ 0 at Re½m2� ¼ 1:5. The variation of Re½mBR�follows that of Re½m2� for small values of f1, when theaerosol is almost completely made up of the secondcomponent.

Fig. 2. The same as Fig. 1 but variations of Im½m2� are considered instead. Here Re½m2� ¼ 1¼ 1:3, and again m1 is fixed to be 1:5þ0:0001i.

Fig. 1. The real (upper panels) and imaginary (lower panels) parts of the effective refractive indices calculated by the four EMAs as functions of the volumefractions (f1) and the real part Re½m2�. In all plots m1 is fixed to be 1:5þ0:0001i and Im½m2 ¼ 0�.

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348334

When jRe½m2��Re½m1�jo0:1, the three comparisonratios are all approximately 1. This indicates that the fourEMAs all produce essentially the same Re½meff �. WhenjRe½m2��Re½m1�j40:1, the differences between valuesfor Re½meff � given by the different EMAs become

more noticeable and increase with an increase ofjRe½m2��Re½m1�j. However, the magnitude of the relativedifferences between the MG results and those given by theBR is no more than 0.8%. Furthermore, the relative differ-ence of Re½mmin� from the BR value is less than 5%, and the

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348 335

relative difference of Re½mmax� from the BR value is lessthan 3%.

With imaginary parts for both components fixed atrelatively small (0.0001 and 0) values, the value forIm½mBR� is almost independent of Re½m2�, and the valuesof Im½meff � given by the BR and MG are relatively close,although the differences do increase as jRe½m2��Re½m1�jincreases. Note from Fig. 1 that the Wiener “minimum”

method can give a value for Im½meff � that is for some flarger than the BR value, and for other f smaller than theBR value. Similarly, the Wiener “maximum” method canalso produce values below as well as above the BR value:relative ratio of Im½mmax� to Im½mBR� ranges between 0.5and 2. Note that we have chosen not to explore absorptionproperties in any detail as part of this study, so only verysmall imaginary parts will be used. Small values are typicalof many atmospheric aerosols at visible wavelengths.However, some atmospheric particles are highly absorp-tive, and the influences of the EMAs on the scatteringproperties of absorptive particles could be an interestingtopic for future study.

Fig. 2 shows variations of the real and imaginary partsof the effective refractive indices, displayed in the samemanner as in Fig. 1, this time with the fixed valueRe½m2� ¼ 1:3 and allowing Im½m2� to vary. The range ofthese variations is indicated on the vertical axis in each ofthe sub-panels. As in Fig. 1, m1 is 1.5þ0.0001i. Althoughthe real part of each component is fixed, the Re½mBR� doesshow some variation with an increase of Im½m2� at fixedvolume fraction f1, and this variation becomes larger whenIm½m2� becomes close to 1. When Im½m2� is smaller than0.2, all four EMAs give almost the same Re½meff �. Thedifferences become more noticeable when Im½m2� is largerthan 0.4. The MG may give either smaller or larger Re½meff �values than the BR by 1%, while much larger differencesare noticed for the values given by the Wiener bounds. TheRe½mmin� can be over 10% larger than the BR result withIm½m2� larger than 0.7 and f1 around 0.5, while theRe½mmax� may be more than 5% smaller than the BR result.For Im½meff �, the values given by the four EMAs differsignificantly, and the differences are sensitive both toIm½m2� and to f1. With the small value chosen for theimaginary part for the component 1, Im½mBR� of coursedecreases with increasing volume faction f1.

For simple mixtures with two components, there are asmany as five variables (i.e. Re½m1�, Im½m1�, Re½m2�, Im½m2�and f1), making it a complicated undertaking to considervariations of all those parameters. This study will insteadjust consider the effects of different volume fractions,while the refractive indices will be fixed. In this considera-tion, Figs. 1 and 2 will become very useful to infer therelative performance of the four EMAs for cases withrefractive indices different from those we discuss.

3. The inhomogeneous particles considered

As is clear from the definitions of effective permittivitiesin the previous section, there is no attention paid to mixingstate or particle shape, and one or another of the four EMAshas been used for more than one kind of particle shapesand/or mixing state. But it is certainly conceivable that the

scattering properties of inhomogeneous particles withdifferent mixing states may be substantially different, andit would seem therefore that the applicability of effectivepermittivity approaches should have their limits. This studyis an exploration of these limits, with a restriction of focusto spherical and spheroidal particles, these being the shapesmost widely used particle geometries for aerosol particles[9,10], with further restriction of the spheroids to thosewith an aspect ratio of 0.5. We will, however, includevariations in size.

What are called “external” and “internal” mixtures arethe most widely used structures to model inhomogeneousaerosols [7,39], e.g. the internally mixed sulfate andorganic particles [40], sea salt and silicate mineral compo-nent [41], and externally mixed mineral dust and blackcarbon [42]. Stratified particles are also very common inthe atmosphere, e.g. aerosol particles surrounded by con-densed water vapor [43] and black carbon coated bysulfate [44]. This study will consider three mixing states:

�

Stratified particles: One component is coated by theother. In our representation the core and mantle partsare concentric, and (for spheroids) they have the sameaspect ratios.

�

Externally mixed: Two parts with different componentsattached to each other, with the connecting surfacebeing planar and perpendicular to the symmetric axisof the spheroids, i.e. a sphere or spheroid separatedinto two parts different refractive indices by a plane.

�

Internally mixed: Elements of the two components aremixed uniformly throughout the entire particle. Thespherical or spheroidal particle is formed by randomlyarranged small elements of the two components, andthe individual elements are small enough to lie in theRayleigh scattering regime.

Fig. 3 illustrates inhomogeneous spheres (upperpanels) and spheroids (lower panels) with the threemixing states. The dark and light regions represent thetwo components with refractive indices of m1 and m2,respectively. For the stratified particles, one componentcan be either the core or the mantle, and the scatteringproperties of both cases will be studied. Of course both themixing states and overall particle shapes we are consider-ing are much simpler than occur in real aerosol particles.But they do include the most important and generalmixing cases seen in atmospheric aerosols, and the con-sideration of only such ideal versions will not, we believe,fundamentally affect the significance of conclusions wewill draw concerning the applicability of the EMAs in casesof these three basic mixing states.

4. The scattering models

In the homogenization approach, equivalent homoge-neous particles with their effective refractive indicescalculated by the EMAs are intended to have similaroptical properties to their inhomogeneous counterparts.With the particle shapes taken to be spherical or spher-oidal, the scattering properties of equivalent homogeneous

m1

m2

Fig. 3. Three models of mixing states of inhomogeneous particles considered in this study: (a) stratified; (b) externally mixed; and (c) internally mixed.

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348336

particles can be calculated efficiently and accurately by theLorenz–Mie [26] and T-matrix [27,28] theories. The accu-racy of the homogeneous approximation can be evaluatedby comparing with the exact scattering properties of theinhomogeneous particles. A number of numerical methodscan be used to calculate the optical properties of inhomo-geneous particles, for example the core–mantle Mie theory[29,30], the finite-difference time domain method (FDTD)[45,46], the PSTD method [31,32,47], or the discrete dipoleapproximation (DDA) [48,49]. We choose to use the core–mantle Mie theory, a rigorous method that can be effi-ciently applied, for example to stratified spheres, and usethe PSTD for other kinds of inhomogeneous particles. ThePSTD is a numerically exact method, where we recall thatin electromagnetic studies the term “numerically exact”refers to a computational method that is based on solvingsome form of Maxwell's equations (hence the adjective“exact”) using a numerically convergent scheme. Otherexamples of numerically exact methods are the DDA andT-matrix methods. This is in contrast to, say, geometricoptics methods, which are based on some form of ray-tracing equations, and hence an “inexact” physical modelthat only applies in a limited range of particle sizes.

The PSTD, which solves Maxwell's curl equations in thetime domain, can simulate the scattering of light byarbitrarily shaped particles [31,50]. The PSTD uses spectralmethods to calculate spatial derivatives and finite-difference methods to calculate temporal derivatives inMaxwell's curl equations. The field in the time domain istransformed to its counterpart in the frequency domainthrough a discrete Fourier transformation, and a surfaceintegral equation is used to transform the near-field valuesinto their far-field counterparts for the scattering proper-ties [51]. By truncating high spatial wavenumber terms,difficulties introduced by the Gibbs phenomena in the

spectral method can be controlled [31,50]. The PSTD hasbeen applied to spheres with size parameters up to 200 atice refractive indices with real parts close to 1.3 [31]. Herewe define the size parameter x for a sphere as

x¼ 2πr=λ; ð6Þwhere r is the radius of the sphere and λ is the incidentwavelength. Liu et al. [32] compared the PSTD with theDDA for homogeneous spheres and spheroids, and foundthe PSTD to be more efficient for numerical simulations ofparticles with refractive indices larger than 1.4 and smallrefractive indices but large sizes. In view of these compar-isons of the relative applicability and efficiency of the twomethods, we have chosen to use primarily the PSTD incalculating the numerically exact scattering properties ofthe inhomogeneous particles, using the DDA only at theoutset for validation, as explained next.

Inhomogeneous particles are numerically specified interms of the spatial distribution of the refractive indices, aspecification that is straight-forward in a discretizedspatial domain. The DDA has been applied for arbitraryparticle shapes and compositions [52]. In this study theAmsterdam DDA implementation (ADDA) is used [53].Because the results mentioned above concerning betterperformance by the PSTD for large particles consideredonly homogeneous particles, we will first verify theapplicability of the PSTD for inhomogeneous particles bycomparing the PSTD results with the core–mantle Mieresults for stratified spheres and results from the DDA forexternally and internally mixed particles.

The PSTD, DDA and core–mantle Mie results of strati-fied spheres with a size parameter of 50 are shown inFig. 4: the volume fractions of the spherical core, i.e. fc, are0.01, 0.1, 0.5, and 0.9. The refractive index of the core is 1.2and that of the mantle is 1.1. For the validation, relatively

Fig. 4. P11 (left panels) and P12/P11 (right panels) for stratified spheres with size parameters of 50 given by the core–mantle Mie, PSTD and DDA methods,and, from upper to lower, the volume fractions of the core part, i.e. fc, are 0.01, 0.1, 0.5, and 0.9. The refractive indices of the core and mantle are 1.2 and 1.1,respectively.

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348 337

small refractive indices will be used because the DDA isvery accurate and efficient when the refractive indices ofthe particles are less than 1.2 [54,55]. The left panels showthe normalized phase function P11, and the right panelsshow the ratios of P12 to P11. Results concerning the othernon-zero phase matrix elements show similar agreementand will not be shown. For all four cases with volumefractions ranging from 0.01 to 0.9, both the PSTD and DDAresults show overall good agreement with the rigoroussolutions, with only slight differences with f c ¼ 0:9 atsome scattering angles. Similar results are found for theratios P12/P11. Fig. 4 indicates both the PSTD and DDA to berobust and accurate methods for calculating the scatteringproperties of inhomogeneous particles with stratifiedstructures and can be applied over the entire range ofthe volume fractions.

The conclusions are a little different if larger refractiveindices are used: Fig. 5 is the same as Fig. 4 but with morerealistic refractive indices of atmospheric components. Therefractive index of the core is 1.5þ0.0001i (mineral dust atvisible wavelength [37]) and of the mantle is 1.3þ0i

(water or ice at visible wavelength [38]). In Fig. 5, it isclear that the PSTD results still agree very well with theanalytic solutions, whereas the DDA does not perform aswell. Furthermore, when fc reaches 0.9, i.e. the compo-nents with refractive index of 1.5þ0.0001i becomes thepredominant component, the DDA converges too slowly,and no DDA result is obtained. This indicates that the PSTDsimulations and their accuracy are less sensitive to therefractive indices of the particles, while the DDA encoun-ters difficulty as the refractive index increases.

Figs. 4 and 5 show the angular-dependent phase matrixelements of the inhomogeneous particles and their inte-gral scattering properties, i.e. the extinction efficiency Qext

and asymmetry factor g; the relative errors are listed inTable 1. The accuracy of both methods is excellent for thecoated spheres with relatively small refractive indices (allrelative errors are less than 1.5%), whereas the relativeerrors of the DDA results become as large as 6.8% for thelarge refractive indices' cases with fc¼0.1. To show therelative efficiency of the PSTD and DDA at the tworefractive index groups, Table 1 also includes the ratios

Fig. 5. The same as Fig. 4 but the refractive indices of the core and mantle are 1:5þ0:0001i and 1:3þ0i, respectively.

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348338

of the computational times used by the PSTD to DDA (allcomputations done on the same machine). From the ratiosof the computational times used by the two methods, wecan see that the PSTD is more efficient for spheres withlarge refractive indices (and also more accurate), whereasthe DDA outperforms the PSTD for the small refractiveindices (i.e. faster with a similar accuracy). Similar resultson relative performance were found in the homogeneouscase [32,56].

Fig. 6 shows P11 and P12/P11 of the externally (upperpanels) and internally (lower panels) mixed spheres givenby the PSTD and DDA. For internally mixed particles theelement size here and below, unless otherwise explicitlymentioned, is D¼0.2 (see definition in Section 5.2). Thesize parameters of the spheres are 50, and the volumefractions of the two components are both 0.5. The refrac-tive indices of the two parts are 1.2 (dark region) and 1.1(light region). The incident directions are illustrated in thefigure. The curves calculated by the PSTD and DDA arealmost undistinguishable for P11 in both cases, but the P12/P11 ratio of the internally mixed sphere differs slightly at afew scattering angles. Again, the excellent agreement

indicates the applicability of the PSTD for externally andinternally mixed particles. The integral scattering proper-ties for the two cases are listed in Table 2, and the relativedifferences of the two methods are less than 1%.

We now turn to consideration of internally mixedparticles. For the construction of these particles, once thevolume fractions of the two components are fixed, thedistribution of each component is randomly determined.Two different realizations of these randomly generatedparticles can appear to be very different, and the effects ofthe particle realization on the scattering properties areunknown. We used a Monte-Carlo method to generate fiveinternally mixed spheres with a chosen volume fractionand size parameter of 50, and Fig. 7 shows the P11 and P12/P11 of the spheres with the different mixing realizations.(More details on this construction are provided in Section5 below.) The volume fractions of the component having arefractive index of 1.5þ0.0001i are 0.1 (upper panels) and0.5 (lower panels) and the other component has a refrac-tive index of 1.3þ0i. The figure clearly shows the fivespheres with the same size and volume fractions, butdifferent mixing realizations, to have almost the same P11

Table 1The integral scattering properties as well as their relative errors of the stratified spheres given by the PSTD and DDA compared with the core–mantle Miesolutions, and the ratios of the CPU times used for the PSTD to DDA calculations.

m fc 0.01 0.1 0.5 0.9

mc ¼ 1:2þ0:0i Qext Core–mantle Mie 2.306 1.778 2.527 1.952mm ¼ 1:1þ0:0i PSTD 2.289 1.765 2.551 1.981

(RE [%]) (�0.73) (�0.84) (0.95) (1.4)DDA 2.306 1.779 2.527 1.946(RE [%]) (0.0) (0.056) (0.0) (0.36)

g Core–mantle Mie 0.9480 0.9202 0.9068 0.8916PSTD 0.9480 0.9177 0.9068 0.8930(RE [%]) (�0.28) (�0.27) (�0.18) (0.16)DDA 0.9493 0.9212 0.9102 0.8986(RE [%]) (0.14) (0.10) (0.20) (0.79)

TPSTD=TDDA 10.6 12.7 13.3 2.30

mc ¼ 1:5þ0:0001i Qext Core–mantle Mie 2.344 1.957 1.976 2.205mm ¼ 1:3þ0:0i PSTD 2.342 1.958 1.978 2.226

(RE [%]) (-0.085) (0.051) (0.10) (0.95)DDA 2.262 2.091 1.978(RE [%]) (�3.5) (6.8) (0.10)

g Core–mantle Mie 0.8582 0.7831 0.7803 0.8380PSTD 0.8642 0.7894 0.7796 0.8357(RE [%]) (0.70) (0.80) (�0.13) (�0.27)DDA 0.8544 0.7984 0.7805(RE [%]) (�0.44) (2.0) (0.026)

TPSTD=TDDA 0.4 0.3 0.2

Fig. 6. P11 (left panels) and P12=P11 (right panels) of the externally and internally mixed spheres with size parameters of 50 given by the ADDA and PSTD.The volume fractions of both components are 0.5. The refractive indices of the two components are 1.1 (light region of the sphere) and 1.2 (dark region).

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and P12/P11. The indication is that the optical properties ofthe internally mixed particles do not depend on themixing realizations, and the light scattering by one reali-zation of the internally mixed particles can be used to

represent these of an ensemble with the same volumefraction.

The previous results indicate that the PSTD is a robustand accurate method to calculate the light scattering

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348340

properties of inhomogeneous particles. Considering therelative efficiency of the PSTD and DDA, as well as therefractive indices considered, this study uses the PSTD asthe exact method to calculate the scattering properties ofthe inhomogeneous particles. Results obtained with thePSTD are then regarded as the “truth” in exploring theapplicability of the EMAs to the simulation of the scatter-ing properties of inhomogeneous particles.

5. Results

This section compares the scattering properties ofinhomogeneous and corresponding equivalent homoge-neous particles. Both single- and bulk-scattering proper-ties will be considered, and the integral scatteringproperties as well as the phase matrix elements will bediscussed. The comparison will focus on two issues: (1) the

Table 2The integral scattering properties of internally and externally mixedspheres given by the PSTD and DDA.

m Mixing state Externally mixed Internally mixed

Qext PSTD 1.967 1.976DDA 1.978 1.977

g PSTD 0.7884 0.8983DDA 0.7932 0.9034

Fig. 7. P11 (left panels) and P12=P11 (right panels) of the internally mixed spheresof the spheres are 50. The volume fractions of components having a refractive inremaining component has a refractive index of 1:3þ0i.

relative performance of the four EMAs, and (2) theperformance of the EMAs to approximate the scatteringproperties of the inhomogeneous particles. The refractiveindices of the two components will be m1 ¼ 1:5þ0:0001iand m2 ¼ 1:3þ0i.

In what follows, when we refer to the volume fraction,we mean the fraction f1 of component with refractiveindex m1 (the volume fraction of the other componentbeing then f 2 ¼ 1� f 1). The two parameters we use todescribe a spheroid with equatorial radius a and semi-length of symmetry axis are the size parameter x and theaspect ratio ρ, defined by

x¼ 2πb=λ; ð7Þ

ρ¼ ab

ð8Þ

We will take ρ¼0.5. It should be noticed that, with thisdefinition spheres and spheroids with the same value forsize parameter can have different volumes. For stratifiedparticles, the term “stratified 1” will refer to the case whenthe component with refractive index m1 forms the core,and “stratified 2” will refer to the case when that compo-nent instead forms the mantle. All scattering propertiesconsidered are for “randomly oriented” particles, whichmeans that averages are taken over a set of randomorientations of the particle with respect to the incidentdirection: in the PSTD simulations, the single-scattering

with different mixing realizations given by the PSTD. The size parametersdex of 1:5þ0:0001i are 0.1 (upper panels) and 0.5 (lower panels), and the

Fig. 8. The integral scattering properties of inhomogeneous spheres and spheroids with a size parameter of 30 as functions of volume fractions. The resultsfor the homogeneous spheres are from a combination of EMAs and Lorenz–Mie theory, and the inhomogeneous spheres from the core–mantle Mie theoryand the PSTD.

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properties of spheroids with three mixing states, as well asthe externally mixed spheres are each averaged over 180scattering planes for 16 particle orientations.

5.1. Scattering from a single particle

Fig. 8 shows Qext (upper panels) and g (lower panels) ofinhomogeneous spheres (left panels) and spheroids (rightpanels) with a size parameter of 30 as functions of thevolume fraction. The markers in the figure indicate thecore–mantle Mie and PSTD results of inhomogeneousparticles with the three mixing states. The values of Qext

and g vary dramatically with the change in volumefraction. Particles with various mixing states have signifi-cantly different integral scattering properties, especiallythe two stratified cases. The four curves in Fig. 8 are theresults given by the EMAs for equivalent homogeneousparticles (combined with the Lorenz–Mie theory forspheres and the T-matrix theory for spheroids). Of thefour EMAs, the MG and BR have almost the same Qext and gfor all volume fractions from 0 to 1. However, when thevolume fraction is between 0.1 and 0.9, the Qext and gbased on the two Wiener bounds are significantly differentfrom those given by the other two, especially for spheres.The discrepancy can be explained by the relatively largedifferences in the effective refractive indices between theWiener bounds and the other two methods. Comparingthe results of the inhomogeneous and equivalent homo-geneous particles, we notice that only in the cases ofinternal mixing do the results from the equivalent

homogeneous cases (based on the BR and MG) agree wellwith those from the PSTD calculations; this agreement isgood over the entire range of volume fractions. Therelative errors given by the MG and BR compared withthe PSTD results for the internally mixed cases are lessthan 5% for the spheres and 2% for the spheroids. However,the PSTD values of Qext and g for the two stratified and theexternally mixed particles are very different from those ofthe equivalent homogeneous particles, and may be over30% larger or smaller at some volume fractions. Thus, theEMAs appear to be able to give accurate approximationsfor only the internally mixed particles, and are very poorfor the stratified and externally mixed particles. Becausethe same results are obtained for both the spheres andspheroids over the entire range of volume fractions, weexpect that similar conclusions regarding the applicabilityof the EMAs is independent of particle shape, but havechosen not to investigate other shapes in this study.

Fig. 9 illustrates the angular-dependent P11 (left panels)and P12/P11 (right panels) of inhomogeneous and equiva-lent homogeneous spheres with volume fractions of 0.5and size parameters of 30. The ratio P12/P11 is a measure ofthe degree of linear polarization of the scattered light. Fora clearer comparison, the results are shown in threepanels: (1) the upper panels are approximations of theequivalent homogeneous spheres given by a combinationof the four EMAs and the Lorenz–Mie method; (2) themiddle panels are results from the two stratified spheresand the homogeneous sphere based on the effectiverefractive index given by the BR; and (3) the lower panels

Fig. 9. P11 (left panels) and P12=P11 (right panels) of the inhomogeneous spheres with size parameters of 30 and f=0.5. The results for the homogeneousspheres are given by a combination of EMAs and Lorenz–Mie theory, and the results for the inhomogeneous spheres are from the core–mantle Mie theoryand the PSTD.

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348342

are the same as the middle ones but for externally andinternally mixed spheres. The upper panels of Fig. 9indicate that the BR and MG give quite similar results forboth phase matrix elements, but these results differ fromthe ones based on the Wiener bounds, especially aroundsome peaks. The middle and the lower panels compare P11and the ratio P12/P11 for the equivalent homogeneous andinhomogeneous particles. The stratified spheres (middlepanels) give totally different phase matrix elements fromthe equivalent homogeneous spheres for both the oscilla-tions and the overall angular dependence. Similar to thecase of integral scattering properties, the results from acombination of the BR and Lorenz–Mie theory agree verywell with those of the internally mixed sphere, except forslight differences in P12/P11 at some scattering angles. Thephase function of the externally mixed sphere follows thesame overall angular dependence as the homogeneoussphere, but shows much weaker oscillations, and the P12/P11 ratio appears very different. Comparisons for othernon-zero phase matrix elements show qualitatively similarpatterns of agreement and disagreement to those of P12/P11, and will not be shown. Overall, we see from Fig. 9 thatthe EMAs, specifically the BR and MG, provide an accurate

approximation for the phase matrix elements P11 and P12/P11 of the internally mixed spheres, but cannot be reliablyapplied to the stratified or externally mixed cases.

Similar to Fig. 9, the P11 and P12/P11 of randomlyoriented spheroids with a size parameter of 30 and volumefractions of 0.5 are given in Fig. 10. Here, the effectiverefractive indices from the EMAs are used for the T-matrixmethod. The agreement between the four EMAs becomesmuch better for spheroids, and the P11 given by the Wienerbounds shows a little difference from the other three atscattering angles about 1701. Furthermore, Fig. 10 indicatesthat EMA applicability for calculating the phase matrix ofnon-spherical particles is also limited to the internallymixed ones, and both P11 and P12/P11 of the stratified andexternally mixed particles are significantly different fromthe ones given by the EMAs.

The previous results are for particles with the same sizeparameter and varying volume fractions. In Fig. 11 weshow the results for Qext and g in the case of inhomoge-neous spheres (left panels) and spheroids (right panels) asfunctions of the particle size parameter. The size para-meter increases from 1 to 100, and the volume fraction isfixed at 0.5. As in Fig. 8, the EMA results are given by

Fig. 10. The same as Fig. 9 but for the spheroids with size parameters of 30. The homogeneous results are given by a combination of the EMAs and the T-matrix theory.

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curves with different markers to indicate the two stratifiedcases as well as the externally and internally mixed cases.When the size parameter is small ðxo5Þ, the Qext and g ofinhomogeneous particles in the three mixing states arequite similar, and the homogeneous approximations basedon the EMAs are close to those of the results of PSTDcalculations for the actual inhomogeneous particles. How-ever, the differences become significant for particles withlarger size parameters, except for the case of internallymixed particles. For particles with x45, the EMAs can giveaccurate approximations only for the internally mixedparticles. For the stratified or externally mixed cases,their scattering properties agree with those of theirhomogeneous counterparts only at some size parameters.

5.2. The scale of internal mixing

It seems that only the scattering properties of theinternally mixed particles can be approximated using theEMAs, whereas the definition of the internally mixedparticles are still incomplete. As we have mentioned inSection 3, the internally mixed particles we consider aremade up of randomly distributed small elements thatindividually have sizes in the Rayleigh scattering range.Earlier studies have shown that the EMAs give reliable

results for particles with such small inclusions [22,23]. Buthow large can the inclusions be before the EMAs lose theirability to represent scattering properties well? The basicelements are defined as cubes, and the size parameter ofan element is defined as

D¼ πdλ; ð9Þ

where d is the length of the cube side.We did calculations over a range of sizes from D¼0.2 to 3

in increments of 0.2 and compared with the results given bythe BR. Fig. 12 shows the comparison of the P11 and P12/P11 ofthe internally mixed spheres with just four different choicesfor element size. The cross sections of four such inhomoge-neous spheres are also illustrated in the figure. To construct arealization of the inhomogeneous sphere with a chosenvolume fraction for one component, we randomly add smallcubic particles with given size in the spherical space untiltheir total volume produces the desired fraction: that totalvolume is assigned to the component and the remainingvolume is assigned to the other component. If only a part of arandomly positioned cube lies inside the edge of the sphere,only that part is used. A volume fraction of 0.5 is used for thesimulation. For P11 and P12/P11, only the sphere withvery small elements (D¼0.2) shows essentially the same

Fig. 11. The integral scattering properties of the inhomogeneous spheres and spheroids as functions of the particle size parameters. The volume fraction is0.5 for the component with refractive index of 1:5þ0:0001i, and the remaining component has the refractive index of 1:3þ0i.

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scattering properties as the BR results, whereas definitedifferences emerge when D becomes 0.4, especially atscattering angles between 801 and 1401. While a choice ofcubes as elementary units may seem arbitrary and special, itmay be expected that, at such small sizes, the shape of theelements is not important: in fact we have found that similarresults to the ones we report here are obtained withspherical basic elements. Generally, as D increases theangular region of disagreement expands, and when inclusionelements have size parameter of 3, the phase function of theinhomogeneous particle becomes very smooth and signifi-cantly different from the equivalent inhomogeneous spherecalculation resulting from BR. A rough qualitative judgmentbased on the results in Fig. 12 would be that mixing mustoccur down to a scale of D� 0:4 for the internally mixedparticles to have the same angular-dependent scatteringproperties as those equivalent homogeneous ones.

5.3. Scattering from a distribution of particles

Previous results indicate that the EMAs cannot be usedto approximate the single-scattering properties of strati-fied and externally mixed particles, and Fig. 13 shows theirperformance for the bulk phase matrix elements forspherical particles, i.e. scattering properties averaged overa collection of spherical particles with a given size dis-tribution. In comparison with single particle results, theaveraging results in scattering angle dependencies ofmatrix elements that are more smooth and relativelyfeatureless. The incident wavelength is 0.6328 μm and

the volume fraction is 0.5. The size distribution of feldspargiven by the ALSD [37] is used, and the ensemble particleshave an effective radius of 1.0 μm and a variance of 1.0.Fig. 13 is organized similar to Figs. 9 and 10. For the bulkphase matrix elements P11 and P12/P11, the four EMAs areequivalent. However, the results of the homogeneousparticles are dramatically different from the two stratifiedcases. For externally mixed particles, the EMA results inthis bulk scattering case are quite close to the exactsolutions given by the PSTD, but some differences arenoticed at scattering angles from 901 to 1501. As expected,the P11 given by the EMAs and Lorenz–Mie theory agreevery well with those for the internally mixed particles,but the agreement becomes relatively poor for the ratioP12/P11.

Fig. 14 is the same as Fig. 13, but the overall particleshape is a spheroid with an aspect ratio of 0.5. The sameincident wavelength and particle size distribution are usedto give the bulk phase matrix. The upper panels of Fig. 14show the consistency of the four EMAs to approximate thebulk scattering properties of the inhomogeneous spher-oids. As seen in the middle panels, neither stratified resultfor P11 or P12/P11 agrees with values given by the combina-tion of the EMAs and T-matrix method, although theoverall angular dependences differ less than is seen inthe case of spherical particles. The lower left panel showsthat the bulk P11 of externally mixed spheroids and thehomogeneous results good agreement for forward scatter-ing, but the homogeneous particle values significantlyunderestimate the backward scattering. Similar results

Fig. 12. Comparison of the P11 (left panels) and P12=P11 (right panels) of the internally mixed spheres with different element sizes. The volume fraction 0.5is used for the simulation; the realizations used for different values of D are illustrated in the top panel.

Fig. 13. The bulk P11 and P12=P11 of an ensemble of spheres with effective radius of 1.0 μm and variance of 1.0.

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Fig. 14. The same as Fig. 13 but for the spheroids.

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was observed at the phase function of the single spheroidwith size parameter of 30 (see Fig. 10). Considering theexact solutions of the inhomogeneous particles with thethree mixing states, only the internally mixed results, bothP11 and P12/P11, coincide with the ones given by thehomogeneous approximations, and the choice of the EMAsis not essential.

Figs. 8–14 compare both the single- and bulk- scatter-ing properties of the inhomogeneous and equivalenthomogeneous particles. The scattering properties of parti-cles with the three different mixing states show significantdifferences. Homogeneous approximations based on theEMAs can only be applied for internally mixed particles,and are not accurate enough for the stratified and exter-nally mixed particles. Once again we stress that we havenot investigated a wider range of geometries than spheresand spheroids, but we nevertheless expect that our qua-litative conclusions will extend to other shapes.

6. Conclusion

This study investigates the applicability of EMAs forcalculating the scattering properties of inhomogeneousatmospheric particles, and four EMAs, three mixing states,and two overall particle shapes are considered. The EMAsare combined with the Lorenz–Mie or T-matrix theories to

approximate the scattering properties of the equivalenthomogeneous particles, and the results are verified bycomparing with the standards-of-truth provided by calcu-lations based on the core–mantle Mie theory or the PSTD,which consider the exact inhomogeneous mixing struc-tures of the particles. The four EMAs give effective refrac-tive indices with only slight differences between eachother. In fact, the four EMAs show essentially the sameability to approximate the bulk-scattering properties of theinhomogeneous particles, and only the single-scatteringproperties given by the Wiener bounds differ from thosebased on the Maxwell–Garnett theory and Bruggemantheory. Considering that the bulk-scattering propertiesare used for most applications and that the Wiener boundsgive the upper and lower limits of the EMAs, the choice ofone EMA rather than another appears not to be essentialfor atmospheric applications. The applicability of the EMAsis independent of the volume fractions of the componentsand size of particles, as well as the (admittedly limited)choice of overall shapes of the particles, but is determinedby the mixing state. The scattering properties of theequivalent homogeneous particles are significantly differ-ent from those of stratified or externally mixed particles,but agree with those of internally mixed particles forwhich the mixing occurs at small scales within the particle(with relative errors in the integral scattering properties

C. Liu et al. / Journal of Quantitative Spectroscopy & Radiative Transfer 146 (2014) 331–348 347

for spheres and spheroids less than 5% and 2% respec-tively). Not surprisingly, there is no clear boundarybetween scales small enough for application of EMAsand scales not small enough, but good representation bythe EMAs seems to require that the scale D characterizingthe degree of mixing should be no larger than approxi-mately Dcrit � 0:4. It should be noted, however, that thisparticular threshold mixing scale D is for the choices ofrefractive indices made in this study, and the thresholdscale may shift somewhat for other refractive indices. Amore complete survey over refractive indices would beneeded to determine what sensitivity the threshold mix-ing scale has to refractive index changes.

We note that this study considers only non-absorptiveor weakly absorptive particles, making the performance ofthe EMAs for particles with significant absorption (e.g.black carbon and metal) an interesting topic for futureresearch. Furthermore, in view of the significant irregular-ity of the atmospheric particles, the particles considered inthis study must be regarded as extremely idealized mod-els, both in terms of mixing states and overall particlegeometry. Nevertheless we believe that the cases consid-ered in this study are sufficient to establish two mainpoints: if the EMAs are to be used as an approximation forthe bulk scattering properties, the particular choice of theEMAs will matter little, and to achieve accurate scatteringproperties of atmospheric particles, the detailed mixingstates have to be considered and only in extremely wellmixed cases will EMAs be reliable.

Acknowledgment

We thank Dr. Michael Mishchenko for providing his T-matrix code, and Dr. M.A. Yurkin and Dr. A.G. Hoekstra fortheir ADDA code. We are grateful for helpful commentsmade by N.V. Voshchinnikov and an anonymous reviewer.This research was partly supported by the National ScienceFoundation (ATMO-0803779 and AGS-1338440). All com-putations were carried out at the Texas A&M UniversitySupercomputing Facility and the authors gratefullyacknowledge the assistance of the Facility staff in portingour codes.

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