Models for integrated and differential scattering optical properties of encapsulated light absorbing carbon aggregates

Michael Kahnert,1,2,∗ Timo Nousiainen,3 and Hannakaisa Lindqvist,3 1Swedish Meteorological and Hydrological Institute (SMHI), Research Department, Folkborgsvägen 17, SE-601 76 Norrköping, Sweden 2 Chalmers University of Technology, Department of Earth and Space Science, SE-412 96 Gothenburg, Sweden 3 Department of Physics, P. O. Box 48, FI-00014 University of Helsinki, Finland ∗ michael.kahnert@smhi.se


Introduction
Understanding the interaction of light absorbing carbon (LAC) aerosols with radiation is essential for interpreting remote sensing observations as well as for modeling the direct climate forcing effect of aerosols.The complex morphology and chemically heterogeneous composition of this class of aerosols presents a major challenge to modeling approaches based on rigorous electromagnetic scattering theory.Freshly emitted LAC particles are fractal-like aggregates that are characterized by the number of primary monomers N s , the monomer radius a, the fractal dimension D f , and the structural prefactor k 0 .The aggregate morphology is described by the scaling relation [1] where the radius of gyration is defined by and r i is the distance of the ith monomer from the aggregate's center of mass.Aging of LAC in the atmosphere results in changes of the morphological and optical properties (e.g.[2][3][4][5][6]).The initially lacy structure of the aggregates collapses (i.e., the fractal dimension increases) on a time scale that is much shorter than the average lifetime of LAC aerosols in the atmosphere (e.g.[7][8][9][10]).At the same time, LAC is partially oxidized and becomes increasingly hydrophillic.This process is accompanied by condensation of various materials onto the LAC aggregate, such as sulfate [11], organic matter, water, or sea salt [12].These liquid-phase materials are characterized by weak optical absorption at visible wavelengths.For this reason, we will collectively refer to the coating materials as "weakly absorbing" materials.Note, however, that at IR wavelengths the imaginary part of the refractive index of the coating materials, such as sulfate, can become comparable to that of LAC.Field measurements confirm that the majority of aged LAC aggregates are encapsulated in a shell of liquid-phase material [13] that is weakly absorbing at visible wavelengths.For a more detailed discussion of the physical and chemical processes that influence the morphology of aged LAC aerosols, see, e.g., [14,15] and references therein.
Electromagnetic scattering computations for such morphologically complex particles are extremely demanding.Computations for realistic model particles are usually limited to a few case studies.On the other hand, operational applications, such as the computation of look-up tables of aerosol optical properties for remote sensing retrieval algorithms or for broadband radiative transfer computations in climate models, require computations for a large range of parameters.This necessitates the introduction of simplifying assumptions.One approach is to introduce simplifications to the physics of the electromagnetic scattering problem.However, scattering of electromagnetic radiation by fractal aggregates is a highly complex process involving coherent near-field interactions among the primary monomers; it has been demonstrated that attempts to by-pass proper solutions to Maxwell's equations can introduce large errors for LAC aggregates (e.g.[16]).(For a general discussion about approximate methods for solving Maxwell's equations, see [17]).Probably the safer approach is to introduce simplifications to the particle morphology, and to validate the simplified model geometries by comparison with morphologically realistic models or with measurements (e.g.[11,15,[18][19][20]).This is the approach that will be followed in the present study.
Various simplified model geometries have been applied in the past.The simplest model treats LAC and weakly absorbing aerosols as externally mixed (i.e.separate) homogeneous spheres, thus ignoring both the complex structure and heterogeneous chemical composition of realistic LAC aerosols internally mixed with weakly absorbing material.The simplest internal mixture model describes these aerosols as internally mixed homogeneous spheres.The homogeneous mixture is characterized by an effective refractive index, which can be computed by simple volume-mixing rules or by effective medium theory (EMT) [21,22].A simple model for an encapsulated geometry is a core-shell model consisting of a spherical LAC core that is concentrically coated with a spherical shell of weakly absorbing material.In the past it has often been tacitly assumed that this model gives a better representation of the optical properties of aged LAC aerosols than the external mixture and homogeneous internal mixture models, because it is, putatively, morphologically more realistic (e.g.[23]).Recent comparisons with morphologically realistic encapsulated aggregate models showed, indeed, that the external mixture model substantially underestimates the absorption cross section C abs , and that the homogeneous internal mixture model overestimates C abs [15].However, it was also found that the core-shell model significantly underestimates C abs ; the error introduced by the core-shell model can even exceed that of the internal homogeneous mixture model [15].An improved model was proposed based on a core-shell-shell geometry, consisting of a weakly absorbing core, an inner LAC shell, and an outer weakly absorbing shell [15].Although this model yields a better representation of the optical properties of encapsulated LAC aggregates than the core-shell or homogeneous internal mixture models, it also has one main shortcoming.The multi-layer sphere model requires significantly more computational resources than a homogeneous sphere or core-shell model, thus making it difficult to use this model for operational purposes.
The work reported in [15] was limited to a few selected wavelengths and to an investigation of integrated optical properties relevant for climate forcing simulations.The present study goes beyond that by (i) extending the investigation to the entire spectral range from the UV-C to the mid-IR; (ii) adding an investigation of remote-sensing relevant quantities, such as the backscattering cross section, the angular distribution of the scattered radiance and polarization, and the absorption Ångström exponent; and (iii) introducing a new model particle (the "coregray shell" model) that achieves an excellent representation of most aerosol optical properties of encapsulated LAC aggregates, while being computationally much less expensive than the previously discussed core-shell-shell model.Section 2 provides an introduction to the morphological models and the numerical methods for solving Maxwell's equations.In Sect. 3 we present and discuss the results of this study, and concluding remarks are offered in Sect. 4.

Model particles
We solve Maxwell's equations for complex encapsulated aggregate morphologies of different sizes and volume fractions, as illustrated in Fig. 1.The parameters characterizing the morphological and dielectric properties of the encapsulated aggregates have been chosen based on field observations and laboratory studies such as to represent morphologically complex, aged LAC aerosols found in the atmosphere.More specifically, we fixed the fractal dimension and structural prefactor at D f =2.6 and k a =1.2, which are based on recent field observations of aged encapsulated LAC aggregates [13].The radius of the primary LAC monomers of our model particles is 25 nm.Sensitivity studies have shown that this value gives the best agreement between modeled and measured single-scattering albedos of pure LAC aggregates [16], and it is consistent with recent field observations of aged encapsulated LAC aggregates [13].The distance D i between the centers of mass of the LAC aggregate and the weakly absorbing coating of radius R shell has been chosen as D i /R shell =0.5, which is based on recent field observations [24].Computations have been performed for a range of particle sizes between R V =0.1-0.5 μm (where R V denotes the volume-equivalent radius).As the total size R V increases, both the number of monomers in the aggregate and the volume of the coating material increase accordingly, such that the volume fraction f remains constant (see Fig. 1 and Table 1).
We limit our study to aggregates coated by a spherical shell.Electron micrographs of encapsulated aggregates with low LAC volume fractions often show that the coating is spherical or nearly spherical (e.g.[11,12,15,25]).The reason for this is that the coating is in the liquid phase.Note, however, that for higher LAC volume fractions the coating may increasingly deviate from spherical shape; in the extreme case of LAC volume fractions approaching unity the coating will more or less assume the shape of a film covering the solid-phase LAC core.Such cases, which are more common for freshly emitted LAC near emission sources, are not Table 1.Number of monomers N s in the LAC aggregates and radius R V of the internally mixed aerosols for LAC volume fractions f =7 % and f =20 %.The radius R V is the radius of a volume-equivalent sphere, where the volume of the encapsulated aggregates is the sum of the volumes of the LAC aggregate and the coating material.The monomer size is 25   considered in our study.
The study of the wavelength dependence of the optical properties focuses on the 14 wavelength bands used in numerical weather prediction model IFS (Integrated Forecasting System) developed at the European Centre for Medium Range Weather Forecast.The bands cover the range of 0.2-12.2μm.
For each particle size we consider two LAC volume fractions, f =7 % and f =20 %.The former is close to values reported in, e.g., [11,24].However, as found in [24], volume fractions can display a high variability; far from emission sources LAC volume fractions are typically only a few percent.The higher value of f =20 % is rather extreme.Such high volume fractions are mostly observed in aerosol plumes that have not traveled far from their emission sources (e.g.[13]).
The simplest model for representing encapsulated LAC aggregates is based on treating the weakly absorbing part and the LAC as externally mixed, i.e. separate, homogeneous spheres, as illustrated in Fig. 2 (left).Another simple model particle is the homogeneous internal mixture model (or "gray-sphere model", Fig. 2, second from left).Here we use the Maxwell Garnett mixing rule [26], which has been shown to work better than the commonly used volume-mixing rules [15], to compute the effective refractive index of the internal mixture.In the Maxwell Garnett rule, it is assumed that LAC inclusions are embedded in a matrix of weakly absorbing material, where the LAC inclusions are assumed to be small compared to the wavelength.A model that is, arguably, morphologically more realistic is the core-shell model (Fig. 2, second from right), in which the LAC is concentrated in a homogeneous sphere and concentrically coated by a shell of weakly absorbing material.
A known problem with all of these simplified model geometries is that they fail to mimic the amount of LAC mass that interacts with the electromagnetic field, which has a strong influence on the absorption cross section.Inside a solid LAC sphere the field is quickly attenuated, so that much of the mass near the core of the particle does not significantly contribute to the 0.3 0.50.absorption process.In an aggregate, a larger fraction of the LAC mass interacts with the field.For this reason, the external mixture and core-shell models tend to strongly underestimate the absorption cross section of encapsulated LAC aggregates.However, in an aged LAC aggregate of relatively high fractal dimension, at least some of the LAC mass is shielded from interacting with the external field.The homogeneous internal mixture model distributes the LAC mass too evenly and allows too high an LAC fraction to interact with the field, thus overestimating the absorption cross section.To overcome these limitations, a core-shell-shell model was recently proposed [15], consisting of a core of weakly absorbing material, an inner shell of LAC, and an outer shell of weakly absorbing material.Although this model was found to provide a good representation of the absorption cross section and single-scattering albedo of encapsulated aggregates, computations with this model are considerably more time consuming than computations with homogeneous sphere models or core-shell geometries.This severely limits the usefulness of this model for building an aerosol optics database that can be used in climate modeling or in inverse modeling of remote sensing observations.Here we propose and test a different model that is computationally much faster than the coreshell-shell model, but also based on the notion that the most important aspect of a versatile model particle is the way in which the LAC mass is distributed within the weakly absorbing host, which determines the amount of LAC that is allowed to interact with the electromagnetic field.The model has a core-shell geometry.But by contrast to the original core-shell model (Fig. 2, second from right), the new model consists of a smaller core that only contains a fraction f core of the LAC mass.The remaining fraction (1 − f core ) of the LAC mass is homogeneously mixed with the weakly absorbing shell, thus "darkening" the shell.This model is illustrated in Fig. 2 (right).For brevity, we shall refer to this geometry as the "core-gray shell" geometry.By reducing the size of the core, less LAC is shielded from interacting with the electromagnetic field.Most of the LAC mass that is homogeneously mixed with the weakly absorbing shell does interact with the field and contributes to absorption.The core fraction f core is a free parameter that can be tuned to fit the reference calculations.This model appears to be sufficiently versatile to mimic the amount of LAC mass that interacts with the electromagnetic field in a realistic encapsulated aggregate, while maintaining a high degree of geometric simplicity and symmetry.Symmetry assumptions are the main factor in expediting numerical computations [27].
The dielectric properties assumed in this study are depicted in Fig. 3.By contrast to many other studies (e.g.[12]) we assume that the refractive index of LAC is dependent on wavelength.The spectral variation of the refractive index m of LAC in Fig. 3 is based on the measurements reported in [28].In a recent review [29] it was found that many refractive indices of LAC found in the literature are unrepresentative of LAC aerosols encountered in the atmosphere.The refractive index values observed in [28] are among the ones that lie closest to the recommendations given in [29] at a wavelength of 550 nm.However, we emphasize that we do not know if the measurements reported in [28] are equally realistic for atmospheric LAC aerosols at UV and IR wavelengths.The weakly absorbing material is assumed to be sulfate.Its spectral variation is based on values taken from the OPAC database [30].As can be seen in the figure, Im(m) of sulfate increases over several orders of magnitude in the near IR part of the spectrum.

Electromagnetic scattering computations
Electromagnetic scattering computations for the encapsulated aggregate model have been performed with the discrete dipole method, using the program DDSCAT described in [31], in conjunction with the Lattice-dispersion polarizability model [32].
Much efforts have been invested into testing the number of dipoles and the number of discrete Euler angles required in the numerical orientational averaging procedures.Among the tests conducted to ensure the correctness of results, we performed comparisons of DDSCAT computations for bare aggregates with numerically exact superposition T-matrix computations (using a program developed by Mackowski and Mishchenko [33]), as well as analyses of computed polarimetric differential scattering cross sections with the reciprocity condition [34].For the latter tests, we employed the polarizability model described in [35].We required that the reciprocity condition be fulfilled with an error of less than 3%.In order to reach such a high accuracy a dipole spacing d was required such that |m|kd ≤ 0.3, where m denotes the complex refractive index of LAC, and k is the wavenumber in vacuum.Note that this dipole spacing is somewhat finer than the commonly recommended value of |m|kd ≤ 0.5.The likely reason is that our encapsulated particle models contain a high degree of geometrical details, and we have a large dielectric contrast between LAC and sulfate.A more detailed account of how to test the accuracy of discrete dipole computations for bare and encapsulated aggregates is given in [15].
For homogeneous spheres we employed a Mie program described in [36].Electromagnetic scattering computations for the core-shell geometries have been performed with the program developed by Wiscombe based on [37].Effective refractive indices for homogeneous mixtures of LAC and sulfate were computed by use of the Maxwell Garnett mixing rule [26].A comparison of this and other effective medium theories as well as volume-mixing rules applied to LAC mixed with weakly absorbing material can be found in, e.g., [15,38,39].

Optical properties relevant for climate forcing of LAC
Figure 4 shows optical properties as a function of particle radius R V and wavelength λ for aerosols with an LAC volume fraction of 7 % (where R V is the volume-equivalent radius of the particle).More specifically, we present the absorption cross section C abs (first row, left), the single-scattering albedo SSA (first row, center), and the asymmetry parameter g (first row, right) for the reference case, i.e. for encapsulated aggregates, as well as corresponding relative differences between these optical properties computed with different simplified model particles and those computed with the reference model (rows 2-5).
As expected, C abs (top row, left panel) is monotonically increasing with R V .The dependence on λ is less trivial; it reflects the dependence of C abs on both the size parameter x = 2πR V /λ and the refractive indices of LAC and sulfate, which, in turn, also vary with wavelength.The external mixture model (second row, left) and, to a lesser extent, the core-shell model (fourth row, left) underestimates C abs , while the homogeneous internal mixture model (third row, left) overestimates C abs .This is most pronounced for large size parameters.The poor performance of these models is related to how the LAC mass is distributed throughout the particle.The external mixture and core-shell models concentrate the LAC mass in a compact homogeneous sphere, in which too high a mass fraction is shielded from interacting with the electromagnetic field.The external mixture model suffers from an additional shortcoming.In a realistic encapsulated aggregate, the coating increases the geometrical cross section of the particle and focuses electromagnetic power onto the LAC aggregate, thus enhancing C abs .While the core-shell model is capable of accounting for this effect, the external mixture model does not.The combined effect of neglecting focusing and concentrating the LAC mass in a compact sphere results in a strong underestimation of C abs (second row, left).On the other hand, the homogeneous internal mixture model distributes the carbon mass too evenly throughout the sulfate host medium, which results in an overestimation of C abs (third row, left).These results help us understand the importance of simulating the optical properties of encapsulated LAC aerosols by using a model particle that attempts to mimic the distribution of LAC mass within the sulfate host particle.The model we propose here has, essentially, a coreshell geometry.However, we only concentrate a fraction f core of the total LAC mass in the core, while the remaining fraction (1 − f core ) is homogeneously mixed with the sulfate shell, using the Maxwell Garnett mixing rule for computing the effective refractive index of the shell.The fraction f core residing in the core is a free parameter that can be adjusted to fit the reference computations.
The bottom left panel in Fig. 4 shows the difference in C abs computed with the core-gray shell model and the reference results obtained for the encapsulated aggregate model.A comparison of the left panels in rows 2-5 clearly shows the superior performance of the core-gray shell model in comparison to the other simplified model geometries.
The second column presents analogous results for the single scattering albedo SSA.This quantity displays a pronounced dependence on wavelength, and a somewhat weaker dependence on size (top center panel) than C abs .Interestingly, the original core-shell model (fourth row, center) does not perform significantly better than the crude external mixture model (second row, center); both overestimate SSA at short wavelengths.By contrast, the homogeneous internal mixture model (third row, center) underestimates SSA at short wavelengths.By far the best results are achieved with the core-gray shell model.The maximum differences between that model and the reference results are almost an order of magnitude smaller than those introduced by the original core-shell model.
Similarly to SSA, the asymmetry parameter g (top right panel) displays a strong dependence on wavelength, and a weaker dependence on particle size.The homogeneous internal mixture model (third row, right) yields somewhat smaller maximum errors than the external mixture (second row, right) and the original core-shell model (fourth row, right).The largest errors are observed for short wavelengths.The core-gray shell model reproduces, again, the reference results most faithfully throughout the size and wavelength spectrum.
Table 2 shows the choices of f core at different wavelengths that yielded the best agreement of the core-gray shell model with the reference computations.f core is found to decrease with wavelength.These values are used throughout the paper for all particle sizes and for both LAC volume fractions, f =7 and 20 %.The fact that f core is independent of f and R V indicates that the optimal choices of the tuning parameter are rather robust.
Figure 5 shows results analogous to those in Fig. 4, but for aerosols containing an LAC volume fraction of 20 %.In general, the new core-gray shell model produces, again, quite satisfactory results.The problems with the original core-shell model become even more pronounced than for a volume fraction of 7 %.The homogeneous internal mixture model produces almost equally good results as the core-gray shell model for C abs and SSA.However, for the asymmetry parameter g the homogeneous internal mixture model displays maximum errors that are somewhat larger than those of the core-gray shell.Note that the choices of f core given in Table 2 were the same for both LAC volume fractions.

Optical properties relevant for remote sensing
The optical properties investigated thus far are those that are needed for modeling the radiative forcing effect of encapsulated LAC aerosols.We now turn to investigating optical properties relevant for remote sensing studies.Figure 6 shows the backscattering cross section C bak as a function of size and wavelength for aerosols with an LAC volume fraction of 7 % (left column) and 20 % (right column).The top row shows the reference results obtained with the encapsulated aggregate model, while the other rows show differences between the reference results and those obtained with various simplified model geometries.By comparing the two top figures, we can see that an increase in the LAC volume fraction from 7 % to 20 % decreases C bak by roughly a factor of 5, which is quite dramatic.For f =7 %, C bak changes by more than a factor of 4 as one increases the size parameter.Thus C bak is about equally sensitive to changes in the LAC volume fraction as to changes in the size parameter.
Among the simple model particles, the external mixture model (second row) clearly performs worst; it produces results for C bak that can be more than an order of magnitude larger than the reference results.This is likely to be caused by the high scattering cross section of the pure sulfate sphere in this model.The internal mixture model (third row) also yields very high errors, although not quite as high as the external mixture model.The original core-shell model (fourth row) and the core-gray shell model (fifth row) give reasonable results for a large range of sizes and wavelengths.The largest errors are observed in the UV, where the core-gray shell model yields significantly lower maximum errors than the original core-shell model.(A closer analysis reveals that these errors are caused by errors in the total scattering cross section, not by errors in the phase function in the backscattering direction.)The overall good representation of C bak by the core-gray shell model is quite remarkable, because the free parameter f core in that model has been optimized to yield a good fit of climate-relevant optical properties, not to reproduce C bak .
In Fig. 7 we compare the Mueller matrix elements log F 11 (left column), −F 12 /F 11 (middle column), and F 33 /F 11 (right column) as a function of scattering angle and particle size at a wavelength of 304 nm.The LAC volume fraction is 7 %.Results are shown for encapsulated aggregates (first row), external mixtures (second row), homogeneous internal mixtures (third row), core-shell particles (forth row), and core-gray shell particles (fifth row).The Mueller matrix elements of the various simple model particles are quite similar to the reference results for encapsulated aggregates.Analogous observations hold for the Mueller matrix elements at other wavelengths and volume fractions (not shown).Thus the relative angular distribution of the scattered intensity is not highly sensitive to the employed particle model, nor is the angular dependence of the linear polarization.However, the magnitude of the scattered intensity will depend on the employed particle model, because the total scattering cross section does.
The reason why most Mueller matrix elements do not differ dramatically among the different particle models can be understood by inspecting Fig. 8.Here we show all six non-zero, independent Mueller matrix elements at a wavelength of 0.5332 μm.We compare computational results for encapsulated aggregates with an LAC volume fraction of 7 % and a total volume-equivalent radius R V =0.5 μm (black), pure volume-equivalent sulfate spheres (red), and bare LAC aggregates (blue).The bare aggregates are exactly the same aggregates as those in the encapsulated geometry, except that they have been stripped of the coating.Apart from F 22 , the Mueller matrix elements of pure sulfate spheres and encapsulated aggregates are very similar.This clearly shows that in the encapsulated aggregates the angular distribution of the scattered radiance and polarization is mostly determined by the sulfate coating (which is assumed to be spherical).The LAC aggregate inside the particle strongly influences, to be sure, the total scattering and absorption cross sections, thus the total scattered radiance, but not its relative angular distribution.Not surprisingly, the exception to this observation is the element F 22 , which gives rise to depolarization.For spherically symmetric particles, F 11 =F 22 holds identically for all scattering angles.The deviation of F 22 /F 11 from unity is a sensitive indicator of non-sphericity.Clearly, none of the simple model particles considered in this study is capable of reproducing the reference results, as all of these models are spherically symmetric particles.The non-sphericity of the LAC aggregate is responsible for the deviation of F 22 /F 11 from unity in the encapsu- One important implication is the following.If most of the Mueller matrix elements are determined by the sulfate coating, this means that the angular distribution of the scattered intensity and polarization is likely to be strongly sensitive to the shape of the coating.In this study we have limited our investigations to spherical coatings.However, the aggregate core can cause the coating to become distorted and deviate from spherical shape, especially for higher LAC volume fractions; this can strongly alter the Mueller matrix.Such effects are not investigated here.
Finally, we study the absorption Ångström exponent (A ÅE), which we define for a wavelength interval [λ 1 , λ 2 ] according to For instance, A ÅE=1 indicates that C abs increases linearly with λ within a wavelength interval, while for A ÅE=0 C abs is independent of λ , and A ÅE<0 in wavelength regions where C abs decreases with λ .The limits of the wavelength intervals we considered are the values given in Table 2.For instance, the first interval at the short-wave end of the spectrum is given by λ 1 =0.2316 μm, λ 1 =0.3040 μm.
In Fig. 9 we plot this quantity as a function of size and of the mid-point wavelength λ = (λ 1 +λ 2 )/2.The first row shows the reference results obtained for encapsulated aggregates with LAC volume fractions of f =7 % (left) and 20 % (right), and rows 2-5 show corresponding differences in the optical properties computed with the simplified model geometries and those obtained with the reference geometries.
We already saw in Figs. 4 and 5 that the wavelength dependence of C abs does not follow a simple λ −1 dependence throughout the wavelength spectrum.As a consequence, A ÅE changes significantly with wavelength.Comparison of the topmost panels in Fig. 9 with the top left panels in Figs. 4 and 5 shows that the spectral variation of C abs is closely reflected by the wavelength dependence of A ÅE.
The dependence of A ÅE on particle size displays the characteristic monotonic decrease.Comparison of the top two panels also reveals that A ÅE depends on the LAC volume fraction.An increase in the LAC volume fraction tends to enhance A ÅE at wavelengths larger than 2 μm and reduce it at shorter wavelengths.In recent measurement campaigns A ÅE values for wavelengths 470-660 nm were reported in the range from 0.86 to 1.47 [40].These observations agree with our computational results for encapsulated aggregates with a low volume fraction of f =7 % (top row, left panel), which is more representative for aged LAC aerosols than the rather extreme case of f =20 %.
The internal mixture (third row) and core-gray shell models (fifth row) yield almost equally good representations of the reference results for high LAC volume fractions.However, for f =7 % the internal mixture model significantly overestimates A ÅE for short wavelengths.The external mixture model (second row) and the original core-shell model (fourth row) mostly underestimate A ÅE throughout the UV and visible part of the spectrum, especially for larger particle sizes.A use of these two models in size retrieval algorithms based on A ÅE observations would therefore have a tendency to overestimate particle size.But note that for λ around 2.5 μm the original core-shell can strongly overestimate A ÅE.The core-gray shell model (fifth row) represents A ÅE reasonably well throughout the considered size and spectral ranges.
Let us try to understand why the external mixture and core-shell models yield lower A ÅE values than the internal mixture and core-gray shell models, especially for particles of larger sizes.To this end, we first inspect the top left panels in Figs. 4 and 5, and we focus on the (C) 2013 OSA larger particle sizes.For both volume fractions, C abs increases from the UV to the visible, followed by a decrease from the visible to the near IR.The external mixture (second row, left) and core-shell models (fourth row, left) display negative errors with a magnitude that increases slightly from the UV to the visible and then sharply decreases to the near IR.This means that the dependence of C abs on wavelength is weaker for the external mixture and core-shell models than for the encapsulated aggregates, resulting in lower values of A ÅE.This is much less pronounced for the internal mixture and core-gray shell models.What the external mixture and core-shell models have in common is that the sulfate is pure, while in the other two models it is homogeneously mixed with at least part of the LAC.In Fig. 3 (bottom) we see that the imaginary part of the refractive index Im(m) of sulfate steeply increases from the visible to the near IR.An increase in wavelength reduces the size parameter of the particle, which results in a decrease in C abs .However, the increase in Im(m) partially compensates for this, thus flattening the decrease of C abs with λ and reducing A ÅE.By contrast, in the internal mixture and coregray shell models the imaginary part of the effective refractive index of the sulfate-LAC mixture increases less rapidly with wavelength, because Im(m) of LAC is almost constant with λ .As a consequence, for a given size, C abs decreases more rapidly with wavelength from the visible to the mid-IR, resulting in higher values of A ÅE.

Summary and conclusions
Modeling the optical properties of morphologically complex LAC aggregates encapsulated in a coating of weakly absorbing compounds is a highly challenging and computationally demanding task.Radiative forcing simulations in climate models as well as look-up tables for remote sensing retrieval algorithms require computations for a large range of aerosol parameters.Such comprehensive computations cannot be performed for realistic encapsulated aggregate models; one needs to rely on simplified model geometries.The present study aims at developing and testing such a model.It it an extension of previous efforts described in [15], in which only a limited set of wavelengths and only climate-relevant optical properties have been considered.
In the present study, we considered the entire spectral range from the UV-C to the mid-IR, and we investigated both climate-and remote sensing-relevant optical properties.The model introduced in [15] was quite accurate, but it proved to be too slow for use in more extensive calculations, as required for generating look-up tables of aerosol optical properties.By contrast, the model introduced in the present study is highly accurate and computationally inexpensive.
The main findings are: • The most critical factor in designing a simple model particle is to mimic the amount of LAC mass that can interact with the electromagnetic field.This is determined by how the LAC mass is distributed within the weakly absorbing host material.Most commonly employed simple model particles fail to take this important factor into account.
• For example, contrary to common believes (e.g.[23]), a simple core-shell geometry does not produce reliable estimates of the optical properties of encapsulated LAC aggregates.In many cases, this model falls short of the even cruder homogeneous internal mixture model.
• Our proposed "core-gray shell" model is found to realistically mimic the amount of LAC that interacts with the electromagnetic field.In this model only part of the LAC resides in the core, while the remaining part is homogeneously mixed with the weakly absorbing material in the concentric coating.For a suitable choice of the fraction f core of LAC residing inside the core, this model is found to give the best overall agreement with the absorption cross section, single-scattering albedo, and asymmetry parameter computed for the encapsulated LAC aggregates.These optical properties are important for accurate estimates of the radiative forcing effect of LAC.
• Although f core in the core-gray shell model has been optimized for reproducing climaterelevant optical properties, we found that the core-gray shell model also gives the best representation of optical properties relevant for remote sensing applications.In reproducing the backscattering cross section and absorption Ångström exponent, it outperforms the conventional core-shell model as well as the external mixture and homogeneous internal mixture models.The elements of the Mueller matrix (except F 22 ) are rather insensitive to the choice of model particle, as they are largely determined by the spherical coating.The F 22 element, being a sensitive indicator of particle nonsphericity, cannot be reproduced by any simplified model particle with perfect spherical symmetry.
• The optimal choices of f core are independent of particle size and LAC volume fraction, which indicates that the tuning of the core-gray shell model is rather robust.However, f core does depend on wavelength.
The improvements achieved with the core-gray shell model are particularly pronounced for low LAC volume fractions, which are most representative for aged, internally mixed LAC aerosols in the atmosphere.Table 3 shows the fractional bias FB and the normalized root mean square error NRMSE for the different models and for different optical properties.These quantities are defined in terms of the reference results x ref i and the results x mod i obtained for different model particles as where x denotes any of the optical properties, and where the index i runs over all sizes, wavelengths, and LAC volume fractions considered in this study.This is a highly aggregated presentation of the detailed results shown in the preceding sections that leaves out a lot of the information about the dependence of model errors on size, wavelength, and volume fraction.However, this summary does provide a concise overview of the overall performance of different models for different optical properties.For instance, the external mixture model consistently underestimates C abs (note the large magnitude of FB combined with the low value of NRMSE).Thus, when performing climate forcing calculations involving averaging over sizes, wavelengths, and LAC volume fractions, this model is guaranteed to underestimate the radiative forcing effect We emphasize, once more, that there are potentially important factors that this study did not account for.First, only a limited number of reference geometries has been used.We did consider aggregates of different sizes and volume fractions, but we did not investigate the variation of the optical properties with fractal parameters, monomer size, or position of the aggregate within the host material.Field measurements indicate that these parameters vary among different samples (e.g.[13,41]).Also, the host material was assumed to be sulfate, while LAC aggregates in nature can be coated with a mixture of different compounds, such as sulfate, organic substances, sea salt, and water.Coatings of different chemical compositions will have different effective refractive indices.Thus we do not claim that our particular choices for the free parameter f core given in Table 2 work equally well under all conditions.Another important limitation of this study is that the coating of the LAC aggregates was assumed to be spherical.The presence of nonspherical coatings will strongly impact the angular distribution of the scattered intensity and polarization.
In summary, the main practical consequence of this investigation is that we have introduced the core-gray shell particle as a fast and flexible model that can realistically mimic the amount of LAC mass that contributes to absorption and scattering in coated LAC aggregates.This model, with suitable choices of the free parameter f core , was found to accurately represent integrated and differential scattering optical properties of morphologically complex encapsulated aggregates.However, our choices for the free parameter in the core-gray shell model may only be valid for the specific reference geometries considered in this study.Therefore, the general recommendation is to further explore the usefulness of the core-gray shell model in more extensive case studies; this should involve modelling studies that consider different coating materials, LAC geometries, and volume fractions, as well as comparison with atmospheric measurements or laboratory experiments (e.g.[42]).

Fig. 3 .
Fig. 3. Spectral variation of the real part (left) and imaginary part (right) of the refractive indices of LAC (blue) and sulfate (red).

CFig. 4 .
Fig.4.C abs (left), SSA (center), and g (right) as a function of particle size and wavelength for particles with an LAC volume fraction of f =7 %.The rows show reference results for encapsulated aggregates (first row), and relative differences between those and external mixtures (second row), internal homogeneous mixtures (third row), core-shell (fourth row), and core-gray shell particles (fifth row).Positive differences indicate that the simplified models overestimate the reference results.

#CFig. 6 .
Fig.6.Backscattering cross section for encapsulated aggregates with LAC volume fractions of f =7 % (top left) and f =20 % (top right), as well as differences between reference computations and results for simplified model particles.

Table 2 .
Choice of the core fraction f core for each of the 14 wavelength bands (where λ denotes the band mid-point).

Table 3 .
Fractional bias FB and, in parentheses, normalized root mean square error NRMSE for the four different model particles and for different optical properties.
#184703 -$15.00USDReceived 1 Feb 2013; revised 18 Mar 2013; accepted 19 Mar 2013; published 26 Mar 2013 (C) 2013 OSAof LAC.On the other hand, the core shell model has a rather low fractional bias for C abs , but a very high value of NRMSE.It is therefore quite unpredictable to what extent averaging over sizes, wavelength, and LAC volume fractions will partially cancel the large positive and negative errors that this model will inevitably introduce.The most important observation is that the new core-gray shell model introduced here consistently yields the lowest fractional biases and normalized root mean square errors.