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

Optics Express, Vol. 21, Issue 7, pp. 7974-7993 (2013)

http://dx.doi.org/10.1364/OE.21.007974

Acrobat PDF (3032 KB)

### Abstract

Optical properties of light absorbing carbon (LAC) aggregates encapsulated in a shell of sulfate are computed for realistic model geometries based on field measurements. Computations are performed for wavelengths from the UV-C to the mid-IR. Both climate- and remote sensing-relevant optical properties are considered. The results are compared to commonly used simplified model geometries, none of which gives a realistic representation of the distribution of the LAC mass within the host material and, as a consequence, fail to predict the optical properties accurately. A new core-gray shell model is introduced, which accurately reproduces the size- and wavelength dependence of the integrated and differential optical properties.

© 2013 OSA

## 1. Introduction

*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

1. A. R. Jones, “Light scattering in combustion,” in *Light Scattering Reviews*, A. Kokhanovsky, ed. (Springer, 2006) [CrossRef] .

*r*is the distance of the

_{i}*i*th monomer from the aggregate’s center of mass.

2. M. Schnaiter, H. Horvath, O. Möhler, K.-H. Naumann, H. Saathoff, and O. W. Schöck, “UV-VIS-NIR spectral optical properties of soot and soot-containing aerosols,” J. Aerosol Sci. **34**, 1421–1444 (2003) [CrossRef] .

6. S. Nyeki and I. Colbeck, “Fractal dimension analysis of single, in-situ, restructured carbonaceous aggregates,” Aerosol Sci. Technol. **23**, 109–120 (1995) [CrossRef] .

7. N. Riemer, H. Vogel, and B. Vogel, “Soot aging time scales in polluted regions during day and night,” Atmos. Chem. Phys. **4**, 1885–1893 (2004) [CrossRef] .

10. R. J. Park, D. J. Jacob, P. I. Palmer, A. D. Clarke, R. J. Weber, M. A. Zondlo, F. L. Eisele, A. R. Bandy, D. C. Thornton, G. W. Sachse, and T. C. Bond, “Export efficiency of black carbon aerosol in continental outflow: Global implications,” J. Geophys. Res. **110**, D11205, (2005) [CrossRef] .

11. A. Worringen, M. Ebert, T. Trautmann, S. Weinbruch, and G. Helas, “Optical properties of internally mixed ammonium sulfate and soot particles–a study of individual aerosol particles and ambient aerosol populations,” Appl. Opt. **47**, 3835–3845 (2008) [CrossRef] [PubMed] .

12. B. Scarnato, S. Vahidinia, D. T. Richard, and T. W. Kirchstetter, “Effects of internal mixing and aggregate morphology on optical properties of black carbon using a discrete dipole approximation model,” Atmos. Chem. Phys. Discuss. **12**, 26401–26434 (2012) [CrossRef] .

13. K. Adachi and P. R. Buseck, “Internally mixed soot, sulfates, and organic matter in aerosol particles from Mexico City,” Atmos. Chem. Phys. **8**, 6469–6481 (2008) [CrossRef] .

14. T. C. Bond, G. Habib, and R. W. Bergstrom, “Limitations in the enhancement of visible light absorption due to mixing state,” J. Geophys. Res. **111**, D20211, (2006) [CrossRef] .

15. M. Kahnert, T. Nousiainen, H. Lindqvist, and M. Ebert, “Optical properties of light absorbing carbon aggregates mixed with sulfate: assessment of different model geometries for climate forcing calculations,” Opt. Express **20**, 10042–10058 (2012) [CrossRef] [PubMed] .

16. M. Kahnert, “On the discrepancy between modelled and measured mass absorption cross sections of light absorbing carbon aerosols,” Aerosol Sci. Technol. **44**, 453–460 (2010) [CrossRef] .

17. M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: Fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transfer **112**, 671–692 (2011) [CrossRef] .

11. A. Worringen, M. Ebert, T. Trautmann, S. Weinbruch, and G. Helas, “Optical properties of internally mixed ammonium sulfate and soot particles–a study of individual aerosol particles and ambient aerosol populations,” Appl. Opt. **47**, 3835–3845 (2008) [CrossRef] [PubMed] .

15. M. Kahnert, T. Nousiainen, H. Lindqvist, and M. Ebert, “Optical properties of light absorbing carbon aggregates mixed with sulfate: assessment of different model geometries for climate forcing calculations,” Opt. Express **20**, 10042–10058 (2012) [CrossRef] [PubMed] .

18. M. Kahnert, “Modelling the optical and radiative properties of freshly emitted light absorbing carbon within an atmospheric chemical transport model,” Atmos. Chem. Phys. **10**, 1403–1416 (2010) [CrossRef] .

20. M. Kahnert and A. Devasthale, “Black carbon fractal morphology and short-wave radiative impact: a modelling study,” Atmos. Chem. Phys. **11**, 11745–11759 (2011) [CrossRef] .

22. G. Videen and P. Chýlek, “Scattering by a composite sphere with an absorbing inclusion and effective medium approximations,” Opt. Commun. **158**, 1–6 (1998) [CrossRef] .

23. M. Z. Jacobson, “Strong radiative heating due to the mixing state of black carbon in atmospheric aerosols,” Nature **409**, 695–697 (2001) [CrossRef] [PubMed] .

*C*

_{abs}, and that the homogeneous internal mixture model overestimates

*C*

_{abs}[15

15. M. Kahnert, T. Nousiainen, H. Lindqvist, and M. Ebert, “Optical properties of light absorbing carbon aggregates mixed with sulfate: assessment of different model geometries for climate forcing calculations,” Opt. Express **20**, 10042–10058 (2012) [CrossRef] [PubMed] .

*C*

_{abs}; the error introduced by the core-shell model can even exceed that of the internal homogeneous mixture model [15

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

## 2. Methods

### 2.1. Model particles

*D*

_{f}=2.6 and

*k*

_{a}=1.2, which are based on recent field observations of aged encapsulated LAC aggregates [13

13. K. Adachi and P. R. Buseck, “Internally mixed soot, sulfates, and organic matter in aerosol particles from Mexico City,” Atmos. Chem. Phys. **8**, 6469–6481 (2008) [CrossRef] .

16. M. Kahnert, “On the discrepancy between modelled and measured mass absorption cross sections of light absorbing carbon aerosols,” Aerosol Sci. Technol. **44**, 453–460 (2010) [CrossRef] .

13. K. Adachi and P. R. Buseck, “Internally mixed soot, sulfates, and organic matter in aerosol particles from Mexico City,” Atmos. Chem. Phys. **8**, 6469–6481 (2008) [CrossRef] .

*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

24. K. Adachi, S. Chung, and P. R. Buseck, “Shapes of soot aerosol particles and implications for their effects on climate,” J. Geophys. Res. **115**, D15206, (2010) [CrossRef] .

*R*

_{V}=0.1–0.5

*μ*m (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).

11. A. Worringen, M. Ebert, T. Trautmann, S. Weinbruch, and G. Helas, “Optical properties of internally mixed ammonium sulfate and soot particles–a study of individual aerosol particles and ambient aerosol populations,” Appl. Opt. **47**, 3835–3845 (2008) [CrossRef] [PubMed] .

12. B. Scarnato, S. Vahidinia, D. T. Richard, and T. W. Kirchstetter, “Effects of internal mixing and aggregate morphology on optical properties of black carbon using a discrete dipole approximation model,” Atmos. Chem. Phys. Discuss. **12**, 26401–26434 (2012) [CrossRef] .

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

25. M. Wentzel, G. Gorzawski, K.-H. Naumann, H. Saathoff, and S. Weinbruch, “Transmission electron microscopical and aerosol dynamical characterization of soot aerosols,” Aerosol Sci. **34**, 1347–1370 (2003) [CrossRef] .

*μ*m.

*f*=7% and

*f*=20%. The former is close to values reported in, e.g., [11

**47**, 3835–3845 (2008) [CrossRef] [PubMed] .

24. K. Adachi, S. Chung, and P. R. Buseck, “Shapes of soot aerosol particles and implications for their effects on climate,” J. Geophys. Res. **115**, D15206, (2010) [CrossRef] .

24. K. Adachi, S. Chung, and P. R. Buseck, “Shapes of soot aerosol particles and implications for their effects on climate,” J. Geophys. Res. **115**, D15206, (2010) [CrossRef] .

*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

**8**, 6469–6481 (2008) [CrossRef] .

26. J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. A **203**, 385–420 (1904) [CrossRef] .

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

*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

27. M. Kahnert, “Irreducible representations of finite groups in the T matrix formulation of the electromagnetic scattering problem,” J. Opt. Soc. Am. A **22**, 1187–1199 (2005) [CrossRef] .

12. B. Scarnato, S. Vahidinia, D. T. Richard, and T. W. Kirchstetter, “Effects of internal mixing and aggregate morphology on optical properties of black carbon using a discrete dipole approximation model,” Atmos. Chem. Phys. Discuss. **12**, 26401–26434 (2012) [CrossRef] .

*m*of LAC in Fig. 3 is based on the measurements reported in [28

28. H. Chang and T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. Lond. A **430**, 577–591 (1990) [CrossRef] .

29. T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: An investigative review,” Aerosol Sci. Technol. **40**, 27–67 (2006) [CrossRef] .

28. H. Chang and T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. Lond. A **430**, 577–591 (1990) [CrossRef] .

29. T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: An investigative review,” Aerosol Sci. Technol. **40**, 27–67 (2006) [CrossRef] .

28. H. Chang and T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. Lond. A **430**, 577–591 (1990) [CrossRef] .

30. M. Hess, P. Koepke, and I. Schult, “Optical properties of aerosols and clouds: The software package OPAC,” Bull. Am. Met. Soc. **79**, 831–844 (1998) [CrossRef] .

*m*) of sulfate increases over several orders of magnitude in the near IR part of the spectrum.

### 2.2. Electromagnetic scattering computations

31. B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A **11**, 1491–1499 (1994) [CrossRef] .

32. B. T. Draine and J. J. Goodman, “Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophysical J. **405**, 685–697 (1993) [CrossRef] .

33. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A **13**, 2266–2278 (1996) [CrossRef] .

34. K. Schmidt, M. Yurkin, and M. Kahnert, “A case study on the reciprocity in light scattering computations,” Opt. Express **20**, 23253–23274 (2012) [CrossRef] [PubMed] .

35. D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” Tech. rep., http://arxiv.org/abs/astro-ph/0403082 (2004).

*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

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

37. O. B. Toon and T. P. Ackermann, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. **20**, 3657–3660 (1981) [CrossRef] [PubMed] .

26. J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. A **203**, 385–420 (1904) [CrossRef] .

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

38. K. A. Fuller, W. C. Malm, and S. M. Kreidenweis, “Effects of mixing on extinction by carbonaceous particles,” J. Geophys. Res. **104**, 15941–15954 (1999) [CrossRef] .

39. G. Lesins, P. Chylek, and U. Lohmann, “A study of internal and external mixing scenarios and its effect on aerosol optical properties and direct radiative forcing,” J. Geophys. Res. **107**(D10), 4094, (2002) [CrossRef] .

## 3. Results and discussion

### 3.1. Optical properties relevant for climate forcing of LAC

*R*and wavelength

_{V}*λ*for aerosols with an LAC volume fraction of 7% (where

*R*is the volume-equivalent radius of the particle). More specifically, we present the absorption cross section

_{V}*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).

*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.

*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.

*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.

*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.

*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*indicates that the optimal choices of the tuning parameter are rather robust.

_{V}*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.

### 3.2. Optical properties relevant for remote sensing

*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.

*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.

*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}.

*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.

*μ*m. We compare computational results for encapsulated aggregates with an LAC volume fraction of 7% and a total volume-equivalent radius

*R*=0.5

_{V}*μ*m (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 encapsulated aggregates, but the spherical coating significantly alters

*F*

_{22}/

*F*

_{11}as compared to the pure aggregates.

*λ*

_{1},

*λ*

_{2}] according to For instance, AÅE=1 indicates that

*C*

_{abs}increases linearly with

*λ*within a wavelength interval, while for AÅE=0

*C*

_{abs}is independent of

*λ*, and AÅE <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

*μ*m,

*λ*

_{1}=0.3040

*μ*m.

*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.

*C*

_{abs}does not follow a simple

*λ*

^{−1}dependence throughout the wavelength spectrum. As a consequence, AÅE 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 AÅE.

*μ*m and reduce it at shorter wavelengths. In recent measurement campaigns AÅE values for wavelengths 470–660 nm were reported in the range from 0.86 to 1.47 [40

40. S. Mogo, V. E. Cachorro, A. de Frutos, and A. Rodrigues, “Absorption ångström exponents of aerosols and light absorbing carbon (lac) obtained from *in situ* data in Covilhã, central Portugal,” J. Environ. Monit. **14**, 3174–3181 (2012) [CrossRef] [PubMed] .

*f*=7% (top row, left panel), which is more representative for aged LAC aerosols than the rather extreme case of

*f*=20%.

*f*=7% the internal mixture model significantly overestimates AÅE for short wavelengths. The external mixture model (second row) and the original core-shell model (fourth row) mostly underestimate AÅE 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 AÅE observations would therefore have a tendency to overestimate particle size. But note that for

*λ*around 2.5

*μ*m the original core-shell can strongly overestimate AÅE. The core-gray shell model (fifth row) represents AÅE reasonably well throughout the considered size and spectral ranges.

*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 AÅE. 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 AÅE. By contrast, in the internal mixture and core-gray 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 AÅE.

## 4. Summary and conclusions

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

**20**, 10042–10058 (2012) [CrossRef] [PubMed] .

- 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.
23. M. Z. Jacobson, “Strong radiative heating due to the mixing state of black carbon in atmospheric aerosols,” Nature

**409**, 695–697 (2001) [CrossRef] [PubMed] . - 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 climate-relevant 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.

*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*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 of 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.

**8**, 6469–6481 (2008) [CrossRef] .

41. K. Adachi, S. H. Chung, H. Friedrich, and P. R. Buseck, “Fractal parameters of individual soot particles determined using electron tomography: Implications for optical properties,” J. Geophys. Res. **112**, D14202, (2007) [CrossRef] .

*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.

*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

42. E. F. Mikhailov, S. S. Vlasenko, I. A. Podgorny, V. Ramanathan, and C. E. Corrigan, “Optical properties of soot-water drop agglomerates: An experimental study,” J. Geophys. Res. **111**, D07209, (2006) [CrossRef] .

## Acknowledgments

## References and links

1. | A. R. Jones, “Light scattering in combustion,” in |

2. | M. Schnaiter, H. Horvath, O. Möhler, K.-H. Naumann, H. Saathoff, and O. W. Schöck, “UV-VIS-NIR spectral optical properties of soot and soot-containing aerosols,” J. Aerosol Sci. |

3. | J. Hallett, J. G. Hudson, and C. F. Rogers, “Characterization of combustion aerosols for haze and cloud formation,” Aerosol Sci. Technol. |

4. | I. Colbeck, L. Appleby, E. J. Hardman, and R. M. Harrison, “The optical properties and morphology of cloud-processed carbonaceous smoke,” J. Aerosol Sci. |

5. | G. Ramachandran and P. C. Reist, “Characterization of morphological changes in agglomerates subject to condensation and evaporation using multiple fractal dimensions,” Aerosol Sci. Technol. |

6. | S. Nyeki and I. Colbeck, “Fractal dimension analysis of single, in-situ, restructured carbonaceous aggregates,” Aerosol Sci. Technol. |

7. | N. Riemer, H. Vogel, and B. Vogel, “Soot aging time scales in polluted regions during day and night,” Atmos. Chem. Phys. |

8. | S. Tsyro, D. Simpson, L. Tarrasón, Z. K. K. Kupiainen, C. Pio, and K. E. Yttri, “Modelling of elemental carbon over Europe,” J. Geophys. Res. |

9. | B. Croft, U. Lohmann, and K. von Salzen, “Black carbon aging in the Canadian Centre for Climate modelling and analysis atmospheric general circulation model,” Atmos. Chem. Phys. |

10. | R. J. Park, D. J. Jacob, P. I. Palmer, A. D. Clarke, R. J. Weber, M. A. Zondlo, F. L. Eisele, A. R. Bandy, D. C. Thornton, G. W. Sachse, and T. C. Bond, “Export efficiency of black carbon aerosol in continental outflow: Global implications,” J. Geophys. Res. |

11. | A. Worringen, M. Ebert, T. Trautmann, S. Weinbruch, and G. Helas, “Optical properties of internally mixed ammonium sulfate and soot particles–a study of individual aerosol particles and ambient aerosol populations,” Appl. Opt. |

12. | B. Scarnato, S. Vahidinia, D. T. Richard, and T. W. Kirchstetter, “Effects of internal mixing and aggregate morphology on optical properties of black carbon using a discrete dipole approximation model,” Atmos. Chem. Phys. Discuss. |

13. | K. Adachi and P. R. Buseck, “Internally mixed soot, sulfates, and organic matter in aerosol particles from Mexico City,” Atmos. Chem. Phys. |

14. | T. C. Bond, G. Habib, and R. W. Bergstrom, “Limitations in the enhancement of visible light absorption due to mixing state,” J. Geophys. Res. |

15. | M. Kahnert, T. Nousiainen, H. Lindqvist, and M. Ebert, “Optical properties of light absorbing carbon aggregates mixed with sulfate: assessment of different model geometries for climate forcing calculations,” Opt. Express |

16. | M. Kahnert, “On the discrepancy between modelled and measured mass absorption cross sections of light absorbing carbon aerosols,” Aerosol Sci. Technol. |

17. | M. I. Mishchenko, V. P. Tishkovets, L. D. Travis, B. Cairns, J. M. Dlugach, L. Liu, V. K. Rosenbush, and N. N. Kiselev, “Electromagnetic scattering by a morphologically complex object: Fundamental concepts and common misconceptions,” J. Quant. Spectrosc. Radiat. Transfer |

18. | M. Kahnert, “Modelling the optical and radiative properties of freshly emitted light absorbing carbon within an atmospheric chemical transport model,” Atmos. Chem. Phys. |

19. | M. Kahnert, “Numerically exact computation of the optical properties of light absorbing carbon aggregates for wavelength of 200 nm V 12.2 |

20. | M. Kahnert and A. Devasthale, “Black carbon fractal morphology and short-wave radiative impact: a modelling study,” Atmos. Chem. Phys. |

21. | P. Chýlek, G. Videen, D. J. W. Geldart, J. S. Dobbie, and H. C. W. Tso, “Effective medium approximations for heterogeneous particles,” in |

22. | G. Videen and P. Chýlek, “Scattering by a composite sphere with an absorbing inclusion and effective medium approximations,” Opt. Commun. |

23. | M. Z. Jacobson, “Strong radiative heating due to the mixing state of black carbon in atmospheric aerosols,” Nature |

24. | K. Adachi, S. Chung, and P. R. Buseck, “Shapes of soot aerosol particles and implications for their effects on climate,” J. Geophys. Res. |

25. | M. Wentzel, G. Gorzawski, K.-H. Naumann, H. Saathoff, and S. Weinbruch, “Transmission electron microscopical and aerosol dynamical characterization of soot aerosols,” Aerosol Sci. |

26. | J. C. Maxwell Garnett, “Colours in metal glasses and in metallic films,” Philos. Trans. R. Soc. A |

27. | M. Kahnert, “Irreducible representations of finite groups in the T matrix formulation of the electromagnetic scattering problem,” J. Opt. Soc. Am. A |

28. | H. Chang and T. T. Charalampopoulos, “Determination of the wavelength dependence of refractive indices of flame soot,” Proc. R. Soc. Lond. A |

29. | T. C. Bond and R. W. Bergstrom, “Light absorption by carbonaceous particles: An investigative review,” Aerosol Sci. Technol. |

30. | M. Hess, P. Koepke, and I. Schult, “Optical properties of aerosols and clouds: The software package OPAC,” Bull. Am. Met. Soc. |

31. | B. T. Draine and P. J. Flatau, “Discrete-dipole approximation for scattering calculations,” J. Opt. Soc. Am. A |

32. | B. T. Draine and J. J. Goodman, “Beyond Clausius-Mossotti: Wave propagation on a polarizable point lattice and the discrete dipole approximation,” Astrophysical J. |

33. | D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A |

34. | K. Schmidt, M. Yurkin, and M. Kahnert, “A case study on the reciprocity in light scattering computations,” Opt. Express |

35. | D. Gutkowicz-Krusin and B. T. Draine, “Propagation of electromagnetic waves on a rectangular lattice of polarizable points,” Tech. rep., http://arxiv.org/abs/astro-ph/0403082 (2004). |

36. | M. I. Mishchenko, L. D. Travis, and A. A. Lacis, |

37. | O. B. Toon and T. P. Ackermann, “Algorithms for the calculation of scattering by stratified spheres,” Appl. Opt. |

38. | K. A. Fuller, W. C. Malm, and S. M. Kreidenweis, “Effects of mixing on extinction by carbonaceous particles,” J. Geophys. Res. |

39. | G. Lesins, P. Chylek, and U. Lohmann, “A study of internal and external mixing scenarios and its effect on aerosol optical properties and direct radiative forcing,” J. Geophys. Res. |

40. | S. Mogo, V. E. Cachorro, A. de Frutos, and A. Rodrigues, “Absorption ångström exponents of aerosols and light absorbing carbon (lac) obtained from |

41. | K. Adachi, S. H. Chung, H. Friedrich, and P. R. Buseck, “Fractal parameters of individual soot particles determined using electron tomography: Implications for optical properties,” J. Geophys. Res. |

42. | E. F. Mikhailov, S. S. Vlasenko, I. A. Podgorny, V. Ramanathan, and C. E. Corrigan, “Optical properties of soot-water drop agglomerates: An experimental study,” J. Geophys. Res. |

**OCIS Codes**

(010.1110) Atmospheric and oceanic optics : Aerosols

(010.1290) Atmospheric and oceanic optics : Atmospheric optics

(010.1310) Atmospheric and oceanic optics : Atmospheric scattering

(290.1350) Scattering : Backscattering

(290.5850) Scattering : Scattering, particles

(290.5825) Scattering : Scattering theory

(010.0280) Atmospheric and oceanic optics : Remote sensing and sensors

**ToC Category:**

Scattering

**History**

Original Manuscript: February 1, 2013

Revised Manuscript: March 18, 2013

Manuscript Accepted: March 19, 2013

Published: March 26, 2013

**Citation**

Michael Kahnert, Timo Nousiainen, and Hannakaisa Lindqvist, "Models for integrated and differential scattering optical properties of encapsulated light absorbing carbon aggregates," Opt. Express **21**, 7974-7993 (2013)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-21-7-7974

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### References

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