Aerosol complex refractive index (ACRI) is an important microphysical
parameter used for the studies of modeling their radiative effects. With
considerable uncertainties related to retrieval based on observations, a
numerical study is a powerful method, if not the only one, to provide a
better and more accurate understanding of retrieved optically effective ACRIs
of aged black carbon (BC) particles. Numerical investigations of the
optically effective ACRIs of polydisperse coated BC aggregates retrieved from
their accurate scattering and absorption properties, which are calculated by
the multiple-sphere T-matrix method (MSTM), without overall particle shape
variations during retrieval, are carried out. The aim of this study is to
evaluate the effects of aerosol microphysics, including

The largest uncertainty in estimates of the effects of atmospheric aerosols on climate stems from uncertainties in the determination of their microphysical properties, which in turn determines their optical properties. As one of the most significant microphysical properties, aerosol complex refractive index (ACRI) should be known for modeling their radiative effects, and the magnitude of radiative forcing is very sensitive to the ACRI, especially the imaginary part (Raut and Chazette, 2008a). The ACRI is determined by particle chemical composition governing its inherent scattering and absorption properties.

Black carbon (BC), emitted from incomplete fossil fuel combustion and biomass burning, can be coated with secondary aerosol species (e.g., organics and sulfate) through the aging process, being one of the largest uncertainties in estimating aerosol radiative forcing due to their complicated geometry and mixing state (Ramanathan and Carmichael, 2008; Myhre, 2009; Bond et al., 2013; Zhang et al., 2015). As a strong absorptive aerosol, pure BC particles have a large ACRI, whereas our understanding of the ACRI of aged BC is still limited because of its internal mixing with weakly absorptive coatings (Shiraiwa et al., 2010; Cui et al., 2016; Peng et al., 2016). The ACRIs of BC internal mixtures, named effective ACRIs, are normally obtained based on the volume-weighted average (VWA) method and effective medium theory (EMT), and the choice of both approaches is driven by high dependency of ACRIs on particle chemical compositions (e.g., Kandler et al., 2009). The state-of-the-art aerosol–climate models employ the VWA method extensively, approximating the effective ACRIs of internally and externally mixed aerosol ensembles at each mode for calculating their optical and radiative properties (e.g., Stier et al., 2005; Kim at al., 2008; Zhang et al., 2012). Nonetheless, the performances of the VWA and EMT are open questions, as several studies have questioned the validity of both approximations in some questions (e.g., Voshchinnikov et al., 2007).

The estimates of the ACRI of coated BC can also be made from observed optical properties, and the ACRI is inferred by obtaining a best fit to numerical simulations with Mie theory assuming a spherical particle shape, which is called optically effective ACRI. For instance, the optically effective ACRIs are retrieved based on simultaneous measurements of surface aerosol scattering and absorption coefficients, as well as size distributions (Abo Riziq et al., 2007; Schkolnik et al., 2007; Mack et al., 2010; Stock et al., 2011). Meanwhile, the airborne in situ measurements of particle optical properties from a particle soot absorption photometer (PSAP), spectral optical absorption photometer (SOAP), sunphotometer, or lidar, combined with a Mie theory-based data analysis scheme, are also applied for the retrieval of optically effective ACRIs (Raut and Chazette, 2008a, b; Petzold et al., 2009; Muller et al., 2009). Muller et al. (2010) even compare retrieved optically effective ACRIs from different techniques and reveal only partly a reasonable agreement with significant differences for the spectra of imaginary part remaining, indicating uncertainties during retrieval. The uncertainties may be that those retrieval methods are based on unrealistic spherical shape assumption, inaccurate numerical modeling, or without considering the errors in aerosol optical measurements, and then sizeable errors in retrieved optically effective ACRIs are posed. Moreover, these uncertainties significantly limit our ability to understand the relationships between the optically effective ACRI and aerosol other microphysical properties, and furthermore to improve radiation simulations in aerosol–climate models. Therefore, a systematic theoretical investigation on optically effective ACRIs of internally mixed particles retrieved from exactly calculated optical properties without particle shapes changed is a must, which is generally missing, and will benefit our understanding of these relationships. For coated BC particles with several chemical compositions, their optically effective ACRIs are not only affected by their compositions, but are also possibly impacted by their other microphysics. However, the effects of coated BC microphysics on their optically effective ACRIs are still under discussion and need more investigation.

Here, numerical investigations of the optically effective ACRIs of
polydisperse coated BC aggregates as examples are systematically presented
based on our current understanding, and the optically effective ACRI
influences are decomposed into that due to particle microphysical
properties, including

Freshly emitted BC particles often exist as loose cluster-like aggregates
with hundreds or even thousands of small spherical monomers (e.g., Li et
al., 2016), and the concept of fractal aggregate has shown great success and
wide applications in representing realistic BC geometries (e.g., Sorensen,
2001). The fractal aggregate can be mathematically described by the
well-known statistic scaling rule following

After being emitted into the atmosphere, BC aggregates tend be coated by
other materials, such as sulfate and organics (e.g., Schwarz et al.,
2008a; Tritscher et al., 2011), through the aging
process, and their chain-like structures tend to collapse into more compact
clusters (Zhang et al., 2008; Coz and Leck, 2011). The aged BC particles can
have BC

Sketch map of the geometry of coated black carbon. An example of fractal black carbon aggregates, containing 200 monomers, is coated by sulfate.

For this inhomogeneous internally mixed particle, the BC aggregates are
generated based on a tunable particle-cluster aggregation algorithm from
Skorupski et al. (2014). The

For ambient atmospheric applications, it is meaningful to consider bulk
particle optical properties averaged over a certain size distribution. This
study explores an ensemble of BC aggregates with different sizes but the same
sulfate coating fraction (i.e., same

The retrieval approach is similar to the methods described in previous studies (e.g., Mack et al., 2010; Stock et al., 2011; Zhang et al., 2013), with the only differences being that the inherent aerosol optical properties are exactly calculated rather than measured and particle overall shapes are not changed during retrieval. Among all particle optical properties, the scattering and absorption are selected for retrieval, since both are basically governed by the real and imaginary parts of the ACRI, respectively. As coated BC models are overall spherical, the optically effective ACRI is determined by an iterative algorithm based on Mie theory, utilizing particle size distributions and calculated scattering and absorption cross sections. Exploiting all calculations, the designed inversion scheme to retrieve the optically effective ACRI follows.

Based on a guess for a real part,

As the optically effective ACRIs of coated BC with fixed microphysical
parameters (such as

This study focuses on the influence of the microphysics of coated BC
aggregates on their optically effective ACRIs, and, therefore, the properties
of the microphysics are our interest. The coated BC optically effective ACRI
depends not only on the particle

To show the effect of BC geometry on coated BC optically effective ACRIs, the
concentric core-shell structures (i.e., mass centers located at the coating
center) with inside BC aggregates exhibiting fractal dimensions of 2.6, 2.8
and 2.98 are considered. Figure 2 compares retrieved optically effective
ACRIs of these coated BC aggregates with different BC geometries at different

The real (

As shown in Fig. 2, in accumulation mode, it is expected that, as

The relative differences of scattering and absorption cross sections
of black carbon aggregates coated by sulfate induced by their optically
effective complex refractive indices (

Unlike accumulation mode, the retrieved optically effective ACRI of
concentric coated BC in coarse mode depicts distinctive patterns, which is
illustrated in Fig. 2b and d. The impact of particle microphysics on the
optically effective ACRIs of coarse concentric coated BC is complicated,
especially for their real parts, which show strong oscillations as a function
of the

The simulations discussed above assume coated BC with a concentric core-shell
structure, which does not always represent realistic aerosols, whereas coated
BC with an off-center core-shell structure may be certainly true for some
ambient particles. Figure 4 portrays retrieved optically effective ACRIs of
coated BC aggregates (BC fractal dimension of 2.8) with the aforementioned
size distributions for two different off-center structures compared to the
concentric core-shell structure. For two off-center core-shell structures
assumed, one is BC aggregates located in the middle of a radius of the
coating sphere and the other is BC in an outer position as close as possible
to the coating boundary. It is evident that coated BC optically effective
ACRIs in accumulation mode decrease with increasing

The real (

Figure 5 illustrates the differences of scattering and absorption cross
sections of coated BC aggregates with different BC inside positions induced
by the VWA, EMT and optically effective ACRIs. The optically effective ACRIs
cause differences of coated BC scattering and absorption within 1 %
compared to its inherent properties in both accumulation and coarse modes,
whereas the VWA and EMT induce large particle scattering and absorption
differences, especially in coarse mode. One can see that, in coarse mode,
the VWA and EMT can overestimate coated BC absorption as high as

The relative differences of scattering and absorption cross sections
of black carbon aggregates coated by sulfate (BC fractal dimension of 2.8)
induced by their optically effective complex refractive indices (

Generally, retrieved optically effective ACRIs of coated BC aggregates show
significantly distinctive patterns in accumulation and coarse modes. In
accumulation mode, besides the

As demonstrated in Bond et al. (2006), particle size distribution affects
coated BC absorption properties and its BC absorption amplification due to
weakly absorbing coatings. Figure 6 illustrates the variations of retrieved
ACRIs of concentric coated BC aggregates (BC fractal dimension of 2.8) with
different particle size distributions at different

The retrieved optically effective aerosol complex refractive indices
of BC aggregates coated by sulfate (BC fractal dimension of 2.8) with a
different

As discussed above, the VWA approximation employed in the state-of-the-art
aerosol–climate models extensively could result in significant errors in the
absorption of thickly coated BC aggregates in coarse mode (specifically,

To demonstrate the performance of the simple expressions in approximating ACRIs of fully coated BC
aggregates in coarse mode with

Comparisons between the relative differences of scattering and
absorption coefficients of coarse coated black carbon aggregates induced by
the refractive indices based on the volume-weighted average method
(

Our theoretical analysis depicts retrieved optically effective ACRI of coated
BC sensitive to its

This study numerically explores the impacts of coating microphysics on the
optically effective ACRIs of polydisperse coated BC particles, which are
retrieved from exactly calculated scattering and absorption properties
without variations in overall particle shapes during retrieval. The numerical
simulations conducted here have multiple controllable microphysical
variables, i.e.,

Our results reveal that retrieved optically effective ACRIs of coated BC aggregates depict significantly different patterns in accumulation and coarse modes. With BC becoming loose or close to coating the boundary, the real parts of retrieved optically effective ACRIs of accumulation-coated BC increase slightly, as opposed to the decrease for the imaginary parts. The retrieved optically effective ACRIs of coated BC in accumulation mode are predominantly influenced by their chemical compositions and composition ratio, which makes it reasonable and looks like the real ACRIs, although it is slightly sensitive to BC geometry, BC position inside the coating and particle size distribution. Nonetheless, retrieved optically effective ACRIs of coarse coated BC are highly complicated functions of particle microphysics, and this challenges conventional beliefs given by the VWA and EMT. The VWA and EMT exhibit acceptable performances for estimating ACRIs of coated BC in accumulation mode, and resulting uncertainties in scattering and absorption are both within approximately 10 %. In coarse mode, the VWA and EMT, nevertheless, produce dramatically higher imaginary parts than those of optically effective ACRIs, and can significantly overestimate coated BC absorption by a factor of nearly 2, especially for heavily coated BC with a large BC fractal dimension or BC close to the coating boundary. This is probably one of the reasons why modeled aerosol optical depth is 20 % larger than observed (Roelofs et al., 2010), as the VWA approximation is widely employed in the state-of-the-art aerosol–climate models.

Although the parameterization of the optically effective ACRI of coarse
coated BC is difficult and challenging, we propose a simple ACRI
parameterization method for heavily coated BC with

The data obtained from this study are available upon
request from Mao Mao (mmao@nuist.edu.cn) or Xiaolin Zhang
(xlnzhang@nuist.edu.cn) and are also available at

The supplement related to this article is available online at:

XZ and MM designed the research plan. YY gave some suggestions for the revision. XZ carried it out, performed the simulations, and prepared the manuscript with contributions from all the co-authors.

The authors declare that they have no conflict of interest.

We particularly acknowledge the source codes of MSTM 3.0 from Daniel W. Mackowski and Michael I. Mishchenko. We also gratefully appreciate the supports from the Special Program for Applied Research on Super Computation of the NSFC-Guangdong Joint Fund (the second phase) under grant no. U1501501.

This work is financially supported by the National Natural Science Foundation of China (NSFC) (nos. 91644224 41505127, and 21406189), the Natural Science Foundation of Jiangsu Province (no. BK20150901), the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (no. 15KJB170009), and the Key Laboratory of Meteorological Disaster, Ministry of Education (no. KLME201810). This work is also supported by the Startup Foundation for introducing Talent of NUIST (nos. 2015r002 and 2014r011), a China Postdoctoral Science Foundation Funded Project (no. 2016M591883), and Jiangsu Planned Projects for Postdoctoral Research Funds (no. 1601262C).

This paper was edited by Yves Balkanski and reviewed by two anonymous referees.