**Research article**
07 Mar 2019

**Research article** | 07 Mar 2019

# Accounting for the effects of nonideal minor structures on the optical properties of black carbon aerosols

Shiwen Teng Chao Liu Martin Schnaiter Rajan K. Chakrabarty and Fengshan Liu

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**Shiwen Teng et al.**Shiwen Teng Chao Liu Martin Schnaiter Rajan K. Chakrabarty and Fengshan Liu

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^{1}Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science & Technology, Nanjing 210044, China^{2}Key Laboratory for Aerosol-Cloud-Precipitation of the China Meteorological Administration, School of Atmospheric Physics, Nanjing University of Information Science & Technology, Nanjing 210044, China^{3}Karlsruhe Institute of Technology, Institute of Meteorology and Climate Research, 76021 Karlsruhe, Germany^{4}Center for Aerosol Science and Engineering, Department of Energy, Environmental and Chemical Engineering, Washington University in St. Louis, St. Louis, Missouri 63130, USA^{5}Black Carbon Metrology, Measurement Science and Standards, National Research Council, Ottawa, Ontario K1A 0R6, Canada

^{1}Collaborative Innovation Center on Forecast and Evaluation of Meteorological Disasters, Nanjing University of Information Science & Technology, Nanjing 210044, China^{2}Key Laboratory for Aerosol-Cloud-Precipitation of the China Meteorological Administration, School of Atmospheric Physics, Nanjing University of Information Science & Technology, Nanjing 210044, China^{3}Karlsruhe Institute of Technology, Institute of Meteorology and Climate Research, 76021 Karlsruhe, Germany^{4}Center for Aerosol Science and Engineering, Department of Energy, Environmental and Chemical Engineering, Washington University in St. Louis, St. Louis, Missouri 63130, USA^{5}Black Carbon Metrology, Measurement Science and Standards, National Research Council, Ottawa, Ontario K1A 0R6, Canada

**Correspondence**: Chao Liu (chao_liu@nuist.edu.cn)

**Correspondence**: Chao Liu (chao_liu@nuist.edu.cn)

Received: 16 Oct 2018 – Discussion started: 22 Nov 2018 – Revised: 28 Jan 2019 – Accepted: 30 Jan 2019 – Published: 07 Mar 2019

Black carbon (BC) aerosol is the strongest sunlight-absorbing aerosol, and its optical properties are fundamental to radiative forcing estimations and retrievals of its size and concentration. BC particles exist as aggregate structures with small monomers and are widely represented by the idealized fractal aggregate model. In reality, BC particles possess complex and nonideal minor structures besides the overall aggregate structure, altering their optical properties in unforeseen ways. This study introduces the parameter “volume variation” to quantify and unify different minor structures and develops an empirical relationship to account for their effects on BC optical properties from those of ideal aggregates. Minor structures considered are as follows: the polydispersity of monomer size, the irregularity and coating of the individual monomer, and necking and overlapping among monomers. The discrete dipole approximation is used to calculate the optical properties of aggregates with these minor structures. Minor structures result in scattering cross-section enhancement slightly more than that of absorption cross section, and their effects on the angle-dependent phase matrix as well as asymmetry factor are negligible. As expected, the effects become weaker with the increase in wavelength. Our results suggest that a correction ratio of 1.05 is necessary to account for the mass or volume normalized absorption and scattering of nonideal aggregates in comparison to ideal ones, which also applies to aggregates with multiple minor structures. In other words, the effects of minor structures are mainly contributed by their influence on particle volume/mass that cannot be ignored, and a relative difference of approximately 5 % is noticed after removing the volume effects. Thus, accurate knowledge and evaluation of BC volume/mass are more important than those of the minor structures themselves. Most importantly, the simulations of optical properties of nonideal aggregates are greatly simplified by applying the empirical relationship because they can be directly obtained from those of the corresponding ideal aggregates, a volume/mass difference parameter, and the correction factor, i.e., 1.05, not the detailed minor structure information. We expect this convenient treatment to find wide applications for the accounting for the effects of nonideal minor structures on BC optical properties.

Black carbon (BC), produced by incomplete combustion of fossil fuels, biofuels, and biomass, is one of the strongest sunlight-absorbing atmospheric constituents (Jacobson, 2001; Andreae and Gelencsér, 2006). BC particles affect the radiative balance at global and regional scales by absorbing solar radiation and reducing the radiation reaching the surface (Crutzen and Andreae, 1990; Menon et al., 2002; Kahnert and Devasthale, 2011). Thus, the optical properties of BC particles are fundamental not only to radiative forcing estimations but also to retrievals of their size and concentration, whereas they are, in turn, highly dependent on the complex and heterogeneous morphology of BC particles.

Fractal aggregates have been widely used to represent BC geometries and to obtain their optical properties (Farias et al., 1996; Liu et al., 2012; Bescond et al., 2013). In the fractal aggregate model, aggregates are formed by numerous perfect same-sized spheres, also called monomers, that are in point contact. Mathematically, fractal aggregates are described by the following statistic scaling rule (Sorensen, 2001):

where *N* is the number of monomers in an aggregate, *a*_{o} is the radius
of primary monomers, and *R*_{g} is the gyration radius. *k*_{f} (fractal
prefactor) and *D*_{f} (fractal dimension) are parameters describing the
aggregate overall structure, i.e., lacy or compact. A large amount of experimental
evidence confirms that freshly emitted BC particles are normally chain-like,
lacy aggregates with *D*_{f} less than 2, whereas aged BC aggregates
tend to be more compact, with *D*_{f} values even close to 3 (Sorensen, 2001;
Chakrabarty et al., 2009; Wang et al., 2017).

BC particles in the ambient atmosphere show significant diversities in their morphologies and are much more complex than the idealized fractal-like aggregates specified by Eq. (1), which only describe their general morphology (Wu et al., 2015a; Pirjola et al., 2017). Figure 1 shows some examples of transmission electron microscope and scanning electron microscope (TEM/SEM) images of BC particles sampled under different real-world scenarios (Gwaze et al., 2006; Kamimoto et al., 2007; Chakrabarty et al., 2009; Yon et al., 2015; Wang et al., 2017). The particles are obviously highly complex, not only in their overall morphology but also concerning their detailed structures. As is apparent, actual BC particles display features that are quite different from the assumptions used for fractal aggregates (e.g., same-sized, perfectly spherical, and point contacting), such as polydispersity of monomer size and necking and overlapping among monomers. Furthermore, BC aggregates also get mixed with organic or inorganic aerosols in varying proportions during transport and aging (e.g., the left column of Fig. 1), and the mixing can alter their morphology as well. Several studies have been carried out to quantify these detailed structures to better represent them in numerical models (Cheng et al., 2014; Moteki, 2016). Among them, the polydispersity of monomer size is the most well studied (Chakrabarty et al., 2006, 2007; Bescond et al., 2014; Wu et al., 2015b), and a complete set of methods have been established to characterize monomer size distribution. Bourrous et al. (2018) developed a semiautomatic analysis to obtain the overlap coefficient and specific surface areas of aggregates. All these studies show clearly and quantitatively the existence of detailed minor structures of actual BC aggregates.

The overall fractal aggregate morphologies and their influences on the optical properties of BC particles have been well studied (Sorensen, 2001; Liu and Mishchenko, 2005; Kahnert and Devasthale, 2011). The minor structures mentioned above, which have even smaller length scales than the aggregate overall size, also gain significant attention for their influence on aggregate optical properties (Farias et al., 1996; Bond et al., 2006; Scarnato et al., 2013). As a consequence, more realistic and detailed models have been developed to improve our knowledge of the optical properties of BC aggregates. Farias et al. (1996) confirmed the effects of particle polydispersity on mean optical cross sections, with an amplification ratio of 1.2 and 1.8 for absorption and scattering, respectively; however, a recent study by Liu et al. (2015) found the ratios to be 1 and 2.5, respectively. The significant differences of the amplification ratios are mainly caused by different monomer size distributions as well as aggregate parameters considered in the studies. Bescond et al. (2013) evaluated the impact of overlapping and necking on the radiative properties of soot aggregates, with the main focus of their effects being on depolarization. Skorupski and Mroczka (2014) extended the investigation to the effects of overlapping and necking on the absorption and scattering properties and found that the effects of “small” necking can be ignored, but when overlap occurs the effects are pronounced. However, Yon et al. (2015) showed that overlapping and necking may significantly affect the absorption and scattering properties with the amplification factor being up to 2 at a wavelength of 266 nm. Meanwhile, small- to moderate-scale coating in different forms have been considered in numerical models to study their effects on the optical properties, and the impact of coating on absorption was found to be, on average, 1.1 in the visible spectral range (Scarnato et al., 2013; Dong et al., 2015; Doner et al., 2017). For a relatively small amount of coating, the extinction and absorption cross sections are approximately enhanced, respectively, by 5 % and 3 % as the volume fraction of coating increases from 0.01 to 0.2, as shown by Liu et al. (2012). However, Dong et al. (2015) indicated that the amplification factors of absorption and scattering increase to 1.15 and 2 for a partial coating with a volume fraction of 0.5.

Table 1 summarizes some of the previous studies investigating the effects of nonideal minor structures on BC optical properties (Farias et al., 1996; Liu et al., 2012; Scarnato et al., 2013; Skorupski and Mrocz, 2014; Yon et al., 2015; Dong et al., 2015; Doner and Liu, 2017; Doner et al., 2017). Key parameters for simulations, including the monomer number, wavelength, structure scale, and the key conclusions are given in the table. The average amplification factor of optical properties caused by these minor structures is used to specify these effects. Obviously, quite different conclusions are found, and this can be attributed to the different assumptions and parameters used in the numerical models. Because of the lack of a unified approach and criterion to quantify the effects of various minor structures, it is difficult to gain an overall quantitative understanding of such effects and to incorporate these effects in numerical models for climate modeling and the retrieval of BC concentrations in optical diagnostics. The relevant investigations will also likely continue to focus on the effects of certain specific structures. Furthermore, with a clear knowledge that the minor structures of aggregates do influence their optical properties, the numerical simulations and applications for radiative properties of BC particles become more difficult because more sophisticated and less computationally efficient numerical models are required to accurately capture the effects of such nonideal structures. Thus, it becomes an open question as to how we should deal with these different structures and apply them to practical applications in the future.

This study systematically investigates the effects of several minor structures on the optical properties of BC aggregates, including the polydispersity of monomer size, monomer surface irregularity, coating, necking, and overlapping and develops a simple method to account for their effects from the optical properties of the corresponding idealized fractal aggregates. This paper is organized as follows. Section 2 introduces the numerical models of BC aggregates with different minor structures and unifies them by introducing a parameter of “volume variation”. The effects of the minor structures and the empirical relationship to consider them in practical applications are discussed in Sect. 3. Section 4 concludes this study.

Actual BC aggregates always possess imperfectly detailed structures, and, to account for their effects on the optical properties of BC aggregates in numerical simulations, accurate numerical models to adequately represent such structures are needed. We define the imperfect geometries, such as monomer size polydispersity, monomer nonsphericity (or irregular monomer surface), thin coating, necking, and overlapping, as “minor structures” in this paper. First, the “minor” indicates that the structures are relevant at the monomer scale and do not strongly alter the overall fractal aggregate structure. Secondly, the structures are assumed to change the aggregate total volume/mass slightly, e.g., a difference less than 20 % from the idealized aggregate. The following introduces the five minor structures that will be considered in this study.

The monomers are often assumed to be same-sized spheres for simplification in
numerical studies on BC optical properties. However, measurements of sampled
BC particles reveal that BC monomers have clear variations in their sizes,
ranging from approximately 10 nm up to even 100 nm (Dankers and Leipertz,
2004; Chakrabarty et al., 2006; Bescond et al., 2014). Thus, the
polydispersity of monomer size is defined as the first minor structure,
i.e., *M*_{1}. Based on various observations (Köylü and Faeth, 1994;
Lehre et al., 2003), a lognormal size distribution provides a close
representation of realistic monomer size distribution:

where *a*_{o} and *c*_{1} are the geometric mean radius and geometric
standard deviation, respectively, and *a*_{1} indicates the radii defined
for *M*_{1}. The *M*_{1} shown in Fig. 2 schematically gives the
two-dimensional illustration of aggregates with polydisperse monomers. As
aggregate size *N* becomes larger, the monomer sizes should follow the
aforementioned size distribution. We fix *a*_{o} to be the same as that of the same-sized monomers, and the geometric standard deviation *c*_{1} is the only variable
to specify the polydispersity. For *c*_{1} larger than 1, the monomer radii
distribute over a wider range. More details of the definition of the monomer
size distribution can be found in Liu et al. (2015).

Secondly, the actual BC monomers in the atmosphere will never be perfectly
spherical, and there must be irregularity or surface roughness to some
degree for BC monomers. Therefore, we consider the irregularity of BC
monomers as the second structure feature *M*_{2} in this study. Various
numerical models, either stochastic or regular, have been developed to
represent the rough surface of aerosols or ice crystals (Li et al., 2004;
Kahnert et al., 2012a), and some of them can also be adopted for BC monomers. The surface roughness is a morphological feature with a random
nature, and the model introduced by Muinonen et al. (1996) is used for each
BC monomer. The model uses multivariate lognormal statistics (Gaussian
random shape), and the monomer surface can be expressed by

Similarly, *a*_{2}(*ϑ*,*φ*) is the function for
defining monomer surface in the model *M*_{2}. *ϑ* and *φ* are the azimuth
and elevation angles for given spherical coordinates. *s*(*ϑ*,*φ*) is the random variable obeying the Gaussian distribution
with the mean value of 0 and a fixed variance of 0.05, and the same
definitions are given in Muinonen et al. (1996). The degree of roughness or
irregularity of each monomer is directly controlled by the standard variance
of Gaussian distribution, i.e., parameter *c*_{2}. With the decrease of
*c*_{2}, the surface of the monomer becomes more rough, and spikes on the
surface become sharper. Following Eq. (3), an example of *M*_{2} for
two monomers is illustrated in Fig. 2, and the spikes on monomer spheres can
be either outward or inward due to its randomness. The changes from that of the perfect aggregate in the
aggregate volume due to irregularity are clearly represented by the red region of *M*_{2} in Fig. 2.

Once emitted into the atmosphere, BC particles are unavoidably mixed with
non-absorbing or weakly absorbing aerosols, such as sulfates, nitrates, and organic
carbon, which significantly affects the absorption and scattering properties
of BC particles (Jacobson, 2000; Schwarz et al., 2008; Liu et al., 2013).
The third minor structure (*M*_{3}) considered is coating. The mixing states
are diverse under different situations, so various complex coating models
have been developed (Scarnato et al., 2013; Dong et al., 2015). Since this
study focuses on monomer minor structures, a thin coating model, i.e.,
*M*_{3} shown in Fig. 2, is considered by wrapping each monomer with a
spherical coating shell. Thus, the radius of the coated sphere can be
described by a coating parameter *c*_{3}:

where *c*_{3} is a dimensionless parameter greater than 1. The red region of
*M*_{3} in Fig. 2 indicates the coating material. As *c*_{3} increases,
the coating becomes thicker and its fraction also increases. Note that we
consider only non-absorbing materials as coating in this study because
absorbing aerosols will introduce additional complications.

Besides the minor structures for individual monomer discussed above,
structures between neighboring monomers also exist and affect the radiative
properties of BC aggregates. The fourth minor structure (*M*_{4}) is defined
as “necking” between neighboring monomers, which is a result of
sintering processes or surface growth around the contact point (Zaitone et
al., 2009; Bescond et al., 2013). We introduce one of the simplest necking
forms, i.e., a cylindrical connector, and, in other words, cylinders are added
around the contact point of two neighboring monomers along their centers.
A dimensionless parameter *c*_{4} is used to quantify the radius of
the circular cylinders *a*_{4}:

The red region of *M*_{4} in Fig. 2 shows the necking between two
neighboring monomers. When *c*_{4}=0, the connection does not exist, and,
for *c*_{4}=1, the radius of the cylinder is equal to the monomer radius.

The last minor structure (*M*_{5}) considered is overlapping between two
connected monomers. Two general ways are used to generate overlapping: one
is to reduce the distance between the centers of two connected monomers
(Brasil et al., 1999; Yon et al., 2015), which will influence the aggregate
overall compactness significantly; another is to enlarge the radius of
primary monomers but keep their centers unchanged (Doner and Liu, 2017), and
the aggregate overall compactness is less influenced. To keep the morphology
of aggregates unchanged and consistent with that of other minor structures
mentioned above, we generate the overlapping using the latter way in this
paper. The enlarged monomer radius is expressed as follows:

When *c*_{5}=1, the monomers are in point contact, as in the ideal fractal
aggregates. With the increase of *c*_{5}, the monomers become larger and
there is overlapping between connected monomers. Although overlapping
leads to a decrease in the aggregate volume, as shown in the red region of
*M*_{5} in Fig. 2, the total volume still increases as the volume of each
monomer is larger.

To summarize, the top row of Fig. 2 schematically illustrates the definitions of the five numerical models in two dimensions. With these basic minor structures defined quantitatively, it is straightforward to build fractal aggregates with these structures. First, idealized fractal aggregates with same-sized spheres in point contact are generated using a tunable cluster–cluster aggregation algorithm (Filippov et al., 2000). From the generated ideal aggregate, we modify each monomer with different minor structures as discussed above to develop the nonideal aggregates. The bottom row of Fig. 2 shows some examples of aggregates with minor structures, which are modified from ideal aggregates generated using fractal dimension and fractal prefactor of 1.8 and 1.2, respectively (Sorensen and Roberts, 1997; Sorensen, 2001; Kahnert and Devasthale, 2011). The aggregates with only 15 monomers are displayed to better highlight the details of minor structures in each case.

Geometrically, different numerical models have to be developed for different
minor structures, which make direct comparisons between these different minor
structures challenging since these structures can lead to different effects
on the optical properties and their effects are seemingly not directly
comparable. Only when these minor structures are defined in a unified manner
can their influences on the BC optical properties be quantitatively compared
and the nature of their effects be better understood. For the five minor
structures considered, five different dimensionless parameters are used to
quantify them independently. To this end, we have to find a unified
parameter that can be connected to the effects of all the five structures in order to quantify
them. Considering that all minor structures influence not only the geometry
but also overall particle volume/mass, we convert geometrical differences
into volume/mass differences in this study. Thus, a unified parameter *β*, called the volume variation, is defined, and it simply represents the
relative volume difference between an aggregate with minor structures and
the corresponding perfect one:

where *V*_{o} is the volume of idealized fractal aggregates, and *V*_{β} is the volume of aggregates with minor structures. Because the particle
volume is one of the most prominent factors affecting
particle optical properties,
especially for particles much smaller than the incident wavelength (e.g.,
single monomer), the parameter of volume variation will not only unify the
definition of the minor structures for comparison but also provide a simple
quantitative method to account for their effects on BC optical properties.

Figure 3 illustrates the relationships between the volume variation *β*
and different structure parameters. The volume of aggregates with minor
structures can be calculated numerically in a discretized space domain with
a sufficiently high spatial resolution, which will also be used for optical
property simulations in the following section. Obviously, the total volume
variation *β* varies rapidly with the change of structure parameters,
and these structure parameters under different volume variations can be
uniquely identified by their clear relationships. When *β* is 0, the BC
aggregates are the original ideal ones. The volume variation range from 0
to 0.2 is considered in this paper. The relationship between the volume
variation and the parameters at a given level of volume variation is also
illustrated in Table 2, in which the five dimensionless parameters leading
to the given *β* values are listed. Here it is seen that these values
correspond to conditions wherein only one minor structure is allowed in each
case. Meanwhile, the relationship may vary slightly for different aggregates
due to their irregular natures, so the *β* value of each aggregate will
be calculated in the following section.

We use the discrete dipole approximation (DDA) method to simulate the
optical properties of BC aggregates with those minor structures, because it
is highly flexible in term of defining particle shape and composition
(Yurkin and Hoekstra, 2007; Kahnert et al., 2012b). The Amsterdam DDA (ADDA) code developed
by Yurkin and Hoekstra (2007, 2011) is used. In the DDA simulations, the
scatterers are discretized into numerous small sub-volumes, namely dipoles,
and particles with complex shapes can be accurately described via such
dipoles as long as the dipoles are sufficiently small in comparison to the
particle size. Note that the surface granularity
inherent to the DDA model may limit the accuracy of the resulting optical
properties, due to the small-sized scale of monomers and the minor structures
(Draine, 1988; Moteki, 2016). To ensure that the detailed monomer structures
are adequately represented in the DDA simulations, we discretize the
particles using 200 dipoles per wavelength (dpl), and this leads to over 900
dipoles per monomer. Liu et al. (2018b) developed a systematic and
comprehensive study to evaluate the performance of the DDA for simulating the
optical properties of BC aggregates by comparing with the multiple-sphere *T* matrix (MSTM) results.
Their results indicate that a systematic relative error (∼3 % with a dpl of 200) does exist between the DDA and MSTM, whereas such a
systematic and small error will not influence our discussions that mainly focus
on the relative values. This paper mainly discusses the optical properties
at an incident wavelength of 500 nm, and the BC refractive index is assumed
to be 1.8+0.6*i*; we will also briefly discuss the effects of refractive
index on our results. The original monomer radius for the ideal aggregate is
assumed to be *a*_{o}=15 nm. For the overall geometry, we consider the
fractal dimension values obtained from Wang et al. (2017) and build both
lacy aggregates (*D*_{f}=1.8) and aged ones with compact structures
(*D*_{f}=2.3) for optical studies. For practical applications, aerosols are
normally considered to be randomly oriented in the ambient atmosphere, and
all the optical properties discussed will be those averaged over a
sufficiently large number of orientations.

Figure 4 compares the relative differences (RDs) between optical properties
of the idealized aggregates and those with different minor structures as a
function of volume variation *β*. Here, aggregates with 200 monomers
are considered, and Fig. 4a and b show results of aggregates
with *D*_{f}=1.8 and *D*_{f}=2.3, respectively. Positive RDs mean that the
minor structures enhance the corresponding optical properties. With the
increase of volume variation, the RDs in the extinction and absorption cross
sections increase almost linearly, and different minor structures tend to
have obvious enhancement in extinction and absorption cross sections to
varying degrees. The effects of necking are largest with ∼35 % RDs as *β* increases to 0.2, while the effects of coating are smallest
with RDs less than 10 %. The RDs of single scattering albedo
(SSA) also become larger with the increase of *β* (up to
∼15 % at *β*=0.2), whereas these differences for
different minor structures are less significant than those in extinction and
absorption cross sections. The results of asymmetry factors for different
minor structures display very small differences, and the RDs, less than 3 %
even for the largest *β* value considered, are close to zero over
the entire range of *β* considered, suggesting that it is nearly
unaffected by the type and magnitude of minor structures. Furthermore,
results for lacy and compact aggregates are similar, and only the
polydispersity of monomer size shows a relatively lower impact on the SSA.

With almost no influence on the asymmetry factors, the effects of these
minor structures on the angle-dependent scattering matrix elements are
also expected to be minor. Figure 5 shows the normalized phase function and
two nonzero scattering matrix elements (*P*_{22} and *P*_{34}) for
aggregates with minor structures at a volume variation of *β*=0.1,
and those of perfect aggregates are given as a direct comparison. Because
minor structures show even lower impacts on other nonzero scattering matrix
elements (*P*_{12}, *P*_{33}, and *P*_{44}), they will not be shown here.
The normalized phase functions and *P*_{34} of aggregates with different
minor structures agree closely with those of idealized aggregates. The
polydispersity (the blue lines) has essentially no influence on *P*_{11} but
shows impacts on backward scattering with scattering angles larger than
90^{∘}, and the effects are slightly larger for compact aggregates.
For *P*_{22}, the necking causes the largest decreases among the five
structures, whereas the overall differences remain small (note that the
range of *P*_{22} values plotted are between 0.96 and 1). To summarize,
minor structures have negligible effects on the angle-dependent scattering,
even at a volume variation of 0.1. Thus, the scattering matrix elements of
aggregates will not be further considered and the following discussion will
focus only on the integral scattering properties (i.e., those shown in Fig. 4).

Different from perfect aggregates, whose optical properties can be calculated conveniently by accurate models such as the MSTM method (Mackowski, 2014) and the generalized multiparticle Mie-solution (GMM) method (Xu, 1995), modeling the optical properties of aggregates with minor structures is much more tedious. This also makes its applications challenging due to the time-consuming simulations and uncertainties in the definition of minor structure parameters. Thus, it is important and necessary to find an empirical relationship between optical properties of aggregates with and without minor structures. The relationship, which can estimate the optical properties of aggregates with minor structures from those of idealized ones, should be general and simple for the purpose of practical application. Fortunately, Figs. 4 and 5 indicate that although the minor structures have different influences on aggregate optical properties, they do have some similar features. For example, their effects on the extinction and absorption cross sections are almost proportional to the volume variation, and the effects on the scattering matrix are ignorable. Those features can be applied for the empirical relationship.

To develop the empirical relationship based on the current DDA results, the significant influence of the volume variation on the cross sections should be considered first. This is reasonable because minor structures only change monomer shapes. In the visible and shortwave infrared spectra ranges, BC monomers are normally in the Rayleigh regime, and the corresponding absorption and scattering cross sections are proportional to the volume and volume square, respectively. Considering the linear relationship shown in the absorption cross section in Fig. 4, it is reasonable to approximate the absorption cross section of aggregates with minor structures as follows:

where ABS_{β} is the absorption cross section of the imperfect
aggregate with a volume of *V*_{β} and volume variation of *β*,
and ABS_{o} and *V*_{o} are the values of the corresponding perfect
aggregate. The 1+*β* term indicates the change in overall volume due to
minor structures, and the parameter *B*_{ABS} can be understood as a
“correction ratio” indicating the direct effects of minor structures on
the absorption cross section. Similarly, the relationship for the scattering
cross section can be given as follows:

Meanwhile, the extinction cross section can be obtained by the sum of the
scattering and absorption cross section. Note that similar relationships
to Eqs. (8) and (9) have been proposed by Farias et al. (1996) to account
for the effect of monomer polydispersity but without the terms *B*_{ABS}
and *B*_{SCA}. Based on the results shown in Figs. 4 and 5, the scattering
matrix and asymmetry factor are unchanged. Because the volume of BC,
absorbing material, is not changed in the coating case, the volume variation
term (1+*β*) in Eq. (8) is removed in the empirical approximation.

The next step is to determine the values of *B*_{ABS} and *B*_{SCA} for
different minor structures. We approximate the two correction ratios based
on the results shown in Fig. 4, and the results are listed in Table 3. All
values in the table are larger than 1, meaning that the minor structures
also enhance the equivalent mass absorption and scattering of aggregates.
Larger values of *B*_{ABS} or *B*_{SCA} mean that the minor structure has a
stronger influence. The *B*_{ABS} of necking is the largest with
a value up to 1.08, and the *B*_{ABS} of polydispersity is the
smallest (1.02). The correction ratios are all in the range between 1.01 and
1.08 for absorption and scattering, so the empirical relationships in Eqs. (8) and (9)
can be further simplified by assigning a constant correction
ratio of ${B}_{\mathrm{ABS}}={B}_{\mathrm{SCA}}=\mathrm{1.05}$ for all cases. In other
words, we specify an average enhancement rate in the scattering and
absorption of aggregates with minor structures after considering volume
variation. The resulting cross sections can be used to obtain the new
single-scattering albedo of aggregates with minor structures as well.
Another advantage of the proposed relationships is that only a volume or
mass variation is needed for the approximation, whereas evaluation of the
geometric parameters of the minor structure is much more challenging.

To evaluate the performance of the empirical relationship, Fig. 6
replots the results of Fig. 4 and directly compares the results using the
aforementioned correction relationships (with the help of those of perfect
aggregates) and those of the rigorous DDA simulations for nonideal
aggregates. To remove the effects of differences in volume in the following
evaluations, mass extinction and absorption cross sections (MEC and MAC),
i.e., values per unit BC mass, are considered instead of absolute cross
sections, and we use a BC density of 1.8 g cm^{−3} (Bond and Bergstrom,
2006). Thus, the relative errors (REs) in Fig. 6 show the relative errors of empirical
relationships for approximating the optical properties of BC aggregates with
minor structures. The gray shaded regions indicate REs of between ±5 %. Again, Fig. 6a shows lacy aggregates, and
Fig. 6b shows
compact aggregates. Obviously, the relative errors between direct DDA
simulations of imperfect aggregates and results from empirical relationships
are well under the ±5 %. This indicates that the empirical
relationships can provide a good approximation of the optical properties of
BC aggregates with minor structures, at least for cases considered in this
study.

All the studies above are based on an assumed refractive index of 1.8+0.6*i*
(but close to the real one), whereas the refractive index is not only one of
the most important parameters for determining BC optical properties but also
one of the most uncertain physical properties (Bond and Bergstrom, 2006; Liu
et al., 2018a). To investigate the influence of BC refractive index on the
minor structure effects, Fig. 7 compares the optical properties of BC
aggregates with minor structures with those from empirical relationships as
functions of the real part (Fig. 7a) and the imaginary part (Fig. 7b) of
refractive index. Aggregates with 200 monomers and a fractal dimension of 1.8
are considered, and the volume variation due to minor structures is set to
0.1. Figure 7 indicates that the refractive indices do affect optical
properties significantly, whereas the relative differences caused by the
minor structures do not change too much. Only the necking causes optical
properties slightly out the ranges given by the empirical results (shadow
regions) due to change of the real part of refractive index. In other words,
the minor structures show similar effects on BC optical properties at
different refractive indices, and our empirical method can be extended to a
wider range of BC refractive indices.

The performance of this simple approximation utilizing the optical properties of perfect aggregates is further evaluated for differently sized aggregates at different wavelengths. Figure 8 compares the optical properties from direct simulations of aggregates with minor structures (markers) and empirical approximations based on properties of perfect aggregates (solid black lines). Here, we consider only lacy aggregates with a fractal dimension of 1.8 and aggregates containing 50 to 1500 monomers are considered. Again, the volume variation is 0.1. Thus, the volume equivalent diameter up to ∼350 nm is included to cover a wide range of BC size distribution (Schnaiter et al., 2005; Reddington et al., 2013). The gray shaded regions indicate ±5 % relative errors within the results from the empirical relationships. As expected, the effects of minor structures are less sensitive to particle size. The MEC and MAC for aggregates with monomer polydispersity and necking are slightly smaller and larger than the empirical results, respectively. Nevertheless, all of the markers, i.e., results from direct simulations of aggregates with minor structures, lie within the ±5 % gray shaded areas. The empirical results of SSA agree more closely with direct simulation results. As expected, the asymmetry factor is least affected by minor structures with average REs of less than 2 %. The monomer polydispersity shows some effects on the asymmetry factor, with REs reaching almost 6.5 % for relatively small aggregates containing 100 monomers or less. Considering the significant uncertainties of BC parameters, a relative error of 5 % is acceptable for most applications; furthermore, Fig. 8 validates that the average volume correction ratio and empirical relationships can be applied confidently to account for the effects of minor structures on the optical properties of BC aggregates. Results for compact aggregates are similar and will not be shown here.

Table 4 shows the bulk optical properties of BC aggregates with minor
structures from direct DDA simulations and those from the empirical
approximations, and the optical properties of ideal fractal aggregates are
also given as a reference. A lognormal size distribution with a geometric
mean diameter and a standard deviation of 120 nm and 1.5, respectively, are used for aggregate sizes (Alexander et al., 2008; Chung et al., 2012). It is clear from
Table 4 that the bulk optical properties of ideal aggregates are
significantly lower than those of aggregates with minor structures,
especially for MEC and MAC. With an average over the aggregate size, the
empirical approximation should give even better approximations. The bulk MEC
and MAC are approximately 7 and 6 m^{2} g^{−1}, respectively, at 500 nm,
which is in reasonable agreement with observations (Bond and Bergstrom,
2006; Kahnert et al., 2010; Liu et al., 2010; Xu et al., 2017). The
empirical results overestimate MEC and MAC of polydispersity by 3.7 % and
4.9 %, respectively, and underestimate those of necking by 3.6 % and
2.8 %. Again, minor structures show less impact on the bulk asymmetry
factor and SSA, and the relative errors from those of idealized aggregates
are always less than 2 %. Clearly, after scaling with the empirical
relationship, results from the perfect aggregates can be readily used to
represent the bulk optical properties of aggregates with minor structures.

The results discussed above are at a wavelength of 500 nm and results at different incident wavelengths between 300 and 700 nm are illustrated in Fig. 9. Aggregates with 200 monomers and the fractal dimension of 1.8 are considered for these simulations at a volume variation of 0.1. Obviously, all the optical properties decrease with the increase of wavelength because the particles become relatively small compared to the longer wavelength. The empirical relationship with the same correction ratio 1.05 is applicable for different wavelengths. Again, all the optical properties of BC aggregates with different minor structures (markers) are within the range of ±5 % from the results of empirical approximations (the solid black lines).

Most previous studies, as well as the aforementioned results, consider minor structures individually; i.e., the BC aggregates deviate from the ideal ones through only one of the five structures, whereas atmospheric BC aggregates in general contain more than one structure simultaneously. To develop more realistic models to represent actual BC particles, we further investigate whether the effects of a combination of different minor structures can still be accounted for by simple empirical relationships. To this end, we include multiple minor structures in a single aggregate and use the volume variation to constrain the combination. For a given total volume variation, the contribution of each minor structure can be randomly generated. Figure 10 shows 10 examples of aggregates containing 15 monomers with random combinations of minor structures to represent their detail morphologies, and 5 examples of aggregates containing 100 monomers, which are more close to actual BC aggregates, are given in the bottom row. Different combinations show clearly different geometries, though they all have the same total volume variation of 0.1 from the corresponding perfect aggregate.

Figure 11 compares the optical properties of aggregates with multiple minor structures calculated by the DDA (blue bars) and those from empirical approximations (yellow bars). For each case, 20 random aggregates are built and calculated, and the same volume variation of 0.1 is imposed for imperfect aggregates. The averaged optical properties as well as their variances (vertical red error bars) are shown in the figure. Surprisingly, even with different combinations of minor structures, the variances are quite small, indicating that the differences between the 20 random cases are minor. The asymmetry factor shows the largest variance, which agrees with the results of ideal aggregates (Liu et al., 2015). Comparing the blue and yellow bars, we can find that the empirical results agree closely with the accurate calculations as all of the relative errors are less than 3 %. Therefore, even for BC aggregates with multiple minor structures, their combined effects can still be accurately accounted for by the empirical relationships proposed in this study due to the consistent volume variation.

To understand the impact of aggregate minor structures on BC radiative
forcing estimations, we compare the direct radiative forcing of BC aerosols
with and without considering minor structures. We perform a simple radiative
transfer simulation for a semiquantitative discussion, and the Santa
Barbara DISORT Atmospheric Radiative Transfer Model is used to calculate BC
forcing at a visible wavelength (Ricchiazzi et al., 1998; Zeng et al.,
2019). Similar to the settings used by Zeng et al. (2019), the midlatitude
summer atmosphere profile is considered, and the surface albedo is set to
0.15. The solar zenith angle is 60^{∘}. We assume BC particles to be
located uniformly at the bottom layer of the atmosphere with a height between
0 and 1 km and a BC number concentration of 10^{10} particles m^{−3}
following Reche et al. (2011). Table 5 compares the input BC properties
(optical properties and aerosol loading) and the corresponding radiative
forcings, including those at the top of the atmosphere (TOA) and the surface
(SFC) and in the atmosphere (ATMOS). The same BC size distribution as well
as the corresponding bulk optical properties given in Table 4 is used. We
consider aggregates with the minor structures that lead to a 10 %
difference in the volume. For the ideal aggregates, we consider two BC
number concentrations: (1) the same as that for BC with minor structures,
and (2) 10 % larger to account for the volume increase due to minor
structures, whereas both cases used the optical properties of ideal
aggregates. Here, the optical properties of aggregates with minor structures
are directly transformed from those of the ideal ones through the empirical
relationship because those from accurate simulations show relative
differences of less than 5 % (see Table 4) and will not change the forcing
values significantly. The table indicates that the effects of aggregate
minor structures on BC radiative forcing are in similar orders to the
effects on the optical properties, i.e., relative differences approximately
∼12 % between the results of Cases 1 and 2. Again, the
differences are mainly contributed by the differences of BC amount/volume
because, after assuming the same BC mass between Cases 1 and 3, the forcings
become close to each other. However the effects are interpreted, the
effects of minor structures can be easily accounted for without the tedious
simulations of the optical properties for particles with minor structures or
without even knowing their details.

This study investigates the effects of different minor structures on the optical properties of BC aggregates and develops a simple empirical treatment to account for their effects from the optical properties of the corresponding ideal fractal aggregates. The structures considered in this study include polydispersity, irregularity, coating, necking, and overlapping, all of which are for monomers and not the aggregate overall structure. The volume variation can be used to unify the effects of different minor structures. Minor structures significantly affect the optical properties of BC aggregates with different fractal dimensions and aggregate sizes, and with the increase of wavelength, these effects become slightly weaker. Among the five minor structures, necking shows the strongest effect. The asymmetry factor and the scattering matrix are almost unaffected by these minor structures. For easy and accurate estimation of BC aggregates involving these minor structures, a simple empirical relationship is developed to account for their effects on the optical properties based on a constant correction ratio of 1.05 and volume variation from that of the corresponding ideal aggregate. Results show that the empirical relationships can accurately represent the optical properties of BC aggregates with minor structures. Additionally, the effects of multiple minor structures are found to be similar to those of a single structure with the same volume variation and the empirical relation is also applicable. Overall, the effects of minor structures on BC aggregate optical properties are mainly caused by their influences on the total overall volume/mass, and the minor structure effects merely result in ∼5 % differences in the mass optical properties. Thus, future studies should pay more attention to BC total volume/mass, and the details of the minor structures may not need to be considered with the empirical relationship developed by this study.

For practical applications, the detailed parameters of fractal aggregates as well as the minor structures should be known, and such knowledge can be obtained via the analysis of aggregate images (Brasil et al., 2000; Chakrabarty et al., 2006). Among the minor structures, the monomer size polydispersity and overlapping have been most widely considered (Bescond et al., 2014; De Temmerman et al., 2014; Bourrous et al., 2018). However, with the proposed empirical relationship, only simple estimations of the volume or mass variation are sufficient to account for the change in optical properties. The methodology proposed in this study enables efficient and accurate prediction of the optical properties of BC aggregates with minor structures based on those of the corresponding ideal aggregates that can be calculated using the MSTM and GMM methods with much less computational effort than the DDA, and the optical properties of those ideal aggregates have been taken from a comprehensive database (Liu et al., 2019). This will make the applications more practical and instructive for experimental analysis.

The data obtained in this study are available at https://github.com/ShiwenTeng/Data_FOR_ACP-2018-1102/ (last access: 25 February 2019).

ST and CL designed the study, carried out the research, and performed data analysis. ST, CL, MS, RC, and FL discussed the results and wrote the paper. All authors gave approval for the final version of the paper.

The authors declare that they have no conflict of interest.

We are deeply thankful to Maxim A. Yurkin and Alfons G. Hoekstra for the
ADDA code. This work was financially supported by the National Key Research
and Development Program of China (2016YFA0602003), the National Natural
Science Foundation of China (41505018), and the Young Elite Scientists
Sponsorship Program by the China Association for Science and Technology (2017QNRC001). The computation of this study was
supported by the National Supercomputer Center in Guangzhou
(NSCC-GZ).

Edited by: Ilona
Riipinen

Reviewed by: three anonymous referees

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