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):

Examples of TEM/SEM images of BC particles from different in situ or laboratory observations (Gwaze et al., 2006; Kamimoto et al., 2007; Chakrabarty et al., 2009; Yon et al., 2015; 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.

Comparison of the previous studies on effects of minor structures on optical properties of fractal aggregates.

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.,

Two- and three-dimensional models of five minor structures. Aggregates with 15 monomers and a fractal dimension of 1.8 are considered for an example.

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

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 (

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 (

The last minor structure (

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

Figure 3 illustrates the relationships between the volume variation

Relationships between geometry parameters and volume variations.

Relationship between different minor structure parameters
(

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

Relative differences of optical properties for aggregates with minor
structures compared to those with perfect fractal aggregate structures at a
wavelength of 500 nm. Aggregates with 200 monomers and a fractal dimension
of 1.8

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

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 (

Three normalized scattering matrix elements (

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:

The next step is to determine the values of

Correction ratio obtained from results in Fig. 4 to give the absorption and scattering cross sections of aggregates with different minor structures.

Relative error of optical properties for aggregates with minor structures between direct DDA simulations and those modified from ideal fractal aggregates using the empirical correction factors and volume/mass variation.

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

Comparison of the optical properties of aggregates with minor structures from direct simulations (markers) and those corrected from empirical relationships (black lines) as functions of refractive indices.

All the studies above are based on an assumed refractive index of

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

Comparison of the optical properties of aggregates with minor structures from direct simulations (markers) and those from ideal aggregates but with corrections accounting for the minor structures (black lines).

Bulk scattering properties for lacy aggregates with minor
structures (

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

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

Same as Fig. 8 but for optical properties of aggregates with 200 monomers at different wavelengths.

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.

Examples of aggregates with random combinations of minor structures. The aggregates are generated to have the same volume variation of 0.1 compared to their perfect counterparts.

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.

Comparison of the optical properties from direct simulations of aggregates with combined multiple minor structures (blue) and those from perfect aggregate structures but with corrections to account for the minor structures (yellow).

BC radiative forcings calculated based on different assumptions of BC optical properties or column amounts.

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

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

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

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