The absorption Ångström exponent (AAE) is an important aerosol
optical parameter used for aerosol characterization and apportionment
studies. The AAE of black carbon (BC) particles is widely accepted to be 1.0,
although observational estimates give quite a wide range of
0.6–1.3. With considerable uncertainties
related to observations, a numerical study is a powerful method, if not the
only one, to provide a better and more accurate understanding on BC AAE. This
study calculates BC AAE using realistic particle geometries based on fractal
aggregate and an accurate numerical optical model (namely the multiple-sphere
T-matrix method), and considers
bulk properties of an ensemble of BC particles following lognormal size
distributions. At odds with the expectations, BC AAE is not 1.0, even when BC
is assumed to have small sizes and a wavelength-independent refractive index.
With a wavelength-independent refractive index, the AAE of fresh BC is
approximately 1.05 and relatively insensitive to particle size. For BC with
geometric mean diameters larger than 0.12

The absorption Ångström exponent (AAE) is an aerosol optical property
describing the wavelength
variation in aerosol absorption. Because aerosol absorption normally
decreases exponentially with wavelength over the visible and near-infrared
spectral region (Ångström, 1929; Bond, 2001; Lewis et al., 2008), the
AAE is defined as

AAE has been widely used for aerosol characterization studies (Russell et al., 2010; Giles et al., 2012). The basis for the AAE in aerosol characterization is that AAE is assumed to be a specific property of each aerosol species. For example, black carbon (BC) aerosols have AAEs around 1.0, and organic aerosols and dust have higher AAE values (Kirchstetter et al., 2004; Russell et al., 2010). Thus, the AAE of an aerosol sample close to 1.0 is considered to be BC-rich aerosol from fossil fuel burning, and larger AAE values are understood to indicate aerosols from biomass/biofuel burning or dust (Russell et al., 2010). AAE has also been quantitatively used to separate brown carbon (BrC) absorption from BC absorption (Kirchstetter and Thatcher, 2012; Lu et al., 2015). In the studies of BC/BrC absorption separation, the BrC absorption from biomass burning aerosols is usually retrieved by assuming that BrC contributes no absorption at near-infrared wavelengths and that BC has an AAE of 1.0 (Ganguly et al., 2005; Kirchstetter and Thatcher, 2012; Lu et al., 2015). Lack and Langridge (2013) quantified the effect of a specified BC AAE value on the absorption apportionment between BC and BrC and demonstrated the importance of BC AAE. Additional uses of AAE include aerosol color. A recent numerical study done by C. Liu et al. (2016), for example, confirms that AAE largely controls the color of aerosols in the ambient atmosphere. They found that aerosols are visually brown if their AAE is larger than approximately 2. In another use, aerosol spectral light absorption is an important parameter for the assessment of the radiation budget of the atmosphere (Schmid et al., 2006).

BC, also known as soot, is an important aerosol species emitted from
incomplete combustion of fossil fuel, biofuel, and biomass (Bond and Sun, 2005; Bond et al., 2013; Chakrabarty et
al., 2014), and it exhibits significant variations in its physical and
chemical properties due to differences in fuels and combustion conditions
(Schnaiter et al., 2006; Bahadur et al., 2012; Reddington et al., 2013). Does
BC indeed have an AAE of 1.0? The AAE of BC from combustion is widely
accepted and used as 1.0 when the particles exist alone and have not
experienced atmospheric aging processes (Bergstrom et al., 2002, 2003;
Schnaiter et al., 2003; Lawless et al., 2004; Bond and Bergstrom, 2006). If
BC particles are much smaller than the incident light wavelength and have a
wavelength-independent refractive index, the Rayleigh approximation does
theoretically derive an AAE of 1.0 (Moosmüller and Arnott, 2009).
Atmospheric BC particles are generally small
compared to the wavelengths of visible light, but it is uncertain that all the BC particles
clearly fall into the Rayleigh regime. Furthermore, there is much uncertainty in the
wavelength dependence of the BC refractive index. In reality, BC aggregates in the ambient
atmosphere may just fall at the edge of the Rayleigh region at visible and
near-infrared wavelengths. Figure 1 visualizes the extent to which the
Rayleigh approximation holds for spheres by comparing with the exact Mie
results. In Fig. 1, the

When the true BC AAE is in doubt, one can alternatively investigate it by measuring the absorption of BC particles in the atmosphere, which turns out to be more challenging. BC AAE has been experimentally investigated in numerous studies (e.g., Schnaiter et al., 2003; Kirchstetter et al., 2004; Bahadur et al., 2012; Chung et al., 2012). In a laboratory study, Schnaiter et al. (2003) found diesel soot to have an AAE of 1.1 and spark-generated carbon nanoparticles to have an AAE of 2.1. The different AAE values were mainly attributed to differences in the wavelength dependence of the refractive index for the two BC materials. In the atmosphere, however, BC particles always co-exist with other aerosol particles. Non-BC particles can affect the total aerosol AAE by containing BrC or mineral dust (which have higher AAE values) and also by coating BC. Coating of BC amplifies the BC absorption, and the amplification of BC absorption is dependent on wavelength. Kirchstetter et al. (2004) measured the absorption of particles near a roadway or inside a tunnel and, after extracting organic carbon (including absorptive BrC), found the AAE to be 0.6–1.3. The locations Kirchstetter et al. (2004) chosen would have yielded AAE without much interference from BrC or coating. However, Kirchstetter et al. (2004) used filter-based instruments to measure the absorption. Filter-based absorption instruments are susceptible to multiple artifacts such as optical interactions between the concentrated particle themselves and that of the particles with the filter substrate (Moosmüller et al., 2009). In addition, filter deposition may alter particle shapes and size distributions greatly (Subramanian et al., 2007), which affects aerosol absorption properties significantly (Li et al., 2016). Weingartner et al. (2003) and Arnott et al. (2005), for example, attempted to address these artifacts. The available correction schemes are not available for every type of BC and furthermore not optimized for adjusting AAE. Chow et al. (2009) showed that a particle soot absorption photometer (PSAP), after an absorption artifact correction, gave AAE values about 20 % less than a filter-free photoacoustic instrument for the aerosols at Fresno, California. Gyawali et al. (2012) and Chakrabarty et al. (2013) generated BC-dominated particles by burning oil and measured the absorption using filter-free photoacoustic instruments. The estimated AAE ranged from 0.8 (kerosene soot) to 0.95–1.1 (mustard oil soot). Because there must be some absorptive organic aerosols (e.g., BrC) in these aerosol samples that have a much larger AAE, the indication from these two studies is that BC AAE is lower than 0.8–1.1.

Comparison of the Rayleigh approximation and Mie theory for the
absorption efficiency of spheres with a refractive index of
1.8

In addition, AAE values could differ due to differences in AAE calculations.
For example, if the absorptions at two wavelengths are observed, the AAE can
be approximated by

Whether BC AAE is exactly 1.0 or not is an issue we will address in this paper. Another issue is whether BC coated with non-absorptive material would have the same AAE as uncoated BC. In the aforementioned BC/BrC absorption separation studies (Kirchstetter and Thatcher, 2012; Lu et al., 2015), the AAE for uncoated BC was implicitly assumed to be the same as that for coated BC. Lack and Cappa (2010) used a core–shell Mie code to investigate how BC changes its AAE value with respect to coating. At realistic particle sizes, they showed that the BC AAE increases to 1.4–1.6 after coating. They computed BC AAE using a group of BC particles where the cores were specified to have a lognormal size distribution. In their study, coating volume fraction was assumed to be fixed for all the particles, an assumption that has no experimental, observational, or theoretical support. When BC particles grow in size by coating, the particle growth is governed by condensation. Theoretically, condensation reduces the diameter spread between big particles and small particles over time (Seinfeld and Pandis, 2016), in contrast to the assumption by Lack and Cappa (2010). Schnaiter et al. (2005) coated BC particles with secondary organic aerosol material in a lab and found that the coating increases particle sizes, while it reduces geometric standard deviation (see Fig. 5 of their paper), as predicted by theoretical calculation of condensation process. Their work provided a meaningful experimental dataset to derive coating fraction variation in BC particles. Furthermore, Schnaiter et al. (2005) also measured the absorption of coated BC particles at 450, 550, and 700 nm (see Fig. 9 of their paper), and from these three wavelengths we see that coating does actually decrease BC AAE (from approximately 1.1 to 0.8), thereby contradicting the coated BC AAE estimates in the study by Lack and Cappa (2010). This indirectly indicates that a fixed coating fraction for different-sized BC may be problematic.

Although observations do not give a clear value of BC AAE, it is safe to say that even accurate observations do not strongly support the theoretical constant of 1.0 for BC AAE. The fact alone that there are different types of soot particles (associated with different refractive indices) points to a range of BC AAE instead of a fixed value. Furthermore, it is not clear if the real BC AAE is 1.0 on average. Despite all these uncertainties, BC AAE has been assumed to be 1.0 in many studies (Lack and Langridge, 2013; Moosmüller et al., 2009; Lack et al., 2008; Kirchstetter et al., 2004; Lewis et al., 2008). Meanwhile, numerical studies on BC AAE have shown neither systematic nor conclusive results to improve our understanding on them (Li et al., 2016; Lack and Langridge, 2013).

This study presents a systematic numerical investigation on the AAE of BC particles and decomposes the AAE influence into that due to each particle microphysical property (e.g., shape, size, refractive indices, and internal mixing). The paper is organized as follows. The properties of BC particles used for absorption simulations are discussed in Sect. 2, and Sect. 3 presents the AAE simulations and decomposes the AAE influences. Section 4 concludes this study.

The absorption of a single particle or an ensemble of particles can be accurately calculated if the particle shape, size, and refractive index are known. With a large amount of observations on BC microphysical and optical properties made in the past decades (Sorensen, 2001; Bond and Bergstrom, 2006), there is much less uncertainty in estimating shape, size, and refractive index for BC compared with AAE estimation. The present study embarks on numerical calculations of the optical properties of an ensemble of BC particles. There have been numerous calculations of BC optical properties on a single wavelength (Sorensen, 2001; Liu and Mishchenko, 2005, 2007; Smith and Grainger, 2014), and the novelty of the present study is considering much wider but realistic ranges of BC properties (especially the particle shape and coating fraction) and focusing on AAE systematically. The following subsections discuss BC geometry, size, and refractive index and explain how these properties are treated in the simulations herein.

BC particles exist in the form of aggregates with hundreds or even thousands
of small spherical particles, called monomers. The concept of the fractal
aggregate (FA) shows great success and wide applications on representing
realistic BC geometries (Sorensen, 2001). The FA is mathematically described
by the statistic scaling rule in the form of

Immediately after being emitted into the atmosphere, BC aggregates exhibit lacy
structures with a small fractal dimension

The optical properties of the mixed BC were investigated by a core–shell
model with a Mie theory in the past, which assumes a spherical BC core in the
center of coating sphere (Chung et al., 2012; Lack et al., 2012; Peng et al.,
2016; Moffet and Prather, 2009), and this may introduce significant
differences on BC optical properties. Meanwhile, some studies have introduced more
complex and realistic geometries to consider the effects of coating on BC
optical properties (Liu et al., 2012; F. Liu et al., 2016; Dong et al.,
2015), whereas the AAE was not explored. For coated BC, this study uses a
compact aggregate as the BC core, and a spherical coating is added as the
coating material following the numerical model developed by C. Liu et
al. (2017). The coating is assumed to be non-absorptive sulfate, the
wavelength-dependent refractive indices of which are obtained from the
well-known aerosol optical property database OPAC (Hess et al., 1998). The
real part of sulfate refractive index decreases slightly from 1.47 to 1.42 as
the wavelength increases from 0.3 to 1.0

However, the amount of coating, another important factor to determine the absorption enhancement, is one of the most poorly investigated issues for coated BC (D. Liu et al., 2017). The key issue here is to develop a relationship between core size and coating amount for a group of different-sized BC particles. In other words, after fresh BC particles become coated over a certain amount of time, do small cores tend to have the same coating amount as big cores? Or do small cores have larger coating fractions than those of big cores? As pointed out in Sect. 1, if coating volume fraction is assumed to be fixed for different core sizes, modeled AAE variation is different from observations (Lack and Cappa, 2010). On the other hand, if we apply aerosol condensation physics, we speculate that small core particles are associated with larger coating amount, since condensation reduces the diameter differences between big and small particles. Thus, a more realistic and observation-based relationship between BC core size and coating amount should be derived.

We choose to use the experiment results by Schnaiter et al. (2005), because
they gave the size distributions of the fresh and coated BC in a closed
chamber environment. They coated diesel soot particles with secondary organic
compounds produced by in situ ozonolysis of

By assuming that, during the 24 h of coating, there was no particle
coalescence or coagulation, the size distribution of fresh BC can be taken
to represent that of BC cores of coated BC. For fresh BC, however, the
mobility diameter deviates substantially from the diameter of equivalent
volume sphere due to the lacy aggregation structures, and the latter is what
we actually need in order to derive the BC core size distribution. To
estimate the equivalent volume diameter of BC core (i.e., fresh BC), we first
convert the mobility diameter of fresh BC aggregates to monomer number, and
this is achieved by applying Eqs. (5) and (21) in Naumann (2003), which
use the fractal geometric parameters of diesel soot aggregates (Schnaiter et
al., 2003). Then, the equivalent volume diameter can be obtained by assuming
the

In addition, we are applying the following two simplifying assumptions to map
the core size distribution into the size distribution of the coated BC.
(1) Large BC cores are still larger than small BC cores after coating, i.e.,
BC cores from the left or right side of the core size distribution will
appear on the left or right side of the coated BC distribution, respectively.
(2) For each single BC core size there exists only one single coating amount,
i.e., applying a “one-to-one” mapping between core size and coating
amount. With those two assumptions, the
resulting equivalent volume diameter distribution of the BC core
(

Geometries of numerically generated BC aggregates with different
particle geometries, i.e., loose aggregate

Figure 3 visualizes some examples of BC particles to illustrate the three BC formats, i.e., geometries considered in this study. A tunable particle-cluster aggregation algorithm is applied to generate the FAs (Filippov et al., 2000; Liu et al., 2012), and the coating sphere is added with its center located at the mass center of compact FA. Three transmission or scanning electron microscope images of BC particles are also given in the figure for comparison (Burr et al., 2012; Lewis et al., 2009; Freney et al., 2010), and we can see that the numerically generated particles have similar structures to those observed ones. Geometric parameters for FA with observational bases have been widely used for numerical studies (Liu and Mishchenko, 2005; Smith and Grainger, 2014; Li et al., 2016), and are also applied here to represent realistic BC particles. Specifically, the diameter of each monomer is set to 30 nm as supported by observations (Brasil et al., 2000; Chakrabarty et al., 2014), and the fractal prefactor of 1.2 that was estimated by Sorensen and Roberts (1997) is used. For the lacy aggregate, a fractal dimension of 1.8, close to the is observed average by Köylü et al. (1995) and Sorensen and Roberts (1997) and both the compact and coated BC particles use a fractal dimension of 2.8 to make the BC particles as compact as we can. With the coating fraction as a function of the BC core size given in Fig. 2d, the aggregates are completely wrapped inside the coating sphere. Nevertheless, to better understand the sensitivity of BC AAE to geometries parameters, we will also explore other monomer diameters (i.e., 20 and 40 nm) and another fractal dimension of 2.3 (i.e., the mean value between the two extreme values and the value derived from observation of Wang et al., 2017).

Particle size distribution is one of the most commonly measured variables
for aerosol studies. Aerosol size measuring instruments (such as the SMPS
mentioned above, and the single-particle soot photometer, i.e., SP2, and
electron microscopy) have repeatedly shown that a lognormal size
distribution is a good fit for realistic BC size distributions (Bond et al.,
2002; Schnaiter et al., 2005; Chakrabarty et al., 2006; Kirchstetter and
Novakov, 2007; Reddington et al., 2013; Wang et al., 2015), and it is also
widely used in numerical calculations of BC radiative properties and forcing
(Moffet and Prather, 2009; Chung et al., 2012; Li et al., 2016). It should
be noted that various different quantities are used to describe BC size
distribution according to the principle used for the measurement. For
example, as noted before, the SMPS measures the mobility equivalent
diameters

To specify and unify the definition, all sizes in this study are referred to
as the diameter of equivalent volume sphere from here on. For fresh or
compact BC, this diameter can be given as

For aggregates with fixed monomer sizes (diameter of 30 nm without special
mention, and 20 and 40 nm used for sensitivity studies), their diameters are
only determined by the number of monomers in the aggregate. We consider
aggregates with the number of monomers ranging from 1 to 2000, corresponding
to diameters of equivalent volume spheres from 0.03 to almost
0.4

The refractive index (RI), a wavelength-dependent complex variable, is one of the most important parameters to determine aerosol AAE, because the absorptions at different wavelengths are significantly influenced by both the real and imaginary parts of RI. However, it is also one of the most uncertain physical properties of BC particles, because it cannot be directly observed. Estimates of BC RI have been normally made from observed absorption, scattering (or extinction), and size distribution of suspended particles, or from reflectance measurements on compressed BC pellets, and the RI is inferred by obtaining a best fit to numerical simulations (either Mie theory by assuming spherical particle shape or the simple Rayleigh–Debye–Gans theory) (Chang and Charalampopoulos, 1990; Schnaiter et al., 2003, 2005; Kirchstetter et al., 2004; Dalzell and Sarofim, 1969; Stagg and Charalampopoulos, 1993; Vanhulle et al., 2002; Moteki et al., 2010). Some of those studies extend RIs at particular wavelengths into the whole spectrum by the dispersion equations or the Kramers–Kronig analysis (Dalzell and Sarofim, 1969; Querry, 1987; Chang and Charalampopoulos, 1990). These retrieval methods based on the unrealistic spherical shape assumption or inaccurate numerical modeling pose sizable errors on estimated RIs, even without considering the error in aerosol optical property measurement. Furthermore, the BC materials from different combustions probably have different RIs, and this was discussed in the past (e.g., Sorensen, 2001; Bond and Bergstrom, 2006). After development over almost half a century, there are numerous datasets available with BC RIs over the entire solar spectrum to obtain optical properties for radiative applications related to BC (d'Almeida et al., 1991; Krekov, 1993; Hess et al., 1998). More details on the BC RIs have been carefully reviewed and summarized by Sorensen (2001) and Bond and Bergstrom (2006).

Figure 4 compares BC RIs from those cited studies. Most observation-based studies give RIs at some specific wavelengths, at which the observations are carried out, and some fitted results with continuous variations are also given. Both real and imaginary parts of the BC particles show quite wide ranges of variations, and we eliminated results with a real part much larger than 2 and an imaginary part much smaller than 0.5. The real part generally varies between 1.5 and 2.0. The imaginary part shows similar range of variation, and values from 0.5 to 1.1 have been retrieved. Furthermore, none of these datasets show a wavelength-independent RI, and quite different variations over the wavelength are shown in the figure. The real part of RI generally increases as the wavelength becoming larger, whereas the slopes of the variations are quite different. However, both increasing and decreasing trends are noticed for the imaginary part of RI. The figure clearly shows the uncertainties and large variations in BC RI, which brings the most significant challenge in approximating its AAE.

The real and imaginary parts of BC refractive indices from various observations. The blue shadow depicts the ranges considered in this study.

In view of Fig. 4, it is difficult to find a single value to represent BC
RI, whereas it is doable to give a reasonable range of variation for
numerical investigation. Considering the significant uncertainty in
estimated BC RI due to differences on combustion fuels and conditions as
well as whether BC is fresh or aged, we consider both wavelength-independent
(i.e., constant) and wavelength-dependent RIs and introduce two parameters
to indicate the variation in the real and imaginary part as wavelength,
respectively. The real and imaginary part are defined as

where

Fitted values for the parameters

With BC shape, size distribution, and RI known, it becomes straightforward to calculate the corresponding optical properties at a given wavelength, and we only consider bulk properties averaged over a given size distribution in this study. Multiple numerical models are available to account for the light scattering properties of a cluster of spheres, where the individual spheres of the cluster do not overlapping, and the multiple-sphere T-matrix method (MSTD) developed by Mackowski and Mishchenko (2011) is used in this study. The MSTM is a numerically exact model for light scattering by multiple spheres, and is widely used to study the scattering properties of BC particles. Due to the high accuracy and efficiency provided by the MSTM, it becomes convenient to consider the optical properties of BC as aggregates of small spherical monomers. In the framework of the MSTM, the BC particles are rigorously treated as FAs shown in Fig. 3 for optical property simulations, so the errors can only be introduced by uncertainties in the particle microphysical properties, and not the numerical model. Furthermore, the MSTM is also capable of considering the interaction among a large sphere and small ones embedded inside the host, which is also the exact configuration for the coated BC case in this study.

The AAE is widely approximated with the absorptions at two wavelengths using
Eq. (2) (Utry et al., 2014; Li et al., 2016), and BC shows notable
different AAE values over different ranges of the wavelength spectrum. To
obtain the most representative AAE value, we use BC absorptions cross
sections at multiple wavelengths between 0.3 and 1.0

Absorption cross sections of fresh BC with a wavelength-independent
refractive index of 1.8

Figure 6 shows the calculated BC AAE with a wavelength-independent RI of
1.8

AAEs of the fresh, compact and coated BC as a function of the volume equivalent GMD of the BC particles (or BC core for coated BC).

The influences of particle RIs on the AAE are illustrated in Fig. 7, and results for the fresh, compact and coated BC are shown from top to bottom panels. Figure 7 considers only wavelength-independent RIs. The left column is for BC particles with a fixed imaginary part of 0.6 but real parts of 1.6, 1.8, and 2.0, and the right column is for those with the same real part (1.8) but different imaginary parts (0.4, 0.6, 0.8, and 1.0). The BC AAE increases as the real part increases or the imaginary part decreases. Although the imaginary part of RI is most directly related to particle absorption, real and imaginary parts affect BC AAE to similar degrees. Again, the sensitivity of BC AAE to its RIs is quite different for particles with different geometries. The AAEs of the fresh BC are less sensitive to RI and show a difference of < 0.1 for the RIs we considered. As the RI real part increases from 1.6 to 2.0, the AAE of the compact BC increases by approximately 0.15, and the changes reach as large as 0.3 for the BC imaginary part between 0.4 and 1.0. However, after coating, the AAE becomes less sensitive to the RI real part, but more sensitive to the RI imaginary part. This means that, with the BC core totally embedded within the non-absorptive coating, the absorption enhancement of coated BC is more sensitive to the imaginary part of BC RI.

Influence of wavelength-independent refractive index on the AAEs of
the fresh BC

All previous studies used a wavelength-independent RI over the entire
spectrum, which may or may not be realistic for BC particles in the
atmosphere. As explained in Sect. 2.3, parameters

Figure 8 illustrates the impacts of wavelength-dependent RIs on BC AAE, and
three rows correspond to results for the fresh (top), compact
(middle), and coated (bottom) BC.
Both

Influence of wavelength-dependent refractive indices on the AAEs of
the fresh BC

With all previous factors considered, it becomes possible to decompose the
influences of BC properties on AAE. The BC properties show clearly monotonic
influence on the AAE but to varying degrees. BC morphology, the most complex
property parameterized by multiple parameters (e.g., fractal parameters,
coating parameters, and coating
amount), cannot be adequately represented by a single variable, whereas it
plays the most important and complex role in determining BC AAE. Thus, the
morphology is considered independently, and its influence can be considered
with other parameters known. To be more specific, the relationships between
the BC AAE and the properties besides shape are simply approximated with
linear equations for each BC geometry, and, thus, the BC AAE is expressed by

To demonstrate the performance of the simple expresses on approximating BC
AAE, Fig. 9 compares BC AAEs from accurate absorption simulations and
Eq. (6). The left panel of the figure shows the cases in which the
approximations show relatively accurate agreement with the simulations,
whereas some examples with relatively poor agreement are illustrated in
the right panel. It is clear that the results for the fresh BC show the best
agreement because of the relatively weak influence of particle size on the
AAE. The relatively poor agreements for the compact and coated BC are mainly
because of the nonlinear variation of the AAE on the GMD. The differences between the accurate
simulations (solid curved) and the approximation (dashed lines) can reach
slightly over 0.1 for small BC particles. However, considering that the BC
are widely observed or considered to have a GMD larger than
0.10

Our results obtained here indicate that BC microphysical properties have a clear influence on BC AAE, and the influence can be quantitatively understood by Eq. (6) and the coefficients listed in Table 2. Considering the obvious variations obtained for BC particle sizes and geometries and the uncertainties on its RI, it is impossible to find a “best” AAE value for all BC aerosols. However, a range of reasonable AAE values can be obtained based on the observations of BC properties and the numerical results. For fresh BC, the AAEs are approximately 1.05, and aging causes AAE of typically sized BC particles to decrease by approximately 0.15 to 0.90. Furthermore, wavelength-dependent RIs make the case much more complicated and give wider ranges of the BC AAE. For coated BC particles, the absorption can also be significantly affected by chemical composition, amount, and geometry of mixing material (Li et al., 2016), and this study introduces a laboratory-based coating amount distribution to reveal the significant effects of coating on BC AAE. With the help of Eq. (6), the BC AAE can be easily calculated if its properties (shape, size, and RI) are known.

Fitted coefficients to show the sensitivities of BC properties on AAE values. Two significant figures are kept for all coefficients.

Comparison between the AAEs given by accurate numerical simulations (solid curves) and those approximated by Eq. (6) and the corresponding coefficients in Table 2 (dashed curves).

We have numerically investigated the AAE of BC aerosols of three states in the atmosphere, i.e., fresh, compact, and coated ones. The numerical computations conducted here have multiple controllable variables (such as BC size distribution) that all effect BC AAE, and we attempted to constrain these variables within the realistic ranges as determined by observation-based studies. The MSTM was used to accurately compute the light absorption of non-spherical particles, and the numerical results were analyzed to better understand the BC AAE values in relation to the controllable variables.

The results challenge conventional beliefs. With a wavelength-independent refractive index, our accurate numerical results show typical BC AAE values of 1.05 and 0.90, instead of 1.0, for fresh and aged BC particles respectively. In reality as revealed by many observational studies, the BC refractive index likely has sizable wavelength dependence, and BC is often coated by non-BC aerosol materials. In these cases, BC AAE can even move beyond a range of 0.5–1.5. As a result, using a flat value of 1.0 for BC AAE could very likely introduce significant errors in aerosol absorption analysis studies.

Our results demonstrate that BC particle shape is the most influential factor in determining BC AAE. The AAE of fresh BC in the form of lacy aggregate is less sensitive to particle size, whereas, after aging processes, the AAE of BC with compact or coated structures may significantly decreases as particle size increases. As the most uncertain particle property, the refractive index cannot be directly measured and thus becomes the most significant challenge in determining BC AAE. To quantify the complicated influences of different BC parameters on AAE, linear approximations for BC AAEs in different conditions were obtained here. Our results clearly demonstrate the importance of various parameters on the BC AAE and the errors of assuming BC AAE as 1.0. However, caution should be taken in interpreting our results as a comprehensive guide or absolute reference, because the closure studies between numerical models and observations on BC properties can be relatively poor (Bond and Bergstrom, 2006; Radney et al., 2014).

The data obtained from this study are available upon request from Chao Liu (chao_liu@nuist.edu.cn).

The authors declare that they have no conflict of interest.

We are deeply thankful to Daniel W. Mackowski and Michael I. Mishchekno for the MSTM code. The authors also gratefully acknowledge the effort by the two anonymous reviewers and Joel C. Corbin to improve the manuscript. This work was financially supported by the National Key Research and Development Program of China (2016YFA0602003), the Natural Science Foundation of China (41505018 and 91644224), the Natural Science Foundation of Jiangsu Province (BK20150899), the Young Elite Scientists Sponsorship Program by CAST (2017QNRC001), the US NSF (award no. AGS-1455759), and the Helmholtz Research Program Atmosphere and Climate. The computation of this study was supported by the National Supercomputer Center in Guangzhou (NSCC-GZ). Edited by: Ari Laaksonen Reviewed by: three anonymous referees