**Research article**| 22 Nov 2022

# Constraining the particle-scale diversity of black carbon light absorption using a unified framework

Payton Beeler and Rajan K. Chakrabarty

**Payton Beeler and Rajan K. Chakrabarty**Payton Beeler and Rajan K. Chakrabarty

- Center for Aerosol Science and Engineering, Department of Energy, Environmental and Chemical Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA

- Center for Aerosol Science and Engineering, Department of Energy, Environmental and Chemical Engineering, Washington University in St. Louis, St. Louis, MO 63130, USA

**Correspondence**: Rajan K. Chakrabarty (chakrabarty@wustl.edu) and
Payton Beeler (beelerpayton@wustl.edu)

**Correspondence**: Rajan K. Chakrabarty (chakrabarty@wustl.edu) and
Payton Beeler (beelerpayton@wustl.edu)

Received: 07 Apr 2022 – Discussion started: 05 May 2022 – Revised: 15 Sep 2022 – Accepted: 22 Sep 2022 – Published: 22 Nov 2022

Atmospheric black carbon (BC), the strongest absorber of
visible solar radiation in the atmosphere, manifests across a wide spectrum
of morphologies and compositional heterogeneity. Phenomenologically, the
distribution of BC among diverse particles of varied composition gives rise
to enhancement of its light absorption capabilities by over twofold in
comparison to that of nascent or unmixed homogeneous BC. This situation has
challenged the modeling community to consider the full complexity and
diversity of BC on a per-particle basis for accurate estimation of its light
absorption. The conventionally adopted core–shell approximation, although
computationally inexpensive, is inadequate not only in estimating but also
capturing absorption trends for ambient BC. Here we develop a unified
framework that encompasses the complex diversity in BC morphology and
composition using a single metric, the phase shift parameter (*ρ*_{BC}),
which quantifies how much phase shift the incoming light waves encounter
across a particle compared to that in its absence. We systematically
investigate variations in *ρ*_{BC} across the multi-space distribution of
BC morphology, mixing state, mass, and composition as reported by field and
laboratory observations. We find that *ρ*_{BC}*>*1 leads to
decreased absorption by BC, which explains the weaker absorption
enhancements observed in certain regional BC compared to laboratory results
of similar mixing state. We formulate universal scaling laws centered on
*ρ*_{BC} and provide physics-based insights regarding core–shell
approximation overestimating BC light absorption. We conclude by packaging
our framework in an open-source Python application to facilitate
community-level use in future BC-related research. The package has two main
functionalities. The first functionality is for forward problems, wherein
experimentally measured BC mixing state and assumed BC morphology are input,
and the aerosol absorption properties are output. The second functionality
is for inverse problems, wherein experimentally measured BC mixing state and
absorption are input, and the morphology of BC is returned. Further, if
absorption is measured at multiple wavelengths, the package facilitates the
estimation of the imaginary refractive index of coating materials by combining
the forward and inverse procedures. Our framework thus provides a
computationally inexpensive source for calculation of absorption by BC and
can be used to constrain light absorption throughout the atmospheric
lifetime of BC.

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The contribution of aerosols to global radiative forcing remains one of the largest sources of uncertainty in current climate models (Reidmiller et al., 2018). Much of this uncertainty stems from disagreements between the predicted and observed radiative forcing by carbonaceous aerosols (Bond et al., 2013; Gustafsson and Ramanathan, 2016; Boucher et al., 2016). One of the most climatically relevant carbonaceous aerosols is black carbon (BC). Black carbon is widely considered to be a predominate light-absorbing atmospheric constituent (Bond et al., 2013; Bond and Bergstrom, 2006). Despite this, light absorption by BC is still significantly underestimated in current climate models, which stems from incorrect parameterization of BC optical properties (Bond et al., 2013; Gustafsson and Ramanathan, 2016; Boucher et al., 2016). Estimation of BC light absorption is particularly complicated, given that BC is often internally mixed with other species, which manifest as external coatings (China et al., 2013).

External coatings enhance light absorption by BC through a “lensing effect”, in which a portion of the incoming light is scattered by the coating into the BC core, where it is then absorbed (Chakrabarty and Heinson, 2018; Cappa et al., 2012; Peng et al., 2016; Saliba et al., 2016; Liu et al., 2017; Shiraiwa et al., 2010). However, previous studies on light absorption enhancement due to the lensing effect have had various results. Some studies find high absorption enhancement (up to a factor of 2.5), while others find little to no absorption enhancement with increasing coating amount (Cappa et al., 2012; Saliba et al., 2016; Shiraiwa et al., 2010; Liu et al., 2015; Cappa et al., 2019; Zhang et al., 2018; Denjean et al., 2020; Zanatta et al., 2018; Xie et al., 2019; Cui et al., 2016). The range of absorption enhancement from previous studies is evident in Fig. 1. Fierce et al. (2020) found that particle-to-particle heterogeneity reconciles a large portion of the observed discrepancies in light absorption enhancement (Fierce et al., 2020). However, even when particle-to-particle heterogeneity is considered, light absorption enhancement is still overestimated, and previous discrepancies cannot be fully resolved. Fierce et al. (2020) have also shown that representation of the complex morphology of BC further improves estimation of its optical properties, but systematic understanding of the effect of BC morphology on light absorption enhancement is understudied.

Here, we take a two-pronged approach to develop a simple yet rigorous unified framework for parameterizing the effects of particle size, morphology, and mixing state on BC light absorption. The first approach involves reducing the aforementioned multivariate space to a single parameter that captures causal relationships between BC's physiochemical properties and corresponding light absorption. Using this parameter, we next develop universal scaling laws for wavelength-dependent BC light absorption as a function of size, morphology, and mixing state. Previously, one would need to make assumptions regarding the morphology of BC, then select an appropriate model to calculate its light absorption properties. Our model is the first to allow for quick calculation of BC optical properties with any morphology. We validate these laws against observational datasets from 11 field campaigns which investigated global trends in BC absorption, as well as laboratory experiments that investigated light absorption enhancement. From the standpoint of practical applications of our framework, we package our scaling laws into open-source Python software which allows researchers to use our results to estimate absorption of BC aerosols based on their size, morphology, and mixing state, as well as to estimate the morphology of BC aerosols based on their size, absorption, and mixing state.

## 2.1 Representation of diverse black carbon morphologies

Black carbon is often modeled assuming a spherical core–shell configuration. However, soon after emission, BC aggregates have been found to have a lacy, fractal-like structure. Surface tension and capillary forces from the buildup of external coatings can cause BC aggregates to collapse and eventually take on a more spherical structure (China et al., 2013; Liu et al., 2017; Fierce et al., 2020; Wang et al., 2017). Recent studies have found that nonsphericity of BC-containing particles (partial encapsulation of BC) can decrease absorption enhancement (Hu et al., 2021, 2022). While these findings are notable, previous studies have not observed a prevalence of partially encapsulated BC, yet decreased light absorption enhancement is still observed (China et al., 2013; Fierce et al., 2020). Therefore, this study is focused on investigating the effects of core restructuring on light absorption enhancement, rather than the effects of partial BC encapsulation.

To model the evolution of BC morphology, we utilize three aggregation models
which represent fresh, partially collapsed, and fully collapsed BC
aggregates. Fresh BC aggregates were created using an off-lattice
diffusion-limited cluster–cluster aggregation model, which has been shown to
accurately represent BC aggregates produced by combustion systems, and have
a fractal dimension (*D*_{f}) of 1.83 ± 0.09
(Meakin, 1983, 1987). Partially and fully collapsed
BC aggregates were respectively simulated with a percolation model and
simple cubic lattice stacking and have *D*_{f} of 2.11 ± 0.22 and
3.0. These particles resemble electron microscope images of moderately and
heavily coated BC, respectively (Fierce et al., 2020). Each
simulated BC particle is comprised of monomers with a radius equal to 20 nm
(Bond et
al., 2013). The amount of coating was quantified by the ratio of coating
mass to BC mass (*R*_{BC}). Under this definition, increased *R*_{BC}
represents increasing coating amount, and *R*_{BC}=0 represents pure BC.
The masses of the BC core and the coating material were determined per their
volume and densities of 1.8 and 1.2 g cm^{−3}, respectively
(Bond and Bergstrom, 2006). This study utilized 345 aggregates,
with BC masses between ∼ 1 and ∼ 70 fg, gyration radius between
∼ 50 and ∼ 300 nm, and *R*_{BC} between 0 and 49. Figure 2
shows examples of simulated aggregates, which represent the range of
observed BC morphology.

## 2.2 Calculation of optical properties

The optical properties of the generated aggregates were calculated using the
Amsterdam Discrete Dipole Approximation (ADDA 1.3b4) algorithm
(Yurkin and Hoekstra, 2011). The ADDA
algorithm operates by breaking complex shapes into subvolumes
small enough compared to the wavelength of light to be treated as point
scatterers which interact with surrounding point scatterers. It is
recommended that the wavelength of light be at least 10 times the size of
individual subvolumes for accurate calculation of optical
properties. For this study, the wavelength of light was approximately 100
times the size of individual subvolumes in order to reduce errors in
calculation of optical properties. The ADDA algorithm calculated the
absorption cross-section of each aggregate, which was then divided by the
mass of the BC core to give the mass absorption cross-section (MAC_{BC}) of
each aggregate. Much of the previous work which investigates light
absorption by internally mixed BC measures absorption enhancement
(*E*_{abs}). Absorption enhancement is commonly defined as absorption by
internally mixed BC divided by absorption by pure BC (Fierce et
al., 2020). There are three common methods for estimating light absorption
by pure BC: direct measurement (using thermodenuders to remove coating
material), extrapolation of best-fit lines of light absorption by internally
mixed BC, and using literature values. All of these methods have challenges
which can ultimately affect the reported value of *E*_{abs}. It has been
found that thermodenuders may not remove low-volatility coating material,
which leads to overestimation of light absorption by pure BC and
underestimation of *E*_{abs} (Shetty et al., 2021). Extrapolation
of absorption measurements by internally mixed BC either assumes that the
morphology of BC does not affect light absorption or that the morphology of
BC remains fixed as coating accumulates. Finally, use of literature values
to approximate light absorption by pure BC assumes that light absorption by
fractal aggregates is equivalent to literature values of absorption by bulk
BC. The Rayleigh–Debye–Gans approximation of light absorption by fractal BC is
significantly lower than commonly used literature values of absorption by
pure BC (Bond and Bergstrom, 2006; Sorensen, 2001), indicating
that use of literature values can also underestimate *E*_{abs}.

In order to avoid the errors associated with measurement of *E*_{abs}, we
instead focus our efforts on quantification of MAC_{BC}. Absorption
cross-section per BC mass is a common input of radiative transfer
algorithms and is vital in converting BC mass concentration to absorption
coefficient
(Bond et
al., 2013). Accurate scaling of MAC_{BC} as a function of aggregate size,
morphology, and mixing state will allow for subsequent calculation of
*E*_{abs}, which accounts for the evolution of BC morphology throughout its
atmospheric lifetime.

## 2.3 Phase shift parameter is a unifying measure of size, morphology, and composition

Previous studies of *E*_{abs} have focused on the effects of a single
dependent variable (absorption) as a function of a single independent
variable (mixing state). However, detailed representation of the
microphysical properties of BC leads to the introduction of several other
measures which describe the size and morphology of BC, increasing the size
of the variable set from two (absorption and mixing state) to four (size,
morphology, absorption, and mixing state).

To reduce the size of the variable set, we utilize the phase shift parameter
(*ρ*), which is a unifying measure of both aggregate size and morphology.
Physically, *ρ* describes the amount of phase shift that light accumulates
when passing through a particle (Heinson and Chakrabarty,
2016; Sorensen and Fischbach, 2000). When *ρ* is less than 1, there is not
a significant amount of phase shift in the incident wave, and the
particle–light interactions are well described by Rayleigh approximations.
Conversely, when *ρ* is greater than 1, the particle–light interactions are
well described by geometric optics (Sorensen and Fischbach, 2000). In
this work, *ρ* is used to describe the size and morphology of the BC core,
not the entire particle (BC core + coating). Therefore, in the remaining
text we refer to the core phase shift parameter (*ρ*_{BC}) to distinguish
from the phase shift parameter of the entire particle. The core phase shift
parameter is given by (Debye, 1958)

where *ρ* is the wavelength of incident light, *R*_{g} is the particle radius
of gyration (size metric), and *m*_{eff} is the effective complex index of
refraction, which in turn is given by

Here, *ϕ* is the BC monomer packing fraction and *m* is the BC complex
refractive index. The BC refractive index is fixed at 1.95 + 0.79*i* at all
wavelengths (Bond and Bergstrom, 2006). The BC monomer packing
fraction was calculated as the volume of BC which lies within a sphere of
radius *R*_{g} (centered at the center of mass) divided by the volume of a
sphere with radius *R*_{g}. It is important to note that all parameters used
in Eqs. (1) and (2) describe the BC core, not the entire particle. It is
also important to note that the dynamic compositional changes which a BC
particle undergoes during atmospheric processing are captured by
*m*_{eff} in Eq. (2) (Heinson et al., 2017). As coating
accumulates on the surface of aggregates, the BC core will begin to collapse
due to surface tension and capillary forces, and *ρ* will increase. This will
affect *m*_{eff} and eventually *ρ*_{BC}. The aggregates in this study have
*ϕ* between 0.029 and 0.52, representing the range observed in coated
aggregates (Zangmeister et al., 2014; Chen et al., 2018). In
general, for aggregates with an equal size parameter, higher *ϕ* leads to higher
*ρ*_{BC}. A plot of *ρ*_{BC} normalized by size parameter can be found in
Supplement Fig. S1.

The consequence of increased *ρ*_{BC} for light absorption can be seen in
Fig. 2b, d, and f, which show internal fields of the BC aggregates
shown in Fig. 1a, c, and e. For fresh aggregates (with *ρ*_{BC}≪1), light is able to fully illuminate the aggregate, and
the entire volume contributes to light absorption. However, for fully
collapsed aggregates (with *ρ*_{BC}*>*1), light is not able to
illuminate the far interior of the particle, leading to areas of decreased
light absorption. Therefore, if *ρ*_{BC} of a particle significantly
increases, its light absorption properties will change significantly. It
should be noted that since *ρ*_{BC} is a function of both the morphology and
size of aggregates, full core collapse will not always lead to *ρ*_{BC}*>*1. Aggregates with a small number of monomers may never achieve
*ρ*_{BC}*>*1, even when the monomer packing fraction reaches
unity.

## 3.1 Sensitivity of MAC_{BC} to coating refractive index

To examine the effects of coating refractive index, we calculate MAC_{BC}
of 30 randomly selected BC aggregates with a coating real refractive index
(*n*_{coat}) of 1.45, 1.55, and 1.65 and a coating imaginary refractive index
(*κ*_{coat}) of 0.00, 0.05, and 0.1. Figure 3 shows the partial derivative
of MAC_{BC} with respect to *n*_{coat} (∂MAC${}_{\mathrm{BC}}/\partial {n}_{\mathrm{coat}}$) and with
respect to *κ*_{coat} (∂MAC${}_{\mathrm{BC}}/\partial {\mathit{\kappa}}_{\mathrm{coat}}$), with constant *ϕ*. We
find that ∂MAC${}_{\mathrm{BC}}/\partial {\mathit{\kappa}}_{\mathrm{coat}}$ is always greater than
∂MAC${}_{\mathrm{BC}}/\partial {n}_{\mathrm{coat}}$ and increases with increased *R*_{BC}. These
results show that the choice of *κ*_{coat} is more important than the
choice of *n*_{coat} when calculating MAC_{BC}. Given this, we further
investigate scaling of MAC_{BC} with *κ*_{coat} between 0.00 and 0.05,
but *n*_{coat} remained fixed at 1.55 (Bond and Bergstrom,
2006). In the context of field and laboratory measurements, particles with
*κ*_{coat}=0.00 are representative of BC which is internally mixed with
nonrefractory material, such as *α*-pinene secondary organic aerosol and
sulfuric acid (Fierce et al., 2020). Particles with *κ*_{coat}*>*0.00 are representative of BC which is internally mixed with
absorbing material, such as brown carbon
(Liu et al., 2015; Lu et al., 2015).

## 3.2 Phase shift parameter controls light absorption

Figure 4 shows MAC_{BC} as a function of *ρ*_{BC} and *R*_{BC} for an
incident wavelength (*λ*) of 532 nm. Figure 4b, d, and f show the
clear emergence of two regimes separated by *ρ*_{BC}=1 (dashed line).
For *ρ*_{BC}≤1, MAC_{BC} increases with increased *R*_{BC} but is
independent of *ρ*_{BC}. For *ρ*_{BC}*>*1, MAC_{BC} decreases
with increased *ρ*_{BC}, and the rate of decrease is dependent on *R*_{BC}.
The finding of decreased light absorption for *ρ*_{BC}*>*1 is
consistent with a recent study which also found decreased MAC_{BC} with
increasing aggregate size (Romshoo et al., 2021). Best-fit lines
for the scaling of MAC_{BC} as a function of *R*_{BC} are shown as solid
lines in Fig. 4 and are summarized by

where *A*, *B*, and *C* are constants, and
*κ*_{coat} is the imaginary part of the coating refractive index. The
fitting parameters *D* and *E* are functions of *R*_{BC}, given by

where *X* generically represents *D* or *E*, and ${x}_{[\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}]}$ denotes
${d}_{[\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}]}$ or ${e}_{[\mathrm{1},\mathrm{2},\mathrm{3},\mathrm{4}]}$.

The boundary condition for Eq. (3a) is that for RBC = 0, MAC_{BC} = MAC_{0}($\mathit{\lambda}/{\mathit{\lambda}}_{\mathrm{0}}$)^{−AAE}, where
MAC_{0} is the average MACBC for uncoated aggregates with *ρ*_{BC}≤1 (6.8 m^{2} g^{−1}), and AAE is the
absorption Ångström exponent for pure BC (1.158). The boundary condition for Eq. (3b) is
that at *ρ*_{BC}=1, MAC_{BC} must be equal to MAC_{BC} calculated using Eq. (3a).

The value of all constants used in Eqs. (3) and (4) can be found in Table 1. Details regarding the acquisition of AAE can be found in Supplement Fig. S2.

We find AAE, which is consistent with previously reported values
(Bond et
al., 2013; Romshoo et al., 2021), and fitting parameter B, which is
consistent with a previous numerical study of coated BC aggregates with
*ρ*_{BC}≤1 (Chakrabarty and Heinson, 2018). The
value of MAC_{0} is less than commonly used literature values of pure BC
(7.75–8.0 m^{2} g^{−1})
(Bond et
al., 2013; Liu et al., 2020) but slightly greater than the Rayleigh–Debye–Gans
approximation for MAC_{BC}, which accounts for the fractal morphology of
BC (5.01 m^{2} g^{−1}) (Sorensen, 2001). Figure 5a shows residual
plots for the fitting of MAC_{BC} using Eq. (3). On average, Eq. (3)
overestimates MAC_{BC} at *λ* = 532 nm by 0.47 %, with a standard
deviation of 8.26 %. Generally, relative errors increase as *R*_{BC}
increases, as shown by Fig. 5b. It should also be noted that for thickly
coated aggregates with *κ*_{coat}*>*0.00, the scaling laws
given in this work overestimate MAC_{BC} for aggregates with large
*ρ*_{BC} and slightly underestimate MAC_{BC} for aggregates with
*ρ*_{BC}*<*1. Underestimation of MAC_{BC} for large *ρ*_{BC} is
likely due to accumulation of phase shift in the incident light as it passes
through the coating material. This finding indicates that the scaling laws
given by Eq. (3) are valid only when it can be assumed that the phase
shift as light passes through coating materials is negligible.

## 3.3 Wavelength dependency and limitations of core–shell Mie theory

Coated BC is conventionally modeled with a core–shell morphology using Mie
theory to calculate its light absorption properties (Bond and
Bergstrom, 2006). Our results indicate that misrepresentation of BC
morphology will inevitably lead to errors in calculation of its light-absorbing properties. To highlight this point, Fig. 6 compares the
accuracy of Eq. (3) in calculating MAC_{BC} of 15 randomly selected
aggregates with *λ* = 405, 532, 880, and 1200 nm to Mie theory
calculations for mass-equivalent spheres (with *κ*_{coat}=0.00). Figure 6 shows that across wavelengths, Eq. (3) accurately calculates
MAC_{BC}, with an average error of 9.08 ± 10.94 %. Figure 6 also shows
that Mie theory is inconsistent in calculating MAC_{BC} for aggregates
with fractal morphologies (*D*_{f}≠3.0). Figure 6b shows that Mie
theory overestimates MAC_{BC} at long wavelengths and underestimates
MAC_{BC} at shorter wavelengths. These results are consistent with
previous findings that Mie theory overestimates absorption by BC, given that
BC absorption is commonly measured at longer wavelengths to avoid absorption
by organic coatings
(Cappa et al.,
2019; Fierce et al., 2020). We also find that the accuracy of Mie theory
improves significantly as *D*_{f} approaches 3, which is analogous to the
morphology of BC approaching that of a sphere. Conversely, Eq. (3) is
more consistent in calculating MAC_{BC} with any morphology. It should be
noted that the development of Eq. (3) involved only data points for *λ* = 532 nm. However, previous work has shown that enhancement of MAC_{BC}
is independent of *ρ* (Chakrabarty and Heinson, 2018),
indicating that results obtained for *λ* = 532 nm are applicable to other
wavelengths. Therefore, Fig. 6c–d also show the utility of Eq. (3)
in calculating MAC_{BC} across wavelengths.

## 3.4 Validation of scaling laws with field and laboratory observations

Figure 7a shows the scaling of MAC_{BC} with *R*_{BC} for aggregates
with *ρ*_{BC}≤1, along with data from studies which find significant
increases in MAC_{BC} with increasing *R*_{BC}
(Yu
et al., 2019; Saliba et al., 2016; Liu et al., 2015; Xie et al., 2019;
Denjean et al., 2020; Zanatta et al., 2018). We find that MAC_{BC} from
these studies closely matches the behavior of Eq. (3a), indicating that
these studies were measuring light absorption properties of aggregates with
*ρ*_{BC}≤1. It has been hypothesized that large values of MAC_{BC}
in these studies could be the result of absorbing coatings. However, we find
that the data from these studies closely match scaling of MAC_{BC} with
*ρ*_{coat} fixed at 0.00. The assumption that *ρ*_{coat}=0.00 is
bolstered by the fact that refractory organics absorb preferentially at
ultraviolet wavelengths
(Chakrabarty
et al., 2010; Sumlin et al., 2018; Kirchstetter et al., 2004; Sengupta et
al., 2018; Shamjad et al., 2018), and we have only included data from
visible and near-infrared wavelengths.

Figure 7b shows a table of previous studies which find little to no
increase in MAC_{BC} with increasing *R*_{BC} and the corresponding
average *ρ*_{BC} which replicates the measured MAC_{BC}. The average
*ρ*_{BC} was found by solving Eq. (3b), inserting each measured *R*_{BC}
and corresponding MAC_{BC}. The importance of *ρ*_{BC} in the estimation of
MAC_{BC} is evident when comparing experimentally measured MAC_{BC} for the
different studies shown in Table 2. For example, Cappa et al. (2019) sampled coated
BC aggregates in Fontana and Fresno, California, and find little to no
increase in MAC_{BC}, even with *R*_{BC}*>*10
(Cappa et al.,
2019). They postulate that the low value of MAC_{BC} is due to unequal
distribution of coating material between BC particles. Separate studies have
shown that uneven distribution of coating can cause decreased MAC_{BC},
but thorough consideration of heterogeneous coating amounts fails to fully
explain low MAC_{BC} observed in the field (Fierce et al., 2020).
Therefore, our results suggest that elevated *ρ*_{BC} due to core
restructuring may be partially responsible for low MAC_{BC} observed by
Cappa et al. (2019), further highlighting the importance of the diversity of BC
morphology and mixing state in estimation of its light absorption properties.

## 3.5 Applications of the developed framework

The scaling laws given in this work allow experimentalists to carry out two
procedures. The first is the forward procedure, wherein
experimentally measured BC mass, mixing state, and coating refractive index
are combined with assumed BC morphology and MAC_{BC} is calculated. The
second is the inverse procedure, wherein experimentally measured BC mass,
mixing state, and MAC_{BC} are inputs and BC morphology is output.
Further, the inverse and forward procedures can be combined to estimate
*κ*_{coat}. We have developed an open-source Python package, called the
“Python BC absorption package” (pyBCabs), which performs the forward and
inverse functions. The following sections provide a brief overview of
pyBCabs, as well as examples of inverse problems and estimation of
*κ*_{coat}. Further details regarding the functionality of the package, as
well as more examples of forward and inverse problems for single BC
particles and distributions of BC particles, can be found at https://pybcabs.readthedocs.io/en/latest/index.html, last access: 28 April 2022.

### 3.5.1 Forward procedure for light absorption properties

In the forward procedure, experimentally measured *R*_{BC} and single-particle BC masses are first combined with the assumed morphology of BC and
*ρ*_{coat}. Then, *ρ*_{BC} is calculated using Eqs. (1) and (2) based on
the assumed morphology and BC mass. Finally, MAC_{BC} is calculated using
Eqs. (3a) or (3b). A flowchart of the forward procedure is shown by the
dashed lines in Fig. 8a. As an example, a fresh BC particle with
mass-equivalent diameter of 300 nm and *R*_{BC}=3.68 is input to the
forward procedure along with the wavelength of interest (405 nm), and
MAC_{BC}=16.703 m^{2} g^{−1} is output. A BC particle with the same
characteristics of the previous example but with a fully collapsed BC core
would have MAC_{BC} of 11.46 m^{2} g^{−1}. This example demonstrates the
utility of the developed framework in evaluating changes in MAC_{BC} as
coating-induced restructuring occurs during atmospheric processing.

### 3.5.2 Inverse procedure for morphology retrieval

In the inverse procedure, experimentally measured *R*_{BC} and *ρ*_{coat}
are first input to Eq. (3a), and MAC_{BC} is calculated. If the
calculated MAC_{BC} replicates the measured MAC_{BC}, then it can be
concluded that the measured BC has *ρ*_{BC}≤1, but the exact
*ρ*_{BC} cannot be determined. If the measured MAC_{BC} is much less than
that predicted by Eq. (3a), then Fig. 8b can be used to estimate
*ρ*_{BC}. Alternatively, Eq. (3b) can be used to calculate *ρ*_{BC}
directly. Finally, the single-particle BC mass and *ρ*_{BC} are combined
with Fig. 8c to give insight into how much restructuring the BC core has
undergone. A flowchart of the inverse procedure is shown by the solid lines
in Fig. 8a. The inverse procedure has been carried out for seven previous
studies
(Cappa
et al., 2012; Saliba et al., 2016; Shiraiwa et al., 2010; Liu et al., 2015;
Zhang et al., 2018; Denjean et al., 2020; Zanatta et al., 2018), and the
results are shown in Fig. 8c. Our results indicate that studies which
find little to no increase in MAC_{BC} with increased *R*_{BC} may be
measuring BC aggregates which have undergone significant coating-induced
restructuring, leading to *ρ*_{BC}*>*1, and may also be
measuring particles which have significant heterogeneity in *R*_{BC}. On the
other hand, studies that find significant increases in MAC_{BC} may be
measuring aggregates which have *ρ*_{BC}*<*1. This does not imply
that these studies are measuring BC which has not been restructured, only that
the product of the size parameter and core packing fraction of BC is not
large enough such that *ρ*_{BC}*>*1.

### 3.5.3 Inverse procedure for coating refractive index retrieval

The inverse and forward procedures can be combined to estimate *ρ*_{coat}
if MAC_{BC} is measured at multiple wavelengths. To accomplish this,
*ρ*_{BC} is first found using the inverse procedure outlined above using
MAC_{BC} measured at a near-infrared wavelength (where *ρ*_{coat} can be
estimated as 0.00). Then, *ρ*_{BC}, *R*_{BC}, and MAC_{BC} can be used to
solve for *ρ*_{coat} at near-ultraviolet and visible wavelengths. This
procedure is outlined in Fig. 9a and has been carried out for the Liu et
al. (2015) study, which measured absorption enhancement for BC which was
internally mixed with absorbing organics (Liu
et al., 2015). We estimate that for this study, *κ*_{coat}=0.056 at
*λ* = 405 nm. Figure 9b shows data collected by Liu et al. (2015) at *λ* = 781 nm
(red points) and *ρ* = 405 nm (blue points). The solid lines show
MAC_{BC} calculated using Eq. (3), inserting the appropriate *λ* and
*κ*_{coat}. Our estimation of *κ*_{coat} is slightly greater than the
reported *κ*_{coat} in Liu et al. (2015). However, Liu et al. (2015) approximated *κ*_{coat} using the
Rayleigh–Debye–Gans approximations, not direct measurement
(Liu et al., 2015). Additionally, our
estimation of *κ*_{coat} is consistent with previous studies of the
refractive index of absorbing organics
(Chakrabarty
et al., 2010; Sumlin et al., 2018; Kirchstetter et al., 2004; Sengupta et
al., 2018; Shamjad et al., 2018).

This study comprehensively investigates the effect of BC morphology on light
absorption, introduces *ρ*_{BC} as a central parameter in accurate
estimation of MAC_{BC}, and develops improved scaling laws for MAC_{BC}.
We find that for aggregates with *ρ*_{BC}≤1, MAC_{BC} increases
with increasing *R*_{BC}. For aggregates with *ρ*_{BC}*>*1,
MAC_{BC} is a function of *R*_{BC} and *ρ*_{BC}. Our work also shows that
as *ρ*_{BC} increases past unity, MAC_{BC} decreases. We then provide a
comparison of the scaling laws presented in this work with Mie theory
calculations for mass-equivalent spheres. We find that Mie theory
consistently overestimates MAC_{BC} of internally mixed BC with *ρ*_{BC}*>*1, which is consistent with previous studies which also find
that Mie theory greatly overestimates absorption by BC
(Cappa
et al., 2012, 2019; Fierce et al., 2020). The scaling laws presented in this
work account for the microphysical properties of BC and provide a new tool
for estimating BC light absorption based on BC morphology.

Finally, we validate our findings with data from 11 previous studies which
measure light absorption enhancement
(Yu
et al., 2019; Cappa et al., 2012; Saliba et al., 2016; Shiraiwa et al.,
2010; Liu et al., 2015; Cappa et al., 2019; Zhang et al., 2018; Xie et al.,
2019; Denjean et al., 2020; Zanatta et al., 2018; Cui et al., 2016). We find
that studies which find significant absorption enhancement with increasing
*R*_{BC} agree well with our scaling laws for BC with *ρ*_{BC}≤1
(Saliba
et al., 2016; Liu et al., 2015; Denjean et al., 2020; Zanatta et al., 2018;
Xie et al., 2019; Yu et al., 2019). We also find that *ρ*_{BC}*>*1 is a possible explanation for studies which find little to no absorption
enhancement
(Cappa
et al., 2012; Shiraiwa et al., 2010; Cappa et al., 2019; Zhang et al., 2018;
Cui et al., 2016). These findings are significant because coating-induced
restructuring of the BC core will lead to increases in the core packing
fraction and consequent increases in *ρ*_{BC}. Our findings suggest that
restructuring of the BC core and increased *ρ*_{BC} can lead to decreased
absorption and may play a role in previous discrepancies in measured
MAC_{BC}. Previous work has shown that heterogeneity in BC mixing state
accounts for a large portion of the discrepancies in measured and modeled
BC but does not fully reconcile previous discrepancies in BC absorption.
Our study shows that particle-resolved mixing state and detailed
representation of BC morphology are both necessary in order to fully
parameterize absorption by internally mixed BC.

In order to make the results of this study readily available to
experimentalists, we conclude by providing an open-source Python module, the
“Python BC absorption package” (pyBCabs). This package has two
functionalities. The first functionality is for forward problems, wherein BC
mass and *R*_{BC} of ambient and laboratory-generated BC are input, and
MAC_{BC} is returned. The second functionality is for inverse problems,
wherein BC mass, *R*_{BC}, and MAC_{BC} of ambient and laboratory-generated
BC are input, and the morphology of BC is returned. The forward and inverse
functionalities can also be combined to estimate the imaginary part of the
coating refractive index if MAC_{BC} is measured at multiple wavelengths.

The inverse functionality of this module allows for in situ inference of BC morphology, as opposed to ex situ methods of determining BC morphology, such as electron microscopy. Use of the inverse functionality of pyBCabs will allow for more detailed studies on the evolution of BC morphology during its atmospheric lifetime. Improved representation of BC morphology, as well as the improved scaling laws developed by this study, can then be incorporated into radiative transfer models and eventually aid in reducing the uncertainty of radiative forcing by carbonaceous aerosols.

All data from ADDA calculations are available for download at https://github.com/beelerpayton/ADDA_datasets (https://doi.org/10.5281/zenodo.7255194, Beeler and Chakrabarty, 2022a). Full details regarding the functionality of the developed Python package can be found at https://pybcabs.readthedocs.io/en/latest/index.html (last access: 26 October 2022, Beeler and Chakrabarty, 2022b).

The Supplement includes methods for converting absorption enhancement from previous field and laboratory studies to a mass absorption cross-section. It also includes two figures showing examples of a modeled partially collapsed BC particle (S1) and data used for calculation of the absorption Ångström exponent (S2). The supplement related to this article is available online at: https://doi.org/10.5194/acp-22-14825-2022-supplement.

RKC and PB conceived of the study and its design. RKC provided guidance and supervision for carrying out the research tasks and interpretation of results, as well as contributing to the preparation of the paper. PB performed the data analysis, developed the figures, and led the preparation of the paper. PB and RKC were involved in the editing and proofreading of the paper.

The contact author has declared that none of the authors has any competing interests.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the US National Science Foundation (AGS-1926817), the NASA ACCDAM program (NNH20ZDA001N), and the US Department of Energy (DE-SC0021011). Funding for collecting data during the 2010 CARES field campaign in California was provided by the Atmospheric Radiation Measurement (ARM) Program sponsored by the US Department of Energy (DOE), Office of Biological and Environmental Research (OBER). The authors thank William Heinson for insightful comments and assisting with getting this project off the ground.

This research has been supported by the US Department of Energy (grant no. DE-SC0021011) and the National Science Foundation (grant no. AGS-1926817).

This paper was edited by Dantong Liu and reviewed by three anonymous referees.

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