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 (Df) 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 Df 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 (RBC). Under this definition, increased RBC represents increasing coating amount, and RBC=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 RBC between 0 and 49. Figure 2 shows examples of simulated aggregates, which represent the range of observed BC morphology.

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 (MACBC) of each aggregate. Much of the previous work which investigates light absorption by internally mixed BC measures absorption enhancement (Eabs). 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 Eabs. 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 Eabs (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 Eabs.

In order to avoid the errors associated with measurement of Eabs, we instead focus our efforts on quantification of MACBC. 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 MACBC as a function of aggregate size, morphology, and mixing state will allow for subsequent calculation of Eabs, which accounts for the evolution of BC morphology throughout its atmospheric lifetime.

Previous studies of Eabs 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 (ρBC) to distinguish from the phase shift parameter of the entire particle. The core phase shift parameter is given by (Debye, 1958) 1ρBC=4πRgλmeff-1, where ρ is the wavelength of incident light, Rg is the particle radius of gyration (size metric), and meff is the effective complex index of refraction, which in turn is given by 2ϕm2-1m2+2=meff2-1meff2+2. 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 + 0.79i 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 Rg (centered at the center of mass) divided by the volume of a sphere with radius Rg. 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 meff 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 meff and eventually ρBC. 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 ρBC. A plot of ρBC normalized by size parameter can be found in Supplement Fig. S1.

The consequence of increased ρBC 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 ρBC≪1), light is able to fully illuminate the aggregate, and the entire volume contributes to light absorption. However, for fully collapsed aggregates (with ρBC>1), light is not able to illuminate the far interior of the particle, leading to areas of decreased light absorption. Therefore, if ρBC of a particle significantly increases, its light absorption properties will change significantly. It should be noted that since ρBC is a function of both the morphology and size of aggregates, full core collapse will not always lead to ρBC>1. Aggregates with a small number of monomers may never achieve ρBC>1, even when the monomer packing fraction reaches unity.

To examine the effects of coating refractive index, we calculate MACBC of 30 randomly selected BC aggregates with a coating real refractive index (ncoat) of 1.45, 1.55, and 1.65 and a coating imaginary refractive index (κcoat) of 0.00, 0.05, and 0.1. Figure 3 shows the partial derivative of MACBC with respect to ncoat (∂MACBC/∂ncoat) and with respect to κcoat (∂MACBC/∂κcoat), with constant ϕ. We find that ∂MACBC/∂κcoat is always greater than ∂MACBC/∂ncoat and increases with increased RBC. These results show that the choice of κcoat is more important than the choice of ncoat when calculating MACBC. Given this, we further investigate scaling of MACBC with κcoat between 0.00 and 0.05, but ncoat remained fixed at 1.55 (Bond and Bergstrom, 2006). In the context of field and laboratory measurements, particles with κcoat=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 κcoat>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).

Figure 4 shows MACBC as a function of ρBC and RBC for an incident wavelength (λ) of 532 nm. Figure 4b, d, and f show the clear emergence of two regimes separated by ρBC=1 (dashed line). For ρBC≤1, MACBC increases with increased RBC but is independent of ρBC. For ρBC>1, MACBC decreases with increased ρBC, and the rate of decrease is dependent on RBC. The finding of decreased light absorption for ρBC>1 is consistent with a recent study which also found decreased MACBC with increasing aggregate size (Romshoo et al., 2021). Best-fit lines for the scaling of MACBC as a function of RBC are shown as solid lines in Fig. 4 and are summarized by ∂MACBC∂RBC=ARBC+1Bexp⁡-CRBC+1+3κcoat,ρBC≤1(3a)D1ρBC2E-1ρBCE,ρBC>1(3b) where A, B, and C are constants, and κcoat is the imaginary part of the coating refractive index. The fitting parameters D and E are functions of RBC, given by 4X=x1+x2-x11+exp⁡x3RBC-x4, where X generically represents D or E, and x[1,2,3,4] denotes d[1,2,3,4] or e[1,2,3,4].

The boundary condition for Eq. (3a) is that for RBC = 0, MACBC = MAC0(λ/λ0)-AAE, where MAC0 is the average MACBC for uncoated aggregates with ρBC≤1 (6.8 m2 g-1), and AAE is the absorption Ångström exponent for pure BC (1.158). The boundary condition for Eq. (3b) is that at ρBC=1, MACBC must be equal to MACBC 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 ρBC≤1 (Chakrabarty and Heinson, 2018). The value of MAC0 is less than commonly used literature values of pure BC (7.75–8.0 m2 g-1) (Bond et al., 2013; Liu et al., 2020) but slightly greater than the Rayleigh–Debye–Gans approximation for MACBC, which accounts for the fractal morphology of BC (5.01 m2 g-1) (Sorensen, 2001). Figure 5a shows residual plots for the fitting of MACBC using Eq. (3). On average, Eq. (3) overestimates MACBC at λ = 532 nm by 0.47 %, with a standard deviation of 8.26 %. Generally, relative errors increase as RBC increases, as shown by Fig. 5b. It should also be noted that for thickly coated aggregates with κcoat>0.00, the scaling laws given in this work overestimate MACBC for aggregates with large ρBC and slightly underestimate MACBC for aggregates with ρBC<1. Underestimation of MACBC for large ρBC 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.

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 MACBC of 15 randomly selected aggregates with λ = 405, 532, 880, and 1200 nm to Mie theory calculations for mass-equivalent spheres (with κcoat=0.00). Figure 6 shows that across wavelengths, Eq. (3) accurately calculates MACBC, with an average error of 9.08 ± 10.94 %. Figure 6 also shows that Mie theory is inconsistent in calculating MACBC for aggregates with fractal morphologies (Df≠3.0). Figure 6b shows that Mie theory overestimates MACBC at long wavelengths and underestimates MACBC 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 Df approaches 3, which is analogous to the morphology of BC approaching that of a sphere. Conversely, Eq. (3) is more consistent in calculating MACBC 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 MACBC is independent of λ (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 MACBC across wavelengths.

Figure 7 shows the scaling of MACBC with RBC for aggregates with ρBC≤1, along with data from studies which find significant increases in MACBC with increasing RBC (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 MACBC from these studies closely matches the behavior of Eq. (3a), indicating that these studies were measuring light absorption properties of aggregates with ρBC≤1. It has been hypothesized that large values of MACBC in these studies could be the result of absorbing coatings. However, we find that the data from these studies closely match scaling of MACBC with κcoat fixed at 0.00. The assumption that κcoat=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.

Table 2 shows previous studies which find little to no increase in MACBC with increasing RBC and the corresponding average ρBC which replicates the measured MACBC. The average ρBC was found by solving Eq. (3b), inserting each measured RBC and corresponding MACBC. The importance of ρBC in the estimation of MACBC is evident when comparing experimentally measured MACBC 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 MACBC, even with RBC>10 (Cappa et al., 2019). They postulate that the low value of MACBC is due to unequal distribution of coating material between BC particles. Separate studies have shown that uneven distribution of coating can cause decreased MACBC, but thorough consideration of heterogeneous coating amounts fails to fully explain low MACBC observed in the field (Fierce et al., 2020). Therefore, our results suggest that elevated ρBC due to core restructuring may be partially responsible for low MACBC 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.

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 MACBC is calculated. The second is the inverse procedure, wherein experimentally measured BC mass, mixing state, and MACBC are inputs and BC morphology is output. Further, the inverse and forward procedures can be combined to estimate κcoat. 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 κcoat. 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.