Constraining the particlescale diversity of black carbon light absorption using a unified framework
 Center for Aerosol Science and Engineering, Department of Energy, Environmental and Chemical Engineering, Washington University in St. Louis, St. Louis, MO 63130
 Center for Aerosol Science and Engineering, Department of Energy, Environmental and Chemical Engineering, Washington University in St. Louis, St. Louis, MO 63130
Abstract. 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 perparticle basis for accurate estimation of its light absorption. The conventionally adopted coreshell approximation, although computationally inexpensive, is inadequate in not only 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 the variations in 𝜌_{BC} across the multispace distribution of BC morphology, mixingstate, mass, and composition as reported by field and laboratory observations. We find that 𝜌_{BC} > 1 leads to decreased absorption enhancement 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 physicsbased insights regarding coreshell approximation overestimating BC light absorption. We conclude by packaging our framework in an opensource Python application to facilitate communitylevel use in future BCrelated research. The package has two main functionalities. The first functionality is for forward problems, where 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, where 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 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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Payton Beeler and Rajan Chakrabarty
Status: final response (author comments only)

RC1: 'Comment on egusphere2022163', Anonymous Referee #1, 25 May 2022
The manuscript “Constraining the particlescale diversity of black carbon light absorption using a unified framework” by Payton Beeler et al. reveals the effect of morphology of BC core on light absorption enhancement of BC due to the “lensing effect”. The study used the Amsterdam Discrete Dipole Approximation (ADDA) method to calculate the particle absorption with varing BC core morphology and different coating compositions. Based on the results of the studies the authors found a factor (phase shift parameter) to describe the increase and decrease of absorption enhancements of BC caused by its morphology, and formulate universal scaling laws centered on the phase shift parameter. This study also provides physicsbased insights regarding coreshell approximation overestimating BC light absorption. The presentation is concise and clear, and the topic fits well into the scope of the journal. The manuscript could be considered for publication with the following concerns being addressed.
 The novelty of the manuscript is not well presented. There have been a number of numerical studies on optical properties of BC with complex morphology using DDA and Tmatrix method. Either the fractal aggregate model or coating scheme has been considered before.. It is recommended to explain the advance specifically for this work at least in the introduction section.
 In this study, ρ_{BC} (the phase shift parameter of BC core) shows the influence of BC core morphology on its light enhancement, but ρ_{BC} was determined not only by the morphology, but the size of BC can also influence ρ_{BC} according to formula (1). In 2.1 section, the authors state to calculate with BC core masses between 1 fg and 70fg, but the BC size calculation was missing in the results. In addition, previous fractal aggregate studies used the fractal dimension (D_{f}) to represent the morphology, what is the D_{f} for the freshly emitted, partially collapsed, and collapsed aggregate in this study?
 The absorption enhancement of BC core through the “lensing effect” was also investigated for lightabsorbing coating materials like BrC, and the author notices that the total particle absorption is very sensitive to the image refractive index of the coating material. The increase of particles absorption with coating increase was a competition between the increase of BrC absorption and the decrease of the BC enhancement due to less light on the BC core. However, MAE_{BC} in this study shows the total absorption of the particle (e.g. in Fig. 4). It is recommend to subtract the absorption by the BrC shell in order to investigate the “lensing effect” of BC.
4 Section 2.1: The discussion about the influence of spherical monomer of BC aggregates on its optical properties is missing. Berry and Percival (1986) discussed that optical properties of fractallike aggregates were determined by the primary spheres. In this study the primary sphere was chosen to be 20nm, Shetty et al., (2021) used 40nm. (https://doi.org/10.1080/02786826.2021.1873909).
 Section 2.2: The settings about the ADDA are not well described. The accuracy of ADDA depended on the size of the subvolume compared to the wavelength of the incident light. What’s the resolution of dipoles per wavelength in this study?
Fig. 2: Y axis label is missing.

AC1: 'Reply on RC1', Rajan Chakrabarty, 15 Sep 2022
The manuscript “Constraining the particlescale diversity of black carbon light absorption using a unified framework” by Payton Beeler et al. reveals the effect of morphology of BC core on light absorption enhancement of BC due to the “lensing effect”. The study used the Amsterdam Discrete Dipole Approximation (ADDA) method to calculate the particle absorption with varying BC core morphology and different coating compositions. Based on the results of the studies the authors found a factor (phase shift parameter) to describe the increase and decrease of absorption enhancements of BC caused by its morphology, and formulate universal scaling laws centered on the phase shift parameter. This study also provides physicsbased insights regarding coreshell approximation overestimating BC light absorption. The presentation is concise and clear, and the topic fits well into the scope of the journal. The manuscript could be considered for publication with the following concerns being addressed.
COMMENT: The novelty of the manuscript is not well presented. There have been a number of numerical studies on optical properties of BC with complex morphology using DDA and Tmatrix method. Either the fractal aggregate model or coating scheme has been considered before. It is recommended to explain the advance specifically for this work at least in the introduction section.
RESPONSE: While Tmatrix and DDA methods have been used extensively to investigate light absorption by BC, this study is the first to provide physicsbased scaling laws for quick calculation of BC optical properties. It is also one of the first studies to provide insight into causes of discrepancies in modeled and measured light absorption enhancement due to internal mixing of BC. Without the use of the scaling laws developed by this work, one would need to rely on computationally expensive methods (Tmatrix or DDA) or incomplete models (such as coreshell Mie theory or scaling laws given by Chakrabarty and Heinson, 2018). Additionally, this study is the first to show (per our knowledge) that measurements of BC light absorption using contemporary instruments can be used to infer the morphology of fractal BC aggregates. We have emphasized these points in the introduction of the revised manuscript.
COMMENT: In this study, ρ_{BC} (the phase shift parameter of BC core) shows the influence of BC core morphology on its light enhancement, but ρ_{BC }was determined not only by the morphology, but the size of BC can also influence ρ_{BC} according to formula (1). In 2.1 section, the authors state to calculate with BC core masses between 1 fg and 70 fg, but the BC size calculation was missing in the results. In addition, previous fractal aggregate studies used the fractal dimension (D_{f}) to represent the morphology, what is the D_{f} for the freshly emitted, partially collapsed, and collapsed aggregate in this study?
RESPONSE: The gyration radius of BC aggregates in this study ranged from 50 – 300 nm. The D_{f} of freshly emitted, partially collapsed, and fully collapsed BC was 1.832 ± 0.089, 2.105 ± 0.223, and 3.0, respectively. This has been added to section 2.1 of the revised manuscript.
COMMENT: The absorption enhancement of BC core through the “lensing effect” was also investigated for lightabsorbing coating materials like BrC, and the author notices that the total particle absorption is very sensitive to the image refractive index of the coating material. The increase of particles absorption with coating increase was a competition between the increase of BrC absorption and the decrease of the BC enhancement due to less light on the BC core. However, MAEBC in this study shows the total absorption of the particle (e.g. in Fig. 4). It is recommended to subtract the absorption by the BrC shell in order to investigate the “lensing effect” of BC.
RESPONSE: The choice was made to focus on total particle absorption as opposed to separating BC absorption due to lensing and BrC absorption because common methods for measuring light absorption by BC will measure total particle absorption, and will likely not be able to parse whether absorption enhancement is the result of absorbing coatings or increased lensing. Therefore, we chose to develop scaling laws based on total particle absorption in order to make the results applicable for experimentalists. The contribution of BrC absorption to absorption enhancement, as well as the change in BC absorption due to the socalled “sunglass effect” is left for future work (Luo et al., 2021).
COMMENT: Section 2.1: The discussion about the influence of spherical monomer of BC aggregates on its optical properties is missing. Berry and Percival (1986) discussed that optical properties of fractallike aggregates were determined by the primary spheres. In this study the primary sphere was chosen to be 20nm, Shetty et al., (2021) used 40nm. (https://doi.org/10.1080/02786826.2021.1873909).
RESPONSE: The radius of BC monomers will affect the radius of gyration, and eventually the phase shift parameter. Therefore, the developed framework is able to account for changing BC monomer size. Shetty et al., (2021) utilized BC spheres with diameter of 40 nm, we have chosen the same monomer size, with radius of 20 nm.
COMMENT: Section 2.2: The settings about the ADDA are not well described. The accuracy of ADDA depended on the size of the subvolume compared to the wavelength of the incident light. What’s the resolution of dipoles per wavelength in this study?
RESPONSE: ADDA recommends 10 dipoles per wavelength for accurate estimation of absorption and scattering properties. Our study utilizes > 100 dipoles per wavelength. This has been added to Section 2.2, for clarity.
COMMENT: Fig. 2: Y axis label is missing.
RESPONSE: This has been corrected.

CC1: 'Blackcarbon phase shift parameter and soot restructuring', J. C. Corbin, 15 Jun 2022
I found Beeler and Chakrabarty (egusphere2022163) very interesting, particularly from the perspective of having just finished a review of soot restructuring studies. That review concluded that only solid coatings or coagulation could allow soot to mix internally without restructuring, and is relevant to the interpretation of the results of this manuscript. I will expand this comment here in the conventional review format for clarity.
In this manuscript, B&C apply the phaseshift parameter $\rho$ to the BC core of internal mixtures. $\rho$ is defined relative to the radius of gyration $R_g$, wavelength $\lambda$, and BC monomer packing fraction $\Phi$ as
\begin{equation}
ρ =
\frac{4 \pi R_g}
{\lambda}
m_\textrm{eff}  1
\end{equation}
\begin{equation}
ρ = 2 x m_\textrm{eff}  1
\end{equation}
Where $ m_\textrm{eff}$ is
\begin{equation}
\Phi \left( \frac{m^2 1}{m^2+2}= \frac{m_\textrm{eff}$ ^2 1}{m_\textrm{eff}$ ^2+2}
\end{equation}
Where $x = 2 \pi R_g / \lambda $ is the size parameter. These equations illustrate that $\rho$ is primarily a function of packing fraction $\Phi$ and size $R_g$ (or $x$). Packing fraction is a morphological parameter, and $m$ is expected to be a function of $R_g$ only for aggregates smaller than those considered here (https://doi.org/10.1016/j.carbon.2022.02.037). So, the main concept in this manuscript is come down to the BCcore packing fraction $\Phi$.
Based on the above concept, the authors present an excellent discussion of the relationship between MAC, $\rho$, and the ratio of coatingtoBCmass, $R_{BC}$, for model soot particles. Then, the authors place their work in the context of the literature by considering whether previous measurements of the relationship between MAC and $R_{BC}$ can be attributed to $\Phi$.
I have two major questions for the authors:
\begin{enumerate}
\item
Could the authors add a more quantitative discussion of $\Phi$? As stated, the entire discussion of the BCcore $\rho$ comes down to $\Phi$, which can be constrained as about 0.1 to 0.4 (Zangmeister et al., 2018, http://www.pnas.org/cgi/doi/10.1073/pnas.1403768111; also Schnitzler et al. 2017 is relevant http://dx.doi.org/10.1016/j.jaerosci.2017.01.005). If there was some reason the authors did not discuss $\Phi$ directly could they please comment? If not,
\begin{enumerate}
\item can the authors calculate $\Phi$ for their model aggregates, and discuss whether the upperlimit packing density identified by Zangmeister et al. 2018 allows the literature trends to be fully explained by $\Phi$?
\item Also, can the authors provide more information about the $\Phi$ of their modelled particles, for example by plotting $\Phi$ versus $\rho$ or MAC?
\end{enumerate}
\item
The authors state that “Our results indicate that studies which find little to no increase in MAC_BC with increased R_BC are measuring BC aggregates which have undergone significant coatinginduced restructuring, while studies that find significant increases in MAC_BC are measuring aggregates which have undergone little to no restructuring.” How confident are the authors that alternatives have been excluded, and that this statement is the most likely given all available evidence? For example:
\begin{enumerate}
\item
Having recently reviewed sootrestructuring studies [Corbin, Modini, and GyselBeer, https://doi.org/10.48550/arXiv.2206.03646] I believe this statement should be reconsidered or discussed in terms of the fundamental physics it implies. To briefly summarize that review, we identified multiple studies that demonstrated unequivocally that liquid condensation typically induces restructuring. These studies used various materials including organics of varying polarity and sulfuric acid. Our review of these studies and complementary laboratory demonstration, showed that condensationcompaction coatings can only be avoided when solid coatings or liquids with very high contact angles (which activated water via the heterogeneous nanodropletactivation mechanism) were used. Examples of such solids include SOA formed at low RH or anthracene (relevant only to the laboratory). Compaction can also be avoided by coagulation. So, if internally mixed BC has not undergone extensive restructuring, it must have mixed by solid deposition or by coagulation.
My impression from the recent studies by Fierce et al. (cited by the authors) is that while nighttime coagulation can be significant, it is unlikely that most soot particles mix by coagulation. But, perhaps the authors’ work implies that this conclusion is inaccurate. My impression is also that solid organic coatings form only rarely, since they require very low RH or low temperatures, while organic vapours are emitted most often at higher temperatures. It seems to me less likely that solid coatings explain the field data on absorption enhancement.
So, the authors’ conclusions can be reconciled with the known mechanisms of soot restructuring by arguing that some studies primarily observe liquidcondensation coatings while other primarily observe soliddeposition or coagulation coatings. I have not reviewed the manuscripts cited by the authors to assess this likelihood.\item On Line 71, the authors state, “Recent studies have found that the nonsphericity of BCcontaining particles (partial encapsulation of BC) can decrease absorption enhancement (Hu et al., 2022, 2021). While these findings are notable, previous studies have not observed a prevalence of partiallyencapsulated BC, yet decreased light absorption is still observed”. Is it possible that the authors have too readily rejected the hypothesis of H1) partially encapsulated, or offcentre mixing states, in favour of H2) condensationwithoutcompaction? Given the abovementioned review, I believe H1 is plausible while H2 is extremely unlikely. I would consider the entire manuscript to remain valid and valuable if H1 is rejected over H2. The only change is that $\Phi$ becomes $\Phi_eff$. (Would the same trends in MAC be observed?)
\item This is more of an editorial comment. The highest $\rho_{BC}$ in Figure 7b were measured at the shortest wavelengths and the two highest studies were both firstauthored by Cappa. Some readers may wonder whether there was a systematic effect here (for wavelength, it is expected by definition; and for the Cappa group, the question is whether they use a unique experimental approach that caused a bias relative to other data sets). I do not believe that these are real issues but they deserve a brief comment.
\end{enumerate}
While reading, I also made various minor notes. I will list them below as suggestions for the authors.
\begin{enumerate}
\item I’d add a row showing partially encapsulated/collapsed examples in Figure 2.
\item
Line 105109 may need clarifying. Why would someone use the RDG MAC when estimating E_abs? To me, a “literature value” would be a measured MAC of mature, openstructured BC (Liu et al., linked at #7 below). Text may not convey your intention here.
\item
What is the role of $\Phi$ in Figure 3? No effect?
\item
Line 161, there may be a better citation for the imaginary refractive index (Sun and Bond?).
\item
Line 163, add SOA after pinene.
\item
Line 164, consider citing Lu et al http://dx.doi.org/10.1021/acs.est.5b00211
\item
Line 180, are the four digits of precision meaningful in 6.819 m2/g? What is the corresponding standard deviation? Also, it may be worth discussing this value in comparison to the measured mean value of 8.0 ± 0.7 m2/g (Liu et al https://doi.org/10.1080/02786826.2019.1676878)
\item
Figure 4 uses both “$\rho$” and “Core Phase Shift Parameter” for the same thing, which confused me initially. Consider harmonizing.
\item
Figure 7a why are there 3 lines? Please label?
\item
Figure 7b consider adding a column of the range of observed R_BC?
\item
Line 272 how could a ‘fresh BC’ particle have $R_{BC} = 3.68$? It seems that ‘fresh’ is ambiguous. Maybe ‘uncompacted’.
\item
Figure 8a consider omitting the instrument lists, which are incomplete and may become outdated in a shorter time than this work will. If you keep it, please revise (e.g. Single particle BC mass can be measured by SPAMS and all of the “MAC” instruments measure absorption, not MAC.)
\item
Figure 8b consider contours, I could not see the contrast on my B&W printout.
\item
Caption of Figure 8 states that the low MAC of Cappa 2012 can be explained by compaction of the BC core, but Cappa 2012 shows lab data (their Fig 3) where compaction was absolutely expected yet absorption enhancement was still observed. (My expectation is based on the review of restructuring mentioned above, which includes a repeat of their same experiments and cites Ghazi and Olfert who also repeated those experiments.)
\item
Figure 9a I found confusing but the caption I found clear. Consider linearizing the figure.
\item
Discussion at end of 3.4.2 may have to change to reflect the restructuring comments above.
\item
What are the uncertainties in k= 0.056 in Section 3.4.3? Does the code include an uncertainty estimation feature? Monte Carlo? This would be helpful as a way to let users know when they have obtained meaningful results.
\item
Why would coating filling the voids in the BC aggregate change $\Phi$, the BC monomer packing fraction? That is only if the BC core collapses, as stated subsequently.
 CC2: 'PDF update.', J. C. Corbin, 15 Jun 2022

AC3: 'Reply on CC1', Rajan Chakrabarty, 15 Sep 2022
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/egusphere2022163/egusphere2022163AC3supplement.pdf

RC2: 'Comment on egusphere2022163', Anonymous Referee #3, 05 Jul 2022
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/egusphere2022163/egusphere2022163RC2supplement.pdf

AC2: 'Reply on RC2', Rajan Chakrabarty, 15 Sep 2022
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/egusphere2022163/egusphere2022163AC2supplement.pdf

AC2: 'Reply on RC2', Rajan Chakrabarty, 15 Sep 2022
Payton Beeler and Rajan Chakrabarty
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