Locating the missing absorption enhancement due to multi‒core black carbon aerosols

Domain: earth_science
Published: 2025-11-19
Editorial outcome: major_revision

Abstract
Black carbon (BC) aerosols, with their strong light-absorbing ability, are major drivers of the global climate. In existing models, BC aerosols are simplified as a single core when determining radiative effects. Here, we found that 21% of BC aerosols contain multiple cores during a wildfire smoke observation. By considering dynamic effective medium approximation (DEMA) with Mie theory and assuming randomly distributed multi-core BC, the light absorption was 1.81 times greater than that under the single-core assumption for particles with overal diameters >400 nm and core diameters >200 nm. A machine learning emulator was developed for DEMA-based absorption enhancements and incorporated into a global atmospheric model. For global aerosol absorption, multi-core BC particles lead to a 19% increase, especially in wildfire-affected regions. This study emphasizes the critical role of multi-core BC particles in amplifying radiative forcing and the necessity to revise models for the simulation of BC climate impact.

Introduction
Black carbon (BC) plays a uniquely critical role in shaping the Earth’s climate system through its radiative absorption ability and aerosol-cloud interactions, which allow it to contribute to slow winds, suppress the planetary boundary layer, alter snow and ice melting conditions, and generate other regional climate and weather changes. Between 25 and 36% of global BC emissions originates from wildfires, with the remainder originating primarily from energy-related combustion processes, such as those associated with residential solid fuels and fossil fuels. The intensity and duration of wildfires are increasing globally due to the higher temperatures, drier vegetation, and longer dry seasons resulting from climate change. When BC aerosols are coated by other nonabsorbing substances, the resulting aerosol-lensing effect can significantly amplify their degree of light absorption, as quantified by the absorption enhancement factor, E_abs. The Core-Shell Mie model, which is widely employed in atmospheric models (e.g., the Weather Research and Forecasting model coupled with Chemistry) can be used to obtain the light absorption enhancement (E_abs) value of BC. In the Core-Shell Mie model, it is assumed that there is one central BC core per particle (n_core = 1), and the BC mixing state is represented via the BC core diameter (D_c) and the particle-to-core diameter ratio (D_p/D_c). The total BC absorption level is represented by the black carbon absorption aerosol optical depth (BC AAOD; Methods), which depends on both E_abs and the BC mass in the atmospheric column. On the basis of the third phase of the Aerosol Comparisons between Observations and Models intercomparison initiative (AeroCom), BC AAOD accounts for 36%–84% of the global annual mean of total absorption aerosol optical depth (AAOD) at the 550-nm wavelength, indicating that BC yields a relatively notable warming effect. However, AeroCom and other global observations indicated that global models frequently underestimate BC AAOD, and a scaling factor of 1.4–2.8 is needed to ensure alignment with global observational data for subsequent radiation assessment. A quantitative review of the BC AAOD is provided in Supplementary Text 1. Some model studies attributed this upscaling factor to uncertainties in emission inventories and BC mass loads; moreover, these researchers focused on constraining the representation of the mixing state of BC in global atmospheric models.

Significant uncertainties exist in the mixing state and E_abs of BC in observations and simulations. Fierce et al. proposed that accounting for particle-to-particle heterogeneity in the mixing state and the complex morphology of BC-containing particles may resolve these uncertainties. Moreover, Liu et al. and Fierce et al. reported that the Core-Shell Mie model might oversimplify the nonspherical morphology of BC particles, leading to potential overestimation of the degree of light absorption at low D_p/D_c values (<~1.4). Chen et al. further proposed that nonsphericity contributes only marginally (3.3%) to the global BC AAOD. In field studies, researchers reported E_abs values varying between 1 and ~1.5; these studies relied solely on single particle soot photometers (SP2) and obtained E_abs values via the Core-Shell Mie model. Direct optical measurements conducted with a photoacoustic spectrometer, a particle soot absorption photometer, and other technologies were employed to determine the average E_abs values, which ranges from 1 to ~2.5 (refer to Supplementary Table 1 for details). It is well known that SP2 measurements and laboratory experiments typically exhibit an upper size limit of D_p <~400 nm, with the detailed range depending on the SP2 configuration. However, field studies roughly estimated that the contribution of BC-containing particles with D_p >400 nm to the bulk BC mass and light absorption level could reach 50%, especially during polluted periods. Detailed comparisons of D_p and D_c across model, field, and laboratory studies are summarized in Supplementary Text 2 and Supplementary Table 2. These findings highlight a gap in the understanding of how the mixing state of large particles (D_p >400 nm) affects E_abs.

Our study is further inspired by considering the maximum E_abs value of ~4, which was reported in certain experiments and field studies and is consistent with the simulations of Fuller, et al. Importantly, the simulations by Fuller, et al. were based on the Dynamic Effective Medium Approximation (DEMA) with Mie theory under the assumption that multiple BC cores (n_core >= 1) are randomly distributed within particles. However, the impacts on E_abs and BC AAOD have been neglected because of a lack of direct evidence for n_core > 1 in DEMA for ambient BC particles. Transmission electron microscopy (TEM) enables detailed analysis of individual particles, and this technique effectively captures the mixing structure of BC-containing particles. Multi-core BC particles have occasionally been reported in previous studies on the basis of TEM (refer to Supplementary Table 3 for details), but their roles in driving radiative forcing and their connection to DEMA remain underexplored. One recent study confirmed that coagulation leads to the occurrence of multi-core BC particles with D_p >400 nm during long-range transport. Some field studies have proposed that coagulation during the long-range transport of wildfire smoke drives the increase in the D_p value of BC-containing particles. Satellite observations show that wildfire smoke produces a dense haze layer at 2000–4000 m above sea level (a.s.l.) from Southeast Asia to Yunnan Province, China. Therefore, these wildfire events provide unique real-world conditions in a laboratory setting for understanding BC light absorption and the changes in mixing states during transport, particularly for multi-core BC aerosols.

To characterize the evolution of mixing states for transported BC, we conducted a field campaign at the Yunya Atmospheric Environment Research Observatory (YAREO) during the wildfire-affected periods. The station is located atop a mountain in Yunnan Province at 2200 m a.s.l. (Fig. 1). We quantified the impact of n_core on the size dependence of E_abs and the subsequent influence on BC AAOD with machine learning algorithms in a global atmospheric model. This study provides the insight into addressing the missing absorption enhancement resulting from multi-core BC aerosols in global atmospheric models.

Results
Characteristic of aerosol particles
Highly polluted periods are defined as when the total mass of elemental carbon, organic matter, dust, and water-soluble inorganic ions exceeds 15 μg m^-3 at the sample station (Supplementary Fig. 1). We found that mass concentrations of these aerosol chemical components during highly polluted periods were approximately 3 times higher than them during less polluted periods (e.g., elemental carbon in Supplementary Table 4). In Supplementary Fig. 2a, BC concentrations increased from 0.15 to 4.4 μg m^-3 over the sampling period, driving BC AAOD to increase from 0.003 to 0.062 based on Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA2). Additionally, Supplementary Fig. 2b-d show that regional BC concentration and BC AAOD during highly polluted periods were about three times higher than them during less polluted periods, while PM10 concentration at the downstream national monitoring station became double during highly polluted periods. Satellite data and air mass trajectories both indicated that south-westerly air masses passed over areas with intense fire activity in Southeast Asia during highly polluted periods, leading to regional pollution events at the downwind sampling site (Fig. 1a). In contrast, western trajectories from South Asia rarely passed over fire-affected regions during less polluted periods (Supplementary Fig. 3). These results suggested that the highly polluted periods were influenced mainly by the long-range transport of wildfire smoke from Southeast Asia, providing ideal real-world conditions for understanding BC ageing mechanisms during long-range transport.

In TEM images, BC particles appeared as fractal-like aggregates of soot, and they were either externally mixed as bare-like BC aerosols (Fig. 1b1) or BC aerosols coated with secondary aerosols (Fig. 1(b2–b3)) after undergoing ageing processes, such as condensation, coagulation, and cloud processing. Among the 3950 analysed particles, 47% were BC-containing particles, comprising 33% coated BC particles and 14% bare-like BC particles (Fig. 1c), whereas the remainder were BC-free particles (Supplementary Fig. 4). In summary, multi-core BC particles accounted for 21% of the BC-containing particles, 14% during the less polluted period and 25% during the highly polluted periods (Fig. 1c). The detailed individual-particle analysis can be found in the Methods section and Supplementary Text 3.

During highly polluted periods, wildfire smoke after long-range transport could enhance the formation of multi-core BC particles (Fig. 1a vs. Supplementary Fig. 3). However, few studies reported the size-dependent characteristics of multi-core BC particles. Instead, some studies considered the two-dimensional data characteristics of D_p/D_c and D_c and the size distribution of D_c to approximately assess the mixing states of BC aerosols in biomass burning smoke and wildfire smoke. To obtain the equivalent volume diameters of individual BC-containing particles and their BC cores (i.e., D_p and D_c, respectively), we calibrated the equivalent circle diameters obtained from TEM images to the equivalent volume diameters via a regression relationship from atomic force microscopy measurements (refer to the Methods). For multi-core BC particles, D_c represents the diameter of a single sphere whose volume is equivalent to the sum of all BC core volumes. Figure 2a and Table 1 both show that multi-core BC particles became the dominant particle type at D_c > 200 nm.

Figure 2b shows that the average D_p/D_c ratio first decreased with increasing D_c, and then remained relatively constant for particles with D_c > 200 nm. This relationship was well fitted by the following equation: D_p/D_c = 2.5 + 6.3 × Exp(-D_c/63) (purple line in Fig. 2b). The fitting results revealed that the average D_p/D_c ratio remained steady at 2.5, mainly for D_p > 400 nm (blue and green dots in Fig. 2b). Similar plateau values of D_p/D_c were reported for Southeast Tibetan Plateau (~2.2) and during a transboundary transport event in Beijing (~2.4). If condensation dominated the BC aging mechanism (n_core = 1), particles with smaller D_c values would exhibit higher D_p/D_c values when coated with the same volume of condensate aerosols. Specifically, D_p/D_c was expected to approach ~1 with increasing D_c value. The blue line shown in Fig. 2b represents the maximum D_p/D_c value under the condensation mechanism described by Chen et al. and Sedlacek et al. Figure 2b shows that particles with D_p > 400 nm (blue and green dots) were above the blue line. Below this blue line, condensation dominated the BC ageing process, maintaining the n_core value at 1 (Fig. 2c). Above the blue line, coagulation dominated the ageing process, resulting in an increase in the n_core value (i.e., n_core > 1) (Fig. 2c).

The Box models developed by Matsui et al. and Riemer et al. revealed that thickly coated BC particles were dominated by coagulation rather than by condensation. They concluded that the number of coagulation events increased with increasing D_p and peaked at D_p > ~400 nm. Several field studies proposed that coagulation could account for most of the increase in particle size (D_p) in wildfire smoke. Theoretically, coagulation between two BC-containing particles could increase n_core and produce a larger D_c value. In this study, multi-core BC particles were observed in the TEM images (Fig. 1b3), thus providing direct evidence of coagulation among BC-containing particles. Figure 2c shows that the range of n_core increased with increasing D_p: for example, n_core ranged from 1 to 5 for D_p between 400 and 600 nm, and from 1 to 8 for D_p >= 800 nm. We observed a critical condition near D_p > 400 nm and when n_core transitioned from 1 to >1 (Fig. 2c). This phenomenon was also observed in the Southeast Tibetan Plateau and Amakusa City, Japan. Table 1 indicates that multi-core BC particles became the dominant type of coated BC particles for D_p > 400 nm and D_c > 200 nm. To facilitate parameterization, we defined two proxy features (M_1 and M_2), as expressed in Eq. (1), to represent the primary size characteristics of multi-core BC particles and their subsequent impact on E_abs. In 1310 coated-BC particles, 20% simultaneously satisfied M_1 = 1 and M_2 = 1 (i.e., M_1 × M_2 = 1), in which the number of multi-core BC particles was three times greater than that of single-core BC particles (Table 1).

Optical absorption enhancement
In this study, we employed an electron-microscopy-to-BC simulation tool, coupled with discrete dipole approximation (EMBS–DDA) to visualize the effect of n_core on E_abs of BC-containing particles. The EMBS tool provides an interactive interface for designs of particles with different sizes and positions of cores that are input into the DDA module to calculate their optical properties. To explain the differences between single-core and multi-core BC particles, we further adopted the near-field calculation algorithm of DDA to simulate the light electric field intensity inside and around scattering particles (i.e., secondary aerosols) and coated BC particles, respectively.

BC-containing particles were assigned different n_core values and positions in the EMBS–DDA simulations, whereas D_p/D_c was set to the average value (i.e., D_p/D_c = 3 in Supplementary Fig. 5) to represent the common scenario for D_p > 400 nm. Under the assumption that BC cores were randomly located in an individual particle, the probability of core occurrence obviously increased from the centre to off-centre positions. Specifically, BC aerosols with a single core or multiple cores at the particle centre represented idealized and specialized scenarios, respectively. Innumerable possible core locations would increase the DDA simulation burden of every configuration excessively, even with fixed input parameters of D_p, D_c, and n_core. Herein, we simulated three representative scenarios using DDA:

DDA_1: Setting n_core = 1 with the core at the centre. The simulated E_abs value was almost equal to that of the Core-Shell Mie model, as shown in Fig. 3a–b.

DDA_2: Setting n_core > 1 with multiple cores concentrated near the centre, as shown in Fig. 3b.

DDA_3: Setting n_core > 1 with cores distributed at various off-centre positions, as shown in Fig. 3b. The detailed settings are provided in Supplementary Text 4, Supplementary Tables 5–6, and Supplementary Fig. 6. In addition, we calculated E_abs via the Core-Shell Mie and DEMA–Mie models on the basis of the same input parameters (D_p, D_c, and n_core) used in DDA. Each core is the same size in the particles in DDA_1–DDA_3, and Supplementary Fig. 7 shows the simulations with different sizes of BC cores (Diff. Size). In Supplementary Fig. 7, Diff. Size shows a similar range of E_abs with DDA_1–DDA_3, which support the assumption in Jacobson: it is ideal to track a size distribution of BC within coated BC particles of each size, but a monodisperse distribution of BC within coated particles was used for quick optical simulations.

We found that the E_abs values for n_core = 1 were 12–222% lower than those for n_core > 1, particularly in DDA_3 scenarios (Supplementary Table 6). Figure 3a shows that the DEMA–Mie could represent the average increasing trend in DDA_3. Notably, the DEMA–Mie model was efficient for simulating the average impact of n_core at random positions on E_abs. Supplementary Fig. 8 shows that the light electric field intensity was enhanced (lensing effect) in the off-centre region of a secondary scattering particle, which is consistent with the experiment findings of Arroyo, et al. and Li, et al. The intensity of the enhanced field, as indicated by colour, was particularly high for D_p > 400 nm (Supplementary Fig. 8). We further modelled the electric field intensity for coated BC particles with various n_core values, as shown in Fig. 3 (c1–c6) for D_p = 600 nm and Supplementary Fig. 9 for D_p = 300 nm. Figure 3 (c1–c6) show the enhanced presence of white and orange areas in the regions marked by black circles, suggesting that DDA_2 and DDA_3 exhibited higher E_abs values than did the DDA_1 simulations. In Supplementary Fig. 9, a smaller colour gradient indicates a weaker lensing effect for D_p = 300 nm, but the difference in E_abs between Core-Shell structure and multi-core structure is still up to 1 (DDA_1 vs. DDA_3). Kahnert et al. also proposed that the energy field rapidly attenuated within a solid BC sphere, which limits the contribution of the core mass near the particle centre to light absorption. Moreover, Beeler and Chakrabarty defined a phase shift parameter to describe a simulation like Fig. 3c2. However, in these previous simulations, multi-core BC particles were not employed as input parameters.

E_abs_DEMA exhibited a quasilinear increase with the logarithm of D_p/D_c under M_1 × M_2 = 0 for all 1310 coated BC particles (black line in Fig. 4a). Notably, the particles with M_1 × M_2 = 1 (red dots in Fig. 4a) exhibited higher E_abs values than did those with M_1 × M_2 = 0 (blue dots) for the same D_p/D_c value. In contrast, the Core-Shell Mie model showed that E_abs first increased with increasing D_p/D_c and then plateaued at ~2 for M_1 × M_2 = 1 (red dots in Fig. 4b). The differences between E_abs_DEMA and E_abs_CS indicated that E_abs_DEMA was 1.17 and 1.81 times greater than E_abs_CS for M_1 × M_2 = 0 and 1, respectively (Fig. 4c).

In summary, these findings suggested that the Core-Shell Mie model could significantly underestimate the light absorption of BC-containing particles when M_1 × M_2 = 1. However, the DEMA–Mie model could be a suitable and efficient tool for simulating the optical properties of multi-core BC particles. Therefore, the developed DEMA–Mie model provides a more favourable balance between computational efficiency and physical representation than the DDA and Core-Shell Mie models.

Machine learning emulator
In previous global simulations, the developers of DEMA used a noninteger n_core value (i.e., D_c^3/1,000,000) to approximate n_core. However, Wang, et al. proposed that a noninteger n_core value was implausible because observational evidence for multi-core BC particles was lacking. Therefore, it was necessary to parameterize the impact of n_core on E_abs on the basis of the real-world size dependence of n_core. We employed a random forest (RF) model to predict E_abs_DEMA and quantify how E_abs_DEMA changed with the mixing state of coated BC. On the basis of 1310 samples (coated BC particles), we developed two RF emulators (RF1 and RF2) for E_abs_DEMA, with input features including D_p/D_c, D_c, D_p, E_abs_CS, n_core, n_core,cal, M_1, M_2, and M_1 × M_2. RF1 included all the input features, whereas RF2 excluded n_core, which was generally unavailable in global models. SHapley Additive ExPlanation (SHAP) values, rooted in coalitional game theory, were employed to distribute the total effect of each input feature on the predicted E_abs_DEMA for individual particles (refer to the Methods). By comparing the prediction performance levels and SHAP values of RF1 and RF2, we determined whether M_1 × M_2 could serve as a proxy for the impact of n_core on E_abs_DEMA.

RF1 performed well for both the training and test datasets, achieving an R^2 value of 0.96 (Supplementary Fig. 10 and Supplementary Table 7). Figure 4d shows that D_p/D_c, D_c, D_p, and E_abs_CS were the primary driving features due to their broad SHAP value distributions. Notably, the SHAP distributions for n_core and M_1 × M_2 exhibited a T shape, indicating that there was a threshold effect on E_abs_DEMA (Fig. 4d). To enhance the interpretability of the RF predictions, we visualized the SHAP values corresponding to each feature value. Figure 4 (e–i) show the driving patterns of these primary features on E_abs_DEMA. Each point represents an individual particle, and its colour is indicative of its M_1 × M_2 value to illustrate how M_1 × M_2 influences the driving patterns of these features (Fig. 4e–i). Figure 4e shows that n_core was positively correlated with its SHAP values only when M_1 × M_2 = 1; otherwise, the SHAP values were approximately 0. These results suggested that the impact of n_core on E_abs_DEMA was the most significant for particles with M_1 × M_2 = 1. Figure 4f shows a positive correlation between D_p/D_c and its SHAP value, with a steeper trend for particles with M_1 × M_2 = 1 (red) that that of particles with M_1 × M_2 = 0 (blue). This result indicated a greater increase in E_abs_DEMA with increasing D_p/D_c for M_1 × M_2 = 1. Figure 4g shows that E_abs_CS was positively correlated with its SHAP value overall, except for particles with E_abs_CS ≈ 2. For these particles, the SHAP value was close to zero when M_1 × M_2 = 1 (red in Fig. 4g), suggesting that E_abs_CS had predictive utility for E_abs_DEMA for M_1 × M_2 = 0 but not for M_1 × M_2 = 1. The increasing D_p value led to a greater increase in E_abs_DEMA with M_1 × M_2 = 1 (Fig. 4h), whereas the transition of M_1 × M_2 from 0 to 1 altered the correlation of D_c with its SHAP value from negative to positive (Fig. 4i). These findings suggested that D_p and D_c exerted a nonlinear driving effect on E_abs_DEMA and that BC cores with M_1 × M_2 = 1 exhibited more notable interactions with light. Similar patterns were observed when n_core (Supplementary Fig. 11) was used to explain the driving patterns in RF1 compared with when M_1 × M_2 was used, as shown in Fig. 4 (f–i). We concluded that M_1 × M_2 could serve as a proxy for the impact of n_core on E_abs_DEMA.

To assess the predictive performance of E_abs_DEMA without n_core as an input feature, we performed additional tests to construct RF2. In RF2, the input features included D_p/D_c, D_c, D_p, E_abs_CS, n_core,cal, M_1, M_2, and M_1 × M_2. RF2 performed comparably to RF1, achieving an R^2 value of 0.95 (vs. 0.96 for RF1) and a root mean square error (RMSE) of 0.14 (vs. 0.17 for RF1) (Supplementary Table 7). The SHAP value distributions remained consistent between RF1 and RF2 (Supplementary Fig. 12 vs. Figure 4), further confirming that M_1 × M_2 could effectively replace n_core for predicting E_abs_DEMA in machine learning models.

Implications for the atmospheric model
Multi-core BC particles are not exclusive to wildfire smoke events but are instead widely observed in other environments, as listed in 11 additional studies in Supplementary Table 3. However, these studies did not focus on multi-core particles or their optical properties. To evaluate the global impact of multi-core BC particles on the bulk optical properties of dry aerosols, we coupled the Community Atmosphere Model with the Aerosol Two-dimensional bin module for formation and Aging Simulation (CAM–ATRAS, a global model) and the RF2 model offline (refer to the Methods and Supplementary Fig. 13). CAM–ATRAS model showed that wildfire sources dominated BC sources in regions such as South Africa and South America (Supplementary Figs. 14a–c). The long-range transport of wildfire emissions from Southeast Asia to southern China frequently increased the BC loadings in March (Supplementary Figs. 14d–e). The global distribution of particles with M_1 × M_2 = 1 overlapped with wildfire hotspots (Supplementary Figs. 15a–e) and regions of intensive anthropogenic emissions (Supplementary Figs. 15f–g). In these areas and downstream regions, such as oceans and the Tibetan Plateau, particles with M_1 × M_2 = 1 accounted for ~2% of the total number and ~30% of the mass of BC in the atmospheric column (Fig. 5a and Supplementary Figs. 16–17). Figure 5b shows that the global average BC AAOD was 2.04 × 10^-3 under the Core-Shell Mie model but increased to 2.42 × 10^-3 under the DEMA–Mie model. These large differences occurred mainly in high-emission regions such as China, India, and South Africa, where the elevated BC AAOD was driven by the number concentrations (N_M1×M2=1) of particles with M_1 × M_2 = 1 in the atmospheric column (Supplementary Fig. 18 and the blue contours in Fig. 5b). Furthermore, the relative difference in the BC AAOD was 19% globally. Hot spots (>19%, red colour in Supplementary Fig. 19) were located downstream of high-emission regions, due to the higher fraction of particles with M_1 × M_2 = 1 (Supplementary Figs. 16-17), which highlighted the role of transported aerosols in determining the BC AAOD.

We highlighted the enhanced light absorption due to multi-core BC particles and acknowledged the potential overestimation of E_abs simulations due to the nonsphericity of BC particles at low D_p/D_c values. In their experiment, Liu et al. measured a value of E_abs near 1 for D_p/D_c < 1.4, whereas the calculated E_abs_CS value increased with increasing D_p/D_c. Our study could address the gap in that of Liu et al., where E_abs and the mixing state were clearly marked as unknown for D_p/D_c > 3 owing to the upper measurement limit of the SP2 instrument (i.e., D_p < ~400 nm). We conducted a sensitivity analysis by setting E_abs at 1 for D_p/D_c < 1.4 and using RF2 in CAM–ATRAS to compare the relative importance due to nonsphericity and multi-core nature of BC particles. We defined a ratio between the DEMA-derived increase and the nonsphericity-derived decrease in the global BC AAOD (refer to the Methods). According to Supplementary Fig. 20, this ratio could increase by up to fourfold across the downstream areas of high-emission regions, particularly in wildfire areas such as Southeast Asia and the sample station of this study.

We evaluated the impacts of multi-core BC particles in dry aerosol particles by using a real-world n_core value rather than the assumed n_core,cal value, integrating microscale individual-particle analysis with optical simulations and global models. BC-containing particles could be activated into cloud condensation nuclei when the humidity increases. Via a rough estimation that involved replacing n_core with n_core,cal for individual cloud droplets, Jacobson reported that the DEMA–Mie model could yield a greater near-surface temperature increase (+33%), greater heating rate (+130%), and a larger number of interstitial aerosol particles (+140%), compared to the Core-Shell Mie model. However, accurately parameterizing the changes in n_core during aerosol-cloud interactions remains a significant challenge, particularly during long-range transport and cloud processes. Therefore, we propose that future global and regional models should incorporate multi-core BC scenarios, potentially for particles with D_p > 400 nm and D_c > 200 nm, to increase the accuracies of BC light absorption and radiative forcing predictions. Our study highlights the missing absorption enhancement of multi-core BC aerosols in ambient air, which helps improve the atmospheric modelling of these aerosols and their potential climatic impacts.

Methods
Aerosol collection and measurement
Aerosol samples were collected from 18 to 29 March 2023 at the Yunya Atmospheric Environment Research Observatory (YAREO) in Yunnan Province, China. We collected individual aerosol particles (particulate matter with an aerodynamic diameter <= 10 μm) on 90-mm-diameter quartz filters for 11.5 h (09:30–21:00 for the daytime; 21:30–09:00 for the nighttime in Beijing time). The chemical composition of PM10 was analysed (refer to Supplementary Text 5 for more details). Hourly PM10 data were downloaded from the China National Environmental Monitoring Centre (https://air.cnemc.cn: 18007/). MERRA2 provided the hourly mass concentrations of BC and the BC AAOD at 550 nm during the sampling periods (https://gmao.gsfc.nasa.gov/reanalysis/MERRA-2/).

Individual particle collection and morphological analysis
A portable DKL-2 sampler (Genstar Electronic Technology, China) equipped with a 0.5-mm jet nozzle was employed to collect individual aerosol particles. The particles were collected on a copper grid coated with a carbon film (carbon Type-B, 300-mesh copper; Tianld Co., China) and silicon wafers. These samples were collected at 3:00, 9:00, 15:00, and 21:00 Beijing time for 15–40 min and stored in plastic capsules under low relative humidity (RH = 20-25%). Twelve samples were selected, and a total of 3950 particles were analysed via TEM (JEOL JEM-2100, Japan) in combination with energy-dispersive X-ray spectroscopy (EDS; INCA X-Max N80T, Oxford Instruments, United Kingdom). The strong electron beam of TEM could easily damage the coating of the coated BC particles. Therefore, we determined the morphology and number of BC cores in an individual particle with this technique, as shown in Supplementary Fig. 21.

RADIUS software (EMSIS GmbH, Germany) was adopted to measure the morphological features of individual particles in the TEM images, e.g., the equivalent circle diameter (ECD) and bearing area (A). Additionally, individual particles on silicon wafers were measured via atomic force microscopy (AFM). Gwyddion 2.64 software (http://gwyddion.net/) was used to automatically obtain the bearing area (A) and bearing volume (V) of 165 individual particles. The ECD and equivalent volume diameter (EVD) of individual particles were calculated according to the following Eqs. (2–3):

A = 1/4 * π * (ECD)^2 (2)

V = 1/6 * π * (EVD)^3 (3)

where A is the bearing area, nm^2; V is the bearing volume, nm^3; ECD and EVD are the equivalent circle diameter and the equivalent volume diameter, nm, respectively.

A bootstrap linear regression analysis was used to obtain the transform ratio between the ECD and EVD. Through bootstrap analysis, we could use a limited dataset to estimate the slope coefficients in the regression equation by resampling 1000 times with replacement, which could increase the credibility and robustness of the results. The final equation was EVD = 0.54 × ECD, with an R^2 value of 0.99 (Supplementary Fig. 22). Consequently, a slope factor (k) of 0.54 was adopted to estimate the EVD from the ECD for all the particles in the TEM images. In this study, D_p was represented by the EVD of the whole particles (coating + BC cores). We used Eq. (4) to obtain D_c for all BC-containing particles.

D_c = (k * (4 * A_core / π)^0.5 * n_core)^1/3 (4)

where D_c is the cumulative equivalent volume diameter of all BC cores in a particle; n_core is the number of BC cores; the subscript i denotes the ith core; A_core is the bearing area obtained from the TEM images; k is the average transform ratio between the ECD and the EVD obtained via AFM, i.e., 0.54 as shown in Supplementary Fig. 22.

Light absorption enhancement (E_abs)
In the DEMA–Mie model, it is assumed that one or multiple spherical BC cores are randomly distributed within one particle. The effective refractive index of the whole BC-containing particle (ϵ) could be obtained via Eq. (5). The Mie model was subsequently used to calculate the optical properties at a wavelength of 550 nm.

ϵ = ϵ_coat * ( (D_p/D_c)^3 + 2 * (ϵ_c - ϵ_coat) / (ϵ_c + 2 * ϵ_coat) ) / ( (D_p/D_c)^3 - (ϵ_c - ϵ_coat) / (ϵ_c + 2 * ϵ_coat) ) (5)

where r is the radius of a BC core, and n(r) is the number concentration of cores per unit radius interval. The multi-BC cores cause complex light interactions and influence E_abs, which was considered as the third term in Eq. (5). Without the third term, Eq. (5) transforms into the Bruggeman mixing rule. Owing to the lack of observational evidence for n_core, the developers of DEMA used a noninteger n_core value (i.e., D_c^3/1,000,000) to approximate n_core. Individual particle analysis provided n_core values using TEM (e.g., n_core = 6 in Fig. 1b3). Notably, λ is the light wavelength, and we selected a value of 550 nm for one case. ϵ_c and ϵ_coat are the dielectric constants of BC and other aerosol components, respectively, which could be transformed via the widely used refractive index (RI) through the procedure outlined in Supplementary Text 6. According to the literature, the RI values are 1.85+0.71j for BC aerosols and 1.53+0j for coating aerosols. Finally, the complex Eq. (5) was solved using the Newton–Raphson method, and the details are provided in Supplementary Text 6.

PyMieScatt, an open-source Python package, was used to calculate E_abs via Eq. (6), according to the Core-Shell Mie (E_abs_CS) and DEMA–Mie models (E_abs_DEMA), respectively.

E_abs = C_abs,p / C_abs,c (6)

where C_abs,p and C_abs,c are the absorption cross sections of the coated BC particles and the bare BC particles, respectively. When the DEMA–Mie model is used, C_abs,c is calculated for spherical BCs with the same D_c. To compare E_abs_CS and E_abs_DEMA, we applied EMBS–DDA model as a reference and constructed different particle shapes with various D_c and n_core values at the same D_p/D_c. Details are provided in Supplementary Text 4.

Random forest (RF) and Shapley additive explanation (SHAP) approach
Both field and modelling studies face challenges in measuring and simulating n_core per particle. Therefore, it is essential to quantify the potential nonlinear impact of n_core on E_abs_DEMA and to develop an emulator for E_abs_DEMA via the use of other features as proxies for n_core. We employed the RF model to create an emulator to predict E_abs_DEMA. We divided 1310 samples (coated BC particles) as follows: 70% were used for the training set, and 30% were used for the test set. Finally, two emulators were developed: RF1 included all input features, such as D_p/D_c, D_c, D_p, E_abs_CS, n_core, n_core,cal, M_1, M_2, and M_1 × M_2; RF2 excluded n_core and included other parameters. Furthermore, the SHAP approach was employed to distribute the total effect on the predicted E_abs_DEMA of each input feature. According to coalitional game theory, the difference in model predictions involving a feature j (e.g., n_core) compared with the predictions without j was attributed to the marginal contribution of this feature. Considering the interactive effects between features, differences were computed for every possible feature subset combination of each sample. For each particle sample, K features were input to generate the prediction for E_abs_DEMA, denoted as f(x_i), including a linear function of feature attribution in Eq. (7).

f(x_i) = Φ_0(f, x) + Σ_{j=1}^K Φ_j(f, x_i) (7)

where K is the total number of features; N is the total samples; f(x_i) is the predicted E_abs_DEMA value for the ith particle via the RF model in this study; and Φ_j(f, x) is the SHAP value, which quantifies the impact of feature j on the prediction for input x_i in the built model. Specifically, the contribution of n_core (i.e., j = n_core) of the ith particle (x_i) to E_abs_DEMA was denoted the SHAP(x_i, j) value in Eq. (7), as shown by the scatter in Fig. 4e. The base value, Φ_0(f, x) was the average value of the predicted results. SHAP calculations were performed using the kernelshap and shapviz R packages, whereas the RF model was trained using the randomForest R package.

Parameterization and global model simulations
In many global models, accumulation-mode aerosols are oversimplified and represented by only a few mixing states. As a result, BC-free particles are often misclassified as a coating on BC particles, leading to an 18–44% overestimation of the BC AAOD (refer to Supplementary Text 1 for details). Although correcting this uncertainty is not our direct objective, we must ensure that BC-free particles do not confound our simulations and examination of the Core-Shell Mie and DEMA–Mie models. To address this issue, we selected the CAM–ATRAS, in which 40 mixing state bins are employed to represent accumulation-mode aerosols, including BC-free particles and various BC-containing particles. The CAM–ATRAS model achieves satisfactory performance against a wide range of surface, aircraft, and satellite observations, particularly within the AeroCom III (Aerosol Comparisons between Observations and Models) framework. AeroCom III is an open international initiative aimed at better understanding global aerosols and their impact. The dataset produced by the 17 models participating in AeroCom III, including the CAM–ATRAS, has been extensively adopted across multidisciplinary studies.

In the CAM–ATRAS, accumulation-mode particles with dry diameters (D_p) ranging from 40 nm to 1250 nm were classified into 5 size bins. Then eight BC mixing state bins were further defined on the basis of the mass ratio of BC to total dry aerosols, ranging from 0% to 100%, as shown in Supplementary Fig. 13. Via the use of these 40 two-dimensional mixing state bins, the CAM–ATRAS simulated various aerosol processes, including emission, new particle formation, condensation, coagulation, activation into cloud droplets, aqueous-phase chemistry, and deposition. The model setup aligned with AeroCom III: emission data were taken from monthly anthropogenic emissions based on the Community Emissions Data System and Global Fire Emissions Database (GFED) version 4.1 (https://www.globalfiredata.org/), in which wildfire emissions were explicitly defined. The simulations were conducted with a horizontal resolution of 1.9° latitude × 2.5° longitude, 30 vertical layers (extending up to ~40 km), and monthly average outputs for the 2008–2015 period.

Via the use of this model, we evaluated changes in the BC AAOD at 550 nm wavelength with and without considering multi-core BC particles. The analysis involved the following steps. (1) D_p and D_c were calculated on the basis of aerosol number concentrations, mass, and density of each aerosol composition of the 40 bins. (2) In the base scenario, D_p and D_c were used to run the Core-Shell Mie model and obtained E_abs_CS. (3) D_p and D_c were transformed into n_core, M_1, M_2, and M_1 × M_2, which were then input into RF2 along with D_p/D_c, D_c, D_p, E_abs_CS, n_core,cal, M_1, M_2, and M_1 × M_2 to obtain E_abs_DEMA for each mixing state bin. (4) E_abs_CS and E_abs_DEMA were substituted into Eq. (8) to calculate the BC AAOD offline, BC AAOD_CS, and BC AAOD_DEMA, respectively. The detailed workflow is shown in Supplementary Fig. 13. Matsui, et al. demonstrated that BC AAOD results from offline calculations closely aligned with those from online simulations using the Core-Shell Mie model.

BC AAOD = Σ_{z=1}^{30} Σ_{k=1}^k (ρ_z N_{z,k} C_abs,c,z,k E_abs,z,k) dh_z (8)

where z denotes the 30 levels for height and k denotes the mixing state bins for BC-containing particles; dh and ρ are the grid cell height and air density for each level, respectively; N is the aerosol number concentration for each mixing state bin; C_abs,c and E_abs are the same as those in Eq. (6).

Sensitivity analysis
We highlight the enhanced light absorption ability resulting from multi-core BC particles while acknowledging the potential overestimation in the findings of E_abs simulations due to the nonsphericity of BC at low D_p/D_c values. An experiment by Liu, et al. revealed that the measured E_abs value approached 1 for D_p/D_c < 1.4, but the modelled E_abs_CS value increased with increasing D_p/D_c. We set E_abs_CS and E_abs_DEMA at 1 for D_p/D_c < 1.4 and then substituted E_abs_CS and E_abs_DEMA into Eq. 8 for D_p/D_c > 1.4 to calculate BC AAOD again; these values were referred to as BC AAOD*_CS and BC AAOD*_DEMA, respectively. We estimated the contribution ratio using Eq. (9) to assess the relative importance of the global BC AAOD simulation due to nonsphericity and multi-core BC.

Contribution ratio = (BC AAOD_DEMA - BC AAOD*_CS) / (BC AAOD_CS - BC AAOD*_CS) (9)

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "Comments are provided in the attachment.", "reviewer_2": "The authors use TEM images of fire-affected air masses arriving to Yunnan province, China. They show that large BC-containing particles are likely to have multiple cores due to coagulation during long-range transport. They use dynamic effective medium approximation-Mie (DEMA-Mie) theory model to calculate absorption enhancement by the BC-containing particles that arrived at the sampling site. They then train a random forest model to predict absorption enhancement based on the DEMA-Mie model. They applied the random forest emulator to a global atmospheric model and found that bulk aerosol absorption increased 19%. This paper introduces a potentially important aspect of BC-containing particles that is not accounted for in global atmospheric models. However, some portions of the TEM analysis are unclear and there is a lack of observational evidence for increased absorption enhancement during the sampling period. I therefore think that this paper needs major revisions to address the following points. Major points: 1. Based on the trajectory analysis presented in Figure 1, it appears that the low-pollution and high-pollution cases are both largely originating in India. Are the low pollution samples affected by BC from urban pollution while the high pollution cases are affected by urban and wildfire BC? Presumably, biomass burning BC will have larger core sizes and be more thickly coated than BC from fossil fuel combustion1. This means that biomass burning BC would be more likely to act as cloud condensation nuclei. Differences in the ncore distribution for low and high pollution cases separately may provide evidence that coagulation is occurring in clouds or near source. Separate analysis of low and high pollution cases may show that multiple-core effects are more important for biomass burning BC than for BC from fossil fuel combustion. 2. The methods used to determine ncore in this study are largely ambiguous. For example, it is not clear that the particle shown in Fig. b3 contains six separate cores. Is coating removed from particles to reveal the distinct number of cores? Are the authors using n ≈ N = D 3/1000000 to approximate n ? It appears that this originates from a paper that is core cal c core studying cloud droplet residuals, but there is no evidence provided that the particles here have undergone cloud processing. 3. Is there a comparison of the DEMA-Mie model with the DDA values? One could presumably calculate the DEMA-Mie absorption enhancement for the same particles that were used in the DDA calculations. I see that there are similar trends in in Fig. 3a and 3c, but there seems to be a sharper decrease in DEMA-Mie absorption enhancement with increasing D for c D < 200 nm than for DDA. c 4. Increased absorption by multi-core particles appears to be an effect of phase shift parameter2,3. Particles with phase shift > 1 experience attenuation of light as it passes through the BC core. The core of multi-core particles may be small enough that they remain in the Rayleigh regime (phase shift parameter < 1). It appears that absorption enhancement reaches a maximum value of approximately 4 for large D /D and for small D /D . The authors briefly touch on this subject on lines c p c 205-208, but I think it warrants more discussion. Is absorption enhancement calculated by the DEMA-Mie model the same as for a single-core particle with equivalent phase shift parameter? 5. Are there any measurements of absorption or AAOD that support the modeling results? The authors hypothesize a nearly 20% change in aerosol absorption. This is certainly notable and would likely show up in measurements or remote sensing. Additionally, previous studies have shown that biomass burning aerosols are too absorbing in climate models4 and that particles with large D tend to have lower D /D , leading to lower absorption enhancement than would be predicted by c p c uniform mixing models5. How do these results fit in with these previous studies? Minor points: 1. Line 89, “Once recent study have confirmed that…” should be changed to “One recent study has confirmed”. 2. Line 245, E should be E . abs_DEAM abs_DEMA References: 1. Schwarz, J. P. et al. Measurement of the mixing state, mass, and optical size of individual black carbon particles in urban and biomass burning emissions. Geophysical Research Letters 35, (2008). 2. Beeler, P. & Chakrabarty, R. K. Constraining the particle-scale diversity of black carbon light absorption using a unified framework. Atmospheric Chemistry and Physics 22, 14825–14836 (2022). 3. Sorensen, C. & Fischbach, D. Patterns in Mie scattering. Optics Communications 173, 145–153 (2000). 4. Brown, H. et al. Biomass burning aerosols in most climate models are too absorbing. Nature communications 12, 277 (2021). 5. Fierce, L. et al. Radiative absorption enhancements by black carbon controlled by particle-to-particle heterogeneity in composition. Proceedings of the National Academy of Sciences 117, 5196–5203 (2020).", "reviewer_3": "Review of paper titled: “Locating the Missing Absorption Enhancement Due to Multi-Core Black Carbon Aerosols” by Chen et al. The paper discusses an important issue in aerosol science, which is how the mixing of black carbon particles affects light absorption, with important potential implications for climate modeling. Overall, the main premise of the paper, meaning the importance of multiple black carbon cores within a single particle, is interesting and a potentially important one. The approach used is quite comprehensive, including microscopy, theoretical considerations, and modeling going all the way from single particles to global-scale simulations, machine learning, etc. The weak point of the paper is the clarity and how the method is described. I found it quite difficult to read. Resorting heavily on SI tables, equations, explanations, and figures made the reading even more difficult. The choice of wording, sentence structure, and logic also often hampered the understanding of the several different steps, acronyms, limitations, etc. Overall, I think the work has very good potential, and it should be shared with the community, but in its present form, it is not ready for publication, especially so in a journal that targets a broader audience such as Nature Communications. I would suggest, therefore, that the authors significantly revise the paper to improve readability. Next, I provide some suggestions and some examples, but there are many other instances where the readability and clarity of the paper can and should be significantly improved. General Comments - A brief description of the difference between EMBS_DDA and DEMA_Mie and how they are calculated early in the paper (instead of waiting for the detailed description in the methods) would be helpful. This can be achieved with just one or two short sentences. - From figure 3, it is difficult for me to evaluate how well DEMA-Mie approximates the DDA calculations, because the two results are shown in different panels. A direct and quantitative comparison in terms of % deviations might be an easier way to demonstrate how good the DEMA_Mie approaches the DDA, as it is done for the comparison of CS with DEMA_Mie in table S5 (and graphically in figure 3(c)). - What index of refraction values have been used, and what’s the impact of the choice? The index of refraction used becomes clear later, which is fine, but perhaps a discussion of the potential uncertainty those choices can have would be useful. - I found the section on the Machine Learning Emulator to be very hard to interpret and quite cryptic. Perhaps this is due to the technical aspect of the techniques, but in my opinion, to reach the broader audience of Nature Communications, much effort should be put into clarifying this section and making interpretations more understandable for a broader audience. Specific Comments - Line 47: Is Eabs_CS indicating the absorption or the absorption enhancement? The notation is confusing. Also, is CS for core-shell? Perhaps it should be explicit. In general, the sentence from line 45 to line 4 I very long. I would suggest breaking it down into at least two or three sentences. - Lines 53-54: The sentence “as a key indicator of BC’s warming effect” reads awkward and is unclear. - Lines 63-66. Very long sentence, please break it down into shorter sentences. Also, I believe Fierce’s paper puts more emphasis on the heterogeneity of coating thickness rather than non-sphericity. It should also be noted that the idea that Eabs might be smaller than what was originally hypothesized was especially sparked by the study by Cappa et al. (Science 2012). Cappa’s conclusions were based on field measurements and optical measurements, from what I recall. So, it seems natural to me that this study should be mentioned here and should be kept into consideration as it contradicts a little bit what the authors seem to imply here that low Eabs has been inferred only through SP2 measurements (lines 66-72) - Line 71 and especially 73: Dp indicates what diameter? Mobility, aerodynamic, area equivalent, optical, mass? Especially for BC, that can make a very large difference. - Line 89: should “once” be “One” or better “A”? Otherwise, I am confused by this sentence. - Line 106 and Table S4, what are the units (not just for K+ but for all reported quantities on the table)? - Equation 1: I the paper the authors use the term BC, and ncore, but in the summation they use nsoot, I would suggest having a single notation. - Line 138: “EVD is obtained in Methods” what does that mean? - Lines 145-147: It is not clear to me why, for condensation alone, Dp/Dc would approach 1 for large Dc. - Line 152: “that we observed the similar phenomenon” reads awkward, please rephrase. - Table S5: I have difficulty understanding the numbers. What do the % represent? The number fraction of particles (in %)? If so, why do the fractions not add up to 100%, and why, instead, is the sum (what sum is that?) always 100%? - Lines 161 to 166: This section is not clear to me. - Line 167: Is the coagulation process alone causing an increase in ncore and Dc, or is it its stochastic nature? I would think the former. - Line 172: “and Dc also increased after coagulation among BC-containing particles” I don’t understand how that could be otherwise. Meaning if mass is conserved, during coagulation Dc has to increase; or am I missing something? - Lines 173-174: What evidence do the authors have to support this statement? I'm not saying I disagree with it, but just saying that it is as such, does not make it true. - Figure 2: This is from the TEM analysis, correct? Perhaps reiterate that in the caption. - Line 189: What is the rationale for using the peak value of 3 for Dp/Dc? - Lines 190 and 191: Perhaps change “under” to “for” - Line 194: I can see how the DEMA-DDA might work, but what’s the difference between DEMA-Mie and Mie core-shell? Does DEMA-Mie assume that all the BC cores are in a single core of the same mass at the center of the coating? If so, that is a core-shell; if not, how is the spherical symmetry necessary in Mie simulations being satisfied in the DEMA-Mie. I am sure this is clear in previous papers or in the method, but a short clarification here would help. - Line 200: “center-off”, what region is that? The center or off the center? If the second, how off is off? From the figure, the impression is that the intensity is particularly enhanced at the forward inside edge of the host particle. - Figure 3, caption: “black line marks the electric field” do the lines mark the electric field, or do they mark the location of the BC ores, and the color within it indicates the electric field intensity? I would think the second. Also, here the wavelength is mentioned as 550 nm; is that the only wavelength that has been explored? I would also mention that information in the text when discussing Figure 3. For figure 3(a), do the circles correspond to Ncore>1 and the squares to Ncore=1? If so, I would use the same color for the squares and for the label “Ncore=1” which, now is much darker. Similar comment for figure 3(c), where the color of the label “Core-Shell Mie model” is much darker than the actual color in the plot - Line 215: What is the meaning of the term “equilibrium state” in this context? Perhaps the authors mean a plateau or a stationary point? - Figure S8: The bubble size indicates what? And the lines? - Figure 4: What does Ncal indicate, I mean what does “cal” stay for? In general, the descriptions in the caption are very cryptic, and the figure is of difficult interpretation. - Line 246: “Ncal is defined as Dc 3/1000000, and details are in Methods” that is fine that details are in the methods section, but a half sentence explaining what this represents and why it is defined as such here would help readability. - Line 246: “RF1”, what is the “1” for? - SHAP: For non-experts, this tool might be difficult to interpret. Perhaps the authors could add a line or two near its introduction, around line 248 or 249 to explain a little bit what that represents and how to interpret it. - Line 253: What is the physical interpretation/origin of such thresholds? - Lines 255-256: “These results suggest that M1×M2 can act as a “threshold effect” feature that improves RF interpretability” how does that improve interpretability? This sentence sounds very mysterious. - Line 258: “illustrating its interaction with the primary driving features.” “Its” refers to what? And what are “the primary features”? The sentence is not very illuminating as it is. - Line 276: Now it seems like the number after RF indicates some sort of scenario, while that might be clear from the methods section, here, this aspect is still confusing. I would suggest adding a sentence about this aspect at the beginning of the session. Also what this sentence seems to suggest is that RF1 uses ncore, while RF2 does not, but the fact that RF1 uses ncore was not clear when RF1 was introduced, so I am not clear did RF1 uses ncore or not; that should be made clear at the first discussion of RF1. - Line 281: What I gather here is that global models cannot provide ncore, which is understandable, but can provide M1XM2, how is that done? - Lines 299-300: “are mainly occurred”, please rephrase. - Lines 304-305: This is probably related to the fact that in such high loading regions, coagulation is enhanced. Is that the correct interpretation? - Line 309-311: This is not necessarily true; meaning, that the sentence it reads as if all the previous work focuses on this single issue. In the literature there is an emphasis on non-spherical shape, that is partially true, but also on other aspects such as the non-uniform distribution of coating on particles, as well as the existence of other mixing configurations (beside coatings), the morphology of the BC particles, the effect of water coatings, the uncertainty on the index of refraction for example while soot is maturing, etc. - Line 312: What does “contribution ratio” mean? - Line 316: “addresses the gap” that might be true, but there could also be many other reasons, so I would add “might” in front of “address” - Line 371: I am not clear how the bootstrap is used and for what, meaning the data were resampled 1000 times, and then what was done with that? - Line 372: Why is the ratio as such? Meaning, why is it half of unity? Is there a plausible explanation for that? Is that due to the change in shape of the particle upon impaction on the substrate? - Line 380: So, this is a Maxwell Garnett mixing rule, correct? I think this should be mentioned explicitly here. - Lines 386-387: I do not quite understand this sentence. - Line 435: What are the eight BC mixing state bins? How are they defined? Do they account for different morphological configurations or just for the core to coating fraction? - Line 464: “necessitated neglecting Eabs” I don’t understand what that means. In general, this sentence is unclear.", "reviewer_1_attached": "Response to Reviewer 1\nWe are grateful for Reviewer 1’s comments. Those comments are all valuable and\nhelpful for improving our paper and English writing. We answered the comments\ncarefully and have made corrections in the submitted manuscript. The corrections and\nthe responses are as following:\nIn the revised manuscript, the red colour was marked as the revised places.\nGeneral comments: The study presented in this manuscript is comprehensive and\nprovides a detailed dataset for BC-containing particles from wildfire events. I applaud\nthe authors for the extensive scope of work covered and the significance of the study to\nthe atmospheric science research community. Unfortunately, the clarity of the\nmanuscript is significantly compromised by very poor grammar and awkward phrasing\nwhich obscure the intended meaning and make it difficult to follow the study’s\noutcomes. The extensive detail is valuable, but the need to frequently consult the\nSupporting Information detracts from the readability of the main text. I strongly\nrecommend that the authors consider using a writing service to help bring the\nmanuscript in line with the language standards expected by Nature Communications.\nAs a result, I am recommending Major Revisions to ensure the clarity and overall\nreadability of the manuscript.\nReply: Thank you for your recognition and comments. We made numerous revisions\nto improve clarity and readability. We also obtained the writing service from Springer\nNature, and Fig. R1 is a certificate. Taking into account the insightful comments from\nreviewers and editors, we restructured our manuscript to minimize reliance on the\nSupporting Information for clarity.\n[REDACTED]\nFigure R1. A certificate confirming that our work was edited by Springer Nature\nEditing Service\nComment 1: Line 54: Either use the range (36-84%) or the approximate value (60%).\nDo not use both. It is confusing.\nReply: Thank you. We correct it in Lines 58-60, and check the whole manuscript:\nMain context: “BC AAOD accounts for 36%–84% of the global annual mean of\ntotal absorption aerosol optical depth (AAOD) at the 550-nm wavelength, indicating\nthat BC yields a relatively notable warming effect”\nComment 2: Lines 65-66: “…current studies…” yet the authors only reference one\nstudy (ref 23). This is misleading.\nReply: Thanks. We corrected it in Lines 74-75: “Chen, et al.1 further proposed that\nnonsphericity contributes only marginally (3.3%) to global BC AAOD.”\nComment 3: Line 74: “over half of the bulk BC mass concentration and light\nabsorption.” I believe both studies referenced reported <50% for light absorption and\n~50% for the bulk BC mass concentration. Do not misrepresent results.\nReply: Thanks for your key comment. We reviewed two observational literatures2,3.\nAlthough two studies focused on different particle diameters (i.e., > 700 nm and 1000\nnm, respectively) in their texts, some charts provided us with detailed information:\n Fig. R2 comes from Zhao, et al. 2. Zhao, et al. 2 proposed that the contribution of\nparticles larger than 700 nm to bulk BC mass and light absorption could both reach\nup to 50% in Fig. R2. Fig. R2 also reflects strong temporal fluctuations in the\npollution case.\n Fig. R3 comes from Wang, et al. 3. Wang, et al. 3 proposed that particles larger than\n1 μm contribute to about 50% of all BC mass and 23% of BC light absorption.\nMoreover, we can find that particles larger than 400 nm may contribute to ~50%\nof BC mass and light absorption.\nAs you suggested, “over half” may not be clear enough for readers to understand\nour focus clearly. Therefore, we revised it as follows:\n“However, field studies roughly estimated that the contribution of BC-containing\nparticles with D > 400 nm to the bulk BC mass and light absorption level could reach\np\n50%, especially during polluted periods” (Details are in Lines 83-86)\n[REDACTED]\nFigure R2. Equivalent BC mass size distribution (eBCMSD) from 31 October to 6\nNovember 2021 in Beijing. (a) The solid black line is the geometric mean diameter.\n(b) the corresponding f (solid red line), f (dashed green line) as well as\nm,>700 ab,>700\nf (dotted black line with triangle marker), which represent the contribution of\nDRF,>700\nparticles larger than 700 nm to mass, light absorption, and direct radiative effect. This\nfigure comes from Zhao, et al. 2.\n[REDACTED]\nFigure R3. (a) Black carbon (BC) size mass distribution in Beijing; (b) Monthly\nmean accumulated fraction of absorption with the particle diameter (D ). The\np\nshading represents quartiles. The red and blue lines are absorption fractions using\ninternal and external mixing assumptions, respectively. The shaded area represents the\n25th to 75th percentile. The dots are calculated by using an external assumption based\non reference data. This figure comes from Wang, et al. 3.\nComment 4: Lines 108-110: “reanalysis data” Is this a reference to MERRA2?\nReply: Thanks for your comment. Yes! This is from MERRA2, and we correct it in\nLines 123-125: “Via the use of Modern-Era Retrospective analysis for Research and\nApplications version 2 (MERRA2), Supplementary Fig. 2 shows that the average BC\nconcentrations increased from 0.54 μg m-3 during less polluted periods to 1.53 μg m-3\nduring highly polluted periods, driving a corresponding increase in BC AAOD from\n0.003 to 0.062.”\nComment 5: Line 133: How do the authors differentiate between a super aggregate or\nagglomerate BC and a multi-core BC?\nReply: We sincerely appreciate your feedback. In our study, a black carbon (BC)\naggregate within an internally mixed particle is defined as a single BC core. Under a\nstrong electron beam in a transmission electron microscope (TEM), we can\nprogressively remove non-refractory aerosol coatings. This process allows us to\ndetermine the independence of each BC aggregate core, confirming that they are not\ninterconnected as shown in Supplementary Figure 19. This technique is widely used to\nanalyse the morphology of coated BC cores, e.g., Ching, et al. 4 and Adachi, et al. 5.\nFor clarity, we added a detailed explanation in the Methods section, specifically on Lines\n445-447, where we elaborate on this procedure: “The strong electron beam of TEM\ncould easily damage the coating of the coated BC particles6. Therefore, we determined\nthe morphology and number of BC cores in an individual particle with this technique,\nas shown in Supplementary Fig. 19.”\nSupplementary Figure 19. Transmission electron microscopy images of aerosol\nparticles at different resolutions. (a) Low magnification and low electron beam\nintensity. (b) a magnified view of the marked particles. (c) Under high-resolution and\nhigh electron beam, this marked particle is identified as a multi-core BC particle. (d)\nThe high electron beam gradually removes the coating and shows that these BC cores\nare independent of each other.\nComment 6: Line 142: Italicize equation parameters so they can be differentiated when\nplaced within text (general comment). For instance, Dp instead of Dp.\nReply: Thanks for your comment. We corrected it and revised the whole manuscript to\nmeet the polices of Nature Communications.\nComment 7: Lines 194, 206-207, 209, and others: The authors need to properly\ndifferentiate their references to core-shell as a mixing state representation and as an\noptical model especially in the context of the DDA simulations. The same applies to\nDEMA. The section on optical enhancement was very confusing because of this. A\nmixing rule or approximation like DEMA (or Maxwell-Garnett) is NOT a mixing state.\nTable S7 should be delineated as single core and multi-core for better inference.\nReply: I truly appreciate your comment. We made the necessary correction to Table S7,\nwhich is now Supplementary Table 6. To clarify the distinction between mixing state\nrepresentation and optical models, we added a clear explanation: the Core-Shell and\nDEMA are mixing state rules, whereas Core-Shell Mie and DEMA-Mie refer to the\noptical models. Furthermore, we emphasized that the DEMA mixing rule does not\ndescribe a mixing state but rather a physical approximation where one or multiple\nspherical BC cores are randomly distributed within a single particle.\nWe restructured the “Optical Absorption Enhancement” section and the associated\nsupplementary material to facilitate easier and more direct comparison of the results\nfrom Core-Shell Mie, DEMA-Mie, and DDA simulations for our readers.\nDetailed information can be found in Lines 225-266 of the manuscript:\n“The EMBS tool provides an interactive interface for constructing complex BC\nmixing structures that are input into the DDA module to calculate their optical\nproperties7. BC-containing particles were assigned different n values and positions\ncore\nin the EMBS–DDA simulations, whereas D /D was set to the average value (i.e.,\np c\nD /D =3 in Supplementary Fig. 5) to represent the common scenario for D > 400 nm.\np c p\nUnder the assumption that BC cores were randomly located in an individual particle,\nthe probability of core occurrence obviously increased from the centre to off-centre\npositions. Specifically, BC aerosols with a single core or multiple cores at the particle\ncentre represented idealized and specialized scenarios, respectively. Innumerable\npossible core locations would increase the DDA simulation burden of every\nconfiguration excessively, even with fixed input parameters of D , D , and n .\np c core\nHerein, we simulated three representative scenarios using DDA:\nDDA : Setting n =1 with the core at the centre. The simulated E value was\n1 core abs\nalmost equal to that of the Core‒Shell Mie model, as shown in Figs. 3a–b.\nDDA : Setting n >1 with multiple cores concentrated near the centre, as shown\n2 core\nin Fig. 3b.\nDDA : Setting n >1 with cores distributed at various off-centre positions, as\n3 core\nshown in Fig. 3b. The detailed settings are provided in Supplementary Text 4,\nSupplementary Tables 5–6, and Supplementary Fig. 6. In addition, we calculated E\nabs\nvia the Core–Shell Mie and DEMA–Mie models on the basis of the same input\nparameters (D , D , and n ) used in DDA.”\np c core\nFigure 3. Light absorption enhancement (E ) at 550 nm of different types of BC-\nabs\ncontaining particles. (a) Simulation results of different methods for different D .\nc\nInput\nparameters and simulation outputs are detailed in Supplementary Tables 5 and 6. (b)\nSchematic diagrams illustrate typical mixing states for DDA , DDA , and DDA\n1 2 3\nsimulations, each with distinct core configurations. (c1-c6) Near-field calculations were\nemployed to determine the electric field intensity at 550 nm for secondary aerosol\nparticles (c1) and various DDA scenarios (DDA : c2"}, "author_response": "Response to Reviewer 1 We are grateful for Reviewer 1’s comments. Those comments are all valuable and helpful for improving our paper and English writing. We answered the comments carefully and have made corrections in the submitted manuscript. The corrections and the responses are as following: In the revised manuscript, the red colour was marked as the revised places. General comments: The study presented in this manuscript is comprehensive and provides a detailed dataset for BC-containing particles from wildfire events. I applaud the authors for the extensive scope of work covered and the significance of the study to the atmospheric science research community. Unfortunately, the clarity of the manuscript is significantly compromised by very poor grammar and awkward phrasing which obscure the intended meaning and make it difficult to follow the study’s outcomes. The extensive detail is valuable, but the need to frequently consult the Supporting Information detracts from the readability of the main text. I strongly recommend that the authors consider using a writing service to help bring the manuscript in line with the language standards expected by Nature Communications. As a result, I am recommending Major Revisions to ensure the clarity and overall readability of the manuscript. Reply: Thank you for your recognition and comments. We made numerous revisions to improve clarity and readability. We also obtained the writing service from Springer Nature, and Fig. R1 is a certificate. Taking into account the insightful comments from reviewers and editors, we restructured our manuscript to minimize reliance on the Supporting Information for clarity. [REDACTED] Figure R1. A certificate confirming that our work was edited by Springer Nature Editing Service Comment 1: Line 54: Either use the range (36-84%) or the approximate value (60%). Do not use both. It is confusing. Reply: Thank you. We correct it in Lines 58-60, and check the whole manuscript: Main context: “BC AAOD accounts for 36%–84% of the global annual mean of total absorption aerosol optical depth (AAOD) at the 550-nm wavelength, indicating that BC yields a relatively notable warming effect” Comment 2: Lines 65-66: “…current studies…” yet the authors only reference one study (ref 23). This is misleading. Reply: Thanks. We corrected it in Lines 74-75: “Chen, et al.1 further proposed that nonsphericity contributes only marginally (3.3%) to global BC AAOD.” Comment 3: Line 74: “over half of the bulk BC mass concentration and light absorption.” I believe both studies referenced reported <50% for light absorption and ~50% for the bulk BC mass concentration. Do not misrepresent results. Reply: Thanks for your key comment. We reviewed two observational literatures2,3. Although two studies focused on different particle diameters (i.e., > 700 nm and 1000 nm, respectively) in their texts, some charts provided us with detailed information:  Fig. R2 comes from Zhao, et al. 2. Zhao, et al. 2 proposed that the contribution of particles larger than 700 nm to bulk BC mass and light absorption could both reach up to 50% in Fig. R2. Fig. R2 also reflects strong temporal fluctuations in the pollution case.  Fig. R3 comes from Wang, et al. 3. Wang, et al. 3 proposed that particles larger than 1 μm contribute to about 50% of all BC mass and 23% of BC light absorption. Moreover, we can find that particles larger than 400 nm may contribute to ~50% of BC mass and light absorption. As you suggested, “over half” may not be clear enough for readers to understand our focus clearly. Therefore, we revised it as follows: “However, field studies roughly estimated that the contribution of BC-containing particles with D > 400 nm to the bulk BC mass and light absorption level could reach p 50%, especially during polluted periods” (Details are in Lines 83-86) [REDACTED] Figure R2. Equivalent BC mass size distribution (eBCMSD) from 31 October to 6 November 2021 in Beijing. (a) The solid black line is the geometric mean diameter. (b) the corresponding f (solid red line), f (dashed green line) as well as m,>700 ab,>700 f (dotted black line with triangle marker), which represent the contribution of DRF,>700 particles larger than 700 nm to mass, light absorption, and direct radiative effect. This figure comes from Zhao, et al. 2. [REDACTED] Figure R3. (a) Black carbon (BC) size mass distribution in Beijing; (b) Monthly mean accumulated fraction of absorption with the particle diameter (D ). The p shading represents quartiles. The red and blue lines are absorption fractions using internal and external mixing assumptions, respectively. The shaded area represents the 25th to 75th percentile. The dots are calculated by using an external assumption based on reference data. This figure comes from Wang, et al. 3. Comment 4: Lines 108-110: “reanalysis data” Is this a reference to MERRA2? Reply: Thanks for your comment. Yes! This is from MERRA2, and we correct it in Lines 123-125: “Via the use of Modern-Era Retrospective analysis for Research and Applications version 2 (MERRA2), Supplementary Fig. 2 shows that the average BC concentrations increased from 0.54 μg m-3 during less polluted periods to 1.53 μg m-3 during highly polluted periods, driving a corresponding increase in BC AAOD from 0.003 to 0.062.” Comment 5: Line 133: How do the authors differentiate between a super aggregate or agglomerate BC and a multi-core BC? Reply: We sincerely appreciate your feedback. In our study, a black carbon (BC) aggregate within an internally mixed particle is defined as a single BC core. Under a strong electron beam in a transmission electron microscope (TEM), we can progressively remove non-refractory aerosol coatings. This process allows us to determine the independence of each BC aggregate core, confirming that they are not interconnected as shown in Supplementary Figure 19. This technique is widely used to analyse the morphology of coated BC cores, e.g., Ching, et al. 4 and Adachi, et al. 5. For clarity, we added a detailed explanation in the Methods section, specifically on Lines 445-447, where we elaborate on this procedure: “The strong electron beam of TEM could easily damage the coating of the coated BC particles6. Therefore, we determined the morphology and number of BC cores in an individual particle with this technique, as shown in Supplementary Fig. 19.” Supplementary Figure 19. Transmission electron microscopy images of aerosol particles at different resolutions. (a) Low magnification and low electron beam intensity. (b) a magnified view of the marked particles. (c) Under high-resolution and high electron beam, this marked particle is identified as a multi-core BC particle. (d) The high electron beam gradually removes the coating and shows that these BC cores are independent of each other. Comment 6: Line 142: Italicize equation parameters so they can be differentiated when placed within text (general comment). For instance, Dp instead of Dp. Reply: Thanks for your comment. We corrected it and revised the whole manuscript to meet the polices of Nature Communications. Comment 7: Lines 194, 206-207, 209, and others: The authors need to properly differentiate their references to core-shell as a mixing state representation and as an optical model especially in the context of the DDA simulations. The same applies to DEMA. The section on optical enhancement was very confusing because of this. A mixing rule or approximation like DEMA (or Maxwell-Garnett) is NOT a mixing state. Table S7 should be delineated as single core and multi-core for better inference. Reply: I truly appreciate your comment. We made the necessary correction to Table S7, which is now Supplementary Table 6. To clarify the distinction between mixing state representation and optical models, we added a clear explanation: the Core-Shell and DEMA are mixing state rules, whereas Core-Shell Mie and DEMA-Mie refer to the optical models. Furthermore, we emphasized that the DEMA mixing rule does not describe a mixing state but rather a physical approximation where one or multiple spherical BC cores are randomly distributed within a single particle. We restructured the “Optical Absorption Enhancement” section and the associated supplementary material to facilitate easier and more direct comparison of the results from Core-Shell Mie, DEMA-Mie, and DDA simulations for our readers. Detailed information can be found in Lines 225-266 of the manuscript: “The EMBS tool provides an interactive interface for constructing complex BC mixing structures that are input into the DDA module to calculate their optical properties7. BC-containing particles were assigned different n values and positions core in the EMBS–DDA simulations, whereas D /D was set to the average value (i.e., p c D /D =3 in Supplementary Fig. 5) to represent the common scenario for D > 400 nm. p c p Under the assumption that BC cores were randomly located in an individual particle, the probability of core occurrence obviously increased from the centre to off-centre positions. Specifically, BC aerosols with a single core or multiple cores at the particle centre represented idealized and specialized scenarios, respectively. Innumerable possible core locations would increase the DDA simulation burden of every configuration excessively, even with fixed input parameters of D , D , and n . Herein, p c core we simulated three representative scenarios using DDA: DDA : Setting n =1 with the core at the centre. The simulated E value was 1 core abs almost equal to that of the Core‒Shell Mie model, as shown in Figs. 3a–b. DDA : Setting n >1 with multiple cores concentrated near the centre, as shown 2 core in Fig. 3b. DDA : Setting n >1 with cores distributed at various off-centre positions, as 3 core shown in Fig. 3b. The detailed settings are provided in Supplementary Text 4, Supplementary Tables 5–6, and Supplementary Fig. 6. In addition, we calculated E abs via the Core–Shell Mie and DEMA–Mie models on the basis of the same input parameters (D , D , and n ) used in DDA.” p c core Figure 3. Light absorption enhancement (E ) at 550 nm of different types of BC- abs containing particles. (a) Simulation results of different methods for different D . Input c parameters and simulation outputs are detailed in Supplementary Tables 5 and 6. (b) Schematic diagrams illustrate typical mixing states for DDA , DDA , and DDA 1 2 3 simulations, each with distinct core configurations. (c1-c6) Near-field calculations were employed to determine the electric field intensity at 550 nm for secondary aerosol particles (c1) and various DDA scenarios (DDA : c2, DDA : c3-4, DDA : c5-6), with 1 2 3 D = 600 nm. White lines represent the border of particles, and black line marks the p location of the BC cores. The colour represents the normalized electric field intensity versus the incident light. “We found that the E values for n = 1 were 12%–205% lower than those for abs core n > 1, particularly in DDA scenarios (Supplementary Table 6). Fig. 3a shows that core 3 the DEMA–Mie could represent the average increasing trend in DDA . Notably, the 3 DEMA–Mie model accounted for scenarios with n ≥ 1 in which the cores were core randomly distributed, but their exact locations were not specified in DDA (Figure 3). We compared the E ranges (from minimum to maximum) obtained via the DEMA– abs Mie and DDA simulations. Fig. 3b shows that the E range from the DDA 3 abs 3 simulations agreed well with that from the DEMA–Mie model. The results further confirmed that DEMA–Mie was efficient for simulating the average impact of n at core random positions on E .” abs Supplementary Table 6. Simulation results of EMBS-DDA Eabs under Eabs under Dp, nm Dc, nm Core-Shell ncore DEMA-Mie Shape model Type Locationb Eabs under DDAa Relative differencec Mie modela modela 300 100 2.46 1 2.90 S11 DDA1 centre 2.44 - 600 200 1.87 1 3.10 S11 DDA1 centre 1.87 - 900 300 1.93 1 4.1 S11 DDA1 centre 1.92 - 1200 400 1.85 1 5.14 S11 DDA1 centre 1.85 - 300 100 - 2 2.90 S21 DDA2 centre-off 3.00 23% 600 200 - 2 3.10 S21 DDA2 centre-off 2.51 34% 900 300 - 2 4.11 S21 DDA2 centre-off 2.49 30% 1200 400 - 2 5.14 S21 DDA2 centre-off 2.53 37% 300 100 - 2 2.88 S22 DDA3 centre-off 3.32 36% 600 200 - 2 3.04 S22 DDA3 centre-off 3.35 79% 900 300 - 2 3.95 S22 DDA3 centre-off 3.70 93% 1200 400 - 2 4.88 S22 DDA3 centre-off 3.79 105% 300 100 - 2 2.88 S23 DDA3 centre-off 3.44 41% 600 200 - 2 3.04 S23 DDA3 centre-off 4.35 133% 900 300 - 2 3.95 S23 DDA3 centre-off 5.58 191% 1200 400 - 2 4.88 S23 DDA3 centre-off 5.64 205% 300 100 - 2 2.88 S24 DDA3 centre-off 3.28 34% 600 200 - 2 3.04 S24 DDA3 centre-off 4.12 121% 900 300 - 2 3.95 S24 DDA3 centre-off 5.04 163% 1200 400 - 2 4.88 S24 DDA3 centre-off 5.42 193% 300 100 - 3 2.88 S31 DDA2 centre-off 2.73 12% 600 200 - 3 3.04 S31 DDA2 centre-off 2.31 24% 900 300 - 3 3.95 S31 DDA2 centre-off 2.34 22% 1200 400 - 3 4.88 S31 DDA2 centre-off 2.42 31% 300 100 - 3 2.87 S32 DDA3 centre-off 2.97 22% 600 200 - 3 3.01 S32 DDA3 centre-off 2.80 50% 900 300 - 3 3.88 S32 DDA3 centre-off 3.04 58% 1200 400 - 3 4.76 S32 DDA3 centre-off 3.30 78% 300 100 - 3 2.87 S33 DDA3 centre-off 2.97 22% 600 200 - 3 3.01 S33 DDA3 centre-off 3.57 91% 900 300 - 3 3.88 S33 DDA3 centre-off 4.65 142% 1200 400 - 3 4.76 S33 DDA3 centre-off 4.92 166% 600 200 - 4 2.87 S41 DDA2 centre-off 2.29 22% 900 300 - 4 3.01 S41 DDA2 centre-off 2.33 21% 1200 400 - 4 3.88 S41 DDA2 centre-off 2.38 29% 600 200 - 4 4.76 S42 DDA3 centre-off 3.10 66% 900 300 - 4 3.00 S42 DDA3 centre-off 3.98 107% 1200 400 - 4 3.84 S42 DDA3 centre-off 4.35 135% 900 300 - 6 4.70 S61 DDA2 centre-off 2.46 28% 1200 400 - 6 3.00 S61 DDA2 centre-off 2.58 39% 900 300 - 6 3.84 S62 DDA3 centre-off 3.30 72% 1200 400 - 6 4.70 S62 DDA3 centre-off 3.74 102% a: Using Dp and Dc to run Core-Shell Mie model; Using Dp, Dc, and ncore to run DEMA-Mie model; Using Dp, Dc, and ncore to build various shape model and then run DDA. b: Location of BC cores. c: The DDA simulations with ncore > 1 are compared to the simulations with ncore = 1. Comment 8: Lines 194-195: There is no increasing trend using the CS-model especially for D > 200nm. This statement is misleading. p Reply: Thanks for this important comment. A similar upward trend in E abs_DEMA was observed for D > 200 nm when simulated with the DEMA-Mie model, whereas E c abs_CS plateaued at approximately 2 for D > 200 nm using the Core-Shell Mie model. c Recognizing potential overlap between Figures 3 and 4, we made adjustments to differentiate them more clearly. Corresponding content has been revised for enhanced clarity and brevity, with specifics detailed on Lines 224-266, which is shown in Comment 7. Comment 9: Lines 218-219: I do not see how this conclusion follows from the preceding sentence. Reply: Thanks for your comments. Overall, this conclusion is based on a comprehensive comparison of DDA, Core Shell Mie, and DEMA Mie. We restructured this section to enhance clarity and facilitate readers’ understanding. We also presented this conclusion in a separate paragraph to emphasize that it is based on a comprehensive summary rather than solely derived from the preceding sentence. Details are in Lines 204-276, so we summarize the main ideas below: The biggest difference between the Core-Shell Mie model and the DEMA-Mie model is whether to keep the E at ~2 for particles with M ×M =1 (i.e., D > 200 nm abs 1 2 c and D > 400 nm). The DDA simulation results indicate that characterizing BC as one p central spherical core cannot effectively absorb light, thus underestimating E . abs However, even after specifying the input parameters D , D , and n , the potential p c core positions of these cores within the particle remain virtually innumerable, precluding DDA from exhaustively simulating all possible scenarios. DDA simulation results show that the DEMA-Mie model can capture the average trend of DDA simulations and has a similar range of E . So we believe that the DEMA-Mie model is a suitable and abs efficient tool for simulating the optical properties of multi-core BC particles, offering a favourable balance between computational efficiency and physical representation 11 compared to the DDA and Core-Shell Mie models. We made adjustments to facilitate readers with a wider range of disciplinary backgrounds to understand and compare the differences between the Core-Shell Mie model, DEMA-Mie model, and DDA model. Details are in Lines 224-266. Figure 3, as well as related supplementary materials, which are shown in Comment 7. Comment 10: Line 245: There are several instances in the manuscript and SI where E is written as E . Check all variables and make sure they are written abs-DEMA abs_DEAM properly. Reply: Thank you! We correct and check it in the whole manuscript. Comment 11: Line 246: Why use N ? The capitalization of the letter “N” suggests cal particle number concentration. Why not n ? Keep the terms consistent. core,cal Reply: Thanks for your useful comment. We finally use n in the whole manuscript. core,cal Comment 12: Line 255: I don’t think “threshold” is the right word to use here. “The M ×M feature improves RF interpretability.” Simple and concise. 1 2 Reply: Thanks! Your suggestion is more concise. We further adjusted this section and added detailed explanations on the interpretability, ensuring that readers can follow our reasoning without needing to refer extensively to supplementary materials. More details can be found in Lines 328-336: “To enhance the interpretability of the RF predictions, we visualized the SHAP values corresponding to each feature value. Figs. 4e–i show the driving patterns of these primary features on E . Each point represents an individual particle, and its abs_DEMA colour is indicative of its M ×M value to illustrate how M ×M influences the driving 1 2 1 2 patterns of these features (Figs. 4e-i). Fig. 4e shows that n was positively correlated core with its SHAP values only when M ×M = 1; otherwise, the SHAP values were 1 2 approximately 0. These results suggested that the impact of n on E was the core abs_DEMA most significant for particles with M ×M = 1.” 1 2 12 Comment 13: Line 270-271: Very subjective statement! The purported “V-shape” is not evident in Figure 3a. Reply: Thank you! We redesigned Fig. 3 in the manuscript and incorporated simulation results of the Core Shell Mie and DEMA Mie models with the same input parameters as DDA for ease of comparison. In the new Fig. 3, we no longer use the very subjective statement “V-shape”, but emphasize that DEMA-Mie can capture the average increasing trend of DDA. Details are in Lines 257-266 of the manuscript: “We found that the E values for n = 1 were 12%–205% lower than those for abs core n > 1, particularly in DDA scenarios (Supplementary Table 6). Fig. 3a shows that core 3 the DEMA–Mie could represent the average increasing trend in DDA . Notably, the 3 DEMA–Mie model accounted for scenarios with n ≥ 1 in which the cores were core randomly distributed, but their exact locations were not specified in DDA (Figure 3). We compared the E ranges (from minimum to maximum) obtained via the DEMA– abs Mie and DDA simulations. Fig. 3b shows that the E range from the DDA 3 abs 3 simulations agreed well with that from the DEMA–Mie model. The results further confirmed that DEMA–Mie was efficient for simulating the average impact of n at core random positions on E .” abs 13 Figure 3. Light absorption enhancement (E ) at 550 nm of different types of BC- abs containing particles. (a) Simulation results of different methods for different D . Input c parameters and simulation outputs are detailed in Supplementary Tables 5 and 6. (b) Schematic diagrams illustrate typical mixing states for DDA , DDA , and DDA 1 2 3 simulations, each with distinct core configurations. (c1-c6) Near-field calculations were employed to determine the electric field intensity at 550 nm for secondary aerosol particles (c1) and various DDA scenarios (DDA : c2, DDA : c3-4, DDA : c5-6), with 1 2 3 D = 600 nm. White lines represent the border of particles, and black line marks the p location of the BC cores. The colour represents the normalized electric field intensity versus the incident light. Comment 14: Line 375: What “tool” is being referred to here? Reply: Thanks for this comment! We believe that “proxy variables” may be easier for readers to understand than “tool features”. In fact, we hope to use simple M and M to 1 2 represent the size characteristics of multi-core BC particles and their subsequent impact on E . Details are as follows: abs  Lines 209-211: “To facilitate parameterization, we defined two proxy features (M 1 and M ), as expressed in Equation (1), to represent the primary size characteristics 2 of multi‒core BC particles and their subsequent impact on E .” abs  Lines 322-323: “By comparing the prediction performance levels and SHAP values of RF1 and RF2, we determined whether M ×M could serve as a proxy for the 1 2 impact of n on E .” core abs_DEMA  Lines 349-350: “We concluded that M ×M could serve as a proxy for the impact 1 2 of n on E .” core abs_DEMA  Lines 500-502: “Therefore, it is essential to quantify the potential nonlinear impact of n on E and to develop an emulator for E via the use of other core abs_DEMA abs_DEMA features as proxies for n .” core 14 Comment 15: Lines 386-387: I do not see how the referenced works relate to this point of a non-integer value for the calculated n . Do fragments of BC particles within these core larger BC-containing particles (D > 400 nm) count as individual cores? p Reply: Thanks for your comment. The non-integer value is not a fragment of an individual BC core, but a hypothesis written by the model developers, because there was no observational evidence of n at the time of model development. Jacobson 8 core coupled DEMA into his global model to simulate the optical properties of cloud, and Fig. R4 shows that n can be calculated as a non-integer value in Jacobson 8. Matsui, core et al. 9 and Fierce, et al. 10 directly used DEMA for BC-containing particles. As shown in Fig. R5, n can also be calculated as a non-integer value. We reconstruct the core statement to make it easier for readers to understand. Details are as follows:  Lines 309-312: “In previous global simulations, the developers of DEMA8,9,11 used a noninteger n value (i.e., D 3/1000000) to approximate n . However, Wang, et al.12 proposed core,cal c core that a noninteger n value was implausible because observational evidence for core,cal multi‒core BC particles was lacking.”  Lines 478-481: “Owing to the lack of observational evidence for n , the developers of core DEMA8,9,11 used a noninteger n value (i.e., D 3/1000000) to approximate n . core,cal c core Individual particle analysis provided n values using TEM (e.g., n = 6 in Fig. 1b3).” core core 15 [REDACTED] Figure R4 A screenshot from Jacobson 8. The red box indicates that n was core calculated as a non-integer value. [REDACTED] Figure R5 A screenshot from Matsui, et al. 9. The red box indicates that n was core calculated as a non-integer value. 16 References: 1 Chen et al. Accounting for black carbon mixing state, nonsphericity, and heterogeneity effects in its optical property parameterization in a climate model. Journal of Geophysical Research: Atmospheres 129, e2024JD041135, (2024). 2 Zhao et al. Measurement report: size-resolved mass concentration of equivalent black carbon- containing particles larger than 700 nm and their role in radiation. Atmospheric Chemistry and Physics 23, 14889-14902, (2023). 3 Wang et al. Significant contribution of coarse black carbon particles to light absorption in North China Plain. Environmental Science & Technology Letters 9, 134-139, (2022). 4 Ching et al. Aerosol mixing state revealed by transmission electron microscopy pertaining to cloud formation and human airway deposition. npj Climate and Atmospheric Science 2, 22, (2019). 5 Adachi et al. Mixing states of light‐absorbing particles measured using a transmission electron microscope and a single‐particle soot photometer in Tokyo, Japan. Journal of Geophysical Research: Atmospheres 121, 9153-9164, (2016). 6 Wang et al. Fractal dimensions and mixing structures of soot particles during atmospheric processing. Environmental Science & Technology Letters 4, 487-493, (2017). 7 Wang et al. Nonlinear enhancement of radiative absorption by black carbon in response to particle mixing structure. Geophysical Research Letters 48, e2021GL096437, (2021). 8 Jacobson. Effects of externally-through-internally-mixed soot inclusions within clouds and precipitation on global climate. Journal of Physical Chemistry A 110, 6860-6873, (2006). 9 Matsui et al. Black carbon radiative effects highly sensitive to emitted particle size when resolving mixing-state diversity. Nature Communications 9, 3446, (2018). 10 Fierce et al. Black carbon absorption at the global scale is affected by particle-scale diversity in composition. Nature Communications 7, 12361, (2016). 11 Fierce et al. Toward reduced representation of mixing state for simulating aerosol effects on climate. Bulletin of the American Meteorological Society 98, 971-980, (2017). 12 Wang et al. Radiative forcing and climate response due to the presence of black carbon in cloud droplets. Journal of Geophysical Research: Atmospheres 118, 3662-3675, (2013). 17 Response to Reviewer 2 We are grateful for Reviewer 2’s comments. Those comments are all valuable and helpful for improving our paper and English writing. We answered the comments carefully and have made corrections in the submitted manuscript. The corrections and the responses are as follows: In the revised manuscript, the red colour is used to mark the revised places. General comments: The authors use TEM images of fire-affected air masses arriving to Yunnan province, China. They show that large BC-containing particles are likely to have multiple cores due to coagulation during long-range transport. They use dynamic effective medium approximation-Mie (DEMA-Mie) theory model to calculate absorption enhancement by the BC-containing particles that arrived at the sampling site. They then train a random forest model to predict absorption enhancement based on the DEMA-Mie model. They applied the random forest emulator to a global atmospheric model and found that bulk aerosol absorption increased 19%. This paper introduces a potentially important aspect of BC-containing particles that is not accounted for in global atmospheric models. However, some portions of the TEM analysis are unclear and there is a lack of observational evidence for increased absorption enhancement during the sampling period. I therefore think that this paper needs major revisions to address the following points. Reply: Thank you for your recognition and comments. We have made numerous revisions to improve clarity and readability. We have also obtained the writing service from Springer Nature, and Fig. R1 is a certificate. Based on the comments of reviewers and editors, we have restructured the manuscript to make the TEM analysis clearer. We improved the quality to be potentially published in Nature Communications. 1 [REDACTED] Figure R1. A certificate confirming that our work was edited by Springer Nature Editing Service Comment 1: Based on the trajectory analysis presented in Figure 1, it appears that the low-pollution and high-pollution cases are both largely originating in India. Are the low pollution samples affected by BC from urban pollution, while the high pollution cases are affected by urban and wildfire BC? Presumably, biomass burning BC will have larger core sizes and be more thickly coated than BC from fossil fuel combustion1. This means that biomass burning BC would be more likely to act as cloud condensation nuclei. Differences in the n distribution for low and high pollution cases separately may core provide evidence that coagulation is occurring in clouds or near source. Separate analysis of low and high pollution cases may show that multiple-core effects are more important for biomass burning BC than for BC from fossil fuel combustion. [1] Schwarz, J. P. et al. Measurement of the mixing state, mass, and optical size of individual black carbon particles in urban and biomass burning emissions. Geophysical Research Letters 35, (2008). Reply: Thank you. Due to space limitations, the original text was unable to present our results logically. We have made extensive modifications to enhance readability and scientific rigour. We addressed this comment in the following aspects:  The impact of local urban emissions 2 The observation station is situated in the Woyun Mountain area to the west of the city, at an altitude approximately 700 meters higher than that of the city (Fig. R2a). The wind roses of the city and observation station (Figs. R2b-c) show that the observation station is located upwind of the city. The surrounding condition of the station is shown in the Figs. R2d-e. Therefore, we proposed that the particles, especially for coated BC, are mainly from long-range transport. The effect of local emissions of BC is limited, as identified by fresh BC using transmission electron microscopy (TEM), which accounts for 14% of the whole analysed particles. This fraction is 18% in less polluted periods and 12% in highly polluted periods. [REDACTED] Figure R2. Environmental conditions near the observation stations. (a) Location. (b) Wind rose. Photographs of the sampling station taken in the west direction during less (d) and highly polluted periods (e), respectively.  Limitations of using D size to identify wildfire impacts and cloud process c during transport events Schwarz, et al. 1 proposed that biomass burning BC could have larger core sizes and be more thickly coated than BC from fossil fuel combustion. However, the aging scenario in this study is more complex: 3 (1)In addition to the urban and biomass burning scenarios, a third scenario exists, namely the background air as reported in Schwarz, et al. 1 and shown in Fig. R3. It should be noted that both urban and biomass burning emissions were subjected to ageing processes for less than 3.5 hours in Schwarz’s study. Furthermore, the results about urban emissions were specifically corrected by eliminating the influence of transported black carbon (BC) particles. Interestingly, the BC particles present in the background aerosols were also found to be coated with a thicker layer compared to those from urban emissions. [REDACTED] Figure R3. A screenshot from Schwarz, et al. 1. Red boxes mark the key details for this comment. Table R1 Geometric mean and geometric standard deviation (SD) for different types of black carbon (BC) Type Fresh BC n = 1 n = 2-3 n > 3 BC-containing core core core Less N total 239 257 64 16 576 polluted Geometric Mean 87.4 111.3 194.4 290.9 110.0 periods Geometric SD 1.6 1.5 1.5 1.5 1.7 Type Fresh BC n = 1 n = 2-3 n > 3 BC-containing core core core Highly N total 330 649 269 56 1304 polluted Geometric Mean 94.8 121.2 180.0 340.6 129.2 periods Geometric SD 1.6 1.5 1.5 1.4 1.7 4 (2) The average coating thickness is not for all BC-containing particles, but just for D from 190 to 210 nm in Schwarz, et al. 1. This is due to the limits of the single c particle soot photometer (SP2), and details are in Schwarz, et al. 1. Near the emission sources, the D of biomass burning BC is larger than vehicle c emission BC2,3. However, as emphasized in this study, D is not fixed as a steady value c but increases with coagulation during the ageing process. To compare less polluted and highly polluted periods, we use geometric mean and geometric standard deviation to describe the D distribution of different types of BC (Table R1). There are a few c differences in the size distribution of D during less/highly polluted periods. For n > c core 3, the geometric mean is 340.6 nm in highly polluted periods, slightly higher than 290.9 nm in less polluted periods. This may be because during highly polluted periods, more BC-containing particles participate in the coagulation during aerosol transport, leading to a greater increase in D . c Coagulation is determined by the size difference and the number concentration between two particles. According to the Brownian Coagulation Theory in Seinfeld and Pandis 4, the coagulation efficiency (K , cm3/h) between Dp and Dp is calculated as cog 1 2 Equation R1. 2𝑘𝑇 𝐷 𝐷 𝑝1 𝑝2 𝐾 = ( + +2)𝛽 (R1) 𝑐𝑜𝑔 3𝜇 𝐷 𝐷 𝑝2 𝑝1 where k is the Boltzmann constant, 1.380649×10-23 J/K. T is the 298.15 K, and we discuss the coagulation efficiency in the room temperature. μ is the viscosity of air, 1.83×10-4 g/(cm×s). β is the correction factor and varies from 0.014 for D = 1 nm, to pi practically 1 for 1000 nm. We neglect the kinetic effect, corresponding to β = 1 for monodisperse aerosols in air. The K is shown in Fig. R4. cog 5 Figure R4. Coagulation efficiency between two particles (Dp and Dp ). 1 2 We agree with you that the cloud process can enhance the coagulation process, because water uptake increases D of the coated BC particles. However, it is difficult p to directly attribute the minor differences between less and highly polluted periods to cloud processes (Table R1). On the other hand, while this study has focused on a typical wildfire transport event to discuss the abundance of multi-core BC particles and their impact on optical properties, it is important to emphasise that the coagulation, as one of the fundamental ageing mechanisms, broadly influences the mixing state of BC across various aging scenarios. This has been demonstrated by previous modelling studies such as Matsui, et al. 5 and Riemer, et al. 6. That is reason that we collaborated with other research groups to provide more detailed evidence on the ageing processes and characteristics of multi-core BC particles. We summarised the above response and added some comparative discussion in Lines 156-167: “During highly polluted periods, wildfire smoke after long-range transport could enhance the formation of multi‒core BC particles (Fig. 1a vs. Supplementary Fig. 3). However, few studies reported the size-dependent characteristics of multi‒core BC 6 particles. Instead, some studies considered the two-dimensional data characteristics of D /D and D and the size distribution of D to approximately assess the mixing states p c c c of BC aerosols in biomass burning smoke and wildfire smoke1,7,8. To obtain the equivalent volume diameters of individual BC-containing particles and their BC cores (i.e., D and D , respectively), we calibrated the equivalent circle diameters obtained p c from TEM images to the equivalent volume diameters via a regression relationship from atomic force microscopy measurements (refer to the Methods). For multi‒core BC particles, D represents the accumulation of all BC cores. Fig. 2a and Table 1 both c show that multi‒core BC particles became the dominant particle type at D > 200 nm.” c Comment 2: The methods used to determine n in this study are largely ambiguous. core For example, it is not clear that the particle shown in Fig. b3 contains six separate cores. Is coating removed from particles to reveal the distinct number of cores? Reply: Thanks for your key comment. In TEM, we use high-energy electron beams to remove the coatings within a few seconds. This process is the physical basis that enables TEM to reveal the internal mixing structures of particles. As shown in Supplementary Figure 19, an image of the particle is captured after the coating has been removed, allowing us to observe the morphology and count the number of BC cores within an individual particle. This method of individual particle analysis is widely used, e.g., Adachi, et al. 9 and Ching, et al. 10. However, these observational studies did not specifically focus on multi-core BC particles or their significant role in coagulation and light absorption processes. We added the necessary information in Lines 421-424: “The strong electron beam of TEM could easily damage the coating of the coated BC particles11. Therefore, we determined the morphology and number of BC cores in an individual particle with this technique, as shown in Supplementary Fig. 19.” 7 Supplementary Figure 19 Transmission electron microscopy images of aerosol particles at different resolutions. (a) Low magnification and low electron beam intensity. (b) a magnified view of the marked particles. (c) Under high-resolution and high electron beam, this marked particle is identified as a multi-core BC particle. (d) The high electron beam gradually removes the coating and shows that these BC cores are independent of each other. Comment 3: Are the authors using n ≈ N = D 3/1000000 to approximate n ? It core cal c core appears that this originates from a paper that is studying cloud droplet residuals, but there is no evidence provided that the particles here have undergone cloud processing. Reply: Thank you. Sorry, our sentence is not organised scientifically enough, which causes you to misunderstand our meaning. DEMA was originally introduced by Chylek, et al. 12 in the context of cloud droplet residuals. However, DEMA has also been applied to aerosol studies, as seen in the works of Fierce, et al. 13, Fierce, et al. 14, and Matsui, et al. 15. The significance of DEMA 8 extends beyond its role in cloud optical simulations. It also serves as a modification of the Bruggemann mixing rule for aerosols. Specifically, Equation R2 represents the DEMA mixing rule. Jacobson 16 proposed that the last term in Equation R2 was introduced to correct the Bruggemann mixing rule (Equation R3), which initially assumed infinitely small black carbon (BC) cores. 𝐷3 ∈ −∈ 𝐷3 ∈ −∈ 2𝜋 2 2 𝑐 𝑐𝑜𝑎𝑡 𝑐 𝑐 (1− )· + · + ( ) ·(∈ −∈) 𝐷3 ∈ +2 ∈ 𝐷3 ∈ +2 ∈ 45 𝜆 𝑐 𝑝 𝑐𝑜𝑎𝑡 𝑝 𝑐 (R2) 5 ∈ 18 ∈ (∈ −2 ∈) ·[1+ + 𝑐 ]·∫𝑟5𝑛(𝑟)𝑑𝑟 2 ∈ +3 ∈ (∈ +2 ∈)2 𝑐 𝑐 = 0 where r is the radius of a BC core, and n(r) is the number concentration of cores per unit radius interval. λ is the light wavelength, and this study selects 550 nm as a case. Here, ∈ and ∈ are the dielectric constant of BC and other aerosol components, 𝑐 𝑐𝑜𝑎𝑡 respectively.  Are the authors using n ≈ N = D 3/1000000 to approximate n ? core cal c core No, we use n from TEM to resolve Equation R2. Previous model study use core D 3/1000000 to approximate n such as Jacobson 16. Matsui, et al. 15 and Fierce, et al. c core 14. As a result, N can be calculated as a non-integer value, because all BC cores have cal the monodisperse size distribution, i.e., the diameter is 100 nm (e.g., Fig. R5). In the real world, n in each particle should be an integer, so the integral in Equation R3 can core be written as Equation R4. 3 𝐷3 𝐸𝑉𝐷 𝑐𝑜𝑟𝑒 = √ 𝑛 𝑐 𝑐𝑜𝑟𝑒 (R4) 𝐸𝑉𝐷5 𝑛 ∫𝑟5𝑛(𝑟)𝑑𝑟 = 𝑐𝑜𝑟𝑒 𝑐𝑜𝑟𝑒 { 32𝐷3 𝑝 where EVD is the average of BC cores. core 9 [REDACTED] Figure R5 A screenshot from Jacobson 16. The red box indicates that n was core calculated as a non-integer value. We reconstruct the statement to make it easier for readers to understand. Details are as follows:  Lines 309-312: “In previous global simulations, the developers of DEMA13,15,16 used a noninteger n value (i.e., D 3/1000000) to approximate n . However, Wang, et al.17 proposed core,cal c core that a noninteger n value was implausible because observational evidence for core,cal multi‒core BC particles was lacking.”  Lines 477-481: “Without the third term, Equation (4) transforms into the Bruggeman mixing rule12,16. Owing to the lack of observational evidence for n , the developers of core 10 DEMA13,15,16 used a noninteger n value (i.e., D 3/1000000) to approximate n . core,cal c core Individual particle analysis provided n values using TEM (e.g., n = 6 in Fig. 1b3).” core core Comment 4: Is there a comparison of the DEMA-Mie model with the DDA values? One could presumably calculate the DEMA-Mie absorption enhancement for the same particles that were used in the DDA calculations. I see that there are similar trends in Fig. 3a and 3c, but there seems to be a sharper decrease in DEMA-Mie absorption enhancement with increasing D for D < 200 nm than for DDA. c c Reply: Thanks for your comment. We reconstructed Fig. 3, as well as the corresponding supplementary materials, for the convenience of readers to compare Core-Shell Mie, DEMA-Mie, and DDA. Details are in Lines 222-266: “In this study, we first employed an electron-microscopy-to-BC simulation tool, coupled with discrete dipole approximation (EMBS–DDA)18 to visualize the effect of n on E of BC-containing particles. The EMBS tool provides an interactive core abs interface for constructing complex BC mixing structures that are input into the DDA module to calculate their optical properties18. BC-containing particles were assigned different n values and positions in the EMBS–DDA simulations, whereas D /D was core p c set to the average value (i.e., D /D =3 in Supplementary Fig. 5) to represent the p c common scenario for D > 400 nm. Under the assumption that BC cores were randomly p located in an individual particle, the probability of core occurrence obviously increased from the centre to off-centre positions. Specifically, BC aerosols with a single core or multiple cores at the particle centre represented idealized and specialized scenarios, respectively. Innumerable possible core locations would increase the DDA simulation burden of every configuration excessively, even with fixed input parameters of D , D , p c and n . Herein, we simulated three representative scenarios using DDA: core DDA : Setting n =1 with the core at the centre. The simulated E value was 1 core abs almost equal to that of the Core‒Shell Mie model, as shown in Figs. 3a–b. DDA : Setting n >1 with multiple cores concentrated near the centre, as shown 2 core in Fig. 3b. 11 DDA : Setting n >1 with cores distributed at various off-centre positions, as 3 core shown in Fig. 3b. The detailed settings are provided in Supplementary Text 4, Supplementary Tables 5–6, and Supplementary Fig. 6. In addition, we calculated E abs via the Core–Shell Mie and DEMA–Mie models on the basis of the same input parameters (D , D , and n ) used in DDA.” p c core Figure 3. Light absorption enhancement (E ) at 550 nm of different types of BC- abs containing particles. (a) Simulation results of different methods for different D . Input c parameters and simulation outputs are detailed in Supplementary Tables 5 and 6. (b) Schematic diagrams illustrate typical mixing states for DDA , DDA , and DDA 1 2 3 simulations, each with distinct core configurations. (c1-c6) Near-field calculations were employed to determine the electric field intensity at 550 nm for secondary aerosol particles (c1) and various DDA scenarios (DDA : c2, DDA : c3-4, DDA : c5-6), with 1 2 3 D = 600 nm. White lines represent the border of particles, and black line marks the p location of the BC cores. The colour represents the normalized electric field intensity versus the incident light. 12 “We found that the E values for n = 1 were 12%–205% lower than those for abs core n > 1, particularly in DDA scenarios (Supplementary Table 6). Fig. 3a shows that core 3 the DEMA–Mie could represent the average increasing trend in DDA . Notably, the 3 DEMA–Mie model accounted for scenarios with n ≥ 1 in which the cores were core randomly distributed, but their exact locations were not specified in DDA (Figure 3). We compared the E ranges (from minimum to maximum) obtained via the DEMA– abs Mie and DDA simulations. Fig. 3b shows that the E range from the DDA 3 abs 3 simulations agreed well with that from the DEMA–Mie model. The results further confirmed that DEMA–Mie was efficient for simulating the average impact of n at core random positions on E .” abs Supplementary Table 6 Simulation results of Core-Shell Mie, DEMA-Mie, and DDA Shape Relative D, nm D, nm E n E Type Locationb E under DDAa p c abs_cs core abs_DEMA abs model differencec 300 100 2.46 1 2.90 S11 DDA centre 2.44 - 1 600 200 1.87 1 3.10 S11 DDA centre 1.87 - 1 900 300 1.93 1 4.1 S11 DDA centre 1.92 - 1 1200 400 1.85 1 5.14 S11 DDA centre 1.85 - 1 centre- 300 100 - 2 2.90 S21 DDA 3.00 23% 2 off centre- 600 200 - 2 3.10 S21 DDA 2.51 34% 2 off centre- 900 300 - 2 4.11 S21 DDA 2.49 30% 2 off centre- 1200 400 - 2 5.14 S21 DDA 2.53 37% 2 off centre- 300 100 - 2 2.88 S22 DDA 3.32 36% 3 off centre- 600 200 - 2 3.04 S22 DDA 3.35 79% 3 off centre- 900 300 - 2 3.95 S22 DDA 3.70 93% 3 off centre- 1200 400 - 2 4.88 S22 DDA 3.79 105% 3 off centre- 300 100 - 2 2.88 S23 DDA 3.44 41% 3 off centre- 600 200 - 2 3.04 S23 DDA 4.35 133% 3 off centre- 900 300 - 2 3.95 S23 DDA 5.58 191% 3 off 13 centre- 1200 400 - 2 4.88 S23 DDA 5.64 205% 3 off centre- 300 100 - 2 2.88 S24 DDA 3.28 34% 3 off centre- 600 200 - 2 3.04 S24 DDA 4.12 121% 3 off centre- 900 300 - 2 3.95 S24 DDA 5.04 163% 3 off centre- 1200 400 - 2 4.88 S24 DDA 5.42 193% 3 off centre- 300 100 - 3 2.88 S31 DDA 2.73 12% 2 off centre- 600 200 - 3 3.04 S31 DDA 2.31 24% 2 off centre- 900 300 - 3 3.95 S31 DDA 2.34 22% 2 off centre- 1200 400 - 3 4.88 S31 DDA 2.42 31% 2 off centre- 300 100 - 3 2.87 S32 DDA 2.97 22% 3 off centre- 600 200 - 3 3.01 S32 DDA 2.80 50% 3 off centre- 900 300 - 3 3.88 S32 DDA 3.04 58% 3 off centre- 1200 400 - 3 4.76 S32 DDA 3.30 78% 3 off centre- 300 100 - 3 2.87 S33 DDA 2.97 22% 3 off centre- 600 200 - 3 3.01 S33 DDA 3.57 91% 3 off centre- 900 300 - 3 3.88 S33 DDA 4.65 142% 3 off centre- 1200 400 - 3 4.76 S33 DDA 4.92 166% 3 off centre- 600 200 - 4 2.87 S41 DDA 2.29 22% 2 off centre- 900 300 - 4 3.01 S41 DDA 2.33 21% 2 off centre- 1200 400 - 4 3.88 S41 DDA 2.38 29% 2 off centre- 600 200 - 4 4.76 S42 DDA 3.10 66% 3 off centre- 900 300 - 4 3.00 S42 DDA 3.98 107% 3 off centre- 1200 400 - 4 3.84 S42 DDA 4.35 135% 3 off centre- 900 300 - 6 4.70 S61 DDA 2.46 28% 2 off centre- 1200 400 - 6 3.00 S61 DDA 2.58 39% 2 off centre- 900 300 - 6 3.84 S62 DDA 3.30 72% 3 off centre- 1200 400 - 6 4.70 S62 DDA 3.74 102% 3 off a: Using Dp and Dc to run Core-Shell Mie model; Using Dp, Dc, and ncore to run DEMA-Mie model; Using Dp, Dc, and ncore to build various shape model and then run DDA. b: Location of BC cores. c: The DDA simulations with ncore > 1 are compared to the simulations with ncore = 1. Comment 5: Increased absorption by multi-core particles appears to be an effect of phase shift parameter2,3. Particles with phase shift > 1 experience attenuation of light as it passes through the BC core. The core of multi-core particles may be small enough that they remain in the Rayleigh regime (phase shift parameter < 1). It appears that absorption enhancement reaches a maximum value of approximately 4 for large D /D p c and for small D /D . The authors briefly touch on this subject on lines 205-208, but I p c think it warrants more discussion. Is absorption enhancement calculated by the DEMA-Mie model the same as for a single-core particle with equivalent phase shift parameter? 2. Beeler, P. & Chakrabarty, R. K. Constraining the particle-scale diversity of black carbon light absorption using a unified framework. Atmospheric Chemistry and Physics 22, 14825–14836 (2022). 3. Sorensen, C. & Fischbach, D. Patterns in Mie scattering. Optics Communications 173, 145– 153 (2000). Reply: Thanks for your references, and Beeler and Chakrabarty 19 have done outstanding work by defining phase shift parameter (ρ in Equation R5), which is used BC to describe the size and morphology of one BC core, as well as its interaction with light. 4𝜋𝑅 𝜌 = 𝑔 |m −1| (R5) 𝐵𝐶 𝜆 𝑒𝑓𝑓 15 where λ is the wavelength of incident light, R is the particle radius of gyration (size g metric) of BC, and m is the effective complex index of refraction, which in turn is eff given by Equation R6. m2 −1 𝑚2 −1 𝑒𝑓𝑓 𝜙( ) = ( ) (R6) m2 +2 𝑚2 +2 𝑒𝑓𝑓 where 𝜙 is the BC monomer packing fraction and m is the BC complex refractive index. The BC monomer packing fraction was calculated as the volume of BC which lies within a sphere of radius R at the centre of mass divided by the volume of a sphere g with radius R . For fresh aggregates (with ρ << 1), light is able to fully illuminate the g BC aggregate, and the entire volume contributes to light absorption. However, light is not able to illuminate the far interior of the particle for fully collapsed aggregates (with ρ > BC 1), leading to areas of decreased light absorption. We emphasize that the BC mass is concentrated at the centre of the particle and may partially fail to interact with light when using the Core-Shell Mie model. This viewpoint is consistent with Beeler and Chakrabarty 19. However, we can see all parameters used in Equations R5-6 describe the one BC core, not the entire particle. Our study emphasises the widespread existence of multiple BC cores within an individual particle, and these cores are separate and unconnected with each other. Multi-core BC particles are not included in Beeler and Chakrabarty 19. Finally, we express our gratitude, and in fact, our electric field simulation is also inspired by Beeler and Chakrabarty 19 and Beeler, et al. 20. We added discussions related to Beeler and Chakrabarty 19 in Lines 260-262: “Moreover, Beeler and Chakrabarty 19 defined a phase shift parameter to describe a simulation like Fig. 3c2. However, in these previous simulations, multi‒core BC particles were not employed as input parameters.” Comment 6: Are there any measurements of absorption or AAOD that support the modelling results? The authors hypothesize a nearly 20% change in aerosol absorption. This is certainly notable and would likely show up in measurements or remote sensing. 16 Reply: Thank you very much for your insightful comment. We appreciate your attention to the importance of supporting our modelling results with measurements of E or BC AAOD. abs Firstly, it is important to note that existing methods for measuring E often abs require multiple large-scale observation instruments. As illustrated in Fig. R2 (Comment 1), our observation station is located on a cliff in a mountainous area, where transportation and power supply are highly inconvenient. In contrast, our single-particle sampler is designed to be portable and can be powered by batteries, making it suitable for such challenging environments. On the other hand, both satellite and ground-based remote sensing techniques currently measure only the bulk optical properties of total aerosols21. When inverting BC AAOD, assumptions about mixing states and other microscopic features are necessary. Fig. R6 highlights a current study on BC inversion, where we circled the two main hypotheses in the workflow22. One hypothesis is that the particles are mixed in a Core-Shell state (n = 1), while the other pertains to the heterogeneity of the mixing core state, which is the focus of your Comment 7. Moreover, there are currently multiple theories regarding E , as you pointed out abs in your other comments, and no research has yet proposed a unified and complete framework for E correction or a sufficiently scientific BC AAOD satellite data abs product. Given these challenges, we hope to leverage interdisciplinary platforms like Nature Communications to raise awareness about multi-core BC particles in this study. We sincerely hope for your understanding and support in this study. 17 [REDACTED] Figure R6. A screenshot from Tiwari, et al. 22. We circle the two main hypotheses in green colour. Comment 7: Additionally, previous studies have shown that biomass burning aerosols are too absorbing in climate models4 and that particles with large D tend to have lower c D /D , leading to lower absorption enhancement than would be predicted by p c uniform mixing models5. How do these results fit in with these previous studies? 4.Brown, H. et al. Biomass burning aerosols in most climate models are too absorbing. Nature communications 12, 277 (2021). 5. Fierce, L. et al. Radiative absorption enhancements by black carbon controlled by particle-to-particle heterogeneity in composition. Proceedings of the National Academy of Sciences 117, 5196–5203 (2020). Reply: Thank you very much for your valuable comments. We agree that E and the abs uncertainty of simulations are indeed hot topics in our field, with numerous studies proposing various theories from different perspectives. We have also taken into account the literature you mentioned. However, due to space limitations, it was challenging to conduct a systematic discussion in a sufficiently clear manner within the manuscript. We addressed each of your points individually: Brown, et al. 23 conducted sensitivity experiments using the Community Atmosphere Model version 5 (CAM) and concluded that primary organic carbon and BC in primary carbon mode should be set as externally mixed. The primary carbon mode is used mainly to represent emissions in CAM5. We concur with the findings of Brown, et al. 23, and in our global model, BC emissions are also set as externally mixed with other aerosol components. Detailed setting of our global model are in Matsui, et al. 15. We should emphasize that, in CAM5, all black carbon (BC) in the accumulation mode is internally mixed with organic matter, sulfates, sea salt, and dust24. However, as you have noted in above comments, many observational studies suggested that particles with larger core diameters (D ) tend to have lower shell-to-core diameter ratios c (D /D )7,8,25. Figs. R7-R9 illustrate this phenomenon, with each point representing an p c individual particle. These particles fall into two distinct regimes: one close to the x-axis, representing local emissions with minimal coating (D /D ≈ 1), and the other showing p c that larger D values are associated with lower D /D ratios. This latter regime is c p c attributed to various sources, including regional haze events in Beijing7,25, suburban wood burning in London25, and long-range transport in Nanjing8. In our study, we also analysed the two-dimensional distribution of D /D and D and identified these two p c c regimes. For freshly emitted BC, D /D is close to 1, while the smooth line represents p c the average trend for coated BC. Our single-particle analysis using TEM provides detailed information on the number of BC cores (n ) for each particle, demonstrating core that coagulation becomes increasingly important as D increases, in addition to the p condensation mechanism. Due to space limitations, the manuscript primarily presents the key conclusions. 19 [REDACTED] Figure R7. A screenshot from Liu, et al. 25. The X-axis is the D of each particle, the c Y-axis is the scattering enhancement (E ), and the grey dashed line represents the sca relative thickness of the coating. [REDACTED] Figure R8. A screenshot from Zhang, et al. 8. 20 [REDACTED] Figure R9. A screenshot from Liu, et al. 7. We added more details in Lines 156-207, and a more comprehensive theoretical derivation and discussion can be found in our previous work (Chen, et al. 26): “During highly polluted periods, wildfire smoke after long-range transport could enhance the formation of multi‒core BC particles (Fig. 1a vs. Supplementary Fig. 3). However, few studies reported the size-dependent characteristics of multi‒core BC particles. Instead, some studies considered the two-dimensional data characteristics of D /D and D and the size distribution of D to approximately assess the mixing states p c c c of BC aerosols in biomass burning smoke and wildfire smoke1,7,8. To obtain the equivalent volume diameters of individual BC-containing particles and their BC cores (i.e., D and D , respectively), we calibrated the equivalent circle diameters obtained p c 21 from TEM images to the equivalent volume diameters via a regression relationship from atomic force microscopy measurements (refer to the Methods). For multi‒core BC particles, D represents the accumulation of all BC cores. Fig. 2a and Table 1 both c show that multi‒core BC particles became the dominant particle type at D > 200 nm.” c Figure 2. Quantifying the size-dependence characteristics of BC mixing state. (a) Different colours represent the number of BC cores (n ) in each particle, and the core histogram is the number distribution of particles with different BC core size (D ). (b) c Two-dimension distribution of particle-to-core diameter ratio (D /D ) and D based on p c c the transmission electron microscopy. The smooth line is used for all coated BC particles (black) and their fitting curve (red). The Boundary line means the max cond D /D versus D in the condensation model, and beyond Boundary line, changes in p c c cond D /D are predominantly influenced by coagulation. The Boundary line is well p c cond described in Chen, et al. 26 and Sedlacek, et al. 27. (c) The range of n in different size core bins of D . p “Fig. 2b shows that the average D /D ratio first decreased with increasing D , and p c c then remained relatively constant for particles with D > 200 nm. This relationship was c well fitted by the following equation: D /D = 2.5+6.3×Exp(-D /63) (red line in Fig. p c c 2b). The fitting results revealed that the average D /D ratio remained steady at 2.5, p c mainly for D > 400 nm (blue and green dots in Fig. 2b). Similar equilibrium states of p 22 D /D were reported for Southeast Tibetan Plateau (~2.2)26 and during a transboundary p c transport event in Beijing (~2.4)7. If condensation dominated the BC aging mechanism (n = 1), particles with smaller D values would exhibit higher D /D values when core c p c coated with the same volume of condensate aerosols27. Specifically, D /D was p c expected to approach ~1 with increasing D value. The blue line shown in Fig. 2b c represents the maximum D /D value under the condensation mechanism described by p c Chen, et al. 26 and Sedlacek, et al. 27. Fig. 2b shows that particles with D > 400 nm p (blue and green dots) were above the blue line. Below this blue line, condensation dominated the BC ageing process, maintaining the n value at 1 (Fig. 2c). Above the core blue line, coagulation dominated the ageing process, resulting in an increase in the n core value (i.e., n >1) (Fig. 2c).” core “Box models developed by Matsui, et al. 5 and Riemer, et al. 6 revealed that thickly coated BC particles were dominated by coagulation rather than by condensation. They concluded that the number of coagulation events increased with increasing D and p peaked at D > ~400 nm5. Several field studies proposed that coagulation could account p for most of the increase in particle size (D ) in wildfire smoke28-31. Theoretically, p coagulation between two BC-containing particles could increase n and produce a core larger D c value5. In this study, multi‒core BC particles were observed in the TEM images (Fig. 1b3), thus providing direct evidence of coagulation among BC-containing particles. Fig. 2c shows that the range of n increased with increasing D : for example, core p n ranged from 1 to 5 for D between 400 and 600 nm, and from 1 to 8 for D ≥ 800 core p p nm. We observed a critical condition near D > 400 nm and when n transitioned from p core 1 to > 1 (Fig. 2c). This phenomenon was also observed in the Southeast Tibetan Plateau26 and Amakusa City, Japan32.” It is evident that the CAM model oversimplifies the mixing state heterogeneity of BC, as seen in the two-dimensional distribution of D /D and D in Figs. R7-R9. This p c c issue has been highlighted by several studies, including Fierce, et al. 33 and other studies15,34, which emphasize that mixing state heterogeneity is a key source of uncertainty in BC optical simulations. For instance, Chen, et al. 35 pointed out that the 23 D /D values in the CAM model are abnormally high due to the misidentification of p c BC-free particles as coatings of BC. To address this limitation, we have chosen to use CAM-ATRAS, which represents accumulation mode aerosols using 40 mixing state bins, as illustrated in Supplementary Figure 11. CAM-ATRAS is a highly valuable global model that can simulate the mixing state heterogeneity of aerosols. It participated in the third phase of the Aerosol Comparisons between Observations and Models intercomparison initiative (AeroCom phase III), which serves as an important foundational dataset for the Intergovernmental Panel on Climate Change (IPCC) Sixth Assessment Report (AR6)36,37. Dr. Fierce also made significant contributions to the application of DEMA in aerosol optical simulations, as demonstrated in Fierce, et al. 14 and Fierce, et al. 13. However, rare studies provide observational evidence demonstrating that multi-core BC particles are prevalent in the real-world atmosphere. Finally, we sincerely thank you for your valuable suggestions and constructive feedback. Supplementary Figure 11. Schematic overview of the offline coupling process between a global atmospheric model and a random forest emulator. 24 Comment 8: Line 89, “Once recent study has confirmed that” should be changed to “One recent study has confirmed”. Reply: Thanks for your comments. We corrected it in Lines 101-103: “One recent study confirmed that coagulation leads to the occurrence of multi‒core BC particles with D > p 400 nm during long-range transport26.” Comment 9: Line 245, E should be E . abs_DEAM abs_DEMA Reply: Thank you! We correct and check it in the whole manuscript. Comment 10: Line 246: Why use N ? The capitalization of the letter “N” suggests cal particle number concentration. Why not n ? Keep the terms consistent. core,cal Reply: Thanks for your useful comment. We finally use n in the whole manuscript. core,cal References 1 Schwarz et al. Measurement of the mixing state, mass, and optical size of individual black carbon particles in urban and biomass burning emissions. Geophysical Research Letters 35, (2008). 2 Yuxuan et al. Particle size and mixing state of freshly emitted black carbon from different combustion sources in China. Environmental Science & Technology 54, 7766-7774, (2020). 3 Pang et al. Morphology and fractal dimension of size‐resolved soot particles emitted from combustion sources. Journal of Geophysical Research: Atmospheres 128, e2022JD037711, (2023). 4 Seinfeld & Pandis. Atmospheric Chemistry and Physics from Air Pollution to Climate Change. (John Wiley & Sons, Inc., 2016). 5 Matsui et al. Development and validation of a black carbon mixing state resolved three- dimensional model: aging processes and radiative impact. Journal of Geophysical Research: Atmospheres 118, 2304-2326, (2013). 6 Riemer et al. Simulating the evolution of soot mixing state with a particle-resolved aerosol model. Journal of Geophysical Research 114, D09202, (2009). 7 Liu et al. Size-resolved mixing state and optical properties of black carbon at an urban site in Beijing. Science of the Total Environment 749, 141523, (2020). 8 Zhang et al. Strong haze‐black carbon‐climate connections observed across Northern and Eastern China. Journal of Geophysical Research: Atmospheres 128, e2023JD038505, (2023). 9 Adachi et al. Mixing states of light‐absorbing particles measured using a transmission electron microscope and a single‐particle soot photometer in Tokyo, Japan. Journal of Geophysical Research: Atmospheres 121, 9153-9164, (2016). 10 Ching et al. Aerosol mixing state revealed by transmission electron microscopy pertaining to cloud formation and human airway deposition. npj Climate and Atmospheric Science 2, 22, (2019). 11 Wang et al. Fractal dimensions and mixing structures of soot particles during atmospheric processing. Environmental Science & Technology Letters 4, 487-493, (2017). 12 Chylek et al. Effect of graphitic carbon on the albedo of clouds. Journal of the Atmospheric 25 Sciences 41, 3076-3084, (1984). 13 Fierce et al. Toward reduced representation of mixing state for simulating aerosol effects on climate. Bulletin of the American Meteorological Society 98, 971-980, (2017). 14 Fierce et al. Black carbon absorption at the global scale is affected by particle-scale diversity in composition. Nature Communications 7, 12361, (2016). 15 Matsui et al. Black carbon radiative effects highly sensitive to emitted particle size when resolving mixing-state diversity. Nature Communications 9, 3446, (2018). 16 Jacobson. Effects of externally-through-internally-mixed soot inclusions within clouds and precipitation on global climate. Journal of Physical Chemistry A 110, 6860-6873, (2006). 17 Wang et al. Radiative forcing and climate response due to the presence of black carbon in cloud droplets. Journal of Geophysical Research: Atmospheres 118, 3662-3675, (2013). 18 Wang et al. Nonlinear enhancement of radiative absorption by black carbon in response to particle mixing structure. Geophysical Research Letters 48, e2021GL096437, (2021). 19 Beeler & Chakrabarty. Constraining the particle-scale diversity of black carbon light absorptionusing a unified framework. Atmospheric Chemistry and Physics 22, 14825-14836, (2022). 20 Beeler et al. Light absorption enhancement of black carbon in a pyrocumulonimbus cloud. Nature Communications 15, 6243, (2024). 21 Bond et al. Bounding the role of black carbon in the climate system: a scientific assessment. Journal of Geophysical Research: Atmospheres 118, 5380-5552, (2013). 22 Tiwari et al. Radiative forcing bias calculation based on COSMO (Core-Shell Mie model Optimization) and AERONET data. npj Climate and Atmospheric Science 6, (2023). 23 Brown et al. Biomass burning aerosols in most climate models are too absorbing. Nature Communications 12, 277, (2021). 24 Liu et al. Description and evaluation of a new four-mode version of the Modal Aerosol Module (MAM4) within version 5.3 of the Community Atmosphere Model. Geoscientific Model Development 9, 505-522, (2016). 25 Liu et al. Contrasting physical properties of black carbon in urban Beijing between winter and summer. Atmospheric Chemistry and Physics 19, 6749-6769, (2019). 26 Chen et al. Quantifying evolution of soot mixing state from transboundary transport of biomass burning emissions. iScience 26, 108125, (2023). 27 Sedlacek et al. Using the black carbon particle mixing state to characterize the lifecycle of biomass burning aerosols. Environmental Science & Technology 56, 14315-14325, (2022). 28 June et al. Aerosol size distribution changes in FIREX-AQ biomass burning plumes: the impact of plume concentration on coagulation and OA condensation/evaporation. Atmospheric Chemistry and Physics 22, 12803-12825, (2022). 29 Sakamoto et al. Aged boreal biomass-burning aerosol size distributions from BORTAS 2011. Atmospheric Chemistry and Physics 15, 1633-1646, (2015). 30 Hodshire et al. Dilution impacts on smoke aging: evidence in Biomass Burning Observation Project (BBOP) data. Atmospheric Chemistry and Physics 21, 6839-6855, (2021). 31 Sakamoto et al. The evolution of biomass-burning aerosol size distributions due to coagulation: dependence on fire and meteorological details and parameterization. Atmospheric Chemistry and Physics 16, 7709-7724, (2016). 32 Xu et al. Tracing the evolution of morphology and mixing state of soot particles along with the 26 movement of an Asian dust storm. Atmospheric Chemistry and Physics 20, 14321-14332, (2020). 33 Fierce et al. Radiative absorption enhancements by black carbon controlled by particle-to- particle heterogeneity in composition. Proceedings of the National Academy of Sciences of the United States of America 117, 5196-5203, (2020). 34 Ching et al. A three-dimensional sectional representation of aerosol mixing state for simulating optical properties and cloud condensation nuclei. Journal of Geophysical Research: Atmospheres 121, 5912-5929, (2016). 35 Chen et al. An aerosol optical module with observation‐constrained black carbon properties for global climate models. Journal of Advances in Modeling Earth Systems 15, e2022MS003501, (2023). 36 Sand et al. Aerosol absorption in global models from AeroCom phase III. Atmospheric Chemistry and Physics 21, 15929-15947, (2021). 37 Gliß et al. AeroCom phase III multi-model evaluation of the aerosol life cycle and optical properties using ground- and space-based remote sensing as well as surface in situ observations. Atmospheric Chemistry and Physics 21, 87-128, (2021). 27 Response to Reviewer 3 We are grateful for Reviewer 3’s comments. Those comments are all valuable and helpful for improving our paper and English writing. We answered the comments carefully and have made corrections in the submitted manuscript. The corrections and the responses are as following: In the revised manuscript, the red colour was marked as the revised places. General comments: The paper discusses an important issue in aerosol science, which is how the mixing of black carbon particles affects light absorption, with important potential implications for climate modelling. Overall, the main premise of the paper, meaning the importance of multiple black carbon cores within a single particle, is interesting and a potentially important one. The approach used is quite comprehensive, including microscopy, theoretical considerations, and modelling going all the way from single particles to global-scale simulations, machine learning, etc. The weak point of the paper is the clarity and how the method is described. I found it quite difficult to read. Resorting heavily on SI tables, equations, explanations, and figures made the reading even more difficult. The choice of wording, sentence structure, and logic also often hampered the understanding of the several different steps, acronyms, limitations, etc. Overall, I think the work has very good potential, and it should be shared with the community, but in its present form, it is not ready for publication, especially so in a journal that targets a broader audience such as Nature Communications. I would suggest, therefore, that the authors significantly revise the paper to improve readability. Next, I provide some suggestions and some examples, but there are many other instances where the readability and clarity of the paper can and should be significantly improved. Reply: Thank you for your recognition and comments. We have made numerous revisions to improve clarity and readability. We have also obtained the writing service from Springer Nature, and Fig. R1 is a certificate. Based on the comments of reviewers and editors, we have restructured the manuscript to minimize the need for frequent consultation of the Supporting Information. We look forward to your further suggestions to improve the potential for publication in Nature Communications. [REDACTED] Figure R1. A certificate confirming that our work was edited by Springer Nature Editing Service Comment 1: A brief description of the difference between EMBS_DDA and DEMA_Mie and how they are calculated early in the paper (instead of waiting for the detailed description in the methods) would be helpful. This can be achieved with just one or two short sentences. Reply: Thanks for your comment. We reconstructed Figures 3, as well as the corresponding supplementary materials, for the convenience of readers to compare Core-Shell Mie, DEMA-Mie, and DDA. Detailed settings are provided in Supplementary Text 4, Supplementary Tables 5-6, and Supplementary Fig. 6. We have modified some text in Lines 222-266: “In this study, we first employed an electron-microscopy-to-BC simulation tool, coupled with discrete dipole approximation (EMBS–DDA)1 to visualize the effect of n on E of BC-containing particles. The EMBS tool provides an interactive core abs interface for constructing complex BC mixing structures that are input into the DDA module to calculate their optical properties1. BC-containing particles were assigned different n values and positions in the EMBS–DDA simulations, whereas D /D was core p c set to the average value (i.e., D /D =3 in Supplementary Fig. 5) to represent the p c common scenario for D > 400 nm. Under the assumption that BC cores were randomly p located in an individual particle, the probability of core occurrence obviously increased from the centre to off-centre positions. Specifically, BC aerosols with a single core or multiple cores at the particle centre represented idealized and specialized scenarios, respectively. Innumerable possible core locations would increase the DDA simulation burden of every configuration excessively, even with fixed input parameters of D , D , p c and n . Herein, we simulated three representative scenarios using DDA: core DDA : Setting n =1 with the core at the centre. The simulated E value was 1 core abs almost equal to that of the Core‒Shell Mie model, as shown in Figs. 3a–b. DDA : Setting n >1 with multiple cores concentrated near the centre, as shown 2 core in Fig. 3b. DDA : Setting n >1 with cores distributed at various off-centre positions, as 3 core shown in Fig. 3b. The detailed settings are provided in Supplementary Text 4, Supplementary Tables 5–6, and Supplementary Fig. 6. In addition, we calculated E abs via the Core–Shell Mie and DEMA–Mie models on the basis of the same input parameters (D , D , and n ) used in DDA.” p c core Figure 3. Light absorption enhancement (E ) at 550 nm of different types of BC- abs containing particles. (a) Simulation results of different methods for different D . Input c parameters and simulation outputs are detailed in Supplementary Tables 5 and 6. (b) Schematic diagrams illustrate typical mixing states for DDA , DDA , and DDA 1 2 3 simulations, each with distinct core configurations. (c1-c6) Near-field calculations were employed to determine the electric field intensity at 550 nm for secondary aerosol particles (c1) and various DDA scenarios (DDA : c2, DDA : c3-4, DDA : c5-6), with 1 2 3 D = 600 nm. White lines represent the border of particles, and black line marks the p location of the BC cores. The colour represents the normalized electric field intensity versus the incident light. “We found that the E values for n = 1 were 12%–205% lower than those for abs core n > 1, particularly in DDA scenarios (Supplementary Table 6). Fig. 3a shows that core 3 the DEMA–Mie could represent the average increasing trend in DDA . Notably, the 3 DEMA–Mie model accounted for scenarios with n ≥ 1 in which the cores were core randomly distributed, but their exact locations were not specified in DDA (Figure 3). We compared the E ranges (from minimum to maximum) obtained via the DEMA– abs Mie and DDA simulations. Fig. 3b shows that the E range from the DDA 3 abs 3 simulations agreed well with that from the DEMA–Mie model. The results further confirmed that DEMA–Mie was efficient for simulating the average impact of n at core random positions on E .” abs  We also display Supplementary Table 6 for your convenience in understanding: Supplementary Table 6. Simulation results of Core-Shell Mie, DEMA-Mie, and DDA Shape E under Relative abs D, nm D, nm E n E Type Locationb E under DDAa p c abs_cs core abs_DEMA abs model differencec 300 100 2.46 1 2.90 S11 DDA centre 2.44 - 1 600 200 1.87 1 3.10 S11 DDA centre 1.87 - 1 900 300 1.93 1 4.1 S11 DDA centre 1.92 - 1 1200 400 1.85 1 5.14 S11 DDA centre 1.85 - 1 centre- 300 100 - 2 2.90 S21 DDA 3.00 23% 2 off centre- 600 200 - 2 3.10 S21 DDA 2.51 34% 2 off centre- 900 300 - 2 4.11 S21 DDA 2.49 30% 2 off centre- 1200 400 - 2 5.14 S21 DDA 2.53 37% 2 off centre- 300 100 - 2 2.88 S22 DDA 3.32 36% 3 off centre- 600 200 - 2 3.04 S22 DDA 3.35 79% 3 off centre- 900 300 - 2 3.95 S22 DDA 3.70 93% 3 off centre- 1200 400 - 2 4.88 S22 DDA 3.79 105% 3 off centre- 300 100 - 2 2.88 S23 DDA 3.44 41% 3 off centre- 600 200 - 2 3.04 S23 DDA 4.35 133% 3 off centre- 900 300 - 2 3.95 S23 DDA 5.58 191% 3 off centre- 1200 400 - 2 4.88 S23 DDA 5.64 205% 3 off centre- 300 100 - 2 2.88 S24 DDA 3.28 34% 3 off centre- 600 200 - 2 3.04 S24 DDA 4.12 121% 3 off centre- 900 300 - 2 3.95 S24 DDA 5.04 163% 3 off centre- 1200 400 - 2 4.88 S24 DDA 5.42 193% 3 off centre- 300 100 - 3 2.88 S31 DDA 2.73 12% 2 off centre- 600 200 - 3 3.04 S31 DDA 2.31 24% 2 off centre- 900 300 - 3 3.95 S31 DDA 2.34 22% 2 off centre- 1200 400 - 3 4.88 S31 DDA 2.42 31% 2 off centre- 300 100 - 3 2.87 S32 DDA 2.97 22% 3 off centre- 600 200 - 3 3.01 S32 DDA 2.80 50% 3 off centre- 900 300 - 3 3.88 S32 DDA 3.04 58% 3 off centre- 1200 400 - 3 4.76 S32 DDA 3.30 78% 3 off centre- 300 100 - 3 2.87 S33 DDA 2.97 22% 3 off centre- 600 200 - 3 3.01 S33 DDA 3.57 91% 3 off centre- 900 300 - 3 3.88 S33 DDA 4.65 142% 3 off centre- 1200 400 - 3 4.76 S33 DDA 4.92 166% 3 off centre- 600 200 - 4 2.87 S41 DDA 2.29 22% 2 off centre- 900 300 - 4 3.01 S41 DDA 2.33 21% 2 off centre- 1200 400 - 4 3.88 S41 DDA 2.38 29% 2 off centre- 600 200 - 4 4.76 S42 DDA 3.10 66% 3 off centre- 900 300 - 4 3.00 S42 DDA 3.98 107% 3 off centre- 1200 400 - 4 3.84 S42 DDA 4.35 135% 3 off centre- 900 300 - 6 4.70 S61 DDA 2.46 28% 2 off centre- 1200 400 - 6 3.00 S61 DDA 2.58 39% 2 off centre- 900 300 - 6 3.84 S62 DDA 3.30 72% 3 off centre- 1200 400 - 6 4.70 S62 DDA 3.74 102% 3 off a: Using Dp and Dc to run Core-Shell Mie model; Using Dp, Dc, and ncore to run DEMA-Mie model; Using Dp, Dc, and ncore to build various shape model and then run DDA. b: Location of BC cores. c: The DDA simulations with ncore > 1 are compared to the simulations with ncore = 1. Comment 2: From figure 3, it is difficult for me to evaluate how well DEMA-Mie approximates the DDA calculations, because the two results are shown in different panels. A direct and quantitative comparison in terms of % deviations might be an easier way to demonstrate how good the DEMA_Mie approaches the DDA, as it is done for the comparison of CS with DEMA_Mie in table S5 (and graphically in figure 3(c)). Reply: Thank you for this suggestion. We addressed a similar question in detail in our response to Comment 1. We believe the information provided there may also be relevant to this comment. In fact, we have almost entirely rewritten the “Optical absorption enhancement”. We realised that part of the content in the original Figs. 3 and 4 might be similar, so we redesigned them to enhance clarity for readers. We have also integrated the relevant supplementary materials with the main text for the convenience of readers to directly compare DDA, Core-Shell Mie, and DEMA-Mie models. Additionally, we have created more intuitive diagrams and provided more comprehensive data to help readers better understand the differences between DDA and DEMA-Mie. Specifically, the DEMA-Mie model assumes that n ≥ 1 and that these cores are core randomly distributed within the particles. Therefore, it only requires three parameters— D , D , and n —to characterise the mixing state of the particles for computation by p c core DEMA-Mie. In contrast, DDA necessitates the construction of specific particle models to fix all core positions within the particles. As a result, we use multiple DDA-based E values corresponding to a single value from the DEMA-Mie model, as abs demonstrated in Supplementary Table 6 in Comment 1. Detailed settings are provided in Supplementary Text 4, Supplementary Tables 5- 6, and Supplementary Fig. 6. We modified some text in Lines 222-266. These details are also described in Comment 1. Comment 3: What index of refraction values have been used, and what’s the impact of the choice? The index of refraction used becomes clear later, which is fine, but perhaps a discussion of the potential uncertainty those choices can have would be useful. Reply: Thanks for your key comment. Through the literature review, the refractive index (RI) is 1.85+0.71j for BC and 1.53+0j for coating aerosol in our study1,2. As you mentioned, some studies use other values, as listed in Table R1. We further employ Monte Carlo simulations, a data-driven approach, using the ranges of values in Table R1 to quantify the potential impact of the uncertainties in the refractive index (RI) on the simulation results of light absorption enhancement (E ). abs We use E and E as examples to illustrate the simulation results. In other abs_DEMA abs_CS words, simulations for the blue line (E ) and the red points (E ) shown in abs_CS abs_DEMA above Fig. 3a in Comment 1 were replicated 500 times. For each replication, different RI and RI were used as input. The potential impact of the RI was then BC coating characterized by analysing the range of final simulation results in Fig. R2. Table R1 Refractive index for BC and coating aerosols in literature. Reference RI * RI * RI * r,BC i,BC r,coating 3 1.85 0.71 1.50 4 1.95 0.79 1.48 1 1.85 0.71 1.53 5 1.95 0.79 1.55 6 1.85 0.71 1.60 7 1.76 0.63 1.48 8 1.76 0.63 1.55 9 1.85 0.71 1.55 *: The subscripts r and i represent the real and imaginary parts of the refractive index (RI), respectively. Typically, RI is set to 0, indicating that absorption by the i,coating coating is not considered. Figure R2. Results of 500 Monte Carlo simulations represent the changes caused by both RI and RI using 1 standard deviation. Note: The Core-Shell Mie coating BC model does not use n as an input parameter. Therefore, E values are the same for core abs the same D and D , regardless of n . p c core In Fig. R2, the uncertainty arising from the differences in the values of RI and coating RI is relatively small. In this study, we primarily focus on the potential impact of BC multi-core BC particles on E and their global effect on BC AAOD. Specifically, we abs aim to highlight the coagulation processes involved in their formation and the distinct optical properties compared to the Core-Shell Mie model. It is common practice to fix RI and RI when discussing the impact of aerosol mixing state changes on E , coating BC abs e.g., the references listed in Table R1. We also emphasize accounting for changes in RI when finally updating atmospheric models or remote sensing inversion algorithms10,11. This, however, represents a large and complex algorithm development project. We hope to leverage the interdisciplinary platform of Nature Communications to raise awareness among relevant research teams about the importance of multi-core BC particles. Comment 4: I found the section on the Machine Learning Emulator to be very hard to interpret and quite cryptic. Perhaps this is due to the technical aspect of the techniques, but in my opinion, to reach the broader audience of Nature Communications, much effort should be put into clarifying this section and making interpretations more understandable for a broader audience. Reply: Thank you for your valuable comment. To ensure that our work is accessible to a broader audience, we have added a comprehensive introduction to the Machine Learning Emulator and reconstructed the relevant results. This should help readers better understand our approach and findings. The detailed changes can be found as follows:  Lines 309-323: “In previous global simulations, the developers of DEMA12-14 used a noninteger n value (i.e., D 3/1000000) to approximate n . However, Wang, et al.15 proposed core,cal c core that a noninteger n value was implausible because observational evidence for core,cal multi‒core BC particles was lacking. Therefore, it was necessary to parameterize the impact of n on E on the basis of the real-world size dependence of n . We core abs core employed a random forest (RF) model to predict E and quantify how E abs_DEMA abs_DEMA changed with the mixing state of coated BC. On the basis of 1310 samples (coated BC particles), we developed two RF emulators (RF1 and RF2) for E , with input abs_DEMA features including D /D , D , D , E , n , n , M , M , and M ×M . RF1 p c c p abs_CS core core,cal 1 2 1 2 included all the input features, whereas RF2 excluded n , which was unavailable in core the global model. SHapley Additive ExPlanation (SHAP) values, rooted in coalitional game theory16, were employed to distribute the total effect of each input feature on the predicted E for individual particles (refer to the Methods). By comparing the abs_DEMA prediction performance levels and SHAP values of RF1 and RF2, we determined whether M ×M could serve as a proxy for the impact of n on E .” 1 2 core abs_DEMA Figure 4. Random Forest Model Emulation of Light Absorption Enhancement (E ) at 550 nm. (a) Fitting of E for particles with M ×M = 0 (blue points) abs abs_DEMA 1 2 using the DEMA-Mie model. (b) Lines showing the increase in E as D /D increases abs p c across various D values under the Core-Shell Mie model (E ). (c) Comparison c abs_CS analysis of differences between E and E for M ×M = 0 and 1, respectively. abs_DEMA abs_CS 1 2 (d) Impacts on E shown as different SHAP values for each input feature via abs_DEMA SHapley Additive ExPlanation (SHAP) approach. Further details on SHAP are in the Methods section. (e-i) Impacts of primary input features on E quantified by the abs_DEMA SHAP values.  Lines 328-350 “To enhance the interpretability of the RF predictions, we visualized the SHAP values corresponding to each feature value. Figs. 4e–i show the driving patterns of these primary features on E . Each point represents an individual particle, and its abs_DEMA colour is indicative of its M ×M value to illustrate how M ×M influences the driving 1 2 1 2 patterns of these features (Figs. 4e-i). Fig. 4e shows that n was positively correlated core with its SHAP values only when M ×M = 1; otherwise, the SHAP values were 1 2 approximately 0. These results suggested that the impact of n on E was the core abs_DEMA most significant for particles with M ×M = 1. Fig. 4f shows a positive correlation 1 2 between D /D and its SHAP value, with a steeper trend for particles with M × M = 1 p c 1 2 (red) that that of particles with M ×M = 0 (blue). This result indicated a greater increase 1 2 in E with increasing D /D for M ×M = 1. Fig. 4g shows that E positively abs_DEMA p c 1 2 abs_CS correlated with its SHAP value overall, except for particles within the grey-shaded region where E ≈ 2. For these particles, the SHAP value was close to zero when abs_CS M ×M = 1 (red in Fig. 4g), suggesting that E had predictive utility for E 1 2 abs_CS abs_DEMA for M ×M = 0 but not for M ×M = 1. The increasing D value led to a greater increase 1 2 1 2 p in E with M ×M = 1 (Fig. 4h), whereas the transition of M ×M from 0 to 1 abs_DEMA 1 2 1 2 altered the correlation of D with its SHAP value from negative to positive (Fig. 4i). c These findings suggested that D and D exerted a nonlinear driving effect on E p c abs_DEMA and that BC cores with M ×M = 1 exhibited more notable interactions with light. 1 2 Similar patterns were observed when n (Supplementary Fig. 9) was used to explain core the driving patterns in RF1 compared with when M ×M was used, as shown in Figs. 1 2 4(f–i). We concluded that M ×M could serve as a proxy for the impact of n on 1 2 core E .” abs_DEMA  Methods section (Lines 502-522): “We employed the RF model to create an emulator to predict E . We abs_DEMA divided 1310 samples (coated BC particles) follows: 70% were used for the training set, and 30% were used for the test set. Finally, two emulators were developed: RF1 included all input features, such as D /D , D , D , E , n , n , M , M , and p c c p abs_CS core core,cal 1 2 M ×M ; RF2 excluded n and included other parameters. Furthermore, the SHAP 1 2 core approach was employed to distribute the total effect on the predicted E of each abs_DEMA input feature16. According to coalitional game theory, the difference in model predictions involving a feature j (e.g., n ) compared with the predictions without j core was attributed to the marginal contribution of this feature. Considering the interactive effects between features, differences were computed for every possible feature subset combination of each sample17. For each particle sample, K features were input to generate the prediction for E , denoted as f(x), including a linear function of abs_DEMA i feature attribution in Equation (6). 𝐾 𝑓(𝑥 )=Φ (𝑓,𝑥)+∑Φ (𝑓,𝑥 ) 𝑖 0 𝑗 𝑖 𝑗=1 (6) 𝑆𝐻𝐴𝑃(𝑥 )=Φ (𝑓,𝑥 ) 𝑖,𝑗 𝑗 𝑖 Φ (𝑓,𝑥)= ∑𝑁 𝑖 𝑓(𝑥 𝑖 ) { 0 𝑁 where K is the total number of features; N is the total samples; f(x)is the predicted i E value for the ith particle via the RF model in this study; and Φ (𝑓,𝑥) is the abs_DEMA 𝑗 SHAP value, which quantifies the impact of feature j on the prediction for input x in i the built model. Specifically, the contribution of n (i.e., j = n ) of the ith particle core core (x) to E was denoted the SHAP (x ) value in Equation (6), as shown by the i abs_DEMA i, j scatter in Fig. 4e. The base value, Φ (𝑓,𝑥) was the average value of the predicted 0 results. SHAP calculations were performed using the kernelshap and shapviz R packages, whereas the RF model was trained using the randomForest R package.” Comment 5: Is E indicating the absorption or the absorption enhancement? The abs_CS notation is confusing. Also, is CS for core-shell? Perhaps it should be explicit. In general, the sentence from line 45 to line 49 very long. I would suggest breaking it down into at least two or three sentences. Reply: I truly appreciate your comment. Yes, E is the absorption enhancement via abs_CS the Core-Shell Mie model. We rewrote this part to make it explicit and clear for readers. Details are in Lines 49-54: “The Core–Shell Mie model, which is widely employed in atmospheric models (e.g., the Weather Research and Forecasting model coupled with Chemistry) can be used to obtain the light absorption enhancement (E ) value of BC. In the Core‒ abs_CS Shell Mie model, it is assumed that there is one central BC core per particle (n = core 1)18,19, and the BC mixing state is represented via the BC core diameter (D ) and the c particle-to-core diameter ratio (D /D ).” p c Comment 6: Lines 53-54: The sentence “as a key indicator of BC’s warming effect” reads awkward and is unclear. Reply: Thanks for your comment. We provide more scientific definition and introduction for the black carbon absorption aerosol optical depth (BC AAOD). Details are as follow and in Lines 54-55: “On the basis of the third phase of the Aerosol Comparisons between Observations and Models intercomparison initiative (AeroCom), BC AAOD accounts for 36%–84% of the global annual mean of total absorption aerosol optical depth (AAOD) at the 550- nm wavelength, indicating that BC yields a relatively notable warming effect20-22.” Comment 7: Lines 63-66. Very long sentence, please break it down into shorter sentences. Also, I believe Fierce’s paper puts more emphasis on the heterogeneity of coating thickness rather than non-sphericity. It should also be noted that the idea that E might be smaller than what was originally hypothesized was especially sparked by abs the study by Cappa et al. (Science 2012). Cappa’s conclusions were based on field measurements and optical measurements, from what I recall. So, it seems natural to me that this study should be mentioned here and should be kept into consideration as it contradicts a little bit what the authors seem to imply here that low E has been inferred abs only through SP2 measurements (lines 66-72) Reply: Thank you very much for your insightful comment. We agree that Dr. Fierce is a representative scientist in the field of particle-to-particle heterogeneity theory. This theory is frequently used to explain discrepancies in absorption enhancement observed across laboratory, field, and modeling studies, as highlighted by Cappa, et al. 23 and others. To enhance clarity for our readers, we revised this section to include a reference to Cappa, et al. 23. We hope this provides a clearer understanding of our considerations, which we outline below: (1)Why do we reference both Liu, et al. 3 and Fierce, et al. 24 to introduce the non- spherical morphology of BC? When the coating thickness is below a certain threshold, the non-spherical morphology of BC particles can result in lower E compared to simulations from the abs Core-Shell Mie model. In our manuscript, we review this threshold value for D /D as p c approximately 1.4, based on Liu, et al. 3. In Fig. R3, the mass ratio (MR) represents the ratio of the mass of non-BC coating to the mass of BC. We marked the lower limit of the grey “Transition” region at an MR of 1.5. Although we did not directly report MR in the manuscript, we converted it to D /D using widely accepted density values to p c facilitate comparison with other studies. For an MR of 1.5, the corresponding D /D is p c approximately 1.4. [REDACTED] Figure R3. A screenshot from Liu, et al. 3. M is the mass ratio between non-BC R coating and BC. We marked the lower limit of the grey “Transition” and “Unknown” at 1.5 and 20, respectively. In the manuscript, we did not report M directly, but R converted it to D /D through widely used density to facilitate comparison of multiple p c literatures. D /D is about 1.4 and 3 for M at 1.5 and 20, respectively. p c R In Fierce, et al. 24, the influence of BC’s non-spherical morphology was also an important factor, although there is relatively little discussion on this topic, compared to particle-to-particle heterogeneity. Fierce, et al. 24 also provided a similar threshold to show the influence range of BC’s non-spherical morphology on E in Fig. R4. In Fig. abs R4, we mark the threshold at R (the volume ratio of non-BC components relative to BC BC) at 4.1, which corresponds to a D /D ratio of about 1.7. It is valuable that Fierce, p c et al. 24 further provided the scanning electron microscopy images at the main stages of ageing experiments, and the box diagram in Fig. R4 is from Cappa, et al. 25. [REDACTED] Figure R4. A screenshot from Fierce, et al. 24. R is the volume ratio of non-BC BC components relative to BC. We marked R at 4.1, which was the threshold based on BC the parameterization by Fierce, et al. 24. We also marked the maximum R at 12. BC Similar to Fig. R3, R was transformed into D /D , that is about 1.7 and 2.4 for R at BC p c BC 4.1 and 12, respectively. (2)The difference and connection between our viewpoint and Cappa, et al. 23,25. Our focus is on multi-core BC particles, which are independent of current theories. In other words, we aim to address the “Unknown” mixing state in Fig. R3, which pertains to M > 20 or R > 12 in Figs. R3-R4. By directly comparing Fig. R4C with R BC images of multi-core BC particles (e.g., Figure 1b3 in the manuscript), it becomes evident that multi-core BC particles are larger and more complex than those depicted in Fig. R4C. We hope to draw more research attention to this area through the interdisciplinary communication platform of Nature Communications. In our future work, we aim to conduct systematic observations of aerosol chemical components, mixing states, and aerosol optics, following the examples of Liu, et al. 3 and Cappa, et al. 23,25. Details are in Lines 68-74: “Significant uncertainties exist in the mixing state and E of BC in observations and simulations23,26. Fierce, et al.24 proposed that accounting abs for particle-to-particle heterogeneity in the mixing state and the complex morphology of BC-containing particles may resolve these uncertainties. Moreover, Liu, et al.3 and Fierce, et al.24 reported that the Core‒Shell Mie model might oversimplify the nonspherical morphology of BC particles, leading to potential overestimation of the degree of light absorption at low D /D values (< ~1.4).” p c Comment 8: Line 71 and especially 73: D indicates what diameter? Mobility, p aerodynamic, area equivalent, optical, mass? Especially for BC, that can make a very large difference. Reply: I truly appreciate this important comment. We agree with your comments that there are significant differences in diameter measurement via different methods. In their Supporting Information, Sedlacek, et al. 27 provided a detailed introduction to the measurement range of SP2: SP2 can measure the mass of BC over the range of ~75-500 nm mass-equivalent diameter (D ), using an assumed BC density of 1.8 g cm-3 (in the SP2 community, this c BC is referred to as refractory black carbon, rBC, to denote its means of determination). The initial scattering signal, before appreciable evaporation has occurred, determines the initial particle size. The mass of the non-refractory coating is derived using the “leading-edge only” (LEO) methodology under the assumption that the particle is in a concentric Core-Shell configuration in which a spherical BC core is surrounded by a homogeneous coating, an assumption justified by the large values of coating thickness obtained. The refractive indices of the core and coating are assumed to be 1.8 - 0.8i and 1.4 - 0i, respectively, and the density of the coating material is assumed to be 1.4 g cm- 3. The range of particle diameters (D ) for which this technique can be applied is p ~180-400 nm. However, we also notice that the detailed definition of D depends on the SP2 p configurations. For example: In Tan, et al. 28, the calibration of the incandescence signal was also conducted by using the combined DMA-SP2 system (DMA, differential mobility analyser), and Aquadag was used as the reference material. A formula was applied to convert the mobility diameter into mass for Aquadag. Considering the different sensitivity of the SP2 response to Aquadag and ambient BC, a correction of 0.75 was applied to derive the BC mass in the ambient particles. The measurement of D are also different in the references in Zhao, et al. 29 and p Wang, et al. 30. Zhao, et al. 29 used an aerosol aerodynamic classifier (AAC) to measure aerodynamic D . Wang, et al. 30 used a micro-orifice cascade impactor (NMCI) with a p quartz filter to measure D . p In the literature review sections, it is particularly challenging to individually determine and compare the definitions of D across different studies. However, we p clearly state in our methodology: 1. Transmission electron microscopy (TEM) measures the equivalent circle diameter. 2. We establish an equation for converting the equivalent circle diameter to an equivalent volume diameter using atomic force microscopy (AFM). 3. Finally, both D and D are represented as equivalent volume diameters in our p c observations. We thoroughly revised the relevant Methods section, which includes but is not limited to refining statements, providing more detailed morphology calculation formulas, and offering 3D scanning images obtained from atomic force microscopy (AFM). This will provide more detailed evidence of the size-dependence and optical characteristics of multi-core BC particles. Currently, aside from TEM, there are no other observational techniques available to determine whether a particle has n >1. We core sincerely hope for your understanding for this point. For specific details, please refer to the Methods section in Lines 448-469: “RADIUS software (EMSIS GmbH, Germany) was adopted to measure the morphological features of individual particles in the TEM images, e.g., the equivalent circle diameter (ECD) and bearing area (A). Additionally, individual particles on silicon wafers were measured via atomic force microscopy (AFM). Gwyddion 2.64 software (http://gwyddion.net/) was used to automatically obtain the bearing area (A) and bearing volume (V) of 165 individual particles. The ECD and equivalent volume diameter (EVD) of individual particles were calculated according to the following Equations (1–2)31: 1 𝐴 = 𝜋(𝐸𝐶𝐷)2 （1） 4 1 𝑉 = 𝜋(𝐸𝑉𝐷)3 （2） 6 where A is the bearing area, nm2; V is the bearing volume, nm3; ECD and EVD are the equivalent circle diameter and the equivalent volume diameter, nm, respectively. A bootstrap linear regression analysis was used to obtain the transform ratio between the ECD and EVD. Through bootstrap analysis, we could use a limited dataset to estimate the slope coefficients in the regression equation by resampling 1000 times with replacement, which could increase the credibility and robustness of the results. The final equation was EVD = 0.54 × ECD, with an R2 value of 0.99 (Supplementary Fig. 20). Consequently, a slope factor (k) of 0.54 was adopted to estimate the EVD from the ECD for all the particles in the TEM images. In this study, D was represented by p the EVD of the whole particles (coating + BC cores). We used Equation (3) to obtain D for all BC-containing particles. c 𝑘×𝐸𝐶𝐷 =𝑘× 2 √ 4𝐴 𝑐𝑜𝑟𝑒 𝑛 =1 𝑐𝑜𝑟𝑒 𝜋 𝑐𝑜𝑟𝑒 𝐷 = （3） 𝑐 2 4∑𝑛𝑐𝑜𝑟𝑒𝐴 𝑘× √ 𝑖 𝑐𝑜𝑟𝑒,𝑖 𝑛 >1 { 𝜋 𝑐𝑜𝑟𝑒 where D is the cumulative equivalent volume diameter of all BC cores in a particle; c n is the number of BC cores; the subscript i denotes the ith core; A is the bearing core core area obtained from the TEM images; k is the average transform ratio between the ECD and the EVD obtained via AFM, i.e., 0.54 as shown in Supplementary Fig. 20.” Supplementary Figure 20. Linear regression between equivalent circle diameter (ECD) and equivalent volume diameter (EVD) of individual particles. (a) Bootstrap method is based on the 1000 resampling with replacement. (b) A particle case is measured by the Atomic Force Microscopy. Comment 9: Line 89, “Once recent study has confirmed that” should be changed to “One recent study has confirmed”. Reply: Thanks for your comments. We corrected it in Lines 101-103: “One recent study confirmed that coagulation leads to the occurrence of multi‒core BC particles with D > p 400 nm during long-range transport26.” Comment 10: Line 245, E should be E . abs-DEMA abs_DEAM Reply: Thank you! We correct and check it in the whole manuscript. Comment 11: Line 246: Why use N ? The capitalization of the letter “N” suggests cal particle number concentration. Why not n ? Keep the terms consistent. core,cal Reply: Thanks for your useful comment. We finally use n in the whole manuscript. core,cal References 1 Schwarz et al. Measurement of the mixing state, mass, and optical size of individual black carbon particles in urban and biomass burning emissions. Geophysical Research Letters 35, (2008). 2 Yuxuan et al. Particle size and mixing state of freshly emitted black carbon from different combustion sources in China. Environmental Science & Technology 54, 7766-7774, (2020). 3 Pang et al. Morphology and fractal dimension of size‐resolved soot particles emitted from combustion sources. Journal of Geophysical Research: Atmospheres 128, e2022JD037711, (2023). 4 Seinfeld & Pandis. Atmospheric Chemistry and Physics from Air Pollution to Climate Change. (John Wiley & Sons, Inc., 2016). 5 Matsui et al. Development and validation of a black carbon mixing state resolved three- dimensional model: aging processes and radiative impact. Journal of Geophysical Research: Atmospheres 118, 2304-2326, (2013). 6 Riemer et al. Simulating the evolution of soot mixing state with a particle-resolved aerosol model. Journal of Geophysical Research 114, D09202, (2009). 7 Liu et al. Size-resolved mixing state and optical properties of black carbon at an urban site in Beijing. Science of the Total Environment 749, 141523, (2020). 8 Zhang et al. Strong haze‐black carbon‐climate connections observed across Northern and Eastern China. Journal of Geophysical Research: Atmospheres 128, e2023JD038505, (2023). 9 Adachi et al. Mixing states of light‐absorbing particles measured using a transmission electron microscope and a single‐particle soot photometer in Tokyo, Japan. Journal of Geophysical Research: Atmospheres 121, 9153-9164, (2016). 10 Ching et al. Aerosol mixing state revealed by transmission electron microscopy pertaining to cloud formation and human airway deposition. npj Climate and Atmospheric Science 2, 22, (2019). 11 Wang et al. Fractal dimensions and mixing structures of soot particles during atmospheric processing. Environmental Science & Technology Letters 4, 487-493, (2017). 12 Chylek et al. Effect of graphitic carbon on the albedo of clouds. Journal of the Atmospheric Sciences 41, 3076-3084, (1984). 13 Fierce et al. Toward reduced representation of mixing state for simulating aerosol effects on climate. Bulletin of the American Meteorological Society 98, 971-980, (2017). 14 Fierce et al. Black carbon absorption at the global scale is affected by particle-scale diversity in composition. Nature Communications 7, 12361, (2016). 15 Matsui et al. Black carbon radiative effects highly sensitive to emitted particle size when resolving mixing-state diversity. Nature Communications 9, 3446, (2018). 16 Jacobson. Effects of externally-through-internally-mixed soot inclusions within clouds and precipitation on global climate. Journal of Physical Chemistry A 110, 6860-6873, (2006). 17 Wang et al. Radiative forcing and climate response due to the presence of black carbon in cloud droplets. Journal of Geophysical Research: Atmospheres 118, 3662-3675, (2013). 18 Wang et al. Nonlinear enhancement of radiative absorption by black carbon in response to particle mixing structure. Geophysical Research Letters 48, e2021GL096437, (2021). 19 Beeler & Chakrabarty. Constraining the particle-scale diversity of black carbon light absorptionusing a unified framework. Atmospheric Chemistry and Physics 22, 14825-14836, (2022). 20 Beeler et al. Light absorption enhancement of black carbon in a pyrocumulonimbus cloud. Nature Communications 15, 6243, (2024). 21 Bond et al. Bounding the role of black carbon in the climate system: a scientific assessment. Journal of Geophysical Research: Atmospheres 118, 5380-5552, (2013). 22 Tiwari et al. Radiative forcing bias calculation based on COSMO (Core-Shell Mie model Optimization) and AERONET data. npj Climate and Atmospheric Science 6, (2023). 23 Brown et al. Biomass burning aerosols in most climate models are too absorbing. Nature Communications 12, 277, (2021). 24 Liu et al. Description and evaluation of a new four-mode version of the Modal Aerosol Module (MAM4) within version 5.3 of the Community Atmosphere Model. Geoscientific Model Development 9, 505-522, (2016). 25 Liu et al. Contrasting physical properties of black carbon in urban Beijing between winter and summer. Atmospheric Chemistry and Physics 19, 6749-6769, (2019). 26 Chen et al. Quantifying evolution of soot mixing state from transboundary transport of biomass burning emissions. iScience 26, 108125, (2023). 27 Sedlacek et al. Using the black carbon particle mixing state to characterize the lifecycle of biomass burning aerosols. Environmental Science & Technology 56, 14315-14325, (2022). 28 June et al. Aerosol size distribution changes in FIREX-AQ biomass burning plumes: the impact of plume concentration on coagulation and OA condensation/evaporation. Atmospheric Chemistry and Physics 22, 12803-12825, (2022). 29 Sakamoto et al. Aged boreal biomass-burning aerosol size distributions from BORTAS 2011. Atmospheric Chemistry and Physics 15, 1633-1646, (2015). 30 Hodshire et al. Dilution impacts on smoke aging: evidence in Biomass Burning Observation Project (BBOP) data. Atmospheric Chemistry and Physics 21, 6839-6855, (2021). 31 Sakamoto et al. The evolution of biomass-burning aerosol size distributions due to coagulation: dependence on fire and meteorological details and parameterization. Atmospheric Chemistry and Physics 16, 7709-7724, (2016). 32 Xu et al. Tracing the evolution of morphology and mixing state of soot particles along with the movement of an Asian dust storm. Atmospheric Chemistry and Physics 20, 14321-14332, (2020). 33 Fierce et al. Radiative absorption enhancements by black carbon controlled by particle-to- particle heterogeneity in composition. Proceedings of the National Academy of Sciences of the United States of America 117, 5196-5203, (2020). 34 Ching et al. A three-dimensional sectional representation of aerosol mixing state for simulating optical properties and cloud condensation nuclei. Journal of Geophysical Research: Atmospheres 121, 5912-5929, (2016). 35 Chen et al. An aerosol optical module with observation‐constrained black carbon properties for global climate models. Journal of Advances in Modeling Earth Systems 15, e2022MS003501, (2023). 36 Sand et al. Aerosol absorption in global models from AeroCom phase III. Atmospheric Chemistry and Physics 21, 15929-15947, (2021). 37 Gliß et al. AeroCom phase III multi-model evaluation of the aerosol life cycle and optical properties using ground- and space-based remote sensing as well as surface in situ observations. Atmospheric Chemistry and Physics 21, 87-128, (2021)."}]

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