Method to quantify black carbon aerosol light absorption enhancement with a mixing state index

Large uncertainties remain when estimating the warming effects of ambient black carbon (BC) aerosols on climate. One of the key challenges in modeling the radiative effects is predicting the BC light absorption enhancement, which is mainly determined by the mass ratio (MR) of non-BC coating material to BC in the population of BCcontaining aerosols. For the same MR, recent research has found that the radiative absorption enhancements by BC are also controlled by its particle-to-particle heterogeneity. In this study, the BC mixing state index (χ ) is developed to quantify the dispersion of ambient black carbon aerosol mixing states based on binary systems of BC and other non-black carbon components. We demonstrate that the BC light absorption enhancement increases with χ for the same MR, which indicates that χ can be employed as a factor to constrain the light absorption enhancement of ambient BC. Our framework can be further used in the model to study the radiative effects of black carbon on climate change.


Introduction
Black carbon (BC) aerosols absorb solar radiation, thus exerting warming effects on the earth's energy system (Bond et al., 2006(Bond et al., , 2013. However, large uncertainties remain when quantifying the BC warming effects (Menon et al., 2002;Koch et al., 2009;Jacobson, 2010;Cui et al., 2016). Most of the BC particles were emitted from incomplete combustion of bio-fossil fuel (Bond et al., 2013). After being initially emitted, the BC particles experience an aging process with some other non-BC components coated on the BC particles (Peng et al., 2016(Peng et al., , 2017. During the aging process, the light absorption of BC aerosols would increase, which is well known as the "lensing effect" (Saleh et al., 2013(Saleh et al., , 2014. One critical challenge in estimating the BC warming effects is quantifying the lensing effects of ambient BC aerosols (Liu et al., 2017).
The light absorption enhancement (E abs ), which is the ratio of light absorption of BC aerosols with the coating to that of bare BC particles, is proposed to quantify the lensing effects. Comprehensive studies have been carried out to study the E abs (Cappa et al., 2012;Liu et al., 2015;Fierce et al., 2016;Peng et al., 2016;Liu et al., 2017;Fierce et al., 2020). However, a large discrepancy remains between the results of E abs from field measurements and laboratory studies. The measured E abs of laboratory-generated monodisperse BC particles can reach up to a factor of 2, which is consistent with the results from the Mie scattering model (Cappa et al., 2012(Cappa et al., , 2019. However, some field measurement shows that the E abs values of ambient BC aerosols are relatively small, with 1.06 at California (Cappa et al., 2012), 1.07 in South China (Lan et al., 2013), and 1.10 in Japan (Nakayama et al., 2014), while the measured E abs of ambient BC reaches 1.59 during summer time in Beijing (Xie et al., 2019).
Many factors, such as the morphology of the BC core, the position of BC core inside coating, the coating thickness, chemical properties of coating materials, and size distribution of the BC, influence the E abs of ambient BC aerosols. Wu et al. (2018) reported that the BC light absorption properties vary significantly for different morphology from the cal- The measured E abs of BC particles from different ambient measurements, including this work (in pink) and lab studies. culation of models. Laboratory studies also find that the light absorption properties of the BC core were tuned due to the change of the BC core morphology (Yuan et al., 2020). Compared with the concentric spherical structure, the off-center coated BC aggregates would lead to up to a 31 % reduction in E abs by the multiple-sphere T-matrix method . It has been well studied that the E abs is highly related with the mass ratio (MR) of coating materials and BC core (Liu et al., , 2017. The coating materials are also critical in regulating the morphology and optical properties as the coating of sulfuric acid has been shown to be more efficient in altering the BC morphology and light absorption (Zhang et al., 2008;Xue et al., 2009b, a). Zhao et al. (2019b) reported that the light absorption properties of ambient BC particles are influenced by BC mass size distribution. In addition, recently, researchers have found that the E abs values are also controlled by particle-to-particle heterogeneity (Fierce et al., 2016(Fierce et al., , 2020. As shown in Fig. 1, the E abs of ambient aerosols for the same MR varies by about 30 %, which is consistent with the results of Fierce et al. (2020). However, there is no study, to the best of our knowledge, that constrains the uncertainties of the E abs for the same MR.
In this study, we developed a BC mixing state index (χ ) to quantify the dispersion of black carbon aerosol mixing states based on binary systems of BC and other non-black carbon components. We demonstrate that the BC E abs increases with χ for the same MR based on the field measurement, which indicates that χ can be employed as a factor to constrain the E abs properties of ambient BC.
2 Data and methods

Field measurements
The field measurements were conducted at a suburban site in Taizhou (119 • 57 E, 32 • 35 N) from 26 May to 18 June.
As shown in Fig. S1, the Taizhou site lies between two large cities of Nanjing and Shanghai, where the aerosols can be seen as representative of those of the Yangtze River Delta area . For more details of the field measurements, the reader is referred to Zhao et al. (2019a). During the field measurements, we placed all of the instruments in a container where the temperature was carefully controlled between 22 and 26 • C. A PM 10 impactor, which is about 5 m above the ground, was mounted on the top of the container. The sample aerosols were drawn from the impactor and then dried by a Nafion dryer tube.
The size-resolved BC core distribution and non-BC coating thickness were measured using a differential mobility analyzer (DMA, model 3081, TSI, USA) in tandem with a single-particle soot photometer (SP2, Droplet Measurement Technologies, USA). For detailed information on the DMA, the reader is referred to Zhao et al. (2019c). SP2 can measure the BC mass concentration from the incandescence signals emitted by the BC particle, which is heated to around 6000 K by a laser with a wavelength of 1064 nm (Zhao et al., 2020b). Along with the measurement of size-resolved BC distributions, a nephelometer (Aurora 300, Ecotech, Australia) (Müller et al., 2011) was employed to measure the aerosol scattering coefficient (σ sca ) at the wavelength of 525 nm.

BC mixing states from the DMA-SP2 system
In this study, the SP2 was placed behind the DMA to measure the size-selected distribution of BC core and non-BC coating thickness. The schematic instrument setup is shown in Fig. S2, and the reader is referred to Sect. 2 in the Supplement for details. The DMA was set to scan the aerosols' D p from 12.3 to 697 nm over a period of 285 s and repeated after a pause of 15 s. After careful calibrations of the SP2 (Sect. 3.1 in the Supplement), transformations of the measured signals to BC mass concentrations (Sect. 3.2 in the Supplement), and multiple charging corrections (Sect. 3.3 in the Supplement), the BC-containing number concentration distribution under different total diameter (D p ) and BC core diameter (D c ) values can be calculated, as shown in Fig. S5b. For the details of the calculation of the size-resolved distribution of BC core and coating thickness from the DMA-SP2 system, the reader is referred to Zhao et al. (2020a). The measured size-resolved distribution of BC core and coating thickness as in Fig. S5b were used for further analysis. It should be mentioned that the measured number distribution of BCcontaining aerosols is two-dimensional d 2 N dlogDp·dlogDc . As noted by Zhao et al. (2020b), the SP2 can only detect these BC-containing aerosols with a core diameter larger than 84 nm. The DMA selects the aerosol in the range between 13.3 and 749.9 nm. In the following discussion, the sizeresolved distribution of BC core and coating thickness is constrained in the range between 84 and 697 nm.

Calculating the aerosol optical properties
2.3.1 Calculating the single-particle aerosol absorption coefficient for a given D p and D c A Mie scattering core-shell model (Bohren et al., 2007) was employed to calculate the aerosol absorption coefficient (σ abs ). When calculating the σ abs of single particles, the Mie scattering model requires the diameter of the core, the coating thickness, the refractive index of the core, and the refractive index of the shell. The refractive index of the core adopted here is 1.67 + 0.67i, which is the mean value calculated by comparing the measured light absorption and calculated light absorption properties (Zhao et al., 2020a). The refractive index of the shell is chosen to be 1.46 + 0i, which is assumed to be that of the non-BC component measured by the DMA-SP2 system (Zhao et al., 2019a, c). With the above information, the σ abs values at a given D p and a given D c can be calculated.

Calculating the aerosol bulk absorption coefficient
We calculate the single-particle σ abs of different D p and D c with the given refractive index of core and shell, and then the ambient aerosol σ abs distributions at different D p and D c

Calculating the aerosol E abs
Along with calculating the σ abs (DpDc) of single particles for different D p and D c , we calculate the corresponding light absorption (σ abs (DcDc)) value for D c without thickness. The corresponding total light absorption of all measured BCcontained aerosols without coating can be calculated by integrating the calculated σ abs (DcDc) among different D p and D c weighted with d 2 N dlogDp·dlogDc . Thus the ambient BC particles without coating (σ abs (D p = D c )) can be calculated. The bulk ambient aerosol E abs can thus be calculated with E abs = σ abs σ abs (D p =D c ) .

Quantifying BC mixing states
In this study, the mass-weighted mixing state index for BCcontaining particles (χ) is developed to investigate the distribution of non-BC material across the BC-containing particle population, which is essentially the same as that of Yu et al. (2020). As for BC particles with known D p and D c , the mass concentration of BC core and coating material can be calculated with the effective density of BC core and coating material. The effective density of the BC core is calculated in detail in Sect. 2.2 in the Supplement. The effective density of the coating material is assumed to be the same as the measured effective density of non-BC aerosols using a centrifugal particle mass analyzer (version 1.53, Cambustion Ltd, UK) in tandem with a scanning mobility particle sizer system (Zhao et al., 2019a), and a mean value of 1.5 g/cm 3 was used here.
For each particle i(i = 1, 2, . . . , N is the measured BCcontaining aerosol number concentration), we can calculate its mass ratio of BC with where m i,BC is the mass concentration of BC, and m i is the total mass concentration of particle i. The mass portion of BC can be calculated as where m BC (the total mass concentration of BC) and m tot (total mass of BC-containing aerosols) can be calculated as The mass portion of particle i to total BC-containing aerosols is calculated as With the definition above, we can calculate the mixing entropy of particle i(H i ) by the average mixing entropy of the population by and the population bulk mixing entropy by Then the average particle species diversity can be calculated by and the bulk population species diversity can be calculated by With the above information, the dispersion of BC particle mixing states can be defined as The basic idea of quantifying the BC particle mixing states is the same as that of Riemer et al. (2013) and Riemer et al. (2019); their framework mainly focuses on the bulk ambient aerosols with about five species (Bondy et al., 2018;Ye et al., 2018). Several different (binary) species definitions for χ have been used in the literature. Ching et al. (2017) used this index to study the impact of mixing of hygroscopic and nonhygroscopic species on cloud condensation nuclei. Dickau et al. (2016) quantified the volatile and nonvolatile species mixing characters. Zheng et al. (2021) compared three different variants for χ , one of which was based on absorbing (BC) and non-absorbing species, and Yu et al. (2020) used a metric that is very related to this paper. Our developed χ is a reduced parameter that only concerns the BC-containing aerosols with two species of BC component and non-BC coating materials.
3 Results and discussions 3.1 BC mixing state diagram A mixing state diagram as shown in Fig. 2 was employed for better understanding of the dispersion of BC mixing states. Nine different aerosol populations are given and summarized in Table 1. For each group, we include six BC-containing particles with different mass concentrations of BC core and non-BC coating material. For group 1, the amounts of BC are very small (near zero), and most of the aerosols are composed of the non-BC compo-nent. The D α and D γ values are 1.00 and 1.00 respectively. These groups can also be described as all of the particles are pure BC particles without coating.
For groups 2, 3, and 4, the mass concentration ratios of the BC component to the non-BC component are 1 : 5, 2 : 4, and 3 : 3 respectively. All of the D α values are 1.00 for groups 2, 3, and 4 because the BC particles are externally mixed. The corresponding D γ values are 1.56, 1.89, and 2.00 respectively. For these three groups, the χ values are all 0.00.
For groups 4, 5, 6, and 7, the mass concentration ratios of the BC component to the non-BC component are all 1 : 1, while the BC component is mixed to a different extent. It is easy to conclude that the BC particles of group 7 are most well mixed among these four groups. The corresponding χ values are 0, 0.26, 0.83, and 1.0 for group 4, 5, 6, and 7, respectively.
As for groups 8 and 9, the mass concentration ratios of the BC component to the non-BC component are 1 : 6.1. The D γ values are 1.5, and the D α values are 1.5 and 1.35 respectively.
From the different groups, the average particle species diversity D γ value is mainly determined by the total mass concentration ratio of the BC component to the non-BC component. It varies between 1 and 2 for different total mass concentration ratios. The D γ increases when the mass ratio approaches 1. The bulk population species diversity D α ranges between 1 and D γ . It denotes the diversity of different BCcontaining particles. Figure S6 gives the time series of our field measurement results. During the field measurements, the σ sca varies between 29 and 1590 Mm −1 . The ranges of H α , H γ , D α , D γ , and χ are 0.10-0.55, 0.42-0.64, 1.32-1.72, 1.52-1.91, and 0.62-0.82 respectively.

Overview of the measurements
For a better understanding of the characteristics of the above parameters, we only present the time series of these parameters during a pollution period between 27 and 30 May in Fig. 3. As shown in Fig. 3, the MR increased from about 2 to 4 when the σ sca increased from 300 to 1200 Mm −1 , which indicates that some secondary aerosol components were coated on the BC particles when the ambient air is more polluted. During the aging process, the H α decreased from 0.51 to 0.38 and H γ decreased from 0.63 to 0.49. The D α decreases from 1.66 to 1.48. The D γ decreases with the MR from 1.86 to 1.66, which is consistent with the results in Sect. 3.1 that the D γ should decrease with the MR when the MR is larger than 1. The χ varies between 0.68 and 0.79. It is worth noting that the χ is not well correlated with the pollution conditions.
The corresponding mean values of BC-containing number size distributions under different D p and D c between the days of 27 and 28, 28 and 29, and 29 and 30 May are shown in Fig. S7. It is obvious that the BC-containing number and coating thickness increase with the pollution levels.  Fig. 2.

Relationship between the χ and E abs from measurements
For each of the measured group of size-resolved distribution of BC core and coating thickness, we calculated the corresponding MR, χ, and E abs . And the relationship between the MR and absorption enhancement is summarized in Fig. 5. It should be noted that the shown BC population is only one of the possible examples with χ equaling 0, 0.81, and 1 respectively. There are many other possible ways the particle composition can be arranged that would give the same mixing state index. Overall, the BC E abs values increase with MR, which is consistent with previous knowledge. For a given value of MR, E abs varies by about 20 %, especially for these conditions with MR larger than 1.0. When MR is larger than 1.0, the E abs increases with the χ. The relationship between the E abs and χ is rather complex when MR is smaller than 1.0. However, only 448 of 6948 groups (6.4 %) of the measured MR values are smaller than 1. Therefore, for most of the con-  ditions, the measured E abs should increase with χ, which indicates that the BC mixing state index χ can be employed as a factor to constrain the E abs of ambient aerosols.
A schematic diagram as shown in Fig. 6 to denote the relationship between the E abs and χ . From Fig. 6, we calculated the E abs and χ under different MRs and then compared the E abs of different bulk aerosols. The first group contains two particles with both the MRs equaling 8. The corresponding χ is 1.00, and E abs is 1.60. Another group of particles contains two particles with MRs equaling 1 and 15, respectively. Thus the second group of particles has a mean MR of 8. The calculated corresponding χ and E abs are 0.79 and 1.42 respectively. Thus, the E abs tends to increase with χ for the same MR, which mainly results from the increasing ratio of E abs (the slope of E abs to MR) decreasing with MR. It is worth noting that the increasing ratio is almost the same when the MR is in the range of 0 and 3. Therefore, the E abs does not tend to increase with the χ when the MR is less than 1, which is consistent with our study, as shown in Fig. 6.

Relationship between the χ and E abs from simulations
A Monte Carlo simulation was carried out for a better understanding of the relationship between χ and E abs . During the simulation, a group of the BC-containing aerosols was generated with the D p and D c meeting the following conditions, and the number of BC-containing particles was assumed to be 30. For each of the BC-containing particles, the core diameter of the BC particle was randomly generated with a geometric mean diameter of 130.7 nm and a geometric standard deviation of 1.5, which are the mean measurement results of the BC core distribution during the field measurements (Zhao et al., 2020b). The corresponding MR of the BC particle is assumed to be randomly distributed in the range between 0.0 (pure BC particles without coating) and 78.0 (particles with a core diameter of 130 nm and a total diameter of 560 nm). For each group of particles, the corresponding aerosol bulk MR, E abs , and χ can be calculated using the core-shell Mie scattering model and the parameterization proposed by Wu et al. (2018) to account for the non-sphericity of the BC aerosols. The simulations were conducted 10 7 times, and the calculated mean and standard deviation of E abs under different MR and χ are summarized in Fig. 7a and b.
From Fig. 7a, the calculated E abs tends to increase with MR for each of the given χ values, which is consistent with previous knowledge of the BC light absorption properties. When the MR is smaller than 2, the calculated E abs does not seem to increase with the χ , which is consistent with the analyzed results from Sect. 3.3 and Fig. 6. When the MR is larger 2, the E abs tends to increase with the χ. The larger the MR is, the more sensitive E abs is to χ. There may be two reasons for this phenomenon. One reason is that the calculated slope of E abs to MR for one particle as shown in Fig. 6 decreases with the MR. Another reason is that the calculated E abs range increases with MR when the χ changes between 0 and 1 as shown in Fig. 5.
As for the uncertainties of simulated E abs , it tends to increase with the MR, which is consistent with the previous discussions that the E abs range tends to increase with MR. Overall, the calculated standard deviations of E abs are smaller than 10 % for different MR and χ. Therefore, the calculated E abs can be well constrained by χ. When the ambient aerosol χ and MR are measured, the corresponding E abs can be estimated from Fig. 7a.

Conclusion
Larger uncertainties remain when estimating the warming effects of ambient BC aerosols due to the poor understanding of the ambient BC light absorption enhance ratio. Previous studies find that the light absorption of ambient aerosols was mainly determined by the morphology of the BC core, the position of the BC core inside coating, the coating thickness, and the size distribution of the BC. We find that there are more than 20 % of uncertainties for the same measured mean coating thickness, i.e. the same measured MR based on the field measurements of the size-resolved distribution of BC core and coating thickness. However, there was no study until now, to the best of our knowledge, that attempts to constrain the uncertainties.
In this study, we developed the BC mixing state index χ based on the mass concentrations of BC components and non-BC material of each BC-containing particle. Results show that the light absorption enhancement ratio E abs tend to increase the χ for the same measured MR. Therefore, our developed parameter χ, which reflects the dispersion of the BC mixing states, can be employed as an effective parameter to constrain the light absorption enhancement of ambient BC-containing aerosols.
The new finding of our study is that the mixing state index can contribute to improvements in the accuracy of simulating the BC radiative effects. In the particle-resolved simulation of ambient aerosols, the particle-to-particle heterogeneity of BC-containing aerosols can be resolved by simply introducing the BC mixing state index χ. The aerosol light absorption enhancement can be better constrained by MR and χ, and then the radiative effects of BC can be estimated. There-fore, our framework can be employed in the model by simply introducing a BC mixing state index for better estimating the BC radiative effects.
Data availability. The research data are available within the paper.
Author contributions. GZ wrote the manuscript. CZ, MH, TT, SG, ZW, YZ, and GZ discussed the results.
Competing interests. The contact author has declared that neither they nor their co-authors have any competing interests.
Disclaimer. Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Review statement. This paper was edited by Manvendra K. Dubey and reviewed by three anonymous referees.