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 (Liu et al., 2020). 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 PM10 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 (σsca) at the wavelength of 525 nm.

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' Dp 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 (Dp) and BC core diameter (Dc) 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 BC-containing aerosols is two-dimensional d2NdlogDp⋅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 size-resolved distribution of BC core and coating thickness is constrained in the range between 84 and 697 nm.

In this study, the mass-weighted mixing state index for BC-containing 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 Dp and Dc, 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/cm3 was used here.

For each particle i(i= 1, 2, …, N is the measured BC-containing aerosol number concentration), we can calculate its mass ratio of BC with 1pi,BC=mi,BCmi, where mi,BC is the mass concentration of BC, and mi is the total mass concentration of particle i. The mass portion of BC can be calculated as 2pBC=mBCmtot, where mBC (the total mass concentration of BC) and mtot (total mass of BC-containing aerosols) can be calculated as mBC=∑i=1Nmi,BC, mtot=∑i=1Nmi. The MR is calculated as 3MR=(mtot-mBC)mBC. The mass portion of particle i to total BC-containing aerosols is calculated as 4pi=mimtot. With the definition above, we can calculate the mixing entropy of particle i(Hi) by 5Hi=-(pi,BClnpi,BC+1-pi,BCln⁡(1-pi,BC), the average mixing entropy of the population by 6Hα=∑i=1NpiHi, and the population bulk mixing entropy by 7Hγ=-(pBCln⁡pBC+1-pBCln(1-pBC). Then the average particle species diversity can be calculated by 8Dα=eHα, and the bulk population species diversity can be calculated by 9Dγ=eHγ. With the above information, the dispersion of BC particle mixing states can be defined as 10χ=Dα-1Dγ-1. 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 non-hygroscopic 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.

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 component. The Dα and Dγ 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 Dα values are 1.00 for groups 2, 3, and 4 because the BC particles are externally mixed. The corresponding Dγ 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 Dγ values are 1.5, and the Dα values are 1.5 and 1.35 respectively.

From the different groups, the average particle species diversity Dγ 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 Dγ increases when the mass ratio approaches 1. The bulk population species diversity Dα ranges between 1 and Dγ. It denotes the diversity of different BC-containing particles.

Figure S6 gives the time series of our field measurement results. During the field measurements, the σsca varies between 29 and 1590 Mm-1. The ranges of Hα, Hγ, Dα, Dγ, 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.

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 σsca 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 Hα decreased from 0.51 to 0.38 and Hγ decreased from 0.63 to 0.49. The Dα decreases from 1.66 to 1.48. The Dγ decreases with the MR from 1.86 to 1.66, which is consistent with the results in Sect. 3.1 that the Dγ 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 Dp and Dc 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. However, the normalized BC core distributions are almost the same for different pollution levels as shown in Fig. S8. The daily variation of σsca, which is highly related to the development of the boundary layer, reaches its maximum value of 525 Mm-1 at 06:00 and a minimum value of 150 Mm at 19:00, as shown in Fig 4. The daily variation of MR is largest at 05:00, with a mean value of 3.16, and reaches its minimum value of 2.56 at 19:00. The daily variation of MR was mainly influenced by the aging process and anthropogenic activities. During the daytime, the newly emitted BC particles due to anthropogenic activities have low MR, and the measured mean MR is lower than that at night. The Dα values, which are anti-correlated with MR, show the opposite trend with MR. As for χ, it is smaller in the daytime than that at night. The lower χ values in the daytime mainly resulted from the mixing of newly emitted BC particles due to anthropogenic activities and some preexisting aged BC particles.

For each of the measured group of size-resolved distribution of BC core and coating thickness, we calculated the corresponding MR, χ, and Eabs. 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 Eabs values increase with MR, which is consistent with previous knowledge. For a given value of MR, Eabs varies by about 20 %, especially for these conditions with MR larger than 1.0. When MR is larger than 1.0, the Eabs increases with the χ. The relationship between the Eabs 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 conditions, the measured Eabs should increase with χ, which indicates that the BC mixing state index χ can be employed as a factor to constrain the Eabs of ambient aerosols.

A schematic diagram as shown in Fig. 6 to denote the relationship between the Eabs and χ. From Fig. 6, we calculated the Eabs and χ under different MRs and then compared the Eabs of different bulk aerosols. The first group contains two particles with both the MRs equaling 8. The corresponding χ is 1.00, and Eabs 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 Eabs are 0.79 and 1.42 respectively. Thus, the Eabs tends to increase with χ for the same MR, which mainly results from the increasing ratio of Eabs (the slope of Eabs 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 Eabs 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.

A Monte Carlo simulation was carried out for a better understanding of the relationship between χ and Eabs. During the simulation, a group of the BC-containing aerosols was generated with the Dp and Dc 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, Eabs, 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 107 times, and the calculated mean and standard deviation of Eabs under different MR and χ are summarized in Fig. 7a and b.

From Fig. 7a, the calculated Eabs 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 Eabs 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 Eabs tends to increase with the χ. The larger the MR is, the more sensitive Eabs is to χ. There may be two reasons for this phenomenon. One reason is that the calculated slope of Eabs to MR for one particle as shown in Fig. 6 decreases with the MR. Another reason is that the calculated Eabs range increases with MR when the χ changes between 0 and 1 as shown in Fig. 5.

As for the uncertainties of simulated Eabs, it tends to increase with the MR, which is consistent with the previous discussions that the Eabs range tends to increase with MR. Overall, the calculated standard deviations of Eabs are smaller than 10 % for different MR and χ. Therefore, the calculated Eabs can be well constrained by χ. When the ambient aerosol χ and MR are measured, the corresponding Eabs can be estimated from Fig. 7a.