**Research article**
09 Dec 2021

**Research article** | 09 Dec 2021

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

Gang Zhao Tianyi Tan Yishu Zhu Min Hu and Chunsheng Zhao

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**Gang Zhao et al.**Gang Zhao Tianyi Tan Yishu Zhu Min Hu and Chunsheng Zhao

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^{1}State Key Joint Laboratory of Environmental Simulation and Pollution Control, International Joint Laboratory for Regional Pollution Control, Ministry of Education, College of Environmental Sciences and Engineering, Peking University, Beijing, 100871, China^{2}Department of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, 100871, China

^{1}State Key Joint Laboratory of Environmental Simulation and Pollution Control, International Joint Laboratory for Regional Pollution Control, Ministry of Education, College of Environmental Sciences and Engineering, Peking University, Beijing, 100871, China^{2}Department of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, 100871, China

**Correspondence**: Chunsheng Zhao (zcs@pku.edu.cn)

**Correspondence**: Chunsheng Zhao (zcs@pku.edu.cn)

Received: 18 Apr 2021 – Discussion started: 21 Apr 2021 – Revised: 25 Oct 2021 – Accepted: 27 Oct 2021 – Published: 09 Dec 2021

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 BC-containing 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.

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Black carbon (BC) aerosols absorb solar radiation, thus exerting warming effects on the earth's energy system (Bond et al., 2006, 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, 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, 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, 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 calculation 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 (Zhang et al., 2017). 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., 2014, 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, 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.

## 2.1 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 (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
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.

## 2.2 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 BC-containing aerosols is two-dimensional
$\left(\frac{{d}^{\mathrm{2}}N}{\mathrm{dlogDp}\cdot \mathrm{dlogDc}}\right)$. 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.

## 2.3 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+0.67*i*, 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+0*i*, 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.

### 2.3.2 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}
$\left(\frac{{d}^{\mathrm{2}}{\mathit{\sigma}}_{\mathrm{abs}}}{\mathrm{dlogDp}\cdot \mathrm{dlogDc}}\right)$ can be calculated by multiplying the number concentrations of the
BC-contained aerosols $\left(\frac{{d}^{\mathrm{2}}N}{\mathrm{dlogDp}\cdot \mathrm{dlogDc}}\right)$. By
integrating the $\frac{{d}^{\mathrm{2}}{\mathit{\sigma}}_{\mathrm{abs}}}{\mathrm{dlogDp}\cdot \mathrm{dlogDc}}$ over different *D*_{c}
values, the ambient aerosol *σ*_{abs} distribution along with
different *D*_{p} $\left(\frac{\mathrm{d}{\mathit{\sigma}}_{\mathrm{abs}}}{\mathrm{dlogDp}}\right)$ can be calculated. The total *σ*_{abs} of the ambient BC-containing aerosols can be calculated by
integrating the $\frac{\mathrm{d}{\mathit{\sigma}}_{\mathrm{abs}}}{\mathrm{dlogDp}}$
over different *D*_{p} values.

### 2.3.3 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 BC-contained aerosols without coating can be calculated by
integrating the calculated *σ*_{abs}(DcDc) among different
*D*_{p} and *D*_{c} weighted with
$\frac{{d}^{\mathrm{2}}N}{\mathrm{dlogDp}\cdot \mathrm{dlogDc}}$. 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
${E}_{\mathrm{abs}}=\frac{{\mathit{\sigma}}_{\mathrm{abs}}}{{\mathit{\sigma}}_{\mathrm{abs}}({D}_{\mathrm{p}}={D}_{\mathrm{c}})}$.

## 2.4 Quantifying BC mixing states

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 *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 BC-containing 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
${m}_{\mathrm{BC}}={\sum}_{i=\mathrm{1}}^{N}{m}_{i,\mathrm{BC}}$, ${m}_{\mathrm{tot}}={\sum}_{i=\mathrm{1}}^{N}{m}_{i}$. The MR is 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 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.

## 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 component. 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 BC-containing particles.

## 3.2 Overview of the measurements

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.

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. However,
the normalized BC core distributions are almost
the same for different pollution levels as shown in Fig. S8. The daily
variation of *σ*_{sca}, 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
*D*_{α} 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.

## 3.3 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
conditions, 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.

## 3.4 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.

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. Therefore, our framework can
be employed in the model by simply introducing a BC mixing state index for
better estimating the BC radiative effects.

The research data are available within the paper.

The supplement related to this article is available online at: https://doi.org/10.5194/acp-21-18055-2021-supplement.

GZ wrote the manuscript. CZ, MH, TT, SG, ZW, YZ, and GZ discussed the results.

The contact author has declared that neither they nor their co-authors have any competing interests.

Publisher’s note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research has been supported by the China Postdoctoral Science Foundation (grant no. 2021M700192), the National Natural Science Foundation of China (grant no. 41590872), and the National Key R&D Program of China (grant no. 2016YFC020000: Task 5).

This paper was edited by Manvendra K. Dubey and reviewed by three anonymous referees.

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