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- Editorial & advisory board
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**Research article**
24 Oct 2019

**Research article** | 24 Oct 2019

Role of black carbon mass size distribution in the direct aerosol radiative forcing

^{1}Department of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, China^{2}Institute for Environmental and Climate Research, Jinan University, Guangzhou 511443, China

^{1}Department of Atmospheric and Oceanic Sciences, School of Physics, Peking University, Beijing, China^{2}Institute for Environmental and Climate Research, Jinan University, Guangzhou 511443, China

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

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

Abstract

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Large uncertainties exist when estimating radiative effects of ambient black carbon (BC) aerosol. Previous studies about the BC aerosol radiative forcing mainly focus on the BC aerosols' mass concentrations and mixing states, while the effects of BC mass size distribution (BCMSD) were not well considered. In this paper, we developed a method of measuring the BCMSD by using a differential mobility analyzer in tandem with an Aethalometer. A comprehensive method of multiple charging corrections was proposed and implemented in measuring the BCMSD. Good agreement was obtained between the BC mass concentration integrated from this system and that measured in the bulk phase, demonstrating the reliability of our proposed method. Characteristics of the BCMSD and corresponding radiative effects were studied based on a field measurement campaign conducted in the North China Plain by using our own measurement system. Results showed that the BCMSD had two modes and the mean peak diameters of the modes were 150 and 503 nm. The BCMSD of the coarser mode varied significantly under different pollution conditions with peak diameter varying between 430 and 580 nm, which gave rise to significant variation in aerosol bulk optical properties. The direct aerosol radiative forcing was estimated to vary by 8.45 % for different measured BCMSDs of the coarser mode, which shared the same magnitude with the variation associated with assuming different aerosol mixing states (10.5 %). Our study reveals that the BCMSD as well as its mixing state in estimating the direct aerosol radiative forcing matters. Knowledge of the BCMSD should be fully considered in climate models.

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How to cite.

Zhao, G., Tao, J., Kuang, Y., Shen, C., Yu, Y., and Zhao, C.: Role of black carbon mass size distribution in the direct aerosol radiative forcing, Atmos. Chem. Phys., 19, 13175–13188, https://doi.org/10.5194/acp-19-13175-2019, 2019.

1 Introduction

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Atmospheric black carbon (BC) is the second strongest absorbing component in the atmosphere (Bond et al., 2013) but the magnitudes of the warming effects are poorly quantified. When emitted to the surroundings, BC particles transform their morphology from fractal to spherical and then grow as fully compact particles with other components depositing on the BC aerosol (Peng et al., 2016). The variation in the shapes of BC aerosols, together with the variation in the mixing states, can lead to substantial changes in aerosol optical properties (Liu et al., 2017; China et al., 2013; Wu et al., 2016, 2018). BC aerosols also have a significant influence on the climate by interacting with clouds (Koch and Del Genio, 2010; Roberts et al., 2008; Stevens and Feingold, 2009), ice and snow (Bond et al., 2013). A recent study shows that the solar absorption of BC can suppress the turbulence in the atmospheric boundary layer (Wilcox et al., 2016). It is found that BC emissions may be responsible for the increase in droughts and floods in China and India (Menon et al., 2002). In addition, BC can pose a serve threat to human health through inhalation (Nichols et al., 2013; Janssen et al., 2011).

Comprehensive studies have been carried out to evaluate the climatic effect
of BC based on the measurement of BC mass concentrations (*m*_{BC})
(Koch et al., 2009; Ramanathan and Carmichael, 2008). The *m*_{BC}
values near the ground have been well characterized (Ramachandran and Rajesh,
2007; Ran et al., 2016b; Reddington et al., 2013; Song et al., 2013), and the
BC vertical distributions are widely measured and evaluated as well (Ran
et al., 2016a; Babu et al., 2011; Ferrero et al., 2011). Despite these
measurements, more insights into the BC microphysical properties can help to
estimate the influence of BC aerosols on visibility (Zhang et al., 2008),
climate (Jacobson, 2001) and human health (Lippmann and
Albert, 1969). These microphysical properties include BC morphology
(Zhang et al., 2016), density (Zhang et al., 2016), complex
refractive index (Bond et al., 2013), hygroscopicity (Zhang et al.,
2008; Peng et al., 2017), mixing states (Moffet et al., 2016; Raatikainen
et al., 2017), and particularly the mass size distribution (BCMSD)
(Cheng et al., 2012; Cheng and Yang, 2016; Gong et al., 2016). Knowledge of
BCMSD is not only helpful to study the mixing state of BC aerosols
(Raatikainen et al., 2017) but also essential to model the role
of BC in evaluating regional and global climate accurately. BC radiative
effects are highly sensitive to the emitted BC particle size distribution
(Matsui et al., 2018). The health impacts of BC are
significantly related to the BCMSD (Turner et al., 2015). Furthermore, the
information of BCMSD can help to study the source, the evolution and the
mixing state of ambient BC aerosols (Yu et al.,
2010).

Many methods have been proposed to measure BCMSD. For instance, the
BCMSD was measured by sampling the aerosol in the size range from about 50 nm to several micrometers on quartz fiber filter substrates using a
micro-orifice uniform deposit impactor (MOUDI) (Venkataraman and
Friedlander, 1994; Guo, 2016). Cheng et al. (2014) developed a method to
measure the BCMSD by employing two Aethalometers in parallel, with one to
measure total *m*_{BC} and the other to measure *m*_{BC} below specific
particle sizes using a size cutoff inlet. The two abovementioned methods measure the
BCMSD corresponding to the aerodynamic diameter. The single-particle soot
photometer (SP2) instrument is developed and widely used because it provides single-particle information, hence the BCMSD and the mixing state of the
atmospheric aerosols can be derived directly (Schwarz et al., 2006; Gao et
al., 2007; Huang et al., 2012; Singh et al., 2016). The BCMSD corresponding to
the ambient aerosol mobility diameter can be measured by using a
differential mobility analyzer (DMA) in tandem with SP2
(Raatikainen et al., 2017). However, the laser-induced
incandescence method cannot provide reliable information about the particles
beyond the range of 70 and 400 nm (Moteki and Kondo,
2010), which results in a lack of the knowledge on the BCMSD
characteristics for these aerosols over 400 nm. The results from MOUDI find
that a large amount of BC locates at the aerodynamic diameter range of
between 370 and 1000 nm (Hu et al., 2012; Huang and Yu, 2008). However,
the measurements of MOUDI cannot give detailed information on the BCMSD
evolution due to the low temporal and diameter resolutions (Hu et al.,
2012; Huang and Yu, 2008). The characteristics of the BCMSD larger than 370 nm is not well studied due to the limitations of the instrument.

Recently, Ning et al. (2013) and Stabile et al. (2012) proposed a new method to measure the BCMSD by using a differential mobility analyzer (DMA) in tandem with an Aethalometer (AE). This method has the potential of measuring the BCMSD from 20 to 584 nm with high time resolution. We develop and validate the BCMSD measurement system based on the works of Ning et al. (2013). The developed measurement system was employed in a field campaign in the North China Plain. The characteristics of the measured BCMSD were studied based on the field measurements. Furthermore, the effects of BCMSD variations on the aerosol optical properties and corresponding direct aerosol radiative properties were evaluated. The aerosol optical properties were calculated by using the Mie scattering theory. The direct aerosol radiative forcing (DARF) was estimated by using the Santa Barbara DISORT (discrete ordinates radiative transfer) Atmospheric Radiative Transfer (SBDART) model.

The structure of this paper is organized as follows. Section 2 gives the information about the instrument setup and field measurements. Section 3 gives the detailed method used in this study, which contains (1) conducting multiple charging corrections when deriving the aerosol BCMSD and (2) evaluating the aerosol optical and radiative properties for different BCMSDs. Results and discussions are shown in Sect. 4. The conclusion is drawn in the last part.

2 Instrument setup

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The measurement system setup was based on the works of Stabile
et al. (2012) and Ning et al. (2013) as schematically shown
in Fig. 1. The ambient sample aerosol particles were firstly dried to below
a relative humidity of 30 % through a Nafion drying tube before passing
through to the DMA (Model 3081, TSI, USA). The DMA scanned aerosol particles
with diameter ranges from 12.3 to 697 nm over a period of 285 s and
started another scanning after a pause of 15 s, so one complete cycle
took 5 min. The sheath and sample flow rates of the DMA were 3 and
0.5 L min^{−1}, respectively. The quasi-monodisperse aerosols that passed through
the DMA were further divided into two flows: one led to an
Aethalometer (AE51, model 51, microAeth, USA) with a flow rate of 0.2 L min^{−1} to
measure the absorption coefficient (*σ*_{abs}) at 1 s time
resolution and the other one with a flow rate of 0.3 L min^{−1} flow directed to a
condensation particle counter (CPC, model 3772, TSI, USA), which counted particle number concentrations at
0.1 s resolution. Clean air with a flow rate of 0.7 L min^{−1} was used to
compensate for the CPC inlet flow, which had a default flow rate of 1 L min^{−1}.
Overall, the combined system of DMA, CPC and AE51 could provide one particle number size distribution (PNSD)
and size-resolved *σ*_{abs} scan every 5 min. If the mass
absorption coefficient (MAC) at a given diameter is known, the BCMSD can be
derived correspondingly.

An Aethalometer (AE33, model 33, Magee, USA) was used to measure the *σ*_{abs} or *m*_{BC} with a time resolution of 1 min.
The mass concentration of particles with diameters smaller than 2.5 µm
(PM_{2.5}) was concurrently measured with a time resolution of 1 min during
the field observations by the tapered element oscillating microbalance
(TEOM) dichotomous ambient particulate monitor (1405-DF), which was an
indicator of the pollution conditions.

From 21 March to 9 April 2017, an intensive field measurement was
conducted to characterize the ambient dry aerosol BCMSD corresponding to
aerosol mobility diameters at the AERONET BEIJING_PKU station
(39^{∘}59^{′} N, 116^{∘}18^{′} E). This station was
located on a roof of Peking University campus in the northwest of
Beijing, China. There were two main streets that surrounding the station: Chengfu Road to the south and Zhongguancun Street to the west. The aerosol
sampled at this station were mainly composed of urban roadside aerosols
(Zhao et al., 2018).

3 Methodologies

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Five steps were involved in calculating the BCMSD using the raw data from the
measurement system: (1) correcting the “loading effect” and “multiple
scattering effect” of *σ*_{abs} measured by AE51; (2) matching the
instrument time between the AE51 and CPC; (3) matching the measured *σ*_{abs} and diameter to get the raw size-resolve *σ*_{abs} that is not
involved in multiple charging corrections; (4) conducting the multiple
charging corrections of the measured raw size-resolved *σ*_{abs}; and (5) transforming the size-resolved *σ*_{abs} into BCMSD.

The Aethalometer (AE51 and AE33) is a well-developed and widely used
instrument to measure the *σ*_{abs} (Drinovec et al., 2015; Hansen
et al., 1984). When absorbing aerosols accumulate on the sample filter of
the Aethalometer continuously, the *σ*_{abs} can be determined by
concurrently measuring the light intensities *I* after the fiber filter and the
light intensities *I*_{0} transmitted through a reference spot, which is free of
aerosol loading. The light attenuation (ATN) is defined as

$$\begin{array}{}\text{(1)}& \mathrm{ATN}=\mathrm{100}\cdot \mathrm{ln}\left({\displaystyle \frac{{I}_{\mathrm{0}}}{I}}\right).\end{array}$$

The total *σ*_{abs} of the loaded particle on the filter is given by

$$\begin{array}{}\text{(2)}& {\mathit{\sigma}}_{\mathrm{abs},\mathrm{tot}}={\displaystyle \frac{A\cdot \mathrm{ATN}}{\mathrm{100}}},\end{array}$$

where *A* is the sample spot area on the filter. The instantaneous *σ*_{abs} can be calculated through the increment of *σ*_{abs,tot}:

$$\begin{array}{}\text{(3)}& {\mathit{\sigma}}_{\mathrm{abs}}={\displaystyle \frac{{\mathit{\sigma}}_{\mathrm{abs},\mathrm{tot}}}{\mathrm{\Delta}t}}={\displaystyle \frac{A\cdot \mathrm{\Delta}\mathrm{ATN}}{\mathrm{100}\cdot F\cdot \mathrm{\Delta}t}},\end{array}$$

where *F* is the flow rate and ΔATN is the ATN variation during the
time period of Δ*t*. The *σ*_{abs,tot} can be transformed to
*m*_{BC} when the mass attenuation cross section (MAC) of BC is known.
Traditionally, a constant MAC of 7.7 m^{2} g^{−1} was used to deduce the
*m*_{BC} (Drinovec et al., 2015).

Corrections to the measured *σ*_{abs} are necessary because a
systematic bias exists due to the prevailingly known “loading effect” and
the multiple scattering effect (Drinovec et al., 2015; Virkkula et al.,
2007, 2015). The AE33 can directly provide the corrected
*σ*_{abs} values through measuring two light intensities of two spots
with different BC load efficiencies (Drinovec et al.,
2015). For AE51, The correcting method in
Virkkula et al. (2007) was adopted:

$$\begin{array}{}\text{(4)}& {\mathit{\sigma}}_{\mathrm{abs},\mathrm{corrected}}=\left(\mathrm{1}+k\times \mathrm{ATN}\right){\mathit{\sigma}}_{\mathrm{abs},\mathrm{uncorrected}},\end{array}$$

where *k* is the correction factor and a constant value of 0.004 is employed
in this study to correct the *σ*_{abs} from AE51. In the first part of
the Supplement, we show that the loading effect corrections
of *σ*_{abs} from AE51 were essential and the value of *σ*_{abs} from AE33 could be used as a reference for the measured BCMSD. As for
the multiple scattering corrections, Zhang et al. (2018) compared the
measured *σ*_{abs} measured by AE33 and by multiangle absorption
photometer (MAAP) at Tsinghua University, which is about 2 km away from our
measurement site. They recommended a compensation factor of 2.6 to be used
and we adopted the same factor in our study.

Time correction was needed because time gaps between voltages implied on the DMA (particle size) and sample particles measured by different instruments were not the same. The time correction procedures were conducted every day during the field measurement to ensure that the time deviations of the CPC and the AE51 were constrained within 2 s.

Figure S3 gives the time series diagram of scanned aerosol diameters by DMA,
measured *σ*_{abs} from AE51 and the aerosol number concentrations
counted by CPC. The aerosol PNSD (or size-resolved *σ*_{abs}) could be
calculated by matching the DMA diameter and the measured aerosol number
concentrations (or measured *σ*_{abs}) by simply using the single-particle charge ratio for each electrical mobility diameter. These measured
PNSD and size-resolved *σ*_{abs} did not consider the effect of
multiple-charging corrections and are labeled as raw aerosol PNSD and raw
aerosol size-resolved *σ*_{abs}.

In the work of Ning et al. (2013), a lot of effort was made to evaluate the performance of the instrument. They considered the diffusion corrections and particle charging corrections. However, the particle charging corrections were limited to single-particle charge ratio as they mentioned that they simplified the particle charge correction by applying the peak electrical mobility for the calculation of representative particle size for each mobility bin and single-particle charge ratio for each primary mobility. They ignored the fact that the aerosol samples selected by the DMA were quasi-monodisperse with different charges and different diameters.

We propose a new algorithm for the multi-charge corrections of the
size-resolved *σ*_{abs}. Multi-charge corrections to the measured size
distribution were prevailing when the DMA was used to scan the aerosol
sizes. When the DMA and CPC are used together to measure the aerosol
particle number size distribution (PNSD), the multi-charging corrected
aerosol PNSD can be significantly different from the raw measured one
(Bau et al., 2014; He and Dhaniyala, 2013; He et al., 2015). As shown in
the results part of this study, the multi-charge corrections of the
size-resolved *σ*_{abs} could cause differences in both the magnitude
and shape of the size-resolved *σ*_{abs}. Therefore, it is necessary
to perform multi-charge corrections on the size-resolved *σ*_{abs}.
This study developed a new algorithm to correct the size-resolved *σ*_{abs} from measured values based on the work of Hagen and
Alofs (2007) and Deng et al. (2011).

When the DMA is charged with a negative voltage, those aerosols with a small
range of electrical mobility (*Z*_{P}) can pass through the DMA:

$$\begin{array}{}\text{(5)}& {Z}_{P}={\displaystyle \frac{{q}_{\mathrm{sh}}}{\mathrm{2}\mathit{\pi}VL}}\mathrm{ln}\left({\displaystyle \frac{{r}_{\mathrm{1}}}{{r}_{\mathrm{2}}}}\right),\end{array}$$

where *q*_{sh} is the sheath air flow rate; *V* is the average voltage on the
inner center rod; *r*_{1} and *r*_{2} are the outer and inner radius of
annular space, respectively. The *Z*_{P} is related to *D*_{p} by
elementary charge (*e*), number of elementary charges on the particle (*n*),
and gas viscosity poise (*μ*) with

$$\begin{array}{}\text{(6)}& {Z}_{p}={\displaystyle \frac{neC\left({D}_{p}\right)}{\mathrm{3}\mathit{\pi}\mathit{\mu}{D}_{p}}},\end{array}$$

where *C*(*D*_{p}) is the Cunningham slip correction:

$$\begin{array}{}\text{(7)}& C=\mathrm{1}+{\displaystyle \frac{\mathrm{2}\mathit{\tau}}{{D}_{p}}}\left(\mathrm{1.142}+\mathrm{0.558}{e}^{-{\scriptscriptstyle \frac{\mathrm{0.999}{D}_{p}}{\mathrm{2}\mathit{\tau}}}}\right),\end{array}$$

where *τ* is the gas mean free path. From Eq. (7), aerosol particles
can have the same *Z*_{P} despite that they have different *n* and *D*_{p}.
At the same time, there exists a relatively larger portion of multiple
charged particles for those particles with diameters between 100 and 400 nm when the ambient aerosols pass through the X-ray
(Tigges et al., 2015; Wiedensohler and Fissan, 1988).
Through the above discussion, the selected aerosols by DMA at a given
electrical mobility can have different charges which will correspond to
different diameters.

When the scan diameter is set as ${{D}_{p}}_{i}$ for the singly charged
particles and the respective voltage of DMA is *V*_{i} ($i=\mathrm{1},\mathrm{2},\mathrm{\dots},I$), aerosol particles with an electrical mobility of *Z*_{p,i} ($i=\mathrm{1},\mathrm{2},\mathrm{\dots},I$) can pass through the DMA and the observed
*σ*_{abs} by AE51 can be expressed as

$$\begin{array}{}\text{(8)}& {R}_{i}=\underset{\mathrm{0}}{\overset{\mathrm{\infty}}{\int}}G\left(i,x\right)A\left(x\right)n\left(x\right)\mathrm{d}x,\end{array}$$

where *x* is the scale parameter with the definition of
$x=\mathrm{log}\left({{D}_{p}}_{i}\right)$; *A*(*x*) is the average *σ*_{abs} of a single particle for scale parameter *x*; and
$n\left(x\right)=\mathrm{d}N/\mathrm{d}\mathrm{log}{D}_{\mathrm{p}}$ is aerosol PNSD that is the multiple charging
corrected results from the measured aerosol PNSD. We define the kernel
function *G*(*i*,*x*), which is crucial to the algorithm, as

$$\begin{array}{}\text{(9)}& G\left(i,x\right)={\sum}_{v=\mathrm{1}}^{\mathrm{\infty}}\left(x,v\right)\mathrm{\Omega}\left(x,v,i\right),\end{array}$$

where (*x*,*v*) is the probability of particles that
are charged with *v* charges at the scale parameter of *x*
(Wiedensohler, 1988). $\mathrm{\Omega}\left(x,v,i\right)$ is the
probability of particles that can pass through the DMA with *v* charges at
the scale parameter *x* (Knutson and Whitby, 1975). In this study,
the maximum value of *v* is 10.

The multiple charging corrections can be expressed as computing the $A\left({x}_{i}^{\ast}\right)$, in which ${x}_{i}^{\ast}$ is the predetermined scale parameter from the DMA. To get the numerical integration results of Eq. (9), the diameter interval that is 1∕50 of the measured diameter is used. Thus, Eq. (9) can be written as

$$\begin{array}{}\text{(10)}& \begin{array}{rl}{R}_{i}& =\underset{\mathrm{0}}{\overset{\mathrm{\infty}}{\int}}G\left(i,x\right)A\left(x\right)n\left(x\right)\mathrm{d}x\\ & =\mathrm{\Delta}{x}_{i}\sum _{j=\mathrm{1}}^{\mathrm{50}}{\mathit{\beta}}_{j}G\left(i,{x}_{i,j}\right)A\left({x}_{i,j}\right)n\left({x}_{i,j}\right),\end{array}\end{array}$$

where $\mathit{\beta}=\left\{\begin{array}{ll}\mathrm{0.5},& j=\mathrm{1},\phantom{\rule{0.125em}{0ex}}J\\ \mathrm{1},& \mathrm{otherwise}\end{array}\right.$; *x*_{i,j} is the *j*th ($j=\mathrm{1},\mathrm{2},\mathrm{\dots},\mathrm{50}$) parameter that locates at the parameter *x*_{i} and *x*_{i+1}; and
*A*(*x*_{i,j}) ($i=\mathrm{1},\mathrm{2},\mathrm{\dots},I$; $j=\mathrm{1},\mathrm{2},\mathrm{\dots},\mathrm{50}$), the BC mass ratio at scale parameter *x*_{i,j}, is
expressed as the linear interpolation of the values at the measured
diameters.

$$\begin{array}{}\text{(11)}& A\left({x}_{i,j}\right)=A\left({x}_{i}\right)+{P}_{i}({x}_{i,j}-{x}_{i}),\end{array}$$

where *P*_{i} is the slope of the linear interpolation result of

$$\begin{array}{}\text{(12)}& A\left({x}_{k}^{\ast}\right)=C+{P}_{i}\cdot {x}_{k}^{\ast}.\end{array}$$

${x}_{k}^{\ast}$ refers to these five diameters that are nearest to the
predetermined scale parameter *x*_{i} and *C* is the intercept of the linear
interpolation result.

With ${H}_{i,j}={\mathit{\beta}}_{j}\mathrm{\Delta}{x}_{i}G\left(i,{x}_{i,j}\right)n\left({x}_{i,j}\right)$, Eq. (11) can be written as

$$\begin{array}{}\text{(13)}& \begin{array}{rl}{R}_{i}& ={\sum}_{j=\mathrm{1}}^{J}{H}_{ij}\left[A\left({x}_{i}\right)+{P}_{i}\left({x}_{i,j}-{x}_{i}\right)\right]\\ & ={\sum}_{j=\mathrm{1}}^{J}{H}_{ij}A\left({x}_{i}\right)+{\sum}_{j=\mathrm{1}}^{J}{H}_{ij}{P}_{i}{x}_{i,j}\\ & -{\sum}_{j=\mathrm{1}}^{J}{H}_{ij}{P}_{i}{x}_{i}\\ & ={\sum}_{k=\mathrm{1}}^{I}\left({\sum}_{j=\mathrm{1}}^{J}{H}_{ij}\mathit{\delta}\left(i-k\right)\right)A\left({x}_{k}^{*}\right)\\ & +{\sum}_{k=\mathrm{1}}^{I}\left({\sum}_{j=\mathrm{1}}^{J}{H}_{ij}{x}_{i,j}\mathit{\delta}\left(i-k\right)\right){P}_{k}\\ & -{\sum}_{k=\mathrm{1}}^{I}\mathit{\delta}\left(i-k\right){P}_{k}{x}_{k}^{*}={\sum}_{k=\mathrm{1}}^{I}{Q}_{ik}A\left({x}_{k}^{\ast}\right)\\ & +{\sum}_{k=\mathrm{1}}^{I}{T}_{ik}{P}_{k}-{\sum}_{k=\mathrm{1}}^{I}{Q}_{ik}{P}_{k}{x}_{k}^{\ast},\end{array}\end{array}$$

where $\mathit{\delta}\left(x\right)=\left\{\begin{array}{l}\mathrm{1},x=\mathrm{0}\\ \mathrm{0},x\ne \mathrm{0}\end{array}\right.$,

$$\begin{array}{}\text{(14)}& {Q}_{ik}={\sum}_{j=\mathrm{1}}^{J}{H}_{ij}\mathit{\delta}\left(i-k\right),\end{array}$$

and

$$\begin{array}{}\text{(15)}& {T}_{ik}={\sum}_{j=\mathrm{1}}^{J}{H}_{ij}{x}_{i,j}\mathit{\delta}\left(i-k\right).\end{array}$$

By letting the

$$\begin{array}{}\text{(16)}& {S}_{i}={R}_{i}-{\sum}_{k=\mathrm{1}}^{I}{T}_{ik}{P}_{k}+{\sum}_{k=\mathrm{1}}^{I}{Q}_{ik}{P}_{k}{x}_{k}^{\ast},\end{array}$$

this equation is then expressed as

$$\begin{array}{}\text{(17)}& {S}_{i}={\sum}_{k=\mathrm{1}}^{I}{Q}_{ik}A\left({x}_{k}^{\ast}\right)\end{array}$$

or

$$\begin{array}{}\text{(18)}& \mathit{S}=\mathbf{Q}\mathit{A},\end{array}$$

where ** S** and

Finally, the *A*(*x*) can be determined and the corresponding size-resolved
*σ*_{abs} that is multiple charging corrected can be calculated.

MAC of different size ranges is necessary when transforming the size-resolved
*σ*_{abs} into BCMSD. The MAC at different sizes should be different.
When the size-resolved *σ*_{abs} is converted into BCMSD with a
constant MAC, the derived BCMSD would be biased.

The size-resolved MAC was calculated using the Mie scattering model
(Bohren and Huffman, 2007). Based on the Mie scattering
theory, MAC values vary for different aerosol core diameter and different
total diameter. Results from SP2 measurements show that the size distribution
of the BC core diameter peaked at around 120 nm in Beijing (Zhang et
al., 2017). For each aerosol diameter, the MAC value with core diameter of
120 nm was used to transform the size-resolved absorption coefficient into the BCMSD. MAC values with core
diameter at 120±15 nm were calculated and shown in Fig. 2. From Fig. 2, the MAC varied significantly between 3.6 and 9.2 m^{2} g^{−1}. The constant
MAC values of 7.7 m^{2} g^{−1} corresponded to the aerosol diameter of 269 nm. The
calculated mean MAC values in Fig. 2 under different diameter were used in
this study.

An example of the multiple charging corrections of the size-resolved *σ*_{abs} is shown in Fig. 3. The corrections of aerosol PNSD are based on
the work of Hagen and Alofs (2007). As shown in Fig. 2a, the
corrected aerosol PNSD was significantly different from the original
uncorrected one. About half of the measured particles have
multiple elementary charge in the size range between 100 and 200 nm. The raw,
uncorrected aerosol PNSD had a peak value of 10 920 cm^{−3} at 98 nm while
the corrected aerosol PNSD reached its peak value of 8450 cm^{−3} at 98 nm. The peak positions of the raw aerosol particle mass size distribution
(PMSD, *d*_{m}∕dlog *D*_{p}) peaked at 322 nm with a peak value of 86.3 µg m^{−3}
and the corrected aerosol PMSD had a peak value of 53 µg m^{−3} at
461 nm. The peak position of the aerosol PMSD shifted a lot before and after
the multiple charging corrections. A similar case for the size-resolved
*σ*_{abs} is shown in Fig. 2b. The shape of size-resolved *σ*_{abs} changed substantially due to the multiple charging corrections.
The measured raw BCMSD had a peak diameter near 320 nm and the magnitude of
size-resolved *σ*_{abs} plateau reached 34.3 Mm^{−1}, which is in
accordance with the results of Ning et al. (2013), where
the multiple charging corrections were not involved. However, the corrected
size-resolved *σ*_{abs} peaks near 410 nm with a peak value of about
29.5 Mm^{−1}. According to the result, a small amount of *σ*_{abs}
remained in particles with diameter between 100 and 200 nm. The measured
size-resolved *σ*_{abs} changed a lot when multiple charging
corrections were implemented, which highlighted the necessity of
the implementation of appropriate multiple charging corrections.

The *σ*_{abs} integrated from measured size-resolved *σ*_{abs} changed after multiple charging corrections. Figure S4 shows the
comparison results of the *σ*_{abs} measured by AE33 and the
*σ*_{abs} integrated from AE51 measurements. The *σ*_{abs}
integrated from uncorrected and corrected size-resolved *σ*_{abs}
versus *σ*_{abs} measured by AE33 are shown in Fig. S4a and
b, respectively. Before multiple charging corrections, the *σ*_{abs} from uncorrected size-resolved *σ*_{abs} increased linearly
with the *σ*_{abs} from AE33, with *R*^{2} equaling 0.87, but it was
2.37 times that of AE33 on average. As a comparison, the overall magnitude of
*σ*_{abs} integrated from corrected size-resolved *σ*_{abs}
agreed better with that from AE33 with a slope of 1.2. With the discussion
above, multiple charging corrections were essential for size-resolved
*σ*_{abs} and BCMSD measurements.

Based on the measurement results, the BCMSD had two modes for most of the conditions. The BCMSDs are assumed to be of two lognormal distributions as

$$\begin{array}{}\text{(19)}& \begin{array}{rl}{m}_{\mathrm{fit},{D}_{p}}& =\sum _{i=\mathrm{1},\mathrm{2}}{\displaystyle \frac{{m}_{i}}{\sqrt{\mathrm{2}\mathit{\pi}\mathrm{log}\left({\mathrm{GSD}}_{i}\right)}}}\\ & \cdot \mathrm{exp}\left(-{\displaystyle \frac{{\left[\mathrm{log}\left({D}_{p}\right)-\mathrm{log}\left({D}_{\mathrm{m},i}\right)\right]}^{\mathrm{2}}}{\mathrm{2}{\mathrm{log}}^{\mathrm{2}}\left({\mathrm{GSD}}_{,i}\right)}}\right),\end{array}\end{array}$$

where *D*_{p} is the diameter of the aerosols; *m*_{i} is the mass of mode *i*
($i=\mathrm{1},\mathrm{2}$); GSD_{i} is the geometric standard deviation at mode *i*
($i=\mathrm{1},\mathrm{2}$); and *D*_{m,i} is the geometric mean diameter of the mode *i*
($i=\mathrm{1},\mathrm{2}$). The GSD_{i} and *D*_{m,i} can be determined by using the
least square method with the objective function as

$$\begin{array}{}\text{(20)}& J=\sum _{i=\mathrm{1},n}{\left({m}_{{D}_{{p}_{i}}}-{m}_{\mathrm{fit},{D}_{{p}_{i}}}({D}_{\mathrm{m}\mathrm{1}},{\mathrm{GSD}}_{\mathrm{1}},{D}_{\mathrm{m}\mathrm{2}},{\mathrm{GSD}}_{\mathrm{2}})\right)}^{\mathrm{2}},\end{array}$$

where ${m}_{{D}_{{p}_{i}}}$ is the measured mass distribution at ${D}_{{p}_{i}}$, while ${m}_{\mathrm{fit},{D}_{{p}_{i}}}$ is the fit mass distribution at ${D}_{{p}_{i}}$.

The Mie scattering model was used to study the influence of the BCMSD
variation on the aerosol optical properties. When running the Mie model,
aerosol PNSD and BC data were necessary. In this study, The BCMSD was assumed to
be a lognormal distributed. *D*_{m} of the BCMSD was set to vary from 100
to 600 nm. Geometric standard deviation (GSD) of the BCMSD was set to be in
the range between 1.3 and 1.8. BC was treated as partially externally mixed
and the remaining aerosol was treated as core-shell mixed. The ratio of
externally mixed *m*_{BC} to core-shell
*m*_{BC} was determined by the method introduced in
Ma et al. (2012) and a mean ratio of
0.51 was used. The density and refractive index of BC were set as 1.5 g cm^{−3}
and 1.8+0.54*i* (Kuang et al., 2015), respectively.
The complex refractive index of non-absorbing aerosols was $\mathrm{1.53}+\mathrm{10}-\mathrm{7}i$
(Wex et al., 2002) at the wavelength of 525 nm. For more details on
calculating the aerosol optical properties by using the aerosol PNSD and
BCMSD, users can refer to Kuang et al. (2016).

The aerosol PNSD and *m*_{BC} used here are the mean result of aerosol PNSD
and *m*_{BC} over the whole field measurement,
respectively. The amount of BC particles adopted in this study is the mean
value of the *m*_{BC} measured by AE33. For each BCMSD,
the extinction coefficient (*σ*_{ext}), the scattering
coefficient (*σ*_{sca}), the single scattering
albedo (SSA) and the asymmetry factor (*g*) could be
obtained from the output of the Mie scattering model.

In this study, the SBDART model (Ricchiazzi et al., 1998)
was employed to estimate the DARF. In our study, the instantaneous DARF for
cloud-free conditions at the top of atmosphere was calculated for an irradiance
wavelength range from 0.25 to 4 µm. Input of the model required the
profiles of aerosol *σ*_{ext}, SSA and
*g*. These profiles were calculated from the parameterization of
the aerosol vertical distributions. For details on calculating the
*σ*_{ext}, SSA and *g* profiles, the user
can refer to Sect. S5 in the Supplement. In brief, the aerosol
*σ*_{ext}, SSA and *g* profiles
were calculated based on the given aerosol PNSD and BCMSD. The DARF can be
estimated using the above aerosol optical profiles.

The aerosol optical properties and the corresponding aerosol optical profiles vary with different BCMSD. Then the DARF should be different for different BCMSDs. By estimating the DARF using different aerosol BCMSDs, the influence of BCMSD on the aerosol radiative properties can be studied.

4 Results and discussions

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The time series of measured PM_{2.5}, aerosol PNSD and BCMSD are shown in Fig. 3. During the observation period, the PM_{2.5} varied from 0.06 to 220 µg m^{−3}, with a mean value of 71.5±52.56 µg m^{−3}. Three
periods of heavy PM_{2.5} loading were observed: (1) PM_{2.5} increased from
around 100 to 200 µg m^{−3} and decreased slowly to 1 µg m^{−3} in the period 21–26 March; (2) the PM_{2.5} accumulated
slowly from 28 to 30 March and dissipated quickly from 30 March to 1 April; (3) the rapid accumulation and dissipation of PM_{2.5} happened during 2
to 5 April. During the last 5 d, PM_{2.5} fluctuated between 20 and 120 µg m^{−3}. For each pollution condition, both the aerosol total number
concentrations and the aerosol median diameter increased. The aerosol median
diameter varied between 31 and 169 nm with a mean value of 78±31 nm.

Our measurements show that the BCMSD had two modes with the coarser mode ranging between 430 and 580 nm in mobility diameter. Many field measurements have revealed that most of the BC mass locates in the aerodynamic diameter range of 320 and 560 nm using the MOUDI (Hu et al., 2012; Huang and Yu, 2008). When the aerodynamic diameter was transformed into mobility diameter with assumption of an aerosol effective density of 1.3, the measured BC aerodynamic diameter range corresponded to mobility diameter range of 280 and 491 nm. Therefore, the measured size range for coarser mode of BCMSD agreed well with the previous measurement.

The measured aerosol in the field site was representative of the urban aerosol. The BC particles emitted by vehicles contributed significantly to the total aerosol BC mass. These BC particles were rarely coated or thinly coated, and the BC core diameter peaked around 120 nm (Zhang et al., 2017). Therefore, the BCMSD of the finer mode measured in our study corresponded to these uncoated or thinly coated particles.

The total *m*_{BC} measured by AE33 ranged from 0.1 to 14 µg m^{−3}
with an average of 5.04±2.64 µg m^{−3}. Good consistence was
achieved between *m*_{BC} measured by AE33 and *m*_{BC} calculated from
measured BCMSD as shown in Fig. 3d.

A lognormal distribution was used to fit each mode of the BCMSD by using the
least square method as introduced in Sect. 3.2. For each mode, the
geometric mean diameter (*D*_{m}) and the geometric standard deviation (GSD)
of the BCMSD were studied.

During the measurement period, both *D*_{m} and GSD of the two modes have
changed significantly as shown in Fig. S7. The *D*_{m} of the first and second
modes varied from 139 to 161 nm and from 420 to 597 nm, respectively. The
corresponding mean *D*_{m} was 151 and 520 nm. The *D*_{m} of the two modes
was found to be positively correlated in Fig. S7a. When the pollution was
released from the beginning to 27 March, the *D*_{m} decreased from 597 to
420 nm and from 160 to 140 nm for the coarser mode and the finer mode,
respectively. The BC-containing aerosols tended to be aged and grew larger
when the surrounding air was polluted.

GSD for the coarser mode and the finer mode showed very different properties as shown in Fig. S7b. For the second mode, GSD varied from around 1.49 to 1.68 with a mean value of 1.57. The GSD became decreased with the changing pollution condition, which indicated that BC-containing aerosols tend to accumulate to a small range of diameters during the aging process. This phenomenon was consistent with the fact that larger particles grew relatively slower in diameter because the growth ratio of small aerosol particles is proportional to the negative power of it's diameter. For the first mode, GSD ranged from 1.41 to around 1.86 with a mean value of 1.63. However, the GSD of the finer mode tended to be larger when the surrounding air was cleaner, which might be related to the complex sources of the BC emission. A small amount of freshly-emitted BC particles can have substantial influence on the mass size distribution of the finer mode because the BC concentrations of the finer mode were small, especially under clean conditions. In general, the GSD of the coarser mode was a good indicator of the BC aging process and that of the finer mode could partially reflect the complex sources of the BC fine particles.

The relationship between the *D*_{m} and the GSD for the coarser mode was
further analyzed by analyzing the distribution of the *D*_{m} and GSD. The
GSD and *D*_{m} had opposite trends as shown in Fig. 5. With the increment of
the *D*_{m} from 420 to 540 nm, the mean value of GSD decreased from around
1.608 to 1.528, while the *m*_{BC} increased with the
*D*_{m}. The statistical relationship between *D*_{m} and GSD offered a
reasonable representation of the BCMSD under different polluting conditions.
In the following work, mean values of the GSD at different *D*_{m} were used
to for further discussion. The *m*_{BC} and GSD is positively correlated.
The *m*_{BC} increased from 2.4 to 8.3 µg m^{−3} when the *D*_{m}
increased from 420 to 540 nm.

Note that the GSD get slightly increased with the increment of *D*_{m} when
*D*_{m} was larger than 520 nm. This might be caused by the limited diameter
range of the BCMSD measuring system, which is from 20 to 680 nm. The multiple
charge corrections applied to the BCMSD could influence the BCMSD when
*D*_{m} of the BCMSD was near the end of the scanned diameter and may lead
to significant uncertainties to the BCMSD. The measurement results indicated
that cases of measured *D*_{m} of BCMSD larger than 520 nm were few,
demonstrating that this multiple correction effect influenced little on
shape of measured BCMSD in most cases.

The aerosol optical parameters using the measured mean aerosol PNSD and mean
*m*_{BC} corresponding to different GSD and *D*_{m} values are shown in
Fig. 6. In Fig. 6a, the aerosol *g* varied from 0.603 to 0.627 (variation of
4 %). Recent work by Zhao et al. (2017) showed that the aerosol *g* value in
the North China Plain may vary at a range of 10 % due to the change of aerosol PNSD.
Aerosol *g* was more sensitive to *D*_{m} when the geometric mean diameter of
the BCMSD was lower than 400 nm. However, when the *D*_{m} was larger than
400 nm, the *g* become sensitive to both the *D*_{m} and the GSD of BCMSD.
Overall, the *g* varied a little bit (0.02 to 0.609) under the representative
conditions during the measurement period. For the aerosol SSA, it was
sensitive to the *D*_{m} over the whole range as shown in Fig. 6b. SSA
varied between 0.90 and 0.94 under the representative measurement
conditions. The *σ*_{sca} had large changes from 325.6 to
364.4 Mm^{−1}. The *σ*_{sca} was quite sensitive to variations in
BCMSD when the *D*_{m} was larger than 450 nm as shown in Fig. 6c. In
addition, variations in *σ*_{sca} relied more on the variations in
*D*_{m} when *D*_{m} was lower than 400 nm. Within the measurement
conditions of BCMSD, the *σ*_{sca} varied from 328 to 345 Mm^{−1}. The measured GSD under different *D*_{m} went along with the
gradient direction of the *σ*_{sca}, which mean that the evolution
of BCMSD in the atmosphere influenced substantially on *σ*_{sca}. As
for the *σ*_{abs}, it changed from 24.06 to 37.27 Mm^{−1} and the corresponding mass absorption cross section (MAC) was
estimated to be in the range of 5.44 to 8.08 m^{2} g^{−1}, suggesting that MAC
of the BC aerosols should be carefully studied under different BCMSD
conditions.

The estimated DARF values for different GSD and *D*_{m} conditions were
estimated. When estimating the DARF, the measured mean aerosol PNSD and mean
BC mass concentration were used. The results of estimated DARF are shown in
Fig. 7a. DARF at the surface varied from −4.3 to −3.59 W m^{−2} for different BCMSD. Within the measured BCMSD range, the DARF
varied from −3.97 to −3.67 W m^{−2}, which corresponding to
8.45 % of variation. The heating rate within the mixed layer was a
powerful indicator of the BC particles' absorbing effects, which may help
evaluate the development of the boundary layer. The calculated mean heating
rate within the mixed layer changed from 2.16 to 2.65 K d^{−1} for
different *D*_{m} and GSD, as shown in Fig. 7b. The heating rate with the
measured BCMSD range could change from 2.24 to 2.50 K d^{−1} with a variation of
11.6 %.

Mixing states of BC play significant roles in calculations of aerosol optical properties and estimations of DARF (Jacobson, 2001). As a comparison, we estimated the DARF under different conditions of BC mixing state: (1) internally mixed, (2) externally mixed and (3) core-shell mixed. Table 1 gives the estimated DARF and mean heating rate within the mixed layer under different mixing state conditions. Results showed that the DARF under different BC mixing state conditions may change by 10.50 %, which shared the same magnitude with 8.45 % variation in DARF caused by BCMSD variations. In addition, the heating rate was estimated to vary by 9.71 %. These results highlighted that the BCMSD plays significant roles in variations of aerosol optical properties and estimations of DARF as well as the air heating rate caused by the existence of BC particles. It was recommended that a real-time measured BCMSD be used when estimating the aerosol DARF instead of a constant one. The BCMSD is as important as that of the BC mixing states.

5 Conclusions

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Knowledge of the BC microphysical properties especially the size-dependent information can help reduce the uncertainties when estimating the aerosol radiative effects. BCMSD is an important quantity in its own right, being directly and indirectly applicable to the determination of the sources, aging processes and mixing states of BC aerosols. In this study, the characteristics of BCMSD were studied from the field measurement results by using our own developed measurement algorithm.

The BCMSD measurement system was developed and validated based on the works
of Ning et al. (2013) by using a differential mobility
analyzer (DMA) in tandem with an Aethalometer (AE). When deriving the BCMSD, a
comprehensive multiple charging correction algorithm was proposed and
implied. This algorithm was validated by closure study between the measured
total *m*_{BC} from AE33 and the *m*_{BC} integrated from the
measured BCMSD using the datasets from field measurements. Results showed
that the multiple charging corrections could significantly change the shapes
and magnitudes of the raw measured BCMSD. Accurate BCMSD characteristics
could be obtained by our proposed method in this paper.

The developed measurement system was employed in a field campaign in the North China Plain from 21 March to 9 April 2017. The BCMSD was found to have two quasi-lognormal modes with peaks at around 150 and 500 nm. These two modes were consistent with the previous measurement results by MOUDI (Wang et al., 2015; Hu et al., 2012). The BC mass concentrations for the coarser-mode peaks were about twice that of the finer mode.

The characteristics of the BCMSD were studied by fitting the shape of BCMSD
with a binormal distribution. The relationships between the fitted *D*_{m}
and GSD were statistically studied. During the aging process, the
opposite trends for *D*_{m} and GSD were found for the coarser mode. This is
the first time that the coarser mode of the BCMSD was synthetically
studied. The BCMSD of the coarser mode varied significantly under different
pollution conditions with peak diameters changing between 430 and 580 nm.
However, the relationship between *D*_{m} and GSD for the finer-mode BC
aerosols was more complex due to the complex sources.

When the BCMSDs changed with the polluting conditions, the corresponding
aerosol optical properties changed significantly. Sensitivity studies found
that the aerosol *g* varies from 0.603 to 0.627 due to the variations in
BCMSD. Aerosol *g* was more sensitive to *D*_{m} when the geometric mean
diameter of the BCMSD is in the range of 300 and 370 nm. The SSA changed from 0.90 to 0.94. The *σ*_{sca} experienced a significant
change with variation in BCMSD from 325.6 to 364.4 Mm^{−1}
and the *σ*_{abs} changed in the range between 24.064 and
37.27 Mm^{−1}. The corresponding BC MAC was calculated to be in the range
between 5.44 and 8.08 m^{2} g^{−1}.

The variations in DARF were estimated to be due to the variations in the BCMSD by
using the SBDART model. Results showed that the DARF can vary by about
8.45 % for different BCMSDs and the heating rate for different measured
BCMSD conditions could change from 2.24 to 2.50 K d^{−1}, corresponding to a
variation of 11.6 %. At the same time, the variations in DARF due to the
variations in the BC mixing state was estimated to be 10.5 % and that of
the heating rate was 8.45 %. Thus, the variations in the BCMSDs may have
significant influence on the aerosol radiative budget, and an accurate
measurement of BCMSD is very necessary.

Data availability

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Data availability.

The data used in this study are available by requesting them from the authors.

Supplement

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Supplement.

The supplement related to this article is available online at: https://doi.org/10.5194/acp-19-13175-2019-supplement.

Author contributions

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Author contributions.

GZ, CZ, JT and YK designed and conducted the experiments; CS, YY, CZ and GZ discussed the results.

Competing interests

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Competing interests.

The authors declare that they have no conflict of interest.

Financial support

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Financial support.

This research has been supported by the National Key R&D Program of China (grant no. 2016YFA0602001) and the National Natural Science Foundation of China (grant no. 41590872).

Review statement

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Review statement.

This paper was edited by Veli-Matti Kerminen and reviewed by two anonymous referees.

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Short summary

Characteristics of the black carbon size distribution (BCMSD) are studied by using our developed measurement system. Results show that the BCMSDs have two modes and the mean peak diameters are 150 nm and 503 nm, respectively. The coarser mode varies significantly under different pollution conditions, which gives rise to significant variation in aerosol bulk optical properties. Our study reveals that the BCMSD as well as the mixing state in estimating aerosol radiative forcing matters.

Characteristics of the black carbon size distribution (BCMSD) are studied by using our developed...

Atmospheric Chemistry and Physics

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