the Creative Commons Attribution 4.0 License.

the Creative Commons Attribution 4.0 License.

# Modeled source apportionment of black carbon particles coated with a light-scattering shell

The Aethalometer model has been used widely for estimating the contributions
of fossil fuel emissions and biomass burning to equivalent black carbon
(eBC). The calculation is based on measured absorption Ångström
exponents (*α*_{abs}). The interpretation of *α*_{abs} is
ambiguous since it is well known that it not only depends on the dominant absorber but also on the size and internal structure of the particles, core
size, and shell thickness. In this work the uncertainties of the
Aethalometer-model-derived apparent fractions of absorption by eBC from
fossil fuel and biomass burning are evaluated with a core–shell Mie model.
Biomass-burning fractions (BB(%)) were calculated for pure and coated
single BC particles for lognormal unimodal and bimodal size distributions
of BC cores coated with ammonium sulfate, a scattering-only material.
BB(%) was very seldom 0 % even though BC was the only absorbing
material in the simulations. The shape of size distribution plays an
important role. Narrow size distributions result in higher *α*_{abs}
and BB(%) values than wide size distributions. The sensitivity of *α*_{abs} and BB(%) to variations in shell volume fractions is the
highest for accumulation-mode particles. This is important because that is
where the largest aerosol mass is. For the interpretation of absorption
Ångström exponents it would be very good to measure BC size
distributions and shell thicknesses together with the wavelength dependency
of absorption.

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Incomplete combustion of organic fuels results in emission of
light-absorbing carbon (LAC) particles that contain both black carbon (BC)
and brown carbon (BrC). BrC is light-absorbing organic matter in atmospheric
aerosols of various origins, e.g., soil humics, humic-like substances
(HULIS), tarry materials from combustion, and bioaerosols (Andreae and
Gelenscer, 2006; Laskin et al., 2015). BrC can significantly absorb solar
radiation in the ultraviolet–visible (UV–Vis) wavelength range (*λ*≈ 300–800 nm). The radiative effects of BC and BrC vary in time
during atmospheric aging. For many combustion sources the absorbance in
fresh emission is almost completely caused by BC particles, but during
atmospheric transport they often get coated with some light-scattering
compounds, for instance ammonium sulfate or light-absorbing organic carbon,
BrC. For some sources (e.g., biomass burning) BrC may contribute
substantially to light absorption already in the directly emitted aerosols
and either increase or decrease during aging. Thus, BrC is highly
time-dependent as its composition and absorption properties change during
atmospheric oxidation processes (Laskin et al., 2015).

The absorption coefficient *σ*_{ap} is approximately proportional to
the power function ${\mathit{\lambda}}^{-{\mathit{\alpha}}_{\mathrm{abs}}}$, where *λ* is the
wavelength and *α*_{abs} is the absorption Ångström
exponent. *α*_{abs} is generally used to distinguish aerosol types: for pure BC particles *α*_{abs}≈ 1, while for other light-absorbing aerosols (BrC, soil dust) it is clearly > 1 (e.g.,
Kirchstetter et al., 2004; Bond and Bergstrom, 2006; Bergstrom et al., 2007;
Moosmüller et al., 2011; Kirchstetter and Thatcher, 2012; Lack et al.,
2012; Bond et al., 2013; Saleh et al., 2013; Laskin et al., 2015; Valenzuela
et al., 2015; Devi et al., 2016). The method has been used not only for in
situ absorption measurements but also for interpreting absorption
coefficients retrieved from remote sensing measurements, such as the Aerosol Robotic Network (AERONET; e.g., Russell et al., 2010; Arola et al., 2011; Chung et al., 2012; Cazorla
et al., 2013; Feng et al., 2013; Schuster et al., 2016; Wang et al., 2016).

One of the instruments used for measuring black carbon concentrations is the
Aethalometer that collects aerosol on a filter tape, measures changes in
light attenuation in the wavelength range of 370–950 nm, and calculates the equivalent black carbon (eBC) concentrations. The data are used also to
calculate *α*_{abs} and to estimate the contributions of fossil fuel
emissions and biomass burning to eBC. The Aethalometer model (Sandradewi et
al., 2008a) is probably the most widely used method for this, and it is even
calculated automatically in the new Aethalometer model AE33. It is there
assumed that the absorption Ångström exponents are *α*_{ff} = 1 and *α*_{bb} = 2 for eBC from fossil fuel and biomass
burning, respectively. These are the default settings in the AE33, but also
different *α*_{ff} and *α*_{bb} values have been used
(Sandradewi et al., 2008b; Herich et al., 2011; Fuller et al., 2014;
Harrison et al., 2013; Healy et al., 2017; Zotter et al., 2017; Helin et
al., 2018)

The interpretation of *α*_{abs} is ambiguous since it not only
depends on the dominant absorber but also on the size and internal structure
of the particles, core size, and shell thickness. For instance, for pure BC
particles, *α*_{abs} may be < 1 and BC particles coated with
non-absorbing material may have *α*_{abs} in the range from
< 1 to ∼ 1.7 (e.g., Gyawali et al., 2009; Lack and
Cappa, 2010; Lack and Langridge, 2013; Schuster et al., 2016; Liu et al.,
2018; Chylek et al., 2019; Zhang et al., 2020). The present paper may be
considered an extension to the abovementioned analyses since they did
not explicitly analyze the effects on the Aethalometer model.

The aim of this study is to estimate uncertainties of the Aethalometer-model-derived fractions of absorption by eBC from fossil fuel and biomass burning when spherical BC cores are coated by some non-absorbing material. To state this more clearly, it is assumed that there is only one type of BC particles that can be called fossil fuel BC in the Aethalometer model terminology. Consequently, any deviations from biomass-burning fraction of BB % = 0 indicate uncertainty in the source appointment. Biomass-burning fractions were calculated for pure and coated single particles for lognormal unimodal and bimodal size distributions. The work is based on modeling only; no measurement data are used.

The BC cores were assumed to be coated with an ammonium sulfate shell by using two approaches. It was assumed (1) that the shell thickness is the same for all particles in a size distribution (Fig. 1a) and (2) that the core volume fraction is the same for all particles in a size distribution (Fig. 1b). The core volume fraction was calculated from

where *V*_{p} is the particle volume, *V*_{core} is the BC core volume,
*D*_{p} is the particle diameter (= *D*_{core}+2*s*), *D*_{core} is the
BC core diameter, and *s* is the shell thickness. The shell volume fraction was then calculated from ${f}_{\mathrm{s}}=\mathrm{1}-{f}_{\mathrm{c}}$. The ratio of the coated particle
diameter to the core diameter is an often used metric for presenting the
coating of particles. *R*, *f*_{c}, and *f*_{s} can be calculated from each other
as

The number-weighted *D*_{p}-to-*D*_{core} ratio is calculated from

where *N*_{i} and *R*_{i} are the number concentration and
*D*_{p}-to-*D*_{core} ratio of the particle diameter *D*_{p,i},
respectively. If *f*_{s} is independent of particle size – which is the
assumption used in some of the simulations below – Eq. (3) simplifies
to ${R}_{N\left({D}_{\mathrm{p}}\right)}$ = *R*.

Lognormal size distributions $n({D}_{\mathrm{p}},{D}_{\mathrm{g}},{\mathit{\sigma}}_{\mathrm{g}})$ were
generated, where *D*_{p} is the particle diameter, *D*_{g} is the geometric
mean diameter, and *σ*_{g} is the geometric standard deviation. The
*D*_{p} range was 3 nm–10 µm. For the unimodal size distributions
the *D*_{g} range was 50 nm–1 µm, and *σ*_{g} was given three
values: 1.4, 1.6, and 1.8 (Fig. 1c and d). Bimodal size distributions were also generated. For the small-particle mode the geometric mean diameter *D*_{g1}
range was 50–100 nm, and the large-particle mode *D*_{g2} range was 100–500 nm. In addition to varying the geometric mean diameters, the
ratios of the number of particles in the two modes were also varied. Two cases
were used for this: (1) *N*_{1}=10*N*_{2}, *σ*_{g1}=1.4,
*σ*_{g2}=1.6 (Fig. 1e) and (2) *N*_{1}=*N*_{2}, *σ*_{g1}=1.6, *σ*_{g2}=1.6 (Fig. 1f).

Absorption coefficients were calculated from

where *Q*_{a} is the absorption efficiency that is a function of the
wavelength *λ*, *D*_{p}, the complex refractive indices of the core and shell, *m*_{core} and *m*_{shell} , respectively, and the shell
thickness *s*. *Q*_{a} was calculated using the N-Mie Fortran code written and described in detail by Voshchinnikov and Mathis (1999). The code is based on the recursive algorithm of Wu and Wang (1991). The code calculates
the extinction, scattering, and absorption efficiency factors for *n*-layered spheres. The complex refractive indices were ${m}_{\mathrm{core}}=\mathrm{1.85}+\mathrm{0.71}i$
(BC as in Lack and Cappa, 2010) and ${m}_{\mathrm{shell}}=\mathrm{1.52}+\mathrm{0}i$ (ammonium
sulfate) for the core and shell, respectively. Absorption coefficients were
calculated for the Aethalometer wavelengths *λ* = 470 and 950 nm,
and *α*_{abs} was calculated from

The wavelengths 470 and 950 nm were used as they are used also in the
AE33 automatic source apportionment. In analyses of aerosol optical depth
data from AERONET, *α*_{abs} is often calculated for the
wavelength pair 440 and 870 nm (Russell et al., 2010; Schuster et al.,
2016). To evaluate the applicability of the simulations of the present work
to AERONET data analyses, *σ*_{ap} was calculated also for these
wavelengths, and the respective *α*_{abs} was calculated from them.
There are size-dependent differences between ${\mathit{\alpha}}_{\mathrm{abs}}(\mathrm{470}/\mathrm{950})$ and
${\mathit{\alpha}}_{\mathrm{abs}}(\mathrm{440}/\mathrm{870})$, but they are not big (see Figs. S1 and S2 in the Supplement), so it may safely be concluded that the results to be presented
below are valid also for the AERONET data.

For the absorption due to particles from wood burning or biomass burning, Zotter et al. (2017) give the equation

where *α*_{ff} and *α*_{bb} are the *α*_{abs} of
fossil fuel and biomass-burning BC in the Aethalometer model. Noting that
${\mathit{\sigma}}_{\mathrm{ap}}\left({\mathit{\lambda}}_{\mathrm{1}}\right)={\mathit{\sigma}}_{\mathrm{ap}}\left({\mathit{\lambda}}_{\mathrm{2}}\right){\left({\mathit{\lambda}}_{\mathrm{1}}/{\mathit{\lambda}}_{\mathrm{2}}\right)}^{-{\mathit{\alpha}}_{\mathrm{abs}}}$, the
fraction of absorption due to biomass burning is

so that BB% depends on the Ångström exponents *α*_{abs}, *α*_{ff}, and *α*_{bb}. Two settings for the constants were
used: the one presented in the AE33 manual, *α*_{ff} = 1 and
*α*_{bb} = 2; and the one presented by Zotter et al. (2017),
*α*_{ff} = 0.9 and *α*_{bb} = 1.68. The symbols and their definitions are presented in Table 1.

## 3.1 Single particles

The absorption Ångström exponent *α*_{abs} and the fraction
of biomass-burning BC for single coated particles are shown in Fig. 2. The
dashed lines in Fig. 2a, c, and e show the core diameter *D*_{core} of particles that have the same diameter *D*_{p} at all shell thicknesses. In Fig. 2b, d, and f the dashed lines show the particle diameter *D*_{p} and *f*_{s} of particles that have the same *D*_{core} at all shell volume
fractions *f*_{s} in the range *f*_{s} ≤ 99 %. The dependence of
*α*_{abs} on core and shell is presented twice. This is apparently
superfluous, but they are visualizations that complement each other.

The first approach (Fig. 2a, c, and e) shows that when *D*_{core} < ∼ 150 nm and *s* > ∼ 25–50 nm the absorption Ångström exponent *α*_{abs} > 1.4. The respective BB fractions are larger than about 40 % or 60 % for
the Aethalometer model parameters of *α*_{ff} = 1, *α*_{bb} = 2 (pair 1) and *α*_{ff} = 0.9, *α*_{bb} = 1.68 (pair 2), respectively. Figure 2a also shows that for *D*_{core} < ∼ 100 nm there are two maxima of the *α*_{abs} when the shell grows thicker. In the second maximum, *α*_{abs} > ∼ 1.6. As a result the BB fractions
would be > 60 % and even > 100 % for the two
Aethalometer model parameter pairs. When *D*_{core} is in the range of
∼ 170–200 nm, *α*_{abs} ≈ 1 and *α*_{abs} decreases with a growing *s*. For larger core diameters the
absorption Ångström exponent is even smaller. When *D*_{core} > 200 nm, *α*_{abs} < 1 and even negative for
*D*_{core} > ∼ 360 nm. Further, when *D*_{core} > 200 nm, *α*_{abs} does not grow essentially at all as a
function of *s*.

The visualization of *α*_{abs} as a function of shell volume
fraction *f*_{s} and particle full diameter *D*_{p} (Fig. 2b) shows some other features. When *D*_{p} < 50 nm, *α*_{abs} varies in the range of 1.0–1.1, and it does not depend on *f*_{s}, but in the *D*_{p} range of about 100–300 nm *α*_{abs} depends strongly on *f*_{s}.
When *D*_{p} ≈ 500 nm, *α*_{abs} < 1 for almost
all shell volume fractions, up to *f*_{s} ∼ 99 %. For larger
particles *α*_{abs} is close to 0 at all shell volume fractions.

The visualization also shows that the *α*_{abs} value of 1, usually
considered an indication of BC, is not a result of an unambiguous
*D*_{core}−*s* (Fig. 2a) or *D*_{p}−*f*_{s} (Fig. 2b) combination.

## 3.2 Unimodal BC core size distributions, same coating thickness for all sizes

For single particles *α*_{abs} depends clearly on both the core size
and the shell thickness. However, in real atmospheric studies the wavelength
dependency of absorption by particle size distributions is measured. Here
these were first modeled by assuming that pure BC particle size
distributions get coated with ammonium sulfate layers so that the shell
thickness is independent of particle size as visualized in Fig. 1a. For
example, the shell thickness on a 50 nm BC particle would be the same as on
a 200 nm particle, which means the shell volume fractions are not the same.
The BC core geometric mean diameter (*D*_{g,core}) was varied from 50 to 200 nm at 10 nm intervals. The geometric standard deviations of the size
distributions were *σ*_{g} = 1.4, *σ*_{g} = 1.6, and
*σ*_{g} = 1.8, representing narrow, average, and wide size
distributions. The shell thickness *s* varied from 0 to 250 nm at 1 nm
intervals. Absorption coefficient and subsequently *α*_{abs} was
calculated for the full size distribution 3–2500 nm.

The results are first shown as a function of *D*_{g,core} and shell
thickness *s* for the three size distribution widths (Fig. 3). There are both
similarities and differences compared with the corresponding relationships
of single particles (Fig. 2). For example, for single particles *α*_{abs} ≈ 1 at *D*_{core} ≈ 180 nm for shell thicknesses
*s* ≈ 0–70 nm as shown by the almost vertical *α*_{abs} = 1 isoline in Fig. 2a, whereas for the size distributions with *σ*_{g,core} > 1 the respective isoline is a strong function of
both *s* and *σ*_{g,core} (Fig. 3a). At all widths of the size
distribution, *α*_{abs} increases with increasing shell thickness and then starts decreasing. For small core sizes (*D*_{g,core} < ∼ 80 nm) *α*_{abs} also has a second maximum when the size distribution is narrow. The width of the size distribution has a clear
effect on the *α*_{abs}: for all core sizes and shell thicknesses
*α*_{abs} decreases with increasing *σ*_{g,core}.

Both for single particles and size distributions the first maximum of
*α*_{abs} is the smaller the larger the *D*_{g,core} and *σ*_{g,core} are (Fig. 4a). The first maximum is reached at shell thickness
*s* ≈ 70 ± 5 nm for all size distribution widths although for
single particles the variability of the shell thickness corresponding to the
first maximum is larger (Fig. 4b). The first maximum *α*_{abs}
results in apparent BB fractions of up to ∼ 100 % for single
particles and in the range from 0 % to ∼ 80 % for the size
distributions, and again the BB(%) is smaller the larger the
*D*_{g,core} and *σ*_{g} are (Fig. 4c and d).

This approach is further followed by plotting the parameters as a function
of shell thickness for three different BC core diameters, 50, 70, and
90 nm of single particles and core size distributions with the geometric
standard deviations of *σ*_{g,core} = 1.4, 1.6, and 1.8 (Fig. 5).
This analysis can be considered a description of what may happen to the
size distribution, *α*_{abs}, and the apparent BB(%) during
condensational growth on fresh small BC cores if the growing shell thickness
were independent of the core diameter, even if this is unrealistic. The
shell volume fraction *f*_{s} increases to > 99.9 % when the
shell thickness grows from *s* = 0 to 250 nm on single 50 nm particles
but to lower fractions for the wider size distributions and larger core
sizes so that for *D*_{g,core} = 90 m and *σ*_{g} = 1.8, *f*_{s} ≈ 98 % even with *s* = 250 nm (Fig. 5a). The geometric mean
diameter *D*_{g} of the whole size distribution grows to ∼ 600 nm when the shell thickness grows to 250 nm, with minimal differences between the
original widths (Fig. 5b). The width of the size distribution, i.e., *σ*_{g}, decreases fast to < 1.2 (Fig. 4c). Such values correspond to
very narrow size distributions, which are not really observed in the real atmosphere.

The number-weighted *D*_{p}-to-*D*_{core} ratio ${R}_{N\left({D}_{\mathrm{p}}\right)}$, Eq. (3), was
calculated for the size range 90–600 nm to present the numbers comparable
with papers that present shell-to-core ratios of refractory BC (rBC)
obtained from SP2 measurements. For instance, Kondo et al. (2011) measured
urban air of Tokyo and obtained the median *R* = 1.1 with a range up to
about 1.3, the mean *D*_{g} = 64 ± 6 nm, and *σ*_{g} = 1.66 ± 0.12. Moteki et al. (2007) conducted SP2 measurements in an
aircraft in urban plumes on the Japanese coast. They fitted the data with
lognormal size distributions with mass median diameters (MMDs) of 190 and 210 nm and *σ*_{g} of 1.55 and 1.45 for fresh and aged rBC,
respectively. The fresh rBC was thinly coated with *R* < 2 and the
aged rBC thickly coated with *R* ∼ 2. The MMD and *σ*_{g} values yield *D*_{g} = 107 and 139 nm. Shiraiwa et al. (2008)
measured the mixing state and size distribution of BC aerosol with an SP2 at
a remote island (Fukue) in Japan. They observed that the BC number median
diameters were in the range of 120–140 nm in every air mass type, and the
median shell $/$ core diameter ratio (*R*) in different air masses varied in the
range of 1.2–1.6. However, they also observed that the fraction of *R*
values in the range 2–3.5 was not negligible either (Fig. 9 of Shiraiwa
et al., 2008). Such values correspond to the range where *α*_{abs}
first grows to > 1.6 for the narrow (*σ*_{g,core} = 1.4) BC core size distribution that has the smallest geometric mean size
(*D*_{g,core} = 50 nm) but to lower values for the wider size
distributions that have larger *D*_{g,core} (Fig. 5c and d). The first
maximum is reached at shell thicknesses of *s* ≈ 70 nm that
correspond to *R* ≈ 2 and shell volume fractions of *f*_{s} ≈ 90 ± 8 % (Fig. 5b). Schwarz et al. (2008) reported statistics of
rBC mass size distributions in urban aerosol: *f*_{s} = 9 ± 6 %, *s* = 20 ± 10 nm, MMD = 170 nm, and *σ*_{g} of 1.71, which
yields *D*_{g} = 72 nm; in biomass-burning emissions: ${f}_{\mathrm{s}}=\mathrm{70}\pm \mathrm{9}$ %, *s* = 65 ± 12 nm, MMD = 210 nm, and *σ*_{g} = 1.43, which yields *D*_{g} = 143 nm; and in background continental
aerosol: *f*_{s} = 46 ± 3 %, *s* = 48 ± 14 nm, MMD = 210 nm, and *σ*_{g}=1.55, which yields *D*_{g} = 118 nm.

The referenced studies show that the *s*, *R*, and *f*_{s} values are in the range observed in ambient measurements studies. What is not realistic in
atmospheric aerosol is the width of the size distribution, which soon
decreases to *σ*_{g} < 1.2 (Fig. 5c).

After reaching the first maximum, *α*_{abs} decreases, and for single
particles and the narrowest core size distributions it starts growing again and
reaches a second maximum at shell thicknesses of *s* ≈ 170 nm, which
corresponds to *R* > 4 and *f*_{s} > 98 %. Such *s* and
*R* values are not in the range observed in the abovementioned studies, nor
are the low geometric standard deviations of *σ*_{g} < 1.1
realistic, so the second maximum can be considered as a theoretical value
only. For size distributions with *D*_{g,core} > 70 nm there is
no second maximum of *α*_{abs}.

As *α*_{abs} increases and decreases, it is clear that this applies to BB(%) as well (Fig. 5d–e). For the smallest core sizes (*D*_{g,core} = 50 nm) and the narrowest size distributions (*σ*_{g,core} = 1.4), the first maximum BB(%) may be as high as ∼ 90 %
when the values of *α*_{ff} = 0.9 and *α*_{bb} = 1.68 are
used in Eq. (7) but lower, ∼ 50 %, when the values of *α*_{ff} = 1 and *α*_{bb} = 2 are used. For the wider core size
distributions the BB(%) fractions are lower. For the widest core size
distributions (*σ*_{g,core} = 1.8) clearly positive BB(%)
values are reached only for the smallest core sizes.

Figure 5 can also be considered a proxy for a time series of the
development of *α*_{abs} and derived BB(%) after emission of BC
particles and their growth by condensation of non-absorbing compounds.
A similar development – *α*_{abs} values increase to > 1.3 and
decrease to < 0.9 during a several-day-long pollution episode during
which the *D*_{g} of the whole size distribution grew possibly by
condensation – was observed at the Station for Observing Regional Processes of the Earth System (SORPES) in Nanjing, China (Fig. 9 of Shen et
al., 2018). There was no SP2 available for the core–shell structure
measurements in that study, so it cannot really be proven that the observed
*α*_{abs} development was due to condensational growth even though
it seems a good explanation and is qualitatively in line with Fig. 5.

## 3.3 Unimodal size distributions with the same BC core fraction for all sizes

The second approach is to assume that the BC core fraction – or
equivalently the shell volume fraction – is the same for all sizes, which
means that the shell thickness increases with size as was visualized in Fig. 1b. This can be considered to be a result of aging of BC by not only
condensational growth but also by cloud processing. The latter would lead to
thick shells on particles activated into cloud droplets that would absorb
for instance SO_{2} and NH_{3} and that would not rain but later get
back into the aerosol phase by evaporation of cloud water. The constant
volume fraction is not realistic but neither is the constant shell
thickness. Both can be considered to be approximations.

In this approach the geometric standard deviations of the whole size
distributions were set to *σ*_{g} = 1.4, 1.6, and 1.8, and the shell
volume fractions *f*_{s} were set to vary from 0 % to 99 %. The resulting *α*_{abs} and BB(%) are presented as a function of *D*_{g}, *f*_{s}, and
*σ*_{g} (Fig. 6). They are comparable with the analogous plots for
single particles, i.e., *σ*_{g} = 1.0 (Fig. 2b, d, and f). Note that from Eq. (2) it follows that the assumption of a constant *f*_{s} means
that also the *D*_{p}-to-*D*_{core} ratio *R* is constant and that the *f*_{s}
range of 0 % to 99 % corresponds to the *R* range of 1 to 4.6. Figure 6
therefore has two *y* axes: one showing the *f*_{s} and the other the
corresponding *R* values.

Several observations can be made from Fig. 6. One of them is that the
isoline of *α*_{abs} = 1 grows with growing *D*_{g} for each of
the size distribution widths (*σ*_{g}) but decreases with growing
*σ*_{g}. Another is that the wider the size distribution is, the
lower the *α*_{abs} and BB(%) are at any given shell volume
fraction. The third one is that for all three widths *α*_{abs} and
BB(%) grows when *f*_{s} grows but that the growth is not uniformly
distributed over the *f*_{s} vs. *D*_{g} space.

The last observation leads to calculations of size-dependent sensitivities
of *α*_{abs} to variations in *f*_{s}. The sensitivity was
calculated as $\mathrm{d}{\mathit{\alpha}}_{\mathrm{abs}}/\mathrm{d}{f}_{\mathrm{s}}$, and its unit is %^{−1}. Figure 7a shows the sensitivities in the whole *f*_{s} range of 1 %–99 % as a
function of *D*_{g} for the three size distribution widths. The sensitivity depends clearly on both *D*_{g} and *σ*_{g} of the size distribution, and it also varies with *f*_{s}. It is very clear that *α*_{abs} is most sensitive to *f*_{s} variations when *D*_{g} of the size distribution
is in the accumulation-mode sizes of 100–200 nm. The sensitivity grows
fairly steadily with growing *f*_{s} until it increases very strongly for *f*_{s} > 90 % – which equals *R* > 2.

Another step for visualizing the sensitivities was taken by calculating
size-dependent average sensitivities of *α*_{abs} and BB(%) in
three *f*_{s} ranges: *f*_{s} = 0 %–50 %, 50 %–90 %, and 90 %–99 %
(Fig. 7b and c).

According to Eq. (2) the *f*_{s} ranges correspond to the *R* ranges of 1–1.3, 1.3–2.2, and 2.2–4.6. The lines in Fig. 7b and c can be used for a rough estimate on a possible effect on *α*_{abs} and BB(%). For instance, if *D*_{g}≈ 140 nm, *σ*_{g} = 1.4, and *f*_{s}
≈ 50 %–90 % , an increase in *f*_{s} from 50 % to 51 % leads
to an *α*_{abs} increase of ∼ 0.01 and consequently to
a BB(%) increase of ∼ 1 % when Aethalometer model
constants of *α*_{ff} = 0.9 and *α*_{bb} = 1.68 are used.

## 3.4 Bimodal size distributions with the same BC core fraction for all sizes in the mode

Finally, bimodal size distributions are examined briefly. The size
distributions consist of two externally mixed modes that have different
shell volume fractions. In both modes the shell volume fractions are
size-independent as in Fig. 1b. Mode 1 is an Aitken mode with the geometric
mean diameter *D*_{g1} in the range 50–100 nm. There are two different
settings for the Aitken mode: in the first case its number concentration is
10 times larger than that of the accumulation mode, i.e., *N*_{1} = 10*N*_{2}; it consists of almost pure fresh BC particles with *f*_{s1} = 5 % (*R* ≈ 1.02); and it is narrow, *σ*_{g1} = 1.4. In the
second setting the number concentrations of the Aitken and accumulation mode
are equal (*N*_{1} = *N*_{2}); the Aitken mode is aged so that *f*_{s1} = 50 % (*R* ≈ 1.3); and it is wider, *σ*_{g1} = 1.6.
Mode 2 is an accumulation mode with the geometric mean diameter *D*_{g2} in
the range 100–500 nm, *σ*_{g2} = 1.6, and it is very aged, with *f*_{s2} = 98 % (*R* ≈ 3.7). The accumulation mode could be the
result of cloud processing as explained above.

The results show that *α*_{abs} is more sensitive to variations of the accumulation mode than of the Aitken mode (Fig. 8a). For instance, if
*D*_{g2} < 250 nm, *α*_{abs} > 1 at all *D*_{g,1}
values. Also, if *D*_{g,1} = 60 nm and *D*_{g2} varies in the whole range of 100–500 nm, *α*_{abs} varies in the range of ∼ 0.4–1.3. When the Aitken mode dominates the number concentration (*N*_{1} = 10*N*_{2}) with the fresh BC particles, the maximum *α*_{abs} ≈ 1.2 at *D*_{g1}≈ 60 nm and *D*_{g2}≈ 140 nm
is smaller than when the two modes have an equal amount of particles. In the
latter case the maximum *α*_{abs} > 1.3. When the Aitken
mode with *f*_{s} = 5 % dominates the number concentration, the whole size distribution moves to the region that is less sensitive to *f*_{s}
variations as discussed above in Sect. 3.3. It is worth noting also that
the maximum *α*_{abs} and BB(%) values (Fig. 8b and c) are
smaller than derived from the unimodal size distributions (Sect. 3.3).

The purpose of this study is not to claim that all Aethalometer model results are wrong but that they have higher uncertainties than have been discussed in the literature. It is clear that there are BrC particles that have absorption Ångström exponents clearly larger than 1, as shown in a very large number of publications. However, the size of light-absorbing particles and their coating even by purely scattering material clearly affect the wavelength dependence of absorption and thus have the potential to affect the Aethalometer model results. Since the wavelength dependency is used for source apportionment, these effects have the potential to result in contributions of wood-burning or fossil fuel emissions that are tens of percent too high or low.

There are some important results. In the modeling, *α*_{abs} equals 1
or 0.9 in very rare cases, and thus BB(%) was very seldom 0 % even
though one type of BC was the only absorbing material in the simulations.
The shape of size distribution plays an important role. Narrow size
distributions result in higher *α*_{abs} and BB(%) values than
wide size distributions. The sensitivity of *α*_{abs} and BB(%)
to variations in shell volume fractions is the highest for accumulation-mode
particles. This is important because that is where the largest aerosol mass
is.

The goal of the paper was not to find out whether some pair of *α*_{ff} and *α*_{bb} is better than the other. Two well-known *α*_{ff} and *α*_{bb} pairs were used and shown how large
the uncertainties may become just for these two pairs even if BC particles
were coated by purely scattering material. The goal was not at all to find a
good pair. On the contrary, the study shows that no constant values are good
since in the real atmosphere BC particle size distributions are not
constant, neither their mean diameter nor the coating of the particles. They
all vary dynamically in the atmosphere. The study shows that any constant
values will undoubtedly lead to large uncertainties of both the BB and FF
fractions if no information on the size of the core or the thickness of the
shell is available, even if purely scattering material is coating BC cores.
As a conclusion, for the interpretation of absorption Ångström
exponents it would be very important to measure BC size distributions and
shell thicknesses together with the wavelength dependency of absorption.

There are obvious limitations in this study. A core–shell Mie model was used only so the work is limited to spherical particles. Fresh BC particles are usually agglomerates. There are studies that show that during aging processes these agglomerate may collapse and become closer to spherical particles, so Mie modeling probably agrees better for aged than fresh BC particles. Further, even if particles were spherical, how well can they be modeled with a Mie code when they are collected on filters? Or does light absorption then follow the spectral absorbance of the bulk materials?

This question could in principle be answered by generating spherical BC
particles, coating them in an aging chamber with some non-absorbing
material – for instance ammonium sulfate, and measuring both light absorption
at multiple wavelengths with an Aethalometer and BC core size distributions
and shell thicknesses with an SP2. If *α*_{abs} increases up to some
maximum value as a function of shell thickness and then starts decreasing
like in the simulations above, then the process agrees with the growth of a
size-independent coating. Or if *α*_{abs} increases steadily then it
suggests that the growth is size-dependent and possibly with a
size-independent shell volume fraction growth rate. If these effects are observed then the uncertainties discussed in this work should be taken seriously.

On the other hand, if none of these effects were observed and the absorption
Ångström exponents of the collected particles were ≈ 1
regardless of core size and shell thickness, it would be safe to say that the
Aethalometer measures the absorption spectra of the bulk materials and that
the Aethalometer model yields correct results. The truth is probably
somewhere between these extremes: when the filter tape is still relatively
clean the particles can be modeled even with a Mie code, and for heavily loaded
filters *α*_{abs} is that of bulk material. This could and
should also be tested experimentally.

The code is described in Voshchinnikov and Mathis (1999) and it is available in http://www.astro.spbu.ru/staff/ilin2/SOFTWARE/nmie0.html.

The data can be generated as described in Sect. 2. No real data were used in this paper.

The supplement related to this article is available online at: https://doi.org/10.5194/amt-14-3707-2021-supplement.

The author declares that there is no conflict of interest.

This article is part of the special issue “Satellite and ground-based remote sensing of aerosol optical, physical, and chemical properties over China”. It is not associated with a conference.

This research has been supported by the Academy of Finland via the projects NABCEA (grant no. 296302) and ACFA (grant no. 335845) and by Business Finland via project BC Footprint (grant no. 528/31/2019).

This paper was edited by Linlu Mei and reviewed by two anonymous referees.

Andreae, M. O. and Gelencsér, A.: Black carbon or brown carbon? The nature of light-absorbing carbonaceous aerosols, Atmos. Chem. Phys., 6, 3131–3148, https://doi.org/10.5194/acp-6-3131-2006, 2006.

Arola, A., Schuster, G., Myhre, G., Kazadzis, S., Dey, S., and Tripathi, S. N.: Inferring absorbing organic carbon content from AERONET data, Atmos. Chem. Phys., 11, 215–225, https://doi.org/10.5194/acp-11-215-2011, 2011.

Bergstrom, R. W., Pilewskie, P., Russell, P. B., Redemann, J., Bond, T. C., Quinn, P. K., and Sierau, B.: Spectral absorption properties of atmospheric aerosols, Atmos. Chem. Phys., 7, 5937–5943, https://doi.org/10.5194/acp-7-5937-2007, 2007.

Bond, T. C. and Bergstrom, R. W.: Light Absorption by Carbonaceous Particles: An Investigative Review, Aerosol Sci. Technol., 40, 27–67, 2006.

Bond, T. C., Doherty, S. J., Fahey, D. W., Forster, P. M., Berntsen, T., DeAngelo, B. J., Flanner, M. G., Ghan, S., Kärcher, B., Koch, D., Kinne, S., Kondo, Y., Quinn, P. K., Sarofim, M. C., Schultz, M. G., Schulz, M., Venkataraman, C., Zhang, H., Zhang, S., Bellouin, N., Guttikunda, S. K., Hopke, P. K., Jacobson, M. Z., Kaiser, J. W., Klimont, Z., Lohmann, U., Schwarz, J. P., Shindell, D., Storelvmo, T., Warren, S. G., and Zender, C.S.: Bounding the role of black carbon in the climate system: A scientific assessment, J. Geophys. Res.-Atmos., 118, 5380–5552, https://doi.org/10.1002/jgrd.50171, 2013.

Cazorla, A., Bahadur, R., Suski, K. J., Cahill, J. F., Chand, D., Schmid, B., Ramanathan, V., and Prather, K. A.: Relating aerosol absorption due to soot, organic carbon, and dust to emission sources determined from in-situ chemical measurements, Atmos. Chem. Phys., 13, 9337–9350, https://doi.org/10.5194/acp-13-9337-2013, 2013.

Chung, C., Ramanathan, V., and Decremer, D.: Observationally constrained estimates of carbonaceous aerosol radiative forcing, P. Natl. Acad. Sci. USA, 109, 11624–11629, 2012.

Chylek, P., Lee, J. E., Romonosky, D. E., Gallo, F., Lou, S., Shrivastava, M., Carrico, C. M., Aiken, A. C., and Dubey, M. K.: Mie scattering captures observed optical properties of ambient biomass burning plumes assuming uniform black, brown, and organic carbon mixtures, J. Geophys. Res.-Atmos., 124, 11406–11427, https://doi.org/10.1029/2019JD031224, 2019.

Devi, J. J., Bergin, M., McKenzie, M., Schauer, J. J., and Weber, R.: Contribution of particulate brown carbon to light absorption in the rural and urban Southeast US, Atmos. Environ., 136, 95–104, 2016.

Feng, Y., Ramanathan, V., and Kotamarthi, V. R.: Brown carbon: a significant atmospheric absorber of solar radiation?, Atmos. Chem. Phys., 13, 8607–8621, https://doi.org/10.5194/acp-13-8607-2013, 2013.

Fuller, G. W., Tremper, A. H., Baker, T. D., Yttri, K. E., and Butterfield, D.: Contribution of wood burning to PM_{10} in London, Atmos. Environ., 87, 87–94, 2014.

Gyawali, M., Arnott, W. P., Lewis, K., and Moosmüller, H.: In situ aerosol optics in Reno, NV, USA during and after the summer 2008 California wildfires and the influence of absorbing and non-absorbing organic coatings on spectral light absorption, Atmos. Chem. Phys., 9, 8007–8015, https://doi.org/10.5194/acp-9-8007-2009, 2009.

Harrison, R. M., Beddows, D. C., Jones, A. M., Calvo, A., Alves, C., and Pio, C.: An evaluation of some issues regarding the use of aethalometers to measure woodsmoke concentrations, Atmos. Environ., 80, 540–548, 2013.

Healy, R., Sofowote, U., Su, Y., Debosz, J., Noble, M., Jeong, C.-H., Wang, J., Hilker, N., Evans, G., and Doerksen, G.: Ambient measurements and source apportionment of fossil fuel and biomass burning black carbon in Ontario, Atmos. Environ., 161, 34–47, 2017.

Helin, A., Niemi, J., Virkkula, A., Pirjola, L., Teinilä, K., Backman, J., Aurela, M., Saarikoski, S., Rönkkö, T., Asmi, E., and Timonen, H.: Characteristics and source apportionment of black carbon in the Helsinki metropolitan area, Finland, Atmos. Environ., 190, 87–98, https://doi.org/10.1016/j.atmosenv.2018.07.022, 2018.

Herich, H., Hueglin, C., and Buchmann, B.: A 2.5 year's source apportionment study of black carbon from wood burning and fossil fuel combustion at urban and rural sites in Switzerland, Atmos. Meas. Tech., 4, 1409–1420, https://doi.org/10.5194/amt-4-1409-2011, 2011.

Kirchstetter, T. W. and Thatcher, T. L.: Contribution of organic carbon to wood smoke particulate matter absorption of solar radiation, Atmos. Chem. Phys., 12, 6067–6072, https://doi.org/10.5194/acp-12-6067-2012, 2012.

Kirchstetter, T. W., Novakov, T., and Hobbs, P. V.: Evidence that the spectral dependence of light absorption by aerosols is affected by organic carbon, J. Geophys. Res., 109, D21208, https://doi.org/10.1029/2004JD004999, 2004.

Kondo, Y., Sahu, L., Moteki, N., Khan, F., Takegawa, N., Liu, X., Koike, M., and Miyakawa, T.: Consistency and traceability of black carbon measurements made by laser-induced incandescence, thermal-optical transmittance, and filter-based photo-absorption techniques, Aerosol Sci. Technol., 45, 295–312, https://doi.org/10.1080/02786826.2010.533215, 2011.

Lack, D. A. and Cappa, C. D.: Impact of brown and clear carbon on light absorption enhancement, single scatter albedo and absorption wavelength dependence of black carbon, Atmos. Chem. Phys., 10, 4207–4220, https://doi.org/10.5194/acp-10-4207-2010, 2010.

Lack, D. A. and Langridge, J. M.: On the attribution of black and brown carbon light absorption using the Ångström exponent, Atmos. Chem. Phys., 13, 10535–10543, https://doi.org/10.5194/acp-13-10535-2013, 2013.

Lack, D. A., Langridge, J. M., Bahreini, R., Cappa, C. D., Middlebrook, A. M., and Schwarz, J. P.: Brown carbon and internal mixing in biomass burning particles, P. Natl. Acad. Sci. USA, 109, 14802–14807, 2012.

Laskin, A., Laskin, J., and Nizkorodov, S. A.: Chemistry of Atmospheric Brown Carbon, Chem. Rev., 115, 4335–4382, https://doi.org/10.1021/cr5006167, 2015.

Liu, C., Chung, C. E., Yin, Y., and Schnaiter, M.: The absorption Ångström exponent of black carbon: from numerical aspects, Atmos. Chem. Phys., 18, 6259–6273, https://doi.org/10.5194/acp-18-6259-2018, 2018.

Moosmüller, H., Chakrabarty, R. K., Ehlers, K. M., and Arnott, W. P.: Absorption Ångström coefficient, brown carbon, and aerosols: basic concepts, bulk matter, and spherical particles, Atmos. Chem. Phys., 11, 1217–1225, https://doi.org/10.5194/acp-11-1217-2011, 2011.

Moteki, N., Kondo, Y., Miyazaki, Y., Takegawa, N., Komazaki, Y., Kurata, G., Shirai, T., Blake, D. R., Miyakawa, T., and Koike, M.: Evolution of mixing state of black carbon particles: Aircraft measurements over the western Pacific in March 2004, Geophys. Res. Lett., 34, L11803, https://doi.org/10.1029/2006GL028943, 2007.

Russell, P. B., Bergstrom, R. W., Shinozuka, Y., Clarke, A. D., DeCarlo, P. F., Jimenez, J. L., Livingston, J. M., Redemann, J., Dubovik, O., and Strawa, A.: Absorption Angstrom Exponent in AERONET and related data as an indicator of aerosol composition, Atmos. Chem. Phys., 10, 1155–1169, https://doi.org/10.5194/acp-10-1155-2010, 2010.

Saleh, R., Hennigan, C. J., McMeeking, G. R., Chuang, W. K., Robinson, E. S., Coe, H., Donahue, N. M., and Robinson, A. L.: Absorptivity of brown carbon in fresh and photo-chemically aged biomass-burning emissions, Atmos. Chem. Phys., 13, 7683–7693, https://doi.org/10.5194/acp-13-7683-2013, 2013.

Sandradewi, J., Prevot, A. S. H., Szidat, S., Perron, N., Alfarra, M. R., Lanz, V. A., Weingartner, E., and Baltensperger, U.: Using aerosol light absorption measurements for the quantitative determination of wood burning and traffic emission contributions to particulate matter, Environ. Sci. Technol., 42, 3316–3323, 2008a.

Sandradewi, J., Prevot, A. S. H., Weingartner, E., Schmidhauser, R., Gysel, M., and Baltensperger, U.: A study of wood burning and traffic aerosols in an Alpine valley using a multi-wavelength Aethalometer, Atmos. Environ., 42, 101–112, 2008b.

Schuster, G. L., Dubovik, O., Arola, A., Eck, T. F., and Holben, B. N.: Remote sensing of soot carbon – Part 2: Understanding the absorption Ångström exponent, Atmos. Chem. Phys., 16, 1587–1602, https://doi.org/10.5194/acp-16-1587-2016, 2016.

Schwarz, J. P., Gao, R. S., Spackman, J. R., Watts, L. A., Thomson, D. S., Fahey, D. W., Ryerson, T. B., Peischl, J., Holloway, J. S., Trainer, M., Frost, G. J., Baynard, T., Lack, D. A., de Gouw, J. A., Warneke, C., and Del Negro, L. A.: Measurement of the mixing state, mass, and optical size of individual black carbon particles in urban and biomass burning emissions, Geophys. Res. Lett., 35, L13810, https://doi.org/10.1029/2008GL033968, 2008.

Shen, Y., Virkkula, A., Ding, A., Wang, J., Chi, X., Nie, W., Qi, X., Huang, X., Liu, Q., Zheng, L., Xu, Z., Petäjä, T., Aalto, P. P., Fu, C., and Kulmala, M.: Aerosol optical properties at SORPES in Nanjing, east China, Atmos. Chem. Phys., 18, 5265–5292, https://doi.org/10.5194/acp-18-5265-2018, 2018.

Shiraiwa, M., Kondo, Y., Moteki, N., Takegawa, N., Sahu, L. K., Takami, A., Hatakeyama, S., Yonemura, S., and Blake, D. R.: Radiative impact of mixing state of black carbon aerosol in Asian outflow, J. Geophys. Res., 113, D24210, https://doi.org/10.1029/2008JD010546, 2008.

Valenzuela, A., Olmo, F. J., Lyamani, H., Antón, M., Titos, G., Cazorla, A., and Alados-Arboledas, L.: Aerosol scattering and absorption Angström exponents as indicators of dust and dust-free days over Granada (Spain), Atmos. Res., 154, 1–13, 2015.

Voshchinnikov, N. V. and Mathis, J. S.: Calculating Cross Sections of Composite Interstellar Grains, Astrophys. J., 526, 257–264, https://doi.org/10.1086/307997, 1999 (code available at: http://www.astro.spbu.ru/staff/ilin2/SOFTWARE/nmie0.html, last access: 21 May 2021).

Wang, X., Heald, C. L., Sedlacek, A. J., de Sá, S. S., Martin, S. T., Alexander, M. L., Watson, T. B., Aiken, A. C., Springston, S. R., and Artaxo, P.: Deriving brown carbon from multiwavelength absorption measurements: method and application to AERONET and Aethalometer observations, Atmos. Chem. Phys., 16, 12733–12752, https://doi.org/10.5194/acp-16-12733-2016, 2016.

Wu, Z. P. and Wang, Y. P.: Electromagnetic scattering for multilayered spheres: recursive algorithms, Radio Science, 26, 1393–1401, 1991.

Zhang, X., Mao, M., Yin, Y., and Tang, S.: The absorption Ångstrom exponent of black carbon with brown coatings: effects of aerosol microphysics and parameterization, Atmos. Chem. Phys., 20, 9701–9711, https://doi.org/10.5194/acp-20-9701-2020, 2020.

Zotter, P., Herich, H., Gysel, M., El-Haddad, I., Zhang, Y., Močnik, G., Hüglin, C., Baltensperger, U., Szidat, S., and Prévôt, A. S. H.: Evaluation of the absorption Ångström exponents for traffic and wood burning in the Aethalometer-based source apportionment using radiocarbon measurements of ambient aerosol, Atmos. Chem. Phys., 17, 4229–4249, https://doi.org/10.5194/acp-17-4229-2017, 2017.