The absorption Ångström exponent αabs 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 Dcore of particles that have the same diameter Dp at all shell thicknesses. In Fig. 2b, d, and f the dashed lines show the particle diameter Dp and fs of particles that have the same Dcore at all shell volume fractions fs in the range fs ≤ 99 %. The dependence of αabs 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 Dcore < ∼ 150 nm and s > ∼ 25–50 nm the absorption Ångström exponent αabs > 1.4. The respective BB fractions are larger than about 40 % or 60 % for the Aethalometer model parameters of αff = 1, αbb = 2 (pair 1) and αff = 0.9, αbb = 1.68 (pair 2), respectively. Figure 2a also shows that for Dcore < ∼ 100 nm there are two maxima of the αabs when the shell grows thicker. In the second maximum, αabs > ∼ 1.6. As a result the BB fractions would be > 60 % and even > 100 % for the two Aethalometer model parameter pairs. When Dcore is in the range of ∼ 170–200 nm, αabs ≈ 1 and αabs decreases with a growing s. For larger core diameters the absorption Ångström exponent is even smaller. When Dcore > 200 nm, αabs < 1 and even negative for Dcore > ∼ 360 nm. Further, when Dcore > 200 nm, αabs does not grow essentially at all as a function of s.

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

The visualization also shows that the αabs value of 1, usually considered an indication of BC, is not a result of an unambiguous Dcore-s (Fig. 2a) or Dp-fs (Fig. 2b) combination.

For single particles αabs 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 (Dg,core) was varied from 50 to 200 nm at 10 nm intervals. The geometric standard deviations of the size distributions were σg = 1.4, σg = 1.6, and σg = 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 αabs was calculated for the full size distribution 3–2500 nm.

The results are first shown as a function of Dg,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 αabs ≈ 1 at Dcore ≈ 180 nm for shell thicknesses s ≈ 0–70 nm as shown by the almost vertical αabs = 1 isoline in Fig. 2a, whereas for the size distributions with σg,core > 1 the respective isoline is a strong function of both s and σg,core (Fig. 3a). At all widths of the size distribution, αabs increases with increasing shell thickness and then starts decreasing. For small core sizes (Dg,core < ∼ 80 nm) αabs also has a second maximum when the size distribution is narrow. The width of the size distribution has a clear effect on the αabs: for all core sizes and shell thicknesses αabs decreases with increasing σg,core.

Both for single particles and size distributions the first maximum of αabs is the smaller the larger the Dg,core and σg,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 αabs 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 Dg,core and σg 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 σg,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, αabs, 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 fs 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 Dg,core = 90 m and σg = 1.8, fs ≈ 98 % even with s = 250 nm (Fig. 5a). The geometric mean diameter Dg 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., σg, 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 Dp-to-Dcore ratio RN(Dp), 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 Dg = 64 ± 6 nm, and σg = 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 σg 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 σg values yield Dg = 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 αabs first grows to > 1.6 for the narrow (σg,core = 1.4) BC core size distribution that has the smallest geometric mean size (Dg,core = 50 nm) but to lower values for the wider size distributions that have larger Dg,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 fs ≈ 90 ± 8 % (Fig. 5b). Schwarz et al. (2008) reported statistics of rBC mass size distributions in urban aerosol: fs = 9 ± 6 %, s = 20 ± 10 nm, MMD = 170 nm, and σg of 1.71, which yields Dg = 72 nm; in biomass-burning emissions: fs=70±9 %, s = 65 ± 12 nm, MMD = 210 nm, and σg = 1.43, which yields Dg = 143 nm; and in background continental aerosol: fs = 46 ± 3 %, s = 48 ± 14 nm, MMD = 210 nm, and σg=1.55, which yields Dg = 118 nm.

The referenced studies show that the s, R, and fs 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 σg < 1.2 (Fig. 5c).

After reaching the first maximum, αabs 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 fs > 98 %. Such s and R values are not in the range observed in the abovementioned studies, nor are the low geometric standard deviations of σg < 1.1 realistic, so the second maximum can be considered as a theoretical value only. For size distributions with Dg,core > 70 nm there is no second maximum of αabs.

As αabs increases and decreases, it is clear that this applies to BB(%) as well (Fig. 5d–e). For the smallest core sizes (Dg,core = 50 nm) and the narrowest size distributions (σg,core = 1.4), the first maximum BB(%) may be as high as ∼ 90 % when the values of αff = 0.9 and αbb = 1.68 are used in Eq. (7) but lower, ∼ 50 %, when the values of αff = 1 and αbb = 2 are used. For the wider core size distributions the BB(%) fractions are lower. For the widest core size distributions (σg,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 αabs and derived BB(%) after emission of BC particles and their growth by condensation of non-absorbing compounds. A similar development – αabs values increase to > 1.3 and decrease to < 0.9 during a several-day-long pollution episode during which the Dg 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 αabs development was due to condensational growth even though it seems a good explanation and is qualitatively in line with Fig. 5.

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 SO2 and NH3 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 σg = 1.4, 1.6, and 1.8, and the shell volume fractions fs were set to vary from 0 % to 99 %. The resulting αabs and BB(%) are presented as a function of Dg, fs, and σg (Fig. 6). They are comparable with the analogous plots for single particles, i.e., σg = 1.0 (Fig. 2b, d, and f). Note that from Eq. (2) it follows that the assumption of a constant fs means that also the Dp-to-Dcore ratio R is constant and that the fs 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 fs and the other the corresponding R values.

Several observations can be made from Fig. 6. One of them is that the isoline of αabs = 1 grows with growing Dg for each of the size distribution widths (σg) but decreases with growing σg. Another is that the wider the size distribution is, the lower the αabs and BB(%) are at any given shell volume fraction. The third one is that for all three widths αabs and BB(%) grows when fs grows but that the growth is not uniformly distributed over the fs vs. Dg space.

The last observation leads to calculations of size-dependent sensitivities of αabs to variations in fs. The sensitivity was calculated as dαabs/dfs, and its unit is %-1. Figure 7a shows the sensitivities in the whole fs range of 1 %–99 % as a function of Dg for the three size distribution widths. The sensitivity depends clearly on both Dg and σg of the size distribution, and it also varies with fs. It is very clear that αabs is most sensitive to fs variations when Dg of the size distribution is in the accumulation-mode sizes of 100–200 nm. The sensitivity grows fairly steadily with growing fs until it increases very strongly for fs > 90 % – which equals R > 2.

Another step for visualizing the sensitivities was taken by calculating size-dependent average sensitivities of αabs and BB(%) in three fs ranges: fs = 0 %–50 %, 50 %–90 %, and 90 %–99 % (Fig. 7b and c).

According to Eq. (2) the fs 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 αabs and BB(%). For instance, if Dg≈ 140 nm, σg = 1.4, and fs ≈ 50 %–90 % , an increase in fs from 50 % to 51 % leads to an αabs increase of ∼ 0.01 and consequently to a BB(%) increase of ∼ 1 % when Aethalometer model constants of αff = 0.9 and αbb = 1.68 are used.

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 Dg1 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., N1 = 10N2; it consists of almost pure fresh BC particles with fs1 = 5 % (R ≈ 1.02); and it is narrow, σg1 = 1.4. In the second setting the number concentrations of the Aitken and accumulation mode are equal (N1 = N2); the Aitken mode is aged so that fs1 = 50 % (R ≈ 1.3); and it is wider, σg1 = 1.6. Mode 2 is an accumulation mode with the geometric mean diameter Dg2 in the range 100–500 nm, σg2 = 1.6, and it is very aged, with fs2 = 98 % (R ≈ 3.7). The accumulation mode could be the result of cloud processing as explained above.

The results show that αabs is more sensitive to variations of the accumulation mode than of the Aitken mode (Fig. 8a). For instance, if Dg2 < 250 nm, αabs > 1 at all Dg,1 values. Also, if Dg,1 = 60 nm and Dg2 varies in the whole range of 100–500 nm, αabs varies in the range of ∼ 0.4–1.3. When the Aitken mode dominates the number concentration (N1 = 10N2) with the fresh BC particles, the maximum αabs ≈ 1.2 at Dg1≈ 60 nm and Dg2≈ 140 nm is smaller than when the two modes have an equal amount of particles. In the latter case the maximum αabs > 1.3. When the Aitken mode with fs = 5 % dominates the number concentration, the whole size distribution moves to the region that is less sensitive to fs variations as discussed above in Sect. 3.3. It is worth noting also that the maximum αabs and BB(%) values (Fig. 8b and c) are smaller than derived from the unimodal size distributions (Sect. 3.3).