The formation of BCFAs from combustion is a process involving several stages. Along with BC, complex mixtures of gas-phase organic compounds with a spectrum of molecular structures are co-emitted during incomplete combustion (Siegmann et al., 2002; Gentner et al., 2017). Depending upon the source of burning, different types of polycyclic aromatic hydrocarbons (PAHs) are considered to be the direct precursors of BCFAs (Bockhorn, 2009). Small PAHs such as acetylene (C2H2) are attached to larger precursor PAHs, resulting in the growth of these elementary structures. It is postulated that the nucleation of two large PAHs leads to the formation of small three-dimensional particles with diameters ranging from 1–2 nm (Calcote, 1981).

Processes like surface growth and coagulation of gaseous-phase molecules or PAHs leads to further growth of these particles. High-resolution transmission electron microscopy (TEM) images revealed these particles to be spherules up to the diameter of 10–30 nm specific to the flame (Homann, 1967). These primary particles show a randomly ordered microstructure of graphite layers (Hess et al., 1969). Following the processes of nucleation and coagulation, the primary particles form larger BCFAs, which subsequently grow by aggregation (Sorensen, 2001). Following this concept of fractal morphology, a mathematical description of fractal aggregates was formulated (Mishchenko et al., 2002): 1Ns=kfRgaoDf, where ao is the radius of primary particles, Ns is the number of primary particles, Df is the fractal dimension, and kf is a fractal prefactor. Rg is the radius of gyration, which characterizes the spatial size of the aggregate. It is defined as the root mean square (rms) distance of the aggregate from its geometrical centre by 2Rg2=1Ns∑i=1Nsri-ro2, where ri is the position vector of the ith primary particle, and ro is the position vector of the centre of mass of an aggregate with radius of gyration Rg.

The size of a BCFA is determined by two parameters, the radius of the primary particle (ao) and number of primary particles (Ns). Both are sensitive to the emission source. BCFAs originating from the combustion of biomass have a radius of the primary particle varying between 15 and 25 nm (Chakrabarty et al., 2006). On the other hand, emissions from aircraft turbines comprise primary particles with a radius of 5 nm (Liati et al., 2014). Aggregates emitted from diesel engines have a radius of the primary particle varying between 10 and 12 nm (Guarieiro et al., 2017). Some experimental studies indicate that in the atmosphere, the radius of the primary particle is polydisperse in nature, varying from 10–100 nm (Bescond et al., 2014). Following these studies, Liu et al. (2015) reported differences in the optical properties of BCFAs due to the monodisperse and polydisperse distribution of the radii of the primary particles. Contrarily, Berry and Percival (1986) showed that light absorption measurements are insensitive to the radii of the primary particles. Additionally, Kahnert (2010b) pointed out that insensitivity is present when the radii of the primary particle fall in the range of 10–25 nm. For the sake of simplicity, aggregates of monodisperse primary particle size were used in this study.

Further, the reshaping of BCFAs into collapsed, sphere-like structures while ageing can be described by the fractal dimension (Df) (Sorensen, 2001). The value of Df increases as an aggregate reshapes into a more spherical particle. A Df value of 3 is the value for a sphere, whereas Df of 1 represents an open-chain-like aggregate. In the early stages of their formation, BCFAs have a fractal dimension (Df) between 1.5 and 1.9 (China et al., 2013; Wentzel et al., 2003). However, as a consequence of atmospheric ageing, the aggregates transform from being bare to partly coated, embedded in coatings. In this case, the fractal dimension can go up to 2.2 (Wang et al., 2017). Exposure to humidity and coatings can collapse the BCFA into a structure having even a larger fractal dimension up to 2.6. (Zhang et al., 2008; Bambha et al., 2013). Hence, studying BC particles under the assumption of aggregate morphology provides a wider range of parameter space (particle size, primary particle size, and morphology). This is limited to only particle size in the case of spherical assumptions.

Aggregates are formed from the random motion of a cluster meeting a cluster (Sorensen, 2001). If the probability of sticking is considered 1, the process of formation is called diffusion-limited cluster aggregation (Witten and Sander, 1983). Following this principle, diffusion-limited algorithms (DLAs) have been developed, which include cluster–cluster aggregation (CCA) (Thouy and Julien, 1994) and particle–cluster aggregation (PCA) methods (Hentschel, 1984). In this study, the tunable diffusion-limited aggregation (DLA) software developed by Woźniak (2012) was used, which iteratively adds the primary particles one by one, preserving the fractal parameters at each step.

Previous modelling studies (Kahnert, 2010a; Smith and Grainger, 2014) investigated the optical properties of pure BCFAs, i.e. without any coating. From the simulated optical properties, parameterizations for pure BCFAs with respect to the number of primary particles at various fractal dimensions and wavelengths were given (Smith and Grainger, 2014). Ouf et al. (2016) conducted near-edge X-ray absorption fine structure (NEXAFS) analysis on BC produced from a diffusion flame-based mini-CAST burner and found that organics (by-products of combustion) get attached to the edge of graphite crystallites without changing the inner structure of the core. This laboratory result can be simulated for coated BC in radiative modelling studies by assuming a spherical coating around each individual primary particle of a BC aggregate (Luo et al., 2018b). It must be noted that the focus of our study is on BCFAs with coatings consisting of non-absorbing organics. If a brown carbon coating was to be included in the study, information and extra computational time regarding refractive indices would be needed. Unfortunately, due to the time-consuming nature of simulations, the generated database could not include BCFAs with brown carbon coating.

For the sake of simplicity and computational limitations, this representation of coated BC shown in Fig. 2 (bottom panel) was chosen for the entire study. In order to simulate such BC aggregates with individually coated primary particles, the inner radius of the primary particle (ai) is fixed to 15 nm, whereas the outer radius of the primary particle (ao) consisting of the organics is varied from 15.1 to 30 nm with the fraction of organics (forganics) changing from 1 % to 90 %, respectively. The relationship between the outer radius of the primary particle (ao), the inner radius of the primary particle (ai), and the fraction of organics (forganics) is shown below: 343πai3=1-forganics43πao3. It must be noted that when the fraction of organics (forganics) is larger than 80 % and the morphology of the aggregate becomes compact, using this coated BC representation results in a practically unrealistic particle (randomly immersed BC primary particles in a spherical coating structure). Therefore, both the composition and morphology of the aggregate play a role when choosing the representation for coated BC. Keeping the above facts in mind, we have limited the use of this coating model only to coated BCFAs with a fractal dimension Df below 2.2. In such cases, in which the BC aggregate does not have a completely compact structure, the results are expected to be reliable (Luo et al., 2018b). Moreover, Kahnert (2017) compared the coating model (closed-cell model) used in this study to a realistic model, which showed good comparability.

Luo et al. (2018b) kept the overall size of aggregates constant to study the sensitivity of optical properties at various numbers of primary particles (Ns) and vice versa. In our study, the size of the BC aggregates is increased gradually to study the subsequent changes in the optical properties. The optical properties of BC aggregates were calculated for various cases following a well-designed description scheme summarized in Fig. 1. All the optical properties are calculated at three wavelengths in the visible range, i.e. 467, 530, and 660 nm. The values are chosen following the availability of the refractive index at these specific wavelengths from Kim et al. (2015). For pure BC aggregates, the optical properties were calculated for 1.5 ≤Df≤ 2.8 in steps of 0.1. In the case of coated BC aggregates, the optical properties are calculated at the above-mentioned wavelengths for 1.5 ≤Df≤ 2.2 in steps of 0.1 and for 1 % ≤forganics≤ 90 % in increments of 5 %. The approach of assuming a spherical coating around each individual BC primary particle results in an unlikely structure for coated BCFAs with Df > 2.2; hence, those cases were omitted in this study. Figure 2 shows a few of the aggregates from the classification at a fixed Df andforganics. The large dataset obtained from the classification helped in further developing the comprehensive parameterization scheme.

In each case of the mentioned classification, the size of the BCFA is changed by incrementing Ns with 5 % and rounded to an integer value, starting from 1 up to 1000. It must be noted that in the results, the size of the BCFA is expressed in terms of mobility diameter (Dmob) instead of the number of primary particles (Ns) using the simple conversion developed by Sorensen (2001) given below: 4Dmob=2ao10-2x+0.92Nsx, where x is the mobility mass scaling exponent given by x=0.51Kn0.043, with 0.46 < x < 0.56 having an estimated error of ±0.02 (Sorensen, 2001). Kn is the Knudsen number, which is the ratio of the molecular free path to the agglomerate mobility radius.

The conversion formula given in Eq. (4) is well founded over the entire range, spanning from the continuum to the free molecular regime. Using pre-calculated values of x, the mobility diameter (Dmob) is derived for the entire dataset. The relationship between the derived mobility diameter (Dmob), number of primary particles (Ns), and volume equivalent diameter (Dequ) for a case of pure BCFA with ao=15 nm is shown in Fig. 3.

BC has a refractive index that is fairly wavelength-independent in the visible and near-visible spectrum range (Bond and Bergstrom, 2006). There are modelling studies which assume a wavelength-independent refractive index of m=1.95+0.79i for BC over the visible spectrum range (Smith and Grainger, 2014; Luo et al., 2018a). For organic carbon, the imaginary part of the refractive index (mi) is highly wavelength-dependent at the shorter wavelengths in the visible and ultraviolet (UV) wavelengths (Moosmüller et al., 2009; Alexander at al., 2008). Contrary to other studies, Kim et al. (2015) concluded that BC shows a fair amount of wavelength dependency and provided refractive indices for BC and organics in the visible spectrum. Following their study, the real (mr) and imaginary (mi) part of the refractive indices used for BC and organics at different wavelengths in this study are summarized in Table 1.

The multi-sphere T-matrix (MSTM) method consists of an algorithm for calculating the time-harmonic electromagnetic properties of a set of arbitrary spheres (Mishchenko et al., 2004; Mackowski and Mishchenko, 2011). MSTM version 3.0 (Mackowski, 2013) calculates the optical properties for fixed and random orientations, the latter being used in this study. MSTM code can calculate the optical properties of coated BCFAs involving nested spheres with the condition that there should be no intersecting surfaces of individual primary particles. Radius and position vectors of the inner and outer primary particle of the BCFA are obtained from the tunable DLA software (Woźniak, 2012), which is coupled to the MSTM code.

The optical properties of the aggregates were modelled at three wavelengths, i.e. 467, 530, and 660 nm. At the wavelengths of 660 and 530 nm, the optical properties from MSTM code are obtained for 1 ≤Ns≤ 1000. Because of the increasing processing time of the MSTM code at lower wavelengths, the calculations are limited to 1 ≤Ns≤ 500 for a wavelength of 467 nm.

For reference purposes, the optical properties were also calculated using the Mie theory and the absorption cross section from Rayleigh–Debye–Gans (RDG) theory. For the Mie theory calculations, spheres with volume equivalent radius of aggregates were taken. In the case of the coated aggregates, a concentric core–shell configuration was used (He at al., 2015). The RDG theory considers the primary particles in the aggregate to be individual Rayleigh scatters, while ignoring the inter-particle scattering (Sorensen, 2001). Therefore, in the RDG theory, the total absorption cross section of the aggregate Cabsagg is the summation of the absorption cross sections Cabspp of individual primary particles (Ns). For a monodisperse distribution, the absorption cross section from the RDG theory is given as 5Cabsagg=NsCabspp.

The radiative parameters calculated from the model are briefly presented below. The MSTM code provides the extinction, absorption, and scattering efficiency (Q), and the asymmetry parameter (g) of BCFAs. The extinction, absorption and scattering cross sections (Cext/abs/sca) are further obtained as the product of efficiency (Q) and geometric cross section (Cgeo) by 6Cext/abs/sca=Qext/abs/sca×Cgeo. In spherical objects with radii (R), the geometric cross section (Cgeo) is related to the radius by 7Cgeo=πR2. Therefore, for a BCFA, the cross sections (Cext/abs/sca) with volume equivalent radius (Rv) are defined as follows: 8Cext/abs/sca=Qext/abs/scaπRv2. The volume equivalent radius (Rv) is calculated by 9Rv=aoNs13. The single-scattering albedo (SSA) is the ratio of scattering efficiency (Qsca) and extinction efficiency (Qext), where Qext is the sum of absorption and scattering efficiency as shown below: 10SSA=QscaQext=QscaQsca+Qabs. Values of ω vary from 0 for a purely absorbing particle to 1 for a completely scattering particle.

The mass absorption cross section (MAC) is calculated from the ratio of absorption cross section (Cabs) and BC mass (mBC) as 11MAC=CabsmBC=Cabs43πRv3ρBC, where ρBC is the density of BC fixed to 1.8 g cm-3 (Bond and Bergstrom, 2006).

The wavelength dependence of light absorption, represented by the Ångström absorption exponent (AAE), is calculated using the absorption cross section (Cabs) at the three wavelengths (λ) of 467, 530, and 660 nm. The AAE value is obtained by 12Cabsλ=467,530,660=bλ-AAE, where b is a constant.

The absorption enhancement factor (Eλ) is defined by the ratio of absorption cross section of coated BCFA (Cabscoated) and pure BCFA (Cabspure) as shown below: 13Eλ=CabscoatedCabspure. This implies that the enhancement is given for particles of different total mass but the same BC mass.

To understand the atmospheric implications, the radiative forcing is estimated using a model for absorbing aerosols given by Chylek and Wong (1995). The black carbon radiative forcing at the top of the atmosphere is calculated as 14ΔFTOA=-So41-NcloudT22τ×(1-a)2βω-2a(1-ω), where So is the solar constant, Ncloud is the cloud fraction, T is the transmittance of the sky above the layer of aerosols, τ is the aerosol optical depth, β is the upward scattering function, a is the surface albedo, and ω is the single-scattering albedo. From Sagan and Pollack (1967), the upward scattering function β is calculated from the asymmetry parameter g by 15β=12(1-g). The model given by Chylek and Wong (1995) for the calculation of top-of-atmosphere (TOA) forcing is a simplified version of the multiple reflection model (Haywood and Shine, 1995; Sheridan and Ogren, 1999) with some implicit approximations. It is important to note that this is an analytical model which can be useful to understand the sensitivities of radiative forcing to various parameters (Chylek and Wong, 1995; Lesins et al., 2002). The simplified version was used in this study to highlight the sensitivity of the TOA forcing towards the morphology and composition of BC. However, the model cannot be used to replace accurate direct radiative forcing calculations.

In the tunable DLA programme, the user-specified values of the number of spheres (Ns), radius of the primary particle (ao), and fractal dimension (Df) are used to generate the fractal aggregate. This gives rise to a possibility of more than one representation of a fractal aggregate satisfying the same fractal dimension (Df), i.e. randomized particle generation. The difference between the various representations is only the different positions of the primary particles constituting the aggregate. This further results in an uncertainty in the radiative results. Depending on the complexity, some studies averaged the radiative results over 5–10 representations (Wu et al., 2016; Luo et al., 2018b), whereas others consider only a single representation (Smith and Grainger, 2014).

Considering the large dataset in this study, the option of taking an average of the multiple representations would be time-consuming. Therefore, the general uncertainty in optical properties for 30 representations of the pure BCFAs is discussed. This is done for various cases of size (Dmob) and morphology (Df). Figure 4 shows the variability in the extinction cross section Cext (first row), absorption cross section Cabs (second row), scattering cross section Csca (third row), and asymmetry parameter g (fourth row) as a function of Df. The results were calculated at a wavelength of 660 nm for pure BCFAs with Dmob values 150, 250, 500, and 1000 nm increasing from left to right in Fig. 4.

The uncertainty in the optical properties was studied for 30 representations of BCFAs with the same value of the fractal dimension. The amount of variability in the optical property at each fractal dimension (x axis) must be seen from the whiskers of the box plot in Fig. 4. The sensitivity of the optical properties with respect to various fractal dimensions can be figured out from the amount of overlap of the y-axis values between adjacent box plots.

For extinction and scattering cross sections (first and third row), the uncertainty is more pronounced at Df < 1.7. This is because of the overlap of extinction and scattering cross sections values at Df < 1.7. The absorption cross section (Cabs) shows the highest uncertainty towards various representations of a BCFA, which can be seen from higher heights of box plots in panels (e), (f), and (g) in Fig. 4. Additionally, at 150 and 250 nm, Cabs is seen to be less sensitive to Df, ranging between 1.5 and 2, whereas for box plots in panel (g) representing a 500 nm BCFA, the Cabs values overlap for Df > 1.8. It may be noted that the Cabs increases with Df for smaller BCFA (panels e and f), whereas the opposite is true for larger BCFA (panels g and h) as also reported by Luo et al. (2018b). This is further explained in detail in Sect. 3.3. The asymmetry parameter (g) shows a similar uncertainty trend as that of the extinction and scattering cross sections, i.e. lower variability but some overlap at certain Df seen in the fourth row. In general, it is observed that the uncertainty of optical properties at larger sizes (Dmob=1000 nm; last column) is comparatively low. The standard deviation in the optical properties is averaged over size and summarized for various cases of Df in Table 2.

The absorption cross section (Cabs) and BC mass absorption cross section (MACBC) have been reported to be insensitive to the radius of the primary particle (ao) for a fixed particle volume (Kahnert, 2010b). Figure 5 shows the optical properties of pure BCFAs with the radius of primary particle (ao) varying between 15nm and 30 nm as a function of Dmob. The results were calculated for a wavelength of 660 nm for pure BCFAs with Df=1.7. The absorption cross section Cabs shown in panel (b) increases by a factor of almost 10 from ao of 15 to 30 nm. Since our results here are represented against Dmob instead of volume equivalent radius (Requ), they are not expected to follow the findings of Kahnert (2010b). The results with respect to Requ are provided in Fig. S1 of the Supplement, which follows the findings of Kahnert (2010b). The asymmetry parameter shows the least dependency on ao as can be seen in panel (d). The single-scattering albedo (SSA) and the BC mass absorption cross section (MACBC) shown in panels (e) and (d) in Fig. 5 show a larger increase at ao > 20 nm for the same Dmob. Acknowledging the effect of changing ao on the optical properties, for the sake of simplicity, in this study the inner radius of the primary particle (ai) was fixed to 15 nm, and the outer radii of the primary particle (ao) were increased with forganics.

Different optical properties as a function of changing Dmob and Df are shown in Fig. 6. The results were calculated for pure BCFAs (forganics=0) at a wavelength of 660 nm. The cross sections (panels a, b, and c) show an increase with Dmob. The cross sections vary from 0.0001 to 0.1 µm2 for BCFA Dmob ranging from 24 to 810 nm. The extinction and scattering cross sections are larger for higher Df, suggesting an increasing coherent scattering for compact morphologies, as also reported by Smith and Grainger (2014). The dependency of the optical cross section on the fractal dimension (Df) was pointed out by Berry and Percival (1986); the change in the cross sections depends on whether the fractal dimension (Df) is less than 2 or greater than 2. The results from Mie calculations for a spherical particle (Df=3) follow the trend of the MSTM results, as seen in Fig. 6.

For smaller BCFAs, the absorption cross section shows negligible dependence on Df. With increasing size, the absorption cross section decreases with Df. This decrease can be interpreted as a shielding effect due to the primary particles on the surface of the aggregate. Further, with Df > 2.5, the absorption cross section increases with Df, showing the highest value for a spherical particle (Df=3). This may be caused by Mie resonances in larger BCFAs. Earlier studies have also reported higher values for the sphere equivalent (Df=3) calculations of BCFA (Liu et al., 2018; Li et al., 2016).

The single-scattering albedo (SSA =Csca/Cext) shown in panel (e) of Fig. 6 has values up to 0.42. The SSA also increases with Dmob and Df; the latter is explained by the decreasing scattering in loosely packed BCFAs. This is due to compact aggregates following a Rayleigh-like polarization curve (Gustafson and Kolokolova, 1999). The asymmetry parameter (g) (panel d) shows a range of values between 0 and 1 for Dmob values between 24 and 810 nm. The asymmetry parameter is higher for chain-like BCFAs with lower Df, indicating larger forward scattering in asymmetrical structures, as also reported by Luo et al. (2018b). When the BCFAs grow larger, the rate of increase in g with size gradually decreases for lower Df because of the scattering tending to the Rayleigh scattering regime.

Black carbon mass absorption cross section (MACBC) values shown in panel (f) fall within the range of findings reported in the literature (Bond and Bergstrom, 2006). The MACBC increases with Dmob, showing a peak at Dmob∼ 250 nm. The dependency of MACBC on Df is similar to that of the absorption cross section; i.e. Mie resonances contribute to the increase at higher Df, explaining the large discrepancy between MSTM and Mie results for MACBC. The above results with respect to Requ are provided in Fig. S2.

Figure 7 shows how the optical properties of BCFAs are influenced by the increasing content of organics. The calculations were done for a BCFA of chain-like morphology with Df = 1.7 at a wavelength of 660 nm. The results are shown as a function of Dmob at various fractions of organics (forganics). The extinction and absorbing cross sections (Fig. 7a and b) decrease steadily with increasing forganics for constant mobility diameters because of the increasing less-absorbing volume fraction in the aggregate. The dependence of the asymmetry parameter g (Fig. 7d) on forganics is very small, meaning that g is more sensitive to morphology than composition. The single-scattering albedo (SSA) increases with forganics, and this is again because of the increasing fraction of less-absorbing material. From the results of black carbon mass absorption cross section (MACBC) values shown in Fig. 7f, a dominating dependence of BCFA on composition is seen in comparison to size and morphology. Similar results for a compact BCFA of Df = 2.2 at a wavelength of 660 nm can be found in Fig. S4 in the Supplement.

Figure 8 is similar to Fig. 6 but shows the dependency of optical properties on the fractal dimension (Df) for organic-coated BCFAs with forganics of 50 % at the wavelength of 660 nm. The cross sections and asymmetry parameter show similar behaviour as that of the pure BCFAs. The SSA has an upper limit of 0.35 at Df= 2.2. The black carbon mass absorption cross section (MACBC) is rather independent of Df but values increase with coating by a factor of 1.2 for coated BCFAs with forganics of 50 % as shown in Fig. 7.

The gradually decreasing impact of the fractal morphology on the optical properties of coated BC particles was shown by Liu et al. (2017). In this study, it is seen that in the case of a non-coated BC particle (Fig. 6c), Csca is more sensitive to Df, whereas when the BC particles are coated (Figs. 7c, 8c), Csca is less sensitive to Df and forganics. It is observed that the Csca and SSA (Fig. 8c, e) become more sensitive to Df when the BCFA grows in size; therefore, the impact of the fractal morphology on the optical properties is also a function of particle size. Moreover, it must be noted that even though there is a decreasing impact of the fractal morphology on optical properties, parameters like Cabs, MACBC, and g showed significant variability towards changes in forganics (Fig. 7a, b, e, and f).

Global models use Mie theory for calculations of BC optical properties (Bond et al., 2013). The Mie theory considers BC to be homogeneously mixed spheres or a core–shell configuration. The results of SSA, g, and MACBC in both Figs. 6 and 8 clearly demonstrate a significant influence of morphology. This is clearly seen from the difference between the coloured lines representing various morphologies of BC as aggregates and the black solid line representing the result when BC is assumed as a core–shell. Therefore, the factor of changing morphology is not adequately considered when using the Mie theory for BC optical properties in global models.

In the sections before, the dependency of BCFA optical properties on size, morphology, and composition was discussed. In this section, besides showing the spectral dependency of BCFA optical properties, it is also demonstrated how this dependency changes with morphology and composition in the visible wavelength range.

Figure 9 shows the changes in the pure BCFA optical properties with wavelength (λ) at various morphologies represented by Df. Pure BCFAs with fixed Dmob equal to 330 nm were taken for this case to demonstrate the effect of morphology. All the optical properties show a decrease with λ in the visible range. Furthermore, whether the rate of decrease might vary for various morphologies was studied. Figure 9 shows that the spectral dependency is insensitive to morphology for the absorption cross section Cabs (panel b) and black carbon mass absorption cross section MACBC (panel f). The spectral dependence of scattering cross section Csca (panel c) is seen to be somewhat sensitive to changes in morphology. The highest sensitivity of spectral dependence to morphology was seen for the asymmetry parameter (g), which is dominant at higher Df, i.e. for compact aggregates.

Figure 10 is provided to illustrate how the spectral dependency of BCFAs changes with composition, i.e. fraction of organics (forganics). For this case, BCFAs are considered with Ns and Df equal to 200 and 1.7, respectively. It must be noted that the size of the BCFAs is also increasing with forganics. Contrary to the results from Fig. 9, all the cross sections (panels a, b, and c) and black carbon mass absorption cross section MACBC (panel f) show a significant increase in the spectral dependency with forganics. The spectral dependency of single-scattering albedo (SSA; panel d) shows a comparatively lower sensitivity to forganics, whereas it is nearly negligible for the asymmetry parameter (g) seen in panel (e). Additionally, the change in the spectral dependency on size is also shown in Fig. S5 of the Supplement.

Figure 11 shows the Ångström absorption exponent (AAE) of a chain-like BCFA (Df=1.7) as a function of mobility diameter (Dmob) and increasing fraction of organics (forganics). The AAE is derived from the slope of Cabs vs. λ at 467, 530, and 660 nm as shown in Eq. (12). As expected, the AAE shows a straightforward dependency on the fraction of organics (forganics). In this case, the values of AAE vary from 1.4 up to 3.6 with an increase in forganics from 1 % until 90 %. The variability in the modelled values of AAE may be attributed to the selection of the refractive indices and wavelengths (Liu et al., 2018). A similar result for the Ångström absorption exponent (AAE) of a more compact BCFA (Df = 2.2) is provided in Fig. S6. Additionally, the impact of morphology or fractal dimension (Df) on the AAE for pure BCFAs is shown in Fig. 12 with values ranging from 1.06 to 1.47. It can be observed that for smaller BCFAs, the AAE increases as the BCFA becomes more compact, whereas for larger BCFA an opposite effect is seen. Figures 11 and 12 could be interpreted as the ageing process of BC in the atmosphere focusing on changing composition and shape, respectively.

Figure 13 shows the trend in absorption enhancement factors (Eλ) as a function of mobility diameter (Dmob) and increasing fraction of organics (forganics) for a BCFA with Df = 1.7. The top row shows the absorption enhancement factors calculated from the results of the MSTM code EMSTMλ, whereas the ones derived from the Mie calculations EMieλ are displayed in the bottom row. In general, the Mie-derived absorption enhancement factors are larger by a factor of 1.1 to 1.5. The enhancement results from both MSTM and Mie calculations are shown for three wavelengths, i.e. 660, 530, and 467 nm (right to left). There is an expected increase in the absorption enhancement factors as the wavelength decreases. The values of the modelled absorption enhancement factors follow the results from various ambient studies which measured enhancement factors ranging from 1.0 to 2.25 at wavelengths between 532 and 678 nm (Cappa et al., 2012; Cui et al., 2016; Wu et al., 2018).

Liu et al. (2017) emphasized the role of the mass ratio of non-BC to BC on the performance of various methods used for simulating the scattering cross section and enhancement factors of BC particles. In this study, it is shown that the Ångström absorption exponent (AAE) calculated from just the MSTM method can show variability of up to a factor of 2 with an increasing non-BC mass fraction larger than 90 %. Similarly, it can be seen that the difference in the enhancement factors calculated from the core–shell theory and fractal assuming MSTM method can be between 1.1 and 1.5.

In this section, the dependence of the black carbon radiative forcing on modifying composition and morphology of BCFAs is discussed. The relative changes in the top-of-the-atmosphere radiative forcing (ΔFTOA) are quantified as a function of fractal dimension (Df) and fraction of organics (forganics). It is a sensitivity analysis applicable mostly to scenarios with an urban pollution and a high mass fraction of combustion aerosols. The black carbon radiative forcing at the top of the atmosphere (ΔFTOA) is estimated using Eq. (14) with fixed values of So = 1368 W m-2, Ncloud= 0.6, T= 0.79, τ= 0.03, and a= 0.1 (Chylek and Wong, 1995; Lesins et al., 2002). To focus primarily on radiative effects of BC, the optical depth τ is taken as 0.03 for smoke aerosol (Penner et al., 1992). The values of β and ω change with fractal dimension (Df) and fraction of organics (forganics), and they are obtained from the MSTM bulk optical properties. The bulk optical properties are calculated at a wavelength of 530 nm over a lognormal polydisperse size distribution with the geometric mean radius (ro) and standard deviation (σ) fixed to 0.12 µm and 1.5, respectively. Details about the bulk optical properties can be found in the Supplement of this work.

Table 3 shows how the values of black carbon radiative forcing change for various morphologies represented by the fractal dimension (Df) for pure black carbon. This can be further understood by the relative change C defined by 16C=ΔFTOA-ΔFTOARefΔFTOARef×100, where ΔFTOARef is the top-of-the-atmosphere radiative forcing for a reference case in which the fractal dimension (Df) is 1.7, i.e. a freshly emitted black carbon particle.

Similarly, the values of black carbon radiative forcing for various compositions represented by fraction of organics (forganics) in a case in which the fractal dimension (Df) is fixed to 2.2 are shown in Table 4. The values of relative change (C) are calculated using Eq. (16) with respect to ΔFTOARef as a reference case of zero fraction of organics (forganics), i.e. pure black carbon particle.

Global models use the simplified core–shell representation for BC (Bond et al., 2013), which is morphologically close to a coated BCFA of Df 2.8. In the case of coated BCFA, there is a relative change (C) of 20 % when Df increases from 1.5 to 2.2. Following the results in Table 4 the relative change (C) in ΔFTOA of coated BCFA is also expected to increase as Df approaches 2.8. Therefore, the influence of morphology on ΔFTOA is adequately considered when using the simplified core–shell representation of BC.

It can be seen from Table 4 that the top-of-the-atmosphere forcing ΔFTOA decreases by up to 55 % as the organic content of the particles increases to 90 %. This result is in agreement with the findings of Zeng et al. (2019), wherein the increasing hygroscopicity of the BC particle results in negative top-of-the-atmosphere forcing. However, it must be noted that in the study of Zeng et al. (2019), the focus was on aged BC particles with 90 %–99 % coating fraction, and the Santa Barbara DISORT Atmospheric Radiative Transfer (SBDART) model was used for estimating the radiative forcing.

Even though the simplified radiative model for absorbing aerosols is used, the results of relative change (C) in Tables 3 and 4 can provide insight about the implications of BC ageing for radiative forcing estimates. It is demonstrated that the radiative forcing results are highly sensitive towards modifications in morphology and composition when using the aggregate representation. It must be noted that these results are of high relevance in BC hotspot regions of Asia: for example, Manilla in the Philippines, where the BC emission shared up to 70 % of calculated PM1 (particulate matter with diameter < 1 µm) mass emission factors (Madueño et al., 2019).

In this section, the optimal fits for the results of the optical properties obtained from the MSTM code are discussed. For the extinction and absorption cross sections, a first-order polynomial on logarithmic scales was found to be the best fit. 17ln⁡Cext=c0+c1ln⁡Dmob18ln⁡Cabs=g0+g1ln⁡Dmob For the fittings of scattering cross section (Csca) and SSA, an equation of the following form was found to fit best. The asymmetry parameter (g) is well captured by a cubic polynomial in a logarithm space of Dmob. 19ln⁡Csca=H0+H1ln⁡Dmob+H2ln⁡ln⁡Dmob20ln⁡SSA=k0+k1ln⁡Dmob+k2ln⁡ln⁡Dmob21ln⁡g=∑n=03snln⁡Dmobn Since the nature of the curve for the mass absorption cross section (MACBC) changes for various Df, it was not possible to find an optimal function representative for the entire dataset. For all the fits, a limitation was found: the smaller particles are not well represented by the above-mentioned functions. Therefore, in order to find an overall good fit, the data are taken for points with Dmob larger than 50 nm. Previous studies have also attempted to fit the optical properties of pure BCFAs with respect to the number of primary particles (Ns) (Smith and Grainger, 2014; Kahnert, 2010b).

In this study, a parameterization scheme is developed for five BC optical properties: the extinction cross section Cext, absorption cross section Cabs, scattering cross section Csca, single-scattering albedo (SSA), and asymmetry parameter g with respect to BC size. In total, the fit coefficients for the five BC optical properties are provided for 192 cases comprising various combinations of wavelengths (λ), fractal dimensions (Df), and fraction of organics (forganics) shown in Fig. 1. For each case, linear regression models were applied individually to the MSTM-modelled optical properties for BC sizes ranging from 10 to 1000 nm. The fit coefficients for the five optical properties in each case are provided in tabular form in the Supplement to this work.

The resultant parameterization scheme provides the user an option to estimate the five optical properties at desired BC size for any of the 192 combinations of λ, Df, and forganics. It must be noted that the MSTM-modelled optical properties were calculated for fixed values of the refractive index because of limited computational resources. Therefore, the parameterization scheme provided in this study is not able to account for variable refractive indices.