Optical modeling of atmospheric black carbon aerosol ensembles with complex particle morphology

Black carbon (BC) aerosol is one of the most important factor in global warming. BC radiative forcing remains unconstrained, mainly because of the uncertain parameterizations of its absorption and scattering properties in the atmosphere. The single sphere model is widely used in current climate assessment of BC aerosols due to its computational convenience, however, their complex morphologies in particle level are excessively simplified which leads to computed inaccuracy. In this study, we present a dynamic model for optical calculations of BC aerosol ensembles considering their complex fractal aggregate morphologies with the constraint of max monomer numbers (N s, max) and radius (a max). We show that the simulation accuracy of the dynamic model with suitable values of N s, max and a max may achieve ∼95% while the computation time may reduce to ∼6%. We find that optical properties of BC aerosol ensembles can be simulated for higher accuracy or faster calculation by performing different selections of monomer numbers and radius in their size distributions. This method enables extensive and accurate optical calculations of BC particles with complex morphologies, which would be useful for the remote sensing inversion and the assessment of climate.


Introduction
Black carbon (BC) is primarily emitted from wildfires, power plants, residential heating and vehicle emissions, contributing the most significant warming effect of aerosols to global climate.The quantification of radiative forcing caused by BC aerosols remains a challenge, primarily due to the issues with their uncertain parameterizations of absorption and scattering properties in the atmosphere [1][2][3][4][5].Microscopy observations have demonstrated that individual BC particles usually aggregate as clusters of small spherical monomers.The complex morphologies of BC particles can be described by the well-known fractal law N s = k 0 ( R g /a ) D f and R 2 g = ∑ Ns i =1 r 2 i /N s , where N s is the monomer number; k 0 is the fractal pre-factor; R g is the radius of gyration calculated by the distance (r i ) from the ith monomer to the center of the aggregate; D f is the fractal dimension, and a is the mean radius of monomers.In the field of remote sensing and climate change, optical modeling of these strongly absorbing aerosols with complex morphology is currently predigested as a single sphere for computational convenience.These complex morphologies of BC particles are commonly simplified to a single sphere with fixed volume-equivalent diameters (D BC = 2a 3  √ N s ), and their optical properties are calculated using the Lorenz-Mie-Debye theory (MIE).The advantage lies in its reduced demand for computer memory and simulation time, however, the excessively simplified basic hypothesis of this method leads to lower computed accuracy.
Recent studies indicated that the simplified assumptions of BC particle morphologies can have a remarkable impact on their optical estimations, leading to large uncertainty in assessments of aerosol radiative forcing [6][7][8].To obtain accurate optical properties of BC aerosols, researchers have employed more realistic models that account for varied particle sizes, chemical components, mixing states and emission sources [9][10][11][12].Previous studies suggested that the scattering errors of the single sphere model using the MIE method may reach to ∼50% [13][14][15].Recently, substantial improvements have been made in the optical modeling of BC particles by incorporating their fractal aggregate morphologies using the multi-sphere superposition T-matrix method (MSTM) and the discrete dipole approximation (DDA).In general, DDA codes provide greater flexibility in modeling particle morphologies while the currently available MSTM codes provide much higher computational efficiency [16,17].The concept of a T matrix was introduced by Waterman in 1971 and has been widely used in studies of electromagnetic scattering and absorption by morphologically complex particles [18].Unlike the approximate methodologies such as DDA, the MSTM is a direct computer solver of the frequency-domain macroscopic Maxwell equations and renders solutions that are both numerically exact and highly efficient [19].The MSTM code developed by Mackowski has extended the formulation to arbitrary configurations of spherical domains wherein any of the spheres can be located at points that are either internal or external to the other spheres, thus it is applicable for both the bare and coated BC particles [20,21].
In current climate models, researchers typically parameterize BC spatiotemporal diversities by considering particle size distributions of aerosol ensembles.Optical properties of BC aerosol ensembles are constructed by the integration of optical properties of the individual BC particles with varied D BC according to the lognormal size distributions of previous observations [22][23][24][25].Single scattering properties of BC particles are calculated for dozens of D BC ranging from 40 to 400 nm.These properties are then integrated with the given size distributions of entire aerosol ensembles.For current modeling of BC aerosol ensembles employing the MIE method, the computation time is negligible.However, it becomes a non-negligible factor when dealing with aggregate models utilizing the MSTM code.For example, the MSTM code requires approximately 100 times more computation time compared to the MIE code for the optical calculations of BC particles with D BC = 300 nm and a = 20 nm, and these time consumptions will exponentially increase for those larger BC particles.The time-consuming properties of these aggregate models, despite offering higher accuracy, significantly limits their practical applications.Therefore, it is necessary to shorten the calculation time and improve the applicability of the aggregate models with a relatively small loss of accuracy.
We present a novel method to dynamically regulate the BC fractal aggregate morphologies in optical modeling by deploying the maximum values of monomer number (N s, max ) and monomer radii (a max ).Basic idea is that it maintains the volumeequivalent diameters (D BC ) of the individual BC particles unchanged, and to modify the monomer radius or monomer number individually.We can simplify the morphology of BC particles continuously between the single sphere model and the aggregate models by the constraint of N s, max and a max .Especially, the widely used sphere model is characterized by N s, max = 1, while the original aggregate model has no limit as N s, max = N s .The goal is to identify suitable values for N s, max and a max that achieve a balance between shorter calculation times and smaller loss of accuracy.

BC dynamic models constrained by N s, max and a max
For a given volume-equivalent diameter (D BC ) of BC particles, their monomer number (N s ) and monomer radius (a) are related (N s = (D BC /2a) 3 ) in the modeling.Figure 1(b) illustrates how the particle morphology simplifications of observed BC depend on the maximums of monomer number (N s, max ) and monomer radius (a max ).The models with N s, max = N s are those BC particles with normal monomers fully estimated by the given volumeequivalent diameters (D BC ) and monomer radius (a), whose N s is always less than or equal to the given N s, max .In the presented BC dynamic model, when the calculated N s is larger than the given N s, max , then we set a new value of monomer number N ′ s equals N s, max , and the monomer radius is increased to a new value of monomer radius a ′ , as follows. if The adjusted monomer radius (a ′ ) on demand according to N s, max is limited by N s, max ⩾ (D BC /2a ′ ) 3 .In the example of BC particles with D BC = 400 nm and a = 20 nm, it is obvious that using a huge value such as N s, max ⩾1000 leads to a relatively tedious computation time, especially for the single BC particles with more than 1000 monomers.
If we assume N s, max to 500, the above BC particles are simulated to be those with D BC = 400 nm, N ′ s = 500, and a ′ = 25.2 nm.According to this morphological simplification, we investigate the typical values of N s, max across aggregate models, ranging from 10 to 1000.The values of a ′ are adjusted to be 92.indicated that the radii of BC monomer varied within the range of 10-25 nm with larger values of up to 40-50 nm in large pool fires [26][27][28].In this way, the simplifications where N s, max ⩽ 50 and a ′ > 50 nm may struggle to accurately reflect the real situation of BC aerosols.Further, the widely used sphere model can be described as representing the BC particles with N s, max = 1 and a max = D BC /2.The corresponding morphological parameters for the above BC particles are D BC = 400 nm, N ′ s = 1, and a ′ = 200 nm.As shown in figure 1(b), the morphological simplifications can be designed for the smaller N s, max in BC modeling.Accordingly, the application of a max is similar to the N s, max , while a max increases for a simpler particle morphology.
To qualify the effect of N s, max and a max on optical properties of BC particles, we investigate the major cases across varied D BC ranging from 40 to 400 nm.Bare BC particles tend to be mixed with other aerosol components through atmospheric processes such as coagulation and condensation over a period ranging from several hours to a week.In these simulations, the fully coated BC particles are assumed to have all BC monomers of the aggregate models embedded in a non-BC spherical host, while they are simplified to be the single core-shell sphere models for N s, max = 1.Volume fraction of pure BC component in a single coated BC particle (F BC = V BC / (V BC + V non-BC )) functions as an indicator of the degree of BC aging [29].Bare BC is a particle with a BC volume fraction of 1, while that of the fully coated BC is assumed to be 0.01.The partly coated and semi-embedded states of BC mixtures are not considered in this study, and they will be conducted in future research.The assumption made regarding the non-BC hosts of the coated BC particles is that they are single spheres, and their diameters remain unchanged with N s, max and vary with D BC because of the constant value of F BC .The fractal dimensions (D f ) of bare and coated BC particles typically vary from 1.8 to 2.8 maintaining a constant fractal prefactor (k 0 ) of 1.2.Previous studies suggested that BC structure becomes more compact after aging but their sizes are unchanged [30,31].In this study, it is assumed that D f for bare and fully coated BC particles are assigned values of 1.8 and 2.8, respectively.Refractive Y Wu et al index of a BC aerosol component was assumed to be 1.95 + 0.79i in the visible and infrared ranges.The non-BC coatings are assumed to be organic materials, and their refractive indices stayed at a constant value of 1.55, considering the assumption of a single non-BC component [32,33].

Optical simulations of the aggregated BC ensembles
Optical properties of BC aerosol ensembles were calculated for individual BC-containing particles varied with N s, max and a max using the MSTM method.This method uses numerically exact solutions of Maxwell's equations, which can be used to calculate the T-matrix descriptions of light scattering from the aggregated BC-containing particles with an appropriate superposition technique.The field arriving at the surface of the ith sphere will consist of the incident field plus scattered fields that originate from all other spheres in the cluster.By use of the addition theorem for vector harmonics these interacting scattered fields can be transformed into expansions about the origin of sphere i, which makes possible an analytic formulation of the boundary conditions at the surface.Thereby, we can analytically obtain the randomorientation cross sections of extinction (C ext ), scattering (C sca ) and absorption (C abs = C ext − C sca ) [34,35].The single scattering albedo (SSA) was described as the ratio of the scattering cross section and the extinction cross section (SSA = C sca /C ext ).Absorption enhancement (E abs ) in atmospheric processes after emissions is determined by the ratio of light absorption of coated BC particles to that by all bare particles (E abs = C abs (Coated)/C abs (Bare)).
In the simulations, optical properties of bare and coated BC particles are computed for the D BC ranging from 40 to 400 nm with intervals of 20 nm, which covers the typical size distributions observed in previous studies.Incident wavelengths (λ) were simulated to cover both the visible and infrared ranges, namely, 440, 532, 670, 865, and 1064 nm.We investigate 12 values of N s, max , namely, 1, 10, 20, 30, 50, 70, 100, 150, 200, 300, 500, and 1000.In the context of the simulated BC particles, the assumption of N s, max = 1000 can be regarded as N s, max = N s without any morphological simplifications.When N s, max = 1, the simulation simplifies BC particles to single spheres.We also investigate 4 values of a max covering the major cases, namely, 20, 30, 40, and 50 nm.Figure 1 depicts that the fewer N s and larger a lead to simpler particle morphologies of bare BC with D BC = 200 nm, introducing larger relative deviations on C sca at 532 nm.As shown in figure 1(c), the BC C sca at the rightmost with N s, max = N s is gradually simplified to the leftmost with N s, max = 1.The corresponding relative deviations of BC C sca with a = 20 nm is shown in figure 1(a), indicating up to ∼140% errors of C sca at 532 nm by single sphere assumption.The purple line with circles in figure 1(c) is the BC C sca without a max assumptions, while the other lines are those with larger a.As shown in figure 1(d), the assumption of larger a may lead to ∼90% errors on BC scattering simulations.Optical calculations of BC particles are significantly influenced by these morphologically simplifications, therefore it is important to select the appropriate assumptions for optical simulations of BC aerosol ensembles.
The computation time is tested by a ThinkStation P920 computer equipped with a Xeon 5218 CPU, and 128GB DDR4 RAM, while the minimum time unit for measurement is minutes.The accuracy of BC optical modeling is described by the relative deviations (RD) of those above optical parameters between the original aggregate models with N s, max = N s (a max = a) and their simplifications with smaller N s, max (larger a max ).For example of C sca , as follows, where N t is the given N s, max , while the other optical parameters remain consistent.The relative deviations caused by the assumption of a max is similar.
The optical properties of entire aerosol ensembles can be computed by integrating optical properties of the individual BC particles with different D BC and the given size distributions, for example of scattering cross sections ⟨C sca ⟩, as follows, According to the observations, the normalized number distributions (N(D BC )) of BC size are typically fitted to be lognormal [32], where σ is the geometric standard deviation and D m is the geometric mean volume-equivalent diameter of the pure BC component in the BC aerosol ensembles.
In this study, we investigate the simulation time and accuracy of different values of N s, max and a max for various BC particle morphologies and mixing states.Further, the optimal values of N s, max and a max for optical modeling of BC aerosol ensembles are discussed, considering different accuracy requirements alongside their corresponding computational costs.Some strategies using the dynamic model are suggested for higher accuracy and faster calculation of optical simulations of BC aerosol ensembles.

Optical simulations of BC particles by dynamic model
Single scattering properties of BC particles are remarkably influenced by their volume-equivalent diameters (D BC ) and max monomer numbers (N s, max ).As shown in figures 2(a) and (c), C sca and SSA of bare BC particles with D BC ranging from 40 to 400 nm are calculated at 532 nm using the models with different N s, max = 1, 10, 100, and 1000, respectively.The original aggregate model with N s, max = N s are morphologically simplified by the models with N s, max < 1000, and the cases of N s, max = 1 corresponding to the commonly used sphere model.The fractal dimensions of these loose aggregate models are constant to be 1.8, and the monomer radii are varied for the fixed D BC and N s, max .Figures 2(b The values of BC C sca increase with larger size, however, this variation is largely influenced by the N s, max .Compared to the aggregate models, the increase in C sca calculated using the sphere model with N s, max = 1 is relatively smaller, which agree with the previous studies [36,37].The sphere model may overestimate BC scattering for smaller particles and underestimate it for larger particles.This uncertain variation poses large challenges in BC optical modeling.The simulations also indicated that the scattering variations calculated using the simplified aggregate models with less N s, max are approximately consistent with the model with N s, max = N s .Thus, maintaining their aggregate morphologies in modeling may effectively reduce simulation errors. As shown in figure 2(b), the sphere model potentially introduce errors up to ∼140% in the simulations of C sca , and these errors are reduced to ∼20% when employing the aggregate model with N s, max = 100.The relative deviations between the aggregate and sphere models increase from D BC = 100 nm to D BC = 200 nm and then decrease with the augment of D BC .Previous simulations have reported the similar variations in relative deviations due to BC monomer numbers [14].The monomer numbers of those simulated BC particles are N s = 16, 125, 422, and 1000 for the cases of D BC = 100, 200, 300, and 400 nm, respectively, assuming their monomer radii are a = 20 nm.The possible reason is that the multiple scattering of BC monomers varies with their numbers in a single aggregate, which is negligible in the sphere model or aggregate model with quite less monomers.Moreover, while the multiple scattering of these BC monomers may remarkably enhance the light scattering of the aggregates, its impact is relatively weakened due to a small amount of monomers for the BC particles of D BC = 100.The direct scattering of BC particles with enough monomers becomes relatively larger than the multiple scattering, in which cases the relative deviations caused by their morphologically simplifications are likely to be constrained.The peak volume-equivalent diameter of the pure BC component in the BC aerosol ensembles (D m ) fall within the range of 150-220 nm by previous observations [22][23][24][25].It is suggested to consider those large errors of C sca due to morphologically simplifications for those BC particles with D BC ≈ 200 nm in the optical modeling of BC aerosol ensembles.This variation is also influenced by the incident wavelength in BC optical calculations.Figure S1 shows that these relative deviations of C sca caused by morphological simplifications tend to increase with larger incident wavelengths.When the incident wavelength is 440 nm, the scattering calculations of BC particles with D BC = 100 nm are most seriously influenced by the model simplifications depend on N s, max , while this effect varies for BC particles with D BC = 300 nm at a wavelength of λ = 1064 nm.
Figure 2(c) shows that BC SSA increases for those larger particles, and the greater simplifications with less N s, max may be more overestimated.The averaged values of SSA range from 0.2 to 0.4 at 532 nm, aligning with previous studies [14,38].The SSA rapidly increases as BC particles become larger, which continues until the SSA reaches a stable value of ∼0.45 when D BC > 200 nm.The RD variations of SSA are similar to the scattering properties, but the values exhibit a slight decrease due to the relatively smaller errors on light absorption investigated in previous studies [39,40].Moreover, figures 2(b) and (d) illustrate that the N s, max can be regarded as a useful parameter for those simplified models gradually approaching the true values of optical properties.The absorption properties of the fully coated BC particles may also be underestimated by those simplified models, and their errors arising from morphologically simplifications are remarkably enhanced by less N s, max , as shown in figure S2.

Effects of N s, max and a max on BC optical simulations
Figure 3 demonstrates that the relative deviations of BC optical properties due to N s, max are also dependent on the incident wavelength.The relative deviations between the original aggregate model and the morphologically simplifications are individually calculated for D BC values ranging from 40 to 400 nm with intervals of 20 nm, and further averaged for each incident wavelength.As the incident wavelength increases, the scattering errors of bare BC particles resulting from morphological simplification tend to escalate, while those of BC absorption diminish.The RD of C sca between the sphere and aggregate model is ∼45% at 440 nm and it grows to ∼200% at 1064 nm.Moreover, lager N s, max may lead to smaller errors of BC optical simulations, within the visible and near-infrared range.Similar variations are investigated in previous studies between the single sphere model with N s, max = 1 and the original aggregate model with N s, max = N s (it is 1000 in this case) [41,42].As shown in figure 3(a), the scattering errors of the sphere model with N s, max = 1 are ∼200% at 1064 nm and these errors decease to ∼150% and ∼70% for the aggregate models with N s, max = 10 and N s, max = 100, respectively.These significant diversities on BC scattering estimations may lead to large uncertainties in the assessment of radiative forcing.Figure 1(b) depicts that the RD of C abs at larger incident wavelength is slightly deceased.Especially, the C abs errors of coated BC calculated using the aggregate models with N s, max > 10 are negligible (<5%) in the range of visible and near-infrared.Generally, the calculation time of the superposition T-matrix method tends to increase with a higher number of monomers in the aggregate models.Thus, in practical application of BC optical simulations, it is necessary to select the minimum monomer number within an acceptable accuracy range to achieve time-saving effects.
Figure 4 depicts that a minor reduction in accuracy leads to a decrease in computation time.The results of BC C sca at 532 nm are compared in figure 4(a) for different morphologically simplified models, including the single sphere model with N s, max = 1, the aggregate model with N s, max = N s /10 = 100, the aggregate model with a max = 2a = 40 nm, the dynamic model with a general assumption (a max = a = 20 nm for 0 < D BC ⩽ 200 nm, a max = 30 nm for 200 < D BC ⩽ 300 nm, a max = 40 nm for 300 < D BC ⩽ 400 nm), and the original aggregate model with N s, max = N s and a max = a.The relative deviations between these morphologically simplified models and the original aggregate model are shown in figure 4(b).The results indicated that the single sphere model may lead to >80% simulation errors in the optical properties of BC particles with D BC ranging from 120 to 220 nm.Moreover, C sca calculations within this range of D BC are also overestimated ∼60% due to the assumption of a larger monomer radius with a max = 2a = 40 nm.For those BC particles with D BC larger than 300 nm, the single sphere may underestimate their scattering properties by <25% relative errors.Both the aggregate model with N s, max = N s /10 = 100 and the The computation time of the aggregate model with N s, max < 200 is only 10% of the original aggregate model with N s, max = N s .Therefore, it would be helpful to optimize the parameters of N s and a using the dynamic model to achieve a balance between accuracy and time of optical simulations of BC aerosol ensembles.

Selection of N s and a for BC aerosol ensembles
Figure 5 demonstrates that optical properties of BC aerosol ensembles can be simulated with a slight accuracy loss and significantly reduced computation time using the dynamic model with suitable values of N s and a.The optical properties of BC aerosol ensembles are integrated by the single scattering properties of the individual BC particles with different D BC according to those size distributions observed by previous studies, as shown in figure S4 [22][23][24][25].The peak volume-equivalent diameter of the pure BC component in the BC aerosol ensembles (D m ) ranges from 100 to 200 nm.The total computation time of these BC aerosol ensembles using the original aggregate model is ∼4800 min.Figure 5 shows that the averaged optical errors of these BC aerosol ensembles simulated by the single sphere model with N s, max = 1 can reach up to ∼80%, while the computation time is less than 5 min.The optical results calculated using the aggregate model with fixed BC N s, max or a max may lead to 20%-30% errors and ∼150 min of computation time.We provide two strategies of the dynamic model, one focuses on accuracy, the other on time.The dynamic model with higher accuracy simulate ∼5% errors on optical properties of BC aerosol ensembles with ∼300 min, which is constructed by different assumptions as a max = a = 20 nm for 0 < D BC ⩽ 240 nm, a max = 30 nm for 240 < D BC ⩽ 320 nm, and a max = 40 nm for 320 < D BC ⩽ 400 nm.For faster calculation, the assumptions with a max = 20 nm for 0 < D BC ⩽ 200 nm, a max = 30 nm for 200 < D BC ⩽ 260 nm, a max = 40 nm for 260 < D BC ⩽ 320 nm, and a max = 50 nm for 320 < D BC ⩽ 400 nm are performed in the dynamic model.Its computation time is reduced to ∼80 min, however, its simulation error is increased to ∼12%.For different applications of the optical calculations of BC aerosol ensembles, it is suggested that the dynamic model with the appropriate amount and radius of monomers may achieve a balance between accuracy and time.
The current methodology for calculating the optical properties of BC aerosol ensembles in the atmosphere relies on the single sphere model, which offers fast computation but low accuracy, leading to a large uncertainty in the climate assessments.One major reason is that the applications of the aggregate models with higher accuracy are greatly constrained by their time-consuming.To solve this problem, a dynamic model considering BC N s, max and a max in aerosol ensembles has been introduced, which is incorporated into the state-of-the-art aggregate models, and specifically applied in optical calculations that involve varied size distributions of BC aerosols.The complex morphologies of BC particles are simplified by the artificially increasing the radii of BC monomer, and thus the monomer numbers in a single aggregate are reducing to a suitable range of calculation time.The results indicated that this dynamic model retains multiple scattering between BC monomers to some extent, despite a slight loss of accuracy.The advantage lies in the significant reduction of computation time.The simulation accuracy of the presented dynamic model with suitable values of N s, max and a max may achieve ∼95% while their computation time may reduce to ∼6%.This method enables extensive optical calculations of BC particles with complex morphologies, which is useful for the optical modeling of BC aerosol ensembles.
The original aggregate model can be improved by the dynamic model with suitable parameterizations of N s, max and a max .Previous microscopic observations indicated that BC monomer radii ranged from 10 to 25 nm under normal conditions and can reach 40-50 nm in large pool fires, with monomers in a single BC particle ranging from 10 to 1000.Such a large value range is not conducive to construct an optical model of BC particles.The restrictions on the number and radius of monomers within a single BC particle could serve as a useful feature for indicating different emission sources and aging processes.Thus, more observations of microscopy and optical properties of BC aerosol ensembles are necessary to understand the relationship between the monomer numbers and the emission states.Quantitative analysis indicates that optical properties of BC aerosol ensembles are strongly influenced by the N s, max and a max , leading to a flexible and effective assessment in the global climate.

Figure 1 .
Figure 1.Comparisons of scattering cross section (Csca) of the fractal aggregate BC particles and their morphologically simplified models.The complex particle morphologies of bare BC with DBC = 200 nm and a = 20 nm can be simplified for both fewer monomer numbers (Ns) and larger monomer radii (a).The max monomer number (Ns, max) of these morphologically simplified models are Ns, max = 1, 10, 20, 30, 50, 100, 150, 200, 300, 500, and 1000 (Ns of the original aggregate model).The assumption of max monomer radii (amax) are 50, 40, 30, and 20 nm.(a) Relative deviation (RD) of Csca due to variations of Ns, max, (b) The morphologically simplified method for the fractal aggregate BC particles, bridging the sphere model and the aggregate model.Note that a indicates the original monomer radius and a ′ is the used monomer radius for modeling.(c) Csca at 532 nm of the fractal aggregate BC particles and their morphologically simplified models.X-axis is the variations of Ns, max and different lines are different assumptions of a.(d) Relative deviation (RD) of Csca due to variations of a.

Figure 2 .
Figure 2. Scattering cross section (Csca) and single scattering albedo (SSA) of bare BC particles calculated using the models with different max monomer numbers (Ns, max).(a) Csca of bare BC particles at 532 nm are calculated using the models with Ns, max = 1, 10, 100, and 1000 (Ns of the original aggregate model) for those with volume-equivalent diameters (DBC) ranged from 40 to 400 nm.The model with Ns, max = 1000 is simulated for the normal monomers as Ns, and the other models are simplified for less monomers and less computation time.(b) Relative deviations of Csca at 532 nm due to model simplifications depend on Ns, max, and BC particles with DBC = 100, 200, 300, and 400 are investigated.(c) Similar to (a) but for the values of SSA.(d) Similar to (b) but for RD of SSA.
) and (d) show relative deviations (RD) of C sca and SSA between the simulations of the original aggregate model and their simplifications with less N s, max for different D BC = 100, 200, 300, and 400 nm, respectively.

Figure 3 .
Figure 3. Relative deviations (RD) of BC optical properties between the general model with normal monomers (max monomer number Ns, max = 1000 as Ns) and the simplified models with artificially reduced monomers (Ns, max = 1, 10, and 100) varied with different incident wavelength, namely, 440, 532, 670, 865, and 1064 nm.(a) Relative deviations of scattering cross sections (Csca) of bare BC particles.(b) Relative deviations of absorption cross sections (C abs ) of coated BC particles.

Figure 4 .
Figure 4. accuracy and computation time (unit: minute) of the single BC particles with different volume-equivalent diameters (DBC) ranged from 40 to 400 nm and different morphologically simplified models, including the single sphere model with Ns, max = 1, the aggregate model with Ns, max = Ns/10 = 100, the aggregate model with amax = 2a = 40 nm, the dynamic model with segmented assumptions (amax = a = 20 nm for 0 < DBC ⩽ 200 nm, amax = 30 nm for 200 < DBC ⩽ 300 nm, amax = 40 nm for 300 < DBC ⩽ 400 nm), and the original aggregate model (a = 20 nm and Ns = (DBC/2a) 3 ).These results are applied for constructing the optical properties of BC aerosol ensembles.(a) Comparisons of scattering cross section of BC particles at 532 nm calculated by these models.(b) Relative deviations (RD) of the scattering properties of BC particles due to their morphologically simplifications in optical modeling.(c) Total calculation time for optical integrations of BC aerosol ensembles (40 DBC ⩽ 400 nm) due to different assumptions of Ns, max.Blue line is labeled as ∼10% time of the original aggregate model.(d) The individual calculation time for different DBC and Ns, max using the original aggregate model.

Figure 5 .
Figure 5. Comparisons of optical calculation time and accuracy of BC aerosol ensembles between different models and strategies.From top to bottom (No. 1-6), the models are the original aggregate model without morphologically simplification (a = 20 nm and Ns = (DBC/2a) 3 ), the dynamic model with higher accuracy (amax = a = 20 nm for 0 < DBC ⩽ 240 nm, amax = 30 nm for 240 < DBC ⩽ 320 nm, amax = 40 nm for 320 < DBC ⩽ 400 nm), the dynamic model with faster calculation (amax = 20 nm for 0 < DBC ⩽ 200 nm, amax = 30 nm for 200 < DBC ⩽ 260 nm, amax = 40 nm for 260 < DBC ⩽ 320 nm, amax = 50 nm for 320 < DBC ⩽ 400 nm), the aggregate model simplified with amax = 2a = 40 nm, the aggregate model simplified with Ns, max = Ns/10 ⩽ 100 (Ns = 1000 for the largest BC particles in optical modeling of the aerosol ensembles), and the single sphere model.The left subfigure is the sketch map indicating the selection of Ns and a in BC size distributions.The middle subfigure is the optical calculation times of BC aerosol ensembles using different models and strategies, and the right subfigure is their simulation accuracy.