## Abstract

Absorption cross section (*C _{abs}*), scattering cross section (

*C*) and asymmetry parameter (

_{sca}*ASY*) of soot particles in different atmospheric aging status were investigated under fixed equivalent volume radius (

*R*) using the numerically exact multiple-sphere T-matrix method. The radiative properties of soot particles would be largely diverse in different aging status even

_{V}*R*is fixed. However, there are many insensitive parameters under different aging status. The

_{V}*C*and

_{abs}*ASY*is insensitive to monomers number (

*N*) when

_{s}*N*is larger than a threshold value. For bare and thinly coated soot aggregates,

_{s}*C*is insensitive to fractal dimension (

_{abs}*D*) when the

_{f}*R*is small, where the relative errors of

_{V}*C*for different

_{abs}*D*are within 2.5%. However, the effects of

_{f}*D*is obvious for large soot due to the shielding effects of large monomers, and the relative errors for different

_{f}*D*can reach to 18% for bare soot. For thinly coated soot, the changes of

_{f}*ASY*with soot volume fraction (

*f*) is small due to the little changes of the fractal structure when the

_{soot}*R*is fixed. In addition, for thickly coated soot,

_{V}*ASY*is insensitive to

*N*due to the unchanged overall spherical structure. Our results give a further understanding of the influences of morphology on radiative properties. It may be helpful for model selection and model simplification.

_{s}© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

As a product of incomplete combustion, soot is the major aerosol that absorbs solar radiation. Detailed knowledge of the radiative properties of soot particles is significant for climate studies [1–3], remote sensing [4, 5] and fire detection [6, 7]. However, there are still large uncertainties in understanding radiative properties of soot particles [8,9].

The radiative properties of soot particles are significantly influenced by the complex morphological factors, such as fractal structure, monomer character and mixing state. Studies have shown that the freshly emitted soot particles are composed of numbers of soot monomers [10]. In addition, the morphology of soot particles varies with different combustion conditions, fuel species and atmospheric aging status. Studies have been conducted to investigate the effects of morphology on radiative properties of fractal-like soot agglomerates [11–13]. However, the soot particles tend to be mixed with other chemical components, which leads to more complex morphology [14,15]. Mie theory [16], a method to calculate the radiative properties of a homogeneous sphere, is commonly applied in climate studies [15, 17, 18]. However, based on situ measurements on urban plumes, the radiative absorption of soot particles has been overestimated in models [14, 19, 20]. The influences of morphology on the radiative properties are still unclear until far.

Although many studies have concerned on the influences of morphology on the radiative properties of soot particles [21–25], to the best of our knowledge, few studies investigate the influences under a fixed particle size. As a matter of fact, this issue is also important. The size distribution of soot particles has great impact on human health [26] and environment [27]. In the climate model, the radiative effects of soot particles are calculated under a given size distribution. The morphology of soot particle is commonly assumed to be spherical, which can be calculated using the Mie theory. However, the radiative properties of soot particles significantly depend on the morphology and mixing status. Therefore, given the size distribution, the calculated radiative effects using Mie theory can introduce large errors.

Nevertheless, it isn’t meant that we should put a morphologically more realistic model to applications due to high computational cost. For example, whether the radiative properties can be calculated using an equivalent volume model with less monomer number than the real morphology is unclear. It is very difficult to reconstruct all the elements of radiative properties with a simplified model. However, it is possible to find a simplified model for a specified problem after knowing the effects of the morphology. To give suggestions to find a relative simple equivalent volume model for the specified problem, there is a need to investigate the effects of morphologies under different fixed equivalent volume radius. In addition, it is also necessary to investigate the sensitivities of morphological parameters on the correlation between radiative properties and particle size.

Although Kahnert [28] has conducted a sensitivity study of the correlation between particle size and radiative properties for bare soot, the monomers radius are among 10–25nm. In physical insights, it is significative because this range is commonly observed in reality. However, in application insights, we are more focused on whether the actual morphology can be simplified as an equivalent volume model with less monomers number, such as a single sphere, which may be not appeared in atmosphere. Therefore, we extended the sensitivity study to morphologies whose monomer radius is not appeared in reality. In addition, the sensitivities of coated soot has not been studied. In this work, we investigated the influences of morphology on the radiative properties of soot particles under fixed particle sizes at first. Then sensitivities of morphologies to radiative properties were analyzed. Recently, the integral optical characteristics has gained increasing attentions [29, 30] due to their essential contributions to calculations of the global radiation forcing. In climate model, bulk *C _{abs}*,

*C*, and

_{sca}*ASY*are commonly calculated by cumulative sum of single particles radiative properties based on a given size distribution. To give suggestions for modeling application,

*C*,

_{abs}*C*and

_{sca}*ASY*were investigated.

The aims of this work is to demonstrate the sensitivity of morphological parameters to radiative properties and provide suggestions for model selection and model simplification. The findings should improve our understanding of the morphological effects and influences of particle size on the radiative properties.

## 2. Methodology

#### 2.1. Generation of soot particles with different morphologies

Studies have shown that freshly emitted soot particles generally present fractal characteristics [31, 32]. The construction of the structure can be described by the well-known fractal laws [17]:

where*N*is the number of the monomers in the cluster,

_{s}*a*is the mean radius of the monomers,

*k*

_{0}is the fractal prefactor,

*D*is the fractal dimension,

_{f}*R*is the radius of gyration, and

_{g}*l*is the distance from the

_{i}*i*th monomer to the centre of the cluster.

Diffusion-limited algorithms (DLA), including particle-cluster aggregation (PCA) [33] and cluster-cluster aggregation (CCA) methods [34], are developed to generate fractal-like aggregates. However, the tunable algorithms are more commonly applied due to the adjustable parameters and fast implementation [35, 36]. A tunable DLA code developed by Wozniak et al. [37] was applied to generate soot aggregates in this work. Different from ordinary DLA code, it preserves fractal parameters at each step of the aggregation, so it can avoid the generation of multi-fractal aggregates [38].

Freshly emitted soot particles tend to be coated with other chemical components in the atmosphere by the coagulation and condensation of secondary aerosol compounds [39–41]. The closed-cell [42–44] structure is an example of where coating material not only covers the outer layers of soot aggregates but also fills the internal voids among primary spherules [45], which can greatly represent the thinly coated soot. However, thickly coated soot are commonly represented by a structure where soot aggregates are completely embedded in a sphere [46, 47]. In order to reflect the influences of morphology on the radiative properties in the whole atmosphere aging process, 3 types of morphologies were considered in this study: (a) bare soot particles, represented by fractal aggregates; (b) thinly coated soot particles, which composed of concentric monomers; (c) thickly coated soot particles, where the soot aggregates are embedded into other chemical components. The typical soot morphologies are shown in Fig. 1. Even though the real morphologies may be more complex, the morphologies considered in this study can greatly reflect the real morphologies and are more likely to be put into climate model. We generated the bare soot aggregates directly by tunable DLA algorithm. For thinly coated soot, the non-absorbing shell were generated by tunable algorithm, then the soot monomers with identical center as non-absorbing shell were added. The details of generations of thinly coated soot were described by Wu et al. [44]. For thickly coated soot, the non-absorbing sphere are covered over the soot aggregates, as the study of Cheng et al. [24]. When investigating the dependences of radiative properties on particle sizes for different fixed morphologies, we preserved the morphologies for different particle sizes by maintain the relative positions of monomers identical. We generated different morphologies under fixed equivalent volume radius by the way shown in Fig. 2. In this process, we keep the equivalent volume radius of soot-containing particles identical. The soot monomers radius were calculated using:

where*a*

_{soot}_{_}

*and*

_{bare}*a*

_{soot}_{_}

*represent the soot monomers radius of bare and coated soot aggregates respectively. When the equivalent volume radius is fixed, the soot monomers radius decreases by increasing*

_{coated}*N*and reducing soot volume fraction. The morphological parameters assumed in this study are shown in Table 1.

_{s}#### 2.2. Multiple-sphere T-matrix method

Recently, numerically exact multiple-sphere T-matrix (MTSM) method [48, 49] has been developed to calculate the arbitrary configurations of spheres without overlapping. Compared with other numerical methods, MSTM method calculates the radiative properties of randomly oriented particles analytically without numerical averaging over particle orientations, so it can calculate the radiative properties of particles with high efficiency. In this study, the MSTM version 3.0 [50] was used in this work. However, the calculations are based on the concept of random orientation particles, which is mathematically defined by Mishchenko et al. [51]. In atmosphere, it is reasonable to assume that the possibility of each particle direction is identical, which rigorously satisfies the definition of random orientation. The spheres positions which generated by tunable DLA algorithm are initialized as the input of MSTM version 3.0 at first. Combined with other parameters, such as refractive index, size parameter, and error tolerance, the absorption, scattering and extinction efficiencies and *ASY* can be obtained from the output file. Due to the fact that there are different configurations satisfy identical fractal law, therefore, the calculation results may be varied over a small range. Many studies attempted to reduce the deviation caused from random clusters by averaging the results over multiple realizations. For example, Wu et al. [52] averaged the radiative properties over 10 realizations and Dong at al. [53] averaged the radiative properties over 5 realizations. In this work, random-oriented radiative properties were averaged over 10 realizations to reflect its general properties.

## 3. Results

#### 3.1. Radiative properties of soot particles under fixed R_{V}

The radiative properties of soot particles were investigated for a visible wavelength at 0.55*um*, and the refractive index was assumed to be 1.95 + 0.79*i* [54]. The coatings were assumed to be organic carbon with a refractive index of constant 1.55 according to Chakrabarty et al. [55].

Figure 3 shows the radiative properties of bare soot particles under fixed *R _{V}*. It indicates that the optical properties can be diverse even the

*R*is fixed.

_{V}*C*decreases with

_{abs}*N*and increases with

_{s}*D*for small particles but the opposite phenomenon is observed for large particles. When the particle size is small, absorbing material is fully exposed to the electromagnetic field, more compact morphology and less monomers number can result in growing contact among absorbing spheres, therefore, lead to larger electromagnetic field interactions among absorbing materials. As a result,

_{f}*C*decreases with

_{abs}*N*and increases with

_{s}*D*. While for large soot, given

_{f}*N*, monomers radius is large. Therefore, the electromagnetic field is not able to penetrate deeply by the shielding effect of the outer layer of monomers. The increases of

_{s}*N*and more fluffy morphology allow more absorbing materials exposed to light. Thus, larger

_{s}*C*is observed.

_{abs}The variation of *C _{sca}* is also dependent on particle size. For small soot, intensifying scattering interaction between the spherules is observed as the soot aggregates become more compact. The results are consistent with the study of Liu et. al [21]. In addition, less

*N*also leads to larger

_{s}*C*as the aggreagtes also turn more compact when

_{sca}*N*is decreased. However, for large soot aggregates, when

_{s}*N*is extremely small,

_{s}*C*may be increased with

_{sca}*N*due to decreasing shielding effects of huge monomers.

_{s}*ASY* increases with *N _{s}* when

*N*is less than a threshold value but changes slowly for more

_{s}*N*. There are deferent dependences on

_{s}*D*for different particle sizes. For small soot, the lower

_{f}*D*result in lager

_{f}*ASY*due to more asymmetrical structures. However, the contrary effects of

*D*are observed for large soot, which may be caused by the shielding effect of large monomers.

_{f}Figure 4 shows the influences of monomers number on the radiative properties of thinly coated soot particles under fixed *R _{V}*. Thinly coated soot shares nearly the same dependence on primary particles number as that of the bare soot. This may be due to the fact that the thinly coated soot particles still present fractal characteristics. Differently, when

*N*is more than a value,

_{s}*C*is always to decrease with

_{abs}*N*due to the fact that the actual absorbing material size is small compared to bare soot.

_{s}Figure 5 demonstrates the variations of radiative properties with *f _{soot}* for different

*D*. With different

_{f}*f*, the relative contents of soot are varied when

_{soot}*R*is fixed. It deserves to be investigated for the reason that the information about the relative contents of coatings and soot is commonly unclear. Therefore, it is significative to understand how the contents of coatings influence the optical properties when the total particle size is known. Given identical

_{V}*R*, larger

_{V}*f*leads to stronger

_{soot}*C*due to increases of absorbing materials.

_{abs}*C*is increases with

_{sca}*f*. It indicates that non-absorbing materials may cause less scattering interaction compared to absorbing materials with identical volume. The dependence of

_{soot}*ASY*on

*f*is very small, this is due to the fact that the fractal structure is not changed.

_{soot}Figure 6 shows how the radiative properties depend on *f _{soot}* for different

*N*. When fixing

_{s}*R*, the radius of primary particles changes with

_{V}*N*. The results show there are nearly identical variations with

_{s}*f*for different

_{soot}*N*. However, the effects of

_{s}*N*are influenced by

_{s}*f*. For small particles, the effects of

_{soot}*N*on

_{s}*C*is small due to the fully exposure to light. However, there are a little increases when increasing

_{abs}*N*. The reason lies that less

_{s}*N*could result more compact structure, which results in more electromagnetic field interactions among absorbing spheres. While for large soot,

_{s}*C*increases with

_{abs}*N*when coatings is thin but decreases if the soot is further coated. This can be explained as below. For large soot where the monomer is large with a identical

_{s}*N*, the outer monomers will block the light deeply into monomers. More

_{s}*N*may result in less shielding effects due to the decreases of monomers radius, so lead to more electromagnetic field interaction among absorbing spheres when the coatings is thin. Therefore, larger

_{s}*C*is caused. However, when the soot are further coated, the increasing

_{abs}*N*leads to smaller

_{s}*C*. This may be due to the fact that the actual soot size is small.

_{abs}The effects of *N _{s}* on the radiative properties of thickly coated soot particles are shown in Fig. 7. Similar to bare soot and thinly coated soot, the effects of

*N*is relative small when

_{s}*N*is over a threshold value. Thus, it is feasible to simplify the thickly coated soot as a model with less

_{s}*N*but by no means core-shell sphere because the optical properties changes obviously with

_{s}*N*if

_{s}*N*is less than the threshold value.

_{s}#### 3.2. Sensitivity of morphologies on radiative properties

Figures 8–9 show the sensitivity of *D _{f}* on radiative properties of bare and thinkly coated soot aggregates. The symbol “

*R*” represents the relative errors of radiative properties of soot with different parameters and “&” is the abbreviation of “and”. For bare and thinly coated soot aggregates, the effects of

*D*on

_{f}*C*is small for small particles, where the relative errors among different

_{abs}*D*are within 2.5%. However, for large soot, the effects

_{f}*D*on

_{f}*C*is obvious. This phenomenon can be explained from physical insights. When the particle is small, the whole particle is nearly fully exposed to the electromagnetic field. Therefore, the changes of

_{abs}*D*leads little variation in electromagnetic interaction among absorbing materials. However, when the soot is large, given identical

_{f}*N*, the monomer radius is large. The soot is difficult to be fully exposed to the electromagnetic field. The changes of compact degree can result in large variation in shielding effects of large monomers, therefore, large variations of

_{s}*C*are observed. In addition, compared with bare soot, the effects of

_{abs}*D*on

_{f}*C*is smaller for thinly coated soot, where the relative errors between different

_{abs}*D*is below 9%. While for bare soot, the relative errors between

_{f}*D*= 2.2 and

_{f}*D*= 2.5 can reach to about 18%.

_{f}*C*and

_{sca}*ASY*is more sensitive to

*D*for small soot.

_{f}Figure 10 shows the sensitivities of *N _{s}* on radiative properties of bare and thinkly coated soot aggregates. For both bare and thinly coated soot, the effects of

*N*on

_{s}*C*is insensitive when

_{abs}*N*is over a threshold value. The relative errors of

_{s}*C*between

_{abs}*N*= 800 and

_{s}*N*= 500 as well as between

_{s}*N*= 500 and

_{s}*N*= 300 are below 1% for bare soot, and below 2.5% for thinly coated soot. The results are consistent with the study of Kahnert [28], who demonstrated that

_{s}*C*is insensitive to monomer radius when the monomer radius are within 15–25

_{abs}*um*. However, when

*N*is further decreased, the relative errors of different

_{s}*N*may be increased. For example, the relative error between

_{s}*N*= 100 and

_{s}*N*= 300 can reach to 5% when

_{s}*R*= 0.2784

_{V}*um*for bare soot. The reason may be due to the blocking effects of large monomers. For large soot, when the coatings are thin, the relative error between

*N*= 100 and

_{s}*N*= 300 is relative small compared to bare soot, which is below 2%. This may be due to the decreases of actual absorbing materials. However, when the soot is further coated, the relative errors are increased, which reach to 9%.

_{s}Relative errors of *C _{sca}* between

*N*= 800 and

_{s}*N*= 500 are below 5%. However, the relative errors between

_{s}*N*= 300 and

_{s}*N*= 100 can be above 15%. It demonstrate again that the

_{s}*C*is insensitive to

_{sca}*N*unless when the

_{s}*N*is above a threshold value. However, the

_{s}*ASY*is more sensitive to

*N*for small soot. This is because the main factor affects

_{s}*ASY*of large soot may be the shielding effects of large monomers.

Figure 11 shows the sensitivities of radiative properties to *f _{soot}* for thinly coated soot. The

*C*is always rather sensitive to

_{abs}*f*. Therefore, when given the total particle size, the contents of different components should be carefully considered.

_{soot}*C*is more sensitive to

_{sca}*f*for small particles. However,

_{soot}*ASY*is insensitive to

*f*, and the relative errors between different

_{soot}*f*are below 1.5%, which indicates the thinly coated soot can be calculated as a homogeneous structure for

_{soot}*ASY*.

Figure 12 shows the sensitivities of *N _{s}* on optical properties of thickly coated soot. The effects of

*N*on

_{s}*C*and

_{abs}*C*is small when

_{sca}*N*is large. For

_{s}*C*, the relative errors between

_{abs}*N*= 200 and

_{s}*N*= 300 and between

_{s}*N*= 100 and

_{s}*N*= 200 is below 1% and 2.5% respectively, and the relative errors of

_{s}*C*are also below 2.5%. However, when the

_{sca}*N*further decreases, the errors are large. For

_{s}*N*= 25 and

_{s}*N*= 100, the relative error of

_{s}*C*can reach to about 18% when

_{abs}*R*= 0.0464

_{V}*um*and the relative error of

*C*can reach to about 10% when

_{sca}*R*= 0.2784

_{V}*um*.

*ASY*is insensitive to

*N*due to the unchanged overall spherical structure, and the relative errors of

_{s}*ASY*for different

*N*are below 2.25%.

_{s}## 4. Summary and conclusions

In this study, a numerical investigation was conducted to understand the morphological effects on radiative properties of soot particles using MSTM method. We have investigated the morphological effects under different fixed *R _{V}*. We keep the

*R*identical, so the soot monomers radius changes for different

_{V}*N*and

_{s}*f*. Our results demonstrate that the effects of different morphologies on the scattering, and absorption properties of soot particles. The radiative properties are significantly affected by morphologies even the particle sizes are fixed. Therefore, we should consider the model errors when using simplified model. However, at a specified range, there are many insensitive parameters under different aging status.

_{soot}For soot in different aging status, under fixed size, the effects of *N _{s}* is insensitive on

*C*when

_{abs}*N*is larger than a threshold value. Therefore, it is possible to simplify the real morphologies as a model with less

_{s}*N*but by no means single core-shell sphere because there are large changes when

_{s}*N*is extremely small, which may be due to the shielding effects of large monomers. For bare and thinly coated soot aggregates,

_{s}*C*is insensitive to

_{abs}*D*when the

_{f}*R*is small, where the relative errors of

_{V}*C*for different

_{abs}*D*are within 2.5%. However, the sensitivity of

_{f}*D*is obvious for large soot due to the shielding effects of large monomers, and the relative errors for different

_{f}*D*can reach to 18% for bare soot. Compared to bare soot, the effects of

_{f}*D*is small for thinly coated soot. In addition, the effects of

_{f}*f*on

_{soot}*C*is obvious.

_{abs}Compared to small *N _{s}*, the effects of

*N*on

_{s}*C*is also relative low for large

_{sca}*N*. Relative errors of

_{s}*C*between

_{sca}*N*= 800 and

_{s}*N*= 500 are below 5% but the relative errors between

_{s}*N*= 300 and

_{s}*N*= 100 can be above 15%. The effects of

_{s}*D*on

_{f}*C*is more obvious for small soot when the

_{sca}*N*is fixed. This is because the main factor affects

_{s}*C*of large soot may be the shielding effects of large monomers.

_{sca}For bare and thinly coated soot, *ASY* is more sensitive to *N _{s}* for small particles. However, the effects of

*N*on

_{s}*ASY*is negligible for thickly coated soot where the relative errors for different

*N*are below 2.25%. The reason may be that the overall spherical structure is unchanged. Compared to small soot, the effects of

_{s}*D*is smaller for large soot when fixed

_{f}*N*. This may be caused by the shielding effects of large monomers. For thinly coated soot,

_{s}*ASY*is insensitive to

*f*due to the unchanged fractal structure and the relative errors between different

_{soot}*f*are below 1.5%.

_{soot}The aims of study are to understand how the morphologic parameters influence the radiative properties under fixed *R _{V}* and investigate the sensitivities of morphological parameters on optical properties of soot particles. It may be helpful for the model simplification. For an example, as shown in Fig. 11, for thinly coated soot, the

*ASY*seems to be insensitive to

*f*, so it may be proper to reconstruct the

_{soot}*ASY*using an aggregate with a homogeneous fractal structure for simplification. In addition, as demonstrated in Fig. 3 and Fig. 4, for both bare and thinly coated soot,

*C*changes slowly with

_{abs}*N*when

_{s}*N*is beyond a threshold value. Therefore, an equivalent volume structure with less

_{s}*N*can be used to reconstruct the

_{s}*C*. Our study may also improve the understanding of the influences of morphology on radiative properties of soot aerosols.

_{abs}## Funding

National Key Research and Development Plan (Grant No. 2016YFC0800100 and 2017YFC0805100); National Natural Science Foundation of China (NSFC) (Grant No. 41675024 and U1733126); Fundamental Research Funds for the Central Universities (Grant No. WK2320000035).

## Acknowledgments

We particularly thank Dr. D. W. Mackowski and Dr. M. I. Mishchenko for the MSTM code and the constructive suggestions of reviewers. We also acknowledge the support of supercomputing center of USTC.

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