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Radiative characteristics of nonspherical particles based on a particle superposition model
School of Aeronautical Science and Engineering, Beihang University, Beijing, China
Corresponding author: Y. Huang, School of Aeronautical Science and Engineering, Beihang University (Beijing University of Aeronautics and Astronautics), Mailbox 505, 37 Xueyuan Road, Haidian District, Beijing, 100191, China. (email@example.com)
 A particle superposition model combined with a Monte Carlo ray tracing method, which is used to study the radiative characteristics of large nonspherical particles, is proposed. The goal of the model is to provide a new approach that not only allows the modeling of large irregular particles of arbitrary shape but also possesses a clear and systematic basic structure that is easy to program. A detailed flow chart and algorithm description are provided in the paper. The data characteristics of four types of irregular particles were calculated and analyzed. The comparison showed that this particle superposition model can be used to calculate the radiative properties of large irregular particles with good accuracy, while at the same time remaining general and extendable to many applications, thus providing a new way to study nonspherical particles.
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 Particle radiation is a comprehensive subject that involves many domains, such as astrophysics, optics, meteorology, and combustion, and that attracts extensive interest and has always been an alluring field in radiation studies [Cai and Liou, 1982a, 1982b; Mishchenko et al., 2000; Kokhanovsky, 2006; Forster, 2007; Shell and Somerville, 2007; Liou et al., 2011]. Currently, much more attention is focused on how to make models closer to the actual situation. Subjects such as nonspherical particle radiation and Gaussian wave incidence [Wu et al., 2005; Han et al., 2012] have drawn the interests of many scholars. There are various particle morphologies in nature, and previous research has shown that most of these particles are nonspherical and have irregular shapes with no particular habits [Reid et al., 2003; Muñoz et al., 2006], which makes the modeling of the particle complex difficult. Ping Yang and colleagues [Yang et al., 2007] indicated that due to the nonspherical effect, the application of the Lorenz-Mie theory may result in a substantial underestimation or overestimation of radiative forcing. Therefore, many improved numerical methods, such as the Discrete Dipole Approximation [Mugnai and Wiscombe, 1986; Draine and Flatau, 1994; Nebeker et al., 1998; Yurkin and Hoekstra, 2007, Yurkin et al., 2007], the T-Matrix [Mishchenko et al., 1997; Yang et al., 2007; Feng et al., 2009], the finite difference time domain [Yang et al., 2000], and the Geometrical Optics (GOM) [Cai and Liou, 1982a, 1982b; Macke, 1993] are often applied. In addition, Yang et al.  showed that spheroids can be seen as the first-order approximation of the overall shapes of nonspherical particles, and previous research [Mishchenko et al., 1997; Dubovik et al., 2006] discussed using shape mixtures of randomly oriented spheroids for modeling desert dust aerosols. However, calculations based on simple assumptions are often inaccurate and previous research [West et al., 1997; Zhao et al., 2003; Wang et al., 2003; Volten et al., 2005] revealed the inevitable differences caused by low-order approximation of the particle shape [Yang et al., 2007]. In this paper, we do not intend to compare the advantages and disadvantages of each method but would like to try to provide a new way of thinking for the modeling of large, complex-shaped particles.
 To model particles with various morphologies, two concerns were mainly considered: (1) the model must be capable of recapitulating the external form of the particle with good accuracy. Early research reduced irregular particles to their equivalent sphere so that the Mie theory could be applied. Alternatively, complex particles were converted into spheroid or cylindrical shapes to simplify the calculation, resulting in the loss of much of the shape information. It is logical to infer that more missing information results in a less accurate in the model. (2) The model must have a certain commonality, which means that it is not proposed to fit for just one particular type of particle but is able to calculate particles of different shapes as well as complex constituents. However, the requirements of the accuracy and commonality of the model actually contradict one another; therefore, a balance must be made between the two requirements.
 In this paper, a particle superposition model combined with the Monte Carlo ray tracing algorithm is proposed to describe scattering by irregularly shaped particles with overall size and features in the geometric optics regime. The application of a geometrical optics approximation dealing with large particle radiation has been discussed by many scholars [Cai and Liou, 1982a, 1982b; Macke, 1993; Yang et al., 2000; Gusarov, 2008, 2010; Mishchenko et al., 2011]. Previous research [Mishchenko et al., 2011] showed the limits of GOM when dealing with spherical aerosols but also showed that it is quite useful when applied to nonspherical particles. In previous research by Gusarov [2008, 2010], rigorous ray-optics calculations were applied (in the case of heterogeneous media) in the limit of geometrical optics and showed no contradiction with the known calculations by ray optics and Monte Carlo simulations and agree with the known experimental data [Gusarov, 2008]. The Monte Carlo method (MCM) has a clear physical concept and possesses the advantage of strong adaptability, which is especially suitable for the study of radiation with multidimensional, complex geometrical shape, and complicated boundary conditions. In our opinion, GOM combined with MCM is a simple and intuitively appealing approximation method with credible accuracy, which showed very good fit in our study. Macke and Mishchenko  offer a publicly available and well-developed Monte Carlo ray tracing model. The model presented in the current study is based on a similar Monte Carlo ray tracing algorithm but, in addition, provides a simple and flexible way to build an irregularly shaped particle (namely, as a superposition of geometric optics regime spheres of different sizes and levels of overlap).
 The basic concept of the model is explained, and a detailed flow chart and a description of the algorithm are provided in section 2. Four different shapes of nonspherical particles were calculated, and the independent radiative characteristics of these particles under monochromatic collimated light were analyzed. Comparisons of data from simple irregular particles were made between the model proposed and the other proven method. The effect of particle orientation under collimated light was also discussed.
2 Particle Superposition Model of Nonspherical Particles
 To meet the requirements discussed above, the irregularly shaped particle was treated differently from before. It was no longer seen as a whole but was broken into several unit spheres overlapping one another. Each unit sphere is an isotropic homogenous particle that can be set with a different radius (within GOM regime), different optical constants, and arbitrary locations according to the morphology of the irregular particle. For a particular nonspherical particle, more unit spheres will result in better performances relative to the real situation, and by adjusting the radius and relative positions of the unit spheres, different shapes of particles can be formed (Figure 1). Theoretically, this method can be used to simulate large particles of any arbitrary shape.
 The ray path in Figure 1 shows that the light ray may experience multiple reflections and refractions through the particle and even re-enter into the medium after it is refracted out. To trace the ray path precisely, the exact relative relation between the light and unit sphere must be judged accurately. Assume that the semitransparent irregular particle is surrounded by a nonattenuating medium with unit refraction index, the unit spheres that formed the particle have the same complex refraction index m = n − i ⋅ κ, and all interfaces of the medium are considered optically smooth. The equations of the unit spheres can be expressed as (x − xi)2 + (y − yi)2 + (z − zi)2 = ri2, where (i = 1, 2, 3 …), and the light tracing vectors are all in the same coordination with these unit spheres, which will avoid the inconvenience caused by intricate coordinate transformation [Cai and Liou, 1982a, 1982b]. For a single light ray that is cast onto the particle, a flow chart of the tracing algorithm is as follows (see also Figure 2).
 1. When a beam casts onto one of the unit spheres, its intersection with the sphere surface P0 and the incident angle can be calculated using a geometrical relationship. The specular reflectivity can be obtained by Fresnel's law:
where θ is the incident angle, and the refraction angle ϕ = arcsin(ni sin θ/nj). A random number is then generated to determine whether reflection or refraction will occur at the interface.
 2-1. If the beam is reflected by the sphere, it could still intersect with other unit spheres. The direction of the reflected ray can be obtained using the following equation [Xia, 1997], in which is the incident vector and is the outward normal vector.
 The probable intersections of the light ray after being reflected can be calculated by substituting the parametric equation of the reflected light into the entire equations of the other unit spheres. If intersections were to exist, a series of tij can be obtained, where i represents the ID of the unit sphere and j = 1, 2. Place tij into the parametric equation, and the one point that satisfies with the minimum |tij| is the re-entry point Pre(xre,yre,zre). Set P0 = Pre, and the algorithm returns to 1. If not, record and the tracing ends.
 2-2-1. If the beam is refracted into the sphere, the direction of the refracted ray can be determined by [Xia, 1997]
with a = n when traveling from an optically thinner medium into a denser medium and a = 1/n when the light is incident in the opposite direction.
 2-2-2. Determine the probable propagation length s by a random number.
 2-2-3-1. Substitute the parametric equation of the light into the unit sphere, where P0 is on, and a virtual intersection P0 ′ can be obtained because it may be within other unit spheres and light will not be reflected nor refracted when passing through it.
 2-2-3-2. Calculate the distances (li) of P0 ′ to the center of other unit spheres Oi and compare li with the corresponding radius ri.
 2-2-4-1. If all other unit spheres li > ri, then this means that P0 ′ is indeed at the interface of the medium and the outer surroundings and P0 ′ is the real intersection Pend. If the distance between P0 and Pendsmax < s, calculate the incident angle and specular reflectivity and determine whether the beam will be reflected or refracted at this point. If the ray is reflected, set P0 = P0 ′ and calculate the reflected direction, and the algorithm goes to 2-2-2. If the ray is refracted, substitute the parametric equation of the refracted ray into the equation of the other unit sphere and search for probable intersections. The determination of the re-entry point is the same with 2-1, and the algorithm goes to 1. However, if no intersections were to exist, then record , and the tracing ends. If smax > s, the light is absorbed by the medium during propagation. Calculate the absorbed point Pab using s and end the tracing.
 2-2-4-2. If the unit sphere li < ri, P0 ′ is within the unit spheres. A series of intersections of the light ray with these spheres can be obtained and the distances sij (i is the ID of the unit sphere and j = 1, 2) of these intersections to P0 are calculated. Then, compare all of the sij, and the one point at max |sij| is the new virtual point Pi ′; the algorithm goes to 2-2-3-2.
3 Calculation and Discussions
 The idea of proposing such a particle superposition model to study the radiative characteristics of nonspherical particles was first inspired during the study of the trafficability of lunar mare terrain [Gao et al., 2009; Li et al., 2010], in which a method was adopted to simulate the dynamic behavior of lunar dust in a triaxial compression simulation. In this paper, the realization of the particle superposition model was also performed according to the morphologies of the simulated lunar dust. As seen from the scanning electron microscope (SEM) photo (Figure 3) [Gao et al., 2009], the shape of simulant lunar dust particles can be summarized into four different shape types: extremely angular particle (AP), subangular particle (SAP), subspherical particle (SSP), and elongated particle (EP) [Li et al., 2010] (Figure 4). The refractive index of real samples of simulant lunar dust varies with the wavelength and the medium of the particle, which also contains bubbles and impurities. However, our goal was only to validate the feasibility of the particle superposition model, not to study the physical properties of lunar dust. Therefore, the particle was treated as homogeneous and semitransparent with a refractive index from 1.1 to 3.0.
 The size parameter χ is an important parameter that affects the radiation of the particle. For nonspherical particles, the size parameter can be approximately obtained by deriving the equivalent radius Requi of a spherical particle that has the same volume as the nonspherical one and χ = 2πRequi/λ. When a nonspherical particle is incident by collimated light, the orientation of the particle will also affect its radiation. Assume the illuminating angle β is the angle between incident light and the negative direction of the long axis of the particle (Figure 5). When discussing the influence of the size parameter and optical constants on the radiation characteristics under collimated incidence, the adopted illuminating angle β = 90o. To determine the size parameter χ of the nonspherical particle, the radius ratio of the unit particle is R1 = R2 = R3 = 0.5R0, and the distances between the centers of the element spheres are D10 = D20 = D30 = R0. The wavelength of incident radiation is 1 µm.
 To validate the Monte Carlo ray tracing algorithm in the current model, a comparison was made with the results of the Monte Carlo ray tracing algorithm presented in Macke and Mishchenko . The authors studied the light scattering properties of cylindrical and ellipsoidal particles. To make our model comparable, the number of unit spheres was reduced to only one, and the shape of the unit sphere was modified to cylinder and spheroid according to the size and optic constants of Macke and Mishchenko , respectively. Figure 6 is the comparison of total absorptivity α with the change of the size parameter, and α is defined as the ratio of the energy absorbed by the medium to the total energy of the incident radiation. As shown in Figure 6, the results calculated by our method are very close to those from the program of Macke and Mishchenko . The data comparison shows that the average difference between the two methods is 0.033 (cylindrical particle) and 0.023 (ellipsoidal particle), with the maximum difference no more than 0.085 (cylindrical particle) and 0.074 (ellipsoidal particle), which shows that the particle superposition model we proposed is valid.
 In Figure 7, we present calculations of the phase functions of the four irregular shape types, AP, SAP, SSP, and EP, along with calculations for spheroidal and cylindrical particles of an equivalent size parameter. The Θ symbol is the scattering angle. According to the data from the calculation, the phase function equations at a given size parameter and optical instant can be fitted and expressed by the following equations:
where Pn is the Legendre polynomials and N is the expansion order. The fitting result is shown in Table 1, where g is the asymmetric factor. Under specular reflection, the scattering of the particle is preferentially forward. Under the same conditions, the scattering curves of all four nonspherical particles are very close to that of the spheroidicity particle while an obvious discrepancy exists between them and the cylindrical particle (Figure 7e). The difference can also be seen from the value of g; it shows that the forward scattering of the cylindrical particle is less strong than other nonspherical particles. The differences between the four complex particles are relatively small: forward scattering of the SSP accounts for up to 89.1% (maximum of the four) of total scattering and the EP accounts for 86.5% (minimum of the four). All four models show a common trend that with an increase of the size parameter, the percentage of forward scattering gradually falls to a stable value, as seen from Figure 7a to 7d (e.g., where the curves at χ = 500 and χ = 1000 are basically the same).
Table 1. Coefficient An of Phase Functions
 The effect of the orientation of the particle under collimated incident light was also studied. The absorption of the particle under collimated radiation at different illuminating angles β is shown in Figure 8. Under the same conditions, when the illuminating angle is 5∘ < β < 175∘, the absorptive capacity under the EP model is stronger than that of all of the other nonspherical particles, and when light is illuminating around 0o and 180o the absorptivities under the SAP, SSP, and AP models are relatively stronger. For asymmetrical particles, the curves of the absorption also showed a strong asymmetry. The difference between the EP model and the AP model is up to 0.13 at the maximum when β = 45o, which means the absorptive ability under the AP model is only 82% of that under the EP model. The minimum difference between them is 0.014 when light is illuminating at β = 180o. The comparisons again verified the significant impact of the particle orientation under collimated light, especially when the particle is much larger and has much more complex morphologies.
 A particle superposition model combined with a validated Monte Carlo ray tracing algorithm for modeling large nonspherical particles was proposed. The method builds an irregularly shaped particle from unit spheres of different sizes and levels of overlap with one another. The radius, material composition, and spatial position of the unit spheres are all set to mimic the complexity of an irregular particle as closely as possible. The differences in scattering properties between the four irregular shape types and spheroids and cylinders of the same size parameter were shown. The comparison indicates that the current model may be able to capture certain scattering features of irregularly shaped particles better than “equivalent” spheres, spheroids, or cylinders. Furthermore, the optical constants of each unit sphere can be assigned in different values. We propose this model as a conceptually simple and easy to use yet extremely flexible method to calculate scattering of electromagnetic radiation by complex irregularly shaped and inhomogeneous particles whose overall size and features are in the geometric optics regime.
radius of the element sphere
the distance between the center of element spheres
reflectivity under specular reflection model
direction vector of the light ray
outward normal vector
the size parameter of the particle
wave length of incident light
surface reflectivity of the particle
the scattering albedo
absorptivity of the particle
0, 1, 2, 3
number of the element sphere
medium on the out and inside of the particle
 This work was supported by the National Science Foundation of China (No. 51376016) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20121102110015).