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Abstract
Alumina particles experience phase transition as an undercooling process along the plume, during which the liquid alumina clusters transform into multiphase, and then into α phase. The phase transformation model was built by an improved diffusion limited aggregation (DLA) algorithm with monomers of stratified structure. The effects of phase transformation on the ultraviolet optical characteristics of alumina clusters were studied using the superposition T-matrix method (STMM). We found that the alumina clusters in phase transition had completely different optical properties compared with the fixed phase ones. Forward scattering, absorption efficiency and asymmetry parameter gradually decreased, whereas backward scattering, scattering efficiency, and single-scattering albedo gradually increased during the phase transformation process. Besides, multiphase alumina clusters were compared with the other two equivalent models, including the sphere model approximated by equivalent volume sphere (EVS) and the equivalent surface sphere (ESS) approaches and single-phase cluster model approximated by Maxwell-Garnett (MG) and Bruggeman (BR) approaches. Generally speaking, the optical properties of the single-phase cluster approximated by MG and BR approaches were relatively close to those of the real multiphase alumina cluster. Whereas the spheres approximated by EVS and ESS had great deviations, especially when the number of monomers in the cluster was 20, the relative error of scattering efficiency calculated by ESS was up to 52%. Therefore, approximate approaches for multiphase clusters should be chosen cautiously. Our results give further the understanding of the optical properties of alumina clusters. As the phase states are usually closely related to the plume radiation and burning process, these kinds of researches will be helpful to aircraft detection, identification, and other related fields.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The interest in alumina (Al2O3) is steadily increasing not only for its porous structure that applied in many applications [1–4], but also for the significant influences on plume radiation [5–11]. To reduce combustion instabilities and increase specific impulse [12–14], larger booster motors use propellants with 14-20% aluminum by weight, resulting in 25-38% of the exhaust plume being condensed alumina particles [15]. Kolb et al. [16] reviewed the factors that affected the ultraviolet radiation from missiles. They thought the thermal emission and scattering of alumina particles were the important sources of ultraviolet radiation of plume. Neel et al. [7] predicted the ultraviolet radiance of plume, he found alumina particles emitted UV and caused scattering inside the plume. The total plume radiant intensity would be underestimated by 10% without considering the alumina scattering properties. Therefore, the detailed knowledge of ultraviolet optical properties of alumina particles is important to predict plume radiation for aircraft detection, identification, and other related fields.
The optical properties of alumina particles are influenced by temperature, particle size distribution, morphology, refractive index, and also phase state. Liquid alumina droplets formed in the high-temperature combustion will crystallize and experience complicated phase transformation as the temperature decrease quickly. Alumina occurs in a variety of crystallized phases, including α, γ, η, κ, θ, and liquid [17,18]. Plastinin et al. [19] found three relatively stable phase states (liquid, γ phase, and α phase) and described the phase transformation of an individual alumina particle during crystallization. As the results showed, liquid alumina particles first transformed to γ phase, then γ phase to α phase. The spectral radiance in UV, visible, and IR regions of alumina particles would vary during the phase transformation process. Bityukov et al. [20,21] measured the intensity of radiation in the visible and near IR spectra upon solidification of melted pure alumina, the results revealed the change of temperature of alumina during the crystallization process. Gimelshein et al. [22] applied direct simulation Monte Carlo (DSMC) method to simulate the rarefied two-phase plume flows that took the phase change of alumina particles into consideration. The results indicated that the particle diameters, temperature, and density would be varied in the phase change processes. The optical properties of alumina particles are sensitive to phase state, however, the influences of phase transformation on optical properties of alumina particles are still unclear due to the complicated morphology.
By scanning electron microscopy (SEM) and transmission electron microscopy (TEM), Jeenu et al. [23] and Pan et al. [24] observed alumina particles of stratified structure with multiphase as shown in Fig. 1(a), that attributed to the different cooling rates between the outer and inner part of particles. Moreover, the alumina particles can aggregate to clusters in the plume [12,25,26], as shown in Fig. 1(b). Hence, several researchers studied the optical properties of alumina particles taking both morphology and phase state into account. Li et al. [27] studied the scattering phase function of α phase, γ phase, and liquid of alumina clusters. The results revealed that the scattering properties of alumina clusters were related with phase state. Li et al. [28] compared the absorption efficiency of an alumina particle between the single-phase alumina particle and the multiphase one, and found that the internal absorption efficiency of multiphase particle reached maximum value at the interface between γ phase and α phase. However, to the best of our knowledge, there is no study concerning the influences of phase transformation on the optical properties of alumina clusters up to now.
Fig. 1. alumina particles. (a) TEM image of an alumina particle with γ phase core and amorphous shell (Ref. [24], Fig. 3); (b) TEM image of an alumina cluster (Ref. [12], Fig. 45); (c) Monomer model.
The primary objective of this paper is to analyze the effects of phase transformation on the ultraviolet optical properties of alumina clusters during crystallization. As a secondary objective, the equivalent models of multiphase alumina clusters were investigated to reduce the computational burden. The phase transformation model of clusters and the method for calculating the optical properties of alumina clusters were introduced in section 2. The random-orientation averaged results were discussed in section 3. Conclusions and summary were given in section 4.
2. Methodology
2.1 Phase transformation process of alumina particle
Alumina particles experience phase transformation from surface to center of the alumina particle during crystallization. The phase transformation, which are highly related to temperature, can generally be described by two stages, “liquid to γ phase” and “γ to α phase” transition as the temperature decreases.
Alumina particles will release crystallization energy during the liquid to solid state (including α phase and γ phase) transition. This is a non-equilibrium process that is described by the relative crystallization front radius $r_\textrm{s}^{\ast }$. A particle in liquid and solid state are denoted separately by $r_\textrm{s}^{\ast }\textrm{ = }1$ and $r_\textrm{s}^{\ast }\textrm{ = }0$, for transitional phase state $0 < r_\textrm{s}^{\ast } < 1$, and the kinetic recession of the crystallization front is described by the equation below [19,29]:
According to the definition of $r_\textrm{s}^{\ast }$, the relationship of r, ${r_L}$, and $r_\textrm{s}^{\ast }$ is described as the following equation:
By substituting Eq. (4) and Eq. (5) into Eq. (1), we can obtain the expression for the liquid radius of alumina particles varying with time as follows:
Therefore, we can get ${r_L}$ at different times during the phase transformation by solving the equation above. Supposing ${f_\alpha }$, ${f_\gamma }$, and ${f_L}$ are the total volume fractions of α phase, γ phase, and liquid respectively in a particle, and satisfy the equation ${f_{\mathrm{\alpha}}} + {f_\mathrm{\gamma }} + {f_\textrm{L}} = 1$. Based on Fig. 1(c), ${f_\textrm{L}}$ can been obtained by the following equations:
${f_{\mathrm{\alpha}}}{\kern 1pt} {\kern 1pt}$ and ${\kern 1pt} {\kern 1pt} {\kern 1pt} {f_\mathrm{\gamma }}{\kern 1pt} {\kern 1pt}$ can be calculated by the following equations [29,30]:2.2 Phase transformation model of alumina cluster
According to the TEM and SEM images of alumina particles [23,24] and the crystallization kinetics [19], the monomer model of stratified structure with multiphase is proposed as shown in Fig. 1(c). The particle consists of three concentric spheres of α phase, γ phase, and liquid alumina respectively from outside to inside.
The DLA algorithm, which was introduced by Witten and Sander [31], includes particle-cluster aggregation (PCA) and cluster-cluster aggregation (CCA) [32–34]. Although the clusters aggregated by CCA algorithm are more similar to the real fractal-like aggregates, the process is complex and time-consuming [35–37]. The PCA is widely used due to its simplicity [38,39], and has been applied herein. The radius of alumina particles satisfy lognormal distribution [15] rather than the same one. Hence, the traditional DLA algorithm [37,38] used to generate clusters with monomer of the same radius is not suitable in this paper. The recent improved DLA algorithm [39], which extends the traditional DLA to aggregate particles with different radiuses, was applied to generate alumina clusters in this paper. Each monomer of the alumina clusters satisfied the model as shown in Fig. 1(c). The construction and morphology of the fractal structure can be described as the well-known fractal laws:
where Ns is the number of the monomers in a cluster, ${k_f}$ is the fractal prefactor, ${R_g}$ is the radius of gyration, a is the mean radius of the monomers, ${D_f}$ is the fractal dimension, and ${l_j}$ is the distance from monomer j to the center of the cluster. ${r_k}$ is the radius of the kth monomer in the cluster. Kim et al. [15] made light transmission measurements at the edge of a plume from a small solid-propellant motor to determine the particle size. He found the size distribution of alumina particles could been assumed as lognormal distribution described as the following equation:2.3 Superposition T-matrix method
T-matrix method has been widely used for the computation of electromagnetic scattering and absorption by single and composite particles [33,40–44]. The recent STMM is ideal for this work because of three qualities. First, it is a direct computer solver of the frequency-domain macroscopic Maxwell equations and involves no approximations. Second, it has been extended to arbitrary configurations of spheres located internally or externally to other spheres, that makes possible to compute the numerically exact computations of electromagnetic scattering and absorption for particles with inclusions [45–47]. Last, the analytical orientation averaging procedure makes possible model the statistically uniform distribution. In this study, we used STMM to calculate the optical properties of alumina clusters, and the detailed descriptions of the STMM has been given by Mackowski and Mishchenko [45,48]. In the standard {I, Q, U, V} representation of polarization state and angular distribution, the normalized stokes scattering matrix of the random orientation averaged particles has the block-diagonal structure [49–51]:
Fig. 2. Phase transformation model of alumina clusters. (a) Initial stage: liquid; (b) Transitional stage: multiphase; (c) Final stage: α phase.
Other useful quantities in the study of plume radiation are scattering efficiency (${Q_{\textrm{sca}}}$), absorption efficiency (${Q_{\textrm{abs}}}$), single-scattering albedo ($\varpi$) and asymmetry parameter ($g$) [52,53]. The g, that illustrates the proportion of forward and back scattering, can be defined by the following equation [50,51]:
3. Results and discussion
According to the proportion of alumina surface’s tension and shear force, alumina clusters will rupture when the fundamental particle number is more than 20∼30 [54]. Also, the effects of fractal dimension on the optical properties of alumina cluster is less [27]. Hence, the alumina clusters of Ns=10 and Ns=20 were randomly simulated by the improved DLA algorithm as shown in Fig. 2. Although alumina clusters occur in a variety of morphologies and different radiuses of alumina particles lead to the different values in Table 2, the changing trend of volume fractions of each phase state are the same during the phase transformation process, the same applies to the changing trend of the optical properties. The ultraviolet radiation of plume was predicted by a generic sensor that was sensitive in the wavelength band 270-290 nm [7]. Therein, 280 nm which belongs to the ultraviolet solar blind spectral band was selected as the incident wavelength in this paper. The complex refractive indices of three phase states of alumina at 280 nm were given in Table 1 [55–57]. The optical properties of all alumina particles were computed by STMM in this work.
Table 1. Refractive index of alumina particle
Table 2. The total volume fractions of three phase states during the phase transformation
3.1 Effects of phase transformation on optical properties of alumina cluster
The results of five moments from t0 to t4, which are in chronological order during the phase transformation process, are shown in this section. The total volume fractions of three phase states at five different times are listed in Table 2 based on the theories in section 2.1, including the initial stage (t0) corresponding to Fig. 2(a), the transitional stage (t1-t3) corresponding to Fig. 2(b), and the final stage (t4) corresponding to Fig. 2(c).
Figure 3 and Table 3 depict scattering phase function, ${Q_{\textrm{abs}}}$, ${Q_{\textrm{sca}}}$, $\varpi$, and g of alumina clusters during the phase transformation process. From the enlarged pictures in Fig. 3, it’s obvious that the forward scattering gradually decreases, while the backward scattering gradually increases with time. Hence, the g concerning the proportion of forward and back scattering will gradually decrease which is consistent with the results shown in Table 3. Backward scattering is important in radar detection. As the results shown in Table 3, ${Q_{\textrm{abs}}}$ and g gradually decrease, whereas ${Q_{\textrm{sca}}}$ and $\varpi$ gradually increase with time. It revealed that liquid alumina leads to stronger forward scattering and ${Q_{\textrm{abs}}}$, and weaker backward scattering and ${Q_{\textrm{sca}}}$.
Table 3. The optical properties of alumina clusters during the phase transformation process
In conclusion, the optical properties of alumina clusters have regular change during the phase transformation, that’s totally different compared with those of the fixed phase ones. This can be attributed to two facts. First, the phase transformation is a process of gradual change with time, that leads to the gradual change of the optical properties. Second, different phase states have different crystal structures and refractive indices, that result in different optical properties. For example, the decrease of ${Q_{\textrm{abs}}}$ during phase transformation is due to the reduction of liquid alumina which is more absorbent than solid state (α and γ phase). Therefore, the deviations will be introduced when the alumina clusters in transition state were treated as invariable phase one in the previous studies [27,39].
3.2 Approximate approaches of alumina cluster with multiphase
Multiphase alumina clusters have complicated structure, therefore, approximate approaches were studied to reduce the computation burden. There are two simplified models of multiphase alumina clusters in the previous studies, sphere model [5,6,11] and single-phase cluster model [27]. Thus, we applied equivalent-spherical approximations [58], EVS and ESS, to simplify a multiphase alumina cluster to a sphere. The traditional effective-medium theories, MG and BR, were used to approximate a multiphase alumina cluster to a single-phase cluster. MG and BR, which are sound theories for the stratified spheres provided that the size parameter is smaller than a thresh hold [59], are appropriate to this work for the small size parameter of alumina monomers [15]. In this section, the optical properties of multiphase alumina clusters, whose volume fractions of three phase states are corresponding to t2 in Table 2, were compared with the corresponding simplified sphere model and single-phase cluster model.
MG approach assumes several types of particles are embedded in a main particle, and is described as the following equation [60]:
According to Fig. 4, scattering phase functions of the spheres approximated by EVS and ESS fluctuate remarkably with scattering angle and show more extrema at side-scattering angles compared with those of the multiphase clusters. The sphere model approximated by ESS gives rise to stronger forward scattering, whereas the sphere model by EVS leads to weaker forward scattering. The backward scattering is overestimate by both EVS and ESS. From Table 4, both EVS and ESS overestimate g, and underestimate ${Q_{\textrm{abs}}}$ and ${Q_{\textrm{sca}}}$ remarkably. When Ns=20, the relative errors of ${Q_{\textrm{abs}}}$ and ${Q_{\textrm{sca}}}$ calculated by EVS are 42% and 45% respectively compared with the corresponding multiphase cluster, and the relative errors of ${Q_{\textrm{abs}}}$ and ${Q_{\textrm{sca}}}$ approximated by ESS are 31% and 52% respectively. It’s easy to understand the discrepancies between the multiphase cluster and simplified sphere. We could view the cluster as a large collection of dipoles. The radius will exceed the radius of the largest monomer in the cluster when a cluster is replaced by a sphere with the same volume or surface. The dipoles constituting the single sphere will be closer to each other than they are in the cluster, in consequence, the interaction will be obviously different which will lead to a different scattering pattern in the far field.
Fig. 4. Scattering phase functions of multiphase alumina clusters and those approximation approaches.
Table 4. Comparisons of the multiphase alumina clusters and other approximate approaches
The optical properties of single-phase clusters approximated by BR and MG approaches are identical. And the relative errors of phase function, g, ${Q_{\textrm{sca}}}$ and $\varpi$ are less than 5% compared with those of the corresponding multiphase cluster, the relative errors of ${Q_{\textrm{abs}}}$ are more than 15%. This is due to the simplification of multiphase cluster into homogeneous single-phase cluster, therefore, the interactions between different phase states of alumina in the cluster are ignored.
In summary, it’s simplistic for previous researchers treating the multiphase alumina clusters as spheres based on the results of EVS and ESS approaches. The optical properties of single-phase cluster approximated by MG and BR approaches are close to those of the multiphase clusters as a whole. However, the relative error of ${Q_{\textrm{abs}}}$ is a little bigger than the other optical properties approximated by MG/BR. According to Table 4, the time required to calculate the optical properties of the single-phase clusters approximated by MG/BR is about half of the time required to calculated those of the corresponding multiphase clusters. The time required to calculate the optical properties of the sphere approximated by EVS and ESS were about a few thousandths of the time required of the corresponding multiphase clusters. To select appropriate approaches for the multiphase clusters, one should take both computation and accuracy into consideration.
4. Conclusions
In this paper, the phase transformation model of alumina was built based on the improved DLA algorithm with monomers of stratified structure, and the influences of phase transformation on the ultraviolet optical properties of alumina clusters were investigated based on STMM. Approximated approaches of transitional alumina cluster with multiphase were studied as well.
As the results shown, the optical properties of alumina clusters have regular changes during the phase transformation process. The forward scattering, ${Q_{\textrm{abs}}}$, and g gradually decrease, while the backward scattering, ${Q_{\textrm{sca}}}$, and $\varpi$ gradually increase with time, which is completely different from those of the fixed phase ones. That’s attributed to the fraction changes of three phase states in the alumina particle. Therefore, the previous studies viewing the alumina clusters just as single-phase ones were simplistic. The optical properties of spheres approximated by EVS and ESS approaches are remarkably different from those of the real multiphase clusters. When Ns=20, the relative errors of ${Q_{\textrm{abs}}}$ and ${Q_{\textrm{sca}}}$ approximated by EVS is up to 42% and 45% compared with the corresponding multiphase alumina cluster respectively. The single-phase clusters approximated by MG and BR approaches are identical, whose relative errors of phase function, g, ${Q_{\textrm{sca}}}$ and $\varpi$ are less than 5% compared with those of the multiphase clusters. However, relative error of ${Q_{\textrm{abs}}}$ is a little bigger than the other properties. The time required to calculate the optical properties by MG/BR is about half of the time required to calculated those of the corresponding multiphase clusters. The time required to calculate the optical properties EVS and ESS were about a few thousandths of those required by the corresponding multiphase clusters. Hence, one should take comprehensive consideration of both computation and accuracy to choose the approximate approaches. In summary, our studies provide further understanding of the influence of phase transformation on ultraviolet optical properties of alumina clusters that are usually related to the plume radiation and burning process, hence the results may be helpful to aircraft detection, identification, and the other related field.
Funding
National Natural Science Foundation of China (61875156).
Acknowledgments
The authors particularly thank Dr. M. I. Mishchenko and Dr. D.W. Mackowski for providing the Fortran code of the superposition T matrix method.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
References
1. C. Wang, P. Qin, D. L. Lv, J. Wan, and H. Ma, “Characterization of anisotropy of the porous anodic alumina by Muller matrix imaging method,” Opt. Express 28(5), 6740–6754 (2020). [CrossRef]
2. Q. Libois, F. Lévesque-Desrosiers, S. Lambert-Girard, S. Thibault, and F. Domine, “Optical porosimetry of weakly absorbing porous materials,” Opt. Express 27(16), 22983–22993 (2019). [CrossRef]
3. D. Kuang, R. Charrière, N. Matsapey, M. Flury, J. Faucheu, and P. Chavel, “Modeling the specular spectral reflectance of partially ordered alumina nanopores on an aluminum substrate,” Opt. Express 23(4), 4506–4516 (2015). [CrossRef]
4. C. H. Huang, H. Y. Lin, S. Chen, C. Y. Liu, and Y. Tzeng, “Electrochemically fabricated self-aligned 2-D silver/alumina arrays as reliable SERS sensors,” Opt. Express 19(12), 11441–11450 (2011). [CrossRef]
5. Q. Binauld, J. M. Lamet, L. Tessé, P. Rivière, and A. Soufiani, “Numerical simulation of radiation in high altitude solid propellant rocket plumes,” Acta Astronaut. 158, 351–360 (2019). [CrossRef]
6. F. P. Neele and H. M. A. Schleijpen, “UV missile-plume signatures,” Proc. SPIE 4718, 369–379 (2002). [CrossRef]
7. F. P. Neele and H. M. A. Schleijpen, “Electro-optical missile plume detection,” Proc. SPIE 5075, 270–280 (2003). [CrossRef]
8. G. Cai, D. Zhu, and X. Zhang, “Numerical simulation of the infrared radiative signatures of liquid and solid rocket plumes,” Aerosp. Sci. Technol. 11(6), 473–480 (2007). [CrossRef]
9. J. Burt and I. Boyd, “A Monte carlo radiation model for simulating rarefied multiphase plume flows,” 38th AIAA Thermophysics Conference, (AIAA, 2005), 2005–4691.
10. H.G. Robert, A.A. Paul, and D.B. John, “Ultraviolet scene simulation for missile approach warning system testing,” Proc. SPIE 3084, 282–291 (1997). [CrossRef]
11. J. Li, Y. Liu, N. Wang, and L. Jin, “DSMC simulation of two-phase plume flow with UV radiation,” Proceedings of the 29th International Symposium on Rarefied Gas Dynamics, (AIP Conf. Proc, 2014), 569–580.
12. D. Sundaram, V. Yang, and R. A. Yetter, “Metal-based nanoenergetic materials: Synthesis, properties, and applications,” Prog. Energy Combust. Sci. 61, 293–365 (2017). [CrossRef]
13. C. Griego, N. Yilmaz, and A. Atmanli, “Analysis of aluminum particle combustion in a downward burning solid rocket propellant,” Fuel 237, 405–412 (2019). [CrossRef]
14. T. Ecker, S. Karl, and K. Hannemann, “Modeling of aluminum particle combustion in solid rocket combustion chambers,” 53rd AIAA/SAE/ASEE Joint Propulsion Conference (AIAA, 2017), 2017–4781.
15. H. O. Kim, D. Laredo, and D. W. Netzer, “Measurement of submicrometer Al2O3 particles in plumes,” Appl. Opt. 32(33), 6834–6840 (1993). [CrossRef]
16. C.E. Kolb, S.B. Ryali, and J.C. Wormhoudt, “The Chemical Physics Of Ultraviolet Rocket Plume Signatures,” ltraviolet Technology II (SPIE, 1988), 2–23.
17. J. Burt and I. Boyd, “Monte carlo simulation of a rarefied multiphase plume Flow,” 43rd AIAA Aerospace Sciences Meeting and Exhibit, (AIAA, 1998), 2005–2964.
18. G. Busca, “The surface of transitional aluminas: A critical review,” Catal. Today 226, 2–13 (2014). [CrossRef]
19. Y. Plastinin, H. Sipatchev, G. Karabadzhak, B. Khmelinin, E. Szhenov, A. Khlebnikov, and Y. Shishkin, “Experimental investigation of alumina particles’ phase transitions and radiation,” 36th Aerospace SciencesMeeting & Exhibit, (AIAA, 1998), 98-0862.
20. V. K. Bityukov and V. A. Petrov, “The radiation of alumina melt in the middle IR spectral region at solidification in the course of free cooling,” High Temp. 49(3), 380–389 (2011). [CrossRef]
21. V. E. Titov, “The radiation of alumina melt in the visible and near IR spectra upon solidification in the process of cooling in ambient air,” High Temp. 46(6), 782–794 (2008). [CrossRef]
22. S. Gimelshein, G. Markelov, and J. Muylaert, “Numerical modeling of low thrust solid propellant nozzles at high altitudes,” 9th AIAA/ASME Joint Thermophysics and Heat Transfer Conference, (AIAA, 2006), 2006–3273.
23. R. Jeenu, K. Pinumalla, and D. Deepak, “Size distribution of particles in combustion products of aluminized composite propellant,” J. Propul. Power 26(4), 715–723 (2010). [CrossRef]
24. C. Pan, P. Shen, and S. Y. Chen, “Condensation, crystallization and coalescence of amorphous Al2O3 nanoparticles,” J. Cryst. Growth 299(2), 393–398 (2007). [CrossRef]
25. F.S. Simmons, Rocket Exhaust Plume Phenomenology, (The Aerospace, 2000).
26. R.A. Reed and V.S. Calia, “Review of aluminum oxide rocket exhaust particles,” 28th Thermophysics Conference, (AIAA, 1993), 93-2819.
27. J. Li, L. Bai, Z. Wu, L. Guo, and Y. Gong, “Ultraviolet scattering properties of alumina particle clusters at three phase states in aircraft plume,” J. Quant. Spectrosc. Radiat. Transfer 191, 40–45 (2017). [CrossRef]
28. J. Y. Li, Q. Li, S. K. Dong, and H. P. Tan, “Radiative properties of radially nonisothermal alumina particles with multiphase,” J. Quant. Spectrosc. Radiat. Transfer 113(4), 318–324 (2012). [CrossRef]
29. A.V. Rodionov, Y.A. Plastinin, J.A. Drakes, M.A. Simmons, and R.S. Hiers, “Modeling of multiphase alumina-loaded jet flow fields,” American Institute of Aeronautics and Astronautics 1–16 (1998).
30. Y.A. Plastinin, N.A. Anfimov, G.G. Baula, G.F. Karabadzhak, and B.A. Khmelinin, “Modeling of aluminum oxide particle radiation in a solid propellant motor exhaust,” 31st Thermophysics Conference, (AIAA, 1996), 96-1879.
31. T.A. Witten and L.M. Sander, “Diffusion-Limited Aggregation, a Kinetic Critical Phenomenon,” Phys. Rev. Lett. 47(19), 1400–1403 (1981). [CrossRef]
32. A. V. Filippov, M. Zurita, and D. E. Rosner, “Fractal-like Aggregates: Relation between Morphology and Physical Properties,” J. Colloid Interface Sci. 229(1), 261–273 (2000). [CrossRef]
33. J. Luo, Y. Zhang, Q. Zhang, F. Wang, J. Liu, and J. Wang, “Sensitivity analysis of morphology on radiative properties of soot aerosols,” Opt. Express 26(10), A420–A432 (2018). [CrossRef]
34. J. Mroczka, N. Riefler, T. Wriedt, and K. Skorupski, “A fast and accurate implementation of tunable algorithms used for generation of fractal-like aggregate models,” Phys. A 404, 106–117 (2014). [CrossRef]
35. R. Thouy and R. Jullien, “A cluster-cluster aggregation model with tunable fractal dimension,” J. Phys. A: Math. Gen. 27(9), 2953–2963 (1994). [CrossRef]
36. C. M. Sorensen, “The Mobility of Fractal Aggregates: A Review,” Aerosol Sci. Technol. 45(7), 765–779 (2011). [CrossRef]
37. W.R. Heinson, C.M. Sorensen, and A. Chakrabarti, “Does Shape Anisotropy Control the Fractal Dimension in Diffusion-Limited Cluster-Cluster Aggregation?” Aerosol Sci. Technol. 44(12), i–iv (2010). [CrossRef]
38. F. L. Braga and M. S. Ribeiro, “Diffusion Limited Aggregation: Algorithm optimization revisited,” Comput. Phys. Commun. 182(8), 1602–1605 (2011). [CrossRef]
39. J. Li, L. Bai, Z. Wu, L. Guo, and Y. Gong, “Scattering properties of alumina particle clusters with different radius of monomers in aerocraft plume,” J. Quant. Spectrosc. Radiat. Transfer 202, 233–239 (2017). [CrossRef]
40. L. Liu and M. I. Mishchenko, “Optics of water microdroplets with soot inclusions: Exact versus approximate results,” J. Quant. Spectrosc. Radiat. Transfer 178, 255–262 (2016). [CrossRef]
41. M.I. Mishchenko, N.T. Zakharova, N.G. Khlebtsov, G. Videen, and T. Wriedt, “Comprehensive thematic T-matrix reference database: A 2015–2017 update,” J. Quant. Spectrosc. Radiat. Transfer 202, 240–246 (2017). [CrossRef]
42. J. Luo, Y. Zhang, F. Wang, and Q. Zhang, “Effects of brown coatings on the absorption enhancement of black carbon: a numerical investigation,” Atmos. Chem. Phys. 18(23), 16897–16914 (2018). [CrossRef]
43. S. Hu, T. Gao, H. Li, M. Chen, and F. Zhang, “Simultaneously simulating the scattering properties of nonspherical aerosol particles with different sizes by the MRTD scattering model,” Opt. Express 25(15), 17872–17891 (2017). [CrossRef]
44. L. Bi, W. Lin, D. Liu, and K. Zhang, “Assessing the depolarization capabilities of nonspherical particles in a super-ellipsoidal shape space,” Opt. Express 26(2), 1726–1742 (2018). [CrossRef]
45. D. W. Mackowski, “A general superposition solution for electromagnetic scattering by multiple spherical domains of optically active media,” J. Quant. Spectrosc. Radiat. Transfer 133, 264–270 (2014). [CrossRef]
46. M. I. Mishchenko, L. Liu, and D. W. Mackowski, “Morphology-dependent resonances of spherical droplets with numerous microscopic inclusions,” Opt. Lett. 39(6), 1701–1704 (2014). [CrossRef]
47. M. Kahnert, “Optical properties of black carbon aerosols encapsulated in a shell of sulfate: comparison of the closed cell model with a coated aggregate model,” Opt. Express 25(20), 24579–24593 (2017). [CrossRef]
48. D. W. Mackowski and M. I. Mishchenko, “A multiple sphere T-matrix Fortran code for use on parallel computer clusters,” J. Quant. Spectrosc. Radiat. Transfer 112(13), 2182–2192 (2011). [CrossRef]
49. M.I. Mishchenko, L.D. Travis, and A.A. Lacis, Scattering, Absorption, and Emission of Light by Small Particles, (Cambridge University, 2002).
50. L. Liu and M. I. Mishchenko, “Scattering and radiative properties of complex soot and soot-containing aggregate particles,” J. Quant. Spectrosc. Radiat. Transfer 106(1-3), 262–273 (2007). [CrossRef]
51. M. I. Mishchenko, L. Liu, L. D. Travis, and A. A. Lacis, “Scattering and radiative properties of semi-external versus external mixtures of different aerosol types,” J. Quant. Spectrosc. Radiat. Transfer 88(1-3), 139–147 (2004). [CrossRef]
52. L. Wang, S. L. Jacques, and L. Zheng, “MCML–Monte Carlo modeling of light transport in multi-layered tissues,” Comput. Methods Programs Biomed. 47(2), 131–146 (1995). [CrossRef]
53. K. Franklin Evans, “The spherical harmonics discrete ordinate method for three-dimensional atmospheric radiative transfer,” J. Atmos. Sci. 55(3), 429–446 (1998). [CrossRef]
54. L. H. Caveny and A. Gany, “Breakup of Al/Al2O3 agglomerates in accelerating flowfields,” AIAA J. 17(12), 1368–1371 (1979). [CrossRef]
55. G. N. Plass, “Temperature dependence of the mie scattering and absorption cross sections for aluminum oxide,” Appl. Opt. 4(12), 1616–1619 (1965). [CrossRef]
56. Y. Plastinin, G. Karabdzhak, B. Khmelinin, G. Baula, and A. Rodionov, “Ultraviolet, visible and infrared spectra modeling for solid and liquid-fuel rocket exhausts,” 39th Aerospace Sciences Meeting & Exhibit, (AIAA, 2001), 2001–0660.
57. D. L. Parry and M. Q. Brewster, “Optical constants of Al2O3 smoke in propellant flames,” J. Thermophys. Heat Transfer 5(2), 142–149 (1991). [CrossRef]
58. H. Li, C. Liu, L. Bi, P. Yang, and G. W. Kattawar, “Numerical accuracy of “equivalent” spherical approximations for computing ensemble-averaged scattering properties of fractal soot aggregates,” J. Quant. Spectrosc. Radiat. Transfer 111(14), 2127–2132 (2010). [CrossRef]
59. C. Liu, P. R. Lee, and P. Yang, “Inhomogeneity structure and the applicability of effective medium approximations in calculating light scattering by inhomogeneous particles,” J. Quant. Spectrosc. Radiat. Transfer 146, 331–348 (2014). [CrossRef]
60. J. C. Maxwell Garnett, ““Colours in metal glasses and in metallic films,” Philosophical Transactions of the Royal Society of London. Series A,” Containing Pap. Math. Phys. Char. 203, 385–420 (1904).
61. D. A. G. Bruggeman, “Berechnung verschiedener physikalischer Konstanten von heterogenen Substanzen. II. Dielektrizitätskonstanten und Leitfähigkeiten von Vielkristallen der nichtregulären Systeme,” Ann. Phys. 417(7), 645–672 (1936). [CrossRef]
