Optical Performance of Sodium Tungsten Bronze Particles in Transparent Matrix: An Ensemble Particle Modeling Study

Different from published works via single particle modeling, the optical performance of multiple sodium tungsten bronze particles (a near-infrared shielding and visible light transparent material) in a transparent matrix was modeled in this study. The effects of particle size, concentration, and shape on the optical performance of particles were investigated. Our particle ensemble modeling showed that multiple light scattering among particles dramatically increased the chance of near-infrared light absorption in a particle-suspended matrix, which differs from that observed in a single particle modeling. The ensemble and single particle modelings evidenced the increase of near-infrared shielding (extinction) with the increase of particle size. For the particle ensemble at a fixed mass concentration, a favored particle size existed for shielding the maximal near-infrared light. It was also found that the attenuation coefficient (normalized by the mass concentration) varied with the mass concentration of particles. The above variation was dependent on the particle size. The near-infrared shielding performance can be improved by cubic-shaped particles due to the localized surface plasmon resonance (LSPR) occurred at the particle corners and edges.


INTRODUCTION
Glass windows with the feature of shielding the NIR (near infrared) light in the solar spectrum have been a solar control strategy, which has been widely applied to reduce the energy consumption of heating, ventilation, and air conditioning (HVAC) systems in buildings and vehicles. The NIR shielding of glass windows is typically realized by low-emissivity coatings made of silver, indiumor antimony-doped tin oxide (ITO and ATO). Although providing good solar heat shielding, complex coating processes are required to place the key metallic layer in a low-emissivity coating. ATO and ITO-based coatings are also excellent for heat shielding, but silver, indium, and antimony costs are expensive.
Alternatively, Tungsten bronze (MxWO3, M = Li, Na, K, Ru, Cs, and NH4) is a promising NIR shielding material in solar energy control applications (Chao et al., 2019). These materials have been synthesized by various methods, for example, hydrothermal (Shi et al., 2014), solvothermal (Liu et al., 2015), fused salt electrolysis (Shanks, 1972), solid-phase reaction (Zhu et al., 2019), flame pyrolysis (Hirano et al., 2018), aerosol-assist spray (Chen and Tu, 2022), ball milling (Song et al., 2020), and sputtering (Sato et al., 2021). For solar control applications, tungsten bronze particles are synthesized and then suspended inside or coated on the surface of transparent media, which could be an either polymer (Guo et al., 2011) or a glass matrix (Chao et al., 2021). Due to the optical properties of tungsten bronze, a particle-suspended coating can shield NIR and be transparent to visible light.
Unlike a solid NIR-shielding material layer, the optical performance of a particle-suspended layer is further influenced by the physical characteristics of tungsten bronze particles, for example, the size, shape, and concentration of particles in carry media (in addition to the optical property of the material itself). It is because of the interaction of particles with light. Due to the sizes of these tungsten bronze particles (typically less than 1 µm in size) being close to the wavelength of electromagnetic radiation in visible and NIR ranges (Visible: 400 nm-750 nm; NIR: 750 nm-2,500 nm), the interaction between the particles and the light in the Vis-NIR wavelength range could be intensified for particles in certain sizes. The above observation has been previously reported. The experimental works (Adachi et al., 2010;Takeda et al., 2008) on lanthanum hexaboride, another NIR shielding material, show that the particle size can highly affect the NIR extinction. In their works, particles in small sizes have a higher extinction in the 900 nm-1200 nm NIR range. The shape of particles also affects the optical performance of particle-suspended media. The modeling (Chao et al., 2021;Sun et al., 2020) via the discrete dipole approximation (DDA) (Draine and Flatau, 2013) showed that the light scattering extinction varied with the size and shape of tungsten bronze particles (CsxWO3, NaxWO3). The valley of absorptance curves shifts with different shapes of particles (when comparing the light absorption curves of spherical, cubic, cylindrical, and tetrahedral particles). When the aspect ratio of particle shape increased, the valley of light absorptance curves for particles was red-shifted in the solar spectrum. More, the valley of the light extinction curves for particles was blue-shifted and broadened as the particle size decreased. The NIR optics of tungsten bronze particles are also affected by the intrinsic properties of materials specifically related to the electron configuration on the surface and crystal structure of particles. For NIR shielding materials, abundant free electrons are presented on the surface of metallic particles (tungsten bronze or lanthanum hexaboride). The localized surface plasmon resonance (LSPR), observed for nano-sized metallic particles, is the interaction between the electromagnetic radiation and the excited surface-free electrons on the particle surfaces ("Localized surface plasmon," 2022). LSPR enables tungsten bronze particles to absorb the specific range of NIR radiation. Adachi's work (Adachi and Asahi, 2012) showed that the ability of cesium tungsten bronze particles to absorb the NIR was due to the plasmon and polaron. The polaron absorption of tungsten bronze particles is due to the doping of alkali ions in the WO3 lattice. The relationship between the concentration and species of doped ions on the optical properties of particles has been investigated by the density functional theory (DFT) method, including the dielectric function, reflectance, and attenuation of tungsten bronze material (Tegg et al., 2017;Yang et al., 2014).
In addition to the modeling via quantum mechanics, the modeling based on the Mie scattering has also been applied to analyze the NIR optics of a single particle. In the Mie theory, the dielectric function (or refractive index) was treated as a constant (as an input variable) at a given wavelength. The extinction cross-section area of a particle in a specific diameter was calculated for a given light wavelength. However, the Mie scattering applies to a particle with a diameter close to the incident light wavelength (Hahn, 2009). The DDA method (Draine and Flatau, 2013) has also been applied to study the optical performance of individual particles via the dipole approximation for the object structure (with the consideration of nanoparticles). The above method modeled the optical characteristics of individual nanoparticles in different morphology (e.g., spherical, cubic, or cylindrical shapes). It is challenging to model super-micrometer-sized particles due to the limit of computational resources, and it also requires the dielectric function of the matter as the input parameter.
Although the research on the NIR optics for tungsten bronze particles has been reported, all the modeling works only focused on individual particles, not particle ensembles. In reality, NIR shielding particles are randomly suspended in a transparent carry matrix (or layer). The effect of particle packing ratio on the optical performance of a tungsten-bronze-particle-suspended matrix has not been investigated.
The objective of this work is thus to study the NIR shielding performance of an ensemble of tungsten bronze particles. In this study, the effects of size, concentration (i.e., packing ratio), and shape of particles on the NIR shielding performance of particles were considered. Maxwell's equations (solved by a finite element method) were used to calculate the NIR optics of sodium tungsten bronze particles randomly dispersed in a transparent carry media. In addition, experiments were performed to validate our experimental findings. For reference, a single tungsten bronze particle's optics were also modeled.

For the Optical Modeling of Particles
The COMSOL software with the optical module was selected for this optical modeling. In the optical module of COMSOL, the Maxwell equations for electromagnetic waves are solved by the finite element method (COMSOL, 2022). Two different computational domains were used in the modeling: one domain is for an ensemble of randomly placed particles (in a cubic domain), and the other is for a single particle (in a spherical domain).
Sodium tungsten bronze particles were selected for the modeling. The dielectric function (or refractive index) of sodium tungsten bronze (Na0.74WO3) in the wavelength ranging from 300 nm to 1,000 nm was obtained from Owen's work (Owen et al., 1978). In the above work, the optical property of sodium tungsten bronze was measured by the polarization modulation ellipsometry on a polished slab of bulk crystalline tungsten bronze synthesized by electrolysis growth from a fused salt melt. A similar process was also applied in measuring other bulk tungsten bronze materials doped by cesium (Adachi and Asahi, 2012) and potassium (Lynch et al., 1973). The dielectric function of sodium tungsten bronze in the wavelength from 1,000 nm to 1,500 nm was obtained from the extrapolation of Owen's data. The above extrapolation was also assisted by the data obtained in Tegg's work's density functional theory calculation (Tegg et al., 2017). The refractive index of sodium tungsten bronze particles used in the calculation is shown in Fig. 1. In this work, the carry media for suspending particles was isopropyl alcohol (IPA). The refractive index of IPA was set to 1.37.
2.1.1 For the optical modeling of a particle ensemble Fig. 2 shows the computational domain for the cases with an ensemble of particles. Particles in a specified total number of n and the diameter of di were randomly distributed in the domain. The incident beam entered the domain from the top and exited from the bottom. The periodic boundary conditions were applied to the sidewalls of the domain. In other words, the domain was irradiated by a plane wave beam from the top. The square cross-section of the domain has the side of w. The path length, l, of the light through the domain, i.e., the thickness of the computational domain, were randomly assigned within the range defined by Eq. (1) for the modeling: Given the total number of particles, n, and their sizes, di (equivalent volumetric diameter, where i is the index for each particle), the packing ratio of particles in the computational domain can be calculated as Eq. (2).
VD is the volume of the computational domain; VpTotal is the total volume of all the particles; p is the volumetric packing ratio of particles in the domain; and w is the width of the square crosssection of the domain. The transmittance of the light passing through the computational domain was calculated based on the ratio of the energy integration on the outgoing surface to that on the incident light surface. The attenuation of the light for the computational domain was calculated from the transmission by Eq. (3).
where T is the transmittance of the incident beam. I0 is the incident beam energy. It is the outgoing beam energy. µatt is the attenuation coefficient of the computational domain. l is the thickness of the computational domain. In addition, the reflectance R can be obtained by the ratio of energy integration for the reflected beam Ir to the incident beam I0 on the incident surface: The absorptance A can be easily obtained as Eq. (5).
The attenuation coefficient µatt can be normalized by the volumetric packing ratio p, marked as µatt/p.
In the modeling, the optical performance of particles in 40, 100, 140, 200, 300, and 400 nm for light in the wavelength ranging from 300 nm to 1500 nm was calculated. Tetrahedral elements were selected in the calculation. The maximum mesh size was kept less than 1/10 of the wavelength in void space and less than 1/7 of the particle size in particle-occupied space. The total number of mesh elements used in each modeling case ranged from 10 6 to 10 7 . Table S1 summarizes the total number of elements used in different cases.
To investigate the effect of particle shape, particles in both spherical and cubic shapes were selected in this study. Note that the sizes indictaed in both sepherical and cubic particle cases are the volumetric equivalent diameters. In other words, individual sepherical and cubic particles with the same size should have the same particle volume. Because of the random nature of particle placement in the computational domain and the domain length, each data shown in the result section was the average of multiple runs (more than 30 runs) for each combination of specific parameters given in above. For particles in the cubic shape, the orientation of a cube was also randomized in addition to the placement and the domain length.
2.1.2 For the optical modeling of a single particle For the comparison, the light scattering by a single particle was also performed. Instead of using Mie theory, the computational domain calculating the light scattering of a single particle is shown in Fig. 3. A spherical particle made of Na0.74WO3 was placed at the center of the spherical domain.
To reduce the computational time, a typical computational domain was divided by a core and a shell layer, i.e., near-and far-fields. The dimensions of each field follow Eq. (6).
where tFF is the thickness of the far-field domain layer; rp is the radius of the spherical particle; λ is the wavelength of the scattering wave; tPML is the thickness of the perfect match layer. The total number of mesh elements used in each modeling of a single particle was 10 4 . Table S2 summarizes the total number of elements used in different cases.
All the geometry is in a background scattering plane wave field, traveling in the x-axis direction (see Fig. 3). The scattering cross section is defined in Eq. (7) (Bohren and Huffman, 2008) as where Csca is the cross section of light scattering; Ws is the scattered energy crosses the particle's surface; Ii is the energy of incident light on the section area of the particle. The absorption crosssection for a particle is the definition (Bohren and Huffman, 2008) as Eq. (8).
where Cabs is the cross-section of absorption; Wa is the net (absorbed in our cases) energy crosses the particle's surface; Ii is the energy rate incident on the section area of the particle. The extinction cross-section Cext can then be calculated (Bohren and Huffman, 2008) as Further, the extinction efficiency Qext, scattering efficiency Qsca, and absorption efficiency Qabs are shown in Eq. (10) (Bohren and Huffman, 2008 The scattering efficiency Qext can be normalized by the effective radius aeff, which is equivalent to the radius rp for a spherical particle. Note that the effective -radius -normalized scattering efficiency Qext⁄aeff has the same unit (m -1 ) as that of the attenuation coefficient, μatt.

For the Validation Experiment
Experiments were also performed to validate the calculated results. The experimental visible-NIR spectrum for particle-suspended media was measured by the method published in our previous work (Tu and Chen, 2023). The method synthesized sodium tungsten bronze particles using an aerosol-assisted route and size-classified by a MOUDI (micro-orifice uniform deposition impactor). An impactor collects particles via particle inertia (characterized by the particle Stokes number). Nine impaction stages are in a MOUDI, numbered from the stage, collecting the particles according to sizes. Most of the produced particles were collected at the 4 th to 9 th stages. The cutoff sizes of particles deposited in the above stages are 36 nm, 77 nm, 160 nm, 320 nm, 620 nm, and 1,100 nm from the 9 th stage to the 4 th stage, respectively. For the optical characterization, particles collected at selected stages were recovered and dispersed in IPA media. All the samples were through the ultrasound vibration to keep particles well suspended in IPA media immediately before the optical characterization. The attenuation coefficient obtained from the measurement can be normalized by the particle mass concentration as μatt⁄c (where c is the mass concentration of particles). Fig. 4 shows the calculated optical performance (i.e., attenuation, absorption, transmission, and reflectance) of an ensemble of sodium tungsten bronze particles in different sizes as a function of the light wavelength. The attenuation coefficient (in Fig. 4(a)) showed that the peak of the curves moved towards a longer wavelength regime (i.e., red-shifted) as the particle size increased, resulting in the improvement of attenuation coefficient for the NIR in a long wavelength (> 1,200 nm) for large particles. The above is observed in the transmittance curves (in Fig. 4(b)), where the light transmittance decreased in a wavelength longer than 1,000 nm. In the meantime, the transmittance of visible light deteriorated with the increase in particle size. The absorptance curves (in Fig. 4(c)) of the particle-suspended matrix indicated that the NIR shielding performance was mostly attributed Fig. 4. Calculated optical performance of an ensemble of sodium tungsten bronze particles in different sizes as the function of light wavelength: (a) for the attenuation coefficient, (b) for the transmittance, (c) for the absorptance, and (d) for the reflectance. The particle packing ratio was 0.1. Note that the transmittance, absorptance, and reflectance were normalized by the light path (the thickness of the domain) as 2,000 nm.

Effect of Particle Size
to the light absorption of sodium tungsten bronze. It is because the light reflection curves (in Fig. 4(d)) showed much less light was reflected than absorbed. Fig. 4(c) also showed the enhanced absorptance of particle-matrix in most of the wavelength range with the increase of particle size. The reflectance curves (in Fig. 4(d)) showed that large particles have high reflectance in the wavelength range above 800 nm (compared with small particles).
To validate the above observation in quality, experiments were performed to characterize the attenuation coefficient of tungsten bronze particles sized-classified by the MOUID. The result of the attenuation coefficient measurement for an ensemble of sodium tungsten bronze particles is given in Fig. 5. It can be found that particles on stage s9 (with the particle cutoff size of 36 nm) have reduced NIR shielding than those collected on stage s8 (with the cutoff size of 77 nm). More, for particles collected at the MOUDI's s4 to s6 stages (with the cutoff sizes of 1,100 nm, 620 nm, and 320 nm, respectively), the NIR shielding of particle-matrix increased with the decrease of particle size. To clearly show the variation of the NIR shielding with the particle size, the fraction for transmitted visible and NIR radiation under the solar irradiation for particles in different particle The mass concentration of particles in the carrying media was kept at 1 mg mL -1 . The vis-NIR spectrum measurement of particle suspensions was based on the method reported in our previous work (Tu and Chen, 2023). Fig. 6. The general trend of both calculated and measured data are consistent in quality. Note that the transmitted fraction of visible and NIR lights under the solar radiation defined herein was based on the definition of solar energy transmittance selectivity (SETS) in Eq. (11) (Hirano et al., 2018;Lee et al., 2014;Zeng et al., 2015).

sizes (both calculated and measured) is shown in
where E(λ) comes from the measurement of solar irradiance in a standard procedure (ASTM International, 2020). Calculated and measured fractions overlapped in the 40-400 nm particle size range. The calculated transmitted visible radiation (in Fig. 6(a)) shows the same downturn tendency observed in the experiments. For the transmitted NIR radiation (in Fig. 6(b)), both calculated and measured data sets have a turning point as the particle size increased from 40 nm to 400 nm. The above indirectly supports the conclusion that the variation of the NIR shielding for particles in different sizes is largely due to the interaction between the particle size and the irradiance wavelength. Therefore, fine particles (cutoff size of 36 nm) having poor NIR shielding are due to the particle size being much less than the NIR irradiance wavelength. For the comparison, we further calculated the optical performance of a single particle under light radiation. Fig. 7(a) shows the extinction efficiency normalized by the effective radius Qext⁄aeff (which equals the radius if the particle is spherical). The unit of Qext⁄aeff, which is m -1 , is the same as that of the attenuation coefficients μatt in the particle ensemble modeling. Compared with the data obtained in the particle ensemble modeling (in Fig. 4), the Qext⁄aeff curves for a single particle (shown in Fig. 7(a)) have less difference in the NIR range, particularly for particles in the 300 nm and 400 nm sizes. The particle ensemble modeling shows higher efficiency in removing NIR at 950 nm-1500 nm in the irradiance for particles of 300 nm to 400 nm. The extinction efficiency Qext of a single sodium tungsten bronze particle is also given in Fig. 7(b). The peaks of the extinction curves moved towards the red wavelength and became flat as the increase of particle size. In other words, large particles easily interact with long wavelengths. The extinction efficiency of particles composes of scattering and absorption efficiency (shown in Figs. 7(c) and 7(d), respectively). It can be found that (1) the scattering efficiency of particles is generally higher than the absorption one, and (2) the difference in the scattering efficiency as the change of particle size is also larger compared to that for the absorption coefficient. The above finding in single particle modeling contradicts that in particle ensemble modeling.
To further evidence the role change of the reflectance (or Qsca) and absorptance (or Qabs) in the ensemble and single particle modeling, the ratio of the light reflectance (or Qsca) to the light absorptance (or Qabs) was calculated in Fig. 8. It can be found that the above ratio is much less in the ensemble particle modeling (Fig. 8(a)) than in the single particle modeling (Fig. 8(b)). It again indicates that the removal of irradiance beam energy is mostly undertaken by the absorption in the ensemble particle modeling, which differs from the energy removal mainly done by scattering in the single-particle modeling. We also found that, in both modelings, particles with large sizes have higher ratio values in the NIR range compared with those for particles with a small size. The above finding indicates that the particle's reflection (or scattering) played an increasing role in the NIR shielding with the particle size increase. One possible explanation for the observed difference between the ensemble and single particle cases is that the multiple scattering among particles dramatically increased the chance of the NIR energy absorption. Moreover, although the NIR absorption for a single particle is very limited, the total NIR absorption of a particle Fig. 7. The calculated optical performance of single tungsten bronze particles in different sizes: (a) extinction efficiency (normalized by the effective radius of the particle); (b) extinction efficiency, (c) scattering efficiency, and (d) absorption efficiency. ensemble can be dramatically varied with the particle size change. It is why the curves of Qext⁄aeff (Fig. 7(a)) in the NIR range show less variation with the change of particle size when compared to the μatt (Fig. 4(a)). The scattered intensity in the far field of a single particle in different sizes can be found in Fig. S1.

Effect of Particle Packing Ratio (or Concentration)
To support the hypothesis of the presence of multiple particles increasing the absorption of NIR scattered energy, we performed the particle ensemble modeling at two different particle densities (i.e., 0.01 and 0.1) for particles in the sizes of 40, 140, and 300 nm. Our calculation result is shown in Fig. 9. Fig. 9(a) shows that, for 40 nm particles, the low packing ratio case has the higher attenuation coefficient curve (μatt⁄p) in the 600-800 nm wavelength range (which partially corresponds to the measured data in Fig. 10(a)) compared with the high-density case. For 300 nm particles, the attenuation in two packing ratio cases is at the same level (close to the experimental data in Fig. 10(b)). It indicates that the level of multiple scattering in different particle packing densities is the reason for the role change in the NIR absorption of particles.  8. (a) The ratio of the light reflectance to light absorptance (R/A) obtained in the ensemble particle modeling (in Figs. 4(d) and 4(c)). (b) The ratio of the light scattering efficiency (Qsca) to the light absorption (Qabs) obtained in a single particle modeling (in Figs 7(c) and 7(d)).
Figs. 9(b), 9(c), and 9(d) show the ratio of light reflectance to absorptance, i.e., R/A ratio, of particles with the sizes of 40, 140, and 400 nm in two packing densities (as a function of light wavelength). It is found that small particles (40 nm) in the low packing ratio (p = 0.01) largely have a lower R/A ratio value than those in the high packing ratio (p = 0.1), particularly in the NIR wavelength range. For particles in the 140 nm size, the curves of the R/A ratio for two different packing densities became close to each other as the wavelength increased and eventually crossed each other in the NIR range. For particles in the 400 nm size, the R/A ratio curve for the p = 0.01 case was higher than that for the p = 0.1 cases in most NIR wavelength ranges. It is thus expected that, for particles with large sizes, the R/A curve for the p = 0.01 case would be much higher than that for the p = 0.1 cases in the NIR range. Given the result shown in Fig. 8(a), it is found that particles with a large size would have a higher fraction of reflection in removing NIR incident energy. From Figs. 4(c) and 4(d), it can be speculated that the effect of multi-scattering strength on the absorption and reflection is varied with the particle sizes. At a specified volumetric concentration of particles, the average distance between particles increases with the particle size. When the average distance among particles is much larger than the irradiance wavelength, the optical behavior of the particle ensemble would be like that of a single particle.
To support the above modeling result, the optical characterization of particle samples collected at the MOUDI stages of s6 and s9 (numbered from the stage collecting the largest particles) were performed at three different particle mass concentrations (i.e., 0.1, 1.0, and 10 mg mL -1 ). The sizes of collected particles on each impaction stage of a cascade impactor range from the cutoff size of the prior stage (having a larger cutoff size) to the cutoff size of this stage. The sample particles collected on stages s6 and s9 are thus around 320 nm and 36 nm, respectively. The selection of the impaction stages was because stage s6 and s9 samples include particles in the sizes of 400 and 40 nm, respectively. The result of the optical characterization of the above particle samples is given in Fig. 10. Due to the particle mixture of different sample sizes, we can only qualitatively compare the modeled and measured results. It is found that the higher the particle mass concentration for selected particle samples, the lower the attenuation coefficients (μatt⁄c) in the detectable light wavelength range. The above confirms that the multiple interactions of particles with light increased with particle concentration. For the samples including 40 nm (i.e., from the stage s9), the attenuation coefficient in the detectable wavelength range is generally consistent with that observed in the modeling: in the 600-800 nm range, a higher attenuation coefficient occurred at a lower mass concentration in the 600-800 nm range and approaches to approximately the same level in the NIR range. Because of particles in sizes different than 40 nm in the samples, Fig. 9. (a) The calculated attenuation coefficient (normalized by the particle packing ratio) of ensemble particles in 40 nm and 300 nm diameters and at the particle packing ratio of p = 0.01 and p = 0.1. The ratio of the reflectance to absorptance (R/A) obtained from the particle multiple ensemble modeling (at the particle packing densities of 0.01 and 0.1) for particles in the diameters of (b) 40 nm, (c) 140 nm, and (d) 300 nm.
the attenuation curves are red-shifted. The attenuation coefficients for the samples containing 400 nm particles are approximately at the same level for three different mass concentrations in the 600-800 nm range and red-shifted compared to the modeled data for 400 nm. In the NIR range, the variation of attenuation coefficient curves at different mass concentrations was slightly increased compared to that in the 600-800nm range (not as obvious as what was observed in the modeling). Considering particles of sizes other than 400 nm in the samples, the experimental observation in the NIR range is consistent with what was obtained in the modeling. Fig. 11 compares the optical performance of sodium tungsten bronze particles (in cubic and spherical shapes. Particles under consideration shared the same volumetric diameter, i.e., 140 nm. The cubic particle matrix shows a higher attenuation on the NIR, as evidenced in Fig. 11(a). The transmittance of particle-matrix (given in Fig. 11(b)) also demonstrates the strong shielding in a wavelength longer than 1000 nm. The particle-matrix's absorptance spectrum shows that most Fig. 10. The measured light attenuation coefficient (normalized by particle mass concentration) for the suspension of ensemble Na0.7WO3 particles collected on the MOUDI stages of s6 and s9. The cutoff sizes of particles for the MOUDI stages s6 and s9 are 320 and 36 nm, respectively. The particle mass concentration in the suspension matrix is in the unit volume of carry media. enhanced NIR shielding came from absorption, not light reflectance. It is because the value of the reflectance coefficient (given in Fig. 11(d)) is much lower than the absorptance (given in Fig. 11(c)). Therefore, the multiple light scattering was more enhanced on the NIR shielding for cubic particles than spherical ones in the matrix. The localized surface plasmon resonance (LSPR), which occurred at the sharp corners and edges of cubic particles, resulted in a more intensified electric field on the surface compared with that of spherical ones (Agrawal et al., 2018), which enables cubic particles absorbing more of the NIR radiation.

CONCLUSION
In this study, the optical performance of a NIR shielding particle ensemble was numerically modeled by solving the Maxwell equations via a finite element method. The particles under investigation were sodium tungsten bronze. COMSOL software with the optical module was used in the modeling. For the comparison, the optical performance of a single NIR shielding particle was also calculated. We compared the optical performance of a particle ensemble and a single particle to investigate the effects of particle size, concentration, and shape on the NIR shielding and visible light transmission. In addition, an experiment was also performed to collect particle samples at different mass concentration using MOUDI, and the optical performance of prepared samples were characterized. The general observations on the light attenuation coefficient curves obtained from experiments were consistent with those in the modelings (under the consideration of multiple-sized particles in the experimental samples).
Our investigation indicated that the ensemble and single particle modelings both show the increase of NIR shielding (extinction) with the increase of particle size in a certain range. The multiple scattering among particles can greatly increase the overall absorption of NIR. Unlike a single particle, in which the scattering is the major mechanism for the NIR extinction, the overall NIR absorption for a particle ensemble is higher than that of the reflection. For a particle ensemble, the ratio of light reflection to absorption in the NIR range increased with the increase in particle size. For the particle ensemble at a fixed mass concentration, a favored particle size existed for shielding the maximal NIR light. Our study also showed that the attenuation coefficient (normalized by the mass concentration) varied with the mass concentration of particles. For small particles (~40 nm), the attenuation coefficient decreased with the increase of the particle mass concentration. The above trend was Fig. 11. Calculated optical performance of an ensemble of sodium tungsten bronze particles in 140 nm as the function of light wavelength: (a) for the attenuation coefficient, (b) for the transmittance, (c) for the absorptance, and (d) for the reflectance. The particle packing ratio was 0.1. Note that the transmittance, absorptance, and reflectance were normalized by the light path (the thickness of the domain) as 2,000 nm.
not observed with the increase in particle size (~400 nm). It is because multiple scattering among particles resulted in the role change from the reflection to absorption in the NIR range as the particle size increased. Due to the sharp edges and vertex, cubic sodium tungsten bronze particles had better NIR shielding performance than spherical ones. The sharp geometry features of cubic particles intensified the surface electric field, enabling more light interaction with particles.

SUPPLEMENTARY MATERIAL
Supplementary material for this article can be found in the online version at https://doi.org/ 10.4209/aaqr.230085 Adachi, K., Asahi, T. (2012). Activation of plasmons and polarons in solar control cesium tungsten