1. Introduction

Optically transparent polymer materials have been widely used as nanoparticulate coatings with different functions, such as optical filters, lenses, antireflection films [1], and thermal barrier coatings [2,3]. Titanium dioxide (TiO₂) and Indium tin oxide (ITO) are proposed as two of the promising candidates of paints to achieve a high transparency in the visible light region. However, these factors represent quite different spectral characteristics in the ultraviolet and infrared wavebands. TiO₂, a typical white pigment, has excellent UV-shielding ability. In contrast, ITO is a conducting oxide with high reflectivity of infrared region.

Many investigations on TiO₂ pigmented coatings focus on the effects of the particle diameter and volume fraction. Vargas et al. [4] optimized the solar reflectance of rutile and anatase pigmented coatings by modulating the size parameters of the pigments and considered the logarithmic size distributions of the pigments in the calculation of diffuse reflectance of rutile pigmented coatings [5]. Maruyama et al. [6] evaluated the radiative properties of TiO₂ particles with different particle sizes and volume concentrations in an absorbing polyethylene resin using the Mie solutions as well as the far field and near field approximations.

Some researchers worked on designing coatings that embrace both the thermal and aesthetic effects, i.e., with high reflectance in the NIR region but low reflectance in the VIS region. Baneshi et al. [7] analyzed the influences of the particle size, volume fraction of the pigments and coating thickness on collimated and diffuse solar irradiation and compared the optimum effects of three white pigments, including TiO₂, ZnO and Al₂O₃. Gonome et al. [8] proposed a cool black-color coating pigmented with CuO particles to weaken the visual discomfort caused by the reflecting solar glare and compared its performance with the coating pigmented with TiO₂. Cheng et al. [9] proposed a double-layer CuO pigmented coating with a higher volume fraction in the upper layer to achieve a better reflectance in the NIR region. Another double-layer coating pigmented with TiO₂ and ZnO was reported by Chai et al. [10] to improve the thermal and aesthetic performances.

ITO film has been studied for a long time and is widely applied as infrared reflecting coating on windows due to its favorable combination of optical, electrical, mechanical and chemical properties. The studies on ITO thin film revolve around its high visible transmittance and high reflectance in the NIR region as well as the electrical properties including the conductivity, mobility and carrier concentration. The quantitative theories for the ITO film’s optical performance due to different fundamental physical processes, mechanisms and applications were reviewed in [11]. The effects of annealing [12], ionized impurity scattering [13] and substrate temperature [14] on thin coatings consisting of ITO nanoparticles were investigated. Furthermore, the spectral transmittance and reflectance of some multilayer doping with other material were discussed, i.e., an ITO-Ag-ITO multilayer [15].

Previous studies that focused on pigmented coatings always assumed the morphology of the particles as homogeneous spheres, or as inhomogeneous spheres with a degree of internal porosity [4]. The cylindrical models were also employed in the simulation of ITO nanowhiskers [16]. The optical properties of the pigment particles were calculated by either Mie theory or Far field approximation (FFA) and Near field approximation (NNA) when the pigment particles dispersed in an absorbing medium (e.g., polyethylene resin) [6]. However, Li et al. [17] proposed a polydisperse submicrometer-sized TiO₂ nanocrystallite aggregate with nanometer-sized crystallites. The advantages of this aggregate lie in amplifying the specific surface area by using the nanometer-sized crystallite as a basic building unit and enhancing the scattering due to the size of the assemblies on the order of submicrometers. The morphology and structural characteristic of TiO₂ aggregates were shown in the SEM and TEM images (see Figs. 1 and 3 in [17]) which may have a strong influence on the light scattering properties of the pigment particles [18].

The individual particles constitute clusters in three connecting ways: point-touch [19,20], overlapping [21–23] or sharing a necking volume between two neighbor particles. Overlapping and volume sharing are always employed in the simulation of sintered aggregates, which can be called “chemically bonded” [24]. The necking structures between the monomers appearing in the TiO₂ and ITO clusters were built by different models, and their impact on the optical properties were deeply studied. Wriedt et al. [25] used the metaball algorithm to characterize the morphology of TiO₂ sintered aggregates. Skorupski et al. [26] applied different sinter neck models for the primary particles of ITO nanoparticle clusters. These studies have revealed that the connecting structures notably influenced the light scattering properties of the sintered aggregates in contrast to the spherical particles with the equal volume-equivalent radius.

We proposed another relatively simple but effective method named the “particle superposition model”. The advantages of this modeling method lie in its fewer modulating parameters and greater convenience for simulating and describing realistic particles or aggregates. Only three parameters are employed to design or simulate the required object: the number, the radius and the center positions of the constituent spherical monomers. The “particle superposition model” method was used to create a surface roughness of the wavelength-size particles with bumps or pits [27] and construct the large irregular particle model for lunar dusts simulation [28]. Compared with other necking simulations, a smaller homogeneous ball is applied to mimic the connection between the particles based on the “particle superposition model”. In this paper, this modeling method was utilized to build TiO₂ and ITO clusters with fractal-like morphology and necking-ball structures to simulate the aggregates, as shown in the SEM and TEM images [17]. The respective complex refractive index $m$ plays an important role in the direct or inverse radiation problems [29,30]. Since TiO₂ and ITO show opposite spectral features of $m$, the main aim of this paper is to compare the scattering and reflection properties between these two clusters. Previous investigations on the pigmented coatings focus on the effects of the particle diameter and volume fraction. In our work, the impacts of different wavelengths of the incident light $\lambda$, ball-necking factor $\eta$ and refractive index of the ambient medium on the light scattering characteristics of TiO₂ and ITO clusters were discussed.

2. Method

2.1 The modeling of TIO and TiO₂ clusters

The fractal geometry and some relevant parameters are normally applied to describe the morphology of the fractal-like TiO₂ and ITO aggregates as follows [31]:

$R_g$is the radius of gyration and presents the deviation of the overall aggregates radius in an aggregate [18]. This variable is defined by the following equation [26]:

A fractal-like cluster with ball-necking connecting structures was modeled by the particle superposition method to represent the sintering aggregate. The validity and reliability for a necking factor $\eta$was presented to denote the degree of necking of sintered aggregates and defined by the Eq. (3):

Similarly, the particle superposition model was used to stimulate the ITO and TiO₂ clusters in the pigmented coatings. The size parameters of the two aggregates are exhibited in Fig. 1. The original monomer radii of the point-touch and ball-necking aggregates were assumed to be $R_1$ and $R_2$, respectively. We conducted a series of simulations of ITO and TiO₂ clusters with the necking balls of various $\eta$. The number of monomers $N_{\text{p}}$ was fixed as 25, and $R_1$ was fixed at 25 nm. To keep the total volume equal to the corresponding point-touch model, a small amount of shrinkage was introduced to the necking model, i.e., $R_2<R_1$ and $R_2$ decreased with the increasing $r$. Note that the distance L between the center of each primary particle was constant instead of being unfixed.

 

Fig. 1 The point-touch and ball-necking aggregate models built by the “particle superposition model”.

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Necking aggregate models with different $\eta$ are illustrated in Fig. 2. It was demonstrated that the scattering properties of different geometrical aggregates with the same fractal and morphological parameters show little discrepancy [26,32]. For this reason, we created only one geometrical configuration of the clusters for each $\eta$. The popular calculation for the volume-equivalent radius $a_{\text{eff}}$ was different between the point-touch aggregates and the sintered ones because the former ones were obtained by $a_{\text{eff}}=N_{\text{p}}^{1/3}\times R_{\text{p}}$, while the latter ones were obtained from the total volume of the dipole cubic lattices with a constant $d$ rather than the real volume of the aggregates [33]. However, in Skorupski’s work [34], $d$ can be changed to fit the real volume of the aggregates in the volume correction procedure. We adopted his method and considered the small amount of shrinkage introduced to the necking models to ensure that the $a_{\text{eff}}$ of the necking aggregates were equal to the point-touch ones. Figure 3 depicts the procedure used to generate the TiO₂ and ITO sintered clusters with ball-necking structure in more detail.

 

Fig. 2 Fractal-like aggregate models composed of $N_p=25$ primary particles with (a) point-touch, $\eta =0$; and with necking connector (b) $\eta =0.2$; (c) $\eta =0.4$; (d) $\eta =0.6$ and (e) $\eta =0.8$.

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Fig. 3 Flow chart of the arithmetic for generating the sintered aggregate with ball-necking structure.

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Figure 4 shows the spectral distributions of the real and imaginary parts of the complex refractive index of ITO [26] and TiO₂ [10] in the range of 0.3-2 μm (the range of the solar spectrum is 0.3-2.5 μm, in this paper the results in the omitted wave band, i.e., 2-2.5 μm, were not depicted for its lack of typical features). The imaginary parts, which are the extinction coefficients $k_{\text{ITO}}$ and $k_{{\text{TiO}}_{\text{2}}}$, change with the wavelength $\lambda$ in opposite ways. $k_{\text{ITO}}$ increases along with $\lambda$, while $k_{{\text{TiO}}_{\text{2}}}$ decreases sharply from 0.3 μm to 0.4μm. The refractive index of TiO₂ reaches its maximum at nearly 0.33 μm, sinks to approximately 2.8 at 0.5 μm, and then exhibits a slight downward trend. In contrast, $n_{\text{ITO}}$ decreases with $\lambda$ smoothly in the range of 0.3-1.2 μm and then remains at approximately 2.5. The distinct differences between $m_{\text{ITO}}$ and $m_{{\text{TiO}}_{\text{2}}}$ were the main reason for the discrepancies in their spectral behaviors.

 

Fig. 4 Complex refractive indices of ITO [26] and TiO₂ [10].

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2.2 Calculation method

The generalized multi-particle Mie (GMM) [35], the superposition T-matrix (STM) [36,37] and the discrete dipole approximation (DDA) [38–40] are widely used in the accurate calculation for the scattering properties of fractal aggregates. Among the three methods, a special version of GMM, which is termed as GMM-PA, can be applied to arbitrary shapes and structures by using a corresponding periodic arrays (PA) of identical point-like components [41,42]. On the other hand, T-matrix can only address aggregates formed by point-touch primary particles, while DDA can overcome the point contact limitation [23]. Yurkin et al. [43] discussed some future improvements in the DDA computer implementations and proposed a highly optimized DDA program termed as Amsterdam DDA (ADDA) [44]. By parallelizing single DDA simulations, ADDA can effectively reduce the computation time and expand the computable range of size parameters. In this paper, DDA was selected for the optical properties calculation. In this method, the total aggregate was divided into a set of discrete dipoles. The dipoles mesh must be able to precisely describe the geometry, whose spacing $d$ (the distance between mesh elements) should meet the requirements $d\le \lambda /(10\left|m\right|)$ [26] and $\left|m\right|kd<1$ [22] for accuracy assurance. In our work, it was $\max (\left|m\right|kd)\approx 0.06$ for the ITO cluster and $\max (\left|m\right|kd)\approx 0.11$ for the TiO₂ cluster. The corresponding numbers of dipoles and the distances $d$ are listed in the Table 1.

Table 1. The morphological parameters of ITO and TiO₂ clusters

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3. Results and discussion

In this section, $C_{\text{ext}}$ (the extinction cross section), $Q_{\text{back,sca}}$ (the back-scattering efficiency) and $g$(the asymmetry factor) of the ITO and TiO₂ clusters were calculated by DDA, and the results of the calculation were averaged over 27 (3 × 3 × 3) orientations.

3.1 Comparison with the Mie theory

The accuracy of DDA decreases and the computational time increases with the increase of the complex refractive index $m$ [45]. It is commonly recommended that the application of the DDA is limited to a range as $\left|m-1\right|<2$ [45]. However, the maximum refractive index of TiO₂ was $m=3.39+1.86i$ at $\lambda =0.3$ μm in our work. In order to verify the accuracy of the calculation outcomes obtained by DDA, a single sphere model with the same parameters (i.e., the same $a_{\text{eff}}$ and $m$) as the TiO₂ cluster was built and divided to dipoles as for the cluster. $Q_{\text{ext}}$ (the extinction factor) of the sphere model calculated by DDA was compared with $Q_{\text{ext}}$ calculated by the Mie theory. The relative errors between the results by DDA and the Mie theory were tabulated (Table. 2). As shown in Table. 2, the maximum of the relative error was $\delta \approx 2\%$, which indicates that the accuracy of the results obtained by DDA in this paper is assured.

Table 2. The relative errors between the extinction factors from DDA and Mie

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3.2 Validation of the ball-necking connector

The comparison of the cylindrical connector ${\text{Y}}_{\text{con}}$ [26] and the ball-necking connector $\eta$ was conducted to validate the effectivity of the necking ball in the simulation of sintering structures. The number and the radius of the ITO monomer were set as $N_{\text{p}}=25$ and $R_{\text{p}}=25$ nm. Figure 5 shows $C_{\text{ext}}$ of the ITO clusters with two kinds of necking structures: ${\text{Y}}_{\text{con}}=0.5,0.75$ (see Fig. 10. in [26]) and $\eta =0.4,0.6,0.8$. The general trends and results exhibited in Fig. 5 remained similar, which demonstrates that the ball-necking model provides similar capabilities for reflecting the effects of a sintered cluster. Therefore, due to its simplicity and ease of modeling, the ball-necking model was recommended in our work.

 

Fig. 5 Dependence of $C_{\text{ext}}$ on $\lambda$ for the ITO clusters with two kinds of necking structures.

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3.3 Impact of the complex refractive index and incident wavelength on the scattering properties of TiO₂ and ITO clusters

_{$C_{\text{ext}}$} associated with the wavelength $\lambda$ for TiO₂ and ITO clusters with the same size parameters is shown in Fig. 6. We did not consider the impacts of the ambient medium in this section; thus, we set $n_{\text{medium}}=1$. According to Fig. 6, both TiO₂ and ITO clusters were affirmed as satisfactory pigments used in transparent thin films due to their high transparence in the visible light region. However, the TiO₂ cluster showed high ultraviolet-insulating properties, whereas ITO had low infrared transmissivity. These significant differences of TiO₂ and ITO clusters are dominated by the spectroscopic discrepancies between $k_{\text{ITO}}$ and $k_{{\text{TiO}}_{\text{2}}}$. A $C_{\text{ext}}$ peak for the TiO₂ cluster occurred at $\lambda \approx 0.34$ μm, where $k_{{\text{TiO}}_{\text{2}}}$ was the highest and $C_{\text{ext}}$ plummeted from approximately 0.12 μm² to approximately 0 μm² as $\lambda$ increased. In contrast, the $C_{\text{ext}}$ maximum for the ITO cluster appeared in the range of 1.3-1.4 μm, which belongs to the NIR region. It was noteworthy that $C_{\text{ext}}$ of the ITO cluster did not keep increasing with $\lambda$ but fell instead. This result could be explained by the impacts of the wavelength $\lambda$ of the incident light on the specific extinction according to the Rayleigh scattering method that reveals $Q_{\text{abs}}$ is proportional to 1/$\lambda$ and $Q_{\text{sca}}$ is proportional to 1/${\lambda }^4$ [21]. The reduction of $Q_{\text{abs}}$ and $Q_{\text{sca}}$ with increasing $\lambda$ attenuated $Q_{\text{ext}}$. Generally, $C_{\text{ext}}$ was mainly governed by two factors: the imaginary part $k$ of the complex refractive index and incident wavelength. It could be deduced that the most effective spectral range of thermal shielding coating pigmented with TiO₂ would be the blue and purple band, more precisely, at the peak of $C_{\text{ext}}$ where $\lambda \approx 0.34$. Compared with TiO₂, to determine the highest extinction position of the pigment whose $k$ increased along with $\lambda$, e.g., ITO, we must consider the comprehensive influence of $k$ and $\lambda$.

 

Fig. 6 Dependence of $C_{\text{ext}}$ on $\lambda$ for ITO and TiO₂ clusters with different $\eta$.

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The back-scattering efficiency $Q_{\text{sca,back}}$ is another crucial parameter of a pigmented thermal barrier coating, which could be used to assess its reflectance. Baneshi et al. [7] suggested that $Q_{\text{sca,back}}$ plays the most important role in the performance of the pigmented coatings. In the DDA method, $Q_{\text{sca,back}}$ could be defined as follows:

Figure 7 displays the dependences of $Q_{\text{sca,back}}$ of the TiO₂ and ITO clusters on the wavelength $\lambda$. As we can see in Fig. 7, the changing trend of $Q_{\text{sca,back}}$ on TiO₂ was similar to that of $C_{\text{ext}}$. However, the maximum $Q_{\text{sca,back}}$ of ITO was not in the NIR range but in the VIS (visible spectroscopy) region. This finding means that within the size range discussed in this paper, the reflecting potential lies in the blue-violet band instead of the NIR region for both the TiO₂ and ITO clusters, even though the values of $Q_{\text{sca,back}}$ were small. In contrast, it was reported by Gonome et al. [8] that $Q_{\text{sca,back}}$ of the CuO particle (whose diameter is over 0.4 μm) can reach a value as high as 5 in the range of 1.5-2.5 μm, meaning that CuO particle with its size greater than 0.4 μm can reflect NIR light effectively. The investigation on the impact of size parameters and materials on $Q_{\text{sca,back}}$ is planned in our next work.

 

Fig. 7 Dependence of $Q_{\text{sca,back}}$ on $\lambda$ for TiO₂ and ITO clusters with different $\eta$.

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Figure 8 shows the asymmetry factor $g$ of the TiO₂ and ITO clusters. The two curves of the TiO₂ and ITO clusters presented an overall downward trend of $g$ with $\lambda$, except for a slight fluctuation in the blue and purple bands. This finding means that the dominant forward-scattering gradually converts to isotropic scattering as $\lambda$ increases. This phenomenon is in agreement with the Rayleigh scattering method, which was reported by Yon et al. [21] and Skorupski et al. [23]. The impact of $g$ on the radiative heat transfer process in the pigmented coatings will be discussed in our next paper.

 

Fig. 8 Dependence of $g$ on $\lambda$ for TiO₂ and ITO clusters with different $\eta$

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3.4 Impact of the ambient medium and ball-necking factor on the scattering properties of the TiO₂ and ITO clusters

It is common to utilize epoxy resin or polyethylene resin [6] as a binder for pigmented coatings. The refractive indices of various kinds of epoxy resins were mainly distributed in the 1.45-1.5 range. For this reason, in most studies, the host medium was set to be non-absorbing, with a refractive index of $n_{\text{medium}}=1.5$ [8–10]. However, to probe the effect of the ambient medium on the scattering properties of the TiO₂ and ITO clusters, three cases of ambient media were conducted for comparison in our work: vacuum, water and epoxy resin. The cases of vacuum and water were chosen for their common applications in the studies on particle scattering. Figure 9 and Fig. 10 show $C_{\text{ext}}$ of the TiO₂ and ITO clusters, respectively, pertaining to different $n_{\text{medium}}$: 1 (vacuum), 1.3 (water) and 1.5 (epoxy resin).

 

Fig. 9 Dependence of $C_{\text{ext}}$ on $\lambda$ for TiO₂ clusters with different $\eta$.

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Fig. 10 Dependence of $C_{\text{ext}}$ on $\lambda$ for ITO clusters with different $\eta$.

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The discrepancies between the TiO₂ and ITO clusters are three-fold, as shown in Fig. 9 and Fig. 10. First, for the TiO₂ cluster, the extinction peaks were slightly suppressed by the growth of $n_{\text{medium}}$ but amplified by the increasing $n_{\text{medium}}$ in the ITO cluster. Second, with the gradually increasing $n_{\text{medium}}$, the position of the maximum $C_{\text{ext}}$ of the ITO cluster obviously moved to a larger $\lambda$, whereas the peak location of $C_{\text{ext}}$ in the TiO₂ cluster almost stayed unchanged. It could be deduced that one of the influence factors for the plasmon resonance range of ITO was the optical characteristics of the ambient medium. Lastly, the effects of $\eta$ on the extinction were opposite for these two materials. The sensitivity of $C_{\text{ext}}$ in the ITO cluster to $\eta$ was much higher than that for TiO₂. On the one hand, a larger $\eta$ roughly led to a higher value of $C_{\text{ext}}$ for TiO₂ but a lower $C_{\text{ext}}$ for ITO. A similar phenomenon was also reported by Skorupski et al. [26] that, for the ITO cluster, the discrepancies caused by different sizes of the necking structures appear in the plasmon resonance range, and $C_{\text{ext}}$ diminishes with larger necking volume. The significant impact of $\eta$ on the extinction of the ITO cluster indicated that the sintering state of the ITO cluster could be probed and assessed by measuring the extinction spectrum. On the other hand, the deviations in $C_{\text{ext}}$caused by different $\eta$ was increased for ITO but decreased for TiO₂ with the increase in $n_{\text{medium}}$.

In fact, the discrepancies from $\eta$ were negligible when $n_{\text{medium}}=1.5$. Wriedt et al. [25] specified that a greater degree of sintering of the TiO₂ aggregate would move the peak of $C_{\text{ext}}$ to larger $\lambda$ and amplify the maximum $C_{\text{ext}}$value. The dependence of $C_{\text{ext}}$ in TiO₂ on the $\lambda$ within the range of 0.3-0.6 μm is profiled in Fig. 11. As $\eta$ increased, the shifting of the maximum $C_{\text{ext}}$ to larger $\lambda$ was not obvious according to our calculation. This discrepancy between our results and Wriedt’s may be caused by the slight differences in the data describing the refractive index of TiO₂ in the two papers and the distinctions between the two modeling methods. However, it is still shown clearly in Fig. 11 that a larger $\eta$ yielded a higher $C_{\text{ext}}$ for a fixed wavelength, or a longer wavelength corresponded to a given $C_{\text{ext}}$. Accordingly, the dependence of $C_{\text{ext}}$ on the wavelength is a potential method for characterizing the degree of sintering of TiO₂.

 

Fig. 11 Dependence of $C_{\text{ext}}$ on $\lambda$ within the range of 0.3-0.6 μm for TiO₂ clusters with different $\eta$.

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4. Conclusions and prospects

TiO₂ and ITO are two pigments and fillers that are commonly applied in functional coatings but show quite different spectral characteristics in the ultraviolet and infrared wavebands. In this paper, sintered fractal aggregates of TiO₂ and ITO with a ball-necking structure quantified by $\eta$ were built based on the “particle superposition model” method. The effectivity and the accuracy of the modeling method was validated by compared with the results by Skorupski [26]. The impact of the spectral dependence of the complex refractive index $m$, the ambient medium and the ball-necking factor $\eta$ on the optical properties (the dependences of the cross sections of extinction, the back-scattering efficiency and the asymmetry on the incident wavelength) of TiO₂ and ITO clusters were compared and investigated. We came to the main conclusions that are listed as follows:

We will evaluate the absorptivity and the reflectivity of the thermal-barrier coating pigmented with TiO₂ and ITO clusters by applying a combination of the Monte Carlo method and the DDA method in our future studies. The effects of the asymmetry factor, the extinction cross section $C_{\text{ext}}$ and the refractive index of the ambient medium $n_{\text{medium}}$ calculated in this paper will be taken into account in the analysis of the heat-insulation performance.