Abstract
Titanium dioxide (TiO2) and indium tin oxide (ITO) are both widely used as pigments for thermal barrier nanoparticulate coatings. However, their respective complex refractive indices exhibit different features from each other. In this paper, TiO2 and ITO sintered clusters with necking-ball structures were generated based on the particle superposition model. The impact of the different incident wavelengths and the ball-necking factor, as well as the refractive index of the ambient medium on their scattering properties, were compared and discussed. The results indicated that because of the distinct spectral characteristics of TiO2 and ITO, the discussed factors (especially the ball-necking factor) displayed quite different effects on the locations and deviations of the maximum extinction cross section in comparison. Though the sensitivity of the ball-necking factor for the ITO cluster to the extinction cross section was higher than that of the TiO2 cluster, the state of sintering could be probed and assessed by measuring the extinction spectrum for both TiO2 and ITO sintered clusters.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
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 plays an important role in the direct or inverse radiation problems [29,30]. Since TiO2 and ITO show opposite spectral features of , 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 , ball-necking factor 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]:
in which is the number of the primary particles, is the radius of the primary particles, and the fractal dimension and the fractal prefactor are the two main factors that characterize the fractal properties, which were assumed as and because they are the typical value used for realistically shaped clusters [26].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]:
in which denotes the position of the ith particle and is the mass center of an aggregate. The deviations of , and , which were due to the necking structures, could be omitted when the only primary particles were monodisperse spheres [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 was presented to denote the degree of necking of sintered aggregates and defined by the Eq. (3):
where is the radius of the necking ball.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 and , respectively. We conducted a series of simulations of ITO and TiO2 clusters with the necking balls of various . The number of monomers was fixed as 25, and 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., and decreased with the increasing . Note that the distance L between the center of each primary particle was constant instead of being unfixed.
Necking aggregate models with different 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 . The popular calculation for the volume-equivalent radius was different between the point-touch aggregates and the sintered ones because the former ones were obtained by , while the latter ones were obtained from the total volume of the dipole cubic lattices with a constant rather than the real volume of the aggregates [33]. However, in Skorupski’s work [34], 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 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.
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 and , change with the wavelength in opposite ways. increases along with , while 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, decreases with smoothly in the range of 0.3-1.2 μm and then remains at approximately 2.5. The distinct differences between and were the main reason for the discrepancies in their spectral behaviors.
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 (the distance between mesh elements) should meet the requirements [26] and [22] for accuracy assurance. In our work, it was for the ITO cluster and for the TiO2 cluster. The corresponding numbers of dipoles and the distances are listed in the Table 1.
3. Results and discussion
In this section, (the extinction cross section), (the back-scattering efficiency) and (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 [45]. It is commonly recommended that the application of the DDA is limited to a range as [45]. However, the maximum refractive index of TiO2 was at 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 and ) as the TiO2 cluster was built and divided to dipoles as for the cluster. (the extinction factor) of the sphere model calculated by DDA was compared with 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 , which indicates that the accuracy of the results obtained by DDA in this paper is assured.
3.2 Validation of the ball-necking connector
The comparison of the cylindrical connector [26] and the ball-necking connector 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 and nm. Figure 5 shows of the ITO clusters with two kinds of necking structures: (see Fig. 10. in [26]) and . 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.
3.3 Impact of the complex refractive index and incident wavelength on the scattering properties of TiO2 and ITO clusters
associated with the wavelength 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 . 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 and . A peak for the TiO2 cluster occurred at μm, where was the highest and plummeted from approximately 0.12 μm2 to approximately 0 μm2 as increased. In contrast, the 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 of the ITO cluster did not keep increasing with but fell instead. This result could be explained by the impacts of the wavelength of the incident light on the specific extinction according to the Rayleigh scattering method that reveals is proportional to 1/ and is proportional to 1/ [21]. The reduction of and with increasing attenuated . Generally, was mainly governed by two factors: the imaginary part 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 where . Compared with TiO2, to determine the highest extinction position of the pigment whose increased along with , e.g., ITO, we must consider the comprehensive influence of and .
The back-scattering efficiency is another crucial parameter of a pigmented thermal barrier coating, which could be used to assess its reflectance. Baneshi et al. [7] suggested that plays the most important role in the performance of the pigmented coatings. In the DDA method, could be defined as follows:
in which denotes the differential cross section for backscattering (area per sr) and is the volume-equivalent radius.Figure 7 displays the dependences of of the TiO2 and ITO clusters on the wavelength . As we can see in Fig. 7, the changing trend of on TiO2 was similar to that of . However, the maximum 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 were small. In contrast, it was reported by Gonome et al. [8] that 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 is planned in our next work.
Figure 8 shows the asymmetry factor of the TiO2 and ITO clusters. The two curves of the TiO2 and ITO clusters presented an overall downward trend of with , 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 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 on the radiative heat transfer process in the pigmented coatings will be discussed in our next paper.
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 [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 of the TiO2 and ITO clusters, respectively, pertaining to different : 1 (vacuum), 1.3 (water) and 1.5 (epoxy resin).
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 but amplified by the increasing in the ITO cluster. Second, with the gradually increasing , the position of the maximum of the ITO cluster obviously moved to a larger , whereas the peak location of 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 on the extinction were opposite for these two materials. The sensitivity of in the ITO cluster to was much higher than that for TiO2. On the one hand, a larger roughly led to a higher value of for TiO2 but a lower 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 diminishes with larger necking volume. The significant impact of 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 caused by different was increased for ITO but decreased for TiO2 with the increase in .
In fact, the discrepancies from were negligible when . Wriedt et al. [25] specified that a greater degree of sintering of the TiO2 aggregate would move the peak of to larger and amplify the maximum value. The dependence of in TiO2 on the within the range of 0.3-0.6 μm is profiled in Fig. 11. As increased, the shifting of the maximum to larger 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 yielded a higher for a fixed wavelength, or a longer wavelength corresponded to a given . Accordingly, the dependence of on the wavelength is a potential method for characterizing the degree of sintering of TiO2.
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 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 , the ambient medium and the ball-necking factor 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:
- (1) was mainly determined by the dependence of the imaginary part of the complex refractive index on the incident wavelength . For pigment particles whose greatest appeared in the UV band, e.g., TiO2, the position of the peak of was the same as the maximum . For materials in which consistently increased with , e.g., ITO, the reached its maximum in the IR band due to the comprehensive impact of and the relation between / and based on the Rayleigh Scattering Method.
- (2) The growth of suppressed the extinction peaks of of the TiO2 cluster but amplified the extinction peaks of of the ITO cluster. However, the gradual increase in obviously moved the greatest of the ITO cluster to a larger , whereas the peak location of of the TiO2 cluster remained almost unchanged.
- (3) A larger roughly led to a lower for ITO but a higher value of for TiO2, and a larger deviation of different for ITO but a smaller one for TiO2. had a significant influence on the of the ITO cluster, indicating that the state of sintering of the ITO cluster could be probed by measuring the extinction spectrum. The sensitivity of to TiO2 was much lower than that for the ITO cluster. The discrepancies of caused by were still observable in the range of 0.3-0.6 μm and demonstrated that spectral features of are a potential way to characterize the degree of sintering of TiO2.
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 and the refractive index of the ambient medium calculated in this paper will be taken into account in the analysis of the heat-insulation performance.
Funding
National Natural Science Foundation of China (NSFC) (51876004, 51705490).
Acknowledgments
Bruce T. Draine and Piotr J. Flatau are acknowledged for making their DDA code publicly available: http://code.google.com/p/ddscat/wiki/downloads?tm=2.
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