Study of optimal filler size for high performance polymer-filler composite optical reflectors

Yue Shao; Yu-Chou Shih; Gunwoo Kim; Frank G. Shi

doi:10.1364/OME.5.000423

1. Introduction

To reach high conversion efficiencies in optical and photonic devices, advanced light diffusion, distribution and guiding mechanisms need to be employed. High performance optical reflectors are key components for many advanced photonic and optical modules such as those for light emitting diode lighting, LCD display backlighting and photovoltatics [1–3]. High quality diffuse reflectors are also critically important for integrating cavities for accurate quantitative characterization of reflectance and absorption of materials [4–6]. In addition, they are also widely employed for illuminated advertising displays, light boxes for inspection purposes, LED emergency signs, traffic signs and vehicle operators. There is an increasing interest in the development of cost-effective and high performance broadband reflectors in order to decrease the price-performance ratio for optical and photonic devices.

Common optical reflectors employed in optoelectronic application are metallic mirrors, conventional dielectric mirrors and diffusive dielectric composites [7]. Metallic mirrors such as silver and aluminum have high reflectance in the visible light range, but also have the serious drawback of poor long-term stability [8,9]. Conventional dielectric mirrors such as distributed Brag reflectors (DBR) cannot provide a random light reflecting due to their well-defined geometries. The limited reflection bandwidth of DBR is another issue for its applicability for solar cells [10]. All-dielectric omnidirectional mirrors can totally reflect electromagnetic waves at all angles with nearly free of loss at optical frequencies, however, their multilayer structures lead to a high production complexity. Diffusive dielectric composites such as polymer-filler reflectors have been demonstrated to be effective reflectors in LED and solar cell technology [11]. These reflectors are embedded with fillers of refractive indices that differing from those of polymeric matrices. In general, polymer-based diffuse reflectors have the advantages of high reflectance over a broad spectral range, low cost, ease of fabrication and high throughput. Recently, silicone-based composites have been used to make high performance reflectors because of their good UV, thermal and environmental stability. Thus, there is a strong need for the development of silicone-based reflecting materials which are more cost-effective and reliable for applications in higher power devices and short wavelength radiation.

Efforts have been made to develop highly reflective polymeric materials for many years because relatively small improvements in the percent reflectivity can have a substantial effect on the amount of energy needed to produce the required lighting [12]. There are many benefits to the optimization of polymer-based reflectors by controlling filler size. With the critical filler size, a reflector can obtain the theoretical maximum reflectance with the lowest filler volume fraction and reflector thickness. Different modeling approaches have already been used to investigate diffuse dielectric reflectors. Monte Carlo simulation is a numerical method with high accuracy but is very costly and time-consuming [13]. The exact radiation transfer theory based on radiation transfer equations (RTE) provides an analytical model, but this model requires heavy calculations and is very difficult to solve without approximations. N-flux one-dimensional models such as Kebulka-Munk model are approximated analytical method providing explicit formulas that are easily utilizable [14]. For industrial application, a simple analytical model is preferred due to its high computationally efficiency and its ability to provide a deeper understanding of physics behind the high reflectance of polymer-filler composites. There are numbers of reports modeling the scattering efficiency of specific fillers such as TiO₂ and ZnO [15–18]. However, based on our best knowledge, there is no report using a simple analytical model to directly predict the critical size of inorganic fillers with low absorption at short wavelength for diffuse polymer-based reflectors.

In this study, we used a simple analytical model to identify the critical filler size to obtain the highest reflectance while also reducing the filler volume fraction and reflector thickness required. The model prediction was then validated with experimental data. Our results demonstrate for the first time that for different inorganic fillers, a critical filler size ranging from submicron to several microns provide the maximum reflectance for polymeric-filler reflectors. The present results have important implications to the targeted formulation for reflective packaging materials.

2. Model and prediction

A simplified schematic diagram of radiation interaction in silicone-filler reflectors is shown in Fig. 1.We assume that the polymer-based diffuse reflector is treated as homogeneous spherical particles of certain radius within the non-absorbing matrix. The fillers used in this experiment are strong scattering and non-absorbing in the wavelength of visible light. Due to the sample preparation method, the specular reflection is negligible and the samples have a largely diffuse appearance.

Fig. 1 Schematic of light scattering in silicone-filler composite reflectors.

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In this study, we use the Kubelka-Munk model of reflectance for analysis of diffuse reflectance for weakly absorbing composites [16]. The model works quite well for optically thick materials with high light reflectivity and low transparency if the material is of a constant finite thickness when illumination is diffuse and homogenous, and when no reflection occurs at the surface of materials. The general equation of Kubelka-Munk theory is given:

S x = \frac{1}{(\frac{1}{R_{\infty}} - R_{\infty})} \ln (\frac{1 - R R_{\infty}}{1 - \frac{R}{R_{\infty}}})

Where x is the thickness of sample, R is the reflectance of sample and $R_{\infty}$ is the reflectivity of sample materials at infinite thickness. The scattering coefficient per unit thickness (S) is a heuristic parameter and can be defined as an expression of scattering efficiency [19]. The scattering behavior of incident light on particles depends on the size parameter (X), which is defined as X = $π$ d/λ. In the present model, the scattering efficiency was calculated from anomalous diffraction theory (ADT) for particles with large size parameters (X≥1), and Rayleigh approximation for small size parameters (X<1). ADT is a very approximate but computationally fast technique to calculate extinction, scattering, and absorption efficiencies for many typical size distributions. For large particles, we calculated the extinction efficiency by:

Q_{e x t} = 2 - \frac{2 λ}{(n - 1) π d} \sin [\frac{2 (n - 1) π d}{λ}] + \frac{4 λ^{2}}{{[2 (n - 1) π d]}^{2}} {1 - \cos [\frac{2 (n - 1) π d}{λ}]}

For small particles in the Rayleigh regime, the scattering efficiency for a single particle is given by:

Q_{s c a} = \frac{8 π^{4}}{3} (\frac{d^{4}}{λ^{4}}) (\frac{n^{2} - 1}{n^{2} + 2}) {(n_{m})}^{4}

Where Q_sca is the scattering efficiency, which is defined as the ratio of the scattering cross section and geometrical cross section πr², λ is the light wavelength, d is the diameter of spherical particle, n is the ratio of refractive indices of fillers and matrices, and n_m is the refractive index of matrix. Based on our assumption, the extinction efficiency (Q_ext) here is equal to the scattering efficiency considering there is no absorption. For spherical particles, the scattering coefficient (S) is usually evaluated as the particle volume fraction times the volumetric scattering cross section of the particle. Here we take into consideration the effects of interfaces between fillers and matrix material and accordingly modified the expression of scattering coefficient as follows:

S = \frac{3 f}{4 V_{p}} C_{s c a} (1 - g) Y = \frac{9 f}{8 d} Q_{s c a} (1 - g) Y

Where f is the volume fraction of particles in composites, C_sca is the scattering cross section of spheres as a function of diameter d, V_p is the volume of a single particle, and g is the asymmetry parameter which can be calculated with Mie theory [16]. Y is the correction factor determined by experimental measurements and is constant for Al₂O₃-silicone composites and TiO₂-silicone composites.

By entering Eqs. (2)–(4) into Kubelka-Munk general Eq. (1), a simple model for analyzing the reflectance of polymer-based diffuse reflectors is developed. For large filler particles, the formula is:

R = \frac{1 - e^{[\frac{9 f}{8 d} {2 - \frac{2 λ}{(n - 1) π d} \sin \frac{2 (n - 1) π d}{λ} + \frac{4 λ^{2}}{{[2 (n - 1) π d]}^{2}} [1 - \cos \frac{2 (n - 1) π d}{λ}]} x (\frac{1}{R_{\infty}} - R_{\infty})] (1 - g) Y}}{R_{\infty} - e^{[\frac{9 f}{8 d} {2 - \frac{2 λ}{(n - 1) π d} \sin \frac{2 (n - 1) π d}{λ} + \frac{4 λ^{2}}{{[2 (n - 1) π d]}^{2}} [1 - \cos \frac{2 (n - 1) π d}{λ}]} x (\frac{1}{R_{\infty}} - R_{\infty})] (1 - g) Y}}

And for small filler particles, the formula is:

R = \frac{1 - e^{{\frac{9 f}{8 d} [\frac{8 π^{3}}{3} (\frac{d^{4}}{λ^{4}}) (\frac{n^{2} - 1}{n^{2} + 2}) {(n_{m})}^{4}] x (\frac{1}{R_{\infty}} - R_{\infty})] (1 - g) Y}}{R_{\infty} - e^{{\frac{9 f}{8 d} [\frac{8 π^{3}}{3} (\frac{d^{4}}{λ^{4}}) (\frac{n^{2} - 1}{n^{2} + 2}) {(n_{m})}^{4}] x (\frac{1}{R_{\infty}} - R_{\infty})] (1 - g) Y}}

The main advantage for this new model is that it provides explicit formulas that are easily utilizable to make a number of predictions. The key factors affecting reflectance are filler size, volume fraction, difference between refractive indices between fillers and matrices, and the thickness of reflector [20]. The model predicts critical filler size for different inorganic fillers as shown in Figs. 2 and 3.RIR in Fig. 3 is the filler-matrix refractive index ratio (RIR = n_filler/n_matrix). Theoretically, a critical filler size for maximum reflectance exists and needs to be investigated in order to reduce the overall costs of reflectors.

Fig. 2 The prediction of reflectance as a function of filler size-wavelength ratio for composite reflectors with different filler-matrix refractive index ratio (RIR = n_filler/n_matrix) at wavelength of 450nm.

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Fig. 3 The prediction of critical filler size as a function of RIR (a). for polymer-based reflectors at different wavelength and (b). for silicone-based reflectors at wavelength of 450nm. The symbols indicate the measured optimized filler size with different fillers (left: Al₂O₃, right: TiO₂) and the line shows the calculated optimized filler size based on model.

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3. Experimental results and discussion

3.1 Experimental

In order to identify the critical filler size for composite reflectors required to obtain the highest reflectance, as well as to compare experimental data versus the modeling prediction, silicone based composites were prepared using inorganic fillers with various particle size at a volume fraction of 0.1. The inorganic fillers used in this experiment are Al₂O₃ (from Inframat Advanced Materials Inc) and TiO₂. The diameters of Al₂O₃ particles was 0.04, 0.08, 0.1, 0.3, 1, 10, 20 and 35 micron. The diameters of TiO₂ particles was 0.04, 0.1, 0.2, 0.5 and 5 micron. The refractive indices at a wavelength of 450nm are 1.78 for α-Al₂O₃ particles and 2.81 for rutile-TiO₂ particles. The matrix material was a high performance silicone formulated by Shi’s lab [21]. The silicone has a refractive index of 1.40 at a wavelength of 450nm and is high in transparency.

The silicone composites were fabricated by mixing silicone resin with various amounts of inorganic fillers using a Shinky high shear mixer at 2500 r/min for 5 minutes. After mixing, the composites were degassed under a vacuum of 10-2 Pa for 30 minutes in order to get rid of the trapped air bubbles introduced during the mixing. The mixture was then transferred into a stainless steel mold and followed by a curing at 150°C for 2.5 hours to solidify the composites. The prepared composites had a plate shape with a diameter of 22mm.

The fabricated composite was repeatedly polished using a South Bay Technology polisher. The light reflectance of the composites, at a wavelength range from 400 to 1000 nm, was measured using a conventional light reflectometer (Filmetrics, Model F20) with an integral sphere.

3.2 Verification of model and optimization of reflectors

The reflectance properties of two types of inorganic fillers were calculated using ADT and Rayleigh scattering approximation coupled with Kubelka-Munk model. To validate the accuracy of model prediction, we compared our calculation results to the experimental measurements. Figure 4 shows the comparison between the model's predictions and experimental data for reflectance of silicone-based reflectors as a function of filler size at a wavelength of 450nm. For better comparison, reflectance is normalized by setting the highest value as 1. The highest value of raw reflectance is 93% for Al₂O₃-silicone composites and 94% for TiO₂-silicone composites. Also shown is the prediction based on the simple analytical model for coating system developed by Wenlan Xu. Xu’s model abandons Mie theory and reverts to a combination of geometrical optics and wave optics to calculate the main scattering and absorption coefficients. The details of this model are given by W. Xu and S.C. Shen [22], and Emslie and Aronson [23]. From the plots in Fig. 4, it is evident that our model better fits the existing experimental data than Xu’s model. Xu's model is able to predict the reflectance for composites with filler sizes similar to and larger than light wavelength. However, it cannot achieve a good fit for fillers at a size smaller than this wavelength and therefore, its applicability for determining the critical filler size of polymer-based reflectors with small size parameters is still questionable.

Fig. 4 Normalized reflectance vs. particle size for 3mm thick (a) Al₂O₃-silicone composites and (b) TiO₂-silicone composites at wavelength of 450nm with 0.1 filler volume fraction. For better comparison, reflectance is normalized by setting the highest value as 1. The circles indicate the measured reflectance and the lines show the calculated reflectance based on model.

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Figure 4 also indicates that the effect of filler size on light reflectance is non-monotonic. For these two fillers, the light reflectance increases as the particle size increases from nano-size to submicron-size, and then decreases as the particle size increases to micron sizes. The critical size for fillers to effectively reflect incident radiation at a 450nm wavelength is 0.8 microns for Al₂O₃ pigment, and 0.2 microns for TiO₂ pigment. The predicted critical filler size of TiO₂ particles shows a good agreement with Vargas's results [12]. In the case of nano-size particles, the effect of refractive index mismatch on light scattering is usually less important than the effect of particle size. A high amount of visible light is transmitted by composites with nano-size particles. The reason for the maximum reflectance of composites with submicron-size fillers is because of its high light scattering coefficient. It is also known from Mie theory that when the size of particle is comparable to the wavelength of incident light, the magnitude of scattering efficiency reaches its maximum. This light reflectance decrease with increasing particle size due to the decrease in effective scattering cross sections. The total surface area for larger particles is lower because at the same volume fraction, the number of smaller particles is much more than that of larger particles. It is demonstrated those modeling results give us a good indication of what particle sizes of fillers are most effective than others for specific fillers at a particular wavelength.

The theoretical critical filler size as a function of the refractive index ratio between filler and matrix for different inorganic fillers at a wavelength of 450nm is shown in Fig. 3b. Overall, there is a good agreement between calculation results and experimental data for samples with different fillers. The results show that the optimized filler size is negatively related to refractive index ratio. The model calculations show that for filler-to-matrix ratios from 1.1 to 2.6, the best reflectance is achieved when the filler sizes used in the reflector are between submicron and several microns. The results provide an important insight for the selection of filler sizes for producing high reflective polymer-based reflectors. In this experiment, we used a silicone matrix and mixed it with different inorganic fillers to produce reflectors. In fact, the same effect shown in the plot can be alternatively made by using other polymer matrices with different refractive indices. Thus, this model can be employed to optimize not only silicone-based reflectors, but also all other polymer-based reflectors in order to increase their reflectance.

Generally, high filler volume fractions and reflector thicknesses are desired for obtaining high reflectance in polymer-based reflectors. However, there is a critical value of filler sizes that enables all inorganic fillers to achieve high reflectance without increasing filler volume fraction and film thickness. The improvements in reflectivity for polymer-based reflectors can be realized by using optimized filler particles. Our results demonstrate for the first time that for different inorganic fillers, a critical filler size ranging from submicron to several microns provides the maximum reflectance. We have demonstrated that our simple analytical model provides a very effective method for optimizing the polymer-based reflectors in LED and solar cell applications. Furthermore, the accuracy loss of the present model is not very significant for the polymer-based diffuse reflectors studied in the paper.

4. Conclusion

In this study, we utilized modeling techniques to investigate the critical sizes of inorganic fillers required to obtain the highest reflectance. The key model predictions were verified by comparisons with experimental data. The light reflectance of tested reflectors increased as the particle size increased from nano-size to submicron-size, and then decreased as the particle size continues to increase to micron-size. The critical size for fillers to effectively reflect incident radiation at 450nm wavelength is around 0.8 microns for Al₂O₃ pigments, and 0.2 microns for TiO₂ pigments. Our results demonstrate for the first time that for different inorganic fillers, a critical filler size that ranges from one submicron to several microns provides the maximum reflectance for polymer-based reflectors. The results provide an empirical basis for selecting filler sizes to produce cost-effective polymer-based diffuse reflectors, while also reducing overall weight.

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Study of optimal filler size for high performance polymer-filler composite optical reflectors

Abstract

1. Introduction

2. Model and prediction

3. Experimental results and discussion

3.1 Experimental

3.2 Verification of model and optimization of reflectors

4. Conclusion

References and links

Cited By

Figures (4)

Equations (6)

Optical Materials Express