Exploring the critical thickness for maximum reflectance of optical reflectors based on polymer-filler composites

Yue Shao; Frank G. Shi

doi:10.1364/OME.6.001106

1. Introduction

High performance optical reflectors for LED lighting, backlighting and solar cells are commonly accomplished by metallic as well as multi-layer dielectric mirrors [1–5]. Polymer-filler composite reflectors have gained great attention as a potential alternative, because of easier manufacturing as well as low cost [6]. In addition, since the polymer-filler composites scatter light in a diffuse pattern, they can improve uniformity and brightness of light distribution, eliminate “dots” and glare for better overall optical aesthetics for LED lighting and LCD backlighting applications [3,7–9].

Among many factors, the reflectance is a function of the thickness of polymer-filler composite. A reduction in thickness of reflectors will be beneficial to the design and manufacturing of display systems in various mobile devices [10,11]. Thus, a deep understanding of reflectance dependence on reflector thickness may accelerate the optimization and development of polymer-filler composite reflectors (see Fig. 1). In this regard, the Kebulka-Munk approximations are widely adopted in modeling the optical reflectance, and so far, none of those modeling investigations suggest a critical thickness at which the optical reflectance of a polymer-filler reflectors reaches its maximum. Therefore, an analytical study of the critical thickness for maximum reflectance is essential for developing cost effective high performance reflectors with optimized thickness.

Fig. 1 The schematic cross-sectional view of a LED package.

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In our previous study, we have developed a simple analytical model with high computational efficiency which is accurate in predicting the filler size dependence of thick polymer-filler reflectors [12]. However, the model does not consider any contribution from substrate and thus it is not applicable for thin polymer-filler optical reflectors on substrates. Thus, the objective of the present work is modify our prior model to include the effect of the substrate. The model predicts the existence of a critical thickness at which the optical reflectance of a polymer-filler composite reaches its maximum value. The dependence of the critical thickness on back reflection of the substrate, filler volume fraction, filler size, as well as the matrix-filler refractive indices difference is investigated. The analytical model developed, which is expected to be valuable for accelerating the design and optimization of polymer-filler composite reflectors is verified experimentally.

2. Experiment

In the experiment, polymer-filler composite reflectors were fabricated by mixing transparent silicone resin with inorganic fillers. The inorganic fillers used for samples are Boron nitride (BN) and Zinc oxide (ZnO) (from Sigma-Aldrich and Inframat Advanced Materials Inc), which are two common inorganic pigments used in coating industry. The properties of materials used in polymer-filler reflectors are listed in Table 1.

Table 1. Properties of materials used for samples

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In order to form a homogenous mixture, various amounts of filler were mixed with high performance silicone 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. The mixture was then coated on to mirror-like Cu or Al substrates using film applicator and cured at 150°C for 2 hours.

The light reflectance of the fabricated reflectors at a wavelength range from 400 to 1000 nm was measured using a conventional light reflectometer (Filmetrics Model F20). The thickness of reflectors was measured using coating thickness gauge (CEM Model DT-156).

3. Analysis

3.1 Model

The interaction of incident radiance with a single particle is characterized in terms of scattering and absorption cross section, C_sca and C_abs respectively. The sum of scattering and absorption cross section is called extinction cross section (C_ext). Anomalous Diffraction Theory (ADT) is an approximate but computationally fast method to calculate above-mentioned efficiencies for particles with a size larger or equal to the wavelength of light [15–18]. Rayleigh Scattering Theory is a more accurate method to describe the scattering behavior of particles which are smaller compared to the wavelength of light [19,20]. Based on these two theories, C_ext can be expressed as a function of geometrical cross section (πr²), light wavelength (λ), size of filler particle (d), the ratio of refractive indices of fillers and matrices (n), and the refractive index of matrix (n_m). Since the absorption of fillers is negligible, the scattering efficiency is equal to the extinction efficiency.

The scattering cross section of large particles is calculated based on ADT:

C_{s c a} = \frac{π d^{2}}{4} {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})}

The scattering cross section for particles in Rayleigh regime is calculated according to Rayleigh scattering theory:

C_{s c a} = \frac{2 π^{5} d^{2}}{3} (\frac{d^{4}}{λ^{4}}) (\frac{n^{2} - 1}{n^{2} + 2}) {(n_{m})}^{4}

In order to better understand and investigate the critical thickness, we developed an analytical model to describe the relationship between reflectance and thickness of reflectors. In this study, the polymer-filler composite reflectors are treated as monosized spherical particles embedded in the non-absorbing matrix. Both BN and ZnO fillers are strong scattering and non-absorbing in the wavelength of visible light. We assume that all of the samples have a largely diffuse appearance and the specular reflection is negligible. The Kubelka-Munk model is employed for describing the optical properties of polymer-filler reflectors. In this two-flux model, the incident light is considered to consist of two isotropic diffuse fluxes travelling both upwards (I_d) and downwards (J_d) through the material. The differential equations for the intensities of two diffuse fluxes are:

\frac{d I_{d}}{d x} = - (S + K) I_{d} + S J_{d}

\frac{d J_{d}}{d x} = (S + K) J_{d} + S I_{d}

Since most of reflective coatings are painted on substrates in real coatings application, we assume there is radiation leak out through the back side of reflectors and back reflected radiation from the substrate need to be included in the model. The solution of diffuse reflection (R) using Kubelka-Muck model is:

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

where x is the thickness of the reflector, which is measured perpendicular to the illuminated side. S and K are scattering and absorption coefficient per unit length, respectively. S for spheres can be expressed as a function of filler volume fraction (f), volumetric scattering cross section (C_sca /V) and correction factor Y. R' is the back reflection from substrates and is the reflectivity of reflector with infinite thickness.

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

Finally, we obtained the expression of reflectance for large fillers as:

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

And the expression for small fillers as:

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

In general, the reflectance (R) increases as back reflection (R'), thickness (x), refractive index ratio between matrices (n), reflectivity of reflectors (R_∞) and filler volume fraction (f) increase. Therefore, it is desired to have high values of all these parameters in order to obtain a high reflectance of polymer-filler reflectors.

3.2 Model prediction

According to Eqs. (7) and (8), the reflectance is strongly dependent on back reflection from substrate, reflector thickness and filler volume fraction. Some key predictions were made and the assumed filler size is 1 µm, refractive index ratio between filler and matrix is 1.3. The targeted wavelength is 450 nm, which is the emission peak of blue LED chips. Our model predicts that the change of reflectance with thickness of reflectors and filler volume fraction as shown in Figs. 2, 3 and 4. Our model also predicts that the critical thickness, where the reflector approach its saturation reflectance, decreases with increasing of the back reflection from substrate and the filler volume fraction, as demonstrated in Figs. 2 and 3. Since the critical thickness is the minimum thickness capable of providing theoretical maximum reflectance, this model can be applied to investigate the critical thickness of reflectors in order to reduce the overall volume and weight of display systems for various mobile devices, such as smart phones, tables and ultrabooks.

Fig. 2 The prediction of reflectance as a function of thickness of reflector for different back reflection from substrate.

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Fig. 3 The prediction of reflectance as a function of reflector thickness of reflector for different filler volume fraction.

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Fig. 4 The prediction of reflectance as a function of volume fraction of reflector.

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4. Result and discussion

In order to investigate the influence of thickness, thin layers of reflective coatings were painted onto the Cu and Al substrates. The experimental data was measured by a reflectometer and an integrating sphere. The comparisons between calculated results and measured reflectance as a function of thickness for BN-silicone and ZnO-silicone reflectors are presented in Fig. 5. All the reflectances are taken at wavelength of 450 nm, which is the emission peak of blue LED chips. There is a good agreement between calculation results and experimental data for samples with different substrates. It is shown that a non-linear relationship between increase in reflectance and thickness of reflector. As can be seen from Fig. 4a, the reflective coating sample on Cu substrate requires a thickness of about 200 µm to reach its saturation reflectance, while for the sample on Al substrate, the corresponding thickness is only about 100 µm. It is concluded that a thinner layer of reflective coating is needed for a substrate with higher reflectance in order to obtain the same effect.

Fig. 5 Reflectance change of (a) BN-silicone coatings and (b) ZnO-silicone coatings on Cu substrate (R’ = 0.07) and Al substrate (R’ = 0.78) at wavelength of 450nm. The filler volume fraction is 0.1. The symbols indicate the measured reflectance with different thickness (x) and the lines show the theoretical reflectance based on model.

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The theoretical and experimental reflectances as a function of reflector thickness for BN-silicone and ZnO-silicone reflectors are shown in Fig. 6. The theoretical results are found to be consistent with experimental data, verifying the optimization trends calculated by the modeling. For both types of reflectors, the reflectance increases rapidly from very small thickness to a critical thickness, and then reaches stabilization as the reflector gets thicker. The scattering interfaces between fillers and matrix within the composite reflectors increases as total thickness increases, resulting in a boost of reflectance for reflectors with a thickness smaller than the critical value. As the thickness further increases, the absorption of light by silicone matrix causes the saturation of reflectance, and therefore it is unnecessary to make a reflector thicker. For BN-silicone reflectors, the critical thickness at which the reflectance becomes stable is around 200 µm; for ZnO-silicone reflectors, the critical thickness to provide stable reflectance is around 100 µm.

Fig. 6 Reflectance vs. reflector thickness for (a) BN-silicone coatings and (b) ZnO-silicone coatings on Cu substrate (R’ = 0.07) at wavelength of 450nm. The symbols indicate the measured reflectance with different filler volume fraction (f) and the lines show the theoretical reflectance based on model.

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Figure 7 presents the reflectance as a function of filler volume fraction for BN-silicone and ZnO-silicone reflectors at wavelength of 450nm. The calculated results are in good agreement with the measurements for all the samples with different thickness. With the same thickness, the reflectance firstly increases as the volume fraction increases from zero to a critical value, and then saturates as the volume fraction continues to increase. The plots indicate that the critical volume fractions for BN and ZnO fillers to effectively reflect incident radiation at 450 nm wavelength are 0.2 and 0.1 for about 200µm thick reflectors. The little deviation in exact value of reflectance could be due to the uncontrolled factors in experimental samples that result in a non-ideal state. However, it still can be seen that the modeling results give us good correlations to determine what filler volume fraction is effective enough for specific fillers at a particular thickness. It is demonstrated from Figs. 5, 6 and 7 that the modeling can be used to determine the critical thickness and filler volume fraction for silicone based reflectors to obtain the maximum reflectance in an approach to reduce the overall weight and manufacturing cost of polymer based reflectors.

Fig. 7 Reflectance vs. filler volume fraction for (a) BN-silicone coatings and (b) ZnO-silicone coatings on Cu substrates at wavelength of 450nm with various thickness. The symbols indicate the measured reflectance at different thickness (x) and the lines show the calculated reflectance based on model.

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Furthermore, in order to evaluate the capability of this model, we predicted the coating reflectance at different wavelength and compared the results with measurements. The wavelengths measured are 450, 550, 630 and 700 nm, which include the emission peaks of blue, green and red single color LEDs. The comparison curves of the reflectors at selected wavelength are shown in Fig. 8. Overall, there is a good agreement between prediction results and experimental data for samples at selected wavelength, indicating the capability of predicting visible light reflectance in our model. We demonstrated that our analytical model presents a useful tool to predict and optimize polymer-filler composite reflectors.

Fig. 8 Reflectance of (a) 110 µm thick BN-silicone coatings and (b) 125 µm thick ZnO-silicone coatings on Cu substrate and Al substrate. The filler volume fraction is 0.1. The symbols indicate the measured reflectance at different wavelength and the lines show the theoretical reflectance based on model.

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5. Conclusion

In this work, we presented a simple analytical model for reflectance of polymer-filler reflector on various substrates by considering the effect of reflector thickness, fillers and back reflection from substrate. This first general analytical results were fully supported by our experimental measurements for the investigation of critical thickness. For the first time, our work, the optimization of reflectors was realized by using the critical thickness of reflector to achieve a theoretical maximum reflectance. Our results indicate the critical thickness can be tailored by controlling the substrate reflection and the filler volume fraction, and our model can be valuable for accelerating the design and optimization of polymer-filler composite reflectors.

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Materials	Refractive index (λ = 450 nm)	Average particle size (µm)
BN powders	1.81 [13]	1
ZnO powders	2.11 [13]	1
Silicone resin	1.42 [14]	-

Exploring the critical thickness for maximum reflectance of optical reflectors based on polymer-filler composites

Abstract

1. Introduction

2. Experiment

3. Analysis

3.1 Model

3.2 Model prediction

4. Result and discussion

5. Conclusion

References and links

Cited By

Figures (8)

Tables (1)

Equations (8)

Optical Materials Express