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PDMS-based subwavelength structures for broadband and wide-angle anti-reflection
Journal Pre-proof PDMS-based subwavelength structures for broadband and wide-angle antireflection
Yunzhen Yin, Yanyan Bu, Xiangfu Wang PII:
S0921-4526(19)30823-3
DOI:
https://doi.org/10.1016/j.physb.2019.411943
Reference:
PHYSB 411943
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
08 October 2019
Accepted Date:
07 December 2019
Please cite this article as: Yunzhen Yin, Yanyan Bu, Xiangfu Wang, PDMS-based subwavelength structures for broadband and wide-angle anti-reflection, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb.2019.411943
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PDMS-based subwavelength structures for broadband and wide-angle anti-reflection Yunzhen Yina, Yanyan Bub*, Xiangfu Wanga,c,* aCollege
of Electronic and Optical Engineering & College of Microelectronics,
Nanjing University of Posts and Telecommunications, Nanjing, 210046, China bCollege
of Science, Nanjing University of Posts and Telecommunications, Nanjing,
210046, China cState
Key Laboratory of Green Building Materials, China Building Materials
Academy, Beijing, China *Correspondence author. E-mail address: [email protected] (Bu), xfwang@njupt. edu.cn (Wang)
Abstract Subwavelength structures were reported to have excellent anti-reflection (AR) performance. However, it is difficult to achieve effective AR in broadband and wideangle. In this work, we report the new subwavelength structures fabricated by polydimethylsiloxane (PDMS) for broadband and wide-angle AR. Two new subwavelength structures, alternating structure and stacking structure, are designed. The admittance recursive method is used to calculate the reflectance of subwavelength structures in the range of incident wavelengths from 200 nm to 2500 nm and incident angles from 0º to 80º. It is found that the combination of the 300 nm stacking structure on the front surface and the 500 nm alternating structure on the back surface has the better AR performance. In the wide spectrum and wide incident angle, the maximum reflectance is reduced to 6.8% and 12.0%. Compared to bare glass, the average transmittance was increased by 6.6% and 12.4%, respectively Key words: Subwavelength structure; Anti-reflection; Polydimethylsiloxane; Admittance recursive method
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1. Introduction Reducing the light reflection on the surface of medium is critical to improving the efficiency of solar energy utilization. Therefore, many researchers designed and manufactured graded refractive index layer systems to control the reflectance of incident spectrum [1–3]. However, graded refractive index layer systems had some disadvantages, such as absorption and scattering loss, low durability, and thermal distortion [4]. Recently, under the inspiration of moth eyes, it was found that the subwavelength structures, avoiding the inherent problems of the graded refractive index layer systems, had good AR performance in a wide spectrum and wide incident angle [2, 5–7]. Moreover, subwavelength structures were much thinner and easier to manufacture [8]. Due to good AR performance, many methods in the visible and nearinfrared range were developed to fabricate subwavelength structures, such as etching, laser interference lithography, and nano-imprint lithography [9-11]. When applied to the surface of glass, including the front and back surfaces, subwavelength structure provided a gradual change in effective refractive index at the air-glass-air interface, making it possible to effectively increase the light transmittance [12, 13].
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In order to reduce the reflectance of the glass surfaces, thin films with subwavelength structures were often loaded onto the glass surfaces instead of directly changing the topography of the glass surfaces. However, the AR performance of subwavelength structures depended on the refractive index of materials [14]. In addition, it was difficult to grow film with subwavelength structures in many cases, since that the lattice mismatch among materials blocked the growth of the film [15]. PDMS, a polymer material with good adhesion, and has good optical properties with the refractive index of 1.43 which is close to the refractive index of the glass (1.52) [11]. Furthermore, due to the strong van der Waals force between PDMS and glass, PDMS film with subwavelength structure can be easily adhered to glass surface without adhesive [16], avoiding the difficulty of lattice matching of materials. Although PDMS has gained wide application due to its good properties, excellent AR performance is limited to a narrow spectral range. For example, Kim et al. used a PDMS film with subwavelength structures to achieve a high-efficiency perovskite solar cell at wavelengths between 300 nm and 800 nm [17]. Lee et al. enhanced the transmittance of sapphire in the wavelength region of 400 nm to 1000 nm by using patterned PDMS
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surface structure [18]. Therefore, the AR performance of PDMS with subwavelength structures still have a large room for improvement in a wide spectrum. When sunlight is incident on the glass, strong reflection result in loss of solar energy due to the gradient of the refractive index at the air-glass-air interface [19]. Especially when the sunlight is incident at a large angle, the reflection of the glass surface will be stronger. Therefore, it is necessary to design new subwavelength structures to achieve low reflectance of the upper and lower surfaces of the glass in a wide incident angle.
Fig. 1. Schematic diagram of a bare glass and a glass with composite structures
In this paper, the composite structure is designed to achieve low reflectance of the glass in a wide spectrum and wide incident angle, which is shown in Fig. 1. The
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admittance recursive method is developed to show the AR performance of subwavelength structures fabricated by PDMS.
2. Model and method The admittance recursive method is used to study the reflectance of subwavelength structures. It is well known that when a film is coated on the substrate, the film and the substrate can be represented by an equivalent interface. The subwavelength structure can be equivalent to an n-layer film on glass, if it is divided into n layers of equal thickness. It is assumed that the film adjacent to the glass is the nth layer, and the film close to the air is the first layer. The thickness, refractive index and extinction coefficient of the jth layer are dj, nj and kj, respectively. The combined admittance of the nth film and the glass is [20]
Yn
Yn 1 cos n i n sin n yn(1) iyn(2) cos n iYn 1 / n sin n
(1)
where 𝛿n and 𝜂n are the effective phase thickness and the modified admittance of the nth layer, and Yn+1 is the admittance of the glass. When gradually recursing to the jth film, the recursion formula for Yj is as follows
Yj
Y j 1 cos j i j sin j cos j iY j 1 / j sin j
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(2) y (1) j iy j
(2)
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After recursing to the first layer, the combined admittance of the subwavelength structure is
Y1 y1(1) iy1(2)
(3)
N 0 Y1 2 ( N 0 y1(1) ) 2 ( y1(2) ) 2 R( ) N 0 Y1 ( N 0 y1(1) ) 2 ( y1(2) ) 2
(4)
The reflection is
where N0 is the admittance of the air. When light is incident on the film surface at a certain angle 𝜃0, the effective phase thickness is
j
2
d j (n 2j k 2j N 02 sin 2 0 2in j k j )1/2
(5)
where λ is the incident wavelength. The incident light can be decomposed into TE waves and TM waves, which are an electric vector perpendicular to the incident surface and an electric vector parallel to the incident surface, respectively. The modified admittance of TE waves is
j n 2j k 2j N 02 sin 2 0 2in j k j The modified admittance of TM waves is
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(6)
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n 2j k 2j 2in j k j (n 2j k 2j N 02 sin 2 0 2in j k j )1/2
(7)
The reflectance of incident light is the average of TE wave reflectance and TM wave reflectance. To verify the feasibility of the admittance recursive method, we simulated the AR performance of a multi-layer nanostructure fabricated by Kuo et al [21]. The reflectance as a function of the incident wavelength and incident angle shown in Fig. 2 is consistent with Fig. 2(b) and Fig. 3 in Ref.17, which means that the admittance recursion method can accurately calculate the reflectance of subwavelength structures.
Fig. 2. Simulation results on AR performance of a multilayer nanostructure. (a)
Reflectance as a function of incident wavelength when incident angle is 8 degrees, (b) reflectance as a function of incident angle when incident wavelengths are 633 nm, 830 nm, and 904 nm, respectively.
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Fig. 3. (a) Schematic diagram of paraboloid, cylinder and cone. (b) The change in the effective refractive index of paraboloid, cylinder and cone with a height of 700 nm, as the incident light travels from air into subwavelength structures.
Since the glass has front and back surfaces, subwavelength structures were first applied to reduce the reflectance of the front surface of glass. The reflectance of the airglass-air interface is then minimized by optimizing the subwavelength structure of the back surface of glass. The admittance recursive method is used to calculate the reflectance of periodic subwavelength structures in the solar spectrum range of 200 nm to 2500 nm and the incident angle range of 0° to 80°. Paraboloid, cylinder and cone shown in Fig. 3(a) were reported to have good AR performance [9, 22, 23]. These subwavelength structures are placed in a hexagonal grid with a side length of 100 nm. The height of the structures is 700 nm and the period is 173 nm. The diameters of the bottom surface of paraboloid, cylinder and cone are fixed at 173 nm. The change of
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effective refractive index of the three structures is shown in Fig. 3(b). The effective refractive index, neff, can be calculated based on [6] neff
fns2 (1 f )ni2
(8)
where ns and ni are the refractive index of PDMS and air. f is the fill factor of PDMS as a function of structure height. It can be observed from Fig. 3(b) that the effective refractive index of the bottom of the paraboloid, cylinder and cone has a small gradient change, while the cylinder also has a large effective refractive index gradient change at the top. It should be noted that in the spectral range studied in this paper, the extinction coefficient of PDMS is very small [24], which has little effect on the results of the study, so the extinction coefficient of PDMS is negligible. In order to compare the AR performance of the three subwavelength structures, the reflectance of the front surface of glass as a function of incident wavelength and incident angle is shown in Fig. 4. As can be seen from Fig. 4(a), when the incident angle is 8º, the reflectance of the cylinder fluctuates between 1.5% and 4.3%. When the incident wavelength is small, the reflectance of the cone has a small fluctuation range. When the incident wavelength increases to 2500 nm, the reflectance of the cone rises
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to 1.1%. The paraboloid, however, always maintains low reflectance in the wavelength range of 200 nm to 2500 nm., with a maximum reflectance of 0.6% occurring at 1360 nm. For sunlight with a wavelength of 495 nm, the reflectance of the three structures is always less than 10% when the incident angle is less than 50º, as shown in Fig. 4(b). When the incident angle continues to increase, the reflectance of cone increases rapidly, while the reflectance of cylinder changes little. In the incident range of 0º to 80º, the maximum reflectance of paraboloid, cylinder and cone is 16.7%, 7.3% and 55.8%, respectively. Since the high reflectance in a wide spectrum, the AR performance of cylinder is inferior to paraboloid. Therefore, the paraboloid has better AR performance in a wide spectrum and wide incident angle.
Fig. 4. For paraboloid, cylinder and cone with a height of 700 nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
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Fig. 5. For paraboloid varying in height from 50 nm to 2000 nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
It is reported that the height of the subwavelength structure plays a dominant role in AR performance [25]. In order to make the paraboloid has better AR performance at the optimal height, the reflectance of paraboloid with different heights (H) is studied, which is shown in Fig. 5. It can be observed from Fig. 5(a) that when the incident angle is 8º, the reflectance decreases in the long wavelength range as the height of paraboloid increases. When H ≥ 600 nm, the reflectance is less than 1% in the entire spectral range. As shown in Fig. 5(b), when the height of the paraboloid is 50 nm and 300 nm, the maximum reflectance is reduced to 8.8% and 11.6% in a wide angle rang, while the reflectance in a wide spectrum is very high. Under the premise that the reflectance is
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less than 1% in the broad spectral range, the paraboloid of 700 nm has better AR performance, with a maximum reflectance of 16.7%. Therefore, paraboloid with height of 700 nm achieves better AR performance in a wide spectrum and wide incident angle. Although paraboloid exhibits excellent AR performance compared to cone and cylinder, the reflectance is still higher at large incident angles, especially when the incident angle is greater than 40º. However, when the incident angle is greater than 40º, the cylinder exhibits good AR performance. It attributed to the difference in the spatial distribution of the effective refractive index of cylinder and paraboloid [26]. Specifically, the effective refractive index of paraboloid varies linearly with the structure height, while the effective refractive index of cylinder varies abruptly. Therefore, in order to achieve excellent AR performance in a wide spectrum and wide incident angle, new structures need to be designed, which has the characteristics of the effective refractive index change of both cylinder and paraboloid. In Fig. 6, two subwavelength structures based on cylinder and paraboloid, alternating structure and stacking structure, are designed, which have the advantages of cylinder and paraboloid
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in AR. The two structures will first be applied to reduce the reflectance of the front surface of the glass.
Fig. 6. Schematic diagram of (a) alternating structure and (b) stacking structure.
For alternating structure, the schematic diagram is shown in Fig. 6(a). The diameters of the bottom surfaces of cylinder and paraboloid are D1 and D2, respectively. The structure height is H1. First, D1 and D2 are both fixed at 173 nm. Then, the effect of height of alternating structure on AR performance is explored by adjusting H1 to vary from 50 nm to 1200 nm. As can be seen from Fig. 7(a), when the structure height H1 is less than 300 nm, the maximum reflectance is greater than 2%. When the structure height is increased to above 300 nm, the reflectance fluctuates between 0% and 2%. As shown in Fig. 7(b), when the structure height is 300 nm, the AR performance is better,
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and the maximum reflectance is reduced to 4.4%. However, in order to ensure good AR performance in a wide spectrum, the structure height should be greater than 300 nm at least. Therefore, for alternating structure, 550nm is the optimal height and the maximum reflectance is 4.6% in wide angle.
Fig. 7. The alternating structure is applied to the front surface of glass. With the alternating structure height varying from 50nm to 1200nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
The schematic diagram of stacking structure is shown in Fig. 6(b). The paraboloid with a height of H2 and a bottom diameter of D3 is divided into two parts of equal height, wherein the upper half is a small paraboloid and the lower half is a round table. Then, the small paraboloid is replaced by a cylinder of the same height. The diameter of the cylinder is the same as the diameter of the bottom surface of the small paraboloid.
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The height of the structure varies from 100 nm to 1400 nm for the better AR performance. The reflectance as a function of incident wavelength and incident angle is shown in Fig.8. As seen from Fig. 8(a), when H2 is less than 300 nm, the reflectance is large in the long wavelength range. When H2 is greater than 300 nm, the reflectance fluctuates between 0 and 2%. In Fig. 8(b), when the structure height is 300 nm, the AR performance is better, and the maximum reflectance is 4.5%. Considering the AR performance in a wide spectrum and wide incident angle, the optimal height of the stacking structure is 300 nm.
Fig. 8. The stacking structure is applied to the front surface of glass. With the stacking structure height varying from 100nm to 1400nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
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Journal Pre-proof Table 1 AR performance of subwavelength structures applied to the front surface of glass are presented, including maximum reflectance and average reflectance in the range of incident wavelengths and angles. Paraboloid
Alternating structure
Stacking structure
Maximum reflectance of wavelength
0.6%
1.8%
2.1%
Average reflectance of wavelength
0.2%
1.0%
1.0%
Maximum reflectance in angle
16.7%
4.6%
4.5%
Average reflectance in angle
5.9%
2.4%
1.9%
Above, for the front surface of glass, the optimal height of the two combined structures is determined. At the optimum height, both subwavelength structures can effectively reduce the reflectance. In Table 1, we summarize, for the sake of comparison, the maximum reflectance and average reflectance of subwavelength structures at incident wavelengths of 200 nm to 2500 nm and incident angles of 0º to 80º. It can be found from Table 1 that the AR performance of the stacking structure in the spectral range is slightly inferior than that of alternating structure. However, the AR performance of the stacking structure in a wide incident angle is higher than that of alternating structure. In particular, the average reflectance of the stacking structure is 16
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0.5% lower than that of alternating structure. Therefore, the stacking structure of 300 nm has better AR performance on the front surface of glass. Next, in order to improve light utilization efficiency, it is also necessary to lower the back surface of glass by applying a subwavelength structure. Since the stacking structure and the alternating structure can effectively reduce the reflectance of the airglass interface, the two structures can be also applied to the back surface of glass. For the convenience of study, the reflectance of air-glass-air interface is calculated with the existence of 300 nm stacking structure on the front surface of glass. For stacking structure of the back surface of glass, the diameter of bottom surface is fixed at 173 nm. The height of the structure (H3) varies from 100 nm to 1400 nm. In Fig. 9(a), the maximum reflectance is between 7% and 8% for different heights in a wide spectral range. The structure with a height of 500 nm has the smallest average reflectance of 2.0%, and the corresponding maximum reflectance is 7.1%. It can be seen from Fig. 9(b) that the maximum reflectance of stacking structure with a height of 300 nm in a wide incident angle is 14.5%, indicating better AR performance. The stacking structures with heights of 200 nm and 500 nm also exhibited the good AR
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performance with maximum reflectance of 16.1% and 17.8%, respectively. However, when the heights are 200 nm and 300 nm, the average reflectance is large in a wide spectrum and incident angle. Therefore, the stacking structure with a height of 500 nm has better AR performance.
Fig. 9. The stacking structure height of the front surface of the glass is 300 nm. For stacking structure of the back surface with height varying from 100 nm to 1400 nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
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Journal Pre-proof Fig. 10. The stacking structure height of the front surface of the glass is 300 nm. For alternating structure of the back surface with height varying from 50 nm to 1200 nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
For the alternating structure adopted on the back surface of glass, the diameter of the bottom surface of cylinder and paraboloid is fixed at 173 nm. The height of alternating structure (H4) varies from 50 nm to 1200 nm. It can be seen from Fig. 10(a) that the structure with a height of 650 nm has better AR performance in a wide spectrum, and its maximum reflectance is 6.2%. When the height is between 450 nm and 650 nm, the maximum reflectance is always less than 7%, which means good AR performance. In Fig. 10(b), the alternating structure with a height of 200 nm has better AR performance in a wide range of angles, with a maximum reflectance of 7.7%. The AR performance of the structures with heights of 250 nm and 500 nm is also good, with maximum reflectance of 11.9% and 12.0%, respectively. However, the alternating structure with a height of 500 nm has the lowest average reflectance, which is 3.8%. Therefore, when the height is 500 nm, the comprehensive AR performance is better in a wide spectrum and wide incident angle.
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Journal Pre-proof Table 2 When subwavelength structures are applied to the back surface of glass, the AR performance are presented with the existence of 300 nm stacking structure on the front surface of glass, including maximum reflectance and average reflectance in the range of incident wavelengths and angles. Stacking structure
Alternating structure
Maximum reflectance of wavelength
7.1%
6.8%
Average reflectance of wavelength
2.0%
1.7%
Maximum reflectance in angle
17.8%
12.0%
Average reflectance in angle
4.2%
3.8%
Fig. 11. (a) The transmittance of bare glass and AR structure as a function of incident wavelength when incident angle is 8º. (b) The transmittance of bare glass as a function of incident angle when incident wavelength is 495 nm and the transmittance of AR structure as a function of incident angle when the incident wavelengths are 375nm, 495nm and 1545nm, respectively.
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In Table 2, the maximum reflectance and average reflectance of glass are summarized when the back surface of glass adopts stacking structure and alternating structure, respectively. Obviously, the alternating structure is superior to the stacking structure when applied to the back surface of glass. Therefore, the reflectance of glass with the stacking structure of 300 nm on the front surface and the alternating structure of 500 nm on the back surface is the lowest. For bare glass and the AR structure, transmittance as a function of incident wavelength and incident angle is shown in Fig.11. It can be seen from Fig. 11(a) that the transmittance of the AR structure is always higher than that of bare glass in the broad spectrum. In order to reveal the wide-angle AR in a broadband range, Fig. 11 (b) shows the change of transmittance with the incident angle when the incident wavelengths are 375nm, 495nm, and 1545nm, respectively. Obviously, the transmittance of the AR structure is always higher than that of bare glass in a wide range of angles, especially when the incident angle is 80º. In addition, to check the theoretical model, a full-wave simulation was performed on the AR structure, which is shown in Fig. 12. It can be found that the AR structure exhibits excellent AR characteristics, which greatly improves the light transmittance of the glass.
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Fig. 12. The full-wave simulation of the AR structure, which includes 300 nm stacked structures on the front of the glass and 500 nm alternating structures on the back of the glass.
Since PDMS is used for AR on glass surfaces, the robustness of the subwavelength AR composite structure designed was evaluated. First, composite structures require high manufacturing accuracy. Specifically, a 300 nm stacking structure array is formed on the front surface of the glass, and a 500 nm alternating structure array is formed on the back surface of the glass. At present, the subwavelength AR structure manufactured by PDMS often uses nano-imprint fabrication technique, which can well replicate the structure on the master to PDMS [11, 17]. It is reported that in the field of nano-imprint lithography systems, the coverage control is close to 2 nm [27], which can fully meet
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the accuracy requirements for manufacturing masters. Secondly, the subwavelength AR composite structure is fragile to mechanical abrasion. To improving the abrasion resistance, a higher proportion of the curing agent can be incorporated into PDMS to harden the surface [28]. Harder PDMS can ensure the integrity of the subwavelength structure for a long time.
3. Conclusions In summary, the application of subwavelength structures on the front and back surface of glass optimizes the light utilization efficiency, achieving low reflectance of glass in a wide spectrum and wide incident angle. Compared with cylinder and cone, paraboloid have better AR performance, except at large incident angles. In order to overcome the shortcomings of high reflectance at large incident angles, two new subwavelength structures based on the combination of paraboloid and cylinder, alternating structure and stacking structure, are designed. When the stacking structure of 300 nm is applied to the front surface and the alternating structure of 500 nm is applied to the back surface, the AR performance is better.
Acknowledgements
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This work is supported by the National Natural Science Foundation of China (11404171), Jiangsu Natural Science Foundation for Excellent Young Scholar (BK20170101), Scientific Research Foundation of Nanjing University of Posts and Telecommunications (NY217037, NY218015), and the Opening Project of State Key Laboratory of Green Building Materials.
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Journal Pre-proof Author Contributions Xiangfu Wang and Yanyan Bu developed the idea and supervised the project. Yunzhen Yin did all the simulation calculation and data processing. All authors
discussed the results and contributed to writing the manuscript.
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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service or company that could be construed as influencing the position presented in this manuscript submitted.
Yunzhen Yin, Yanyan Bu, Xiangfu Wang PII:
S0921-4526(19)30823-3
DOI:
https://doi.org/10.1016/j.physb.2019.411943
Reference:
PHYSB 411943
To appear in:
Physica B: Physics of Condensed Matter
Received Date:
08 October 2019
Accepted Date:
07 December 2019
Please cite this article as: Yunzhen Yin, Yanyan Bu, Xiangfu Wang, PDMS-based subwavelength structures for broadband and wide-angle anti-reflection, Physica B: Physics of Condensed Matter (2019), https://doi.org/10.1016/j.physb.2019.411943
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PDMS-based subwavelength structures for broadband and wide-angle anti-reflection Yunzhen Yina, Yanyan Bub*, Xiangfu Wanga,c,* aCollege
of Electronic and Optical Engineering & College of Microelectronics,
Nanjing University of Posts and Telecommunications, Nanjing, 210046, China bCollege
of Science, Nanjing University of Posts and Telecommunications, Nanjing,
210046, China cState
Key Laboratory of Green Building Materials, China Building Materials
Academy, Beijing, China *Correspondence author. E-mail address: [email protected] (Bu), xfwang@njupt. edu.cn (Wang)
Abstract Subwavelength structures were reported to have excellent anti-reflection (AR) performance. However, it is difficult to achieve effective AR in broadband and wideangle. In this work, we report the new subwavelength structures fabricated by polydimethylsiloxane (PDMS) for broadband and wide-angle AR. Two new subwavelength structures, alternating structure and stacking structure, are designed. The admittance recursive method is used to calculate the reflectance of subwavelength structures in the range of incident wavelengths from 200 nm to 2500 nm and incident angles from 0º to 80º. It is found that the combination of the 300 nm stacking structure on the front surface and the 500 nm alternating structure on the back surface has the better AR performance. In the wide spectrum and wide incident angle, the maximum reflectance is reduced to 6.8% and 12.0%. Compared to bare glass, the average transmittance was increased by 6.6% and 12.4%, respectively Key words: Subwavelength structure; Anti-reflection; Polydimethylsiloxane; Admittance recursive method
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1. Introduction Reducing the light reflection on the surface of medium is critical to improving the efficiency of solar energy utilization. Therefore, many researchers designed and manufactured graded refractive index layer systems to control the reflectance of incident spectrum [1–3]. However, graded refractive index layer systems had some disadvantages, such as absorption and scattering loss, low durability, and thermal distortion [4]. Recently, under the inspiration of moth eyes, it was found that the subwavelength structures, avoiding the inherent problems of the graded refractive index layer systems, had good AR performance in a wide spectrum and wide incident angle [2, 5–7]. Moreover, subwavelength structures were much thinner and easier to manufacture [8]. Due to good AR performance, many methods in the visible and nearinfrared range were developed to fabricate subwavelength structures, such as etching, laser interference lithography, and nano-imprint lithography [9-11]. When applied to the surface of glass, including the front and back surfaces, subwavelength structure provided a gradual change in effective refractive index at the air-glass-air interface, making it possible to effectively increase the light transmittance [12, 13].
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In order to reduce the reflectance of the glass surfaces, thin films with subwavelength structures were often loaded onto the glass surfaces instead of directly changing the topography of the glass surfaces. However, the AR performance of subwavelength structures depended on the refractive index of materials [14]. In addition, it was difficult to grow film with subwavelength structures in many cases, since that the lattice mismatch among materials blocked the growth of the film [15]. PDMS, a polymer material with good adhesion, and has good optical properties with the refractive index of 1.43 which is close to the refractive index of the glass (1.52) [11]. Furthermore, due to the strong van der Waals force between PDMS and glass, PDMS film with subwavelength structure can be easily adhered to glass surface without adhesive [16], avoiding the difficulty of lattice matching of materials. Although PDMS has gained wide application due to its good properties, excellent AR performance is limited to a narrow spectral range. For example, Kim et al. used a PDMS film with subwavelength structures to achieve a high-efficiency perovskite solar cell at wavelengths between 300 nm and 800 nm [17]. Lee et al. enhanced the transmittance of sapphire in the wavelength region of 400 nm to 1000 nm by using patterned PDMS
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surface structure [18]. Therefore, the AR performance of PDMS with subwavelength structures still have a large room for improvement in a wide spectrum. When sunlight is incident on the glass, strong reflection result in loss of solar energy due to the gradient of the refractive index at the air-glass-air interface [19]. Especially when the sunlight is incident at a large angle, the reflection of the glass surface will be stronger. Therefore, it is necessary to design new subwavelength structures to achieve low reflectance of the upper and lower surfaces of the glass in a wide incident angle.
Fig. 1. Schematic diagram of a bare glass and a glass with composite structures
In this paper, the composite structure is designed to achieve low reflectance of the glass in a wide spectrum and wide incident angle, which is shown in Fig. 1. The
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admittance recursive method is developed to show the AR performance of subwavelength structures fabricated by PDMS.
2. Model and method The admittance recursive method is used to study the reflectance of subwavelength structures. It is well known that when a film is coated on the substrate, the film and the substrate can be represented by an equivalent interface. The subwavelength structure can be equivalent to an n-layer film on glass, if it is divided into n layers of equal thickness. It is assumed that the film adjacent to the glass is the nth layer, and the film close to the air is the first layer. The thickness, refractive index and extinction coefficient of the jth layer are dj, nj and kj, respectively. The combined admittance of the nth film and the glass is [20]
Yn
Yn 1 cos n i n sin n yn(1) iyn(2) cos n iYn 1 / n sin n
(1)
where 𝛿n and 𝜂n are the effective phase thickness and the modified admittance of the nth layer, and Yn+1 is the admittance of the glass. When gradually recursing to the jth film, the recursion formula for Yj is as follows
Yj
Y j 1 cos j i j sin j cos j iY j 1 / j sin j
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(2) y (1) j iy j
(2)
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After recursing to the first layer, the combined admittance of the subwavelength structure is
Y1 y1(1) iy1(2)
(3)
N 0 Y1 2 ( N 0 y1(1) ) 2 ( y1(2) ) 2 R( ) N 0 Y1 ( N 0 y1(1) ) 2 ( y1(2) ) 2
(4)
The reflection is
where N0 is the admittance of the air. When light is incident on the film surface at a certain angle 𝜃0, the effective phase thickness is
j
2
d j (n 2j k 2j N 02 sin 2 0 2in j k j )1/2
(5)
where λ is the incident wavelength. The incident light can be decomposed into TE waves and TM waves, which are an electric vector perpendicular to the incident surface and an electric vector parallel to the incident surface, respectively. The modified admittance of TE waves is
j n 2j k 2j N 02 sin 2 0 2in j k j The modified admittance of TM waves is
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(6)
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n 2j k 2j 2in j k j (n 2j k 2j N 02 sin 2 0 2in j k j )1/2
(7)
The reflectance of incident light is the average of TE wave reflectance and TM wave reflectance. To verify the feasibility of the admittance recursive method, we simulated the AR performance of a multi-layer nanostructure fabricated by Kuo et al [21]. The reflectance as a function of the incident wavelength and incident angle shown in Fig. 2 is consistent with Fig. 2(b) and Fig. 3 in Ref.17, which means that the admittance recursion method can accurately calculate the reflectance of subwavelength structures.
Fig. 2. Simulation results on AR performance of a multilayer nanostructure. (a)
Reflectance as a function of incident wavelength when incident angle is 8 degrees, (b) reflectance as a function of incident angle when incident wavelengths are 633 nm, 830 nm, and 904 nm, respectively.
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Fig. 3. (a) Schematic diagram of paraboloid, cylinder and cone. (b) The change in the effective refractive index of paraboloid, cylinder and cone with a height of 700 nm, as the incident light travels from air into subwavelength structures.
Since the glass has front and back surfaces, subwavelength structures were first applied to reduce the reflectance of the front surface of glass. The reflectance of the airglass-air interface is then minimized by optimizing the subwavelength structure of the back surface of glass. The admittance recursive method is used to calculate the reflectance of periodic subwavelength structures in the solar spectrum range of 200 nm to 2500 nm and the incident angle range of 0° to 80°. Paraboloid, cylinder and cone shown in Fig. 3(a) were reported to have good AR performance [9, 22, 23]. These subwavelength structures are placed in a hexagonal grid with a side length of 100 nm. The height of the structures is 700 nm and the period is 173 nm. The diameters of the bottom surface of paraboloid, cylinder and cone are fixed at 173 nm. The change of
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effective refractive index of the three structures is shown in Fig. 3(b). The effective refractive index, neff, can be calculated based on [6] neff
fns2 (1 f )ni2
(8)
where ns and ni are the refractive index of PDMS and air. f is the fill factor of PDMS as a function of structure height. It can be observed from Fig. 3(b) that the effective refractive index of the bottom of the paraboloid, cylinder and cone has a small gradient change, while the cylinder also has a large effective refractive index gradient change at the top. It should be noted that in the spectral range studied in this paper, the extinction coefficient of PDMS is very small [24], which has little effect on the results of the study, so the extinction coefficient of PDMS is negligible. In order to compare the AR performance of the three subwavelength structures, the reflectance of the front surface of glass as a function of incident wavelength and incident angle is shown in Fig. 4. As can be seen from Fig. 4(a), when the incident angle is 8º, the reflectance of the cylinder fluctuates between 1.5% and 4.3%. When the incident wavelength is small, the reflectance of the cone has a small fluctuation range. When the incident wavelength increases to 2500 nm, the reflectance of the cone rises
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to 1.1%. The paraboloid, however, always maintains low reflectance in the wavelength range of 200 nm to 2500 nm., with a maximum reflectance of 0.6% occurring at 1360 nm. For sunlight with a wavelength of 495 nm, the reflectance of the three structures is always less than 10% when the incident angle is less than 50º, as shown in Fig. 4(b). When the incident angle continues to increase, the reflectance of cone increases rapidly, while the reflectance of cylinder changes little. In the incident range of 0º to 80º, the maximum reflectance of paraboloid, cylinder and cone is 16.7%, 7.3% and 55.8%, respectively. Since the high reflectance in a wide spectrum, the AR performance of cylinder is inferior to paraboloid. Therefore, the paraboloid has better AR performance in a wide spectrum and wide incident angle.
Fig. 4. For paraboloid, cylinder and cone with a height of 700 nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
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Fig. 5. For paraboloid varying in height from 50 nm to 2000 nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
It is reported that the height of the subwavelength structure plays a dominant role in AR performance [25]. In order to make the paraboloid has better AR performance at the optimal height, the reflectance of paraboloid with different heights (H) is studied, which is shown in Fig. 5. It can be observed from Fig. 5(a) that when the incident angle is 8º, the reflectance decreases in the long wavelength range as the height of paraboloid increases. When H ≥ 600 nm, the reflectance is less than 1% in the entire spectral range. As shown in Fig. 5(b), when the height of the paraboloid is 50 nm and 300 nm, the maximum reflectance is reduced to 8.8% and 11.6% in a wide angle rang, while the reflectance in a wide spectrum is very high. Under the premise that the reflectance is
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less than 1% in the broad spectral range, the paraboloid of 700 nm has better AR performance, with a maximum reflectance of 16.7%. Therefore, paraboloid with height of 700 nm achieves better AR performance in a wide spectrum and wide incident angle. Although paraboloid exhibits excellent AR performance compared to cone and cylinder, the reflectance is still higher at large incident angles, especially when the incident angle is greater than 40º. However, when the incident angle is greater than 40º, the cylinder exhibits good AR performance. It attributed to the difference in the spatial distribution of the effective refractive index of cylinder and paraboloid [26]. Specifically, the effective refractive index of paraboloid varies linearly with the structure height, while the effective refractive index of cylinder varies abruptly. Therefore, in order to achieve excellent AR performance in a wide spectrum and wide incident angle, new structures need to be designed, which has the characteristics of the effective refractive index change of both cylinder and paraboloid. In Fig. 6, two subwavelength structures based on cylinder and paraboloid, alternating structure and stacking structure, are designed, which have the advantages of cylinder and paraboloid
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in AR. The two structures will first be applied to reduce the reflectance of the front surface of the glass.
Fig. 6. Schematic diagram of (a) alternating structure and (b) stacking structure.
For alternating structure, the schematic diagram is shown in Fig. 6(a). The diameters of the bottom surfaces of cylinder and paraboloid are D1 and D2, respectively. The structure height is H1. First, D1 and D2 are both fixed at 173 nm. Then, the effect of height of alternating structure on AR performance is explored by adjusting H1 to vary from 50 nm to 1200 nm. As can be seen from Fig. 7(a), when the structure height H1 is less than 300 nm, the maximum reflectance is greater than 2%. When the structure height is increased to above 300 nm, the reflectance fluctuates between 0% and 2%. As shown in Fig. 7(b), when the structure height is 300 nm, the AR performance is better,
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and the maximum reflectance is reduced to 4.4%. However, in order to ensure good AR performance in a wide spectrum, the structure height should be greater than 300 nm at least. Therefore, for alternating structure, 550nm is the optimal height and the maximum reflectance is 4.6% in wide angle.
Fig. 7. The alternating structure is applied to the front surface of glass. With the alternating structure height varying from 50nm to 1200nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
The schematic diagram of stacking structure is shown in Fig. 6(b). The paraboloid with a height of H2 and a bottom diameter of D3 is divided into two parts of equal height, wherein the upper half is a small paraboloid and the lower half is a round table. Then, the small paraboloid is replaced by a cylinder of the same height. The diameter of the cylinder is the same as the diameter of the bottom surface of the small paraboloid.
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The height of the structure varies from 100 nm to 1400 nm for the better AR performance. The reflectance as a function of incident wavelength and incident angle is shown in Fig.8. As seen from Fig. 8(a), when H2 is less than 300 nm, the reflectance is large in the long wavelength range. When H2 is greater than 300 nm, the reflectance fluctuates between 0 and 2%. In Fig. 8(b), when the structure height is 300 nm, the AR performance is better, and the maximum reflectance is 4.5%. Considering the AR performance in a wide spectrum and wide incident angle, the optimal height of the stacking structure is 300 nm.
Fig. 8. The stacking structure is applied to the front surface of glass. With the stacking structure height varying from 100nm to 1400nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
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Journal Pre-proof Table 1 AR performance of subwavelength structures applied to the front surface of glass are presented, including maximum reflectance and average reflectance in the range of incident wavelengths and angles. Paraboloid
Alternating structure
Stacking structure
Maximum reflectance of wavelength
0.6%
1.8%
2.1%
Average reflectance of wavelength
0.2%
1.0%
1.0%
Maximum reflectance in angle
16.7%
4.6%
4.5%
Average reflectance in angle
5.9%
2.4%
1.9%
Above, for the front surface of glass, the optimal height of the two combined structures is determined. At the optimum height, both subwavelength structures can effectively reduce the reflectance. In Table 1, we summarize, for the sake of comparison, the maximum reflectance and average reflectance of subwavelength structures at incident wavelengths of 200 nm to 2500 nm and incident angles of 0º to 80º. It can be found from Table 1 that the AR performance of the stacking structure in the spectral range is slightly inferior than that of alternating structure. However, the AR performance of the stacking structure in a wide incident angle is higher than that of alternating structure. In particular, the average reflectance of the stacking structure is 16
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0.5% lower than that of alternating structure. Therefore, the stacking structure of 300 nm has better AR performance on the front surface of glass. Next, in order to improve light utilization efficiency, it is also necessary to lower the back surface of glass by applying a subwavelength structure. Since the stacking structure and the alternating structure can effectively reduce the reflectance of the airglass interface, the two structures can be also applied to the back surface of glass. For the convenience of study, the reflectance of air-glass-air interface is calculated with the existence of 300 nm stacking structure on the front surface of glass. For stacking structure of the back surface of glass, the diameter of bottom surface is fixed at 173 nm. The height of the structure (H3) varies from 100 nm to 1400 nm. In Fig. 9(a), the maximum reflectance is between 7% and 8% for different heights in a wide spectral range. The structure with a height of 500 nm has the smallest average reflectance of 2.0%, and the corresponding maximum reflectance is 7.1%. It can be seen from Fig. 9(b) that the maximum reflectance of stacking structure with a height of 300 nm in a wide incident angle is 14.5%, indicating better AR performance. The stacking structures with heights of 200 nm and 500 nm also exhibited the good AR
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performance with maximum reflectance of 16.1% and 17.8%, respectively. However, when the heights are 200 nm and 300 nm, the average reflectance is large in a wide spectrum and incident angle. Therefore, the stacking structure with a height of 500 nm has better AR performance.
Fig. 9. The stacking structure height of the front surface of the glass is 300 nm. For stacking structure of the back surface with height varying from 100 nm to 1400 nm, (a) reflectance as a function of incident wavelength when incident angle is 8º, (b) reflectance as a function of incident angle when incident wavelength is 495 nm.
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For the alternating structure adopted on the back surface of glass, the diameter of the bottom surface of cylinder and paraboloid is fixed at 173 nm. The height of alternating structure (H4) varies from 50 nm to 1200 nm. It can be seen from Fig. 10(a) that the structure with a height of 650 nm has better AR performance in a wide spectrum, and its maximum reflectance is 6.2%. When the height is between 450 nm and 650 nm, the maximum reflectance is always less than 7%, which means good AR performance. In Fig. 10(b), the alternating structure with a height of 200 nm has better AR performance in a wide range of angles, with a maximum reflectance of 7.7%. The AR performance of the structures with heights of 250 nm and 500 nm is also good, with maximum reflectance of 11.9% and 12.0%, respectively. However, the alternating structure with a height of 500 nm has the lowest average reflectance, which is 3.8%. Therefore, when the height is 500 nm, the comprehensive AR performance is better in a wide spectrum and wide incident angle.
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Journal Pre-proof Table 2 When subwavelength structures are applied to the back surface of glass, the AR performance are presented with the existence of 300 nm stacking structure on the front surface of glass, including maximum reflectance and average reflectance in the range of incident wavelengths and angles. Stacking structure
Alternating structure
Maximum reflectance of wavelength
7.1%
6.8%
Average reflectance of wavelength
2.0%
1.7%
Maximum reflectance in angle
17.8%
12.0%
Average reflectance in angle
4.2%
3.8%
Fig. 11. (a) The transmittance of bare glass and AR structure as a function of incident wavelength when incident angle is 8º. (b) The transmittance of bare glass as a function of incident angle when incident wavelength is 495 nm and the transmittance of AR structure as a function of incident angle when the incident wavelengths are 375nm, 495nm and 1545nm, respectively.
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In Table 2, the maximum reflectance and average reflectance of glass are summarized when the back surface of glass adopts stacking structure and alternating structure, respectively. Obviously, the alternating structure is superior to the stacking structure when applied to the back surface of glass. Therefore, the reflectance of glass with the stacking structure of 300 nm on the front surface and the alternating structure of 500 nm on the back surface is the lowest. For bare glass and the AR structure, transmittance as a function of incident wavelength and incident angle is shown in Fig.11. It can be seen from Fig. 11(a) that the transmittance of the AR structure is always higher than that of bare glass in the broad spectrum. In order to reveal the wide-angle AR in a broadband range, Fig. 11 (b) shows the change of transmittance with the incident angle when the incident wavelengths are 375nm, 495nm, and 1545nm, respectively. Obviously, the transmittance of the AR structure is always higher than that of bare glass in a wide range of angles, especially when the incident angle is 80º. In addition, to check the theoretical model, a full-wave simulation was performed on the AR structure, which is shown in Fig. 12. It can be found that the AR structure exhibits excellent AR characteristics, which greatly improves the light transmittance of the glass.
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Fig. 12. The full-wave simulation of the AR structure, which includes 300 nm stacked structures on the front of the glass and 500 nm alternating structures on the back of the glass.
Since PDMS is used for AR on glass surfaces, the robustness of the subwavelength AR composite structure designed was evaluated. First, composite structures require high manufacturing accuracy. Specifically, a 300 nm stacking structure array is formed on the front surface of the glass, and a 500 nm alternating structure array is formed on the back surface of the glass. At present, the subwavelength AR structure manufactured by PDMS often uses nano-imprint fabrication technique, which can well replicate the structure on the master to PDMS [11, 17]. It is reported that in the field of nano-imprint lithography systems, the coverage control is close to 2 nm [27], which can fully meet
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the accuracy requirements for manufacturing masters. Secondly, the subwavelength AR composite structure is fragile to mechanical abrasion. To improving the abrasion resistance, a higher proportion of the curing agent can be incorporated into PDMS to harden the surface [28]. Harder PDMS can ensure the integrity of the subwavelength structure for a long time.
3. Conclusions In summary, the application of subwavelength structures on the front and back surface of glass optimizes the light utilization efficiency, achieving low reflectance of glass in a wide spectrum and wide incident angle. Compared with cylinder and cone, paraboloid have better AR performance, except at large incident angles. In order to overcome the shortcomings of high reflectance at large incident angles, two new subwavelength structures based on the combination of paraboloid and cylinder, alternating structure and stacking structure, are designed. When the stacking structure of 300 nm is applied to the front surface and the alternating structure of 500 nm is applied to the back surface, the AR performance is better.
Acknowledgements
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This work is supported by the National Natural Science Foundation of China (11404171), Jiangsu Natural Science Foundation for Excellent Young Scholar (BK20170101), Scientific Research Foundation of Nanjing University of Posts and Telecommunications (NY217037, NY218015), and the Opening Project of State Key Laboratory of Green Building Materials.
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Journal Pre-proof Author Contributions Xiangfu Wang and Yanyan Bu developed the idea and supervised the project. Yunzhen Yin did all the simulation calculation and data processing. All authors
discussed the results and contributed to writing the manuscript.
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We declare that we have no financial and personal relationships with other people or organizations that can inappropriately influence our work, there is no professional or other personal interest of any nature or kind in any product, service or company that could be construed as influencing the position presented in this manuscript submitted.