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Abstract
In the temperate latitudes, high-reflectivity exterior surfaces save energy spent on ventilation and cooling during summer, but cost energy on heating in winter. Angularly selective surfaces that adjust their reflectivities by sun position allow beneficial effects in both seasons — high reflectivity in summer and high absorption in winter. Here we show how a planar microstructured surface can produce such an angularly selective behavior and estimate its energy efficiency under direct solar irradiance at 35° N. Results show that such an ideal angularly selective surface has the potential to improve efficiency by up to 43.2% compared to a conventional concrete surface. Numerical results for an aluminum one-dimensional periodic structure indicate that it achieves a 25.7% improvement of efficiency. Finally, we validate the designed structure by measuring the reflectivity of the fabricated surface at a series of angles.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
During the 1971-2015 period, global energy consumption doubled, and 30 percent of this increase has been for controlling the environment within buildings [1]. Forty to fifty percent of the energy consumed in the USA is for the heating, ventilation and air conditioning (HVAC) of buildings [2]. One idea for reducing this energy consumption is to use a “cool roof” [3], in which a high reflectivity paint is applied to exterior surfaces to act as a sunlight reflector in summer. While this has been shown to reduce heat conduction through the roof [4], the use of these paints also creates an energy penalty in winter, especially in the temperate latitudes [5]. Reducing energy consumption in winter requires a low reflectivity (absorptive) surface. A single surface which can perform both tasks can provide energy benefits in both seasons. Such a surface can also improve the thermal environment of fixed facilities and outdoor machines such as used in telecommunications, heat exchangers, etc.
Akbari et al. have investigated the use of heterogeneous directional reflective materials (DRM) to realize a surface with seasonally varying reflectivity [6]. This DRM surface has a periodic triangular structure such that faces pointing towards the sun have high reflectivity and faces pointing away from the sun have low reflectivity. The total reflectivity of the DRM is calculated from the reflectivity ratio of the two faces, the view factor of each surface, and the solar illumination angle. Akbari et al. showed that this triangular DRM has an effective reflectivity of approximately 48 percent in summer and 38 percent in winter, when using 90 percent and 4 percent reflectivity on the two faces, respectively. This seasonal difference in reflectivity is disappointingly small; a surface with a larger change can provide better energy efficiency. We explore the question of whether a periodic microstructured surface can provide the sharper angular dependence needed to achieve this. We also provide an ideal angular reflectivity profile, and show how to make use of this to quantitatively evaluate the effectiveness of any given surface.
In the discussion below, we start by modeling the angular reflectivity distribution around the sun position trajectories distributed over the course of a year. Using this model, we use rigorous coupled wave analysis (RCWA) to estimate the amount of reflected and absorbed sunlight experienced by one-dimensional periodic structure made of aluminum. After fabricating the designed structure, we show reflectivity measurements and compare with the values expected from the RCWA model.
2. Principles
From any point on the Earth’s surface, we can track the position of the sun at any given time using its coordinates $\def\upalpha{\unicode[Times]{x03B1}}\theta$ and $\phi$, where $\theta$ is the sun zenith angle and $\phi$ is the azimuthal angle. The sun position $(\theta ,\phi )$ is given by [7]
where $\Delta$ is the latitude, $\omega$ is the time angle in degrees, $\delta$ is the sun declination in degrees, $\Lambda$ is the ecliptic longitude, $\tau$ is the local time in hours. Figure 1 shows a view of these coordinates and parameters.
Fig. 1. (a) A view of the sun position $(\theta , \phi )$ in ecliptic coordinates with respect to a viewer standing at an arbitrary point on the earth. (b) An arbitrary point on the Earth at latitude $\Delta$. (c) Sun position with respect to a viewer at the arbitrary latitude.
If we consider an opaque horizontal surface with angularly-varying reflectivity $R(\theta ,\phi )$, the time-averaged reflected power $P_{r}$ [W] and absorbed power $P_{\upalpha}$ [W] from a unit area of this surface can be expressed as
The amount of direct normal sunlight onto the unit area $P_{d}(\theta ,\phi )$ can be obtained by [8]
where $H_{o}(\theta ,\phi )$ is the extraterrestrial sunlight irradiance [W/m2], $U$ is the exposed area [m2], and $T$ is the atmospheric transmittance.A quantity that we call the “total positive power” $P$ [W] gives the improvement in building insolation by adding the (desirable) decrease in absorbed power $P_\textrm {r}$ in summer with the (also desirable) increase in absorbed power in winter $P_{\upalpha }$. The actual period of time for summer and winter have to be determined to calculate $P_\textrm {r}$ and $P_{\upalpha }$, and this can be based on the local statistical report of the HVAC system usage. The energy efficiency can be used to set the importance of summer reflectivity and/or winter absorptivity by weighing the seasonal amount of sunlight [9]. In this case, we can replace $P_\textrm {d}$ in Eq. (5) and Eq. (6) with $P_\textrm {wd} = w(t) \, P_\textrm {d}$, for weighting function $w(t)$. We use an equal weight for summer reflectivity and winter absorptivity in the discussion below.
We select 35° N as our location for this work as it is roughly the location of the city of Tokyo, and is also a latitude near which one can find a number of large cities in North America. The Sun positions $(\theta ,\phi )$ as viewed from 35° N over the course of a year, obtained from Eq. (2) and Eq. (3), together with the direct normal sunlight $P_\textrm {d}$ for unit area $U = {1}\;\textrm{m}^2$, are shown in Fig. 2. The lowest and highest Sun zenith are 13° and 60°, respectively. Since seasonal changes affect the hours of daylight but have negligible affect on the azimuth $\phi$, we only consider the Sun’s zenith angle $\theta$ when estimating the total positive power $P$ of the surface.
Fig. 2. The range of Sun positions over the course of a day (following along a curve) and the course of seasonal change (moving perpendicular to the curves) for a viewer at 35° N. The direct normal sunlight for unit area $P_\textrm {d}$ is displayed as a grayscalemap. Due to the cosine effect at varying angles of incidence, the irradiance at summer solstice noon is twice that at winter solstice noon.
3. Analysis of angular reflectivity model
3.1 Numerical method
We define a parametrization of $R (\theta )$ that provides a continuous family of curves going from a discontinuous curve for only winter and summer to a continuous curve that includes all of the seasons. These curves indicate the transition from high reflectivity to low reflectivity at the boundary incident angle, considering that HVAC systems in temperate latitudes would be used for cooling in summer and heating in winter. We discuss the discontinuous profile in Sec. 4. below; the other continuous curves are usable if one considers neither perfect reflective or absorptive properties for spring and fall. Thus it is useful to allow the ideal angular reflectivity curve to vary depending on the range of seasons considered. If we write the maximum $\theta$ for each season $\theta _\textrm {Summer}$, $\theta _\textrm {Spring/Fall}$, and $\theta _\textrm {Winter}$ (visible in Fig. 2), from the known relations among the maximum zenith angles $\theta _\textrm {Summer} \leq \theta _\textrm {Spring/Fall} \leq \theta _\textrm {Winter}$, the reflectivity requirement of 50% for $R (\theta )$ during spring/fall can be expressed as
A parametrized sigmoid curve fulfills the requirement of continuous adaptability such that the angular reflectivity $R(\theta ; \rho )$ of the surface varies from $R_\textrm {min}$ to $R_\textrm {max}$ and passes through the 50% point at the “transition” angle of incidence $\theta _\textrm {trans}$ by
Fig. 3. Reflectivity profile of a model angular selective surface for transition angle $\theta _\textrm {trans} = 50^{\circ }$.
Next we calculate the total positive power $P$ for an angular profile model, given as the sum of $P_\textrm {r}$ in summer and $P_{\alpha }$ in winter for 35° N, and analyze the dependence of $P$ on each of the surface design parameters. We also analyze a concrete surface having 60% reflectivity, independent of incidence angle, as the reference roofing material [10]. The Sun position $(\theta , \phi )$ and direct normal sunlight $P_\textrm {d}$ values at 35° N are shown in Fig. 2. The variation in total positive power $P$ — the thing we want to optimize — is performed first for $R_\textrm {min}$ and $R_\textrm {diff}$ (Fig. 4(a)), and then for $\theta _\textrm {trans}$ and $\rho$ (Fig. 4(b)), assuming that June through August belong to summer, and November through March to winter. The rest are swing-season months and thus are ignored when calculating $P$.
Fig. 4. Total positive power $P$ dependence of(a) $R_\textrm {diff}$ and $R_\textrm {min}$, (b) $\theta _\textrm {trans}$ and $\rho$.
3.2 Total power of angular reflectivity model
Figure 4 shows the dependence of the total positive power $P$ on the reflectivity profile model parameters (Eq. (5) and Fig. 3) $R_\textrm {min}$ & $R_\textrm {diff}$, shape factor $\rho$, and reflectivity transition angle $\theta _\textrm {trans}$. The maximum value of $P$ from Fig. 4(a) is obtained as 261.5 W where $R_\textrm {diff} = 100\%$. In contrast, the total positive power $P$ for an angularly independent 100% reflective surface is found as $P = 230.8\;\textrm{W}$.
The dependence of $\theta _\textrm {trans}$ and $\rho$ is also shown in Fig. 4(b), where we see that the maximum $P$ is obtained at $\theta _\textrm {trans} = 50^{\circ }$. Figure 5 shows the $P$ value obtained for various angular reflectivity profiles; for reference, we also note that concrete has $P = 182.6\;\textrm{W}$.
Fig. 5. Angular reflectivity profile of various condition, optimum angular selective surface (red), angular change of 50 percent reflectivity (purple), perfectly reflective surface (green), concrete roof (black).
4. Numerical analysis of one-dimensional periodic structure
4.1 Numerical method
Figure 5 shows that an angular reflectivity profile whose reflectivity approaches to zero for incidence angle of 50–90° has high $P$. Periodic structure has been shown to be capable of achieving broad angular reflectivity profiles that are appropriate for this. For example, Popov et al. have shown that a structure that is periodic at a frequency close to $n \lambda / 2$ for any integer $n$ and wavelength $\lambda$ has a broad angular absorptivity [11]. The angular absorptivity range and the peak of these structures can be adjusted by changing the surface’s shape parameters so that the broad angular absorptivity is centered within the incidence angle of 50–90°. We chose a one-dimensional periodic structure to simplify the simulation parameters and validate the basic performance of angular reflectivity.
In order to locate the surface structure’s shape parameters that maximize the total positive power $P$, we use commercial rigorous coupled-wave analysis software (DiffractMod, Rsoft Inc.). The one-dimensional periodic structure has a rectangular cross-section as shown in Fig. 6, with shape parameters $h$ (height), $A$ (period), and $D$ (duty cycle). High reflectivity is desirable for the incident angle ranges of $\theta\;<\;\theta _\textrm {trans}$ to reduce the summer sunlight absorption. We use aluminum as the structure material because it has a high reflectivity over the wavelength range containing most sunlight energy. In order to simplify the analysis, we perform the RCWA simulations at reference wavelength $\lambda = 0.8$ µm, at which the optical constants of aluminum are $N = n + \textrm {i} k = 2.80 + \textrm {i} 8.45$ [12]. We initially varied the period in the range $0.1\;<\;A / \lambda\;<\;10$, but results quickly showed that useful values of reflectivity are only obtained within $0.5 \leq A/\lambda \leq 5$. The calculations also varied the aspect ratio in the range $0.2 \leq h/w \leq 2$, while keeping the duty cycle constant at $D=0.67$. The reflectivity calculations are performed separately for both TE- and TM-mode polarization, and then combined to get the mean reflectivity
4.2 Numerical results of one-dimensional periodic structure
Surfaces that have seasonal tuning of reflectivity can be classified by the ratio of total absorbed power $P_{\upalpha }$ to total positive power $P$, calculated via Eq. (5) for the mean reflectivity $R(\theta ,\phi )$. The ratio of $P_{\upalpha }$ to $P$ indicates the contribution of winter absorption to the total amount of reflection and absorption over the course of a year. Thus the value of $P_{\upalpha }/P$ ratio and $P$ can be the figure of merit of seasonal tuning of sunlight. For example, an uncoated aluminum surface at a wavelength of 0.8 µm gives $P = 227.9\;\textrm{W}$ and $P_{\upalpha } = 16.8\;\textrm{W}$, so that $P_{\upalpha } / P = 0.07$. This means that the aluminum surface has a low angular dependence and its winter absorption fraction will be low. Figure 7 shows the contour map of the $P_{\upalpha }$ to $P$ ratio for parameters $A$ and $h/w$. Within the domain shown in Fig. 7, the minimum and maximum values for $P$ are 200.2 W and 229.5 W, with the maximum value of $P_{\upalpha }/P = 0.23$ located at $A/\lambda = 1.5$, $h/w=1$, and $D=0.67$. At this location, the structure has $P_\textrm {r} = 171.4\;\textrm{W}$, $P_{\upalpha } = 51.5\;\textrm{W}$, and $P = 222.9\;\textrm{W}$. Figure 8 shows the effective reflectivity $R_\textrm {e}$ of this structure as calculated by Eq. (7).
Fig. 7. The ratio of absorptivity $P_{\upalpha }$ to total positive power $P$ for duty cycle $D=0.67$, at $\lambda = 0.8$ µm.
Fig. 8. (a) Angular reflectivity profile of the structure having period $A = 1.5 \lambda$, aspect ratio $h/w=1$, and duty cycle $D=0.67$, calculated at $\lambda = 0.8$ µm. (b) Seasonal change of effective reflectivity $R_{e}$ in each month, for this structure.
5. Experiment
5.1 Fabrication and measurement
We demonstrate the angular reflectivity of a one-dimensional periodic structure. Fabricable structures are limited to have lower aspect ratios of $h/w\;<\;0.5$ because of our manufacturing facilities. Thus, we analyze ratio $P_{\upalpha }/P$, and positive power $P$, for structures under the limitation of low aspect ratio $h/w=0.5$, and with duty cycle $D=0.5$ and period $A/\lambda =2$. The reference wavelength is 0.8 µm.
Fabrication involves a lift-off process as follows. An opaque, thick aluminum film is deposited onto a silicon wafer substrate. An electron-beam (EB) resist (ZEP-520A) is spin-coated to achieve a uniform thickness of over 0.5 µm as a sacrificial layer. The line and space pattern is exposed via EB lithography and then developed using ZED-N50. An 0.4 µm aluminum thin film is then evaporated onto the resist. And then resist and aluminum is removed using acetone.
The angular reflectivity is measured by a gonio-photometer, with the double-arm experimental setup shown in Fig. 9. A laser diode of wavelength $\lambda = 0.8$ µm (Thorlabs, CPS808S) is used as the light source. The reflectivity is measured over incidence angles of 10–70° in 5° steps. The reflected light power at each angle is measured by a laser power meter (Thorlabs, S-120C), and the TE and TM polarizations measured separately by using a Glan-prism polarizer. We also calculate the $P$ and $P_{\upalpha }$ by interpolating the measured data.
Cross sectional images of the fabricated structure are taken using a focused ion beam (FIB) – scanning electron microscope (SEM) (XVision 200TB, SII nanotechnology Inc.). A carbon protective layer is deposited onto the structure before FIB etching. The structure geometry is first measured from the SEM image, but since the milling direction angle and the SEM imaging angle differ by 54°, the structure height $h$ is divided by $\sin 54^{\circ }$ to get the actual physical height.
5.2 Experimental results
The structure design (constrained by the limits of what we can manufacture in our facility) is determined from searching for the parameter set achieving the highest $P_{\upalpha }/P$ values within Fig. 10, from which we obtain an optimum at $A/\lambda =2$, $h/w=0.5$.
Fig. 10. The calculated ratio of absorptivity $P_{\upalpha }$ to total positive power $P$, using structure parameters for the as-fabricated structure: $h/w \leq 0.5$ and $D=0.5$.
Figure 11(a) shows measurements of this manufactured structure. Numerical simulations for the original design $A/\lambda =2$, $h/w=0.5$ shows its reflectivity should be 80% at 10°, and 50% at 70°. The minimum reflectivity is found at about $\theta = 50^{\circ }$ for the manufactured structure (red squares) but at $\theta = 70^{\circ }$ for the original design (blue line). Also, the manufactured structure shows 10% less reflectivity for incident angles below 40°. The $P$ and $P_{\upalpha }/P$ of this structure are 204.0 W, 0.22, respectively. This is an 11% improvement above the reference concrete roof. An SEM image of the fabricated structure (see Fig. 11(b)) shows that the structure has a trapezoidal cross-sectional shape where the bottom width $w_\textrm {bot}=$ 0.75 µm, upper width $w_\textrm {top}=$ 0.55 µm. The other geometry parameters of duty cycle and period are approximately the same for the manufactured structure as for the design. Because the simulated and experimental structures are not quite the same, we perform numerical simulations to analyze the optical effects of the manufactured structure’s cross-sectional shape. Figure 11(a) (red line) shows the calculated result using the SEM-based cross-sectional shape for input. We see that the minimum reflectivity angle is shifted from $\theta = 70^{\circ }$ for the original design (blue line) to $\theta = 55^{\circ }$ for the SEM-based model (red line) and that the overall reflectivity of the SEM-based model shows an overall 10% difference in reflectivity from the measured reflectivity.
Fig. 11. (a) Reflectivity profile of the structure, where the measured reflectivity is shown in red squares, the simulated reflectivity based on the RCWA model and parameters estimated from the SEM image (red line), and the simulated reflectivity based on the original design parameters (blue line). (b) Cross sectional SEM image for the fabricated structure.
6. Discussion
As we noted above, one can estimate the energy efficiency of roofing materials by the total positive power $P$, which is the summation of reflected sunlight in summer and absorbed sunlight in winter. For an angular-selective surface, this $P$ is proportional to $R_\textrm {diff}$, the difference between reflectivities below and above the transition angle. The resulting $P$ for a concrete roof is obtained as 182.6 W (Fig. 5). In contrast, the ideal angular-selective surface has $P = 261.5\;\textrm{W}$. This is a 43.2% improvement, indicating that such angular-selective surfaces have the potential to save significant energy.
Statistical research from the Energy Information Administration shows that the annual average energy usage for residential heating is around two times larger than for air conditioning, e.g. for 35°N latitudes in the United States [13]. Also, the EIA report for monthly residential energy consumption showed that energy consumption in December is around 30% higher compared to July [14]. Thus the energy input in winter can be more beneficial rather than summer reflectivity.
Numerical simulation of a one-dimensional periodic surface (Fig. 7) shows that such structures have a mean value of $P= 217.4\;\textrm{W}$. This means that these structures have a better energy performance (total positive power $P$) compared to concrete surfaces. In particular, a surface with period $A/\lambda = 1.5$ and aspect ratio $h/w=1$ gives $P = 222.9\;\textrm{W}$, and $P_{\upalpha }/P = 0.23$. This is a 25.7% improvement. This structure demonstrates a larger annual reflectivity difference compared to the 90%/4% heterogeneous directional reflective surface characterized by Akbari et al. [6].
The experimental results show that the fabricated surface has an angular dependence of reflectivity but that the reflectivity difference, between angles below and above the transition angle, is less than the simulation. Whereas the simulation gives $R_\textrm {diff} \sim 30\%$, the fabricated surface achieved only $R_\textrm {diff} \sim 15\%$, and its reflectivity peak angle is shifted from $70^{\circ }$ to $50^{\circ }$ (Fig. 11(a)). These differences between measurement and simulation are likely caused by alterations in the designed shape during fabrication. We can see in Fig. 11(b) that the cross-sectional shape of the fabricated structure has a rounded trapezoidal shape because of the angular divergence of aluminum atoms through evaporation [15]. In order to verify that the trapezoidal shape is causing the difference in angular reflectivity from the designed profile, we adapted the RCWA model cross-sectional shape (Fig. 12) to more closely representing the measured shape. The resulting simulated reflectivity profile more closely reproduces the measured reflectivity peak shift and shape.
Fig. 12. Angular reflectivity for cross-sectional shapes of a rectangule (black), a trapezoid (red), and a triangle (blue). The heights of each structure is held fixed.
Another reason for measurement-simulation differences is surface roughness. Since the aluminum thin-film is deposited by simple vacuum evaporation, surface roughness is increased compared to the other deposition techniques. This roughness causes surface scattering which does not contribute to the measured reflected light. To obtain an optically smooth surface, we can employ electron-beam evaporation or sputtering with substrate heating. Surface corrosion due to outdoor exposure can also cause affect the surface roughness. However, aluminum naturally forms a protective oxide layer when it is exposed to air, and this oxide layer will help to protect the aluminum surface from corrosion. This protection will not work when the surface is covered by the other materials. The structure will also be affected by surface contamination such as weather, human access, dust/debris accumulation, etc. Transparent medium layers can be used to protect its surface by filling the structure surface to avoid the contact of the other materials. When one uses a protective layer, each of the structure parameters has to be divided by the refractive index of the fill material, since the wavelength of light in the protective layer will be $\lambda /n$.
We have used electron-beam lithography in this study to achieve structure accuracy rather than manufacturing speed. These structures can also be fabricated by other techniques such as nano-imprint lithography and photolithography, both of which are scalable to produce large area molds [16] and which are cost-effective compared to electron-beam lithography since molds can be used repeatedly.
Although the calculation wavelength was restricted to 0.8 µm in order to simplify the analysis, we can also estimate the equivalent results for sunlight by averaging the spectral angular profile of reflectivity. We use the AM1.5G spectrum [8] and wavelength-dependent optical constants from 0.28 µm to 2.5 µm [12]. The AM1.5G spectrum is a standard spectrum, measured when the Sun is 41° above the horizon — a spectrum that is commonly used in evaluate solar cell evaluation. Incorporating the shape of the solar spectrum, and averaging the reflected and absorbed power over wavelength, while time-averaging the solar angle, we obtain $P_{\upalpha }/P = 0.08$ and 0.01 for the original design and bulk aluminum, respectively. As we noted above, the value of $P_{\upalpha }/P$ indicates the amount of seasonal tuning of sunlight interaction. Also, the values of $P_{\upalpha }/P$ for numerical results and bulk aluminum at $\lambda = 0.8$ µm were 0.23 and 0.07, respectively. Thus, the angularly selective properties of the one-dimensional periodic structure are retained, although significantly reduced, for full-spectrum solar light.
7. Conclusion
In the temperate latitudes, microstructured surfaces can be used to optimize angular selectivity to tune seasonal changes in reflectivity. Using a model of the sunlight intensity and angle of incidence over each day and over each season, we estimate that an ideal angularly-selective design can achieve a 43.2% improvement in total positive power — the average power saved in summer by increased reflection plus the average power saved in winter by increased absorption. Building a numerical simulation of a one-dimensional periodic aluminum microstructure, we estimate that it has the potential to achieve a 25.7% improvement in total positive power compared to a concrete roof. We have also demonstrated that such a surface can be fabricated through a lift-off process, and have shown the fabricated surface to have rough correspondence to the simulation. We designed the structure for $\lambda = 0.8$ µm light and measured it experimentally at that wavelength, but in future work we plan to design surfaces for a broader spectrum and evaluate them over the full solar spectral range.
Disclosures
The authors declare no conflicts of interest.
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