Ideal spectral emissivity for radiative cooling of earthbound objects

We investigate the fundamental limit of radiative cooling of objects on the Earth's surfaces under general conditions including nonradiative heat transfer. We deduce the lowest steady-state temperature attainable and highest net radiative cooling power density available as a function of temperature. We present the exact spectral emissivity that can reach such limiting values, and show that the previously used 8–13 μm atmospheric window is highly inappropriate in low-temperature cases. The critical need for materials with simultaneously optimized optical and thermal properties is also identified. These results provide a reference against which radiative coolers can be benchmarked.

Scientific RepoRtS | (2020) 10:13038 | https://doi.org/10.1038/s41598-020-70105-y www.nature.com/scientificreports/ where P cooler (T) is the radiant exitance of the cooler, P sun (Ω sun , T amb , T, α), P space (T amb , T, α), and P atm (T amb , T, α) are the absorbed irradiance on the cooler from the sun, the outer space, and the atmosphere, respectively, and P non-rad (T amb , T) is the absorbed non-radiative power density from the surroundings; these all have a unit of W/ m 2 . P space (T amb , T, α) can be usually ignored because the cosmic background is much cooler than T amb or T. The rest of the terms in Eq. (1) are expressed as where d� = 2π 0 dφ π/2 0 dθ sinθ is the hemispherical integration and ∼ I BB ( , T) = 2hc 2 5 1 exp(hc/ k B T)−1 is the spectral radiance of an ideal blackbody following Plank's law (h, c, k B , and λ are the Plank constant, the velocity of light in vacuum, the Boltzmann constant, and wavelength, respectively). ε c (λ, Ω, T) represents the spectral and directional emissivity of the cooler. I sun (λ, Ω sun , T amb , α) is the spectral solar irradiance at a mid-latitude, sea-level location in the northern hemisphere when the sunlight is incident from angle Ω sun . In Eqs. (2b and 2c), the absorptivity of the cooler is replaced by its emissivity using Kirchhoff 's law. In Eq. (2d), the non-radiative absorption is expressed by an effective non-radiative heat transfer coefficient h c , which depends on environmental conditions, such as wind speeds, as well as on how well the cooler and the object are thermally insulated from the environment 19 . ∼ I atm ( , �, T amb , α) is the angle-dependent spectral radiance from the atmosphere, obtained by multiplying the black-body radiation spectrum with the atmospheric emissivity ε atm . The latter has been approximated as ε atm (λ, Ω, T amb , α) = 1 − t(λ, T amb , α) AM(θ)20 , where t(λ, T amb , α) is the atmospheric transmittance from the sea level toward the outer space in the zenith direction (data from MODTRAN 6 21 with 2 nm spectral resolution) and AM(θ) accounts for larger attenuations at a non-zero zenith angle θ. In doing so, we applied a spherical shell model for AM(θ) (see Supplementary Methods and Fig. S1) instead of the flat-earth model of AM(θ) = 1/cos θ 20 .

Results and discussion
Now, let us derive the ideal spectral emissivity of a radiative cooler at a specific temperature for maximum net cooling power density assuming it is isotropic at all angles. We first rearrange the net cooling power density in Eq. (1) into radiative and non-radiative parts as  ( , T) and I atm ( , T amb , α) = d�cosθ ∼ I atm ( , �, T amb , α) are spectral irradiances and P rad (Ω sun , T amb , T, α) = P cooler (T) − P sun (Ω sun , T amb , T, α) -P atm (T amb , T, α) is the net radiative power density. We note that ε c (λ, T) works as a scaling factor for the net radiative heat exchange at each wavelength. For maximal P net (Ω sun , T amb , T, α) and P rad (Ω sun , T amb , T, α) at a given temperature, ε c (λ, T) can be suitably adjusted between 0 and 1 depending on the sign of the term I rad,BB = I BB (λ, T) -I sun (λ, Ω sun , T amb , α) -I atm (λ, T amb , α). For wavelengths at which the sign is positive (i.e., net radiative emission occurring), ε c (λ, T) = 1 should be chosen to maximize the radiation whereas, for other wavelengths, ε c (λ, T) = 0 should be chosen to minimize absorption. This emissivity design rule is summarized as is the sign function, whose value at x = 0 is undefined but does not affect the results in this situation. We emphasize that ε ideal is sensitively determined by the temperature of the cooler. This aspect was not fully investigated in previous studies. Two types of emissivity patterns were usually considered as optimal: a broadband emissivity pattern for above-ambient cooling and a rectangular (1 over 8-13 μm and 0 for other wavelengths) emissivity pattern for below-ambient cooling 6,7,14,[22][23][24][25][26] . However, the actual spectral regions contributing to cooling and heating via radiation change dramatically as the temperature drops below the ambient temperature. Thus, emissivity spectra designed for a temperature range around ambient temperature are no longer an optimal solution at lower temperatures.
Another important fact revealed by Eq. (4) is that the temperatures achievable through radiative cooling have a fundamental lower bound. For P rad (Ω sun , T amb , T, α) to be positive, I rad,BB must be positive at least for one wavelength; otherwise ε ideal is zero at all wavelengths by Eq. (4) and net radiation will not occur. This imposes the ultimate lower bound, T ideal,min , for the temperature of a radiatively cooled object, where T ideal,min is the temperature at which I rad,BB is zero at one (or more) wavelengths and negative at all other wavelengths.
The temperature of the cooler in a steady state can be identified by finding T that satisfies P net (Ω sun , T amb , T, α) = 0 using the ideal emissivity specified by Eq. (4). First, we investigate the most extreme case of h c = 0. Without non-radiative heat transfer, P net (Ω sun , T amb , T, α) for ε ideal is always positive if T > T ideal,min . Thus, the ideal radiative cooler cools down until the temperature approaches T ideal,min . For example, at Daejeon city (36.35 ○ N in latitude), T ideal,min is 243.6 K and 180.5 K at noon in summer and winter, respectively, as shown in Fig. 1b,c. This corresponds to 56.4 K and 92.5 K drops below ambient temperature (T amb is assumed to be 300 K and 273 K in summer and winter, respectively). Owing to seasonal variations in atmospheric emissivity, the temperature drop is much larger in dry winter than in humid summer. As expected from Eq. (4), the ideal emissivity at these temperatures is a needle-like function centered at 8.96 μm in summer and 11.61 μm in winter; each single wavelength is the only one at which I rad,BB is non-negative. Considering the lowest measured temperature in previous studies (approximately 243 K under vacuum conditions in winter) 5 , our results suggest that temperature can be further lowered by more than 50 K in principle. We note that this needle-like spectrum is a limiting case of ε ideal at T ideal,min and has zero net radiative power. It is still the ideal spectrum in terms of net radiative power since all other emitter design would have negative net radiative power at this temperature. It is also the ideal spectrum in terms of the steady-state temperature because no other design can reach this temperature at a steady state. Nonetheless, this design mainly serves as a theoretical limiting case and ε ideal for higher temperatures is not a needle, with non-zero net radiative power (see Supplementary Fig. S2). It indicates that a temperature-sensitive ε ideal , if realized, would enable fast radiative cooling, due to its initial high net radiative power, down to a very low steady-state temperature (See Supplementary Methods, Figs. S3 and S4).
In reality, the presence of conduction or convection (h c ≠ 0) means that the steady-state temperature, T ideal (h c ), of an ideal cooler designed for a given non-zero h c value, is now a function of h c and rises above T ideal,min for nonzero h c ; i.e., T ideal,min defined above as the zero net radiative power temperature is the lower bound of T ideal (h c ) and can be reached if and only if there is no non-radiative heat transfer (T ideal,min = T ideal (h c = 0)). The steadystate temperature of any radiative cooler with temperature-independent spectral emissivity can be easily found graphically because P rad and P non-rad are monotonically increasing and decreasing functions of T, respectively, and they cross each other at the steady-state temperature. For example, P rad 's for an 8-13 μm emitter (ε 8-13 , black dashed line) and a broadband emitter (ε Full , black dotted line) are plotted in Fig. 2a, under summer atmospheric conditions assuming solar irradiance corresponding to AM1.5 and an average solar zenith angle of 48.2°. The 8-13 μm emitter and the broadband emitter have a unity emissivity for wavelengths between 8 and 13 μm and for all wavelengths longer than 4 μm, respectively. Both hypothetical emitters have exactly zero emissivity for all other wavelengths. The red solid lines in Fig. 2a represent P non-rad for several h c values. As expected, the steadystate temperature for either emitter rises for higher h c values.
In general, the net radiative power density of any radiative cooler cannot exceed the values indicated by the black solid line in Fig. 2a, representing the performance of an ideal radiative cooler at each cooler temperature as designed with Eq. (4). In previous studies, the 8-13 μm emitter was considered as an almost ideal radiative cooler for below-ambient cooling cases. However, it can be seen that the net radiative power density of this cooler (black dashed line in Fig. 2a) never touches the black solid line at any temperature. In other words, there is always a better spectral design than unit emissivity from 8 to 13 μm at any target temperature. For example, at 273.15 K, P net (� sun , T amb , T, α) = P rad (� sun , T amb , T, α) − P non−rad (T amb , T)  , the lower bound of the steady-state temperature, which is 271.10 K, is below the freezing temperature of water. The ideal cooler, which can reach this bound, has a highly selective spectral emissivity with many disjointed sets of wavelengths over which the emissivity is unity, as shown in Fig. 2b. Thus, the ideal spectral emissivity appears as a collection of several non-overlapping rectangular functions of unity amplitude added together. Thermal insulation on the level of h c = 0.5 W/(m 2 K) can be achieved with a 7 cm-thick polystyrene foam at the back and an infrared-transmitting composite window at the front. Inside the previously accepted 8-13 μm transparency window, it is possible to identify several important wavelength ranges for which the emissivity must be minimized. One such wavelength range is around 9.5 μm, where there are multiple atmospheric absorption At lower h c values, the difference becomes more dramatic. T ideal and ε ideal are plotted for a range of h c values in Fig. 2c,d. As h c decreases, the difference between T ideal and the steady-state temperature of the 8-13 μm emitter (T 8-13 ) becomes larger, reaching 24.67 K at h c = 0, as shown in Fig. 2c. The width of the wavelength ranges for which the emissivity should be unity diminishes as h c is reduced and become highly selective, as illustrated in Fig. 2d. These results imply that, in a properly insulated system, it is possible to achieve a much lower steady-state temperature than T 8-13 if the spectral emissivity is optimally designed to benefit from the lower h c .
In practice, however, it might be challenging to realize such a highly selective spectrum. Thus, we also consider a simper, single-band emitter with unit emissivity over a single wavelength range from λ short to λ long and optimize λ short and λ long for best performance. For h c = 0.5 W/(m 2 K), the steady-state temperature is shown in Fig. 3a for different combinations of λ short and λ long . Among the various potential designs, the optimal design is λ short = 8.30 μm and λ long = 12.38 μm, with a corresponding steady-state temperature of 274.40 K. Whereas the ideal, multi-band emitters depicted in Fig. 2 have non-negative net radiative emission at all wavelengths, singleband emitters have net radiative absorption at some wavelength regions, as shown in Fig. 3b with a red color. Nonetheless, an optimally designed single-band emitter can exhibit a considerably lower steady-state temperature www.nature.com/scientificreports/ (T ideal,SB ) than T 8-13 for highly insulated systems (Fig. 3c). In particular, at the perfect insulation limit, T ideal,SB and T 8-13 converge to 243.64 K and 268.31 K, respectively, exhibiting a difference of 24.67 K. Even in a more realistic case of h c = 0.13 W/(m 2 K), T ideal,SB (265.60 K) is 5 K lower than T [8][9][10][11][12][13] and the corresponding emission band is from 9.98 to 12.26 μm. Figure 3d illustrates the optimal emission band for a wide range of h c values. At high h c values, the ideal emission band for a single-band radiative cooler is similar to previous designs of 8-13 μm. However, at low h c values, the optimal emission band narrows down considerably. In particular, it can be seen from Fig. 3c,d that it is better to abandon the wavelength range from 8 to 10 μm if the target steady-state temperature is lower than the freezing point of water. Of course, if dual or multi-band designs are permissible, a part of this wavelength range can be used for radiation to decrease the steady-state temperature further or to increase the net radiative power density.
For direct comparison, we present Table 1 that shows the steady-state temperatures of ideal and non-ideal coolers for various h c conditions. The spectrally-selective and single-band based ideal coolers outperform other non-ideal coolers by many degrees for small h c . Even for h c = 2 W/(m 2 K) that is practically realizable even without vacuum sealing 26 , the ideal cooler shows noticeable advantage.
In conclusion, we presented a systematic method to calculate the ultimate lower bound of a radiatively cooled object's steady-state temperature as well as the ultimate upper bound of net radiative power density at a given cooler temperature under general environmental conditions with an arbitrary effective non-radiative heat transfer coefficient. We also derived the ideal spectral emissivities that can reach such bounds. Unlike the often-adopted contiguous emission window of 8-13 μm used in previous radiative coolers, the ideal radiative cooler exhibits unity emissivity over disjointed sets of wavelengths. We also investigated the ideal emission band for a singleband emitter and found that the optimal band narrows down considerably at lower temperatures. The proposed scheme may serve as a basic guideline for designing the emissive properties of extreme radiative coolers as well as for estimating the amount of thermal insulation required for them.