1. Introduction

The atmosphere of the Earth is relatively transparent to light in particular infrared wavelength ranges [Fig. 1(a)]. Radiative coolers utilize this transparency window for uninhibited outward radiation toward the cold cosmic background, thereby achieving below-ambient cooling at night [1–4] and even at daytime with strong solar reflection [5–16]; this is promising for designing zero-energy houses [17–21] and novel energy harvesting systems [22–24]. To achieve a high cooling power, the emissivity of the cooler has been designed with isotropic but spectrally selective in the 8–13 µm range for below-ambient cooling [5,16,25–28] or spectrally broad at wavelengths from 4 µm to tens of microns for above-ambient cooling [6,29–32]. Recently, in contrast with these approaches, a few studies have shown that the widely considered 8–13 µm wavelength range is not the optimal spectral range of emissivity at below-ambient temperatures [33]; instead, wavelength- and angle-selective emissivity can yield optimal emitters [34] for energy harvesting [24]. These studies have revealed that designing the emissivity at the expected operation temperature, rather than at the ambient temperature, is crucial for optimal radiative cooling. However, the angular aspect of emissivity for optimal cooling, which is deeply relevant to temperature-based spectral selection, has not been thoroughly examined. Most studies on radiative cooling have assumed, in their design phase, isotropic emissivity without explicit consideration of the actual operating temperature of the cooler, which can be substantially different from the ambient temperature [5,6,8,10–12,14,16,25–28]. Thus, the current state of research naturally prompts important questions about the impact of directionality on cooling and about the ideal wavelength and angular dependence of emissivity of radiative coolers as a function of their actual temperature as well as the ambient temperature.

 

Fig. 1. Spectral angular properties of the atmosphere. (a) Atmospheric transmittance. (b) Absorbed spectral radiance from the atmosphere (colored border line) and emitting spectral radiance for a perfect emitter (black solid line). For easy visualization, we convoluted the fine spectral features with a Gaussian smoothing function, g = exp(-λ²/(2σ²)) where σ = 5 nm in (b).

Download Full Size | PDF

In this letter, we explicitly and quantitatively demonstrate that the directionality of emission, in addition to spectral selectivity, is critically important for high-power, below-ambient cooling. Directional radiation by thermal emitters can enhance the net radiative power density, and their temperature can drop further below that of the best omni-directional emitters. Our angle-resolved investigation reveals that radiative heat exchange at high zenith angles causes heating rather than cooling at below-ambient temperatures, even within a commonly considered “transparency” window of 8–13 µm, owing to the dominant atmospheric radiation over the outward radiation from the cooler. The heating power and the angular range over which heating occurs increase as the temperature drops further below the ambient temperature. We show that popular emitters for radiative cooling, with emission bands in the 8–13 µm range or in a broad mid infrared (mid-IR) region, can exhibit significantly better cooling performance if their emission and absorption at high angles are simply blocked. Based on the selective emissivity design rule [24,33,34], we present the ideal spectral directional emissivity that yields the maximal allowed net radiative power at a given temperature and achieves the theoretical lower bound on the steady-state temperature, outperforming all other possible radiative cooling surface designs. We found that the steady-state temperature under a negligible non-radiative heat transfer condition can be much lower than the freezing point of mercury (−38.8 °C) and more than 17 K lower than that of the best isotropic emitter under the same atmospheric conditions. This explains why better cooling performance was observed in some previous experiments with implicitly imposed directionality [9,35–37] than that observed for other designs. In addition, our results indicate that directional radiation is practically essential when the surface of the radiative cooler does not point to the zenith direction, such as in complicated rooftops and façades of modern buildings.

2. Theoretical model

In our model, a radiative cooler covers the top surface of the object to be cooled, and the cooler and the object are in the thermal equilibrium at temperature T. The cooler is at the sea level and radiates toward the sky. We first consider horizontally placed coolers with a sky-view angle of 2π steradians, but later generalize to an arbitrary angular orientation. The cooler exchanges radiation with the sun, the outer space, and the atmosphere, and exchanges heat non-radiatively (i.e., via conduction and convection) with the ambient air and other abutting objects. The radiation through the atmosphere is subject to various conditions, such as the season [38], humidity [39], and temperature variation along the altitude [40]. We capture these environmental effects by a collective variable α; the zenith angle of incident sunlight is θ_sun. We also assume that the temperatures of non-radiatively interacting objects are the same as the ambient air temperature T_amb. Then, the net cooling power density, which can be divided into radiative and non-radiative components, is

To solve Eqs. (2b)–(2c), we approximate ${\tilde{I}_{\textrm{sun}}}$(${\tilde{I}_{\textrm{atm}}}$) as a multiplication of the atmospheric transmittance (atmospheric emissivity) and extraterrestrial solar radiance (radiative spectrum of an ideal blackbody at the ambient temperature). The atmospheric transmittance and its emissivity are related via ɛ_atm(λ, Ω, T_amb, α) = 1 – t_atm(λ, Ω, T_amb, α) [5,42]. The attenuation in t_atm can be obtained from the effective path length through atmosphere, such relation can be arranged as t_atm(λ, Ω, T_amb, α) = t₀^AM(θ), where t₀(λ, T_amb, α) is the transmittance in the zenith direction (data from MODTRAN 6 [43]) and AM(θ) is angular attenuation coefficient [33]. In this work, we use AM(θ) of the spherical shell model for the atmosphere [33,44], which is more accurate than AM(θ) = 1/cosθ of the flat earth model especially at high zenith angles [19]. The other parameters and their values in our calculations were as follows: the season, latitude, Ω_sun, and T_amb are summer, 36.35°N, 12.9° (zenith angle at noon), and 300 K, respectively.

3. Results and discussion

As a motivation for a more detailed analysis, we begin by comparing the spectral radiance of an isotropic blackbody and that of the atmosphere, which correspond to ${\tilde{I}_{\textrm{BB}}}(\lambda ,T)$ and ${\tilde{I}_{\textrm{atm}}}(\lambda ,\Omega ,{T_{\textrm{amb}}},\alpha )$ in Eqs. (2a) and (2c), respectively, at several emitter temperatures and zenith angles. The spectral radiance of a blackbody depends on its temperature, but is angle-independent owing to the isotropy assumption. On the other hand, the spectral radiance of the atmosphere is independent of the emitter temperature, but strongly depends on the zenith angle [24,34] because of the effective thickness of the atmosphere and, hence, its transparency are determined by the zenith angle [Fig. 1(a)]. For large zenith angles approaching 90°, it is apparent that the atmospheric transmittance diminishes at all wavelengths, indicating increased absorption. According to reciprocity, this means that the atmospheric emission is intense at these large zenith angles, and it can surpass the emission from a perfect thermal emitter at below-ambient temperatures [Fig. 1(b)]. Shown in Fig. 1(b) are ${\tilde{I}_{\textrm{BB}}}(\lambda ,T)$ at three emitter temperatures (black lines) and ${\tilde{I}_{\textrm{atm}}}(\lambda ,\Omega ,{T_{\textrm{amb}}},\alpha )$ at three zenith angles (colored graphs). When the black line is above the colored graph, the net spectral radiance (${\tilde{I}_{\textrm{BB}}} - {\tilde{I}_{\textrm{atm}}}$) is positive at that specific wavelength, emitter temperature, and zenith angle. If the emitter is at the same temperature (300 K) as T_atm, the net radiance is positive at all wavelengths at all zenith angles, meaning that radiative channels at all wavelengths and angles can contribute to cooling. However, if the emitter temperature drops to 280 K, the net radiation is negative for most wavelengths even within the 8–13 µm transparency window at the zenith angle of 80°, indicating that radiative channels near the zenith angle of 80° or larger actually contribute to heating rather than cooling. If the temperature decreases further to 265 K, a significant part of the transparency window (∼ 60%) contributes to heating even at 60°. These examples show that as the temperature of the emitter decreases, the atmospheric radiance begins to dominate over the radiance of the emitter at progressively smaller zenith angles. The deficiency of isotropic emitters as sub-ambient radiative coolers is evident, especially when the target temperature is well below the ambient temperature.

Building on these initial findings, we quantitatively analyzed the performance of two popular isotropic emitter designs at all zenith angles, over a wide range of cooler temperatures. The two most widely used isotropic emitter designs have unit emissivity values either in the 8–13 µm range (their spectral emissivity is denoted hereafter $\varepsilon _{8 - 13}^{(\textrm{iso})}$) or in the 4–20 µm range or longer wavelengths (denoted $\varepsilon _{\textrm{broad}}^{(\textrm{iso})}$), while the emissivity in the other spectral regions is zero. In this case, we assume that the emitters are positioned flat on the ground, facing directly the zenith direction. Their net radiance in reference to the emitter surface area, instead of the transverse area normal to the direction of radiation, is $\cos (\theta )\,{\tilde{P}_{\textrm{rad}}} = \cos \theta \int {\textrm{d}\lambda \,{{\tilde{I}}_{\textrm{rad}}}{\varepsilon _\textrm{c}}}$ and has the unit W/(m²·sr). At the ambient temperature, the emitters exhibit non-negative net radiance in all upper hemispherical directions; thus, radiative channels at all zenith angles contribute to cooling [Figs. 2(a)–2(b)]. However, at below-ambient temperatures, net radiances become negative at large zenith angles [Figs. 2(c)–2(f)], which originates from two facts: the gradual increase of the atmospheric radiance at higher zenith angles and the gradual decrease of the blackbody radiance at lower temperatures, as illustrated earlier in Fig. 1(b). These are also related to the reasons why radiative cooling is not highly effective in reducing the surface temperature to a habitable level on Venus with its hot and dense atmosphere. Note that the emissivity of the cooler is kept constant regardless of the zenith angle or the temperature. Among the two emitter designs, the broadband emitter suffers more severe heating, and at 280 K, net radiative heating occurs at all angles, as shown in Fig. 2(f). More quantitatively, the azimuthally integrated power densities per unit zenith angle in Figs. 2(g)–2(h) reveal that the emitters encounter stronger radiative heating at high zenith angles when they cool further below the ambient temperature. Owing to this effect, isotropic emitters cannot passively cool down past the dashed lines in Figs. 2(g)–2(h), which indicate the temperatures for the zero net radiative power density (268.5 K and 286.9 K for $\varepsilon _{8 - 13}^{(\textrm{iso})}$ and $\varepsilon _{\textrm{broad}}^{(\textrm{iso})}$, respectively).

 

Fig. 2. Net radiance dependence on temperature as (a, b) 300 K, (c, d) 290 K, and (e, f) 280 K, for horizontally placed (a, c, e) isotropic 8–13 µm emitter and (b, d, f) isotropic broadband emitter. Net radiance per unit zenith angle for (g) isotropic 8–13 µm emitter and (h) isotropic broadband emitter. Dashed lines in (g, h) indicate steady-state temperature with zero radiative power density.

Download Full Size | PDF

Now, we demonstrate that by suppressing radiative channels at high zenith angles, emitters can be designed for reaching temperatures below the temperature limit of isotropic emitters. We focus on directional emitters that possess the same spectral emissivities as isotropic emitters within a cut-off angle θ_c and zero emissivity above θ_c [Figs. 3(a)–3(b)]. In other words, they are angle-limited versions of isotropic emitters. Then, the steady-state temperature can be obtained by solving Eq. (1) for P_net = 0 depending on θ_c and h_c [Figs. 3(c)–3(d)]. The optimal θ_c for the lowest possible steady-state temperature can be found numerically and is shown as red triangular points in Figs. 3(c)–3(d), for different hc values. The optimal θ_c is smaller for broadband emitters than for selective emitters. For example, at h_c = 3 W/(m²K), θ_c is 72.3° and 61.4° for the 8–13 µm emitters ($\varepsilon _{\textrm{8 - 13}}^{(\textrm{dir})}$) and broadband emitters ($\varepsilon _{\textrm{broad}}^{(\textrm{dir})}$), respectively. As h_c decreases, the optimal θ_c becomes even smaller, and the difference between the steady-state temperature of optimal directional emitters and isotropic emitters increases. For instance, at h_c = 0.4 W/(m²K), optimal directional emitters $\varepsilon _{\textrm{8 - 13}}^{(\textrm{dir})}$ can freeze water at the steady-state temperature of 271.8 K, whereas isotropic emitters with $\varepsilon _{\textrm{8 - 13}}^{(\textrm{iso})}$ cannot.

 

Fig. 3. Directional radiation for limited high angular emissivity. (a) Schematic of directional radiation. (b) Angularly limited emissivities of 8–13 µm emitter and broadband mid-IR emitter. (c, d) Steady-state temperature for various cutoff angles and non-radiative heat transfer coefficients.

Download Full Size | PDF

Now, we derive the ideal spectral directional emissivity that maximizes P_rad at a given emitter temperature by implementing a selective emissivity design rule that we developed in Ref. [33] and generalized here to include the angle dependence. In Eq. (3), the emissivity (ε_c) affects P_rad as a non-negative multiplicative factor to the net spectral radiance (${\tilde{I}_{\textrm{rad}}}$) at each (λ,ω) combination. Thus, P_rad can be maximized by assigning the largest allowed value of emissivity (i.e., unity) to (λ,ω) combinations with ${\tilde{I}_{\textrm{rad}}} > 0$ and the smallest allowed emissivity (i.e., zero) to (λ,ω) combinations with ${\tilde{I}_{\textrm{rad}}} < 0$. Such a design scheme can be expressed simply as

 

Fig. 4. Radiative cooling for horizontally placed emitters. (a) Ideal spectral directional emissivity depending on the cooler’s temperature, where the colored region presents unit emissivity. (b) Radiative (black, blue, and red) and non-radiative (green) power densities as a function of the cooler’s temperature. (c) The steady-state temperature and temperature difference between the isotropic emitter and the optimal directional emitter, as a function of the non-radiative heat transfer coefficient.

Download Full Size | PDF

The ideal spectral directional emissivity derived above imposes an upper bound on P_rad at a given temperature, which can never be exceeded by any other emitting surface in reciprocal systems [Fig. 4(b)]. Even when some additional constraints are imposed on the spectral directional emissivity, Eq. (3) can still be used for deriving an optimal solution under given constraints. In those cases, however, the resulting P_rad will be lower than what can be achieved by $\varepsilon _{\textrm{ideal}}^{(\textrm{dir})}$ without such constraints. For example, one can impose the isotropy constraint (i.e., the emissivity is not a function of direction), and the optimal spectral emissivity (hereinafter referred to as $\varepsilon _{\textrm{ideal}}^{(\textrm{iso})}$) has an expression similar to Eq. (5), but with ${\tilde{I}_{\textrm{rad}}}$ replaced by its directionally integrated value. This spectral emissivity places an upper bound on P_rad for all isotropic designs, but this bound is lower than that imposed by $\varepsilon _{\textrm{ideal}}^{(\textrm{dir})}$. Instead of constraints on directional properties, one can also consider constraints on spectral features and a “single-band” constraint is such an example, in which non-zero emissivity only occurs over a single, continuous band of wavelengths. Two well-known cases are the 8–13 µm selective emitter and the broadband emitter that we have already reviewed above, and optimal directional emitters with such constraints ($\varepsilon _{\textrm{8 - 13}}^{(\textrm{dir})}$ and $\varepsilon _{\textrm{broad}}^{(\textrm{dir})}$) are in fact angle-limited emitters that we derived in Fig. 3; these outperform the corresponding isotropic single-band emitters ($\varepsilon _{\textrm{8 - 13}}^{(\textrm{iso})}$ and $\varepsilon _{\textrm{broad}}^{(\textrm{iso})}$, respectively) in terms of P_rad. These examples reveal that the directional selectivity and spectral selectivity of emissivity are crucial for obtaining the highest possible P_rad.

In addition to P_rad at a given temperature, another important indicator of the cooler performance is the steady-state temperature of the cooler under given atmospheric conditions and for a given solar zenith angle. In contrast to P_rad, the steady-state temperature strongly depends on how well the cooled object is insulated to reduce conductive and convective heat exchange with surroundings (as phenomenologically captured by h_c in our model). With perfect insulation (i.e., h_c = 0), the steady-state temperature of an object radiatively cooled with $\varepsilon _{\textrm{ideal}}^{(\textrm{dir})}$ can be as low as 226.5 K at mid-day in summer. If it is placed horizontally on the ground at latitude 36.35°N, this temperature serves as the lower bound on the steady-state temperature for any spectral directional emissivity design. Below this temperature, $\varepsilon _{\textrm{ideal}}^{(\textrm{dir})}$ in Eq. (5) is zero because ${\tilde{I}_{\textrm{rad}}}$ is negative for any (λ,Ω) combination, and any thermal emitter with non-zero emissivity is radiatively heated owing to a negative P_rad, as illustrated in Fig. 4(b). Similarly, the steady-state temperature of $\varepsilon _{\textrm{ideal}}^{(\textrm{iso})}$ is also bounded from below, but the isotropy constraint places the bound at a higher level of 243.6 K, as reported in Ref. [33].

In general, the steady-state condition, P_net = 0, can be solved graphically by finding a crossing point between P_rad and P_non-rad (green solid lines) in Fig. 4(b). Under negligible non-radiative heat transfer conditions (h_c = 0), the directional emitters with $\varepsilon _{\textrm{ideal}}^{(\textrm{dir})}$, $\varepsilon _{\textrm{8 - 13}}^{(\textrm{dir})}$, and $\varepsilon _{\textrm{broad}}^{(\textrm{dir})}$ exhibit steady-state temperatures lower than those of their isotropic counterparts, by 17.1 K, 18.5 K, and 36.9 K, respectively. In the presence of a non-radiative effect (h_c ≠ 0), the emitters are stabilized at higher temperatures, but the directional selectivity helps to attain lower temperatures. This also means that, when cooling an object down to a specific target temperature, a directional emitter requires less heat insulation than the corresponding isotropic emitter. For example, coolers with $\varepsilon _{\textrm{8 - 13}}^{(\textrm{dir})}$ can reach the freezing point of water when h_c ≤ 0.49 W/(m²K), in contrast to isotropic coolers $\varepsilon _{\textrm{8 - 13}}^{(\textrm{iso})}$ that require h_c ≤ 0.29 W/(m²K), which means that the insulation materials for isotropic coolers should be at least ∼70% thicker than those for directional coolers.

In actual applications of radiative cooling, the surfaces of target objects (such as buildings and vehicles) typically are not fully parallel to the ground: building façades and slanted rooftops of houses are such examples. In these cases, in which the surface normal direction of the emitter attached to the surface is not aligned with the zenith, directional selectivity becomes even more important and the effect can be analyzed by modifying Eq. (3) as

 

Fig. 5. Cooling performance for slanted emitters. (a) Radiative (black, blue, and red) and non-radiative (green) power densities as a function of the cooler’s temperature. (b) The steady-state temperature and temperature difference between the isotropic emitter and the optimal directional emitter as a function of the non-radiative heat transfer coefficient. (c) Net radiance dependence on temperature.

Download Full Size | PDF

4. Conclusion

In conclusion, we investigated the directional aspect of emissivity for improving the net radiative power density as well as the steady-state temperatures of radiative coolers. We revealed that previous isotropic emitters can be cooled further by simply restricting the maximal angle of radiation while maintaining their spectral properties. We also presented the ideal spectral directional emissivity for general environmental conditions and derived the ultimate upper bound on the net radiative power density and the lower bound on the steady-state temperature. In particular, we showed that the steady-state temperature can be in principle extended down to 226.5 K, which is 17.1 K lower than that of the ideal isotropic emitter. The results were generalized to slanted emitters, making them useful for the optimal design of directional emissivity in practical situations with complex surface morphologies.