# Colored Radiative Cooling Coatings with Nanoparticles

Refet Ali Yalçın, Etienne Blandre, Karl Joulain, Jérémie Drevillon

► To cite this version:

Refet Ali Yalçın, Etienne Blandre, Karl Joulain, Jérémie Drevillon. Colored Radiative Cooling Coatings with Nanoparticles. ACS photonics, 2020, 7 (5), pp.1312-1322. 10.1021/acsphotonics.0c00513. hal-03783896

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Submitted on 24 Mar 2025

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Distributed under a Creative Commons Attribution 4.0 International License
COLORED RADIATIVE COOLING COATINGS WITH NANOPARTICLES

Refet Ali Yalçın<fnref n="1" />, Etienne Blandre, Karl Joulain and Jérémie Drévillon

Institut Pprime, CNRS, Université de Poitiers, ISAE-ENSMA, F-86962 Futuroscope Chasseneuil,
France

Abstract

Daytime radiative cooling under sunlight can be achieved by reflecting solar irradiation and emitting infrared thermal radiation in the transparency regions of earth atmosphere. For the sake of aestheticism, control over coloration would be a clear added-value, for automobile body or walls of a building. Since coloration decreases inevitably cooling performances because of solar absorption in the visible range, mitigating the loss in radiative cooling power requires selective narrowband absorption in the visible range. In order to make a compromise between aestheticism and energetic efficiency, we aim in this paper to obtain colored surfaces with mitigated sunlight absorption and high emission in the infrared. By this numerical study, we demonstrate that appropriate optical properties can be achieved with plasmonic core-shell nanoparticles (silica core, silver shell) and plain silver nanoparticles, embedded in a silica matrix. Spectral properties of the composites are obtained by Monte Carlo method. Varying the radii of the core and shell allows tuning the spectral position of the surface plasmon resonance, leading to different colors. It is possible to obtain multiple absorption peaks and therefore additional colors by mixing particles with different geometries. Eventually, increasing the volume fraction of nanoparticles leads to more vivid colors but also to a decrease of radiative cooling power. In conclusion, the structure that is proposed can be used for daytime radiative cooling with large control over the coloration.

TOC Graphic

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1. Introduction

Thermal load of structures whose exterior parts are exposed to the sun, such as cars and buildings, have a high impact in terms of energy efficiency. Indeed, thermal load inevitably increase under direct solar irradiance and as a consequence, 48% of the household energy consumption in USA

<footnote n="1">¹ Corresponding author email: refet.ali.yalcin@univ-poitiers.fr</footnote>
occurs as a result of heating ventilating and air-conditioning¹. Similarly, air conditioning increases fuel consumption up to 25% in cars². Solutions to reduce these thermal loads exist. Among them, radiative cooling³ is a very interesting one. It can be achieved by combining high reflection in the solar spectrum with large infrared emission. Under these conditions, objects can even be cooled below ambient temperature by exchanging radiation with the cold outer space through the transparency windows of earth's atmosphere in the infrared.

Efficient designs with high radiative cooling power have already been achieved⁴,⁵ in the past. However, they appear whitish since most of the incident solar intensity is reflected in order to optimize the performance. For the sake of aestheticism, colors other than white **are needed** including but not limited to car paints, building design, urbanism and signaling.

Color can be obtained by absorption in the visible spectrum which results of an increase of the radiative heat load. Although absorption of light is unavoidable, its amount can be mitigated by absorbing selectively in narrow spectral bands in the visible range and keeping high reflection in the infrared and UV parts of the solar spectrum. As a result, colored surface with a low heat load compared to common colored coatings or even with positive net radiative cooling can be obtained⁶.

Different studies about the relation between color and temperature can be found in the literature. Li *et al.*⁷ showed that two different layered photonic structures, that have the same color appearance, can have 47.6 °C difference under sunlight due to metamerism, infrared solar absorption and radiative cooling. As an alternative to layered structures, nanoparticles can also be used to absorb light and color a coating. Smith *et al.*⁸ used flake particles in binder to make colorful energy efficient coatings that reduce solar absorptance with respect to other commercial paints that gives same color output. Use of plain and core-shell nanoparticles to produce color is studied by Yong *et al.*⁹. Tunability of plasmon resonances in the visible wavelength for core-shell particle has been further inspected by Kado *et al.*¹⁰. In this study, we demonstrate that colored surfaces with positive net radiative cooling power can be achieved by embedding plain and/or core-shell nanoparticles in a silica matrix.

To obtain radiative cooling power and estimate the color, reflectance of the coating has to be estimated first. There are various methods to numerically estimate the reflectance of a composite material, namely; finite element method, discrete dipole approximation, finite difference time domain and Monte Carlo method. Considering the fact that modelling a thin shell requires a fine meshing and the fact that using metallic particle and thick coating results in a long computation time, Monte Carlo method is the most robust and feasible method among the mentioned ones¹¹. In Monte Carlo method, parameters to define scattering and absorption properties of the medium are calculated using the Mie theory which determines analytically scattering and absorption in the case of an incident wave on a sphere. These parameters are used for ray tracing inside the medium and are valid as long as the nanoparticles are far from each other such that scattering and absorbing fields of particles are not overlapped. The metallic nanoparticle volume fraction to obtain color at a 500 µm thick silica coating with reflector is of the order of 10⁻⁶, which is in a confident region¹² to use Monte Carlo method as described in detail by Lee¹³. After obtaining reflectance of the
coating, radiative cooling power at chosen environment conditions can be easily calculated as well as visual outcome. The color of the coating can also be estimated using the international standards International Commission on Illumination (CIE) model¹⁴.

In the following, we numerically investigate in the next sections the radiative cooling properties and appearance of a coating composed of nanoparticle embedded silica medium applied on a silver reflective layer. The spectral reflection of the coating is calculated with a Monte-Carlo method which is modified to take into account a second type of particle and refraction in a lossy medium. The optical properties of the coating are then used to estimate its radiative cooling power and color. A parametric study on size and concentration is performed to find the appropriate color and radiative cooling power.

## 2. Problem Statement

To obtain a good reflectivity of solar radiation, we chose a silver substrate because it exhibits a high reflectivity in the solar spectrum range. To obtain large infrared thermal emission in the atmospheric transparency window, a 500 µm thick silica layer is considered over the reflecting silver substrate as it is represented in Fig. 1. Such a silica layer has a high transmissivity in the solar spectrum range and high emissivity at atmospheric transparency window.

In order to obtain color, absorption should occur within the visible spectrum. Nanoparticles are embedded inside the silica matrix to obtain narrow absorption peaks provided by localized surface plasmon resonances. Silver nanoparticles are also considered, since absorption by free electrons for this material is dominant in the visible range, allowing localized surface plasmon resonance to occur in the desired spectral region. Two types of nanoparticles are used; the first type is the plain spherical nano-particles, the second type is the silica core - silver shell nanoparticles which are represented in Fig. 1.

<figure><img src="image_3.png" /><figcaption>Figure 1 – The coating composed of silver reflector and nanoparticle embedded silica.</figcaption></figure>
Finally, note that convection effects are not taken into account for simplicity by taking the temperatures of the surroundings and the coating as equal to 300 K. The column water vapor and air mass are chosen to be 1 mm and 1.5 AM respectively. The resulting atmospheric transmissivity and solar intensity is represented in Supplementary Fig. 1 along with blackbody intensity at T=300 K.

### 3. Optical properties of silver nanoparticles embedded silica coatings

The bulk optical properties (n, k) of silica and silver are taken from Kitamura et al. ¹⁵ and Yang et al. ¹⁶ respectively and represented in Supplementary Fig. 2. Because the characteristic dimensions of the silver nanoparticles are smaller than the electron mean free path, the optical properties of silver for nanoparticles have to be corrected to account increase electron scattering due to collisions at the boundaries. The corrective terms are calculated using Eq. (1) as suggested by Mendoza Herrera et. al. ¹⁷. Parameters in the case of silver required in Eq. (1) are given in the Supplementary Table 1.

$$ \Delta\varepsilon_{free}(\omega, r_{eff}) = -\omega_p^2 \sum_{n=1}^{\infty} (-1)^n \frac{\left(\frac{i\omega A v_F}{r_{eff}}\right)^n}{(\omega^2 + i\omega\gamma_{free})^{n+1}} \quad (1-a) $$

$$ \varepsilon(\omega, r_{eff}) = \varepsilon_{bulk}(\omega) + \Delta\varepsilon_{free}(\omega, r_{eff}) \quad (1-b) $$

where

$$ r_{eff} = \frac{2(D_o^3 - d_i^3)}{3(D_o^2 + d_i^2)} \quad \text{for core-shell structures} $$

$$ r_{eff} = D/2 \quad \text{for simple spherical nanoparticles} $$

The modelling of the dielectric function correction is firstly validated with data from the literature. The results are presented in Supplementary Fig. 3 and then used in Mie theory to calculate optical properties of nanoparticles in Supplementary Fig. 4. The result shows that using bulk permittivity values overestimates absorptance and that it is necessary to modify the refractive indices, especially for thin shell core-shell nanoparticles.

Scattering efficiency, absorbing efficiency and asymmetry parameter of a single nanoparticle are required to estimate the reflectance of the coating when using Monte Carlo method. Analytical solutions of scattering by plain spheres (Mie theory) and core-shell structures are available in the literature ¹⁸ and provided in the Supplementary Note 4. Mie parameters are calculated using the size dependent modified refractive indices as described before.

Absorption efficiencies of plain and core-shell particles are calculated and represented in Fig. 2(a) and Fig. 2(b) respectively. Corresponding results shows that plain silver and silica core - silver
shell nanoparticles are able to provide required absorption peaks in the visible spectrum which is expected to produce color when embedded inside a silica matrix deposited over a silver reflector. Plain silver nanoparticles have absorption peaks around 0.4 µm. Parametric study by altering the plain particle size in Fig 2 (a) shows that; although the magnitude of absorption strongly varies with the particle size, the absorption peak shift is limited. Besides, the peak location can be altered within a broader spectrum for core-shell particles as it could be seen in Fig. 3 (b) where the absorption peak is shifted from infrared to 0.55µm by changing the shell thickness with a constant core size, dᵢ=40nm.

Similar absorption peak locations could be observed by using different core-shell size pairs as it is represented in Fig. 3. All core-shell particles in Fig. 3 have a dipole Mie mode absorptance peak around 0.6 µm. However, aside from that peak, quadrupole Mie mode around 0.47µm is also observable for larger pigments as it could also be seen in Figs. 2(b) beside Fig. 3. The second narrower absorption peak at shorter wavelengths that arises from quadrupole mode is unwanted since it is less controllable compare to dipole peak and produce extra absorption that reduce net cooling. Therefore, smaller core-shell particles are chosen (i.e. dᵢ=24 nm, D₀=30 nm from Fig 3) if possible, to avoid the second peak from quadrupole Mie mode.

<figure><img src="image_4.png" /><figcaption>Figure 2 - Absorption efficiency of (a) plain silver at various diameters and (b) core-shell particle with changing shell thickness at constant inner diameter, dᵢ=40 nm.</figcaption></figure>

Monte Carlo method uses medium properties such as absorption and scattering coefficients. Therefore, the optical properties of the medium from single particle properties are calculated by using the volume fraction and the size of the particle as;

$$ \sigma_{s,\lambda} = \frac{3f_v Q_{sca}}{4r} \qquad (2) $$

$$ \kappa_{\lambda} = \frac{3f_v Q_{abs}}{4r} + \frac{4\pi k_{\lambda,si02}}{\lambda} \qquad (3) $$
$$ \omega_{0,\lambda} = \frac{\sigma_{s,\lambda}}{\sigma_{s,\lambda} + \kappa_{\lambda}} \qquad (4) $$

where outer radius, $D_o/2$, is taken as $r$ for the core-shell structures and second term of the absorption coefficient, $\kappa_\lambda$, is the absorption contribution from matrix.

<figure><img src="image_5.png" /><figcaption>Figure 3 - Absorption efficiency for the similar color outcomes at different particle size.</figcaption></figure>

As it is discussed in the next sections, two types of particles can be mixed to obtain wider color range. In the case of a second type of particle in the medium, scattering efficiency, absorbing efficiency and asymmetry parameter are calculated separately for both types and corresponding parameters are defined with subscripts 1 and 2 to refer their types such as $\kappa_{\lambda,1}$ and $\kappa_{\lambda,2}$.

To estimate the reflectance of the coating, RTE is solved by Monte Carlo method.

$$ \frac{dI_{\lambda}}{ds} = \kappa_{\lambda}I_{b,\lambda} - \kappa_{\lambda}I_{\lambda} - \sigma_{s,\lambda}I_{\lambda} + \frac{\sigma_{s,\lambda}}{4\pi} \int I_{\lambda} \Phi_{\lambda}(\Omega', \Omega) d\Omega' \qquad (5) $$

Implementation of Monte Carlo method to solve RTE in isotropic medium has been described in the literature¹¹ in details.

Validation of the Monte Carlo method is performed with another radiative cooling study¹⁹ where the method was used. Comparison is represented in Supplementary Fig. 6 where one can note a very good agreement between the two calculations.

Calculation of refraction angle at air-medium interface is important for ray tracing since the pathlength of the ray within the medium changes with the entrance angle $\theta'$. Imaginary refractive
index of silica, $k_2$, is negligible at visible wavelengths, thus refraction of ray, $\theta \to \theta'$, at the air - silica interface can be handled by simple Snell-Descartes's law, $\sin(\theta)=n_2 \sin(\theta')$, where $n_2$ is real refractive index of silica medium, $\theta$ is incidence angle and $\theta'$ is refraction angle. However, $k_2$ is significant especially in atmospheric sky window spectrum and therefore refracting angle $\theta'$ is calculated from $^{20}$;

$$ \theta' = \tan^{-1} \left( \frac{\sin \theta}{\sqrt{\frac{1}{2} \left[ \sqrt{n_2^2 - k_2^2 - (\sin \theta)^2} + 4n_2^2 k_2^2 - (n_2^2 - k_2^2 - (\sin \theta)^2) \right]}} \right) \quad (6) $$

Normally, an effective refractive index for the medium is calculated and used when finding the specular reflectance at the boundaries. However, since the volume fraction of nanoparticles are in the order of $10^{-6}$, effective medium theory is not implemented and refractive indices of medium are simply taken as silica's refractive indices.

<figure><img src="image_6.png" /><figcaption>Figure 4 - Spectral reflectances of the coating with core-shell pigments ($D_o=30$ nm – $d_i=24$ nm) resulting cyan color in 500 µm silica with varying $f_v$.</figcaption></figure>

An example of spectral hemispherical reflectance of coating with core-shell nanoparticles (dimensions) is presented in Fig. 4, for different volume fractions of particles. It is observed that the coating display large emissivity in the infrared, except for two sharp reflection peaks corresponding to the phonon-polaritons of silica. Overall, the emissivity in the transparency regions of earth atmosphere is still large. Furthermore, it is worth mentioning that the reflectivity in the Reststrahlen bands can be significantly decreased by using surface gratings, but this is
beyond the scope of this study. When no nanoparticles are considered, reflectivity in the spectral range where the solar spectrum extends is large, meaning that the plain silica on silver structure had appropriate optical properties for radiative cooling. When nanoparticles are considered, an absorption peak appear in the visible range, whose depth increases with volume fraction. As a conclusion for this section, we demonstrated that metallic nanoparticles dispersed in a silica matrix display excellent properties for radiative cooling with coloration. In the following section, evaluation of the radiative cooling efficiency and of the coloration are discussed.

## 4. Radiative cooling performances and coloration

One of the objectives of the study is to estimate net radiative cooling of colored coatings from the calculated absorptance. The proposed coating dissipates heat by emitting the thermal radiative heat flux, $P''_{rad}$, whereas it gains heat by absorbing solar power, $P''_{solar}$, and absorbed atmospheric radiation, $P''_{atm}$. Therefore, its net cooling power reads:

$$P''_{net} = P''_{rad} - P''_{atm} - P''_{solar}. \qquad (7)$$

With the help of Kirchhoff's law, $\alpha(\lambda, \theta)=\varepsilon(\lambda, \theta)$, components of the net radiative cooling in Eq. 7 can be calculated as;

$$P''_{rad} = \int_{2\pi} \int_{0}^{\infty} I_{bb}(\lambda, T)_{T=T_{sur}} \varepsilon(\lambda, \theta) \cos(\theta) d\lambda d\Omega \qquad (8)$$

$$P''_{atm} = \int_{2\pi} \int_{0}^{\infty} I_{bb}(\lambda, T)_{T=T_{amb}} \alpha(\lambda, \theta) \varepsilon_{atm}(\lambda, \theta) \cos(\theta) d\lambda d\Omega \qquad (9)$$

$$P''_{solar} = \int_{0}^{\infty} I_{solar}(\lambda) \alpha(\lambda, \theta)_{\theta=0} d\lambda \qquad (10)$$

where,

$$\varepsilon_{atm}(\lambda, \theta) = 1 - t(\lambda)^{1/\cos(\theta)} \qquad (11)$$

Here $t(\lambda)$ is the atmospheric transmittance, $I_{solar}$ is normal direct irradiance, $I_{bb,Tsur}$ is the blackbody intensity at the surface temperature which are represented in Supplementary Fig. 1.

Estimation of the color is performed by converting a reflection spectrum to a color. The reflected spectral intensity is calculated using the calculated reflectivity spectra of the coatings the spectral intensity of the source of light. For the source, D65 illumination, a CIE standard representing the outside illumination accounting for the environment, is used in this study and its normalized spectral intensity is represented in Supplementary Fig. 7. It is multiplied with the reflectance of the coating to find the reflected spectral intensity. Spectrum to color conversion is then performed by using the CIE model. Color matching functions relating the spectral reflected intensity and the
color, that are established by CIE ¹⁴ are represented in Supplementary Fig. 8. These functions are related to the way that human eye detects color, i.e. by means of three detectors (cone cells) able to detect red, blue and green color. Using the spectral reflected intensity and the color matching functions, parameters X, Y, Z, which corresponds to coordinates in the CIE color space, can be calculated ⁷;

$$X = 100 \frac{\int_{360nm}^{830nm} I_{D65}(\lambda) R(0, \lambda) \bar{x}(\lambda) d\lambda}{\int_{360nm}^{830nm} I_{D65}(\lambda) \bar{y}(\lambda) d\lambda} \quad (12)$$

$$Y = 100 \frac{\int_{360nm}^{830nm} I_{D65}(\lambda) R(0, \lambda) \bar{y}(\lambda) d\lambda}{\int_{360nm}^{830nm} I_{D65}(\lambda) \bar{y}(\lambda) d\lambda} \quad (13)$$

$$Z = 100 \frac{\int_{360nm}^{830nm} I_{D65}(\lambda) R(0, \lambda) \bar{z}(\lambda) d\lambda}{\int_{360nm}^{830nm} I_{D65}(\lambda) \bar{y}(\lambda) d\lambda} \quad (14)$$

Normalized x, y, z parameters can be used to identify the color in chromacity diagram which is represented in Supplementary Fig. 9 are calculated as ⁷;

$$x = \frac{X}{X + Y + Z} \quad (15)$$

$$y = \frac{Y}{X + Y + Z} \quad (16)$$

$$z = \frac{Z}{X + Y + Z} \quad (17)$$

Here, to define how vivid the color is, saturation S is defined as ⁷;

$$S = \frac{\sqrt{a^2 + b^2}}{L} \quad (18)$$

where CIE LAB colour space parameters lightness, L, redness - greenness, a, and yellowness - blueness, b, are given explicitly in appendix 6.

## 5. Results

The baseline coating is composed of silver reflector and silica emitter which does not contain embedded nanoparticles. Its spectral reflectance at normal incidence is represented in Fig. 5. At the conditions described in the problem statement, baseline coating has a whitish appearance with
a 114 Wm⁻² net cooling power. Although modifications such as structuring the surface²¹ can improve the net cooling power, baseline coating is kept simple in order to focus on the objective of the study which is to observe effect of size and concentration of nanoparticles on color and radiative cooling performance.

**5.1. Coating containing a Single Type of Particle**

We now focus on colored coatings made of plain silver or silica core - silver shell particles embedded into silica. Firstly, the coating with plain nanoparticle is investigated. Change of coating color and radiative cooling power by varying nanoparticles size and volume fraction can be observed in Fig. 5. The color gamut that can be obtained is limited due to the fact that for plain nanoparticles, absorption peaks cannot occur in the whole visible range (as seen in Fig. 2(a)). Indeed, plain silver nanoparticles can produce mainly yellow color, which corresponds to absorption of blue light around 0.45 µm. It results in reflected light which is a mix of red and green, producing yellow color. It is also observed that increasing the volume fraction turns the color more vivid, but also decreases the radiative cooling power of the coating. When the volume fraction is increased, the depth of the absorption peak becomes larger, therefore more sunlight is absorbed by the structure, hence the decrease in radiative cooling power.

<figure><img src="image_7.png" /><figcaption>Figure 5 - Silver nanoparticles at various diameters in 500 µm silica coating. Numbers in boxes corresponds to P<sub>net</sub> [W m⁻²].</figcaption></figure>

To increase the color gamut that can be obtained, core-shell nanoparticles are now considered. As it can be observed in Fig. 2(b), plasmon resonances for core-shell nanoparticles can occur in the whole visible range. To inspect the core-shell particles effect on the color, we explore variations
of the inner and outer radius of the particles for a constant volume fraction $f_v = 2 \times 10^{-6}$.
Corresponding colors and radiative cooling powers are calculated and shown in Fig. 6. The black part of Fig. 6 represents geometrically non-physical case where inner diameter is greater than outer diameter. When the shell is thin (i.e. when the outer diameter is close to the inner one), resonance occur outside the visible, leading to white coloration. Note that in this case, since absorption of sunlight still occur, the radiative cooling power is less than the one without nanoparticles. As the shell gets thicker for a constant inner diameter the color becomes cyan, magenta and then yellow due to the shift of the absorption peak. As it has been discussed earlier, absorption peaks with the same spectral locations could be observed by different core-shell size pairs. When looking at the diagonal boxes of Fig. 6, one can see that there are various inner and outer diameter pair options to choose a similar looking color.

Cyan can be achieved by a core-shell nanoparticle with $D_o=30$ nm – $d_i=24$ nm. Corresponding thermal performance and saturation change with the increasing volume fraction is also represented in Fig. 7. For a saturation of 0.65, the radiative cooling power becomes nil.

<figure><img src="image_8.png" /><figcaption>Figure 6 - Silica core, silver shell at 500 µm silica coating with $f_v = 2 \times 10^{-6}$. Numbers in boxes corresponds to $P_{net}$ [W m$^{-2}$]</figcaption></figure>

The spectral reflectance of resulting cyan ($D_o=30$ nm – $d_i=24$ nm), magenta ($D_o=24$ nm – $d_i=17$ nm) and yellow ($D=40$ nm) coatings with single type of nanoparticles are represented in Fig. 8. For the three types of particles, a different spectral position of the localized plasmon resonance occur. As shown in Fig. 8, absorption of blue color around 0.45 µm results in a mix of reflected red and green color, which produces yellow color. Absorption of green light around 0.55 µm
produces magenta color, which is a mix of red and blue. Similarly, around 0.6 µm, cyan color is produced by the mix of green and blue color resulting from the absorption of red color.

As a conclusion for this section, we have demonstrated that the proposed coating can produce three different colors, cyan magenta and yellow, by tuning the spectral position of the plasmon resonance when varying the geometry of the nanoparticles. Positive value of radiative cooling can be obtained along with coloration, but a compromise between the vividity of the color and the radiative cooling power has to be set, since increasing one leads to a decrease of the other.

In order to obtain more colors, we propose in the next section to mix different nanoparticles inside the silica matrix.

<figure><img src="image_9.png" /><figcaption>Figure 7 - Color and thermal properties of 500µm thick silica coating with core-shell pigments ($D_o=30$ nm – $d_i=24$ nm) for varying $f_v$.</figcaption></figure>

<figure><img src="image_10.png" /><figcaption>Figure 8- Spectral reflectances of cyan, magenta and yellow in 500 µm silica coating with $f_v$ = $2 \times 10^{-6}$. (Line colors correspond to estimated coating colors)</figcaption></figure>
## 5.2 Coating with Two Types of Particles

In order to get colors like red, blue and green, multiple absorption peaks are needed as mentioned earlier. To obtain such optical properties, different particles displaying different spectral position of plasmon resonance be mixed inside the matrix. We therefore investigate in this section coloration and radiative cooling power of coatings containing two types of nanoparticles with different geometry. Monte Carlo method is modified to handle such two-type particle mixture. The flowchart is represented in Supplementary Fig. 5. To handle a second type of particle; an extinction coefficient of medium is defined as total extinction coefficient, $β_T$, which is the summation of the extinction coefficient contributions of two types and is formulated as $β_T = β_1 + β_2$. To decide which type of particle is interacting for the traced ray, a random number is generated and compared with $l_{β,1}/l_{β,T}$. where $l_β$ is excitation length as its formulation is presented in Supplementary Fig. 5. If the random number is smaller than $l_{β,1}/l_{β,T}$, then the incoming ray is interacting with a particle of type 1, otherwise by a particle of type 2. The algorithm simply handles single type of particle in a binder if volume fraction of second type is given as zero.

Resulting color and radiative cooling power of coatings with two types of nanoparticles are presented in Figs. 9-11, as a function of the volume fraction of both types of nanoparticles. In Fig. 9, particles that are used to absorb yellow color and blue color are mixed. When the volume fractions of both nanoparticles are comparable, the coating produce red color. In the same way, green color can be obtained by mixing nanoparticles absorbing blue light and red light, and blue color can be obtained with particles absorbing red light and green light. It is worth noticing that when varying the volume fractions of the two types of nanoparticles independently, various other color can be obtained, such as turquoise, orange and violet. In Fig. 12, the spectral reflectance of different mix of nanoparticles are plotted. In each case, the two absorption peaks corresponding to both types of nanoparticles can be observed. Eventually, mixing two types of particles results in an increase of sunlight radiation absorbed by the coatings, and therefore of the radiative cooling power.

<table><caption>Table 1- Ingridents of the Nanoparticle Mixture with Particle 1 (P1) and Particle 2 (P2) in 500µm silica coating.</caption><thead><tr><th>Color Outcome</th><th>P<sub>1</sub> f<sub>v</sub> [x10<sup>-6</sup>]</th><th>P<sub>1</sub> type</th><th>P<sub>1</sub> size</th><th>P<sub>1</sub> color</th><th>P<sub>2</sub> f<sub>v</sub> [x10<sup>-6</sup>]</th><th>P<sub>2</sub> type</th><th>P<sub>2</sub> size</th><th>P<sub>2</sub> color</th></tr></thead><tbody><tr><td>Red</td><td>1</td><td>Simple</td><td>D=50nm</td><td>Yellow</td><td>1</td><td>Core-Shell</td><td>D<sub>o</sub>=24 nm – d<sub>i</sub> 17 nm</td><td>Magenta</td></tr><tr><td>Green</td><td>1</td><td>Simple</td><td>D=50nm</td><td>Yellow</td><td>1</td><td>Core-Shell</td><td>D<sub>o</sub>=30 nm – d<sub>i</sub> 24 nm</td><td>Cyan</td></tr><tr><td>Blue</td><td>0.5</td><td>Core-Shell</td><td>D<sub>o</sub>=24 nm – d<sub>i</sub> 18 nm</td><td>Magenta / Purple</td><td>1</td><td>Core-Shell</td><td>D<sub>o</sub>=30 nm – d<sub>i</sub> 24 nm</td><td>Cyan</td></tr></tbody></table>
<figure><img src="image_11.png" /><figcaption>Figure 9 – Red color production by mixing yellow (D=50 nm) and magenta (Do=24 nm – di=17 nm) at different concentrations in 500 µm silica coating.</figcaption></figure>

<figure><img src="image_12.png" /><figcaption>Figure 10 – Green color production by mixing yellow (D=50 nm) and cyan (Do=30 nm – di=24 nm) at different concentrations in 500 µm silica coating.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>Figure 11 – Blue color production by mixing cyan ($D_o=30$ nm – $d_i=24$ nm) and magenta ($D_o=24$ nm – $d_i=18$ nm) at different concentrations in 500µm silica coating.</figcaption></figure>

<figure><img src="image_14.png" /><figcaption>Figure 12 – Spectral reflectances of red, green and blue particles mixtures given in Table 1. (Line colors correspond to estimated coating colors)</figcaption></figure>

As a conclusion for this section, we demonstrated that by mixing two types of particles with different geometries, the coloration of the coating can be controlled and tuned over a large gamut.
## 5.3 Angular dependence of optical properties and color

Another optical property that affects the appearance of an object is the directional distribution of the reflected light. In the case of specular reflection, objects tend to have a metallic appearance, while for diffuse reflection, they tend to have a matt appearance. Using the Monte-Carlo approach, we are able to probe the directionality of the reflected light, which is plotted in Fig. 13. The reflection is highly specular due to low scattering albedo of the metallic nanospheres, therefore the coating is expected to have a metallic appearance.

<figure><img src="image_15.png" /><figcaption>Figure 13 – Average reflectance in the visible spectrum of the red coating in Table 1 under normal illumination.</figcaption></figure>

## 5.4 Effect of size variation in synthesized coating

Color outcome of the coating with core-shell particles highly is dependent on the nanoparticle size as it is presented in Fig. 6. Strong variation of color occur for small variation of shell thickness. Since all known core-shell nanoparticle synthesizing methods²² result in a variation in size, we analyze the impact of distributing the size of the nanoparticles on the coloration. Core size of the core-shell nanoparticle, *d*ᵢ, is taken constant as 40 nm, and color outcome with net radiative cooling amount is investigated by varying standard deviations for different mean outer diameters. Corresponding results are shown in Fig. 14. According to the results σ<sub>Do</sub> = 2 nm is acceptable which is also achievable to synthesize ²².

Change of extinction coefficients with varying standard deviation for D<sub>o</sub>=62 nm and D<sub>o</sub>=50 nm are represented in Supplementary Figs 10 and 11 respectively. Since cyan arises from thinner shell, it is easily affected by particle distribution and absorption peak shifts. For magenta, location of absorption peak stays still, however since the coating starts to absorb at a wider range, it also starts to look grayish like cyan and lose its vividness with increasing standard deviation. Therefore, it is
concluded that precise control of the nanoparticle size is required to control the coloration of the coating.

<figure><img src="image_16.png" /><figcaption>Figure 14 – Effect of size distribution of silica core, silver shell nanoparticle with $d_i=40$ nm at 500 µm silica coating with $f_v = 2 \times 10^{-6}$</figcaption></figure>

## 6. Conclusions

We have presented a numerical study on colored surface with positive radiative cooling power. The proposed coating consists in a silica matrix with dispersed silver nanoparticles, deposited on a silver reflector. Results show that it is possible to obtain a radiative cooling structure with a colored coating. Colors like cyan, magenta and yellow can be obtained using a single type of nanoparticle. However, in order to obtain red, green and blue, two types of particles have to be mixed. Coating has a metallic appearance due to dominating specular reflectance. Plain silver nanoparticle that gives yellow color has a higher manufacturing tolerance for its size compared to silica core – silver shell nanoparticles. Core-shell structures on the other hand, should be manufactured within +-2 nm error margin to produce the numerically estimated color correctly. Concentration of nanoparticles also affects the color. Increasing volume fraction makes the coating more vivid but it also reduces the net radiative cooling. A compromise between color vividity and radiative cooling power has to be made depending on the application. It is important to note that even if a colored structure displays negative values of radiative cooling power, the thermal load resulting from solar absorption is still rather small since absorption of sunlight radiation occur only in narrow spectral band. Therefore, compared to a standard color coating with broadband absorption, the proposed coatings can be a good alternative for thermal management of colored objects. In conclusion, this work paves the way for both efficient and aesthetic surfaces exposed to the sky, such as wall buildings, car bodies and many others.
References

(1) https://www.eia.gov/consumption/residential/data/2009/index.php?view=consumption.

(2) http://www.nrel.gov/docs/fy00osti/28960.pdf.

(3) Rephaeli, E.; Raman, A.; Fan, S. Ultrabroadband Photonic Structures To Achieve High-Performance Daytime Radiative Cooling. *Nano Lett.* **2013**, *13* (4), 1457–1461.
https://doi.org/10.1021/nl4004283.

(4) Kou, J.; Jurado, Z.; Chen, Z.; Fan, S.; Minnich, A. J. Daytime Radiative Cooling Using Near-Black Infrared Emitters. *ACS Photonics* **2017**, *4* (3), 626–630.
https://doi.org/10.1021/acsphotonics.6b00991.

(5) Yalçın, R. A.; Blandre, E.; Joulain, K.; Drévillon, J. Daytime Radiative Cooling with Silica Fiber Network. *Sol. Energy Mater. Sol. Cells* **2019**, *110320*.
https://doi.org/10.1016/j.solmat.2019.110320.

(6) Zhu, L.; Raman, A.; Fan, S. Color-Preserving Daytime Radiative Cooling. *Appl. Phys. Lett.* **2013**, *103* (22), 223902. https://doi.org/10.1063/1.4835995.

(7) Li, W.; Shi, Y.; Chen, Z.; Fan, S. Photonic Thermal Management of Coloured Objects. *Nat. Commun.* **2018**, *9* (1), 4240. https://doi.org/10.1038/s41467-018-06535-0.

(8) Smith, G. B.; Gentle, A.; Swift, P. D.; Earp, A.; Mronga, N. Coloured Paints Based on Iron Oxide and Silicon Oxide Coated Flakes of Aluminium as the Pigment, for Energy Efficient Paint: Optical and Thermal Experiments. *Sol. Energy Mater. Sol. Cells* **2003**, *79* (2), 179–197. https://doi.org/10.1016/S0927-0248(02)00410-5.

(9) Yong, K.-T.; Sahoo, Y.; Swihart, M. T.; Prasad, P. N. Synthesis and Plasmonic Properties of Silver and Gold Nanoshells on Polystyrene Cores of Different Size and of Gold–Silver Core–Shell Nanostructures. *Colloids Surfaces A Physicochem. Eng. Asp.* **2006**, *290* (1–3), 89–105. https://doi.org/10.1016/j.colsurfa.2006.05.004.

(10) Kado, S.; Yokomine, S.; Kimura, K. Widely Tunable Plasmon Resonances from Visible to Near-Infrared of Hollow Silver Nanoshells. *Bull. Chem. Soc. Jpn.* **2017**, *90* (5), 537–545.
https://doi.org/10.1246/bcsj.20160389.

(11) Ertürk, H.; Howell, J. R. Monte Carlo Methods for Radiative Transfer. In *Handbook of Thermal Science and Engineering*; Kulacki, F. A., Ed.; Springer International Publishing: Cham, 2017; pp 1–43. https://doi.org/10.1007/978-3-319-32003-8_57-1.

(12) Brewster, M. Q.; Tien, C. L. Radiative Transfer in Packed Fluidized Beds: Dependent Versus Independent Scattering. *J. Heat Transfer* **1982**, *104* (4), 573–579.
https://doi.org/10.1115/1.3245170.

(13) Lee, S.-C. Dependent vs Independent Scattering in Fibrous Composites Containing Parallel Fibers. *J. Thermophys. Heat Transf.* **1994**, *8* (4), 641–646.
https://doi.org/10.2514/3.593.

(14) Fairman, H. S.; Brill, M. H.; Hemmendinger, H. How the CIE 1931 Color-Matching Functions Were Derived from Wright-Guild Data. *Color Res. Appl.* **1997**, *22* (1), 11–23.
https://doi.org/10.1002/(SICI)1520-6378(199702)22:1<11::AID-COL4>3.0.CO;2-7.

(15) Kitamura, R.; Pilon, L.; Jonasz, M. Optical Constants of Silica Glass from Extreme Ultraviolet to Far Infrared at near Room Temperature. *Appl. Opt.* **2007**, *46* (33), 8118.
https://doi.org/10.1364/AO.46.008118.

(16) Yang, H. U.; D'Archangel, J.; Sundheimer, M. L.; Tucker, E.; Boreman, G. D.; Raschke, M. B. Optical Dielectric Function of Silver. *Phys. Rev. B* **2015**, *91* (23), 235137.
https://doi.org/10.1103/PhysRevB.91.235137.
(17) Mendoza Herrera, L. J.; Arboleda, D. M.; Schinca, D. C.; Scaffardi, L. B. Determination of Plasma Frequency, Damping Constant, and Size Distribution from the Complex Dielectric Function of Noble Metal Nanoparticles. *J. Appl. Phys.* **2014**, *116* (23), 233105.
https://doi.org/10.1063/1.4904349.

(18) *Absorption and Scattering of Light by Small Particles*; Bohren, C. F., Huffman, D. R., Eds.; Wiley-VCH Verlag GmbH: Weinheim, Germany, Germany, 1998.
https://doi.org/10.1002/9783527618156.

(19) Huang, Z.; Ruan, X. Nanoparticle Embedded Double-Layer Coating for Daytime Radiative Cooling. *Int. J. Heat Mass Transf.* **2017**, *104*, 890–896.
https://doi.org/10.1016/j.ijheatmasstransfer.2016.08.009.

(20) Modest, M. F. *Radiative Heat Transfer*; 2013. https://doi.org/10.1016/C2010-0-65874-3.

(21) Hervé, A.; Drévillon, J.; Ezzahri, Y.; Joulain, K. Radiative Cooling by Tailoring Surfaces with Microstructures: Association of a Grating and a Multi-Layer Structure. *J. Quant. Spectrosc. Radiat. Transf.* **2018**. https://doi.org/10.1016/j.jqsrt.2018.09.015.

(22) Jackson, J. B.; Halas, N. J. Silver Nanoshells: Variations in Morphologies and Optical Properties. *J. Phys. Chem. B* **2001**, *105* (14), 2743–2746.
https://doi.org/10.1021/jp003868k.

(23) Charles Kittel. *Introduction to Solid State Physics*; Wiley: New York, 2004.

(24) Okamoto, T. Near-Field Spectral Analysis of Metallic Beads. In *Near-Field Optics and Surface Plasmon Polaritons*; Kawata, S., Ed.; Springer: Berlin, Heidelberg; pp 97–123.
# SUPPLEMENTARY INFORMATION

## Supplementary Note 1: Environment Conditions

Radiative cooling highly depends on environment conditions such as atmospheric transmissivity and incoming solar heat flux. The resulting atmospheric transmissivity and solar intensity is represented in Supplementary Fig. 1 along with blackbody intensity at T=300K.

<figure><img src="image_17.png" /><figcaption>Supplementary Figure 1- Atmospheric transmissivity, normalized solar and blackbody intensity</figcaption></figure>

## Supplementary Note 2: Refractive indices of materials

Bulk refractive indices of silica and silver used in this study is represented in Supplementary Fig. 2.

<figure><img src="image_18.png" /><figcaption>Supplementary Figure 2 Refractive indices of silver and silica used in the study.</figcaption></figure>
**Supplimentary Note 3: Size dependent dielectric constant**

The validation represented in Supplementary Fig. 3 is made using gold nanoparticles with the given size. Refractive index and other gold parameters like fermi velocity are taken from the literature ¹⁷.

<figure><img src="image_19.png" /><figcaption>Supplementary Figure 3 Validation of size dependent $\epsilon''$ calculation at various gold size.</figcaption></figure>

After calculating the size dependent dielectric function, its effect on calculating absorption efficiency is inspected to emphasize the effect of modification. As it could be seen in Supplementary Fig. 4 although the location of absorption peaks is not affected, the magnitude of absorption is different and using bulk optical data overestimates the absorption at the absorption peak. Also, it should be noted that the difference between bulk and size dependent data decreases as the shell thickness increases.

<table><caption>Supplementary Table 1 – Parameters of silver</caption><tr><td>ω<sub>p</sub><sup>17</sup></td><td>Plasma Frequency</td><td>1.375 x 10<sup>16</sup> [s<sup>-1</sup>]</td></tr><tr><td>γ<sub>free</sub><sup>17</sup></td><td>Damping Constant</td><td>3.12 x 10<sup>13</sup> [s<sup>-1</sup>]</td></tr><tr><td>ν<sub>f</sub><sup>23</sup></td><td>Fermi Velocity</td><td>1.39 x 10<sup>6</sup> [m s<sup>-1</sup>]</td></tr><tr><td>A<sup>24</sup></td><td>Scattering constant</td><td>1</td></tr></table>
<figure><img src="image_20.png" /><figcaption>Supplementary Figure 4 Calculation of absorption efficiency with bulk and size dependent dielectric functions.</figcaption></figure>

### Supplementary Note 4: Analytical solution of scattering by a nanosphere (Mie theory)

To solve Monte Carlo method, scattering efficiency, absorption efficiency and asymmetry parameter are the required that is found from Mie theory. Using the Mie parameters a_n and b_n, they are calculated as;

$$Q_{sca} = \frac{2}{x^2} \sum_{n=1}^{\infty} (2n + 1) (|a_n|^2 + |b_n|^2)$$

$$Q_{ext} = \frac{2}{x^2} \sum_{n=1}^{\infty} (2n + 1) Re(a_n + b_n)$$

$$g = \frac{4}{Q_{sca}x^2} \left[ \sum_{n} \frac{n(n+2)}{n+1} Re\{a_n a_{n+1}^* + b_n b_{n+1}^*\} + \sum_{n} \frac{2n+1}{n(n+1)} Re\{a_n b_{n+1}^*\} \right]$$

Here x is size parameter and defined as $x = 2\pi r n_{medium}/\lambda$ where r is radius for plain, outer-radius for core-shell particles.

Calculation of Mie parameters a_n and b_n differs for plain and core-shell particles and presented explicitly in the following subsections.

#### 4.1 Calculation of Mie parameters a_n and b_n for plain spherical particles

$$a_n = \frac{m\psi_n(mx)\psi'_n(x) - \psi_n(x)\psi'_n(mx)}{m\psi_n(mx)\xi'_n(x) - \xi_n(x)\psi'_n(mx)}$$
$$b_n = \frac{\psi_n(mx)\psi'_n(x) - m\psi_n(x)\psi'_n(mx)}{\psi_n(mx)\xi'_n(x) - m\xi_n(x)\psi'_n(mx)}$$

$$\psi_n(\rho) = \rho j_n(\rho)$$

$$\xi_n(\rho) = \rho h_n^{(1)}(\rho)$$

Where $j_n$ and $h_n$ are spherical bessel functions

$$m = \frac{n_{sphere} + k_{sphere}}{n_{medium}}$$

## 4.2 Calculation of Mie parameters $a_n$ and $b_n$ for core-shell spherical particles

$$a_n = \frac{(\tilde{D}_n/m_2 + n/y)\psi_n(y) - \psi_{n-1}(y)}{(\tilde{D}_n/m_2 + n/y)\xi_n(y) - \xi_{n-1}(y)}$$

$$b_n = \frac{(m_2 \tilde{G}_n + n/y)\psi_n(y) - \psi_{n-1}(y)}{(m_2 \tilde{G}_n + n/y)\xi_n(y) - \xi_{n-1}(y)}$$

where;

$$\tilde{D}_n = \frac{D_n(m_2 y) - A_n \chi'_n(m_2 y) / \psi_n(m_2 y)}{1 - A_n \chi_n(m_2 y) / \psi_n(m_2 y)}$$

$$\tilde{G}_n = \frac{D_n(m_2 y) - B_n \chi'_n(m_2 y) / \psi_n(m_2 y)}{1 - B_n \chi_n(m_2 y) / \psi_n(m_2 y)}$$

$$A_n = \psi_n(m_2 x) \frac{m D_n(m_1 x) - D_n(m_2 x)}{m D_n(m_1 x) \chi_n(m_2 x) - \chi'_n(m_2 x)}$$

$$B_n = \psi_n(m_2 x) \frac{m D_n(m_2 x)/m - D_n(m_1 x)}{m \chi'_n(m_2 x) - D_n(m_1 x) \chi_n(m_2 x)}$$

$$\frac{1}{\psi_n} = \chi_n D_n - \chi'_n$$

$$\chi'_n(z) = \chi_{n-1}(z) - \frac{n\chi_n(z)}{z}$$

$$m_1 = \frac{n_{core} + k_{core}}{n_{medium}}$$
$$m_2 = \frac{n_{shell} + k_{shell}}{n_{medium}}$$

$$m = \frac{m_2}{m_1}$$

## Supplementary Note 5: Monte Carlo method

During ray tracing, the deviation $\theta$ from the incoming direction $\mathbf{s} = s_x \mathbf{i} + s_y \mathbf{j} + s_z \mathbf{k}$ to a new direction $\mathbf{s}_n = s_{n,x} \mathbf{i} + s_{n,y} \mathbf{j} + s_{n,z} \mathbf{k}$, after scattering is determined by Henyey Greenstein phase function that is given by:

$$\cos(\theta) = \begin{cases} 1, & g = 1 \\ 2 \text{Rnd} - 1, & g = 0 \\ \frac{1}{2g} \left[ 1 + g^2 - \left( \frac{1 - g^2}{1 - g + 2g} \right)^2 \right], & x \ge 0 \end{cases}$$

Then $\mathbf{s}_n$ is calculated by,

$$s_{n,x} = \sin(\theta) \frac{(s_x s_z \cos(\phi) - s_y \sin(\phi))}{\sqrt{1 - s_z^2}} + s_x \cos(\theta)$$

$$s_{n,y} = \sin(\theta) \frac{(s_y s_z \cos(\phi) + s_x \sin(\phi))}{\sqrt{1 - s_z^2}} + s_y \cos(\theta)$$

$$s_{n,z} = -\sin(\theta) \cos(\phi) \sqrt{1 - s_z^2} + s_x \cos(\theta)$$

where

$$\phi = 2\pi * \text{Rnd}$$
<figure><img src="image_21.png" /><figcaption>Supplementary Figure 5 - Flowchart of Monte Carlo method that is used in this study.</figcaption></figure>
Validation of Monte Carlo method is performed at a 500µm thick coating over an ideal black surface where refractive index of the polymer binder is taken constant, 1.5. Volume fraction of the TiO₂ particles is 0.04. Reflectance at normal incidence is represented at Fig A.2.

<figure><img src="image_22.png" /><figcaption>Supplementary Figure 6 – Validation of Monte Carlo method for different size of TiO₂ particles in a 500 mm thick polymeric binder with $n_{medium}=1.5$ applied on an ideal black surface.</figcaption></figure>

Monte Carlo method is a statistical method that launches thousands of rays and trace them to solve radiative transfer equation. Since those rays are independent one of each other, the method is suitable for parallelization. 10⁵ rays are launched for 30 angles (from 0° to 90°) and 24701 wavelengths (from 0.3 µm to 25 µm). Therefore, approximately 7.4 x 10⁹ rays are launched in order to find reflectance of a coating. It takes 1875 seconds to find reflectance with CPU (1 core, 3.4 GHz) while GPU (1152 core, 1.7 GHz) can find the result within 37 seconds. This 50-fold difference make a big impact on high number of calculations. For example, to present Fig.6, 741 coatings with different core-shell size are estimated, instead of 16 days with CPU, the data is calculated less than 8 hours thanks to GPU. In this study, three different modifications are performed in the Monte Carlo code to encounter;

- Second type of nanoparticles

- Refraction at lossy medium

- GPU acceleration for an approximately 50 times faster execution

The resulting codes is provided as a supplementary material and will also be available at https://www.github.com/refetaliyalcin

**Supplementary Note 6: CIE Color Parameters**

$$ 
\begin{aligned}
L &= 116 f(Y/Y_n) - 16 \\
a &= 500[f(X/X_n) - f(Y/Y_n)] \\
b &= 200[f(Y/Y) - f(Z/Z_n)]
\end{aligned}
$$

where,
$$f(s) = s^{1/3} \text{ if } s > (24/116)^3$$

$$f(s) = (841/108)s + 16/116 \text{ if } s(24/116)^3$$

$X_n=95.047, Y_n=100, Z_n=108.883;$

<figure><img src="image_23.png" /><figcaption>Supplementary Figure 7 - Standard D65 light spectrum for representation</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>Supplementary Figure 8 - CIE color matching functions</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>Supplementary Figure 9 - Chromaticity diagram</figcaption></figure>

**Supplementary Note 7: Optical properties at various particle size distribution**

<figure><img src="image_26.png" /><figcaption>Supplementary Figure 10 - Extinction coefficients of coating with core-shell nanoparticle $d_i=40$ nm, $D_o=62$ nm at 500µm silica coating with $f_v = 2 \times 10^{-6}$.</figcaption></figure>
<figure><img src="image_27.png" /><figcaption>Supplementary Figure 11 - Extinction coefficients of coating with core-shell nanoparticle *d*<i>*i*</i>=40 nm, *D*<i>*o*</i>=50 nm at 500 µm silica coating with *f*<sub>v</sub> = 2x10<sup>-6</sup>.</figcaption></figure>