Polymer coating with gradient-dispersed dielectric nanoparticles for enhanced daytime radiative cooling

Polymeric coatings with randomly distributed dielectric nanoparticles have attracted intensive attention in the passive daytime radiative cooling application. Here, we propose a modified Monte – Carlo method for investigating the spectral response and cooling performance of polymer coating with gradient-dispersed nanoparticles. Using this method, we carry out a quantitative analysis on the solar reflectance, infrared emittance and cooling power of four categories of gradient structures. It is shown that the gradient profile of particle distribution at the near-surface region has a significant influence on the overall performance of the coatings. Compared to a randomly distributed structure, the downward size-gradient structure exhibits superiority in both solar reflectance and cooling power. The presented gradient design, also applicable to porous structures, provides an effective and universal strategy for significantly improving the cooling performance of radiative cooling coatings.


| INTRODUCTION
Passive daytime radiative cooling as a promising energy conservation strategy has shown great potential for saving the building energy, mitigating the urban heat island effect and combatting global warming. 1,2Compared to conventional active refrigeration technologies, it takes advantages of the passive nature of thermal radiation and the transparent atmospheric window (8-13 μm) to dissipate heat from a terrestrial object (~300 K) to the cold universe (~3 K).Anti-intuitive sub-ambient radiative cooling without any energy consumption and green-house gas emission can be achieved even under direct sunlight. 3,4To maximize the cooling effect, the radiative cooler should be entitled with high emissivity within the atmospheric window and high solar reflectance simultaneously.Recently, photonic structures, 3,[5][6][7] metamaterials, [8][9][10][11] particle dispersed polymeric coatings, [12][13][14][15] and porous structures [16][17][18][19][20][21] have been investigated to simultaneously achieve high solar reflectance as well as high infrared radiation.Among these methods, polymeric coatings have shown the best potential for real-world large-scale applications for its low cost, ease in production and excellent long-term durability. 22enerally, polymeric coatings consist of a polymer matrix and randomly distributed micro/nano dielectric particles as fillers.The high thermal radiation originates mainly from the intrinsic vibrational modes of functional groups/bonds of the polymer matrix and partially from the phonon resonances of filled particles, 9,23 while the high solar reflectance of the coatings benefits from the multiple interfacial Mie scattering between the polymer matrix and the filled particles. 24Therefore, materials with a high refractive index and large band gap are preferred for use as fillers to enhance Mie scattering and thus the overall solar reflectance.7][28] In other words, a heterogeneous distribution may be formed instead of an ideal homogeneous one during the formation of the polymeric nanocomposite coating.In general, the gradient distribution caused by sedimentation was utilized to separate different species with different mass and sizes. 29][32][33][34][35] In this study, we numerically investigate the optical properties of a polymeric coating from ultra-violet to mid-far infrared regions, as well as its cooling potential, using both density-and size-gradient structures.An effective multilayer Monte-Carlo method is established to solve the radiative transfer equation for the gradient structures, of which the solar reflectance, infrared emittance and net cooling power are investigated.We have observed an obvious impact of gradient distribution on the cooling performance of the coating and the downward size-gradient structure is superior to the randomly distributed one.

| Materials and optical constants
Many kinds of polymers, including polydimethylsiloxane (PDMS), polymethyl methacrylate (PMMA), polyvinylidene fluoride (PVDF), polymethyl pentene (TPX), etc., have been used as the polymer matrix that makes the radiative cooler feature low solar absorption and strong thermal emissivity ascribing to the functional groups such as C O, C F, C H, and O H. In this study, PDMS is chosen as the polymer matrix due to its lower refractive index (~1.4within the solar band) and good infrared emissivity within the atmospheric window. 36The refractive index and extinction coefficient are shown in the top panel of Figure 1 according to the measurement by Zhang et al. 37,38 Nano-sized dielectric fillers can significantly boost the solar reflectivity through Mie scattering; therefore, an appropriate filler should be selected with consideration of its complex refractive index.The real part, refractive index n, should be high enough than the polymer matrix to achieve an excellent scattering efficiency according to Mie theory, while the imaginary part, extinction coefficient k, needs to be as low as possible to diminish the parasitic absorption under illumination.Therefore, dielectric materials with large bandgap (over 6 eV), such as BaSO 4 , Al 2 O 3 , and CaCO 3 , are preferred in order to reduce solar absorption.Nevertheless, the refractive indices of these dielectric materials are around 1.6-1.7 over the solar spectrum, which are slightly higher than most polymer matrices.The rutile TiO 2 , however, retains the refractive index over 2.4 within the solar spectrum and are widely used as a commercial coating component, as shown in the bottom panel of Figure 1. 39Hence, we chose TiO 2 as the filler to take full advantage of Mie scattering.Though TiO 2 bears intrinsic UV absorption owing to a bandgap of ~3 eV, it can be minimized by a recently proposed Purcell-effect-enhancedfluorescence method, in which the fluorescent materials compete with TiO 2 in UV absorption that can be converted and re-emitted as visible light. 22

| Theories and methods
To investigate the spectral solar reflectivity and infrared emissivity, a statistical Monte-Carlo method, in which a F I G U R E 1 Real part (dark) and imaginary part (blue) of complex refractive indices of PDMS (top panel, Adapted with permission from: Copyright 2020, Elsevier 37 and Copyright 2020, Optical Society of America 38 ) and TiO 2 (bottom panel, solid line for parallel polarization and dashed line for perpendicular polarization) 39 mass number of light paths are traced, rather than finite element method is applied to reduce the computational efforts.The required scattering and absorption properties of both medium and particles will be obtained by Mie theory, and then coupled with the radiative transfer equations (RTE) which will be solved in the Monte-Carlo model.

| Mie theory
For each single particle in the medium, the scattering efficiency, extinction efficiency and asymmetry parameter can be calculated by solving Maxwell equations 40,41 : where x is the size parameter defined by the particle size, wavelength and refractive index of polymer matrix, a n , b n , a Ã n , and b Ã n are the Mie coefficients and their conjugates.In the Monte-Carlo calculation for a nanoparticlepolymer composite, the uniform distribution leads to a layer of effective medium.The effective properties of the coating can be directly obtained by accumulating all the parameters of fillers. 12,24However, the gradient distribution of particles creates a continuous but non-homogenous phase that cannot be regarded as a single layer effective medium.Here, we treat the gradient structure as many thinner sublayers, and every sublayer could be considered as homogeneous.Then the effective scattering and absorption properties of the ith sublayer become: where the subscript j represents the jth kind of particle, f i,j is the corresponding volume fraction in ith sublayer, r j is corresponding radius, k mat is the extinction coefficient of polymer matrix.Since TiO 2 particles shows different complex refractive indices for parallel and perpendicular polarization, here the same amount of TiO 2 particles is assumed for both polarizations in the coating. 12

| Modified Monte-Carlo method
The spectral reflectance and emittance of the coating can be estimated by solving the quasi-steady form of radiative transfer equation 41 : where I λ and I b,λ are the spectral radiative intensities of the coating and a blackbody, s denotes the geometric path of radiative transfer, Φ λ ŝi ,ŝ ð Þ is the scattering phase function describing the scattering probability from direction ŝi to direction ŝ of the incoming heat flux at solid angle of dΩ 0 , κ λ and σ λ are the effective properties obtained by Equations ( 4) and ( 5).
As mentioned above, Monte-Carlo method was implemented to solve RTE in coatings with random particle distribution and achieved good feasibility for photon transport estimation.However, to evaluate the optical properties of gradient structures, a new calculation method should be developed to quantitatively estimate the spectral responses of the coating considering such inhomogeneity.In addition to the multi-sublayer assumption, the interfacial effect is neglected when the light propagates from one sublayer to another due to their same matrix.The flowchart of modified Monte-Carlo method used for gradient structures is shown in Figure 2. Random numbers (RN) are used in the model to statistically trace the light path.Effective properties of sublayers and scattering features of particles are marked using the layer flag i and particle type flag j, respectively.

| Radiative cooling performance
Two key parameters, net cooling power at ambient temperature P net and sub-ambient temperature difference ΔT at the thermodynamic equilibrium state, were usually deployed to evaluate the cooling performance of the radiative cooler.While for a typical application (e.g., on building envelope), the coatings are always at near or above the ambient temperature due to thermal interaction with surrounding urban environment and indoor heat generation. 23Hence, here we calculate the net cooling power only for estimating the cooling performance of the gradient structured coating.Considering all the thermal exchange processes, the net cooling power P net of a radiative cooler at temperature T can be expressed as 3 : where P rad is the radiated power by the cooler, P atm , P sol , and P nonÀrad are the absorbed power of atmospheric radiation, solar irradiation and non-radiative (conductive and convective) thermal energy, respectively, T amb (= 300 K) is the typical temperature of an ambient environment.Each thermal exchange process can be described as: where ε λ, θ, T ð Þ is the angular spectral emissivity of the cooler, I BB T, λ ð Þ is the spectral radiance of blackbody at temperature T, ε atm λ, θ ð Þ is the angular dependent emissivity of the atmosphere that is related to the atmospheric transparency in the zenith direction. 42θ sun is the angle between the sun and zenith direction.h eff is the effective coefficient of non-radiative heat transfer.Here, the temperature of the cooler is assumed to be the same as the ambient, thus the non-radiative contribution will be neglected, that is, P nonÀrad ¼ 0.

| Validation of modified Monte-Carlo method
First, for validating the effeteness of our developed Monte-Carlo method, the optical properties of random particle distribution coating are calculated under both F I G U R E 2 Flowchart of the modified Monte-Carlo method for analyzing the spectral response of gradient structures with multi-sublayer and multi-sized particles methods (i.e., the traditional one and the modified one) for comparison, where the coating is assumed to consist of 100 μm thick PDMS and randomly distributed TiO 2 nanoparticles with a volume fraction of f ¼ 0:05 and uniform radius of r ¼ 200 nm.The obtained reflectivity at solar range and emissivity at infrared band from both methods matched perfectly each other, as shown in Figure 3, which proves the feasibility of the multisublayer assumption and neglecting the interfacial effect.Here, the 100 μm thick coating is divided into 5, 10, and 20 sublayers with the sublayer thickness of 20, 10, and 5 μm for the modified Monte-Carlo method, respectively.

| Density-gradient and size-gradient structures
In practice, the sedimentation effect may lead to the formation of four gradient distributions of particles, incorporating two types of gradients (i.e., density-gradient and size-gradient) and two directions of gradients (i.e., downward and upward).As shown in Figure 4, downward gradient manifests that the size or volume fraction gradually increases from the top boundary to the bottom boundary of the coating while the upward gradient features the opposite trend.To explore the optical properties and cooling performance of the density-gradient coating, the overall solar reflectance and infrared emittance are calculated using the modified Monte-Carlo method and are shown in Figure 5A,B, where the 300 μm thick PDMS coating is filled with TiO 2 nanoparticles at a uniform radius of r ¼ 200 nm.By increasing the volume fractions at the boundaries (f top , at the top boundary and f bot , at the bottom boundary) from 0.01 to 0.3, we can clearly see that both the reflectance and emittance are significantly boosted.Therefore, the best cooling performance can be obtained at the highest volume fraction (i.e., f top ¼ f bot ¼ 0:3), as shown in Figure 5C.Moreover, it should be noticed that the upward gradient coating outperforms the downward gradient one in the solar reflectance since a higher volume fraction of TiO 2 at the top boundary can scatter the sunlight more efficiently.
Further, the optical and cooling characteristics of size-gradient coating is investigated in Figure 5D-F, where the particle radii at the boundaries (r top at top boundary and r bot at bottom boundary) vary from 100 to 1000 nm.It can be observed that the downward sizegradient coating exhibits a superior solar reflectance compared to its upward size-gradient counterpart.Since the scattering efficiency of TiO 2 nanoparticles approaches the maximum at the radius around 100-200 nm and gradually falls down at larger sizes, 25 downward size-gradient coating that has smaller particles at the top always scatters the sunlight more efficiently than the upward size-gradient coating except for the case denoted by a red triangle in Figure 5D, in which the particles at the top (r top ¼ 100 nm) has a slightly lower scattering efficiency than the bottom (r top ¼ 200 nm).Finally, the spectral properties and cooling performance of the coating with random particle distribution are calculated in Figure 5G-I through the traditional Monte-Carlo method for comparison.It can be seen that the solar reflectance manifests a similar reduction with increasing the size of particles.However, we can see obvious differences between the random distribution and upward/downward size-gradient structures.For example, comparing to the downward sizegradient structure, the randomly distributed structure exhibits a larger reduction of solar reflectance and a marginal improvement of infrared emissivity when the size distribution is broadened.An opposite trend is observed between the randomly distributed and upward size-gradient structures.
Detailed comparisons of the spectral properties and cooling performance are demonstrated in Figure 6.It can be observed that at a broad size distribution (r ¼ 100 -1000 nm), the downward size-gradient structure leads to a superior solar reflectivity and considerable infrared emissivity as shown in Figure 6A.The randomly distributed structure exhibits a better infrared emissivity at the wavelength below 6 μm, while the downward size-gradient structure leads to a higher infrared emissivity at the range of 6-9 μm and 13-20 μm.Thus, although the downward size-gradient structure does not perform the best infrared emissivity, its superior solar reflectivity enables an extra cooling power up to ~36 W/m 2 than the randomly distributed structure, as shown in Figure 6B.On the contrary, the upward gradient distribution shows poorer performance for both solar reflectivity and infrared emissivity, and thus the cooling power.It should be noticed that there is an exception for cooling power difference in Figure 6B, which corresponds to the reflectance exception in Figure 5D.Based on our calculation results, the downward size gradient, which could be naturally formed during the production of an actual coating, may enable further enhancement of the cooling performance than the random size distribution.Such structure can also be intentionally utilized to configure the coating for better cooling performance in future.

| CONCLUSION
Considering the sedimentation effect during curing of polymer coating, we have explored the spectral properties and cooling potentials for gradient structures under two types and two directions of gradients.Traditional Monte-Carlo method is modified through multiple sublayers simplification while ignoring the interfacial effect.A significant impact of gradient distribution on the coating performance is observed.The upward density-gradient and downward size-gradient perform better in all four studied gradient structures because the light interacts with particles at the top of the coating first.Compared to the randomly distributed structure, the downward size-gradient structure leads to an improvement of the cooling power up to ~36 W/m 2 at a broad size distribution, while upward size-gradient structure behaves the opposite.Therefore, though the sedimentation effect was not considered in previous studies, the construction procedure may produce the size gradient to an extent, resulting in a better cooling performance than the prediction based on the random distribution assumption.The findings arisen from the present theoretical study has shed light on how to optimize the performance of a radiative cooling coating through introducing gradientdistributed nanoparticles into the polymer matrix.

F
I G U R E 3 Reflectivity (left y axis) and emissivity (right y axis) of 100 μm thick PDMS-TiO 2 coating calculated by both traditional Monte-Carlo method (black line) and the modified one (magenta, blue and red lines for 5, 10, and 20 sublayers, respectively) F I G U R E 4 Schematics of (A) downward, (B) upward densitygradient structures, and (C) downward, (D) upward sizegradient structures F I G U R E 5 (A-C) Calculated solar reflectance (A), infrared emittance (B) and net cooling power (C) of density-gradient structures at ambient temperature.The size of TiO 2 is fixed as r = 200 nm.(D-F) Solar reflectance (D), infrared emittance (E) and net cooling power (F) of size-gradient structures at ambient temperature.The volume fraction is fixed as f ¼ 0:05.(G-I) Solar reflectance (G), infrared emittance (H) and net cooling power (I) of randomly distributed structures at ambient temperature.The volume fraction is also fixed as f ¼ 0:05.The thickness of all the coatings is 300 μm