Cementitious materials as promising radiative coolers for solar cells

iScience Article Cementitious materials as promising radiative coolers for solar cells Cementitious materials Micro-structure Molecular simulations Electromagnetic properties Radiative cooling of solar cells Effective medium R T 1 Reflector Cement paste Solar cell Matteo Cagnoni, Alberto Tibaldi, Jorge S. Dolado, Federica Cappelluti [email protected] Highlights A multi-scale approach predicts dielectric properties of cementitious materials Cementitious slabs can emit thermal radiation in the atmospheric window Low-cost cementitious materials can be effective radiative coolers for solar cells Cement-based radiative cooling could significantly extend solar cells lifetime Cagnoni et al., iScience 25, 105320 November 18, 2022 ª2022 The Author(s). https://doi.org/10.1016/ j.isci.2022.105320 ll OPEN ACCESS iScience Article Cementitious materials as promising radiative coolers for solar cells Matteo Cagnoni, 1,5, *Alberto Tibaldi, 1,2 Jorge S. Dolado, 3,4 and Federica Cappelluti 1 SUMMARY Nowadays, radiative coolers are extensively investigated for the thermal management of solar cells with the aim of improving their performance and lifetime. Current solutions rely on meta-materials with scarce elements or complex fabrication processes, or organic polymers possibly affected by UV degradation. Here, the potential of innovative cement-based solutions as a more sustainable and cost-effective alternative is reported. By combining chemical kinetics, molecular mechanics and electromagnetic simulations, it is shown that the most common cements, i.e., Portland cements, can be equipped with excellent radiative cooling properties, which might enable a reduction of the operating temperature of solar cells by up to 20 K, with outstanding efficiency and lifetime gains. This study represents a first step toward the realization of a novel class of energy-efficient, economically viable and robust radiative coolers, based on cheap and available cementitious materials. INTRODUCTION Reducing the operating temperature is a key challenge in solar cells technology. A lower temperature not only increases the power conversion efficiency, by about 0:5%=K in silicon-based devices (Skoplaki and Palyvos, 2009), but also extends the system lifetime, by roughly 23=10 K (Dupre ´et al., 2017). Motivated by the possibility of achieving significant gains in performance, researchers have proposed severalcoolingsolutionsovertheyearswhicharebasedondiverseconcepts(Chandrasekar et al., 2015). Among these, radiative cooling has been attracting much attention lately, not only for the thermal management of solar cells (Li et al., 2017,2021;Perrakis et al., 2020,2021;Safi and Munday, 2015;Wang et al., 2020), but also for application in buildings (Hossain and Gu, 2016;Li and Fan, 2019;Zhao et al., 2019). This technology stands out thanks to its remarkable potential in terms of energetic efficiency, economical viability, environmental friendliness, and reliability, fostered by its completely passive nature, effectiveness, systemic simplicity and absence of moving parts. Radiativecoolersarebodiesdesignedtostronglyemitthermalradiationwithintheatmospheretransparency window (AW) between 8 and 13 mm(seeFigure S1)(Catalanotti et al., 1975). Radiation ejected through this channel dodges the bounce-back effect of the atmosphere and reaches outer space without returning to the sender. This uncompensated energy removal reduces the temperature of radiative coolers. Moreover, if strict spectral requirements are fulfilled, sub-ambient temperature can be reached even under direct sunlight, as experimentally proven only recently (Raman et al., 2014). Thanks to these characteristics, radiative coolers can act as effective heat sinks when thermally coupled to a warming body, such as a solar cell. Indeed, the excess heat generated within the cell on sunlight absorption is going to flow toward the colder radiative cooler. Then, the latter is going to permanently remove it from the system in the form of thermal radiation through the atmospheric window. Remarkably, this process can reduce the operating temperature of silicon-based devices by up to 18:5K(Zhu et al., 2014), roughly leading to a 9 % efficiency gain (Skoplaki and Palyvos, 2009) and 360 % of lifetime (Dupre ´et al., 2017). Different kinds of radiative coolers were discovered in the last few years (Hossain and Gu, 2016). The most common ones are meta-materials made of vertically stacked thin films (Li et al., 2017;Raman et al., 2014;Kecebas et al., 2017), or thick layers with a micro-patterned surface (Perrakis et al., 2021;Zhu et al., 2014;Hossain et al., 2015;Kong etal., 2019).Yet,itisunclearwhetherthesetechnologiesareadaptableto large-scale manufacturing, 1 Department of Electronics and Telecommunications, Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy 2 Istituto di Elettronica e di Ingegneria dell’Informazione e delle Telecomunicazioni, Consiglio Nazionale delle Ricerche c/o Politecnico di Torino, Corso Duca degli Abruzzi 24, Torino 10129, Italy 3 Centro de Fı ´sica de Materiales, CSIC-UPV/EHU, Paseo Manuel de Lardizabal 5, San Sebastia ´n 20018, Spain 4 Donostia International Physics Center, Paseo Manuel de Lardizabal 4, San Sebastia ´n 20018, Spain 5 Lead contact *Correspondence: [email protected] https://doi.org/10.1016/j.isci. 2022.105320 iScience 25, 105320, November 18, 2022 ª2022 The Author(s). This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/). 1 ll OPEN ACCESS because of their reliance on scarce materials such as Ag or Hf (EuChemS, 2021), or complex deposition and patterning methods. To overcome these issues, organic materials such as hierarchical porous polymers have been proposed as a low cost alternative (Carlosena et al., 2021;Wang et al., 2021b,a;Li et al., 2019;Mandal et al., 2018), but their use might be jeopardized by UV degradation (Zhao et al., 2019). This impasse is forcing researchers into a trade-off between performance, cost, and reliability, and calls for prompt action to identify alternative classes of radiative coolers capable of fulfilling all these requirements simultaneously. Faced with this challenge, we have considered several options and identified (meta-)concretes as a very promising class of cheap and scalable (meta-)materials. Conventional concrete is made by gluing together aggregates such as sand or gravel with a cement paste (binder) (Allen and Iano, 2019). These aggregates canbereplacedwithmore‘‘exotic’’inclusionstoformmeta-concrete, a concrete-like meta-material that canbeequippedwithunconventionalproperties(Mitchell et al., 2014). A first hint at the possible application of these materials as radiative coolers is provided by their multi-scale porous structure (Dolado and Van Breugel, 2011), which strongly resembles the one of the aforementioned hierarchical porous polymers. Furthermore, there exist many recipes for the cement paste and many possible choices for the aggregates, which lead to concretes with very different chemistry and micro-structure (Aı ¨tcin, 2000;Bensted and Barnes, 2002;Bohnet and Ullmann, 2003). As a matter of fact, these materials form an extremely broad class, which can be used not only in buildings (Gagg, 2014),butalsoinclinicalapplicationssuchasboneprostheses (Kenny and Buggy, 2003) and tooth restoration (Chadwick and Evans, 2007). This characteristic provides researchers with many knobs to tune properties. Finally, concretes are already being investigated in the context of buildings as a structural material equipped with radiative cooling capacities (European Commission, 2021), with recent experiments confirming their strong thermal emissivity in the atmospheric window (Lu et al., 2021) and high reflectance at sunlight wavelengths (Levinson and Akbari, 2002). Encouraged by these observations, we have transferred for the first time with this work the idea of cementand concrete-based radiative coolers from buildings to solar cells. In particular, we have investigated the suitability of ordinary Portland cements (OPC) (Taylor, 2004),whicharethemostcommonlyusedtypeof binder, for their thermal management. Remarkably, we have discovered that they can be equipped with dielectric properties suitable for the thermal management of solar cells and potentially capable of providing outstanding gains in performance. These findings may represent a major breakthrough in radiative cooling research, because the main elements found in these cements, such as Ca, Si, O, and H, are among the cheapest and most available on Earth (EuChemS, 2021). At the same time, the stability and reliability of cementand concrete-based solutions is something that we experience every day. With this article, we demonstrate that cements and concretes can also be equipped with the properties needed for the effective radiative cooling of solar cells and can become the ultimate radiative coolers, capable of fulfilling performance, cost and reliability requirements at the same time. These findings call for further research aimed at realizing an ultimate photovoltaic system design and fabrication protocol. To reach these conclusions, we have defined a multi-scale interdisciplinary simulation workflow that calculates the cement paste electromagnetic properties from scratch and uses them in a power balance model to estimate the solar cell temperature reduction driven by its coupling to the cement-based radiative cooler. The essentials of the workflow are depicted in Figure 1. First, several cement micro-structures are generated by modeling the cement paste formation with methods from chemical kinetics. At the same time, the IR dielectric properties of the basic components of the heterogeneous cement paste are calculated by atomistic simulations. Next, these micro-structural and dielectric data are combined into a suitable effective medium theory to convert the space-dependent dielectric function into an equivalent homogeneous dielectric function. Then, this is plugged into an electromagnetic simulator to calculate the cement slab emissivity. Finally, this slab is ‘‘attached’’ to the solar cell and the operating temperature of the resulting device is calculated by the detailed balance model. More information on each of these methods and the corresponding findings is provided in the results and discussion section, with additional details in the STAR methods section and the supplemental information. RESULTS AND DISCUSSION We start our discussion with the hypothetical planar structure depicted in Figure 2A, based on the typical model used for the performance assessment of radiative coolers made of meta-materials (Perrakis et al., ll OPEN ACCESS 2iScience 25, 105320, November 18, 2022 iScienc e Article 2020,2021;Safi and Munday, 2015;Zhu et al., 2014;Cagnoni et al., 2022). The device consists of a stack made by a reflector, a cement-based radiative cooler and a bifacial (Guerrero-Lemus et al., 2016) solar cell; the solar cell top surface is facing the Sun. This structure could be realized, for example, by building a thin film solar cell onto a cement-based substrate by sputtering, evaporation or solution deposition techniques. Experiments can be found in the literature, where thin-film solar cells have been placed onto structural elements of buildings such as roof tiles and concrete blocks (A ´guas et al., 2011;Iencinella et al., 2009;Hosseini et al., 2013). By design, the solar cell and the radiative cooler are thermally coupled but mutually transparent. Indeed, the former absorbs sunlight in the UV-visible spectral range, whereas the latter emits thermal radiation in theIRspectralrange,wheretheAWisfound.Because absorbance and emissivity spectra of a body are equal according to Kirchhoff’s law of thermal radiation (Balaji, 2014) (they will be used interchangeably from this point onward), the solar cell and the radiative cooler do not exchange energy with each other IR permittivity of cement paste components Chemical kinetics (µic) Atomistic simulations (GULP) Cement paste microstructure H2O Effective medium theory (Slovick) Homogenized cement paste IR permittivity Electromagnetic simulations (generalized TMM) Cement slab absorbance/emissivity Detailed-balance model Solar cell operating temperature vs band-gap Ideal mirror Cement paste Solar cell Figure 1. Workflow for the assessment of cement-based radiative coolers for the thermal management of solar cells ll OPEN ACCESS iScience 25, 105320, November 18, 2022 3 iScienc e Article electromagnetically, but do so only with the Sun and the atmosphere, separately. These observations are clarified in Figure 2B, where the most significant spectra are depicted (see Figure S1 for atmosphere transmittance spectra). According to the considerations above, the device can be modeled as a single body at temperature T, whose electromagnetic properties are the ones of the solar cell in the UV-visible range and the ones of the radiative cooler in the IR spectral range (Safi and Munday, 2015). Then, one can calculate the net power density (power per top surface unit area) PnetðTÞexiting the device as function of Tand determine the operating temperature by solving PnetðTÞ=0, which corresponds to the stationary state of the system. Pnet consists of several terms, as depicted in Figure 2A: PnetðTÞ=Pcell rad T;EgPSunESun e;l;Eg |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} UV visible spectrum +PelecT;Eg+Pcool rad T;Acool U;lPatmT0;Aatm U;l;Acool U;l |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} IR spectrum +PconðT;T0;hcÞ(Equation 1) PSun,Pcell rad and Pelec are the power densities that the solar cell absorbs from the Sun, emits as radiation and delivers to the end-user load at maximum power point (MPP), respectively; they have been calculated according to the Shockley-Queisser model (Shockley and Queisser, 1961). On the other hand, Patm and Pcool rad are the power densities that the radiative cooler absorbs from the atmosphere and emits as thermal radiation, respectively. Finally, Pcon is an empirical term to account for conduction and convection phenomena between the device and the environment. All these terms are widely discussed in the literature (Perrakis et al., 2020,2021;Safi and Munday, 2015;Zhu et al., 2014). Therefore, we refer the reader to the STAR methods section for their explicit formulas and we show here only their parametric dependencies, to highlight which information must be known to obtain Pnet as a function of Tonly. In particular, some electromagnetic spectra must be supplied as a function of wavelength land zenith angle q(the system is invariant with respect to the azimuth angle). The first one is the Sun spectral irradiance ESun e;lðlÞ, modeled using the global standard spectrum AM1.5g (ASTM International, 2022). The next one is the atmosphere spectral directional emissivity Aatm U;lðl;qÞ,obtainedaccordingtotheformulaAatm U;lðl;qÞ= 1Tatm 0;lðlÞ1=cosðqÞ(Perrakis et al., 2020), Tatm 0;lðlÞbeing the zero-zenith spectral directional transmittance Reflector (ideal mirror) Radiative cooler (cement paste) Solar cell (Shockley-Queisser) Atmosphere Outer space AW AB Figure 2. Illustration of the detailed balance model employed (A) Planar structure used to assess the suitability of a radiative cooler for the thermal management of solar cells. The power density terms representing the channels through which the device, the environment and the end-user load exchange energy are also depicted, together with the atmosphere radiation shielding effect outside of its transparency window (AW). (B) Comparison between the main electromagnetic spectra involved in the operation of a solar cell equipped with a radiative cooler. ESun e;lis the normalized Sun spectral irradiance (AM1.5g), Eatm e;lis the normalized atmosphere spectral irradiance, and AW are the atmosphere transparency windows. It is shown that solar cell and Sun are electromagnetically active in a spectral range different from the one of radiative cooler and atmosphere. ll OPEN ACCESS 4iScience 25, 105320, November 18, 2022 iScienc e Article calculated for the summer season with the software LOWTRAN (Hirsch, 2016). The last one is the spectral directional absorbance of the cement-based radiative cooler Acool U;lðl;qÞ, calculated according to the workflow outlined in the Introduction and depicted in Figure 1. The results originating from this workflow are the core subject of this section. However, before moving to their discussion, a few parameters of Equation 1 still need explanation. In particular, Egis the band-gap of the solar cell semiconductor, for which we have considered values in the range between 1 and 3 eV, whereas T0and hcare the ambient temperature (set to 293:15 K) and the conduction/convection coefficient (set to 10:6Wm 2K1to represent average winds (Perrakis et al., 2020)); different values of these two do not affect our findings, hence they are not considered here. Back to Acool U;l, this can be readily determined by the transfer-matrix method (TMM) for a planar structure made of layers with known homogeneous permittivity (Born and Wolf, 2019). However, as already anticipated, common cement pastes are made of a heterogeneous mixture of chemical species arranged into a complex multi-scale porous structure (Dolado and Van Breugel, 2011). This apparent incompatibility can be lifted by resorting to a suitable effective medium theory (Choy, 2016), which enables us to substitute the microscopically inhomogeneous permittivity with a homogeneous one that provides equivalent electromagnetic properties at the macro-scale. This procedure requires knowledge of the cement paste micro-structure and of the complex permittivity of its homogeneous components, as well as a proper choice of the homogenization (effective medium) model. We are now going to discuss these aspects one by one. Micro-structure of the cement paste Common cement pastes are prepared by mixing a fine powder (clinker) with water. This mixture undergoes a hydration process whose products form the cement paste, which gradually hardens over time. For OPCs, thepowderismadeofalite,whosechemicalformulaisCa 3SiO5(C3S in cement chemist notation), by up to 70 % (Taylor, 2004). Accordingly, we have considered a cement paste made by hydrating alite only, to simplify the model while capturing all the essential features of OPCs. Hydration time A B CDE Figure 3. Simulation of the alite hydration process for an initial particle size distribution corresponding to N0= 105particles per ð100 mmÞ3 (A) Sketch of the hydration process of a single alite particle representing the model used in this work. (B) Definition of the radii used to quantify the particles size in this work. (C) Simulation snapshot of the initial cement powder. (D) Simulation snapshot of the partially hydrated cement paste. (E) Simulation snapshot of the fully hydrated cement paste. ll OPEN ACCESS iScience 25, 105320, November 18, 2022 5 iScienc e Article The surface of the alite powder particles, which are typically assumed to be spherical (Navi and Pignat, 1996), dissolves on reaction with water. The dissolution products form shells of calcium silicate hydrate, whose chemical formula is ðCaOÞ3ðSiO2Þ2ðH2OÞ4(CSH in cement chemist notation), around the original particles, or form new particles made of portlandite, whose chemical formula is CaðOHÞ2(CH in cement chemist notation), by nucleation and growth in the interstitial regions. This process is sketched in Figure 3A and follows the volumetric formula (Pignat et al., 2005): 1:0VC3S+1:318 VH2O/1:57 VCSH +0:596 VCH (Equation 2) The resulting cement paste is made of a disordered ensemble of two kinds of domains, namely the ones with a core of C3S and a shell of CSH, and the ones made of CH only, characterized by a pseudo-spherical shape. To generate plausible cement micro-structures, we have simulated the alite hydration process with the open source package mic (Bishnoi and Scrivener, 2009a). In particular, we have applied the well-established model from Pignat et al. (2005) to a specimen filled with a 0.4 water/C3Smassratio,inlinewithcommon cement recipes, and considered initial particle size distributions (PSDs) for the alite powder corresponding to N0=102;103;104;105;106;107particles per ð100 mmÞ3, to unravel the interplay between properties and micro-structure. The implementation of this model in mic is well described in the doctoral thesis by Bishnoi (2008); for the reader’s convenience, we have reported a detailed description in the STAR methods section. Table 1 reports information concerning the final chemical composition, characterized by the volume fractions f, and the resulting particle size statistics, quantified by the expectation value mand the standard deviation sof the radii of the C3S cores, the CSH shells, and the CH particles; the definition of the radii is shown in Figure 3B. Figure 3 also shows three simulation snapshots for N0=105particles per ð100 mmÞ3, corresponding to the initial cement powder (C), the partially hydrated cement paste (D) and the fully hydrated cement paste (E). As expected, a larger value of N0leads to smaller final sub-domains. More importantly, the size of these sub-domains is comparable to the thermal radiation wavelengths. Therefore, micro-structural size effects are expected to impact the emissivity properties of the samples, hence providing a knob to tune the dielectric response. Finally, it can be seen from the values of fC3S(%5 %) that alite is consumed almost completely on full hydration and can be neglected in the homogenization step. Complex permittivity of the cement paste components To apply effective medium theory, the micro-structural information obtained above must be combined with the complex permittivity of the cement paste homogeneous components, namely C3S, CSH and CH. Because atomic vibrations are responsible for the dielectric response in the IR spectral range where the radiative cooler operates, we have resorted to molecular simulations to obtain this information. We have performed these simulations with the General Utility Lattice Program (GULP) (Gale, 1997) according to the force field method (Leach, 2001). Under this atomistic scheme, the interaction between atoms is described by parameterized interatomic potentials so that the system energy can be traced back from their positions. In particular, we have adopted a well-tested polarizable force field, which is known to describe Table 1. Initial number of particles per ð100 mmÞ3N0,volumefractionsfand statistics (expectation values mand standard deviations s) of the particle radii Rof the simulated samples after full hydration N0 ð100 mmÞ3 fC3S % fCSH % fCH % m½RC3S mm m½RCSH mm m½RCH mm s½RC3S mm s½RCSH mm s½RCH mm 1025 61 23 3.32 12.51 21.94 2.80 1.82 2.49 1035 62 24 0.71 5.20 11.33 1.34 1.89 2.00 1044 63 24 0.11 1.85 5.04 0.42 1.12 1.18 1054 63 24 0.01 0.58 2.27 0.12 0.52 0.58 1064 64 24 0.00 0.17 0.86 0.03 0.21 0.56 1072 66 25 0.00 0.10 0.36 0.01 0.09 0.28 The sum of the volume fractions is not unity because the remaining space consists of empty pores. The definition of the radii is depicted in Figure 3B. ll OPEN ACCESS 6iScience 25, 105320, November 18, 2022 iScienc e Article correctly the structure and elastic properties of most cementitious phases (Manzano et al., 2009). Later, we have calculated the complex permittivity of C3S, CSH and CH by following the method employed in (Dolado et al., 2020) for studying the response of cement-based materials in the THz regime. The corresponding absorption coefficient is reported in Figure 4A for all components. Details about molecular simulations and crystal structures can be found in the STAR methods section, whereas Figure S2 - drawn with VESTA (Momma and Izumi, 2011)-andTable S1 report information on the unit cells. Remarkably, the broad absorption maximum of CSH, which is the component with the highest volume fraction (see Table 1), overlaps significantly with the AW, as desired. At the same time, all the components exhibit significant absorption outside the AW. This is an advantage for solar cell applications, where the supraambient device operating temperature ensures that the radiative cooler is going to eject more energy than the one received from the atmosphere also outside of the AW, hence enhancing the cooling performance (Zhao et al., 2019). Finally, although the absorption coefficient (104cm1) is not as strong as the one of typical thin-film absorbers (106cm1), the possible realization of thick geometries still allows us to achieve high absorbance, hence removing this apparent weakness. Complex permittivity of the cement paste The ingredients required to estimate the complex permittivity of the generated cement pastes by a suitable effective medium theory are now available. Care should be taken because Bruggeman’s formula (Bruggeman, 1935), which is the most commonly used model for heterogeneous materials, does not account for micro-structure size effects. Indeed, it is derived in the long-wavelength (LW) limit, i.e.,under the assumption that the particles size is much smaller than the wavelengths of interest. However, we have seen above that the size of the cement sub-domains is comparable to the wavelengths around the AW. Therefore, a model capable of capturing the main effects of micro-structure on the dielectric properties is needed to properly estimate the effective complex permittivity of our cement pastes. Accordingly, we have used a recent model proposed by Slovick (2017) for a disordered ensemble of spherical particles with a space filling matrix of infinitesimally small spheres. This is representative of our cement paste, made of a high-density disordered network of CSH and CH spheres with a mean radius dependent on the alite initial PSD according to Table 1, separated by empty interstices (pores). According to Slovick’s model, the effective complex permittivity εeff of this system can be obtained by solving the equation: fCSH εCSH FðkCSH RCSHÞεeff εCSH FðkCSH RCSHÞ+2εeff +fCH εCH FðkCH RCHÞεeff εCH FðkCH RCHÞ+2εeff +fair 1εeff 1+2εeff =0 (Equation 3) where f,εand kare volume fraction, complex permittivity and complex wave vector, and FðxÞ=2ðsinx xcosxÞ=½xcosx+ðx21Þsinx(Belyaev and Tyurnev, 2018). It is worth noting that Equation 3 becomes A B Figure 4. Absorption spectra of the materials studied (A) Calculated absorption coefficient of the homogeneous components of cement pastes made by hydrating alite powder. (B) Effective absorption coefficient of the generated cement pastes as a function of the mean particle (sub-domain) radii ðRCSH;RCHÞ. ll OPEN ACCESS iScience 25, 105320, November 18, 2022 7 iScienc e Article Bruggeman’s formula in the LW limit becauseFðx/0Þ/1. Furthermore, by making no distinction between background medium and inclusions, the formula is valid for any volume fraction, contrarily to MaxwellGarnett formula, which is valid only for a low volume fraction of the inclusions (Markel, 2016). Figure 4B reports the obtained cement paste absorption coefficient as a function of the mean particle (subdomain) radii ðRCSH;RCHÞ. The spectra exhibit increased absorption at wavelengths comparable to the particles size, because of the multiple reflections occurring at the particle boundaries, leading to interference effects and enhanced absorption lengths, similarly to the case of planar layers. As the particles become smaller, the additional absorption band shifts toward the UV-visible range (shorter wavelength) and, eventually, size effects become negligible in the IR range, which can then be described in the LW limit (see the case ðRCSH;RCHÞ=ð0:1;0:4Þmm). The eventual additional emissivity in the UV-visible range is not going to impact the radiative cooler performance, because its radiated spectral power density is given by the product of its emissivity spectrum (%1) with the spectral power density radiated by a black-body (Balaji, 2014), which is negligible in the UV-visible spectrum at terrestrial temperatures. These results show that the absorption properties of the cement paste can be tuned by modifying its microstructure and tailored to the spectral requirements of radiative cooling. As a matter of fact, a similar approach has been applied to hierarchical porous polymers mentioned in the Introduction (Mandal et al., 2018). Although this is very promising, care should be taken. Indeed, modeling the cement paste micro-structure as an ensemble of air-embedded spherical inclusions is a geometrical approximation because these are eventually going to ‘‘collide’’ during growth and partially ‘‘deform’’. In addition, effective medium theories assume that all particles are subject to the same mean field. Deviations might occur close to the percolation threshold. These aspects should be investigated in future studies. These results are extremely encouraging, but not enough. Indeed, a larger absorption coefficient does not necessarily imply a stronger absorbance because it stems from a larger extinction coefficient that might also increase the layer reflectance at the same time. Therefore, more radiation could be lost by reflection before even entering the cement paste, hence leading to a reduced absorbance, as we shall see in a moment. Emissivity of the cement paste We have used the effective complex permittivity εeff calculated above to determine the spectral directional absorbance Acool U;lðl;qÞof layers made of the generated cements by the transfer-matrix method. We have considered cement slabs with a thickness of 100 mm. Indeed, based on the absorption coefficient values reported in Figure 4B, nanometer-scale thicknesses, although experimentally feasible (Rheinheimer and Casanova, 2012), are unsuitable to obtain large absorbance/emissivity because of the lack of absorption strength. This is clearly not an issue with cement-based solutions, for which very thick geometries are usually preferred. As a matter of fact, we are opting for a rather thin cement layer, which corresponds to a worst case scenario and makes our assessment stronger. Furthermore, this choice let us draw conclusions also about concrete, where aggregates are going to break the cement paste continuity and make the effective thickness smaller. Finally, this thickness has been experimentally achieved in the literature (Zhang et al., 2016). Implementation-wise, we have adopted a generalized TMM capable of describing incoherent propagation through layers having rough surfaces with Gaussian disorder (Katsidis and Siapkas, 2002;Centurioni, 2005). This model is representative of cement slabs, typically characterized by randomly disordered surfaces with roughness values from a few hundred nanometers to a few micrometers, depending on polishing (Apedo et al., 2015). This kind of rough surfaces can provide an advantage in terms of emissivity, because they have been shown to reduce reflection and increase absorption of an incoming electromagnetic wave by randomizing its direction and enlarging the effective absorption length (light-trapping) (Yablonovitch, 1982;Kowalczewski et al., 2012;Cappelluti et al., 2018). Our calculations are in agreement with these statements and show that a rougher surface slightly increases the absorbance of our cement slabs. Because of their supplementary nature, these results on the effect of surface roughness are shown in Figure S3 and only the worst case of a flat surface is considered in the main text. The STAR methods section contains a detailed description of the generalized TMM used. ll OPEN ACCESS 8iScience 25, 105320, November 18, 2022 iScienc e Article where LBB e;U;lis the spectral directional radiance of a black-body at temperature Twith an applied voltage V (Wu ¨rfel, 1982): LBB e;U;lðl;T;VÞ=2hc 2 l5 1 exphc =lqV kT 1 JMPP and VMPP are the solar cell electric current density and output voltage at maximum power point (MPP), respectively, calculated with the Shockley-Queisser model. l,U,h,c,kand qare wavelength, solid angle, Planck’s constant, speed of light in vacuum, Boltzmann’s constant and elementary charge, respectively. The other symbols have been defined in the main text. The reader is referred to the literature for more details concerning the definition of the radiometry quantities introduced above (Balaji, 2014). Alite hydration model We have simulated the alite hydration process depicted in Figure 3 by implementing the model from Pignat et al. (Pignat et al., 2005) into the open-source chemical kinetics package mic (Bishnoi and Scrivener, 2009a), as done by Bishnoi in his doctoral thesis (Bishnoi, 2008). We have considered a cubic specimen of 100 mm side length with periodic boundary conditions initially filled with a continuum of water and discrete C3S spherical particles in a water/C3S mass ratio of 0.4 (Bishnoi and Scrivener, 2009b), in line with common cement recipes. We have considered different initial particle size distributions (PSDs), which were provided with the software, corresponding to an initial number of alite particles N0within the 100 mm-side-length cube equal to 102,10 3,10 4,10 5,10 6and 107, so that we could investigate the role of micro-structure in thedeterminationoftheradiativecoolingproperties.Therateofthehydrationprocess,whichisdescribed by the volumetric formula given in Equation 2 under a mass density of 3:15 g cm3for C3S, 2:0gcm 3for CSH and 2:24 g cm3for CH, is controlled by the decrease in size of the C3S particles. In turn, this is related to the formation rate of CSH and CH. CSH shells form onto the C3S particles surface by the combination of three mechanisms, namely a nucleation and growth mechanism, a phase boundary mechanism, and a diffusion controlled mechanism. The corresponding equations are: dRC3S dt=3k1t2expk1t3 dRC3S dt=k2 dRC3S dt=k3 RCSH RC3S where k1=1:14 3104h3,k2equals the minimum of the right-hand-side of the first equation, and k3= 0:01 mm2h1(Pignat et al., 2005;Bishnoi, 2008). At the same time, new CH particles form at an exponentially decreasing nucleation rate in the interstitial regions of the hydrating cement paste according to the formula nðtÞ=nmax1expðatÞ where nmax is set to one-fifth of N0(Navi and Pignat, 1996)anda=0:213 h1(Jennings and Parrott, 1986). Their growth occurs randomly but constrained by the amount of product available according to the hydration reaction rate. Molecular simulations and crystal structures To obtain the IR dielectric properties of the cement paste components, we have performed molecular simulations with the General Utility Lattice Program (GULP) (Gale, 1997) according to the force field method (Leach, 2001), by adopting a well-tested polarizable force field, which is known to describe correctly the structure and elastic properties of most cementitious phases (Manzano et al., 2009). For the atomistic structure of the almost amorphous CSH, we have employed the model proposed in (Dolado et al., 2020)and(Duque Redondo, 2018), which corresponds to a very large system whose exact stoichiometry is ðCaOÞ254ðSiO2Þ152ðH2OÞ306. For the crystalline structures of C3S and CH, we have relaxed the experimental unit cells measured in (Mumme, 1995)and(Desgranges et al., 1993), respectively. The ll OPEN ACCESS iScience 25, 105320, November 18, 2022 15 iScienc e Article stoichiometry and the final simulation cell parameters are disclosed in Table S1. The unit cells drawn with the software VESTA (Momma and Izumi, 2011) are depicted in Figure S2. Later, we have calculated the complex permittivity of C3S, CSH and CH by following the method employed in (Dolado et al., 2020) for studying the response of cement-based materials in the THz regime. The method resorts to expressing the dielectric tensor in terms of the oscillator strengths of the vibrational modes as εijðuÞ=εijð+NÞ+4p VX m Um ij u2 mu2 where uis the angular frequency, Vis the unit cell volume, misthephononmoderank,andumare the mode-specific frequencies. The oscillator strength tensor for each vibrational mode mdepends on the Born effective charges qBand the eigenvector eij forthatmodeaccordingto Uab = X N i qB iajeij ffiffiffiffiffiffi mi p! X N i qB ibjeij ffiffiffiffiffiffi mi p! with midenoting the ion masses. In the practical computational implementation, we have used a small damping term dof 10 cm1by the substitution u2/uðu+idÞ. Finally, we have averaged the principal components of the dielectric tensor, i.e.,wehavetakenεðuÞ=P3 i=1εiðuÞ=3, in agreement with the disordered micro-structural nature of cements, which lifts off any preferred orientation. Generalized transfer-matrix-method We have used a generalized form of the transfer-matrix method capable of describing incoherent propagation and rough surfaces. In particular, the surface height variation is supposed to follow a Gaussian probability distribution with a given root-mean-square value (RMS) (Katsidis and Siapkas, 2002;Centurioni, 2005). The amplitude Eof the electric fields propagating from left to right ( +)andrighttoleft()ontheleft(L) and right (R) sides of the layer are related by the formula: 0 @E+ L2 EL21 A=ðDLPD RÞ0 @E+ R2 ER21 A where DLand DRdescribe the field propagation across the left and right interfaces between the layer and vacuum, respectively, and Pacross the layer. Their formulas are D=1 jtlrj21jrrlj2 jrlrj2jtlr trlj2jrlr rrlj2 P= exp ðidÞ20 0exp ðidÞ2! with rlr =rð0Þ lr exp"22pRMS Nl l2# tlr =tð0Þ lr exp"1 22pRMS l2 ðNrNlÞ2# d=2pNdcosq l The subscripts l and r denote left and right sides of the interface, while lr means from the left to the right side of the interface. A graphical representation of the quantities introduced can be found in Figure S5. ll OPEN ACCESS 16 iScience 25, 105320, November 18, 2022 iScienc e Article rð0Þ lr and tð0Þ lr are the Fresnel reflection and transmission coefficients of a smooth planar surface (for which we have used the average between the sand ppolarization values), Nis the complex refractive index, dis the layer thickness, and qis the refraction angle. The spectral directional reflectance, transmittance and absorbance are readily obtained, respectively, from the following formulas: RU;lðl;qÞ=ðDLPD RÞ21 ðDLPD RÞ11 TU;lðl;qÞ=1 ðDLPD RÞ11 AU;lðl;qÞ=1RU;lðl;qÞTU;lðl;qÞ The (hemispherical) spectral reflectance, transmittance and absorbance are simply the angular average of their spectral directional counterparts. For example, the spectral reflectance is calculated as follow for a system with azimuthal invariance such as the device depicted in Figure 2A: RlðlÞ=2Zp=2 0 dqRU;lðl;qÞcosðqÞsinðqÞ ll OPEN ACCESS iScience 25, 105320, November 18, 2022 17 iScienc e Article