Sreyash Sarkar
email: sreyash.sarkar@esiee.fr

Elyes Nefzaoui
email: elyes.nefzaoui@esiee.fr

Philippe Basset

Tarik Bourouina

Far-field Radiative Thermal Rectification with Bulk Materials

Keywords: radiative thermal rectification, bulk materials, far-field

In this paper, we explore the far-field radiative thermal rectification potential of common materials such as metals,ceramics and doped semi-conductors using radiative and thermo-radiative properties extracted from literature. Seventeen different materials are considered. The rectification coefficient is then calculated for 136 pairs of materials; each pair can be used for the two terminals of a radiative thermal diode. A thermal bias of 200 K is considered. The choice of materials and thermal bias value are only bound by data availability in literature. Obtained results, highlight new candidate materials for far-field radiative thermal rectification. They also highlight materials where thermal rectification is not negligible and should be considered with care in heat transfer calculations when considering systems subject to a comparable thermal bias and where these materials are used. Among the materials studied, undoped Indium Arsenide (InAs) shows great promise to be employed for thermal rectification, with a thermal rectification ratio reaching 96.35% in combination with other materials. Obtained results pave the way for an optimized design of thermal radiation control and management devices such as thermal diodes.

Introduction

Thermal rectification (TR) can be defined as an asymmetry in the heat flux when the temperature difference between two interacting thermal reservoirs is reversed. Therefore, a two-terminal thermal device exhibits thermal rectification if it transports heat in one direction with more ease than in the reverse direction. Thermal rectification has been a subject of intrigue since its underlying test perception in 1936 by Starr [START_REF] Starr | The copper oxide rectifier[END_REF] because of its aptitude to open up innovation in heat transport control, inspiring analogies to the tremendous advancement in the electronics industry following the invention of such nonlinear elements as the transistor and the diode.

Other potential applications of thermal rectification include thermal barrier coatings [START_REF] Miller | Thermal rectification by ballistic phonons in asymmetric nanostructures[END_REF], enhanced efficiency thermo-electric devices [START_REF] Saha | Monte carlo simulation of phonon backscattering in a nanowire[END_REF][START_REF] Benenti | From thermal rectifiers to thermoelectric devices[END_REF], temperature variation driven heat engines [START_REF] Sothmann | Rectification of thermal fluctuations in a chaotic cavity heat engine[END_REF] and radiative cooling [START_REF] Raman | Passive radiative cooling below ambient air temperature under direct sunlight[END_REF][START_REF] Kecebas | Passive radiative cooling design with broadband optical thin-film filters[END_REF]. From that point forward, several investigations have been performed to understand which systems can exhibit thermal rectification [START_REF] Roberts | A review of thermal rectification observations and models in solid materials[END_REF] and have introduced the concepts of numerous innovative devices like thermal transistors [START_REF] Ordonez-Miranda | Radiative thermal memristor[END_REF][START_REF] Ben-Abdallah | Near-field thermal transistor[END_REF][START_REF] Joulain | Modulation and amplification of radiative far field heat transfer: Towards a simple radiative thermal transistor[END_REF][START_REF] Joulain | Quantum thermal transistor[END_REF], thermal logic circuits [START_REF] Christoph | Scalable radiative thermal logic gates based on nanoparticle networks[END_REF][START_REF] Casati | Device physics: The heat is on-and off[END_REF][START_REF] Murad | A thermal logic device based on fluid-solid interfaces[END_REF][START_REF] Nefzaoui | Tunable radiative thermal rectifiers: Toward thermal logical circuits[END_REF][START_REF] Kathmann | Scalable radiative thermal logic gates based on nanoparticle networks[END_REF] and thermal diodes [START_REF] Martínez-Pérez | Rectification of electronic heat current by a hybrid thermal diode[END_REF][START_REF] Ben-Abdallah | Phase-change radiative thermal diode[END_REF][START_REF] Ordonez-Miranda | Photonic thermal diode based on superconductors[END_REF][START_REF] Ott | Radiative thermal diode driven by nonreciprocal surface waves[END_REF][START_REF] Ordonez-Miranda | Conductive thermal diode based on the thermal hysteresis of vo2 and nitinol[END_REF][START_REF] Chen | A photon thermal diode[END_REF].

Recent investigations of thermal rectification have covered different heat transfer(HT) modes including conduction [START_REF] Zhang | Thermal conduction and rectification in few-layer graphene y junctions[END_REF][START_REF] Avanessian | Adsorption-based thermal rectifier[END_REF], convection [START_REF] Reis | Convective thermal rectification in an air-filled parallelogramic cavity[END_REF] and radiation [START_REF] Rogers | Heat transfer at the interface of dissimilar metals[END_REF][START_REF] Clausing | Heat transfer at the interface of dissimilar metals-the influence of thermal strain[END_REF][START_REF] Lewis | Heat transfer at the interface of stainless steel and aluminum-the influence of surface conditions on the directional effect[END_REF][START_REF] Yang | Carbon nanocone: A promising thermal rectifier[END_REF][START_REF] Li | Thermal diode: Rectification of heat flux[END_REF][START_REF] Otey | Thermal rectification through vacuum[END_REF][START_REF] Francoeur | Electric and magnetic surface polariton mediated near-field radiative heat transfer between metamaterials made of silicon carbide particles[END_REF][START_REF] Iizuka | Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates[END_REF]. We describe the main current trends in the following paragraphs and more extensive reviews can be found in [START_REF] Roberts | A review of thermal rectification observations and models in solid materials[END_REF][START_REF] Ben-Abdallah | Contactless heat flux control with photonic devices[END_REF].

To achieve conductive thermal rectification, several mechanisms have been proposed including thermal potential barrier between material contacts [START_REF] Rogers | Heat transfer at the interface of dissimilar metals[END_REF], thermal strain/warping at interfaces of two materials [START_REF] Clausing | Heat transfer at the interface of dissimilar metals-the influence of thermal strain[END_REF], nanostructured geometric asymmetry [START_REF] Shen | High-performance noncontact thermal diode via asymmetric nanostructures[END_REF] and anharmonic lattices [START_REF] Li | Thermal diode: Rectification of heat flux[END_REF]. The asymmetry at the interface between two materials due to the difference of their thermal conductivity temperature dependence has also been shown to be the main driving mechanism for conductive thermal rectification by Marucha et. al [START_REF] Marucha | Heat flow rectification in inhomogeneous gaas[END_REF] and subsequently used in the models of Hoff et. al [START_REF] Hoff | Asymmetrical heat conduction in inhomogeneous materials[END_REF], Sun et. al [START_REF] Sun | Evaluation of thermal rectification at the interface of dissimilar solids by phonon heat transfer[END_REF] and in Go et. al [START_REF] Go | On the condition for thermal rectification using bulk materials[END_REF]. Based on the same principle, Hu et al [START_REF] Hu | Thermal conductivity and thermal rectification in graphene nanoribbons: a molecular dynamics study[END_REF] and Zhang et. al [START_REF] Zhang | Thermal conduction and rectification in few-layer graphene y junctions[END_REF] presented thermal rectifiers based on different thermal conductivities of dissimilar graphene nano-ribbons or Y junctions. At the micro and nano-scales, the asymmetry at the origin of TR can be widely tuned and engineered in nano-structured materials [START_REF] Yang | The effect of structural asymmetry on thermal rectification in nanostructures[END_REF] which led to many contributions using several phenomena observed in a large diversity of nano-structures such as ballistic phonons anisotropy at large thermal bias [START_REF] Miller | Thermal rectification by ballistic phonons in asymmetric nanostructures[END_REF], phonons lateral confinement [START_REF] Wang | Phonon lateral confinement enables thermal rectification in asymmetric singlematerial nanostructures[END_REF] and the temperature dependence of lattice vibrations density of states [START_REF] Lee | Thermal rectification in three-dimensional asymmetric nanostructure[END_REF]. Solid-state thermal rectifiers have also been proposed taking benefit of nonlinear atomic vibrations [START_REF] Li | Thermal diode: Rectification of heat flux[END_REF] [START_REF] Chang | Solid-state thermal rectifier[END_REF], nonlinear dispersion relations of the electron gas in metals [START_REF] Segal | Single mode heat rectifier: Controlling energy flow between electronic conductors[END_REF], and thermal streams through Josephson junctions [START_REF] Martínez-Pérez | Efficient phase-tunable Josephson thermal rectifier[END_REF] or metal-superconductor nano-intersections [START_REF] Giazotto | Thermal rectification of electrons in hybrid normal metal-superconductor nanojunctions[END_REF]. At even smaller scales, TR in quantum systems has recently lured researchers such as in the work of Landi et.al [START_REF] Landi | Flux rectification in the quantum x x z chain[END_REF] on TR using a quantum XXZ chain subject to an inhomogeneous field and the research of Chi et. al [START_REF] Landi | Flux rectification in the quantum x x z chain[END_REF] discussing TR in a system of a single level quantum-dot connected to ferromagnetic leads. On the other hand, Scheibner et al [START_REF] Scheibner | Quantum dot as thermal rectifier[END_REF] experimentally demonstrated a TR behavior in quantum dots subject to high in-plane magnetic fields [START_REF] Scheibner | Quantum dot as thermal rectifier[END_REF] and most recently in the work of Senior et. al [START_REF] Senior | Heat rectification via a superconducting artificial atom[END_REF], it has been shown that a superconducting quantum bit coupled to two superconducting resonators can achieve magnetic flux-tunable photonic heat rectification between 0 and 10%.

Radiative heat transfer is another heat transfer mode that has been actively investigated for thermal rectification application in the past decade after the seminal works of Ruokola et. al [START_REF] Ruokola | Thermal rectification in nonlinear quantum circuits[END_REF] and Otey [START_REF] Otey | Thermal rectification through vacuum[END_REF]. Two main paths have been considered. On one hand, near field (NF) Radiative Thermal Rectification (RTR) between materials supporting resonating surface waves separated by wavelength scale or sub-wavelength gaps. Plasmonic materials supporting surface plasmon polaritons and dieletrics supporting surface phonon polaritons have been mainly considered. Used materials for NF RTR include gold and silicon [START_REF] Wang | Thermal rectification enabled by near-field radiative heat transfer between intrinsic silicon and a dissimilar material[END_REF][START_REF] Wen | Ultrahigh thermal rectification based on near-field thermal radiation between dissimilar nanoparticles[END_REF] with temperatures between 300K and 600K, doped silicon [START_REF] Basu | Near-field radiative transfer based thermal rectification using doped silicon[END_REF], SiC [START_REF] Otey | Thermal rectification through vacuum[END_REF][START_REF] Francoeur | Electric and magnetic surface polariton mediated near-field radiative heat transfer between metamaterials made of silicon carbide particles[END_REF][START_REF] Zhu | Ultrahigh-contrast and large-bandwidth thermal rectification in near-field electromagnetic thermal transfer between nanoparticles[END_REF], SiC and SiO 2 [START_REF] Joulain | Radiative thermal rectification between SiC and SiO 2[END_REF][START_REF] Iizuka | Rectification of evanescent heat transfer between dielectric-coated and uncoated silicon carbide plates[END_REF] , InSb and graphene-coated SiO2 [START_REF] Xu | Highly efficient near-field thermal rectification between insb and graphene-coated sio2[END_REF]. Many works on NF RTR also used phase change materials and mainly Vanadium Dioxyde (V O 2 ) [START_REF] Ghanekar | High-rectification near-field thermal diode using phase change periodic nanostructure[END_REF][START_REF] Zheng | Graphene-assisted near-field radiative thermal rectifier based on phase transition of vanadium dioxide (vo2)[END_REF][START_REF] Fiorino | A thermal diode based on nanoscale thermal radiation[END_REF][START_REF] Gu | Thermal switch and thermal rectification enabled by near-field radiative heat transfer between three slabs[END_REF], and more rarely, thermochromic materials [START_REF] Huang | Thermal rectification based on thermochromic materials[END_REF]. In addition to the diversity of used materials, a large variety of geometries and sizes has been explored as in the work of Shen et. al [START_REF] Shen | High-performance noncontact thermal diode via asymmetric nanostructures[END_REF], which elucidated non-contact thermal diodes using asymmetric nanostructures while [START_REF] St-Gelais | Demonstration of strong near-field radiative heat transfer between integrated nanostructures[END_REF][START_REF] Shen | Near-field thermal radiation between nanostructures of natural anisotropic material[END_REF] reported on Near field RTR in the deep sub-wavelength regime between planar surfaces separated by nanometre-scale distances. Zhou et. al [START_REF] Zhou | Radiation-based near-field thermal rectification via asymmetric nanostructures of the single material[END_REF] proposed for instance a thermal diode using a nanoporous plate and a plate, both made of silicon carbide and where the two terminals are separated by a nanometric vacuum gap while Ott et. al [START_REF] Ott | Radiative thermal diode driven by nonreciprocal surface waves[END_REF] explored a radiative thermal diode made of two nanoparticles coupled with the nonreciprocal surface modes of a magneto-optical substrate.

On the other hand, far-field (FF) radiative heat transfer also enables RTR and has been an active area of interest in the past decade. Compared to NF RTR, FF RTR provides the advantage of simplicity and ease of fabrication since it does not require the combination and alignment of objects separated by micro-metric or nano-metric gaps. Two main solutions have been reported for this purpose : the use of phase change materials (PCM) [START_REF] Joulain | Modulation and amplification of radiative far field heat transfer: Towards a simple radiative thermal transistor[END_REF][START_REF] Nefzaoui | Radiative thermal rectification using superconducting materials[END_REF][START_REF] Ordonez-Miranda | Photonic thermal diode based on superconductors[END_REF] and tunable metamaterials often used as selective radiation emitters and absorbers [START_REF] Nefzaoui | Simple far-field radiative thermal rectifier using Fabry-Perot cavities based infrared selective emitters[END_REF].

The use of phase change materials, and V O 2 in particular, is predominant in both conceptual [START_REF] Audhkhasi | Design of far-field thermal rectifiers using gold-vanadium dioxide micro-gratings[END_REF] and experimental works [START_REF] Ito | Experimental investigation of radiative thermal rectifier using vanadium dioxide[END_REF][START_REF] Jia | Far-field radiative thermal rectifier based on nanostructures with vanadium dioxide[END_REF]. Recently, Ghanekar et. al [START_REF] Ghanekar | High contrast far-field radiative thermal diode[END_REF] proposed the concept of a far-field radiative thermal rectifier combining both PCM, V O 2 for instance, and a Fabry-Perot cavity based meta-material.

Beyond thermal rectification, various radiative heat transfer control devices have been proposed such as thermal transistors [START_REF] Prod'homme | Dynamical behaviour of a far-field radiative thermal transistor[END_REF][START_REF] Ordonez-Miranda | Radiative thermal memristor[END_REF][START_REF] Ben-Abdallah | Near-field thermal transistor[END_REF][START_REF] Joulain | Modulation and amplification of radiative far field heat transfer: Towards a simple radiative thermal transistor[END_REF][START_REF] Joulain | Quantum thermal transistor[END_REF] and photonic thermal memristors [START_REF] Ordonez-Miranda | Radiative thermal memristor[END_REF] . The large majority of those PCM based far-field radiation control devices use V O 2 since the seminal work of Van Zwol et. al [START_REF] Van Zwol | Phonon polaritons enhance near-field thermal transfer across the phase transition of vo 2[END_REF] who provided a comprehensive characterization of V O 2 radiative and thermo-radiative properties (TRP) around its metal-insulator transition temperature. Surprisingly, the good performances of V O 2 for RTR seem to have prevented authors from exploring other materials, contrarily to the what have been done and reported for NF RTR. Indeed, the large variety of existing materials have not been systematically considered to assess their potential use in FF RTR. This can be legitimately explained by the scarcity of literature on materials' radiative properties in the wavelength range of interest for thermal radiation and their subsequent temperature dependence. However, a thorough investigation of commonly available materials performances with respect to thermal rectification has not been realized up until now.

In the present work, we present an evaluation of RTR potential of different materials commonly used in micro-fabricated devices such as Indium Arsenide, Gallium Arsenide, Gallium Antimonide, Germanium, Zinc Sulphide, Silicon, metals such as Platinum,Copper, Nickel, Gold, refractory metal such as Tungsten and ceramics like Titanium Nitride, Ruby, Corundum and rare-earth material such as Neodymium Gallate, based on radiative and thermo-radiative properties available in literature. The paper is organized as follows: in the present section, we introduce the concept of thermal rectification along with a brief review of the works accomplished so far and delineate the pertinence of the present work; next in section 2 , we explain the principle of radiative thermal rectification and introduce the main quantities that govern and characterize a radiative thermal rectification behavior. We also, describe the used radiative and thermo-radiative properties of the considered materials and the process of relevant data extraction from literature. Finally, we report and discuss the main results on the radiative thermal rectification potential of the considered materials in section 3.

Method

Theory

Variation of the heat flux magnitude when the sign of the temperature gradient between two points is reversed is the basic definition of thermal rectification. Let us consider two thermal reservoirs 1 and 2 at two different temperature T 1 and T 2 , respectively, with T 2 > T 1 . The temperature difference between the two reservoirs is noted ∆T = T 2 -T 1 . This temperature difference results in heat flux Φ F . We will refer to this initial configuration as forward bias configuration. If we invert the temperature difference between the two reservoirs, i.e if we set reservoir 1 temperature to T 2 and reservoir 2 temperature to T 1 , a reverse bias flux Φ R is observed. Thermal rectification occurs when the heat fluxes in forward Φ F and reverse bias Φ R , under the same magnitude of the thermal potential difference, i.e for a constant |∆T | but with opposite ∆T signs, are unequal. Thermal rectification is generally quantified, even though it is not the only indicator encountered in the literature, by means of the normalized rectification coefficient [START_REF] Starr | The copper oxide rectifier[END_REF][START_REF] Roberts | A review of thermal rectification observations and models in solid materials[END_REF] which can be defined as:

R = |Φ F -Φ R | max(Φ F , Φ R ) (1)
where, Φ F and Φ R denote the heat flux under forward and backward bias, respectively. In this paper, the definition of normalized rectification coefficient given in equation 1 is preferred because it provides bound rectification coefficient values between zero and one, as opposed to alternative definitions adopted by some authors [START_REF] Sawaki | Thermal rectification in bulk materials with asymmetric shape[END_REF][START_REF] Giazotto | Thermal rectification of electrons in hybrid normal metal-superconductor nanojunctions[END_REF] which lead to values from zero to infinity. A schematic of two thermal reservoirs exchanging heat flux in forward and reverse bias configurations is given in 1.

In this paper, we consider the simple case of far-field radiative thermal rectification between two semi-infinite planes, separated by a vacuum gap of thickness d, characterized by their radiative optical properties (their temperature dependent emissivities and reflectivities). Therefore, the distance separating the two planes exchanging heat by radiation is much larger than Thermal Bath 2

T2

Thermal Bath 1 the characteristic wavelength of thermal radiation at the considered bodies temperatures given by Wien's law : d>λ W (T ), where λ W (T ) = hc 2 5k B T , where h, c, k B and T are Planck's constant, the speed of light in vacuum, Boltzmann constant and the absolute temperature respectively. For simplicity's sake, we also assume they are both Lambertian sources, meaning that their emissivities and reflectivities ( (λ, T ) and ρ(λ, T ) respectively) are direction independent [START_REF] Modest | Radiative heat transfer[END_REF]. Considering these assumptions, the net radiative heat flux (RHF) exchanged by the two bodies resumes then to the far field contribution, which can be written as [START_REF] Joulain | Radiative thermal rectification between SiC and SiO 2[END_REF]:

T1 Φ F d Thermal Bath 2 Thermal Bath 1 Φ R d (a) (b) T 2 >T 1 T2 T1
Φ(T 1 , T 2 ) = π ∞ λ=0 [I 0 (λ, T 1 )I 0 (λ, T 2 )]τ (λ, T 1 , T 2 )dλ (2)
where,

I 0 (λ, T ) = hc 2 λ 5 1 e hc k B T -1 (3)
is the black body intensity at a temperature T and wavelength λ, where h, c, k B and T are Planck's constant, the speed of light in vacuum, Boltzmann constant and the absolute temperature respectively [START_REF] Bose | Planck's law and light quantum hypothesis[END_REF] and

τ (λ, T 1 , T 2 ) = (λ, T 1 ) (λ, T 2 ) 1 -ρ(λ, T 1 )ρ(λ, T 2 ) (4)
is the monochromatic RHF density transmission coefficient between 1 and 2, where (λ, T ) and ρ(λ, T ) is the monochromatic emissivity and reflectivity at a given temperature respectively. The RHF density transmission coefficient is governed by the radiative properties of the two considered thermal baths, in particular their emissivities and reflectivities and the temperature dependence of these properties. These properties are completely governed by the considered bodies dielectric functions and geometries. In the case of opaque bodies, energy conservation [START_REF] Modest | Radiative heat transfer[END_REF] and Kirchhoff's laws [START_REF] Modest | Radiative heat transfer[END_REF] combination leads to the following relation between the monochromatic emissivity, reflectivity and transmittivity at a given temperature:

(λ, T ) = 1 -ρ(λ, T ) -t(λ, T ) (5)
Therefore, to evaluate the thermal rectification potential of a given pair of materials, we need to know the emissivity, reflectivity and transmittivity of each terminal of the considered thermal diode and its temperature dependence.

Since the emissivity and reflectivity depend on the dielectric function, knowing the dielectric permittivity or the complex refractive index of the considered materials is also sufficient to estimate their potential for thermal rectification.

In our case, these data have been harvested from literature. [START_REF] Palik | Handbook of Optical Constants of Solids, Five-Volume Set: Handbook of Thermo-Optic Coefficients of Optical Materials with Applications[END_REF] and that of Weber [START_REF] Weber | Handbook of optical materials[END_REF], provide an exhaustive collection of material optical properties, unfortunately, available data is not temperature dependent.

Radiative properties data extraction

Radiative properties of common materials at different temperatures, in the required infrared range, are scantily available in literature. Incidentally, the majority of relevant data found, regarding the works dedicated to this sub-area of research, are pertaining to the study of semiconductors. The optical properties of metals such as, Gold [START_REF] Beran | The reflectance behaviour of gold at temperatures up to 500°C[END_REF][START_REF] Aksyutov | Temperature dependence of the optical constants of tungsten and gold[END_REF], Aluminum [START_REF] Ujihara | Reflectivity of metals at high temperatures[END_REF][START_REF] De | Temperature and thickness dependence of the resistivity of thin polycrystalline aluminium, cobalt, nickel, palladium, silver and gold films[END_REF], Tungsten [START_REF] Aksyutov | Temperature dependence of the optical constants of tungsten and gold[END_REF],

Molybdenum [START_REF] Veal | Optical properties of molybdenum. i. experiment and kramers-kronig analysis[END_REF], Silver [START_REF] Ehrenreich | Optical properties of ag and cu[END_REF], Copper [START_REF] Setién-Fernández | Spectral emissivity of copper and nickel in the mid-infrared range between 250 and 900 c[END_REF], Nickel [START_REF] Setién-Fernández | Spectral emissivity of copper and nickel in the mid-infrared range between 250 and 900 c[END_REF] have been studied, but unfortunately researchers have not been particularly interested in exploring the aforesaid optical properties at higher temperatures and at larger wavelengths, adherent to the IR range, which is an imperative in this study. Materials like Si [START_REF] Francoeur | Electric and magnetic surface polariton mediated near-field radiative heat transfer between metamaterials made of silicon carbide particles[END_REF] and Ge [START_REF] Esaki | Properties of thermally treated germanium[END_REF] have been considered broadly, beginning in the 1950's.

GaAs, InAs, InP, and GaSb make a case for volumes of research of their own and have been utilized as a part of light emanating and optoelectronic devices [START_REF] Harris | Optical properties of Si, Ge, GaAs, GaSb, InAs, and InP at elevated temperatures[END_REF]. Material properties were learned at temperatures extending from low temperatures (liquid helium, 4.2 K, or liquid nitrogen, 77 K) to about room temperature, 298 K [START_REF] Harris | Optical properties of Si, Ge, GaAs, GaSb, InAs, and InP at elevated temperatures[END_REF]. The 2010 work of Thomas R. Harris [START_REF] Harris | Optical properties of Si, Ge, GaAs, GaSb, InAs, and InP at elevated temperatures[END_REF] provides valuable experimental data on the study of optical properties of Germanium, Gallium and Indium derivatives and on Bulk Silicon, at various temperatures.

Optical properties of Silicon Carbide [START_REF] Ng | Temperature-dependent reflectivity of silicon carbide[END_REF][START_REF] Pitman | Optical properties and applications of silicon carbide in astrophysics[END_REF], and on Zinc Sulphide [START_REF] Harris | Thermal, structural, and optical properties of Cleartran® multispectral zinc sulfide[END_REF] have also been reported in the past decades. The use of numerical simulation data for bulk silicon generated using a Drude Model was also suggested in [START_REF] Chen | Heavily doped silicon complex gratings as wavelength-selective absorbing surfaces[END_REF][START_REF] Basu | Near-field radiative heat transfer between doped silicon nanowire arrays[END_REF],

and we took use of this model to produce temperature dependent radiative properties data for bulk silicon. The list of materials considered in the present study is indicated in Table .2.

In the work of Thomas R. Harris [START_REF] Harris | Optical properties of Si, Ge, GaAs, GaSb, InAs, and InP at elevated temperatures[END_REF], temperature dependent transmission measurements were carried out for Si, Ge, GaAs, GaSb, InAs, and InP from 0.6 to 25 µm at temperatures ranging from 295 up to 900 K. Band gap shifts were observed as temperature changed and were compared to previous works. General agreement was observed in the trend of the change in the band gap with temperature, however, the actual band gap energy values deviated from the expected-on average by about 10 %. The reflectivity maximum increased in magnitude with increasing temperature, with successful measurements being done up to 517 K [START_REF] Harris | Optical properties of Si, Ge, GaAs, GaSb, InAs, and InP at elevated temperatures[END_REF].

To completely gather temperature dependent reflectivity data of GaAs, GaSb, InAs, Ge another set of works were adhered to such as the work of Skauli et. al [START_REF] Skauli | Improved dispersion relations for gaas and applications to nonlinear optics[END_REF], where the refractive index of GaAs has been measured. From 1979-1984, H. H Li [START_REF] Li | Refractive index of silicon and germanium and its wavelength and temperature derivatives[END_REF][START_REF] Li | Refractive index of zns, znse, and znte and its wavelength and temperature derivatives[END_REF] prepared a comprehensive report on the temperature dependence of the refractive indices of Si, Ge, ZnS, ZnSe, ZnTe etc., where he provided near non-existent temperature-dependent data on these particular semi-conductors refractive index in a wide spectral range including mid and far infra-red.

For SiC in the IR wavelength range from 1 µm to 25 µm, many data are available in literature and because much of SiC optical properties depend on its crystallographic orientation [START_REF] Spitzer | Infrared properties of hexagonal silicon carbide[END_REF][START_REF] Spitzer | Infrared properties of cubic silicon carbide films[END_REF], they have been compared to other semiconductor relevant data with care. Some widely cited sources are that of Bohren et. al [START_REF] Bohren | Absorption and scattering of light by small particles[END_REF], Mutschke et al. [START_REF] Mutschke | Infrared properties of SiC particles[END_REF][START_REF] Clément | New laboratory spectra of isolated -SiC nanoparticles: Comparison with spectra taken by the Infrared Space Observatory[END_REF], the measurements of Daniel Ng [START_REF] Ng | Temperature-dependent reflectivity of silicon carbide[END_REF] and Joulain et. al [START_REF] Joulain | Radiative thermal rectification between SiC and SiO 2[END_REF]. Cagran et. al [START_REF] Cagran | Temperature-resolved infrared spectral emissivity of sic and pt-10rh for temperatures up to 900°c[END_REF] provided a comprehensive study on the temperature-resolved infrared spectral emissivity of SiC and this data was finally employed in the present study due to its reportage of wide and flexible temperature dependence of emissivity.

On the other hand, the measurements on Zinc Sulphide (ZnS) reported in [START_REF] Harris | Thermal, structural, and optical properties of Cleartran® multispectral zinc sulfide[END_REF][START_REF] Leviton | Temperature-dependent refractive index of cleartran zns to cryogenic temperatures[END_REF][START_REF] Hawkins | The temperature-dependent spectral properties of filter substrate materials in the far-infrared (6-40 µm)[END_REF] provide valuable insight into the thermal, structural and optical properties of a commercially available sample of multispectral ZnS. The hemispherical transmission, reflectivity, absorptivity, emissivity of unpolarized light at normal incidence on ZnS at 24 °C, 100 °C and 200 °C are reported in [START_REF] Harris | Thermal, structural, and optical properties of Cleartran® multispectral zinc sulfide[END_REF].

While the work of Bauer et. al [START_REF] Bauer | Emissivities of ceramic materials for for high temperature processes[END_REF], depicts the spectral emissivities of various ceramic materials at different temperatures in the spectral range from 0.8 to 25 µm, the work of Meneses et al. [START_REF] Meneses | Apparatus for measuring the emittance of materials from far infrared to visible wavelengths in extreme conditions of temperature[END_REF], provides the temperature dependent emissivities of Ruby and a rare earth material, Neodymium Gallate(N dGaO 3 ).

Results and Discussion

A simple Matlab program based on equations 1 to 5, was implemented to quantify the thermal rectification coefficient for different pairs of materials, which takes the emissivity data-set of two different materials and their temperature as input and calculates the forward and reverse bias fluxes, thereby finally giving the value of RTR coefficient as defined in equation 1.

Radiative optical properties of the materials under consideration were studied at a temperature difference of 200 K, with T 1 =300K and T 2 =500K while the spectral range taken into account is 1 -25 µm. This temperature range was considered under the constraints of data availability.

Table.2 shows the list of materials considered in this study, divided into subcategories: metals, semiconductors, ceramics and rare-earth materials for a classified approach. The respective unequal RHF density transmission coefficients in forward and reverse directions for considered pairs of materials are illustrated in Fig. 3, while Table 4 In the following paragraphs, we discuss the performances of the different considered materials subcategories, i.e semiconductors, metals and ceramic materials with respect to RTR.

Among the semiconductors, InAs exhibits the highest levels of RTR, in combination with most considered materials. This is due to the temperature dependence of its emissivity. As depicted in Fig. 2(b), the emissivity increases with increasing temperature and the material becomes more uniformly emissive, contrary to its behaviour at 300 K, where it is more reflective from 1-5µm, than other wavelength ranges. Consequently, the radiative heat transfer is lower in this spectral range at low temperatures. Its large emissivity increase when temperature increases directly contributes to material combinations with InAs exhibiting higher RTR. The emissivity of InAs has been shown

to strongly depend on its plasmonic properties [START_REF] Park | Dynamic thermal emission control with inasbased plasmonic metasurfaces[END_REF]. On the other hand, it has also been shown that the plasmon energy, i.e the plasma frequency, of InAs significantly decreases with increasing temperature [START_REF] Bell | Plasmon excitations and accumulation layers in heavily doped inas (001)[END_REF]. This significant temperature dependence of the plasma frequency in InAs is probably behind the emissivity variation of InAs when temperature increases, hence the large RTR obtained in combination with other materials. For SiC, we observe in Fig. 2(e) a slight shift and broadening of the resonance peak of emissivity at around 12µm when temperature increases. This is due to anharmonicity effects responsible for the phonon-phonon interactions increase with the temperature [START_REF] Sham | The electron-phonon interaction[END_REF]. This broadening of the emissivity peak of SiC has certainly an impact on the exchanged radiative heat transfer. However, it is negligible compared to the effect of InAs emissivity temperature dependence when we consider an SiC-InAs based rectifier. For other semi-conductors like Ge, GaSb and GaAs, which do not exhibit a significant emissivity variation as a function of temperature, no marked variation in the radiative heat transfer is observed with these materials when they are considered for RTR.

Among the metals, Ni shows non-negligible RTR of 60% in most material combinations as shown in Table .4. The level of emissivity of Ni as depicted in the work of de Arrieta et al [START_REF] De Arrieta | Evolution of the infrared emissivity of ni during thermal oxidation until oxide layer opacity[END_REF], increases with increasing temperature. For most part of the IR wavelengths, Ni remains highly reflective.

The curves show the usual behaviour for metallic emissivity, decreasing with wavelength and increasing with temperature. Besides, it can be seen that the temperature dependence is greater in the ferromagnetic phase,(T<354°C) than in the para-magnetic one. This marked difference in its emissivity behaviours at higher temperatures enables Ni in combination with other materials to yield RTR of 60%. The other metals like Pt [START_REF] Orosco | Temperature-dependent infrared optical and radiative properties of platinum[END_REF], Cu [START_REF] Setién-Fernández | Spectral emissivity of copper and nickel in the mid-infrared range between 250 and 900 c[END_REF] and the refractory metal W [START_REF] Aksyutov | Normal spectral emissivity of gold, platinum, and tungsten[END_REF] exhibit negligible variation of their emissivity as a function of temperature, though showing the same as any metallic emissivity behaviour, elucidating the reason behind such low RTR(%)<5% with most material combinations. This is however not the case for Au [START_REF] Aksyutov | Normal spectral emissivity of gold, platinum, and tungsten[END_REF], where the temperature rise decreases the already low emissivity of the material significantly, making it highly reflective. This is probably why most material combinations with Au have at least RTR(%)>30%.

Among the ceramic materials, Al 2 O 3 is highly emissive from 6-11µm

and then the emissivity drops to lower levels and also has multiple resonance peaks above 11µm, which broaden as temperatures increase [START_REF] Bauer | Emissivities of ceramic materials for for high temperature processes[END_REF]. This behaviour is not observed in either Ruby [START_REF] Meneses | Apparatus for measuring the emittance of materials from far infrared to visible wavelengths in extreme conditions of temperature[END_REF] or TiN [START_REF] Briggs | Temperaturedependent optical properties of titanium nitride[END_REF], where there is almost no difference in emissivity as temperatures increase.

Taking the overall average for each subcategory of materials, the semiconductors fair the best in terms of RTR(%), closely followed by metals, ceramic materials and rare-earth material. On the other hand, to achieve non-negligible RTR between a pair of materials, at least one of them should exhibit a significant variation of its emissivity as a function of temperature.

Due to constraints of data availability on the materials' thermoradiative properties, a thermal bias |∆T = 200K| was chosen which is the most common temperature difference that we found in literature. To enable the generalization of obtained results to other magnitudes of the thermal bias, normalized RTR coefficients were also calculated for all the considered materials.

We show in materials which exhibit a significant change of their radiative properties with a very small temperature variation around their critical temperature which would lead to an normalized RTR.

In Table 6, we present the Temperature Coefficient of total Emissivity (TCE) that can be defined by d t dT , where, t is a spectral average with the spectral emissive power as a weighting factor and can be defined as [START_REF] Modest | Radiative heat transfer[END_REF]:

t = 1 n 2 σT 4 ∞ 0 λ (T, λ)E bλ (T, λ)dλ (6)
where, σ is the Stefan-Boltzmann constant, T is the absolute temperature, λ (T, λ) is the spectral hemispherical emissivity and E bλ (T, λ) is the blackbody spectral emissive power. This quantity(TCE) has been defined in a similar manner to the refractive index temperature coefficient, dn dT commonly used in optics [START_REF] Bertolotti | Temperature dependence of the refractive index in semiconductors[END_REF]. Here due to data availability constraints, a thermal bias of 200 K was considered. TCE can be used as a key indicator to identify the best candidate materials for thermal rectification since a large variation of radiative properties with respect to temperature is required to obtain a large thermal rectification coefficient. As can be observed in Table 6, the greater the TCE, the more susceptible the materials are for thermal rectification, thus reaffirming the best case of a far field radiative thermal rectifier composed of InAs and SiC.

Results reported in the present manuscript could thus serve as a basic guiding step for researchers and engineers interested in the design of radiative thermal rectifiers for the choice of best candidate materials.

Conclusion

We reported in this paper on the thermal rectification potential of 136 pairs of materials commonly used in several applications such as the microelectronics industry. At first, the theoretical concept of thermal rectification is introduced and the key quantities are defined. Then, used radiative properties, a key parameter for RTR coefficients calculation, are presented. Finally, results of the RTR coefficient of 136 pairs of materials are reported.Obtained results depict that semiconductors show larger RTR coefficient than metals which are in turn better than ceramic materials, due to a non-negligible temperature dependence of the plasmonic properties in such materials which induces a significant emissivity variation as a function of temperature. In particular, InAs-Ruby along with InAs-TiN,InAs-W, InAS-SiC can be very good candidates for a far-field radiative thermal rectifier when combined together with a rectification ratio up to 96.35% for a thermal bias of 200 K. We also show that several pairs of materials provide a relatively high rectification coefficient, larger than 50% . This suggests that they can also be used for thermal rectifiers. A corollary of this result is that non negligible thermal rectification occurs when these pairs of materials are combined in a system with a thermal bias of the order of hundreds of kelvins. Taking into account the temperature dependence of the materials radiative properties is therefore mandatory for an accurate calculation of the exchanged heat flux between these materials in such systems. Presented results may be helpful for thermal management applications and in the advancement of the research and engineering of thermal rectifiers, thermal logical circuits and in thermal energy harvesting.

Supplementary Materials II: Mapping RTR coefficient of considered materials As depicted in Table 4, RTR coefficients for different combinations of considered materials, belong to different ranges from 0-91%. In an attempt to streamline and to make each material combination identifiable for an appropriate RTR coefficient, majority of RTR coefficients are mapped in

Figure 1 :

 1 Figure 1: (a)Forward and (b) Reverse bias conditions of HT between two semi-infinite planes acting as thermal baths separated by a vacuum gap of thickness, d. Thermal rectification occurs only when, Φ F and Φ R denoting the heat flux in the forward and reverse operating modes under two temperatures (T 2 >T 1 ), are unequal. The gap width (d) is assumed to be much larger than the dominant thermal radiation wavelength (Wien wavelength, λ W (T )).

  Thermo-radiative properties, i.e materials radiative properties and their temperature dependence, are the main input data required to assess the radiative thermal rectification ability of a given material. Difficulty to obtain such data, especially in the wavelength range of thermal radiation from room temperature to a few hundred Kelvins above room temperature, i.e. in the mid and far-infrared, perhaps explains the dearth of literature on thermal rectification with the large variety of existing materials and the strong concentration of literature on VO 2 . Although the most important material libraries in this regard such as The Handbook of Optical Materials by Edward D. Palik

Figure 2 :

 2 Figure 2: Emissivity of a few common materials found in literature-(a) GaSb (b) InAs (c) GaAs (d) Ge (e) SiC (f) ZnS at T 1 =300K and T 2 =500K.

Figure 2 :

 2 Figure 2: Emissivity of a few common materials found in literature-(a) GaSb (b) InAs (c) GaAs (d) Ge (e) SiC (f) ZnS at T 1 =300K and T 2 =500K.

Figure 3 :

 3 Figure 3: Obtained RHF density transmission coefficients in forward and reverse bias configurations between two materials-(a) SiC/DBuSi(1 × 10 20 cm -3 ) ; (b) GaAs/GaSb; (c) SiC/GaSb; (d) SiC/Au; (e) InAs/Cu (f) InAs/SiC at T 1 =300K and T 2 =500K.

0 50 100 Figure 4 :

 1004 Figure 4: RTR(%) coefficients of thermal rectifiers made of pairs of considered materials for a thermal bias of 200 K.

Figure 5 :

 5 Figure 5: Normalized RTR(%) coefficient i.e RTR coefficients of thermal rectifiers made of pairs of considered materials normalized by the thermal bias.

Figure 6 :

 6 Figure 6: TCE(%) of thermal rectifiers made of pairs of considered materials.

Figure . 7 :

 7 Figure .7: Mapping RTR coefficients of thermal rectifiers made of pairs of considered materials for a thermal bias of 200 K based on the table in Fig.4

Figure. 7

 7 Figure.7 in different ranges namely: 0-5%, 5-5.5%, 5.5-15.2%, 15.2-16%, 16-20%, 20-40%, 40-50%, 50-100%. These ranges have been ascertained relying on the overall data range in Table 4. Each bar height represents the value of the RTR coefficient and the bar width represents the distribution of entries for each data range. The bar color map corresponds to the amplitude of RTR i.e the lowest values starting from green to red being the highest value obtained. As seen in Figure.7, there are highest no of entries in the range from 16-20% and 0-5%. There are 7 material combinations in the range of 50-100%, which suggests that these combinations have non negligible thermal rectification.

Table 1 :

 1 Considered Materials

	Materials	Symbols
	Semiconductors
	Germanium[89, 95]	Ge
	Gallium Derivatives[89, 96]	GaAs,GaSb
	Undoped Indium Arsenide[89, 97, 98]	InAs
	Zinc Sulphide [92, 99, 100, 101]	ZnS
	Silicon Carbide[57, 90]	SiC
	Bulk Doped N-type Silicon(1 × 10 20 cm -3 )	DBuSi 1
	Bulk Doped N-type Silicon(3 × 10 20 cm -3 )	DBuSi 3
	Metals
	Gold [81, 82]	Au
	Copper[87]	Cu
	Platinum[102]	Pt
	Nickel[103]	Ni
	Tungsten[104]	W
	Ceramic Materials
	Titanium Nitride[105]	TiN
	Corundum[106]	Al 2 O 3
	Ruby[107]	Al 2 O 3 : Cr
	Rare Earth Materials
	Neodymium Gallate[107]	N dGaO 3
	transmission measurements for Ge were taken up to approximately 650 K. The
	data was taken in near IR and mid IR wavelength ranges. Comparison with
	older literature[108] shows good agreement, indicating that the measured data
	are reliable. Similarly, temperature dependent transmission measurements were
	taken up to approximately 850 K for undoped GaAs and up to approximately
	550 K for n-type GaAs doped with silicon. In the case for undoped GaSb,
	measurements were taken up to 650 K. Unfortunately, for sulphur doped

  Al 2 O 3 : Cr), as shown in Table.4. The highest value of RTR with materials other than InAs-Al 2 O 3 : Cr, can invariably be attributed to InAs-TiN which records a value of 95.54%. As can be seen in Table.4, all materials in combination with InAs record more than 50% RTR except Au, which has an

	summarises the obtained RTR coefficient
	for all possible pairs of materials with 136 pairs of materials considered in this
	study.
	Among all materials under consideration, Indium Arsenide (InAs)
	shows the largest values of RTR coefficient, up to 96.35% when used with
	Ruby(

Table 5

 5

	, the normalized RTR coefficient which is defined as
	a ratio of RTR to the magnitude of the considered thermal potential difference
	|∆T |. Note that this definition of normalized RTR enables comparison
	of different pairs of materials operating under different thermal potential
	differences. However, this particular definition is not applicable for PCM

Acknowledgement

This work was supported by SATT IDF Innov (now Erganeo) in the framework of the project DIOTHER and by the I-SITE FUTURE Initiative (reference ANR-16-IDEX-0003) in the framework of the project NANO-4-WATER.

Supplementary Materials I: Radiative properties extraction of considered materials


Mathilde Berard

Cécile Rattaz

Marianne Peries

Julie Loubersac

Kerim Munir

Amaria Baghdadli

Impact of containment and mitigation measures on children and youth with SD during the COVID-19 pandemic: Report from the ELENA cohort

Keywords: COVID-19, Containment, Autism Spectrum Disorders, Children behaviors, France Autism Spectrum Disorders (ASD), Children and Youth (CaY)

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

The coronavirus disease 2019 , first identified in December 2019 in Wuhan, China, spread rapidly across the globe, and was declared a pandemic by the World Health Organization on March 11, 2020(World Health Organization, 2020). On March 16, 2020, population measures were implemented across France to control the community transmission of COVID-19 including containment, involving social distancing with restriction of movements, and mitigation, also known referred to as lockdown, involving the closure of nurseries, school, universities, and cancellation of public venues such as sports and concerts, introduction of restricted work hours and teleworking, use of face coverings when distancing was not possible, and implementation of hygiene measures. As the number of reported cases and deaths from COVID-19 increased in ensuing weeks, the French population was subjected to several months of social isolation, and allowed to leave only for essential sustenance and medical needs [START_REF] Casagrande | The enemy who sealed the world: Effects quarantine due to the COVID-19 on sleep quality, anxiety, and psychological distress in the Italian population[END_REF]. In this paper we use the terms containment and mitigation that represented an extended period of restriction to prevent widespread community transmission, instead of the term quarantine that refers to a period of isolation of persons potentially exposed to COVID-19.

These restrictive measures have had a profound impact on people's lives [START_REF] Brondino | Effective Strategies for Managing COVID-19 Emergency Restrictions for Adults with Severe ASD in a Daycare Center in Italy[END_REF][START_REF] Colizzi | Psychosocial and Behavioral Impact of COVID-19 in Autism Spectrum Disorder: An Online Parent Survey[END_REF][START_REF] Pisano | A qualitative report on exploratory data on the possible emotional/behavioral correlates of Covid-19 lockdown in 4-10 years children in Italy[END_REF]. For some children, adolescents and adults alike, the sudden disruption of daily routines and family lives represented a distressing event.

In fact, some authors [START_REF] Pfefferbaum | Mental Health and the Covid-19 Pandemic[END_REF][START_REF] Rubin | The psychological effects of quarantining a city[END_REF] have suggested that, in addition to the stress associated with the fear of contracting the illness, containment and mitigation themselves have presented additional psychological burden for the population [START_REF] Casagrande | The enemy who sealed the world: Effects quarantine due to the COVID-19 on sleep quality, anxiety, and psychological distress in the Italian population[END_REF][START_REF] Qiao | A Survey Analysis of Psychological Impact on Chinese during Quarantine against COVID-19[END_REF][START_REF] Yan | Prevalence and risk factors of psychological impacts among the quarantined population during the COVID-19 pandemic in China[END_REF].

Regarding children and youth, the effects of quarantine in Italy involved parental stress and psychological problems in families facing the COVID-19 [START_REF] Spinelli | Parents' Stress and Children's Psychological Problems in Families Facing the COVID-19 Outbreak in Italy[END_REF] with children and youth staying at home without the opportunity to engage in familiar activities that helped enrich their daily lives [START_REF] Spinelli | Parents' Stress and Children's Psychological Problems in Families Facing the COVID-19 Outbreak in Italy[END_REF]. In an Indian study [START_REF] Saurabh | Compliance and Psychological Impact of Quarantine in Children and Adolescents due to Covid-19 Pandemic[END_REF], most confined children and adolescents during the COVID-19 pandemic were reported to exhibit psychological distress much higher than the non-confined group. In China, [START_REF] Jiao | Behavioral and emotional disorders in children during the COVID-19 epidemic[END_REF] reported the presence of psychological difficulties in children during the COVID-19 pandemic, with fear, inattention and irritability being the most severe symptoms in children. A further study of confinement in Italy and Spain (2020), showed that 85.7% of the parents perceived changes in their children´s emotional state and behavior, the most frequent symptoms being difficulty concentrating, boredom, irritability, restlessness, loneliness, and worries. Similarly, a qualitative study of emotional and behavioral correlates of confinement of 4 to 10 years children in Italy [START_REF] Pisano | A qualitative report on exploratory data on the possible emotional/behavioral correlates of Covid-19 lockdown in 4-10 years children in Italy[END_REF], also showed marked effects, with half the children exhibiting greater irritability, intolerance of rules and excessive demands. Concerning the sleep patterns of children, a study by [START_REF] Orgilés | Immediate Psychological Effects of COVID-19 Quarantine in Youth from Italy and Spain[END_REF] noted more hours of sleep during confinement, whereas a study by [START_REF] Pisano | A qualitative report on exploratory data on the possible emotional/behavioral correlates of Covid-19 lockdown in 4-10 years children in Italy[END_REF] showed that one in five children present sleep problems including difficulty falling sleep as well as frequent waking up. Finally, a mixed-methods study showed that diseasecontainment measures during the pandemic leads to post-traumatic stress disorder (PTSD) with 30% of isolated children meeting PTSD criteria based on parent report, and 25% met PTSD criteria based on self-reports [START_REF] Sprang | Posttraumatic stress disorder in parents and youth after healthrelated disasters[END_REF].

The specific vulnerabilities of risk groups have been highlighted during the COVID-19 outbreak, including the elderly, persons with pre-existing medical comorbidities, pregnant women, as well as the homeless [START_REF] Courtenay | COVID-19 and People with Intellectual Disability: impacts of a pandemic[END_REF][START_REF] Kirby | Efforts escalate to protect homeless people from COVID-19 in UK[END_REF][START_REF] Qiao | A Survey Analysis of Psychological Impact on Chinese during Quarantine against COVID-19[END_REF]World Health Organization, 2020). Less attention has been paid to people with neurodevelopmental disorders, although it is increasingly recognized that people with pre-existing mental conditions need extra support during confinement [START_REF] Brooks | The psychological impact of quarantine and how to reduce it: Rapid review of the evidence[END_REF] and have more risk of psychological distress following disaster-related traumas [START_REF] Alvarez | Risk and resilience in canine search and rescue handlers after 9/11[END_REF][START_REF] Cavallera | Children and Adolescents with Developmental Disabilities in Humanitarian Settings, Child, Adolescent and Family Refugee Mental Health[END_REF][START_REF] Cukor | Prevalence and predictors of posttraumatic stress symptoms in utility workers deployed to the World Trade Center following the attacks of September 11, 2001[END_REF].

In persons with neurodevelopmental disorders, including those with Autism Spectrum Disorder (ASD), already with compromised deficits in social communication and interactions and restricted patterns of behavior (American Psychiatric Association, 2013), vulnerability to COVID-19 has been considerable [START_REF] Frankova | The impact of COVID-19 on people with autism, learning disabilities and mental health conditions[END_REF]. Especially since persons with ASD follow routines and need to be prepared for changes and transitions, unexpected events lead to behavioral challenges [START_REF] Courtenay | COVID-19 and People with Intellectual Disability: impacts of a pandemic[END_REF][START_REF] Frankova | The impact of COVID-19 on people with autism, learning disabilities and mental health conditions[END_REF] and suffering [START_REF] Narzisi | Handle the Autism Spectrum Condition During Coronavirus (COVID-19) Stay At Home period: Ten Tips for Helping Parents and Caregivers of Young Children[END_REF]. Moreover, given that repetitive thoughts and anxiety are common among persons with ASD [START_REF] Meier | Obsessive-compulsive disorder and autism spectrum disorders: longitudinal and offspring risk[END_REF], they are likely to be overwhelmed by negative information related to COVID-19 in the media. Persons with ASD also represent a highly heterogeneous group that may have sensory aversion to tolerate protective covering or comply with requests for screening. These factors therefore heighten their vulnerabilities, fears and guarded thinking [START_REF] Courtenay | COVID-19 and People with Intellectual Disability: impacts of a pandemic[END_REF], increase their stereotypies and may even lead to emergence of extreme challenging behaviors such as self-injury [START_REF] Narzisi | Handle the Autism Spectrum Condition During Coronavirus (COVID-19) Stay At Home period: Ten Tips for Helping Parents and Caregivers of Young Children[END_REF]. Since children and youth with ASD are expected to undergo interventions for several hours a week with special therapists at home or in dedicated centers or in clinics and institutes [START_REF] Narzisi | Handle the Autism Spectrum Condition During Coronavirus (COVID-19) Stay At Home period: Ten Tips for Helping Parents and Caregivers of Young Children[END_REF], the ensuing interruption or modification of level of interventions due to containment measures compounds their well-being. As noted by [START_REF] Courtenay | COVID-19 and People with Intellectual Disability: impacts of a pandemic[END_REF], the impact of confinement on children and youth with ASD has been especially heightened where the usual support of residential schools, day services, or respite care centers have been withdrawn due to the pandemic.

In the general population, Brooks and colleagues (2020) highlighted as main stressors the duration of isolation, fear of infection, frustration, boredom, not having adequate information or clear guidelines from authorities, financial problems and stigma related to exposure or illness. Other factors include the configuration of the home environment in terms of indoor space (room per person ratio) or access to outdoor safe play areas. According to [START_REF] Orgilés | Immediate Psychological Effects of COVID-19 Quarantine in Youth from Italy and Spain[END_REF], psychological effects were noted to be more prominent among children in Spain compared to those in Italy, due to greater availability of home gardens and more relaxed confinement rules in the latter setting. The sensory calming effect of peaceful indoor spaces, as well as access to outdoors, with opportunity for exercise and participatory activities was salient. In addition, the interruption of interventions in the familiar setting was likely to be disproportionately disruptive for children and youth with ASD. Other risk factors could be linked to age, severity of ASD symptoms as well as co-occurrence of other mental disorder and specifiers including intellectual disorders.

Aims of the study

The main aim of the current study was to examine the behavioral effects of containment and mitigation during the COVID-19 pandemic on a sample of children and youth with ASD. Specifically, the current study examined changes in sleep, nutrition, communication, challenging behaviors and stereotyped behaviors. A secondary aim was to explore the risk and protective factors related to changes in behaviors related to age, ASD symptom severity, sociodemographic factors, living conditions and interventions continuity.

Materials and Method

Study design

The present study is a cross-sectional descriptive and analytical survey carried out between 27 April and 13 May 2020, during the Stage 3 maximum containment and mitigation level measures in France (President of the French Republic, 2020). The source population for this sub study was all participants of the ELENA Cohort, for whom parents have answered to the specific COVID-19 questionnaire. The ELENA cohort is a French prospective and multicenter study of ASD subjects ascertained between ages 2-16 years in which patients are followed for a period of 6 years. By collecting clinical, social, environmental, and genetic data, as well as data relating to the parental quality of life, the main objectives of the ELENA cohort are (i) to identify patterns in outcome trajectories for a large sample of children and adolescents with ASD, (ii) to examine determinants of these outcome trajectories at different follow-up stages, and (iii) to constitute an open French database on ASD that promotes collaboration in translational studies [START_REF] Baghdadli | Investigating the natural history and prognostic factors of ASD in children: the multicEntric Longitudinal study of childrEN with ASD-the ELENA study protocol[END_REF][START_REF] Rattaz | School Inclusion in Children and Adolescents with Autism Spectrum Disorders in France: Report from the ELENA French Cohort Study[END_REF]. Complete details of the protocol have been published in detail elsewhere [START_REF] Baghdadli | Investigating the natural history and prognostic factors of ASD in children: the multicEntric Longitudinal study of childrEN with ASD-the ELENA study protocol[END_REF]. The current study was approved by the Internal Review Board of University Hospital of Montpellier.

Data collection

A letter was sent to the parents in the ELENA cohort via the database electronic system inviting them to participate to the present study by filling out an online questionnaire. Only one questionnaire was completed per child by one parent or both parents serving as joint informant. The COVID-19 parent structured questionnaire comprised of 4 parts: (i) Family environment: house size, number of rooms, household number and composition, and outdoor accessibility; (ii) Parent professional status: loss or reduction of work and teleworking; (iii); Level of information on COVID-19; and (iv) Child status: health, special education interventions, medical care and any need for care (related or unrelated to COVID-19). The parents were also asked to describe their children's behavior during the pandemic on a 3-point Likert scale (unchanged, improved, worse) for 5 domains: nutrition, sleep, challenging behaviors, communicative abilities, and stereotyped behaviors. The clinical characteristics were collected before confinement by licensed psychologists using standardized tools during assessments scheduled in the context of the ELENA protocol. Child and youth adaptive skills were assessed through the Vineland Adaptive Behavior Scales, Second Edition (VABS-II) [START_REF] Sparrow | Vineland adaptive behavior scales: Survey forms manual[END_REF]. Symptom severity was measured using the calibrated severity score (CSS) of the Autism Diagnostic Observation Schedule-2 (ADOS-2) [START_REF] Gotham | A replication of the Autism Diagnostic Observation Schedule (ADOS) revised algorithms[END_REF][START_REF] Lansing | PEP-3 Profil psycho-éducatif: Psychoeducational profile[END_REF]. A best estimate intellectual functioning was calculated using standardized and validated instruments (Brunet-Lézine R [START_REF] Brunet | Brunet-Lézine révisé: échelle de développement psychomoteur de la première enfance: BLR[END_REF]; BECS [START_REF] Adrien | BECS : Batterie d'évaluation cognitive et socio-émotionnelle. Pratiques psychologiques et recherches cliniques auprès des enfants atteints de TED[END_REF]; PEP-3 [START_REF] Lansing | PEP-3 Profil psycho-éducatif: Psychoeducational profile[END_REF];WPPSI-IV (Wechsler, 2014b); WISC-V (Wechsler, 2014a) ; WAIS-IV [START_REF] Wechsler | Wechsler adult intelligence scale-Fourth Edition (WAIS-IV)[END_REF]; K-ABC [START_REF] Kaufman | K-ABC: Kaufman assessment battery for children: Interpretive manual[END_REF])) selected in function of child's age and level, according to the approach of [START_REF] Howlin | Cognitive and language skills in adults with autism: a 40-year follow-up[END_REF]. The tools and measures are thoroughly described in the cohort study profile [START_REF] Baghdadli | Investigating the natural history and prognostic factors of ASD in children: the multicEntric Longitudinal study of childrEN with ASD-the ELENA study protocol[END_REF].

Statistical analysis

According to the normality of the continuous variables, mean with standard deviation (SD) or median with interquartile range (IQR) were reported. Categorical variables were described

Results

Study sample

Among the 892 participants included in ELENA cohort, there were 240 children and youth (CaY), 2 to 21 years of age, whose parents completed the COVID-19 questionnaire. One child was excluded because he didn't live with his parents at the time of the confinement. The 239 CaY who were involved in the analyses had higher scores regarding best estimate IQ (p=0.01), VABS daily living skills (p=0.03) and communication (p=0.003) scores than the others participants from the ELENA cohort, but no difference regarding ASD severity at the ADOS were found (Supplementary Table S1, available as supplementary data). The questionnaire was filled for 188 children by the mothers (78.7%), for 18 by the fathers (7.5%), for 32 by both parents together (13.4%) and for 1 child by another adult in the child's entourage who lived with him or her (0.4%). Among the 239 children and adolescents, 190 were males (79.5%; 4-1 male to female ratio). Children's mean age was 9.1 years (SD=4.0).

More details about participants' characteristics are in Table 1.

[Insert Table 1 about here]

Family and sociodemographic characteristics

Participants' familial and sociodemographic characteristics are described in Table 2. About 24% of the children belonged to single-parent families. Majority of families had only one child with ASD or another neurodevelopmental disorder living in the house (81.6%, n=195), whereas 17.2% of families had two (n=41) and 1.3% had three children with ASD or another neurodevelopmental living in the house (n=3). A decline in household income was reported by 99 respondents (n=41.6%) who attributed it to total or partial unemployment due to COVID-19.

[Insert Table 2 about here]

Living conditions

The average household size was 4 (SD = 1.2), with two children per household on average (SD = 1.0). On average there were 4.5 rooms per household (SD = 1.4). For 78.7% of respondents (n=188), the number of rooms in the home was at least equal to the number of occupants. Most respondents in our sample had a garden (64.9%, n=155). Around 25 % of respondents (n=60), reported that at least one parent teleworked. Overall, 67.0% of CaY went able to access outdoors regularly during confinement, while 33.0% were strictly confined at home (n=78). Among the most frequent reasons for strict confinement were fear of contamination (n=48), refusal by the child or youth (n=12) and having access to a garden (n=7). The majority of the respondents (83.9%, n=198) stated that they were aware of the specific conditions of authorization of outings for people with disabilities.

Interventions received for ASD

Most of the CaY (86.0%, n=203) were in school before confinement and mitigation measures; 61.4% (n=145), were receiving special education or care services, 78.8% (n=186) were in therapy with a professional (mostly speech or physical), and 5.5% (n=13) were receiving no interventions at all. During the confinement, interventions were maintained for 75.8 % (n=169) and were interrupted for 24.2% (n=54). For those whom interventions were maintained, these took the form of phone interviews with the parents (44.4%, n=103), videoconferences with the child (18.1%, n=42) or appointment with the child (8.6%, n=20).

Sleep, communication, nutrition, stereotyped and challenging behaviors

With regards sleep, communicative abilities and stereotyped behaviors, about half the parents (respectively 55.5% (n=131), 57.2% (n=135) and 54.7% (n=129)) reported no changes during confinement. When a change was reported in these domains, the behavior was more often perceived as worsening than improving, except for communication in which more parents reported progress (28.8%, n=68) than regression (14%, n=33). Most parents (71.6%, n=169) reported no change in nutrition behaviors, with only one fifth of them reported a worsening.

Most parents (64.4%, n=152) reported increase in challenging behaviors during confinement (see Figure 1).

[Insert Figure 1 Adaptive level and living conditions were not significantly related to the changes in behaviors during COVID-19. Finally, changes in nutrition and stereotyped behaviors were not related to any of the variables above.

Multivariate analysis

In the multivariate analysis and after adjustment, higher ADOS score was associated with decreased probability of improvement of sleep [OR 0.70 (95%CI 0.53-0.91)] compared to the group without change in sleep. Belonging to a single parent family was a risk factor for the sleep worsening ].

Then, an increase in VABS-II daily living skills score increased the risk to belong to the group with improvement in challenging behaviors [OR 1.47 (95%CI 1.07-2.03)] compared to the group without change, suggesting that CaY with better daily living skills were those whose challenging behaviors decreased the most.

Finally, an increase in subject age decreased the probability to belong to improved group membership with reported improved communicative abilities [OR 0.80 (95%CI 0.71-0.89)] compared to the group without change. In other words, younger children were those for whom more progress in communication were observed by the parents.

[Insert Table 3 about here]

Discussion

It is remarkable that during the Stage 3 phase of the pandemic, the results showed that while majority of CaY with ASD were able to go out regularly, one-third remained strictly confined at home as their parents feared transmission. Further, the interventions from special education services or private professionals were maintained for three quarters of the CaY; nevertheless, the interventions were adapted and mainly provided via telephone and telehealth services.

Although not optimal, this level of interventions bodes well for the care of the CaY enrolled in the ELENA cohort during the COVID-19 pandemic but is not necessarily reflective of the country at large.

Concerning the impact of confinement on CaY in terms of sleep, nutrition, communication, as well as stereotypical and challenging behaviors, the parents reported mostly deterioration in challenging behaviors, with some worsening of sleep and stereotypical repetitive behaviors.

These results suggest that CaY with ASD are likely to be adversely affected by containment measures, a finding that is congruent with studies of neurotypical CaY who show psychological effects of confinement including fear, irritability or sleep problems during the pandemic [START_REF] Jiao | Behavioral and emotional disorders in children during the COVID-19 epidemic[END_REF][START_REF] Orgilés | Immediate Psychological Effects of COVID-19 Quarantine in Youth from Italy and Spain[END_REF][START_REF] Pisano | A qualitative report on exploratory data on the possible emotional/behavioral correlates of Covid-19 lockdown in 4-10 years children in Italy[END_REF].

In terms of communication, the parents reported more progress than regression: almost one third of the parents reported improvement in language or social communication and interaction of their CaY. This finding can be interpreted in a number of ways. First, it is conceivable that the containment emergency measures can have a positive impact on CaY with ASD who are no longer exposed to stressful situations in their daily life, with reduced transitions between home and school, reduced aversive or sensory interactions and academic pressures. A previous study of the ELENA cohort on school inclusion in children and adolescents with ASD [START_REF] Rattaz | School Inclusion in Children and Adolescents with Autism Spectrum Disorders in France: Report from the ELENA French Cohort Study[END_REF] found that there were more anxiety symptoms in children and adolescents who were attending school on a full-time basis as compared to those who were attending school part-time. An entire day of school may represent a considerable challenge for children with ASD, requiring them to maintain sustained attention, tolerate interactions with other students, as well as manage their sensory environment. Second, we hypothesized that being confined at home might alleviate parental stress, which might have an effect on improved perception of communication in the CaY. It would be congruent with previous studies showing alleviation of stress in children confined at home, who appeared as calmer [START_REF] Pisano | A qualitative report on exploratory data on the possible emotional/behavioral correlates of Covid-19 lockdown in 4-10 years children in Italy[END_REF]. Positive effects of containment also included greater proximity of the CaY to their parents, who were themselves more likely to be sequestered at home, with enhanced opportunity to engage with them.

In terms of the secondary study aim related to the risk and protective factors, greater improvement in sleep was reported for younger subjects, and in those with lower ASD symptom severity. Sleep disorders are one of the most burdensome and profound complaints among parents and caregivers of CaY with ASD, described in about 40-80% of individuals with ASD [START_REF] Cohen | The relationship between sleep and behavior in autism spectrum disorder (ASD): a review[END_REF], and are highly variable across time [START_REF] Mazurek | Course and predictors of sleep and co-occurring problems in children with autism spectrum disorder[END_REF].

The change in challenging behaviors were also significantly linked to subjects' age; parents reported greater improvement in challenging behaviors for younger subjects. Such a relationship between age and improvement can also be viewed in the light of studies on developmental trajectories in ASD, with a higher curve in challenging behaviors in the earlier ages [START_REF] Baghdadli | Developmental trajectories of adaptive behaviors from early childhood to adolescence in a cohort of 152 children with autism spectrum disorders[END_REF].

Regarding family and sociodemographic factors, our results suggest that living in a single parent family is related to more variability (improvement or worsening) in sleep during confinement as well as less progress in communication. We can hypothesize that living in a single family, and consequently not having respite for household chores, professional activity, and taking care of the CaY with ASD, would constitute a supplementary source of stress during confinement, possibly leading to more difficulties for the CaY. We found no significant relationships between living conditions during confinement and parent reported behavior changes among the CaY, a finding not supported by previous results in neurotypical children ( 2020).

An important positive finding regarding continuity of interventions provided to the CaY with ASD, was related to greater progress in communication. [START_REF] Courtenay | COVID-19 and People with Intellectual Disability: impacts of a pandemic[END_REF] indicated that the impact of confinement on families is especially heightened where the usual supports of residential schools, day services, or respite care have been withdrawn due to the pandemic.

Consistent with this finding, the CaY in the present study who progressed in communication were those who continued to benefit from specialized interventions during confinement, even if these interventions were mostly provided via telehealth tools.

In terms of risk and protective factors, the multivariate analysis demonstrated that CaY with greater ASD symptom severity scores, and those living in a single parent family, were the ones for whom their parents reported less sleep improvement, or more sleep worsening, respectively. These results suggest that ASD severity, and family characteristics, are key factors enabling sleep regulation in CaY with ASD.

Moreover, we found that decrease in challenging behaviors was more important in CaY with better daily living skills. In summary, CaY with ASD who are more dependent were less likely to adapt to changes in the home environment and more likely to experience difficulties in adjusting to their newly imposed confined lifestyle, potentially causing a heightened level of stress and expression of challenging behaviors. Such an indirect relationship between adaptive abilities and challenging behaviors is consistent with prior literature [START_REF] Baeza-Velasco | Are Aberrant Behavioral patterns associated with the adaptive behavior trajectories of teenagers with Autism Spectrum Disorders?[END_REF][START_REF] Franchini | Early adaptive functioning trajectories in preschoolers with autism spectrum disorders[END_REF].

Finally, even though several negative effects of the emergency containment and mitigation measures on the behaviors of CaY were reported, such as sleep and challenging behaviors, favorable effects were noted by the parents for some subjects in the communication domain.

Greater progress in communication skills were more likely to be noted for younger subjects compared to older ones. This observation underscores the potential for early development of communicative abilities in children with ASD, and the need for early intervention to prevent delays [START_REF] Landa | Early communication development and intervention for children with autism[END_REF]. Although confinement could represent a distressing event for CaY with ASD impacting their behaviors, it may also reduce level of stress in some subjects in view of diminished opportunities for aversive sensory and social stimulation, that so often can contribute to stress and anxiety in some subjects [START_REF] Rattaz | School Inclusion in Children and Adolescents with Autism Spectrum Disorders in France: Report from the ELENA French Cohort Study[END_REF]. Understanding the reactions of CaY with ASD, and identifying risk and protective factors, is therefore essential to more selectively and appropriately address their needs, and to tailor present, and future interventions [START_REF] Sprang | Posttraumatic stress disorder in parents and youth after healthrelated disasters[END_REF]. This is likely to empower CaY with ASD, as well as their parents, as they are likely to face future recurrence of COVID-19 and other outbreaks [START_REF] Courtenay | COVID-19 and People with Intellectual Disability: impacts of a pandemic[END_REF].

Several limitations should be note when interpreting our findings. First, the sample of subjects involved in the analyses had higher scores regarding cognitive adaptive levels than the other participants in the ELENA cohort, which limits the generalization of our results. The impact of containment and mitigation measures in CaY with ASD might be more important than what we observed in this study. Second, most of questionnaires were filled by only one parent (86.2%), however parents may have differing perspectives regarding the behaviors of their CaY with ASD. Further, a parent report can be influenced by their own stressful experience of the pandemic based on the home life and work circumstances, as well as time available with children, resources and supports. Likewise, the level of external supports and service

  with frequency and percentage. To determine first the factors linked to each child or youth behavior change domain during confinement (i.e., communicative abilities, sleep, nutrition, challenging and stereotyped behaviors), we used Pearson chi-square tests for categorical variables and Anova regressions or Kruskal-Wallis tests for continuous variables. Multivariate polytomic regressions were then used to determine factors significantly associated for each child or youth behavior change domain during confinement. Variables with P value <0.20 in univariate analysis were included in the multivariate model and we presented OR with 95% confidence intervals. Backward selection was used. Multivariate regressions were adjusted with the child's best estimate IQ. Following factors were considered in the analysis: child and youth age during confinement, last ELENA follow-up ADOS-2 comparison score (CSS), last VABS-II standard scores in communication, socialization and daily living skills, number of children living in home, outing during confinement, continuation of care during confinement, adequation between rooms number and inhabitants, access to an outdoor, single parent family, questionnaire responder and child' gender. P-values below 0.05 were considered statistically significant. Analyses were performed using SAS software v9.3.

  For sleep, the univariate analysis first showed that subjects who were noted to have improved sleep were significantly younger (median = 7.0,.0], n=28) than those for whom no change in sleep was reported (median = 9.0, IQR[6][7][8][9][10][11][12]), n=131, p=0.01). CaY with reported improved sleep had a lower severity score (median 6.0, (IQR: [5.0 -7.0], n=23) than those who worsened their sleep (median 8.0, IQR: [6.0-9.0], n=70, p=0.003) or those for whom no change was reported by their parents (median 7.0, IQR: [6.0-9.0], n=120, p=0.01).Finally, there is more single parent families among CaY who improved or worsened their sleep (33.8%, n=26 and 32.1%, n=9 respectively) than among those for whom sleep didn't change during COVID-19 pandemic (16.8%, n=22), p=0.01.Challenging behaviors were also significantly linked to the subject age. CaY whose parents reported decrease in challenging behaviors were younger (median=7.0, IQR= [5.0-10.0], n=46) than those whose parents reported no change (median=9.0, IQR= [7.0-12.0], n=84), p=0.004.Regarding changes in communicative abilities, subjects whose parents reported improvement in communicative abilities were younger (median=6.0, IQR= [5.0-9.0], n=68) than those with parent report of regression (median=9.0, IQR= [6.0-14.0], n=33) or no change in communicative abilities (median=9.0, IQR= [7.0-12.0], n=135), p<0.01. Results showed that there were more subjects for whom interventions were maintained during COVID-19 among those who progressed in communication (86.8%, n=59) than among those whose communicative abilities didn't change (65.9%, n=89) or regressed (63.6%, n=21). Finally, there were more single parent families among CaY whose communicative abilities regressed (30.3%, n=10) or didn't change (28.1%, n=38) than among those whose communicative abilities progressed (13.2%, n=9), p=0.04.

Fig. 1 :

 1 Fig.1: Parental perception of their children's behaviors during confinement (%)

Table 1 : ASD children and youth characteristics

 1 Data are given in mean (SD), median (IQR), or n (%). °° 3 missing value, °4 missing values * 9 missing values, ** 24 missing values.

	Tables
		All children
		N=239
	Sex
	Male	190 (79.50%)
	Female	49 (20.50%)
	Age during quarantine (years)	9.11 (±4.0)
	Intellectual level
	Best estimate t IQ
	<70	82 (35.65%)*
	≥70	148 (64.35%)
	ADOS
	Comparison score	7.242 (±1.818)**
	Vineland Adaptive Behavior Scales
	(VABS II)
	Communication SS	72.13 (±16.16)°
	Daily living skills SS	72.20 (±14.07)°
	Socialization SS	68.96 (±13.19)°

Table 2 : Family and sociodemographic characteristics

 2 1 missing value, °° 3 missing values, £ 33 missing values, ££ 29 missing values

	N=239

¤

Acknowledgments

We thank the contributing families, the ELENA cohort staff (Ela Miniarikova, Flore Couty, Florine Dellapiazza, Lee Audras-Torrent, Cécile Michelon, Myriam Soussana, Laetitia Ferrando and Christelle Vernhet) and the clinical team of Montpellier hospital. We express gratitude to the CNSA and DGOS for funding this research.

availability for CaY with ASD varies across settings that may influence parent perspectives.

Moreover, longitudinal studies are required to assess the longer lasting effects of confinement and mitigation on CaY with ASD during pandemics. Finally, the lack of a pre-lockdown baseline complicates the interpretation of the effects reported, however it would have been complicated to measure it due to unpredictability of epidemics.