Marianne Balat-Pichelin
email: marianne.balat@promes.cnrs.fr

Jean-Louis Sans

Eric Bêche

Ludovic Charpentier

Alain Ferrière

Sébastien Chomette

Emissivity at high temperature of Ni-based superalloys for the design of solar receivers for future tower power plants

Keywords:

In the framework of the H2020 European project POLYPHEM, whose main objective is to improve the flexibility and performance of small-scale solar power plants, studies have been conducted on various nickel-based alloys for the design of the solar receiver. Inconel 600 alloy is used as a reference material in the advanced manufacturing of high temperature solar receivers but other oxidation resistant alloys have been selected such as Inconel 617, Haynes 230, Haynes HR120 and Hastelloy X and the best candidate will be used for making the prototype. Thus, two essential quantities, namely the solar absorptivity αthe higher the betterand the total hemispherical emissivity εthe lower the betterwere measured in high vacuum and under atmospheric air up to 1400 K in the MEDIASE reactor installed at the focus of the 1000 kW solar furnace. The objective is to reach the best solar absorptivity and the α/ε ratio the highest as possible. The emissivity measurement was made by a direct method requiring the measurement of the true temperature of the material using a bi-chromatic pyroreflectometer and that of its radiance by spectro-radiometry. Total hemispherical emissivity data are presented at the same time as microstructural (SEM, XRD) and topographic (3D profilometry) characterizations allowing the interpretation of the emissivity evolution with temperature and the growth of the oxide layer. Finally, the alloy Haynes 230 was chosen to build the solar receiver.

Introduction and context

Solar thermal electricity with integrated thermal energy storage system has demonstrated a high potential to increase the share of renewable electricity in the energy mix. The maximum temperature of liquid heat transfer fluids implemented in Concentrated Solar Power (CSP) plants (830 K) currently limits the solar-to-electric conversion efficiency at about 25%. The development of central solar receivers working at very high temperature (1020 K and more) paves the way to the utilization of engines offering efficiency higher than steam Rankine turbines, like combined cycles or supercritical CO 2 Brayton cycles. The research project POLYPHEM addresses some of the major challenges raised by the development of the next generation of solar tower technologies (Fig. 1). A prototype plant of small-scale solar thermal combined cycle is designed, implemented and operated. The top cycle is a solar-driven micro gas-turbine of 70 kW and the bottom cycle is an Organic Rankine Cycle of 25 kW. A thermal energy storage of 2.6 MWh using thermocline tank is integrated between both cycles.

Conventional fuel is substituted by solar energy in the gas-turbine through the integration of the solar receiver in the air Brayton cycle. The pressurized air solar receiver is a key component, it must be reliable, efficient, cost-competitive and it must offer a long lifetime. Its development addresses material issues, fluid distribution and fluid flow issues, and heat transfer issues. The implementation of metallic materials is preferred. CNRS and CEA have already proposed a new concept of efficient pressurized air solar absorber and have validated it at laboratory scale [START_REF] Bellard | Experimental Characterization of a High Temperature Pressurized Air Solar Absorber for the PEGASE Project[END_REF]. The core technology developed is a bundle of thin nickel-based alloy tubes (diameter < 8 mm) arranged in parallel rows and embedded into a highly conductive material (Cu alloy). An external thin plate of nickel-based alloy prevents oxidation and corrosion of the Cu alloy. Both metallic materials are assembled by diffusion bonding using High Isostatic Pressure (HIP) fabrication process. This process yields no thermal resistance at interfaces. Additional R&D work is necessary to optimize the concept and to adapt it to the specifications of POLYPHEM. The required heat rate is 500 kW under standard solar direct normal irradiance of 1000 W/m 2 . The temperature of the air is increased from 460 K at the inlet up to 1020 K at the outlet, with a maximum allowed pressure drop below 170 hPa (e.g. 5% of the operating pressure of 3320 hPa). The targeted thermal efficiency of the receiver exceeds 80%. The manufacturing cost target will be 0.4 €/W when TRL 9 is achieved.

The material exposed to concentrated solar energy is submitted to strong thermo-mechanical stresses due to temperature gradients increased by the non-uniform solar flux density distribution. The selection of the most reliable alloy for the fabrication of the solar receiver is a critical step of the project. To this end, oxidation resistant metallic alloys were pre-selected. Their optical properties were measured and thermo-mechanical characterizations were also carried out from room temperature up to service representative temperature of the solar receiver.

The literature about oxidation and emissivity at high temperature of Ni-based superalloys is scarce. Main of the studies was carried out in the frame of the development of the Very High Temperature nuclear Reactors (VHTR) [START_REF] King | Impact of corrosion on the emissivity od advanced reactor structural alloys[END_REF][START_REF] Maynard | Total hemispherical emissivity of potential structural materials for very high temperature reactor systems: Hastelloy X[END_REF][START_REF] Maynard | Hemispherical total emissivity of potential structural materials for very high temperature reactor systems: Haynes 230[END_REF][START_REF] Marchetti | Photoelectrochemical study of nickel base alloys oxide films formed at high temperature and high pressure water[END_REF][START_REF] Kim | Oxidation characteristics and oxide layer evolution of alloy 617 and Haynes 230 at 900 • C and 1100 • C[END_REF][START_REF] Kong | Normal spectral emissivity of GH536 (Hastelloy X) in three surface conditions[END_REF][START_REF] Ledoux | Oxide growth characterization during short-time oxidation of a commercially available chromia-forming alloy (HR-120) in air at 1050 • C[END_REF] on Inconel 617, Haynes 230, Hastelloy X and HR 120.

Inconel 617

Oxidation of Inconel 617 was studied up to 1370 K in air by Kim et al. [START_REF] Kim | Oxidation characteristics and oxide layer evolution of alloy 617 and Haynes 230 at 900 • C and 1100 • C[END_REF] and they have observed a NiO-Cr 2 O 3 double-layer that was formed at 1170 K with some isolated triple layer NiO-NiCr 2 O 4 -Cr 2 O 3 structures. At 1370 K, a double layer of inner Cr 2 O 3 and outer TiO 2 was formed with some spallation of this last phase.

The normal spectral (3-18 μm) emissivity of Inconel 617 at 570 K was measured by King et al. [START_REF] King | Impact of corrosion on the emissivity od advanced reactor structural alloys[END_REF] on non-oxidized samples, and on oxidized ones in a mixture He/H 2 O during 500 h at 1300 K. The normal spectral emissivity decreases from 0.20 at 3 μm to 0.10 at 18 μm for the reference non-oxidized sample (Ra = 0.02 μm) and after oxidation with the formation of a thick black layer with an increased roughness (Ra = 1.6 μm), it increases from 0.80 up to a maximum of 0.92 around 13.5 μm and then decreases to 0.30 at 16 and 18 μm.

Haynes 230

About Haynes 230, oxidation at 1170 K and 1370 K has led to the formation of the mixed oxide as for Inconel 617, with an inner layer of Cr 2 O 3 and an outer one of MnCr 2 O 4 respectively. It was observed that H230 is less oxidized than Inconel 617 and this was attributed to the decreasing diffusion of oxygen through the mixed oxide layer, and for both the alloys the weight change followed a parabolic law [START_REF] Kim | Oxidation characteristics and oxide layer evolution of alloy 617 and Haynes 230 at 900 • C and 1100 • C[END_REF].

The normal spectral emissivity of H230 measured in the same conditions as for Inconel 617 by King et al. has given nearly the same results with an emissivity going from 0.25 at 3 μm down to 0.10 at 18 μm for the reference non-oxidized sample and after high temperature oxidation during 500 h, the emissivity goes from 0.70 to 0.85 from 3 to 13.5 μm then drops down up to 16 μm, this being correlated to the two absorption bands of Cr 2 O 3 between 15 and 18 μm [START_REF] King | Impact of corrosion on the emissivity od advanced reactor structural alloys[END_REF]. Another work of Maynard et al. has given some values for the total hemispherical emissivity of H230 [START_REF] Maynard | Hemispherical total emissivity of potential structural materials for very high temperature reactor systems: Haynes 230[END_REF]. For the as-received non-oxidized sample, the emissivity measured in secondary vacuum (10 -4 -10 -5 Pa) increases from 0.18 around 600 K up to 0.24 around 1400 K, this value being in agreement with the one of Cockeram et al. (0.23 at 1370 K) [START_REF] Cockeram | The development and testing of emissivity enhancement coatings for thermophotovoltaic (TPV) radiator applications[END_REF]. After 15 min oxidation in air around 1200 K, the total hemispherical emissivity increases from 0.20 at 600 K up to 0.50 at 1270 K with a sudden increase in the range 1070-1100 K and the main oxide present detected by XRD was NiO.

Hastelloy X

The spectral emissivity of Hastelloy X was measured by Kong et al. [START_REF] Kong | Normal spectral emissivity of GH536 (Hastelloy X) in three surface conditions[END_REF] and its total hemispherical emissivity by Maynard et al. [START_REF] Maynard | Total hemispherical emissivity of potential structural materials for very high temperature reactor systems: Hastelloy X[END_REF]. Kong et al. measured the normal spectral emissivity in the wavelength range 1-15 μm from 400 to 1200 K on reference and oxidized samples. For the non-oxidized sample, the emissivity decreases from 0.40 down to around 0.20 when the wavelength increases from 1 to 15 μm, according to the electromagnetic theory for metals and increases linearly with the temperature. For the oxidized sample at 1170 K during 720 min, the emissivity is decreasing from 0.80 around 1 μm down to 0.60 at 15 μm with the absorption bands corresponding to the presence of Cr 2 O around 11-14 μm.

Maynard et al. have measured the emissivity on as-received samples and have obtained values going from 0.18 at 470 K up to 0.28 at 1500 K.

Haynes HR120

We have found only one work on the oxidation of HR120 alloy at 1320 K in air from 30 min up to 100 h [START_REF] Ledoux | Oxide growth characterization during short-time oxidation of a commercially available chromia-forming alloy (HR-120) in air at 1050 • C[END_REF]. After oxidation, a multi-layered oxide was formed and composed of SiO 2 -Cr 2 O 3 -XCr 2 O with X = Mn and/or Fe, Ni from the metal-oxide interface to the top of the oxide scale. For shorter oxidation duration, at 34 min, the oxide layer is only composed of the spinel MnCr 2 O 4 and Cr 2 O 3 and then after 42 min, the second spinel FeCr 2 O 4 grew and this is valid up to 100 h oxidation. No data have been found for the emissivity of the HR120 alloy.

Finally, very few data are available in the literature for non-oxidized Ni-based alloys and for oxidized ones in air at high temperature. The following sections will present the materials and the experimental results obtained for the measurement of emissivity of such alloys to be used as solar receivers in CSP plants.

Materials and experimental section

The different alloys selected for the benchmark and the ones of the final selection to be studied are presented. The method for the emissivity Fig. 1. Scheme of the POLYPHEM concept. determination, the experimental set-up and the experimental protocol are detailed.

Materials

The solar receiver considered in the European research project POLYPHEM will be manufactured using two types of alloys assembled using diffusion bonding made in a HIP apparatus. The first one is used for the surface in contact with air at high temperature (outer + inner surface of the solar receiver modules and manifolds). The second alloy is used inside the receiver with high thermal conductivity (copper alloy, not studied here). For the first type of material, the selected one needs high temperature oxidation resistance but also high temperature mechanical properties (inside pressure, thermo-mechanical fatigue, creep …). Previous receiver modules were manufactured by diffusion bonding with only nickel-based alloy Inconel 600 parts. However, it was decided to make a material benchmark in the POLYPHEM project in order to find a potential alloy with higher performance as an alternative to alloy Inconel 600. The selection of the benchmark alloys is done thanks to their high temperature oxidation resistance. These alloys are listed in Table 1.

The alloy benchmark is principally based on the comparison of the material properties extracted from alloy manufacturer's documentation at 1150 K. This temperature corresponds to the highest one recorded on previous solar receiver modules made by diffusion bonding and tested under solar radiation. The selected physical properties are: high temperature oxidation and nitridation, thermal conductivity and thermal expansion. For mechanical properties, the study focuses on the tensile yield strength at 0.2% offset, tensile elongation to rupture, fatigue, creep strain to obtain 1% of deformation after different duration. The study is completed with an alloy cost comparison based on the supply of plates with a thickness of 5 mm (material manufacturers or retailers alloy cost per kilogram).

As the selected material will be used to manufacture a solar receiver, one property has not the same importance as another one. Consequently, the alloys have been classified using a weighting for each property depending on its significance (Figure of Merit). As some properties are not available for all alloys, it was decided to quote to 0 for the corresponding alloy property as no complementary tests are planned to evaluate it. Concerning physical properties, the most important is the high temperature oxidation resistance while the least one is the nitridation resistance. The thermal conductivity and expansion behavior have an average rating as a low thickness of alloy is expected for manufacturing the solar receiver modules. For mechanical properties, the highest weightings have been put on the tensile yield strength and the fatigue response of the alloys. The elongation and the creep behavior (to obtain 1% of deformation after 10000 h) have an average rating, as the expected pressure inside the module is relatively low. Creep properties for reduced time to obtain a deformation of 1% are not important. A graphic representation of the benchmark results has been made and normalized with the highest score and is reported in Fig. 2.

In order to complete the investigations on potential alloys for manufacturing the solar receiver modules, complementary tests are on the four best alloys (manufacturer) of the previous benchmark: Haynes 230 (Haynes), Inconel 617 (Special Metals), Inconel 600 (ThyssenKrupp VDM) and Haynes Hastelloy X (Haynes). The alloy Haynes HR120 (Haynes) has also been tested in order to compare the optical properties of an alloy at the middle of the ranking with the best ones.

Method for the measurement of the emissivity

The direct method used at PROMES-CNRS laboratory is the one that corresponds directly to the definition of emissivity: the directional spectral radiance (L' λ ) of the material is measured as well as its temperature to calculate the spectral radiance of the blackbody (L 0 λ ) at the same temperature [START_REF] Balat-Pichelin | Emissivity measurements on carboncarbon composites at high temperature under high vacuum[END_REF][START_REF] Brodu | Influence of roughness and composition on the total emissivity of tungsten, rhenium and tungsten-25% rhenium alloy at high temperature[END_REF][START_REF] Brodu | Efficiency and behavior of textured high emissivity metallic coatings at high temperature[END_REF]. The ratio of the radiances gives the spectral directional emissivity: ελ

′ = Lλ ′ Lλ0.
The corresponding hemispherical emissivity ελ∩ is then calculated by integration of the directional values:ελ∩ = 0π2ελ

′ sin2θ dθ

The instrument used for radiance measurement is a spectroradiometer that allows measurements at multiple discrete wavelengths, leading to a finer analysis of the behavior of materials, as for example during oxidation of alloys at high temperature.

The front face of the sample is exposed to the concentrated solar radiation and measurements are made on the rear side of the sample, away from incident solar radiation, in an environment at room temperature. It should be noticed that this causes a temperature gradient in the sample: the insolated face must be brought to a temperature significantly higher than that of the measured face, the thermal gradient depending essentially on the thermal conductivity and the thickness of the sample.

Experimental set-up

The experimental set-up MEDIASE (Fig. 3) consists of a stainless steel vessel with a capacity of about 60 l, equipped with a turbo-molecular pumping system. A hemispherical silica window, 35 cm in diameter, is placed in front of this vessel and allows concentrated solar radiation to heat the sample (up to 10 MW/m 2 ). The sample (40 mm in diameter, 2 mm thickness) is exposed to concentrated solar radiation by the gradual opening of the doors of the solar furnace. The temperature of the sample is measured on the rear face using the two-color pyro-reflectometer developed at PROMES-CNRS laboratory [START_REF] Hernandez | Experimental validation of a pyroreflectometric method to determine the true temperature on opaque surface without hampering reflections[END_REF]. This device gives the temperature of the sample without contact with it. Unlike conventional optical pyrometers, this device does not require knowing the emissivity of the sample to obtain its real temperature. To do this, the two-color pyroreflectometer is made up of two parts: one measuring the thermal emission of the sample, the other measuring the bidirectional reflectivities at the two wavelengths. The combination of these two types of measurement allows calculating a common diffusion factor for the two wavelengths that will be used to determine the true temperature of the material. The pyroreflectometer requires two calibrations: one in front of a blackbody (up to 2000 K) equipped with a standard pyrometer, the other in front of a reflectivity standard to determine the reflectivities of the sample. We obtain a relative uncertainty of the order of 2% on the temperature of the sample.

The radiance measurements are carried out using a spectroradiometer (CI systems, SR-5000 N) equipped with an optic allowing, at the measurement distance used, to observe an area of 9 mm diameter at normal incidence in the center of the sample. The spectral domain studied (0.4-14 μm) does not make it possible to work with a single detector as two systems are present: a spectro-photometer working from 0.4 to 1 μm and a spectro-radiometer operating from 1.3 to 14 μm. We have a blind zone between 1 and 1.3 μm. It produces signals spaced about 30 nm in the range 1.3-14 μm and 3 nm in the range 0.4-1 μm, respectively 315 and 1060 wavelengths which is enough to calculate the

Experimental procedure

The samples studied are discs of 40 ± 0.25 mm diameter and 2-3 mm thickness. They were delivered by CEA (LITEN/DTBH/SCTR/LCA).

Five alloys (melting temperature) are available: Haynes 230 (1574-1644 K), Haynes HR120 (1573 K), Haynes Hastelloy X (1533-1628 K), Inconel 600 (1627-1686 K) and Inconel 617 (1605-1653 K), 4 discs per grade were received. One as-received sample was kept as reference for surface characterization by SEM, XRD and 3D profilometry.

Measurements were made from 1050 K to 1400 K approximately, in high vacuum and under air at atmospheric pressure, by steps of about 100 K, and for angles of incidence between 0 and 80 • . The measured spectral range is between 1.3 and 14 μm.

The sample to be measured is placed in a cooled sample-holder and the chamber is conditioned according to the required measurement conditions. The sample is progressively submitted to concentrated solar radiation until the temperature of the rear face is close to the desired temperature. After adjusting and stabilizing the temperature, the rotation of the goniometer from 0 to 80 • is started at the same time as the recording of the spectro-radiometer signal by means of a data acquisition device. Each radiance measurement is preceded and followed by a temperature measurement. The process is repeated at each temperature level. The subsequent data processing consists of:

-extracting the spectro-radiometer signals for each angle of incidence, -calculating the radiance fractions of the blackbody under the same conditions of temperature and wavelength, -performing the radiance ratio to calculate the total and/or spectral directional emissivity.

The corresponding hemispherical emissivity is then calculated by integration of the directional values.

Experimental results and discussion

The experimental measurement of the radiance up to 1400 K, in the two interested wavelength ranges 1.3-2.8 μm -approximating the solar absorptivity -and 1.3-14 μm for the total emissivity, for the five alloys studied was carried out and the data are reported in the following sections according to the atmosphere surrounding the samples.

In high vacuum

The emissivity in high vacuum up to 1300 K was measured for all the alloys to have a reference data for non-oxidized materials. Fig. 4 presents the results and Table 2 gives the coefficients a and b of the linear regression: ε = a. T+ b.

When the alloys are compared, the results show that except the data for the 617 at 1300 K, the alloys follow nearly the same trend concerning the total hemispherical emissivity versus temperature with a slight decrease when the temperature increases (Fig. 4). The emissivity goes almost from 0.35 to 0.40 around 1100 K down to 0.25 at 1300 K. Only the alloy 617 still keeps a constant behavior with a total emissivity comprised between 0.30 and 0.35.

In fact, the emissivity decreases when the temperature increases due to the smoothing of the surface at high temperature, this being in relation with the coarsening of the grains. This phenomenon can be observed when looking at the surface morphology of the samples acquired by SEM (Fig. 5) and surface roughness measured by 3D profilometry and reported in Table 3. For all the alloys except for 617, the mean square surface roughness Sq decreases when the temperature increases. Once again, alloy 617 does not show the same trend as the surface roughness increases when exposed to high temperature going from Sq = 1.33 ± 0.03 μm for as received samples up to Sq = 1.47 ± 0.07 μm for the sample heated up to 1300 K.

XRD characterization was carried out on the samples before and after heat treatment in high vacuum and an example of XRD pattern is given for alloys HX and H230 in Fig. 6. One crystalline phase was identified. The diffraction peaks observed for heated samples revealed higher peak intensities and thinner peak FWHM in comparison with the diffractograms collected for the as-received reference samples. This result indicates a higher crystallinity of the heated sample in vacuum. A decreasing of the volume of the lattice is also observed. No metal or alloy oxide was detected for the heated samples (1300 K) in vacuum.

At atmospheric pressure

Concerning the measurement of the directional emissivity during in situ oxidation, the example of results obtained for the alloy HR120, shown in Fig. 7, prove that between angles of incidence of 0 • and 60 • , no difference can be observed whatever the wavelength in the studied range (1.3-14 μm). For higher angles, the emissivity decreases when the angle increases. Another result is that oxidation increases the emissivity for long wavelengths in the infrared domain (same samples tested at different temperature levels).

It can be noticed that for such oxidized sample, no oscillation of the spectral emissivity can be visible for the short wavelengths as the oxide layer is too thick and opaque to see this effect. This is confirmed later by Fig. 5. SEM micrographs of the three alloys studied, on the left column, as-received samples and on the right column, after exposure in high vacuum up to 1300 K.

Table 3

Surface roughness of the alloys before (as received) and after exposure in high vacuum up to 1300 K (Sa arithmetic and Sq mean square surface roughness). the SEM images presented in Fig. 9 and the XRD patterns in Fig. 10.

From the directional measurement shown in Fig. 7, we can obtain by integration the hemispherical emissivity in the two wavelength ranges:

1.3-2.8 μm approximating the solar absorptivity and 1.3-14 μm for the total emissivity. The measurement was performed up to 1200 K for one sample of each alloy and up to 1400 K for another one. Fig. 8 reports the results obtained for the five alloys, and Table 4 gives the parameters obtained for the linear regression of the total hemispherical emissivity: ε = a. T + b.

Looking at Fig. 8, it clearly appears that the total hemispherical emissivity is increasing with temperature and this is due to the formation of oxide compounds on the surface as already observed for other metallic materials [START_REF] Balat-Pichelin | Oxidation and emissivity of Inconel 718 alloy as potential space debris during its atmospheric entry[END_REF][START_REF] Barka | Oxidation and emissivity of Invar 36 alloy in air plasma at high temperatures[END_REF]. On the contrary, the approximated solar absorptivity is quite stable in the temperature range 1100-1400 K.

When the total hemispherical emissivity in the wavelength range 1.3-14 μm and the solar absorptivity in the wavelength range 1.3-2.8 μm for alloys 600, HX, H230 and HR120 are compared at different temperatures, some differences can be observed with a higher emissivity and absorptivity for alloy HX than for the others, for example. However, the comparison of the α/ε ratio shows no real differences, sign of similar behavior for these alloys (Table 5).

In the same way, the SEM images of the surfaces of the alloys 600, HX, H230 and HR120 after oxidation up to 1200 K do not show significant differences, but some differences can be observed up to 1400 K even if the grain size is close (Fig. 9). In all cases, the oxidized surface is full of small and numerous thin flat grains for heat treatment up to 1200 K. This surface morphology evolves to bigger shape oxides at 1400 K. Only the grain size varies from one alloy to another. The behavior of the alloy 617 is slightly different compared to the one of other materials. Indeed, Fig. 8 shows that the calculated solar absorptivity is above the level obtained for other alloys (close to 0.90 when others are near or lower than 0.80). However, the total hemispherical emissivity is very high too. That is why the α/ε ratio is reduced for alloy 617 from 1100 to 1300 K compared to the other materials but similar at 1400 K (Table 5). Observing the oxidized surface of alloy 617 samples, the thin flat grains formed during oxidation in air up to 1200 K for all previous alloys are also visible on the 617 samples (image on the left, Fig. 9). Nevertheless, while these tips disappeared for oxidation up to 1400 K on previous materials, they are still present on the surface of the 617 samples (image on the right, Fig. 9). The different behavior of the alloy 617 compared to all the other materials could be explained by the fact that the 617 samples were machined in a bar while all the others come from plates. The thermo-mechanical treatments are different for bars and for plates during the manufacturing process that can lead to different initial microstructure and internal stress. Moreover, the surface roughness of the as-received 617 sample is higher than the one for the other alloys (Table 3).

Table 6 gives the results obtained for the surface roughness measurement carried out on the oxidized surface after heat treatment at 1400 K.

Looking at the results presented in Table 6, increasing the temperature leads to an increase of the oxidation layer that induces a slight increase of the surface roughness of the samples, except for the alloy 600. The increase of the surface roughness with temperature has an important effect on the emissivity as emissivity is always increasing with the roughness [START_REF] Maynard | Total hemispherical emissivity of potential structural materials for very high temperature reactor systems: Hastelloy X[END_REF][START_REF] Brodu | Influence of roughness and composition on the total emissivity of tungsten, rhenium and tungsten-25% rhenium alloy at high temperature[END_REF][START_REF] Brodu | Efficiency and behavior of textured high emissivity metallic coatings at high temperature[END_REF][START_REF] Balat-Pichelin | Oxidation and emissivity of Inconel 718 alloy as potential space debris during its atmospheric entry[END_REF][START_REF] Barka | Oxidation and emissivity of Invar 36 alloy in air plasma at high temperatures[END_REF][START_REF] Jo | Spectral emissivity of oxidized and roughened metal surfaces[END_REF]. XRD analysis was carried out to identify the oxide compounds present on the surface after oxidation (2θ from 20 • to 100 • ). A part of the XRD patterns (zoom in the range 20 • -45 • ) is presented in Fig. 10 for all the samples studied after heat treatment at 1200 and 1400 K.

For the alloys 600 and 617, three metal oxide crystalline phases were identified: chromium (III) oxide, Cr 2 O 3 (ICDD 73-6214), iron oxide, magnetite Fe 3 O 4 (ICDD 89-0688) and iron (III) oxide, hematite, alpha-Fe 2 O 3 (ICDD 84-0311). Higher is the temperature, higher is the amount of Cr 2 O 3 (and Fe 3 O 4 ) compounds. The Fe 3 O 4 crystalline phase was only detected for the sample heated at 1400 K. A minor Fe 2 O 3 phase was observed for the sample heated at 1200 K. Higher is the temperature, higher is the thickness of the oxide layer, the metal oxides being mainly composed of Cr 2 O 3 compound.

For the alloy H230, four metal oxide phases were identified: a majority of Cr For HX alloy, as for the alloys 600 and 617, three main metal oxides were detected: Cr 2 O 3 , Fe 2.93 O 4 (ICDD 86-1353) and Fe 2 O 3 . Again, higher is the temperature, higher is the amount of Cr 2 O 3 , lower is the amount of Fe 2 O 3 that was not detected for the sample heated at 1400 K. The Fe 2.936 O 4 crystalline phase was only detected for the sample heated at 1400 K. Again, higher is the temperature, higher is the thickness of the oxide layers, the metal oxides being composed of Cr 2 O 3 and Fe 2.936 O 4 compounds.

Finally, for the HR120 alloy, three main metal oxides were detected: Cr 2 O 3 , Fe 2 O 3 and less Fe 2.93 O 4 . Again, higher is the temperature, higher is the amount of Cr 2 O 3 , lower is the amount of Fe 2 O 3 not detected for the sample heated at 1400 K. The Fe 2.936 O 4 crystalline phase was only detected for sample heated at 1400 K. Higher is the temperature, higher is the thickness of the oxide layers visible by the decreasing intensity of the alloy peak and the increasing one of the peak of the metal oxides composed of Cr 2 O 3 and Fe 2.936 O 4 compounds.

Table 7 reports the main oxides present on the surface of the different alloys. Clearly, the chromium oxide is the main one formed and on all the samples, no NiO nor mixed oxides was detected in agreement with the results of Marchetti et al. [START_REF] Marchetti | Photoelectrochemical study of nickel base alloys oxide films formed at high temperature and high pressure water[END_REF], King et al. [START_REF] King | Impact of corrosion on the emissivity od advanced reactor structural alloys[END_REF] and Kong et al. [START_REF] Kong | Normal spectral emissivity of GH536 (Hastelloy X) in three surface conditions[END_REF], and on the contrary to the results of Maynard et al. [START_REF] Maynard | Hemispherical total emissivity of potential structural materials for very high temperature reactor systems: Haynes 230[END_REF] and Kim et al. [START_REF] Kim | Oxidation characteristics and oxide layer evolution of alloy 617 and Haynes 230 at 900 • C and 1100 • C[END_REF].

Conclusion

The benchmark concerning the potential alloy used for manufacturing the solar receiver module part in contact with high temperature air leads to the characterization of five superalloys and the final selection of the nickel-based alloy H230. However, the reference alloy 600 employed for making previous solar receiver is also well ranked. Optical properties evaluation in real conditions does not change the results.

The optical measurements carried out at high temperature on alloys 600, HX, H230 and HR120 samples show low differences in emissivity and solar absorptivity. Only alloy 617 properties are different (worse performance) that could be explained by the oxide morphology but also by the alloy supply (from a rod and not from a plate compared to the others). The evolution of the emissivity with temperature was explained through the characterization or the material surface using SEM, XRD and 3D profilometry, and linear laws were obtained for the total hemispherical emissivity up to 1300 K in high vacuum and up to 1400 K in air at atmospheric pressure, the solar absorptivity being quite constant in the same temperature range.

Table 6

Surface roughness of the alloys after exposure up to 1200 and 1400 K (Sa arithmetic and Sq mean square surface roughness).

  radiances to the spectral bands chosen.The measurements are carried out through an original 3-mirrors angular positioner (goniometer) developed at PROMES-CNRS -the sample, the spectro-radiometer and the optical window remaining fixed -at incidence angles from 0 • to 80 • by 10 • step plus 45 • and 75 • for samples of 40 mm diameter. The nature of the optical window is adapted to the wavelength range studied and in that case it is made of thallium iodo-bromide (KRS5). The spectro-radiometer -goniometer -optical window assembly is calibrated before the series of tests, in front of a blackbody whose temperature is measured with a standard pyrometer.

Fig. 2 .

 2 Fig. 2. Result of the benchmark of selected alloys.

Fig. 3 .

 3 Fig. 3. Photo and scheme of the experimental set-up MEDIASE at the focus of the 1000 kW solar furnace in the configuration for emissivity measurement.

Fig. 4 .

 4 Fig. 4. Total hemispherical emissivity versus temperature for the five alloys studied in high vacuum.

Table 2

 2 Parameters for the linear regression of emissivity versus temperature.

Fig. 6 .

 6 Fig. 6. Diffraction patterns of samples HX and H230 before (ref) and after heat treatment in high vacuum at T max = 1300 K.

Fig. 7 .

 7 Fig. 7. Directional emissivity versus wavelength during in situ oxidation -Example of alloy HR120 at 1080 K (up) and 1410 K (down).

Fig. 8 .

 8 Fig. 8. Hemispherical emissivity versus temperature during in situ oxidation of the several alloys up to 1400 K (plain dots for total hemispherical emissivity and open dots for solar absorptivity; triangles for alloy 600 and dots for alloy 617 in the upper graph).

Fig. 9 .

 9 Fig. 9. SEM images for the five alloys showing the evolution of the oxidized surfaces after high temperature exposure in air (on the left column, after exposure at 1200 K and on the right column after 1400 K).

Fig. 10 .

 10 Fig. 10. Smoothed diffraction patterns (20 • -45 • ) collected for heated samples in air (high: at 1400 K, low: at 1200 K). The symbols are associated to the following species: black triangle with tip up: alloy; cross: Cr 2 O 3 ; full dot: Fe 2 O 3 ; star: Fe 2.93 O 4 or Fe 3 O 4 ; black triangle with tip down: MoO 3 ; black diamond: W 3 O.

  2 O 3 and Fe 3 O 4 , molybdenum oxide Mo 3 O (ICDD 065-4549) and tungsten oxide W 3 O (ICDD 73-2526). The Fe 3 O 4 crystalline phase was mainly detected for the sample heated at 1400 K. For the sample heated at 1200 K, Fe 3 O 4 (oriented 311, 2θ = 35.38 • ) is not detected because the diffraction peak located at about 30.05 • (220) is missing. The feature located at 35.38 • also contains the contribution of the W 3 O diffraction peak (about 20%). Fe 2 O 3 was not clearly observed for the sample heated at 1200 K but two other metal oxide phases, W 3 O and Mo 3 O, were mainly detected at 1200 K.

Table 1

 1 List of the superalloys selected for the benchmark.

	Alloy type	Commercial	ISO	Major chemical
		reference	reference	elements
	Iron base alloys	800H	1.4876	Fe-Ni-Cr
		330	1.4886	Fe-Ni-Cr
	Nickel base	602 CA (6025HT)	2.4633	Ni-Cr-Fe-Al
	alloys	690	2.4642	Ni-Cr-Fe
		214	2.4646	Ni-Cr-Al
		617	2.4663	Ni-Cr-Co-Mo
		X	2.4665	Ni-Cr-Fe-Mo
		230	2.4733	Ni-Cr-W-Mo
		600 (ref.)	2.4816	Ni-Cr-Fe
		601	2.4851	Ni-Cr-Fe
		HR-120	2.4854	Ni-Fe-Cr
		HR-160	2.4880	Ni-Co-Cr-Si

Table 4

 4 Parameters for the linear regression for the total hemispherical emissivity.

	Alloys	Coefficient a	Coefficient b	R 2
	600	9 × 10 -4	-0.5681	0.8852
	617 HX H230 HR120	10 -3 9 × 10 -4 6 × 10 -4 6 × 10 -4	-0.5059 -0.4192 -0.1815 -0.2114	0.9036 0.8707 0.9532 0.7579

Table 5

 5 Ratio of the solar absorptivity to the total emissivity α/ε for four temperature levels.

	T (K)	600	617	H230	HX	HR120
	1100	1.63	1.37	1.53	1.55	1.52
	1200	1.48	1.20	1.30	1.43	1.43
	1300	1.23	1.05	1.25	1.24	1.25
	1400	1.09	1.04	1.10	1.05	1.06

Table 7

 7 Main oxide compounds present on the surface of the different alloys treated up to 1400 K.

	Alloy	600	617	H230	HX	HR120
		1200 K/1400 K	1200 K/1400 K	1200 K/1400 K	1200 K/1400 K	1200 K/1400 K
	Sa (μm)	1.39/1.38	1.46/1.53	1.30/1.36	1.44/1.52	1.24/1.33
	Sq (μm)	1.71/1.70	1.83/1.90	1.57/1.69	1.81/1.93	1.56/1.67
				Alloys	Main Crystalline phases
				HX	Cr 2 O 3 + Fe 2.936 O 4
				H120	Cr 2 O 3 (+Fe 2 O 3 or Fe 2.936 O 4 )
				H230	Cr 2 O 3 (+Fe 2 O 3 -Fe 3 O 4 )
				600	Cr 2 O 3
				617	Cr 2 O 3

Acknowledgments

The Polyphem project has received funding from the European Union's Horizon 2020 research and innovation programme under grant agreement n • 764048.

The authors want to also thank the programme "Investissements d'Avenir" (Investment for the Future) of the Agence Nationale de la Recherche (National Agency for Research) of the French State under award number ANR-10-EQPX-49-SOCRATE.

CRediT authorship contribution statement

Declaration of competing interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.


Won June Kim
email: wjkim@changwon.ac.kr

Adel Mesbah

Xavier Deschanels

Samuel Bernard

Sébastien Lebègue

First principles investigations of the optical selectivity of titanium carbide-based materials for concentrating solar power applications †

To improve the efficiency of concentrated solar power devices and to reach a high conversion rate of the solar energy into heat, a rational design of new high temperature absorber materials is a challenge of prime importance. Ultra-high temperature ceramics can fit the requirements to act as absorber materials, provided that their chemical composition and their properties are finely tuned. Within this context, we investigated the optical selectivity of titanium carbide modified with B, N, O impurity atoms or carbon vacancies, by first principles simulations. By computing the dielectric function and the reflectivity spectra of each composition, we found that modifying TiC can have important consequence on the optical properties of the bulk material, leading in certain cases to an enhancement of the optical selectivity. Then, we assess the performance of nanocomposites made of the corresponding nanoparticles embedded in a silicon carbide media by using Mie scattering theory fed by our ab initio dielectric functions. In this case, we found that while carbon vacancies are detrimental to selectivity, alloying TiC with boron improves notably the performance of the composite material, with an increase in selectivity of about 50 %.

Introduction

As the consideration about the global energy crisis and climate changes due to the use of fossil fuels increases, the development and use of efficient, renewable, and sustainable energy sources has become a great issue. 1 Among the various ways to solve such problems, solar energy is highly considered owing to the possibility that it offers for a massive generation of energy, for instance through concentrating solar power (CSP) technologies. 2,3 In CSP technologies, the sunlight is firstly concentrated on the absorber with the aid of several heliostats, followed by the conversion of the absorbed solar radiation into heat. Then, the converted heat is a Laboratoire de Physique et Chimie Théoriques LPCT (UMR CNRS 7019), Faculté des Sciences et Techniques, Université de Lorraine, BP70239, Boulevard des Aiguillettes, 54506 Vandoeuvre-lès-Nancy CEDEX, France. E-mail: sebastien.lebegue@univlorraine.fr transferred into a fluid to operate the turbine of the power plant. Therefore, the design of an efficient absorber material is a keypoint for a successful CSP technology, with the aim to maximize absorption of sunlight and to minimize infrared (IR) emittance at the same time. Furthermore, since the Carnot efficiency increases as the working temperature increases, therefore a high mechanical stability and resistance to oxidation are also required for good solar absorber materials.

In this regard, silicon carbide (SiC) has been commonly used as the absorber material due to its high absorptance and high resistance to oxidation up to 1400 • C. However, despite good sunlight absorption, SiC has a high thermal emittance, leading to a poor optical selectivity. 4,5 Thus, the main challenge is the development of novel absorber materials able to resist to damage at very high temperatures while keeping good spectral selectivity, good thermal conductivity and favorable radiative properties at the operating temperatures. Ultra-high temperature ceramics (UHTCs) could be the most appropriate candidates as solar absorber materials. UHTCs correspond to borides, carbides, and nitrides of transition metals in column IV, with particular chemical properties that show covalent and metallic characters simultaneously. 6 Because of this bonding character, UHTC materials show both a high reflectivity in the IR region and a high absorptance in the UV-vis-NIR region, and the resulting optical properties are therefore spectrally selective. Also, their covalent bonding leads to a high melting temperature and makes them physically very stable. Among various UHTC materials, titanium carbide (TiC) is one of the most important compound. However, the major limitation of TiC is its low resistance to oxidation, therefore it usually required to protect it with a matrix while forming composite materials. 7 In particular, embedding TiC nanoparticles in a protective silicon carbide (SiC) matrix can compensate the limitations of the two materials as solar absorbers working at high temperature.

In spite of the many experimental studies on the optical properties of different UHTC materials and their nanocomposites, theoretical studies so far focused on the stability and electronic properties of such materials and their alloys. [8][9][10][11][12][13][14][15][16][17][18][19][20][21][22] To our knowledge, only few theoretical studies have been conducted to explore the optical properties of pure UHTC materials, [23][24][25][26][27] and there are no theoretical studies on the effect of alloying elements or impurities on their optical properties. In this paper, we study the optical properties of TiC and the effect of alloying elements B, N, O or carbon vacancy using first principles calculations. Our choice is motivated by the fact that TiC has a strong tendency to form non stoichiometric compositions, 28 while B, N, and O atoms have a size close to the one of carbon, and therefore upon alloying the crystal structure of TiC will be mostly preserved. We searched various compositions based on TiC up to four-component alloy systems, then assessed their performance for CSP absorbers by investigating their reflectivity spectra. Then, to be as close as possible to experimental conditions, we studied their spectral selectivity when they are embedded as nanoparticles in a SiC substrate.

Computational Details

Our calculations have been performed using density functional theory (DFT) with the projector augmented wave method as implemented in the Vienna ab initio simulation package (VASP), 29,30 except for the calculation of the electron-phonon interaction part of the electron scattering rate (see below). The generalized gradient approximation (GGA) designed by Perdew, Burke, and Ernzerhof (PBE) was used for the exchangecorrelation functional, 31 and an energy cutoff of 550 eV was chosen for the the plane wave basis set. The electronic total energy has been converged until the total energy difference between electronic steps became smaller than a threshold of 10 -7 eV, and a geometry optimization was performed until all atomic forces became smaller than 5.0 × 10 -3 eV/Å. For the geometry relaxation, we used a Γ-centered k-mesh with a grid step of 0.2 Å -1 , which corresponds to 13 × 13 × 13 and 7 × 7 × 7 k-points for the primitive cell of TiC and its 2 × 2 × 2 supercell, respectively.

Experimentally, it has been shown that carbon vacancies 28,32 or the incorporation of other 2p-block elements [32][33][34][35] can be realized in TiC. Following these facts, our structural models have been designed using the following procedure: firstly, we obtained the equilibrium lattice parameter of TiC with the rock-salt structure (space group No. 225, Fm 3m). Our calculated lattice parameter a = 4.33 Å is close to the experimental value of 4.329 Å. 36 Then to study the effect of alloying elements on the optical properties of TiC, we constructed a 2 × 2 × 2 supercell to model the following systems: 1) singly doped systems (see Figure 1

V C -V C , B-B, N-N, O-O, V C -B, V C -N, V C -O, B-N, B-O, or N-O pairs of atoms.
For the doubly doped systems, when considering the crystal symmetry of TiC, there are only two possible configuration for the dopants: nearest neighbor (NN) or next nearest neighbor (2NN), respectively. We relaxed the two possible geometries for each system, and then the most stable configuration was selected for the calculation of the optical properties.

To study the optical properties of TiC and its alloys, we calculated the frequency dependent dielectric function, ε(ω) = ε (1) (ω) + iε (2) (ω). For metallic systems, the total dielectric function can be divided into interband and intraband terms, ε(ω) = ε inter (ω) + ε intra (ω). The first term is written in the independent particle approximation (IPA) and in the limit of long wavelength as:

ε (2),inter αβ (ω) = 4π 2 e 2 Ω lim q→0 1 q 2 ∑ c,v,k 2w k δ (E ck -E vk -ω) × u ck+e α q |u vk u vk |u ck+e β q (1)
where e is the electron charge, Ω is the cell volume, w k is the weight of the k-points, e i is a unit vector in cartesian coordinates, and c and v are the indices for the conduction and valence states, whereas u jk and E jk are the one-electron Blöch functions and eigenvalues, respectively. Then we can apply a Kramers-Kronig relation to calculate the real part, ε (1),inter (ω), by using:

ε (1),inter αβ (ω) = 1 + 2 π P ∞ 0 ε (2),inter (ω )ω ω 2 -ω 2 + iη dω ( 2
)
where P is the principal value and η is an infinitesimal complex shift. It is noteworthy to mention that our calculated dielectric function could be affected by the level of theory used, within DFT or beyond, like with the GW approximation. Here, the metallic nature of TiC makes the effect of the self-energy correction on optical properties negligible, with a GW density of states very close to the one obtained with GGA (data not shown here). A similar finding has been made by M. Kumar et al. 26 , for Au and TiN. Therefore, all the dielectric functions in this study have been calculated within the GGA formalism.

For the intraband contribution to the dielectric function, we employed the Drude expression: 26,37,38 for a given plasma frequency, ω p , and a scattering rate, γ, ε intra (ω) can be approximated by the following equation:

ε intra (ω) = 1 - ω 2 p ω 2 + iγω (3)
where the plasma frequency can be obtained from the electronic structure by tracing the following tensor: 38

ω 2 p,αβ = - 4πe 2 Ω ∑ v,k 2 ∂ w k (E v ) ∂ E v ∂ E vk ∂ k e α ∂ E vk ∂ k e β (4)
The second parameter, the electron scattering rate, can be calculated from the imaginary part of the electron-phonon selfenergy, Σ nk (T ). 27,38,39 From the electronic band structure and phonon dispersion, the self-energy is expressed as: 40

Σ nk (T ) = 2π h ∑ qν,m w q |g SE mn,ν (k, q)| 2 × n qν (T ) + f mk+q (T ) E nk -E mk + ω qν -iη + n qν (T ) + 1 + f mk+q (T ) E nk -E mk -ω qν -iη (5)
where ω qν is the phonon frequency of branch ν and wave vector q, n qν (T ) is the Bose-Einstein distribution, and f mk+q (T ) is the Fermi-Dirac distribution. g SE mn,ν (k, q) is the electron-phonon matrix element. Then the scattering rate of electron in the band n and wave vector k is given by γ ep (nk, T ) = 2ImΣ nk (T ). For the calculation of the dielectric function, we averaged γ ep (nk, T ) over the Fermi surface as follows: 41,42

γ ep (T ) = +∞ -∞ γ ep (ε, T ) - ∂ f ∂ ε dε ( 6
)
where γ ep (ε, T ) is the energy-dependent scattering rate obtained by integrating γ ep (nk, T ) over the first Brillouin zone. An other contribution to the electron scattering rate is due to electron-electron scattering, which can be modeled from the selfenergy in the GW approximation. In the study of M. Xu el al., 38 the effect of electron-electron scattering rate on the total dielectric functions of noble metals (Au, Ag, and Cu) has been shown to be negligible. Likewise, our test showed that the contribution of the electron-electron scattering rate in pure TiC is also small. Therefore, we neglected this contribution in the calculations of the other compounds.

The calculation of the electron-phonon interaction has been done by using the Quantum Espresso code 43,44 and the EPW 40,45 package. The plane waves were expanded up to a kinetic energy of 70 Ry, and the same k-grid as in the VASP calculations was used for the self-consistent calculations. The phonon dispersion has been obtained with a 3 × 3 × 3 q-grid, then the electronphonon coupling has been calculated with a finer BZ sampling

Results and Discussion

Bulk electronic and dielectric properties

Table 1 shows our calculated plasma frequencies and scattering rates for TiC and its alloys. The plasma frequency of pure TiC is 3.55 eV, which is similar to the previous result of Kumar et al. 26 Compared with pure TiC, the compounds with a carbon vacancy (V C ), i.e., Ti 8 C 7 , Ti 8 C 6 and Ti 8 C 6 X (X = B, N, O) show smaller ω p values due to the deficiency of electrons. On the other hand, there are two behaviors when the carbon atoms are substituted with other elements without a vacancy: the plasma frequencies of Ti 8 C 7 X and Ti 8 C 6 X 2 are larger than the one of pure TiC, whereas the substitution of two different elements (BN, BO, and NO pairs) does not present a systematic trend. For singly-doped systems, the plasma frequency increases as the number of electrons increases, i.e., from Ti 8 C 7 B to Ti 8 C 7 O. For doubly-doped systems with different dopants, the same trend is seen with ω p increasing from BN to BO and NO. Concerning the scattering rates, it is seen from our results that they are always larger for the alloys than for pure TiC, a fact that can be intuitively understood by the increased disorder in the alloys in comparison with the perfect TiC crystal.

Using the parameters presented in table 1 and the interband contribution of the dielectric function, we calculated the total dielectric function for the whole set of compounds investigated here, the results being presented in Figure 2. Among the compositions that we studied, the dielectric functions of pure TiC and (V C )-containing TiC compares well with available experimental data: for instance, Pflüger and co-workers have measured the bulk dielectric function of stoichiometric TiC by electron-energyloss spectroscopy (EELS). 46 Also, Koide and co-workers studied the effect of the concentration of (V C ) on the dielectric properties of TiC. 28 Here, Ti 8 C 6 has a vacancy concentration close to the TiC 0.6 sample of their work.

In terms of the optical selectivity of the photothermal absorber, an important parameter is the frequency where the total reflec- 1 shows the value of ω c for each system. The crossover frequency of pure TiC, 0.70 eV, is in good agreement with the value (0.68 eV) of Kumar et al. 26 By comparing with pure TiC, the effect of alloying can be classified following three patterns. Firstly, when one or two carbon vacancies are introduced, the crossover frequency is either decreased or stable, with Ti 8 C 6 O, which has a value of ω c of 0.93 eV, being an exception. Secondly, the crossover frequency is increased when the carbon atoms are substituted by an other single type of element (Ti 8 C 7 X and Ti 8 C 6 X 2 ), with a larger increase of ω c by doping with B or O than with N. The last case is when doping with two different elements, the crossover frequency increases in the order BN < BO < NO, following the trends of both the plasma frequency and the scattering rate.

Figure 3 shows our calculated reflectivity spectra for TiC and its alloys, with the frequency-dependent reflectivity calculated as :

R (ω) = ε 1/2 (ω) -1 ε 1/2 (ω) + 1 2 (7)
Henning and Mutschke have measured the reflectivity spectrum of bulk TiC using infrared reflectivity spectroscopy. 47 More recently, Aréna and co-workers conducted more elaborate measurements of the reflectivity spectra, as varying the relative density of the TiC sample. 48 Our calculated reflectivity spectrum of TiC shows a good agreement with their experimental data (See figure S1 in the ESI † for direct comparison.). For photothermal applications, an optimal selectivity at room temperature (which is the temperature at which our phonons calculations were conducted) is obtained when a relatively sharp drop of the reflectivat around 2.5 µm is observed. 1 While certain systems (Ti 8 C 6 , Ti 8 C 6 B, Ti 8 C 6 N, or Ti 8 C 6 O) show a reflectivity in the IR region which is roughly similar to the one of TiC, for Ti 8 C 7 and Ti 8 C 6 BN) it appears to be significantly lower. However, for Ti 8 C 7 B, and Ti 8 C 6 BO a small increase of the reflectivity around λ = 2.5 µm can be observed, while for the remaining compounds (Ti

8 C 7 N, Ti 8 C 7 O, Ti 8 C 6 B 2 , Ti 8 C 6 N 2 , Ti 8 C 6 O 2 ,
and Ti 8 C 6 NO), the changes are more drastic, with either a displacement of the transition towards shorter wavelengths and/or an increased value of the IR reflectivity, showing that a modification of TiC can affect significantly its reflectivity.

Optical properties of nanoparticles

For applications in concentrating solar power (CSP) units, transition metal carbides are usually included in the form of nanoparticles in a matrix, such as SiC. 48,49 Since such systems cannot be directly investigated using density functional theory for computational reasons, we have investigated the optical properties of the nanoparticles using the scattering theory of Mie, 50,51 which we outline briefly below. When the size a of a nanoparticle is comparably smaller than the wavelength of the incident light λ , Mie theory states that the extinction, scattering, and absorption cross sections (Q ext , Q sca , and Q abs , respectively) of this particle can be calculated using the following equations:

Q ext = 2 x 2 ∞ ∑ L=1 (2L + 1) Re [a L + b L ] (8)
Q sca = 2 x 2 ∞ ∑ L=1 (2L + 1) |a L | 2 + |b L | 2 (9)
Q abs = Q ext -Q sca (10)
In the equations above, the coefficients a L and b L are written by using spherical Riccati-Bessel functions Ψ L and ξ L and their derivatives:

a L = mΨ L (mx) Ψ L (x) -Ψ L (x) Ψ L (mx) mΨ L (mx) ξ L (x) -ξ L (x) Ψ L (mx) (11)
b L = Ψ L (mx) Ψ L (x) -mΨ L (x) Ψ L (mx) Ψ L (mx) ξ L (x) -mξ L (x) Ψ L (mx) (12)
where x = 2πa/λ is the reduced particle size and m = n met /n die is the ratio between the refraction index of the nanoparticle and that of the surrounding dielectric media. With the help of the package PyMieScatt, [START_REF] Sumlin | [END_REF] the scattering and absorption cross section, Q sca and Q abs , have been calculated by using the complex refractive index obtained form the dielectric function presented earlier in figure 2. For the host material surrounding the nanoparticle, we set ε ∞ = 2.63, which corresponds to SiC. The diameter of the nanoparticles was set to 50 nm.

Figure 4 shows our calculated scattering and absorption cross section of nanoparticles of TiC and its alloys. Comparing with the reflectivity spectra in figure 3, two patterns are observed in the behavior of Q abs . Firstly, if the reflectivity increases slowly, as in pure TiC, Q abs has only one peak below 1 µm. Ti 8 C 6 BN and the systems with a C-vacancy (except for Ti 8 C 6 N) show this behavior. On the other hand, if the reflectivity decreases rapidly, there is a second peak between 1 µm and 2.5 µm. This second peak is observed when nitrogen or oxygen (or both of them) is present in the composition. Some systems, including boron-doped compositions and Ti 8 C 6 N, show an intermediate behavior: only a small shoulder appears after the first peak. In terms of reflectivity, this behavior is related to the small increase of the reflectivity in the IR region compared to that of pure TiC.

The selectivity of the different compounds can be directly accessed if we use a parameter that can quantify the absorbance in the UV/visible versus the absorbance in the IR region. This can be achieved by integrating Q abs , weighted by either the solar and or blackbody intensities as:

I solar abs = λ 2 λ 1 Q abs (λ ) P (λ ) dλ (13)
I BB abs = λ 2 λ 1 Q abs (λ ) B (λ ) dλ (14)
where P (λ ) and B (λ ) are the solar and blackbody spectrum, respectively. To cover both the UV/visible and IR regions, the in- tegration has been done for a range between 0.28 and 16 µm, and with a blackbody temperature of T = 300 K, which corresponds mostly to infrared radiations. Then, a large I solar abs /I BB abs will correspond to a high selectivity .

Figure 5 shows I solar abs /I BB abs for TiC and its alloys in the form of nanoparticles. Compared with pure TiC, the alloys containing carbon vacancies as well as Ti 8 C 6 BN show a very low optical selectivity. This is due to two factors: 1) a lower value of the maximum of Q abs compared to TiC and 2) a more smeared Q abs above 2.5 µm compared to TiC (see figure 4). While most of the substituted alloys shows a selectivity similar to that of TiC, Ti 8 C 6 B 2 shows a noticeably larger selectivity. In this case, Q abs is on the overall larger than for TiC in the UV/visible region (λ < 2.5 µm), and also shows very small area in the IR region (λ > 2.5 µm). These facts leads to both a larger UV/visible absorbance and a lower IR emission, making the composition Ti 8 C 6 B 2 the best candidate for a photothermal absorber material among the formulations that we have investigated, with a selectivity which is roughly 50 % better than the one of TiC.

Conclusion

In this work, we have studied the effect of doping on the optical properties of TiC using first principles calculations. By substituting carbon by B, N, O atoms or by introducing a carbon vacancy, we calculated the dielectric function and reflectivity spectra of TiC and its alloys in the bulk phase. For the 15 different compositions of TiC and its alloys, the crossover frequency for the real part of dielectric function is highly related to the transition of reflectivity near λ = 2.5 µm, at which the reflectivity should be truned off for the ideal photothermal absorbers; the higher the crossover frequency is, the better the reflectivity behavior is. From this analysis, we found that the IR reflectivity decreases when the carbon vacancy is introduced.

Then, the overall performance of the different compounds has been accessed through Mie scattering theory by simulating the properties of the corresponding nanoparticles embedded in a SiC host. It was shown that while carbon vacancies are detrimental to the selectivty of the material, the Ti 8 C 6 B 2 composition shows a selectivity which is 50 % larger than the one of TiC, a fact which is promising in the aim of increasing the efficiency of concentrating solar power devices, and can serve as a guideline for future experimental studies in this field.

  (a)) in which one carbon atom is replaced with a vacancy (V C ) or a B, N, or O atom,

Fig. 1

 1 Fig. 1 Atomic structures of (a) singly and (b) doubly doped configurations of TiC used in the present study.

Fig. 2

 2 Fig. 2 Calculated total dielectric functions of TiC and its alloys. The black and red solid lines correspond to the real and imaginary parts of the dielectric function, respectively. The magenta dashed line is to guide to the eyes corresponding to zero.

Fig. 3

 3 Fig. 3 Calculated reflectivity spectra of TiC and its alloys, as a function of wavelength. The magenta dashed line in each plot is the wavelength where the reflectivity should turn off for an ideal photothermal absorber (λ = 2.5 µm.)

Fig. 4 Q

 4 Fig.4 Q abs (red solid lines) and Q sca (black solid lines) spectra for a 50 nm nanoparticle of TiC-based compound embedded in a SiC matrix. The magenta dashed line in each plot has the same meaning as in the Figure3

Fig. 5

 5 Fig. 5 Values of I solar abs /I BB abs for different TiC-based nanoparticles embedded in SiC matrix at 300 K.

Table 1

 1 Configuration, plasma frequencies, scattering rate, and crossover frequency of TiC and its alloys.Composition Configuration ω p (eV) γ ep (meV) ω c (eV)

	Ti 8 C 8	-	3.55	17.09	0.70
	Ti 8 C 7	-	2.22	26.34	0.31
	Ti 8 C 7 B	-	3.93	31.24	1.11
	Ti 8 C 7 N	-	3.99	28.74	0.90
	Ti 8 C 7 O	-	4.43	45.91	1.11
	Ti 8 C 6	NN	3.38	54.86	0.79
	Ti 8 C 6 B 2	NN	4.88	35.71	1.21
	Ti 8 C 6 N 2	2NN	4.63	48.85	1.14
	Ti 8 C 6 O 2	NN	4.71	51.42	1.54
	Ti 8 C 6 B	NN	3.45	41.51	0.72
	Ti 8 C 6 N	NN	3.37	57.01	0.54
	Ti 8 C 6 O	NN	3.40	53.20	0.93
	Ti 8 C 6 BN	NN	2.92	23.80	0.70
	Ti 8 C 6 BO	NN	3.64	28.25	0.95
	Ti 8 C 6 NO	NN	4.70	52.66	1.36

(12 × 12 × 12 k-points and 20 × 20 × 20 q-points) through a Wannier functions representation.

| 1-8

| 1-8

K. Xu, M. Du, L. Hao, J. Mi, Q. Yu and S. Li, J. Materiomics,

2020, 6, 167-182. 2 C. Granqvist, Adv. Mater., 2003, 15, 1789-1803. 3 Y. Tian and C. Zhao, Applied Energy, 2013, 104, 538-553.

J o u r n a l N a me , [ y e a r ] , [ v o l . ] ,

Acknowledgements

The authors acknowledge financial support from the Agence Nationale de la Recherche within the CARAPASS ("Carbide & Carbonitride nanocomposite based photothermal solar absorbers") project (Grant No. ANR-16-CE08-0026).

Notes and references

Author contributions

W.J.K. carried out the calculations. W.J.K. and S. L wrote the first version of the manuscript. All authors discussed the results and contributed to the final manuscript.

Conflicts of interest

There are no conflicts to declare.