Sébastien Quenard
email: sebastien.quenard@cea.fr

Marilyne Roumanie
email: marilyne.roumanie@cea.fr

A Simple Method for a Protective Coating on Stainless Steel against Molten Aluminum Alloy Comprising Polymer-Derived Ceramics, Oxides and Refractory Ceramics

Keywords: coating, polymer-derived ceramics, high temperature, corrosion barrier, liquid aluminum alloy

A new coating based on polymer-derived ceramics (PDC), oxides and refractory ceramic with a thickness of around 50 μm has been developed to improve the resistance corrosion of stainless steel substrate against molten aluminum alloy in a thermal energy storage (TES) system designed to run at high temperature (up to 600 °C). These coatings implemented by straightforward methods, like tape casting or paintbrush, were coated on planar and cylindrical stainless-steel substrates, pyrolyzed at 700 °C before being plunged for 600 and 1200 h in molten AlSi12 at 700 °C. The stainless-steel substrate appears healthy without intermetallic compounds, characteristic of molten aluminum alloy corrosion. The protective coating against AlSi12 corrosion shows excellent performance and appears interesting for TES applications.

Introduction

Renewable energies such as solar energy are intermittent resources that can induce a mismatch between supply and demand and constitute a limit to their use. Thermal energy storage (TES) is essential in increasing the supply and use of renewable energy and reducing the carbon footprint. The integration and utilization of latent thermal energy storage (LTES) with heat recovery systems is the most potential and cost-effective solution. Due to the large energy storage density of metallic phase change materials (PCM), combining solar power plants with LTES is the most effective method to provide flexible electricity to the grid and supply large-scale power services [START_REF] Li | Applications and technological challenges for heat recovery, storage and utilisation with latent thermal energy storage[END_REF].

Many PCMs have been reported in the literature for concentrating solar power (CSP) such as Mg-51% Zn [START_REF] Blanco-Rodríguez | Thermophysical characterization of Mg-51%Zn eutectic metal alloy: A phase change material for thermal energy storage in direct steam generation applications[END_REF], light-weight alloys based on Mg-Zn-Al [START_REF] Risueño | Experimental investigation of Mg-Zn-Al metal alloys for latent heat storage application[END_REF], CaSi (melting temperature (Tm) of 782 °C) [START_REF] Andraka | Metallic Phase Change Material Thermal Storage for Dish Stirling[END_REF], CuMgSi (Tm of 742 °C) [START_REF] Andraka | Metallic Phase Change Material Thermal Storage for Dish Stirling[END_REF], eutectic alloy compositions based on 88Al-12Si (AlSi12-Tm of 577 °C) [START_REF] Khare | Selection of materials for high temperature latent heat energy storage[END_REF], and 60Al-34-Mg-6Zn (Tm of 454 °C) [START_REF] Sun | Thermal reliability test of Al-34%Mg-6%Zn alloy as latent heat storage material and corrosion of metal with respect to thermal cycling[END_REF].

AlSi12 alloy is considered to be a promising metallic PCM [START_REF] Kotzé | High Temperature Thermal Energy Storage Utilizing Metallic Phase Change Materials and Metallic Heat Transfer Fluids[END_REF]. It has a low melting temperature [START_REF] Wang | Experimental research on a kind of novel high temperature phase change storage heater[END_REF], is stable during heating and cooling cycles [START_REF] Li | The Influence of Heating-Cooling Cycles on the Thermal Storage Performances of Al-17 wt.% Si Alloy[END_REF], has high thermal conductivity (190 W.m -1 .K -1 at 577 °C [START_REF] Kotzé | High Temperature Thermal Energy Storage Utilizing Metallic Phase Change Materials and Metallic Heat Transfer Fluids[END_REF]), high latent heat of fusion of 548.6 J.g -1 at 577 °C [START_REF] Li | The Influence of Heating-Cooling Cycles on the Thermal Storage Performances of Al-17 wt.% Si Alloy[END_REF] and is low cost.

Liquid aluminum alloys are materials known to be extremely corrosive to most metals and metal oxides [START_REF] Deqing | Phase evolution of an aluminized steel by oxidation treatment[END_REF], which is critical in a TES system designed in stainless steel and working at a temperature above 600 °C. Corrosion issues in TES systems conditions are hardly addressed in the literature [START_REF] Kotzé | Thermal Energy Storage in Metallic Phase Change Materials[END_REF], although the literature is well documented concerning the development of corrosion-resistant materials. Among the investigations, boronized carbon steel showed good resistance to corrosion [START_REF] Tsipas | Degradation behaviour of boronized carbon and high alloy steels in molten aluminium and zinc[END_REF] following 120 h at 630 °C in molten aluminum. This working time is, however, limited compared to that of the TES systems.

Fukahori et al. [START_REF] Fukahori | Thermal analysis of Al-Si alloys as high-temperature phase-change material and their corrosion properties with ceramic materials[END_REF] introduce the high corrosion resistance of ceramic substrates such as alumina (Al2O3), aluminum nitride (AlN), and silicon nitride (Si3N4) to molten Al-Si alloys. However, these interesting materials for the LTES are brittle, expensive, and difficult to deposit with low-cost processes. To make these materials suitable for the application, a coating of iron container surface with a protective layer of Al2O3 has been proposed [START_REF] Kotzé | Thermal Energy Storage in Metallic Phase Change Materials[END_REF]. The method consists of hot dipping the steel into molten aluminum. Molten aluminum in contact with iron creates iron aluminide (intermetallic compound). The surface is then oxidized to obtain the protective ceramic layer [START_REF] Kotzé | Thermal Energy Storage in Metallic Phase Change Materials[END_REF]. Although the Al2O3 layer shows excellent corrosion resistance at 250 °C for 250 h, no information is given for applications working above 600 °C. The sol-gel process enables the synthesis of ceramic material as silica (SiO2), zirconia (ZrO2), Al2O3, titanium oxide (TiO2). These materials, easily deposited on surfaces with inexpensive processes, showed excellent chemical stability, and improve the corrosion resistance of metal substrates (steel, aluminum, and their alloys) at low temperatures [START_REF] Wang | Sol-gel coatings on metals for corrosion protection[END_REF].

Polymer-derived ceramics (PDC) [START_REF] Colombo | Polymer-Derived Ceramics: 40 Years of Research and Innovation in Advanced Ceramics[END_REF] are organic/inorganic polymers, forming amorphous ceramics after pyrolysis at temperatures between 600 and 1000 °C. PDC are easy to apply on substrates of any shape by dip coating, spray-coating, spin-coating, or tape casting. PDC composite coatings are described in literature as promising candidates to be used as alternative environmental barrier coatings for corrosion and oxidation protection of metals at elevated temperatures (600-1000 °C) [START_REF] Günthner | Advanced coatings on the basis of Si(C)N precursors for protection of steel against oxidation[END_REF]. The main drawback of PDC technology is the unavoidable shrinkage which occurs due to the large density change when the polymer precursor with a typical density of 1 g.cm -3 , is converted to the ceramic product, often with a density above 2 g•cm -3 [START_REF] Nguyen | Novel polymer-derived ceramic environmental barrier coating system for carbon steel in oxidizing environments[END_REF]. The volume shrinkage can be greater than 50%. Active or passive fillers are, therefore, added to limit shrinkage. In the literature based on oxygen corrosion, the addition of glass in the PDC slurry can be noticed [START_REF] Nguyen | Novel polymer-derived ceramic environmental barrier coating system for carbon steel in oxidizing environments[END_REF][START_REF] Parchovianský | PDC Glass/Ceramic Coatings Applied to Differently Pretreated AISI441 Stainless Steel Substrates[END_REF][START_REF] Parchovianský | Passive filler loaded polysilazane-derived glass/ceramic coating system applied to AISI 441 stainless steel, part 2: Oxidation behavior in synthetic air[END_REF]. This glass improves the adhesion of the PDC-containing layer to the substrate and thus improves the corrosion protection of the metal. Al2O3 [START_REF] Nguyen | Novel polymer-derived ceramic environmental barrier coating system for carbon steel in oxidizing environments[END_REF] or Al2O3-Y2O3-ZrO2 [START_REF] Parchovianský | PDC Glass/Ceramic Coatings Applied to Differently Pretreated AISI441 Stainless Steel Substrates[END_REF][START_REF] Parchovianský | Passive filler loaded polysilazane-derived glass/ceramic coating system applied to AISI 441 stainless steel, part 2: Oxidation behavior in synthetic air[END_REF] passive fillers are also blended to minimize the shrinkage and for its good antioxidation properties at high temperature. These, although interesting, composite PDCs have not been evaluated in the presence of molten aluminum.

Commercial corrosion protection solutions have also been developed for the aluminum foundry industry (Condat, Dycote, Aluminium Martigny, Chimilin, France). These materials are based on graphite, boron nitride, and refractory powders. However, these solutions have not been listed in the TES literature. A preliminary study based on commercial materials was, therefore, carried out and detailed in the Appendix A. These materials are easy to process and inexpensive. An investigation at 700 °C in molten AlSi12 of these materials showed poor adhesion, delamination, or lack of corrosion resistance of the coating on the stainless-steel TES substrate. However, this study confirmed the interest in using the boron nitride material to fight against corrosion of aluminum alloys.

The primary object of this study is to develop a coating on 304L stainless steel with a high corrosion protection in presence of molten AlSi12 at 700 °C for 600-1200 h. Besides, to be easy to implement and economically attractive, the relevant coating needs to have the following properties: (i) good adhesion at high temperature, delamination-and crack-free; (ii) low wettability to molten aluminum alloy (AlSi12); (iii) good durability in molten aluminum alloy (AlSi12).

The composition of the protective layer is based on polymer-derived ceramic, glass frit and passive filler with low interaction with molten aluminum. The thermal behavior, wettability and thermal conductivity of the protective material were first characterized. This material was then applied to 304L stainless steel substrates, and its behavior in the presence of molten AlSi12 was evaluated.

Materials and Methods

Protective Material

Polysilsesquioxane Silres MK (23 wt% Wacker AG, Munich, Germany), a polymerderived ceramic (PDC) in the solid-state used as a binder in the formulation, was dissolved in a solvent (43 wt% Diestone ® Socomore, Vannes, France). The mixture was blended with a glass frit (28 wt%) based on zinc oxide (major elements) and a boron nitride powder having a mean particle size of 0.5 μm (6 wt% Momentive Grade AC6111, NY, USA). The PDC composite slurry was homogenized for 1 h by rotation in a closed container. This PDC has a melting temperature of around 46 °C, which facilitates its shaping. Finally, this preceramic polymer crosslinks from 200 °C and up to 300 °C to become an infusible material. The ZnO-based glass frit was chosen for its high ZnO content (> 30 wt%-ZnO is known to have a good elasticity behavior) and for having a low glass transition temperature (Tg = 475 °C). Preliminary tests with PDC and glass frit layer were carried out and are described in Appendix A. Despite a good adhesion, this corrosion barrier was too reactive with molten AlSi12 and was not effective (total loss of the initial layer). To overcome the issue, boron nitride filler was blended to the PDC and the glass frit. BN is known for its high thermal conductivity and for low wettability to the molten metal.

Methods

Protective Material Characterization

The PDC composite slurry was spread on a non-wetting polyethylene terephthalate (Mylar) and dried at room temperature. Dried pieces of PDC composite material were then ground using a planetary ball mill (Pulverisette 6, FritschGmbH, Idar-Oberstein, Germany), for 10 min with a fixed speed rate of 200 rpm. The jar and the grinding media were made of TZ3Y (polycrystalline tetragonal zirconia stabilized with 3 mol% of Y2O3). Grinding media were 2 mm diameter balls and the volume of the jar was 500 mL. PDC composite powder was then sieved at 200 μm. The resulting powder was shaped by thermo-pressing at a temperature of 205 °C and a pressure of 300 bar. The pellets were then pyrolyzed in argon at 700 °C for 1 h. The coefficient of thermal expansion (CTE) and the thermal conductivity of the protective material were determined from these pellets.

The CTE, the glass transition temperature, and the softening point of the PDC composite were performed by dilatometry (SETARAM Kep technologies, Caluire, France -SETSYS thermomechanical analyzer) on the temperature range from 20 to 700 °C with a rate of 3 °C.min -1 . Two CTE measurements were done, the first time after one thermal cycle up to 700 °C and the second time after five thermal cycles up to 700 °C on a pellet (diameter of 8 mm/height of 7 mm). The CTE was calculated from the following Equation (1) between 20 and 500 °C:

𝛼 1 𝐿 𝐿 𝐿 𝑇 𝑇 (1)
where 𝛼 (10 -6 •K -1 ) is the coefficient of thermal expansion, 𝑇 and 𝑇 (K) the initial temperature and the final temperature in the selected temperature range, and 𝐿 and 𝐿 (mm), respectively, thicknesses at 𝑇 and at 𝑇.

Thermal conductivity was measured by the Hot Disk Transient Plan Source (TPS) method according to ISO 22007-2 by the company Thermoconcept [20]. This method described in detail by He et al. [START_REF] He | Rapid thermal conductivity measurement with a hot disk sensor: Part 1. Theoretical considerations[END_REF] is a nondestructive, direct, and fast technique. The Hot disk sensor is placed between two PDC composite pellets (diameter of 25 mm, height of 8 mm). This sensor serves both as a heat source and a temperature sensor. Specifically, the variation of temperature is determined from the change in resistance via a Wheatstone bridge (2):

𝑅 𝑡 𝑅 1 𝛼∆𝑇 𝑡 (2)
where 𝑅 is the total electrical resistance at time 𝑡, 𝑅 the initial resistance at 𝑡 0, 𝛼 the temperature coefficient resistivity of the nickel (material of the sensor), and the ∆𝑇 change in temperature between 𝑡 0 and 𝑡.

The average temperature increases in the sensor surface following Equation (3):

∆𝑇 𝜏 𝑃 𝜋 𝑎𝜆 𝐷 𝜏 ( 3
)
where 𝑃 is the power output of the sensor, a the radius of the largest ring, 𝜆 the thermal conductivity, and 𝐷 𝜏 is a function proportional to the temperature rise depending on a dimensionless parameter 𝜏 √𝜅𝑡𝑎 with 𝜅 the thermal diffusivity.

The thermal conductivity of protective material was determined at room temperature, 300 °C and 600 °C in air.

Protective Coating Preparation and Tests in Molten AlSi12

The slurry was spread on a 304L stainless steel substrate, a material used for storage applications, to assess the protective coating behavior in operation. The stainless-steel samples had a surface roughness of 3.3 μm. The metallic substrates were previously degreased with acetone and immersed in an ultrasonic ethanol bath. The protective layer, in the first step, was deposited by a doctor blade on planar 304L substrate. The layer was crosslinked at 200 °C and pyrolyzed for 1 h at 700 °C in argon. Stainless-steel planar substrates coated with the protective layer were inserted in an alumina crucible containing AlSi12 metal to assess the corrosion behavior (Figure 1). The crucible was sealed with hightemperature ceramic glue and then placed in a furnace. The temperature of the furnace was set at 700 °C and the heating rate at 5 °C per minute. At this temperature, the AlSi12 is molten. The contact of the liquid aluminum with the sample was performed for 600 and 1200 h. In the second step, the protective layer was applied by paintbrush on cylindrical components, representative of the application (diameter of 54 mm, height of 200 mm). The layer inside the tube was homogenized by rotation using a roller system. The crosslinking at 200 °C in air and the pyrolysis at 700 °C in argon of the layer was performed on these different parts. This process, inexpensive and straightforward, should be easily implemented into a storage system. The tube was filled with AlSi12 alloy, closed with the lids, and treated for 600 h at 700 °C in a furnace (Figure 2).

Characterizations

The microstructure of the protective layer after coating on the 304L substrate and after static immersion in AlSi12 during 600 and 1200 h was investigated by scanning electron microscopy (Philipps XL30 SEM, Eindhoven, The Netherlands). In parallel to SEM observations, energy dispersive X-ray spectroscopy (EDS, Esprit, Bruker Nano GmbH, Berlin, Germany) analyses in map mode were completed on samples in contact with AlSi12 molten alloy. This was especially retained to investigate the aluminum distribution. The surface topography and roughness, evaluated as a root-mean-square (RMS) surface roughness, were studied by microscopy confocal (infiniteFocusmagnification X 50, Bruker Alicona, Graz, Austria ).

The adhesion of the protective layer on the 304L substrate was performed with automatic adhesion tests [START_REF] Vakili | The corrosion performance and adhesion properties of the epoxy coating applied on the steel substrates treated by cerium-based conversion coatings[END_REF][START_REF] Zheng | Adhesion Tests and Failure Modes Study on Structural Steel Coatings[END_REF] related to ASTM 4541. These tests were carried out with the Elcometer 510 automatic pull-off adhesion gauge (La Chapelle Saint Mesmin, France). Aluminum dollies (diameter of 20 mm) were glued with the Araldite 2011 adhesive. Samples were kept at ambient temperature for 24 h before tests. Dollies were pulled at a speed of 0.2 MPa.s -1 in the normal direction to the coating surface. The adhesion force was measured once the dollie was no longer in contact with the substrate (Figure A2 in Appendix A). Three modes of fracture result from the characterization: (i) cohesive break (interfacial fracture), (ii) adhesive break (a break between the substrate and the layer), and (iii) glue break (coating adhesion higher than glue adhesion). Tests were carried out using three replicates to ensure the repeatability of the measurement.

The surface free energies of the protective layer crosslinked at 200 °C and pyrolyzed at 700 °C were determined by measuring the contact angle. This measurement allows the determination of the solid/liquid interactions, essential parameters in TES application where the AlSi12 in liquid state interacts with the protective layer. The contact angles were measured by depositing 3 μL of a drop of liquid on the layer surface through a calibrated microsyringe and a programmable pump system. The polar and dispersive contributions to the surface energy were obtained through three test liquids (water, diiodomethane, ethylene glycol). Recorded images were analyzed to assess the contact angle. The surface free energies were calculated using the Owens and Wendt, Rabel, and Kaelble (OWRK) method [START_REF] Owens | Estimation of the surface free energy of polymers[END_REF]. This method assumes that the SFE (𝛾 ) is a sum of a polar component (𝛾 ) and a dispersive one (𝛾 ) (4):

𝛾 𝛾 𝛾 (4)
The probe liquids used (water, diiodomethane, and ethylene glycol) cover a wide range of properties from very polar water to very dispersive diiodomethane. Table 1 reports the characteristics of the three liquids [START_REF] Spiesser | Surface Free Energy of Cubic Boron Nitride Films Deposited on Nanodiamond[END_REF].

Results and Discussion

Protective Material Properties

The thermomechanical behavior of the material containing polysiloxane, glass frit, and boron nitride was determined by dilatometry (Figure 3). As we can see, the thermomechanical behavior appears similar between 20 and 500 °C after several cycles up to 700 °C. For each measurement, a material expansion of 25 μm was observed in this temperature range, and the calculated CTE is to 6.4 K -1 (Figure 3a) and 7.3 K -1 (Figure 3b). In Figure 3a, a slope change was observed at 200 °C. This temperature corresponds to the crosslinking temperature of the PDC. The samples being thick (7 mm), a first thermal cycling for 1 h at 700 °C before the study does not seem sufficient to have a complete crosslinking of the PDC. However, this time is sufficient to crosslink thin layers used in the TES application. This expansion can be related to the out-gassing and modification of the PDC. Above this temperature, the behavior of the protective material changes regarding the cycles. A significant shrinkage is observed up to 700 °C after the first thermal cycling. This shrinkage can result from two phenomena. The first one can be related to the presence of the glass frit that has a transition temperature from the vitreous to the liquid state close to 500 °C. The second one concerns the polymer-derived ceramic (PDC). The PDCs show a substantial shrinkage when heated to temperatures above 600 °C, as the organic side chains evaporate, and a porous silicon-oxy-carbide glass is formed [START_REF] Colombo | Advances in Polymer Derived Ceramics and Composites[END_REF].

After five cyclings in temperature (Figure 3b), a glass transition temperature (Tg) is also observed at 500 °C. Above Tg, glasses become soft and capable of deformation without fracture. The softening temperature is observed at 560 °C. Having a part of the material molten in temperature and solidified by cooling can help absorb thermomechanical stresses and limit layer cracking risk during future storage applications. After five thermal cyclings up to 700 °C, the material seems to be stabilized with a glass behavior. For temperature energy storage applications, it is interesting that the materials used are thermally conductive to promote exchanges and limit thermal barriers. AlSi12 has a thermal conductivity of 190 W.m -1 .K -1 at 577 °C and 304L stainless steel used for TES applications of 16.2 W.m -1 .K -1 . The thermal conductivity of the protective material was measured between 0.5 and 0.8 W.m -1 .K -1 for a temperature range between 20 and 600 °C (Figure 4). These experimental data could be interesting for future storage system modeling, even if the thin layer would probably have little impact on global thermal exchange. As a remark, the thermal properties of the commercial protective coating materials listed in Appendix A are not provided.

Protective Coating on Planar 304L Samples

Topology and Adhesion

Figure 5 shows a cross-section of a stainless-steel substrate coated with the protective layer pyrolyzed one time at 700 °C. A crack-free layer appears homogeneously distributed on the surface of the substrate on the different areas observed. This layer presents pores that appear trapped in the PDC composite. These pores are probably due to the offgassing associated with the beginning of the ceramization of the PDC into silicon oxycarbide. The glass frit has a glass transition temperature of about 500 °C. Therefore, the heat treatment at 700 °C led to a liquid glass which, during cooling, froze and trapped the gas bubbles. The layer surface appears highly irregular, even suggesting that the layer is cracked (Figure 6a). A confocal microscopic topography of the sample surface (Figure 6b) confirms that the deposited layer is strongly wavy. The root mean squares (RMS) surface roughness, sensitive to large peaks and valleys, is 5 μm confirming the previous observations. The protective layer appears continuous. Considering that the thickness measured on the SEM image is a mean point of the layer. The thickness of the layer is therefore about between 30 and 50 μm. This continuous protective layer over the entire 304L planar substrate has an adhesion greater than 3 MPa. This value is in line with the adhesion values of the commercial coatings listed in Appendix A. The fracture took place in the glue between the dollie and the layer (glue break), proving that the coating adhesion is strong on the substrate and in the layer (coating adhesion higher than glue adhesion).

Surface Energy

Measurements of contact angles of specific liquids are used to determine the surface free energy (SFE) of the protective layer. The contact angles of distilled water, diiodomethane, and ethylene glycol measured on the protective layer are displayed in Table 2. The contact angles were measured below 90° using distilled water as the probe liquid. The layer has, therefore, a partial hydrophilic behavior. The hydrophilic character appears to increase after the pyrolysis of the layer. The surface free energies are 22.6 and 26.4 mJ•m -2 , respectively, for a crosslinked layer and a crosslinked/pyrolyzed layer. This protective layer has low surface energy close to plastics as polyvinyl fluoride (PVF = 30.3 mJ•m -2 ) and polyethylene (PE = 32.4 mJ•m -2 ) and lower than that of stainless steel, 40-60 mJ•m -2 [START_REF] Matjie | Tailored surface energy of stainless steel plate coupons to reduce the adhesion of aluminium silicate deposit[END_REF][START_REF] Tang | Surface free energy changes of stainless steel after one atmospheric pressure plasma treatment[END_REF].

It can be noticed that with temperature, the dispersive and polar components are reversed. Thus, stronger interactions appear when the PDC is in a ceramic state since the 𝛾 increases at the expense of the dispersive component. The chemical nature of the PDC evolves between processing temperatures. The PDC in the crosslinked state contains organic bonds that break during pyrolysis. At this stage, there are essentially weak Van der Waals-type interactions due to the higher dispersive component. At 700 °C, the PDC is being ceramized, organic groups split off, and the hydrogen bonds are likely accessible in the matrix. These hydrogen bonds evacuate as hydrogen up to 1000 °C. The presence of these bonds leads to a higher wettability with water and polar liquid [START_REF] Schelm | Mechanical and Surface-Chemical Properties of Polymer Derived Ceramic Replica Foams[END_REF]. The dispersive component after pyrolysis of the protective layer is low, the wettability to nonpolar material is therefore low.

The low surface free energy combined with a low dispersion component should favor a nonadhesion of metallic liquid in TES application.

TES Application

Planar 304L Substrate

Figure 7 and Figure 8 show cross-sectional interfacial morphologies and EDS analyses of the samples after molten aluminum corrosion, respectively, 600 and 1200 h. The interface exhibited a sandwich structure after 600 h, consisting of a 304L substrate with a protective layer and above aluminum. After 1200 h, the configuration is similar. A gap between the aluminum and the protective layer is, however, observed. This gap is due to the nonadhesion of aluminum after removing the substrate from the AlSi12. The protective layer appears continuous over the entire substrate without any cracks after treatment. The pores or gas bubbles previously observed no longer appear to be present in the layer. The glass frit is liquid at a temperature of 700 °C. The pores can, therefore, be filled, and gas bubbles evacuated. The protective layer is intact and does not appear to have been affected by the presence of the molten AlSi12.

The 304L substrate appears healthy, suggesting the absence of the Fe-Al intermetallics, characteristic of corrosion. The elemental analyses carried out by EDS show a protective layer represented by the Si, O, and C elements, 304L substrate by Fe, and the aluminum alloy consisting of Al and Si. The Al element is only observed above the surface of the protective layer. These observations performed after 600 and 1200 h confirm the protective aspect of this layer against molten AlSi12.

Components for TES Application

Figure 9 shows the tube after aging in AlSi12 during 600 h at 700 °C and after taking a sample to carry out characterizations. Figure 10 displays a cross-sectional morphology and EDS analysis of the stainlesssteel tube coated with the protective layer after liquid aluminum aging during 600 h. After removing the samples from the representative TES component, the absence of adhesion between AlSi12 alloy and container was noticed. An aluminum-free area above the protective layer filled by the SEM resin is, therefore, observed on the SEM image. The thickness of this layer is between 20 and 50 μm. The protective layer has also non-open cavities on the stainless-steel substrate. The cavities have elongated shapes with rough surfaces or boot-like shapes. AlSi12 does not seem to penetrate into the layer, otherwise solid aluminum would have torn off the layer when the sample was taken. The presence of cavities is probably due to the coating process combined with higher thermomechanical stresses in cylindrical substrate compared to planar substrate. The thickness variation of the layer does not seem to affect the effectiveness of this protective layer since the 304L substrate is healthy without any corrosion points. EDS analysis did not detect any aluminum in the layer or in the substrate. The protective layer can, therefore, be considered an excellent barrier against AlSi12 corrosion and looks promising for TES applications.

Conclusions

A comparative study of different anticorrosive barriers for 304L stainless steel against molten AlSi12 showed low performance over long periods. A composite based on polymerderived ceramics, ZnO-based glass frit and boron nitride powder was, therefore, developed and applied to stainless steel surfaces by tape casting and brushing. Characterizations of wettability and adhesion, aging tests of the coating in contact with an aluminum alloy (AlSi12) at 700 °C for 600 and 1200 h, show high performance to the already studied and existing solutions with a significant increase in durability. This solution also makes it possible to envisage low-cost technical processes for future applications such as energy storage based on metallic phase change materials and applications in aluminum alloy foundry. This solution should be further optimized in the future, especially for the industrial application process of the protective layer (spraying, etc.). The result of these several tests is summarized in Table A1. Coatings with a poor adhesion (> at 15% of affected area), and with fragility and friability after thermal treatment were not selected for the other evaluations. As the evaluation of Figure A2, automatic adhesion tests for selected coatings were also done according to ASTM 4541. These tests were carried out with the Elcometer 510 automatic pull-off adhesion gauge. Aluminum dollies were glued with the Araldite 2011 adhesive. Samples were kept at ambient temperature for 24 h before tests. Then, a slot was made around dollies and they were pulled at a speed of 0.2 MPa/s normal to the coating surface until the coating was detached from the steel substrate (Figure A2). Three modes of fracture exist: cohesive break (interfacial fracture), adhesive break (a break between the substrate and the layer) and glue break (coating adhesion higher than glue adhesion). All tests were carried out using three replicates to ensure the repeatability of the measurement. All the results are summarized in Table A1. After a thermal treatment at 620 °C, Conductal ® (Aluminium Martigny, Chimilin, France) and DR87 ® (Foseco, Staffordshire, UK) coatings were not selected (not a sufficient adhesion, delamination of the layer).

Aging tests were realized on samples in 304L stainless steel with these identified coatings: Gstart ® (Graphite spray from Condat, Chasse sur Rhône, France), Antonol NB 20 ® (Aluminium Martigny, Chimilin, France) , TiAlN and AlCrN PVD, Aluminizing method by pack cementation, Composite of Polymer Derived Ceramic based SiO2 and glass frit. Coated samples were inserted in melted AlSi12 during 600 and 1200 h in order to assess the interest and the performance of the coating.

After aging, the samples were cut, embedded in a transparent resin and polished, allowing the visualization of the layers and interfaces by scanning electronic microscopy (Philipps XL30 SEM, Eindhoven, The Netherlands).

To sum up all these results: The Graphite spray was not selected because the layer did not have good adhesion after aging (delamination of the layer).

After contact with molten aluminum, all coated substrates in stainless steel (304L) decreased in thickness, also with PVD coatings (AlCrN layer and TiAlN layer, see Figure A3). PDC composite (polymer derived ceramics with glass frit) was too reactive in these conditions with AlSi12. Despite a good adhesion, this barrier of diffusion is not effective (total loss of the initial layer).

Coating based on boron nitride seems interesting, no AlSi12 wettability on the 304L substrate but not a good adhesion after 1200 h of aging and presence of many attack points of corrosion.

The preliminary tests are summarized in Table A1.

Figure 1 .

 1 Figure 1. Corrosion resistance of a 304L stainless steel coated with a protective layer in molten AlSi12 during 600 and 1200 h at 700 °C (a) schematic view and (b) ceramic containers in the furnace.

Figure 2 .

 2 Figure 2. Small representative component coated with protective layer and filled with molten AlSi12.

Figure 3 .

 3 Figure 3. Displacement measurement vs. temperature on SETSYS thermomechanical analyzer (a) after one cycling in temperature up to 700 °C and (b) after five cyclings in temperature up to 700 °C.

Figure 4 .

 4 Figure 4. Thermal conductivity of the protective material at 20, 300, and 600 °C.

Figure 5 .

 5 Figure 5. A cross-sectional SEM image of the protective layer coated on 304L stainless steel substrate, crosslinked at 200 °C and pyrolyzed at 700 °C in argon.

Figure 6 .

 6 Figure 6. (a) SEM image of the surface of protective layer coated on 304L stainless steel substrate, crosslinked at 200 °C and pyrolyzed at 700 °C in argon; (b) 3D topography measurement of the layer.

Figure 7 .

 7 Figure 7. (a) Cross-sectional SEM observation of the protective layer coated on the 304L substrate after 600 h at 700 °C in AlSi12 (300 X magnification) and EDS analysis of Al (b), Si (c), O (d), Fe (e), and C (f).

Figure 8 .

 8 Figure 8. (a)Cross-sectional SEM observation of the protective layer coated on the 304L substrate after 1200 h at 700 °C in AlSi12 (300 X magnification) and EDS analysis of Al (b), Si (c), O (d), Fe (e), and C (f).

Figure 9 .

 9 Figure 9. Small representative component coated with protective layer after test in AlSi12.

Figure 10 .

 10 Figure 10. (a) SEM image of the protective layer on the 304 L tube after aging during 600 h at 700 °C in liquid AlSi12 (300 X magnification) and EDS analysis of Al (b), Si (c), O (d), Fe (e), and C (f).

Figure A1 .

 A1 Figure A1. Image analysis example for the composite with polymer derived ceramics after crosscut test.

Figure A2 .

 A2 Figure A2. Automatic adhesion tests.

Figure A4 .

 A4 Figure A3. Example-SEM micrograph after aging (TiAlN coating-initial substrate thickness = 3, 7 mm).The effects of these different diffusion barriers are not efficient. As you can see in FigureA4, aluminum diffuses into the substrate and creates intermetallic layers based on Fe-Al (Fe-Al-Si-Cr).

Table 1 .

 1 Dispersion and polar components and surface free energy of water, ethylene glycol and diiodomethane.

	Liquid	Dispersion Component 𝜸 𝑺 𝒅	Polar Component 𝜸 𝑺 𝑷	Surface Free Energy 𝜸 𝑺 (mJ m -2 )
	Water	21.8	51	72.8
	Ethylene glycol	29	19	46.1
	Diiodomethane	46.6	4.2	50.8

Table 2 .

 2 Surface free energy 𝛾 and its components of the protective layer following different surface treatments.

	Surface Treatment	Contact Angle Ɵ (°) Water Diiodome thane Ethylene Glycol	Surface Free Energy and its Components (mJ/m 2 ) 𝛄 𝐒 𝛄 𝐒 𝐝 𝛄 𝐒 𝐏
	Crosslinking at 200 °C in air 88.1	72.7	82.0	22.6 19.2	3.4
	Pyrolysis at 700 °C in Ar	69.4	89	82.6	26.4	9.8	16.6

Table A1 .

 A1 Preliminary tests of several coatings.

				Crosscut
	Coating	Process	Thickness	Adhesion Test % Area	Pull-Off Adhesion Test	Aging in Liquid AlSi12 at 700 °C 600 and 1200 h
				Removed
	CONDAT Bstop ® Based on BN	Spray Drying at 25 °C	225 μm	26%	Not realized Adhesion not effective
	CONDAT Gstart ® Based on graphite	Spray Drying at 25 °C	80 μm	15%	<at 1 MPa (cohesive break) After TTh (620 °C)	Some attack points at 600 °C Delamination of the layer after 1200H
	DYCOTE Sub layerDR87 ®	Dip-coating Drying at 75 °C	220 μm	11%	5.3 MPa without TTh (cohesive break) No adhesion of the layer after TTh (620 °C)
	DYCOTE Holcote 110 ® Based on ZrSiO4	Dip-coating Drying at 75 °C	650 μm	70%	Not realized Adhesion not effective
	Aluminum Martigny Conductal 8M ®	Brushing Drying at 25 °C	200 μm	8%	Delamination of the layer after TTh (620 °C)	Not realized Adhesion not effective
	Aluminum					Many attack points of
	Martigny Antonol NB 20 ®	Brushing Drying at 75 °C	70 μm	10%	4.3 MPa (adhesive break) after TTh (620 °C)	corrosion and fragility of the layer adhesion
	Based on BN					after 1200 h
						Total loss of the initial layer
	Composite of	Dip-coating			1.8 MPa without TTh	Decreasing of the initial
	PDC based on SiO2 and glass	Drying at 75 °C and curing at	160 μm	12%	(adhesive break) 4.8 MPa (cohesive break)	substrate thickness (2 mm instead of 3.8 mm)
	frit	620 °C			after TTh (620 °C)	Two intermetallic layers
						created
	Balinit futurea ®	PVD TiAlN	3 μm	8%	>5.5 MPa (glue break) after TTh (620 °C)	Decreasing of the initial substrate thickness (3.3 mm instead of 3.8 mm)
	Balinit alcrona pro ®	PVD AlCrN	3 μm	2%	>5.1 MPa (glue break) after TTh (620 °C)	Decreasing of the initial substrate thickness (2.6 mm instead of 3.8 mm)
	Aluminizing pack cementation with Al	CVD: Al2O3 + NH4Cl activator + Al oxidation in air at 1150 °C (1H)	180 μm of intermetallic layer + oxide layer of 6 μm	Not realized but good adhesion observation	Decreasing of the initial substrate thickness (2 mm instead of 3.8 mm) Two intermetallic layers created

Funding: This project has received funding from the European Union's Horizon 2020 research and innovation programme under IN-POWER programme (Developing Highly Efficient Concentrated Solar Power Architecture) with the grant agreement N°720749. Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable.

Data Availability Statement: Data sharing is not applicable for this article.

Conflicts of Interest:

The authors declare no conflict of interest.

Author Contributions: methodology, investigation, analysis, writing-original draft preparation, writing-revision S.Q.; analysis, writing-original draft preparation, writing-revision, M.R. All authors have read and agreed to the published version of the manuscript.

Appendix A

This section contains details about first preliminary tests and benchmark concerning different coatings and technical solutions:  Industrial technical used for casting molds of aluminum based on graphite or boron nitride (commercial solutions, CONDAT, DYCOTE and ALUMINUM MARTIGNY);  PVD coating with TiAlN or AlCrN (BALINIT commercial coatings named Alcrona pro ® and futurea ® Oerlikon balzers Ferrières-en-Brie France);  Protective layer with a Polymer-Derived Si-Based Ceramics (coating based on commercial raw materials with polysiloxane and glass frit);  CVD coating, aluminization by pack cementation as mentioned in [START_REF] Xiang | Low temperature aluminisation of alloy steels by pack cementation process[END_REF].

In the coating industries, adhesion testing is often used to determine if the coating will adhere correctly to the substrate to which they are applied.

First, a qualitative test was realized on these several samples (crosscut adhesion test with a crosshatch adhesion tester), according to method B in ASTM D3359 standard. Adhesive criterion is correct if less than 15% of area is not affected. Images analysis for calculating the percent area removed were realized for all the tested coatings (Figure A1).


Franc ¸ois Gardes
email: gardes@univ-paris1.fr.

A SOLUTION TO THE ESTIMATION OF AN ENLARGED GDP INCLUDING DOMESTIC PRODUCTION: AN ESTIMATION ON MICRO DATA

Keywords: Domestic Production, Time Allocation, GDP, Opportunity Cost of Time. JEL: D31, J22

Using a generalization of the Becker's time allocation model to estimate the shadow price of time, the article proposes to evaluate the component of home production which could be substituted by market goods and services. An enlarged households' total expenditure including the production of the informal sector and the value of domestic production and the corresponding enlarged GDP are compared to their monetary counterparts in five developed and under-developed countries. Domestic production substituable to market goods corresponds to 23 to 41% of the GDP in Canada, France, Poland and the US, but much less in Burkina Faso because of much lower opportunity cost of time and elasticities of substitution. Finally, the enlarged GDP including the informal sector is larger by 40% than the official GDP in the three developed countries, 34% in Poland and by 54% in Burkina Faso.

I Introduction

The exclusion of the value of non-market activities has been raised as a serious issue since the introduction of the time allocation theory (sometimes referred to as the household production theory) by [START_REF] Alpman | Statistical Matching for Combining Time-Use Surveys with Consumer Expenditure Surveys: An Evaluation on Real Data[END_REF]. Indeed, the size of the home sector is quite large (home production equals on average two-thirds of a household's total monetary income according to [START_REF] Gronau | Home Production-A Forgotten Industry[END_REF]) 1 . The Stiglitz report (2009) considers that the estimation of an enlarged GDP containing the value of the production in informal markets and the value of households' home production (together with the cost of environmental externalities) is crucial to discuss the changes of the households' well-being. The inclusion of informal markets adds generally less that 10% to the GDP in developed countries2 and does not vary much, except during national structural changes such as those experienced in previously communist states 3 . The size of the informal sector can be captured through surveys on employment or comparing actual households' expenditures to those which are predicted by increased incomes from black markets (see Pissarides and Weber, 1989[START_REF] Lyssiotou | Estimates of the Black Economy based on Consumer Demand Approaches[END_REF], or Gardes-Starzec, 2010). On the contrary, the estimation of home production needs new methods to value home production and to define precisely the type of domestic products which can be added to the market product. I propose to add to market goods only those home produced final goods which could be acquired on the market (the cake home produced but not a conversation with one's spouse).

The estimation made by Gronau (as well as [START_REF] Bonke | Distribution of Economic Resources: Implications of Including Household Production[END_REF][START_REF] Frazis | How Does Household Production Affect Measured Income Inequality?[END_REF][START_REF] Jenkins | Household Income Plus Household Production: The Distribution of Extended Income in the U.K[END_REF] used the households' average market wage as the opportunity cost of time. Direct surveys as well as recent estimation show that this opportunity cost is rather, at an individual level, around one half of household's average wage rate and indexed on it with an rather high

The full income, which includes the value of home production, is also more equally distributed than the monetary income [START_REF] Frick | The Impact of Home Production on Economic Inequality in Germany[END_REF][START_REF] Frazis | How Does Household Production Affect Measured Income Inequality?[END_REF][START_REF] Bonke | Distribution of Economic Resources: Implications of Including Household Production[END_REF][START_REF] Aaktuna-Gunes | Informal Markets, Domestic Production and Demand Elasticities: A Case Study for Turkey[END_REF] Only 5.2% in Canada for instance according to Fortin, Lacroix and Montmarquette, 2009, in a study based on individual data. In Poland for instance, the size of black markets increased temporally till 30% of the GDP after the liberalization of the economy in 1989. elasticity, between one third and two third according to the country [START_REF] Gardes | The Estimation of Price Elasticities and the Value of Time in a Domestic Production Framework: An Application using French Micro-Data[END_REF]. This usual calibration by an average (or the minimum) wage rate suffers both from its over-estimation and from its lack of relation with income, the individual wage rate or the family structure. For that purpose, we use a home production model where the consumer combines time with market goods to produce activities (final goods) that generate utility. Utility maximization implies that the value of time is given by the ratio of the marginal utility of time to the marginal utility of market goods. The method proposed by [START_REF] Gardes | On the Value of Time and Human Life[END_REF][START_REF] Gardes | The Estimation of Price Elasticities and the Value of Time in a Domestic Production Framework: An Application using French Micro-Data[END_REF] shows that this ratio can be structurally estimated provided that data on time and income are available. To overcome the lack of this type of data, individual data containing both informations at the household level are built by a matching of a Households Budget survey with a Time Use survey for the same population and period (the matching procedure is detailed in Appendix A, I).

Section 2 presents the model defining substituable chores, section 3 present the model used for the estimation of the opportunity cost of time and section 4 provides empirical results for five developed and under-developed countries. The statistical matching procedure that we use to combine the Time use survey with a Family expenditure survey and the datasets are presented in Appendix.

II Estimation of the proportion of substituable chores Two indicators of full income

The usual full income concept y f 0 is the maximum monetary income which could be earned when working during all disposable time T, valued at the market wage rate net of taxes W:

y f 0 = W T +V.
(1)

with V other incomes unrelated to market work (financial revenues, social distribution...).

Another indicator of household's full resources, adding the opportunity cost of domestic production to monetary income from market work and other sources, can be defined, first valuing the time used in home production by the estimated opportunity cost and second, considering that only a part of domestic production could be afforded by market goods and services. The non-substituable component of domestic production (speaking with one's spouse for instance)

would not be added to the full estimate of household's resources. Indicating by π i the proportion of home product i which can be replaced by a market product or service, this indicator of full resources writes:

y f = Wt w +V + ω ∑ π i t i .
(2)

The next section proposes a method to calculate these proportions π i .

Substituable chores

Chores (domestic dayly works) are generally defined as the minimum amount of home production the value of which could be added to the value of the agent's production working on the labor market. Pure leisure activity would for instance be excluded. Home production related to food would be considered as a chore only if it corresponds to the normal amount of food for a typical similar agent, the individual surplus of a given agent being considered as linked to its particular constraints or preferences to be excluded from the chores. But this minimum amount is difficult to estimate, either directly on a survey (choosing for instance the first tercile of the distribution of time use for food among similar individuals) or by estimation of a demand system such as the Linear Expenditures System.

I propose a new method estimating the proportion of time devoted to some home activity which can be considered as a chore, by means of the degree of substitution with market alternative goods or services: the value of chores for a given domestic production (for instance of transport) is the cost to acquire a substitute on the market. The integration of that value in the market production thus proceeds from the possibility to obtain the same final good on the market at a given price which is attributed to that component of the domestic production.

This requires that the elasticity of substitution between market factors of production and time must be large, so that it is possible to produce the same amount of final good with these market factors substituted to time use.

Suppose that the domestic production of some final good i is divided between 'substituable' chores 1 produced under possible substituability with market goods or services (supposed to be governed by a unitary elasticity of substitution between monetary and time inputs, corresponding to a Cobb-Douglas domestic production function: : El t i,1 /ω = 1) and non-substituable chores 2 characterized by a zero elasticity of substitution with market goods

(σ i,2 = δ [(t i,2 )/m i ] δ (ω)
= 0). The same division applies to the total time devoted to the home production of the final good: t i = t i,1 +t i,2 , with t i,1 and t i,2 the times spent in substituable (respectively non-substituable) home work producing the final good i. The elasticity of substitution of the aggregate time t i over the opportunity cost of time ω writes, with m i the expenditure made for market goods and services used producing the final good:

σ i = δ [(ti,1+ti,2)/mi] δ (ω) [(t i,1 + t i,2 )/m i ] /ω = δ [ti,1/mi] δ (ω) (t i,1 /m i )/ω t i,1 (t i,1 + t i,2 ) = σ i,1 t i,1 t i,1 + t i,2 = - t i,1 t i,1 + t i,2 = -π i (3)
The proportion of chores in this domestic activity can then be measured by the absolute value of the elasticity of substitution of the aggregate, which is smaller than one for all activities in our french dataset : 0.36 for food, 0.30 for clothing, 0.66 for leisure activities, 0.30 for All Other activities (see Canelas et al., 2018). The total value of chores v i in each domestic activity will be calculated by means of these elasticities of substitution and added to the monetary value of expenditures made to produce the corresponding final good:

v i = ω i .t i,1 = |σ i |.ω i .t i . A
generalization to the case of multiple components of chores with different elasticities of substitution is straightforward.

The household's full resources are therefore the sum of monetary income and of that component of substituable chores valued by the estimated opportunity cost of the household:

y f = Wt w +V -ω ∑ σ i t i = Wt w +V + ω|σ |(T -t w ) (4)
with σ the weighted average (weighted by the time budget shares) of the elasticities of substitution σ i . The permanent income is estimated at the households' level, then aggregated over the population.

III Estimation of the opportunity cost of time 4

In order to estimate the shadow price of time, I assume that the consumer combines time with monetary expenditures to produce activities that generate utility in a model where the market work time is valued by the consumer's wage rate while the remaining time (e.g., time allocated to leisure or non-market work) is valued by the shadow price of time that may differ from the wage rate. It is assumed that the consumer's utility function is given by u

(Q) = ∏ i a i Q γ i
i where a i is a positive parameter and Q i is the quantity of the activity i produced by the combination of monetary and time inputs denoted m i and t i , respectively:

Q i = b i m α i i t β i
i where m i = x i p i with x i the quantity of the market goods i, p i its price, and b i a positive parameter. 5 The choice of the Cobb-Douglas forms allows the parameters to be identifiable. As we estimate the parameters locally (i.e., for each observation in the dataset), the Cobb-Douglas specifications imply simply constant substitution between time and monetary resources only in the neighborhood of each individual's equilibrium point. As discussed in [START_REF] Gardes | On the Value of Time and Human Life[END_REF][START_REF] Gardes | The Estimation of Price Elasticities and the Value of Time in a Domestic Production Framework: An Application using French Micro-Data[END_REF], the Cobb-Douglas specification allows identifying a unique shadow price of time for all activities while a more general model based on CES household production functions allows identifying only activity-specific shadow prices of time.

Combining the utility and the production functions allows to write the utility in terms of inputs:

u(m,t) = Π i (a i b γ i i ) ∏ i m α i γ i ∑ α i γ i i ∑ α i γ i ∏ i t β i γ i ∑ β i γ i i ∑ β i γ i (5) = A m ∑ αγ i t ∑ β i γ i (6)
where m and t are geometric weighted means of the monetary and time inputs and A ≡ Π i a i b

γ i i .
In this framework, the consumer is subject to an income constraint, ∑ m i = wt w +V ≡ Y , and to a time constraint, ∑ i t i + t w = T , where V is other income and t w is the time allocated to market work. Utility maximization implies that the shadow price of time, denoted ω, is given by

ω = ∂ u ∂t ∂t ∂ (∑t i ) ∂ u ∂ m ∂ m ∂Y = m ∑ β i γ i ∂t ∂ (∑t i ) t ∑ α i γ i ∂ m ∂Y (7)
The shadow price of time differs from the market wage rate when, for instance, there exists some market imperfections, transaction costs, and constraints on the labor market or in the home sector. The shadow price of time can be estimated provided that estimates of α i , β i , and γ i are available which is obtained by means of substitutions of monetary and time expenditures in the production of each final goods and by substitution of time or money between activities (see [START_REF] Gardes | The Estimation of Price Elasticities and the Value of Time in a Domestic Production Framework: An Application using French Micro-Data[END_REF].

IV Aggregate results for five countries

Table 1 present estimations for three developed countries (Canada, France and the US), Poland after the liberalization of the economy (in 1989) and Burkina Faso.

The value of domestic production is similar in France and the U.S. (aroud 40% of the GDP), while it is smaller in Canada and negligeable in Burkina Faso. The low proportion of domestic production in Burkina Faso procedes from a very low value of time and also a low elasticity of substitution between the factors of domestic production (-0.083)6 . That possibility Note: * [START_REF] Hassan | Size and Development of the Shadow Economies of 157 Countries Worldwilde: Updated and New Measures from 1999 to 2013[END_REF] estimates.

to substitute market goods and services to home production is thus a very important feature of the allocation of time by households and it generates the relative importance of domestic production in the enlarged GDP. In Canada, the opportunity cost of time has been estimated on a survey containing only family composed of two aduts without children, which would imply, as shown in other similar countries (France and the US), an under-estimation of 10 to 20% compared to the estimate on the whole population.

Finally, the enlarged GDP is larger than the GDP by 40% in the three developed countries and by 34% in Poland (because of the low level of domestic production), while it is much larger in Burkina Faso because of the extension of the informal sector.

Two results appear clearly in a dynamic study on Poland between 1997 and 2000 for which a unique Time Use survey (2003)(2004) is matched with a panel of four years: the dispersion of the value of domestic production (coefficient of variation equal to 2.53) is larger than the dispersion of households' income (0.77) and expenditures (0.67). Second, the change of these resources between 1997 and 2000 shows that their evolution are not parallel, specially considering short term changes: for instance, households real income increases between 1997

and 1998 by 8.35% while the change for enlarged income (income+domestic production) is +16.99%. On the contrary, the corresponding changes between 1998 and 1999 are +8.01% vs 7.63%, and 7.92% vs 1.21% between 1999 and 2000. On the whole 1997-2000 period, the changes for income, expenditures, domestic production, Income+Domestic Production and Expenditures+Domestic Production are respectively: +26.29%, +15.95%, +28.27%, +23.16% and +27.44%. This shows that it is important to examine the amounts and evolutions of these monetary and time resources over a longer period as they can be substituable to a large extent unitary elasticities of substitution, i.e. without the correction for the degree of substitution of home production with market goods and services, is greater by 25% in France, which shows the importance of that correction in the short run.

The inequality of the households' home production across the population is somewhat larger than the corresponding inequality for households' income or total expenditure. Its logarithmic standard error is for instance 101.9% in Poland, while it amounts to 61.6% for households' income (45.0% for households' total expenditure) and 67.1% for their enlarged income.

This inequality depends both on the volatility of the opportunity cost of time (logarithmic standard error equal to 96.5%) and on the differences in times allocated for each domestic activity.

It would even be greater if the elasticities of substitution were estimated at the individual level, for each household or household type.

V Conclusion

This paper provides a method to estimate the component of households' domestic production which can be added to GDP because it corresponds to the production of commodities which could have been purchased on the market. The definition of these "substituable chores" is an essential element of the computation of an enlarged GDP. The results show that incorporating the value of domestic production changes a lot the total resources of the household, both in level and as concerns their dispersion among the population. The evolutions of households monetary income and enlarged income including the value of their domestic production also differ largely, specially in the short term, as proved by an analysis of the results on a four years polish panel. all adults in the family, while expenditures concern the whole family, including children (the numbers of adults and children are on average 5.36 and 5.72 respectively). The hypothesis is thus made that only adults contribute to the domestic productions. Time uses for activities such as gardening or cattle breeding are both for domestic use and for selling products or services on the market. We have no information on this repartition so that we made the assumption that 70% of time uses corresponds to consumption by the household and 30% to a production cation time and education expenditures, health care time and health expenditures, leisure time and leisure expenditures, transport time and transport expenditures, miscellaneous time use and miscellaneous goods and services. Time uses for all selected activities are regressed on the households' characteristics for all observation units in FTB survey and these estimations serve to predict the time spent on these activities for the corresponding units in the FES survey.

Poland

The Polish panel of family expenditures contains 3052 households over four years .

This panel is matched with one Time use survey conducted in 2003-2004 over approximately 10000 households (20000 individuals). The matching procedure is made by regression over a common set of socio-economic characteristics of households which are present in both surveys.

The estimated coefficients are used to predict these times for each household in the Family Expenditure survey. We define 6 types of activities or time use types compatible with the available data both from FES and BDT: food, housing, clothing, transport, leisure and various expenditures (including health services).

USA

The Consumer Expenditure (CE) survey is performed each quarter over 5000 to 7000 households. Usual expenditures are recorded on a weekly basis, some less frequent for one month.

It is matched to the American Time Use survey (ATUS) which gives detailed information on time budgets for one year (see a presentation of these data in Aguiar et al., 2013). Both surveys are delivered each year since 2003. A continuous matching has been made between 2003 and 2011 (see Alpman and Gardes, 2015) using the Rubin's method.

TABLE 1

 1 Enlarged GDP (Per capita value in US dollars 2015)

	Country	ω	ω w	GDP	In f ormal GDP *	Home production	Enlarged GDP	EnlargedGDP GDP
	Burkina Faso	0.098	0.30	671.7	52.6%	4.1	1029	1.53
	Canada	10.93	0.69	50304	15.5%	9910	68011	1.35
	France	7.01	0.52	42503	14.3%	14413	62994	1.48
	Poland	4.26	0.43	12702	30.2%	3446	19984	1.57
	U.S.A.	12.75	0.74	53042	9.5%	19995	78016	1.47

  time uses for three final goods(Food, Domestic Activities and Other) and check that Rubin's procedure gives imputed values of time uses close to the true values. Its average annual economic growth has been 5% since 2000, the unemployment rate is 3%, but 83% of the population is below the poverty line according to the UNDP multidimensional index and 40.1% according to the National Institute for Statistics and Demography (INSD, 2015), with 92% of the poor in rural areas. The data used for this study are taken from the 2008 round of the farm household survey conducted by the Ministry of Agriculture of

	II. Countries
	Burkina Faso
	Burkina Faso is located in West Africa. The country accounts for 16.9 millions of citizens hav-
	ing an average life expectancy of 56 years. The population is very young (46% of young less
	than 15 years). The country is ranked 181th over 187 countries based on its human development
	(United Nations Development Program, 2014). The per capita GDP is 720 US dollars, mainly
	concentrated in the service sector (52%), industry and agriculture representing only 26% and
	22%. Burkina Faso. The survey covers 71 villages in the 45 provinces, with a total of 6941 house-
	holds surveyed. It contains information on family characteristics (incomes from agriculture or
	other activities, age of the head and the spouses, number and age of children, education level,
	accessibility to social services, income, financial situation, equipment...), households' expen-
	ditures (over 40 goods and services) and time use over 14 activities: unproductive activity, rain
	fed agriculture, vegetable farming, arboriculture, livestock farming, fishing, gathering, wood
	harvesting for selling on a market, wood harvesting for household needs, search for water, mar-
	ket work, other domestic activities, personal activities, other activities. Times are recorded for

Documents de travail du Centre d'Economie de la Sorbonne -2021.24

Documents de travail du Centre d'Economie de la Sorbonne -2021.24

Based on Gardes, 2019.

Q is assumed to depend on m (rather than x) because the dataset informs only expenditures. This approach yields consistent results when all households face the same prices.

Documents de travail du Centre d'Economie de la Sorbonne -2021.24

Note however that the domestic production assuming a unitary elasticity of substitution is also low in Burkina Faso, at

70, corresponding to only 10% of the GDP. On the contrary, the estimation of the full GDP in France with 7 Documents de travail du Centre d'Economie de la Sorbonne -2021.24

Rubin's procedure does not assume implicitly that the variables to be matched are conditionally independent, which is implicitly supposed by other statistical matching methods. This assumption leads to highly biased results when it is not verified by the dataset (see[START_REF] Rubin | Statistical Matching Using File Concatenation with Adjusted Weights and Multiple Imputations[END_REF] the Stata program in Alpman, 2017).

Appendix : Datasets I. Matching Procedure

The definition of comparable good and time groups of expenditure is a difficult and sometimes arbitrary operation. This rather difficult exercise needs some arbitrary assumptions about the substitution between time use and monetary expenditures (see Gronau and Hammermesh 2006 for a discussion). The commodity consumption structure does not correspond exactly to what is very often used as a standard classification of time uses even if differences can be limited. The reason is that not all time use activities can have a clear work equivalent. This is particularly the case of the leisure time. However, comparing our classification with other similar approaches [START_REF] Gronau | Time vs. Goods: the Value of Measuring Production Technologies[END_REF] we obtain similar patterns of what these authors call "relative goods/time intensity" defined as a ratio of good to time inputs relative to total amount of goods and time allocated to commodity production. For France, like for Israel and United States in Gronau-Hamermesh paper, the goods/time intensity is relatively high for Dwelling, Health, Clothing and to the less extent for Transport. However, the Eating item differs considerably between France and these countries, being weakly good intensive in France (0.57) and highly good intensive in the US and Israel (1.62, 1.82 respectively).

The matching procedure of a Family Expenditures survey with a Time-Use survey is either made by a regression of the time-uses observed in the matched survey using common covariates observed both surveys (such as education level and the age of the head, the family demographic structure or location), or using the Rubin's Multiple Imputation method (1986) which takes into account the correlations (conditional to covariates) for each variable which is matched between the two surveys (see Alpman, 2017, and Alpman, Gardes and Thiombiano, 2015 7 ). A matching of the Consumer Expenditure and the American Time use surveys using both methods shows that simple matching by regression gives often similar estimates. On the other hand, Alpman (2017) compares regression based imputation with Rubin's multiple imputation for simulated data and shows that the later furnishes imputed data which are closer to the original dataset.

In [START_REF] Alpman | Statistical Matching for Combining Time-Use Surveys with Consumer Expenditure Surveys: An Evaluation on Real Data[END_REF], we use a survey containing both monetary expenditures and which is sold on the market. In this paper, the monetary expenditures and the time use have been grouped into three common domestic activities: (i) food (ii) domestic activities and (iii) leisure and other activities. Expenditures are recorded for one week for food and one quarter for other expenditures, while time uses correspond to one week. All have been transformed into yearly values. As family size can be very large, time uses corresponding to all adults in the household may be performed in fact by a small part of these households (say two or three). The descriptive analysis in Gardes and Thiombinao (2017, Table 1) indicates indeed that couples with two adults have a significantly greater ratio of monetary expenditures to time uses than singles, which shows that their time are not the fact of all adults in the family. In order to correct for this probable bias, time uses have been multiplied by the ratio of the OECD equivalence scale (one for the first adult, 0.7 for other adults) over the number of adults (which perhaps still overstates the true number of adults corresponding to recorded time uses).

Canada

The Survey of Household Spending for 1998 is matched with the survey of time use budgets contained in the General Social survey for the same year. The family size and the number of children are not informed sufficiently so that the matched sampling contains only couples with zero or one child less of 14. The categorization of time use in the SHS is very similar to the categorization of spending in the SHS simplifying our task considerably. The matching is made using the Rubin's method.

France

The French dataset from INSEE combines at the individual level the monetary and time expenditures into a common, unique goods and services consumption structure by a statistical match of the information contained in two surveys: the Family Expenditure Survey (FES, INSEE BDF 2001) and the Family Time Budget (FTB, INSEE BDT 1999). I define 8 types of activities or time use types compatible with the available data both from FES and BDT: Eating and cooking time (FTB) and food consumption (FES), cleaning and home maintenance and dwelling expenditures (including imputed rent), clothing maintenance and clothing expenditures, edu-