M Romedenne

F Rouillard

D Hamon

B Malard

D Monceau

Carburization of austenitic and ferritic stainless steels in liquid sodium: Comparison between experimental observations and simulations

work of Toulouse researchers and makes it freely available over the web where possible

Introduction

In the framework of the development of a Sodium-cooled Fast nuclear Reactor (SFR) prototype for industrial demonstration called ASTRID (Advanced Sodium Technological Reactor for Industrial Demonstration), the French Alternative Energies and Atomic Energy Commission and its industrial partners have launched intensive research programs on several reactor components where improvements are needed. In particular, in the core of the reactor, the control rods are prone to carburization, a phenomenon responsible for their embrittlement and thus, reduced lifetime [START_REF] Hugot | Prolongation de la durée de vie des EA de la filière N.R. Bilan des examens réalisés sur les gaines en 316Ti hypertrempé et 1515Ti écroui[END_REF]. The nature and kinetics of caburization of steels in liquid sodium at 450-650 °C, where the activity of carbon in sodium is close to unity, were studied in the past. These studies led to the development of simplified carburization models predicting the evolution of the carburization kinetics of stainless steels in sodium [START_REF] Anderson | Carburization of austenitic stainless steel in liquid sodium[END_REF][START_REF] Gwyther | Carburization of austenitic and ferritic alloys in carbon-saturated sodium at 916 K[END_REF][START_REF] Aubert | Carburation d'acier 316 par transfert de carbone dans le sodium[END_REF][START_REF] Thorley | Carbon in Sodium -A Review of Work in the UK[END_REF][START_REF] Brunel | Proposition d'une loi enveloppe de carburation des gaines d'éléments absorbants[END_REF][START_REF] Dickson | Carburisation of Stainless Steel Immersed in Carbon-Contaminated Sodium: A New Technique for Estimating the Diffusion Coefficients[END_REF]. In these approaches, apparent diffusion coefficients of carbon D app were calculated by fitting the carbon concentration profiles with the usual solution of the Fick's second law [START_REF] Crank | Mathematics of Diffusion[END_REF]. In most of the studies, this parameter was neither clearly related to the diffusion of carbon in the matrix (ferrite or austenite) nor related to an effective grain boundary coefficient [START_REF] Anderson | Carburization of austenitic stainless steel in liquid sodium[END_REF][START_REF] Gwyther | Carburization of austenitic and ferritic alloys in carbon-saturated sodium at 916 K[END_REF][START_REF] Aubert | Carburation d'acier 316 par transfert de carbone dans le sodium[END_REF][START_REF] Thorley | Carbon in Sodium -A Review of Work in the UK[END_REF][START_REF] Brunel | Proposition d'une loi enveloppe de carburation des gaines d'éléments absorbants[END_REF]. Only Dickson et al. [START_REF] Dickson | Carburisation of Stainless Steel Immersed in Carbon-Contaminated Sodium: A New Technique for Estimating the Diffusion Coefficients[END_REF] proposed a prediction of the carbon concentration profiles using two different apparent diffusion coefficients, D app , one for the apparent diffusion of carbon in the grains and another one for the apparent grain boundary diffusion of carbon. In most of the cases, D app was a fitted parameter used to predict the maximum thickness affected by the carburization process. It is obvious that the maximum carburized thickness depends on the material (carbon diffusion coefficient in the matrix and at grain boundaries and the amount of precipitated phases (carbides)). As a consequence, D app is strongly dependent on the nature of the steel. It cannot be used to predict the carburized thickness of another steel with a different chemical composition and microstructure. Thus, a more accurate lifetime model is needed.

In the present study, out-of-pile carburization experiments were performed in liquid sodium at 600 and 650 °C (maximum temperatures measured in the core). Three stainless steels grades coupons were exposed and their extent of carburization and kinetics were followed. Moreover, the carburization kinetics in liquid sodium was simulated using DICTRA-ThermoCalc software. DICTRA is a module of the commercial software ThermoCalc which allows simulations of diffusion controlled transformations in multi-component systems [START_REF] Borgenstam | Dictra, a tool for simulation of diffusional transformations in alloys[END_REF]. DICTRA is commonly used to simulate carburization or nitriding kinetics [START_REF] Engström | Computer simulation of diffusion in multiphase systems[END_REF][START_REF] Turpin | Carbon diffusion and phase transformations during gas carburizing of high-alloyed stainless steels: Experimental study and theoretical modeling[END_REF][START_REF] Sudha | Carburization behavior of aisi 316ln austenitic stainless steelexperimental studies and modeling[END_REF][START_REF] Garcia | Experimental investigations and dictra simulations on formation of diffusion-controlled fcc-rich surface layers on cemented carbides[END_REF][START_REF] Rong | Thermodynamic simulation of low temperature colossal carburization of austenitic stainless steel[END_REF]. The first published DICTRA simulations, reported by A. Engström et al. [START_REF] Engström | Computer simulation of diffusion in multiphase systems[END_REF], simulated the carburization behavior of Ni-Cr and Ni-Cr-Fe at 850 °C after 1000 h exposure under 10% CH 4 and a C = 1 and Fe-25Cr-20Ni alloys after carburization for 200 h at 1100 °C. They compared the results with literature data [START_REF] Bongartz | A finite difference model describing carburization in high-temperature alloys[END_REF][START_REF] Bongartz | A mathematical model describing carburization in multielement alloy systems[END_REF][START_REF] Grabke | Aufkohlung von chrom-nickel-eisenstählen in der kohlenstoffpackung[END_REF]. For the first set of experiments they simulated with reasonable accuracy the evolution of M 3 C 2 , M 7 C 3 and M 23 C 6 carbides and the carbon concentration profiles with a C = 1 as boundary condition and with a labyrinth factor equal to f 2 (the meaning of this parameter will be detailed later in the manuscript). For the second set of experiments, good agreement was achieved between the simulated and experimental observations assuming the formation of M 7 C 3 and M 23 C 6 carbides, f 2 as labyrinth factor and an adjusted boundary condition at the gas-metal interface (a C = 0.45). Turpin et al. [START_REF] Turpin | Carbon diffusion and phase transformations during gas carburizing of high-alloyed stainless steels: Experimental study and theoretical modeling[END_REF] compared the experimental and calculated carbon profiles of Fe-13Cr-5Co-3Ni-2Mo-0.07C grades exposed at 955 °C for 23, 53 and 145 min. The predicted carbon profiles agreed well with the experimental carbon concentration profiles using the measured experimental carbon concentration at the gas-metal interface and, again, a labyrinth factor equal to f 2 . Garcia et al. well predicted nitriding of alloy W-Ti-Ta-Nb-Co-C at 1400 °C [START_REF] Garcia | Experimental investigations and dictra simulations on formation of diffusion-controlled fcc-rich surface layers on cemented carbides[END_REF]. All the previous DICTRA simulations observed a good agreement with experimental observations at high temperatures (T > 850 °C). Few DICTRA simulations were reported for the carburization of steels at lower temperatures (T < 850 °C) [START_REF] Sudha | Carburization behavior of aisi 316ln austenitic stainless steelexperimental studies and modeling[END_REF][START_REF] Rong | Thermodynamic simulation of low temperature colossal carburization of austenitic stainless steel[END_REF]. Sudha et al. attempted modelling the carburization behavior of 316L(N) steel at 525 °C after 16,000 h exposure without a clear description of their procedure [START_REF] Sudha | Carburization behavior of aisi 316ln austenitic stainless steelexperimental studies and modeling[END_REF]. Finally, Rong et al. attempted to simulate a carburization process at low temperature for which the precipitation of chromium rich carbides was prevented. Although the carburization depth was in agreement with the experimental results, the predicted carbon concentration at the surface was not in accordance with the experimental value [START_REF] Rong | Thermodynamic simulation of low temperature colossal carburization of austenitic stainless steel[END_REF].

The purpose of the present study is twofold. First, the extent of carburization and kinetics of three steel grades exposed in carbon-saturated sodium at 600 and 650 °C are characterized. Second, the experimental results are compared with simulations using the DICTRA module in order to discuss the processes which control the kinetics. In that purpose, an accurate description of the carburized samples, was crucial for the comparison with the DICTRA simulations. Electron Probe MicroAnalysis (EPMA), Transmission Electron Microscopy (TEM) and X-Ray Synchrotron Diffraction (XRSD) were used to thoroughly characterize the extent of carburization of the steels.

Materials and experimental procedure

Materials

Two austenitic steel grades, AIM1 (strain-hardened 25%) and 316L (solution-annealed), and one ferritic-martensitic steel, EM10 (strainhardened 25%), were studied. Detailed chemical composition of the three steels is given in Table 1. The austenitic steels were constituted of relatively equiaxed grains with a grain size of 40 and 10 μm for AIM1 and 316L respectively. The microstructure of the EM10 was formed of martensite laths of 0.3-0.4 μm width.

Experimental procedure

The carburization experiments were performed in a mild steel crucible (iron containing 0.38 wt. % of carbon) filled with 2 liters of nuclear grade liquid sodium previously purified from oxygen (composition reported in Table 2) with the use of Zr foils. A carbon activity higher than unity (a C > 1) was obtained in the liquid sodium by the rapid dissolution of carbon from the crucible into the sodium. A carbon activity higher than unity can be achieved by the formation of Fe 3 C considering the metastable Fe-C system. In liquid sodium, high carbon activities were measured where the formation of acetylide Na 2 C 2 in liquid sodium was observed [START_REF] Johnson | Enthalpy of formation of disodium acetylide and of monosodium acetylide at 298.15 k, heat capacity of disodium acetylide from 5 to 350 k, and some derived thermodynamic properties[END_REF][START_REF] Ainsley | The solubility of carbon in sodium[END_REF].

Coupons of the three steels grades (20 mm × 20 mm × 1 mm) with as-received surface finish were cleaned in ethanol and weighed prior to exposure in high temperature liquid sodium. The specimens were exposed under two temperatures. The first experiment was performed at 600 °C for various exposure times up to 5000 h. In the second experiment the specimens were exposed at 650 °C for 200 h.

Specimens of 1 mm thickness were used to evaluate the carburization kinetics. Additional 150 μm thick strips of the respective steel grades were simultaneously exposed under the same conditions to measure the value of the carbon concentration reached in the samples at thermodynamic equilibrium with the carbon activity in sodium. Weight gain measurements were carried out after 500 h, 1000 h, 3000 h and 5000 h after three cleaning steps in ethanol to remove all metallic sodium from the surface of the samples. The total carbon content of the specimens was measured by infrared-inductive carbon combustion. Then, cross sections were polished and etched with either di-ammonium persulfate (10 g (NH 4 ) 2 S 2 O 8 , 100 mL water) or oxalic acid (10 g of C 2 H 2 O 4 , 100 mL water, 6 volts) for metallographic examination.

The carbon concentration profiles within the depth of the samples were measured on unetched cross sections by EPMA (SX 100 CAMECA) operated at 15 kV and 20 nA using the phi-rho-z method. The analytical crystal for Kα C was LPC2. Band scan mode (2 × 50 μm 2 ) was used to average the carbon concentrations over several grains of the sample. Three quantitative profiles (step 4 μm) were realized in each sample and showed a relative deviation of 5%. The total carbon content deduced from the EPMA profiles was in accordance with the values obtained using carbon combustion analysis and in agreement with the mass gain measurements within ± 5% [START_REF] Romedenne | Carburization of austenitic and ferritic steels in carbon-saturated sodium: preliminary results on the diffusion coefficient of carbon at 873 k[END_REF].

Transmission XRSD was performed on the ID11 beamline at the European Synchrotron Radiation Facility (ESRF). A single-scan approach was used to evaluate the chemical and structural gradients within the cross section of the samples. The experimental set-up is represented in Fig. 1. Similar set-up and experiments have already been used to study the phase ratio and stress gradient in ferritic thin films with a 150 nm depth resolution [START_REF] Vaxelaire | Sub-micrometre depth-gradient measurements of phase, strain and texture in polycrystalline thin films: a nano-pencil beam diffraction approach[END_REF], or to measure the lattice parameter evolution in α-titanium alloy after oxidation (beam dimensions of 300 μm length and 7 μm height) [START_REF] Baillieux | Observation using synchrotron x-ray diffraction of the crystallographic evolution of alpha-titanium after oxygen diffusion[END_REF]. The samples were illuminated at 65 keV with a beam of 5 μm height in the y direction and 100 to 800 μm length in the x direction. A large length (800 μm) was necessary to maximize the number of grains in the analyzed volume. Careful attention was given to align the beam parallel to the surface of the samples. For this purpose, the signal of a photodiode was used. Calibration of the distance between the sample and the detector and of the instrument resolution was performed using a standardized CeO 2 powder (NIST 674b). The powder was inserted inside a capillary and glued above the middle of the sample. After integrating the data on the entire 2D diffractogram, the 1D data were analyzed using the Rietveld method within the MAUD software [START_REF] Lutterotti | Rietveld texture analysis from synchrotron diffraction images. I. Calibration and basic analysis[END_REF]. With this approach, the volume fractions of the phases and their lattice parameters were refined using a full pattern matching approach.

Table 1

Composition of studied alloys (in wt. %) analyzed by inductively coupled plasma-optical emission spectrometry (ICP-OES) and combustion analysis for carbon.

Modelling procedure in DICTRA

The carburization kinetics was simulated with the DICTRA module (database MOBFE3) implemented in the ThermoCalc software (database TCFE8). The Homogenization model was used for long-range diffusion occurring in each phase (matrix and carbides here) for which mobility data are available [START_REF] Larsson | A homogenization approach to diffusion simulations applied to α + γ Fe-Cr-Ni diffusion couples[END_REF][START_REF] Larsson | Multiphase diffusion simulations in 1D using the dictra homogenization model[END_REF]. In this model, elemental concentration profiles of all elements are calculated in one dimension. For carbon it is assumed that diffusion occurs only in a matrix which is the austenite or ferrite and that no diffusion occurs through the carbides as no mobility data in the carbides were available in MOBFE3.

In this coupled thermodynamic-kinetic approach, the flux J k of component k in the z direction is given as function of x k and V S , the mole fraction and volume of substitutional elements only. The effective diffusion flux of carbon J C eff can be written as follow

= - ∂ ∂ J V f M x µ z 1 C S q C C C eff tot tot (1)
where,

= + ′ - ′ x x f x x ( ) k k k k tot tot (2)
• ′

x k tot , total mole fraction of k after the diffusion step;

• x k tot , total mole fraction of k before the diffusion step;

• ′

x k , mole fraction of k in the matrix after the diffusion step;

• x k , mole fraction of k in the matrix before the diffusion step;

• f, volume fraction of matrix.

The product M x C C

tot tot corresponds to the average mobility and composition of carbon and μ C refers to the chemical potential of carbon. The function f q (with f the volume fraction of the matrix and q a number), is called the labyrinth factor and, is implemented in the software. This function is used to consider that the presence and the geometry of carbides can modify the diffusion of carbon in the matrix. In carburization studies, the value of q is usually set at 2 [START_REF] Engström | Computer simulation of diffusion in multiphase systems[END_REF][START_REF] Turpin | Carbon diffusion and phase transformations during gas carburizing of high-alloyed stainless steels: Experimental study and theoretical modeling[END_REF][START_REF] Sudha | Carburization behavior of aisi 316ln austenitic stainless steelexperimental studies and modeling[END_REF][START_REF] Garcia | Experimental investigations and dictra simulations on formation of diffusion-controlled fcc-rich surface layers on cemented carbides[END_REF][START_REF] Rong | Thermodynamic simulation of low temperature colossal carburization of austenitic stainless steel[END_REF]. In other studies, f q , with q equal to 1 [START_REF] Frykholm | A new labyrinth factor for modelling the effect of binder volume fraction on gradient sintering of cemented carbides[END_REF] or 2 [START_REF] Ekroth | Gradient zones in wc-ti(c,n)-co-based cemented carbides: experimental study and computer simulations[END_REF] was used to simulate the effect of volume fraction binder on the sintering of carbides.

The sample geometry was discretized into a suitable number of volume cells. In the present work, the following phases considered for the calculations were γ-FCC or α-BCC, M 7 C 3 and M 23 C 6 . As graphite and cementite were not observed experimentally, their formation was prevented (dormant phases). This allowed to obtain a carbon activity greater than 1 at the surface of the steels. For durations up to 3000 h, the carbon concentrations measured by EPMA at the surface of the thick samples were used as boundary conditions at the sodium -metal interface (3.5, 4.0 and 2.7 wt. % for AIM1, 316L and EM10 respectively). These values were equal to their respective homogeneous carbon concentration measured in the foils. For 5000 h exposure, boundaries values equal to 3.7, 4.3 and 2.7 wt. % were used for AIM1, 316L and EM10 steels. The assumption of symmetry at the center of the samples provided the second spatial boundary condition of zero flux for all elements. The measured chemical compositions of the steels in the asreceived state were used as the initial composition at time t = 0. The model provided element concentrations in the matrix (γ-FCC phase for AIM1 and 316L steels or α-BCC phase for EM10 steel), average element concentrations in all phases and phase fractions as a function of time and distance.

Results and discussion

Exposure at 600 and 650 °C

For both austenitic steels, three carburized zones were observed after 500 h exposure at 600 °C (Fig. 2). The outer dark zone (called intragranular carburization zone) was composed of a dense precipitation of carbides in grains and at grain boundaries as shown on the TEM images in Fig. 3. This dense precipitation could explain why these areas of the specimen were strongly revealed by the etching (zone entirely dark in Fig. 2). The carbides were enriched in Cr and Fe and some of them were identified as M 23 C 6 carbides by diffraction TEM as shown in Fig. 4. The presence of graphite or cementite at the surface of the steel could not be identified by TEM analyses. Underneath this first zone, the grains were partially revealed but all grain boundaries were etched (Fig. 2). Finally, this second zone was followed by grain boundaries strongly marked by etching suggesting the precipitation of carbides in grain boundaries. For the ferritic EM10 steel, one single carburized zone was observed in Fig. 2 (intragranular carburization zone). Similar carburized zones were observed at 650 °C (not shown here).

Micrographs and TEM observations did not provide enough information about the nature and the volume fractions of carbides within the depth of the steels. For this purpose, the Rietveld refinements of the diffractograms, obtained every 10 μm with transmission XRSD, were analyzed. The results are presented in Fig. 5, for the three studied steels. For each diffractogram the following parameters were adjusted one at a time and always in the same order: number and type of phases (austenite or ferrite, M 23 C 6 and M 7 C 3 ), lattice parameters, crystallite size and micro strain, the volume fractions of each phase and texture.

The two last parameters are mainly related to the intensities of the peaks and significantly influenced the quality of the Rietveld refinement and thus the calculated volume fractions of carbides. The experimental diffractograms did not allow to measure and simulate the texture of the specimens. In order to correctly predict the volume fractions of phases, an arbitrary texture was used in the MAUD software. This procedure achieved a relative deviation of 5% for the volume fractions of carbides for the AIM1 and EM10 steels. For the 316L steel, a significant relative deviation (around 20%) was observed for depths higher than 250 μm. This effect was also observed for a few diffractograms of AIM1 and EM10 steels, where carbides could be identified but not quantified. This observation was due to the low number of peaks. The broadening of the peaks was induced by the small size of the crystallites or an increase of microstrain inside the crystallites. As a consequence, for the 316L steel, the volume fractions of carbides were not reported at depths higher than 250 μm. For AIM1 and EM10 steels, only reliable values of the volume fraction of carbides, extracted from the diffractograms, were reported. Examples of diffractograms used to obtain the aformentioned results are reported in Fig. 6 for each alloy. These diffractograms were obtained at 40 μm from the surface. Initial diffractograms and after 5000 h exposure at 600 °C are superimposed.

After exposure, M 7 C 3 and M 23 C 6 carbides (with M = Fe, Cr) were identified in the first 50 μm. In agreement with the TEM measurements, neither graphite nor cementite was found. According to the Rietveld refinements in Fig. 5, large volume fractions of carbides (M 7 C 3 + M 23 C 6 ) were measured, around 50, 80 and 40% at 10 μm depth for AIM1, 316L and EM10 respectively. These values were in agreement with the high density of carbides observed on the TEM images. In Fig. 5, the internal precipitation front was observed to progress with time resulting in the deeper precipitation of M 23 C 6 located at the grain boundaries of the steels revealed on the micrographs.

The precipitates nature and volume fraction slightly differed between the steels. On one hand, for 316L steel, about 20% of M 7 C 3 and 50 to 60% of M 23 C 6 were measured within the first 30 μm with no noticeable evolution between 1000 and 5000 h. On the other hand, for AIM1 steel, slightly higher volume fractions of M 7 C 3 were observed over a larger thickness. The thickness increased with time (from 100 to 170 μm between 1000 and 5000 h). Such a large difference in the volume fractions of M 23 C 6 and M 7 C 3 carbides between 316L and AIM1 steel was not predicted by the ThermoCalc calculations (not shown here) as the two steels have rather similar chemical compositions. The larger volume fractions of carbides for AIM1 steel could be induced by the higher density of dislocations formed during its cold-working. These defects in addition to the presence of grain boundaries could have promoted the transformation of M 23 C 6 into M 7 C 3 carbides. For the EM10 steel, much deeper precipitation of M 23 C 6 and M 7 C 3 was measured and again the thickness increased with time (from 150 to 180 μm between 1000 and 5000 h). This observation can be explained by the greater permeability of carbon in ferrite (4 10 -12 wt. % cm 2 s -1 ) than in austenite (5 10 -14 and 9 10 -14 wt. % cm 2 s -1 for AIM1 and 316L steels respectively) at 600 °C and by its finer microstructure (more grain boundaries acting as diffusion paths).

The difference in the precipitation behavior between the steels was also reflected on the carbon concentration profiles (Fig. 7). In Fig. 7, the measured carbon concentrations were representative of the carbon dissolved in the matrix and the carbon trapped in carbide precipitates. The amount of carbon trapped in the precipitates (C carbides ) was calculated using the measured carbide fractions using Eq. (3) [START_REF] Pillai | Carbides in an aluminised single crystal superalloy: tracing the source of carbon[END_REF], with ρ i the densities of M 7 C 3 and M 23 C 6 carbides taken from literature [START_REF] Haynes | CRC Handbook of Chemistry and Physics[END_REF][START_REF] Pearson | Interstitial Alloys[END_REF], f i the carbide volume fraction from the Rietveld measurements and w C i the weight fraction of carbon in the carbides.

= + + C f ρ w f ρ w f ρ f ρ (wt. %)/100 carbides M C M C C M C M C M C C M C M C M C M C M C
In Fig. 7, considering the relative deviation of the XRSD measurements, a good agreement was observed between the carbon concentration profiles measured by EPMA and the ones calculated with the Rietveld refinement values. This observation suggested that the measured carbon concentrations corresponded mainly to carbon trapped in precipitates. At 650 °C, the measured carbon concentrations values were close to the one obtained at 600 °C after 500 h exposure (Fig. 8), suggesting as well a large precipitation of carbides in the grains and at the grain boundaries of the steels. Finally, in Fig. 7, the carbon concentration measured at the surface of the specimens was independent of time and corresponded to the homogeneous carbon concentration measured in the foils. This observation indicated that the carbon activity reached at the liquid-metal interface was, at any time, equal to the carbon activity imposed by the carburizing sodium. As mentioned previously, this value was used as a boundary condition for the DICTRA simulations described further in the article. Interestingly, after carburization, the lattice parameters of austenite or ferrite (reported in Fig. 9) evolved along the carbon concentration profiles (relative deviation of 0.002 Å). In the intragranular carburization zone, the lattice parameter value was lower than the lattice parameter value measured before the experiment. Then, the value increased within the intermediate carburization zone (intra and intergranular carburization zones) to finally reach the initial lattice parameter value deeper in the steel. The evolution of the lattice parameter is usually induced by modifications of the chemical composition and/or by mechanical stresses in the metallic matrix. In this case, the effect of mechanical stresses on the lattice parameter could be neglected because the integration from 2D to 1D of all the positions of the matrix peaks (austenitic or ferritic) were averaged along the Debye-Scherrer rings obtained on the 2D detector of the ID11 beamline. As a consequence, the decrease of the lattice parameters of austenite or ferrite was likely induced by a modification of the chemical composition. From the literature, the concentration of chromium and the concentration of carbon in the matrix can impact the lattice parameter value. The lattice parameter of austenite increases with addition of chromium in austenite [START_REF] Dyson | Effect of alloying addition on the lattice parameter of austenite[END_REF] and also with carbon in supersaturation in austenite [START_REF] Sun | Structural characteristics of low temperature plasma carburised austenitic stainless steel[END_REF][START_REF] Cao | Colossal carbon supersaturation in austenitic stainless steels carburized at low temperature[END_REF]. As demonstrated previously (TEM and XRSD measurements), the carbon was mainly trapped into carbides. Thus, the carbon concentration in the matrix of the studied steels were relatively low and no strong impact on the lattice parameter is expected. Thus, the evolution of the lattice parameter might be rather induced by an evolution of the chromium concentration in the matrix. The following scenario can be proposed. When the chromium carbides were formed, the matrix was depleted in chromium, consequently decreasing the lattice parameter. This decrease was both a function of the degree of chromium enrichment in the carbides and a function of the volume fraction of carbides. In Fig. 9, for AIM1 steel, two "plateau" values were observed. They corresponded to the presence of M 7 C 3 and M 23 C 6 in the zone closer to the surface and then to the presence of M 23 C 6 underneath. A higher lattice parameter value was observed in the first zone, near the surface, in good agreement with a higher iron enrichment of carbides (lower chromium depletion) where the carbon activity/concentration is higher (see Fig. 10). Deeper in the steel, the volume fraction of M 7 C 3 started to decrease while the volume fraction of M 23 C 6 increased. This resulted in a higher chromium depletion in the substrate, since the M 23 C 6 carbides were richer in metallic elements than M 7 C 3 (the total carbide volume fraction was roughly the same). This can explain the observed larger decrease of the lattice parameter (2nd "plateau"). Finally, at the end of the intragranular carburization zone, the carbide volume fraction decreased. The extent of chromium depletion decreased and the lattice parameter increased again to reach a value close to the initial lattice parameter value. For 316L steel, the two "plateau" were less pronounced due the lower volume fraction of M 7 C 3 . For EM10, the lattice parameter was lower than the initial value through the whole sample thickness as the sample was carburized entirely.

Thermodynamic and kinetic modeling of carbon diffusion

After thoroughly investigating the extent of carburization in the three steels, numerical simulations were performed to discuss the mechanisms which control the nature and kinetics of carburization.

Influence of grain boundary diffusion and carbides geometry on the carburization kinetics

In the simulations, both carbides M 23 C 6 and M 7 C 3 were considered to be stable. As graphite and cementite were not observed experimentally, their formation was prevented (dormant phases). A comparison of the temporal evolution of the carbon concentration profiles is presented in Fig. 11 for the three steels after 500 and 3000 h exposure at 600 °C. For the austenitic steels in Fig. 11a andb, the experimental carbon profile could not be simulated entirely. As a consequence, the carbon concentration profile was divided in two parts: the intragranular and intergranular carburization zones. In the intragranular carburization zone, a good agreement between the predicted and the measured carbon profile was obtained. Nevertheless, this agreement strongly depended on the chosen q value for the labyrinth factor, f q . In Fig. 11a and b, it was not possible to simulate the carbon concentration profiles that corresponded to the intragranular zone with the commonly used labyrinth factor (neither f 1 or f 2 as in the literature [START_REF] Engström | Computer simulation of diffusion in multiphase systems[END_REF][START_REF] Turpin | Carbon diffusion and phase transformations during gas carburizing of high-alloyed stainless steels: Experimental study and theoretical modeling[END_REF][START_REF] Sudha | Carburization behavior of aisi 316ln austenitic stainless steelexperimental studies and modeling[END_REF][START_REF] Garcia | Experimental investigations and dictra simulations on formation of diffusion-controlled fcc-rich surface layers on cemented carbides[END_REF][START_REF] Rong | Thermodynamic simulation of low temperature colossal carburization of austenitic stainless steel[END_REF][START_REF] Ekroth | Gradient zones in wc-ti(c,n)-co-based cemented carbides: experimental study and computer simulations[END_REF][START_REF] Frykholm | A new labyrinth factor for modelling the effect of binder volume fraction on gradient sintering of cemented carbides[END_REF] nor without one (f 0 = 1)). The most accurate simulations were achieved with labyrinth factors equal to f 4 after 500 h and f 5 after 3000 h exposure. For the ferritic steel EM10 (Fig. 11c), the values were lower and equal to f 2 after 500 h and between f 2 and f 3 after 3000 h. All the exponents of the labyrinth factor values were greater than the ones used in the literature (f 1 or f 2 [START_REF] Engström | Computer simulation of diffusion in multiphase systems[END_REF][START_REF] Turpin | Carbon diffusion and phase transformations during gas carburizing of high-alloyed stainless steels: Experimental study and theoretical modeling[END_REF][START_REF] Sudha | Carburization behavior of aisi 316ln austenitic stainless steelexperimental studies and modeling[END_REF][START_REF] Garcia | Experimental investigations and dictra simulations on formation of diffusion-controlled fcc-rich surface layers on cemented carbides[END_REF][START_REF] Rong | Thermodynamic simulation of low temperature colossal carburization of austenitic stainless steel[END_REF][START_REF] Ekroth | Gradient zones in wc-ti(c,n)-co-based cemented carbides: experimental study and computer simulations[END_REF][START_REF] Frykholm | A new labyrinth factor for modelling the effect of binder volume fraction on gradient sintering of cemented carbides[END_REF]). Furthermore, for the three steels, the q value increased with the exposure time. This observation suggested a progressive decrease of the diffusion rate of carbon which can be linked to the decrease of the matrix volume fraction (increase of the carbide volume fractions) and to the evolution of the carbide geometry. An evolution of the morphology of the carbides into a more complex and blocking geometry, such as platelets perpendicular to the inwards flux of carbon, could explain a decrease in the carbon diffusion in the alloy. This evolution could not be clearly observed on the etched cross-sections.

At 650 °C, the simulations are superimposed to the carbon concentration profiles in Fig. 8 after 200 h exposure. Again, the labyrinth factors were adjusted in order to fit correctly the intragranular carburization zone. For the austenitic steels, a factor equal to f 4 was used whereas no labyrinth factor was necessary for the ferritic steel EM10 (f 0 ).

Finally, the experimental intergranular carburization zone could not be simulated with the DICTRA model. The disagreement between the experimental and simulated carbon concentration profiles in this zone was clearly visible for the two austenitic steels in Figs. 8 and11. For these steels, the transition from the intragranular to the intergranular carburization zone was marked by the presence of a carbon diffusion tail deep in the substrate. For the ferritic steel EM10, the effect of grain boundary diffusion was marked after 500 h and no longer visible after 3000 h, where only one "mixed" carburized zone was observed experimentally. As a consequence, the whole carbon concentration profile observed after 3000 h could be simulated with the model with an adjusted labyrinth factor value. The absence of simulated carbon concentration profile in the intergranular zone was of course expected since the diffusion at grain boundaries was not considered in the model used in this study. The model used underestimated of the carburization depths reached in the steels. The predicted carburization depths were about 2 times lower than the one measured experimentally for the austenitic steels and 1.5 times lower for the ferritic steel after 500 h exposure.

Non-equilibrium carburization state

In order to understand why higher values of q were needed for f q in our study, compared to the ones usually used in published simulation of carburization kinetics, the experimental and simulated volume fractions of carbides formed in the steels grades were compared. The results are plotted in Figs. 12 and13a. Two distinct zones were observed on the simulated profiles whereas three different zones were identified on the experimental results. The results are summarized in Table 3.

The last zone in the experimental carbon profile corresponded to the intergranular carburization. As mentionned previously, this zone could not be predicted by the model. Regarding the intragranular carburization zone (zone 1 and 2 in Table 3), the amounts and type of carbides predicted by the DICTRA simulations were different than the one measured experimentally. For the austenitic steels (AIM1 in Fig. 12 and 316L in Fig. 13a), the simulations predicted a large amount of M 7 C 3 and a small amount of M 23 C 6 within the first 200 μm. These predictions differed strongly with the experimental results. Indeed, a large volume fraction of M 23 C 6 carbides and a smaller volume fraction of M 7 C 3 was observed. For EM10 steel, both M 7 C 3 and M 23 C 6 carbides were formed over 200 μm whereas DICTRA only predicted the precipitation of M 7 C 3 in the whole thickness of the EM10 steel. Both comparisons indicated that thermodynamic equilibrium of precipitation was not reached in the steels. At 600 °C, the conversion of M 23 C 6 into M 7 C 3 might have been not fast enough to reach thermodynamic equilibrium. As a consequence, the volume fraction of M 23 C 6 and M 7 C 3 carbides could not be precisely predicted with DICTRA. In this model the reactions of formation of carbides are assumed to be fast enough to be at thermodynamic equilibrium. With the aim of predicting correctly the nature and amount of carbides precipitated in the steels, new simulations were performed for the carburization of 316L steel. Only the precipitation of M 23 C 6 carbides was considered since their volume fraction was larger than the volume fraction of M 7 C 3 (the formation of M 7 C 3 was prevented, dormant phase). This assumption was relevant for the 316L steel for which a very low amount of M 7 C 3 was observed experimentally.

In Fig. 13b, a good agreement between the simulated and experimental carbide fractions was observed, except within the first 10 μm where a non-negligible volume fraction of M 7 C 3 carbides was observed experimentally. It is important to mention that, for the DICTRA simulations shown in Fig. 13b, a new labyrinth factor equal to f 2 (which differs from the previously used f 4 considering both carbides M 7 C 3 and M 23 C 6 ) was needed in order to fit correctly the EPMA carbon concentration profiles. This new labyrinth factor value is now similar to values of the common labyrinth factors used in the literature [START_REF] Engström | Computer simulation of diffusion in multiphase systems[END_REF][START_REF] Turpin | Carbon diffusion and phase transformations during gas carburizing of high-alloyed stainless steels: Experimental study and theoretical modeling[END_REF][START_REF] Sudha | Carburization behavior of aisi 316ln austenitic stainless steelexperimental studies and modeling[END_REF][START_REF] Garcia | Experimental investigations and dictra simulations on formation of diffusion-controlled fcc-rich surface layers on cemented carbides[END_REF][START_REF] Rong | Thermodynamic simulation of low temperature colossal carburization of austenitic stainless steel[END_REF][START_REF] Ekroth | Gradient zones in wc-ti(c,n)-co-based cemented carbides: experimental study and computer simulations[END_REF][START_REF] Frykholm | A new labyrinth factor for modelling the effect of binder volume fraction on gradient sintering of cemented carbides[END_REF].

Conclusions

The extent of carburization of three different steels exposed in liquid sodium at 600 and 650 °C up to 5000 h was thoroughly characterized.

Precipitation of M 23 C 6 and M 7 C 3 was identified with the use of transmission XRSD and TEM in the grains and at grain boundaries of the steels. The larger amount of M 7 C 3 observed in the AIM1 steel (strainhardened) than in the 316L steel (annealed) was associated with a higher density of dislocations. These defects, in addition to the presence of grain boundaries, could have allowed a faster conversion of M 23 C 6 into M 7 C 3 carbides. The deeper precipitation of carbides observed in the EM10 steel was induced by its fine microstructure and the higher solubility product compared to the austenite.

At 600 and 650 °C, several attempts were made to simulate the entire experimental carburization profiles obtained via EPMA measurements in the thickness of the three different steels with DICTRA. Firstly, the simulated carbon profiles strongly underestimated the maximum carburization depths. Half of the carbon concentration profiles measured was induced by the carbon diffusion and precipitation at grain boundaries. This process could not be considered and simulated with the chosen model. New simulations should be performed with the grain boundary model in DICTRA in order to improve the accuracy of the simulations. Secondly, the carbon concentration profiles in the intragranular zone could be predicted correctly but needed a careful adjustment of two input parameters: the labyrinth factor f q and the chosen carbide phases. At these intermediate temperatures, it is clear that local thermodynamic equilibrium, such as predicted by DICTRA, was not reached in the steels. As a consequence, the nature and evolution of the amount of the precipitated carbides could not be predicted with accuracy. In the simulations, for the 316L steel, the declared phases were chosen in accordance with the experimental observations. This indicated that the rate of precipitation of carbides must be considered for correct simulations which is, currently, not possible with the DICTRA model.

Unfortunately, even if a more accurate description of the phases could be used for one specimen, the value of f q could not be determined experimentally. As a consequence, the value was adjusted to the experimental profiles. As indicated in the literature [START_REF] Engström | Computer simulation of diffusion in multiphase systems[END_REF][START_REF] Turpin | Carbon diffusion and phase transformations during gas carburizing of high-alloyed stainless steels: Experimental study and theoretical modeling[END_REF][START_REF] Sudha | Carburization behavior of aisi 316ln austenitic stainless steelexperimental studies and modeling[END_REF][START_REF] Garcia | Experimental investigations and dictra simulations on formation of diffusion-controlled fcc-rich surface layers on cemented carbides[END_REF][START_REF] Rong | Thermodynamic simulation of low temperature colossal carburization of austenitic stainless steel[END_REF], the role of the labyrinth factor is to decrease the carbon flux in the matrix when the precipitate volume fraction increases. At 600 and 650 °C, the blocking effect of the carbides could not be neglected and was observed to evolve with time. Thus, the description of the labyrinth factor in the model should be improved. For example, by considering the evolution of the morphology of carbides with time.

Table 3

Differences between experimental and simulated carbide fraction profiles after 5000 h carburization at 600 °C and a C > 1.

Fig. 1 .

 1 Fig. 1. (a) Experimental set-up of the X-ray diffraction measurements at ESRF. (b) Schematic view of a sample cross section annotated with the successive measurements.

Fig. 2 .

 2 Fig. 2. Microstructure after etching of specimens (a) AIM1, (b) 316L et (c) EM10 after 500 h exposure at 600 °C and a C > 1.

Fig. 3 .

 3 Fig. 3. (a) BF and (b) HAADF images of carbides in grain and grain boundaries at 50 μm from the surface of AIM1 steel after 1000 h exposure at 600 °C and a C > 1.

Fig. 4 .

 4 Fig. 4. BF image and diffraction pattern or one carbide located in a grain boundary of AIM1 steel after 5000 h exposure at 600 °C and a C > 1.

Fig. 5 .

 5 Fig. 5. Carbide volume fractions in (a) AIM1, (b) 316L and (c) EM10 steels after 1000 and 5000 h exposure at 600 °C and a C > 1.

Fig. 6 .

 6 Fig. 6. Diffractogram at 40 μm from the surface of (a) AIM1, (b) 316L and (c) EM10 steels after 5000 h exposure at 600 °C and a C > 1 (▽ austenite ★ M 7 C 3 ⊡ M 23 C 6 ) superimposed with the diffractogram before exposure.

Fig. 7 .

 7 Fig. 7. EPMA carbon concentration profiles of (a) AIM1, (b) 316L and (c) EM10 measured after 1000 and 5000 h exposure at 600 °C and a C > 1 and calculated carbon concentration profiles from the volume fractions of the carbides measured by XRD.

Fig. 8 .

 8 Fig. 8. EPMA carbon concentration profiles (symbols) and DICTRA simulations (lines) and respective labyrinth factors f q for AIM1, 316L and EM10 after 200 h exposure at 650 °C and a C > 1.

Fig. 9 .

 9 Fig. 9. Evolution of the crystal parameter of (a) AIM1, (b) 316L and (c) EM10 steel as a function of time and depth after 1000 and 5000 h exposure at 600 °C and a C > 1.

Fig. 10 .

 10 Fig. 10. Evolution of the composition of carbides as function of the total carbon concentration in the steel. Calculated with ThermoCalc at 600 °C with TCFE8 database and AIM1 steel composition.

Fig. 11 .

 11 Fig. 11. EPMA carbon concentration profiles (symbols) and DICTRA simulations (lines) with different labyrinth factors f q for (a) AIM1, (b) 316L and (c) EM10 steels after 500 and 3000 h exposure at 600 °C and a C > 1.

Fig. 12 .

 12 Fig. 12. Experimental and simulated volume fraction of carbides measured in (a) AIM1 and (b) EM10 steels after 5000 h exposure at 600 °C and a C > 1.

Fig. 13 .

 13 Fig. 13. (a) Experimental and simulated volume fraction of carbides measured in 316L steel after 5000 h exposure at 600 °C and a C > 1 considering the precipitation of M 23 C 6 and M 7 C 3 . (b) Experimental and simulated volume fraction of carbides measured in 316L steel after 5000 h exposure at 600 °C and a C > 1 considering the precipitation of M 23 C 6 only f 2 as labyrinth factor.

Zones XRD experimental results Simulations Zone 1 6 Zone 3

 163 Presence of M 7 C 3 and M 23 C 6 with almost constant volume fractions Presence of M 7 C 3 of decreasing volume fraction and M 23 C 6 of increasing volume fraction (for AIM1 and 316L only) Zone 2 Presence of M 7 C 3 of decreasing volume fraction and M 23 C 6 of increasing volume fraction Rapid decrease of the volume fraction of M 23 C Slow decrease of the volume fraction of M 23 C 6 Not predicted

Table 2

 2 Composition of the nuclear grade sodium (industrial).

	Na	Ca	Cl	Fe	K	O
	99.95 wt. %	2 ppm	4 ppm	1 ppm	4 ppm	< 5 ppm

Acknowledgements

The authors are grateful to the ID11 beamline team: Pavel Sedmak and Jonathan Wright (ESRF Grenoble, France) for their assistance in carrying out the XRSD experiments. Prof. Daniel Chateigner (CRISMAT, Caen, France) is gratefully acknowledged for his help with the MAUD software and Rietveld analyses. Authors are also thankful to Rishi Pillai (ORNL, United States) for his advice with the DICTRA simulations.


domain: spi phys

text:
A Bablet

F Viallefont-Robinet

S Jacquemoud
email: jacquemoud@ipgp.fr

S Fabre

X Briottet
email: xavier.briottet@onera.fr

High-resolution mapping of in-depth soil moisture content through a laboratory experiment coupling a spectroradiometer and two hyperspectral cameras

Keywords: Hyperspectral remote sensing, Soil moisture content, High-resolution water vertical profile, MARMIT model Highlights

 A lab experiment is designed to measure surface and in-depth soil reflectance  The MARMIT model is used to associate surface moisture content to reflectance  Vertical maps of soil moisture content are generated on clay loam and sandy soils  The corresponding profiles are calculated for both soils

Introduction

Soil moisture content (SMC) is an important variable of the critical zone (Folke et al., 2004) that plays a key role in weather and climate predictions by controlling exchange and partitioning of water, gas, and energy fluxes between the land surface and the atmosphere [START_REF] Vereecken | On the value of soil moisture measurements in vadose zone hydrology: A review[END_REF][START_REF] Kornelsen | Root-zone soil moisture estimation using data-driven methods[END_REF]. Accurate estimation at various spatial and temporal scales is also important for strategic management of water resources [START_REF] Vereecken | Modeling soil processes: Review, key challenges, and new perspectives[END_REF], flood prediction [START_REF] Ochsner | State of the art in large-scale soil moisture monitoring[END_REF][START_REF] Schreiner-Mcgraw | Closing the water balance with cosmicray soil moisture measurements and assessing their relation to evapotranspiration in two semiarid watersheds[END_REF], drought monitoring [START_REF] Bolten | Evaluating the utility of remotely sensed soil moisture retrievals for operational agricultural drought monitoring[END_REF] and yield forecasting [START_REF] De Wit | Crop model data assimilation with the Ensemble Kalman filter for improving regional crop yield forecasts[END_REF] in agriculture. However, SMC greatly varies in time and space due to inherent variations in soil hydraulic properties [START_REF] Montzka | Hydraulic parameter estimation by remotely-sensed top soil moisture observations with the particle filter[END_REF], precipitation rates [START_REF] Koster | Regions of strong coupling between soil moisture and precipitation[END_REF][START_REF] Rosenbaum | Seasonal and event dynamics of spatial soil moisture patterns at the small catchment scale[END_REF], topography [START_REF] Joshi | Physical controls of near-surface soil moisture across varying spatial scales in an agricultural landscape during SMEX02[END_REF][START_REF] Jana | A comparative study of multiple approaches to soil hydraulic parameter scaling applied at the hillslope scale[END_REF][START_REF] Schröter | Estimation of catchment-scale soil moisture patterns based on terrain data and sparse TDR measurements using a fuzzy c-means clustering approach[END_REF], and vegetation characteristics [START_REF] Famiglietti | Variability in surface moisture content along a hillslope transect: Rattlesnake Hill, Texas[END_REF][START_REF] Gaur | Evolution of physical controls for soil moisture in humid and subhumid watersheds[END_REF].

Depending on the wavelength range, remote sensing is likely to provide information about surface or in-depth water content. Indeed, in the solar domain, sunlight only penetrates through the first micrometers or millimeters of the soil (e.g., [START_REF] Perkins | Penetration of light into littoral soils[END_REF][START_REF] Woolley | Light penetration and light-induced seed germination in soil[END_REF][START_REF] Tester | The penetration of light through soil[END_REF][START_REF] Liang | Supernormal vision and high-resolution retinal imaging through adaptive optics[END_REF][START_REF] Ciani | Light penetration in soil and particulate minerals[END_REF], thus the information collected is surface SMC.

According to the frequency, microwaves go deeper into the ground, thus the information collected is volume SMC. In situ measurements generally also correspond to volume SMC. Both surface and volume SMC are valuable information, according to the application, but the remote sensing community has seldom jointly studied them because the two wavelengths domains are separated in terms of signal processing. [START_REF] Sadeghi | High-resolution shortwave infrared imaging of water infiltration into dry soil[END_REF] pioneered the measurement of high-resolution soil water content profiles over seven soils covering a wide range of textures. They set up a laboratory experiment involving a broadband camera operating in the shortwave infrared (900-1700 nm) and a novel TDR array. They applied a relationship derived from the Kubelka-Munk theory to study water flux profiles with an unprecedented accuracy.

This article presents a laboratory experiment imagined nearly at the same period to assess both surface SMC and in depth SMC profiles thanks to a spectrometer, two hyperspectral cameras, and a radiative transfer model called MARMIT [START_REF] Bablet | MARMIT: A multilayer radiative transfer model of soil reflectance to estimate surface soil moisture content in the solar domain (400-2500 nm)[END_REF]. We poured controlled amounts of water on two different soils and measured the evolution of the SMC profiles with high spatial resolution.

Section 2 presents the laboratory experiment and the calibration of the radiometric data. In Section 3, the spectral reflectance of the soils measured on the surface is compared to the one imaged on the side of the aquarium; we also quantify the evolution of the SMC in time thanks to the MARMITforSMC method. The results are discussed in Section 4.

Material, measurements and processing

Experimental set-up

As a first trial, two different soils, a clay loam and a sand, have been selected for this experiment.

Their main characteristics are summarized in Table 1. Although the fraction passing a 2 mm sieve was collected for each soil, the sand was mostly composed of large white grains of homogeneous size while the clay loam consisted of smaller brown grains of heterogeneous sizes. Both soils were successively poured into an aquarium (dimensions: 27.5 cm long  13 cm wide  20 cm high) with glass walls 1.5 cm thick in their center, the corners being slightly thicker than the faces. The aquarium was put on a linear motion platform. The spectral radiance was measured from above (surface measurement) and from one side of the aquarium (in-depth measurement). An ASD FieldSpec 3 HR spectroradiometer equipped with a 5° aperture lens was used for the nadir acquisitions. The detector being located at a distance of 37 cm from the soil, the footprint is a disk of about 3.2 cm in diameter. Two halogen lamps positioned on each side of the spectroradiometer lit the soil from above with an incidence angle of 10°. A third halogen lamp lit the side of the aquarium. Two hyperspectral cameras fixedly positioned in front of the aquarium were used for the lateral acquisitions. They measure the radiance of a vertical pixel line in the visible-near infrared (HySpex VNIR-1600) and in the shortwave infrared (HySpex SWIR-320m-e). The pixel size of the VNIR camera is 0.06 mm × 0.12 mm while that of the SWIR camera is 0.287 mm × 0.287 mm.

The two cameras imaged the scene line-by-line while the aquarium passed in front of them, like a pushbroom scanner. The images consist of 720 vertical lines in the VNIR and 360 vertical lines in the SWIR.

Their co-registration was helped by small pieces of duct tape on the outer glass wall of the aquarium. A common area 10.2 cm long and 9.1 cm wide was finally delimited to study the water profiles. The experimental setup is illustrated in Fig. 1 and a summary of the main characteristics of the optical sensors is provided in Table 2.

ASD FieldSpec 3 Hr

HySpex cameras

Linear motion platform

Finally, after each scan, two TDR (Time Domain Reflectometry) probes were inserted in the soil from above to measure volume moisture content at two different depths: the Trime-Pico 64/32 probe (5 cm long pins) and the FieldScout TDR 100 probe (3.7 cm long pins). According to the user manuals, soil moisture is detected up to a distance of 3 cm around the pins, which corresponds to a surveyed cylinder shape volume of 141 cm 3 for the Trime-Pico probe and 104.6 cm 3 for the FieldScout TDR. These values are to compare to the volume of the aquarium that is 7150 cm 3 . The probe accuracy provided by the manufacturers is ± 3% moisture by volume.

Measurement protocol

The measurements are carried out in four stages:

1) A known amount of water is uniformly and softly poured onto the soil using a plastic box drilled with 2 mm diameter holes. It limits the change in appearance of the soil surface, and it prevents water from flowing along the walls of the aquarium. After moistening, the aquarium is put back on the linear motion platform in exactly the same position.

2) The radiance spectrum of the soil surface is measured with the ASD spectroradiometer. Each spectrum is actually an average of 50 spectra. During acquisition, the lamp illuminating the aquarium from the side and used for the HySpex measurements is obscured by an opaque shield.

3) Hyperspectral images of the soil are acquired with the line-scan HySpex cameras by translating the aquarium laterally.

4) The two TDR probes are then inserted into the soil at one side of the aquarium to measure volumetric soil moisture content.

Once the cycle is over, a given amount of water (~100 g) is added and new measurements are performed (steps 1 to 4) after the water front stabilizes. A total of nine cycles of measurements were made corresponding to the dry soil plus eight moisture levels (Table 3). The waiting time between two cycles was much longer for the clay loam soil (up to 50 minutes) than for the sandy soil (up to 15 minutes) so the duration of one cycle varies according to soil type and moistening level. The water mass (g) was translated into rainfall values (mm) for comparison with natural precipitations. For instance, Météo-France, the French national meteorological service, indicates a monthly cumulative rainfall of 55 mm in Paris: a light rain corresponds to a rainfall of 1 to 3 mm h 1 , an average rain to a rainfall of 4-7 mm h 1 , and a heavy rain to a rainfall of 8 mm h 1 or more. During extreme events, these values may exceed 100 mm h 1 . The quantities of water poured into the aquarium during our experience consequently ranged from light to heavy rainfalls in a temperate region.

Data pre-processing chain

Data production requires a series of pre-processing operations. The first step is the coregistration of the VNIR and SWIR images acquired by the two cameras in order to cover the entire reflective domain.

Their instantaneous fields of view are different, so are the spatial resolutions of the images. Thus, preprocessing begins with the degradation of the spatial resolution of the VNIR image in order to match that of the SWIR image. This resampling is achieved by using the nearest neighbor filter that preserves the spectral information. The two images are then co-registered using GeFolki, a multi-scale registration software developed by ONERA that exploits the few overlapping spectral bands between the two images [START_REF] Brigot | Adaptation and evaluation of an optical flow method applied to coregistration of forest remote sensing images[END_REF]. It results in a hyperspectral data cube containing 320 × 300 pixels with a side length of 0.287 mm (9.1 cm × 8.5 cm image) in the reflective domain. Each pixel contains a continuous radiance spectrum made of 408 narrow bands between 414 nm and 2400 nm (Fig. 2). The spectral sampling is 3.26 nm for the VNIR camera and 5.45 nm for the SWIR camera. The last eight spectral bands of the VNIR camera, also measured by the SWIR camera, are removed in order to avoid spectral domain overlap. Fig. 2 displays the radiance of a pixel extracted from the dry sand image. A jump in the radiance spectrum is clearly visible at the wavelength of 966 nm corresponding to the boundary between the VNIR and SWIR images (Fig. 2). That problem, common in radiometry, is expected to disappear when calculating the reflectance factor of the soil-aquarium system:

𝑅 𝑠-𝑎 = 𝐿 𝑠-𝑎 𝐿 𝑟𝑝 × 𝑅 𝑟𝑝 (1)
with 𝑅 𝑠-𝑎 and 𝐿 𝑠-𝑎 the reflectance and the radiance of the soil-aquarium system, 𝑅 𝑟𝑝 and 𝐿 𝑟𝑝 the reflectance and the radiance of the reference panel (Spectralon®). The relevant quantity is nevertheless the soil reflectance. To remove the contribution of the aquarium glass wall, we placed a sheet of white paper, the optical properties of which are known, successively inside and outside the aquarium. To first order, the transmittance of the aquarium glass wall equals:

𝜏(𝜆) = 𝑅 𝑝𝑎𝑝𝑒𝑟+𝑔𝑙𝑎𝑠𝑠 𝑅 𝑝𝑎𝑝𝑒𝑟 (2)
with 𝑅 𝑝𝑎𝑝𝑒𝑟+𝑔𝑙𝑎𝑠𝑠 and 𝑅 𝑝𝑎𝑝𝑒𝑟 respectively the reflectance of the paper behind and in front of the glass wall. Fig. 3 shows the spectral variation of 𝜏(𝜆). Then each reflectance spectrum of the soil-aquarium system is divided by 𝜏(𝜆) to calculate the soil reflectance only. However, although reduced after conversion to reflectance and correction of the aquarium transmittance, the jump at 966 nm was still noticeable (Fig. 4). This artifact may have several causes including camera instability or non-nominal shutter operation of one of the two cameras. To remove it, we compared the reflectance of the dry sand acquired vertically with the ASD with that acquired laterally with the cameras. The sand was used rather than the clay loam because the grain size, which is known to control soil reflectance (e.g., [START_REF] Okin | Effect of grain size on remotely sensed spectral reflectance of sandy desert surfaces[END_REF][START_REF] Sadeghi | Particle size effects on soil reflectance explained by an analytical radiative transfer model[END_REF], is more homogeneous. Fig. 4 shows a good agreement in the SWIR but an overestimation of the reflectance spectrum by the HySpex VNIR camera. According to the pixel, this overestimation varies between 0% and 5%. Therefore, the reflectance ratio 𝐶 defined as

𝐶 = 𝑅(𝜆 = 963.3 nm) 𝑅(𝜆 = 966.7 nm) (3)
is calculated for each pixel of the image at the junction between the two cameras and the VNIR spectrum is divided by 𝐶 to obtain a continuous spectrum (Fig. 4). This pre-processing is a prerequisite for analyzing the spectra acquired by the HySpex cameras.

Results

Reflectance analysis

Stability

The impact on the reflectance of a variation in illumination during the experiment was first assessed.

Two phenomena are likely to occur: 1) a drift of the luminous intensity of the lamps, and 2) a variation of the ambient light of the laboratory that was in the half-light. These effects can be theoretically corrected by regular measurements of the radiance of the Spectralon® panel but, due to experimental constraints, only one spectrum was acquired at the beginning of the experiment. In order to appraise this impact, we considered a deep zone in the sand that remains dry from measurements #1 to #5 (Fig. 5a), as well as one of the three pieces of duct tape (Fig. 5b). In both cases, the change in reflectance was less than 4%.

Vertical variation

The change in soil reflectance as a function of depth provides us with information on its variation in relation to surface reflectance. To study that relationship, we divided the images into twenty contiguous rectangles (78 pixels × 15 pixels) having the same surface area (2.24 cm × 0.43 cm) as illustrated by Fig. 6. Prior to comparing the reflectance of wet and dry areas, one can analyze the reflectance of the dry soils. Fig. 7 superimposes the reflectance spectrum of the surface on the average reflectance spectrum measured in each layer. The variability of dry soil reflectance is higher for the clay loam than for the Wavelength (nm) Wavelength (nm)

sand, probably due to a larger range of grain size. For the various moisture levels, the differences in reflectance levels can be only attributed to soil water content since the other soil properties influencing reflectance (mineralogical composition, organic matter, texture, homogeneity ...) are invariant during the experiment. For the clay loam, the lowest reflectance spectrum that corresponds to the wettest soil layer remains close to the surface and it matches closely the ASD reflectance spectrum. For the sand, it is found deeper in the aquarium as water is poured in. We also plotted the standard deviation of the reflectance of all pixels in a zone (Fig. 8). It is higher for the dry clay loam (8% maximum for Zone 12) than for the dry sand (3% maximum for Zone 3). One can also note that it is lowest at high moisture levels (saturated soil). In the transition zones where the soil is partially wet, it is higher for the sand than for the clay loam, suggesting that the distribution of water in the sand is highly variable within the same zone. The changes in reflectance due to the penetration of water into both soils are analyzed qualitatively. To go further, the next section will focus on soil moisture variation during infiltration. On the left, the clay loam and on the right, the sand.

Water content and moisture profile estimation

The MARMIT model and the MARMITforSMC approach

MARMIT is a radiative transfer model that simulates the reflectance spectrum of a wet soil in the solar domain assuming that the reflectance spectrum of the dry soil is known [START_REF] Bablet | MARMIT: A multilayer radiative transfer model of soil reflectance to estimate surface soil moisture content in the solar domain (400-2500 nm)[END_REF]. It considers a wet soil as a dry soil covered with a thin film of water characterized by two parameters: the thickness of the film 𝐿 and the efficiency 𝜀 defined as the proportion of the ground covered by water.

These two parameters are retrieved by fitting MARMIT to the measured reflectance spectrum. A calibration of the model is required for each soil: it consists in measuring reflectance spectra at various SMC and in linking SMC and the mean water thickness 𝜑 = 𝐿 × 𝜀 by a sigmoid function (Fig. 9):

𝑆𝑀𝐶 𝑐𝑙𝑎𝑦 𝑙𝑜𝑎𝑚 = 45.3 1 + 9.4 × 𝑒 -403×𝜑 (4)
𝑆𝑀𝐶 𝑠𝑎𝑛𝑑 = 23.6 1 + 9.0 × 𝑒 -39.4×𝜑

(5) 4) and ( 5) respectively.

Note that the soil samples used in this calibration phase were placed in Petri dishes 1 cm deep, therefore, the estimated SMC is strictly speaking not a surface attribute but actually a volume attribute.

Thanks to Eqs. ( 4) and ( 5), one can map SMC in the aquarium using the hyperspectral cameras, and at the same time, assess SMC on the soil surface using the ASD spectroradiometer. These two quantities have never been measured simultaneously before.

Check for consistency

Applying Eqs. ( 4) and ( 5) for the aquarium experiment implies that the water distribution is the same as in the Petri dishes. Moreover, since the measurement protocols differ in the two experiments (illumination angle and distance of the sensors from the soil sample), it is imperative to verify that the reflectance spectra of the dry soils are similar both for the calibration and experiment samples. A difference up to 10% is observed for the clay loam (Fig. 10a), while they match closely for the sand (Fig. 10b). The discrepancy is more important in the near-infrared and in the short-wave infrared than in the visible region of the electromagnetic spectrum. It has been shown that the influence of the grain size on reflectance is minimum in the visible and maximum in the infrared (e.g., [START_REF] Okin | Effect of grain size on remotely sensed spectral reflectance of sandy desert surfaces[END_REF][START_REF] Sadeghi | Particle size effects on soil reflectance explained by an analytical radiative transfer model[END_REF]. We conclude from this that the differences in reflectance for the clay loam are probably due to inhomogeneities in the grain size, and that the protocol has little or no influence on the measured reflectance.

ⓐ ⓑ

Fig. 10. Differences between the spectral reflectances acquired with the ASD for soil in the Petri dish and in the aquarium (dash-dot line in Fig. 7) for (a) the dry clay loam and (b) the dry sand.

SMC maps and profiles versus TDR measurements

The input parameters 𝐿 and 𝜀 of MARMIT are first estimated by inversion of the model on the soil reflectance spectra acquired both on the surface and the side of the aquarium. Then, the values of 𝜑 = 𝐿 × 𝜀 are injected into Eqs (4) and ( 5) to calculate SMC (Figs 11 and 12). The estimation error is about 3% [START_REF] Bablet | MARMIT: A multilayer radiative transfer model of soil reflectance to estimate surface soil moisture content in the solar domain (400-2500 nm)[END_REF]. Note that the calibration equations linking SMC and φ do not necessarily pass through the origin (Fig. 9), which may lead to uncertainties for low water content (SMC < 5% for the clay loam and SMC < 3% for the sand). Water content profiles are then easily obtained by averaging the SMC values on each horizontal pixel line of the hyperspectral images. The integrated values of soil moisture content (SMCi) determined using the water content maps can be compared with the values measured by the FieldScout TDR 100 (SMCfs) and the Trime-Pico 64/32 (SMCtp) probes just after the radiometric measurements. These measurements enable us to assess integrated SMC between 0 and 3.7 cm and between 0 and 5 cm, respectively (Table 4). The values of the two probes agree for the clay loam but are somewhat different for the sand. Such a difference was expected but the evolution of the measurements with the successive "rain falls" doesn't follow the increase and decrease suggested by the profiles, with a decrease occurring earlier for the FieldScout. Moreover, there is no apparent relationship between SMCi and the probe measurements. Their representativeness may be poor in such a context due to the successive driving and digging up of the probes, which disturb the soil layers. These effects may vary from one soil to another.

The profiles extracted from the images agree with the literature: after moistening, water is evenly distributed in the clay loam when the sand displays a wide disparity, as noticed by [START_REF] Edwards | Comparison of surface moisture measurements with depthintegrated moisture measurements on a fine-grained beach[END_REF].

At the highest moisture level (#8), SMC is less than 5% in the first 2cm of the sand whereas it equals 17% on average below. For the clay loam, the water content equals 45% at surface and at depth when the soil is saturated with water. 2). S = piece of duct tape. The star points the SMC value measured at the top of the soil.

For the sand, we also noticed that SMC determined with the ASD spectroradiometer did not exactly match SMC determined with the HySpex cameras at the top of the profile (Figs 11 and 12). This difference is due to the unevenness of surface moistening: the central part of the aquarium is wetter than the edges, where water infiltration is monitored by cameras. In addition, SMC at the top of the soil is difficult to determine accurately from the images.

Despite the issues mentioned, with regard to water content profiles, we recognize the three regimes that have been identified by Al-Hamdan and Cruise (2010):

1) the soil has just been watered, it is wet on the surface but dry in depth (moisture levels #1 to #7 for the loam clay and moisture level #1 for the sand);

2) water has entered into the soil, which is dry on the surface but wet in depth (moisture levels #6 to #8 for the sand);

3) the soil is still dry in depth, but and has become dry again on surface after infiltration or evaporation (moisture levels #2 to # 4 for the sand).

Such a dynamic (or transitional) regime typically occurs several hours after a rain.

Various techniques can be used in the laboratory to assess soil moisture profiles. [START_REF] Sadeghi | High-resolution shortwave infrared imaging of water infiltration into dry soil[END_REF] who reviewed the most common ones (X-ray micro-computed tomography, neutron radiography, and magnetic resonance imaging) explain that optical methods are easier, faster, and cheaper to implement.

In particular, the spatial resolution is higher for equal soil volume. Moreover, our experiment enables to assess surface and in-depth SMC nearly simultaneously.

If the soil physical parameters are available, profile can also be computed thanks to theoretical hydrological models such as the HYDRUS computer software packages (Šimůnek et al., 2016).

Although this approach is a powerful way to connect surface and subsurface SMC in a large number of cases, it is beyond the scope of this article. However, our experiment provide specialists with unique experimental data to run such models.

Conclusion

We designed a new experiment to measure simultaneously soil surface reflectance and soil in-depth reflectance, and to study their evolution over space and time while pouring water onto a soil. We used the MARMITforSMC method to obtain high spatial resolution maps of SMC and the associated water profiles. We observed profiles commonly described in the literature, but with a spatial resolution rarely achieved.

As observed of [START_REF] Edwards | Comparison of surface moisture measurements with depthintegrated moisture measurements on a fine-grained beach[END_REF] on sand, surface SMC is much lower than SMC at 3 cm depth about 15 to 20 min after a heavy rainfall (20-30 mm). For the clay loam, water penetrates uniformly into the soil. We also tried to compare the SMC profiles with measurements provided by TDR probes.

While the agreement is good for the clay loam, there are discrepancies in the results for the sand. It would be interesting to perform complementary experiments with TDR arrays [START_REF] Sheng | A TDR array probe for monitoring near-surface soil moisture distribution[END_REF] or to assess the shape of SMC profiles when evaporation occurs. Our experiment also indicates that, at least for some soils, it is necessary to improve the link between surface reflectance and SMC established using Petri dishes.

Studying other soils covering the texture triangle, like in [START_REF] Sadeghi | High-resolution shortwave infrared imaging of water infiltration into dry soil[END_REF], would be worthwhile, while keeping in mind that a large amount of matter (~ 350 cm 3 ) is needed to fill the aquarium. Several spaceborne hyperspectral sensors are under development (e.g., EnMap, HyspIRI, Shalom) or already operational (e.g., PRISMA) so that new remote sensing data will be soon available. The validation of semi-empirical relationships or radiative transfer models for SMC retrieval using these data is an important step before producing SMC maps routinely. In such a context, the comparison between surface SMC and in-depth SMC is a key point regarding the link between hyperspectral imagery and in situ measurements.

Fig. 1 .

 1 Fig. 1. Experimental set-up showing the two hyperspectral cameras, the spectroradiometer, the two lamps illuminating the soil from above, and the aquarium on the linear motion platform.

Fig. 2 .

 2 Fig. 2. (Left) image of the dry sand in the aquarium obtained with the HySpex cameras at 966 nm with the pieces of duct tape in grey and (right) radiance spectrum of a pixel of the image expressed in W m 2 sr 1 µm 1 .

Fig. 3 .

 3 Fig. 3. Spectral transmittance τ of the aquarium wall.

Fig. 4 .

 4 Fig. 4. Uncorrected reflectance spectrum of the sand acquired by the cameras compared to the reflectance spectrum acquired on the surface by the ASD and to the corrected spectrum of the same soil in the aquarium.

Fig. 5 .

 5 Fig. 5. Reflectance variations of (a) dry sand and (b) duct tape at moisture levels #1 to #5.

Fig. 6 .

 6 Fig. 6. Representation of the twenty rectangles on the RGB image of the dry clay loam. The dark blue rectangle, referred as Zone 1, is the closest to the surface while the yellow rectangle, referred as Zone 20, is the farthest. In light grey, the three pieces of duct tape.

Fig. 7 .

 7 Fig. 7. Average reflectance spectra of the twenty rectangular zones for the dry and wet soils at moisture levels #2, #5, and #8. In dark blue, the closest to the surface (Zone 1) and in yellow, the farthest (Zone 20). On the left, the clay loam and on the right, the sand.

Fig. 8 .

 8 Fig. 8. Standard deviation of the reflectance spectra in the twenty rectangular zones for the dry and wet soils at moisture levels #2, #5, and #8. In dark blue, the closest to the surface (Zone 1) and in yellow, the farthest (Zone 20).On the left, the clay loam and on the right, the sand.

Fig. 9 .

 9 Fig. 9. Variation of SMC as a function of the mean water thickness for the clay loam and the sand. The lines correspond to Eqs. (4) and (5) respectively.

Fig. 11 .

 11 Fig. 11. SMC maps of the clay loam as a function of the amount of water poured and associated water content profiles. From top left to bottom right: moisture levels #1 to #8 (Table2). S = piece of duct tape. The star points the SMC value measured at the top of the soil.

Fig. 12 .

 12 Fig. 12. SMC maps of the sand as a function of the amount of water poured and associated water content profiles. From top left to bottom right: moisture levels #1 to #8 (Table2). S = piece of duct tape. The star points the SMC value measured at the top of the soil.

Table 1 .

 1 Some characteristics of the soil samples.

	Soil sample		Texture		Composition	Bulk density
		Clay	Silt	Sand
	Clay loam	35.8 %	40.3 %	23.9 %
	Sand	0 %	0 %	100 %	SiO 2 (98.5 %)	1.5 g cm 3

Table 2 .

 2 Main characteristics of the optical sensors.

	Sensor	Spectral range	Spectral sampling	Number of spectral	Aperture angle (°) or field
		(nm)	interval (nm)	bands	of view (mrad)
	ASD	350-2500	1.4-2	1512	5°
	VNIR camera	414-992	3.7	160	0.18 mrad
	SWIR camera	966-2500	6	256	0.75 mrad

Table 3 .

 3 Quantity of water poured over the two soils at each moistening stage. ∆t is the elapsed time between the addition of water and the measurement with the cameras.

	Moisture			Clay loam				Sand
	level	t	Added water	Cumulative	Cumulative	t	Added water	Cumulative	Cumulative
		(min)	mass (g)	water (g)	water (mm)	(min)	mass (g)	water (g)	water (mm)
	#1	18	96.5	96.5	2.7	9	100.3	100.3	2.8
	#2	8	98.2	194.7	5.5	8	105.9	206.2	5.8
	#3	9	108.2	302.9	8.5	7	99.1	305.3	8.5
	#4	16	99.6	402.5	11.3	6	107.0	412.3	11.5
	#5	17	115.3	517.8	14.5	10	112.5	524.8	14.7
	#6	14	127.2	645.0	18.1	10	101.1	625.9	17.5
	#7	48	180.1	825.1	23.1	12	109.6	735.5	20.6
	#8	50	208.1	1033.2	28.9	15	209.7	945.2	26.5

Table 4 .

 4 Volume soil moisture content (cm cm 3 × 100) measured by the two TDR probes: SMCfs for the FieldScout TDR 100 (3.7 cm pins) and SMCtp for the Trime-Pico 64/32 (5 cm pins).

	Moisture level	Clay loam		Sand
		SMCfs	SMCtp	SMCfs	SMCtp
	Dry soil	0.0	0.3	0.0	0.3
	#1	8.4	4.3	1.9	1.3
	#2	13.8	12.2	2.2	5.1
	#3	28.0	29.9	3.1	6.5
	#4	34.3	32.5	5.4	17.0
	#5	35.7	37.2	5.1	21.5
	#6	36.5	38.9	4.8	18.7
	#7	37.0	39.0	4.5	19.1
	#8	40.0	44.5	4.6	20.7

Acknowledgements

The PhD thesis of Aurélien Bablet was funded by ONERA and IPGP. This work was also supported by the PNTS (Programme national de télédétection spatiale) in the frame of the SOILSPECT-2 project.

The authors would like to thank Philippe Déliot (ONERA) for his invaluable help during the measurements with the HySpex cameras.

Discussion

The water front takes longer to stabilize in the clay loam than in the sand, therefore the time elapsed between the addition of water on the soil and the radiance measurement is higher for the first soil (~50 minutes max) than for the second one (~15 minutes max) (Table 3). One reason is that water penetrates faster in the sand. This agrees with Darcy's law that describes the flow of a fluid through a porous medium. It stipulates that the hydraulic conductivity 𝐾 is much higher in coarse-grained soils (e.g., the sand) than in small-grained soils (e.g., the clay loam) due to the large intra-granular distance. Capillary forces are very low and the force of gravity prevails, so water tends to infiltrate without stagnating.

Therefore, movement of water across the sand is much faster. On the contrary, in a clay loam soil, the capillary forces that are greater due to the short intra-granular distance counterbalance the gravitational forces applied on the water. In short, water naturally seeps into the sand and it penetrates into the clay only because of the increased pressure due to the addition of water to the water column. Another reason may be evaporation that dries the surface: the first stage occurs earlier for the sand than for the clay loam (e.g., [START_REF] Han | Dynamics of soil water evaporation during soil drying: Laboratory experiment and numerical analysis[END_REF][START_REF] Or | Advances in soil evaporation physics-A review[END_REF]Song et al., 2014;[START_REF] Merz | Moisture profiles of the upper soil layer during evaporation monitored by NMR[END_REF]. However, only the capillary force explains the observations at a depth of 2 cm or below for the sand in such a short time.

The link between the hydraulic conductivity and the grain size may also explain the high standard deviation values in the sand (Fig. 8): for example, at moisture level #5, one can see rapid variations in SMC at about 4.5 cm depth. This anomaly is certainly linked to a different particle size in this zone. Indeed, the larger the grains, the greater the percolation. Therefore, SMC is smaller in this area.

Another issue concerns the deep pixels where a non-zero SMC is predicted while water did not arrive at this depth yet (see SMC #1 in Fig. 11). These pixels may correspond to pores in the dry soil, which create shadows that decrease the reflectance. Note that soil compaction is also likely to occur between two experiments, either because the aquarium is moved meanwhile, or because of the moistening itself.

The model assimilates such low-reflectance pixels to a wet soil.