# Carburization and Stress Profiles Characterized by High-Energy X-ray Diffraction in 316L Austenitic Stainless Steel After Exposure at 500 °C and 600 °C in Carburizing Liquid Sodium

Mohamed Fares Slim, Guillaume Geandier, Marie Romedenne, F. Rouillard, Benoit Malard

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Mohamed Fares Slim, Guillaume Geandier, Marie Romedenne, F. Rouillard, Benoit Malard. Carburization and Stress Profiles Characterized by High-Energy X-ray Diffraction in 316L Austenitic Stainless Steel After Exposure at 500 °C and 600 °C in Carburizing Liquid Sodium. Oxidation of Metals, 2021, 96 (3-4), pp.185-199. 10.1007/s11085-021-10039-6. hal-03799768

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# HTCPM Focus Issue / ID 1116

## Carburization and Stress Profiles Characterized by High-Energy X-ray Diffraction in 316L Austenitic Stainless Steel After Exposure At 500 °C and 600 °C in Carburizing Liquid Sodium

Mohamed Fares Slim<sup>a</sup>, Guillaume Geandier<sup>a</sup>, Marie Romedenne<sup>b</sup>, Fabien Rouillard<sup>c</sup>, Benoit Malard<sup>d</sup>

mohamed-fares.slim@univ-lorraine.fr, guillaume.geandier@univ-lorraine.fr, rmdmarie@gmail.com, fabien.rouillard@cea.fr, benoit.malard@toulouse-inp.fr

### Abstract

316L austenitic stainless steel was carburized in sodium containing a high carbon activity at two different temperatures, 500 °C and 600 °C for 1000 h. The carbon profile, carbide mass fraction and residual stress tensor profile were determined using electron probe microanalysis and high-energy X-ray diffraction. The carbon profile and carbides mass fractions were also predicted using a thermodynamic and kinetic modelling tool (DICTRA). At 600 °C, the experimental results and predictions suggested that the carbon absorbed by the sample was mainly trapped to form $M_{23}C_6$ and $M_7C_3$ carbides. The residual stress profile in austenite was strongly dependent on the $M_{23}C_6$ precipitation and the induced modification of the substrate chemical composition. At 500 °C, results and predictions suggested that the carbon was mainly dissolved in austenite. The residual stress profile was governed by the formation of expanded austenite at the sample surface.

**Keywords:** Carburization, austenitic stainless steel, high-energy X-ray diffraction, residual stress tensor

## 1. Introduction

In the framework of the development of a new generation of sodium-cooled fast nuclear reactor (SFR), the French alternative energies and atomic energy commission has launched a research program to optimize the control rods lifetime in reactor. In the current French SFR concept, the control rods consist of stainless steel cladding encapsulating $B_4C$ pellets. Post-mortem analysis of stainless steel cladding after exposure in former French SFRs revealed the formation of a deep and highly carburized zone [1]. Carburization has an impact on the mechanical properties of the stainless steel cladding: it causes a reduction in ductility and induces a brittle behavior [2, 3]. It is also susceptible to induce residual stresses (RS) [4, 5]. Thus, the characterization of the carburization and RS profiles is essential. In this work, the carburization and RS profiles of 316L austenitic stainless steel were characterized using electron probe microanalysis and high-energy X-ray diffraction after carburization at 500 °C and 600 °C for 1000 h.

## 2. Experimental setup

The detailed chemical composition of the studied specimen is given in Table 1. The studied material is a 316L austenitic stainless steel (solution-annealed) constituted of relatively equiaxed grains with an average size of 10 µm. Two steel samples were carburized at two different temperatures, 500 °C and 600 °C for 1000 h. Specimens of 20 x 20 x 1 mm³ were cut and immersed in a XC38 mild steel grade vessel filled of about 2.5 liters of nuclear grade liquid sodium. More details about the carburization experimental procedure can be found in previous works [6 - 8]. After carburization, the samples were cut. The carbon concentration profile, the phases mass fractions and the RS profile were characterized across the sample thickness. The carbon concentration profiles were obtained by electron probe microanalysis (EPMA) at 15 kV and 20 nA using the phi-rho-z method. The phi-rho-z method was used for the calibration of the analytical crystal for LPC2 $K_{αC}$ from a SiC standard as described in [6]. The mass fractions and RS profiles in carbides and austenite were determined by high-energy X-ray diffraction (HEXRD). The HEXRD experiments were performed in transmission mode at the PETRA P07-EH2 beamline (DESY-PETRA III, Hamburg, Germany). A schematic presentation of the experimental set-up is given in Fig. 1. The high-energy monochromatic rectangle-shaped beam (E = 103.4 keV, λ = 0.119907 Å, 300 x 2 µm²) allowed

<footnote><sup>a</sup> Université de Lorraine, CNRS, IJL, F - 54000 Nancy, France</footnote>

<footnote><sup>b</sup> Oak Ridge Natl Lab, Mat Sci & Technol Div, Oak Ridge, TN 37831 USA</footnote>

<footnote><sup>c</sup> Université Paris-Saclay, CEA, Service de la Corrosion et du Comportement des Matériaux dans leur Environnement, 91191, Gif-sur-Yvette, France</footnote>

<footnote><sup>d</sup> CIRIMAT, Université de Toulouse, CNRS, INPT, UPS, ENSIACET 4, allée Émile Monso, 31030 Toulouse, France</footnote>
working in transmission mode through the sample of 3 x 3 x 1 mm³ dimensions. During the experiments, the sample was moved along the z axis with a velocity V = 10 µm/s. A 2D Perkin Elmer detector (2048 x 2048 pixels, pixel size of 200 x 200 µm²) with high acquisition rate (10 Hz) was coupled with a continuous translational movement of the sample allowed to record 2D images of a diffracted volume of 300 x 3000 x 3 µm³ at every micrometer through all the traveled thickness. To center the specimen and align the beam with the top surface of the sample, a series of absorption and reflective scans were made using the direct beam and a photodiode positioned behind the sample. The 2D detector to sample distance and the detector tilts were calibrated using a cerium dioxide powder diffraction standard from NIST (CeO₂ – SRM 674b). The measured samples were all positioned at the same position (±50 µm) thanks to a camera. For phase quantification, the 2D diffraction images were integrated all around the rings using PyFAI software [9]. The 1D diffractograms (Intensity vs 2Θ) were corrected from instrumental aberration and analyzed via a full Rietveld refinement using MAUD software [10]. The Rietveld treatments allowed to determine the evolution of the mass fraction of the phases and their crystallographic parameters along the sample thickness. The carbon mass fraction trapped in carbides was calculated using Eq. 1 with f^X the mass fraction of phase X and w_C^X the carbon mass fraction in phase X.

$$w_C^{\text{Carbides}}(\text{wt.}\%) = f^{M_{23}C_6} \cdot w_C^{M_{23}C_6} + f^{M_7C_3} \cdot w_C^{M_7C_3} \qquad \text{Eq. 1}$$

<table><caption>Table 1: Composition of 316L studied alloy (in wt. %) analyzed by inductively coupled plasma-optical emission spectroscopy (ICP-OES). Carbon was quantified by IR - combustion analysis.</caption><thead><tr><th>Cr</th><th>Ni</th><th>Mo</th><th>C</th><th>Mn</th><th>Co</th><th>Si</th><th>Cu</th><th>Al</th><th>Fe</th></tr></thead><tbody><tr><td>16.55</td><td>10.52</td><td>2.05</td><td>0.03</td><td>1.55</td><td>0.12</td><td>0.18</td><td>0.24</td><td>0.02</td><td>Base</td></tr></tbody></table>

<figure><img src="image_3.png" /><figcaption>Fig. 1: a) Experimental set-up of the HEXRD, b) Schematic view of the sample cross section.</figcaption></figure>

For the RS profile determination, the 2D diffraction images were integrated along the Debye-Scherrer ring over 2° sectors using PyFAI software [9]. The RS evaluation in the austenite was carried out by probing the shift of the (220) austenite peak (γFe) along the Debye-Scherrer ring using an automatic Python procedure. The peaks line profiles were fitted using a Pearson VII function to determine the 2Θ positions, full width at half maximum (FWHM), intensity, shape factor and background parameters. The FWHM is sensitive to stress-strain accumulation (plasticity: dislocation density), microstrains (strain heterogeneities inside grains), and microstructural evolution (essentially chemical gradient in our case). The full RS tensor determination was carried out using the sin²Ψ method following the methodology developed in the work by Geandier et al. [11]. The strain in the measurement direction ε<sub>ΦΨ</sub>, with Φ and Ψ the angles defining the measurement direction in the sample reference system, is defined as following [12, 13]:

$$\varepsilon_{\Phi\psi} = \ln\left(\frac{\sin\theta_0^{hkl}}{\sin\theta^{hkl}}\right) = \frac{1}{2}S_2^{hkl}(\sigma_{\Phi} - \sigma_{33})\sin^2\psi + \frac{1}{2}S_2^{hkl}\tau_{\Phi}\sin(2\psi) + \frac{1}{2}S_2^{hkl}\sigma_{33} + S_1^{hkl}\mathrm{Tr}(\bar{\sigma}) \qquad \text{Eq. 2}$$

$$\sigma_{\Phi} = \sigma_{11}\cos^2\Phi + \sigma_{12}\sin2\Phi + \sigma_{22}\sin^2\Phi \qquad \text{Eq. 3}$$

$$\tau_{\Phi} = \sigma_{13}\cos\Phi + \sigma_{23}\sin\Phi \qquad \text{Eq. 4}$$

$$\mathrm{Tr}(\bar{\sigma}) = \sigma_{11} + \sigma_{22} + \sigma_{33} \qquad \text{Eq. 5}$$
with $S_1^{hkl}$ and $\frac{1}{2}S_2^{hkl}$, the X-ray elasticity constants (XEC) of plane (hkl), $\theta_0^{hkl}$, the strain-free diffraction angle of the diffracting plane, $\theta^{hkl}$, the diffraction angle of the stressed plane, $\bar{\sigma}$, the stress tensor and $\sigma_{ij}$, the stress tensor components. In the case of a material under triaxial stress state with an isotropic macroscopic elastic behavior, an ellipse is obtained. The slope of the ellipse axis presents the stress difference ($\sigma_\phi - \sigma_{33}$) in the plane of measurement. The opening of the ellipse allows to determine the shear stresses. In our study, two different directions were probed, $\Phi = 0^\circ$ and $\Phi = 90^\circ$ which allows to extract the stress differences ($\sigma_{11} - \sigma_{33}$) and ($\sigma_{22} - \sigma_{33}$). Moreover, no ellipse was observed meaning that no shear stresses were present in the samples. To determine the RS profile induced in austenite due to carbon enrichment and formation of new phases such as carbides, the strain-free lattice parameter of austenite along the sample thickness must be known first. The strain-free lattice parameter is dependent on the material microstructure, chemical gradient within the depth of the sample in our case. So, the chemical composition of austenite must be determined at every depth in accordance with carbon and phases evolution. The chemical composition of austenite at each depth was calculated by Thermo-Calc® using the carbon concentration measured by EPMA. Then, the austenite strain-free lattice parameter was determined from Eq. 6 using the composition (mainly in Cr, Mo and C) predicted from the previous simulations. The coefficients used in Eq. 6 were taken from the works of Kahn et al. [14] for carbon and Dyson and Holmes for the metallic elements [15].

$$ a_Y = a_0 + 0.041(\pm 0.011)\%C^\gamma + 0.00095(\pm 0.00015)\%Mn^\gamma - 0.0002(\pm 0.00004)\%Ni^\gamma + 0.0006(\pm 0.0003)\%Cr^\gamma + 0.0056(\pm 0.0007)\%Al^\gamma - 0.0004(\pm 0.0001)\%Co^\gamma + 0.0015(\pm 0.0005)\%Cu^\gamma + 0.0031(\pm 0.0004)\%Mo^\gamma \quad \text{Eq. 6} $$

with $a_0$ the lattice parameter at ambient temperature in the absence of carbon and alloying elements and $\%X^\gamma$ the wt. % of the element X in the austenite matrix as predicted by Thermo-Calc®. $a_0$ was obtained by subtracting the lattice expansion caused by dissolved carbon and alloying elements in the as-received steel from the initial lattice parameter ($a_Y^{initial}$). This last value was determined on a released stress sample: $a_Y^{initial} = 3.5955 \pm 0.0005$ Å.

### 3. Carburization kinetics modeling using DICTRA

The carburization profile in 316L austenitic stainless steel at 600 °C was predicted using a thermodynamic and kinetic modelling tool (DICTRA module) using TCFE9 and MOBFE2 databases as described in [8]. A one-dimension model based on the homogenization approach was used assuming that carbon diffusion occurs only in the austenitic matrix and that no diffusion occurred through the carbides. The labyrinth factor was kept constant for all the simulations and equal to $f^2$ as usually used in literature [16 - 20]. The labyrinth factor is implemented in DICTRA software and used to consider the influence of the presence of carbides on the carbon diffusion in the matrix. A planar one-dimensional unit cell with a length equal to the half sample thickness was considered since the diffusion of carbon occurs uniformly from both sides of the sample (symmetry assumption). In the present work, three phases, austenite matrix and the two $M_{23}C_6$ and $M_7C_3$ carbides were considered. A constant carbon activity was used as a boundary condition at the sodium-metal interface ($a_C = 1.4$). The assumption of symmetry at the center of the samples provided the second spatial boundary condition of zero flux for all elements.

### 4. Results

#### 4.1 316L carburized at 600 °C for 1000 h

##### 4.1.1. Phases mass fractions

Fig. 2 shows the evolution of phases mass fractions, measured by HEXRD, in 316L alloy after carburization at 600 °C for 1000 h. In agreement with results presented in [7, 8], $M_{23}C_6$ carbides were detected in the whole sample thickness and $M_7C_3$ carbides were only present within the first 50 µm below the sample surface. Small amounts of ferrite (< 3 %) were also detected between 100 µm and 500 µm.
<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /><figcaption>**Fig. 2:** Evolution of the phases mass fractions after carburization at 600 °C for 1000 h: a) austenite, b) ferrite, c) $M_{23}C_6$ and d) $M_7C_3$ (zoom on its existence domain).</figcaption></figure>

### 4.1.2. Carbon concentration profile

Fig. 3 shows a comparison between the carbon profile measured by EPMA and the carbon profile calculated from the data shown in Fig. 2 using Eq. 1. The carbon profile measured by EPMA represents the carbon concentration in all phases (matrix + carbides) and the carbon profile calculated using Eq. 1 corresponds to the carbon concentration trapped into carbides. The measured and calculated profiles were in excellent accord and in very good agreement with earlier observations from carburized 316L steel after 1000 h in sodium at 600 °C. This observations demonstrates that most of the carbon transferred into the sample was trapped into carbides as previously reported in [7, 8].

<figure><img src="image_6.png" /><figcaption>**Fig. 3:** Comparison between the carbon profiles determined by EPMA and from Rietveld analysis on HEXRD data with Eq. 1 after carburization at 600 °C for 1000 h.</figcaption></figure>

### 4.1.3. Kinetic modelling of carburization

Fig. 4-a and c show a comparison between the experimental carbide mass fractions and carbon profile and the ones predicted by DICTRA simulations. From Fig. 4-c, as reported in [7, 8], the experimental carbon profile was correctly predicted by the model for up to 120 µm in depth. However, in that region, the predicted carbides mass fractions were not in accordance with the HEXRD measurements (Fig. 4-a). The model overestimated and underestimated the $M_7C_3$ and $M_{23}C_6$ mass fractions respectively as discussed in [7,8]. The $M_7C_3/M_{23}C_6$ mass fraction ratio predicted by DICTRA was much higher than the one measured experimentally. Based on this
<figure><img src="image_7.png" /></figure>

statement and previous works [7, 8], new simulation was made by preventing the formation of M₇C₃ carbides and were then compared with experimental results. The comparison between the predicted and experimental results is again shown in Fig. 4-b and d. As in previous works [7, 8], excellent agreements were obtained between the predicted and measured carbon profile and predicted and experimental M₂₃C₆ mass fractions, except near the surface of the sample where M₇C₃ were formed in non-negligible amount (20 % in Fig. 4-a). As a consequence, for the RS measurements presented later, the chemical composition of the austenite cannot be obtained in the region where M₇C₃ carbides were formed. As reported in [7,8] it is noticeable in Fig. 4-d that carbon diffused much deeper in the 316L specimen than predicted by DICTRA. The carbon concentration decreased drastically from about 1.5 wt.% to the initial carbon concentration at around 120 µm of depth. The difference between the measured and predicted carbon diffusion profiles from 120 µm was due to the diffusion of carbon through substrate grain boundaries as evidenced by optical microscopy images of the carburization zone in [6 - 8] a phenomenon which was not considered in the predictions.

<figure><img src="image_8.png" /><figcaption>**Fig. 4:** Comparison between the experimental and simulated carbon profile and carbides mass fractions after carburization at 600 °C for 1000 h. a) and c) simulation with M₇C₃ formation, b) and d) simulation without M₇C₃ formation.</figcaption></figure>

### 4.1.4. Residual stress profile

Fig. 5a shows the evolution of the normal RS components ($\sigma_{11}=\sigma_{xx}$, $\sigma_{22}=\sigma_{yy}$, $\sigma_{33}=\sigma_{zz}$), the von Mises stress (calculated using the von Mises yield criterion, Fig. 5b), the phases mass fractions (Fig. 5c) and the FWHM of the austenite matrix (Fig. 5d) in 316L alloy carburized at 600 °C for 1000 h. No RS profile could be determined within the first 50 µm because the M₇C₃ peaks overlapped with the peaks of the austenite phase and the determination of the austenite strain-free lattice parameter profile in that zone was not possible. In fact, the chemical composition of the austenite matrix cannot be accurately predicted by Thermo-Calc® in that zone (§ 4.1.3). The RS state was close to an hydrostatic stress state (all $\sigma_{ii}$ are almost similar except in the depth range between 120 µm and 140 µm). The RS, increased from the heart of the sample ($\sigma_{(11,22,33)} \approx 500 \pm 200$ MPa) to 150 µm in depth where they reached maximum values, $\sigma_{(11, 22, 33)} \approx 1800 \pm 100$ MPa. This RS increase followed the increase of M₂₃C₆ mass fraction. Then, from 140 µm to 50 µm, a relaxation of the RS was observed. Between 500 and 150 µm, the von Mises stresses were quite constant (lower limit close to zero) and lower than the tensile yield stress, $\sigma_y$, of the as-received 316L ($\sigma_y = 330$ MPa). This last parameter was measured independently from tensile tests at room temperature [2]. From 140 to 120 µm in depth, the von Mises stresses increased ($\sigma_{von \ Mises} \approx 370 \pm 150$ MPa) and exceeded the tensile yield stress of the as-received steel meaning that the RS state induced a plastic deformation at this depth. Then, at lower depth the $\sigma_{von \ Mises}$ decreased below $\sigma_y$ of the as-received steel. The FWHM was almost constant between 500 µm and 200 µm. After that, it increased until 120 µm in depth then decreased between 120 and 50 µm to a value higher than the one measured in the core of the material.
<figure><img src="image_9.png" /><figcaption>**Fig. 5:** 316L alloy carburized at 600 °C for 1000 h: a) Normal residual stresses components in austenite, b) von Mises stress in austenite, c) Mass fractions of all the diffracted phases, d) FWHM of the austenite.</figcaption></figure>

## 4.2 316L carburized at 500 °C for 1000 h

### 4.2.1 Phases mass fractions

Fig. 6 shows the evolution of phases mass fractions, measured by HEXRD in 316L alloy after carburization at 500 °C for 1000 h. Two types of austenite with two different lattice parameters were detected near the sample surface: an austenite with a higher lattice parameter which was assumed to be due to carbon enrichment ($\gamma_{Fe}^{C}$, named expanded austenite, see Fig. 6a) and an austenite with a smaller lattice parameter ($\gamma_{Fe}$, see Fig. 6b). Only $M_{23}C_{6}$ carbides were formed in the alloy after 1000 h (see Fig. 6c) which were present in the alloy from the surface to 100 µm in depth, with non-negligible quantity, $\ge$ 2 wt. % until 50 µm. Small amount of ferrite was also detected (see Fig. 6d). From 50 µm going up to the surface of the sample, a respectively parallel decrease and increase of the mass fractions of expanded austenite $\gamma_{Fe}^{C}$ and $M_{23}C_{6}$ + austenite $\gamma_{Fe}$ occurred. This evolution suggests that a progressive transformation of expanded austenite into $M_{23}C_{6}$ carbide + austenite occurred in that region due to a high carbon concentration and/or a prolonged treatment time. This scenario is in good agreement with the fact that expanded austenite decomposes into carbides and austenite by heat treatment at temperature higher than 380 °C and that carbides formed when carbon concentration exceeds the limit of carbon solubility in expanded austenite [21].
<figure><img src="image_10.png" /><figcaption>**Fig. 6:** Evolution of the phases mass fractions in 316L alloy carburized at 500 °C for 1000 h: a) expanded austenite, b) austenite, c) the evolution of M<sub>23</sub>C<sub>6</sub> mass fraction is presented with a zoom on its existence domain and d) ferrite.</figcaption></figure>

### 4.2.2 Carbon concentration profile

Fig. 7 shows a comparison between the carbon profile determined by EPMA and the carbon profile from the data shown in Fig. 6 using Eq. 1. A carbon enrichment was observed in the sample up to 180 µm in depth. In the first 10 µm, both carbon profiles were in relatively good agreement meaning that carbon was mainly trapped into M<sub>23</sub>C<sub>6</sub> carbides. Between 10 and 50 µm from the surface, carbon in the sample was shared between M<sub>23</sub>C<sub>6</sub> and austenite (Fig. 6a-c). After 50 µm, very small amount of M<sub>23</sub>C<sub>6</sub> was detected (< 2 wt. %) and almost all carbon was dissolved in austenite. Since, the carbon concentration measured in the carbon enriched zone was much higher than the carbon solubility limit at 500 °C (0.006 wt. % calculated using Thermo-Calc at 500°C), carbon supersaturation of the austenite lattice occurred. At 500 °C, DICTRA failed as previously in [7] to predict the carburization profile of the sample, carbon concentration, carbide mass fractions formed at the sample surface and carbon supersaturation of the austenite lattice. Consequently, as presented later in the manuscript, the chemical composition of the austenite cannot be obtained at the region where carbide formed.

<figure><img src="image_11.png" /><figcaption>**Fig. 7:** Comparison between the carbon profiles determined by EPMA and from Rietveld analysis on HEXRD data with Eq. 1 after carburization at 500 °C for 1000 h.</figcaption></figure>

### 4.2.3 Residual stress profile
<figure><img src="image_12.png" /></figure>

Fig. 8a shows the evolution of the normal RS components, the von Mises stress (Fig. 8b), the carbon profile (Fig. 8c) and the FWHM (Fig. 8d) in the austenite matrix in 316L alloy carburized at 500 °C for 1000 h. The RS profile was not determined in the zone where two types of austenite and a non-negligible amount of $M_{23}C_6$ were formed (< 50 µm) because the determination of the strain-free lattice parameter profile in that zone was not possible. Indeed, the chemical composition of the matrix could not be accurately determined in that zone. Deeper than 50 µm, an equibiaxial stress state ($\sigma_{11} = \sigma_{22}$) was observed in the sample plane (x, y), see Fig. 5a. Near the sample surface (from 50 to 160 µm), compressive and tensile RS were respectively developed in the (x, y) plane and z direction. After 160 µm, a transition from compressive to tensile RS was observed in the (x, y) plane. From 120 µm to the surface of the sample, the von Mises stresses were higher than the tensile yield stress, $\sigma_y$, of the as-received 316L and very close to the tensile yield stress, 605 MPa, of the carburized 316L sample at 500 °C for 1000 h [2]. It was even higher in the depth range between 70 and 80 µm ($\sigma_{von Mises} \approx 650$ MPa). The FWHM was almost constant from 500 to 160 µm then slightly increased from 0.022° to 0.028° between 160 µm and 50 µm. A high carbon concentration gradient profile was observed in that last region (region I).

<figure><img src="image_13.png" /><figcaption>**Fig. 8:** 316L alloy carburized at 500 °C for 1000 h from 50 µm in depth to the half thickness: a) Normal residual stresses components in austenite, b) von Mises stresses in austenite, c) Carbon profile in the sample and d) FWHM of the austenite phase. The continuous lines indicate the zones depending on the formed phases. The dashed line shows the limit of regions I and II.</figcaption></figure>

# 5 Discussion

At 600 °C, the carbon absorbed by the sample was trapped into $M_{23}C_6$ and $M_7C_3$ carbides in good agreement with thermodynamic and kinetic predictions. Nevertheless, the experimentally observed carbides mass fractions were not in agreement with the predictions. The measured $M_{23}C_6/M_7C_3$ mass fraction ratio was higher. Similar observations were made from carburized 316L after 1000, 3000 and 5000 h in liquid sodium at 600°C [7, 8]. It was suggested that the local thermodynamic equilibrium was not completely reached in the carburization zone: the transformation of $M_{23}C_6$ into $M_7C_3$ was not complete [8]. This was also in agreement with other observations at similar temperatures [22 - 24]. A direct consequence of the precipitation of carbides in 316L sample was the formation of RS. From 500 µm to 150 µm in depth, the RS increased with the increase of the $M_{23}C_6$ mass fraction as a consequence of the increase in the elastic energy due to the elastic accommodation of the misfit between the matrix and the carbides. Interestingly, in the region from 150 to 50 µm, the $M_{23}C_6$ mass fraction increased as well but a relaxation of RS was observed. This RS relaxation could be explained by a loss of coherency between the austenite matrix and the $M_{23}C_6$ precipitates which results in a drop in the elastic energy and therefore a decrease of the RS. Wen et al. [25] and Ding et al. [26] have noted that with the coarsening of $M_{23}C_6$, incoherent interfaces between carbide and matrix were formed. As said previously (§ 2), the FWHM evolution is related to different phenomena. The increase observed between 200 and 120 µm was in agreement with the presence of: i) plastic strain ($\sigma_{von Mises} > \sigma_y$), ii) chemical gradient (depletion in Cr and Mo of the austenite matrix due to the carbide formation) and iii) strain heterogeneities inside grains (massive presence of carbides in grains and at grain
boundaries). The decrease of the FWHM observed between 120 and 50 µm can be due to the release of plasticity ($\sigma_{von Mises} < \sigma_y$). Thus, the residual FWHM in that region can be attributed only to the presence of chemical gradient and strain heterogeneities. It is very important to note that the RS determined in the austenite phase (Fig. 5a) were tensile throughout the whole sample thickness. That was the consequence of the depletion of the austenite matrix in carbide former metallic elements, mainly Cr and Mo. To determine the macroscopic RS developed within the sample, an evaluation of the RS in carbides must be done as well. Unfortunately, more data on carbides such as the evolution of its lattice parameter as a function of its composition are necessary and are not currently available. To satisfy the condition of stress equilibrium in the sample, compressive RS are expected to develop in the carbides.

At 500 °C, carbides were formed only near the sample surface (with a mass fraction ≥ 2 wt. % up to 50 µm) in agreement with [8]. The HEXRD results suggested that carbon supersaturation of austenite lattice occurred in the rest of the carburized zone. Furthermore, at this temperature, DICTRA failed to predict the carbon supersaturation of the austenite lattice as previously in [7]. These observations suggest that thermodynamic equilibrium was not reached in the 316L after 1000 h in carburizing sodium at 500 °C. The carbon supersaturation of austenite has been previously reported by different authors [5, 27 - 30]. At this temperature, the Cr atoms have a lower mobility than the C atoms. In consequence, the C atoms dissolve in the octahedral interstices of austenite and prevent the formation of chromium carbides. That phenomenon was described as 'colossal' carbon supersaturation in [31]. The massive carbon insertion in austenite lattice resulted in its expansion but because of displacement restriction by the non-carburized subsurface layer, formation of compressive RS mainly in the (x, y) plane occurred. Unlike in the case of carburization at 600 °C, an equibiaxial RS state is developed instead of a hydrostatic RS state. It could be explained by the coexistence of carbide precipitation at the sample surface (matrix depletion in metallic elements) and formation of expanded austenite but the exact reason is currently unknown. The formation of compressive RS at the sample surface owing to the growth of expanded austenite was extensively reported in literature [4, 5, 30]. The von Mises stress (about 650 MPa) between 70 and 80 µm was higher than the tensile yield stress of the carburized sample meaning that the RS state induced a plastic deformation at this depth. As mentioned above (see § 2), the FWHM evolution is proportional to different phenomena. In the region between 160 and 50 µm, plastic strains ($\sigma_{von Mises} > \text{tensile yield stress}$) and chemical gradient (C gradient profile) were present. This may explain the increase of the FWHM in that region. The macroscopic RS in the region between the sample surface and 50 µm (undetermined in this work, see Fig. 8a) were expected to be compressive to satisfy the equilibrium stress assumption in the whole sample.

## 6 Conclusions

* At 600 °C, the carbon transferred into 316L was mainly transformed into $M_{23}C_6$ and $M_7C_3$ carbides. The carbon profile was correctly predicted using DICTRA thermodynamic and kinetic modelling except in the deepest region where carbon diffusion into the substrate grain boundaries became the main diffusion process. The lower measured $M_7C_3/M_{23}C_6$ mass fraction ratio than the one predicted by thermodynamic equilibrium indicated that the transformation of $M_{23}C_6$ into $M_7C_3$ could not be considered as infinitely fast at this temperature.

* The normal residual stresses profile at 600 °C was strongly related to the $M_{23}C_6$ mass fraction and the coherence between austenite and $M_{23}C_6$ precipitates. High tensile residual stresses were generated in austenite throughout the whole sample thickness. That was the consequence of the depletion of the austenite matrix in metallic elements, mainly Cr and Mo. The determined von Mises stresses were higher than the tensile yield stress of the carburized sample in localized region.

* At 500 °C, experimental results and DICTRA modelling suggested the formation of carbides was rate limited and that the carbon transferred into 316L was dissolved in the octahedral interstices of austenite lattice.

* At 500 °C, the formation of expanded austenite resulted in the development of compressive residual stresses in the carburized austenite zone and tensile residual stresses in the sample subsurface. The evolution of the normal residual stress profile followed the evolution of the carbon concentration profile. Again, the determined von Mises stresses were higher than the tensile yield stress of the carburized sample in localized region.
**Funding:**

This work was supported by the French Alternative Energies and Atomic Energy Commission and the French State through the program "Investment in the future" operated by the National Research Agency (ANR) and referenced by ANR-11-LABX-0008-01 [32].

**Acknowledgements**

The authors gratefully acknowledge the Deutsches Elektronen-Synchrotron (DESY-Petra III, Hamburg, Germany) for provision of beamtime at the PETRA P07-EH2 beamline.

**References:**

[1] F. Rouillard, M. Romedenne, Etat de l'art sur l'interaction B4C – acier de gaine et loi de durée de vie des gaines des éléments absorbants, NT DPC/SCCME 17-789-A 2017 (2017).

[2] F. Rouillard, Influence de la carburation sur le comportement mécanique des aciers : Données d'entrée pour la loi de durée de vie des gaines des Eléments Absorbants et Réflexions, NT DPC/SCCME 18-872-A (2017).

[3] A. Thorley, C. Tyzack, British Nuclear Energy Society, Conference Proceedings (1971).

[4] Y. Peng, Z. Liu, Y. Jiang, B. Wang, J. Gong, M. A. J. Somers, Scripta Materialia, 157, 106 – 109 (2018).

[5] T. Christiansen, M. A. J. Somers, Metallurgical and Materials Transactions A, 40A, 1791 – 1798 (2009).

[6] M. Romedenne, F. Rouillard, B. Duprey, D. Hamon, M. Tabarant, D. Monceau, Oxidation of Metals. 87, 643–653 (2017).

[7] M. Romedenne, Thèse de doctorat de l'Université de Toulouse (2018).

[8] M. Romedenne, F. Rouillard, D. Hamon, B. Malard, D. Monceau, Corrosion. Science. 159, 108147 (2019).

[9] J. Kieffer, D. Karkoulis, Journal of Physics: Conference Series, 425, 202012 (2013).

[10] L. Lutterotti, S. Matthies, H. R. Wenk, Proceeding of the 12th International Conference on Textures of Materials (ICOTOM-12), vol. 1, 1999, p. 1599.

[11] G. Geandier, L. Vautrot, B. Denand, S. Denis, Materials, 11, 1415 (2018).

[12] V. Hauk, Elsevier, Amsterdam (1997).

[13] I. Noyan, J. Cohen, Springer Verlag, New York (1987).

[14] H. Kahn, G. M. Michal, F. Ernst, A. H. Heuer, Metallurgical and Materials Transactions A, 40A, 1799 – 1804 (2009).

[15] D. J. Dyson, B. Holmes, Journal of The Iron and Steel Institute, 208, 469 – 474 (1970).

[16] A. Engström, L. Höglund, J. Ågren, Metallurgical and Materials Transactions A, 25 (6), 1127–1134 (1994).

[17] T. Turpin, J. Dulcy, M. Gantois, Metallurgical and Materials Transactions A, 36 (10), 2751–2760 (2005).

[18] C. Sudha, N. S. Bharasi, R. Anand, H. Shaikh, R. Dayal, M. Vijayalakshmi, Journal of Nuclear Materials, 402 (2), 186–195 (2010).

[19] J. Garcia, O. Prat, Applied Surface Science, 257 (21), 8894 – 8900 (2011).

[20] D. Rong, J. Gong, Y. Jiang, Procedia Engineering, 130, 676 – 684 (2015).

[21] T. L. Christiansen, K. Stahl, B. K. Brink, M. A. J. Somers, Steel Research International, 87 (11), 1935 – 1405 (2016).

[22] H.J. Christ, Materials and Corrosion, 49 (4), 258–265 (1998).

[23] S.I. Ford, P.R. Munroe, D.J. Young, in: P.F. Tortorelli, I.G. Wright, P.Y. Hou (Eds.), John Stringer Symposium on High Temperature Corrosion, ASM International, 2003, pp. 77–85.

[24] Y. Gong, D. J. Young, C. Atkinson, T. Olszewski, W. J. Quadakkers, R. C. Reed, Corrosion Science, 173, 108699 (2020).

[25] H. Wen, B. Zhao, X. Dong, F. Sun, L. Zhang, Materials Letters, 261, 126984 (2020).

[26] Z. Ding, B. Liang, Z. Xu, L. Dong, American Chemical Society Applied Materials and Interfaces, 12, 19235 – 19242 (2020).

[27] Y. Sun, X. Li, T. Bell, Materials Science and Technology, 15, 1171 – 1178 (1999).

[28] Y. Cao, F. Ernst, G. M. Michal, Acta Materialia, 51, 4171 – 4181 (2003).

[29] F. Ernst, Y. Cao, G. M. Michal, Acta Materialia, 52, 1469 – 1477 (2004).

[30] T. L. Christiansen, T. S; Hummelshoj, M. A. J. Somers, Surface Engineering, 26, 242 – 247 (2010).

[31] G. M. Michal, F. Ernst, H. Kahn, Y. Cao, F. Oba, N. Agarwal, A. H. Heuer, Acta Materialia, 54, 1597 – 1606 (2006).

[32] LabEx DAMAS. Available online: labex-damas.univ-lorraine.fr.