Mekha Vimal

Michele Natile

Jean-François Lupi

Florent Guichard

Dominique Descamps

Marc Hanna
email: marc.hanna@institutoptique.fr

Patrick Georges

Nonlinear post-compression of a hybrid vortex mode in a gas-filled capillary

We demonstrate nonlinear temporal compression of a vortex beam by propagation in a gas-filled capillary. Starting from an ytterbium-based laser delivering 700 µJ 640 fs pulses at 100 kHz repetition rate, the vortex beam is generated using a spiral phase plate and coupled to a capillary where it excites a set of four modes that have an overlap integral of 97% with a Laguerre-Gauss LG 10 mode. Nonlinear propagation of this hybrid, orbital angular momentum (OAM)-carrying mode results in temporal compression down to 74 fs at the output. Beam and pulse characterizations are carried out to determine the spatial profile and temporal duration of the compressed pulses. This result in multimode nonlinear optics paves the way towards the generation of OAM-carrying few-cycle pulses, isolated attosecond XUV pulses, and tunable UV pulses through resonant dispersive wave emission.

INTRODUCTION

The existence of orbital angular momentum (OAM) of light and its transfer from light to matter was first demonstrated three decades ago [1]. An azimuthally varying, quantized phase, e iℓθ is connected to photonic energy flow, implying a quantized, optical OAM in units of ℓh. Such beams have a point of zero intensity in the transverse plane and are sometimes referred to as vortex beams. A natural basis for such beams are the Laguerre-Gaussian (LG) modes, in cylindrical coordinates

LG ℓp (r, θ, z) = LG 00 L |ℓ| p r 2 w 2 r w |ℓ| e -i(|ℓ|+2p)ζ-iℓθ , (1)
where LG 00 is the Gaussian beam, p ∈ N, ℓ ∈ Z, w(z) and ζ(z) are the Gaussian beam radius and Gouy phase, and L |ℓ| p are generalized Laguerre polynomials.

Femtosecond vortex beams have been gaining interest in the ultrafast community, for instance in atomic spectroscopy [2] and dichroism of chiral molecules [3,4] . To this end, the transfer of OAM from an infrared driver to the extreme ultraviolet through high harmonic generation was studied extensively [5,6]. Recent studies show the potential of OAM beams to study chiral and/or topological magnetic materials using helicoidal dichroism in the EUV and X-ray domains [7].

Although structured light beams carrying OAM have been produced using optical elements like spiral phase plates (SPP) [8] for decades, these methods cannot be applied with few-cycle pulses because of chromatic effects. Another possibility to produce few-cycle vortices is to perform nonlinear temporal compression on a previously converted vortex beam, the generation of which would even allow the generation of isolated attosecond vortices [9]. In free space post-compression setups, the use of vortex beams was studied theoretically [10,11] and demonstrated experimentally [12,13] both for multi-plate schemes and multipass cells, in the context of energy scaling. In setups using guiding structures, although vortex beams have been extensively studied in optical fibers [14], there has been no demonstration of post-compression in a gas-filled capillary with a vortex beam. However it has been achieved with radially polarized beams [15]. Several recent studies also point towards exploiting the nonlinear propagation of multimode [16] or higher-order mode [17] fields in such guiding structures, and the field of multimode nonlinear photonics is expanding rapidly [18].

In this Letter, we show that it is possible to perform nonlinear temporal compression on a set of capillary modes synthesizing an LG 10 free space mode. We first describe the hybrid OAM capillary modes to which an LG 10 mode optimally couples before presenting the experimental results of spectral broadening with a guided vortex beam. This work paves the way to the generation of few-cycle OAM-carrying pulses, isolated attosecond vortices, and tunable deep and vacuum ultraviolet (V/DUV) OAM pulses through the resonant dispersive wave generation mechanism [19]. Using an optimized design based on a large core capillary, it could also be a technique for energy scaling of capillary-based compression setups.

The modes of a hollow-core capillary can be categorized into three families, namely the TE 0m , TM 0m and EH nm modes [20]. Similar to the case of optical fibers [14], it is possible to synthesize a field that is very close to an LG 10 beam with linear polarization and a helical phase front by a linear combination of the TE 01 , 21 . The two modes in (e) and (f) when added with a phase shift of π/2 yield the linearly polarized hybrid OAM mode (g).

TM 01 , and EH 21 modes, as illustrated in Fig. 1.

OAM = [TE 01 + EH 21 ] + i TM 01 + EH π/4 21 OAM x = 0 OAM y = 2J 1 u 11 r a e iθ , (2)
where J 1 is the first-order Bessel function of the first kind, u 11 ≊ 3.83 is its first zero, and a is the capillary radius. These capillary modes share the same propagation constant

β = 2π λ 1 - 1 2 u 11 λ 2πa 2 , (3)
while their attenuation coefficient (α) depends on the nature of the mode.

α = u 11 2π 2 λ 2 a 3          1 √ ν 2 -1
, for TE 01

ν 2 √ ν 2 -1 , for TM 01 1 2 (ν 2 +1) √ ν 2 -1 , for EH 21 ( 4
)
where ν is the refractive index of the capillary glass, and λ is the wavelength. Note that it is also possible to synthesize left-or right-circularly polarized LG 10 -like modes using combinations of only two modes EH 21 + iEH π/4 21 or TE 01 + iTM 01 . We now examine the coupling and propagation losses properties of the hybrid OAM mode. Fig. 2.(a) shows the overlap integral between this mode and a LG 10 beam at its waist as a function of the waist size. For reference, the coupling of a Gaussian beam to the EH 11 mode is also shown. The calculated optimal coupling efficiency is 97.2% for a waist radius equal to 0.56a, to be compared to 98.1% for a waist radius of 0.64a in the Gaussian case. This shows that coupling to this hybrid mode can be almost as efficient as in the fundamental mode with a Gaussian beam. However, the coupling tolerance for variations in spot size is lesser for the LG 10 case, making it experimentally more difficult to achieve. From Eq. (3) and Eq. ( 4), we can see that the modes that contribute to the hybrid OAM mode are degenerate in propagation constant, but not in attenuation coefficients. This difference in propagation losses modifies the intensity, phase, and polarization state pattern of the hybrid OAM mode at the output. For example, in Fig. 2.(b), we plot the transmission of the three modes for capillary core diameters of 400 µm and 200 µm, at a wavelength of 1.03 µm. Since the optical losses scale as 1/a 3 , they are both higher and spread over a larger relative interval for the smaller capillary. As a result, their effect on the beam output intensity profile is more visible when the core radius of the capillary is small, as also displayed in Fig. 2

.(b).

The OAM-carrying output field can be derived analytically by taking into account the different propagation losses. Simple algebra yields

|OAM y (z)| = J 1 u 11 r a e -X 2 z + e -X 2 ν 2 z
× cos 2 θ e -Xz + sin 2 θ e -Xν 2 z (5)

arg OAM y (z) = arctan tan θ e -X 2 (ν 2 -1)z (6) |OAM x (z)| = J 1 u 11 r a e -X 2 ν 2 z -e -X 2 z × cos 2 θ e -Xν 2 z + sin 2 θ e -Xz (7
)

arg [OAM x (z)] = arctan 1 tan θ e X 2 (ν 2 +1)z (8)
with X = u 11 2π 2 λ 2 a 3 1 √ ν 2 -1 .
The deviation from the input in terms of intensity, phase, and polarization profiles increases as the core diameter decreases. But the fidelity of the output field is high in the case of high transmission, corresponding to most practical situations. This linear analysis of the modal properties leads us to use a capillary with a core diameter of 400 µm in the experiments.

The schematic of the experimental set up is show in Fig. 3. The experiment is carried out using a high-energy Ytterbiumbased femtosecond laser source (Tangor, Amplitude) delivering 700 µJ, 640 fs pulses at a central wavelength of 1030 nm and a repetition rate of 100 kHz, corresponding to an average power of 70 W. The Gaussian spatial mode at the output of the laser is The LG 10 -like mode thus generated is coupled into a hollow glass 1 m-long 400 µm-diameter capillary filled with Xenon gas at 2.4 bar. Beam coupling is achieved using the following procedure. Starting with a Gaussian beam (removing the SPP) we optimize coupling to the capillary using a waist radius around 0.64a. We then insert the SPP, and find that this ensures reliable excitation of the hybrid OAM-carrying mode. This procedure results in a theoretical transmission of 51%, taking into account the imperfect coupling due to the non-optimal beam size, the fact that using a SPP with a Gaussian beam does not generate a perfect LG 10 beam, and a theoretical transmission of 86% for the OAM-carrying hybrid mode. The measured overall (coupling and propagation) transmission in 2.4 bar Xe at maximum input power is 80% and 38% for the Gaussian and LG 10 modes respectively with output powers of 56 W and 27 W.

The output beam is collimated with a spherical mirror and reflected from a glass wedge to the imaging camera (see line 1 in Fig. 3). On the same line is mounted a cylindrical lens on a flip mount for spatial characterization of the output beam. Astigmatic transformation pattern at the focus of the cylindrical lens for the input vortex beam (e) and the capillary output at 1 W input power (f) and 70 W input power (g). The single dark fringe and the similar tilt orientation in (e), (f) and (g) indicate that the OAM value ℓ = 1 is conserved upon linear and nonlinear propagation. Beam profile at the capillary output for a Gaussian mode (h).

In order to remove the positive group delay dispersion (GDD) induced by the self phase modulation accumulated during the nonlinear propagation in Xe filled capillary, a set of chirped mirrors is used with a total GDD of -8550 fs 2 and > 95% reflectivity. The spectral and temporal characterization of the compressed pulses is then performed at the output of chirped mirrors. The temporal profile of the pulse is measured with a Second Harmonic Generation Frequency Resolved Optical Gating apparatus (SHG-FROG).

Fig. 5 shows a comparison between the spectral broadening of Gaussian and LG 10 spatial modes in the capillary at the maximum average power of 70 W and with 2.4 bar of Xe gas. The measured spectra correspond to Fourier-Transform Limited (FTL) pulses of 40 fs FWHM and 70 fs FWHM for the Gaussian and the LG 10 mode respectively, with a compression ratio that is 1.6 times higher for the Gaussian mode. This is in line Fig. 5. Measured spectra at the output of the capillary when coupling Gaussian (red) and OAM-carrying beams (blue). The shaded region corresponds to the input spectrum. with the fact that the effective area for the Gaussian and OAMcarrying beams can be approximated by A G eff = π × (0.64a) 2 and A LG eff = 2π × (0.56a) 2 , yielding a ratio of approximately 1.5. The coupling efficiency and propagation losses also affect the compression ratio. Estimated values of B-integral are 10 rad for the vortex beam and 17 rad for the Gaussian beam.

The FROG temporal characterization results of the compressed OAM-carrying pulse are shown in Fig. 6. The retrieved spectrum matches well the independently measured spectrum. The FTL of the retrieved spectrum is 74 fs, very close to the pulse duration of 70 fs obtained from simulations. These numerical simulations solve the vectorial multimode nonlinear Schrödinger equation including losses, dispersion, self-phase modulation, self-steepening, and ionization-related effects. The FROG-measured input laser intensity and phase profiles are set as the input conditions. The simulated spectrum and compressed pulse intensity profile are plotted in Fig. 6, with features in good agreement with the experimental FROG result. The energy contained in the main temporal lobe is estimated to be 200 µJ, 72% of the total available energy at the output.

In summary, we have demonstrated nonlinear postcompression of a hybrid vortex beam in a gas-filled capillary at a pulse energy level of 700 µJ. The spatial intensity and phase profile of the vortex beam are conserved upon nonlinear propagation, while ensuring a compression ratio of 9 for the chosen experimental parameters. This confirms the robust nature of the hybrid capillary mode for carrying a topological charge. In the near future, such a setup combined with a first nonlinear pulse compression stage based on multipass cell could become a robust source for applications requiring few-cycle pulses carrying an OAM.

Fig. 1 .

 1 Fig. 1. Vector representation of the capillary modes that synthesize an OAM-carrying beam. Each vector represents qualitatively the intensity and direction of the electric field. (a)TE 01 (b)EH 21 (c)TM 01 (d) EH π/4 21 (e) mode mixing of TE 01 + EH 21 and (f) TM 01 + EH π/421 . The two modes in (e) and (f) when added with a phase shift of π/2 yield the linearly polarized hybrid OAM mode (g).

Fig. 2 .

 2 Fig. 2. (a) Coupling efficiency as a function of the ratio of beam waist w 0 to capillary core radius a between a perfect Gaussian beam and EH 11 capillary mode (red) and between a LG 10 beam and the hybrid OAM mode (blue). (b) Optical power transmission as a function of propagation distance for the four modes in a capillary of core diameter 400 µm (solid curves) and 200 µm (dashed curves), with the corresponding simulated intensity profiles at the output.

Fig. 3 .

 3 Fig. 3. Schematic of the experimental set-up. SPP: Spiral Phase Plate, M1: Mirror 1 on flip mount, M2: Mirror 2, W: glass wedge on flip mount, CL: Cylindrical Lens on flip mount, CM : Chirped Mirror.

Fig. 4 .

 4 (b),(c) show the output beam profile and Fig.4.(f),(g) the corresponding focal pattern of the cylindrical lens. The results show that the output beam has a LG 10 -like intensity profile with topological charge ℓ = 1, similar to the input both at low and high levels of nonlinearity. As an additional check, we reconvert the output vortex beam with an SPP with ℓ = 1 at full power, before chirped mirrors. The obtained beam is shown after focusing by a f=150 mm lens in Fig.4.(d), and features a Gaussian profile, confirming that the vortex mode is preserved upon nonlinear propagation in the capillary at full energy. For comparison, the output of the capillary when coupling a Gaussian mode is shown in Fig 4.(h).

Fig. 4 .

 4 Fig. 4. Beam profile before (a) and after the capillary at 1 W input power (b) and 70 W input power (c). Near field beam profile (d) of the reconverted Gaussian beam at the output.Astigmatic transformation pattern at the focus of the cylindrical lens for the input vortex beam (e) and the capillary output at 1 W input power (f) and 70 W input power (g). The single dark fringe and the similar tilt orientation in (e), (f) and (g) indicate that the OAM value ℓ = 1 is conserved upon linear and nonlinear propagation. Beam profile at the capillary output for a Gaussian mode (h).

Fig. 6 .

 6 Fig. 6. Temporal characterization of the output compressed OAM-carrying pulses obtained with a capillary filled with 2.4 bar of Xe. (a) Spectral information. Shaded area: independently measured spectrum, blue curve: retrieved from FROG measurement, dashed red curve: simulation, green curve: spectral phase retrieved from FROG. (b) Temporal information. Blue curve: temporal intensity profile retrieved from FROG, dashed red curve: simulation. The legend shows the corresponding FWHM values. (c) Measured and (d) retrieved FROG traces.

Funding. Agence Nationale de la Recherche (ANR-21-CE30-0038-SHOTIME).

Data Availability Statement. Data underlying the results presented in this article are not publicly available at this time but may be obtained from the authors upon reasonable request.

Disclosures.

The authors declare no conflicts of interest.


Florian Le Bourdais
email: florian.lebourdais@cea.fr

Olivier Mesnil

Oscar D'almeida

Machine-learning based temperature compensation for Guided Wave Imaging in Structural Health Monitoring

. More specifically, the inspection of the unknown state must be conducted under the same EOCs as the ones of the pristine state. Besides measuring baselines on the structure under all the EOCs of interest, which is prohibitively expensive, a potential solution is to compensate the EOC effects on the measured signals.

Several solutions in the literature, such as Baseline Signal Stretch [2] and Dynamic Time Warping [3], have been proposed to solve this problem, but are somewhat limited in terms of amplitude of compensation or range of application. In this paper, a model-based machine learning procedure to compensate the measured signals in an unknown state and known EOCs is presented. The compensation model is trained on experimental data acquired at various temperatures on a structure representative of the one of interest. In other words, an experiment under various EOCs must be conducted on a simplified version of the structure with at least two transducers. Material and transducers of the simplified experiment must be identical to the ones of the real structure, but the actual geometry might differ. The compensated signals are then used to conduct guided wave imaging, allowing immediate defect detection and localization. Results are shown for both aluminum and composite panels.

Introduction

In current applications, Guided Wave (GW) based Structural Health Monitoring (SHM) measurements are usually obtained from a set of piezoelectric transducers mounted on the structure to monitor [START_REF] Giurgiutiu | Structural health monitoring: with piezoelectric wafer active sensors[END_REF]. Since the structure under scrutiny is exposed to an uncontrolled environment, the Environmental and Operational Conditions (OECs) can significantly affect the measurement. In the particular case of GWs, modifications of the EOCs usually influence both the amplitude and the times of the wave packets travelling through the structure. This makes it difficult to separate wave packets linked to structural defects from others. For this reason, a baseline, i.e. a set of measurements on the inspected structure in a pristine state, is usually recorded. This later serves to quantify the deviation from the undamaged state by highlighting the response of the flaw. More specifically, the inspection of the unknown state must be compared to the baseline obtained under the same EOCs. Besides measuring baselines on the structure under all the EOCs of interest, which is prohibitively expensive, a potential solution is to compensate the EOC effects on the measured signals. Several solutions in the literature, such as Baseline Signal Stretch (BSS) [START_REF] Croxford | Efficient temperature compensation strategies for guided wave structural health monitoring[END_REF] and Dynamic Time Warping [START_REF] Douglass | Dynamic Time Warping Temperature Compensation for Guided Wave Structural Health Monitoring[END_REF], have been proposed to solve this problem, but are somewhat limited in terms of amplitude of compensation or range of application. Baseline signal stretch can suffer from a frequency distortion due to resampling [START_REF] Douglass | Dynamic Time Warping Temperature Compensation for Guided Wave Structural Health Monitoring[END_REF]. Dynamic Time Warping on the other hand, while better suited for large amplitude temperature variation, may remove the signature of the defect while compensating for the temperature [START_REF] Kulakovskyi | Development of a SHM system by elastic guided waves applied to aeronautic structures[END_REF]. In this paper, we present a temperature compensation model relying on a training phase based on experimental measurements at various temperatures and present results obtained on aluminum and composite plates in a varying temperature environment.

Temperature compensation model

This section introduces the temperature dependency model of the guided wave signals.

Model

The model used to compensate guided wave signals for the temperature variation is described by the following equation:

𝑠(𝑡, 𝑇) = 𝛼 𝑠(𝛽𝑡, 𝑇 𝑟𝑒𝑓 ) (1)
In the above equation, 𝑠 is the guided wave signal, 𝑇 is the temperature, 𝑇 𝑟𝑒𝑓 is the reference temperature, 𝛼 the coefficient describing guided wave amplitude change as a function of temperature, 𝛽 the coefficient describing phase velocity change as a function of temperature. 𝑠̂ is the signal measured at temperature 𝑇 but transformed back as if it had been obtained at temperature 𝑇 𝑟𝑒𝑓 . In the present work, it is assumed that the parameters alpha and beta are dependent on the inspected structure and the corresponding SHM system. In other word, these parameters must be uniquely defined for each use-case of interest. The sources of variations of alpha and beta include the following: change of elastic properties of the propagation media and the coupling medium (epoxy), change of piezoelectric properties. Due to the potentially large number of influencing variables on these parameters, the choice is made to measure them experimentally. This leads to the following cost between a signal measured at a temperature 𝑇 and a signal measured at temperature 𝑇 𝑟𝑒𝑓 , with a number of samples defined by 𝑁:

𝑐𝑜𝑠𝑡 (𝑠 𝑇 ̂, 𝑠 𝑇 𝑟𝑒𝑓 ) = ∑ (𝑠 𝑇 ̂[𝑡 𝑖 ] -𝑠 𝑇 𝑟𝑒𝑓 [𝑡 𝑖 ]) 𝑁-1 𝑖=0 2 𝑐𝑜𝑠𝑡 (𝑠 𝑇 ̂, 𝑠 𝑇 𝑟𝑒𝑓 ) = ∑ (𝛼𝑠 𝑇 [𝑡 𝑖′ = 𝛽𝑡 𝑖 ] -𝑠 𝑇 𝑟𝑒𝑓 [𝑡 𝑖 ]) 𝑁-1 𝑖=0 2 (2)
This naturally leads to a cost minimization procedure that allows, for each temperature, to search the 𝛼 and 𝛽 coefficients that minimize the cost with respect to a reference temperature.

An interpolation scheme is used to compute the transformed signals. It uses time domain linear interpolation to project from one set of time coordinates to another.

Training phase

To train the model, experimentally measured guided wave signals are pre-processed before the minimization algorithm is applied. First a narrow bandpass filter is applied to limit the training to a limited frequency band and the reflected wave packets are removed. The unconstrained BFGS (Broyden-Fletcher-Goldfarb-Shanno) minimization algorithm is applied for each emit-receive pair to estimate the optimum 𝛼 and 𝛽 that best describe the data transformation from the current temperature to the reference temperature.

Model learning on experimental data

Dataset

An experimental dataset consisting of guided wave measurements was obtained for an aluminum plate and a composite plate for various temperatures using a climatic chamber. With the objective to conduct guided wave imaging, the plates are instrumented with a sparse array of piezoelectric transducers, each acting sequentially as emitter and receiver of the guided waves. The data acquisition process is conducted at temperatures between -40 and +87 °C. The center frequency of the emitted signal is 40 kHz. Signal were acquired both in a pristine state and in a damaged state. The damage created in both cases is a 10 mm diameter circular through hole. The plate configurations are shown below:

Model training

The measurements acquired in the pristine configuration at various temperatures are used to train the temperature transformation previously described. The model coefficients obtained for the aluminum plate as a function of temperature are as follows: A linear evolution of β is observed while the linear domain of α is limited to temperatures below 60°C. Overall, for this panel, the spread of the data is limited and the two parameters are relatively constant for all the emittedreceiver paths. It should be noted that the slope of the above stretch parameter is consistent with known experimental data. Indeed, a negative slope of the stretch coefficient implies that waves travel more slowly at higher temperatures, which is intuitively consistent with the aluminum getting less stiff. The results for the composite plate are shown in Figure 4. As can be seen, the evolution of the β factor is linear and negative, on average, which is coherent with the decrease of group velocity with the temperature. However, the α behavior is not nearly as clear. Moreover in both cases, large data variability is shown for each emitter-receiver paths, which means that the guided wave paths are not influenced in the same way by the temperature. This may be explained by several factors. First, the temperature may not be as homogeneous in the composite panel than in the aluminum panel, due to lower thermal conductivity. Second, the composite panel is anisotropic. Hence, a directivity dependency of the guided wave path is expected. Third, the bounding operation of the sensors on the structure is not as repeatable on the composite due to the rougher surface, inducing additional variability. Finally, the measured signals are of lower signal to noise ratio in the composite panel due to attenuation, leading to potentially poor fits.

Model exploitation for temperature compensation

After learning the right temperature transformation on the data, it is used to compensate the guided wave signals in the damaged configuration. An example of this is shown in the next figure. Judging from several time traces like the one above, the output of the model is able to compensate over a wide range of temperatures. To evaluate whether this leads to better performance on SHM applications, the obtained signals were used to perform Delay And Sum (DAS) imaging on the two plates, using the obtained image as the evaluation metric.

Application to guided wave imaging 4.1. DAS imaging

In order to highlight the response of the defect, the difference between the pristine and damaged signals is computed. This residual signal is then used to image the whole plate domain, using the Delay And Sum (DAS) algorithm with constant group velocity [START_REF] Michaels | Detection, localization and characterization of damage in plates with an in-situ array of spatially distributed ultrasonic sensors[END_REF]. The DAS algorithm relies on a backprojection method that sums the signals originating from a certain plate location, given a path to compute the time of flight needed to reach that location.

Results

The obtained images were compared with and without temperature compensation. An example is shown below for the aluminum plate: As can be seen, the image obtained without temperature compensation is highly affected by the temperature variation and fails to detect the defect for a temperature difference of 15°C, while the compensated image is very close to the reference image with no temperature difference. In this case, the improvement in detection capability is therefore important. In the composite case, the results are less favorable than in the aluminum case, since the signal to noise ratio is lower and the reconstruction model does not take into account the variability observed in Figure 4. Still, the image obtained with a 15 degree Celsius difference is sufficient to detect and locate the defect. the output of the images obtained at various temperatures shows that the transformation learned from the data is robust in an interval of environmental temperature changing between -20 and + 30 °C (relative to the reference temperature) for the aluminum plate and -15 + 25°C (relative to the reference temperature) for the composite plate.

Conclusions

This paper outlines a machine-learning based temperature compensation scheme that can be used for guided-wave based SHM. The model was trained using both an aluminum and a composite sample. The compensation model operates on the signals received by piezoelectric transducers positioned on the plates. Based on reference temperature signals, the model was trained to transform signals acquired at any temperature back to the time-scale of the reference temperature. The compensated signals were validated by producing DAS images that showed enhanced coherence of the DAS images over a large temperature range. Another aspect of this work was to investigate whether an average transformation instead of each emit-receive pair would give similar results. Results showed that the average model coefficients, not accounting for probe locations, was also useful for reconstructing images. This opens up interesting possibilities for laboratory or small-scale temperature calibration of SHM systems before transferring them to production. Perspectives of this work include statistical assessment of the temperature transform (for example using ROC curves) as well as further work on automatic detection and image quality criteria.

Figure 1 .

 1 Figure 1. Aluminum plate (left) and composite plate (right) with dimensions indicated in millimeters. The aluminum plate is instrumented with 6 piezoelectric transducers, while the composite plate is instrumented with 10 piezoelectric transducers. Dimensions are expressed in mm.

Figure 2

 2 Figure 2 shows an excerpt of the guided wave signals from the dataset for both plate types.

Figure 2 :

 2 Figure 2: guided wave signals for aluminum plate (top) a composite plate (bottom). Each plot shows two panels: an individual A-scan signal (overlaid with reference temperature signal), and a time-frequency representation of that A-scan.

Figure 3 .

 3 Figure 3. Model coefficients for the aluminum plate, as a function of temperature. Each dot represents an emitter/receiver pair. The red line represents the average coefficient as a function of the temperature.

Figure 4 .

 4 Figure 4. Model coefficients for the composite plate, as a function of temperature. Each dot represents an emitter/receiver pair. The red line represents the average coefficient as a function of the temperature.

Figure 5 .

 5 Figure 5. Comparison of signals before (left column) and after applying learned temperature correction (right column). The top row shows signals from the aluminum plate while the bottom row shows signals from the composite plate.

Figure 6 .

 6 Figure 6. Delay and sum images obtained on the aluminum plate. The reference images are in the top row while the images obtained with a temperature difference of 15 degrees are shown in the bottom row. The pink circles represents the transducers while the red circle represents the defect.

Figure 7 .

 7 Figure 7. Delay and sum images obtained on the composite plate. The reference images are in the top row while the images obtained with a temperature difference of 15 degrees are shown in the bottom row.

Acknowledgements

The authors acknowledge the help of Christian Bolzmacher and Andrii Kulakovskyii for the data acquisition.


Oriane Senninger
email: oriane.senninger@northwestern.edu

Frédéric Soisson

Enrique Martínez

Maylise Nastar

Chu-Chun Fu

Yves Bréchet

Modeling radiation induced segregation in Iron-Chromium alloys

Keywords: Fe-Cr alloys, Monte Carlo simulation, Diffusion, Irradiation effect, Segregation

Radiation induced segregation in ferritic Fe-Cr alloys is studied by Atomistic Kinetic Monte Carlo simulations that include diffusion of chemical species by vacancy and interstitial migration, recombination, and elimination at sinks. The parameters of the diffusion model are fitted to DFT calculations. Transport coefficients that control the coupling between diffusion of defects and chemical species are measured in dilute and concentrated alloys. Radiation induced segregation near grain boundaries is directly simulated with this model. We find that the diffusion of vacancies toward sinks leads to a Cr depletion. Meanwhile, the diffusion of self-interstitials causes an enrichment of Cr in the vicinity of sinks. For concentrations lower than 15%Cr, we predict that sinks will be enriched with Cr for temperatures lower than a threshold. When the temperature is above this threshold value, the sinks will be depleted in Cr. These results are compared to previous experimental studies and models. Cases of radiation induced precipitation and radiation accelerated precipitation are considered.

Introduction

Ferritic Fe-Cr steels are good candidates as structural materials for the next generation of nuclear power plants (generation IV and fusion reactors) [START_REF] Yvon | Structural materials challenges for advanced reactor systems[END_REF][START_REF] Zinkle | Structural materials for fission and fusion energy[END_REF].

The addition of Cr prevents corrosion and the Fe-Cr ferritic form is weakly sensitive to the swelling phenomenon. However, the irradiation flux induces an increase of point defect concentrations -vacancies and self-interstitial atoms (I) -that migrate toward sinks where they are eliminated. These defect fluxes may induce a variation in the alloy composition in the vicinity of sinks called Radiation Induced Segregation (RIS) [START_REF] Nastar | Radiation-induced segregation[END_REF]. If this causes a Cr depletion, the alloy can become sensitive to corrosion. On the other hand, if this leads to a Cr enrichment, the local concentration can reach the solubility limit, causing Radiation Induced Precipitation (RIP), which could lead to embrittlement. RIS has been extensively studied in austenitic steels where Cr depletion and Ni enrichment are frequently observed at sinks. These tendencies are relatively well-understood, even if the interstitial contribution is still under discussion [START_REF] Nastar | Radiation-induced segregation[END_REF]. In ferritic steels, the situation is far from being clear: approximately 15 experiments have been reviewed by Lu et al. [START_REF] Lu | Irradiation-induced grain boundary chromium microchemistry in high alloy ferritic steels[END_REF] in very different materials and conditions, showing both depletions and enrichments in the vicinity of sinks. However, it is difficult to draw clear conclusions on how irradiation conditions and materials properties control the tendency towards enrichment or depletion. A systematic study, focusing on industrial steels with Cr content between 8 and 12% has been performed by Was et al. [START_REF] Was | Assessment of radiation-induced segregation mechanisms in austenitic and ferritic-martensitic alloys[END_REF]. They observed that for given irradiation conditions, sinks tend to be enriched in Cr at low temperature and to be depleted in Cr at high temperature. Moreover, the Cr enrichment tends to decrease with the alloy Cr content. These tendencies are reproduced with the inverse Kirkendall model of Wharry et al. [START_REF] Wharry | Application of the inverse kirkendall model of radiation-induced segregation to ferritic-martensitic alloys[END_REF] and are attributed to a positive coupling of Cr with self-interstitials (dominant at low temperatures) and a negative coupling with vacancies (dominant at high temperatures). Other factors, such as grain boundary misorientation [START_REF] Hu | Effect of grain boundary orientation on radiation-induced segregation in a fe-15.2 at.% cr alloy[END_REF], interactions with C or other impurities [START_REF] Hu | Effect of grain boundary orientation on radiation-induced segregation in a fe-15.2 at.% cr alloy[END_REF][START_REF] Marquis | A systematic approach for the study of radiation-induced segregation/depletion at grain boundaries in steels[END_REF] may influence the segregation of Cr at sinks.

In the present study, we limit ourselves to the Fe-Cr binary system in order to understand basic mechanisms controlling RIS at the atomic scale. Thermodynamics of Irreversible Processes (TIP) shows that a complete description of RIS requires a full determination of the Onsager coefficients L ij , or alternatively, of partial diffusion coefficients d ij [START_REF] Nastar | Radiation-induced segregation[END_REF]. Such a consistent description remains to be established for concentrated Fe-Cr alloys. Existing models are based on Density Functional Theory (DFT) calculations and multi-frequency models that are only valid in the dilute limit [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF][START_REF] Messina | Exact ab initio transport coefficients in bcc Fe-x (x = Cr, Cu, Mn, Ni, P, Si) dilute alloys[END_REF], or have been extended to concentrated alloys using additional approximations [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF][START_REF] Was | Assessment of radiation-induced segregation mechanisms in austenitic and ferritic-martensitic alloys[END_REF][START_REF] Wharry | Application of the inverse kirkendall model of radiation-induced segregation to ferritic-martensitic alloys[END_REF]. Others use Molecular Dynamics (MD) simulations that rely on empirical potentials and only determine the self-interstitial contribution [START_REF] Terentyev | On the migration and trapping of single self-interstitial atoms in dilute and concentrated fecr alloys: Atomistic study and comparison with resistivity recovery experiments[END_REF][START_REF] Pechenkin | On the radiationinduced segregation: Contribution of interstitial mechanism in fe-cr alloys[END_REF].

We present here a multiscale approach of RIS, starting from DFT calculations of the point defect migration energies. A diffusion model describing the variation of point defect properties with local composition is developed. Atomistic Kinetic Monte Carlo (AKMC) simulations are performed in order to (i) determine the L ij coefficients and predict the RIS tendencies; and (ii) model the evolution of point defect and Cr concentration profiles under irradiation in the vicinity of a grain boundary (GB). AKMC simulations include the effect of non-homogeneous concentration fields and account for the stochastic nature of point defect migration. It also enables the simulation of RIP, when the RIS leads to a strong enrichment of Cr near GBs and therefore to nucleation of α' precipitates.

In section II we present a reminder of RIS principles in the framework of TIP. In section III, the diffusion model and the AKMC method are introduced.

The fitting of parameters to DFT calculations is detailed and relevant point defects properties are discussed. In section IV, the point defect tracer diffusion coefficients and Onsager coefficients are analyzed. The effect of composition and temperature on these Onsager coefficients is studied. Simulations of RIS and RIP near grain boundaries are also shown. We discuss these results by comparing them to previous models and experiments. We conclude by commenting on the limitations of our model, possible improvements and perspectives.

Thermodynamics of Irreversible Processes and Radiation Induced

Segregation

RIS is due to fluxes of vacancies (J V ) and self-interstitials (J I ) towards point defect sinks that, in an A-B alloy, induce fluxes of chemical species (J A and J B ). TIP describes the fluxes J as linear combinations of thermodynamic driving forces, i.e. chemical potential gradients [START_REF] Nastar | Radiation-induced segregation[END_REF]:

J α = -Σ β L αβ X β (1)
with X β = ∇µ β /(k B T ). L αβ are the Onsager coefficients, µ β the chemical potentials, k B the Boltzmann constant and T the temperature.

In steady state, under the conditions where there is no precipitation and L αβ coefficients are independent of the local composition, the gradient of B and V near a sink are related by:

∇C B = - L AI L AV C V (L AI D B + L BI D A ) L BV L AV - L BI L AI ∇C V (2)
where D A and D B are intrinsic diffusion coefficients. This profile may also be expressed in term of partial diffusion coefficients: [START_REF] Wiedersich | A theory of radiation-induced segregation in concentrated alloys[END_REF][START_REF] Wolfer | Drift forces on vacancies and interstitials in alloys with radiation-induced segregation[END_REF].

d αV = L αV /C α C V and
d αI = L αI /C α C I
The sign of the L αβ coefficients corresponds to the direction of couplings. For example, in the A-B alloy if L BV > 0 fluxes of B and V are in the same direction and if L BV < 0 they are in opposite directions. In simple cases, general rules of concentration variations can be derived from point defect diffusion coefficients.

RIS is thus often analyzed in terms of Inverse Kirkendall effect: for the vacancies, if B is the fast diffusing species (ie D B > D A ), a depletion of B at sinks is expected (note that it is only true when L BV < 0); for interstitials, if B is the fast diffusion species, an enrichment of B is expected (L BI is always positive) [START_REF] Nastar | Radiation-induced segregation[END_REF].

On the other hand, in the case of significant attraction between solute atoms and vacancies, L BV may be positive, leading to solute atoms dragged by vacancies and a solute enrichment at sinks (this is called the drag effect). Beyond such interpretations, one must know the full set of L ij coefficients in Eq. 2 and how they depend on temperature and concentration in order to determine the RIS profiles.

The L ij coefficients are very difficult to extract from diffusion experiments but they can be in principle easily deduced from atomic scale simulations. This requires: (i) a good description of jump frequencies and their variation with respect to the local environment and (ii) a good method to compute the L ij coefficients from the jump frequencies. For the second point, two strategies are possible:

1. A numerical determination of the L ij (or d ij ) coefficients by MD or AKMC simulations. MD simulations with an empirical potential have been used by Terentyev, Pechenkin et al. [START_REF] Terentyev | On the migration and trapping of single self-interstitial atoms in dilute and concentrated fecr alloys: Atomistic study and comparison with resistivity recovery experiments[END_REF][START_REF] Pechenkin | On the radiationinduced segregation: Contribution of interstitial mechanism in fe-cr alloys[END_REF] for the coefficients of self-interstitials.

Here we use AKMC simulations to directly measure the L ij coefficients of both vacancies and self-interstitials.

2. An analytical calculation of the L ij coefficients (or alternatively, of the partial diffusion coeffcients) from point defects jump frequencies, using multi-frequency diffusion models valid for dilute systems. This approach has been applied to Fe-Cr alloys by Choudhury et al. [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF] and by Was, Wharry et al. [START_REF] Was | Assessment of radiation-induced segregation mechanisms in austenitic and ferritic-martensitic alloys[END_REF][START_REF] Wharry | Application of the inverse kirkendall model of radiation-induced segregation to ferritic-martensitic alloys[END_REF]. Those diffusion models were limited to short-range interacting alloys and the contribution to the L ij coefficients of some vacancy jumps beyond the first nearest neighbor sites of the solute atom had to be neglected. Recently, long-range interaction models of the L ij coefficients have been developed [START_REF] Garnier | Solute drag by vacancies in body-centered cubic alloys[END_REF][START_REF] Messina | Ab initio modelling of vacancysolute dragging in dilute irradiated iron-based alloys[END_REF]). In concentrated alloys, due to the high complexity, approximate mean field methods are used to determine the L ij coefficients [START_REF] Nastar | Radiation-induced segregation[END_REF].

Diffusion model and AKMC simulations

Diffusion model and effective pair interactions

The calculation of point defects jump frequencies is based on a rigid lattice model using effective pair interactions hereafter referred to as the "AKMC model". We start from a previous version [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF] that only includes vacancy diffusion. It assumes that the free enthalpy of a given atomic configuration can be written as a sum of effective pair interactions on a rigid bcc lattice.

Such "broken bond" models with constant pair interactions are widely used in AKMC simulations to model diffusive phase transformations in alloys [START_REF] Pareige | Ordering and phase separation in ni-cr-al: Monte carlo simulations vs three-dimensional atom probe[END_REF][START_REF] Booth-Morrison | Effect of er additions on ambient and high-temperature strength of precipitation-strengthened al?zr?sc?si alloys[END_REF][START_REF] Mao | The mechanism of morphogenesis in a phase-separating concentrated multicomponent alloy[END_REF][START_REF] Clouet | Complex precipitation pathways in multicomponent alloys[END_REF]. However, Fe-Cr alloys have a special thermodynamic behaviour, with strong vibrational and magnetic contributions. The latter are especially important: they lead to a change in the sign of the alloy mixing energy around 10 at.% Cr. In ref [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF], magnetic and vibrational contributions are taken into account by introducing a temperature dependence on all interactions and by making Fe-Cr interactions on lattice sites dependent on local concentration (see Refs. [START_REF] Levesque | A simple concentration-dependent pair interaction model for large-scale simulations of fe-cr alloys[END_REF][START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF]). This model generates solubility limits in good agreement with recently updated Fe-Cr phase diagrams [START_REF] Bonny | New contribution to the thermodynamics of fe-cr alloys as base for ferritic steels[END_REF][START_REF] Xiong | Phase equilibria and thermodynamic properties in the fe-cr system[END_REF], showing an asymmetrical miscibility gap (see fig. 1 in Ref [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF]). It gives diffusion coefficients and kinetics of precipitation in good agreement with experimental data, in the range of temperatures and compositions considered here (300-950 K and up to 20% of Cr).

1

In the present study, the diffusion of self-interstitials is modeled in a similar manner. As we will see in the next section, dumbbells with a 110 orientation have the most stable interstitial configuration in Fe and dilute Fe-Cr alloys. We therefore only consider this configuration. The migration occurs through the Johnson mechanism [START_REF] Fu | Stability and mobility of mono-and di-interstitials in alpha-fe[END_REF]: a jump of the dumbbell towards a nearest-neighbor atom, with a rotation of 60 • . Two types of nearest neighbor sites can be distinguished : four sites in tension (T) and four sites in compression (C) [START_REF] Olsson | Ab initio study of interstitial migration in fecr alloys[END_REF].

Self-interstitial atoms can only jump to compression sites.

The jump frequency for an exchange between an atom A (= Fe, Cr) and a self-interstitial I is given by Γ AI = ν exp(-∆G mig AI /k B T ), where ν = 10 13 s -1 is the Debye frequency and ∆G mig AI the migration barrier :

∆G mig AI = Σ j gAIj -Σ k,n g (n) Ak -Σ l g Il -g Ii -Σ pq,n ∆g (n) pq (3)
where the first term on the right hand side corresponds to interactions created by the atom of the interstitial jumping at the saddle point (Σ j gAIj ). The second and third terms correspond to broken interactions around the targeted atom A and around the interstitial. g Ii is the interaction broken within the dumbbell and the last term corresponds to pq bounds that are not broken but experience a change in their local concentration and consequently a change in their interactions [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF].

In a first approximation, the interactions between self-interstitials and atoms in the lattice (g Il ) are limited to those between first nearest neighbors. The dumbbell composition (Fe-Fe, Fe-Cr or Cr-Cr) is taken into account and specific interactions are introduced to control its formation, binding and migration enthalpies. For this, we take into account interactions between the atoms within the dumbbell and distinguish interactions of dumbbell atoms with their neighbors on C sites and on T sites. For mixed Fe-Cr dumbbells, the interactions with atoms on C sites depend on the dumbbell direction (see Table 2). The detailed balance conditions are verified and symmetrical relations are taken into account.

Ab initio calculations of point defects properties

Vacancy properties

The details of the vacancy properties has been presented in Ref. [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF]. We only recall here the results relevant for the analysis of Onsager coefficients.

Martinez et al. have computed the 10 migration barriers controlling the Fe

and Cr tracer diffusion coefficients in Fe (according to Le Claire's model [START_REF] Leclaire | Correlation effects in diffusion in solids[END_REF]).

The results are very similar to those of Choudhury et al. [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF] and Messina et al. [START_REF] Messina | Ab initio modelling of vacancysolute dragging in dilute irradiated iron-based alloys[END_REF]. These studies show a weak Cr-V interaction and a low migration barrier for the Cr-V exchange (corresponding to the W 2 frequency using the usual LeClaire's notation) for the Cr-V exchange, in comparison with the other vacancy jumps. This results in a rapid diffusion of Cr in Fe (tracer diffusion coefficients D Cr * > D F e * ) [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF].

The pair interactions of the AKMC model have been fitted to some of these DFT results. In most cases, migration barriers and binding enthalpies generated by this model are in good agreement with the DFT calculations. The migration barriers of jump frequencies W 3 and W 4 are slightly higher than the DFT values. Meanwhile, the binding energy between Cr and the vacancy is slightly negative (almost zero) in the AKMC model, while it is slightly positive in the DFT calculations. W 3 and W 4 jump frequencies may affect the rotation of the vacancy around solute atoms and therefore the drag effect [START_REF] Garnier | Solute drag by vacancies in body-centered cubic alloys[END_REF][START_REF] Messina | Exact ab initio transport coefficients in bcc Fe-x (x = Cr, Cu, Mn, Ni, P, Si) dilute alloys[END_REF]. In spite of these differences, the model gives tracer and interdiffusion coefficients in good agreement with experimental data in ferromagnetic alloys [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF][START_REF] Senninger | Atomistic simulations of the decomposition kinetics in fecr alloys: Influence of magnetism[END_REF].

Self-interstitial atoms properties

Self-interstitial properties have been studied by Olsson et al. [START_REF] Olsson | Ab initio study of cr interactions with point defects in bcc fe[END_REF] in dilute Fe-Cr alloys using DFT calculations and by Terentyev et al. [START_REF] Terentyev | On the migration and trapping of single self-interstitial atoms in dilute and concentrated fecr alloys: Atomistic study and comparison with resistivity recovery experiments[END_REF] in concentrated Fe-Cr alloys using the 2BM empirical potential.

We present here a DFT study of self-interstitial properties (formation, migration and binding energies) in pure metals, dilute and more concentrated Fe-Cr alloys, with the objective of fitting the corresponding AKMC parameters.

DFT calculations were performed using the PWSCF code [START_REF] Giannozzi | Quantum espresso: a modular and open-source software project for quantum simulations of materials[END_REF]. We performed spin-polarized calculations within the generalized gradient approximation (GGA). Accurate projector augmented wave (PAW) potentials were used, where the 3d and the 4s electrons were considered as valence electrons. The kinetic energy cutoff chosen for the plane-wave basis set was 544 eV. All the calculations were carried out with 128-atom supercells, using a (3 × 3 × 3)kpoint grid and the Methfessel-Paxton broadening scheme with a 0.3 eV width.

To calculate the energetics of interstitial atoms, we have adopted a constant volume-per-atom approach. For instance, the lattice parameter of a cubic supercell containing N Fe lattice atoms and one Fe interstitial was rescaled in order to give a total volume equal to the sum of N + 1 times the Fe atomic volume in a defect-free bcc lattice. The atomic positions were fully relaxed with a residual value of less than 0.04eV / Å for each force component of all atoms.

The binding energy of a configuration containing n point defects noted d i is

given by:

E b (config) = Σ n i=1 E(d i ) -E(config) -(n -1)E ref (4)
where E ref is the total energy of a supercell of pure Fe, E(d i ) is the total energy of a supercell containing the single defect d i and E(config) is the total energy of the supercell containing all the defects (d i ). A positive binding energy refers then to an attractive interaction.

The migration barriers have been determined using the drag method as in our previous work [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF].

Pure metals. In the 52 configurations of Fig. 1 the 110 dumbbell is more stable than the 111 , except for two cases. In these two cases, the 111 configuration is an energy slightly lower than the 110 configuration, but is still negative with a strongly negative energies (respectively -0.63 and -0.80 eV) and therefore have negligible probabilities of occurrence.

The mixed interstitial (line 1 in Table 2) is stable in pure Fe, with a positive binding energy (+0.022 eV). The interactions between a Fe-Fe 110 dumbbell and a Cr atom on tension site (line 2) are repulsive (-0.038 eV). The ones with a Cr atom on a compression site (line 3) are attractive (+0.016 eV).

Cr-Cr dumbbells are strongly repulsive in pure iron (line 11, -0.554 eV) and more generally in all the dilute Fe-Cr configurations that we have considered.

The binding energy of a mixed interstitial with a Cr atom on a tension site is negative (line 7) whereas the one with a Cr atom on a compression site depends on the position of the Cr atom within the dumbbell (lines 8 and 9).

Configurations with more Cr atoms (e.g. lines 10, 11 and 12) usually have very negative binding energies. This results from the magnetic frustration occurring between Cr neighbors in Fe. These frustrations explain the ordering tendency of dilute Fe-Cr alloys [START_REF] Klaver | Magnetism and thermodynamics of defect-free fe-cr alloys[END_REF]. However, some configurations with two or more Cr atoms have positive binding energies (especially with Cr on compression sites, e.g. on lines 4, 6, 9) and may tend to decrease the mobility of dumbbells.

In pure Cr, the mixed and Fe-Fe 110 dumbbells (lines 13 and 14) are very attractive, with high binding energies respectively close to 1 and 2 eV.

In general, our results are similar to the previous DFT results of Olsson et al. [START_REF] Olsson | Ab initio study of cr interactions with point defects in bcc fe[END_REF]. The results obtained by Terentyev et al. [START_REF] Terentyev | On the migration and trapping of single self-interstitial atoms in dilute and concentrated fecr alloys: Atomistic study and comparison with resistivity recovery experiments[END_REF] with the 2BM potential tend to overestimate the attraction between 110 dumbbells and Cr atoms compared to DFT calculations. As mentioned in [START_REF] Terentyev | On the migration and trapping of single self-interstitial atoms in dilute and concentrated fecr alloys: Atomistic study and comparison with resistivity recovery experiments[END_REF], the binding energies between 111 dumbbells and Cr atoms are positive (lines 15 and 16), but they are lower than the difference between the formation energies of 111 and 110 dumbbells in pure Fe (approximately 0.8 eV): these 111 configurations are therefore not probable and should not be able to trap the self-interstitials.

The binding energies of the AKMC model are usually in good agreement with the corresponding DFT values (Fig. 1), especially for configurations involving only 1 or 2 Cr atoms. In the case of configurations including 3 and 4 Cr atoms, the AKMC model reproduces the general trend, i.e. strongly negative binding energies.

Migration energies in Fe-Cr alloys

DFT calculations of migration barriers between configurations involving one 110 dumbbell and between 1 and 4 Cr atoms are given in Fig. 2.

The present calculations give a migration energy of 0.34 eV for the 110 dumbbell in pure iron, in agreement with previous studies [START_REF] Fu | Stability and mobility of mono-and di-interstitials in alpha-fe[END_REF][START_REF] Olsson | Ab initio study of interstitial migration in fecr alloys[END_REF][START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF]. In alloys, our DFT study shows that this mechanism is associated with the lowest migration energy of the 110 dumbbell. The mixed interstitial has a migration barrier significantly lower than the Fe-Fe dumbbell in pure iron: 0.23 instead of 0.34 eV.

The migration barriers of the Cr-Cr dumbbells are even lower (∼ 0.05 eV) but configurations involved in these processes are very repulsive (E b 0).

In configurations with one Cr atom, the migration barriers are close to those of Choudhury et al. [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF] and Olsson et al. [START_REF] Olsson | Ab initio study of interstitial migration in fecr alloys[END_REF]. Choudhury et al. [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF] have also computed migration barriers for self-interstitials jumps without rotation. Their migration barriers are systematically higher when compared to the Johnson mechanism, which further validates our choice of migration mechanism.

For migration barriers with 1 or 2 Cr atoms, the AKMC model is in good agreement with the DFT calculations (Fig. 2). For higher Cr concentrations, the discrepancy can be as high as 0.4 eV. However, they correspond to configurations with negative binding energies, i.e. low probabilities. A better agreement could be obtained by introducing pair interactions between dumbbells and atoms at positions further than first nearest neighbors and a composition dependence to the saddle-point pair interactions. However, as this would slow down the AKMC code, we prefer to leave this improvement for a future study.

Finally, the attempt frequency of vacancy jumps is assumed to be ν 0 = 10 13 s -1 and the migration barriers include a migration entropy of 2.1 k B [START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF].

The migration entropy of interstitials has never been experimentally measured.

For the sake of simplicity, it has been chosen here to take the same value as the vacancies.

To summarize, key self-interstitials properties revealed by DFT calculations and reproduced by the AKMC model are: the moderate attraction between interstitials and Cr atoms in dilute Fe, the stability of mixed interstitials in dilute Fe and the rapid migration of the mixed interstitial.

AKMC simulations

Two kinds of AKMC simulations have been performed: measurements of transport coefficients and simulations of RIS near a grain boundary.

For the first ones, one point defect (V or I) is introduced in a simulation box with N = 4096 bcc sites, periodic boundary conditions and up to 15% of Cr.

The jump frequencies are computed with our diffusion model and a residence time algorithm (RTA) is used to choose the jumps and to increment the time [START_REF] Bortz | New algorithm for Monte-Carlo simulation of ising spin systems[END_REF]. Tracer diffusion coefficients of point defects d = I, V and L ij coefficients are computed with generalized Einstein's relations [START_REF] Allnatt | Atomic transport in solids[END_REF]:

D d = r 2 d 6t (5)
and

L ij = 1 6V tot k B T R i R j t , (6)
where r d is the average displacement of defect d during time t. R i is the total displacement of species or defect i during time t and V tot is the total volume of the system. To obtain sufficient statistics, average values are calculated on 10 5 series of 10 5 Monte Carlo steps. Measurements are performed in homogeneous solid solutions, i.e. outside the miscibility gap or slightly inside -but in the latter case checking that α precipitation does not occur during the measurements.

Simulations of RIS at grain boundaries are performed in samples with N = 512 × 64 × 64 bcc sites, with a GB located in the middle of the long dimension.

These simulations take into account the following events [START_REF] Soisson | Kinetic monte carlo simulations of radiation induced segregation and precipitation[END_REF][START_REF] Soisson | Atomistic simulations of copper precipitation and radiation induced segregation in -iron[END_REF]:

-the creation of Frenkel pairs (I-V) with a production rate K 0 . I-V pairs are created randomly within the simulation box, along sequences of 10 displacements in 111 directions. This mechanism corresponds to electron rather than neutron or heavy ion irradiation.

-the migration of I and V, according to the diffusion model, -the recombination of I and V pairs below a recombination radius R c of 4 nearest neighbors.

-the annihilation of I and V when they jump on a GB. Within our rigid approach, the GB is simply modeled as a planar perfect sink.

Results

Point defect tracer diffusion coefficients

We present first diffusion coefficients of point defects which determine the (Table 2 in ref. [START_REF] Terentyev | On the migration and trapping of single self-interstitial atoms in dilute and concentrated fecr alloys: Atomistic study and comparison with resistivity recovery experiments[END_REF]). However, in their MD simulations, the threshold temperature is lower (T s ∼ 400 K). This may be related to the overestimation of the I-Cr attraction by the 2BM potential, as the trapping effect becomes dominant at lower temperatures.

Onsager coefficients

We have measured that for concentrations ranging from 0.25 to 15 %Cr and temperatures between 300 and 1500 K, the L F eI and L CrI coefficients are positive and the L F eV and L CrV coefficients are negative. Fe and Cr atoms diffuse thus in the opposite direction than vacancies and in the same direction as interstitials.

In the dilute limit, one observes the theoretical result L F eV /L F eF e = -1.

The L CrV /L CrCr ratio is close to -2 which indicates a strong negative coupling between vacancies and Cr atoms displacements, as could be expected from the small migration barrier ∆H 2 for the Cr-V exchange in Fe and from the absence of drag effect. This ratio is lower than in the studies of Choudhury et al. [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF] and Messina et al. [START_REF] Messina | Ab initio modelling of vacancysolute dragging in dilute irradiated iron-based alloys[END_REF] that give a value close to -1 in the dilute limit (one Cr atom).

The results of Messina at al. suggest a vacancy drag effect (i.e. that L CrV becomes positive) at very low temperatures (below 300 K). These discrepancies are due to the overestimation by the AKMC model (see 3.2.1) of the migration barriers, ∆H 3 and ∆H 4 , that control the rotation of the vacancy around the Cr atoms [START_REF] Garnier | Solute drag by vacancies in body-centered cubic alloys[END_REF][START_REF] Messina | Ab initio modelling of vacancysolute dragging in dilute irradiated iron-based alloys[END_REF]. This illustrates how sensitive the Onsager coefficients, and therefore the RIS phenomena, are to small differences in the migration barriers.

Intrinsic diffusion coefficients D F e and D Cr coefficients are positive outside of the miscibility gap. Therefore:

- L F eI L F eV C V (L F eI D Cr + L CrI D F e ) > 0.
and the segregation tendency (enrichment or depletion) depends on the sign of (L CrV /L F eV ) -(L CrI /L F eI ) according to Eq. 2. These ratios are shown in The L CrI /L F eI and L CrV /L F eV ratio both decrease with temperature, but not at the same rate. The interstitial contribution is dominant (L CrI /L F eI > L CrV /L F eV ) below a threshold temperature while the vacancy contribution predominates above that temperature. Between 5 and 15%Cr the alloy should tend to be enriched in Cr in the vicinity of point defect sinks at low temperature and to be depleted at high temperature. One should notice that both contributions are quite close, especially at high temperature (Fig. 4). The temperature threshold depends on the alloys composition: it increases between 0.25 and 5%

Cr, then decreases between 5 and 15% (in the Fe-15%Cr alloy, the α' precipitation occurs rapidly below 600 K, preventing measurement of the Onsager coefficients, but the threshold value can be roughly estimated to be 400-500 K by extrapolation). These two features are in qualitative agreement with the experimental observations and simulations of Wharry et al. [START_REF] Wharry | Application of the inverse kirkendall model of radiation-induced segregation to ferritic-martensitic alloys[END_REF] performed in alloys with concentrations between 8 and 12%Cr.

For self-interstitials, these results can be compared with the MD study by Pechenkin et al. [START_REF] Pechenkin | On the radiationinduced segregation: Contribution of interstitial mechanism in fe-cr alloys[END_REF], performed in a similar range of concentrations. They give a ratio of partial diffusion coefficients d CrI /d F eI decreasing from ∼ 3.7 to ∼ 2.2 in Fe-5%Cr between 600 and 1000 K, and from ∼ 1.8 to ∼ 1.2 in Fe-10%Cr. With

d CrI /d F eI = (C F e L CrI )/(C Cr L F eI
), our AKMC simulations give higher ratios and higher variations: from ∼ 8 to ∼ 3 in both alloys. Penchenckin et al. [START_REF] Pechenkin | On the radiationinduced segregation: Contribution of interstitial mechanism in fe-cr alloys[END_REF] have measured the tracer diffusion coefficients of Fe and Cr by self-interstitial diffusion in Fe-Cr alloys through MD simulations and have derived an estimation of the partial diffusion coefficients based on the Manning's approximation. It is thus difficult to determine if the discrepancy comes from this approximation or from differences in the self-interstitials jump frequencies. Nevertheless, the present study confirms the main conclusions of these approaches, without relying on similar approximations.

Radiation induced segregation at grain boundaries

AKMC simulations of RIS near point defect sinks have been performed to check the predictions deduced from the measurements of the L ij coefficients.

Contrary to Eq. 2 they take into account the variation of transport coefficients with the solute concentration in the vicinity of the sink. Moreover, they can model precipitation as well as segregation phenomena. We first present steady state concentration profiles when no precipitation is observed and a kinetic study of RIS and then analysis of precipitation in undersaturated and oversaturated alloys. In these simulations, Cr segregation profiles across the grain boundary are characterized by the concentration on the sink plane (C GB Cr ) and by the total quantity of Cr segregated (S tot Cr = ( C Cr C F e (x) -

C 0 Cr C 0 F e
)dx where x is the direction perpendicular to the sink and C 0 Cr/F e is the Cr/Fe concentration in the bulk) [START_REF] Wagner | Radiation-induced segregation in ni-cu alloys[END_REF].

Simulations have been performed for alloy compositions from 1 to 15%Cr, at 650 K and 950 K. Simulations at 650 K were performed with a flux of 10 -6 dpa.s -1 and the one at 950K with a flux of 10 -3 dpa.s -1 .

Point defects and Cr concentration profiles obtained in simulations at 10%Cr and at the two temperatures are reported in Fig. 5. Steady state point defect concentrations far from sinks are given by:

C V = - K I 2R + K 2 I 4R 2 + K 0 K I RK V (7)
C I = - K V 2R + K 2 V 4R 2 + K 0 K V RK I (8)
with vacancies (see Fig. 3), they reach the GB more rapidly and their concentration in the bulk is lower (Fig. 5) : at steady state

K I = k Is D I , K V = k V s D V ,
C I D I = C V D V .
At 950 K one observes a depletion of Cr near the GB (see Fig. 5 as an example) for all compositions. At 650K, the observed evolutions depend on the alloy concentration. At 1%Cr, we observe no evolution of the alloy composition at sinks. At 5 and 10%Cr, the GB tends to be enriched in Cr with a larger concentration increase for the alloy at 5%Cr (C GB Cr goes from 5% to 11%) than for the one at 10%Cr (C GB Cr goes from 10% to 13%). This RIS behavior is in agreement with the L ij measurements summarized in Fig. 4.

The kinetic evolution of Cr segregation profile according to irradiation dose has been analyzed for the simulation at 10%Cr and 650K using a box of 256×64× 64 bcc sites (see Fig. 6). We observe that the concentration at sinks reaches a steady state very fast and that the total quantity of Cr segregated evolves much slower (an exponential fit of these evolutions gives a relaxation dose of 0.023 dpa for C GB Cr and 0.107 dpa for S tot Cr ). It has to be noted that the difference in kinetics between C GB Cr and S tot Cr observed in our simulations are in agreement with experimental observations in Ni-Cu [START_REF] Grandjean | Kinetic model for equilibrium and nonequilibrium segregation in concentrated alloys under irradiation[END_REF] and Ni-Si [START_REF] Rehn | Dose dependence of radiation-induced segregation in ni-1 at% si[END_REF] alloys. Moreover, by multiplying and dividing the density of sinks by a factor 2, we observe that

S tot

Cr decreases with the density of sinks but that C GB Cr does not evolve much with it. From our best knowledge, kinetic evolutions of concentration profiles at sinks have never been observed in the Fe-Cr alloy at such low doses. Wharry et al. [START_REF] Wharry | The mechanism of radiation-induced segregation in ferriticmartensitic alloys[END_REF] have nevertheless computed with their Inverse Kirkendall model a significant enrichment at sinks at low doses (lower than 10 -3 dpa) and predict a steady state of Cr enrichment which is reached at 1 dpa or less. However, their measurements on GBs show that concentration profiles keep evolving in doses higher than 1 dpa [START_REF] Wharry | The mechanism of radiation-induced segregation in ferriticmartensitic alloys[END_REF]. This evolution is attributed to microstructural evolutions during irradiation. In particular, the flux of ions induces an evolution of the dislocation loop density which leads to a change of defect concentration and so to an evolution of RIS profiles.

At low temperatures, the Cr enrichment may be sufficient to reach the solubility limit, even if the alloy is undersaturated. In such cases one observes a local Radiation Induced Precipitation near the GB. An example is given in an alloy at 9%Cr and at 563K on Fig. 7. In this case, Eqs. [START_REF] Hu | Effect of grain boundary orientation on radiation-induced segregation in a fe-15.2 at.% cr alloy[END_REF] and [START_REF] Marquis | A systematic approach for the study of radiation-induced segregation/depletion at grain boundaries in steels[END_REF] give the concentrations of vacancies and self-interstitials respectively of 1.5 × 10 -10 and 4.5 × 10 -13 in agreement with the simulation.

The RIS phenomenon can also be associated with radiation accelerated precipitation. Indeed, at 15%Cr and 650K, the analytic RIS model predicts a depletion of Cr at sinks (see Fig. 4). However, at this composition and temperature, the alloy is in the miscibility gap and decomposes into α and α phases.

As shown in Fig. 8, we observe the formation of α precipitates and a small depletion of Cr in the vicinity of the sink. This decomposition is accelerated compared to thermal equilibrium as point defect concentrations are increased by irradiation in the bulk. In the GB vicinity, point defect concentrations de-crease (Fig. 8(a)) and the nucleation, growth and coarsening of α precipitates is much slower than in the bulk: one observes a precipitate free zone around a GB. Such precipitate free zones have been experimentally observed by 3D Atom Probe during the α precipitation near grain boundary in ferritic steels [START_REF] Kuksenko | Cr precipitation in neutron irradiated industrial purity fecr model alloys[END_REF].

In Figs. 7 and8 we observe some small peaks in self-interstitial concentration profiles. These peaks are due to trapping of self-interstitials in configurations highly concentrated in Cr and at α/α interfaces.

Conclusions

In this study we have analyzed radiation-induced segregation (RIS) in the binary bcc Fe-Cr alloy with a new Atomistic Kinetic Monte Carlo model. The previous RIS model of Wharry et al. [41] uses partial diffusion coefficients defined by single effective diffusion barriers for each element-defect exchange fitted to a mean of migration barriers in a dilute configuration. This model does not exactly take into account the correlations between successive point defect jumps. It is nevertheless efficient for Fe-Cr alloys, as diffusion properties are quite simple, due to the weak Cr-V interactions. It would probably be less precise in case of drag effect. Meanwhile, our model gives a coherent description of jump frequencies and transport coefficients in concentrated alloys and has the advantage that it can be also used to study radiation-accelerated precipitation (RAP) and radiationinduced precipitation (RIP). This model could however still be improved by correcting the overestimation of the negative coupling between Cr atoms and vacancy diffusion. It is difficult to fully reproduce the evolution of self-interstitial migration properties according to the local environment. Improved diffusion models with interactions at longer range and/or concentration dependent saddlepoint interactions would certainly help to reproduce these variations better.

Nevertheless we have confirmed qualitative conclusions of previous studies : the migration of vacancies toward sinks induces a depletion of Cr in the vicinity of sinks. The stability of mixed interstitials and their fast migration induces a tendency towards Cr enrichment at sinks. Measures of coupling coefficients and simulations of RIS have shown that, for concentrations and temperatures outside of the miscibility gap, the alloy tends to be enriched in Cr at sinks at temperatures lower than a threshold value and to be depleted in Cr at sinks at higher temperatures. The threshold temperature depends on the bulk concentration in a non-monotonic way.

We however note that the experimental observations in ferritic steels are far less simple [START_REF] Lu | Irradiation-induced grain boundary chromium microchemistry in high alloy ferritic steels[END_REF] as already noticed in [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF]. The main reason is probably that RIS results from a delicate balance between the enrichment due to self-interstitials and the depletion due to the vacancy in these ferritic alloys. The situation is therefore very different from the case of austenitic alloys where the elimination of vacancies and interstitials both contribute to a depletion of Cr and enrichment of Ni [START_REF] Nastar | Role of interstitial and interstitial-impurity interaction on irradiation-induced segregation in 675 austenitic steels[END_REF][START_REF] Nastar | A mean field theory for diffusion in a dilute multi-component alloy: a new model for the effect of solutes on self-diffusion[END_REF]. In ferritic steels, small variations in migration barriers can induce an inversion of tendencies [START_REF] Choudhury | Ab-initio based modeling of diffusion in dilute bcc fe-ni and fe-cr alloys and implications for radiation induced segregation[END_REF]. These variations can be in the order of the uncertainties in DFT calculations (see also Messina et al. [16]), making such multiscale approaches to RIS phenomena especially challenging. Furthermore, any contribution affecting the migration barriers and the L ij coefficients could affect the segregation behavior:

-the influence of impurities, e.g. carbon, on the experimental profiles [START_REF] Hu | Effect of grain boundary orientation on radiation-induced segregation in a fe-15.2 at.% cr alloy[END_REF][START_REF] Marquis | A systematic approach for the study of radiation-induced segregation/depletion at grain boundaries in steels[END_REF] that are not included here. The carbon effect will be integrated in our model in future works.

-the influence of equilibrium segregation tendencies, that may reinforce or oppose the radiation induced ones, and will strongly depend on the interaction between Cr atoms and various point defects sinks.

-long range elastic interactions between sinks (e.g. dislocations) and solute atoms.

These contributions may be introduced in AKMC simulations and give further insights into the mechanisms controlling RIS in ferritic steels.

The results presented in this paper may suggest that systematic experimental studies of RIS in pure binary Fe-Cr alloys (especially with minimum C content) depending on Cr content outside of the miscibility gap would help to better understand couplings between defects and chemical species evolutions with Cr content. Moreover, most of RIS profiles have been observed at 1 dpa and higher doses. To separate short term evolution (the segregation observed for a given density of point defect sinks), from later evolution due to microstructural changes, alloys could be analyzed at low doses (below 1 dpa, or even 0.1 dpa). By comparison with ion or neutron irradiations that lead to the creation of high densities of small dislocation loops in displacement cascades, electron irradiation would probably delay the evolution of the sink density. This may also help to separate the evolution at low and high doses.

  steady state point defect concentration under irradiation. The tracer diffusion coefficients of one vacancy and one interstitial in alloys at nominal concentrations up to 15%Cr and at temperatures between 300 K and 1500 K are presented in Figures 3. The vacancy diffusion coefficients slightly increase with the Cr concentration, due to the low Cr-V exchange barrier in Fe. Self-interstitial diffusion coefficients are higher (up to 5 orders of magnitude at 300 K) than the vacancy diffusion coefficients. It has to be noted that a change of interstitial migration entropy in the energetic model would change the I diffusion coefficients. The effect of Cr concentration on the I diffusion coefficient depends on temperature. At temperatures lower than the threshold value T s ∼ 600 K, the addition of Cr induces an acceleration of self-interstitial diffusion. At T > T s the inverse tendency is observed. The increase of the interstitial diffusion coefficient with concentration at low temperature is due to the stability and the low migration barrier of the mixed dumbbell. The influence of Cr concentration on self-interstitial diffusion at high temperature can be explained by the dissociation of the mixed interstitials and the trapping of I on Cr atoms. These results are in qualitative agreement with the MD simulations of Terentyev, Pechenkin et al., using the 2BM empirical potential [11, 12]. A fit of the self-interstitial diffusion coefficients by an Arrhenius law: D I = D 0 I exp(-∆H mig I /(k b T ) gives H mig I = 0.34, 0.29, 0.27, 0.25 eV for 0.25, 5, 10 and 15%Cr respectively, close to the values of Terentyev et al.

Fig. 4 ,

 4 Fig. 4, for different compositions and temperatures. L CrI /L F eI > C Cr /C F e , indicating a preferential transport of Cr by self-interstitials which implies a systematic tendency for enrichment of Cr near the point defect sinks by selfinterstitial diffusion. L CrV /L F eV > C Cr /C F e also indicates a preferential transport of Cr by vacancies which implies a systematic tendency for depletion of Cr near the point defect sinks by vacancy diffusion.

  where k Is and k V s terms are the sink strengths for point defects elimination[START_REF] Sizmann | The effect of radiation upon diffusion in metals[END_REF], R = 4πR c (D I + D V )/Ω with R c the critical distance of recombination, K 0 the point defect production rate and Ω the alloy atomic volume. These equations predict concentrations of vacancies and selfinterstitials respectively of 1.5 × 10 -11 and 2 × 10 -13 at 650 K. At 950 K, these concentrations change to 4.5 × 10 -10 and 4.5 × 10 -11 . Our simulations show a fairly good agreement with these values. As interstitials diffuse faster than

Figure 1 :

 1 Figure 1: Binding energies between one 110 dumbbell and Cr atoms in Fe. Comparison between the AKMC model and the DFT calculations.

Figure 2 :

 2 Figure 2: Migration energies of a 110 dumbbell, with up to four Cr neighbors in Fe. Comparison between the AKMC model and the DFT calculations.

Figure 3 :

 3 Figure 3: Vacancy and interstitial tracer diffusion coefficients in Fe-Cr alloys at various nominal compositions. AKMC simulations 10 5 × 10 5 MCS in stable and metastable solutions

Figure 4 :

 4 Figure 4: evolution of L CrI /L F eI and L CrV /L F eV with T for different alloy compositions. AKMC measurements: 10 5 × 10 5 MCS, in stable and metastable solid solutions.

Figure 5 :

 5 Figure 5: Steady-State point defects and Cr concentration profiles in the vicinity of a perfect planar sink for an alloy at 10%Cr left) at 650 K irradiated by a flux of 10 -6 dpa.s -1 (at 0.007 and 0.294 dpa) right) at 950 K irradiated by a flux of 10 -3 dpa.s -1 (at 0.012 and 0.636 dpa).

Figure 6 :

 6 Figure 6: Concentration of Cr at sink (C GB Cr ) and total quantity of Cr segregated (S tot Cr ) as a function of dose. Simulations for an alloy at 10%Cr and 650 K using a simulation box of 256 × 64 × 64 bcc sites.

Figure 7 :

 7 Figure 7: AKMC simulations of RIS and RIP near the grain boundary in a Fe-9%Cr alloy at 563K under a dose rate of 10 -6 dpa.s -1 and a dose of 0.200 dpa. (left) point defects and Cr concentration profiles and (right) map of Cr atoms in the simulation box

Table 1 :

 1 Self-interstitial atoms formation energies (in eV) in pure Cr, computed in DFT-PAW calculations

	configuration Cr-NM	Cr-AFM
	100	6.63	6.84
	110	5.58	6.08
	111	5.64	6.20
	Octahedral	6.76	7.04
	Tetrahedral	6.31	6.60
	In pure iron, our DFT calculations give a formation enthalpy of 3.96 eV
	for the 110 dumbbell, the most stable orientation in pure iron (in agreement
	with previous studies [31, 25]). The 111 dumbbell has a formation enthalpy
	of 4.74 eV.
	Formation energies of different self-interstitial configurations in pure Cr, with
	a non-magnetic (NM) or anti-ferromagnetic (AFM) state, are reported in

Table 1 .

 1 Olsson et al. [29] have obtained values slightly smaller but with almost the same relative stability between the different configurations. The magnetic state has a very limited effect on the relative stability of the different configurations.The 110 configuration is the most stable one in each case, but the formation energy of 111 is very close. The formation energies of the 100 dumbbell, octahedral and the tetrahedral configurations are significantly higher.

	Binding energies in Fe-Cr alloys.
	Binding energies have been calculated for 52 configurations involving a 110
	dumbbell in Fe, with up to four Cr atoms in first and second nearest neighbor
	positions. Table 2 presents a selection of binding energies computed in DFT
	and with our AKMC model. Fig. 1 summarizes the comparison between the

DFT calculations and the AKMC model for all configurations. A few binding energies between Fe atoms and 110 dumbbells in Cr, and between Cr and 111 dumbbells in Fe are also given on Table

.

2.

In Fe-Cr solid solutions, the ferro-to-paramagnetic transition leads to an increase of the diffusion coefficients that can be modeled by a correction of the vacancy migration barriers[START_REF] Martinez | Decomposition kinetics of fe-cr solid solutions during thermal aging[END_REF]. It is shown to significantly accelerate the precipitation kinetics in alloys with high Cr contents (above typically

25%). As we limit our study to concentrations lower than ∼ 15%, such effects are not included in the present model.

Acknowledgments

We would like to thank E. Clouet for many fruitful discussions. This research has received partial funding from the Eurofusion IReMEV program and from the European Atomic Energy Community 7th Framework Program (FP7/2007-2011), under Grant agreement No. 212175 (GetMat project). This work also contributes to the Joint Program on Nuclear Materials (JPNM) of the European Energy Research Alliance (EERA). DFT calculations were performed using resources from DARI within project x2015096020. E. M. gratefully acknowledges the support of the US Department of Energy through the LANL/LDRD Program for this work.