Matteo Mancinelli
email: matteo.mancinelli@uniroma3.it

Peter Jordan
email: peter.jordan@univ-poitiers.fr

Anton Lebedev
email: anton.lebedev@univ-poitiers.fr

Real-Time Estimation of Jet-Surface Interaction Noise

Keywords: aeroacoustics, jet noise, jet-surface interaction

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Introduction

Jet noise is one of the main noise sources of aircraft engine noise, especially during take off when engines are at maximum power. The increasingly stringent laws for noise emissions with the aim of reducing their impact on the community impose severe constraints to the development of commercial aircraft industry. According to the well-known Lighthill's eight power law [START_REF] Lighthill | On sound generated aerodynamically i. general theory[END_REF], jet noise emissions are substantially reduced when the jet velocity exhausting from the engine is decreased. The velocity reduction has to be compensated by the increase of the mass flow and, hence, of the fan/nacelle diameter, to maintain the same thrust. This technical solution led to the development of the Ultra-High By-Pass Ratio (UHBPR) engines. As a drawback, the larger size of the new engines leads to a stronger interaction between the jet flow and the wing/flap surfaces and the consequent generation of a noise source associated with the jet-wing interaction. As outlined by [START_REF] Huber | Characterization of installation effects for hbpr engine, part 4: Assessment of jet acoustics[END_REF] and [START_REF] Huber | Large-scale jet noise testing, reduction and methods validation" exejet": 1. project overview and focus on installation[END_REF], the increase of acoustic emissions due to the jet-surface interaction (JSI) may jeopardise the benefit of noise decrease associated with the jet velocity reduction. Installation effects must consequently be mitigated in future aircraft configurations.

As shown by [START_REF] Cavalieri | Scattering of wavepackets by a flat plate in the vicinity of a turbulent jet[END_REF], [START_REF] Piantanida | Scattering of turbulent-jet wavepackets by a swept trailing edge[END_REF], [START_REF] Nogueira | Acoustic radiation of subsonic jets in the vicinity of an inclined flat plate[END_REF] and [START_REF] Nogueira | Acoustic radiation of subsonic jets in the vicinity of an inclined flat plate[END_REF], JSI noise can be modelled by the diffraction of turbulent wavepackets by the trailing edge (TE) of the surface. In these works, the authors showed that non-linear aspects described by jittering were not essential to obtain accurate predictions of installed jet noise. The effectiveness of a linear model to describe jet-surface interaction noise was further underlined by [START_REF] Faranosov | The modeling of jet-plate interaction noise in the presence of co-flow[END_REF], who showed that installed far-field noise could be accurately predicted using a Wiener-Hopf model based on near-field pressure fluctuations provided by parabolised stability equations. The linearity of the JSI noise mechanism was exploited by [START_REF] Kopiev | Active control of jetplate interaction noise for excited jets by plasma actuators[END_REF] and [START_REF] Bychkov | Plasma-based active closed-loop control of instability waves in unexcited turbulent jet. part 2. installed jet[END_REF] to implement harmonic control strategies in an installed configuration for a forced and unforced jet using a linear reactive control framework.

Plasma actuators at the nozzle exit were driven by near-field sensors via a feedback control scheme to reduce the wavepacket amplitude in the flow, thus reducing the noise scattered by the TE of the surface. Correct positioning of near-field sensors in the jet plume is essential for correct estimation at the observer position and, thus, for control effectiveness in such approaches [START_REF] Maia | Real-time reactive control of stochastic disturbances in forced turbulent jets[END_REF].

The above discussion motivates the present work where we perform noise estimation in an installed configuration using linear transfer functions of the kind used in the real-time, reactive control strategies [START_REF] Maia | Real-time reactive control of stochastic disturbances in forced turbulent jets[END_REF]. We investigate the jet-surface interaction phenomena on a simplified configuration consisting of a subsonic jet grazing a flat plate parallel to the nozzle axis. Sensor and observer readings in the near field and far field of the jet, respectively, are used to build empirical transfer functions to estimate the noise radiated in the far field. This approach has been used by [START_REF] Kleine | Real-time supersonic jet noise predictions from near-field sensors with a wavepacket model[END_REF] to perform estimations of supersonic jet noise. Both velocity and pressure measurements are used in the near field to derive the empirical transfer functions that can be used for real-time noise estimation. The transfer function as well as the accuracy of the estimation performed are characterised as a function of the type of the near-field sensor used, the streamwise location of the sensors, the radial distance between the jet and the flat plate and the jet flow conditions. This analysis is fundamental for the guidance of sensor positioning for closed-loop control of JSI noise (Sasaki et al, 2018a,b).

The remainder of the present document is organised as follows. In 2 we describe the experimental test campaign and the instrumentation adopted. A steel flat plate was installed parallel to the jet at different radial distances H = 0.6D, 0.75D and 1D from the nozzle axis with the trailing edge positioned at an axial distance x/D = 2. We used both a hot wire and a microphone array as near-field sensors. A 6-microphone azimuthal near-field array was placed at a radial distance of r = 0.9 D from the nozzle axis to measure the pressure fluctuations associated with the azimuthal Fourier mode m = 0, which is the most efficient mode in terms of noise emissions in an installed configuration [START_REF] Cavalieri | Scattering of wavepackets by a flat plate in the vicinity of a turbulent jet[END_REF]. We remind the reader that the m = 0 mode extracted in the installed configuration is substantially different from that determined in the free-jet configuration, because it contains the signature of the acoustic field scattered from the surface in addition to the incident field. The accuracy of the azimuthal Fourier decomposition for the frequencies of interest where the JSI noise is dominant was checked by computing the coherence between neighbouring microphones [START_REF] Cavalieri | Scattering of wavepackets by a flat plate in the vicinity of a turbulent jet[END_REF][START_REF] Mancinelli | A complex-valued resonance model for axisymmetric screech tones in supersonic jets[END_REF]. The near-field array was free to move along the streamwise direction. A hot-wire (HW) probe was placed on the nozzle axis and was free to move along both the streamwise and radial directions. Two far-field microphones were placed at the TE axial position at a radial distance of 26 D from the trailing edge. The microphone observers were installed at azimuthal angles θ = 0 • and θ = 180 • in the shielded and unshielded sides, respectively, of the acoustic field where the dipole TE noise source has its peak radiation.

Measurements were carried out in isothermal conditions for different jet Mach numbers ranging from M = 0.15 to M = 0.75. The low Mach numbers, although far from the flow conditions found in real industrial applications, are herein considered given that our primary interest is to explore the possibility of building a real-time, laboratory-scale control demonstrator. Pressure fluctuations were measured by PCB 46BP microphones which are characterised by a flat frequency response in the range 4 Hz-70 kHz. Hot-wire measurements were performed using a 55P11 Dantec probe connected to a Streamline Pro anemometer with a frequency response up to 30 kHz. Data were acquired using

(t) = +∞ 0 h ηζ (τ ) η (t -τ ) dτ , (1)
where h ηζ (t) is the inverse Fourier transform of the transfer function H ηζ (f ), which is given by

H ηζ (f ) = ⟨S ηζ (f )⟩ ⟨S ηη (f )⟩ , (2)
where S ηζ is the cross-spectrum between the near-field sensor and the far-field observer, S ηη is the auto-spectrum of the near-field sensor, f is the frequency and ⟨•⟩ indicates an ensemble average performed on 550 blocks of 8192 samples.

(1) considers only positive τ , thus providing a causal estimation.

In what follows, we characterise the transfer function between near-field sensors η (t) and far-field observers ζ (t) as a function of jet Mach number, spatial location of the near-field sensors and radial distance between the jet and the plate. We consider both the HW and the microphone array as near-field sensors. Previous to that, we identify the frequency range for which installed jet noise is dominant by comparing the far-field spectra for both free-jet and installed configurations in the shielded side for different jet Mach numbers.

Figure 2 shows that the typical noise bump associated with JSI noise is dom-

inant in the St-number range St = [0.1, 0.8],
where the Strouhal number is calculated on the basis of the nozzle diameter and the jet velocity. Hence, all the results presented hereafter are obtained for time signals band-pass filtered in this frequency range. We remind the reader that the use of a band-pass filter might introduce a potentially significant time delay in the real-time control application and so this procedure should be used only for the off-line control kernel calculation. We further observe that the noise increase due to the JSI reduces with increasing Mach number. This trend is consistent with the results reported by [START_REF] Piantanida | Scattering of turbulent-jet wavepackets by a swept trailing edge[END_REF] and can be explained by the progressive dominance of the 'direct' jet noise over the installed jet noise scattered by the trailing edge as M increases, given that the two noise components scale with M 8 and M 5 , respectively [START_REF] Lighthill | On sound generated aerodynamically i. general theory[END_REF][START_REF] Williams | Sound generation by turbulence and surfaces in arbitrary motion[END_REF][START_REF] Piantanida | Scattering of turbulent-jet wavepackets by a swept trailing edge[END_REF].

For what concerns the near field of the jet, typical noise and velocity spectra are shown in appendix A.

Cross-statistical analysis

As shown by [START_REF] Maia | Real-time reactive control of stochastic disturbances in forced turbulent jets[END_REF], the coherence between sensors and observers is a key quantity where control effectiveness is concerned. We therefore use two-point coherence between near-field sensors η and far-field observers ζ as a means to characterise the transfer function. The coherence is defined as,

γ (f ) = |⟨S ηζ (f )⟩| 2 ⟨|S ηη (f ) |⟩⟨|S ζζ (f ) |⟩ , (3)
where S ζζ is the observer auto-spectrum. We use as near-field sensor signal η (t) both the azimuthal mode m = 0 extracted from the microphone array measurements and the velocity measured at different streamwise and radial positions. ference between hydrodynamic and acoustic components of the near pressure field, as studied by [START_REF] Coiffet | Coherent structures in subsonic jets: a quasi-irrotational source mechanism?[END_REF] and [START_REF] Guitton | Velocity dependence of the near pressure field of subsonic jets: understanding the associated source mechanisms[END_REF].

Microphone array measurements

We now consider the sensor/observer cross-statistics in the time domain by calculating the cross-correlation coefficient

ρ ηζ (τ ) = R ηζ (τ ) σ η σ ζ , (4)
where σ η and σ ζ are the standard deviations of near-field sensor and far-field observer time signals, respectively, and R ηζ (τ ) is the cross-correlation defined as

R ηζ (τ ) = +∞ -∞ η (t) ζ (t + τ ) dt.
(5)

τ H+A = r T E,η U c + r T E,ζ c ∞ , (6a)
τ A = r T E,ζ -r T E,η c ∞ , (6b)
where U c is the convection velocity taken as 0.8 U j , as supported by linear stability analysis calculation reported in B, and c ∞ is the ambient speed of sound. (2022) would allow such positions to be considered without deteriorating the control efficiency.

Hot-wire measurements

Motivated by the contribution of the back-scattered acoustic pressure to the coherence between near-field sensors and far-field observers and the problems that this would pose for closed-loop, reactive control (Sasaki et al, 2018b,a;[START_REF] Maia | Real-time reactive control of stochastic disturbances in forced turbulent jets[END_REF], we conducted a series of experimental tests using a HW probe to more deeply investigate the relation between the hydrodynamic component in the near field and the scattered sound in the far field. We remind the reader that a HW installed on the jet axis purely measures axisymmetric velocity fluctuations and the information from higher order modes become more significant when the HW is moved towards the shear layer of the jet. We previously verified that the HW effect on far-field noise is negligible for the Stnumber range and far-field microphone position we investigated herein, that is, St = [0.1, 0.8] and shielded side, respectively. These results are not reported for the sake of conciseness. Figure 6 shows the coherence maps between the far-field noise on the shielded side and the HW on the jet axis as a function of

St and streamwise position of the HW probe for jet Mach numbers M = 0.2, 0.3 and 0.4 and jet-plate distance H = 1D. We note that the coherence level is much lower than that observed between the m = 0 mode extracted from the near-field microphone array and the far-field noise, thus confirming that the back-scattered acoustic pressure makes a significant contribution to the coherence between the m = 0 mode and the far-field noise. The coherence between the HW and the far-field microphone is higher in the proximity of the TE consistent with the results obtained using the near-field microphone array for which the correlation peak for positions close to the TE occurred for time delays related to the hydrodynamic convection plus acoustic scattering phenomenon. We finally note that, consistent with what was observed using the microphone array, the coherence level decreases with increasing M . We now explore the coherence between HW and far-field noise as a function of the radial position of the HW. As an example, we select the jet Mach number M = 0.2, the jet-plate distance H = 1D and a streamwise position in the proximity of the TE at x/D = 1.68. Figure 7 shows the coherence map between the HW and the far-field microphone on the shielded side as a function of St and radial distance z, revealing that the coherence exhibits higher levels on the nozzle axis. This behaviour is consistent with the physics of jet installation noise, whose generation is mostly underpinned by the m = 0 mode [START_REF] Cavalieri | Scattering of wavepackets by a flat plate in the vicinity of a turbulent jet[END_REF], which is better captured when the HW probe is positioned on the jet axis. Moving away from the jet axis towards the shear layer, higherorder azimuthal modes dominate the jet dynamics, thus explaining the loss of coherence between the HW and far-field microphone signals observed for these radial positions.

The effect of the jet-plate distance on the coherence between HW and farfield microphone is illustrated in figure 8, which shows the coherence maps as a function of St and streamwise position of the probe for z/D = 0 and for jetplate distances H = 0.6D, 0.75D and 1D. We note that the coherence map becomes peakier for decreasing jet-plate distances, although no remarkable changes of the coherence level are appreciated. (1). Specifically, the estimation performed using only the information related to the hydrodynamic convection of the turbulent wavepacket plus acoustic scattering from the TE, that is, considering h ηζ for τ ≥ τ H+A , is hereafter denoted 'H+A'. The estimation achieved by considering the transfer function for τ > 0 is hereafter denoted 'T', where 'T' stands for 'total', given that it exploits both the information from the hydrodynamic convection plus acoustic scattering and acoustic back-scattering. This estimation can be defined as a causal estimation on the basis of a mere mathematical definition of causality as opposed to the 'H+A' estimation which is based on the notion of a phenomenological causality given that the hydrodynamic wavepacket is the cause of the scattered sound. Hence, the estimations performed using the portion of the transfer function for 0 < τ < τ H+A exploit phenomenological non-causal information involving the sound scattered by the TE. This may potentially pose a problem for control implementation given that the controller will be unable to act prior to the generation mechanism, i.e., the scattering at the TE. Nevertheless, as inferred by the results obtained by [START_REF] Bychkov | Plasma-based active closed-loop control of instability waves in unexcited turbulent jet. part 2. installed jet[END_REF] and [START_REF] Kopiev | Active control of jetplate interaction noise for excited jets by plasma actuators[END_REF] and as suggested by [START_REF] Sipp | Linear closed-loop control of fluid instabilities and noise-induced perturbations: a review of approaches and tools[END_REF], actuators can have a certain control effectiveness even when they act a little after the cause as a result of a certain spatio-temporal coherence of the scattering Figure 10 shows the comparison between the measured and estimated noise on the shielded side for M = 0.2, H = 1D and x/D = 0. As an example, we

Far-field noise estimation

Pressure-based estimations

show the results obtained considering the 'total' causal estimation. We note that the far-field noise appears to be accurately estimated for the streamwise position considered. In what follows, we will quantify the accuracy of the estimation performed.

Figure 11 shows the comparison between the measured and estimated farfield spectra on the shielded side for the same configuration considered above for all the types of estimation defined above. We note that the use of the hydrodynamic+acoustic component provides a poor estimation of the far-field for all the types of estimation exhibits a similar trend. Specifically, the total estimation shows the highest correlation level at the nozzle exit, the correlation then drops for axial positions in between 0.5D and 1D and then rises for larger streamwise positions. On the contrary, the H+A estimation exhibits a low correlation with the measured signal at the nozzle exit, a further drop up to x/D = 0.5 and then a rise with a maximum around x/D = 1.5, where the correlation level is comparable with that of the total estimation.

We remind the reader that high cross-correlation levels between two signals imply a shape similarity between the two signals but not necessarily their Jet-surface interaction noise estimation energy being the same. For this reason, we compute the ∆OASP L between the measured and estimated noise signals for all the streamwise positions of the near-field array as an additional metrics to further quantify the accuracy of the estimation performed. The OASP L is calculated only in the frequency band for which the JSI noise is dominant, that is, St = [0.1, 0.8]. The Band-Limited OverAll Sound Pressure Level (BLOASP L) is computed according to the formula

BLOASP L = 20 log 10 σ p ref , ( 7
)
where σ is the standard deviation of the pressure signal and p ref = 20µ P a is a reference pressure. The standard deviation is computed from the variance, which is evaluated as

σ 2 = f2 f1 P SD P a 2 /Hz df , (8)
where f 1 and f 2 are the frequency limits corresponding to the Strouhal number range [0.1, 0.8]. Figure 13 In conclusion, the characterisation of the estimation accuracy herein performed revealed that the near-nozzle region is dominated by the contribution of the scattered sound, whereas the causal component related to the hydrodynamic convection plus acoustic scattering emerges moving downstream in the jet plume. We finally note that, although the effect was not identified by the analysis we carried out, for large streamwise positions the scattering from the microphone array itself might play a role in the determination of the sensor/observer coherence and, hence, in the accuracy of the far-field noise estimation.

Velocity-based estimations

Figure 14 shows the ∆ BLOASP L between the measured and estimated farfield noise on the shielded side for different jet Mach numbers and jet-plate distances. The estimation performed using the HW positioned on the jet axis (which comprises only the hydrodynamic fluctuations) is much less accurate than that achieved using the microphone array. Specifically, the estimation discrepancy in dB from the measured data achieved using the HW is twice that obtained using the microphone array for the same M . We note that the

Conclusions

In this work we dealt with the real-time estimation of the noise radiated by a jet interacting with a flat plate. We considered the estimations obtained using linear transfer functions built from near-field sensor and far-field observer readings. We characterised the transfer function and the accuracy of the farfield noise estimations as a function of jet flow conditions, radial distance between the jet and the plate and streamwise position of the near-field sensors.

We considered both an azimuthal microphone array and a hot wire as near-field sensors. This analysis is essential to guide the positioning of near-field sensors for the implementation of real-time closed-loop control strategies of jet-surface interaction noise. We considered the coherence as a means to characterise the sensor/observer transfer function and as an indicator of the linearity of the jet installation noise phenomenon: the higher the coherence level, the more accurate the noise estimation obtained using linear transfer functions.

We observed that the sensor/observer coherence level is much higher when using the microphone array rather than the HW probe as near-field sensor.

Specifically, the coherence as well as the noise estimation accuracy is high for streamwise positions of the microphone array close to the nozzle exit, where the coherence is mainly dominated by the scattered acoustic pressure information.

This feature could potentially represent an issue when implementing real-time control given that the control kernel will be calculated on the basis of the effect that is sought to control. The noise estimations performed using the linear transfer function revealed an accuracy of 1 dB and 7 dB in terms of OASP L

for the lowest and highest jet Mach numbers, respectively. High and moderate coherence levels were found in the vicinity of the trailing edge of the surface when using as near-field sensors the microphone array and the hot-wire probe, respectively. For these axial positions the coherence is mainly dominated by the hydrodynamic component. The noise estimation accuracy achieved using the microphone array is comparable to that observed for streamwise positions close to the nozzle exit and drastically drops when hot-wire readings are used to perform the estimation. We finally observed a reduction of the coherence level, and hence a deterioration of the accuracy of the noise estimation, for increasing jet Mach number.

We finally note that, in order to implement closed-loop control and achieve a reduction of the observer intensity, the characteristic time scale of the loop controller has to be smaller of the characteristic time scale of the flow phenomenon sought to control. This work reveals that, in the case of installed jet noise control, the characteristic flow time scale does not only depend on the spatial location of the near-field sensors but also on the nature of the information -either hydrodynamic or acoustic -exploited to perform the observer estimation and drive the actuation process. As suggested by [START_REF] Bychkov | Plasma-based active closed-loop control of instability waves in unexcited turbulent jet. part 2. installed jet[END_REF], [START_REF] Kopiev | Active control of jetplate interaction noise for excited jets by plasma actuators[END_REF] and [START_REF] Sipp | Linear closed-loop control of fluid instabilities and noise-induced perturbations: a review of approaches and tools[END_REF], actuators could however have a certain effectiveness even if driven by phenomenologically noncausal acoustic information as a consequence of the spatio-temporal coherence of the hydrodynamic wavepackets. Furthermore, in the case of actuators placed at the trailing edge of the surface, the use of the Wiener-Hopf control approach developed by [START_REF] Martini | Resolvent-based tools for optimal estimation and control via the wiener-hopf formalism[END_REF] could permit to place near-field sensors at streamwise positions close to the surface edge where the causal hydrodynamic information was found to be dominant significantly reducing the causality issues associated with the vicinity between sensors and actuators. was used to non-uniformly distribute the 350 grid points along the radial direction to efficiently solve the shear layer of the jet. The analytic hyperbolic tangent profile proposed by [START_REF] Lesshafft | Linear impulse response in hot round jets[END_REF],

U = 1 2 M 1 + tanh R 4δ R r - r R , (B2)
was used as base flow, where R = 0.5 is the radius and δ is the shear-layer thickness. Consistent with HW measurements, the shear-layer thickness was chosen so that R/δ = 25. Figure B3 shows the evolution along the St of the K-H mode phase speed for M = 0.15, 0.35, 0.55 and 0.75. Consistent with what previously reported in the literature (see e.g. [START_REF] Michalke | Survey on jet instability theory[END_REF]), the phase speed is frequency dependent and its value decreases for increasing M . The plot shows that the value of 0.8 U j can be accurately adopted as a preliminary and simplified evaluation of the convection speed independent of both frequency and flow conditions. Furthermore, the results reported in figure B3 provide a possible explanation for the reduction of noise estimation accuracy observed

  3 is dedicated to the characterisation of the sensor/observer transfer function and the estimation of far-field noise based on sensor readings. Conclusions are provided in 4. 2 Experimental set-up An experimental test campaign has been performed in the Bruit & Vent anechoic facility at Prométée, the technological platform of Institut Pprime in Poitiers. The anechoic chamber is 12.6 × 10.6 × 7.85 m 3 in size and is equipped with absorbing acoustic foam leading to a cut-off frequency of 200 Hz. The feed-line consists of a compressed dry-air duct supplied by a screw compressor delivering a continuous mass flow rate up to 1 kgs -1 at 3.5 bar. An electrically driven valve downstream of the compressor permits regulation of the jet velocity by controlling the incoming mass flow. A cooling system is installed downstream of the regulating valve to provide a jet flow in isothermal conditions. Stagnation temperature and pressure in the plenum chamber are monitored using a thermocouple and a pressure transducer, respectively. The jet issues from a convergent nozzle of diameter D = 0.05 m. The boundary layer inside the nozzle is tripped by a strip of carborundum particles placed 2.5 diameters upstream of the nozzle exit so as to generate an initially turbulent jet.

Fig. 1

 1 Fig. 1 Sketch and picture of the experimental setup and sensors disposition. (a) near-field microphone array, (b) hot wire installation.

Fig. 2

 2 Fig. 2 Comparison between far-field spectra in free-jet and installed configurations on the shielded side. Blue line refers to the free-jet configuration, red line to the installed one. The installed configuration refers to the jet-plate distance H = 1D.

Figure 3

 3 Figure 3 shows the coherence maps as a function of Strouhal number and axial position of the near-field array between the far-field noise measured on the shielded side and the near-field azimuthal mode m = 0 for jet Mach numbers M = 0.15, 0.35, 0.55 and 0.75 and jet-plate distance H = 1D. The coherence level is the highest close to the nozzle exhaust and in the vicinity of the trailing edge of the plate, suggesting that these regions are best suited for sensors placement in the implementation of real-time control. Furthermore, we note that the coherence level decreases significantly for increasing jet Mach number, this trend follows that of the reduction of JSI noise shown in figure 2. The localised 'holes' in the coherence maps can be attributed to destructive inter-

Fig. 3

 3 Fig. 3 Coherence maps as a function of St and x/D between far-field noise on the shielded side and near-field azimuthal mode m = 0 for different jet Mach numbers and jet-plate distance H = 1D.

Figure 4

 4 Figure4shows the cross-correlation coefficient map as a function of the time delay and the near-field array axial position between the far-field noise on the shielded side and the azimuthal mode m = 0. As an example, we show the map for M = 0.35 and H = 1D. We note the presence of a correlation peak for a near-constant time delay for axial positions close to the nozzle exhaust and a correlation peak, whose intensity increases as the axial position approaches the trailing edge, with a variable time delay as a function of the streamwise location considered. This behaviour can be ascribed to the fact that the nearfield microphones sense both the signature of the hydrodynamic wavepacket and the scattered acoustic pressure from the TE. Hence, the correlation pattern described above is the result of two different phenomena characterised by different time delays: (i) the hydrodynamic convection of the wavepacket from the near-field sensors to the TE and the subsequent acoustic scattering from the TE to the far-field microphones; (ii) the scattering of the acoustic pressure from the TE to both the near-field sensors and the far-field observers.These phenomena are schematically represented in figure5. Denoting the distance between the near-field array and the trailing edge r T E,η and the distance

  These characteristic time delays are superimposed on the cross-correlation map in figure4and found to match the time delays of the correlation peaks. The estimated time delays shed light on the nature of the coherence in the regions close to the nozzle exit and in the vicinity of the trailing edge. Specifically, the high coherence observed close to the nozzle exhaust is mainly related to the back-scattered acoustic pressure from the TE, whereas the high correlation level detected in the proximity of the TE is associated with the hydrodynamic wavepacket. This component is the one we would be interested in to control installed jet noise using a feed-forward closed-loop control strategy where actuators placed on the TE are driven by near-field upstream sensors to reduce the noise diffracted by the surface. Hence, the exploitation of the hydrodynamic wavepacket convection information to drive the control would lead to the installation of near-field sensors in the close vicinity of the TE and, thus, of the actuators. Despite this might generate non-causal components in the control kernel, the Wiener-Hopf control approach developed byMartini et al

Fig. 4

 4 Fig. 4 Cross-correlation coefficient map as a function of time delay and streamwise position of the near-field array between far-field noise on the shielded side and near-field azimuthal mode m = 0 for jet Mach number M = 0.35 and jet-plate distance H = 1D. Green dashed line refers to the acoustic scattering time delay (τ A ), black dashed line to the hydrodynamic convection plus acoustic scattering time delay (τ H+A ).

Fig. 5

 5 Fig. 5 Scheme of the two different phenomena measured by the near-field sensors and farfield observers: (i) hydrodynamic convection of the wavepacket from the near-field sensors to the TE plus acoustic scattering from the TE to the far-field observers; (ii) acoustic scattering from the TE to both the far-field observers and near-field sensors.

Fig. 6

 6 Fig. 6 Coherence maps as a function of St and streamwise position of the HW probe between far-field observer in the shielded side and velocity fluctuations on the jet axis.

Fig. 7

 7 Fig. 7 Coherence map as a function of St and transversal position of the HW probe between far-field observer in the shielded side and velocity fluctuations for M = 0.2 and x/D = 1.68.

Fig. 8

 8 Fig. 8 Coherence maps between HW and far-field microphone in the shielded side as a function of St and streamwise position x/D for jet Mach number M = 0.2 and HW transversal position z/D = 0.

Figure 9

 9 Figure 9 shows an example of the transfer function h ηζ (t) in the time domain between m = 0 mode and far-field noise on the shielded side for M = 0.2,

Fig. 9

 9 Fig. 9 Transfer function in the time domain between near-field m = 0 and far-field noise on the shielded side for M = 0.2, streamwise position x/D = 0.84 and jet-plate distance H = 1D. The time delays corresponding to the two causal estimations (total and 'H+A') considered in the present work are superimposed on the plot. The dashed green line indicating the back-scattering time delay τ A defined in (6) is indicated as well for completeness.

Fig. 10

 10 Fig. 10 Comparison between measured and estimated far-field noise on the shielded side for M = 0.2, H = 1D and x/D = 0. Blue line refers to the measured signal, red line to the estimated one. Both signals are normalised by their standard deviation σ. The causal estimation is performed considering the transfer function for positive τ .

Fig. 11

 11 Fig. 11 Comparison between measured and estimated far-field noise spectra on the shielded side for M = 0.2, H = 1D and x/D = 0. Estimation is performed using the azimuthal microphone array as near-field sensor. Blue line refers to the measured data, red line to the total causal estimation and black line to the H+A-based estimation.

Fig. 12

 12 Fig.12Cross-correlation coefficient peak between measured and estimated far-field noise signals in the shielded side. Estimation is performed using the microphone array as nearfield sensor. Red ⋄ refer to total estimation and black △ to H+A estimation.

  shows the ∆BLOASP L between measured and estimated far-field noise for all the streamwise positions of the near-field microphone array, H = 1D and M = 0.15, 0.35, 0.55 and 0.75. The total estimation shows the maximum accuracy close to the nozzle exit and its accuracy decreases with the increasing Mach number. Concerning the estimation using the H+A component, the accuracy is poor for streamwise positions close to the nozzle exit and is comparable with that obtained using the total causal component for 1 ≤ x/D ≤ 1.5. The estimation accuracy degradation observed for M = 0.15 and large x/D is ascribed to an under-estimation of the convection speed (as shown in B) and a likely local acceleration of the grazing flow due to Coanda effect, as previously suggested by[START_REF] Cavalieri | Scattering of wavepackets by a flat plate in the vicinity of a turbulent jet[END_REF] and[START_REF] Proença | Investigation into the turbulence statistics of installed jets using hot-wire anemometry[END_REF].

Fig. 13 ∆

 13 Fig. 13 ∆ BLOASP L between measured and estimated JSI noise in the St-number range [0.1, 0.8]. Red ⋄ refer to total estimation and black △ to the H+A estimation.

Fig. 14 ∆

 14 Fig. 14 ∆ BLOASP L between measured and estimated far-field noise on the shielded side for different jet-plate distances and jet Mach numbers. Estimations are computed using the HW as near-field sensor.

Figure 15

 15 Figure15shows the cross-correlation coefficient peak between estimated and measured far-field noise signals on the shielded side for different jet Mach numbers and jet-plate distances. The results are consistent with the estimation accuracy in dB shown in figure14. We note that the maximum correlation level detected is slightly lower than 0.6 close to the TE. These moderate correlation levels, together with the 5 dB estimation accuracy at best observed above, imply that while elimination of JSI noise would not be possible, a sound reduction should be achieved using a closed-loop control strategy. The quantification of this noise reduction by implementing a linear reactive control strategy will be the subject of a future work.

Fig. 15

 15 Fig. 15 Cross-correlation coefficient peak between measured and estimated far-field noise in the shielded side for different jet Mach numbers and jet-plate distances. Estimations are computed using the HW as near-field sensor.

Fig. A1

 A1 Fig. A1 Streamwise evolution of the pressure spectra of the near-field m = 0 mode for M = 0.55 and jet-plate distance H = 1D.

Fig. A2

 A2 Fig. A2 Streamwise evolution of the velocity spectra on the nozzle axis for M = 0.4 and jet-plate distance H = 1D.

Fig

  Fig. B3 Kelvin-Helmholtz phase speed evolution as a function of the Strouhal number calculated from linear stability analysis. The stability problem is solved using as base flow the hyperbolic tangent velocity profile reported in (B2) with R/δ = 25. Results refer to azimuthal mode m = 0 and isothermal flow conditions.

Acknowledgments. This work has been supported by EU project DJINN funded by the European Union's Horizon 2020 research and innovation programme under grant agreement No 861438.

Redouane Kari is acknowledged for the support in the experimental setup preparation.

Funding This work has been supported by EU project DJINN funded by the European Union's Horizon 2020 research and innovation programme under grant agreement No 861438.

Declarations

Author contribution M.M. and A.L. carried out the experimental tests, M.M. performed the data analysis, M.M. and P.J. discussed the data interpretation, M.M. wrote the first draft of the manuscript, M.M. and P.J. reviewed and edited the text of the manuscript. All the authors read and approved the final version of the manuscript.

Conflict of interest

The authors report no conflict of interest.

Ethical approval

Not applicable.

Informed consent

Not applicable.

Appendix A Near-field spectra

We here characterise the spectral content of the near-field m = 0 mode and streamwise velocity measured by the azimuthal microphone array and the hot wire, respectively. Figure A1 shows the streamwise evolution of the near-field spectra of the m = 0 mode for jet Mach number M = 0.55 and jet-plate distance H = 1D. Axial positions x/D = 0.06, 0.44, 0.82, 1.2, 1.58 and 1.96 are herein considered. We note that the axisymmetric mode exhibits the typical spectral behaviour of the near pressure field with a low-frequency hump related to the dominance of hydrodynamic fluctuations (see among many [START_REF] Arndt | The proper orthogonal decomposition of pressure fluctuations surrounding a turbulent jet[END_REF] and [START_REF] Mancinelli | Wavelet decomposition of hydrodynamic and acoustic pressures in the near field of the jet[END_REF]). Consistent with the typical instabilitywave behaviour, the energy hump amplitude increases for large x/D and is less significant for small x/D, thus confirming the results presented above where we show that the acoustic contribution is dominant for axial positions in the vicinity of the nozzle exhaust.

Figure A2 shows the streamwise evolution of the velocity spectra on the nozzle axis for M = 0.4, H = 1D and same axial positions considered above.

The spectra are normalised using as a reference time scale D/U j and as a reference velocity U j . Consistent with the near-field pressure spectra evolution shown above, we note an energy hump associated with the Kelvin-Helmholtz (K-H) instability for x/D ≥ 1.5 and for St ≈ [0.3, 0.7], that is, the most amplified frequencies of the axisymmetric K-H wavepacket [START_REF] Michalke | Survey on jet instability theory[END_REF].

Appendix B Convection velocity evaluation

We here apply local-parallel flow linear stability theory to a finite thickness model to evaluate the convection speed of the hydrodynamic wavepacket via the phase speed of the K-H mode. Writing the linearised Euler equations in terms of pressure, we obtain the compressible Rayleigh equation, which, for an isothermal flow of the kind we consider herein, reads


Mohamed-Amine Guerboukha

Matthieu Petit
email: matthieu.petit@univ-amu.fr

Aurélie Spiesser

Alain Portavoce

Omar Abbes

Vasile Heresanu

Sylvain Bertaina

Cyril Coudreau

Lisa Michez

Tuning the Mn 5 Ge 3 and Mn 11 Ge 8 thin films phase formation on Ge(111) via growth process

Keywords: Thin film, Germanide, Phase formation, epitaxy, Molecular beam epitaxy, Mn 5 Ge 3, Mn 11 Ge 8

We have studied the stability of manganese germanide thin films grown on Ge(111) substrates by three different growth methods: solid phase epitaxy, reactive deposition epitaxy and co-deposition.

By combining X-ray diffraction and magnetic measurements, we demonstrate that we can form either Mn 5 Ge 3 or Mn 11 Ge 8 thin films depending on the growth processes. In the case of solid phase epitaxy, we explain the phase formation sequence by taking into consideration the kinetic and thermodynamic processes involved. Tuning phase formation of the manganese germanide thin films was determined using in situ X-ray diffraction and the effect of the thin film thickness on the Mn 5 Ge 3 thermal stability was investigated.

Introduction

Magnetic thin films are extensively used in data storage, magnetic sensors, and other spintronics technologies [START_REF] Vedmedenko | The 2020 magnetism roadmap[END_REF]. To achieve the desired magnetic properties, a precise control of phase formation during thin film growth is usually required. The driving forces that govern phase formation include several factors such as nucleation barrier, interface energy and diffusion flux [START_REF] Gösele | Growth kinetics of planar binary diffusion couples: "Thin-film case" versus "bulk cases[END_REF][START_REF] Gaudet | Thin film reaction of transition metals with germanium[END_REF]. Also phases generally appear sequentially and not simultaneously, and also strongly depend on the film thickness [START_REF] Churms | Study of Pd/Ge interaction in a lateral diffusion couple by microbeam Rutherford backscattering spectrometry[END_REF]. Thus, understanding the mechanisms of phase formation as a function of the film thickness is an important advantage for device applications.

The interest in the manganese germanide compounds mainly stems from the Mn 5 Ge 3 alloy which exhibits interesting features for spintronic applications. This ferromagnetic compound has a Curie temperature (T C ) of 296 K that can be increased by carbon insertion up to 450 K. Its hexagonal D8 8type crystal structure (a hex = 7.18 Å, c hex = 5.05 Å, prototype structure Mn 5 Si 3 ) enables the epitaxial growth of thin films on Ge(111) substrates with the following relationships: Mn 5 Ge 3 (0001)//Ge(111) and Mn 5 Ge 3 //Ge with a lattice mismatch of -3.75% [START_REF] Zeng | Epitaxial ferromagnetic Mn5Ge3 on Ge(111)[END_REF][START_REF] Gajdzik | Strongly enhanced Curie temperature in carbon-doped Mn5Ge3 films[END_REF][START_REF] Tawara | On the magnetic anisotropy of single crystal of Mn5Ge3[END_REF][START_REF] Spiesser | Magnetic anisotropy in epitaxial Mn5Ge3 films[END_REF]. Studies about the magnetic and electrical properties and the several growth methods of Mn 5 Ge 3 thin films on Ge(111) substrates have demonstrated the assets of this intermetallic alloy as a ferromagnetic electrode for spin injection for instance [START_REF] Spiesser | Large spin accumulation voltages in epitaxial Mn5Ge3 contacts on Ge without an oxide tunnel barrier[END_REF][10][START_REF] Petit | Very low-temperature epitaxial growth of Mn5Ge3 and Mn5Ge3C0.2 films on Ge(111) using molecular beam epitaxy[END_REF][START_REF] Olive-Mendez | Epitaxial growth of Mn5Ge3/Ge(111) heterostructures for spin injection[END_REF][START_REF] Michez | Magnetic reversal in Mn5Ge3 thin films: an extensive study[END_REF]. Moreover Mn 11 Ge 8 , sometimes referred as Mn 3 Ge 2 , is another manganese germanide phase exhibiting interesting magnetic properties. Mn 11 Ge 8 is a non-collinear antiferromagnet below 150 K and a ferromagnet up to 274 K [START_REF] Yamada | Magnetic properties of intermetallic compoung Mn11Ge8[END_REF]. Its orthorhombic crystal structure is closed to the Mn 5 Ge 3 structure with a = 13.21 Å, b = 15.88 Å, c = 5.09 Å (prototype structure Cr 11 Ge 8 ). Theoretical predictions suggest the possibility of epitaxial growth on Ge(111) [START_REF] Arras | Phase diagram, structure, and magnetic properties of the Ge-Mn system: A first-principles study[END_REF].

The present work combines X-ray diffraction (XRD) and magnetic measurements to study the formation and stability of epitaxial Mn 5 Ge 3 and Mn 11 Ge 8 thin films on Ge(111) substrates via different growth methods: solid phase epitaxy (SPE), reactive deposition epitaxy (RDE) or codeposition epitaxy. Emphasis is given on the SPE process to determine the formation sequence of germanide phases and the stability of the Mn 5 Ge 3 films as a function of the film thickness.

Growth methods and experimental details

Thin epitaxial films of Mn 5 Ge 3 and Mn 11 Ge 8 were grown on high-purity Ge(111) substrates in a molecular beam epitaxy chamber with a base pressure better than 2 × 10 -7 Pa. To obtain a flat, smooth, and oxide-free surface, we cleaned the Ge substrates following this method: first, the samples were immersed in an acetone solution and sonicated for 10 minutes, then immersed in an isopropanol alcohol ultrasonic bath for another 10 minutes in order to degrease the surface and finally immersed in a last bath containing a 50% HF acid -50% H 2 O mixture to remove the native oxide. Subsequently, we performed in situ thermal annealing for several minutes at 720 K followed by flash annealing at 1020 K to remove the residual Ge surface oxide, which can be formed during sample transfer into high vacuum. Eventually a 50 nm thick Ge buffer is deposited to bury any remaining surface impurities and to ensure that the starting surface before the deposition is perfectly cleaned. In this present article, the films were grown using three techniques, schematized on fig. 1.

The first technique is the SPE technique used to synthesize the first Mn 5 Ge 3 thin films on a Ge(111) substrate. It consists in the deposition of a Mn layer at room temperature (RT) onto the Ge(111) substrate, followed by a thermal anneal at a temperature around 720 K to activate the interdiffusion of Mn and Ge atoms and the nucleation of the phases. The second method is the RDE: the substrate is heated up to 720 K and kept at this temperature during the deposition of Mn. The third growth technique is the co-deposition of Mn and Ge with the 5:3 stoichiometric flux and at a given substrate temperature. The last two growth methods enable the controlled nucleation of the phase formation [START_REF] Vantomme | Concentrationcontrolled phase selection of silicide formation during reactive deposition[END_REF]. The chamber is fitted out with a reflection high-energy electron diffraction (RHEED) system to monitor and check in situ these growth processes. Mn and Ge atoms are evaporated from Knudsen cell. The flux are carefully calibrated using a quartz crystal microbalance and RHEED oscillations. Structural characterizations were carried out by XRD. XRD patterns were collected using a rotating anode Rigaku RU200BH (non-monochromatic CuKα1 radiation λ=1.541Å) equipped with a 2D Mar345 detector. Data were collected over a 2θ range of 20-65°. The XRD curves (2θ versus intensity) are drawn by integrating the intensity of the 2D XRD maps.

Magnetic characterizations of the grown films were performed using a superconducting quantum interference device magnetometer Quantum Design MPMSXL in a temperature range varying from 5 to 350 K. The temperature dependence of the magnetization has been recorded using a constant 0.1 T external in-plane magnetic field.

Mn 5 Ge 3 and Mn 11 Ge 8 heats of formation

Phase formation has been widely studied in the case of thin films growth taking into account equilibrium, diffusion processes as well as interface and stress energy [START_REF] Walser | First phase nucleation in silicon-transition-metal planar interfaces[END_REF][START_REF] Ottaviani | Metallurgical aspects of the formation of silicides[END_REF][START_REF] Pretorius | Prediction of phase formation sequence and phase stability in binary metal-aluminum thin-film systems using the effective heat of formation rule[END_REF][START_REF] Schutter | Texture in thin film silicides and germanides: A review[END_REF][START_REF] Portavoce | Theoretical investigation of the influence of reaction and diffusion kinetics upon thin-film reactive diffusion[END_REF][START_REF] Bené | A kinetic model for solid-state silicide nucleation[END_REF]. According to the bulk Ge-Mn phase diagram, Mn 11 Ge 8 is the Ge richer -non-congruent-phase which undergoes a peritectic transition into Mn 5 Ge 3 and liquid around 1042 K. Mn 5 Ge 3 is a congruent phase melting at ~1245 K and the closest one to the lowest eutectic (993 K, Ge concentration = 57%at) [START_REF] Gokhale | The Ge-Mn (Germanium-Manganese) System[END_REF]. This lowest eutectic is also the eutectic closest to the center of the Mn-Ge diagram.

The effective heat of formation (EHF) model is a useful tools to explain the phase formed according to the growth techniques [START_REF] Pretorius | Phase sequence of silicide formation at metal-silicon interfaces[END_REF][START_REF] Pretorius | Thin film compound phase formation sequence: An effective heat of formation model[END_REF][START_REF] Pretorius | Compound phase formation in thin film structures[END_REF]. It was initially developed to predict the sequence of the phase formation during film growth in the case of a solid phase interaction. The driving force of a reaction is the decrease of the Gibbs free energy (ΔG = ΔH -TΔS, T the temperature, ΔH and ΔS the enthalpy and entropy variations respectively). As we considered only solid-state reaction, ΔS is negligible, thus ΔH is a relevant approximation for the variation of ΔG. However, the local concentrations of atoms at the interface where the reaction occurs must be taken into account since it represents the real amount of atoms locally available for the reaction. These local concentrations depend on the different atoms flux caused by the diffusion phenomena and by the evaporation flux from the effusion cells. The expression of the effective heat of formation ΔH' can be found in ref. [START_REF] Pretorius | Phase sequence of silicide formation at metal-silicon interfaces[END_REF]. The heat of formation ΔH 0 of the two germanide intermetallic compounds are close: Berche et al. proposed ΔH 0 = -17.5 ± 0.3 kJ.(mol.at) -1 for Mn 11 Ge 8 and ΔH 0 = -18.4 ± 0.3 kJ.(mol.at) -1 for Mn 5 Ge 3 [START_REF] Berche | Thermodynamic modeling of the germanium-manganese system[END_REF]. Injecting these values into the effective heat of formation (EHF) model, we can draw the triangular diagram (cf. Fig. 2) representing the effective heats of formation ΔH ' of Mn 5 Ge 3 and Mn 11 Ge 8 as a function of the Mn concentration.

If the relative concentration of Mn is higher than 58.5%at, the formation of the Mn 5 Ge 3 phase is more favorable than the Mn 11 Ge 8 one, germanium being the limiting element in the phase formation. These results are consistent with the first principle study on the Mn-Ge systems made by Arras et al. [START_REF] Arras | Phase diagram, structure, and magnetic properties of the Ge-Mn system: A first-principles study[END_REF]. At the growth interface, the local concentrations of Mn and Ge atoms can be changed according to the flux of diffusion of both species or according to the deposition flux, thus moving the equilibrium. However, the EHF model does not take into account the nucleation barrier of the phases nor the interface energy. These aspects depend on the congruency or non-congruency of the phases near the lowest eutectic. Usually, a low eutectic means low interfacial energy and low nucleation barrier. The Walser-Bené (W-B) rule enriched by Ronay and Bené deals with these considerations. It was established to predict the first nucleated phase and the subsequent one at planar interfaces [START_REF] Walser | First phase nucleation in silicon-transition-metal planar interfaces[END_REF][START_REF] Bené | A kinetic model for solid-state silicide nucleation[END_REF][START_REF] Ronay | Reinvestigation of first phase nucleation in planar metal-Si reaction couples[END_REF]. Merging the W-B rule and the EHF model, the first phase to form at a planar metal-germanium interface in a solid state reaction should be the congruent phase neighboring the eutectic closest to the center of the phase diagram (in at%) and with the lowest effective heat of formation. This phase is Mn 5 Ge 3 in the case of a Mn thin film on a Ge substrate according to the binary Mn-Ge phase diagram.

Results and discussion

Growth modes and phase formation

For each growth modes, the initial surfaces consist of the Ge buffer layer ones. These surfaces are 2D and are checked by RHEED prior to any further growth processes. The in situ RHEED monitoring of thin films synthesized according to the different growth methods show that the corresponding films surfaces remain 2D. Mn 5 Ge 3 thin films were historically first synthesized using the technique of the SPE [START_REF] Zeng | Epitaxial ferromagnetic Mn5Ge3 on Ge(111)[END_REF]. For 70 nm thick films and an annealing temperature of 720 K, this technique produces mono-crystalline thin films where only the Mn 5 Ge 3 phase is detected either using XRD measurements (fig. 3 a) and g)) or magnetization versus temperature (M(T)) technique (fig. 3 d)). The image of Fig3.g) is a typical 2D-XRD map recording with the Mar345 2D detector and is consistent with a single crystal Mn 5 Ge 3 layer with the c-axis normal to the sample surface, the position of the diffraction spots being in good agreement with the reflections (as indicated on the map) mentioned in the reference card of Mn 5 Ge 3 (ICSD-01-089-4887). Especially the peak at 2θ = 35.48° is attributed to the Mn 5 Ge 3 (002) plans. The M(T) curve gradually decreases to reach almost after 299 K which is typical of a ferromagnetic material with a T C = 299 K.

On the contrary, in the case of the RDE growth mode, XRD shows the predominantly existence of the Mn 11 Ge 8 phase with the (002) peak at 2θ = 35.40°. M(T) analysis displays an antiferromagnetic transition at 145 K and two ferromagnetic transitions at 270 and 293 K, attributed to the coexistence of the Mn 5 Ge 3 and Mn 11 Ge 8 compound in the thin film. As for the SPE synthesis, the W-B rule prevails at the interface between the Ge substrate and the Mn metallic film: Mn 5 Ge 3 nucleates first, being the closest congruent phase to the eutectic in the Mn-Ge binary diagram. The RDE involves two concomitant phenomena, the interdiffusion between the Ge atoms of the substrate and the atoms of the Mn film as well as the phase nucleation. The Mn flux emitted by the Knudsen cell being rather low (~0.3 nm.s -1 ), every Mn atoms reaching the sample surface is immediately consumed: the nucleation of the phase with the lowest Mn concentration is favored, leading to the formation of Mn 11 Ge 8 as predicted by the values of the effective enthalpy of formation. The deposition rate which determines the supply of Mn atoms control the selection of the formed phase.

The detection of the Mn 5 Ge 3 phase is explained by the concept of the critical thickness of the first forming phase before a second phase could start growing [START_REF] Heurle | Kinetics of formation of silicides: A review[END_REF][START_REF] Gas | Formation of silicide thin films by solid state reaction[END_REF][START_REF] Portavoce | Physical origin of thickness-controlled sequential phase formation during reactive diffusion: Atomistic modeling[END_REF]. As the deposition time increases, the thickness of the Mn 11 Ge 8 thin film also increases. The Ge atoms coming from the substrate then diffuse through this film which slows down the Ge supply at the now Mn 11 Ge 8 surface. The local relative concentration shifts toward Mn richer environment, favoring the nucleation of Mn 5 Ge 3 according to fig. 2.

Regarding the co-deposition, two thin films were synthesized using two different substrate temperatures: RT and 720 K. In the case of T sub = RT, XRD and M(T) measurements only reveal the presence of the Mn 5 Ge 3 phase as a mono-crystalline thin film. The accurate control of the stoichiometric ratio of the Mn and Ge fluxes combined with the low interface energy (γ Mn5Ge3/Ge(111) = 0.53 J.m -2 ) of the Mn 5 Ge 3 /Ge(111) system promote the Mn 5 Ge 3 phase formation [START_REF] Arras | Phase diagram, structure, and magnetic properties of the Ge-Mn system: A first-principles study[END_REF][START_REF] Ronay | Reinvestigation of first phase nucleation in planar metal-Si reaction couples[END_REF][START_REF] Arras | Étude théorique de la structure et de la stabilité des alliages GeMn dans le cadre de la spintronique. Un prototype de semiconducteur magnétique confronté aux résultats expérimentaux[END_REF]. At T sub = 720 K, the Mn 11 Ge 8 phase is detected and the thin film is polycrystalline. The peak around 2θ = 35° exhibits two contributions attributed to the (002) plans of Mn 5 Ge 3 and Mn 11 Ge 8 . Like the RDE experiments, the M(T) curve shows the antiferromagnetic transition around 150 K and two Curie temperature around 270 and 295 K indicating the presence of both Mn 5 Ge 3 and Mn 11 Ge 8 alloys.

Since the deposition flux of Mn and Ge atoms are carefully kept in the Mn 5 Ge 3 stoichiometric ratio, the diffusion of the Ge atoms coming from the substrate is a governing factor explaining the difference since at 720 K the Ge flux of diffusion being higher, the local Ge concentration at the substrate/thin film interface leads to the preferential nucleation of Mn 11 Ge 8 .

Sequence of the formation of the Mn 5 Ge 3 and Mn 11 Ge 8 phase in the SPE case

A 50 nm thick Mn film was deposited on a Ge(111) substrate, followed by an isochronic thermal annealing at 5 K.min -1 up to 860 K. The normalized and integrated intensities of the Mn(330), The evolution of the peaks intensities confirms the sequential phase formation. Mn 5 Ge 3 is the first phase to be nucleated at the sub-eutectic temperature of 500 K. These observation can be compared to the same experiment conducted on a Ge(001) substrate [START_REF] Wittmer | The first phase to nucleate in planar transition metalgermanium interfaces[END_REF]. The shape of the variation of the Mn 5 Ge 3 (002) peak intensity exhibits a linear part from 500 K to 600 K. According to kinetic Monte Carlo simulation, this linear behavior may be explained by the prevalence of the nucleation process on other processes such as diffusion ones [START_REF] Portavoce | Theoretical investigation of the influence of reaction and diffusion kinetics upon thin-film reactive diffusion[END_REF]. In solid phase reaction, intermixing at the interface is expected to take place at the concentration of the central eutectic point. At this concentration, Mn 5 Ge 3 is not the compound with the lowest effective heat of formation (fig. 2) but is the closest congruent phase to the eutectic meaning the easiest phase to nucleate. The nucleation is the driving phenomenon in the formation of Mn 5 Ge 3 in the case of the SPE growth mode. Earlier works have studied the SPE growth of submonolayer to one monolayer thick Mn on Ge(111) [START_REF] Zeng | Initial stages of Mn adsorption on Ge(111)[END_REF]. The authors showed that the first formed Mn:Ge islands are composed of Mn 5 Ge 3 except for any other phase. As the temperature increases and reaches 740 K, the Mn 5 Ge 3 film starts to decompose and Mn 11 Ge 8 starts to form. The growth of a new phase occurs only it reduces the Gibbs free energy of the whole system. In the experimental range of temperature, diffusion processes are non-negligible. The finite size of the Mn-rich film compared to the Ge(111) substrate may induce an asymmetric Mn/Ge interdiffusion profile leading to a Ge enrichment at the interface and thus to Mn 11 Ge 8 becoming more favorable. In order to bring elements of discussion, we can mention that the Manganese diffusion in Ge(001) substrate is vacancy-mediated with a diffusivity close to the Ge self-diffusivity (the activation energy is around 2.37 eV) [START_REF] Portavoce | Manganese diffusion in monocrystalline germanium[END_REF]. A low diffusion flux of Mn atoms into the Ge substrate is thus expected. On the contrary the epitaxial relationship between the Mn 5 Ge 3 thin films and Ge(111) leads to chains of octahedral interstitial sites along the Mn 5 Ge 3 [001] direction i.e. normal to the surface of the samples [START_REF] De Padova | Mn5Ge3 films grown on Ge(1 1 1)-c(2 × 8)[END_REF]. These chains may establish preferential diffusion path for the Ge atoms from the substrate to the thin film. Previous studies dealt with the sequence of phases formation of a Mn film deposited on an amorphous Ge (a-Ge) layer [START_REF] Abbes | Phase formation during Mn thin film reaction with Ge: Self-aligned germanide process for spintronics[END_REF]. The comparison between fig. 4 and fig. 2 of ref. [START_REF] Abbes | Phase formation during Mn thin film reaction with Ge: Self-aligned germanide process for spintronics[END_REF] shows several differences. First the kinetics of the formation of Mn 5 Ge 3 is faster on an a-Ge layer than on a single crystalline Ge(111) substrate. The faster Mn 5 Ge 3 growth rate on a-Ge can be explained by the different diffusion paths used by Ge and Mn atoms in a polycrystalline Mn 5 Ge 3 layer grown on a-Ge where grain boundaries diffusion may take place and in a monocrystalline Mn 5 Ge 3 layer grown on Ge(111) where the lattice diffusion process is prevailing. Secondly the range of stability of the Mn 5 Ge 3 phase, before the nucleation of Mn 11 Ge 8 , is wider on a Ge(111) substrate (ΔT = 320 K versus 150 K on a-Ge). Together with the different diffusion paths, an epitaxial interface offering a low interface energy and few defects is expected to delay the germination of Mn 11 Ge 8 since the phase transformation (nucleation of a new phase) depends on the induced change in interface energy [START_REF] Heurle | Kinetics of formation of silicides: A review[END_REF]. Thirdly the Mn 5 Ge 3 dissolution rate is slower in the case of a Ge(111) layer. As the Mn 11 Ge 8 layer is polycrystalline in both cases, the discrepancy in the rates rather comes from the crystalline quality of the Mn 5 Ge 3 films. We should note that surprisingly in ref [START_REF] Wittmer | The first phase to nucleate in planar transition metalgermanium interfaces[END_REF], no other phase than Mn 5 Ge 3 is detected even at temperatures as high as 870 K.

Thin films thickness and thermal stability of the Mn 5 Ge 3 phase

Since the thickness of the Mn 5 Ge 3 thin films is a parameter influencing the magnetic properties of the film, the phase formation have been examined on two type of samples having tow different thicknesses [START_REF] Michez | Magnetic reversal in Mn5Ge3 thin films: an extensive study[END_REF]. The SPE growth was used to elaborate Mn 5 Ge 3 thin films on Ge(111) substrates with thicknesses of 20 and 70 nm. The annealing temperature for the SPE growth process was set at 720 K. The formed Mn 5 Ge 3 films were then post-annealed at two different temperatures T reheat of 720 K and 970 K during 25 min. XRD measurements and M(T) curves were recorded on each postannealed samples and are displayed on figure 5. The XRD measurements of the 20 nm films only clearly show the Mn 5 Ge 3 (002) peaks. The low amount of available matter makes it difficult to detect usable curves on the whole 2θ range. The M sat (T) curves exhibit a similar law of decrease. However a small plateau on the curve corresponding to the 970 K re-heating can be seen between 100 and 160 K. This indicates the presence of a small quantity of the Mn 11 Ge 8 phase. Regarding the results for the 70 nm thick samples, the 2D-XRD maps (not shown here) exhibit partially ring-shapped patterns meaning the manganese germanide films are polycrystalline. The XRD spectrum of the 720 K re-heated film only exhibits the Mn 5 Ge 3 phase, while the Mn 11 Ge 8 phase is clearly identified in the 970 K post-annealed films. The M sat (T) curves support the XRD observations. The 70 nm / 720 K curve reveals only the Mn 5 Ge 3 ferromagnetic signature while the 70 nm / 970 K is characterized by the Mn 11 Ge 8 anti-ferromagnetic transition at 150 K and lower Curie temperature coming from the sum of the ferromagnetic behaviour of the Mn 5 Ge 3 and Mn 11 Ge 8 phases. The combination of an increased thickness and post-annealings is detrimental to the crystallinity of the thin films and to the Mn 5 Ge 3 phase. A post-annealing tends to lead to the formation of the Mn 11 Ge 8 compound which agrees with a diffusion controlled phase formation. Indeed the thermal treatments entail a diffusive Ge flux coming from the substrate. Moreover, a previous work have shown that Mn diffusion along the [111] orientation of Ge is energetically unfavorable [START_REF] Zhu | Contrasting Growth Modes of Mn on Ge(100) and Ge(111) Surfaces: Subsurface Segregation versus Intermixing[END_REF]. This asymmetric interdiffusion profile shifts the equilibrium towards Mn 11 Ge 8 favorable conditions. However the small proportion of the Mn 11 Ge 8 phase in the 20 nm / 970 K film cannot be explained considering only the diffusion phenomena. Two points may be contemplated.

First, the difference between the 20 nm and 70 nm series is the annealing time (cf. Fig. 1 SPE step 2) needed to form the Mn 5 Ge 3 phase. This annealing time is longer for a sample thickness of 70 nm, as there are more Mn atoms to consume and transform into Mn 5 Ge 3 . Hence, the germination of the Mn 11 Ge 8 phase could have already occurred at the Mn 5 Ge 3 /Ge(111) interface for the thicker film during this first annealing step, the growth of the Mn 11 Ge 8 phase remaining only during the postheating stage. Secondly the free energy ΔG has a component related to the interfacial energy. As already mentioned, γ Mn5Ge3/Ge(111) is low. According to the thickness and the resulting diffusion flux, the weighted importance of γ Mn5Ge3/Ge(111) in ΔG may lead to the swing of the thermodynamic equilibrium between maintaining a Mn 5 Ge 3 thin film on Ge(111) with the corresponding interface energy and forming the Mn 11 Ge 8 phase at the interface since ΔH' Mn11Ge8 < ΔH' Mn5Ge3 for Ge rich conditions.

Conclusion

The EHF model provides a suitable framework to explain the formation of the Mn 5 Ge 3 and Mn 11 Ge 8 phase in the manganese germanide thin films on Ge(111) substrates according to the used growth methods. The model takes into account the different factors interplaying at the substrate/thin film interface. The key parameters of the growth process -substrate temperature, atoms fluxes-highlight the major role of the diffusion phenomena in the phase formation. The selection of the growth method -SPE, RDE or co-deposition-enables to control the phase formation of the manganese germanide thin films. In the case of the SPE, In situ X-ray diffraction shows a sequential formation of the phase. Mn 5 Ge 3 is the first formed phase, driven by the nucleation process. Then the diffusion phenomena shift the equilibrium, allowing the subsequent formation of Mn 11 Ge 8 by consuming Mn 5 Ge 3 . A comparison with a previous work dealing with the SPE growth on an a-Ge layer and the study of the effect of the film thickness show the role of the interface on the stability of the Mn 5 Ge 3 phase. The epitaxy of Mn 5 Ge 3 on Ge(111) expands the range of the thermal stability of this phase which is a key point for a potential use of Mn 5 Ge 3 as a ferromagnetic metallic contact for spin injection.

Figure 1 :

 1 Figure 1: Three different growth methods were used to form the Mn5Ge3 or Mn11Ge8 thin films on Ge(111) substrates: the solid phase epitaxy (SPE), the reactive deposition epitaxy (RDE) and the co-deposition of Mn and Ge atoms.

Figure 2 :

 2 Figure 2: The effective heats of formation ΔH ' of the phases Mn5Ge3 and Mn11Ge8 versus the Mn concentration.

Figure 3 :

 3 Figure 3: a-c) X-ray diffraction measurements of manganese germanide thin films grown on Ge(111) substrates according to the three growth methods: SPE, RDE and co-deposition from top to bottom. Yellow dashed and dots lines index the Mn5Ge3 related peaks, the cyan dashed line index the Mn11Ge8 peaks. g) 2D-XRD map of a Mn5Ge3 film grown by SPE. The black indexes correspond to the Ge(111) substrate, the yellow ones to the Mn5Ge3 film. d-f) Normalized magnetizations versus temperature for the Mn5Ge3 thin films according to the used growth methods. The insets display the corresponding derivative curves dM/dT. The inflection points of the M(T) curves are deduced from the minima of the curves, and corresponds to the Curie temperatures.

Mn 5

 5 Ge 3 (002) and Mn 11 Ge 8 (223) diffraction peaks were recorded in real time during this annealing by in situ XRD measurements and shown in fig.4. Only one peak for each phase is plotted for clarity purpose.

Figure 4 :

 4 Figure 4: Variations of integrated and normalized XRD peaks intensities versus annealing temperature in the case of the SPE growth of a Mn5Ge3 thin film on a Ge(111) substrate.

Figure 5 :

 5 Figure 5: XRD curves and normalized magnetization recorded on 20 (a-b) resp.) and 70 nm (c-d) resp.) thick Mn5Ge3 films synthesized by the SPE growth mode and re-heated at 720 and 970 K in order to study the role of the thickness of the film on the thermal stability of the phase.

Acknowledgment

Part of this work was done with the help of the PICS n°200835 TRANPOLSPIN (CINaM-CNRS/AIST) project.


Sarah Iaquinta

Jonathan Barés

Estefania Peña

A S Caro-Bretelle

Mesures d'indentation pour la détection non invasive de l'endommagement du périnée pendant l'accouchement

teaching and research institutions in France or abroad, or from public or private research centers.

  permettant de quantifier l'endommagement du périnée avant l'apparition de déchirures au cours de l'accouchement 2. Méthode Conception d'un dispositif d'Indentation -Relaxation -Recouvrance (IRR) Dispositif d'IRR automatisée Points de mesure Protocole: • Indentation Enfoncement de l'indenteur de 5 mm. Indicateur : 𝛼𝛼 • Relaxation Maintien de l'indenteur en position. Indicateurs : 𝛽𝛽 et Δ𝐹𝐹 • Recouvrance Retrait rapide de l'indenteur et mesure de la remontée du tissu avec un laser. Indicateurs : 𝐴𝐴 et Δ𝑑𝑑 • Durée de l'essai : 'endommagement 𝑑𝑑 fonction de l'allongement 𝜆𝜆 • Génération de données pour indentifier 𝑑𝑑 et 𝜆𝜆 par analyse inverse (A.-S. Caro, 2023, JAPS) Échantillonage quasi Monte Carlo (entraînement et validation) Effet de l'allongement 𝜆𝜆 et de l'endommagement D sur les indicateurs 3. Résultats • Identification inverse de 𝑑𝑑 et 𝜆𝜆 possible par régression linéaire (𝑅𝑅 2 = 0.97) 4. Conclusion Les résultats obtenus numériquement sur le silicone sont encourageants concernant la possibilité d'utiliser les indicateurs issus d'une mesure IRR pour quantifier l'endommagement du périnée pendant l'accouchement Modèle numérique Prochaines étapes : Caractérisation des tissus (modèle porcin) et réitérer l'étude numérique Études cliniques sur cohortes (hôpitaux partenaires) Allongement λ IRR A