Sorelle Hilary

Nguedjang Kouakeuo

Aurelie Solignac

Ruth Sabariego

Laurent Morel

Marie-Ange Raulet

Borel Toutsop

Pierre Tsafack

Benjamin Ducharne
email: benjamin.ducharne@insa-lyon.fr.

Aurélie Solignac

Marie-Ange Raulet

Aurélie Solignac

Marie- Ange Raulet

Borel Toutop

Internal Characterization of Magnetic Cores, Comparison to Finite Element Simulations: A Route for Dimensioning and Condition Monitoring

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Abstract-Magnetic cores are typically used in every stage of an electrical energy production and distribution chain. Local defects, including edge burrs and interlaminar faults, are detrimental for system performance and should be detected swiftly to ensure high reliability and durability. Real-time magnetic core condition monitoring is a promising solution for rapid fault detection. However, similar to dimensioning or performance evaluation, this monitoring has always been performed through averaged magnetic properties, which limits efficiency. In this domain, significant progress is forecasted by achieving precise local measurements. In this study, an innovative solution is proposed to measure local magnetic properties, which is adapted to real-time monitoring and magnetic circuit evaluation. The proposed sensor is a one-piece device that is flat enough to be placed noninvasively between the laminations at distinct positions in the magnetic core. It measures the magnetic excitation and induction fields in two dimensions, which can be used as local inputs for realtime condition monitoring. In this manuscript, the proposed sensor was first detailed, experimental results were provided, and finite-element simulations were performed. The sensor capability was validated both experimentally and through simulation. The results of this study contribute to the development of an intelligent magnetic core that includes an effective real-time monitoring system. The study provides a local validation of the most advanced highfidelity simulation method, which could be used to predict the responses of magnetic cores and electromagnetic converters to defects of various natures, as well as geometrical and property changes.

Index Terms-Finite elements simulation, local defect, magnetic laminations stack, magnetic sensor, predictive maintenance.

I. INTRODUCTION

nergy efficiency can play a crucial role in mitigating climate change. The rise in carbon dioxide emissions and the limited time available to achieve mitigation targets has aggravated the climate change problem. Improving the efficiency of electromagnetic devices can considerably contribute to achieving this target [START_REF] Silveyra | Soft magnetic materials for a sustainable and electrified word[END_REF].

Numerous strategies, including novel geometrical designs, materials, technologies, have been proposed to improve electrical energy conversion rates. In this study, we focused on the magnetic conversion step, and particularly on the real-time control of the magnetic properties in the magnetic core at which energy transformation occurs [START_REF] Fiorillo | Measurements of magnetic materials[END_REF], [START_REF] Batista | An automated measurement system for core loss characterization[END_REF].

Most magnetic cores used to achieve energy conversions are composed of electrical steel sheet stacks. Electrical steel is an iron alloy designed for specific magnetic properties such as low power loss, high permeability. Even if some grades contain up to 6.5 wt% silicon, the content of most commercialized specimens remains close to 3 wt%. The silicon adjunction increases electrical resistivity, which reduces the induced eddy currents and magnetic losses (up to three times that of conventional steel) [START_REF] Gyselinck | Calculation of eddy currents and associated losses in electrical steel laminations[END_REF], [START_REF] Bertotti | Hysteresis in magnetism[END_REF].

Nonoriented electrical steel (NO FeSi) is produced without additional treatment to control crystal orientation. Close magnetic properties can be observed independently from the magnetization direction (quasi-isotropic behavior).

Electrical steel can be manufactured to develop optimal magnetic behavior in the rolling direction (RD). Anisotropy is generated through tight crystal orientation control (Goss texture [START_REF] Goss | [END_REF]). After this treatment, iron silicon alloy becomes grainoriented electrical steel (GO FeSi). NO FeSi is cheaper than GO FeSi. NO FeSi is preferred when the cost is more critical than efficiency or for applications in which the magnetization direction is not set (such as electric motors and generators). GO FeSi is used in power and distribution transformers for maximum efficiency [START_REF] Bozorth | Magnetism[END_REF].

Electrical steel is produced in cold-rolled strips less than 2mm thick. These strips are cut, coated, and stacked to form laminated magnetic cores. Various types of coating, including organic and inorganic, have been proposed. The coating layer maximizes the electrical resistance, limits the eddy current propagation, and provides corrosion resistance [START_REF] Beatty | Standard Handbook for Electrical Engineers[END_REF]. Stacking and coating electrical steel laminations have been used to considerably improve the efficiency of ferromagnetic cores. However, the technique should be improved considerably; in 2014, the world databank estimated that 8% of the world's electrical energy production was lost between generation and consumption [START_REF]The world databank[END_REF]. A considerable portion of this loss was attributed to magnetic conversion, hysteresis, and eddy current losses. Reducing losses from such conversions is critical for improving energy conversions.

During the machining processes, mechanical stresses can considerably affect magnetic properties. Furthermore, undesirable edge burrs around the cut edges and punched holes can be created, which results in interlaminar faults (ILFs) and negatively affects magnetic core efficiency [START_REF] Schulz | Short Circuit Current Measurements Between Transformer Sheets[END_REF]- [START_REF] Shah | Thermographic measurement and simulation of power losses due to interlaminar contacts in electrical sheets[END_REF]. ILFs can also occur during an operation because of multiple reasons, such as the vibration of loose windings, arcing from winding failure, transportation, and installation.

Magnetic core condition monitoring (MCCM) has been proposed to improve performance and durability [START_REF] Hamzehbahmani | A Phenomenological Approach for Condition Monitoring of Magnetic Cores Based on the Hysteresis Phenomenon[END_REF], [START_REF] Hamzehbahmani | An experimental approach for condition monitoring of magnetic cores with grain-oriented electrical steels[END_REF]. MCCM consists of a real-time inspection of the magnetic circuit performances to detect defects and predict a decrease in electromagnetic converter efficiency. MCCM has been performed by controlling the input and output electrical signals [START_REF] Hamzehbahmani | A Phenomenological Approach for Condition Monitoring of Magnetic Cores Based on the Hysteresis Phenomenon[END_REF]. This simple approach limits the magnetic core diagnostic to average quantities and renders it impossible to geo-localize eventual defects. Considerable progress has been achieved in local magnetic measurements.

Numerous studies have focused on the conversion efficiency of magnetic cores (less than a year for [START_REF] Plotnikov | Determination of eddy-current and hysteresis losses in the magnetic circuits of electrical machines[END_REF]- [START_REF] Ducharne | A universal method based on fractional derivatives for modeling magnetic losses under alternating and rotational magnetization conditions[END_REF]). However, most of these manuscripts are limited to simulation results. Some of the proposed techniques can be used to predict the local distribution of the magnetic fields, and like MCCM, these results were validated by comparing with averaged results. Furthermore, performance can be improved by locally measuring the magnetic behavior in the magnetic core.

For geometrical problems, current laminated magnetic core characterizations are limited to either global information (averaged through the cross-section) or surface observations [START_REF] Fiorillo | Measurements of magnetic materials[END_REF], [START_REF] Shah | Thermographic measurement and simulation of power losses due to interlaminar contacts in electrical sheets[END_REF], [START_REF] Imamori | Influence of interlocking on magnetic properties of electrical steel laminations[END_REF]. Embedding any instrumentation system inside a laminated magnetic core was difficult. In [START_REF] Nguedjang Kouakeuo | Embedded printed magnetic needle probes sensor for the real-time control of the local induction state through a laminated magnetic core[END_REF] and [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF], printed electronics have been used to perform such measurements.

In [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF], we specifically focused on embedding a B ⃗ sensor (where B ⃗ is the magnetic flux density (induction field) averaged through the tested specimen cross-section). We used the printed magnetic needle probes method (PMNPM) (also described in [START_REF] Ducharne | Directional Barkhausen noise magnetic measurement using the magnetic needle probe method[END_REF]), in which the conventional needle probe [START_REF] Yamaguchi | Theoretical analysis of localized magnetic flux measurement by needle probe[END_REF], [START_REF] Loisos | Demonstration of a new method for magnetic flux measurement in the interior of a magnetic material[END_REF] is replaced with a printed probe. The resulting sensor exhibits a thickness of a conductive ink layer (≈30 μm). In [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF], a 270-μm thick chip composed by a giant magneto resistance (GMR) sensor [START_REF] Moulin | Optimizing magnetoresistive sensor signal-tonoise via pinning field tuning[END_REF], [START_REF] Hauser | Giant magnetoimpedance sensors[END_REF] was positioned next to the PMNPM sensor to measure the surface tangent magnetic excitation field H ⃗ locally. Local hysteresis cycles B (H ) were plotted for each lamination (all sensors were placed in positions where B ⃗ and H ⃗ are presented collinearly). Experimental characterizations were validated by summing all local measurements and comparing them to averaged values obtained with wound coils. Although excellent results were obtained, some concerns, including the thickness of the GMR sensor (it was still too high compared to the thickness of the coating and of the electrical steel sheets), remained. Furthermore, electrical contacts between the printed part and the acquisition device were ensured using a 100-µm thick bondable terminals, which created a small bump on the lamination edge.

In this study, we performed magnetic core local characterization using a novel method. A novel sensing solution was proposed to address the problems of the previous methods. This solution was described and validated in the second section of this manuscript. In the third section, the measurements were validated using local simulation results obtained using the open-source Onelab® software suite [START_REF] Geuzaine | ONELAB Open numerical engineering laboratory[END_REF] (finite-element software and mesh generator).

In Onelab®, finite elements are combined with vector hysteresis material laws to realize the most advanced highfidelity simulation method.

An experimental validation of the simulation method was performed [START_REF] Sullivan | Electromagnetic simulation using FDTD method[END_REF] to confirm the predictive capability of the proposed method. Discussions and conclusions are provided in the last section of the manuscript.

II. BA(HSURF) LOCAL CHARACTERIZATION OF AN ELECTRIC STEEL MAGNETIC CORE USING EMBEDDED SENSORS.

A. Magnetic Induction Measurement

The magnetic needle probe method (MNPM) is used to record magnetic flux variations in a ferromagnetic laminated core by surface measurement. MNPM is based on the potential difference measured by two conductive needles in contact with the surface of a ferromagnetic laminated core (Fig. 1). The potential difference can be attributed to eddy currents circulating in the lamination cross-section during the magnetization process [START_REF] Loisos | Critical evaluation and limitations of localized flux density measurements in electrical steels[END_REF].

From its first description by Werner in 1949 [36] and up to its most recent published version (2020-21 [START_REF] Mazgaj | Approximation of hysteresis changes in electrical steel sheets[END_REF]- [START_REF] Ducharne | A universal method based on fractional derivatives for modeling magnetic losses under alternating and rotational magnetization conditions[END_REF]), numerous studies have focused on an accurate description of the ideal experimental conditions to ensure accuracy and MNPM reproducibility: _ As described in [START_REF] Loisos | Critical evaluation and limitations of localized flux density measurements in electrical steels[END_REF], the voltage drop between positions 1 and 2 in Fig. 1 below can be approximated by using the following expression:

V ≈ ∫ • dS , (1)
where B1234 and S1234 are the induction magnetic field and the cross-section through positions 1, 2, 3, and 4 (blue mesh in Fig. 1), respectively. In order to correctly use the needle probe sensor, the following specifications must be met: 1) For accurate measurements, the distance 1-2 should be higher than 10 mm [START_REF] Yamaguchi | Theoretical analysis of localized magnetic flux measurement by needle probe[END_REF].

2) The distance from the edges should be higher than half of the specimen thickness (d/2) [START_REF] Yamaguchi | Theoretical analysis of localized magnetic flux measurement by needle probe[END_REF].

3 ) The leakage flux in the enclosed near-surface magnetized air should be negligible [START_REF] Senda | Localized magnetic properties in grain-oriented silicon steel measured by stylus probe method[END_REF], [START_REF] Senda | Localized magnetic properties in grain-oriented electrical steel measured by needle probe method[END_REF]. 4) The probe's distance to the specimen ratio (1-2/d) should be significant [START_REF] Matsubara | A novel method of measurements of magnetic flux in silicon steel sheet with magnetic flux leakage[END_REF].

The application of MNPM is limited because of weak electromotive forces, especially in the low-frequency range. However, considerable advances have been achieved in analog and digital electronics. Improvements in the MNPM have resulted in numerous applications, such as local nonuniform electromagnetic field investigations [START_REF] Enokizono | The problem of simplified two-dimensional magnetic measurement apparatus[END_REF], [START_REF] Abdallh | Magnetic Material Identification in Geometries With Non-Uniform Electromagnetic Fields Using Global and Local Magnetic Measurements[END_REF], cutting-edge stress [START_REF] Abdalh | Local magnetic measurements in magnetic circuits with highly non-uniform electromagnetic fields[END_REF], [START_REF] Loisos | Novel flux density measurement methods of examining the influence of cutting on magnetic properties of electrical steels[END_REF], stress-induced peening processes [START_REF] Crevecoeur | Local identification of magnetic hysteresis properties near cutting edges of electrical steel sheets[END_REF], and homogenization analysis in lamination stacks [START_REF] Tene Deffo | Local measurement of peening-induced residual stresses on Iron Nickel material using needle probes technique[END_REF]. PMNPM, as described in [START_REF] Nguedjang Kouakeuo | Embedded printed magnetic needle probes sensor for the real-time control of the local induction state through a laminated magnetic core[END_REF], satisfies all the aforementioned conditions. The test probe can be replaced with a printed conductive layer, which results in a leakage flux that is negligible. The thickness of the whole sensor (a conductive ink layer, ≈30 μm) also becomes irrelevant.

B. Magnetic Excitation Measurement

The PMNPM is a nonintrusive method for measuring a local directional flux variation through the cross-section of a laminated core. Here, Ba is obtained from time-based integration. For a complete hysteresis cycle plot, Hsurf should be recorded using small dimensions GMR sensors. GMRs can be used to detect up to less than a few nanoteslas beyond substantial frequency and temperature ranges [START_REF] Moulin | Optimizing magnetoresistive sensor signal-tonoise via pinning field tuning[END_REF]. GMRs consist of multiple thin layers, including two ferromagnetic deposited thin films isolated by a metallic nonmagnetic spacer. The total active part thickness is typically smaller than 100 nm.

The first ferromagnetic layer, which is the reference layer (or fixed layer), functions as a permanent magnet. The second ferromagnetic deposit (the free or nonfixed layer) is magnetically soft. The magnetic state of the layer fluctuates under the influence of external magnetic excitations. These variations induce overall resistance fluctuations because of the spin-dependent charge transport phenomena [START_REF] Chopin | Magnetoresistive sensor in two-dimension on a 25 μm thick silicon substrate for in vivo neuronal measurements[END_REF]- [START_REF] Tsymbal | Perspectives of giant magnetoresistance[END_REF]. The linearity of the GMR is set by the shape anisotropy in the free layer. This anisotropy induces an easy magnetization direction at 90° of the fixed layer magnetization [START_REF] Moulin | Optimizing magnetoresistive sensor signal-tonoise via pinning field tuning[END_REF]. The GMR sensor is sensitive to the in-plane field and along the magnetization of the fixed layer.

For additional information on the GMR working principles, please refer to [START_REF] Tsymbal | Perspectives of giant magnetoresistance[END_REF].

C. Complete Low-Thickness Sensing Solution

For GMR deposition, the roughness of the substrate should be less than 1 nm rms. Under such a condition, a direct implementation is not possible on a ferromagnetic electrical steel lamination whose roughness is approximately tens of nanometers in the best cases. In [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF], a 270-μm thick silicon substrate was used to develop a noninvasive solution. Such a high thickness is not feasible. However, polishing treatments or fabrications process can be performed [START_REF] Chopin | Magnetoresistive sensor in two-dimension on a 25 μm thick silicon substrate for in vivo neuronal measurements[END_REF]. Although these methods were not tested in [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF], the resulting substrate thickness can decrease to a few tens of micrometers.

As mentioned in the Introduction, PMNPM deposition is associated with thickness concerns on the edges of the lamination stack because of electrical contacts. In this study, we solved this problem and improved the overall electrical connections by depositing the GMR and printing the needles probe on the same patch. Fig. 2 (right) displays the layout of the resulting sensor, including both the magnetic induction and excitation measurement in RD and transverse direction (TD). As displayed in the three-dimensional (3D) illustrations in Fig. 2 (left) and confirmed in the photographs in Fig. 3, all the electrical contacts (large pads) were placed on the bottom part of the sensor patch located outside the lamination stack. An additional local layer of conductive ink deposed in a final fabrication step ensured needle probe electrical contacts. The coating was removed locally on the bottom face of the tested lamination. Contact resistances were characterized from 20 Hz to 1 kHz and for temperatures varying from 20 to 200 °C. However, these results were inferior to 0.5 Ω for all the tests and minimal compared with the 10 6 Ω acquisition device input impedance, which render their influence negligible. The Si/SiO2 (500)//Ta (3)/Ni80Fe20 (6.3)/Co90Fe10 (2.1)/Cu (2.9)/Co90Fe10 (2)/Ru (0.85)/Co90Fe10 (2)/Ir22Mn78 (7.5)/Ru (0.4)/Ta (5) (thicknesses in nanometers) GMRs were deposited through sputtering (Rotaris-Singulus) on 270-µm thick oxidized silicon wafers. Standard UV lithography techniques were used to fabricate meander-shaped GMR structures with metallic contacts (100 nm of Ta/cu/Ta), as displayed in the enlarged image of Fig. 2. The probe contains two magnetic tips and two GMR sensors for vector field measurements. The axis of sensitivity of the GMR could be changed by varying local courant pulse heating under a field [START_REF] Chopin | Magnetoresistive sensor in two-dimension on a 25 μm thick silicon substrate for in vivo neuronal measurements[END_REF]. The thickness of the patch was 270 µm (Fig. 4), but it could be easily reduced through CMP polishing or silicon on insulator process [START_REF] Chopin | Magnetoresistive sensor in two-dimension on a 25 μm thick silicon substrate for in vivo neuronal measurements[END_REF] to less than 30 µm. However, because of the extreme fragility of the resulting sensor, processing must be performed meticulously.

The sensitivity and linearity of the GMRs have been evaluated in an initialization stage using Helmholtz coils. The results of this precharacterization are displayed in Fig. 5.

The proximity of the GMR to the tested ferromagnetic sheet in the final configuration is also crucial. Commercially available hall effect sensors are used for performing tangent surface excitation field measurements in classical approaches. However, the field gradients above the specimen surfaces are critical, and the hall sensors used are too bulky to provide reliable information. This concern has been discussed in many publications, including [START_REF] Lewis | Development and application of measurement techniques for evaluating localized magnetic properties in electrical steel[END_REF], [START_REF] Stupakov | Applicability of local magnetic measurements[END_REF], and complex alternative solutions have been proposed [START_REF] Stupakov | Controllable magnetic hysteresis measurement of electrical steels in a single-yoke open configuration[END_REF]. The GMR sensors described in this study provide close measurements and ensure excellent Hsurf observation.

D. Experimental Validation

The experimental setup used to validate the new sensing solution has been comprehensively described in [START_REF] Nguedjang Kouakeuo | Embedded printed magnetic needle probes sensor for the real-time control of the local induction state through a laminated magnetic core[END_REF] and [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF]. The IEC 60404-3 standard defines the use of a single-sheet tester to characterize electrical steel lamination [START_REF]Magnetic materials -Part 3: Methods of measurement of the magnetic properties of electrical strip and sheet by means of a single sheet tested[END_REF], even if differences exist, such as in the use of a single yoke, the yoke dimensions, the primary winding being wound around the yoke instead of the tested specimens, and a stack of 10 laminations replacing the single sheet of electrical steel for stack tests. The experimental setup of this study was established using this standard.

All the GO FeSi specimens tested were picked up from the same batch. As provided by the manufacturer and previously characterized by the team [START_REF] Nguedjang Kouakeuo | Embedded printed magnetic needle probes sensor for the real-time control of the local induction state through a laminated magnetic core[END_REF], [START_REF] Nguedjang Kouakeuo | Non-invasive local magnetic hysteresis characterization of a ferromagnetic laminated core[END_REF], their physical properties at room temperature are displayed in Table 1. Here μrmax, Hc, Js, Tc, and λs are the relative permeability, coercivity, saturation polarization, the Curie temperature, and the magnetostriction coefficient, respectively. Here, l is the initial length and Δl is the length variation at saturation. In the first experimental tests, a single instrumented ferromagnetic sheet was laid on the yoke. Measurements with the sensor patch were monitored at various frequencies and amplitudes. The experimental results are displayed in Fig. 6. The sensor size allows measurement in the middle of the lamination far from the edges at which inconsistency in the MNPM could affect measurements [START_REF] Yamaguchi | Theoretical analysis of localized magnetic flux measurement by needle probe[END_REF]. Thus, all measurements were performed in the central zone of the laminated sheet. The 2D capability of the sensing solution was tested subsequently by positioning the sensor patch at 45° (Fig. 7). As displayed in Fig. 8, when the angle corrections were processed (projections), similar results were obtained in the xand y-direction, which confirmed the 2D performance of the sensor patch. This first series of tests were completed to verify the cohesion between the experimental results and the manufacturer information. Next, a pile of 10 electrical steel laminations, including the instrumented laminations, was used to validate the sensor in a stack configuration. The instrumented sheet was moved in every stack position, and various excitation levels and frequencies were tested. Some of these results are included in the next section. All tests were performed at room temperature.

Table I Typical FeSi GO magnetic Properties

III. INTERNAL MAGNETIC BEHAVIOR OF A LAMINATED CORE: COMPARISON TO FINITE ELEMENTS SIMULATION RESULTS

Numerous simulation methods have been developed to predict the magnetic behavior and electric characteristics of electromagnetic devices. Simulations can be used in a phenomenological manner to improve the knowledge and interpretation based on experimental results [START_REF] Annakkage | A current transformer model based on the Jiles-Atherton theory of ferromagnetic hysteresis[END_REF]. Simulations can also be used to predict quantities and information beyond reach, including the effect of potential defects on the overall electromagnetic converter behavior [START_REF] Wang | Finite element analysis of internal winding faults in distribution transformers[END_REF]. Numerical computations can avoid fastidious (and often expensive) experimental campaigns during the dimensioning phase. 7 for experimental conditions illustration). In electromagnetic devices design, simulation methods based on the space discretization to solve partial differential equations, such as the finite elements method (FEM), are highly promoted because of features such as their versatility, complex geometries, local material characteristics, and nonlinearities. In the specific case of electrical steel lamination stacks, FEM has typically been used to predict the local distribution of the magnetic fields [START_REF] Gyselinck | Calculation of eddy currents and associated losses in electrical steel laminations[END_REF]- [START_REF] Lamprecht | Investigations of eddy current losses in laminated cores due to the impact of various stacking processes[END_REF], including the magnetic core where previous instrumentation could not reach. In these studies, the experimental validation was not systematic and averaged results at the core level (usually, the magnetic losses) are always used, as displayed in Fig. 26 of [START_REF] Lamprecht | Investigations of eddy current losses in laminated cores due to the impact of various stacking processes[END_REF] or Fig. 5 of [START_REF] Kaimori | FEM computation of magnetic field and iron loss in laminated iron core using homogenization method[END_REF].

In the previous section, a sensing solution sufficiently thin to be embedded between the laminations of a magnetic core and return local information has been described. In this section, FEM simulation results are analyzed. By comparing the experimental and numerical results, we validated both the practical and numerical methods.

A. Description of the Simulation Method

The magnetic yoke, lamination stack, and excitation coil were investigated. We developed a 2D planar FEM magneto dynamic model (Fig. 9) based on the classical magnetic vector potential formulation within the open-source Onelab® software suite [START_REF] Geuzaine | ONELAB Open numerical engineering laboratory[END_REF]. We used the symmetry (half electromagnet circuit) and a shell transformation (no truncation) to detail their infinity. A homogeneous current density crossed the excitation coil j (𝑥, 𝑦, 𝑡). A set of 10 laminations was considered. The vectorial Jiles-Atherton model described in [START_REF] Gyselinck | Complementary twodimensional finite element formulations with inclusion of a vectorized Jiles-Atherton model[END_REF], a variant of the model first proposed by Bergqvist [START_REF] Bergqvist | A simple vector generalization of the Jiles-Atherton model of hysteresis[END_REF], was applied as the material law in the ferromagnetic parts. This extension was not feasible for anisotropic materials such as FeSi GO tested in the study.

However, the comparisons with the experimental observations were limited to the X-axis direction and far from the yoke legs, where Ba and Hsurf are assumed in RD and colinear. The hysteresis simulation parameters are presented in Table 2. These parameters were set according to the RD behavior in pre-processing through optimization and comparisons with quasi-static experimental results. According to the Jiles-Atherton theory of ferromagnetism, α quantifies the interdomain coupling, a is the domain walls density, k is the average energy to break the pinning sites, and c is magnetization reversibility [START_REF] Jiles | Theory of ferromagnetic hysteresis[END_REF].

Table 2 Jiles-Atherton (J-A) simulation parameters for FeSi GO (RD).

B. Simulation Results and Comparisons With Experimental Results

To test the simulation and the measurement methods in the worst-case scenario, we intentionally set the electrical current in the excitation coil to produce a significant gradient of magnetic induction between the first and the last laminated sheet of the stack. The simulated evolution of the magnetic induction in the magnetic stack when Hsurf varies from 0 to 100 A•m -1 is given in Fig. 10. Even if the resulting hysteresis cycles are not identical (probably because of an imperfect estimation of the J-A model parameters or to the simulation hypothesis), the close evolution of the magnetic gradient is notable. The predictive capability of the simulation method is evident in Fig. 13, in which comparisons were proposed for the variation of a series of hysteresis cycle indicators (Fig. 12) as a function of the magnetic sheet position. A depression was observed in the hysteresis cycles of the first, second, and third positions. Such behavior has already been noted in literature and associated with the dynamic losses during the magnetization process [START_REF] Zirka | Implementation of inverse hysteresis model into EMTP-Part I: Static model[END_REF], [START_REF] Zirka | Implementation of inverse hysteresis model into EMTP-Part II: Dynamic model[END_REF].

Finally, the uncertainty marker defined in Eq. 2 was applied to the five tested indicators to verify the global performance of the proposed simulation method. The results are presented in Table 3.

Table 3 Quantitative evaluation of the simulation method (m = 10 is the laminations number).

Table 3 details the large errors between the simulation predictions and the experimental results. However, the Eq. 2 error definition is the average value of the relative error calculated on the ten sheet positions. Considerably lower error percentages could have been obtained by limiting those calculations to the bottom laminations at which the magnetic flux density is concentrated. The error is small for the first sheets (positions 1, 2, and 3). The measured magnetic quantities exhibit significant values for these positions and lead to a consequent ratio between the valuable signal and ambient noise. From laminations 4 to upwards, the induction levels are in the same range as PMNP uncertainty. For these positions, the comparisons are inaccurate and considerably alter quantitative evaluation. Different paths can be followed to improve these results: from the simulation perspective, alternative vectorial hysteresis models have been described in scientific literature [START_REF] Ducharne | A universal method based on fractional derivatives for modeling magnetic losses under alternating and rotational magnetization conditions[END_REF], [START_REF] Yue | A modified inverse vector hysteresis model for nonoriented electrical steels considering anisotropy for FEA[END_REF]- [START_REF] Ducharne | An anisotropic vector hysteresis model of ferromagnetic behavior under alternating and rotational magnetic field[END_REF]. Accuracy was claimed to be improved and will be evaluated in the future. Furthermore, the evaluation of the simulation parameters can be improved. Eventually, from the practicality perspective, the calibration of the GMR sensors and the signal treatment of PMNP can be improved.

IV. CONCLUSION

Energy efficiency and reliability have attracted considerable research attention. Magnetic cores are essential elements of electrical energy production and distribution. Even if these methods have been thoroughly studied, further improvements is necessary.

Local characterizations of the magnetic behavior in the magnetic core should be investigated. Technical limitations have constrained such measurement. However, simulation tools have been proposed for prediction. Moreover, validation have been compared with average or surface observations, and local verifications were expected. A 100-µm thick patch sensor (whose thickness can be reduced) combining MNPM and GMR sensors was proposed to provide in-plan magnetic characterization inside a ferromagnetic laminated sheet stack. The patch was tested first on a single ferromagnetic laminated sheet. A 2D measurement was proposed by comparing the two pairs of sensors in an experimental condition that required the sensor to provide the same information. Next, stack tests were performed by positioning the sensor patch between every laminated sheet. Subsequently, comparisons with time-domain FEM simulations were proposed and used as validation for both the experimental and the simulation tools.

Even if the proposed patch sensor provided accurate local measurements in all the performed tests, the authors have noted the following limitations: 1) The PMNP electrical contacts were ensured from an additional layer of conductive ink processed in a further fabrication step. However, this solution is not optimal, and alternatives should be determined.

2) The fragility of the sensor patch was the most critical concern for high technology readiness level (TRL) developments. The thinned patch should be handled meticulously, especially during polish treatments. The use of flexible substrates is plausible but constitutes time-consuming developments beyond the scope of this project. A flexible substrate is the long-term objective of this study. The following short-term prospects exist: 1) To test the sensor patch in a real-life situation. An instrumented power transformer was designed, including multiple sensor patches dispatched all over the transformer laminated magnetic core.

2) To evaluate the detection of ILFs or edge burrs with the sensor patch. A specific experimental setup including wellknown defects should be designed for this purpose.

Fig. 1

 1 Fig. 1 Schematic of the magnetic needle probe method (MNPM).

Fig. 3 Fig. 3

 33 Fig. 2 MNPM/GMR sensor; left: illustration of the electrical contacts, right: whole sensor layout.

Fig. 4

 4 Fig. 4 Sensor patch resulting thickness.

Fig. 5

 5 Fig. 5 Precharacterization of GMR sensors (R0 = 630 Ω), linearity, and sensitivity estimation.

Fig. 6

 6 Fig. 6 Single lamination tests (left: vs. frequency, max (Hsurf) = 120 A•m -1 , right: vs. Hsurf amplitude, f = 50 Hz).

Fig. 7

 7 Fig. 7 Two-dimensional measurement verification and illustration of the sensor positioning.

Fig. 8

 8 Fig. 8 Superimposition of the hysteresis cycles measured on both axis O ⃗ , O ⃗ (see Fig.7for experimental conditions illustration). In electromagnetic devices design, simulation methods based on the space discretization to solve partial differential equations, such as the finite elements method (FEM), are highly promoted because of features such as their versatility, complex geometries, local material characteristics, and nonlinearities. In the specific case of electrical steel lamination stacks, FEM has typically been used to predict the local distribution of the magnetic fields[START_REF] Gyselinck | Calculation of eddy currents and associated losses in electrical steel laminations[END_REF]-[START_REF] Lamprecht | Investigations of eddy current losses in laminated cores due to the impact of various stacking processes[END_REF], including the magnetic core where previous instrumentation could not reach. In these studies, the experimental validation was not systematic and averaged results at the core level (usually, the magnetic losses) are always used, as displayed in Fig.26of[START_REF] Lamprecht | Investigations of eddy current losses in laminated cores due to the impact of various stacking processes[END_REF] or Fig.5of[START_REF] Kaimori | FEM computation of magnetic field and iron loss in laminated iron core using homogenization method[END_REF].In the previous section, a sensing solution sufficiently thin to be embedded between the laminations of a magnetic core and return local information has been described. In this section, FEM simulation results are analyzed. By comparing the experimental and numerical results, we validated both the practical and numerical methods.

Fig. 9

 9 Fig. 9 Two-dimensional planar, symmetric simulation domain, and its mesh (left) with spherical shell (outer ring) to account for infinity; detail of the electromagnet circuit (right).

6 Fig. 10

 610 Fig. 10 Simulation of the magnetic induction (B) in the stack of laminated sheets, Hsurf varying from 0 to 100 A•m -1 .

Fig. 11

 11 Fig. 11 Comparisons of the Onelab® simulations and experimental results in local tests.

Fig. 11

 11 Fig. 11 presents a comparison of the magnetic hysteresis cycles simulated and measured for every ferromagnetic lamination during a full magnetization cycle.Even if the resulting hysteresis cycles are not identical (probably because of an imperfect estimation of the J-A model parameters or to the simulation hypothesis), the close evolution of the magnetic gradient is notable. The predictive capability of the simulation method is evident in Fig.13, in which comparisons were proposed for the variation of a series of hysteresis cycle indicators (Fig.12) as a function of the magnetic sheet position.

Fig. 12

 12 Fig. 12 Series of hysteresis cycle indicators tested.

Fig. 13

 13 Fig. 13 Comparison measurements/simulations for the main hysteresis cycle indicators versus the magnetic sheet position.


Susan Sananes Israel

Diane Rébiscoul

† Michael Odorico

Valérie Flaud

André Ayral

Surface properties of alkoxysilane layers grafted in supercritical carbon dioxide

Keywords: Surface functionalization, alkoxysilane, silica, supercritical carbon dioxide

Silicon oxide surface properties can be easily modified by grafting alkoxysilane molecules. Here, we studied the structure and the morphology of ultrathin layers prepared by the grafting of alkoxysilanes having different head groups (thiol, amine and iodo) in supercritical carbon dioxide (SC CO2) on model plane silicon oxide surfaces. Several characterization techniques (X-ray reflectivity -XRR, water contact angle, X-ray photoelectron spectroscopy -XPS and atomic force microscopy -AFM) were used to determine the physicochemical properties of the layers prepared at different temperatures. Moreover, for the first time, AFM peak force measurements were used to delve deeper into the determination of the structure of these ultrathin alkoxysilane layers. The results show that the grafting temperature and the nature of the head group strongly affect the morphology and the structure of the grafted layers. Dense monolayers, polycondensed layers and dense bilayers are obtained with 3-(mercaptopropyl)trimethoxysilane (MPTMS) at 60 °C, with [3-(aminoethylamino)propyl]trimethoxysilane (AEAPTMS) between 40°C and 120 °C, and with 3-(iodopropyl)triethoxysilane (IPTES) at 120 °C, respectively.

Introduction

Functionalized silica materials are widely used in microelectronic [START_REF] Perrut | Alkoxysilane Layers Compatible with Copper Deposition for Advanced Semiconductor Device Applications[END_REF][START_REF] Rébiscoul | Alkoxysilane Layers Deposited by SC CO 2 Process on Silicon Oxide for Microelectronics Applications[END_REF][START_REF] Weibel | An Overview of Supercritical CO2applications in Microelectronics Processing[END_REF][START_REF] Ciaramella | Crosslinking Impact of Mesoporous MSQ Films Used in Microelectronic Interconnections on Mechanical Properties[END_REF] , decontamination [START_REF] Zub | Removal of Pb ( II ) Ions from Aqueous Systems Using Thiol-Functionalized Cobalt-Ferrite Magnetic Nanoparticles[END_REF][START_REF] Benhamou | Aqueous Heavy Metals Removal on Amine-Functionalized Si-MCM-41 and Si-MCM-48[END_REF][START_REF] Xue | Removal of Cu(II) from Aqueous Solution by Adsorption onto Functionalized SBA-16 Mesoporous Silica[END_REF][START_REF] Gascón | Aqueous Heavy Metals Removal by Adsorption on Amine-Functionalized Mesoporous Silica[END_REF][START_REF] Bois | Functionalized Silica for Heavy Metal Ions Adsorption[END_REF][START_REF] Lee | Synthesis of Functionalized Porous Silicas via Templating Method as Heavy Metal Ion Adsorbents: The Introduction of Surface Hydrophilicity onto the Surface of Adsorbents[END_REF][START_REF] Mureseanu | Modified SBA-15 Mesoporous Silica for Heavy Metal Ions Remediation[END_REF][START_REF] Mercier | Heavy Metal Ion Adsorbents Formed by the Grafting of a Thiol Functionality to Mesoporous Silica Molecular Sieves: Factors Affecting Hg(II) Uptake[END_REF][START_REF] Makowski | Mesoporous Materials in the Field of Nuclear Industry: Applications and Perspectives[END_REF] , medical [START_REF] Ichiyanagi | Functional Magnetic Nanoparticles for Medical Application[END_REF][START_REF] Sanchez | Medical Applications of Organic-inorganic Hybrid Materials within the Field of Silica-Based Bioceramicsw[END_REF][START_REF] Vallet-Regí | Revisiting Silica Based Ordered Mesoporous Materials: Medical Applications[END_REF] , catalysis [17][START_REF] Conley | Mesostructured Hybrid Organic -Silica Materials: Ideal Supports for Well-De Fi Ned Heterogeneous Organometallic Catalysts[END_REF][START_REF] Shin | Facile Synthesis of Silver-Deposited Silanized Magnetite Nanoparticles and Their Application for Catalytic Reduction of Nitrophenols[END_REF][START_REF] Yokoi | Applied Catalysis A : General Amino-Functionalized Mesoporous Silica as Base Catalyst and Adsorbent[END_REF] and biosensor [START_REF] Karimi | A Single Hybrid Optical Sensor Based on Nanoporous Silica Type SBA-15 for Detection of Pb2+ and I-in Aqueous Media[END_REF][START_REF] Slowing | Mesoporous Silica Nanoparticles for Drug Delivery and Biosensing Applications[END_REF][START_REF] Sarkar | Selective Zinc(II)-Ion Fluorescence Sensing by a Functionalized Mesoporous Material Covalently Grafted with a Fluorescent Chromophore and Consequent Biological Applications[END_REF] applications. Indeed, the surface of silica can be easily tuned, leading to materials with various properties, depending on the molecule grafted. For instance, the ion retention capacity on silicon oxide surfaces can be enhanced by the grafting of alkoxysilane molecules [START_REF] Perrut | Alkoxysilane Layers Compatible with Copper Deposition for Advanced Semiconductor Device Applications[END_REF][START_REF] Rébiscoul | Alkoxysilane Layers Deposited by SC CO 2 Process on Silicon Oxide for Microelectronics Applications[END_REF][START_REF] Bois | Functionalized Silica for Heavy Metal Ions Adsorption[END_REF][START_REF] Ruckenstein | Surface Modification and Functionalization through the Self-Assembled Monolayer and Graft Polymerization[END_REF] . The surface morphology and the structure of the grafted layers are the main drivers of processes occurring in various applications of functionalized silica (such as extraction, sensing….). Conventional organic solvent such as toluene can be used for the silanization reaction [START_REF] Arkles | Tailoring Surfaces with Silanes[END_REF][START_REF] Brzoska | Silanization of Solid Substrates: A Step toward Reproducibility[END_REF] . Nevertheless these solvents present economic and environmental drawbacks and supercritical fluids have emerged as a greener alternative [START_REF] Aswal | Self Assembled Monolayers on Silicon for Molecular Electronics[END_REF] . Different works focused on the functionalization of silicon oxide surfaces by alkoxysilanes in SC CO2 and more particularly in porous silica and silica nanoparticles [START_REF] Perrut | Alkoxysilane Layers Compatible with Copper Deposition for Advanced Semiconductor Device Applications[END_REF][START_REF] Rébiscoul | Alkoxysilane Layers Deposited by SC CO 2 Process on Silicon Oxide for Microelectronics Applications[END_REF][START_REF] Stojanovic | Preparation of MEMO Silane-Coated SiO2 Nanoparticles under High Pressure of Carbon Dioxide and Ethanol[END_REF][START_REF] Domingo | Grafting of Trialkoxysilane on the Surface of Nanoparticles by Conventional Wet Alcoholic and Supercritical Carbon Dioxide Deposition Methods[END_REF][START_REF] Lopez-Periago | Chemical Modification of Nanometric TiO2 Particles by Anchoring Functional Silane Molecules in Supercritical CO2[END_REF][START_REF] Purcar | Surface Modification of Silica Particles Assisted by CO2[END_REF][START_REF] Sanz-Moral | Gradual Hydrophobic Surface Functionalization of Dry Silica Aerogels by Reaction with Silane Precursors Dissolved in Supercritical Carbon Dioxide[END_REF][START_REF] García-Gonzaĺez | Measurements and Correlation of Octyltriethoxysilane Solubility in Supercritical CO2 and Assembly of Functional Silane Monolayers on the Surface of Nanometric Particles[END_REF][START_REF] Zemanian | Deposition of Self-Assembled Monolayers in Mesoporous Silica from Supercritical Fluids[END_REF][START_REF] Tenorio | Thiol Group Functionalization of Mesoporous SiO2 SBA-15 Using Supercritical CO2[END_REF][START_REF] Sánchez-Vicente | A New Sustainable Route in Supercritical CO2 to Functionalize Silica SBA-15 with 3-Aminopropyltrimethoxysilane as Material for Carbon Capture[END_REF][START_REF] López-Aranguren | Regenerable Solid CO2 Sorbents Prepared by Supercritical Grafting of Aminoalkoxysilane into Low-Cost Mesoporous Silica[END_REF][START_REF] López-Aranguren | Sorption of Tryalkoxysilane in Low-Cost Porous Silicates Using a Supercritical CO2 Method[END_REF][START_REF] Sun | Surface Sol-Gel Modification of Mesoporous Silica Molecular Sieve SBA-15 with TiO2 in Supercritical CO2[END_REF][START_REF] Builes | Alkylsilane-Functionalized Microporous and Mesoporous Materials: Molecular Simulation and Experimental Analysis of Gas Adsorption[END_REF][START_REF] Ashu-Arrah | Synthesis and Characterisation of Bonded Mercaptopropyl Silica Intermediate Stationary Phases Prepared Using Multifunctional Alkoxysilanes in Supercritical Carbon Dioxide as a Reaction Solvent[END_REF] . In this study, we focused on the grafting of different alkoxysilanes in SC CO2 on model plane silicon oxide surfaces. This green solvent has already proved its efficiency for the functionalization of silicon oxide surfaces [START_REF] Kendall | Polymerizations in Supercritical Carbon Dioxide[END_REF][START_REF] Li | Grafting of 2-Hydroxyethyl Methacrylate onto Isotactic Poly ( Propylene ) Using Supercritical CO 2 as a Solvent and Swelling Agent[END_REF][START_REF] Sanli | Silylation from Supercritical Carbon Dioxide : A Powerful Technique for Modification of Surfaces[END_REF] . Above its critical point (T=31 ºC P=74 bars), carbon dioxide is in a supercritical state, with physicochemical properties between those of the liquid and the gas states, such as zero surface tension, high diffusivity and low density. Moreover, alkoxysilanes are soluble in SC CO2. These properties minimize the polycondensation of alkoxysilanes favoring the molecule transport to the SC CO2/SiO2 interface where the grafting process occurs.

Usually, the grafting of alkoxysilanes is done by silanization, resulting from different phenomena [START_REF] Sanli | Silylation from Supercritical Carbon Dioxide : A Powerful Technique for Modification of Surfaces[END_REF] : (i) the dissolution of the alkoxysilane in SC CO2 and the molecule transport to the silicon oxide surface; (ii) the alkoxysilane hydrolysis by the interfacial water; (iii.1) the condensation and self-assembly of the hydrolyzed molecule with the surface silanols; and/or (iii.2) the self-condensation of the molecule and its physisorption at the silicon oxide surface. These different reactions (Figure S1 in the Supporting Information) are simultaneous and depend on the alkoxysilane used and on the experimental parameters [START_REF] López-Aranguren | Sorption of Tryalkoxysilane in Low-Cost Porous Silicates Using a Supercritical CO2 Method[END_REF][START_REF] García-Gonzaĺez | Measurements and Correlation of Octyltriethoxysilane Solubility in Supercritical CO2 and Assembly of Functional Silane Monolayers on the Surface of Nanometric Particles[END_REF] .

In this work, three organic molecules were chosen and grafted in the SC CO2 process: 3-(mercaptopropyl)trimethoxysilane (MPTMS), which is already known to form self-assembled monolayers (SAMs) on the silicon oxide surface 2 ; [3-(aminoethylamino)propyl]trimethoxysilane (AEAPTMS); and for the first time, 3-(iodopropyl)triethoxysilane (IPTES). Table S1 summarizes some physicochemical properties of these three molecules. The grafting of alkoxysilanes in SC CO2 has already been studied and some structures have been proposed for a grafting of a monolayer of MPTMS 29 at 100 bars. Thus, in this study, a pressure of 100 bars was used during the grafting process. Indeed, this pressure is sufficient to graft MPTMS on silica planar surface. This is not the case in porous media, which for the best parameters to form MPTMS monolayers in mesoporous materials were found at T = 80 ºC and P = 250 bars [START_REF] Ashu-Arrah | Synthesis and Characterisation of Bonded Mercaptopropyl Silica Intermediate Stationary Phases Prepared Using Multifunctional Alkoxysilanes in Supercritical Carbon Dioxide as a Reaction Solvent[END_REF] .

Generally, the structure and morphology of these alkoxysilane layers are difficult to characterize precisely because of the small thickness of the layers. In this study, we propose a new methodology coupling several surface characterization techniques to determine the structure and morphology of layers prepared in silicon oxide flat surfaces. First, the classical properties of the layers, such as density, thickness, roughness, chemical composition and hydrophilicity, were determined using Xray reflectivity (XRR), atomic force microscopy (AFM), X-ray photoelectron spectroscopy (XPS) and contact angle measurements. Second, their spatial organization was explored through a pioneer study by AFM in peak force mode. This method is currently used to obtain information about the adhesion parameters of polymers and thin films of 2 to 10 nm thick [START_REF] Loi | Rupture of Molecular Thin Films Observed in Atomic Force Microscopy. II. Experiment[END_REF][START_REF] Butt | Rupture of Molecular Thin Films Observed in Atomic Force Microscopy. I. Theory[END_REF][START_REF] Sababi | Structural and Nanomechanical Properties of Paperboard Coatings Studied by Peak Force Tapping Atomic Force Microscopy[END_REF][48] . Here, it was used to delve deeper into the determination of the structure of ultrathin layers between 1 and 2 nm. Finally, we summarize the effects of the temperature and the functional group of the molecule on the layer morphology.

Experimental section

Material preparation

After the comparison of the contact angle obtained on the silicon surface pre-treated with a Caro solution and a 10 wt% nitric acid solution under reflux for 24 h (Table S3), the nitric acid preparation was selected. Thus, the surface of silicon wafers (<100> p-doped from MEMC Electronic Materials) was activated using a 10 wt% nitric acid solution under reflux for 24 h and rinsed with ultrapure water. The obtained hydrated silicon oxide layer was approximately 7 to 9 Å thick (determined from XRR measurements). MPTMS and AEAPTMS were provided by Sigma Aldrich. IPTES was obtained by the halogen exchange of (3-chloroproyl)triethoxysilane and NaI according to the procedure described in the literature 49 .

The SC CO2 grafting process was performed with a SEPAREX supercritical fluid extractor (See Figure S2). The activated silicon wafer was placed in the stainless-steel reactor. Then, the process described in Figure S3 was applied at 40°C, 60°C, 80°C, 100°C and 120°C for each molecule: SC CO2 was flowed at 30 g/min, and a pressure of 100 bars was maintained. The reaction pressure has been set to 100 bars according to previous results on silicon oxide thin layers 2 . After 5 minutes, a solution containing 1 wt% of the organic molecule (MPTMS or IPTES) diluted in acetone was mixed with the SC CO2 at 1.75 g/min and flowed in the reactor for 15 min. Due to the gelation of AEAPTMS in contact with SC CO2, a 1 mL glass vial of AEAPTMS was placed directly in the reactor before the beginning of the process. Finally, samples were rinsed with SC CO2 for 5 minutes. At the end of the process, samples were stored under vacuum. Samples were named X-Y, with X equal to MPT, AEA and IPT for MPTMS, AEAPTMS and IPTES, respectively, and Y equal to the grafting process temperature (in degree Celsius). The samples corresponding to the ungrafted silicon wafer with a hydrated silica layer were referenced Si-OH.

Sample characterization

X-ray reflectivity (XRR) analysis was used to determine the electron density profile of the samples. Measurements were carried out using a Bruker D8 diffractometer with CuK1 ( = 0.154056 nm) radiation. Standard -2 scans for the data collections were taken from 2 = 0º to 6º with an angular resolution of 0.01º. Reflectivity curves are usually presented as the evolution of the logarithm of intensity received by the detector as a function of the wave vector transfer

   sin 4  q
with  as the incident angle. Experimental curves were fitted using Firefx4c_6 software based on the Parratt algorithm in order to obtain electron density profiles and thus the thickness and the electron density of the prepared layer.

X-ray photoelectron spectrometry was performed using an ESCALAB 250 apparatus from ThermoElectron with a monochromatic Al Kα =1486.6 eV electron source. The takeoff angle for this spectrometer is 90º. Photoelectron spectra were calibrated to the C1s energy of the C-C bond at 284.8 eV. The background signal was obtained by the Shirley method [START_REF] Shirley | High-Resolution X-Ray Photoemission Spectrum of the Valence Bands of Gold[END_REF] . The Si2p XPS spectra were decomposed with two contributions for the reference sample Si-OH and three contributions for the grafted samples [START_REF] O'hare | Development of a Methodology for XPS Curve-Fitting of the Si 2p Core Level of Siloxane Materials[END_REF] . It has been performed with the software Advantages from ThermoElectron, considering the silicon wafer Si-Si (99.0 eV), the SiO2 layer Si(-O)4 (102.7 eV)

and the silicon from the grafted molecules Si(-O)3 (102.1 eV). The spectra were fitted with two Gauss-Lorentzian functions per contribution, corresponding to Si2p ½ ant Si2p 3/2 orbitals. These two peaks were fitted taking into account several conditions:

-The Full Width at Half Maximum is the same for the two Gauss-Lorentzian functions;

-The Area (A) of the Gauss-Lorentzian function Si2p3/2 is twice the area of the Gauss-Lorentzian function Si2p1/2: A Si2p3/2=2 A Si2p1/2;

-For binding energy (BE) BE Si2p3/2= BE Si2p1/2 -0.6 eV.

Moreover, the FWHM and BE for the contributions Si-Si and Si(-O)4 were fixed for all samples.

Water contact angle measurements were done using a Kruss Drop Shape Analyzer DSA100. A 10 µL drop was deposited on the sample, and the contact angle was measured using the Young-Laplace method. The measurements were repeated 3 times.

The roughness of the grafted silicon oxide surfaces was obtained by atomic force microscopy using an AFM Multimod8 with Nanoscope V from Brüker. The peak force measurements were performed with the same triangular SNL silicon tip from Brüker (spring constant of 350 pN/nm).

All the measurements were performed in the same laboratory atmosphere (same relative humidity.) in order to avoid artefacts of the water meniscus [START_REF] Weeks | Direct Imaging of Meniscus Formation in Atomic Force Microscopy Using Environmental Scanning Electron Microscopy[END_REF] . Cantilever calibration was performed by the thermal tune method with a hard "naked" silicon substrate and the spring constant was set to 280 pN/nm. For each sample, force curves were measured on 30 points. The resulting force curves were divided on an approach curve and a retract curve related to the tip-surface attraction and tip-surface adhesion, respectively. We focus on two different data extracted from the retract curve: the adhesion length (L) and the adhesion force (F). The principle of the peak force measurements is explained in Figure S4. The profile of the force curve depends on the adhesion forces between the tip and the elaborated layer [START_REF] Heinz | Spatially Resolved Force Spectroscopy of Biological Surfaces[END_REF] .

Results and Discussion

Figures 1 and2 present the evolution of the contact angle, the electron density and the thickness of the grafted layer obtained from the fits of the X-ray-reflectivity curves and the AFM images, respectively. In the non-grafted sample, the presence of some organic contaminations can be characterized by XPS (Figure S7). The experimental and fitted X-ray reflectivity curves of the samples and the electron density profiles obtained from the fit are presented in Figure S5 and the parameters extracted from the grafted layer (thickness, electron density, roughness and contact angle) are present in the Table S2 of the supplementary data To determine the grafted molecules density in the layer, we have used the methodology presented in the experimental part. The results of the fit of the Si2p XPS spectra are reported in Table 1. The fitted curves are also available in

where th is the thickness (Å) of the layer resulting from the fit of the reflectivity data. S1.

MPTMS grafting

The grafting of MPTMS leads to a linear increase in the layer thickness with the grafting temperature (Figure 1.a.). The electron density is lower than the theoretical value of liquid MPTMS (0.33 e-.Å -3 ) except for MPT-60. At temperatures lower than 60°C, the layer is less dense than a grafted monolayer and close to the monolayer thickness. This may be due to a partial filling of the monolayer. Above 60°C, polycondensed layers are obtained as they are thicker and less dense than the monolayer value. Additionally, the measured contact angles are higher than those for the MPTMS SAM (69-71º) [START_REF] Aswal | Self Assembled Monolayers on Silicon for Molecular Electronics[END_REF] . These results enhance the possibility of a polycondensed layer formation. For grafting at 60°C, both the electron density and the thickness of the grafted layer are consistent with the density and the length of the grafted molecule (10 Å) [START_REF] Rébiscoul | Alkoxysilane Layers Deposited by SC CO 2 Process on Silicon Oxide for Microelectronics Applications[END_REF] . This is also confirmed by the contact angle value approximately 70°2 [START_REF] Xue | Removal of Cu(II) from Aqueous Solution by Adsorption onto Functionalized SBA-16 Mesoporous Silica[END_REF] Moreover, the structure of the MPT-60 is confirmed by XPS with a peak of weak intensity from the S2s binding energy, confirming the presence of the MPTMS molecule (Figure 3). Its low intensity can be explained by the monolayer structure, since sulfur atoms account for 0.01 % to 0.02 % of the probed zone (from 5 to 10 nm). In addition, as previous results suggest that a monolayer is obtained, the calculated ratio of r equal to 0.034 Å -1 (Table 1) will be used as a reference for a monolayer.

AEAPTMS grafting

For process temperatures ranging from 60°C to 120°C, the grafted layer thickness increases, but its polycondensed structure remains the same. Indeed, all the AEAPTMS grafted layers have similar electron densities below the theoretical electron density (0.32 e-. Å -3 ) and a thickness higher than the length of the AEAPTMS molecule (20 Å) above 60ºC (Figure 1.b). Moreover, the roughness obtained by the fit of the experimental X-ray reflectivity curves increases with the grafting process temperature. The contact angle remains between 73° and 83°. The values obtained are similar to those from a previous work 2 but are not consistent with a monolayer containing surface amino groups, as the contact angle expected for an amine is near 36º [START_REF] Aswal | Self Assembled Monolayers on Silicon for Molecular Electronics[END_REF] . The results suggest that AEAPTMS grafted layers are rather polycondensed.

These results are confirmed by the AFM images of the AEA-60 sample (Figure 2.c), showing a higher roughness than the MPT-60 sample. Moreover, AEA-60 has a lower grafting ratio r than MPT-60 (0.026 Å -1 vs 0.034 Å -1 ) (Table 1). This result confirms that the density of alkoxysilanes in the layer is lower than in a monolayer. Furthermore, the N1s peak was also decomposed into three contributions 2 (Figure 5.a.): the amine from the head group -NH2 (399.3 eV), the amine inside the alkyl chain -NH-(400.5 eV) and the amine hydrogen-bonded to a surface silanol -NH2…HO-(401.6 eV). This last contribution is assigned to the hydrogen bond of the amine with surface silanols or with other hydroxylated molecules. Its presence indicates that AEAPTMS is not selfaligned at the silicon oxide surface and that some of the amine head groups are probably oriented towards the silicon oxide layer, exposing hydrolyzed silanol groups on the surface 54 . This fact probably accelerates the polycondensation of the layer.

Moreover, the AFM peak force analysis performed on the AEA-60 sample (Figure 6.a) shows that the adhesion force is weaker (F= -1.84 nN) and the adhesion length is higher (L=17.4 nm) than those of the MPT-60 sample. This means that the grafted AEAPTMS layer is less linked to the surface than the MPTMS in a monolayer. As the prepared layer is polycondensed, the inhomogeneous tip-surface adhesion leads to random bond breaking, resulting from a larger retract curve (Figure 6.b).

IPTES grafting

Regardless of the temperature, the IPTES grafted layers present thicknesses close to twice the molecule length (Table S1), and the densities are lower than that of liquid IPTES (0.42 e-.Å -3 ), except for IPT-120 (Figure 1. c). Moreover, the roughness obtained by the fit of the experimental X-ray reflectivity curves is constant whatever the grafting process temperature and close to the one of the monolayer of MPTMS. The AFM images show that the IPT-120 sample is not modified after the grafting process, with a roughness similar to that of the non-grafted sample (Figure 2). For the IPT-120 sample, the XRR and AFM results could suggest the presence of a grafted bilayer, which has the density of liquid IPTES with the presence of scattered nanoagglomerates at the surface.

Moreover, the I3d XPS spectra of IPT-80 and IPT-120 samples (Figure 7) show the presence of the iodine peak of the IPTES molecule. . However, r obtained from XPS is lower for IPT-80 (0.022 Å -1 ) than for IPT-120 (0.041 Å -1 ) (Table 1), attesting to the lower density of IPTES in the IPT-80 than in the IPT-120 sample. For IPT-120, r is higher than the r of the monolayer of the MPT-60 sample. According to this result, it is possible that a bilayer presenting an interpenetration of two monolayers was obtained. In addition, contact angles ranged from 58° to 66º for the different prepared layers except for IPT-120 (75°). This increase in contact angle for IPT-120 may suggest the formation of a dense and interpenetrated bilayer. We have also to precise that the few scattered nanoaglomerates at the surface could also increase the contact angle.

When comparing the different AFM peak force curves of the IPT-80 and IPT-120 samples, only small differences in the adhesion length and adhesion force values are observed (Figure 9): Z=13.8 nm; F=-1.83 nN for IPT-80 vs. Z=12.1 nm; F=-2.36 nN for IPT-120 sample. The increase in the adhesion force with temperature may indicate that the layer is more bonded to the silicon oxide surface for IPT-120. Nevertheless, the adhesion force in MPT-60 sample is higher than that for IPT-120. This can indicate that the molecule is not as linked to the surface as in MPT-60.

Summary on the effects of temperature and functional group on the morphology of the grafted layer

Here, we discuss the structure of the grafted layer, which depends on the functional groups and the temperature. Figure 10 illustrates the different layer morphologies obtained as a function of temperature. The MPTMS monolayer is probably formed through the classical silanization reaction [START_REF] Rébiscoul | Alkoxysilane Layers Deposited by SC CO 2 Process on Silicon Oxide for Microelectronics Applications[END_REF][START_REF] Sanli | Silylation from Supercritical Carbon Dioxide : A Powerful Technique for Modification of Surfaces[END_REF]55 . First, alkoxysilanes are hydrolyzed by the water on the extreme silicon oxide surface. Then, the molecule bonds to surface silanols. Finally, the self-assembly between the molecules gives a homogeneous monolayer. When the temperature is above 60ºC, molecules self-react before they reach the O-H surface groups, leading to disorganized and low-density layers.

In the case of AEAPTMS, in addition to the self-condensation of the grafted molecules with an increase in temperature leading to thicker layers, there is also an effect of the attraction between the amine and the silicon oxide surface. This attraction of the head group to the silicon oxide surface favors the formation of a disorganized layer regardless of the temperature. Nevertheless, the increase of the layer thickness indicates that the self-polymerization of the alkoxysilane is promoted at high temperatures.

With IPTES, the CO2 SC grafting process probably leads to a bilayer presenting an interpenetration of two monolayers. The density of this bilayer increases with the temperature until its complete filling at 120°C.

Summary and conclusions

For the first time, the morphology and the structure of the alkoxysilane grafted layer on silica plane surface were finely characterized by implementing different complementary techniques and the AFM peak force measurements. The SC CO2 grafting of IPTES has also been reported as a novelty. The experimental data show that the grafting temperature and the nature of the head group are the main parameters affecting their morphology and structure. While dense monolayers are grafted with MPTMS at 60ºC, polycondensed layers with AEAPTMS and dense bilayers with IPTES at 120ºC are obtained. Some ab initio calculations are ongoing to determine the possible conformations in a case of a bilayer. While the alkoxysilane grafting in SC CO2 was already studied in porous materials [START_REF] Perrut | Alkoxysilane Layers Compatible with Copper Deposition for Advanced Semiconductor Device Applications[END_REF][START_REF] Rébiscoul | Alkoxysilane Layers Deposited by SC CO 2 Process on Silicon Oxide for Microelectronics Applications[END_REF][START_REF] Stojanovic | Preparation of MEMO Silane-Coated SiO2 Nanoparticles under High Pressure of Carbon Dioxide and Ethanol[END_REF][START_REF] Domingo | Grafting of Trialkoxysilane on the Surface of Nanoparticles by Conventional Wet Alcoholic and Supercritical Carbon Dioxide Deposition Methods[END_REF][START_REF] Lopez-Periago | Chemical Modification of Nanometric TiO2 Particles by Anchoring Functional Silane Molecules in Supercritical CO2[END_REF][START_REF] Purcar | Surface Modification of Silica Particles Assisted by CO2[END_REF][START_REF] Sanz-Moral | Gradual Hydrophobic Surface Functionalization of Dry Silica Aerogels by Reaction with Silane Precursors Dissolved in Supercritical Carbon Dioxide[END_REF][START_REF] García-Gonzaĺez | Measurements and Correlation of Octyltriethoxysilane Solubility in Supercritical CO2 and Assembly of Functional Silane Monolayers on the Surface of Nanometric Particles[END_REF][START_REF] Zemanian | Deposition of Self-Assembled Monolayers in Mesoporous Silica from Supercritical Fluids[END_REF][START_REF] Tenorio | Thiol Group Functionalization of Mesoporous SiO2 SBA-15 Using Supercritical CO2[END_REF][START_REF] Sánchez-Vicente | A New Sustainable Route in Supercritical CO2 to Functionalize Silica SBA-15 with 3-Aminopropyltrimethoxysilane as Material for Carbon Capture[END_REF][START_REF] López-Aranguren | Regenerable Solid CO2 Sorbents Prepared by Supercritical Grafting of Aminoalkoxysilane into Low-Cost Mesoporous Silica[END_REF][START_REF] López-Aranguren | Sorption of Tryalkoxysilane in Low-Cost Porous Silicates Using a Supercritical CO2 Method[END_REF][START_REF] Sun | Surface Sol-Gel Modification of Mesoporous Silica Molecular Sieve SBA-15 with TiO2 in Supercritical CO2[END_REF][START_REF] Builes | Alkylsilane-Functionalized Microporous and Mesoporous Materials: Molecular Simulation and Experimental Analysis of Gas Adsorption[END_REF][START_REF] Ashu-Arrah | Synthesis and Characterisation of Bonded Mercaptopropyl Silica Intermediate Stationary Phases Prepared Using Multifunctional Alkoxysilanes in Supercritical Carbon Dioxide as a Reaction Solvent[END_REF] , the impact of the confinement was not determined. In order to investigate this confinement effect, nanochannels 56 made of two silicon oxide plane surfaces spaced of 3 and 5 nm will be grafted with the same molecules and will be then characterized using X-ray reflectivity. The obtained results should help us to evidence a possible confinement effect (pore size and diffusion path) on the grafting efficiency and also on the morphology of the grafted layer. The influence of the pressure on the morphology of these layers can also be of great interest for evaluating the impact of changes in the SC CO2 properties such as diffusion coefficient and density.

Figure S6 .

 S6 Figure S6. The ratio between Si-Si and Si(-O)4 𝑅 = 𝑎𝑡𝑜𝑚% 𝑆𝑖-𝑆𝑖 𝑎𝑡𝑜𝑚% 𝑆𝑖(-𝑂)4 remains around 0.2 for all the

Figure 1 .

 1 Figure 1. Evolutions of the electron density and the thickness of the grafted layers of (a) MPTMS,

Figure 2 .

 2 Figure 2. AFM images and topography profiles of Si-OH, MPT-60, AEA-60 and IPT-120 samples.

Figure 3 .

 3 Figure 3. S2s XPS spectra of MPT-60.

Figure 4 .

 4 Figure 4. AFM peak force curves of MPT-60 (a) and schematic representation of possible

Figure 5 .

 5 Figure 5. N1s XPS spectra of AEA-60 sample.

Figure 6 .

 6 Figure 6. AFM peak force curves of AEA-60 (a) and schematic representation of possible

Figure 7 .

 7 Figure 7. I3d XPS spectra (a) (b) of IPT-80 and IPT-120 samples.

Figure 8 .

 8 Figure 8. Possible conformations for the IPTES molecules in IPT-120 sample. R = OH or OCH3

Figure 9 .

 9 Figure 9. AFM peak force curves of IPT-80 (green) and IPT-120 (blue) samples (a). One retract

Figure 10 .

 10 Figure 10. Schematic representations of the morphology for the prepared layers of MPTMS (top),

Table 1 .

 1 Results of the decomposition of the Si2p peak for Si-OH, MPT-60, AEA-60, IPT-80 and IPT-120 samples.

	Bond type	Binding Energy Si-OH MPT-60 AEA-60 IPT-80 IPT-120
		eV	%at	%at	%at	%at	%at
	Si-Si	99.0	86.5	77.7	79.3	82.2	64.8
	Si(-O)4	102.7	13.5	15.2	16.0	13.1	19.4
	Si(-O)3	102.1		7.2	4.6	4.7	15.8
	R		0.2	0.2	0.2	0.2	0.3
	r (Å -1 )			0.034	0.026	0.022	0.041

  and the AFM images (Figures 2.a and 2.b) showing no surface modification after silanization. The roughness of the Si-OH sample (Ra=0.275 nm) is in agreement with values obtained in a previous work on silicon oxide surfaces (Ra=0.280 nm) 2 .

Acknowledgments. This work was supported by Université de Montpellier and Commissariat à l'Energie Atomique. We are grateful to Bruno Corso for XRR maintenance. Supporting Information. Physicochemical properties of the grafted alkoxysilanes, explanations on the experimental set-up, XRR and XPS curves and the results of the fit can be found in the supporting information

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