Tao Wu

Guillaume Lahiner

Christophe Tenailleau

Benjamin Reig

Teresa Hungria

Alain Esteve

Carole Rossi

Unexpected Enhanced Reactivity of Aluminized Nanothermites by Accelerated Aging

Keywords: Nanothermites, Aging, DSC, kinetics studies, Friedman isoconventional method, shell-life

We examine the thermal aging of four Al based thermites chosen among the most commonly used in microenergetic systems: Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4. For each nanothermite system, we applied the modified Friedmann isoconversional method from DSC signals to calculate the kinetic parameters of reactions and the function of the reaction progress in air and argon. As result, it is found that all thermites should be thermally stable for one century when stored at temperatures below 200 °C, except for Al/Fe3O4 which readily converts into Al/Fe2O3 before the thermite reaction onset. Then, we designed annealing experiments to a fixed reaction progress : (i) as simulated after 100 years storage at the ambient temperature, (ii) up to 5%. The pressure development and burning rate of aged thermites are compared with as-prepared ones. After annealing at 200 °C in air and 400 °C in argon, an unforeseen faster reaction and pressurization rates are observed for Al/CuO and Al/Fe2O3 thermites. STEM and EDX show that the annealing provokes a modification of the aluminum particles structure accompanied with a softening of the alumina which explains the enhanced pressure performances and burn rates. Whereas the same nanothermites, Al/CuO and Al/Fe2O3, annealed at 400 °C in air, feature a reduced reactivity due to the over-oxidations of the aluminum core and the formation of crystalline alumina shell. These results reveal the high complexity of aging processes in powdered nanothermites as it provokes, not only a consumption of the heat reservoir, but also a modification of the Al/oxidizer interface morphology and chemistry which has a greater impact on the material reactivity than cumulative heat release and reaction progress evolution.

Introduction

Thermite materials feature redox reactions between a metallic fuel such as aluminum and a metal oxide acting as an oxidizer. They attract a lot of interest as they are known to be stable energetic mixtures possessing high volumetric energy density. [START_REF] Fischer | Theoretical energy release of thermites, intermetallics, and combustible metals[END_REF][START_REF] Rossi | Engineering of Al/CuO reactive multilayer thin films for tunable initiation and actuation[END_REF][START_REF] Koch | Thermites -Versatile Materials[END_REF][START_REF] Rossi | Nanoenergetic materials for MEMS: A review[END_REF] Traditional thermite composed of particles at the micrometer scale do not manage a fully self-sustained reaction and have, thus, been limited to their original applications such as railroad welding. The development of metallic nanopowders in the last three decades, with regular improvements of their fabrication processes, has permitted the onset of nanothermites [START_REF] Aumann | Oxidation Behavior of Aluminum Nanopowders[END_REF][START_REF] Chaalane | The formulation and testing of new solid propellant mixture (DB plus x%BP) for a new MEMS-based microthruster[END_REF][START_REF] Dreizin | Metal-based reactive nanomaterials[END_REF][START_REF] Yetter | Metal particle combustion and nanotechnology[END_REF][START_REF] Martirosyan | Nanoenergetic Gas-Generators: principles and applications[END_REF][START_REF] Rossi | Nanoscale energetic materials[END_REF][START_REF] Wu | Performance of iodine oxides/iodic acids as oxidizers in thermite systems[END_REF] defined as a metallic fuel/oxidizer mixture in which each component has at least one dimension less than 100 nm. In certain stoichiometric and dimensional conditions, the reaction speeds can be as high as 2500 m/s. [START_REF] Martirosyan | Novel nanoenergetic system based on iodine pentoxide[END_REF][START_REF] Martirosyan | Synthesis and performance of bismuth trioxide nanoparticles for high energy gas generator use[END_REF][START_REF] Cheng | Kinetic study of thermal-and impact-initiated reactions in Al-Fe2O3 nanothermite[END_REF] Consequently, nanothermite mixtures such as Al/CuO, Al/MoO3, Al/Fe2O3, Al/Bi2O3, Al/Co3O4, etc. can replace advantageously other CHNO materials in microscale energetic (or microenergetic) systems [START_REF] Stamatis | Thermal Initiation of Al-MoO3 Nanocomposite Materials Prepared by Different Methods[END_REF][START_REF] Rossi | Solid propellant microthrusters on silicon: Design, modeling, fabrication, and testing[END_REF][START_REF] Pezous | Fabrication, assembly and tests of a MEMS-based safe, arm and fire device[END_REF][START_REF] Nicollet | Investigation of Al/CuO multilayered thermite ignition[END_REF][START_REF] Sundaram | Metal-based nanoenergetic materials: Synthesis, properties, and applications[END_REF][START_REF] Wang | Doped delta-bismuth oxides to investigate oxygen ion transport as a metric for condensed phase thermite ignition[END_REF][START_REF] Sullivan | Reactive sintering: An important component in the combustion of nanocomposite thermites[END_REF][START_REF] Jian | Nanothermite reactions: Is gas phase oxygen generation from the oxygen carrier an essential prerequisite to ignition?[END_REF][START_REF] Julien | Integration of Gold Nanoparticles to Modulate the Ignitability of Nanothermite Films[END_REF][START_REF] Glavier | Nanothermite/RDX-Based Miniature Device for Impact Ignition of High Explosives[END_REF][START_REF] Baijot | Modeling the Pressure Generation in Aluminum-Based Thermites[END_REF][START_REF] Nicollet | Fast circuit breaker based on integration of Al/CuO nanothermites[END_REF][START_REF] Wu | New coordination complexes-based gas-generating energetic composites[END_REF] for applications in civil engineering, and aerospace industry. [START_REF] Pennarun | Design, fabrication and characterization of a MEMS safe pyrotechnical igniter integrating arming, disarming and sterilization functions[END_REF][START_REF] Larangot | Solid Propellant Micro Rockets -Towards a New Type of Power MEMS in: AIAA (Ed.) NanoTech 2002 -"At the Edge of Revolution[END_REF] Despite the active experimental research effort into nanothermites over the three last decades, the literature specifically dedicated to aging studies is notably rare whereas this is a crucial point for their further deployment in current and future applications, not only for safe handling and storage but also to anticipate the aging effects through the design in order to avoid malfunctioning of microenegetic systems incorporating nanothermites. Only two exploratory experimental works [START_REF] Nie | Combustion characteristic and aging behavior of bimetal thermite powders[END_REF][START_REF] Wang | Thermodynamics and performance of Al/CuO nanothermite with different storage time[END_REF] showed that aging effects do exist in aluminized nanothermites, but the literature lacks of a rigorous study of the effects of temperature and the ambient air on the thermites' energetic performances. Wang et al. [START_REF] Wang | Thermodynamics and performance of Al/CuO nanothermite with different storage time[END_REF] experimentally investigated the shortterm storage stability of Al/CuO nanothermite prepared by electrospray method, and observed modifications of the mixture performance after 13 months aging at the ambient. They mentioned a slight lowering of the pressure generation from 686 down to 626 kPa, and an increase of the initiation temperature (from 754 to 775 °C) after observing a slight decrease at mid-term aging with further understanding. These observations focusing on a storage of 13 months at the ambient cannot be extended to other thermite couples nor aging conditions. The scope of this present study is to set up a complete study of the thermal aging behavior of four Al based thermites, prepared by nanopowders mixing, chosen among the most used in microenergetic systems: Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4. Their thermite reaction is basically governed by intrinsic properties, characterized by an ensemble of decomposition mechanisms upon heating and driven by kinetic parameters, mass transport processes (diffusion), and thermal properties (specific heat, thermal conductivity). Rigorously, all these kinetic parameters in addition to the physical and thermodynamic parameters governing reaction enthalpies and phase transitions are needed to properly define the function that gives the decomposition progress to predict the aging of nanothermites under different storage temperatures and environmental atmospheres. This is a challenging objective. A traditional and easier route is to perform a thermokinetic analysis of the material based on differential scanning calorimetry (DSC) experiments with multiple heating rates [START_REF] Simon | Isoconversional methods -Fundamentals, meaning and application[END_REF][START_REF] Roduit | The simulation of the thermal behavior of energetic materials based on DSC and HFC signals[END_REF].

Herein, we will carry out a critical study of the application of such a thermokinetic analysis, based on a Friedmann isoconversional method to evaluate the thermal aging of the four aluminized nanothermites, i.e. their energetic property changes during storage at various temperature, their shelf life and operative guidelines for achieving virtual thermal aging experiments. After the presentation of the materials preparation and their main energetic properties, the study is divided into two main parts. A first part presents the isoconversional kinetic analysis of DSC data in both air and Ar atmospheres implemented to obtain the variable activation energies of each thermite system, and further predict their lifetime at moderate temperatures (ambient -300 °C range) in air. Simulations show that, below 200 °C, Al/CuO, Al/Fe2O3 and Al/Co3O4 are thermally stable hundreds of years, whereas Al/Fe3O4 readily starts oxidizing into Fe2O3 at 200 °C. The second part aims at investigating how the aging in nanothermites affects their combustion properties. Hence, using predicted kinetic data from the DSC signals, annealing experiments are designed to reach either the reaction progress that would be obtained after 100 years storage at the ambient temperature, or a fixed reaction progress of 5%. Then, the pressure development and burn rate are systematically characterized for aged samples and compared to fresh ones. As unforeseen results, Al/CuO and Al/Fe2O3 which underwent mild aging (~15 days at 200 o C, α ≈ 1.1%) in air and high temperature aging (~440 o C, 2.6 hours, α = 5%) in argon exhibit higher peak pressure, higher pressurization rates and faster propagation speeds. These results are opposite of traditional data that typically report negative impacts of aging during several months at the ambient or a few days at 200 °C on the thermite reactivity [START_REF] Pennarun | Design, fabrication and characterization of a MEMS safe pyrotechnical igniter integrating arming, disarming and sterilization functions[END_REF][START_REF] Larangot | Solid Propellant Micro Rockets -Towards a New Type of Power MEMS in: AIAA (Ed.) NanoTech 2002 -"At the Edge of Revolution[END_REF][START_REF] Simon | Isoconversional methods -Fundamentals, meaning and application[END_REF]. A careful analysis of structure and phase changes of the nanothermites upon annealing using scanning transmission electron microscopy (STEM) and energy dispersive x-ray analysis (EDX), indicate that the annealing at certain temperatures and/or environmental conditions enhances the contact between aluminum and metal oxides and modifies the Al nanoparticles structure by softening the alumina shell, which enhances the thermite reactivity.

Materials and methods

2.1.Materials

The aluminum nanopowders (Al) (80 nm) were purchased from Novacentrix (Texas, USA) and stored in a glove box. The active Al content was 69% by mass, calculated from thermogravimetric analysis before use. Copper oxide (CuO, ~100 nm), iron oxide (Fe2O3, ~50 nm), iron oxide (Fe3O4, 50 -100 nm) and cobalt oxide (Co3O4, ~50 nm) purchased from Sigma-Aldrich were directly used as received. Mixtures of different nanothermite materials were prepared using a classic sonication method where Al nanoparticles were mixed with CuO, Fe2O3, Fe3O4 and Co3O4 nanoparticles, respectively, with an equivalence ratio of 1.2 in hexane followed by a 20 minutes sonication. SEM images (Figure S1) and XRD patterns (Figure S2) of the four starting nanothermite materials (Al/CuO, Al/Fe2O3, Al/Fe3O4, and Al/Co3O4) are given and described in supplementary file S1.

2.2.Aging procedure

About 100 mg of each nanothermite material were aged by an isothermal annealing (temperatures are given in section 3.3 after the kinetic analysis) in a furnace under Ar or air (2 L/min gas flow rate). After a natural cooling down to room temperature, the aged materials are collected and stored sealed in a glass vial for future use.

Characterization methods

For each prepared nanothermite material, the morphology, particle size, agglomeration was observed by SEM using a Hitachi S-4800 (cold FEG) operating at 2.0 kV, and by STEM using a JEOL cold-FEG JEM-ARM200F operated at 200 kV (energy resolution 0.3 eV) and equipped with a probe Cs corrector with a maximum spatial resolution of 0.078 nm. EDX spectra were recorded with a JEOL CENTURIO SDD detector. The composition and crystallographic structure of the nanothermites was characterized using a SEIFERT XRD 3000 TT XRD with Cu-Ka radiation ( = 1.54059 Å) fitted with a diffracted-beam graphite monochromator. The diffraction angle (2) was scanned from 10 to 100°. Thermal analysis was performed in both argon and air environments at several heating rates (5, 10, 20 and 30 °C/min) by Differential Scanning Calorimetry using a NETZSCH DSC 404 F3 Pegasus device equipped with a DSC-Cp sensor type P over a temperature ranging from ambient to 950 °C. Experiments were performed with ~3 mg of material in platinum crucibles in Ar (99.998% pure) or air atmosphere at a flow rate of 50 mL/min. The traces were normalized by the mass of thermite material and the plotted traces were baseline-corrected by subtracting the signal recorded during a second heating of the same sample. The pressure data were acquired by a high-frequency pressure transducer (Kistler 6215 type) : for all tests, about 13 mg of nanothermite material were packed in a pellet placed in a stainless steel, high pressure resistant cylindrical reactor [START_REF] Wang | Doped delta-bismuth oxides to investigate oxygen ion transport as a metric for condensed phase thermite ignition[END_REF] which has a total volume of 25 mm 3 and ignited with a simple NiCr wire. The mean linear burn rate was recorded using a high-speed camera SA3 Photron at a speed of 75,000 frames per second, with a 128 × 32 image resolution [START_REF] Roduit | The simulation of the thermal behavior of energetic materials based on DSC and HFC signals[END_REF] : about 25 mg of each materials were loosely deposited in polycarbonate trench (30 mm long, 1 mm wide and 1 mm deep) and ignited at one end.

2.4.Processing of DSC data for kinetics prediction

We used a model-free approach to evaluate the reaction kinetic parameters from the DSC data.

The general and effective reaction progress rate equation has the following well-known expression:

𝑑 𝑑𝑡 = 𝑘 𝑜 𝑒𝑥𝑝 (- 𝐸 𝑎 𝑅𝑇 ) . 𝑓(𝛼) (1)
R being the ideal gas constant, f(α) assumes the type of basic reactive phenomena taking place during the combustion is known. Ea and 𝑘 𝑜 are the activation energy and pre-factor associated with the thermite reaction, respectively. The most frequently used approach, if several measurements with different heating rates and/or different temperatures ranges are performed, is the method developed by Kissinger [START_REF] Kissinger | Reaction Kinetics in Differential Thermal Analysis[END_REF] assuming the occurrence of first order reactions, i.e.

𝑓(𝛼) = (1 -𝛼), limited to independent reactions occurring successively after each other, within specific temperature ranges. However, the thermite reactions, which combines chemical oxido-reduction reactions with mass transport through barrier layers [START_REF] Lahiner | A redox reaction model for self-heating and aging prediction of Al/CuO multilayers[END_REF][START_REF] Esteve | How Thermal Aging Affects Ignition and Combustion Properties of Reactive Al/CuO Nanolaminates: A Joint Theoretical/Experimental Study[END_REF][START_REF] Vyazovkin | ICTAC Kinetics Committee recommendations for performing kinetic computations on thermal analysis data[END_REF][START_REF] Abdallah | Structure and Chemical Characterization at the Atomic Level of Reactions in Al/CuO Multilayers[END_REF] that have dynamical changes in their structures over temperature cannot rigorously be reduced to a single Arrhenius law activation. Thus, we opted for the use of a differential isoconversional method based on the Friedmann method. [START_REF] Lahiner | A redox reaction model for self-heating and aging prediction of Al/CuO multilayers[END_REF][START_REF] Friedman | Kinetics of thermal degradation of char-forming plastics from thermogravimetry. Application to a phenolic plastic[END_REF][START_REF] Dreizin | Experimental study of stages in aluminum particle combustion in air[END_REF][START_REF] Stamatis | A multi-step reaction model for ignition of fullydense Al-CuO nanocomposite powders[END_REF] In contrast to Kissinger, this methodology states that the reaction rate is only a function of temperature at constant reaction progress. This can be easily demonstrated by taking the logarithmic derivative of the reaction rate equation (Eq.

(

)) at α constant. ⌈ ∂ln (dα dt) ⁄ ∂T -1 ⌉ α = [ ∂ln k o (T) ∂T -1 ] α + [ ∂ln f(α) ∂T -1 ] α 1
α being constant, f(α) is also constant, Eq. ( 2) becomes:

⌈ ∂ln (dα dt) ⁄ ∂T -1 ⌉ α = - E a R (3)
Thus, the temperature dependence of the isoconversional rate can be used to evaluate isoconversional values of the activation energy, Ea , which are now defined at each point of the reaction progress, without assuming any particular form of the reaction model. Therefore, to reconstruct an accurate temperature dependent baseline, it was assumed that the degree of conversion was the similar for all heating rates when the first and strongest peak occurred in the DSC signals. The initial and final slopes of the DSC curve recorded at 10 °C/min were used to construct a smooth initial baseline for this measurement and determine the respective degree of conversion corresponding to the position of the first peak. This degree of conversion corresponded to 5%, and it was used consistently to adjust the baseline representing the weighted averages of the initial and final slopes for other used heating rates.

Results

3.1.Thermal analysis and kinetic parameters of as-prepared thermite materials

Figure 1 shows the DSC traces of the four nanothermite materials, Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4, collected at a heating rate of 10 °C/min in Ar (Figure 1a) and air (Figure 1b).

To be noted, the heat flow values on Y axis are offset here (See Figure S3 for non-offset curves). The DSC result of pure Al is also shown in Figure S4. For all nanothermite systems two main exothermic peaks (upward peaks) are observed: first one occurs prior to the melting of Al, in a range of 550 -600 °C and the second, less intense and broader, occurs in the range of 700 -950 °C. The endothermic peak at ~660 °C corresponds to the aluminum core melting as all systems are prepared in fuel rich condition (1.2 stoichiometry). It is interesting to note that temperatures at which the first exothermic peaks occur are comparable for the four materials systems: ~525 °C and ~575 °C, for the onset (Tonset) and peak temperature (Tpeak)

respectively. This indicates that these four thermite materials, when being heated to 600 °C undergo a similar thermal decomposition pathways based on solid-state transport mechanisms : oxygen diffusion [START_REF] Jian | Nanothermite reactions: Is gas phase oxygen generation from the oxygen carrier an essential prerequisite to ignition?[END_REF] and counter diffusion of Al ions through alumina shell [START_REF] Stamatis | A multi-step reaction model for ignition of fullydense Al-CuO nanocomposite powders[END_REF]. In addition, both Al/iron oxides (Al/Fe2O3 and Al/Fe3O4) feature the less intense first exotherm which agrees with their higher decomposition temperatures (compared to other considered oxidizers) together with their high activation barrier for oxygen diffusion [START_REF] Sullivan | Simultaneous Pressure and Optical Measurements of Nanoaluminum Thermites: Investigating the Reaction Mechanism[END_REF][START_REF] Glavier | Nanoenergetics as pressure generator for nontoxic impact primers: Comparison of Al/Bi2O3, Al/CuO, Al/MoO3 nanothermites and Al/PTFE[END_REF]. This is also consistent with the fact that iron oxides are often used as moderators (weak oxidizer) to reduce the reactivity and sensitivity of other nanothermites [START_REF] Sullivan | Reactive sintering: An important component in the combustion of nanocomposite thermites[END_REF][START_REF] Stamatis | A multi-step reaction model for ignition of fullydense Al-CuO nanocomposite powders[END_REF]. It is also to be noted that the Al melting endotherm is more intense in Al/iron oxide materials, which indicates that less aluminum is being consumed by the oxides before its melting compare to the two others, CuO and Co3O4.

For each thermite systems, the heat release, ΔHexp, is calculated by integrating the DSC traces over a temperature range from 400 to 950 °C (Table 1 and Table 2). It is important to mention that while the exothermicity can be detected reliably at defined temperatures and relative peak sizes can be compared, an absolute and unambiguous value of heat release cannot be reported with accuracy. Despite this fact, the heat release, ΔHexp is an important metric often used to qualitatively compare energetic performance of different thermite couples. From the highest to the lowest ΔHexp, the ranking is Al/CuO, Al/Fe2O3, Al/Fe3O4, Al/Co3O4. Al/CuO nanothermite demonstrates the highest heat release closed to its theoretical one (4 kJ/g), whereas Al/Co3O4 material releases only 58% of its theoretical potential (~4 kJ/g). Al/Fe2O3 and Al/Fe3O4 materials release only 70% and 68% of their theoretical values, respectively. Sintering process that may happen upon heating iron and cobalt oxides [START_REF] Sullivan | Reactive sintering: An important component in the combustion of nanocomposite thermites[END_REF][START_REF] Jian | Nanothermite reactions: Is gas phase oxygen generation from the oxygen carrier an essential prerequisite to ignition?[END_REF] can explain why ΔHexp is well below the theoretical values for Al/Fe2O3, Al/Fe3O4 and Al/Co3O4 materials. Indeed, sintering process traps the oxygen inside leading to a long period of oxygen releasing time and therefore limiting the exothermicity of iron oxides and cobalt oxides-based nanothermites [START_REF] Pezous | Fabrication, assembly and tests of a MEMS-based safe, arm and fire device[END_REF]. Jian et al. also reported that the ignition temperature of Al/Co3O4 is 400 °C higher than the oxygen release temperature of Co3O4, which to some extent implies the redox reaction between Al and Co3O4 is much harder to initiate and the oxygen release from Co3O4 is much less effective than CuO [START_REF] Jian | Nanothermite reactions: Is gas phase oxygen generation from the oxygen carrier an essential prerequisite to ignition?[END_REF].

Another observation from the DSC traces is that the environment, being oxidizing or inert, does not impact the onset of reactions: temperatures at which the first exothermic peaks occur are similar in air and in Ar for all four nanothermite materials (Table 1 and Table 2). This indicates that the condensed phase still plays the dominant role in this first reactional step. However, the presence of oxygen in the environment enhances the heat released in the first exotherm: for all material systems, we note that the energy released before 600 °C is much greater in air than in argon (see Table 1 and Table 2). ΔHexp prior Al melting is increased by ~1.4, ~1.6 and ~4 for Al/CuO, Al/Co3O4 and Al/iron oxides cases, respectively. We deduce that ambient molecular oxygen contributes to the thermite reaction in all cases, and greatly when for iron oxides as they are poor oxygen providers at this temperature range. Besides, the endotherm peaks at 660 °C for all four nanothermites in oxidizing air are weaker than those in inert argon atmosphere which is in accordance with precedent comment: more aluminum is being consumed during the first exothermic step due to the presence of gaseous O2 in air. Now, looking further at the Al/Co3O4 thermite case, unlike the three other thermites, the atmospheric environment does not change the aluminum melting endotherm intensity/area. This means that the same amount of aluminum core is consumed in the first exothermic event in either inert or oxidizing environment. Al/Co3O4 powders annealed at 635 and 890 °C in air and Ar at 10 °C/min were analyzed by XRD (Figure S5) to determine their phases prior and after Al melting. Even though the characteristic peaks of aluminum are overlapped with those of Co3O4, the intensities of aluminum peaks are greatly reduced in the product collected at 635 °C (Al peak at 2θ 83° disappeared) indicating that some aluminum was consumed, which is also asserted by the increase in the heat release upon air exposure. Since the characteristic XRD peaks (2θ 10 -100°) of Co3O4 and Al2CoO4 are overlapped, it is difficult to firmly determine the phases of the sample annealed at 890 °C. However, we highly suspect that the majority of cobalt elements is present in Al2CoO4 in these two annealed samples, where some aluminum is trapped during the first exothermic event and thus leading to the inefficiency of Al/Co3O4 thermite reaction (Al/Co3O4 thermite releases only 58% of its theoretical total heat). Also, it could also explain why under air exposure, the heat release is increased by a factor of 1.6 suggesting a re-oxidation of reduced cobalt by the air, rather than oxidation of the aluminum reservoir. The data collected from DSC experiments at various heating rates (see

3.2.Prediction of heat released under isothermal conditions

The above kinetic parameters allow us to simulate systematically the behavior of each nanothermite materials in terms of the reaction progress and cumulative heat release over time and under isothermal conditions (Figure 3). For all four nanothermite materials, there are less than 1.2% of energy lost after 100 years at temperatures below 200 °C, except for Al/Fe3O4, where Fe3O4 oxidizes into Fe2O3 after several hours at 200 °C, and then follows the similar reactional pathway as to Al/Fe2O3. We can conclude that the aluminized thermites except Al/ Fe3O4, are thermally stable materials at the ambient and low temperature (< 200 °C). The only physicochemical process that is activated at low temperature does penalize only a negligible part of the energetic reservoir.

3.3.Impact of thermal aging on reactivity and combustion performances

This section aims at investigating how the aging in nanothermites affects their combustion properties. Accelerated aging i.e. annealing experiments, were carried out to simulate various aging conditions, and the combustion performance and reactivity change of the aged thermites were quantified in terms of linear burn rate, peak pressure and pressurization rate within a closed chamber.

From the kinetics parameters plotted in Figure 2, we calculated the annealing temperature-time couples in order to:

1. Obtain an equivalent aging of nanothermites for a 100-years storage under ambient condition, i.e. a reaction progress α equal to 1.2% for Al/CuO, 1.1% Al/Fe2O3, 5.5%

Al/Fe3O4 and 1% Al/Co3O4. To achieve such reaction progresses, each nanothermite was annealed at 200 °C in air for ~15 days (according to predictions in Figure 3).

2. Achieve a reaction progress of 5% which corresponds to a heat release of 204, 198, 184 and 212 J/g for Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4, respectively. Fixing the annealing time at 2.6 h, the corresponding annealing temperature was calculated from the thermokinetic analysis, and summarized in Table 3. Table 4 presents the close bomb tests results (peak pressure, pressurization rate) and linear burn rates for the four nanothermite couples after the different aging conditions and compared with as-prepared materials (fresh samples). The pressure traces and high-speed video images of the combustion of all samples tested are provided in Figure 4 and Al/Fe3O4 and Al/Co3O4 do not produce much pressure, which was expected as they are known to exhibit gasless combustion. Consistently, it has to be noted that large spherical metal (Fe and Co) balls (~500 µm to 1 mm) were collected from the closed bomb chamber after the tests indicating that the combustion occurs in the condensed phase. Both pressurization and pressure values are small, in addition to what none of them featured sustained combustion. Even the reference samples do not sustain propagation, so that discussing the effect of aging on their performances is not relevant. Another worth mentioning point is that the color of Al/Fe3O4 powder changed from black to red after aging and XRD analysis (Figure S10) indicated an oxygen state shifted from Fe3O4 to Fe2O3.

For Al/CuO thermite, the measured peak pressure values do not vary much comparing with the as prepared material, after aging at 200 °C in air or ~400 °C in argon ( For Al/Fe2O3, both peak pressure and pressurization rate values (Table 4) are much smaller than those of Al/CuO (50 vs 700 kPa and 20 vs 5400 kPa/µs) in agreement with the previous studies on these two nanothermite materials [START_REF] Glavier | Nanoenergetics as pressure generator for nontoxic impact primers: Comparison of Al/Bi2O3, Al/CuO, Al/MoO3 nanothermites and Al/PTFE[END_REF][START_REF] Wu | Silver ferrite: a superior oxidizer for thermite-driven biocidal nanoenergetic materials[END_REF] In agreement with the pressure development results, the burn rate (Table 4, last column) is increased by a factor ~2 to reach 100 and 75 m/s when Al/CuO is annealed in air at 200 o C for ~15 days and in Argon at ~400°C for 2.6 h, respectively. Although the burn rates from Al/Fe2O3 samples are significantly slower than those of Al/CuO, the same trend is found: the annealing at 200 o C in air and at ~435 o C in argon induces a beneficial effect.

Discussion

The peak pressure and pressurization rate values are higher in the Al/CuO (and a lesser extent Al/Fe2O3) thermites annealed ~15 days at 200 °C in air or 2.6 h at ~400 °C in Ar compared to fresh ones. Since the difference among the combustion data from Al/Fe2O3 samples are not significant enough to draw strong arguments, we focus now on Al/CuO cases to further understand the mechanisms at play in these observations. TEM and EDX were performed on Al/CuO ref , Al/CuO 200C_air and Al/CuO 425C_air samples (Figure 5) in order to identify changes in the structure and composition upon aging. Al/CuO 425C_air is not included here because it does not demonstrate an improving effect as strong as to Al/CuO 200C_air and thus diminished its importance in this study. 5i-k), that the alumina shell is now ill-defined and features non-uniform shapes; the outside alumina shell appears to be in jagged shapes with different thicknesses (2.5 -10 nm) around the aluminum core. This inhomogeneity in thickness and shape may be the reason for a higher rate of mass transport of species across the shell, leading to the increase in reactivity. From these observations, we suggest that, upon Al melting, a better oxidizer wetting process is achieved, that may explain this unforeseen faster reaction rate after annealing despite the reduction of the energy reservoir.

In all the TEM images of

A similar phenomenon was reported by Jacob et al. [START_REF] Jacob | Pre-stressing aluminum nanoparticles as a strategy to enhance reactivity of nanothermite composites[END_REF] where they argued that a soften effect on the shell from pre-annealing of the aluminum could lead to faster release of aluminum. Thus, the improved reactivity of Al/CuO 200C_air results from these two beneficial effects.

Looking now at powders annealed in air at 425 °C, referred to as Al/CuO 425C_air , we observe a better contact between Al and CuO than in Al/CuO ref (Figure 5m-q). However, the alumina shell thickened up to ~50 nm (Figure 5r). In certain TEM images (Figure 5o-q), we can also observe some alumina shells being peeled off from the aluminum nanoparticles (marked by red arrows), exposing the core to air that leads to further oxidation of the aluminum. Even though such alumina pilling phenomenon was also found in Al/CuO 200C_air (Figure 4j), it is much severe for Al/CuO 425C_air since there are way more alumina scattered around the aluminum particles especially in Figure 5o and5q. The EDX analysis in Figure 5r also shows significant oxidation of the aluminum nanoparticles, which might explain the reduced combustion performance. At some place, the alumina shell also shows crystalline structures (marked by arrows in Figure 5q). Previous studies have shown that transitioning from amorphous to crystalline alumina shell in Al-based energetic materials can reduce their flame propagation speeds by as much as 45% depending on the sizes of aluminum [START_REF] Walzel | Aluminum particle reactivity as a function of alumina shell structure: Amorphous versus crystalline[END_REF]. The reduction of combustion performances of Al/CuO 425C_air is therfore attributed to the thickening, and presumably difference in the alumina structure alumina (shell transitions from amorphous to crystalline), as well as the consumption of the aluminum reservoir.

Conclusions

We evaluated a modified Friedman isoconventional method as a tool to predict the shelf life of four nanothermite mixtures (Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4) at the ambient temperature and operate virtual thermal aging. Unexpectedly, results show that certain aging conditions can boost the combustion performance while sacrificing portion of the energetic reservoir. Another message drawn from these results is that when considering nanothermite materials, contrary to CHNO energetics, aging is not only a matter of the energetic reservoir consumption. It may provoke a modification of the reactant structures accompanied with a modification of the Al/oxidizer interface chemistry which has a greater impact on the material reactivity than cumulative heat release and reaction progress evolution. Obviously, caution must be taken, due to many assumptions associated with the isoconversional methods, but even more from the subtle low temperature aging mechanisms that do not leave an exploitable thermal signature in the DSC traces. Therefore, thermokinetic study must be combined with a precise evaluation of morphological and structural changes at the nanoscale in the nanothemites upon heating as it was recently proposed for nanolaminate systems [START_REF] Esteve | How Thermal Aging Affects Ignition and Combustion Properties of Reactive Al/CuO Nanolaminates: A Joint Theoretical/Experimental Study[END_REF][START_REF] Abdallah | Structure and Chemical Characterization at the Atomic Level of Reactions in Al/CuO Multilayers[END_REF]. Such an approach would make it possible to identify the main reactional steps of low temperature, as recorded in the DSC but without any a priori knowledge of their fundaments.

  [START_REF] Abdallah | Structure and Chemical Characterization at the Atomic Level of Reactions in Al/CuO Multilayers[END_REF] Concretely, for each prepared nanothermite material, a set of DSC experiments were conducted in both argon and air at 5, 10, 20, 30 °C/min in the temperature range of ambient -950 °C. DSC data are presented and analyzed in supplementary file S2. It has to be noted that because the temperature ranges used in the experiments are broad (20 -950 °C), the baselines are not linear.

Figure 1 .

 1 Figure 1. DSC traces collected for Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4 as prepared

Figure S6- 7 )

 7 were then processed in order to plot the dependence of the activation energy against the reaction progress or cumulative heat release (Figure2) in air and argon environment. In argon, the dependency of activation energy on the reaction progress indicates, for Al/CuO, Al/Fe2O3 and Al/Co3O4 nanothermite materials, a large plateau confirming that the exothermic events are dominated by a single activated process. The value of the activation energy remains close to 200, 150, and 200 kJ/mol until a reaction progress of about 25%, for Al/CuO, Al/Fe2O3, and Al/Co3O4 respectively. However, the activation energy is not constant for Fe3O4 oxidizer: we distinguish two values for Al/Fe3O4, 65 and 200 kJ/mol. The first exothermic peak could be deconvoluted into three main contributions, which results in the splitting of the activation barrier into two well stabilized parts: a first plateau is obtained at about 65 kJ/mol, while a second portion stabilizes in the range of 180 -230 kJ/mol. This might correspond to the two remaining DSC peak contributions. In contrast, in air, all DSC peaks (FigureS7) appear sharper and exhibit a single plateau as seen in Figure2, even in the case of Al/Fe3O4 material. The activation energy is around 250 kJ/mol until a reaction progress reaches ~40%, for Al/CuO, Al/Fe2O3, Al/Fe3O4, and Al/Co3O4 respectively. Despite the same temperature range, the reaction progress doubles when in air which clearly points to the role of the atmosphere at low heating ramp, which leads to faster and higher degree of the energy reservoir consumption.

Figure 2 .

 2 Figure 2. Activation energy (kJ/mol) of Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4 in Ar (solid

Figure 3

 3 Figure3shows that an aging step is reached at 300 °C. After one-year storage at such

Figure 3 .

 3 Figure 3. Simulated heat release and reaction progress of each nanothermite system at various

Figure S8 ,

 S8 and Figure S9, respectively.

Figure 4 .

 4 Figure 4. Pressure vs time for Al/CuO (left) and Al/Fe2O3 (right) under different conditions, Reference; Annealed at 200 o C for 15 days; Aged in argon; Aged in air. Results of Al/Fe3O4

  . Among the four Al/Fe2O3 samples, Al/Fe2O3 200C_air , annealed at 200 °C, features the highest peak pressure and pressurization rate values, 90 kPa and 170 kPa/µs respectively, against 50 kPa and 20 kPa/µs obtained for Al/Fe2O3 ref . The Al/Fe2O3 sample annealed at 430 °C temperature in air (Al/Fe2O3 430C_air ) features the lowest peak pressure (20 kPa) and pressurization rate values (10 kPa/µs). Unlike Al/CuO, the pressurization rate of Al/Fe2O3 is also weakened after being aged at high temperature in an inert environment (Al/Fe2O3 430C_argon ).

Figure 5 ,

 5 particles with darker color and irregular shapes are CuO nanoparticles whereas spherical particles with lighter color are aluminum ones. For Al/CuO ref , both aluminum and CuO nanoparticles are aggregated within their own clusters and then align together to create the contact between fuel and oxide particles. Both TEM and EDX analysis show uniform and flawless amorphous alumina shell in surrounding aluminum nanoparticle cores, with a thickness of 4.5 ± 0.8 nm (Figure 5d-e, thicknesses are measured by ImageJ). After annealing at 200 °C, referred to as Al/CuO 200C_air , the mixing between Al and CuO is improved compared to Al/CuO ref since the majority of aluminum nanoparticles examined are in contact to CuO nanoparticles (Figure 5g-k), which is not the case for Al/CuO ref . We also clearly observe in the TEM images of Al/CuO 200C_air (Figure

Figure 5 .

 5 Figure 5. TEM images and EDX mapping results of Al/CuO ref (a-f), Al/CuO 200C_air (g-l) and Al/CuO 425C_air (m-r).

Table 1 .

 1 Onset temperature, peak temperature and heat released collected for Al/CuO,

	Al/Fe2O3, Al/Fe3O4 and Al/Co3O4 systems in Ar environment.
		Tonset ( o C)	Tpeak ( o C)		Heat released ΔHexp (J/g)
	Material
	system	1 st	2 nd	1 st	2 nd	Total Theoretical	1 st	Endotherm
		exotherm	exotherm	exotherm	exotherm		total [1]	exotherm
	Al/CuO	528	691	581	743	3945	4072	1567	26
	Al/Fe2O3	514	733	575	701	2791	3956	470	47
	Al/Fe3O4	527	770	578	853	2516	3677	580	51
	Al/Co3O4	534	698	575	794	2457	4234	1063	8

Table 2 .

 2 Onset temperature, peak temperature and heat released collected for Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4 systems in air environment.

		Tonset ( o C)		Tpeak ( o C)		Heat released ΔHexp (J/g)
	Material					Total	Theoretical
	system	1 st exotherm	2 nd exotherm	1 st exotherm	2 nd exotherm	ΔHexp	total (J/g)	1 st exotherm	Endotherm
						(J/g)	[1]
	Al/CuO	538	697	581	772	5125	4072	2241	17
	Al/Fe2O3 539	705	581	785	4214	3956	2114	11
	Al/Fe3O4 540	710	583	796	4302	3677	2217	11
	Al/Co3O4 536	702	579	780	3958	4234	1756	10

Table 3 .

 3 Corresponding annealing temperature to obtain 5% of reaction progress after 2.6 hours for each nanothermite in argon and air.To make the discussion easier, the labelling of the four materials under different aging conditions goes as the following, taking Al/CuO for example: Al/CuO ref , Al/CuO 200C_air , Al/CuO 400C_argon , and Al/CuO 425C_air , refer to as the Al/CuO nanothermite as-prepared, annealed ~15 days at 200 °C, annealed 2.6 h at temperature value reported in the Table3in Ar and in

		Calculated annealing temperature ( o C)
	Nanothermite material
		In Ar	In air
	Al/CuO	400	425
	Al/Fe2O3	430	430
	Al/Fe3O4	450	320
	Al/Co3O4	450	440

air, respectively. The same labeling rule was applied to Al/Fe2O3, Al/Fe3O4 and Al/Co3O4 as well.

Table 4 .

 4 Results of peak pressure, pressurization rate and burn rate for Al/CuO, Al/Fe2O3, Al/Fe3O4 and Al/Co3O4 nanothermites (average value and deviation when possible are calculated on 3 experiments for each sample).

			Peak
				Pressurization	Burn rate
	Samples		pressure
				rate (kPa/µs)	(m/s)
			(kPa)
		Al/CuO ref	700 ± 100	5400 ± 2700	40 ± 2
		Al/CuO 200C_air	600 ± 10	18900 ± 7200	100 ± 5
	Al/CuO
		Al/CuO 400C_argon 600 ± 100	9000 ± 4100	75 ±5
		Al/CuO 425C_air	300 ± 10	4200 ± 400	30 ± 2
		Al/Fe2O3 ref	50 ± 10	20 ± 2	0.4 ± 0.1
		Al/Fe2O3 200C_air 90 ± 20	170 ± 10	0.6 ± 0.2
	Al/Fe2O3
		Al/Fe2O3 430C_argon 30 ± 10	10 ± 2	0.3 ± 0.1
		Al/Fe2O3 430C_air 20 ± 10	10 ± 2	0.2 ± 0.1
		Al/Fe3O4 ref	2	2
		Al/Fe3O4 200C_air 10	10
	Al/Fe3O4				No
		Al/Fe3O 450C_argon 10	10
					propagation
		Al/Fe3O4 320C_air 30	30
	Al/Co3O4	Al/Co3O4 ref	10	10

Table 4 ,

 4 peak pressure values in 600 -700 kPa range for Al/CuO ref , Al/CuO 200C_air and Al/CuO 400C_argon ). Whereas the peak pressure reduces by 50% after aging at high temperature in air (Al/CuO 425C_air ) which demonstrates that air plays a part in reducing its pressure generation performance. Surprisingly, the highest pressurization rate (~2 × 10 4 kPa/µs) among all four Al/CuO samples is generated by the Al/CuO nanothermite being annealed in air at 200 o C for ~15 days. Then, the second highest value is generated by Al/CuO 400C_argon with ~1×10 4 kPa/µs roughly the double compared to Al/CuO ref (~5.5×10 4 kPa/µs). Therefore, the annealing at 200 o C in air and at ~400 o C in argon have both a beneficial effect on the Al/CuO reactivity and combustion performance (see discussion section).

Acknowledgment

The authors acknowledge support from the European Research Council (H2020 Excellent Science) Researcher Award (grant 832889 -PyroSafe) and the Occitanie Region / European Union for their FEDER support (THERMIE grant). We thank as well Sandrine Assie-Souleille for her great support with experimental set up.

Funding Sources C.R. received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation program (grand agreement No. 832889 -PyroSafe).

Author contributions

T. W. prepared all samples, performed all characterization measurements, and drafted the manuscript. G.L. assisted T. W. in aging predictions. C. T. performed XRD measurements. B. R. assisted T.W. in DSC experiments. T. H. performed all TEM observations and EDX analysis.

A.E and C.R. provided support for the manuscript preparation. C.R. supervised this research.

Competing financial interests

The authors declare no competing financial interests.

Supplementary materials

Supplementary materials associated with this article are available in a separate document.


Alexey Bobtsov
email: bobtsov@mail.ru

Nikolay Nikolaev
email: nanikolaev@itmo.ru

Romeo Ortega
email: romeo.ortega@itam.mx

Denis Efimov
email: denis.efimov@inria.fr

State Observation of LTV Systems with Delayed Measurements: A Parameter Estimation-based Approach with Fixed Convergence Time

Keywords: Linear time-varying systems, state observer, delay systems, parameter estimation

, the estimated state converges in fixed-time with extremely weak excitation assumptions.

Introduction and Problem Formulation

It is common in control applications that the measuring devices introduce time-varying delays. This fact complicates the problem of estimating the state of a dynamical system since the output signal will be subject to delays, a problem that has been studied by many authors. For linear time invariant (LTI) systems, such a difficulty is well understood and the convergence of the observer is usually verified checking the feasibility of a linear matrix inequality [START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF]. On the other hand, the problem for LTV systems is widely open-see the literature review and the list of references in the recent papers [START_REF] Juan G Rueda-Escobedo | Gramian-based uniform convergent observer for stable ltv systems with delayed measurements[END_REF][START_REF] Sanz | Observation and stabilization of LTV systems with timevarying measurement delay[END_REF]. We consider in this paper an LTV system

ẋ(t) = A(t)x(t) + B(t)u(t) y(t) = C(ϕ(t))x(ϕ(t)), (1)
where x(t) ∈ R n , u(t) ∈ R m , y(t) ∈ R q , ϕ(t) is a continuous known nonnegative delay function that codifies the measurement delay, that is

ϕ(t) := t -d(t),
where d(t) is the time-varying delay verifying

0 ≤ d(t) ≤ d M , (2)
and the state initial condition x(0) = x 0 ∈ R n .

All system matrices are continuous and known with C(ϕ(t)) bounded and A(t), B(t) verifying the following.

Assumption 1

The state transition matrix of the homogeneous part of the system (1), denoted Φ(t, τ ), verifies

Φ(t, τ ) ≤ c 1 , ∀t ≥ τ. Assumption 2 t τ Φ(t, τ )B(τ ) dτ ≤ c 2 , ∀t ≥ τ.
x(t) = x(t), ∀t ≥ t c , (3)
for some t c ∈ (0, ∞) and for all x 0 ∈ R n , χ(0) ∈ R nχ and all continuous bounded u(t).

Main result

To streamline the presentation of our observer we recall from [START_REF] Kreisselmeier | Richness and excitation on an interval-with application to continuoustime adaptive control[END_REF] the following.

Definition 1 A bounded signal ∆(t) ∈ R is an interval exciting (IE) if there exists a time t IE ∈ (0, ∞) such that tIE 0 ∆ 2 (s)ds ≥ ρ, (4)
for some ρ > 0.

Proposition 1 Consider the system (1) satisfying Assumptions 1 and 2 and d(t) verifying ( 2) with the generalized PEBO+DREM

ξ(t) = A(t)ξ(t) + B(t)u(t) (5a) ΦA (t) = A(t)Φ A (t), Φ A (0) = I n (5b) Ẏ (t) = -λY (t) + λΨ (t)z(t) (5c) Ω(t) = -λΩ(t) + λΨ (t)Ψ(t) (5d)
with λ > 0 and the gradient parameter estimator1

θ(t) = -γ∆(t)[∆(t) θ(t) -Y(t)] (6)
with γ > 0, and the definitions

Ψ(t) := C(ϕ(t))Φ A (ϕ(t)) (7a) z(t) := C(ϕ(t))ξ(ϕ(t)) -y(t) (7b) Y(t) := adj{Ω(t)}Y (t) (7c) ∆(t) := det{Ω(t)}, (7d)
where adj{•} is the adjugate matrix. Define the state estimate as

x(t) = ξ(t)-Φ A (t) 1 1 -w c (t) [ θ(t) -w c (t) θ(0)], (8)
with

ẇ(t) = -γ∆ 2 (t)w(t), w(0) = 1, (9)
and w c (t) defined via the clipping function

w c (t) = w(t) if w(t) ≤ 1 -µ 1 -µ if w(t) > 1 -µ, , (10)
where µ ∈ (0, 1) is a designer chosen parameter. Assume ∆(t) verifies ( 4) with

ρ = - 1 γ ln(1 -µ). (11)
Then, ( 3) is ensured for some t c ≥ t IE , and all signals remain bounded.

PROOF. First, we prove the signal boundedness claim. Due to Assumptions 1 and 2 and boundedness of u(t), ξ(t) and the fundamental matrix Φ A (t) are bounded. This, together with boundedness of C(ϕ(t)), guarantees that z(t) and Ψ(t) are bounded. Finally, positivity of λ ensures that Y (t) and Ω(t) are also bounded.

Second, we apply the PEBO technique to derive a (vector) linear regression equation (LRE). For this purpose, define the error signal e(t) := ξ(t) -x(t), which satisfies ė(t) = A(t)e(t), hence e(t) = Φ A (t)θ, with θ := e(0). Consequently,

x(t) = ξ(t) -Φ A (t)θ.
The output of the system (1) then satisfies

y(t) = C(ϕ(t)) [ξ(ϕ(t)) -Φ A (ϕ(t))θ] .
From which we get a LRE, that allows us to identify θ, as z(t) = Ψ(t)θ, (12) where we used the definitions in (7a) and (7b). Notice that these signals are well defined because the assumption (2) ensures that ϕ(t) is lower bounded. Third, we apply the DREM procedure to (12) to generate n scalar LREs. Towards this end, we consider the following chain of implications

(12) ⇒ Ψ (t)z(t) = Ψ (t)Ψ(t)θ (⇐ Ψ (t)×) ⇒ Y (t) = Ω(t)θ ⇐ λ p + λ [•] and (5c), (5d)
⇒ Y(t) = ∆(t)θ, (⇐ adj{Ω} × and (7c), (7d)), with p = d dt , where we have used the fact that for any, possibly singular, n × n matrix L we have adj{L}L = det{L}I n in the last line.

The last step is the analysis of the gradient estimator [START_REF] Ortega | Adaptive control of linear multivariable systems using dynamic regressor extension and mixing estimators: Removing the high-frequency gain assumptions[END_REF] with the FCT observer ( 8)- [START_REF] Sastry | Adaptive control: Stability, Convergence and Robustness[END_REF]. Replacing the latter identity in (6) yields the error equation

θ(t) = -γ∆ 2 (t) θ(t),
where θ(t) := θ(t) -θ. The solution of this equation is given by θ

(t) = e -γ t 0 ∆ 2 (s)ds θ(0).
Notice that the solution of ( 9) is

w(t) = e -γ t 0 ∆ 2 (s)ds , hence we have that θ(t) = w(t) θ(0).
Clearly, this is equivalent to [1 -w(t)]θ = θ(t) -w(t) θ(0).

On the other hand, if ∆(t) is IE with ρ satisfying (11), we have that there exists a t c ≥ t 0 such that

w(t) = w c (t) < 1, ∀t ≥ t c .
Consequently, we conclude that

1 1 -w c (t) [ θ(t) -w c (t) θ(0)] = θ, ∀t ≥ t c .
Replacing this identity in (8) completes the proof.

Remark 2 Note that the upper bound on the delay implicitly impacts the value of t c given in ( 4), thus, it predefines the fixed time of convergence.

Remark 3 Another, more complex, solution to this problem that requires strict positivity of the delay d(t) and the knowledge of its derivative is reported in [START_REF] Sanz | Observation and stabilization of LTV systems with timevarying measurement delay[END_REF] under the classical assumption of existence of an exponentially stable Luenberger observer for the LTV system (1) with ϕ(t) = t, i.e. [9, Assumption 2]. The estimator is based on-the now classical-PDE representation of the delay, with an observer designed for the coupled LTV-PDE system.

Remark 4 In [START_REF] Juan G Rueda-Escobedo | Gramian-based uniform convergent observer for stable ltv systems with delayed measurements[END_REF] a Kalman-Bucy-like observer with fractional powers is proposed, which ensures that the state estimation enters a residual set in a finite time provided the pair (C(t), A(t)) is uniformly completely observable.

As it is well known [START_REF] Wilson | Linear System Theory[END_REF], this assumption is a sufficient condition for the verification of [9, Assumption 2]. It is obvious that both assumptions are strictly stronger than our IE condition.

Simulation Results

Consider the LTV system (1) with m = q = 1, n = 2 and

A = 0 1 -sin 2 (t) 0 , B = 0 1 , C = 1 0
For simulations consider the initial conditions x(0) = col [START_REF] Aranovskiy | Performance enhancement of parameter estimators via dynamic regressor extension and mixing[END_REF][START_REF] Fridman | Introduction to time-delay systems: Analysis and control[END_REF] and zero for all the other states.

We consider three cases:

C1 ϕ(t) = t (Fig. 1 and Fig. 2); C2 ϕ(t) = ϕ(t-τ ), τ = 1+0.9 sin(t) (Fig. 3 and Fig. 4); C3 ϕ(t) = ϕ(t -τ ), τ = 0.1 + cos 2 (3t) (Fig. 5 and Fig. 6). it ensures convergence in fixed time under the extremely weak assumption of IE. Following the PEBO approach, the observation of the state is carried out via the estima-tion of some suitable initial conditions and, in contrast to [START_REF] Sanz | Observation and stabilization of LTV systems with timevarying measurement delay[END_REF], we do not require any kind of "delay compensation". To the best of our knowledge this is the strongest result available for this problem.

Current research is under way to relax the strict assumption of exact knowledge of the system dynamics and the time delay function.

Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .Fig. 5 .Fig. 6 .

 123456 Fig. 1. Error transients x1(t) -x1(t) for different γ and case C1

Preprint submitted to Automatica

See[START_REF] Sastry | Adaptive control: Stability, Convergence and Robustness[END_REF] Chapter

2] for the definition of a gradient estimator.

Acknowledgements: The authors would like to express their gratitude to the anonymous reviewers that provided insightful remarks that were instrumental to improve the quality of the paper. This work was supported by the Ministry of Science and Higher Education of Russian Federation, passport of goszadanie no. 2019-0898.


L'impact des activités de fonderie sur les sols environnant un site industriel a été évalué dans ce travail. Il s'agissait en priorité d'établir l'état de contamination métallique des sols, pouvant résulter des activités industrielles historiques et actuelles. Notre objectif était ensuite d'identifier l'origine des métaux (géologique ou anthropogène) et les risques environnementaux qu'ils pourraient induire (contamination des eaux souterraines et risque pour la santé humaine). Dans ce contexte, une stratégie d'échantillonnage appropriée a été développée, et le raisonnement suivi est explicité ici. Des cartographies de la pollution métallique ont été réalisées par SIG, et l'étude de la répartition spatiale des métaux a permis de mettre en évidence une pollution anthropique et historique des sols.

The assessment of metal pollution in soils around a metallurgical factory was the aim of the present paper.The main objectives were to identify the origin (geological or anthropogenic) of some metals and the environmental risks they could induce (ground water contamination and human health risk).A suitable sampling strategy was developed and all the tools used in this procedure were chosen according to European guidelines and recommendations. For soil analyses, ISO protocols were mainly adopted. Cartographies of metal contents were used to represent both the horizontal and vertical distribution of metals.The results highlighted an anthropic and historical pollution of top soils around the site. No significant vertical migration was observed for the metals typical of the present industrial activities so that no groundwater contamination is to be feared at the present time. However, groundwater analyses are needed to check the migration of the tracer from the past industrial activities. The sampling strategy developed seemed to be well suited to our context. However, there is no universal sampling procedure and each one must be "purpose-fit", i.e. developed according to clearly defined objectives.

Mots clés : sols pollués; métaux; industries métallurgiques; stratégie d'échantillonnage

INTRODUCTION

Les métaux se trouvent naturellement à l'état de traces dans l'environnement et dans les sols. Cependant, les activités historiques ou actuelles des industries métallurgiques ont conduit à la contamination des horizons supérieurs des sols environnants, en raison de l'émission dans l'atmosphère de particules riches en métaux. La présence de ces éléments dans les sols entraîne un risque de contamination des horizons profonds et par conséquent des eaux souterraines par l'intermédiaire de la percolation d'eau de pluie (mobilité), ainsi que la contamination des chaînes alimentaires (biodisponibilité). De nombreuses études [START_REF] Baize | Les eléments traces métalliques dans les sols, approches fonctionnelles et spatiales[END_REF] ont mis en évidence une importante variabilité spatiale dans la distribution des métaux, selon la source de pollution (diffuse ou localisée) et la nature du rejet dans l'environnement (fumée, épandage…). Pour réaliser une évaluation appropriée des risques pour la santé humaine, une caractérisation et une quantification de l'état de pollution des sols sont nécessaires [START_REF] Malherbe | Designing a contaminated soil sampling strategy for human health risk assessment[END_REF]. La mise en oeuvre d'études appropriées sur l'impact d'une pollution métallique exige une approche raisonnée des prélèvements de sols, grâce à une stratégie d'échantillonnage qui doit être définie et adaptée en fonction des objectifs de l'étude. Beaucoup d'études ont été publiées sur la pollution métallique des sols, provoquée par des activités industrielles, mais peu d'entre elles ont décrit et discuté la méthodologie employée pour définir leur stratégie d'échantillonnage et son importance [START_REF] Ariza | Metal sequential extraction procedure optimized for heavily polluted and iron oxide rich sediments[END_REF][START_REF] Sutherland | Metal phase associations in soils from an urban watershed, Honolulu, Hawaï[END_REF][START_REF] Li | Chemical partitioning of trace and major elements in soils contaminated by mining and smelting activities[END_REF]. Cependant, plusieurs documents peuvent servir de référence pour aider à faire des choix. La norme ISO 10381-5 (2002) fournit par exemple des directives sur les stratégies de prélèvement et de traitement des échantillons. Le projet CEEM (2001) avait quant à lui pour objectif principal de systématiquement noter, étudier, évaluer et comparer les directives de prélèvement de sol utilisées dans chaque pays européen. Le but de cette étude était également de proposer des procédures opératoires standard qui pourraient être appliquées au niveau européen pour la surveillance des sols, les études de contaminations métalliques et l'évaluation des risques environnementaux. Le site étudié est composé de trois zones différentes : des habitations, des forêts et des prairies ou parcelles agricoles (Fig. 1). Concernant la géologie, les échantillons localisés autour du site ainsi que la référence locale (définie ci-après) reposent sur deux principales roches mères : des gneiss et des granites.

Mise au point d'une stratégie d'échantillonnage (Fig. 1) :

La délimitation spatiale de la zone d'étude (diamètre maximum de 3 km) a été basée sur une étude de dispersion des polluants atmosphériques émis par les cheminées du site. Etant donné le mode de contamination principal (voie aérienne, émetteur proche) il a été décidé de faire un schéma d'échantillonnage systématique à maille carrée de 500 m de côté. Cet intervalle correspond à environ 30 points de prélèvements (échantillons simples), et constitue un compromis entre une couverture optimale de la zone d'étude et la faisabilité technique et financière.

Les horizons de surface ayant des propriétés différentes des horizons profonds et les polluants étant supposés se propager par dispersion atmosphérique, 2 profondeurs ont été collectées pour chaque point d'échantillonnage : 0-10 cm et 10-50 cm. Afin d'évaluer le fond pédogéochimique de la région, une référence locale a été prélevée dans une prairie naturelle, exempte de toute pollution historique connue et dont la nature du sol est proche des sols environnant le site industriel.

Caractérisation analytique des sols :

Avant analyse, tous les échantillons de sols ont été conditionnés et prétraités suivant la norme NF ISO 11464 (AFNOR, 1994). Les principales caractéristiques pédologiques étudiées sont la capacité d'échange cationique (CEC), le pH et la teneur en matière organique [START_REF] Afnor | Evaluation de la qualité des sols, 1, Méthodes d'analyse chimique[END_REF]. Les teneurs totales en métaux ont été déterminées après minéralisation des échantillons de sols (minéralisation acide par four micro-ondes, modèle "CEM Mars 5"). Selon les concentrations, les métaux ont été déterminés avec un spectromètre d'absorption atomique par flamme (Varian SpectrAA 220, équipé d'une flamme air-acétylène et de la correction de bruit de fond par lampe deutérium) ou un spectromètre d'absorption atomique par four graphite (Varian SpectrAA 800 équipé de la correction d'effet Zeeman).Toutes les valeurs analytiques reportées dans cette étude tiennent compte d'une correction de masse vis à-vis de l'humidité résiduelle du sol, de sorte que les unités sont exprimées en mg/kg de sol sec (ou ppm). La précision pour chaque valeur représente l'écart-type obtenu sur l'analyse de trois réplicats expérimentaux.

RESULTATS / DISCUSSION

Principales caractéristiques pédologiques

L'ensemble des résultats des analyses pédologiques (Tableau 1) met en évidence une bonne homogénéité des caractéristiques pédologiques des sols pour les deux profondeurs prélevées, quel que soit le point de prélèvement considéré. Cette homogénéité des propriétés pédologiques simplifiera la discussion sur les teneurs en métaux mesurées dans les sols. Les horizons de surface ont des propriétés différentes des horizons profonds ; ils sont notamment plus riches en matière organique et ont une CEC plus élevée.

Impact des industries métallurgiques

Répartition spatiale des métaux dans les horizons de surface et en profondeur

La distribution horizontale des métaux (Fig. 2) permet dans un premier temps d'identifier rapidement s'il y a une contamination des sols environnant le site étudié, mais aussi d'évaluer l'étendue et l'intensité de cette contamination.

Traceurs de l'activité actuelle : M2, M3 et M4

Dans le cas des trois traceurs de l'activité actuelle, aucune relation n'est observée entre les teneurs en métaux mesurées dans les sols et la distance par rapport à la source d'émission (Fig. 2). Pour le métal M4, les valeurs sont réparties de part et d'autre de la référence locale et sont homogènes pour les deux profondeurs échantillonnées. Ces résultats montrent d'une part la bonne représentativité de la référence locale choisie visà-vis des sols environnant le site industriel, mais aussi l'origine naturelle de ce métal. Pour le métal M2, l'ensemble des valeurs sont faibles. Elles sont majoritairement inférieures à la référence locale et sont homogènes suivant la profondeur. M2 a donc une origine géogène, la référence surestimant légèrement les teneurs naturelles autour du site. Pour le métal M3, les valeurs sont faibles mais supérieures à la référence locale.

Elles sont soit équilibrées entre les deux horizons, soit plus élevées dans les horizons plus profonds. Ce léger gradient (augmentation de la concentration avec la profondeur) et l'absence de relation entre les teneurs mesurées et la distance par rapport au site laissent penser que M3 a une origine naturelle. Dans ce cas, la référence choisie sousestime le fond pédogéochimique de la zone échantillonnée. Ces résultats montrent l'absence de contamination par les métaux M2, M3 et M4 (traceurs de l'activité actuelle). Les teneurs en métaux mesurées ont donc une origine géologique naturelle. Des différences entre les sols échantillonnés et la référence locale sont observées. Elles proviennent de variations dans la nature des sols et donc principalement dans la nature des formations géologiques desquels ils sont issus.

Traceur des activités passées : M1

On observe une relation marquée entre la concentration en métal M1 dans les sols et l'éloignement du site (Fig. 2). Dans un rayon de 600 m autour du site, les teneurs sont élevées (jusqu'à 250 ppm) et nettement supérieures à la référence locale. Au-delà de 600 m, les teneurs diminuent avec l'éloignement du site jusqu'à atteindre la référence locale. La nature de la roche mère ne pouvant expliquer de telles teneurs en métal M1, une situation de contamination anthropique est clairement observée. De plus, ces résultats illustrent que la référence locale apparaît représentative des teneurs naturelles en métal M1 des sols environnant le site industriel. La répartition des métaux entre les deux horizons montre un gradient de concentration suivant la profondeur, les concentrations totales diminuant avec l'augmentation de la profondeur. Cet écart entre les teneurs mesurées dans les deux profondeurs de sol diminue lorsque l'on s'éloigne du site, jusqu'à l'équilibre entre ces teneurs.

  , Edit Fabiàn, François Bordas, Jean-Claude Bollinger Laboratoire des Sciences de l'Eau et de l'Environnement (LSEE), Faculté des Sciences & Techniques, 123, avenue Albert Thomas, 87 060 Limoges (France) tél : 05 55 45 75 89 ; fax : 05 55 45 74 59 * email: penilla_sonia@yahoo.fr Le travail présenté ici traite de la pollution des sols au voisinage d'un site métallurgique. Pour développer notre stratégie d'échantillonnage, nous avons décidé non pas de comparer plusieurs outils entre eux, mais de choisir, conformément aux recommandations du projet CEEM, les paramètres les plus adaptés à notre contexte et à nos objectifs qui étaient les suivants :-définir dans un premier temps si les teneurs en métal total mesurées dans les sols sont d'origine naturelle ou dues à une contamination anthropique, -s'il s'agit d'une contamination, est-elle historique ou provient-elle des activités industrielles actuelles ?, -finalement, évaluer s'il existe une possibilité de migration verticale de la pollution, ce qui pourrait entraîner une contamination des eaux souterraines.Les publications spécialisées qui visent à quantifier l'incertitude totale de telles données environnementales (par exemple[START_REF] Ramsey | Sampling the environment : Twelve key questions that need answers[END_REF], ont mis en évidence qu'il est quasiment impossible d'éliminer, voire d'estimer, toutes les sources d'erreurs liées à l'échantillonnage et à la préparation des échantillons. Il conviendra donc d'interpréter les valeurs analytiques obtenues avec précaution, en ayant conscience des limites de fiabilité que l'on pourra apporter à ces valeurs. MATERIELS ET METHODES Description du site : L'ensemble du projet dont une partie des résultats est présentée dans cet article est soumis à confidentialité. Les quatre métaux étudiés seront donc nommés M1, M2 M3 et M4. Le site industriel est exploité depuis 1920 pour la métallurgie du métal M1. Le nouvel exploitant du site (rachat en 2000) a pour principale activité la production, par des procédés de fusion, d'alliages des métaux M2, M3 et M4. Les principales sources d'émission des polluants, historiques comme actuelles, étant localisées (cheminées), notre hypothèse initiale est une contamination des sols par dépôt atmosphérique.

Figure 1 :

 1 Figure 1 : Schéma récapitulatif de la stratégie d'échantillonnage développée

6

  Figure 2 : Relation entre les teneurs en métaux M1 (traceur de l'activité passée du site) et M2-4 (traceurs de l'activité actuelle) et la distance au site U

  La répartition dans l'espace des métaux étudiés nous a permis d'aboutir aux conclusions suivantes : i) une contamination par le métal M1 apparaît nettement autour du site, et cette contamination est due aux activités industrielles passées ; ii) pour les taux de contamination les plus importants (distance au site ≤ 600 m) le métal M1 a migré en profondeur dans les sols ; iii) aucune contamination par les métaux M2, M3 et M4 n'est observée autour du site. La stratégie d'échantillonnage développée dans cette étude s'est avérée adaptée à l'évaluation de la contamination des sols situés à proximité d'un site métallurgique dans le cas d'une pollution localisée et d'une dissémination des polluants métalliques par voie atmosphérique.De plus, cette campagne d'échantillonnage aura permis de récolter des valeurs de référence représentant un état initial du site au moment de sa reprise par le nouvel exploitant. Ces données pourront également être comparées aux résultats d'investigations ultérieures, ce qui permettra d'identifier et d'évaluer les risques pour l'environnement ainsi que l'effet d'un usage continu du site sur la qualité des sols. Cela permettra également d'identifier les risques liés à la présence de métaux dans les sols en fonction de l'usage de ces sols[START_REF] Mate | Guide Gestion des sites pollués : Diagnostic approfondi -Evaluations détaillées des risques[END_REF].

	pH H2O + 0,1 Ainsi, la source principale de M1 est due à des apports pH KC1 + 0,1 --superficiels. Afin de confirmer la source d'émission de cette contamination,	CEC meq/100g + 0,2 -	% MO + 0,2 -
	min max les courbes d'isoconcentrations en métal M1 (25, 50 et > 150 4,2 6,3 échantillon de sol ppm) sont présentées sur une carte simplifiée (Fig. 3). Cette 0-10 cm carte met en évidence l'orientation nord/nord-est du panache	3,7 4,9	7,3 21,3	2,8 11,5
	min de pollution, ce qui correspond aux vents dominants. Ces 4,6 échantillon de sol résultats confirment notre hypothèse précédente au sujet de	3,7	4,5	0,5
	max la source de contamination du métal M1, qui résulte d'un 6,4 10-50 cm	4,9	17,6	9,2
	dépôt atmosphérique par diffusion des fumées suivant les vents
	dominants.	référence locale	6,4	4,4	12,2	4,4
	Migration verticale des métaux et risque de
	contamination des eaux souterraines
	Les teneurs en métaux M2, M3 et M4 mesurées dans les sols
	ont principalement une origine géologique naturelle. De plus,
	l'étude de la répartition des métaux M2, M3 et M4 entre les
	deux horizons, ne met en évidence aucune relation entre les
	teneurs en ces métaux et la profondeur. Il n'y a donc
	actuellement pas de contamination possible des eaux
	souterraines par ces trois métaux.
	Pour M1, les teneurs mesurées en surface décroissent avec
	l'éloignement du site (de 250 à 40 ppm), tandis qu'à trois
	exceptions près les valeurs mesurées en profondeur sont
	équivalentes à la référence locale ( 40 ppm). Ces trois
	exceptions correspondent aux points les plus contaminés en
	surface. Une contamination par M1 des sols de surface apparaît
	nettement autour du site et en certains points (situés à moins
	de 600 m du site) cette contamination de surface a migré en
	profondeur. Etant donné que M1 a été produit et traité de façon
	intensive depuis le début du 20° siècle, nous pouvons alors nous
	demander pourquoi cette contamination n'est pas plus étendue
	(en quantité, en distance et en profondeur). Du fait de la
	mobilité avérée de ce métal, ceci pourrait s'expliquer par une
	atténuation naturelle avec le temps par lessivage,
	exportation…Une migration verticale ayant été observée dans
	les sols à proximité du site, il serait nécessaire d'effectuer des
	analyses d'eaux souterraines avant de pouvoir conclure sur
	un risque potentiel de contamination des eaux.

CONCLUSION GENERALE

Remerciements

Ces travaux s'inscrivent dans le cadre d'un programme de recherches financé par le Contrat de plan Etat -Région Limousin et par le Conseil Régional du Limousin, et coordonné par le LSEE. La contribution de Edit Fabiàn constitue son stage du DESS "Pollutions Chimiques et Environnement" de l'université d'Orsay (2004).