Lucie Blondeau
email: lucie.blondeau@cea.fr

Eddy Foy

Hicham Khodja

Magali Gauthier
email: magali.gauthier@cea.fr

Unexpected Behavior of InSb Alloy for Mg-ion Batteries: Unlocking the Reversibility of Sb

Electrochemical behavior and performance of negative electrodes in metal batteries can be modified and improved by combining different elements. Herein, a beneficial coupling of In and Sb in the alloying reaction with Mg was considered through the preparation of the InSb alloy by mechanochemical synthesis. Despite a strong inactivity of Sb as a sole element in Mg-ion batteries, the combination of Sb with In partially unlocks the reversibility of the alloying reaction of Sb with Mg to form Mg3Sb2. For the first time, this beneficial effect is not only observed during the first magnesiation but along few tens of cycles. The analysis of the behavior of InSb through electrochemical and X-ray diffraction measurements also revealed a more complex path than reported in the literature. Uncommonly a preferential electrochemically-driven amorphization of MgIn is suggested in standard galvanostatic measurements. Crystallization of MgIn is however observed through a galvanostatic intermittent titration technique, suggesting strong kinetic effects on the microstructure, strain or disorder in the InSb phase upon magnesiation.

Introduction

The Li-ion batteries production might face some sustainability issues in a near future due to the relatively low abundance and uneven concentration of lithium on the Earth crust and more importantly due to the cobalt criticality. Magnesium (Mg)-based batteries are a promising alternative thanks to Mg high abundance, low price, safety features and its attractive theoretical capacities (2200 mAh g -1 and 3800 mAh cm -3 ). [START_REF] Song | Mapping the Challenges of Magnesium Battery[END_REF][START_REF] Yoo | Mg Rechargeable Batteries: An on-Going Challenge[END_REF][START_REF] Muldoon | Quest for Nonaqueous Multivalent Secondary Batteries: Magnesium and Beyond[END_REF][START_REF] Bucur | Confession of a Magnesium Battery[END_REF] Alike lithium, magnesium metal has a tendency to react with conventional electrolytes. However magnesium transport is limited by the formation of a blocking passivation layer on its surface instead of an ions conducting solidelectrolyte interphase (SEI) for lithium. [START_REF] Song | Mapping the Challenges of Magnesium Battery[END_REF][START_REF] Yoo | Mg Rechargeable Batteries: An on-Going Challenge[END_REF][START_REF] Bucur | Confession of a Magnesium Battery[END_REF][START_REF] Shao | Highly Reversible Mg Insertion in Nanostructured Bi for Mg Ion Batteries[END_REF][START_REF] Aurbach | On the Mechanisms of Reversible Magnesium Deposition Processes[END_REF][START_REF] Mizrahi | Electrolyte Solutions with a Wide Electrochemical Window for Rechargeable Magnesium Batteries[END_REF] This layer impedes the Mg 2+ ions path to the Mg negative electrode, and thus dramatically limits the reversible stripping/plating electrochemical processes. Aurbach's [START_REF] Aurbach | On the Mechanisms of Reversible Magnesium Deposition Processes[END_REF][START_REF] Mizrahi | Electrolyte Solutions with a Wide Electrochemical Window for Rechargeable Magnesium Batteries[END_REF][START_REF] Aurbach | Electrolyte Solutions for Rechargeable Magnesium Batteries Based on Organomagnesium Chloroaluminate Complexes[END_REF] group, among others, has overcome this problem in the 2000s, using Grignard reagents in an organohaluminate-based electrolyte in ethereal solution. Unfortunately, the difficult preparation conditions, their extreme air-sensitive and corrosive characters as well as their instability, limit their use in industry. [START_REF] Yoo | Mg Rechargeable Batteries: An on-Going Challenge[END_REF][START_REF] Bucur | Confession of a Magnesium Battery[END_REF][START_REF] Mohtadi | Magnesium Batteries: Current State of the Art, Issues and Future Perspectives[END_REF][START_REF] Aurbach | Prototype Systems for Rechargeable Magnesium Batteries[END_REF] Above all, the narrow electrochemical window of organohaluminate-based electrolytes makes it difficult to develop high-voltage positive electrode materials. [START_REF] Aurbach | On the Mechanisms of Reversible Magnesium Deposition Processes[END_REF][START_REF] Aurbach | Electrolyte Solutions for Rechargeable Magnesium Batteries Based on Organomagnesium Chloroaluminate Complexes[END_REF] An interesting alternative for the development of magnesium batteries is to replace magnesium metal with a negative electrode material, which should be compatible with conventional electrolytes. Such Mg-ion cells could be a reliable alternative to lithium-based systems. Recently, some studies showed that p-block elements (Sn, [START_REF] Parent | Realizing the Full Potential of Insertion Anodes for Mg-Ion Batteries Through the Nanostructuring of Sn[END_REF][START_REF] Cheng | Interface Promoted Reversible Mg Insertion in Nanostructured Tin-Antimony Alloys[END_REF][START_REF] Singh | A High Energy-Density Tin Anode for Rechargeable Magnesium-Ion Batteries[END_REF][START_REF] Singh | Bismuth-Tin Binary Anodes for Rechargeable Magnesium-Ion Batteries[END_REF][START_REF] Kitada | Electrochemical Reactivity of Magnesium Ions with Sn-Based Binary Alloys (Cu-Sn, Pb-Sn, and In-Sn)[END_REF][START_REF] Yaghoobnejad Asl | In Situ Dealloying of Bulk Mg2Sn in Mg-Ion Half Cell as an Effective Route to Nanostructured Sn for High Performance Mg-Ion Battery Anodes[END_REF][START_REF] Singh | Indium-Tin Binary Anodes for Rechargeable Magnesium-Ion Batteries[END_REF] Sb, [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] In, [START_REF] Singh | Indium-Tin Binary Anodes for Rechargeable Magnesium-Ion Batteries[END_REF][START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF][START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF] Pb, [START_REF] Periyapperuma | The Reversible Magnesiation of Pb[END_REF] Bi, [START_REF] Shao | Highly Reversible Mg Insertion in Nanostructured Bi for Mg Ion Batteries[END_REF][START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Ramanathan | A Porous Electrode Model for the Magnesiation and Demagnesiation of a Bismuth Electrode in Rechargeable Magnesium-Ion Cells[END_REF][START_REF] Dileo | Composite Anodes for Secondary Magnesium Ion Batteries Prepared via Electrodeposition of Nanostructured Bismuth on Carbon Nanotube Substrates[END_REF][START_REF] Wang | A Novel Bismuth-Based Anode Material with a Stable Alloying Process by the Space Confinement of an in Situ Conversion Reaction for a Rechargeable Magnesium Ion Battery[END_REF][START_REF] Murgia | Insight into the Electrochemical Behavior of Micrometric Bi and Mg3Bi2 as High Performance Negative Electrodes for Mg Batteries[END_REF][START_REF] Zhang | High Energy Density Electrode Materials for Rechargeable Magnesium Batteries[END_REF] ...) can electrochemically alloy with Mg and can be stable with conventional electrolyte such as Mg(TFSI)2 dissolved in a glyme solvent. [START_REF] Shao | Highly Reversible Mg Insertion in Nanostructured Bi for Mg Ion Batteries[END_REF][START_REF] Murgia | Insight into the Electrochemical Behavior of Micrometric Bi and Mg3Bi2 as High Performance Negative Electrodes for Mg Batteries[END_REF] Despite their interesting electrochemical reactivity, these alloys present lower theoretical gravimetric capacities than Mg metal (300 to 900 mAh g-1 ) [START_REF] Parent | Realizing the Full Potential of Insertion Anodes for Mg-Ion Batteries Through the Nanostructuring of Sn[END_REF][START_REF] Cheng | Interface Promoted Reversible Mg Insertion in Nanostructured Tin-Antimony Alloys[END_REF][START_REF] Singh | A High Energy-Density Tin Anode for Rechargeable Magnesium-Ion Batteries[END_REF][START_REF] Singh | Bismuth-Tin Binary Anodes for Rechargeable Magnesium-Ion Batteries[END_REF][START_REF] Kitada | Electrochemical Reactivity of Magnesium Ions with Sn-Based Binary Alloys (Cu-Sn, Pb-Sn, and In-Sn)[END_REF][START_REF] Yaghoobnejad Asl | In Situ Dealloying of Bulk Mg2Sn in Mg-Ion Half Cell as an Effective Route to Nanostructured Sn for High Performance Mg-Ion Battery Anodes[END_REF][START_REF] Singh | Indium-Tin Binary Anodes for Rechargeable Magnesium-Ion Batteries[END_REF][START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF][START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF][START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF] . However, their theoretical volumetric capacity can compete with Li or Mg metal, with for example a value of ~ 1900 mAh cm -3 for bismuth. Among these elements, indium presents the lowest alloying potential (0.09VMg) and a good reversibility but suffers from kinetics limitations at high rates and scarce resources. [START_REF] Singh | Indium-Tin Binary Anodes for Rechargeable Magnesium-Ion Batteries[END_REF][START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF] Differently, antimony presents one of the highest theoretical capacity (660mAh g -1 ), [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] despite its lack of electrochemical activity as pure element. Magnesiation of pure Sb to form Mg3Sb2 was only observed in thin films, [START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] while poor-if anyalloying occurs for pure micrometric Sb particles. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF] In any case, Sb electrodes present a strong irreversibility and capacity fading. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] Bi1-xSbx and SnSb alloys were respectively investigated by Murgia et al. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF] and Parent et al. [START_REF] Parent | Realizing the Full Potential of Insertion Anodes for Mg-Ion Batteries Through the Nanostructuring of Sn[END_REF] as negative electrodes for Mg-ion batteries to seek for a synergy between the different elements.

The first authors evidenced that antimony can be activated when coupled with bismuth.

However, this beneficial effect is only observed on the first magnesiation as Mg 2+ ions cannot be pulled out of the Mg3Sb2 structure in the subsequent demagnesiation. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF] Differently, Parent et al. [START_REF] Parent | Realizing the Full Potential of Insertion Anodes for Mg-Ion Batteries Through the Nanostructuring of Sn[END_REF] revealed that the electrochemical reaction with Mg modifies SnSb particles into a network of Sn and Sb subparticles, where only Sn remains electroactive. The presence of Sb, inactive towards Mg, enhances the performance of Sn through the formation of an interface stabilizing Sn into its cubic phase. A beneficial coupling of In and Bi in the InBi alloy was also explored by Murgia et al. [START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF] No clear synergy exists in InBi but a complex conversion-type electrochemical mechanism with multiple intermediate phases was evidenced. [START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF] In this work, a synergy between In and Sb in the InSb alloy was investigated by electrochemical analysis and ex situ X-Ray Diffraction (XRD). In a new way, Sb is shown to reversibly react in part with Mg. An electrochemically-driven amorphization [START_REF] Limthongkul | Electrochemically-Driven Solid-State Amorphization in Lithium-Silicon Alloys and Implications for Lithium Storage[END_REF] is here demonstrated for the first time for MgIn in Mg-ion batteries, by analogy with other alloys such as silicon in Li-ion batteries [START_REF] Limthongkul | Electrochemically-Driven Solid-State Amorphization in Lithium-Silicon Alloys and Implications for Lithium Storage[END_REF] or SnSb in Na-ion batteries. [START_REF] Baggetto | The Reaction Mechanism of SnSb and Sb Thin Film Anodes for Na-Ion Batteries Studied by X-Ray Diffraction, 119 Sn and 121 Sb Mössbauer Spectroscopies[END_REF]

Methods

In (Alpha Aesar, 99.99%) and Sb (Alpha Aesar, 99.5%) powders were purchased and used as received. The InSb compound was prepared by planetary ball milling in a Fritsch Pulverisette with 3 stainless steel balls in a stainless-steel vial. In and Sb powders were introduced in stoichiometric amount with a ball-to-powder ratio of 1:70 and milled for 6 h under inert atmosphere (Ar). A milling yield (corresponding to the ratio of powder masses after and before milling) higher than 95% was obtained, demonstrating poor welding between the powder and the balls/vial. Powders samples were characterized by XRD on a RU-200B rotating anode X-ray generator using monochromotized Mo radiation (λ=0.7093Å). Scanning Electron Microscopy (SEM) was performed on a SEM-FEG Zeiss Ultra 55 model operating at 3kV in order to study the morphology of the InSb powder.

Composite electrodes were formulated by mixing 80 wt.% of the InSb ball-milled active material with 10 wt.% of carbon (Csp, 99+ %, Alfa Aesar) and 10 wt.% of polyvinylidene fluoride (PVDF, Solvay Solef) with N-methyl-2-pyrrolidone (NMP, 99 %, Acros Organics) as the solvent. The slurry was stirred for 30 min before being cast onto a Cu foil (99.8%, Alfa Aesar). The film was dried at room temperature during 24 h and subsequently for 24 h at 120°C under vacuum. Electrode disks were cut out and stored in a glove box. The average active material loading was around 0.35 -0.52 mg.cm -2 .

Electrochemical tests were performed in a half-cell configuration using 2-electrodes Swageloktype cells with a Mg disc (99.95%, Gallium Source) as both reference and negative electrode.

Whatman GF/A glass microfiber filters were used as separators. The electrolyte consists of a mixture of ethylmagnesium chloride (EtMgCl, 2.0 M in THF, Sigma-Aldrich) and diethylaluminium chloride (Et2AlCl, 97 %, Sigma-Aldrich) in a 1:1 molar ratio solubilized in anhydrous tetrahydrofuran (THF, 99.9 %, Carlo-Erba). The final concentration of the organohaluminate complex "EtMgCl-Et2AlCl" is estimated ~ 0.35 mol. L -1 . All the operations of cell assembly were made in an Ar-filled glove box. Galvanostatic tests were performed at room temperature using a research-grade potentiostat (VMP3, Biologic) between 0.005 and 0.8 VMg (vs. Mg 2+ /Mg). The electrochemical tests were performed at C/50, if not otherwise mentioned, where C/n rate means that the full magnesiation is reached in n hours. C/50 rate corresponds here to a current density of 11 mA g -1 . A galvanostatic intermittent titrate technique (GITT), which consists of a series of current pulses (CCV : constant current voltage), each followed by a relaxation time (OCV : open-circuit voltage), was used to study the electrochemical mechanisms.

In our case pulse periods of 1 h at a C/50 rate followed by open circuit courant (OCV) periods for 2 h were used. Every electrochemical test was repeated at least three times to verify the reproducibility.

Ex situ XRD characterization after cycling was performed on electrodes previously washed with THF to avoid any trace of the electrolyte. To avoid reactivity with air, all XRD measurements were carried out by protecting the sample with Kapton tape in the glove box.

Results and discussion

High energy mechanochemical synthesis is known as an efficient method to synthesize intermetallic compounds [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF][START_REF] Suryanarayana | Mechanical Alloying and Milling[END_REF][START_REF] Fecht | Nanocrystalline Metals Prepared by High-Energy Ball Milling[END_REF][START_REF] Matsui | Active Material for Rechargeable Battery[END_REF] such as InSb. [START_REF] Johnson | Electrochemistry and In-Situ X-Ray Diffraction of InSb in Lithium Batteries[END_REF] The XRD pattern (Figure 1) and the profile matching (Supplementary Information, Figure S1) collected on the synthesized powder shows the formation of a cubic phase corresponding to the InSb compound with crystallographic parameters in agreement with the literature (space group F4 ̅ 3m, a = 6.476 Å ±0.003 34 ). A small diffraction reflex at 18° is also detected and is explained by iron impurities from the vial and balls during the milling process. The SEM image (Figure 1b) evidences the formation of large micrometric aggregates of around 1 to 10 μm made of sub-micrometric particles roughly welded together.

Electrochemical mechanism of InSb-Mg alloying: competition between crystallization and electrochemically-driven amorphization

To determine if a beneficial coupling between In and Sb exists, the electrochemical properties of the InSb alloy were first compared with the behaviour of a simple mixture of pure micrometric In and Sb powders in stoichiometric amounts. The first magnesiation of the In+Sb blend (Supplementary Information, Figure S2) presents only one electrochemical plateau at ~ 0.09 VMg and only 0.6 Mg 2+ are inserted in the electrode. The voltage value of the plateau is characteristic of the reactivity of In with Mg. [START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF] Therefore, only In is reacting while Sb is completely inactive. This is expected as no electrochemical activity for Sb micrometric particles was observed previously with a fast decrease of potential down to 0VMg. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] On the contrary, the InSb composite electrode (Figure 2a) is able to host 1.9 moles of Mg 2+ in its structure, demonstrating a beneficial effect between the two elements. The electrochemical profile suggests that Sb is electrochemically activated by the presence of In. This behavior seems comparable to what was already showed by Murgia et al. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF] for the Bi1-xSbx where antimony is activated in the presence of Bi for the first magnesiation, and will be discussed in more details below. 2a). However, the presence of two electrochemical reactions appears more clearly on the representation of the incremental capacity (Figure 2b), where two peaks at approximatively 0.16 VMg and 0.07 VMg can be noted. As already reported in the literature, pure

In electrodes display a flat plateau around 0.09 VMg related to the two-phase reaction leading the formation of crystalline MgIn (Figure S3). [START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF] For the Sb electrodes, the two-phase reaction with Mg is predicted at 0.31 VMg to form Mg3Sb2. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] Therefore, the alloying region around 0.16 VMg (before ~1.25), should be linked to the formation of Mg3Sb2. The second region occurring at a lower potential around 0.07 VMg should be associated with the reaction of In with Mg to form MgIn. This agrees well with the electrochemical reactivity of the pure In and Sb elements, [START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF][START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF] and can be summarized as follows:

𝐼𝑛𝑆𝑏 + 2.5𝑀𝑔 2+ + 5𝑒 -→ 1 2 𝑀𝑔 3 𝑆𝑏 2 + 𝑀𝑔𝐼𝑛 (1)
At the beginning of the first magnesiation, the cell voltage drops rapidly down to ~ 0.07 VMg (this value varies between 0.05-0.1 VMg depending on the cell) but then rises again to finally reach the first alloying plateau at 0.16 VMg. As already seen in other materials [START_REF] Shao | Highly Reversible Mg Insertion in Nanostructured Bi for Mg Ion Batteries[END_REF][START_REF] Singh | A High Energy-Density Tin Anode for Rechargeable Magnesium-Ion Batteries[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF][START_REF] Periyapperuma | The Reversible Magnesiation of Pb[END_REF][START_REF] Dileo | Composite Anodes for Secondary Magnesium Ion Batteries Prepared via Electrodeposition of Nanostructured Bismuth on Carbon Nanotube Substrates[END_REF] this can be related to the equilibrium between first the nucleation of the Mg3Sb2 grains and second the growth of the particles. The Mg-In/Sb electrochemical alloying potentials are not really stable in the first magnesiation but, at the end, the system reaches a capacity of 535 mAh g -1 which is very close to the theoretical capacity of 566 mAh g -1 (calculated based on the formation of Mg3Sb2

and MgIn).

The first demagnesiation (Figure 2a, b) shows two well-defined plateaus at 0.39 VMg and 0.14

VMg. Considering the dealloying potential values of Mg3Sb2 [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] and MgIn, [START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF][START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF] the higher plateau is attributed to the dealloying of Mg3Sb2, while the second is assigned to MgIn dealloying. Mg dealloying reaction occurs at 0.14 and 0.39 VMg during charge so the system presents a consequential polarization of about 230 mV (for the alloy Mg3Sb2) and 80 mV (for MgIn). The system is poorly reversible at the first cycle, with a capacity retention of only 39 % (Figure 2a).

Few factors can explain this low reversibility. The large volume expansion during the first cycle (theoretical volume expansion estimated ~100%) could lead to the disconnection of some active material particles from the global matrix, or Mg 2+ diffusion can be impeded in some large particles due to the sluggish diffusion of Mg 2+ in solids. [START_REF] Shao | Highly Reversible Mg Insertion in Nanostructured Bi for Mg Ion Batteries[END_REF][START_REF] Aurbach | On the Mechanisms of Reversible Magnesium Deposition Processes[END_REF][START_REF] Mizrahi | Electrolyte Solutions with a Wide Electrochemical Window for Rechargeable Magnesium Batteries[END_REF][START_REF] Aurbach | Electrolyte Solutions for Rechargeable Magnesium Batteries Based on Organomagnesium Chloroaluminate Complexes[END_REF][START_REF] Mohtadi | Magnesium Batteries: Current State of the Art, Issues and Future Perspectives[END_REF] A reactivity between the electrode and the electrolyte could also irreversibly consume electrons and/or Mg 2+ ions, implying a poor stability of the alloy in the electrolyte, in contradiction to the expected stability of THF based solutions. [START_REF] Lu | On the Electrochemical Behavior of Magnesium Electrodes in Polar Aprotic Electrolyte Solutions[END_REF] Most probably, we believe that the irreversibility could arise from the poor ability of the Mg3Sb2 to be demagnesiated. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] In contrast with the first cycle, the second magnesiation (Figure 2a In order to better understand the mechanism of Mg alloying/de-alloying in InSb, a galvanostatic intermittent titration technique (GITT) was applied (Figure 3) and compared with the conventional galvanostatic cycling described previously (Figure 2a). The procedure allows the electrode to reach its equilibrium potential and can provide insights on the kinetics phenomena. Following the OCV points on the GITT, it is possible to see a different behavior upon alloying than the GCPL and a similar trend during the dealloying. In contrast to the conventional cycling, the thermodynamic potentials of alloying processes are clearly defined with two plateaus upon the magnesiation in GITT (Figure 3). The plateaus of the first magnesiation are observed around 0. The phase evolutions on the first magnesiation are discussed in the following points:

At the beginning of the magnesiation, up to the reaction of ~1 Mg 2+ , a large decrease of the InSb reflex and an increase of new reflexes at 10.3°, 11.2° and 11.7° are observed. These new reflexes are characteristic of Mg3Sb2 [START_REF] Martinez-Ripoll | The Crystal Structure of α-Mg3Sb2[END_REF] and therefore confirms the magnesiation of Sb to form Mg3Sb2, in agreement with the electrochemical data. Simultaneously another diffraction reflex corresponding to pure metallic indium 37 appears at 15.0°, indicating that In is extruded from the InSb alloy, as already seen in the lithiation of InSb. [START_REF] Vaughey | Structural and Mechanistic Features of Intermetallic Materials for Lithium Batteries[END_REF] After the alloying of around 1.5 Mg 2+ within the material, Mg3Sb2 and pure In are still detected, while the intensity of the characteristic reflex of InSb at 10.9° decreases.

After full magnesiation, Mg3Sb2 and pure In are still observed while some InSb remains, revealing that not all the active material has reacted. Surprisingly, while the electrochemical profile and ex situ diffraction data of In electrodes will suggest the formation of crystalline MgIn (Figure S3), only a small contribution of crystalline MgIn is supposedly detected with the small reflexes appearing at 9.3°, 12.8° and 15.6° ( P4/mmm MgIn) [START_REF] Ino | X-Ray and Thermal Analyses of the Order-Disorder Transition in Magnesium-Indium Alloys[END_REF] . It is worth mentioning that the diffraction reflex of Mg3Sb2 at 15.2° is close to the main reflex at 15.6° of MgIn, rendering the analysis of the patterns relatively complex. Above all, it is important to note that the characteristic reflexes of MgIn were not always observed on fully magnesiated samples when the tests were reproduced (Figure S5), suggesting the presence of MgIn in an amorphous state. To better identify the phases present during Mg alloying in InSb, XRD patterns from samples fully magnesiated either at a C/100 rate or in GITT mode were compared (Figure 5, complete patterns on Figure S6). The XRD pattern of the GITT sample collected at the end of the first magnesiation shows a different profile in comparison with the GCPL sample pattern.

Interestingly the XRD pattern of the GITT magnesiated sample presents sharper and better-defined reflexes, denoting more highly-crystallized phases. More importantly, the main difference is the unambiguous presence of three sharp reflexes at 9.3°, 12.8° and 15.6° which corresponds to the P4/mmm MgIn phase. MgIn crystallization is clearly evidenced on the GITT sample -in which the electrode is allowed to reach a steady state-while the crystallization is circumvented with the GCPL sample. The presence of amorphous MgIn seems counterintuitive as the magnesiation of pure In 20 (Figure S3) and of an InBi alloy [START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF] leads to crystalline MgIn.

However the capacity extracted from InSb (Figure 2) can not only be justified by the alloying/dealloying process of Sb. The relatively small proportion of crystalline MgIn, if any, observed in the GCPL electrode, compared to what is expected from the electrochemical profile and capacity recovered, indirectly suggests an electrochemically-driven amorphization of the MgIn phase. The crystallization of MgIn is thus kinetically dependent, implying a struggle between amorphization and crystallization depending on the rate applied. What frustrates the crystallization of MgIn has still to be fully rationalized. However, we can suggest that the extrusion of In from InSb during the first part of the magnesiation could cause dramatic changes of morphology and/or enhance the atomic disorder and create nanosized domains of In (electrochemical grinding). The formation of highly reactive In nanoparticles could completely modify the reaction path of In with Mg and conducts to the formation of a nanosized amorphous MgIn phase. Moreover, the sluggish diffusion of Mg 2+ could slow down the atomic rearrangement in the MgIn structure, while an inner strain could also hinder the formation of crystalline MgIn. [START_REF] Baggetto | The Reaction Mechanism of SnSb and Sb Thin Film Anodes for Na-Ion Batteries Studied by X-Ray Diffraction, 119 Sn and 121 Sb Mössbauer Spectroscopies[END_REF] In order to confirm our hypotheses, some in situ XRD and in situ X-ray absorption experiments should be performed to identify the phases formed at a more local scale.

By coupling the XRD characterization and electrochemical data, we thus propose the following reaction pathway for the first magnesiation of InSb, based on two successive biphasic reactions governed by the magnesiation of Sb and In, with first an extrusion of In and then an incomplete transformation of the InSb material which leads to the crystallization of Mg3Sb2 and the electrochemically-driven amorphization of MgIn using galvanostatic cycling:

𝑰𝒏𝑺𝒃 + 3 2 𝑥 𝑀𝑔 2+ + 𝑥𝑒 -→ (1 -𝑥)𝐼𝑛𝑆𝑏 + 𝑥 ( 𝟏 𝟐 𝑴𝒈 𝟑 𝑺𝒃 𝟐 ) + 𝑥𝐼𝑛 (2)
𝑥𝐼𝑛 + 𝑦𝑒 -+ 𝑦 𝑀𝑔 2+ → 𝑦𝑴𝒈𝑰𝒏 + (𝑥 -𝑦)𝐼𝑛 (3)
Finally, we also investigated the phases formed at the end of the first cycle. The XRD pattern acquired on a completely demagnesiated electrode is shown in Figure 5. Interestingly, the first demagnesiation does not produce Sb and In as pure elements from the demagnesiation of MgIn and Mg3Sb2 phases, but instead the intermetallic compound InSb is reformed as the major product. Nevertheless, some Mg3Sb2 and In still remain which demonstrates an uncompleted demagnesiation of the overall electrode and explain for the most part the large irreversibility observed in the first cycle.

Cycling performance and evidence for Sb electrochemical activation

The study of InSb-based composite electrodes was firstly proposed to investigate a possible synergy between In and Sb. The alloy could present a better capacity due to the high capacity of the antimony, namely 660 mAh g -1 , [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] and the lowest working potential of indium among the other p-block elements. [START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF][START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF] It is interesting, from a fundamental point of view, to evaluate the electrochemical performance and to compare them with In and Sb-based electrodes already reported in the literature. [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF] Electrochemical cycling tests at a C/50 rate were performed upon more than 40 cycles on an InSb-based electrode (Figure 6). Despite the lack of activity of the pure Sb reported previously, the InSb alloy presents capacities higher than indium composite electrodes (first magnesiation capacity around 460 mAh g -1 (theoretical capacity 467 mAh g -1 ) [START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF] at least for the first cycle.

After the first cycle a sharp decrease of the capacity is observed and then the capacity stabilizes for both alloying and dealloying reactions with a slight gradual increase from 270 to 350 mAh g - until a C/25 rate while it then regularly decreases to reach almost no capacity at 1C. These results manifest kinetics limitations during the InSb magnesiation/demagnesiation processes. This emphasizes the sluggish diffusion of Mg 2+ in the solid and is probably accentuated by the reactivity of antimony [START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] in the alloy and the high stability of the ionic Mg-Sb bonds [START_REF] Watson | On the Electronic Structure of the Semiconducting Compounds Mg3Bi2 and Mg3Sb2[END_REF] in the Mg3Sb2 phase. However, we want to stress out that the present study was first aimed at understanding the fundamental electrochemical behavior of InSb, thus formulations of the electrode and the electrolyte have not been optimized. We believe that nanostructuration of the material could improve this performance by reducing the diffusion paths for Mg 2+ ions.

Conclusion

The electrochemical reactivity and performance of the InSb alloy as a negative electrode for and some subsequent capacities around 300 mAh g -1 . Some questions remain on the mechanisms leading to the activation of Sb in the InSb alloy, whether it can involve a peculiar interface in the material or be related to the microstructure. This requires further investigation for example with in situ XRD/X-ray absorption to understand how to further trigger the full reactivity of Sb in Mgion batteries.

Figure 1 .

 1 Figure 1. (a) XRD pattern and (b) SEM picture of the InSb powder obtained by high energy ball-

Figure 2 .

 2 Figure 2. (a) Galvanostatic cycling of an InSb/Mg battery at a C/50 rate and (b) corresponding

  , b) exhibits two clear plateaus around 0.3 and 0.08 VMg in agreement with the reactivity of Sb[START_REF] Murgia | Electrochemical Mg Alloying Properties along the Sb1-xBix Solid Solution[END_REF][START_REF] Arthur | Electrodeposited Bi, Sb and Bi1-xSbx Alloys as Anodes for Mg-Ion Batteries[END_REF] and In[START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF] and a small third flexion in between at 0.21 VMg. The small third flexion may highlight a change in the electrochemical pathway, with for example a reorganization of the material structure upon alloying, or an electrochemical reactivity of some impurities in the powder. The presence of the higher plateau at 0.3 VMg demonstrates that Sb is not only active at the first cycle but can partially be reversible upon cycling. This confirms the peculiar beneficial effect provided by the presence of In with Sb. Strong morphology modification in the active material[START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF] upon the first cycle could explain the large profile difference observed between the first and the second magnesiation. In fact, the electrochemical reaction can induce a complete disaggregation of the initial micrometric grains of active materials, inducing a grinding of the material explaining the different electrochemical behavior between the first and second cycle.[START_REF] Murgia | Electrochemical Magnesiation of the Intermetallic InBi through Conversion-Alloying Mechanism[END_REF]

Figure 3 .

 3 Figure 3. GITT profile (grey) at the first cycle obtained from an InSb/Mg battery (1 h of

  3 and 0.06 VMg compared to 0.16 and 0.07 VMg for the GCPL. The successive formations of Mg3Sb2 and MgIn upon magnesiation of InSb are further confirmed. The polarization is reduced to 40 mV for both Mg3Sb2 and MgIn compared to 230 mVMg and 80 mVMg in the conventional galvanostatic test respectively. The large difference observed between CCV and OCV value in the Mg3Sb2 formation region reveals a kinetic limitation at the first cycle, which will be discussed later on. The conclusions drawn above on the formation of Mg3Sb2 and MgIn as products of the alloying reactions of InSb with Mg were confronted with ex situ XRD measurements on InSb composite electrodes cycled at C/100 rate. To follow the formation of the Mg alloys and to better understand the reaction mechanisms, cells were stopped and electrodes removed at different states of the magnesiation. The corresponding XRD patterns are presented on Figure 4 (complete patterns on Figure S4).

Figure 4 .

 4 Figure 4. Ex situ XRD patterns collected on InSb-based composite electrodes stopped at

Figure 5 .

 5 Figure 5. Ex situ XRD patterns collected on the pristine InSb powder, and on electrodes stopped

Figure 6 .

 6 Figure 6. (a) Evolution of the capacity of magnesiation and demagnesiation of InSb/Mg battery

1 .

 1 The capacity recovery upon cycling can be related to an in situ nanostructuration of the active material during cycling allowing a deeper diffusion of Mg 2+ ions into the electrode and therefore generating a more important proportion of InSb being active towards Mg 2+ . However, electrolyte's degradation along cycling can also possibly explain the extra-capacity observed and give the impression of a global increase of the capacity. These capacities observed are nevertheless lower than those reached with pure In electrodes. Murgia et al.[START_REF] Murgia | First Investigation of Indium-Based Electrode in Mg Battery[END_REF] showed a first magnesiation capacity of 460 mAh g -1 with then capacities stabilizing approximatively around 440 mAh g -1 . Nevertheless, we evidence for the first time a partial reversibility of Sb in an electrode for Mg-ion batteries. The proportion of the magnesiation capacity due to Mg3Sb2 was calculated based on the capacity obtained on the first alloying plateau (Figure7). The formation of Mg3Sb2 is responsible for ~ 42% of the capacity at the first magnesiation while the value then slowly decreases to reach ~ 20% after 20 cycles and ~ 10% after 40 cycles. This confirms that Sb is electrochemically activated by the presence of In and that its contribution to the total capacity is not negligible. Despite a large irreversible capacity at the first cycle, we show that InSb cycling promotes the reactivity of Sb, not only at the first magnesiation, but also upon further cycling. The activation of Sb in InSb could be explained by the formation of a peculiar interface favoring Sb reactivity. This has already been observed in the promotion of Sn reactivity in SnSb by the presence of Sb, despite the inactivity of Sb in the material.[START_REF] Parent | Realizing the Full Potential of Insertion Anodes for Mg-Ion Batteries Through the Nanostructuring of Sn[END_REF][START_REF] Cheng | Interface Promoted Reversible Mg Insertion in Nanostructured Tin-Antimony Alloys[END_REF] Finally, we also suggest a possible effect of a different microstructure in the InSb material compared to a pure micrometric Sb powder. The different microstructure inside the active material could stimulate the reactivity of Sb and create a highly reactive Mg3Sb2 phase more prone to demagnesiation. These hypotheses need to be confirmed with further investigations.

Figure 7 .

 7 Figure 7. Evolution of the full capacity of magnesiation of InSb/Mg battery and of the part of the

  Mg-ion batteries have been investigated. As expected from the literature on pure In and Sb, the potential profile of the first magnesiation evidences two successive two-phase reactions corresponding to the formation of Mg3Sb2 and MgIn. However, surprisingly, XRD patterns collected at different magnesiation states suggest a kinetically dependent electrochemicallydriven amorphization of MgIn. This phenomenon seems unique to the InSb phase as crystalline MgIn has always been detected in the literature in the case of pure In or InBi electrodes. This behavior suggests a possible competition between crystallization and amorphization in the material which could be explained by an increase of disorder at the atomic scale in the material, an internal strain or modification of the microstructure upon reaction. Further studies are required to understand the driving force the amorphization of MgIn, and how it can influence the electrochemical behavior and performance of InSb, and in a broader context of alloy materials in metal batteries. Additionally, we also revealed a strong synergy between In and Sb in InSb, with the promotion of the electrochemical activity of Sb towards magnesiation. In contrast to what was already reported in other studies, Sb is shown for the first time to be able to partially react reversibly with Mg along few cycles. The first magnesiation reaches a capacity > 500 mAh g-1

Acknowledgment

Financial funding from the Agence Nationale de la Recherche (ANR) of France under the contract ANR-16-CE05-0004 is acknowledged.

Supporting Information

Supplementary electrochemical data and ex situ X-Ray Diffraction data

Conflicts of interest

There are no conflicts to declare.


Anaïs Vergne

Laurent Decreusefond

Philippe Martins

Computing the k-coverage of a wireless network

Coverage is one of the main quality of service of a wireless network. k-coverage, that is to be covered simultaneously by k network nodes, is synonym of reliability and numerous applications such as multiple site MIMO features, or handovers. We introduce here a new algorithm for computing the k-coverage of a wireless network. Our method is based on the observation that k-coverage can be interpreted as k layers of 1-coverage, or simply coverage. We use simplicial homology to compute the network's topology and a reduction algorithm to indentify the layers of 1-coverage. We provide figures and simulation results to illustrate our algorithm.

I. INTRODUCTION

Wireless networks encompass cellular networks, WiFi access points, sensor networks, and so on. With the increasing usage of high data rates mobile devices such as smartphones and tablets, and the development of the Internet of Things (IoT), they have become indispensable in our everyday lives. A common key quality of service of this type of networks is the coverage. The coverage of a wireless network is the set of points that are in the sensing range of at least one network node. The greater the covered area, the more mobile devices can have access to it. For cellular networks, coverage can even be a governmental obligation. The absence of coverage holes inside the covered area is needed to offer a continuous access to services.

However, network nodes are often not regularly deployed on lattice or according to the hexagonal model in practice, see [START_REF] Gomez | A Case Study on Regularity in Cellular Network Deployment[END_REF] for cellular networks in France for example. And, deciding whether a set of network nodes does cover a given area is not that easy for arbitrary deployments. Simplicial homology can help us do that, considering the network nodes GPS positions and their coverage ranges, it is possible to build a purely combinatorial object, namely an abstract simplicial complex, of which it is possible to compute the topology. Basically an abstract simplicial complex is the generalization of the concept of graph, it is made of k-simplices where 0-simplices are vertices, 1-simplices are edges, 2-simplices are triangles, 3simplices are tetrahedron and so on. In particular, geometrical simplicial complexes such as the Cech complex and the Vietoris-Rips complex, represent exactly and approximatively respectively, the topology of the union of the coverage disks as stated in [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF]. Then algebraic topology, [START_REF] Hatcher | Algebraic Topology[END_REF], is a mathematical tool that can compute the number of connected components, of coverage holes, and of 3D voids, namely the Betti numbers of the simplicial complex representing the network, as explained in [START_REF] Silva | Coordinate-free Coverage in Sensor Networks with Controlled Boundaries Via Homology[END_REF]. Since, the computational time to obtain the Betti numbers can explode with the size of the simplicial complex, many works focus on faster ways to compute them, for instance in a decentralized way [START_REF] Muhammad | Decentralized Computation of Homology Groups in Networks by Gossip[END_REF], using persistent homology [START_REF] Zomorodian | Computing Persistent Homology[END_REF], thanks to chain complexes reduction [START_REF] Kaczyński | Homology computation by reduction of chain complexes[END_REF] , or with witness complexes reduction [START_REF] Silva | Topological estimation using witness complexes[END_REF]. In our work we use simplicial complex reduction to reduce a simplicial complex to the minimum number of points needed to provide 1-coverage. Precisely, we use the reduction algorithm presented in [START_REF] Vergne | Reduction algorithm for simplicial complexes[END_REF], that can also be found for coverage hole detection in [START_REF] Yan | Homology-Based Distributed Coverage Hole Detection in Wireless Sensor Networks[END_REF] and for energy efficiency in cellular networs in [START_REF] Vergne | Simplicial Homology for Future Cellular Networks[END_REF].

Coverage can thus be computed mathematically thanks to algebraic topology. However k-coverage computing is not that simple. Indeed, a point is said to be k-covered when it is in the covered area of at least k network nodes. Consequently, an area is k-covered whenever every point in it is k-covered. The expected k-coverage in wireless networks has been studied in [START_REF] Yen | Expected k-coverage in wireless sensor networks[END_REF] in order to propose a node scheduling scheme that conserves energy while retaining network coverage. In [START_REF] Li | Ensuring k-coverage in low-power wide area networks for internet of things[END_REF], the authors use k-order Voronoï diagrams to compute the density of network nodes required to achieve k-coverage.

In this article, we propose a method and an algorithm for computing the k-coverage of a given wireless network. Theoretically it is easy to compute the probability for a point to be k-covered for a wireless network generated by a Poisson point process. However it is more difficult to apprehend the kcoverage of a whole area. Moreover, probabilistic results can not be applied to every wireless network. That is why we need simplicial homology representation to compute the topology of a given network as a whole. We then exhibit that the kcoverage can be seen as k layers of 1-coverage and give an algorithm that compute the k-coverage of a wireless network. For operating purposes, the k layers of network nodes that ensure each 1-coverage are returned by our algorithm.

First in Section II, we define the k-coverage and discuss its application for wireless networks such as IoT sensor networks and cellular networks. We present a probabilistic network model and give some theoretical results in Section III, and introduce few needed mathetical tools in Section IV. Then in Section V, we give our algorithm for computing k-coverage, and simulation results in Section VI. Finally we conclude in Section VII.

II. k-COVERAGE

Cellular networks, Wireless Local Area Networks (WLANs), and sensor networks take part in the family of wireless networks. In these networks, coverage define the utility of the network. In cellular networks or WLANs, users can access the service only if they are covered by a network node. In sensor networks, sensor can communicate only if they are in the sensing range of each other. In a wireless network, a point is said to be covered if it is in the sensing range of a network node, that is to say if it is in the coverage of this node. An area is then covered, when every point of it is covered. By extension, an area is k-covered when every point of the area is in the coverage of at least k network nodes. A wireless network providing k-coverage for an area with a large k is then a densely deployed network.

In the literature, k-coverage is more often used for sensor networks such as Low-Power Wide Area Networks (LP-WANs), and Internet of Things (IoT) [START_REF] Yen | Expected k-coverage in wireless sensor networks[END_REF], [START_REF] Li | Ensuring k-coverage in low-power wide area networks for internet of things[END_REF]. Indeed the benefits of k-coverage include better reliability, better accuracy in sensor measurements, greater throughput by using multiple channels, etc. And these uses concern primarily sensor nodes. Moreover, sensor nodes are small, live on battery, and are cheap to buy and replace, so they can be deployed in large quantities, thus providing easily k-coverage with a great k.

However, k-coverage can also be of interest for cellular networks, where it becomes synonym of multi-site transmitter. The first application is the handover. Actually, a handover is performed when a user changes cells during a communication. It is called a soft-handover when the user is connected simultaneously to multiple cells for the transition between cells, or for interference mitigation in dense area in 3G networks. Therefore, a user in a k-covered area would have the possibility to have handovers with k different cells, which means that telecommunication operators could make trafic off-loading decisions by directing users to less-busy cells, or offer better radio channels thanks to antenna diversity. In 4G and later networks, k-coverage means also that MIMO transmissions from multiple base stations to a user can be performed. That is the basis of the Coordinated Multi-Point Joint Processing (CoMP JP) scheme that allows great capacity gains [START_REF] Vergne | Evaluating the capacity gains from coordinated multipoint transmission and reception[END_REF]. In CoMP JP, two or more base stations can cooperate to serve simultaneously a user leading to a throughput multiplied by 2 or more.

III. PROBABILISTIC ANALYSIS

We represent the wireless network nodes by a Poisson point process:

Definition 1. Let dµ = λdx be the Lebesgue measure on E ⊆ R 2 , N is a spatial Poisson point process of intensity λ > 0 on E if: • N (A) the number of points that fall in A ⊂ E follows a
Poisson law

P[N (A) = k] = e -µ(A) µ(A) k k! • If A, B ⊂ E such that A ∩ B = ∅, then N (A) and N (B) are independant.
We can note that µ(A) = λS(A) where S(A) is the area of A. Moreover, conditionnally to N (A) = n for A ⊂ E, then the points of N are independantly and uniformly distributed on A.

We suppose that every network node has the same sensing range r. Thus its coverage area is a disk of radius r > 0. Then for x ∈ E, the probability for x to be k-covered is given by:

P[x k-covered] = P[∃y 1 , . . . , y k ∈ N |x ∈ k i=1 B(y i , r)],
where B(y, r) is the ball of center y ∈ E and radius r > 0. We immediatly have that:

P[x k-covered] = P[∃y 1 , . . . , y k ∈ N |x ∈ k i=1 B(y i , r)] = P[∃y 1 , . . . , y k ∈ N |∀i, y i ∈ B(x, r)] = P[N (B(x, r)) ≥ k] = 1 -P[N (B(x, r)) < k] = 1 - k-1 i=0 e -λπr 2 (λπr 2 ) i i! .
We can see in Fig. 1 the probability for a point to be kcovered for k = 1, . . . , 6, depending on λπr 2 . As r is fixed, only λ varies. Logically, as λ grows, the probability to be k-covered tends to 1, and the greater k is, the smaller the probability to be k-covered is. The probability that a point x ∈ E is exactly k-covered and not (k + 1)-covered is then:

P[x exactly k-covered] = P[N (B(x, r)) = k] = e -λπr 2 (λπr 2 ) k k! .
We can derive the mean k for which a point x ∈ E is k-covered and not (k + 1)-covered:

E[k] = ∞ k=1 kP[x exactly k-covered] = ∞ k=1 e -λπr 2 (λπr 2 ) k (k -1)! = e -λπr 2 ∞ k=0 (λπr 2 ) k k! (λπr 2 ) = λπr 2 .
Therefore, the mean k for which a point x ∈ E is k-covered and not (k -1)-covered is directly proportional to λ. However, considering the probability for a point to be kcovered is not sufficient to benefit from the advantages of kcoverage. Indeed, in wireless networks, reception devices such as phones are mobile. Therefore, one needs a whole area to be k-covered to offer k-coverage applications such as joint processing or handover. The probability of k-coverage of one point is an upper-bound of the probability of a whole area to be completely k-covered without any hole. Indeed it is easier to ensure that a point is covered, than a whole area without any hole. So probabilistic results can not be used for engineering purposes. That is why we need to consider another approach : to study and compute the coverage of the network as a whole, that is mathematically to study the topology of the network.

IV. SIMPLICIAL HOMOLOGY AND ALGEBRAIC TOPOLOGY

Considering a set of points representing network nodes, the first idea to apprehend the topology of the network would be to look at the neighbors graph: if the distance between two points is less than a given parameter then an edge is drawn between them. However this representation is too limited to transpose the network's topology. First, only 2-by-2 relationships are represented in the graph, there is no way to grasp interactions between three or more nodes. Moreover, there is no concept of coverage in a graph. That is why we are interested in more complex objects.

Indeed, graphs can be generalized to more generic combinatorial objects known as simplicial complexes. While graphs model binary relations, simplicial complexes can represent higher order relations. A simplicial complex is thus a combinatorial object made up of vertices, edges, triangles, tetrahedra, and their n-dimensional counterparts. Given a set of vertices X and an integer k, a k-simplex is an unordered subset of k + 1 vertices {x 0 , . . . , x k } where x i ∈ X, ∀i ∈ {0, . . . , k} and x i = x j for all i = j. Thus, a 0-simplex is a vertex, a 1-simplex an edge, a 2-simplex a triangle, a 3-simplex a tetrahedron, etc. See Fig. 2 for instance.

Any subset of vertices included in the set of the k+1 vertices of a k-simplex is a face of this k-simplex. A k-face is then a face that is a k-simplex. The inverse notion of face is coface. An abstract simplicial complex is a set of simplices such that all faces of these simplices are also in the set of simplices. In this article, we are intersted in representing the topology of a wireless network, we introduce the two following abstract simplicial complexes: Definition 2 ( Cech complex). Let ω be a finite set of points in R 2 , and r a real positive number. The Cech complex of parameter r of ω, C r (ω), is the abstract simplicial complex whose k-simplices correspond to the unordered (k + 1)-tuples of vertices in ω such that the intersection of the k + 1 balls centered on them is non empty.

Definition 3 (Vietoris-Rips complex). Let ω be a finite set of points in R 2 , and a real positive number. The Vietoris-Rips complex of parameter of ω, R (ω), is the abstract simplicial complex whose k-simplices correspond to the unordered (k + 1)-tuples of vertices in ω which are pairwise within distance less than of each other.

The Cech complex provides the representation of the exact topology of the network (see the Nerve lemma in [START_REF] Ghrist | Coverage and hole-detection in sensor networks via homology[END_REF]) but can be tricky to compute due to the check of whether three disks intersect or not. One can see easily that the Vietoris-Rips complex R 2r (ω) is an approximation of the Cech complex C r (ω) that is way easier to compute since it is a clique complex based only on the neighbors graph information. This approximation is quite good: in the case of a random uncorrelated deployment with network nodes deployed according to a Poisson point process the error is less than 0.06% in the computation of the covered area [START_REF] Yan | Accuracy of Homology based Approaches for Coverage Hole Detection in Wireless Sensor Networks[END_REF]. An example of a Cech complex representing a wireless network can be seen in Fig. 3. We can see 4 coverage holes in the network that are highlighted in the simplicial complex representation. Fig. 3. A Cech complex representing a wireless network Given an abstract simplicial complex, one can define an orientation on the simplices by defining an order on the vertices, where a change in the orientation, that is a swap between two vertices, corresponds to a change in the sign. Then let us define the vector spaces of the k-simplices of a simplicial complex, and the associated boundary maps: Definition 4. Let S be an abstract simplicial complex.

For any integer k, C k (S) is the vector space spanned by the set of oriented k-simplices of S. Definition 5. Let S be an abstract simplicial complex and C k (S) the vector space of its k-simplices for any k integer.

The boundary map ∂ k is defined as the linear transformation ∂ k : C k (S) → C k-1 (S) which acts on the basis elements [x 0 , . . . , x k ] of C k (S) via:

∂ k [x 0 , . . . , x k ] = k i=0 (-1) i [x 0 , . . . , x i-1 , x i+1 , . . . , x k ].
For example, for a 2-simplex we have:

x 0 x 1 x 2 x 0 x 1 x 2 ∂2([x0, x1, x2]) = [x1, x2] -[x0, x2] + [x0, x1]
As its name indicates, the boundary map applied to a linear combination of simplices gives its boundary. The boundary of a boundary is the null application. Therefore the following theorem can be easily demonstrated (see [START_REF] Hatcher | Algebraic Topology[END_REF] for instance):

Theorem 1. For any k integer, ∂ k • ∂ k+1 = 0.
Let S be an abstract simplicial complex. Then we can denote the k-th boundary group of S as B k (S) = im ∂ k+1 , and the k-th cycle group of S as Z k (S) = ker ∂ k . We have B k (S) ⊂ Z k (S). We are now able to define the k-th homology group and its dimension: Definition 6. The k-th homology group of an abstract simplicial complex S is the quotient vector space:

H k (S) = Z k (S) B k (S) .
The k-th Betti number of the abstract simplicial complex S is:

β k (S) = dim H k (S).
According to its definition, the k-th Betti number counts the number of cycles of k-simplices that are not boundaries of (k + 1)-simplices, that are the k-th dimensional holes. In small dimensions, they have a geometrical interpretation:

• β 0 is the number of connected components, • β 1 is the number of coverage holes, • β 2 is the number of 3D-voids. For any k ≥ d where d is the dimension, we have β k = 0.

For further reading on algebraic topology, see [START_REF] Hatcher | Algebraic Topology[END_REF].

V. ALGORITHM

Thanks to simplicial homology, we have a representation for a wireless network that allows the computation of the network's topology, that is its coverage, or 1-coverage. Computing the k-coverage is another problem. In order to do that, we choose to view the k-coverage as k layers of coverage: Lemma 1. An area is k-covered, for k integer, if there exists k sets of network nodes without any common nodes such that each set provides 1-coverage on the area.

Proof. If there exists k sets of network nodes that provide 1coverage, then let x be any point in the area, x is covered by each layer. Thus, there exists k nodes, one per layer, that cover x. And the area is k-covered.

Reciprocally, it is not possible to find k sets of nodes such that each provide 1-coverage. We can suppose that there exists k -1 sets of nodes that provide exactly 1-coverage, and a last set with the remaining nodes that do not provide 1-coverage. That is there exists at least one coverage hole in the coverage provided by the k-th set. Then let x be a point in this coverage hole, then x is in the coverage range of exactly one node in each of the first k -1 sets, since these sets of node provide exactly 1-coverage. The point x is inside a coverage hole of the remaining nodes, then there is no other node which coverage range covers x. And x is not k-covered.

Therefore, to compute the k-coverage of a wireless network, we intend to count the number of 1-coverage layers. To slice the network in layers, we use the simplicial complex reduction algorithm that we presented in [START_REF] Vergne | Reduction algorithm for simplicial complexes[END_REF]. This reduction algorithm takes as input a simplicial complex, then removes points and their cofaces (that is the simplices they are part of) until it is no more possible without creating neither a coverage hole nor a disconnectivity in the network. At the end, we obtain a simplicial complex that provides the same coverage as the initial complex with a minimal set of points. Then the set of network nodes associated to these points provide at least 1coverage on the whole covered area, but not 2-coverage on the whole area or more points could be removed and the simplicial complex could be further reduced. However, locally 2-coverage is provided by the reduced complex, since we consider coverage disks and disks can not tile the plane, there will always exist intersection of disks. It is important to note that the reduction algorithm needs the definition of a boundary (via a list of points) to delimit the area to be covered, here it is the boundary of the area where one need to compute the k-coverage. We can see an example of the reduction algorithm on a Vietoris-Rips simplicial complex in Fig. 4.

Our algorithm for computing k-coverage then takes as input the positions of the network nodes, compute the simplicial complex to represent their topology. Then, the reduction algorithm is applied, its result constitutes of the first layer of 1-coverage. This first layer is then discarded, the simplicial complex is built on the remaining points and the we re-apply the reduction algorithm on it. We continue while the number of Fig. 4. A complex reduced by the reduction algorithm presented in [START_REF] Vergne | Reduction algorithm for simplicial complexes[END_REF] connected components stays at 1, and the number of coverage holes stays at 0. At the end, our algorithm provides the k index of k-coverage of the wireless network, and also supplies the k sets of points/network nodes that are the k layers of coverage. The pseudo-code of the algorithm is given in Alg. 1. Our algorithm provides a lower-bound for k-coverage, that is that k-coverage is guaranteed in every point of the area. More specifically, when the algorithm returns the value k for a wireless networks on the area A, that means that:

• ∀x ∈ A, x is k-covered, • ∃x ∈ A, x is not (k + 1)

-covered,

• There may exist some x ∈ A that are l-covered with l > k (at the intersection of coverage disks).

VI. SIMULATION RESULTS

In this section, we give some figures illustrating the functioning of our algorithm and present some simulation results on the k-coverage of a wireless network simulated by a Poisson point process. We can see an example of the execution of our k-coverage computation algorithm for the wireless network represented by a Vietoris-Rips complex of Fig. 5. This network was simulated with N = 40 points randomly placed in a square of size 10, plus a boundary of fixed points on the square to delimit the area of the square in which we want to compute the kcoverage. The coverage radius is set to 2.5. In order to obtain nicer figures, the process used to draw the points positions is of hard-core type, that is it is forbidden for 2 points to be too close to each other. We can see in Fig. 6 that our algorithm exhibits 3 layers of coverage, that means that the wireless network provides 3-coverage. The last layer presents a coverage hole in the bottom right corner, so 4-coverage is not available in that part, and thus on the square. In each subfigure, points in red are the remaining points that are not yet part of a layer. On the left of each subfigure is the wireless network representation with the coverage disks, and on the right is the Vietoris-Rips representation. We also provide an example of our k-coverage algorithm running of the Cech complex of Fig. 7. The configuration set-up is the same as before, except the number of points is initially set to N = 50. This network has more points and provides 4 layers of coverage that is 4-coverage as we can see in Fig. 8. Finally, we provide some simulation results on the kcoverage of a wireless network generated by a Poisson point process. We look at the value of k, where k is the maximum index such that the network provides k-coverage, depending on the intensity of the process, that is the mean number of points by surface unit, and whether the topology is computed via a Vietoris-Rips or a Cech complex.

We consider a set of points generated with a Poisson point process of intensity λ on a square with side of size 10. We add points on the boundary of the square to delimit the area to be covered. The coverage radius is set to 2.5, one can note that πr 2 ≈ 20. We compute k the mean value of k such that the complex provides k-coverage and not (k + 1)-coverage, on average on 1000 simulations. We can see in Table I the result of the simulations. Moreover, these results are plotted in the graph of Fig. 9. We can compare these results to the theoretic ones of the mean k for which a point x is k-covered: E[k] = λπr 2 . We can see that, as expected, the simulated values are below the theoretic ones. This is because, theoretically we are only capable of computing the probability of a point to be k-covered, but not the probability that a whole area is k-covered without any coverage holes. The second one, that we can approach by simulation, is smaller than the first one. Furthermore, our algorithm guarantees k-coverage, but locally points may be l-covered with l > k.

VII. CONCLUSION

In this article, we propose a method and an algorithm for computing the k-coverage of a wireless network. The kcoverage is the fact for a point to be covered by k network nodes, this definition can be extended to a whole area: an area is k-covered if every point in it is k-covered. Theoretically it is easy to compute the probability for a point to be k-covered for a wireless network generated by a Poisson point process. However it is more difficult to apprehend the k-coverage for a whole area, and we need simplicial homology representation to compute the topology of the network as a whole. We then exhibit that the k-coverage can be seen as k layers of 1coverage and give an algorithm that compute the k-coverage of a wireless network. We provide some figures and simulation results to illustrate our algorithm.

Fig. 1 .

 1 Fig. 1. Probability for a point to be k-covered depending on λπr 2

Fig. 2 .

 2 Fig. 2. Examples of k-simplices.

Algorithm 1 k

 1 -coverage computing algorithm. Require: set V of n vertices, coverage radius r. S := R 2r (V ) or C r (V ) Computation of β 0 (S) and β 1 (S) k := 0 while β 0 (S) = 1 and β 1 (S) = 0 do k = k + 1 Apply reduction algorithm to S Save reduced complex as k-th layer Save list of discarded vertices as V S := R 2r (V ) or C r (V ) Computation of β 0 (S) and β 1 (S) end while return k and the k layers of coverage

Fig. 5 .

 5 Fig. 5. A wireless network and its Vietoris-Rips representation

Fig. 6 .

 6 Fig. 6. The 3 layers of coverage and the last incomplete layer.

Fig. 7 .

 7 Fig. 7. A wireless network and its Cech representation

Fig. 8 .

 8 Fig. 8. The 4 layers of coverage and the last incomplete layer.

Fig. 9 .

 9 Fig. 9. Mean k computed by our algorithm.

TABLE I MEAN

 I k FOR A VIETORIS-RIPS AND FOR FOR A CECH COMPLEX

	λ	0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40
	λπr 2	1	2	3	4	5	6	7	8
	k	0.004	0.081	0.268	0.590	0.967	1.396	1.820
	λ	0.05	0.10	0.15	0.20	0.25	0.30	0.35	0.40
	λπr 2	1	2	3	4	5	6	7	8
	k	0.002	0.079	0.237	0.540	0.859	1.297	1.695