Alexia Lemoine

Ronan Invernizzi

Germain Salvato Vallverdu

Jacob Olchowka

Liliane Guerlou-Demourgues

Isabelle Baraille

Delphine Flahaut

Relating the electrochemical behavior of Birnessite to the morphology and specific surface: interest of studying the surface reactivity

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Relying the electrochemical behavior of birnessite to the morphology and specific surface: interest of studying the surface reactivity.

Alexia Lemoine [START_REF] González | Review on Supercapacitors: Technologies and Materials[END_REF] , Ronan Invernizzi [START_REF] Simon | Materials for Electrochemical Capacitors[END_REF][START_REF] Choudhary | Asymmetric Supercapacitor Electrodes and Devices[END_REF] , Germain Salvato Vallverdu [START_REF] González | Review on Supercapacitors: Technologies and Materials[END_REF][START_REF] Choudhary | Asymmetric Supercapacitor Electrodes and Devices[END_REF] , Jacob Olchowka [START_REF] Simon | Materials for Electrochemical Capacitors[END_REF][START_REF] Choudhary | Asymmetric Supercapacitor Electrodes and Devices[END_REF][START_REF] Shao | Design and Mechanisms of Asymmetric Supercapacitors[END_REF] , Liliane Guerlou-Demourgues [START_REF] Simon | Materials for Electrochemical Capacitors[END_REF][START_REF] Choudhary | Asymmetric Supercapacitor Electrodes and Devices[END_REF][START_REF] Shao | Design and Mechanisms of Asymmetric Supercapacitors[END_REF] *, Isabelle Baraille [START_REF] González | Review on Supercapacitors: Technologies and Materials[END_REF][START_REF] Choudhary | Asymmetric Supercapacitor Electrodes and Devices[END_REF] , Delphine Flahaut [START_REF] González | Review on Supercapacitors: Technologies and Materials[END_REF][START_REF] Choudhary | Asymmetric Supercapacitor Electrodes and Devices[END_REF] This article focuses on the understanding of the link between morphologies, specific surface of well-known birnessites with electrochemical performances by studying the surface reactivity of each material. Our study is especially dedicated to point out the impact of the materials surface on the faradaic and pseudocapacitive mechanisms involved in the energy storage of the supercapacitors. For this purpose, a multiscale study was carried out on three birnessites, nonprotonated (Na-MnO2, K-MnO2 and HT-MnO2) and protonated (HNa-MnO2, HK-MnO2 and HHT-MnO2), to ensure the investigation of the surface reactivity on birnessites for each step of the nanocomposite conception based on birnessite. Scanning electron microscopy (SEM) was used to characterize the morphology of the materials. Coupling X-ray photoemission spectroscopy (XPS) and SO2 gas probe adsorption are especially devoted to determine the nature of active sites and the electronic structure of the materials surface. Thus, we put in evidence that the surface properties, as specific surface area and surface active sites, are linked to the electrochemical mechanism (faradaic or pseudocapacitive) of storage depending of the scan rate. We show that the sites concentration determined by XPS can be directly linked to the contribution of the pseudocapacitive mechanism observed at low and moderate scan rates. As for the capacitive mechanism is discriminated at high scan rate, its contribution is related to the BET surface. The largest specific surface is obtained for the HK-MnO2 with its eroded veails (85 m²/g). Redox reactivity have been evaluated from XPS quantification comparing the S/M values, i.e. at. % of S / at% of Mn. HT-MnO2, its protonated derivative HHT-MnO2 and Na-MnO2 exhibit the largest redox reactivity, with respectively, S/M values of 0.49, 0.46 and 0.31, according to their smooth platelets morphology.

Except for the high temperature birnessite, the change of morphology after the protonation makes decrease the concentration of redox active sites while BET surface is increasing. This work point out that the HK-MnO2 presents the best performances, 46.6 F/g at 100 mV/s, to be used as an electrode material in supercapacitor storage systems.

INTRODUCTION

With the increasing demand for energy storage devices, especially in applications that require high power density, supercapacitors appear as perfectly adapted storage systems [START_REF] González | Review on Supercapacitors: Technologies and Materials[END_REF][START_REF] Simon | Materials for Electrochemical Capacitors[END_REF] . The most widespread systems currently used are electric double layer capacitors (EDLCs), however they have limits in terms of energy density delivered (3-10 W h/kg), which is a brake for a wider application [START_REF] Choudhary | Asymmetric Supercapacitor Electrodes and Devices[END_REF][START_REF] Shao | Design and Mechanisms of Asymmetric Supercapacitors[END_REF] . This limitation is mainly caused by the charge storage mode, which only consists in adsorbing and desorbing ions from the electrolyte on the surface of carbon electrodes, so that the capacity is dependent on the accessible specific surface area (SSA). One of the solutions to increase the energy density is to involve another storage mechanism in addition to the electrostatic one by replacing one of the carbon electrodes by a faradaic or pseudo-capacitive electrode material, which leads to hybrid or asymmetric supercapacitors. [START_REF] Jiang | Definitions of Pseudocapacitive Materials: A Brief Review[END_REF][START_REF] Nguyen | Metal Oxide and Hydroxide-Based Aqueous Supercapacitors: From Charge Storage Mechanisms and Functional Electrode Engineering to Need-Tailored Devices[END_REF][START_REF] Lin | Materials for Supercapacitors: When Li-Ion Battery Power Is Not Enough[END_REF] A pseudo-capacitive material involves charge storage through rapid and reversible redox reactions at the surface of the material. This process allows to increase the energy density while keeping a high power density. In the literature, several metal oxides are known for their pseudo-capacitive property such as RuO2, FeWO4 or MnO2 [START_REF] Xia | A Symmetric RuO2∕RuO2 Supercapacitor Operating at 1.6 V by Using a Neutral Aqueous Electrolyte[END_REF][START_REF] Brousse | Crystalline MnO[Sub 2] as Possible Alternatives to Amorphous Compounds in Electrochemical Supercapacitors[END_REF][START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF][START_REF] Zhu | Birnessite based nanostructures for supercapacitors: challenges, strategies and prospects[END_REF] . The nanostructuration of the material is an essential criterion to increase the electrolyte/electrode interface and facilitate the access of the surface active sites. Since the oxidation-reduction reactions are essentially performed on the surface, this is a way to optimize the electrochemical performances. In the goal to elaborate a material with optimized performance, we need to understand the parameters that influence the surface reactions types that directly affect the electrochemical performance. Several investigations [START_REF] De | Synthesis, chemical modification and electrochemical behaviour of layered sodium manganese dioxide[END_REF] are performed in this study such as surface and morphology characterization, as well as surface reactivity.

Birnessite δ-MnO2 is well known for its interesting properties as positive electrode material in supercapacitors due to its promising pseudo-capacitive behaviour [START_REF] Toupin | Charge Storage Mechanism of MnO 2 Electrode Used in Aqueous Electrochemical Capacitor[END_REF] . Its lamellar structure, based on a stacking of MnO2 slabs, is very versatile and can lead to a large panel of composition for the interlamellar space as well as for the metallic slab (possibility of vacancies and various Mn oxidation states). In this work, a complete study is carried out on three birnessites with various compositions and morphologies designated as Na-MnO2, K-MnO2 and HT-MnO2, as well as on their derivatives after protonation for use in a nanocomposite. We have to note that protonataed birnessites (monolayers birnessites) are one of the building blocks used to obtained nanocomposite supercapacitors 10-14-15 . Thus, surface reactivity investigation of protonated birnessite with different morphology and thickness are the key to rely the electrochemical performances and storage mechanisms (surface charge storage by faradaic reaction and bulk pseudocapacitive reaction redox reactions associated to faradaic mechanism and surface adsorption associated to capacitive mechanism) to the birnessite structural criteria (nature of the surface active sites (redox and/or acidbase), specific surface area, etc..).

To get deeper in the understanding of the factors that impact the capacity of the material, a multiscale study was conducted to investigate in detail the morphology, surface reactivity, nature of active sites and identification of mechanisms involved in the storage of charges for each material. The materials will be compared characteristic by characteristic in order to understand the impact of each on the electrochemical properties. To do this, scanning electron microscopy (SEM) was used to highlight the morphology of the materials. SO2 gas probe adsorptions were performed to probe the surface active sites of the birnessites. X-ray photoelectron spectroscopy (XPS) was used to both characterize the surface of the materials before and after adsorption and to determine and quantify the adsorption modes of SO2 (redox-acid/base).

EXPERIMENTAL SECTION

Synthesis and bulk characterization

Synthesis method: Na-birnessite (Na-MnO2) and K-birnessite (K-MnO2), corresponding to manganese oxide layers with respectively sodium and potassium ions intercalated in between, are synthesized by precipitation [START_REF] Ching | Synthesis and Characterization of a Layered Manganese Oxide: Materials Chemistry for the Inorganic or Instrumental Methods Lab[END_REF] . First, 5 mmol of MnSO4•H2O is dissolved in 15 mL of distilled water, followed by drop by drop addition of 15 mL of a strong alkaline solution (6M NaOH or KOH) under vigorous stirring. A spontaneous precipitation of a tan coloured slurry of Mn(OH)2 is formed, which is then oxidized with the addition of K2S2O8 (3.5 mmol) in order to obtain the Nabirnessite or K-birnessite. The resulting brown-black slurry is stirred for an additional 30 min, before being centrifuged and washed five times with distilled water. The brownish/black solid corresponding to Na-birnessite (or K-birnessite) is then dried at 70°C for 24h under air.

HT-MnO2 was obtained by thermal decomposition of KMnO4. 3 g of the reactant were placed in an alumina crucible and heated in a furnace under air at 800°C for 5h [START_REF] Kim | Preparation of Layered MnO 2 via Thermal Decomposition of KMnO 4 and Its Electrochemical Characterizations[END_REF][START_REF] Gaillot | Structure of Synthetic K-Rich Birnessite Obtained by High-Temperature Decomposition of KMnO 4 . I. Two-Layer Polytype from 800 °C Experiment[END_REF] . Then, the recovered powder was centrifuged 3 times in distilled water to eliminate the soluble by-products and was dried at 50°C for 24h.

The protonation step consists in a topotactic ion-exchange reaction, during which the MnO2 slabs are preserved and the alkaline ions replaced by protons. To do so, the polycrystalline powder is suspended and stirred in HCl aqueous solution (0.1 M) for 4 days, before being recovered by centrifugation and washed with distilled water. Finally, the dark brown solid is then dried at 50°C in air for one day. The phases obtained after protonation of Na-MnO2, K-MnO2 and HT-MnO2 are named HNa-MnO2, HK-MnO2 and HHT-MnO2 respectively.

X-ray diffraction: The X-ray powder diffraction patterns are recorded with a Philips Panalytical X'Pert Pro diffractometer using Cu-Kα radiation (λKα2= 1.540 Å, λKα1= 1.544 Å). The powder diffraction patterns are recorded in the 10°-80° (2θ) angular range, with a 0.0167° (2θ) step size and a 2.022° (2θ) active width in the detector (200 s per step) for 2 hours.

ICP: Alkaline (Na and K) and Mn elements are quantified in the birnessites materials with a Varian 720ES ICP-OES spectrometer. The samples (25 mg of powder) are first dissolved into HCl, the solutions being heated in 5 mL of HCl until full dissolution of powder. Then solutions with a concentration of target element between 1 and 200 mg/L are prepared and introduced in a nebulization chamber along with an argon flow to create an aerosol. To ensure accuracy of our results, five measurements are performed for each sample. CHNS: A Thermo Flash EA 111E Series apparatus is used to determine the carbon, hydrogen, nitrogen and sulfur elemental composition. Exactly 1.5 mg of powder is placed in the tin foil. Two measurements are performed for each sample.

Iodometric titration: The mean oxidation state of Mn is determined by using iodometric titration.

We first dissolve around 30 mg of powder in a solution composed of 5 mL distilled water, 10 mL of KI solution at 10 g/L and 5 mL HCl solution at 12M (37%). Then the solution is heated to ensure the total dissolution of powder (yellowish color). Just after this step, the solution is titrated by sodium thiosulfate solution to obtain Veq, which represented the volume of added sodium thiosulfate when the yellow solution turns to transparent. The principle of iodometric titration is explained in ref [START_REF] Invernizzi | Controlled Nanostructuration of Cobalt Oxyhydroxide Electrode Material for Hybrid Supercapacitors[END_REF] .

BET: The analyses are performed on ASAP2010 (Micromeritics Corp., Norcross, GA, USA).

The specific surfaces areas of materials are conducted by N2 adsorption at 77K. The Brunauer-Emmett-Teller (BET) model was used to calculate the specific surface SBET [START_REF] Emmet | The Measurement of Surface Areas of Soils and Soil Colloids by the Use of Low Temperature Van Der Waals Adsorption Isotherms[END_REF] .

Surface analysis

XPS: X-ray Photoelectron Spectroscopy (XPS) analysis are performed on a Thermo K-alpha spectrometer using a focused monochromatized Al Kα radiation (hν = 1486.6eV). A Cu double tape was used to fix on the sample holder the powder materials to be analyzed. In order to avoid contaminating the samples and being in contact with the air after performing the 𝑆𝑆𝑆𝑆 2 adsorption, the samples are prepared in a glove box and transferred into the glove box which is directly connected to the spectrometer (𝐻𝐻 2 𝑆𝑆/𝑆𝑆 2 < 5ppm) using a transfer vessel. A pass energy of 20 eV and an energy step of 0.1 eV are used to record the core peaks. The analysis depth is about 5 nm and the analysis area is of 200 x 400 µ𝑚𝑚 2 . A neutralizer gun is used to minimize the surface charging and therefore charge compensation. XPS spectra processing and quantification are obtained using CASA XPS software with Shirley-type background subtraction. Scofield's relative sensitivity factors are used for the quantification of the surface composition [START_REF] Foix | X-Ray Photoemission Spectroscopy Study of Cationic and Anionic Redox Processes in High-Capacity Li-Ion Battery Layered-Oxide Electrodes[END_REF] . SEM: Scanning electron microscopy (SEM) For Scanning electron microscopy (SEM) an Auger JEOL JAMP 9500F spectrometer was used. To analyze the surface and the core of the bare material, samples are cut with a JEOL IB-09010CP cross polisher under a controlled atmosphere of N2. The cross polisher is located in a glove box where the atmosphere is also controlled. For realised the cross section, the sample powders are mixed with a silver conducting epoxy resin to ensure a low charge effect during acquisition and then deposited on a silicon wafer at about 250 µm thickness. The wafer and the powder-epoxy mixture are then cut with Argon plasma of 6 kV and 150 µA at 1.10 -4 Pa during 4 hours. The cutting depth of the sample is about 150 µm. The cross-sectioned samples are then transferred from the glove box to the Auger spectrometer for analysis using a transfer vessel.

The SEM images highlight the bulk morphology of birnessite materials. The SEM is carried out under Ultra High Vacuum conditions (< 2 x 10 -7 Pa) and an acceleration voltage of 20 keV and a current beam of 4 nA has been used. The SEM images are performed at 30° tilt to prevent charging effect.

Gas probe adsorption

The 𝑆𝑆𝑆𝑆 2 adsorption is carried out with a Micromeritics Autochem 2920 Analyzer. The adsorptions of SO2 gas probes are performed following four steps: the first step consists in cleaning the reactor from potentially adsorbed species by exposing it for one hour to a Helium flow at a temperature of 500°C. Then, the sample powders are introduced into the reactor (mpowder = 0.050 g). The second step consists in heating the device following a temperature ramp up to 80°C in order to keep the presence of water molecules between the nanosheets, still under Helium flow.

During the third step, the sample is exposed to pulses of a He(80%)/(SO2(5%)-He)(20%) gas mixture. The gas pulses of 0.5 cm 3 volume are sent every 40 seconds in a number of 35 to 120 pulses depending on the sample, until the amount of gas sent at the device inlet is the same at the outlet. This means that the SO2 molecules are no more adsorbed on the sample and therefore that all its active sites have reacted. Finally, the last step consists in cooling down the device to the ambient temperature. After adsorption, the sample powders are directly transferred and stored in a glove box under controlled atmosphere until it is analysed. The adsorption of SO2 gas probes allows to quantify and to determine the nature of the active sites on the surface, either acid-base or redox. For this, if SO 2 molecules are adsorbed on the surface as sulfite species (SO 3 2-), the active sites are basic and if SO 2 molecules are adsorbed as sulfate species (SO 4 2-), the active sites are oxidizing. The quantity of adsorbed species is determined from XPS analysis from the area ratio between the S 2p and the Co 2p core peaks. The adsorption mode is assigned depending on the S 2p core peak binding energy. A binding energy (B.E.) of the S 2𝑝𝑝 3/2 core peak at 167.5 eV is characteristic of sulfite species and a B.E. of 169.0 eV is characteristic of sulfate species [START_REF] Andreu | XPS Investigation of Surface Reactivity of Electrode Materials: Effect of the Transition Metal[END_REF] . Due to the ultra-high vacuum conditions of XPS analyses, only the chemisorbed species are detected since the physisorbed species involve interactions that are too weak for these analytical conditions.

Electrochemical tests

The electrodes were made by mixing the active material, carbon black and PTFE following the weight ratio 80/15/5 into a motar. All the components were mixed until the formation of a selfsupported black paste. This later was further rolled and a electrode was cut using a round cutter.

The average mass loading of the electrode is 10 mg/cm². Electrochemical measurements were carried out in a solution of 0.5 M K2SO4 in a 3-electrodes configuration at 25°C. Ag/AgCl electrode and a platinum wire were respectively used as the reference electrode and the counter electrode.

In this manuscript, the capacitance values were calculated by EC-lab software from cyclic voltammetry by integrating the area under the CV charge or discharge curve. We also verified the capacity values with our own calculations.

The capacity using cyclic voltammetry can be obtained thanks to this formula:

Cp = A / (2k.m.ΔV)
where A is the area under the curve (mA.V), k is the scan rate (mV/s), m the mass of active material (mg) and ΔV the potential windows (V).

The CV curves were plotted on Origin and the area under the curve was integrated using Origin software. The values obtained were in good agreement with those directly calculated by EcLab software, then we considered these latter capacity values.

RESULTS

Crystallographic structure

Birnessite is the common name given to the layered manganese oxide polymorph, δ-MnO2. Its structure consists of layers made of edge-sharing MnO6 octahedra separated by interlamellar alkaline cations (K + , Na + ), protons and water molecules. X-ray diffraction patterns were recorded for the prepared materials in order to check the structure in comparison with data that were already published by numerous works [START_REF] Drits | Structure of Synthetic Monoclinic Na-Rich Birnessite and Hexagonal Birnessite: I. Results from X-Ray Diffraction and Selected-Area Electron Diffraction[END_REF][START_REF] Gaillot | Caractérisation Structurale de La Birnessite: Influence Du Protocole de Synthese[END_REF] . The indexation of the XRD patterns is reported in Figure 1 but the structural study of the materials, already performed by many authors, was not the purpose of this paper. The cell parameters and space group are listed in the Table 1. Na-MnO2 and K-MnO2 crystallize in a monoclinic cell with the C2/m space group while HT-MnO2 crystallizes in a hexagonal cell with the P63/mmc space group. The diagram of K-MnO2 exhibits wider and less defined diffraction peaks 10 than for the two other alkaline birnessites. This can be correlated to the fact that, for K-MnO2, the structure is disordered and the coherent domains are significantly smaller comparing with Na-MnO2 and HT-MnO2, as reported in section 2. As shown by the inter-reticular distance of the line at the lowest angle (001) corresponding to the interlamellar distance, the interslab distance in K-MnO2 and HT-MnO2, both containing intercalated K + ions, is slightly shorter than in Na-MnO2. This is at first sight counter intuitive because, in size, potassium ions are much bigger than sodium ions: their ionic radius is 1.51 Å, compared to 0.99 Å for sodium [START_REF] Shannon | Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides[END_REF] . However, the polarizing character, which is estimated by charge/size ratio, is higher for Na + than for K + . This induces a larger hydration shell around the Na + cation, which conducts to an enlargement of the interslab space.

The protonation step does not affect the general shape of the X-ray diffraction patterns. We may thus suggest that the lamellar structure of the birnessite materials are maintained, while the size of coherent domains, linked to the width of the diffraction lines, is barely modified. The protons are exchanged with Na + and K + cations in the interlayers, which leads to the increase of the inter-layer distances (0.12 Å starting from Na-MnO2 and K-MnO2, and 0.10 Å starting from HT-MnO2). An increase of the peak width has also to be noticed after protonation for Na-MnO2, which suggests size decrease of the coherent domains for this material.

The accurate compositions of the birnessites series have been determined by combining ICP-OES, iodometric titrations and CHNS analyses which allow to determine respectively the cationic content (Na, K, Mn), the manganese mean oxidation state and the hydrogen content (proton + water). In the case of Na-MnO2 and K-MnO2 (Table 2), the results confirm that, during the protonation reaction, the Na(K)/Mn ratio drops while the H/Mn ratio increases, showing the efficient exchange of alkaline ions for protons. Moreover, the average oxidation state of manganese is increasing during protonation, from 3.53 to 3.65 in the case of Na-MnO2 and from 3.52 to 3.63 in the case of K-MnO2. The behaviour of the HT-MnO2 is completely different from the previous phases. The potassium/proton exchange is only partial, and the increase of the manganese oxidation state is not significant.

Morphology

For all the studied compounds, the morphologies and the arrangement of particles within their aggregates are illustrated in Figure 2. For K-MnO2 presented in figure 2.a, the SEM image shows a morphology that looks like folded and aggregated veils between them [START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF] . Due to the structure of these aggregates, the determination of the size of the veils is difficult. In order to go deeper into the morphological study of these materials, SEM images were made on samples cut with a cross- and Na-MnO2 respectively; g), h), i) SEM images and j),k),l) cross section SEM images of the HK-MnO2, HHT-MnO2 and HNa-MnO2 protonated phases respectively.

After protonation, the impact on the morphology of the materials depends of the composition.

In the case of HK-MnO2, the veil-like morphology of the material is maintained but the particles seem eroded (figure 2.g). The cross section image (figure 2.j) brings out the aggregation of the veils to form small clusters of several tens of micrometers after the protonation. For HHT-MnO2, the structure is completely modified. The initial three-dimensional network has been broken. Thus, the particles are now disconnected from each other (figure 2.h). However, we can observed on the cross section image (figure 2.k) that the size of the primary particles are preserved.

In contrast, no major modification is observed for HNa-MnO2, except the erosion of the edges of the platelets particles (figure 2.i) such as for the HK-MnO2 veils. The erosion comes from the dissolution of the edges by the acidic medium during protonation. From the cross section images (Figure 2.l), we can easily observed that the particles become randomly oriented after protonation and appear more porous.

To resume, the protonation step has an impact on the initial morphology of the materials. For HK-MnO2 and HNa-MnO2, the protonation erodes the edge of the veils/platelets, due to the acidic environment in which the protonation is performed, and turns the arrangement of the veils/platelets from stacked particles along the same direction toward randomly orientated within aggregates.

Dealing with large particles as for HHT-MnO2, the protonation breaks the 3D network formed by the particles without affecting the morphology of the primary particles. Figure 3 presents the N2 adsorption/desorption isotherms obtained for non-protonated and protonated manganese oxides. According to the classification established by Sing et al. [START_REF] Lykiema | Prepared for Publication by the Subcommittee on Reporting Gas Adsorption Data Consisting of[END_REF] , the isotherm curves of non-protonated and protonated Na-MnO2 and K-MnO2 can be classified in type IV category. Thus, it would correspond to mesoporous materials (pore size from 2 to 50 nm). On the other hand, HT-MnO2 and HHT-MnO2 are identified as non-porous materials, which might be a factor explaining the difficulty to protonate HT-MnO2. From these curves, the specific surface of each material can be determined with the BET method (Table 2).

Among the three alkaline birnessites, K-MnO2 (45 m²/g) exhibits the higher specific surface.

Specific area of 35 m²/g is obtained for Na-MnO2. As it could be expected from the high temperature synthesis conditions, HT-MnO2 presents a specific surface area almost 10 times smaller (4 m²/g) than the two other birnessites, in accordance with its very different morphology.

After protonation, the specific surfaces areas are strongly increased for all birnessites, almost twice. Tang et al. [START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF] have explained those results based on the partial delamination of the particles during the protonation process. Moreover, the effect of an acidic medium, which tends to "nibble" the surface of the particles, has also to be taken into account. From our SEM investigation, the increase of BET surface area is correlated with the break of 3D network for the HT-MnO2, the erosion and the disordered stacking of the veils and platelets particles, i.e. HK-MnO2 and HNa-MnO2. Finally, it is observed that the protonation does not really impact the porosity of platelet birnessites (Na-MnO2 and HT-MnO2) but slightly increases the mesoporosity of veil birnessite by creating a population of small mesopore (Figure S1).

Electrochemical performance

Figure 4 presents the cyclic voltammetry curves for the δ-MnO2 (a), c) and e)) series and the corresponding protonated phases (b), d) and f)) measured in 0.5M K2SO4 aqueous solution at various sweep rates. Figure 5 summarizes the evolution of the specific capacity for all the electrode materials depending on the applied sweep rate.

At 0.5 mV/s, the three alkaline birnessite electrode materials exhibit a pseudo rectangular profile typical of pseudocapacitive materials in which reversible redox peaks can be distinguished (Figure 4a). These peaks are currently assigned to intercalation/deintercalation of alkaline ions into/from the interslab space through a faradaic process. These peaks are sharper and more intense for HT-MnO2, suggesting a more important "bulk" redox contribution than for the K-MnO2 and Na-MnO2, which leads to a slightly higher capacity for the manganese oxide obtained at high temperature (143.4 F/g for HT-MnO2 versus 131.6 F/g and 125.7 F/g for K-MnO2 and Na-MnO2 respectively (Figure 5). Although the redox peaks become broader at 5 mV/s (Figure 4 c)), the same trend is observed within the series. At higher rate (Figure 4 e)), the rectangular shape turns to an ellipse for all alkaline birnessites and the redox peaks could not be distinguished anymore due to the resistance becoming more dominant in the electrochemical response. Moreover, the poor conductivity of the birnessites prevent fast intercalation and thus, bulk redox reactions. At 100 mV/s, Na-MnO2 exhibits the best capacity (36.6 F/g), above HT-MnO2 (26.3 F/g) and K-MnO2 (23.1 F/g) (Figure 5). These results clearly show that although surface reactions (redox and adsorption mechanisms) are preponderant in the charge storage, the electrochemical performances are not directly proportional to the specific surface area measured previously. Indeed, HT-MnO2 possesses more than 10 times lower BET surface area than K-MnO2 whereas it delivers higher specific capacity in similar experimental conditions, including at high sweep rate. After protonation, all electrode materials exhibit a well-defined rectangular shape without any redox peak, at low and moderate sweep rates (figures 4b) and 4d)). Besides the change of CV curve profile, the capacity of the protonated HK-MnO2 and HNa-MnO2 dropped by 20-30 % compared to their alkaline counterparts. More surprisingly, the specific capacity of HT-MnO2 is divided, after protonation, by more than a factor 4 (e.g 143.4 F/g vs 28.5 F/g and 125.9 F/g vs 20.7 F/g at 0.5 mV/s and 5 mV/s, respectively). This huge capacity drop is completely relied with the important structural/morphological changes reported in the previous part. Finally, HK-MnO2 and HNa-MnO2 deliver comparable capacities, much higher than that measured for HHT-MnO2. At 100 mV/s, HK-MnO2 shows the best performance, and even better than for the corresponding non-protonated material. Its cyclic voltammetry profile remains rather rectangular whereas that of HNa-MnO2 becomes an ellipse. Concerning HHT-MnO2, its profile remains unchanged. The results presented in Figure 5 confirm that protonation globally leads to decrease the performance at low and moderate rates, even if it increases the interlayered spacing, promoting thus a possible cationic intercalation, and increases the specific surface area. On the other hand, at sweep rates higher than 50 mV/s, HK-MnO2 exhibits the best capacities most probably thanks to its high specific surface area. The galvanostatic charge and discharge tests are presented Figure S2 and lead to the same conclusions. The alkaline birnessite deliver higher specific capacity at low current density whereas from 5 A/g, HK-MnO2 exhibits a much higher specific capacity than all the other birnessites.

In conclusion, the electrochemical study shows that, at high scan rate, the material that presents the best electrochemical performances after protonation is HK-MnO2 with a capacitance of 40 F/g as a reason of its highest specific surface.

The next part of the study aims at determining the active sites engaged in the storage mechanism through the coupling between gaseous probe adsorption and XPS experiments. A correlation between the surface reactivity of the samples, the specific surfaces and the electrochemical properties will then be attempted.

Surface characterization

In this part, we will present the surface characterization of materials before and after protonation.

Before protonation

Before studying the surface reactivity of the different materials, they were first characterized using the XPS technique, before and after protonation. The XPS spectra obtained for the materials before protonation are presented in figure 6 and the spectra of the materials after protonation in figure S3. All spectra in this study were calibrated from the contamination carbon at 285.0 eV present in the C 1s spectra (not shown here). For each sample, the XPS spectra of Mn 2p, O 1s and Na 1s or K 2p are presented. For the Mn 2p spectra of the materials before protonation (figure 6), three peaks are observed, two resulting from spin-orbit coupling at 642.5 eV for Mn 2p3/2 and 654.2 eV for Mn 2p1/2. The third peak at 665.0 eV corresponds to the "shake-up" satellite peak associated with Mn 2p1/2 while that of Mn 2p3/2 is merged with the Mn 2p1/2 peak. A shoulder is visible on the Mn 2p3/2 at 641.5 eV (*) which suggests the presence of a mixed valence Mn 4+/3+ . It is possible to determine the oxidation state by using known references for manganese in a octahedral environment with an oxidation state of +IV, +III or +II [START_REF] Brabers | X-Ray Photoelectron Spectroscopy Study of the Cation Valencies in Nickel Manganite[END_REF][START_REF] Allen | A study of a number of mixed transition metal oxide spinels using X-ray photoelectron spectroscopy[END_REF] . The decomposition of the Mn 2p spectrum with these references allows to confirm the mixed valence Mn 3+ and Mn 4+ 29 (see figure S4). The Mn 3+ /Mn 4+ ratios determined for all samples are given in Table 3. The Mn mixed valence can also been confirmed by the B.E. splitting value of the Mn 3s core peak. It is known that the ΔB.E.Mn3s value of the multiplet splitting is directly related to the oxidation state, i.e., the ΔB.E.Mn3s value for Mn 4+ , Mn 3+ and Mn 2+ are 4.5 eV, 5.5 eV and 6.5 eV, respectively. In our birnessite materials, the splitting value in the Mn 3s spectra (see figure S5) are: ΔB.E. Mn3s = 4.8 eV for Na-MnO2, ΔB.E.Mn3s = 4.7

eV for K-MnO2 and ΔB.E.Mn3s = 4.6 eV for HT-MnO2. As a result, the mixed Mn 4+ /Mn 3+ valence is confirmed, with a larger contribution of the Mn 4+ in the HT-MnO2 material. The manganese oxidation states of the materials before protonation could also be determined by XPS with the Mn 2p spectra and are as follows: Na-Mn +3.56 O2, K-Mn +3.58 O2, HT-Mn +3.66 O2. These results are in agreement with the tendencies obtained with the Mn 3s spectra. Moreover, these results extracted from exploitation of Mn 2p spectra corroborate the oxidation states values from ICP measurements for which we obtained the following values: Na-Mn +3.53 O2, K-Mn +3.52 O2, HT-Mn +3.70 O2.

For the O 1s spectra, for all materials, three components are observed at 530.0 eV, 531.6 eV and 533.4 eV. The component at 530.0 eV is attributed to the bulk O 2-oxygens, the one at 531.6 eV to the adsorbed -OH species as well as to the under-coordinated surface oxygens and finally the component at 533.5 eV is assigned to the adsorbed species and to water molecules that are trapped between the nanosheets of the materials. For Na-MnO2, the presence of sodium ion is detected as expected. The Auger peak of Na 1s is present at 536.0 eV on the O 1s spectrum (pink component).

After protonation

The Mn 2p, O 1s and Na 1s or K 2p XPS spectra of the materials after protonation are shown in figure S1. The same accuracy method as mentioned just before was used to determine the manganese oxidation state, in order to follow the protonation impact on the material and their electronic structure. The Mn 4+ /Mn ratios are presented in table 3. The ratios (0.72 for HK-MnO2, 0.69 for HNa-MnO2 and 0.66 for HHT-MnO2, against 0.58 for K-MnO2, 0.56 for Na-MnO2 and 0.66 for HT-MnO2) indicate that the amount of Mn 4+ increases after protonation of K-MnO2 and Na-MnO2, while it is maintained for HT-MnO2. The tendency toward an increase of the oxidation state after protonation is confirmed by the evolution of the ΔB.E.Mn3s splitting values, from 4.8 eV and 4.7 eV for Na-MnO2 and K-MnO2 respectively toward 4.5 eV for HNa-MnO2 and HK-MnO2. This means, according to the references, that the protonation induces an increase of the Mn 4+ /Mn 3+ ratios, as it was deduced by iodometric titration. One of the hypotheses to explain this slight oxidation takes place in the dissolution of the edges of the sheets of these two materials when there are placed in the acidic environment of the protonation step. On the other hand, for HHT-MnO2 the large size of platelets inhibits the dissolution process and therefore the oxidation.

The success of the protonation step is confirmed by the decrease of the Na/Mn and K/Mn ratio values after protonation (almost divided by 10 (table S1)). We can notice also that protonation increases the contribution of the components associated to the under-coordinated oxygens and the adsorbed hydroxide to the whole O 1s signal. In fact, an increase of the -OH groups is expected after protonation. Especially, the atomic percentages of O531.7 component give an increase of 100%

for HNa-MnO2 and HK-MnO2 but only 30% for HHT-MnO2. In that case, the protonation does not impact in a same manner the three birnessites. In particularly, the HHT-MnO2 particles are not eroded after protonation.

To sum up, the protonation method is more effective for HK-MnO2 and HNa-MnO2 than for HHT-MnO2, which leads to a stronger impact in terms of oxidation of the surface manganese ions and of increase of surface hydroxyl groups amount.

Investigation of surface reactivity

To study the surface reactivity, gas probe adsorptions of SO2 were performed. The objective of this reactivity study is to investigate 1) the correlation between the adsorption capacity to the birnessite morphology and the impact of the protonation on it and 2) the adsorption mechanism and the active site concentration in chemisorption conditions.

Before protonation

First, adsorption of SO2 molecules with a number of pulses until saturation of the actives sites has been performed on the pristine birnessites. In that case, the number of pulses depends on the whole active sites available at the surface engaged in both physisorption or chemisorption mechanism. We obtain that the number of pulses required to saturate the sample surface differs from one material to another, these results are presented in table 3. For the materials before protonation, 14 pulses of gas probe were required to reach the surface saturation for HT-MnO2 samples, 20 pulses for Na-MnO2 and 36 pulses for K-MnO2. As expected, the larger the BET value, the higher the required number of pulses.

After adsorption of the gas probes, the samples were analyzed by XPS to quantify the active sites and determine the adsorption mechanism directly linked to the nature of the active sites. In this part, only the chemisorption process will be taken into account as the analysis chamber is under ultra-vacuum. As the adsorption does not impact the Mn 2p spectra, only the O 1s and S 2p spectra will be presented, in Figure 7 for the materials before protonation and in figure 8 after protonation. For all materials after adsorption, well defined S 2p core peaks can be detected, confirming the SO2 adsorption as chemisorbed species.

First of all, for the Na-MnO2 and K-MnO2 materials, 95% of the adsorbed species are sulphates (B.E. S2p3/2 = 168.7 eV in figure 7) and 5% are sulfites (B.E. S 2p3/2 = 167.0 eV), while 100% sulphate species are observed for HT-MnO2.

Figure 7. O 1s (green: oxide anion, red: adsorbed hydroxide, oxygen from SO2 adsorbed and under coordinated oxide, blue: water and adsorbed species and pink: Na 1s Auger) and S 2p (blue: sulfate, orange: sulfite) core peaks of Na-MnO2, K-MnO2 and HT-MnO2 materials after SO2 adsorption.

Moreover, an increase of the O1s531.7 component is observed according to the surface oxygen engaged in a strong bond with sulphur issued from SO2 molecules. We found that the atomic percentage of the component located at 531.7 eV after adsorption corresponds roughly to twice the detected atomic percentage of sulphur. Indeed, for sulfate species (major contribution in our case), one sulphur is surrounded by three and four oxygens, located at 531.7 eV, including two oxygens coming from the SO2 molecules

In terms of quantity of active sites, the S/Mn ratios are 0.31, 0.24, 0.49 for Na-MnO2, K-MnO2 and HT-MnO2 respectively. As a result, before protonation, the HT-MnO2 material, though exhibiting the smallest BET surface (5 m 2 /g), has the largest surface redox sites concentration while the most divided material, i.e. K-MnO2, exhibits the smallest redox response. This behaviour, quite counterintuitive at first sight, shows that there is no direct relation between the BET surface value and the surface reactivity of the material.

After protonation

As already observed, the number of pulses required to saturate the surface increases with the BET surface, i.e. 23 pulses for HHT-MnO2 with a BET surface of 18 m 2 /g, 58 pulses for HNa-MnO2 and a BET surface of 60 m 2 /g and finally 96 pulses for HK-MnO2 with 85 m 2 /g.

Performing adsorption of SO2 gas probes coupled with XPS on HNa-MnO2, HK-MnO2, and HHT-MnO2 materials allows following the impact of the protonation on the surface reactivity and especially the nature and concentration of active sites. The XPS spectra of O 1s and S 2p after adsorption are shown in figure 8. The active sites concentration that relies on the S/Mn ratio is reported in table 3 for all birnessites. We note a slight change in terms of adsorption mechanism after protonation since the acid-base adsorption mechanism associated to formation of sulphite species is not detected, only sulfate species are indeed identified. The S/Mn ratios are evaluated as 0.21 for HNa-MnO2, 0.20 for HK-MnO2 and 0.46 for HHT-MnO2, respectively, against 0.31, 0.24 and 0.49 for the phases before protonation. The amounts of active sites are therefore lowered after protonation. As compared with the other two materials, in relative value, the decrease of sites concentration (S/Mn) for the HT-MnO2 ceramic birnessite appears lower, suggesting a minor change in the surface properties for this material after protonation. We can argue that, for this material, which exhibits a weak specific surface, the exchange of protons for interlamellar K + cations is only partial, as revealed by the variation of K/Mn ratio before and after protonation. This behaviour can be related to the morphology, constituted of wide micrometric platelets, which develop basal surface (00l) higher than the surface corresponding to the slab edges. Because the exchange of the potassium ions by protons is more favoured on the edges of the slabs and therefore of the platelets than on the basal surface, it cannot be completed within the bulk and the platelet basal surface is not really modified, leading to a minor change in the total redox surface reactivity of the material.

From a general point of view, the tendency is that, after protonation, the number of pulses needed to saturate the surface is increasing while the amount of redox active sites (S/Mn ratio) is decreasing. We can thus conclude that the protonation i) increases the number of sites for physisorption and therefore the porosity according to the volume of the gaseous probe but ii) affects the redox adsorption mode.

DISCUSSION

It is commonly accepted that charge storage mechanisms can be divided into 3 different types: i) faradaic storage due to redox reactions linked to intercalation/deintercalation processes within the bulk of the materials, ii) pseudocapacitive storage due to redox reactions coupled to ionic adsorption on multiple sites at the surface / near surface of the materials, iii) capacitive storage corresponding to the exchange of electrostatic charges following the electrochemical double layer model [START_REF] Zhu | Birnessite based nanostructures for supercapacitors: challenges, strategies and prospects[END_REF] . The faradaic storage is not influenced by the specific surface, while the pseudocapacitive and capacitive storages are expected to be strongly dependent on it. Moreover, pseudocapacitive mechanims depends to the nature of surface active sites, especially redox sites. The present work aims at investigating the potentialities of surface reactivity measurements (by adsorption of gaseous probes coupled to XPS studies) as a key tool to predict or understand the charge storage mechanism of powdered materials.

For that purpose, six birnessite type phases with various morphologies and slightly different compositions [START_REF] Thèse | Exfoliation and restacking of manganese and cobalt based lamellar oxides for supercapacitor electrodes HAL[END_REF] were studied in the present work: 3 pristine birnessites Na-MnO2 (nanoplatelets), K-MnO2 (veils) and HT-MnO2 (large platelets of some micrometers that form a 3D network), and their 3 derivatives obtained by exchange of alkaline ions for protons, namely HNa-MnO2 (eroded nanoplatelets), HK-MnO2 (eroded veils) and HHT-MnO2 (still large platelets, but the 3D network linking them is destroyed). In terms of porosity, Na-MnO2 and K-MnO2, as well as their protonated derivatives, can be considered as mesoporous materials, while not for HT-MnO2 and its derivative.

The largest specific surface goes to HK-MnO2 with its eroded veils (85 m 2 /g), and the smallest to HT-MnO2 (5 m 2 /g) with its interconnected large platelets.

The adsorption of SO2 gaseous probe performed on the various materials allows to deduce surface reactivity. First, regarding the adsorption of SO2 molecules with a pulse mode until saturation of the surface actives sites, the number of pulses depends on the whole active sites (physisorption or chemisorption mechanism) available at the surface. It should be noted that, experimentally, for the whole series of materials, the order of the number of pulses follows quite well the order of the specific surface values. Second, regarding the S/Mn ratio determined by XPS analysis of the surface after adsorption, as a reason of the ultra-vacuum experimental conditions, it corresponds to the concentration of the surface redox sites (chemisorption mechanism). The values of this ratio appear to be the largest for, in the order, HT-MnO2 (0.49), its protonated derivative HHT-MnO2 (0.46) and Na-MnO2 (0.31), probably because these materials are constituted of integral and smooth platelets, favouring thus the presence of (001) surfaces. As soon as the platelets are eroded by the protonation acidic medium, which is the case of HNa-MnO2, or when the platelet morphology is replaced by veils, as in the case of K-MnO2 and HK-MnO2, the concentration of redox active sites decreases (S/M ratio close to 0.2).

As far as the electrochemical properties are concerned, let us try to correlate the evolution of the capacities to the properties and characteristics listed above. At low scan rate, as shown by the shape of the voltammograms in Figure 4, the charge storage process is dominated for the 3 pristine materials by a pseudocapacitive behaviour (rectangular part) and a significant faradaic one (redox peaks) due to intercalation/deintercalation of the alkaline ions [START_REF] Zhu | Birnessite based nanostructures for supercapacitors: challenges, strategies and prospects[END_REF][START_REF] Thèse | Exfoliation and restacking of manganese and cobalt based lamellar oxides for supercapacitor electrodes HAL[END_REF] . For the protonated derivatives, the faradaic contribution disappears because the intercalated protons hinder the intercalation/deintercalation process of the alkaline ions. That is the reason why the capacities delivered by the protonated materials are significantly lower than those of their non protonated precursors. At this low rate, the best capacity is obtained for the material HT-MnO2, which exhibits an intense faradaic contribution (high surface of the redox peaks), and the largest concentration of redox sites (S/M = 0.49), which can reasonably be linked to the pseudocapacitive component. It should be noted that this material exhibits thus the highest capacity, in spite of its lowest specific surface (5 m 2 /g). On the contrary, at high rate, as an example for 100 mV/s, the best capacity is for the HK-MnO2 phase (46 F/g), which exhibits the highest specific surface (85 m 2 /g). This behaviour is in accordance with the fact that, at such rate, the charge storage process is dominated by a capacitive process, which is all the more promoted as the specific surface is high [START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF][START_REF] Zhu | Birnessite based nanostructures for supercapacitors: challenges, strategies and prospects[END_REF] . It should be noticed that the capacity of HHT-MnO2 is particularly low with regard to other materials on the whole scan rate range. This bad performance can be assigned to the fact that i) the disconnection of the large platelets is likely to limit the inter-particle ionic and electronic transfers, thus inhibiting both the pseudocapacitive behaviour, in spite of a good concentration of redox sites (S/Mn=0.46, close to the 0.49 value observed for HT-MnO2), and the capacitive behaviour, ii) the specific surface (18 m 2 /g) is quite low, which is particularly detrimental to the capacitive behaviour iii) the faradaic intercalation-deintercalation of alkaline ions between the slabs is completely prevented by the presence of protons at the edges of the interlamellar spaces.

The main mechanisms involved after protonation of the materials are then the surface redox reactions (pseudocapacitive) and electrostatic surface adsorption (capacitive). These two mechanisms could be studied in a distinct way by varying the scan rate during the cycling as seen previously. Moreover, the results of the adsorption experiments, which allowed to determine the amount of surface active sites, can be directly linked to the contribution of the pseudocapacitive mechanism observed at low and moderate scan rates. As for the capacitive mechanism is discriminated at high scan rate, its contribution is related to the BET surface of the material as well as to the quantity of gas probe pulses sent during the adsorption experiments. The larger the BET surface area, the larger the number of pulses sent to probe the material surface. On the other hand, the contribution of the capacitive phenomenon cannot be quantified with the XPS technique because the analyses are carried out under ultra-vacuum, thus only the chemisorbed species are still present. The BET surface is an important criterion for the capacitive phenomenon but has a moderate impact on the concentration of active redox sites on the surface, it does not directly influence the pseudocapacitive phenomenon. The electrochemically active surface appears thus different from the BET surface.

CONCLUSIONS

The objective of this work is to establish a link between the morphology and the specific surface of birnessite materials Na-MnO2, K-MnO2 and HT-MnO2 and after protonation with its electrochemical behaviour. Our approach was to study the surface reactivity of the materials through adsorption of SO2 gas probes, with XPS monitoring to identify and quantify the adsorbed species present at the materials surfaces.

It was shown that the three birnessites have different morphologies, veil-like for K-MnO2, platelet-like for Na-MnO2 and large platelets forming a 3D network for HT-MnO2. By performing cyclic voltammetry curves of these materials at different scan rates, it could be observed from the shape of the curves that surface redox reactions (pseudocapacitive), bulk redox reactions (faradaic) and electrostatic adsorption (capacitive) occur. After protonation of these same materials, an evolution of the morphology is observed. Indeed, for HNa-MnO2 and HK-MnO2, a slight erosion of the edges of the veils and platelets occurred due to the acidic environment in which the protonation was performed. Thanks to the probes of the surface chemical reactivity before and after protonation it could be confirmed that this erosion leads to a decrease in the quantity of redox active sites. It was also possible to observe a disappearance of bulk redox reactions following this protonation.

As for the HHT-MnO2 material, the protonation led to a rupture of the 3D network and consequently the capacity was strongly reduced, well below that of HNa-MnO2 and HK-MnO2.

Due to its veil-like morphology, the BET surface of HK-MnO2 is larger than those of the other materials (85 m 2 /g vs 60 m 2 /g for HNa-MnO2 and 18 m²/g for HHT-MnO2). Thus, the contribution of the capacitive mechanism is the largest in the case of HK-MnO2 material. The surface chemical reactivity has revealed that HK-MnO2 and HNa-MnO2 materials exhibit an equivalent amount of surface redox active sites, the difference in electrochemical performance can be explained by the BET surface of HK-MnO2 which allows a larger electrostatic charge storage. This explains why after protonation the capacity of HK-MnO2 is the highest.

In conclusion, the most suitable material for use as an electrode material in supercapacitor storage systems is HK-MnO2 because of its surface chemical reactivity, BET surface area and specific capacity.

ASSOCIATED CONTENT Supporting Information. The Supporting information contains figures, tables and description to complete and precise some points in the main manuscript. This material is available free of charge via the Internet at http://pubs.acs.org.

Pore size distribution of the six studied birnessite. Galvanostatic charge and discharge curves obtained in 0.5M K2SO4 at 0.5 A/g (left) and (right) specific capacity evolution for all birnessite

* 1 .

 1 Université de Pau et des Pays de l'Adour, E2S UPPA, CNRS, IPREM, UMR 5254, 64000 Pau, France 2. Univ. Bordeaux, CNRS, Bordeaux INP, ICMCB, UMR 5026, F-33600 Pessac, France 3. RS2E, Réseau Français sur le Stockage Electrochimique de l'Energie, FR CNRS 3459, F-80039 Amiens Cedex 1, France 4. ALISTORE-ERI European Research Institute, FR CNRS 3104, F-80039 Amiens Cedex 1, France KEYWORDS: supercapacitors, energy storage, MnO2, surface reactivity, morphology ABSTRACT.

Figure 1 .

 1 Figure 1. X-ray diffraction patterns of pristine Na-MnO2, HT-MnO2 and K-MnO2, and their

  polisher. The coupling of the two methods allows having information on both the surface and the core of the materials. The cross sections image of the K-MnO2 material (figure 2.d) shows the veils rolled up one with each other, which makes it them difficult to be distinguished. Compared to the SEM image made on an uncut sample, the cross section image still shows a relatively compact arrangement in the core of the aggregates. On the other hand, the morphology is very different for the HT-MnO2 material (figure 2.b). The SEM image reveals a macroporous three-dimension network. The objects are much denser and larger than in the previous material with a particle size between 1 and 15 µm. Moreover, we have also found large particles (dark grey) as reported in the SEM image on cross section (figure 2.e) of about 15 µm in diameter. The platelets are not identifiable and no porosity is observed in agreement with the BET surface (4 µm 2 /g). The light grey particles correspond to the metallic epoxy used during the preparation of the sample to make the cross section. A third morphology is observed for Na-MnO2. The corresponding SEM image (figure2.c) shows platelets of irregular size and shape that are essentially of the order of several hundred nanometers in length[START_REF] Tang | Synergy of Mn and Co in Slab-Based Nanocomposites for Hybrid Supercapacitors: Impact of Restacking Process on Electrochemical Properties[END_REF] . The cross section image (figure2.f) highlights the regular stacking of the MnO2 platelets for the edges of the platelets are clearly visible.

Figure 2 .

 2 Figure 2. a), b), c) SEM images and d), e), f) cross section SEM images of K-MnO2, HT-MnO2

Figure 3 .

 3 Figure 3. N2 adsorption/desorption isotherms at 77K for bare birnessites (a): Na-MnO2, HT-MnO2

Figure 4 .

 4 Figure 4. Cyclic voltammograms of non-protonated and protonated birnessites obtained at 0.5

Figure 5 .

 5 Figure 5. Variation of capacitance vs scan rate obtained for non-protonated and protonated

Figure 6 .

 6 Figure 6. Mn 2p, O 1s (green: oxide anion, red: adsorbed hydroxide and under coordinated oxide,

Figure 8 .

 8 Figure 8. O 1s (green: oxide anion, red: adsorbed hydroxide, oxygen from SO2 adsorbed and under

Table 1 .

 1 Cell parameters of the pristine synthesized birnessites and the protonated derivatives.

		Space group	a (Å)	b (Å)	c (Å)	β (°)
	K-MnO2	C2/m	5.166(7)	2.911(6)	7.089(7)	90.8(1)
	HK-MnO2	C2/m	2.836(3)	2.836(3)	7.32(2)
	Na-MnO2	C2/m	5.223(6)	2.750(3)	7.286(6)	102.29(9)
	HNa-MnO2	C2/m	2.839(2)	2.839(2)	7.27(1)
	HT-MnO2	P63/mmc	2.860(4)	2.860(4)	14.24(4)
	HHT-MnO2	P63/mmc	2.837(1)	2.837(1)	14.40(8)

Table 2 .

 2 Chemical composition and specific area of the pristine synthesized birnessites and the

	protonated derivatives.
	Materials	Chemical composition	Specific area (m²/g)
	Na-MnO2	H0.15 + Na0.32 + (H2O)0.45Mn 3.53+ O2	35
	HNa-MnO2	H0.34 + Na0.01 + (H2O)0.39Mn 3.65+ O2	60
	K-MnO2	H0.23 + K0.25 + (H2O)0.34Mn 3.52+ O2	45
	HK-MnO2	H0.35 + K0.02 + (H2O)0.35Mn 3.63+ O2	85
	HT-MnO2	K0.30 + (H2O)0.32Mn 3.70+ O2	4
	HHT-MnO2	H0.20 + K0.08 + (H2O)0.42Mn 3.72+ O2	18

Table 3 .

 3 Summary of the results

		Number of SO2 pulses to saturate the surface	BET (m 2 /g)	S(at.%)/ Mn(at.%)	Sulfate/ Mn	Sulfite/ Mn	at. % S	Mn 4+ /Mn before adsorption	Mn 4+ /Mn after adsorption	Redox peak at 0.5 mV/s	specific capacity at 100 mV/s (F/g)
	Na-MnO2	20	35	0.31	0.3	0.01	4.9	0.56	0.36	yes	42.7
	HNa-MnO2	58	60	0.21	0.21	0	4.3	0.69	0.57	no	42.7
	K-MnO2	36	45	0.24	0.23	0.01	4.5	0.58	0.56	yes	23.1
	HK-MnO2	96	85	0.20	0.19	0.01	4	0.72	0.59	no	46.6
	HT-MnO2	14	5	0.49	0.49	0	6.3	0.66	0.59	yes	26.3
	HHT-MnO2	23	18	0.46	0.46	0	6.6	0.66	0.69	no	9.2

in function of current density. Mn 2p, O 1s, Na 1s or K 2p XPS spectra of protonated birnessites; Deconvolution of the Mn 2p spectrum of HNa-MnO2 material from Mn 3+ and Mn 4+ reference materials; Mn 3s core peaks of non-protonated and protonated birnessites; detailed XPS results (identifications and quantifications) before and after protonation and after an exposure to the SO 2 pulses.

ACKNOWLEDGMENTS

The authors acknowledge Saft and Région Nouvelle-Aquitaine for the funding of the project.

AUTHOR INFORMATION

Corresponding Author

*Delphine Flahaut delphine.flahaut@univ-pau.fr *Liliane Guerlou-Demourgues liliane.Guerlou-Demourgues@enscbp.fr


Z Awada

B Nedjar

Finite viscoelastic modeling of yeast cells with an axisymmetrical shell approach

Keywords: Cell-wall, Axisymmetric shells, Finite viscoelasticity, Internal cell-volume conservation, Finite element method

Based on the use of shell kinematics, a theoretical model is developed to predict time-dependent responses of Saccharomyces cerevisiae yeasts. These cells are herein described as thin-walled shells with axisymmetrical shapes and, due to their soft nature, the finite strain range is a priori adopted. Accordingly, a finite viscoelastic modeling is assumed for the cell-wall. An Ogden-type energy function is used together with a creep-like potential, this latter to describe the way overstresses vanish at thermodynamic equilibrium. The cell-volume conservation due to the inner incompressible fluid is accounted for via an update procedure of the Uzawa type. Numerical simulations highlight the efficiency of the proposed framework. As a further outcome, the emergence of a necessary cell-wall representation by at least two sets of layers with sensitively different mechanical properties is established.

Introduction

Over the past decades, research in the field of cell mechanics has received considerable attention relative to other biomechanical fields. The investigations began experimentally on animal cells around the 1930's. Later on, improvements were made to be able to study the mechanical behaviour of walled cells such as, to mention a few, plant cells, tomato cells, yeast cells, and sea-urchin eggs. In what follows, the interest is on Saccharomyces cerevisiae yeasts. This kind of cells plays a crucial role in our lifes as it presents medical, veterinary, economical and cosmetic advantages. The medical, benefits are such as, for example, the production of vitamins, antibiotics and insulin [START_REF] Hatoum | Antimicrobial and probiotic properties of yeasts: From fundamental to novel applications[END_REF]. Economically speaking, yeasts are used in the fabrication of fermented foods and beverages especially bread, wine, beer, among others, e.g. [START_REF] Lahue | History and domestication of saccharomyces cerevisiae in bread baking[END_REF][START_REF] Parapouli | Saccharomyces cerevisiae and its industrial applications[END_REF]. These cells are in general of spherical, or at least close to spherical shapes, surrounded by a multi-functional wall. Firstly, this latter constitutes a protective barrier against the invasion of bacteria and microbes that can cause infections [START_REF] Moeendarbary | Cell mechanics: principles, practices, and prospects[END_REF]. It is then essential to understand the disruption process and the extraction of intracellular products [START_REF] Smith | The mechanical properties of saccharomyces cerevisiae[END_REF][START_REF] Mashmoushy | Micromanipulation measurement of the mechanical properties of baker's cells[END_REF]. In addition, the wall is primordial to determine the cell shape and to understand its growth [START_REF] Dague | An atomic force microscopy analysis of yeast mutant defective in cell wall architecture[END_REF]. This has been mainly our goal toward the cell-wall characterization and modeling.

During the last years, mathematical models have been initiated with results in [START_REF] Feng | On the contact problem of an inflated spherical nonlinear membrane[END_REF] on hollow spherical membrane compression together with semianalytical solution strategies, see [START_REF] Wang | Modelling the mechanical properties of single suspension-cultured tomato cells[END_REF]. Later on, the modeling has been improved in [START_REF] Smith | The mechanical properties of saccharomyces cerevisiae[END_REF][START_REF] Lardner | Compression of spherical cells[END_REF] by representing cells as thin walled, liquid-filled spheres. Several constitutive relationships were assumed for the cell-wall, e.g. Mooney-Rivlin is used in [START_REF] Skalak | Strain energy function of red blood cell membranes[END_REF][START_REF] Lardner | Compression of spherical cells[END_REF] for red blood cells and sea-urchin eggs, respectively. A linear-elastic constitutive relationship for yeast cells has been applied in [START_REF] Smith | The mechanical properties of saccharomyces cerevisiae[END_REF]. Recently, authors in [START_REF] Stenson | Modelling the mechanical properties of yeast cells[END_REF] proposed a model by using a Hencky-type law. In reality, there are time-dependent effects such as permeability and intrinsic viscoelasticity of the cellwall that influence the whole cell response. Those are just few examples of the many models reported in the literature.

In the present contribution, a theoretical and numerical modeling framework is derived for the time-dependent cell response. Yeasts are herein described as axisymmetrical, impermeable, liquidfilled shells. In view of the viscoelastic character of the cell-wall and its soft nature, a finite viscoelastic approach is adopted. The kinematical choice is based on the multiplicative decomposition of the deformation gradient into an elastically relaxing part and a viscous part, e.g. [START_REF] Sidoroff | Un modèle viscoélastique non linéaire avec configuration intermédiaire[END_REF][START_REF] Reese | A theory of finite viscoelasticity and numerical aspects[END_REF]. Use is made of an Ogden-type constitutive relation that we extend toward a finite viscoelastic modeling. In a first approach, we assume that the cell-wall is homogenous, isotropic and incompressible. Furthermore, the internal cell-volume conservation related to the incompressibilty of the fluid is solved via an Uzawatype update algorithm.

Axisymmetric shell modeling

Yeasts can be regarded as thin-walled shell structures and, due to their soft nature, a formulation within the finite strain range is adopted. We first recall the basic kinematics that we extended toward finite viscoelasticity. The balance equation is then given that accounts for the action of the liquid inside a typical closed cell. We next derive the constitutive equations that will be used later on for numerical simulations.

Finite viscoelastic kinematics

Within the finite strain range, we denote as usual by F the deformation gradient, and we recall the definition of the Green-Lagrange strain tensor E as,

E = 1 2 F T F -1 , (1)
where 1 is the second-order identity tensor. In this study, use is made of the quasi-Kirchhoff-type theory for thin shells of revolution as derived in [START_REF] Wagner | A finite element model for non-linear shells of revolution with finite rotations[END_REF][START_REF] Wriggers | An axisymmetrical quasi-Kirchhoff shell element for large plastic deformations[END_REF]. The non-zero components of the above strain reduce to the meridional and circumferential ones, here respectively E 11 ≡ E 1 and E 22 ≡ E 2 . These latter being regarded as principal strains, the principal stretches λ γ , γ = 1, 2, that are relative to the deformation gradient F , follow from (1) as,

E γ = 1 2 λ 2 γ -1 ⇒ λ γ = 2E γ + 1 , (2)
together with λ 3 = 1. Now for the extension toward finite viscoelasticity, use is made in this work of the multiplicative decomposition of the deformation gradient F into an elastically relaxing part F e and a viscous part F v , e.g. see [START_REF] Reese | A theory of finite viscoelasticity and numerical aspects[END_REF][START_REF] Nedjar | An anisotropic viscoelastic fibre-matrix model at finite strains: Continuum formulation and computational aspects[END_REF]: F = F e F v , and that we adapt to the present shell theory. A multiplicative decomposition of the stretches is then deduced as well,

λ i = λ e i λ v i , (3)
where λ e i and λ v i are the principal stretches relative to F e and F v , respectively. For later use, we define the logarithmic strains of the above stretches as,

ε i = ln λ i , ε e i = ln λ e i , ε v i = ln λ v i , (4
) and, further, the multiplicative decomposition implies,

ε i = ε e i + ε v i . (5)

Mechanical balance

Within a Lagrangian formulation, the mechanical balance of a yeast cell that occupies the domain B in its reference configuration can be given in terms of the second Piola-Kirchhoff stress tensor S by the following weak form:

B S : δE dV = - ∂Bt pn.δu da , ∀δu , (6)
where δu is a variation of the displacement field u. Here we consider follower loads pn acting in the direction n normal to the deformed shell B t , and the integration of which is more easily performed with respect to the current configuration, [START_REF] Wriggers | Nonlinear Finite Element Methods[END_REF][START_REF] Nedjar | A coupled BEM-FEM method for finite strain magneto-elastic boundary-value problems[END_REF] among others. For an axisymmetric quasi-Kirchhoff shell of mid-surface reference M and thickness h, Eq. ( 6) is rewritten as, 2π

M h S γ δE γ rdξds + M h E 13 δE 13 rdξds = - ∂Bt pn.δu da, (7) with ξ ∈ [-h/2, +h/2],
and where is the penalty parameter for the transverse shear part, i.e. the constraint E 13 = 0. Accordingly, S γ are the principal second Piola-Kirchhoff stresses that are work conjugate with the principal strains

E γ , γ = 1, 2.
In what follows, the constitutive equations will be given in terms of the Kirchhoff stress tensor τ that is related to S by the push-forward procedure τ = F SF T . One has the following connection:

S γ = τ γ λ 2 γ , (8)
where τ γ are principal Kirchhoff stresses.

Constitutive equation: finite viscoelasticity

Motivated by the generalized Maxwell model, see Fig. 1, the free energy ψ is additively split into an equilibrium part ψ eq , and a non-equilibrium part ψ neq . While the former depends on the total strains, the latter depends on the elastically relaxing ones. Herein, we choose the following incompressible N=1-Ogden-type model written in terms of principal stretches as,

ψ = 2µ α 2 λ α 1 + λ α 2 + λ α 3 -3 ψ eq + f 2µ α 2 λ e α 1 + λ e α 2 + λ e α 3 - 3
ψ neq , (9)
where µ denotes the long-term shear modulus, α the Ogden's coefficient, and the adimensional factor f is a weighting parameter for the overstress contribution, i.e. the instantaneous shear modulus is here evaluated as (1 + f )µ. A Neo-Hooke-type model is retrieved with α = 2. Notice that other choices instead of the form ( 9) can be used as well, e.g. [START_REF] Steinmann | Hyperelastic models for rubber-like materials: consistent tangent operators and suitability for Treloar's data[END_REF][START_REF] Nedjar | A finite element implementation of the isotropic exponentiated Hencky-logarithmic model and simulation of the eversion of elastic tubes[END_REF].

Next, the principal Kirchhoff stresses are given by,

τ i = λ i ∂ψ eq ∂λ i + λ e i ∂ψ neq ∂λ e i + ς, (10)
where ς denotes the pressure related to the material incompressibility. It is deduced from the plane stress assumption τ 3 = 0, and when replaced back into Eq. ( 10), we end-up with the following constitutive relation for the present axisymmetric shell theory:

τ γ = 2µ α λ α γ -(λ 1 λ 2 ) -α τ eq γ + f 2µ α λ e α γ -(λ e 1 λ e 2 ) -α τ neq γ , ( 11
) for γ = 1, 2.
In parallel, it is crucial to develop the way the non-equilibrium stress τ neq vanishes at thermodynamic equilibrium. The dissipation related to viscoelasticity is given by, e.g. [START_REF] Reese | A theory of finite viscoelasticity and numerical aspects[END_REF][START_REF] Nedjar | Frameworks for finite strain viscoelasticplasticity based on multiplicative decompositions. Part I: Continuum formulations[END_REF][START_REF] Nedjar | A coupled BEM-FEM method for finite strain magneto-elastic boundary-value problems[END_REF] :

D = τ neq : - 1 2 £ v b e b e -1 ≥ 0 , (12)
where

£ v b e = F dC v -1 dt F T (13)
E eq ≡ E > 0 is the Lie derivative of the elastically relaxing left Cauchy-Green tensor b e = F e F e T ≡ F C v -1 F T , and where

E neq ≡ f E η > 0 ε v ε σ = σ eq + σ neq
C v = F v T F v is the viscous right Cauchy-Green tensor.
A simple way to satisfy the inequality [START_REF] Stenson | Modelling the mechanical properties of yeast cells[END_REF] for any admissible process is to postulate the existence of a creep-like potential that we denote here by φ vis (τ neq ), a convex function such that,

- 1 2 £ v b e b e -1 = ∂φ vis (τ neq ) ∂τ neq . (14)
Further, convenient for the present formulation, φ vis is equivalently expressed in terms of principal stresses; φ vis (τ neq ) ≡ φvis (τ neq 1 , τ neq 2 ), and to make matters as concrete as possible, the following quadratic form will be used in the present work,

φvis = 1 2η τ 2 neq 1 + τ 2 neq 2 , ( 15
)
where η is the viscosity parameter such that the associated relaxation time τ is given by,

τ = η f µ . ( 16
)
When written in principal components, and taking into account [START_REF] Wagner | A finite element model for non-linear shells of revolution with finite rotations[END_REF], the tensorial form ( 14) of the evolution equation becomes,

εv γ = τ neq γ η γ = 1, 2 , (17)
and the numerical resolution of which is nowadays classical, e.g. [START_REF] Reese | A theory of finite viscoelasticity and numerical aspects[END_REF][START_REF] Nedjar | Frameworks for finite strain viscoelasticplasticity based on multiplicative decompositions[END_REF][START_REF] Nedjar | A coupled BEM-FEM method for finite strain magneto-elastic boundary-value problems[END_REF]. Thus, in summary, four mechanical parameters are needed to identify the cell-wall viscoelastic behaviour: (i) the long-term shear modulus µ, (ii) the N=1-Ogden coefficient α, (iii) the adimensional factor that characterizes the level of the nonequilibrium overstresses f , and (iv) the relaxation time of the viscous evolution τ .

Intrinsic cell-wall parameters

Few studies have reported time-dependent mechanical properties of yeast cell-walls, most probably because of the difficulties related to the cellular dimensions. Here in the absence of an ad-hoc experimental campaign which would have made it possible to carry out a method to identify the material parameters, use is made of direct results found in [START_REF] Ahmad | Nanoindentation methods to measure viscoelastic properties of single cells using sharp, flat, and buckling tips inside ESEM[END_REF], see also [START_REF] Fukuda | Micro-Nanorobotic Manipulation Systems and Their Applications[END_REF]. So far, it is noticiable that too few results have been reported in the literature on measuring viscoelastic properties of single yeast cells.

The key point is that, as for any model within the finite strain range, the one of Section 2.3 tends to its linear viscoelastic version at the limiting case of an infinitesimal perturbation. This property is here exploited accordingly. Thereby, the rheological model of Fig. 1 is nowadays classical and its apparatent Young modulus is given by,

E = E(1 + f e - t τ ), (18)
from whichone can identify the long-and shortterm moduli E and (1 + f )E, respectively, in complete agreement with the constitutive relation [START_REF] Skalak | Strain energy function of red blood cell membranes[END_REF].

In the reference [START_REF] Ahmad | Nanoindentation methods to measure viscoelastic properties of single cells using sharp, flat, and buckling tips inside ESEM[END_REF], the values found for the short-term modulus were about 3.67 ± 0.27 MPa, while for the long-term modulus, they were about 1.16±0.22 MPa. The relaxation time was about 6 to 7 s. Observe that the diversity of some viscoelastic properties may depend, on several biological factors such as, growth phases, biological culture, and so on. Hence, with regards to the expression [START_REF] Wriggers | Nonlinear Finite Element Methods[END_REF], this gives the following estimates within the present framework:

E = 1.16 ± 0.22 MPa, f = 2.325 ± 0.865, τ = 6.5 ± 0.5 s. ( 19
)
Further, interesting observations must be pointed-out at this stage: (i) the relaxation time τ is of the order of few seconds, (ii) the overstress factor f is far from negligible, i.e. here about 230%, and (iii) whether the long-or short-term moduli, their values are in the range of earlier reported Young moduli, as in [START_REF] Dague | An atomic force microscopy analysis of yeast mutant defective in cell wall architecture[END_REF][START_REF] Goldenbogen | Dynamics of cell wall elasticity pattern shapes during yeast mating morphogenesis[END_REF], but different by two orders of magnitude from values acquiered by other authors, for instance in [START_REF] Smith | On uniquely determining cell-wall material properties with the compression experiment[END_REF][START_REF] Stenson | Modelling the mechanical properties of yeast cells[END_REF]. This point will be brought up and discussed again in Section 5.1.1.

Internal cell-volume conservation during loading

Inside the closed cell, the liquid resists to the applied outer loads. It is represented by a normal pressure on the inner surface. Thus, focus is made on the external surface load term in Eq. ( 7) that we rewrite here for the sake of clarity:

g p (u, δu) = - ∂Bt δu.pn da , (20)
where the deformation-dependent unit normal vector n is chosen pointing toward the interior of the cell. The pressure magnitude p can increase or decrease due to the conserved cell-volume during compression as the filled liquid is taken to be incompressible: p constitutes then an additional unknown of the problem at hand. We propose in this section a method to numerically solve for it. Let denote by V int0 the initial volume inside the cell. During a global iterative process, say at iteration (k) where the mechanical equilibrium is solved with a fixed current estimate of the internal pressure p(k) , one computes the resulting internal volume V (k) int within the post-processing. Then, if V (k) int < V int0 , the pressure amplitude must be increased, while it must be decreased if

V (k) int > V int0 .
Herein, this update procedure is encompassed by the following Uzawa-like algorithm,

p(k+1) = p(k) + 1 - V (k) int V int0 , (21)
where is a penalty parameter.

Numerical simulations

In this set of simulations, we consider examples where yeasts are compressed between parallel flat plates. The first example treats the classical case of spherical cells with noteworthy observations. Then, pushing further the possibilities of the present framework, a non spherical geometry is considered in a second example.

Compressions of a spherical yeast

We consider a spherical yeast of external diameter 5 µm and thickness h = 150 nm. The initial mid-plane diameter is then D = 4.85 µm. For the finite element simulations, 200 shell elements are used. The (rigid) plates are modeled by implicit functions. A frictionless contact is adopted through a penalty formulation, see for example [START_REF] Wriggers | Nonlinear Finite Element Methods[END_REF]. For the cell-wall, we use the set of parameters ( 19) together with Ogden's coefficient α = 4: In parallel, to show the efficiency of the nested algorithm derived in Section 4, the corresponding evolutions of the relative internal volumes are shown in Fig. 3. For all the computations, they at most deviate at less than 0.15% of the initial internal volume V int0 . Finally, and for illustrative purposes, Fig. 4 shows typical deformed configurations at different compression levels, here for the case of the compression at the velocity of 2 µm/s.

µ = 0.

Comments and discussions

Even if the shapes of the curves in Fig. 2 are qualitatively in good agreement with those found in the literature, quantitatively this is not the case. The reaction forces are here lower by at least one order of magnitude from values obtained in, e.g. [START_REF] Smith | On uniquely determining cell-wall material properties with the compression experiment[END_REF][START_REF] Stenson | Determining the mechanical properties of yeast cell walls[END_REF]. The reason is most probably because, so far, we have supposed the cell-wall as homogeneous. However, the biological composition of S. cerevisiae cellis nowadays well understood. Its two major components are β-glucan and mannoproteins which respectively represent about 50-60% and 40-50% of the cell-wall mass, see [START_REF] Dague | An atomic force microscopy analysis of yeast mutant defective in cell wall architecture[END_REF]. A third component is the chitin that, even at low amounts (1-3%), has stiff mechanical properties. From the modeling point of view, it would be reasonable to represent the cellwall as layered through its thickness, with at least two sets of layers: (i) a soft outer layer mostly made of mannoproteins, and (ii) and a stiff inner layer mostly made of β-glucan and chitin.

Indeed, and as it was demonstrated in [START_REF] Mercadé-Prieto | Mechanical double layer model for Saccharomyces Cerevisiae cell wall[END_REF] for elasticity, while AFM puts to contribution the soft part of the cell-wall, micromanipulation mostly puts to contibution its stiffer part. For illustrative purposes, Fig. 5 shows results of similar computations as for Fig. 2, but this time with the following material parameters though the cell-wall thickness:

µ = 0.39 MPa, for ξ ∈ [0, h 2 ], µ = 39 MPa, for ξ ∈ [- h 2 , 0], (23)
that is, with a long-term modulus two orders higher in magnitude for the inner half wall, see [START_REF] Smith | The mechanical properties of saccharomyces cerevisiae[END_REF][START_REF] Stenson | Determining the mechanical properties of yeast cell walls[END_REF]. The rest of the viscoelastic parameters are kept identical due to the lack of informations: α = 4, f = 2.325, and τ = 6.5 s. Here one can observe that, quantitatively, the levels of reaction forces are of the order of those obtained in [START_REF] Smith | On uniquely determining cell-wall material properties with the compression experiment[END_REF][START_REF] Stenson | Modelling the mechanical properties of yeast cells[END_REF].

Compression of ellipsoidal yeasts

In this last example, we consider an ellipsoidal yeast cell with a prolate spheroid shape. The inimid-plane is of major diameter D = 4.85 µm and minor diameter d = 3 µm, and the wall thickness is taken as h = 150 nm. For the finite element simulations, 100 shell elements are used, and the compression between the flat plates is here again modeled by frictionless contact with implicit functions. For the cell-wall, we use the above two-layers model, this time with a Neo-Hooke-type model:

µ = 0.39 MPa, for ξ ∈ [0, h 2 ], µ = 39 MPa, for ξ ∈ [- h 2 , 0], (24)
together with α = 2, f = 2.325, and τ = 6.5 s for both layers. Fig. 6 presents the results of computations at different compression speeds: (i) very slow at 0.001 µm/s, (ii) very fast, and (iii) at intermediate velocities of 1 and 2 µm/s. The reaction forces are plotted versus the fractional deformations z/D, where z is the downward vertical displacement of the upper plate. Notice here again the consequence of the cell-wall viscoelasticity on the time-dependent response of the whole cell. In the end, Fig. 7 shows typical deformed configurations at different compression levels, here once more for the case of the compression at the velocity of 2 µm/s.

Conclusions and perspectives

The aim of this contribution was to derive a viscoelastic modeling framework within the finite strain range that is adapted for the modeling of S. cerevisiae yeast cells. Within a shell theory approach, an Ogden-type model has been proposed. The viscoelastic part is motivated by the generalized Maxwell model in conjunction with a multiplicative decomposition of the deformation gradient. An internal variable approach has then been adopted, the evolution of which satisfies the continuum thermodynamics requirements. Further, the very important internal cell-volume conservation is taken into account as well. For this, we have derived a nested algorithmic strategy in conjunction with an Uzawa-like update procedure to adjust the amplitudes of the unknown follower pressure. The numerical simulations have shown the ability of the present framework to predict the responses under different loading situations and at various speeds.

We believe that this framework can trigger deeper research. For instance, this modeling can be used to capture the influence of the culture conditions on the mechanical responses. Further, an extension toward general shells will increase the field of applications, i.e. for any cell geometries.

Figure 1 :

 1 Figure 1: Motivation: one-dimensional generalized Maxwell model.

  Fig. 2 presents the results of computations at different compression speeds: (i) very slow, (ii) very fast, and (iii) at intermediate velocities of 0.4 and 2 µm/s. The reaction forces are plotted versus the fractional deformations z/D, where z is the downward vertical displacement of the upper plate. Notice the consequence of the cell-wall viscoelasticy on the time-dependent response of the whole cell.In parallel, to show the efficiency of the nested algorithm derived in Section 4, the corresponding evolutions of the relative internal volumes are shown in Fig.3. For all the computations, they at most deviate at less than 0.15% of the initial internal volume V int0 . Finally, and for illustrative purposes, Fig.4shows typical deformed configurations at different compression levels, here for the

  at loading velocity 2 µm/s at loading velocity 0.4 µm/s no viscosity (very slow loading)

Figure 2 :

 2 Figure 2: Compression of a spherical cell at different loading velocities.

  displacement (z/D)very fast loading at loading velocity 2 µm/s at loading velocity 0.4 µm/s no viscosity (very slow loading)

Figure 3 :

 3 Figure 3: Internal cell-volume variations for the different loading velocities.

Figure 4 :

 4 Figure 4: Initial and different deformed configurations of the spherical cell for the compressive loading at 2 µm/s.

Figure 5 :

 5 Figure 5: Compression of a layered spherical cell at different loading velocities.

  at loading velocity 2 µm/s at loading velocity 1 µm/s very slow loading at 0.001 µm/s

Figure 6 :

 6 Figure 6: Compression of an ellipsoidal cell at different loading velocities.

Figure 7 :

 7 Figure 7: Initial and different deformed configurations of the ellipsoidal cell for the compressive loading at 2 µm/s.

Acknowledgement

The authors would like to thank the french ANR agency for the financial support within the frame of Rheolife project (ANR-18-CE45-0012).