Parth Desai

Juan Forero-Saboya

Valentin Meunier

Gwenaëlle Rousse

Michael Deschamps

Artem M Abakumov

Jean-Marie Tarascon
email: jean-marie.tarascon@college-de-france.fr

Sathiya Mariyappan
email: sathiya.mariyappan@college-de-france.fr

Mastering the synergy between Na 3 V 2 (PO 4 ) 2 F 3 electrode and electrolyte: A must for Na-ion cells

Keywords: Na-ion batteries, transition metal dissolution, hard carbon poisoning, surface coating, high temperature cycling

Broader context

The huge success of lithium-ion batteries in the last decades has fueled the boom of portable electronics and has paved the way for the energy transition from fossil fuels to more renewable energy sources. However, a full decarbonization will require the development of complementary energy storage technologies relying on more sustainable and more abundant raw materials. Among the different next-generation batteries being explored, sodium-ion is the most mature, given the easy technology transfer from the existing Li-ion processes. Indeed, the commercialization of Na-ion cells is imminent, with several companies worldwide showing a growing interest in their development and optimization. In such thriving research ecosystem, studying the cell degradation is crucial as its understanding will allow the development of long-cycle life cells. In this study we tackle the problem of transition metal dissolution pertaining to the promising sodium-ion technology (Na 2 V 2 (PO 4 ) 2 F 3hard carbon), that can either result in cell degradation and eventually cell dead. We show that this issue can be cured by combining the specific design of electrolytes together with coating approaches, hence enhancing the high temperature performances of the NVPF-HC technology that is a must to reach a highly demanding energy market.

Introduction:

Even though lithium-ion batteries (LIBs) have achieved unprecedented commercial success in the previous decades, uneven distribution and poor abundance of lithium lead to raw material supply tensions that pose a severe challenge for their use in large-scale applications. 1,[START_REF] Tian | [END_REF] In this regard, sodium-ion batteries (NIB) are an attractive complementary technology that shares a similar working principle and cell engineering to LIBs. 3,4 Sodium has numerous advantages over Li in terms of abundance (~23,000 ppm vs. 20 ppm in the earth's crust), distribution, and cost, offering economic sustainability. 5 However, one of the significant drawbacks of NIBs is their lower energy density than LIBs. In turn, sodium ion chemistries such as Na 3 V 2 (PO 4 ) 2 F 3 (NVPF)-hard carbon (HC) and Prussian blue analogs (PBA)-hard carbon offer high-power capabilities making them more attractive and cheaper alternatives for high-power applications [6][7][8][9][10] such as hybrid grid storage, power tools, fast (dis)charging electromotive, military equipment, etc.

In this context, our group in collaboration with the Na-ion battery start-up 'TIAMAT Energy (France)' is actively working to develop high-power NVPF -HC based Na-ion cells. NVPF is a structurally robust material, which usually shows a reversible capacity of ~120 mAh/g (Supporting Fig. S1) and exhibits excellent cycling performance at ambient temperature. [11][12][13] The NVPF voltage profile displays two distinct steps (Supporting Fig. S1) at 3.7 V and 4.2 V, corresponding to the subsequent (de)insertion of two sodium ions from one formula unit. This high redox potential of 4.2 V vs. Na + /Na 0 (or equivalent to 4.53 V vs. Li + /Li 0 ) creates severe issues of electrolyte oxidation that could further trigger parasitic reactions, especially at a high temperature of 55 °C. 6,8 We have previously shown that one way to alleviate this issue involves stabilizing the electrolyte against oxidation or reduction via several additives. 8,[14][15][16] For instance, by adding vinylene carbonate, sodium oxalatodifluoroborate, succinonitrile, and tri-methyl silyl phosphite (VC + NaODFB+ SN + TMSPi) to the control (1M NaPF 6 in EC+PC+DMC) electrolyte, 8,16 we could drastically reduce but not fully eliminate the capacity loss at 55 °C. Thus, improving further the high-temperature cyclability of NVPF-HC cells without losing their high-power capability, calls for a greater understanding of the aging mechanism of NVPF-HC cells at both high and ambient temperatures.

Learning from the Li-ion battery technology, another source of cell degradation besides electrolyte oxidation is rooted in the transition metal (TM) dissolution as exemplified with the LiMn 2 O 4 /Graphite chemistry, whose practical aspect has long been plagued by Mn dissolution. [17][18][19][20] Dissolution-migration of TM from the positive electrode and its deposition on the negative electrode is considered as the cause of capacity fade. 21,22 Moreover, such a TM dissolution is not specific to the Li-ion chemistry as it was recently reported for Na-based positive electrodes such as sodium layered oxides. [23][24][25][26] Thus, we decided to study the V-dissolution thoroughly in NVPF-HC cells depending on the electrolyte formulation (solvent and salt), temperature, and state of charge (SOC) of the NVPF material.

From post-mortem analyses of the NVPF-HC cycled cell, we find that the positive electrode suffers from vanadium dissolution that in turn poisons the electrolyte and the HC negative electrode. Further digging into the origin of this dissolution issue, we thus study its dependence on both electrolyte decomposition and structural evolution upon cycling. Using combined TEM, NMR, and IR analyses, together with electrochemical analyses of the different electrolyte/solvent leached samples, we observe a strong influence of the electrolyte formulation on V-dissolution that triggers structural changes in the NVPF material. Subsequently, by carrying out electrochemical analyses on the vanadium-containing electrolyte, we identify, besides the structural aspects, the reactivity of dissolved vanadium in both positive and negative electrode surfaces that led to further cell deterioration. Finally, we provide evidence that such effects can be cured by the use of well-designed carbon coatings together with optimized electrolyte formulation.

Experimental Section:

Cell assembly and electrochemical tests: The Na 3 V 2 (PO 4 ) 2 F 3 (NVPF) sample containing ~5% carbon coating is used throughout the studies except for studies on 18650 cells where the NVPF containing either 3 -7% carbon coating were used. The NVPF and HC electrodes coated (one side) on Al foil for coin cell experiments as well as the dry 18650 cells without electrolyte for prototype cell assembly were received from TIAMAT, Amiens. The electrochemical tests on Na 3 V 2 (PO 4 ) 2 F 3 (NVPF) were carried out in either Na-metal half-cell using Na metal as a counter (negative) electrode, or Na-ion full cell where hard carbon casted on aluminum foil was used as the negative electrode. Either 2032 coin type cells or 18650 cylindrical type cells (exclusively for full cell analyses) were used for the study. The electrolyte used is 1M NaPF 6 in Ethylene carbonate (EC)-Propylene carbonate (PC) -Dimethyl Carbonate (DMC) in 1:1:2 volume ratio, with or without electrolyte additives. The electrolyte additives used are 3 wt. % succinonitrile (SN), 1 wt. % vinylene carbonate (VC), 0.5 wt. % sodium oxalatodifluoroborate (NaODFB) and 0.5 wt.% trimethylsilylphosphite (TMSPi). A mixture of 0.7M NaPF 6 + 0.3M NaTFSI is used as an electrolyte salt for some selected experiments.

Coin cell assembly was carried out in an argon-filled glove box using a glass fiber separator, whereas 18650 cells were filled with electrolyte and sealed in a dry room. For analyzing the electrochemical performance of the Na 1 V 2 (PO 4 ) 2 F 3 samples after treatment with different electrolytes, Na-metal half cells were assembled with 1M NaPF 6 in PC as electrolyte. Three-electrode cyclic voltammetry tests were performed with glassy carbon as the working electrode, Pt wire as the counter electrode, and Ag/Ag + organic reference electrode (calibrated against Na metal). Leached electrolyte for cyclic voltammetry (CV) was recovered as explained in the leaching procedure section next or by washing the NVPF recovered from 18650 after current interruption device (CID) break (mentioned as 'green electrolyte' as the prepared electrolyte was greenish due to the presence of dissolved vanadium in it).

All electrochemical measurements were carried out using a Biologic battery cycler/potentistatgalvanostat, at a specific cycling rate and temperature as specified in the main text.

Leaching procedure: Firstly, 300 mg of Na 3 V 2 (PO 4 ) 2 F 3 powder (90% active mass-10% carbon black Super P) was charged in a homemade big cell to the Na 1 V 2 (PO 4 ) 2 F 3 composition using 1M NaPF 6 in PC electrolyte. The powder was recovered from the cell and washed several times with DMC before drying in the glovebox antechamber. For leaching, either a solvent mixture (EC -PC-DMC) or the electrolytes containing 1M salt from NaPF 6 , NaClO 4 , NaBF 4 , NaFSI, and NaTFSI in EC-PC-DMC (1:1:1 by volume) were used. All the salts were dried overnight in a Büchi oven under vacuum at 80 °C before transferring into the glove box. All the solvents were dried over molecular sieves till the water content of less than 10 ppm, as measured by Karl-Fischer titration. Secondly, the electrolyte or solvent (3 mL) was mixed with Na 1 V 2 (PO 4 ) 2 F 3 (20 mg) in a 15 mL polypropylene vial and sealed in an aluminum bag inside an argon-filled glovebox. The aluminum bag was then transferred to a 55 °C oven for 21 days of leaching. After leaching, the Al bags were transferred to an argon-filled glovebox, and the samples were recovered. The leached NVPF powder was separated from the electrolyte by centrifugation and washed several times with DMC before drying under vacuum. The recovered electrolyte was filtered using a 0.2 μm polypropylene syringe filter (VWR international) to ensure the removal of any NVPF particles in the recovered electrolyte for analyses.

The separated solid (leached NVPF) and the liquid (leached electrolyte) were used for further characterization.

Quantifying vanadium dissolution using ICP-MS:

For the 2032 coin cell mode, HC electrodes and the separator touching the HC electrodes were recovered after the specified charging/storage procedure. HC + separator was digested in 10.000 mL of 2 wt. % nitric acid (Suprapur, 65 % HNO 3 , Merck, vanadium content ≤ 0.5 ppb ), and solutions were sonicated and rested overnight. Solutions were filtered with polypropylene 0.2 μm PP syringe filters and diluted with 2 wt. % nitric acid solution until final vanadium concentrations between 1 ppb to 1 ppm were reached. The vanadium amount was quantified using inductively coupled plasma mass spectroscopy (ICP-MS, Nexion 2000, Perkin Elmer). The calibration was obtained from three diluted vanadium standard solutions (TraceCERT, 1 mg.L -1 V in nitric acid, Sigma-Aldrich). For the leached electrolyte obtained in the leaching procedure section, the 200.00 μL of electrolyte was diluted until final vanadium concentrations between 1 ppb and 1 ppm were obtained and measured with ICP-MS.

X-ray diffraction (XRD):

Synchrotron XRD patterns (λ = 0.4589 Å) were collected via the mail-in service of the 11-BM beamline at the Advanced Photon Source at Argonne National Laboratory. The leached and reference NVPF powders were sealed in glass capillaries and kept inside the kapton holder. The Rietveld refinement of the synchrotron XRD pattern (for solvent leached NVPF) was performed using the FullProf program. 27 Operando X-ray measurements were taken by a BRUKER D8 Advance diffractometer with Cu Kα radiation (λKα 1 = 1.54056 Å, λKα 2 = 1.54439 Å) and a Lynxeye XE 583 detector. The operando XRD analyses were conducted using a homemade airtight XRD cell equipped with a beryllium window.

Transmission electron microscopy (TEM): Samples for transmission electron microscopy (TEM)

were prepared in an Ar-filled glove box by grinding the leached NVPF powder in an agate mortar in dimethyl carbonate and depositing drops of suspension onto a copper TEM grid with a holey carbon support layer. The sample was transported to the TEM column with a Gatan vacuum transfer holder avoiding contact with air and moisture. High angle annular dark field scanning transmission electron microscopy (HAADF-STEM) images, energy-dispersive X-ray (EDX) spectra, STEM-EDX maps, electron energy loss spectra (EELS) and STEM-EELS maps were collected with a probe-corrected Titan Themis Z electron microscope operated at 200 kV equipped with a Super-X EDX detector and a Gatan Quantum ER965 spectrometer. The energy resolution, measured from the full width at half maximum of the zero-loss peak, was 0.175 eV.

Scanning Electron Microscopy (SEM): SEM images of the NVPF with different carbon coating

were obtained on an FEI Magellan scanning electron microscope equipped with an Oxford Instruments energy dispersive X-ray spectroscopy (EDX) detector. EDX analyses of the NVPF particles were carried out using an acceleration voltage of 20 kV.

Fourier-Transform Infrared (FTIR) Spectroscopy: The leached NVPF powders were discharged to 2 V (0% state of charge) vs. Na metal for FTIR measurement. FTIR spectra were recorded on a Nicolet iS5 FTIR spectrometer, mounted with a diamond Attenuated Total Reflectance accessory (iD1 ATR). All the spectra were taken using 16 scans with a 4 cm -1 resolution from 2000 to 400 cm - 1 . The background correction is performed by measuring the ambient atmosphere under the same conditions as for the leached NVPF.

Mass Spectroscopy (MS):

Mass spectra were recorded for the gas inside the pouch bags collected during the leaching process of Na 1 VPF with NaTFSI or NaPF 6 for 21 days at 55 °C. The pouch bags with samples containing polypropylene tubes were bulged up due to gassing after 21 days of storage at 55 °C. A needle was pierced to the pouch and the gas was transported to the mass spectrometer via a gastight syringe. The gas samples were then injected into an ExQ gas analysis system (Hiden Analytical, USA) composed of a HAL (Hiden Analytical) series quadrupole mass spectrometer, an ultra-high vacuum (UHV) mass spectrometer vacuum chamber, a vacuum pumping system and a QIC series capillary inlet. The partial pressures are eventually determined for each gas based on their mass to charge ratio (m/z), after ionization in the ionization source of the MS.

Nuclear magnetic resonance (NMR) spectroscopy:

The leached NVPF powders were discharged to 2 V (0% state of charge) vs Na metal for solid state NMR analyses. Solid-state 31 P and 23 Na NMR spectra were recorded on electrode materials with a 4.7 T (200 MHz for 1 H) Bruker WB spectrometer equiped with an Avance HD console and a 1.3 mm Magic Angle Spinning (MAS) double resonance 1 H- 19 F/ 31 P- 15 N probe. The rotors were filled inside an argon filled glovebox and the rotor were spinning at  R = 60 kHz under 100% N 2 atmosphere. The solid-state NMR spectra were recorded with a rotor synchronized Hahn echo sequence (90° -1/ R -180°-1/ Racquisition), and the chemical shift were referenced with 85% H 3 PO 4 in water for 31 P and 1 M NaCl in water for 23 Na, both at 0 ppm. Full relaxation of the magnetization was achieved with a 50 ms recovery delay between each transient in both case. 1 024 000 transients were recorded for 31 P and 10240 for 23 Na.

Liquid-state spectra were recorded in the same spectrometer with a double resonance probe equiped with 10 mm inserts, tuned to 31 P or 51 V. Liquid-state spectra were recorded for all samples inside 5 mm glass tubes filled in the glove box, and sealed with parafilm. An N 2 flow of 500 L/h was sent inside the probe to shield the sample from the air. The spectra were recorded with a 90°-acquisition sequence. 102 400 transients were recorded for 31 P (recovery delay 5s) and 1 024 000 transients were recorded for 51 V (recovery delay 50 ms).

The presence of HF in the electrolyte was evaluated by 19 F liquid-state NMR spectra, recorded in a Bruker 7.046 T (282 MHz for 19 F) Advance III NMR spectrometer mounted with a 5 mm HX(F) probehead. The samples were meassured in a 5 mm glass tube equiped with a D 2 O-filled inset for signal lock. The spectra were recorded with a 90° adquisition sequence, over 16 scans (5 s adquisition time each) and 10 s recovery delay, and were referenced with the expected signal of the anion (PF 6 -or TFSI -accordingly).

Results and Discussion:

Post-mortem analysis of NVPF-HC cells to understand the degradation mechanism:

Dry 18650 NVPF-HC cells were received from TIAMAT and filled with 1M NaPF 6 in EC-PC-DMC

(1:1:1 by volume) electrolyte. The cell formation and cycling were performed at 55 °C using a cycling rate of C/5 (1C= 128 mAh/g). Fig. 1a shows the capacity retention plot, with 93 % of the original capacity retained after 28 cycles. Although the cell does not show a big capacity loss, the current interruption device (CID) broke after ~10 days (28 cycles) at 55 °C, which is indicative of a huge pressure increase (~ 15 bars) inside the cell due to parasitic reactions enlisting gaseous byproducts generation.

After the CID break, the cell was opened and individual electrodes were recovered for analysis.

Upon washing with EC-PC-DMC, the NVPF electrode releases some yellow/ slight green coloration into the solvent (Fig. 1b), that indicates the dissolution of some vanadium as confirmed by inductively coupled plasma-mass spectroscopy (ICP-MS). This suggests the presence of easily soluble vanadium species in the NVPF electrode, contrary to the pristine NVPF electrode, which

shows no dissolution in this solvent mixture. Additionally, by SEM-EDX analysis, we spotted (Fig. 1c) the presence of deposited vanadium on the recovered HC electrode surface as well. Altogether, these results unambiguously indicate the presence of V-dissolution in the cycled NVPF/HC cell, with part of the soluble vanadium species going into the electrolyte that further deposits to hard carbon negative electrode.

Quantification of vanadium dissolution:

In order to quantify the amount of vanadium dissolved at different cycling conditions, a series of experiments were performed in coin cells employing fresh electrodes (single side casted NVPF and HC electrodes on Al foil), two glass fiber separators soaked in 1M NaPF 6 in EC-PC-DMC electrolyte. The cells were cycled to a particular state of charge (SOC), and maintained at that potential for 24 hrs. or 1 week prior being disassembled. The experiments were done either at 25°C or 55°C.

The hard carbon electrode together with one separator next to HC, were recovered from the cells and digested separately or together using 2% HNO 3 . Worth mentioning that, we have not used the separator next to the NVPF because of some remaining attached NVPF particles that would have rendered our measurements meaningless. The obtained solutions were filtered and analyzed by ICP-MS for vanadium content with the corresponding detected amounts indicated in Fig. 2a. Note that the mathematical sum of the vanadium detected from the carbon electrode (~0.1 %) and separator (~0.44 %) alone is nearly equal to the experimental value obtained (~0.6 %) by analyzing both at the same time, hence explaining why from now on we solely report the total amount of vanadium in both HC+ separator at once for meaningful comparisons. Following this protocol, we show (Fig. 2b) that the Vdissolution increases with increasing oxidation potential/state of charge of the NVPF electrode, with maximum dissolution observed for 100% state of charge (4.3 V). Further, increasing either the resting period at high potential or the cycling temperature promotes V-dissolution (Fig. 2c).

Moreover, Fig. 2d shows that the amount of dissolved vanadium keeps increasing upon cycling, however, at a lower speed with increasing cycle numbers. At this stage, a legitimate question regards whether such V-dissolution is limited to NaPF 6 based electrolytes? To address this point, we next studied the effect of different electrolyte formulations by solely changing the nature of the salt and not the solvent. The experiments were done by placing the fully charged NVPF powders (4.3 V vs. Na, having the Na 1 V 2 (PO 4 ) 2 F 3 composition) in the presence of the studied electrolytes and stored at 55 °C for 21 days (see Supporting Fig. S2). After this period, the supernatant liquid and solid products were separated by centrifugation and filtration. The liquid supernatants were analyzed via ICP-MS for their vanadium content and the solids were analyzed by XRD to eventually spot the structural evolutions. From now on, the charged material leached with EC-PC-DMC solvents (with no salt) is named as SL-NVPF (Solvent Leached-NVPF), whereas those leached with electrolyte bearing salts are referred using the salt anion, so that the sample leached with NaPF 6 is denoted as PF 6 -NVPF (and so on TFSI-NVPF, ClO 4 -NVPF, etc.).

The amounts of dissolved vanadium are shown in Fig. 2e > NaPF 6 > solvent > NaTFSI. Interestingly, the NaTFSI-based electrolyte, with the lower tendency for HF formation [32][33][34] , exhibits the least (almost nil) V-dissolution from 'charged NVPF'. Note that similar results (almost no V-dissolution) were obtained with NaFSI-based electrolyte as well (supporting Fig. S3). The results suggest that the formation of acidic by-products from electrolyte oxidation might be behind the observed V-dissolution sequence, however, it is not the only cause as NaClO 4 and NaBF 4 electrolytes that are known to generate less acidic species showed more Vdissolution than NaPF 6 electrolyte [35][36][37] . This suggests that the salt anion or its oxidized by-products most likely play a role in the V-dissolution. Lastly and surprisingly, we found that a pure solvent mixture (EC-PC-DMC) also leads to some V-dissolution, although it cannot be explained either by the presence of any acidic impurity or oxidized salt anion species. This suggests a copious reactivity of oxidized NVPF material with the carbonate solvent itself, causing some vanadium losses, further complexing the V-dissolution mechanism.

Overall, we have observed that a significant amount of vanadium is dissolved from the NVPF material into the electrolyte, particularly when it is fully charged (Na 1 V 2 (PO 4 ) 2 F 3 ). In the following sections we will explore the effect of this dissolution on the NVPF material, and on the general aging of the NVPF | HC cells upon cycling.

Structural evolutions of NVPF due to vanadium dissolution:

To interrogate the effect of V-dissolution on the material's electrochemical performances, Na-NVPF half cells were assembled using the leached NVPF materials as positive electrodes and 1M NaPF 6 in PC as electrolyte and the cells were cycled by starting on oxidation (Fig. 3a). For comparison purpose, the as-charged Na 1 V 2 (PO 4 ) 2 F 3 material that is stored in the argon atmosphere without any treatment with solvent/electrolyte is also analyzed. Asides from the cell based on the ascharged NVPF (no leaching), all the other cells have a starting potential (OCV) falling within the low voltage plateau (< 3.9 V), implying the uptake of Na during the leaching process, with this effect being more pronounced with NaTFSI. Note that solvent-treated NVPF (SL-NVPF), where there is no extra sodium for uptake in the EC-PC-DMC blend, also showed some oxidation in the first charge.

In the subsequent cycle, charge and discharge curves (Fig. 3b) show each stair-case profiles reminiscent of NVPF, although it is relatively less pronounced for the SL-NVPF with a profile being nearly alike that of oxygen substituted NVPF phase (Supporting Fig. S4). 38 Moreover, except for the SL-NVPF, PF 6 -H 2 O-NVPF, and TFSI-NVPF specimens, the measured reversible capacity is lower than the usual capacity of NVPF (~120 mAh/g), implying that the leaching process is associated with V-dissolution (leading for example to NaV 2-x □ x (PO 4 ) 2 F 3 ) or to structural deterioration of NVPF. The latter is evidenced by the fact that the observed loss in capacity (Fig. 3b) does not correlate with the calculated capacity loss if we would simply remove vanadium during leaching (Supporting Fig. S5).

Therefore, NVPF is decomposing into one electrochemically active and another inactive phase(s)

with the latter being responsible for the electrochemical dead weight and reduced capacity. From the open circuit voltage of the leached samples in Fig. 3a, one could hypothesize that the structure should consist of a mixture of Na 2 V 2 (PO 4 ) 2 F 3 and Na 2+x V 2 (PO 4 ) 2 F 3 phases. To check this hypothesis, synchrotron XRDs were collected for all the leached samples and compared with the patterns of Na 1 V 2 (PO 4 ) 2 F 3 , Na 2 V 2 (PO 4 ) 2 F 3 and Na 3 V 2 (PO 4 ) 2 F 3 in Fig. 4a and supporting Fig. S6. For the sake of clarity, we solely plot the 2=[4.6-8.5°] region which is very sensitive to the sodiumvacancy ordering and Na content (x). 13,39 Interestingly, the PF 6 -H 2 O-NVPF sample looks single phase with an XRD pattern alike Na 2 V 2 (PO 4 ) 2 F 3 (Fig. 4a). However, both are slightly different from the previously reported XRD pattern for Na 2 V 2 (PO 4 ) 2 F 3 , which was indexed in the I 4/m m m space group with a≈6.30 Å and c≈10.81 Å by Bianchini et al 39 . Our synchrotron patterns for the PF 6 -H 2 O-NVPF sample and for Na 2 V 2 (PO 4 ) 2 F 3 cannot be indexed using this tetragonal cell, as the (2 0 0) peak is split into two peaks at 2 ≈ 8.335°, so that we engaged into the structural resolution of this phase.

The S1). This phase is denoted 'P' hereafter and dotted lines are used to highlight its main Bragg peaks in Fig. 4a.

The phase 'P' together with another 'A' phase is present in the BF 4 -, TFSI-, and PF 6 -NVPF samples. In contrast, this P phase is not present for the ClO 4 -NVPF sample at the expense of the 'B' and 'C' phases that became the majority. By comparing with the operando XRD of NVPF (Supporting Fig. S7), the 'A' and 'B' phases could be closely matched to Na ~2.2 V 2 (PO 4 ) 2 F 3 with slightly differing sodium stoichiometry and 'C' phase with Na ~3V 2 (PO 4 ) 2 F 3 . As a result, the major peaks in all the electrolyte leached samples could be indexed with NVPF structure with difference in sodium content. Moreover, worth mentioning that Rietveld refinement attempts for the collected XRD pattern of SL-NVPF fails as the peaks are too broad, indicative of structural disorder driven by the leaching process. Lastly, it looks like this Na uptake reaction is the only one occurring for the TFSI-NVPF as it is structurally unaffected with almost no capacity loss (in Fig. 3b). Altogether, the XRD and cycling results show that all the leached samples self-discharged via an uptake of Na ions. With the exception of SL-NVPF, all samples maintain the NVPF structure with or without extra phases due to NVPF decomposition during leaching process.

Microscopic analyses on the leached samples:

To grasp further insights into those competing degradation-dissolution reactions, we relied on transmission electron microscopy (TEM) by focusing solely on the SL-NVPF and PF 6 -NVPF samples for which XRD could not identify the phases(s) present. HAADF-STEM images of the SL-NVPF crystals (supporting Fig. S8) shows numerous elongated pits and cracks confirming the leaching process. Moreover, STEM-EDX maps and EDX spectra (Fig. 5a, supporting Fig. S9) indicate a core-shell type structure with the sodium-rich core (Na 1.9(2) : V 1.999(8) : P 2.001 (8) , as quantified from integrated EDX spectra) while the shell contains virtually no Na and F (Na 0.4(2) : V 2.00(1) : P 2.00(1) ). A difference between the core and shell is also reflected with the EELS spectra (Fig. 5b) of the V-L 2,3 and O-K edges. In the EELS spectra from the core area there is a clear splitting of the V-L 3 and V-L 2 peaks indicative of coexisting V 3+ (V-L 3 at ~518.1 eV) and V 5+ (V-L 3 peak at ~519.6

eV) while the V-L 2,3 edge for the shell consists of two broad peaks centered at ~519.3 eV and 525.9

eV that is between the energy losses characteristic of V 4+ and V 5+ . It should be noted that the V cations in both core and shell are significantly more oxidized compared to vanadium in the pristine NVPF, which is in exclusively +3 oxidation state (Supporting Fig. S10).

The O-K edge EELS allows for differentiating the chemical nature of oxygen: the onset of a pre-peak at ~531 eV is related to the O2p-V3d hybridization representing "free" oxide ions (O 2-that are not bonded to the PO 4 group) 40 while the intense peak at 539.4 eV is due to mixed O2p and P3sp states (O 2-in the PO 4 group). The intense O2p-V3d pre-peak for the shell indicates that substantial amount of oxygen does not belong to the PO 4 groups. In contrast, the observed intensity of pre-peak at 531 eV is very low for the core in agreement with the Na 3-x V 2 (PO 4 ) 2 F 3 formula, taking into account that the spectrum from the core unavoidably includes the shell contribution. Combined with a low fluorine content, the chemical formula for the shell phase can be suggested as Na 0.4 V 2 O y (PO 4 ) 2 F z . The presence of a vanadyl bond in the SL-NVPF was also confirmed by NMR and IR analyses (supporting Fig. S11). Thus, one can view the SL-NVPF sample as consisting of the core that is more reduced with high sodium content than the expected 1Na for 'charged NVPF' (Na 1.9 V 2 (PO 4 ) 2 F 3 ) and the surface that is more oxidized, strongly desodiated, and richer in oxygen Na 0.4 V 2 O y (PO 4 ) 2 F z . Turning to the crystallites of the PF 6 -NVPF sample, they are largely disintegrated, forming numerous small loose, shapeless particles (Fig. 6a). EDX analyses on different particles show the presence of at least two different types (Fig. 6a, Supporting Fig. S12 a,b) exhibiting different sodium stoichiometry confined to the Na 1.7(3) : V 2.00(2) : P 2.00(2) and Na 2.4(2) : V 2.00(1) : P 2.00(1) formulas.

Additionally, a core-shell type structure is observed (Fig. 6b) with the surface layer containing almost no Na and significantly different V:P ≈ 1:2 ratio (Na 0.09(2) : V 1.00(2) : P 2.00(1) , Fig. S12c)

indicating highly depleted vanadium regions. This suggests the feasibility of non-homogeneous vanadium dissolution in the PF 6 -NVPF material. Moreover, the EELS spectra taken over the Vdepleted shell regions (Supporting Fig. S13) demonstrate the V 3+ signal exceeding that of V 5+

indicating that the latter is leached predominantly. Overall, as depicted in Fig. 7, both SL-and PF 6 -NVPF samples exhibit a core-shell type structure, with the core being richer in Na than in the starting phase Na 1 VPF. In the presence of Na + -free solvent, the observed Na excess in the SL-NVPF core can solely result from Na diffusion from surface to the core that is coherent with the Na-deficient surface/shell seen by TEM in this sample.

This explains why the surface NVPF is more oxidized (contains V 4+ and V 5+ ) than the core (V 3+ and V 5+ ) together with the observed capacity of ~50 mAh/g in the first charge (SL-NVPF Fig. 3a), owing to its Na-rich core. It is worth to note that the charge capacity for SL-NVPF is the lowest (in Fig. 3a)

among the leached samples due to the Na + depleted from the surface in absence of Na + in the leaching media.

Nevertheless, a surprising observation with SL-NVPF is the presence of vanadyl (V=O) bond in the shell of the particles. The oxygen needed to form this V=O bond does not come from the solvent molecules since O-free solvent alike acetonitrile shows identical results (Supporting Fig. S14).

Hence, we must suspect NVPF itself via a process that enlists the breaking of O-PO 3 bonds that release phosphate anions into the solvent to form vanadyl bond together with the sodium migration from surface to the bulk of NVPF. This explanation is substantiated by NMR studies that reveal the presence of phosphate anion in the recovered solvents from the leaching reaction leading to SL-NVPF. In contrast, no phosphate species were present in the ones recovered from the leaching reaction leading to PF 6 -NVPF (supporting Fig. S15).

Altogether, we observed a different leaching mechanism associated to either solvent or NaPF 6 electrolyte. For SL-NVPF, the process is homogeneous with particles undergoing internal redox process leaving a V=O rich and Na-depleted shell. In the case of PF 6 -NVPF, the leaching is not homogeneous, and leads to the formation of V-deficient shells, which account for the inactive phases observed before by XRD and electrochemical cycling.

Mechanism of the self-discharge-degradation process as a function of the solvent/ electrolyte media:

Based on the aforementioned observations we propose a reaction scheme (Fig. 8) to account for the formation of PF 6 -NVPF, TFSI-NVPF, and SL-NVPF samples. In all cases, a self-discharge phenomenon is evidenced by the capacity recorded during the first charge after recovering the materials (Fig. 3b). However, the reducing agent donating the required electrons to fuel the selfdischarge process differs in each case, resulting in different reaction pathways, as we detailed below. Firstly, in the presence of NaPF 6 electrolyte (Fig. 8a), Na + cations are inserted in Na 1 VPF, while the carbonate solvent (EC/ PC/ DMC) is oxidized [41][42][43][44] . This oxidation is promoted by the interaction with PF 6 -anions 45 , forming PF 5 and HF (confirmed by 19 F-NMR, supporting Fig. S16), the latter being responsible for the V-dissolution observed by ICP. Additionally, the organic radical formed after solvent dehydrogenation eventually decomposes into formaldehyde, CO 2, and a methyl radical that can attack other electrolyte components and generate CH 4 as supported by mass spectroscopy (Supporting Fig. S17 and table S2).

A different scenario is proposed with the NaTFSI electrolyte where the TFSI -anions get oxidized together with solvent 45 , without the formation of HF (confirmed by NMR, supporting Fig. S18), thus avoiding the V-dissolution. The radical formed by the oxidation of TFSI -is stabilized by the high charge delocalization of the anion, although it can eventually decompose/polymerize with or without the evolution of gaseous by-products. The absence of HF release through this anion decomposition explains the barely detectable V-dissolution observed. Nevertheless, the charged NVPF in this electrolyte still shows large self-discharge, associated with electrolyte oxidation that we must address before using NaTFSI-based electrolytes for Na-ion cells.

Turning to the SL-NVPF sample, it shows the more pronounced structural changes associated to a reduction of the core of the particles with the surface acting as a reductant (See EELS data in Fig. 5b). This redox process affects the homogeneous distribution of V 3+ -V 5+ bi-octahedra in Na 1 VPF material with the core evolving towards V 3+ -V 4+ (which eventually disproportionate back to V 3+ -V 5+ ) while the surface changes into V 4+ -V 5+ . As V 4+ is not stable in this coordination environment, the ligands must reorganize, producing some dissolution of either PO 3 -groups ( 31 P NMR of collected leached solvent shows the presence of P in solution) or F -together with a small amount of vanadium.

These findings lead to the proposed reaction scheme (Fig. 8c) to account for the formation of the vanadyl bonds (V=O) in the SL-NVPF material, while a more detailed reaction mechanism requires further operando experiments.

Implication of dissolved vanadium on the electrochemistry of NVPF-HC cells:

After having studied the structural and electrochemical modification suffered by the NVPF material due to some vanadium dissolution, we now turn our attention to the effect of such dissolved vanadium in the electrochemical reactivity of the electrolyte. As a first step, we analyzed the nature of vanadium present in the NaPF 6 based electrolyte after leaching process, termed hereafter as PF 6leached electrolyte (see the 51 V-liquid NMR result in supporting Fig. 19). The absence of an NMR signal for PF 6 -leached electrolyte indicates that vanadium is present in either +2, +3, or +4 oxidation states.

Next, the PF 6 -leached electrolyte was tested by cyclic voltammetry using the glassy carbon and platinum mesh as working and counter electrodes respectively and an organic Ag + /Ag reference electrode. 6,15 The anodic sweep shows an oxidation peak around 3.9 V vs. Na + /Na 0 (Fig. 9a) and some reduction current is observed in the subsequent cathodic sweep although without a clear reduction peak. Upon further reduction to 1 V vs. Na + /Na 0 (supporting Fig. 20), the vanadium reduction process is observed at 1.36 V vs. Na + /Na 0 . The CV results show that the PF 6 -leached electrolyte containing dissolved vanadium is redox-active and could undergo oxidation-reduction on NVPF and HC electrode surfaces, which operate at potentials between 3-4.3 V and 2-0.0 V vs.

Na + /Na 0 , respectively, in a full NVFP-HC cell.

A similar CV experiment was carried out using the soluble vanadium compound recovered from 18650 cell after the CID break by washing the NVPF electrode with EC-PC-DMC (shown in Fig. 1b) and using the resultant solution for making the electrolyte (green electrolyte as mentioned in experimental section) by adding 1M NaPF 6 . The anodic sweep between 2 to 4.3 V vs. Na + /Na 0 in Fig. 9b indicates oxidation (3.34 V and 3.9 V) and reduction (3.2 and 3.6 V) processes that are most probably associated to the soluble vanadium species.

In light of this finding, one would expect upon cycling of a NVPF-HC full cell, the emergence of some redox activity features that should increase with the amount of dissolved vanadium. Fig. 9c shows the cycling profile of NVPF-HC 18650 Na-ion cells cycled at high temperatures, and remarkably there is appearance upon long cycling (30 cycles) of an oxidation event occurring in the similar potential range (3.9 V; see arrow) as observed via cyclic voltammetry. This feature is further emphasized in Fig. 9d by plotting the derivative dQ/dV which shows nicely the appearance of the oxidation peak at around 3.4 and 3.9 V, which is partly/ fully reversible on reduction with the appearance of reduction processes between 3 and 3.7 V. The peaks could be correlated with the redox reaction of dissolved vanadium on the NVPF electrode surface (by comparing with CV in Fig. 9a and Fig. 9b). Its slow evolution from cycle to cycle (supporting Figure S21) indicates increasing concentration of soluble vanadium in the electrolyte during cycling, in agreement with the findings by ICP-MS discussed before. Therefore, these redox peaks could be used as finger prints for following the amount of dissolved vanadium during cycling. Our cross experiments using the fresh NVPF-HC cell with the electrolyte containing dissolved vanadium re-confirm the observations of vanadium oxidation and the changes in the cycling profile due to dissolved vanadium (supporting Fig. S22).

At the same time, the dissolved vanadium can also be reduced in the negative electrode, as hard carbon spends most of its time at very low potentials (~0.05 V vs. Na +/ Na 0 ). If the vanadium reduction leads to its deposition on the HC surface, it could lead eventually to surface passivation, active material loss and impedance increase (note the polarization increase in the low voltage region of cycling in Fig. 9c). It is important to note here that the acidic impurities (HF, HPO 2 F 2 etc.) formed during the electrolyte oxidation, as mentioned in previous reports [45][46][47] , can also play a role in dissolution and recreation of interphases (SEI, CEI) leading to Na-inventory loss as well as increasing cell impedance. The influence of acidic impurities in the performance of NVPF | HC cells is a subject that requires deeper studies, which will be presented in future reports. A consequence of the evolution of these redox processes and of the increased polarization in the low voltage region (2-3.6 V) was always concomitant with the pressure increase (e.g, CID break) of the 18650 cells, although we don't fully master the associated parasitic reactions yet. Meanwhile, it is essential to reduce this V-dissolution from NVPF in order to improve the cell life of the NVPF-HC cells. Hence, we studied the following strategies to reduce and possibly eliminate V-dissolution completely.

Suppressing partially the vanadium dissolution problem:

As our first strategy, we explore using NaTFSI as the electrolyte salt, as it prevents the V-dissolution into the electrolyte (Fig. 2). However, NaTFSI solely as electrolyte salt corrodes the Al current collector at high potential 48 , therefore a mixture of NaPF 6 and NaTFSI must be used. Fig. 10a compares the discharge capacity (%) at 55°C with 1M NaPF 6 and 0.7M NaPF 6 -0.3M NaTFSI in 1:1:2 mixture of EC-PC-DMC. The cells are allowed to rest after 10 cycles at 100% SOC for 1 week to promote parasitic reactions. The capacity loss during self-discharge is almost similar in both cells (Fig. 10b). However, the capacity recovery in the subsequent cycle is higher (less irreversible capacity loss during self-discharge) with NaTFSI containing electrolyte. The capacity retention (Fig. 10a) after self-discharge is also higher for NaTFSI containing electrolyte with less changes in cell polarization (Fig. 10c) compared to a huge increase in cell resistance and poor capacity retention for the 1M NaPF 6 electrolyte. After 850 h of cycling, the cells were disassembled and the hard carbon electrode and one separator were analyzed for V-content by ICP. The ICP results in Fig. 10d show 5.45% vanadium dissolution for the cells containing 1M NaPF 6 electrolyte, whereas only 2.17% for the NaPF 6 :NaTFSI mixed electrolyte. The results are encouraging, although we still observed selfdischarge loss of 25% in both cells indicates some degree of electrolyte oxidation. This calls for further curing tricks that are being tried in our group at present. Meanwhile, in the second approach, we tested solely NaPF 6 as electrolyte salt and introduced electrolyte additives that protect the NVPF surface. To do so we used the electrolyte formulation reported in our previous work that consists of 1M NaPF 6 in an EC-PC-DMC mixture with 4 additives, namely NaODFB, VC, SN, and TMSPi. Fig. 11 a shows that the amount of dissolved vanadium after 1 week storage of NVPF-HC coin cells at 55 °C in 100 % SOC is reduced by nearly 50 % in presence of additives. Encouraged by these results, 18650 cylindrical cells loaded with this electrolyte were cycled. They exhibit over four months of tests (Fig. 11b) at 55 °C a better capacity retention without any CID break, in comparison to the cells without any electrolyte additive that dies just after 10 days (same cell as used in Fig. 1). This confirms the beneficial effect of the electrolyte additives. Interestingly, the derivative plot shows upon cycling the appearance of a barely noticeable 3.9 V peak (arrow Fig. 11c left) for the electrolyte containing additives as opposed to a well pronounced feature for the electrolyte without additives. Bearing in mind that this peak is the fingerprint of V-dissolution, these results unambiguously confirm the positive attribute of using electrolyte additives to reduce vanadium dissolution and improves the cycling performance at high temperature, but not to fully eliminate it. The third strategy has consisted in exploring the coating of NVPF particles by preparing two batches of NVPF having different amounts of carbon coating as determined by TGA analyses (Fig. 12a).

Batches 1 and 2 show carbons contents of ~7% and ~3%, respectively, with a more uniform Ccoating for batch 1 as deduced from EDX mapping (supporting Fig. S23). Both batches were then tested for vanadium dissolution in coin cells (1M NaPF 6 in EC-PC-DMC as electrolyte) by storing them at 100 % SOC for one week at 55 °C, and the results in Fig. 12b show a higher amount of vanadium dissolution for the NVPF sample having the lowest amount of coated C (Batch2).

Moreover, coin cells using both batches as the positive electrode and additive-free electrolyte were cycled at 55 °C. Their capacity retention and polarization evolution upon cycling, Fig. 12 c and12d, respectively, show that NVPF with higher (7%) carbon coating presents less self-discharge loss during rest (1 week) together with a lower polarization after the self-discharge process, when compared to the NVPF having solely 3 % C-coating. Altogether, the results confirm the positive attributes of carbon coating towards reducing vanadium dissolution during rest at 100 % SOC.

To ensure the robustness of this C-coating effect we compared (Fig. 12e) the capacity retention plots at 55 °C of 18650 Na-ion cells, comprising NVPF positive electrode with 3 and 7% carbon coating, hard carbon negative electrode and 1M NaPF 6 in EC-PC-DMC with 4 additives (NaODFB, VC, SN and TMSPi) as the electrolyte. Though the capacity retentions are nearly similar for both cells, the one containing 7% C coated NVPF displays slightly higher efficiency indicative of less parasitic reactions than the cell containing 3% C coated NVPF. The derivative plot in Fig. 12f further supports the above trend since a lower evolution of the redox process at ~3.9 V (the fingerprint region as explained in Figure 9) and ~3.2 V for NVPF is observed for the higher carbon content (green plots Fig. 12f) indicative of a lower vanadium dissolution for the 7% C coated sample. In short, combining the positive effect of additives and carbon coating of NVPF helps reducing the NVPF-electrolyte interface reactivity. However, a negative side effect of increasing the carbon content (e.g, thickening of the coating) is a decrease of the material specific energy and possibly the power rate capability. Thus, proper tuning of surface coating together with a careful choice of electrolyte formulation combined with the establishment of an optimized formation cycle is crucial to minimize the extent of V-dissolution and, therefore, to obtain, at the industrial-level, high performing NVPF/HC Na-ion cells.

Conclusion:

Transition metal dissolution is a significant problem for various battery chemistries and is not fully resolved for even matured Li-ion battery technology after 25 years of research. In this study, we provide evidence that 3d metal dissolution is equally a severe problem for sodium-ion batteries and more so for the NVPF/HC chemistry since the redox potential of NVPF (~4.2 V vs. Na/Na + ) falls beyond the thermodynamic stability of the electrolytes. Vanadium from charged NVPF dissolves at high temperature in conventional electrolytes. This dissolved vanadium is redox active and can trigger redox shuttle reactions in both positive and negative electrodes, thereby deteriorating the

Fig. 1 :

 1 Fig. 1: Electrochemical characterization and post-mortem analysis of cylindrical 18650-type NVPF|HC cell cycled with C/5 rate at 55 °C. (a) Discharge capacity retention plot of NVPF and HC cell cycled at 55 °C. The data from formation cycles are not included here for clarity purpose. The inset shows the NVPF and HC electrodes recovered after the CID break. (b) Green/yellow colored solution recovered by washing 360 mg of cycled NVPF electrode with 3 mL of EC-PC-DMC; ICP results showed 115 ppm dissolved vanadium in this solution. (c) SEM/EDX analysis of recovered HC with SEM image (left), EDX elemental mapping of vanadium (middle) and EDX spectra (right).

Fig. 2 :

 2 Fig. 2: Different factors affecting vanadium dissolution. The amount of vanadium dissolved from NVPF detected by ICP-MS. For better comparison, the obtained ppm values are converted to %V where the dissolved vanadium is normalized with respect to total vanadium present in the NVPF electrode taken for the experiment. The %V detected from (a) Hard carbon electrode, separator separately and hard carbon + separator together recovered from NVPF-HC coin cells stored for 1 week at 4.3 V and 55 °C. (b) Hard carbon+ separator recovered from cells cycled to different upper cutoff voltages of 2.7 V, 3.7 V and 4.3 V and stored for 1 week at 55°C. (c) HC+ separator recovered from cells charged to 4.3 V and stored at 25°C and 55°C for 1 day and 1 week. (d) Amount of dissolved vanadium and discharge capacity vs. cycle number for NVPF-HC coin cells cycled at 25 °C. In all the experiments from a-d, the electrolyte used is

Fig. 3 :

 3 Fig. 3: Electrochemical performance of leached/ unleached Na 1 VPF in Na-half cells: (a) First charging curve. The OCV after the cell assembly and observed charge capacity is mentioned in figure. (b) The second cycle. All leaching experiments were carried out at 55 °C for 21 days in the mentioned electrolyte or solvents. The salt used is mentioned in the Fig. 3a and 'S' represents solvents mixture EC-PC-DMC.

  pattern of the PF 6 -H 2 O-NVPF sample can be indexed using a body-centered orthorhombic unit cell with space group I m m m and lattice parameters a = 6.311646(6) Å, b = 6.318539(6) Å and c = 10.817618(11) Å, which is a distortion of the tetragonal I 4/m m m cell. The Rietveld refinement of PF 6 -H 2 O-NVPF is shown in Fig. 4b, and the Na content was refined to 0.489(12) leading to a Na 2 V 2 (PO 4 ) 2 F 3 composition (the refined structural parameters are given in Supporting Information, Table

Fig. 4 :

 4 Fig. 4: X-ray diffraction pattern of leached Na 1 VPF. (a) The 11BM synchrotron powder XRD pattern of the recovered Na 1 VPF powder after leaching treatment in the mentioned electrolyte at 55 °C for 21 days. For the comparison, synchrotron patterns of unleached Na 1 V 2 (PO 4 ) 2 F 3 , Na 2 V 2 (PO 4 ) 2 F 3 and Na 3 V 2 (PO 4 ) 2 F 3 are also presented. (b) Rietveld refinement of the PF 6 -H 2 O-NVPF sample where the experimental, calculated and difference patterns are marked by green circle points, black line and blue line, respectively. The vertical tick marks correspond to Bragg reflections in the I m m m orthorhombic cell (see text) (c) Zoomed (in y-axis) image of Fig.4ashowing the extra peaks that are not indexed by NVPF structure and must be due to the decomposed products of NVPF during the leaching process.

Fig. 5 :

 5 Fig. 5: Identifying chemical changes in solvent-leached SL-NVPF using TEM. (a) HAADF-STEM image with EDX compositional maps demonstrating Na-depleted core (looks red in the mixed Na/P map); (b) EELS spectra of the shell and core in SL-NVPF along with the characteristic spectra of vanadium oxides.

Fig. 6 :

 6 Fig. 6: Identifying compositional changes in the electrolyte-leached NVPF using TEM. (a) HAADF-STEM image and EDX compositional maps of the crystallites with different Na content visible in the mixed Na/P map due to different balance of green and red colors; (b) HAADF-STEM image and EDX compositional maps of the crystallite demonstrating a Nadepleted shell (looks red in the mixed Na/P map).

Fig. 7 :

 7 Fig. 7: Schematic of structural degradation in SL-and PF 6 -NVPF. Structural changes undergone by Na 1 VPF particle because of (a) Solvents leached NVPF leading to core-shell type structure with a thick shell containing oxygenated material. (b) Electrolyte (1M NaPF 6 in EC-PC-DMC) leached NVPF leading to core-shell type structure with a thin shell depleted in vanadium.

Fig. 8 :

 8 Fig. 8: Mechanism representing structural degradation of Na 1 VPF in NaPF 6 , NaTFSI, and pure carbonate solvent. (a) For NaPF 6 based electrolyte, simultaneous oxidation of DMC and Na + insertion in Na 1 VPF with the formation of HF and its attack on Na 1 VPF particle surface. (b) For NaTFSI based electrolyte, simultaneous Na + insertion in Na 1 VPF particle and oxidation of TFSI -anion leading to TFSI decomposition. (c) The Na 1 VPF -solvent interaction proceeds with internal phase separation in Na 1 VPF with simultaneous solvent oxidation. Note that oxidation of DMC molecule is used here as a representative example, whereas, in reality EC and PC could also undergo similar oxidation pathways.

Fig. 9 :

 9 Fig. 9: Effect of dissolved vanadium on NVPF-HC cell electrochemistry. Three-electrode cyclic voltammetry of (a) NaPF 6 electrolyte recovered after NVPF leaching, cycled in the voltage window of 2.27-4.3 V vs. Na + /Na 0 with scan rate of 20 mV/s. (b) CV of the electrolyte prepared by washing NVPF from 18650 with EC-PC-DMC (shown in Fig.1b) and then adding 1M NaPF 6 in 2.27-4.3V vs. Na + /Na 0 with scan rate of 20 mV/s. (c) Electrochemical profile and (d) corresponding dQ/dV profile for 2 nd and 30 th cycle of NVPF-HC 18650 cell cycled at 55 °C with C/5 rate.

Fig. 10 :

 10 Fig. 10: Effect of NaTFSI in protecting the NVPF surface. 1M NaPF 6 or 0.7M NaPF 6 -0.3M NaTFSI in EC-PC-DMC (1:1:2) was used as an electrolyte for NVPF-HC coin cells cycled at 55 °C at C/5 rate. (a) Discharge capacity retention. (b) selfdischarge (Q 9 /Q 10 ) and recovery(Q 11 /Q 10 ) process during the storage period. (c) Polarization evolution (in terms of ΔV= average charge voltage -average discharge voltage). (d) Amount of vanadium in the coin cells after 100 cycles measured with ICP-MS.

Fig. 11 :

 11 Fig. 11: Effect of electrolyte additives in protecting the NVPF surface. a. ICP analyses on hard carbon+ 1 separator recovered from the cells cycled using 1M NaPF 6 in EC-PC-DMC with and without electrolyte additives. The cells were stored at 100 % SOC, for one week at 55°C before decrimping to do ICP analyses. (b) Percentage capacity retention of

Fig. 12 :

 12 Fig. 12: Effect of carbon coating on vanadium dissolution and cell electrochemistry. a. TGA analysis of Batch-1 and Batch-2 for amount of carbon coating. b. Amount of vanadium dissolved from NVPF for 7 and 3% C-coated NVPF when NVPF-HC cell was stored at 55 °C for 1 week. c. Discharge capacity retention of 7 and 3% C-coated NVPF in NVPF-HC full cell mode at 55 °C with C/5 rate. d. Corresponding cell polarization in terms of ΔV= (average charge voltage -average discharge voltage) e. Discharge capacity retention for commercial 18650 type NVPF-HC cell containing 7 and 3% Ccoated NVPF with the 1M NaPF 6 in EC-PC-DMC and VC + NaODFB + SN + TMSPi additives. f. Corresponding dQ/dV profile of 18650 cells with the arrows indicating the changes in the curve. Note that the hard carbon used and the positive to negative electrode materials balancing between the 18650 cells used in Fig. 11 and in Fig. 12 are slightly different, hence the observed difference in the cycling profile and corresponding derivative plots.

Acknowledgments:

The authors thank the RS2E Network for funding as well as the financial support of Région Nouvelle Aquitaine, of the French National Research Agency (STORE-EX Labex Project ANR-10-LABX-76-01). S. M. acknowledges Horizon 2020 research and innovation program under grant agreement No 875629-NAIMA. P. D. thanks RS2E for PhD funding. TIAMAT, France is gratefully acknowledged for providing the NVPF/HC 18650 cells specifically designed for R&D purposes as well as the NVPF, HC electrodes for the coin cell studies. Use of the 11-BM mail-in service of the APS at

electrochemical performance of the cell. Suppressing vanadium dissolution is an essential step to improve the cycle life and slow down cell aging.

In this study we report on our efforts to prevent the V-dissolution in NVPF|HC cells. First, we show that a NaTFSI-based electrolyte prevents this process to happen, but still undergo degradation at high potentials. Therefore, a better strategy must be designed to avoid NaTFSI oxidation and protect the aluminum current collector on high voltage cycling of NVPF-HC cells. Second, we demonstrated that another attractive strategy to reduce vanadium dissolution is rooted in the coating of NVPF particles by carbon and surface protection using electrolyte additives. Overall, we show the importance of mastering the synergy between electrode and electrolyte to reach the best optimum performance of the NVPF/HC Na-ion cells.

Technology-wise, the gained knowledge from this work will further help in tuning the cell chemistry and engineering as well as formation protocol to reduce/eliminate the vanadium dissolution, hence fasten the commercialization of NVPF-HC based Na-ion cells. From a fundamental point of view, we believe the study provides a key understanding of how transition metal dissolution happens, its relation with the different salts, and the role of dissolved transition metal in altering the electrochemical performance of the cell. Similar studies could be extended to any new chemistry to pave the way to improve the system-level performance from fundamental understanding.

Argonne National Laboratory was supported by the US Department of Energy under contract No. DE-AC02-06CH11357 and is gratefully acknowledged. Access to TEM facilities was granted by the Advance Imaging Core Facility of Skoltech. The authors acknowledge Damien DEGOULANGE (Collège de France) for the help with mass spectrometry analyses.

Conflict of interest

The authors declare no conflict of interest.

Associated Content

*Supporting Information

The Supporting Information containing additional cycling, XRD, NMR, IR and TEM images is available.


C Péralle

R Charrière

J Faucheu

VO 2 -polymer nanostructured coatings for smart windows: a numerical study

Thermochromic vanadium dioxide loaded into a polymer opal photonic crystal is studied through optical simulations. The possibility of using this material as flexible transparent energy efficient smart material is explored.

Introduction

Vanadium dioxide (VO2) undergoes a phase transition with temperature that switches its behavior from semiconducting to metallic and thus modifies its optical properties. Its thermo-active characteristics have been studied for coating applications related to solar-irradiation on substrates that can be either transparent or absorbent materials (fluid absorber, smart windows, smart tiles). Among the various processes that have been explored to produce VO2 glazings, a promising approach relies on the mixing of VO2 nanoparticles with a glass or a polymer matrix [START_REF] Faucheu | A Review of Vanadium Dioxide as an Actor of Nanothermochromism: Challenges and Perspectives for Polymer Nanocomposites[END_REF]. This work is a numerical study of a model smart glazing made of a VO2 nanoparticles mixed with a polymer latex that could enable to form a VO2-polymer composites with an opal-like nanostructure [START_REF] Noël | Tunable architecture for flexible and highly conductive graphene-polymer composites[END_REF]. The performances of such model glazing in terms of visible and IR transmittance are evaluated using Maxwell-Garnett [START_REF] Markel | Introduction to the Maxwell Garnett approximation: tutorial[END_REF] (MG) approximation and Fourier Modal Method [START_REF] Li | Fourier Modal Method[END_REF] (FMM). Considering these two complementary modelling approaches enables to highlight the impact of introducing photonic affects related to highly ordered nanostructured (opals) compared to more disordered nanostructured.

Materials and process

Nanostructure of the model material

Our model material is a closed-packed opal made of hard polystyrene (PS) spheres (refractive index n=1.58) in a soft polyethylacrylate (PEA) matrix (n=1.47). The spacing between each PS sphere is set to a = 500nm. VO2 nanoparticles have been loaded into the matrix. VO2 refractive indices used in these calculations are extracted from [START_REF] Mlyuka | Thermochromic multilayer films of VO2 and TiO2 with enhanced transmittance[END_REF] Four different VO2 concentrations have been considered: 0.1vol%, 0.5vol%, 1vol% and 1.3vol%. Low concentrations are required for the material to stay transparent to visible light.

Homogeneous equivalent material: Abeles Matrices Model

A first approach consists in using the MG approximation to model the coating as an average material having an effective permittivity εeff depending on f the volumic fraction of the nanoparticles, εp and εm the dielectric permittivity of the nanoparticles and surrounding material here, the model material is a monolayer, and the transmittance characteristics are computed through an Abeles Matrices Model (AMM). [START_REF] Abelès | La théorie générale des couches minces[END_REF] . In this direction, the structure is modelled by 15 periods, each of them are 866nm thick. The material total thickness is then 12.99µm. The interstices containing VO2 nanoparticles and PEA have been modelled using the Maxwell-Garnett approximation. The material is surrounded by air. FMM is usually limited to 1D or 2D photonic crystals. In order to extend to 3D crystals, we approximated each sphere by a superposition of cylinders of constant heights. The spatial resolution is set so that each period is modelled by 24 layers.

Nanostructured material: Fourier modal method

Criteria used to evaluate the materials performances

Optimizing such smart glazings for smart windows applications relies on maximizing the visible transmittance 𝑇 𝑣𝑖𝑠 in both warm and cold conditions and maximizing the infrared (IR) transmittance modulation Δ𝑇 𝐼𝑅 as defined herafter.

Results and discussion

Fig. 2: Transmission spectra of the polymer opal loaded with different VO2 concentrations computed with FMM (a) in cold state (inset: comparison with an equivalent homogeneous material computed with AMM, in the case of 1vol% VO2) and (b) hot state

Using MG and FMM, transmission spectra of the nanostructured material are calculated over the 400-2500nm wavelength range for different VO2 concentrations in both VO2 phases (fig. 2). The oscillations observed at high wavelengths are due to thin film interferences, incident light reflected at the sample surface interfering with the light reflected at the bottom of the sample. Note that for the VO2 hot state at higher concentration (fig. 2b), the incident light is absorbed before reaching the substrate, leading to the oscillations disappearance. The transmission minimum observed in all transmittance spectra around 900 nm is due to the photonic bandgap (PBG), characteristic of ordered photonic crystals which is consistent with Bragg's law at normal incidence. As the VO2 concentration increases, the material transmittance significantly decreases due to the absorption by the VO2 nanoparticles. In fig. 2b, the transmission decrease around 1100 nm is due to the plasmonic absorption in metallic VO2 nanoparticles. To confirm these assumptions, we compared the transmission spectra computed with FMM with those obtained for the equivalent homogeneous material using AMM, as presented in the inset of Figure 2a in the case of 1vol%. As expected the oscillations observed for the equivalent homogeneous thin film perfectly fit those in our opal photonic crystal, confirming their origin as the thin film interference. We also observe the disappearance of the minimum around 900 nm for the homogeneous thin film, confirming that its origin comes from the material ordering.

The corresponding solar visible and infrared transmittances in both cold and warm conditions as a function of VO2 concentration are presented in fig. 3a and3b. In order to use this coating for smart energy efficient material, we are looking for a high visible transmittance in both states of the material with no or little differences between both states (low Δ𝑇 𝑣𝑖𝑠 ), and a large difference in infrared transmittance between both states (high Δ𝑇 𝐼𝑅 ). Fig. 3c gives these transmission modulations as a function of the VO2 concentration. The optimized VO2 concentration is obtained for the highest difference between the infrared and visible transmission modulations, shown in fig. 3d. A high infrared transmission modulation (Δ𝑇 𝐼𝑅 = 0.42) and low visible modulation (Δ𝑇 𝑣𝑖𝑠 = 0.26) is obtained for a 1vol% VO2 concentration.

Conclusions and prospects

Optical numerical simulations have been performed on a polymer opal photonic crystal loaded with VO2 nanoparticles. As expected, the transmittance in both visible and IR regions decreases as the VO2 concentration increases in both cold and warm conditions. The transmittance modulation increases as the VO2 concentration increases. Thus an optimal value that maximizes the transmittance modulations is found at 1vol% VO2 concentration, in the case of a light injection along the (111) crystallographic direction. In order to conclude on the coating global performance as a smart transparent energy efficient material, further studies considering all incidence angles would be required. Also, the influence of the coating thickness and possibility of a multilayer are still open questions.

Fig. 1 :

 1 Fig. 1: Numerical representation of a close packed polymer opal made of PS spheres in PEA filled with VO2 nanoparticles, and light incident over the (111) crystallographic directionLight is injected along the (111) crystallographic direction of the considered polymer opal photonic crystal (Fig1). In this direction, the structure is modelled by 15 periods, each of them are 866nm thick. The material total thickness is then 12.99µm. The interstices containing VO2 nanoparticles and PEA have been modelled using the Maxwell-Garnett

  𝐸 𝜆 is the weight coefficient for solar direct normal irradiance extracted from ASTM G173-03 and 𝑇(𝜆) is the material spectral transmittance. The transmittance modulation in IR is then defined as the difference in the infrared transmittance between both states of the thermochromic VO2. Δ𝑇 𝑣𝑖𝑠 = | 𝑇 𝑣𝑖𝑠 (𝑐𝑜𝑙𝑑) -𝑇 𝑣𝑖𝑠 (𝑤𝑎𝑟𝑚) | and Δ𝑇 𝐼𝑅 = | 𝑇 𝐼𝑅 (𝑐𝑜𝑙𝑑) -𝑇 𝐼𝑅 (𝑤𝑎𝑟𝑚) | (2)

Figure 3 :

 3 Figure 3: Impact of the VO2 concentration in solar transmittances for both cold and warm states of the material, in the visible (a) and infrared region (b), their modulations between both states of the material (c) and difference between infrared and visible transmission modulations (d)

Acknowledgements

This work has been funded by La Région Auvergne-Rhône-Alpes and supported by the LABEX MANUTECH-SISE (ANR-10-LABX-0075) of Université de Lyon, within the program "Investissements d'Avenir" (ANR-11-IDEX-0007) operated by the French National Research Agency (ANR) and ANR Thermocoat (ANR-16-CE22-0014).


M Bonforte

J Dolbeault

G Grillo

J L Vázquez

Sharp rates of decay of solutions to the nonlinear fast diffusion equation via functional inequalities

The goal of this note is to state the optimal decay rate for solutions of the nonlinear fast diffusion equation and, in self-similar variables, the optimal convergence rates to Barenblatt self-similar profiles and their generalizations. It relies on the identification of the optimal constants in some related Hardy-Poincaré inequalities and concludes a long series of papers devoted to generalized entropies, functional inequalities and rates for nonlinear diffusion equations. Fast diffusion equation | porous media equation | Barenblatt solutions | Hardy-Poincaré inequalities | large time behaviour | asymptotic expansion | intermediate asymptotics | sharp rates | optimal constants

Introduction

The evolution equation

∂u ∂τ = ∇ • (u m-1 ∇u) = 1 m ∆u m [ 1 ]
with m = 1 is a simple example of a nonlinear diffusion equation which generalizes the heat equation, and appears in a wide number of applications. Solutions differ from the linear case in many respects, notably concerning existence, regularity and large-time behaviour. We consider positive solutions u(τ, y) of this equation posed for τ ≥ 0 and y ∈ R d , d ≥ 1. The parameter m can be any real number. The equation makes sense even in the limit case m = 0, where u m /m has to be replaced by log u, and is formally parabolic for all m ∈ R. Notice that [ 1 ] is degenerate at the level u = 0 when m > 1 and singular when m < 1. We consider the initial-value problem with nonnegative datum u(τ = 0, •) = u0 ∈ L 1 loc (dx), where dx denotes Lebesgue's measure on R d . Further assumptions on u0 are needed and will be specified later.

The description of the asymptotic behaviour of the solutions of [ 1 ] as τ → ∞ is a classical and very active subject. If m = 1, the convergence of solutions of the heat equation with u0 ∈ L 1 + (dx) to the Gaussian kernel (up to a mass factor) is a cornerstone of the theory. In the case of equation [ 1 ] with m > 1, known in the literature as the porous medium equation, the study of asymptotic behaviour goes back to [1]. The result extends to the exponents m ∈ (mc, 1) with mc := (d -2)/d; see [2]. In these results the Gaussian kernel is replaced by some special self-similar solutions UD,T known as the Barenblatt solutions (see [3]) given by UD,T (τ, y)

:= 1 R(τ ) d " D + 1-m 2 d |m-mc | ˛y R(τ ) ˛2" -1 1-m + [ 2 ]
whenever m > mc and m = 1, with R(τ ) := (T + τ )

1 d (m-mc ) ,
where T ≥ 0 and D > 0 are free parameters. To some extent, these solutions play the role of the fundamental solution of the linear diffusion equations, since limτ→0 UD,0(τ, y) = M δ, where δ is the Dirac delta distribution, and M depends on D. Notice that the Barenblatt solutions converge as m → 1 to the fundamental solution of the heat equation, up to the mass factor M . The results of [1,2] say that UD,T also describes the large time asymptotics of the solutions of equation [ 1 ] as τ → ∞ provided M = R R d u0 dy is finite, a condition that uniquely determines D = D(M ). Notice that in the range m ≥ mc, solutions of [ 1 ] with u0 ∈ L 1 + (dx) exist globally in time and mass is conserved: R R d u(τ, y) dy = M for any τ ≥ 0. On the other hand, when m < mc, a natural extension for the Barenblatt functions can be achieved by considering the same expression [ 2 ], but a different form for R, that is

R(τ ) := (T -τ ) - 1 d (mc-m) .
The parameter T now denotes the extinction time, a new and important feature. The limit case m = mc is covered by

R(τ ) = e τ , UD,T (τ, y) = e -d τ `D + e -2τ |y| 2 /d
´-d/2 . See [4,[START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF] for more detailed considerations.

In this note, we shall focus our attention on the case m < 1 which has been much less studied. In this regime, [ 1 ] is known as the fast diffusion equation. We do not even need to assume m > 0. We shall summarize and extend a series of recent results on the basin of attraction of the family of generalized Barenblatt solutions and establish the optimal rates of convergence of the solutions of [ 1 ] towards a unique attracting limit state in that family. To state such a result, it is more convenient to rescale the flow and rewrite [ 1 ] in self-similar variables by introducing for m = mc the time-dependent change of variables

t := 1-m 2 log " R(τ ) R(0) « and x := q 1-m 2 d |m-mc | y R(τ )
, [ 3 ] with R as above. If m = mc, we take t = τ /d and x = e -τ y/ √ d. In these new variables, the generalized Barenblatt functions UD,T (τ, y) are transformed into generalized Barenblatt profiles VD(x), which are stationary:

VD(x) := `D + |x| 2 ´1 m-1 x ∈ R d . [ 4 ]
If u is a solution to [ 1 ], the function

v(t, x) := R(τ ) d u(τ, y)
solves the equation

∂v ∂t = ∇ • » v ∇ " v m-1 -V m-1 D m -1 «- t > 0 , x ∈ R d , [ 5 ] with initial condition v(t = 0, x) = v0(x) := R(0) -d u0(y)
where x and y are related according to [ 3 ] with τ = 0. This nonlinear Fokker-Planck equation can also be written as

∂v ∂t = 1 m ∆v m + 2 1 -m ∇ • (x v) t > 0 , x ∈ R d .

Main results

Our main result is concerned with the sharp rate at which a solution v of the rescaled equation [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF] converges to the generalized Barenblatt profile VD given by formula [ 4 ] in the whole July 17, 2009 range m < 1. Convergence is measured in terms of the relative entropy given by the formula

E [v] := 1 m -1 Z R d » v m -V m D m -V m-1 D (v -VD)
dx for all m = 0 (modified as mentioned for m = 0). In order to get such convergence we need the following assumptions on the initial datum v0 associated to [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF]:

(H1) VD 0 ≤ v0 ≤ VD 1 for some D0 > D1 > 0, (
E [v(t, •)] ≤ C e -2 Λ t ∀ t ≥ 0 . [ 6 ]
The sharp decay rate Λ is equal to the best constant Λ α,d > 0 in the Hardy-Poincaré inequality of Theorem 2 with α := 1/(m -1) < 0. Moreover, the constant C > 0 depends only on m, d, D0, D1, D and E [v0].

The precise meaning of what sharp rate means will be discussed at the end of this paper. As in [4], we can deduce from Theorem 1 rates of convergence in more standard norms, namely, in L q (dx) for q ≥ max{1,

d (1 -m)/ [2 (2 - m) + d (1 -m)]},
or in C k by interpolation. Moreover, by undoing the time-dependent change of variables [ 3 ], we can also deduce results on the intermediate asymptotics for the solution of equation [ 1 ]; to be precise, we can get rates of decay of u(τ, y) -R(τ

) -d UD,T (τ, y) as τ → +∞ if m ∈ [mc, 1), or as τ → T if m ∈ (-∞, mc).
It is worth spending some words on the basin of attraction of the Barenblatt solutions UD,T given by [ 2 ]. Such profiles have two parameters: D corresponds to the mass while T has the meaning of the extinction time of the solution for m < mc and of a time-delay parameter otherwise. Fix T and D, and consider first the case m * < m < 1. The basin of attraction of UD,T contains all solutions corresponding to data which are trapped between two Barenblatt profiles UD 0 ,T (0, •), UD 1 ,T (0, •) for the same value of T and such that R R d [u0 -UD,T (0, •)] dy = 0 for some D ∈ [D1, D0]. If m < m * the basin of attraction of a Barenblatt solution contains all solutions corresponding to data which, besides being trapped between UD 0 ,T and UD 1 ,T , are integrable perturbations of UD,T (0, •). Now, let us give an idea of the proof of Theorem 1. First assume that D = 1 (this entails no loss of generality). On R d , we shall therefore consider the measure dµα := hα dx, where the weight hα is the Barenblatt profile, defined by hα(x) := (1 + |x| 2 ) α , with α = 1/(m -1) < 0, and study on the weighted space L 2 (dµα) the operator

L α,d := -h1-α div [ hα ∇• ] which is such that R R d f (L α,d f ) dµα-1 = R R d |∇f | 2 dµα
. This operator appears in the linearization of [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF] if, at a formal level, we expand

v(t, x) = hα(x) ˆ1 + ε f (t, x) h 1-m α (x) ĩn
terms of ε, small, and only keep the first order terms: We have m * < mc < m2 < m1 < 1. Similar definitions for d = 2 give m * = -∞ so that α * = 0, as well as mc = 0, and m1 = m2 = 1/2. For the convenience of the reader, a table summarizing the key values of the parameter m and the corresponding values of α is given in the Appendix.

∂f ∂t + L α,d f = 0 .
Theorem 2. (Sharp Hardy-Poincaré inequalities) Let d ≥ 3. For any α ∈ (-∞, 0) \ {α * }, there is a positive constant Λ α,d such that Λ α,d Z R d |f | 2 dµα-1 ≤ Z R d |∇f | 2 dµα ∀ f ∈ H 1 (dµα) [ 7 ] under the additional condition R R d f dµα-1 = 0 if α < α * . Moreover, the sharp constant Λ α,d is given by Λ α,d = 8 > > < > > : 1 4 (d -2 + 2 α) 2 if α ∈ ˆ-d+2 2 , α * ´∪ (α * , 0) , -4 α -2 d if α ∈ ˆ-d, -d+2 2
´,

-2 α if α ∈ (-∞, -d) .
For d = 2, inequality [START_REF] Hardy | Inequalities[END_REF] holds for all α < 0, with the corresponding values of the best constant

Λα,2 = α 2 for α ∈ [-2, 0) and Λα,2 = -2α for α ∈ (-∞, -2). For d = 1, [ 7 ] holds, but the values of Λα,1 are given by Λα,1 = -2 α if α < -1/2 and Λα,1 = (α -1/2) 2 if α ∈ [-1/2, 0).
The Hardy-Poincaré inequalities [START_REF] Hardy | Inequalities[END_REF] share many properties with Hardy's inequalities, because of homogeneity reasons. A simple scaling argument indeed shows that

Λ α,d Z R d |f | 2 (D + |x| 2 ) α-1 dx ≤ Z R d |∇f | 2 (D + |x| 2 ) α dx holds for any f ∈ H 1 ((D + |x| 2 ) α dx) and any D ≥ 0, un- der the additional conditions R R d f (D + |x| 2 ) α-1 dx = 0 and D > 0 if α < α * .
In other words, the optimal constant, Λ α,d , does not depend on D > 0 and the assumption D = 1 can be dropped without consequences. In the limit D → 0, they yield weighted Hardy type inequalities, cf. [START_REF] Caffarelli | [END_REF][START_REF] Hardy | Inequalities[END_REF].

Theorem 2 has been proved in [4] for m < m * . The main improvement of this note compared to [START_REF] Blanchet | [END_REF]4] is that we are able to give the value of the sharp constants also in the range (m * , 1). These constants are deduced from the spectrum of the operator L α,d , that we shall study below.

It is relatively easy to obtain the classical decay rates of the linear case in the limit m → 1 by a careful rescaling such that weights become proportional to powers of the modified expression (1 + (1m) |x| 2 ) -1/(1-m) . In the limit case, we obtain the Poincaré inequality for the Gaussian weight. As for the evolution equation, the time also has to be rescaled by a factor (1-m). We leave the details to the reader. See [START_REF] Bartier | Improved intermediate asymptotics for the heat equation[END_REF] for further considerations on associated functional inequalities.

A brief historical overview

The search for sharp decay rates in fast diffusion equations has been extremely active over the last three decades. Once plain convergence of the suitably rescaled flow towards an asymptotic profile is established (cf. [1,2] for m > mc and [4,[START_REF] Daskalopoulos | [END_REF] for m ≤ mc), getting the rates is the next step in the asymptotic analysis. An important progress was achieved by M. Del Pino and J. Dolbeault in [11] by identifying sharp rates of decay for the relative entropy, that had been introduced earlier by J. Ralston and W.I. Newman in [12,13]. The analysis in [11] uses the optimal constants in Gagliardo-Nirenberg inequalities, and these constants are computed. J.A. Carrillo and G. Toscani in [14] gave a proof of decay based on the entropy/entropy-production method of D. Bakry and M. Emery, and established an analogue of the Csiszár-Kullback inequality which allows to control the convergence in L 1 (dx), in case m > 1. F. Otto then made the link with gradient flows with respect to the Wasserstein distance, see [15], and D. Cordero-Erausquin, B. Nazaret and C. Villani gave a proof of Gagliardo-Nirenberg inequalities using mass transportation techniques in [16].

The condition m ≥ (d-1)/d =: m1 was definitely a strong limitation to these first approaches, except maybe for the entropy/entropy-production method. Gagliardo-Nirenberg inequalities degenerate into a critical Sobolev inequality for m = m1, while the displacement convexity condition requires m ≥ m1. It was a puzzling question to understand what was going on in the range mc < m < m1, and this has been the subject of many contributions. Since one is interested in understanding the convergence towards Barenblatt profiles, a key issue is the integrability of these profiles and their moments, in terms of m. To work with Wasserstein's distance, it is crucial to have second moments bounded, which amounts to request m > d/(d + 2) =: m2 for the Barenblatt profiles. The contribution of J. Denzler and R. McCann in [START_REF] Denzler | Proc. Natl. Acad. Sci[END_REF][START_REF] Denzler | [END_REF] enters in this context. Another, weaker, limitation appears when one only requires the integrability of the Barenblatt profiles, namely m > mc. Notice that the range [mc, 1) is also the range for which L 1 (dx) initial data give rise to solutions which preserve the mass and globally exist, see for instance [1,[START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF].

It was therefore natural to investigate the range m ∈ (mc, 1) with entropy estimates. This has been done first by linearizing around the Barenblatt profiles in [19,20], and then a full proof for the nonlinear flow was done by J.A. Carrillo and J.L. Vázquez in [21]. A detailed account for these contributions and their motivations can be found in the survey paper [22]. Compared to classical approaches based on comparison, as in the book [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF], a major advantage of entropy techniques is that they combine very well with L 1 (dx) estimates if m > mc, or relative mass estimates otherwise, see [4].

The picture for m ≤ mc turns out to be entirely different and more complicated, and it was not considered until quite recently. First of all, many classes of solutions vanish in finite time, which is a striking property that forces us to change the concept of asymptotic behaviour from large-time behaviour to behaviour near the extinction time. On the other hand, L 1 (dx) solutions lose mass as time evolves. Moreover, the natural extensions of Barenblatt's profiles make sense but these profiles have two novel properties: they vanish in finite time and they do not have finite mass.

There is a large variety of possible behaviours and many results have been achieved, such as the ones described in [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF] for data which decay strongly as |x| → ∞. However, as long as one is interested in solutions converging towards Barenblatt profiles in self-similar variables, there were some recent results on plain convergence: a paper of P. Daskalopoulos and N. Sesum, [START_REF] Daskalopoulos | [END_REF], using comparison techniques, and in two contributions involving the authors of this note, using relative entropy methods, see [START_REF] Blanchet | [END_REF]4]. This last approach proceeds further into the description of the convergence by identifying a suitable weighted linearization of the relative entropy. In the appropriate space, L 2 (dµα-1), with the notations of Theorem 2, it gives rise to an exponential convergence after rescal-ing. This justifies the heuristic computation which relates Theorems 1 and 2, and allows to identify the sharp rates of convergence. The point of this note is to explicitly state and prove such rates in the whole range m < 1.

Relative entropy and linearization

The strategy developed in [4] is based on the extension of the relative entropy of J. Ralston and W.I. Newmann, which can be written in terms of w = v/VD as

F[w] := 1 1 -m Z R d » w -1 - 1 m `wm -1 ´-V m D dx .
For simplicity, assume m = 0. Notice that

F[w] = E [v]. Let I[w] := Z R d ˛1 m -1 ∇ ˆ(w m-1 -1) V m-1 D ˜˛2 v dx
be the generalized relative Fisher information. If v is a solution of [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF], then

d dt F[w(t, •)] = -I[w(t, •)] ∀ t > 0 [ 8 ]
and, as a consequence, limt→+∞ F[w(t, •)] = 0 for all m < 1.

The method is based on Theorem 2 and uniform estimates that relate linear and nonlinear quantities. Following [4,23] we can first estimate from below and above the entropy F in terms of its linearization, which appears in [START_REF] Hardy | Inequalities[END_REF]:

h m-2 Z R d |f | 2 V 2-m D dx ≤ 2 F[w] ≤ h 2-m Z R d |f | 2 V 2-m D dx [ 9 ]
where

f := (w -1) V m-1 D , h1(t) := inf R d w(t, •), h2(t) := sup R d w(t,
•) and h := max{h2, 1/h1}. We notice that h(t) → 1 as t → +∞. Similarly, the generalized Fisher information satisfies the bounds Z

R d |∇f | 2 VD dx ≤ [1 + X(h)] I[w] + Y (h) Z R d |f | 2 V 2-m D dx [ 10 ] where h 2(2-m) 2 /h1 ≤ h 5-2m =: 1 + X(h) and d (1 -m) ˆ(h2/h1) 2(2-m) -1 ˜≤ d (1 -m) ˆh4(2-m) -1 ˜=: Y (h).
Notice that X(1) = Y (1) = 0. Joining these inequalities with the Hardy-Poincaré inequality of Theorem 2 gives

F[w] ≤ h 2-m [1 + X(h)] 2 ˆΛα,d -Y (h) ˜I[w] [ 11 ]
as soon as 0 < h < h * := min{h > 0 : Λ α,d -Y (h) ≥ 0}. On the other hand, uniform relative estimates hold, according to [23], formula (5.33): for some C = C(d, m, D, D0, D1),

0 ≤ h -1 ≤ C F 1-m d+2-(d+1)m . [ 12 ]
Summarizing, we end up with a system of nonlinear differential inequalities, with h as above and, at least for any t > t * , t * > 0 large enough,

d dt F[w(t, •)] ≤ -2 Λ α,d -Y (h) ˆ1 + X(h) ˜h2-m F[w(t, •)] .
[ 13 ]

Gronwall type estimates then show that lim sup

t→∞ e 2 Λ α,d t F[w(t, •)] < +∞ .
This completes the proof of Theorem 1 for m = 0. The adaptation to the logarithmic nonlinearity is left to the reader.

Results in [4] are improved in two ways: a time-dependent estimate of h is used in place of h(0), and the precise expression of the rate is established. One can actually get a slightly more precise estimate by coupling [ 12 ] and [ 13 ].

Corollary 3. Under the assumptions of Theorem 1, if h(0) < h * , then F[w(t, •)] ≤ G `t, h(0), F[w(0, •)] ´for any t ≥ 0, where G is the unique solution of the nonlinear ODE

dG dt = -2 Λ α,d -Y (h) [1 + X(h)] h 2-m G with h = 1 + C G 1-m d+2-(d+1)m and initial condition G(0) = F[w(0, •)].
Operator equivalence. The spectrum of L α,d

An important point of this note is the computation of the spectrum of L α,d for any α < 0. This spectrum was only partially understood in [4,[START_REF] Blanchet | [END_REF]. In particular, the existence of a spectral gap was established for all α = α * = (2d)/2, but its value was not stated for all values of α.

J. Denzler and R.J. McCann in [START_REF] Denzler | Proc. Natl. Acad. Sci[END_REF][START_REF] Denzler | [END_REF] formally linearized the fast diffusion flow (considered as a gradient flow of the entropy with respect to the Wasserstein distance) in the framework of mass transportation, in order to guess the asymptotic behaviour of the solutions of [ 1 ]. This leads to a different functional setting, with a different linearized operator, H α,d . They performed the detailed analysis of its spectrum for all m ∈ (mc, 1), but the justification of the nonlinear asymptotics could not be completed due to the difficulties of the functional setting, especially in the very fast diffusion range.

Our approach is based on relative entropy estimates and the Hardy-Poincaré inequalities of Theorem 2. The asymptotics have readily been justified in [4]. The operator L α,d = -h1-α div [ hα ∇• ] can be initially defined on D(R d ). To construct a self-adjoint extension of such an operator, one can consider the quadratic form f → (f, L α,d f ), where ( • , • ) denotes the scalar product on L 2 (dµα-1). Standard results show that such a quadratic form is closable, so that its closure defines a unique self-adjoint operator, its Friedrich's extension, still denoted by the same symbol for brevity. The operator H α,d is different: it is obtained by taking the operator closure of L α,d , initially defined on D(R d ), in the Hilbert space H 1, * (dµα

) := ˘f ∈ L 2 (dµα-1) : R R d |∇f | 2 dµα < ∞ and R R d f dµα-1 = 0 if α < α * ¯, so that H α,d f := h1-α ∇ • ˆhα ∇ `h1-α ∇ • (hα ∇f ) ´˜.
Because of the Hardy-Poincaré inequality, R R d |∇f | 2 dµα defines a norm. Denote by • , • the corresponding scalar product and notice that f , g = ( f , L α,d g ) for any f, g ∈ D(R d ).

Proposition 4. The operator L α,d on L 2 (dµα-1) has the same spectrum as the operators H α,d on H 1, * (dµα).

The proof is based on the construction of a suitable unitary operator U :

H 1, * (dµα) → L 2 (dµα-1), such that U H α,d U -1 = L α,d . We claim that U = p L α,d
is the requested unitary operator. By definition, D(R d ) is a form core of L α,d , and as a consequence, the identity has to be established only for functions

f ∈ D(R d ). Since U f 2 = `f , L α,d f ´= R R d |∇f | 2 dµα, we get `U H α,d U -1 f , g ´= ˙Hα,d U -1 f , U -1 g = `Lα,d U -1 f , L α,d U -1 g ´= `U f , U g ´= ˙f , g where
we have used the properties U * = U -1 and U 2 = L α,d . This unitary equivalence between L α,d and H α,d implies the identity of their spectra.

We may now proceed with the presentation of the actual values of the spectrum by extending the results of [START_REF] Denzler | [END_REF]. According to [START_REF] Berger | Le spectre d'une variété riemannienne[END_REF], the spectrum of the Laplace-Beltrami operator on S d-1 is described by

-∆ S d-1 Y ℓµ = ℓ (ℓ + d -2) Y ℓµ with ℓ = 0, 1, 2, . . . and µ = 1, 2, . . . M ℓ := (d+ℓ-3)! (d+2ℓ-2) ℓ! (d-2)!
with the convention M0 = 1, and M1 = 1 if d = 1. Using spherical coordinates and separation of variables, the discrete spectrum of L α,d is therefore made of the values of λ for which

v ′′ + " d-1 r + 2 α r 1+r 2 " v ′ + " λ 1+r 2 -ℓ (ℓ+d-2) r 2 " v = 0 [ 14 ]
s (1 -s) y ′′ + [ c -(a + b + 1) s ] y ′ -a b y = 0 ,
see [25]. Based on [START_REF] Blanchet | [END_REF][START_REF] Denzler | [END_REF], we can state the following result.

Proposition 5. The bottom of the continuous spectrum of the operator

L α,d on L 2 (dµα-1) is λ cont α,d := 1 4 (d + 2 α -2) 2 .
Moreover, L α,d has some discrete spectrum only for m > m2 = d/(d + 2). For d ≥ 2, the discrete spectrum is made of the eigenvalues

λ ℓk = -2 α (ℓ + 2 k) -4 k " k + ℓ + d 2 -1 « [ 15 ]
with ℓ, k = 0, 1, . . . provided (ℓ, k) = (0, 0) and ℓ + 2k

-1 < -(d + 2 α)/2. If d = 1, the discrete spectrum is made of the eigenvalues λ k = k (1 -2 α -k) with k ∈ N ∩ [1, 1/2 -α].
Using Persson's characterization of the continuous spectrum, see [26,[START_REF] Blanchet | [END_REF], one can indeed prove that λ cont α,d is the optimal constant in the following inequality: for any f ∈ D(R d \ {0}), The condition that the solution of [ 14 ] is in the domain of L α,d determines the eigenvalues. A more complete discussion of this topic can be found in [START_REF] Denzler | [END_REF], which justifies the expression of the discrete spectrum. Since α = 1/(m -1), we may notice that for d ≥ 2, α = -d

λ cont α,d Z R d |f | 2 |x| 2(α-1) dx ≤ Z R d |∇f | 2 |x| 2 α dx . λ01 λ10 λ11 λ02 λ20 Essential spectrum of L α,d λ cont α ,d := 1 4 (d + 2 α -2) 2 -d -d+2 2 -d-2 2 α 0 (m = m 1 ) (m = m 2 ) (m = m * )
(corresponding to -2 α = λ10 = λ01 = -4 α -2 d) and α = -(d+2)/2 (corresponding to λ01 = λ cont 0 := 1 4 (d+2 α-2) 2 ) re- spectively mean m = m1 = (d-1)/d and m = m2 = d/(d+2).
The above spectral results hold exactly in the same form when d = 2, see [START_REF] Denzler | [END_REF]. Notice in particular that λ cont α,d=2 = α 2 so that there is no equivalent of m * for d = 2. With the notations of Theorem 2, α * = 0. All results of Theorem 1 hold true under the sole assumption (H1).

In dimension d = 1, the spectral results are different, see [START_REF] Denzler | [END_REF]. The discrete spectrum is nonempty whenever α ≤ -1/2, that is m ≥ -1.

The critical case

Since the spectral gap of L α,d tends to zero as m → m * , the previous strategy fails when m = m * and one might expect a slower decay to equilibrium, sometimes referred as slow asymptotics. The following result has been proved in [23].

Theorem 6. Assume that d ≥ 3, let v be a solution of [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF] with m = m * , and suppose that (H1)-(H2) hold. If |v0 -VD| is bounded a.e. by a radial L 1 (dx) function, then there exists a positive constant C * such that

E [v(t, •)] ≤ C * t -1/2 ∀ t ≥ 0 , [ 16 ]
where C * depends only on m, d, D0, D1, D and E [v0].

Rates of convergence in L q (dx), q ∈ (1, ∞] follow. Notice that in dimension d = 3 and 4, we have respectively m * = -1 and m * = 0. In the last case, Theorem 6 applies to the logarithmic diffusion.

The proof relies on identifying first the asymptotics of the linearized evolution. In this case, the bottom of the continuous spectrum of L α * ,d is zero. This difficulty is overcome by noticing that the operator L α * ,d on R d can be identified with the Laplace-Beltrami operator for a suitable conformally flat metric on R d , having positive Ricci curvature. Then the on-diagonal heat kernel of the linearized generator behaves like t -d/2 for small t and like t -1/2 for large t. The Hardy-Poincaré inequality is replaced by a weighted Nash inequality: there exists a positive continuous and monotone function N on R + such that for any nonnegative smooth function f with

M = R R d f dµ -d/2 (recall that α * -1 = -d/2), 1 M 2 Z R d |f | 2 dµ -d/2 ≤ N " 1 M 2 Z R d |∇f | 2 dµ (2-d)/2
« .

The function N behaves as follows: lim s→0 + s -1/3 N (s) = c1 > 0 and lims→∞ s -d/(d+2) N (s) = c2 > 0. Only the first limit matters for the asymptotic behaviour. Up to technicalities, inequality [ 13 ] is replaced by (F[w(t,

•)]) 3 ≤ K I[w(t, •)]
for some K > 0, t ≥ t0 large enough, which allows to complete the proof.

Faster convergence

A very natural issue is the question of improving the rates of convergence by imposing restrictions on the initial data. Results of this nature have been observed in [19] in case of radially symmetric solutions, and are carefully commented in [START_REF] Denzler | [END_REF]. By locating the center of mass at zero, we are able to give an answer, which amounts to kill the λ10 mode, whose eigenspace is generated by x → xi, i = 1, 2. . . d. This is an improvement compared to the first result in this direction, which has been obtained by R. McCann and D. Slepčev in [27], since we obtain an improved sharp rate of convergence of the solution of [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF], as a consequence of the following improved Hardy-Poincaré inequality.

Lemma 1. Let e Λ α,d := -4 α -2 d if α < -d and e Λ α,d := λ cont α,d if α ∈ [-d, -d/2). If d ≥ 2, for any α ∈ (-∞, -d), we have e Λ α,d Z R d |f | 2 dµα-1 ≤ Z R d |∇f | 2 dµα ∀ f ∈ H 1 (dµα) under the conditions R R d f dµα-1 = 0 and R R d x f dµα-1 = 0. The constant e Λ α,

A variational approach of sharpness

Recall that (d -2)/d = mc < m1 = (d -1)/d. The entropy / entropy production inequality obtained in [11] in the range m ∈ [m1, 1) can be written as F ≤ 1 2 I and it is known to be sharp as a consequence of the optimality case in Gagliardo-Nirenberg inequalities. Moreover, equality is achieved if and only if v = VD. The inequality has been extended in [21] to the range m ∈ (mc, 1) using the Bakry-Emery method, with the same constant 1/2, and again equality is achieved if and only if v = VD, but sharpness of 1/2 is not as straightforward for m ∈ (mc, m1) as it is for m ∈ [m1, 1). The question of the optimality of the constant can be reformulated as a variational problem, namely to identify the value of the positive constant

C = inf I[v] E [v]
where the infimum is taken over the set of all functions such that v ∈ D(R d ) and R R d v dx = M . Rephrasing the sharpness results, we know that C = 2 if m ∈ (m1, 1) and C ≥ 2 if m ∈ (mc, m1). By taking vn

= VD (1 + 1 n f V 1-m D ) and letting n → ∞, we get lim n→∞ I[vn] E [vn] = R R d |∇f | 2 VD dx R R d |f | 2 V 2-m D dx .
With the optimal choice for f , the above limit is less or equal than 2. Since we already know that C ≥ 2, this shows that C = 2 for any m > mc. It is quite enlightening to observe that optimality in the quotient gives rise to indetermination since both numerator and denominator are equal to zero when v = VD. This also explains why it is the first order correction which determines the value of C, and, as a consequence, why the optimal constant, C = 2, is determined by the linearized problem.

When m ≤ mc, the variational approach is less clear since the problem has to be constrained by a uniform estimate. Proving that any minimizing sequence (vn) n∈N is such that vn/VD -1 converges, up to a rescaling factor, to a function f associated to the Hardy-Poincaré inequalities would be a significant step, except that one has to deal with compactness issues, test functions associated to the continuous spectrum and a uniform constraint.

Sharp rates of convergence and conjectures

In Theorem 1, we have obtained that the rate exp(-Λ α,d t) is sharp. The precise meaning of this claim is that Λ α,d = lim inf h→0 + inf w∈S h

I[w] F[w]

,

where the infimum is taken on the set S h of smooth, nonnegative bounded functions w such that w -1 L ∞ (dx) ≤ h and such that R R d (w -1) VD dx is zero if d = 1, 2 and m < 1, or if d ≥ 3 and m * < m < 1, and it is finite if d ≥ 3 and m < m * . Since, for a solution v(t, x) = w(t, x) VD(x) of [START_REF] Vázquez | Smoothing and decay estimates for nonlinear diffusion equations[END_REF], [START_REF] Blanchet | [END_REF] holds, by sharp rate we mean the best possible rate, which is uniform in t ≥ 0. In other words, for any λ > Λ α,d , one can find some initial datum in S h such that the estimate F[w(t, •)] ≤ F[w(0, •)] exp(-λ t) is wrong for some t > 0. We did not prove that the rate exp(-Λ α,d t) is globally sharp in the sense that for some special initial data, F[w(t, •)] decays exactly at this rate, or that lim inft→∞ exp(Λ α,d t) F[w(t, •)] > 0, which is possibly less restrictive.

However, if m ∈ (m1, 1), m1 = (d -1)/d, then exp(-Λ α,d t) is also a globally sharp rate, in the sense that the solution with initial datum u0(x) = VD(x + x0) for any x0 ∈ R d \ {0} is such that F[w(t, •)] decays exactly like exp(-Λ α,d t). This formally answers the dilation-persistence conjecture as formulated in [START_REF] Denzler | [END_REF]. The question is still open when m ≤ m1.

Another interesting issue is to understand if improved rates, that is rates of the order of exp(-λ ℓk t) with (ℓ, k) = (0, 0), (0, 1), (1,0)

Appendix. A table of correspondence

For the convenience of the reader, a table of definitions of the key values of m when d ≥ 3 is provided below with the correspondence for the values of α = 1/(m -1). This note is restricted to the case m ∈ (-∞, 1), that is α ∈ (-∞, 0).

m = -∞ m * mc m 2 m 1 1 m = -∞ d-4 d-2 d-2 d d d+2 d-1 d 1 α = 0 -d-2 2 -d 2 -d+2 2 -d -∞

  H2) if d ≥ 3 and m ≤ m * , (v0 -VD) is integrable for a suitable D ∈ [D1, D0]. The case m = m * := (d-4)/(d-2) will be discussed later. Besides, if m > m * , we define D as the unique value in [D1, D0] such that R R d (v0 -VD) dx = 0. Theorem 1. Under the above assumptions, if m < 1 and m = m * , the entropy decays according to

  The convergence result of Theorem 1 follows from the energy analysis of this equation based on the Hardy-Poincaré inequalities that are described below. Let us fix some notations. For d ≥ 3, let us define α * := -(d -2)/2 corresponding to m = m * ; two other exponents will appear in the analysis, namely, m1 := (d -1)/d with corresponding α1 = -d, and m2 := d/(d + 2) with corresponding α2 = -(d + 2)/2.

  has a solution on R + ∋ r, in the domain of L α,d . The change of variables v(r) = r ℓ w(-r 2 ) allows to express w in terms of the hypergeometric function 2F1(a, b, c; z) with c = ℓ + d/2, a + b + 1 = ℓ + α + d/2 and a b = (2 ℓ α + λ)/4, as the solution for s = -r 2 of

Fig. 1 .

 1 Fig. 1. Spectrum of L α,d as a function of α, for d = 5.

Theorem 7 .

 7 d is sharp. This covers the range m ∈ (m1, 1) with m1 = (d -1)/d. Assume that m ∈ (m1, 1), d ≥ 3. Under Assumption (H1), if v is a solution of [ 5 ] with initial datum v0 such that R R d x v0 dx = 0 and if D is chosen so that R R d (v0 -VD) dx = 0, then there exists a positive constant e C depending only on m, d, D0, D1, D and E [v0] such that the relative entropy decays like E [v(t, •)] ≤ e C e -e Λ α,d t ∀ t ≥ 0 .

  are sharp or globally sharp under additional moment-like conditions on the initial data. It is also open to decide whether exp(e Λ α,d t) is sharp or globally sharp under the extra condition R R d x v0 dx = 0.

ACKNOWLEDGMENTS. This work has been supported by the ANR-08-BLAN-0333-01 project CBDif-Fr and the exchange program of University Paris-Dauphine and Universidad Autónoma de Madrid. MB and JLV partially supported by Project MTM2008-06326-C02-01 (Spain). MB, GG and JLV partially supported by HI2008-0178 (Italy-Spain).