Keywords: alkali transition metal layered oxide, honeycomb ordering, DFT, solid-state NMR, stacking faults

The reaction between P2-type honeycomb layered oxides Na2Ni2TeO6 and K2Ni2TeO6 enables the formation of NaKNi2TeO6. The compound is characterized by X-ray diffraction and 23 Na solid-state nuclear magnetic resonance spectroscopy and the structure is discussed through density functional theory calculations. In addition to the honeycomb Ni/Te cationic ordering, NaKNi2TeO6 exhibits a unique example of alternation of sodium and potassium layers instead of a random alkali-mixed occupancy. Stacking faults simulations underline the impact of the successive position of the Ni/Te honeycomb layers and validate the presence of multiple stacking sequences within the powder material, in proportions that evolve with the synthesis conditions. In a broader context, this work participates to a better understanding of the alkalimixed layered compounds.

Stacking versatility in alkali-mixed honeycomb layered NaKNi2TeO6

Romain Berthelot, 1,2,* Jon Serrano-Sevillano, [START_REF] Delmas | The Layered Oxides in Lithium and Sodium-Ion Batteries: A Solid-State Chemistry Approach[END_REF][START_REF] Sathiya | Reversible Anionic Redox Chemistry in High-Capacity Layered-Oxide Electrodes[END_REF][START_REF] Sathiya | High Performance Li2Ru1-yMnyO3 (0.2 ≤ y ≤ 0.8) Cathode Materials for Rechargeable Lithium-Ion Batteries: Their Understanding[END_REF] Bernard Fraisse, [START_REF] Abakumov | Solid State Chemistry for Developing Better Metal-Ion Batteries[END_REF] François Fauth, [START_REF] Saubanère | An Intuitive and Efficient Method for Cell Voltage Prediction of Lithium and Sodium-Ion Batteries[END_REF] François Weill, [START_REF] Sathiya | Reversible Anionic Redox Chemistry in High-Capacity Layered-Oxide Electrodes[END_REF] Danielle Laurencin, [START_REF] Abakumov | Solid State Chemistry for Developing Better Metal-Ion Batteries[END_REF] Montse Casas-Cabanas, [START_REF] Delmas | The Layered Oxides in Lithium and Sodium-Ion Batteries: A Solid-State Chemistry Approach[END_REF][START_REF] Sathiya | High Performance Li2Ru1-yMnyO3 (0.2 ≤ y ≤ 0.8) Cathode Materials for Rechargeable Lithium-Ion Batteries: Their Understanding[END_REF][START_REF] Mccalla | Visualization of O-O Peroxo-like Dimers in High-Capacity Layered Oxides for Li-Ion Batteries[END_REF] Dany Carlier, [START_REF] Kubota | Electrochemistry and Solid-State Chemistry of NaMeO2 (Me = 3d Transition Metals)[END_REF][START_REF] Sathiya | Reversible Anionic Redox Chemistry in High-Capacity Layered-Oxide Electrodes[END_REF][START_REF] Sathiya | High Performance Li2Ru1-yMnyO3 (0.2 ≤ y ≤ 0.8) Cathode Materials for Rechargeable Lithium-Ion Batteries: Their Understanding[END_REF] Gwenaëlle Rousse, [START_REF] Kubota | Electrochemistry and Solid-State Chemistry of NaMeO2 (Me = 3d Transition Metals)[END_REF][START_REF] Pearce | Evidence for Anionic Redox Activity in a Tridimensional-Ordered Li-Rich Positive Electrode β-Li2IrO3[END_REF][START_REF] Yahia | Unified Picture of Anionic Redox in Li/Na-Ion Batteries[END_REF] Marie-Liesse Doublet [START_REF] Abakumov | Solid State Chemistry for Developing Better Metal-Ion Batteries[END_REF][START_REF] Kubota | Electrochemistry and Solid-State Chemistry of NaMeO2 (Me = 3d Transition Metals)[END_REF] 1 : ICGM, Univ Montpellier, CNRS, ENSCM, Montpellier, France 2 : Réseau sur le Stockage Électrochimique de l'Énergie (RS2E), CNRS, Amiens, France Dr. Romain Berthelot: romain.berthelot@umontpellier.fr Institut Charles Gerhardt de Montpellier, CC1502 -2 place Bataillon -34095 Montpellier, France

Introduction

Layered alkali transition metal oxides gather a wide range of chemical compositions and various correlated physical properties. Their crystallographic structure is usually described as a stacking of edge-shared MO6 octahedra layers with alkali ions being inserted in the interlayer space. Cationic substitutions within the transition metal layers are generally easy to achieve thanks to the very close ionic radii in octahedral symmetry of a wide range of elements (transition metals, and also s-and p-block elements).This chemical diversity offers a wonderful playground for chemists and physicists to design multi-metal layered oxides and to tune their electronic properties, as witnessed by the tremendous efforts devoted to studying this family of materials for electrochemical applications. [START_REF] Abakumov | Solid State Chemistry for Developing Better Metal-Ion Batteries[END_REF][START_REF] Kubota | Electrochemistry and Solid-State Chemistry of NaMeO2 (Me = 3d Transition Metals)[END_REF][START_REF] Delmas | The Layered Oxides in Lithium and Sodium-Ion Batteries: A Solid-State Chemistry Approach[END_REF] For particular substitutions leading to AxM2/3M'1/3O2 compositions wit M and M' being cations of different size, a honeycomb-like cationic ordering might appear within the transition metal layer, with each M'O6 octahedron being surrounded by six other MO6 octahedra. In this group of compositions, O3-type Li-rich layered compounds Li2MO3 (Li[Li1/3M2/3]O2) have been brought to light for promoting anionic redox chemistry in high-capacity electrodes. [START_REF] Sathiya | Reversible Anionic Redox Chemistry in High-Capacity Layered-Oxide Electrodes[END_REF][START_REF] Sathiya | High Performance Li2Ru1-yMnyO3 (0.2 ≤ y ≤ 0.8) Cathode Materials for Rechargeable Lithium-Ion Batteries: Their Understanding[END_REF][START_REF] Saubanère | An Intuitive and Efficient Method for Cell Voltage Prediction of Lithium and Sodium-Ion Batteries[END_REF][START_REF] Mccalla | Visualization of O-O Peroxo-like Dimers in High-Capacity Layered Oxides for Li-Ion Batteries[END_REF][START_REF] Pearce | Evidence for Anionic Redox Activity in a Tridimensional-Ordered Li-Rich Positive Electrode β-Li2IrO3[END_REF][START_REF] Yahia | Unified Picture of Anionic Redox in Li/Na-Ion Batteries[END_REF] The magnetic behavior of honeycomb-ordered layered compounds is also widely investigated. [START_REF] Kanyolo | Honeycomb Layered Oxides: Structure, Energy Storage, Transport, Topology and Relevant Insights[END_REF] In 2011, Evstigneeva and coworkers reported on the fast sodium ionic conductivity in the family of Na2M2TeO6 -Na2/3(M2/3Te1/3)O2compounds, with M being Ni, Co, Zn or Mg. [START_REF] Evstigneeva | A New Family of Fast Sodium Ion Conductors: Na2M2TeO6 (M = Ni, Co, Zn, Mg)[END_REF] Keeping the P2-type oxygen stacking [START_REF] Delmas | Structural Classification and Properties of the Layered Oxides[END_REF] and the in-plane Ni/Te honeycomb cationic ordering (Figure 1a), the structural difference between all the compositions lies on the relative position of the tellurium cations from one layer to the next (Figure 1b). For Na2Ni2TeO6, straight chains of tellurium and nickel are observed along the hexagonal chex.-axis, whereas "zigzag" tellurium sequences characterize the other compositions. The latter was confirmed by the group of Cava in a parallel study focused on Na2Co2TeO6. [START_REF] Viciu | Structure and Basic Magnetic Properties of the Honeycomb Lattice Compounds Na2Co2TeO6 and Na3Co2SbO6[END_REF] Nickel substitutions by other metals are possible through a solid-solution behavior and induce a progressive shift from one stacking sequence to the other. [START_REF] Berthelot | Studies on Solid Solutions Based on Layered Honeycomb-Ordered Phases P2-Na2M2TeO6 (M=Co, Ni, Zn)[END_REF] In the alkali interlayer space, various trigonal prismatic sites are possible for the sodium cations depending on whether the NaO6 prism shares faces with MO6 and/or TeO6 octahedra (Figure 1c). Na2Ni2TeO6 was also investigated as electrode material for Na-ion battery. [START_REF] Gupta | Na2Ni2TeO6: Evaluation as a Cathode for Sodium Battery[END_REF][START_REF] Bera | Temperature-Dependent Na-Ion Conduction and Its Pathways in the Crystal Structure of the Layered Battery Material Na2Ni2TeO6[END_REF] Since 2018, Masese and coworkers have been investigating the potassium layered analogs K2M2TeO6 and underlined their interesting electrochemical behavior in K-ion batteries. [START_REF] Masese | Rechargeable Potassium-Ion Batteries with Honeycomb-Layered Tellurates as High Voltage Cathodes and Fast Potassium-Ion Conductors[END_REF][START_REF] Masese | A High Voltage Honeycomb Layered Cathode Framework for Rechargeable Potassium-Ion Battery: P2-Type K2/3Ni1/3Co1/3Te1/3O2[END_REF][START_REF] Masese | Topological Defects and Unique Stacking Disorders in Honeycomb Layered Oxide K2Ni2TeO6 Nanomaterials: Implications for Rechargeable Batteries[END_REF] The crystallographic structure of K2Ni2TeO6 was solved using the Na2Ni2TeO6 structural model resulting in the expected increase of the cell parameters mainly induced by the thicker alkali interlayer, and in a significant difference in the potassium cations distribution (Table S1).

In the family of alkali transition metal layered oxides, there are only few examples of alkalimixed compositions. In 1994, Balsys and Davis reported on the LixNayCoO2 phase and described its layered structure as an alternation of O3-LiCoO2 and P2-NaxCoO2 blocks. [START_REF] Balsys | The Structure of Li0.43Na0.36CoO1.96 Using Neutron Powder Diffraction[END_REF] This unique OP4 stacking was later studied by some of us and others especially to prepare the O4type form of LiCoO2 by topotactic ionic exchange. [START_REF] Komaba | A New Polymorph of Layered LiCoO2[END_REF][START_REF] Berthelot | Synthesis and Investigations on an O4-LiCoO2 Polytype[END_REF][START_REF] Yabuuchi | A Comparative Study of LiCoO2 Polymorphs: Structural and Electrochemical Characterization of O2-, O3-, and O4-Type Phases[END_REF] Searching for other alkali-mixed compositions, in this work we investigated the solid-solid reaction between honeycomb layered oxides Na2Ni2TeO6 and K2Ni2TeO6. The formation of a new lamellar phase NaKNi2TeO6 is evidenced by powder diffraction techniques, coupled with [START_REF] Yabuuchi | A Comparative Study of LiCoO2 Polymorphs: Structural and Electrochemical Characterization of O2-, O3-, and O4-Type Phases[END_REF] Na solid-state nuclear magnetic resonance spectroscopy and density functional theory calculations. The structure of NaKNi2TeO6 exhibits an alternation of sodium and potassium blocks; however, the stacking disorder does not allow solving the complete crystallographic structure. Additionally, stacking faults simulations enabled to validate the presence of multiple possible stacking sequences in the powder material.

Experimental

Na2Ni2TeO6 and K2Ni2TeO6 precursors were prepared by conventional solid-state reaction, by adapting previously reported procedures. [START_REF] Evstigneeva | A New Family of Fast Sodium Ion Conductors: Na2M2TeO6 (M = Ni, Co, Zn, Mg)[END_REF][START_REF] Berthelot | Studies on Solid Solutions Based on Layered Honeycomb-Ordered Phases P2-Na2M2TeO6 (M=Co, Ni, Zn)[END_REF][START_REF] Masese | Rechargeable Potassium-Ion Batteries with Honeycomb-Layered Tellurates as High Voltage Cathodes and Fast Potassium-Ion Conductors[END_REF][START_REF] Masese | A High Voltage Honeycomb Layered Cathode Framework for Rechargeable Potassium-Ion Battery: P2-Type K2/3Ni1/3Co1/3Te1/3O2[END_REF][START_REF] Nalbandyan | Heterovalent Substitutions in Na2M2TeO6 Family: Crystal Structure, Fast Sodium Ion Conduction and Phase Transition of Na2LiFeTeO6[END_REF] Anhydrous alkali carbonates (Na2CO3, Sigma-Aldrich, ≥ 99.5 % or K2CO3, Sigma-Aldrich, ≥ 99.0 %), tellurium oxide (TeO2, Alfa-Aesar, 99.99%) and nickel oxide (NiO, Alfa-Aesar, 99 %) were mixed together, pelletized and then heated in a gold crucible. A nominal 10 wt.% excess of alkali carbonates was used to balance the high-temperature loss of alkali oxides. In the literature, Na2Ni2TeO6 and K2Ni2TeO6 are usually prepared at 900 and 800 °C. As our attempts to prepare K2Ni2TeO6 at 900 °C were unsuccessful, we choose to prepare the precursors at 800 °C (9 hours in air with heating rate of 2.5 °C/min). To synthetize NaKNi2TeO6, stoichiometric blends of the as-prepared precursors were then heated under five various conditions of dwell temperatures and durations (9h at 200, 400, 600 or 800 °C, and also 4 days at 800 °C. To avoid any moisture contamination, sample powders were taken out of the furnace at 150 °C on cooling and transferred to an argon-filled glovebox.

Laboratory X-ray powder diffraction (XRPD) analyses were performed with a PANalytical X'Pert θ/θ diffractometer with Cu Kα radiation in an airtight sample holder for preliminary profile matching refinement. The powder sample of NaKNi2TeO6 obtained after 9 hours at 800 °C was also analyzed by high angular resolution synchrotron X-ray powder diffraction (SXRPD) collected on BL04_MSPD beamline of the ALBA synchrotron (Cerdanyola del Vallès, Spain). [START_REF] Fauth | The New Material Science Powder Diffraction Beamline at ALBA Synchrotron[END_REF][START_REF] Fauth | The Crystallography Stations at the Alba Synchrotron[END_REF] Here, the sample powder was enclosed in a 0.7 mm diameter quartz capillary sealed in argon atmosphere, and data were collected in transmission geometry at a wavelength of λ = 0.49589 Å. Profile matching refinement and Rietveld attempts were carried out using the FullProf suite. [START_REF] Rodríguez-Carvajal | Recent Advances in Magnetic Structure Determination by Neutron Powder Diffraction[END_REF] Electron diffraction was performed on the same powder. Patterns were obtained using a JEOL 2100 transmission electron microscope equipped with an Orius 200D camera from GATAN. Prior to the observation, the sample was crushed in alcohol and a drop of this suspension was deposited on a carbon-supported film on a copper grid. [START_REF] Yabuuchi | A Comparative Study of LiCoO2 Polymorphs: Structural and Electrochemical Characterization of O2-, O3-, and O4-Type Phases[END_REF] Na solid-state nuclear magnetic resonance spectroscopy (NMR) was performed on powder samples of Na2Ni2TeO6 and NaKNi2TeO6 using a Bruker Avance spectrometer at 79.4 MHz (7.05 T magnet). The samples are introduced in 2.5 mm zirconia rotors inside a glovebox. [START_REF] Yabuuchi | A Comparative Study of LiCoO2 Polymorphs: Structural and Electrochemical Characterization of O2-, O3-, and O4-Type Phases[END_REF] Na Magic Angle Spinning Nuclear Magnetic Resonance (MAS-NMR) experiments were carried out using a 30 kHz spinning frequency. The spectral width was set to 1 MHz, and the recycle time to D0 = 0.2 s, which was long enough to avoid the T1 saturation effects. A one rotor period synchronized Hahn echo sequence (π/2-delay-π) with a π/2 = 2 µs pulse duration was used.

The center of the experimental window was set to 680 ppm, in the region of the maximum intensities observed, with 0.1 M NaCl aqueous solution as external reference. Attempts to record [START_REF] Dudarev | Electron-Energy-Loss Spectra and the Structural Stability of Nickel Oxide: An LSDA+U Study[END_REF] K solid-state NMR spectra on both K2Ni2TeO6 and NaKNi2TeO6 were performed at 9.4 and 14.1 T. However, due to the severe acoustic ringing of the probes used at the very low resonance frequency of [START_REF] Dudarev | Electron-Energy-Loss Spectra and the Structural Stability of Nickel Oxide: An LSDA+U Study[END_REF] K, and to the observation of background [START_REF] Dudarev | Electron-Energy-Loss Spectra and the Structural Stability of Nickel Oxide: An LSDA+U Study[END_REF] K resonances, no exploitable data could be obtained even after several days of acquisition.

DIFFax and FAULTS programs [START_REF] Treacy | A General Recursion Method for Calculating Diffracted Intensities from Crystals Containing Planar Faults[END_REF][START_REF] Casas-Cabanas | A Program for Refinement of Structures with Extended Defects[END_REF] have been widely used to better understand stacking faults of alkali transition metal oxides. [START_REF] Casas-Cabanas | Microstructural Characterisation of Battery Materials Using Powder Diffraction Data: DIFFaX, FAULTS and SH-FullProf Approaches[END_REF][START_REF] Boulineau | Structure of Li2MnO3 with Different Degrees of Defects[END_REF][START_REF] Boulineau | Thermal Stability of Li2MnO3: From Localized Defects to the Spinel Phase[END_REF][START_REF] Serrano-Sevillano | Enhanced Electrochemical Performance of Li-Rich Cathode Materials through Microstructural Control[END_REF][START_REF] Bianchini | From LiNiO2 to Li2NiO3: Synthesis, Structures and Electrochemical Mechanisms in Li-Rich Nickel Oxides[END_REF][START_REF] Shunmugasundaram | A Study of Stacking Faults and Superlattice Ordering in Some Li-Rich Layered Transition Metal Oxide Positive Electrode Materials[END_REF][START_REF] Mortemard De Boisse | Coulombic Self-Ordering upon Charging a Large-Capacity Layered Cathode Material for Rechargeable Batteries[END_REF] In this work, they were used to build up various stacking sequences of NaKNi2TeO6 with a perfect Na/K alternation, and then stacking faults could be generated by statistical distribution of these stacking configurations. In doing so, X-ray diffraction powder patterns could be simulated and then compared with experimental ones.

Practically, sodium and potassium blockswith a honeycomb Ni/Te slab and an alkali layer abovewere created from the structural models of Na2Ni2TeO6 and K2Ni2TeO6, as well as different stacking vectors to alternately superimpose the blocks (see Supporting Information section for details).

DFT calculations including full structural relaxations were performed for Na2Ni2TeO6 with the crystallographic structure provided by Evstigneeva et al. [START_REF] Evstigneeva | A New Family of Fast Sodium Ion Conductors: Na2M2TeO6 (M = Ni, Co, Zn, Mg)[END_REF] as the starting model. Structures were built up with various sodium ion distributions to account for their multiple environments (edge-sharing and/or face-sharing trigonal prisms, Figure S1). Additional structures were then generated with "zigzag" tellurium sequences by substituting one nickel for tellurium cations at z = 0.5. In the crystallographic structures of Na2Ni2TeO6 and K2Ni2TeO6 (Table S1), the alkali positions are the same and the only difference is the occupancy of the three alkali sites. It was thus assumed that no particular alkali/vacancy ordering could be formed and consequently K2Ni2TeO6 input structures were obtained by replacing sodium by potassium cations from all the previously generated Na2Ni2TeO6 configurations.

Secondly, random intralayer alkali distributions were considered for different Na2-xKxNi2TeO6 compositions assuming a theoretical solid-solution behavior, i.e. sodium to potassium substitution within the same alkali layer, with an equivalent degree of substitution for each layer. To do so, 121 (236 atoms) and 221 (472 atoms) supercells were necessary.

Lastly, for the particular NaKNi2TeO6 compositions, the perfect alternation of pure sodium and pure potassium layers was compared to configurations with a progressive random Na/K interlayer mixing.

All these crystallographic structures were relaxed using the VASP program package 37 using PAW pseudopotentials [START_REF] Kresse | From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method[END_REF] for the atomic description and the rotationally invariant DFT+U formalism of Dudarev [START_REF] Dudarev | Electron-Energy-Loss Spectra and the Structural Stability of Nickel Oxide: An LSDA+U Study[END_REF] within the PBE generalized gradient approximation [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] for the exchange and correlation potential and Ueff = 6.2 eV for Ni. The convergence criteria were set with atomic forces smaller than 2.10 -3 eV/Å -1 , an energy cut-off of 600 eV for the plane-wave basis set and a k-point grid density for Brillouin zone integration of at least 1000/at. The relative energies of all these structures are given here in meV per A2Ni2TeO6.

The Fermi contact shifts were obtained from the computed isotropic hyperfine coupling parameters, as discussed in our previous works. [START_REF] Castets | Multinuclear NMR and DFT Calculations on the LiFePO4.OH and FePO4.H2O Homeotypic Phases[END_REF][START_REF] Nguyen | Density Functional Theory-Assisted 31 P and 23 Na Magic-Angle Spinning Nuclear Magnetic Resonance Study of the Na3V2(PO4)2F3-Na3V2(PO4)2FO2 Solid Solution: Unraveling Its Local and Electronic Structures[END_REF][START_REF] Serrano-Sevillano | DFT-Assisted Solid-State NMR Characterization of Defects in Li2MnO3[END_REF][START_REF] Bamine | Understanding Local Defects in Li-Ion Battery Electrodes through Combined DFT/NMR Studies: Application to LiVPO4F[END_REF] The Curie temperature was taken from a previous measurement for Na2Ni2TeO6 (34 K) [START_REF] Berthelot | Studies on Solid Solutions Based on Layered Honeycomb-Ordered Phases P2-Na2M2TeO6 (M=Co, Ni, Zn)[END_REF] and it was assumed to be the same for all compounds. The theoretical value of molar magnetic susceptibility calculated by Curie's law was used for the Fermi contact shift calculations. The temperature at which the Fermi contact shifts were calculated, was chosen to be 320 K, which is the approximate temperature inside the rotor at a spinning rate of 30 kHz.

Results

The synthesis of P2-type Na2Ni2TeO6 and K2Ni2TeO6 precursors is first verified by laboratory X-ray powder diffraction (Figure S2). For the potassium layered compound, the pattern is in good agreement with the literature [START_REF] Masese | Rechargeable Potassium-Ion Batteries with Honeycomb-Layered Tellurates as High Voltage Cathodes and Fast Potassium-Ion Conductors[END_REF] and can be indexed with the P63/mcm space group, indicating straight chains of tellurium cations. For the sodium analog, with the above-mentioned synthesis conditions the P6322 space group appears more suitable to index the pattern, showing here that the "zigzag" sequence is preferred like in Na2Co2TeO6. [START_REF] Viciu | Structure and Basic Magnetic Properties of the Honeycomb Lattice Compounds Na2Co2TeO6 and Na3Co2SbO6[END_REF] Figure 2 gathers the XRPD patterns collected after heating an equimolar mixture of Na2Ni2TeO6 and K2Ni2TeO6 in different heating conditions. After 9 hours at 200 °C, the main observation is the decrease of the (002) peaks of the layered precursors and the in-between growth of an intense peak at an averaged d-space value. As no more trace of the precursors is visible for the syntheses performed at 400 °C and above, the reaction could be considered complete.

Focusing now on the details of the XRPD patterns, the appearance of a superstructure peak at 2θ = 7 ° corresponds to a d-space value twice larger than the one of the main peak at 2θ = 15 °.

A doubled periodicity along the chex.-axis is logically suggested, a could be explained by an alternation of sodium and potassium layers as it has been previously observed in OP4-LixNayCoO2 phase. [START_REF] Balsys | The Structure of Li0.43Na0.36CoO1.96 Using Neutron Powder Diffraction[END_REF][START_REF] Ren | Enhanced Thermopower in an Intergrowth Cobalt Oxide Li 0.48Na0.35CoO2[END_REF][START_REF] Berthelot | Reinvestigation of the OP4-(Li/Na)CoO2-Layered System and First Evidence of the (Li/Na/Na)CoO2 Phase with OPP9 Oxygen Stacking[END_REF] Then it is important to observe the changes in the range 2θ = 18-29 ° with various heating temperatures (inset of Figure 2). Logically, there is a progressive increase in crystallinity with both the temperature and the duration of the heat treatment. Interestingly the relative intensity of the two close peaks around 19.5 ° evolves with the synthesis conditions.

With higher temperature and longer heating, the second peak becomes more important. This evolution calls for a change of the stacking sequence of the honeycomb layer by analogy with previous works on P2-type A2M2TeO6 compounds [START_REF] Evstigneeva | A New Family of Fast Sodium Ion Conductors: Na2M2TeO6 (M = Ni, Co, Zn, Mg)[END_REF][START_REF] Berthelot | Studies on Solid Solutions Based on Layered Honeycomb-Ordered Phases P2-Na2M2TeO6 (M=Co, Ni, Zn)[END_REF] Overall, from the XRPD patterns it seems reasonable to assume the formation a new layered compound NaKNi2TeO6 with alternated sodium and potassium layers. In order to obtain other Na2-xKxNi2TeO6 compositions, the initial proportions of the layered precursors have been tuned.

However, in all our attempts a biphasic product was finally obtained (NaKNi2TeO6 and Na2Ni2TeO6 or K2Ni2TeO6 whether the nominal mixture was sodium-rich or potassium rich, respectively).

Figure 2: XRPD patterns of Na2Ni2TeO6 and K2Ni2TeO6 (black lines) and obtained after heating stoichiometric mixtures in different conditions (colored lines). Intensities are normalized for a better comparison. Symbol (*) indicates the superstructure peak at low angle accounting for the alternation of sodium and potassium blocks. The inset highlights the other superstructure peaks, with a dashed rectangle focusing on the intensity evolution.

The NaKNi2TeO6 powder sample obtained after heating 9 hours at 800 °C was analyzed using SXRPD, so to principally enable full structure resolution. The smallest hexagonal cell allowing the indexation of both the main and the superstructure peaks has lattice parameters a = 5.225(1) Å and c = 35.403(1) Å (Figure S3). These lattice parameters correspond to an average of those of Na2Ni2TeO6 and K2Ni2TeO6, with however a tripled chex.-axis cell parameter (6 layers), likely indicating a complex stacking ordering. However, our attempts to build a complete structural description accounting for all superstructure peaks' intensities were unsuccessful, whatever the space group and atomic distributions we considered. To understand why no meaningful structural model enables a good Rietveld refinement of the SXRPD pattern, electron diffraction was carried out on the same sample (Figure S4). The main spots can be indexed in a hexagonal cell with parameters values a = 5.22 Å and c = 11.8 Å. Additional weak reflections are also observed and need a 2  a and 3  c superstructure. The values of a and 3  c are in line with the SXRPD refinement, but the 2  a superstructure is not explained and thus confirms the complexity of this crystallographic structure. Note that very recently a similar 2  a superstructure was also proposed by high-resolution TEM for Na2Ni2TeO6. [START_REF] Masese | Unveiling Structural Disorders in Honeycomb Layered Oxide: Na2Ni2TeO6[END_REF] Additionally, [START_REF] Yabuuchi | A Comparative Study of LiCoO2 Polymorphs: Structural and Electrochemical Characterization of O2-, O3-, and O4-Type Phases[END_REF] Na solid-state NMR analyses were performed on the Na2Ni2TeO6 and NaKNi2TeO6 in order to get more insight into the local structure of the sodium ions. This technique is extensively being used to investigate alkali transition metal layered oxides and other battery electrode materials, in order to specifically determine the insertion site of the alkali ions during reversible electrochemical cycling. [START_REF] Grey | Lithium MAS NMR Studies of Cathode Materials for Lithium-Ion Batteries[END_REF][START_REF] Pecher | Methods: NMR in Battery Research[END_REF] As shown in Figure 3, the NMR spectrum of NaNi2TeO6 shows a major isotropic peak at δ = 677 ppm. Although the sodium ions occupy different sites (Figure 1c), their high mobility results in an averaged unique peak as it has been already observed in other P2-and P3-type sodium transition metal oxides. [START_REF] Kalapsazova | Combined Use of EPR and 23 Na MAS NMR Spectroscopy for Assessing the Properties of the Mixed Cobalt-Nickel-Manganese Layers of: P3-NayCo1-2xNixMnxO2[END_REF][START_REF] Carlier | Sodium Ion Mobility in NaxCoO2 (0.6 < x < 0.75) Cobaltites Studied by 23Na MAS NMR[END_REF][START_REF] Carlier | The P2-Na2/3Co2/3Mn1/3O2 phase: Structure, Physical Properties and Electrochemical Behavior as Positive Electrode in Sodium Battery[END_REF][START_REF] Blangero | High-Temperature Phase Transition in the Three-Layered Sodium Cobaltite P'3-NaxCoO2 (x~0.62)[END_REF] A unique isotropic signal located at δ = 825 ppm was observed for NaKNi2TeO6. The presence of a single resonance for NaKNi2TeO6 confirms that the reaction leading to the alkali-mixed compositions is indeed complete (absence of residual reagent signature at 677 ppm), and that no crystalline or amorphous by-products were formed.

Discussion

Heating a mixture of Na2Ni2TeO6 and K2Ni2TeO6 leads to a new phase NaKNi2TeO6. Thanks to XRPD, an alternation of sodium and potassium layers is proposed. Very recently the group of Masese investigated the same compound and confirmed the ordering of the alkali layers with high-resolution TEM. [START_REF] Masese | Mixed Alkali-Ion Transport and Storage in Atomic-Disordered Honeycomb Layered NaKNi2TeO6[END_REF] The section hereafter aims at going further into the details of this layered structure by confronting experimental results (powder diffraction and solid-state NMR spectra) with a theoretical approach based on DFT calculations and stacking faults simulations.

First-principles DFT calculations were first performed on the Na2Ni2TeO6 and K2Ni2TeO6. In both cases, the most stable configurations are obtained when AO6 prisms only share edges with the above and below NiO6 and TeO6 octahedra. Structures with a fraction of prisms sharing faces are systematically less stable, in particular when TeO6 octahedra are facing (Figure S5).

Focusing on configurations with AO6 prisms sharing edges only, structures with straight sequences of tellurium are favored with, however, negligible energy stabilization (within the thermal activation energy) compared to the "zigzag" sequences. This suggests no impact of the Ni/Te interlayer sequence on the edge-sharing preference of the AO6 prisms which allows restricting the number of DFT calculations for alkali-mixed compositions to configurations with only edge-sharing AO6 prisms and straight sequences of tellurium cations.

In Figure 4, the formation energy of the relaxed configurations Na2-xKxNi2TeO6 (0 ≤ x ≤ 2) is plotted with respect to the proportional mixture of Na2Ni2TeO6 and K2Ni2TeO6 (considering their most stable configurations). The positive bell-shape curve indicates a significant energy destabilization, even for limited amount of substitution (from either the sodium or the potassium analog). The analysis of the relaxed structures reveals that random Na + /K + distributions within the same alkali layer induces significant distortions of the NaO6 octahedra, and to a lower extent the KO6 octahedra, while the local structure of Ni and Te are not altered. Hence, the difference in Na + and K + ionic radii induces local mechanical stress which results in a volume expansion of the alkali layer, therefore penalizing the interlayer electrostatic energy (Figure S6). In contrast, structures in which Na and K lie in distinct layers are preferred and any mixing leads to a loss of electrostatic interactions (Figure S7). This interpretation agrees with the similar line width of the NMR spectra of both Na2Ni2TeO6 and NaKNi2TeO6. Indeed, the hypothesis of simultaneous occupation of both sodium and potassium cations in the same layer would have induced more local distortions and sodium environment distributions and consequently a larger width, as observed for O3-type Li3-xNaxNi2SbO6 solid-solution. [START_REF] Vallée | Alkali-Glass Behavior in Honeycomb-Type Layered Li3-xNaxNi2SbO6 Solid Solution[END_REF] Interestingly, no quenching at the end of the thermal treatment is necessary to avoid phase segregation in contrast to what is observed for the alkali-ordered OP4-LixNayCoO2 under slow cooling procedure. [START_REF] Berthelot | Reinvestigation of the OP4-(Li/Na)CoO2-Layered System and First Evidence of the (Li/Na/Na)CoO2 Phase with OPP9 Oxygen Stacking[END_REF] , NaKNi2TeO6 can therefore be considered as thermodynamically stable with respect to segregation. This experimental fact is consistent with the phase stability diagram of Figure 4 showing that structures with alternating Na and K layers lie only ~10meV/f.u. above the segregation reference. Although positive, such a small formation energy can easily switch to negative values once entropy contributions are considered. Indeed, configuration entropy is expected to be larger in alkali-mixed compositions (Na/K) than single-alkali compositions (Na or K), therefore stabilizing the former over the latter. It is also important to remind here that different occupancies of the three available alkali cations sites are reported from the structural description of Na2Ni2TeO6 and K2Ni2TeO6 (Table S1). As the heat treatment leading to the mixed NaKNi2TeO6 could be seen as a second sintering of the precursors, it is reasonable to assume a higher fraction of alkali ions in the most stable sites (edge-sharing prisms) in NaKNi2TeO6 compared to the Na2Ni2TeO6 and K2Ni2TeO6 precursors Accordingly, the Na2Ni2TeO6 and K2Ni2TeO6 edge-sharing configurations taken as references in the phase stability diagram are artificially stabilized with respect to mixed edge-face-sharing configurations, as corroborated by our DFT calculations (see Figure S5). Raising up the reference energy of the phase stability diagram of Figure 4 by considering configurations with fraction of face sharing alkali sites should therefore contribute to stabilizing the alkali-mixed NaKNi2TeO6 phase with respect to segregation. Once the alternation of sodium and potassium layers is validated, the next question lies on the thicknesses of these layers in NaKNi2TeO6. In the work of Masese et al., the alkali layers thickness in NaKNi2TeO6 is found similar to those in Na2Ni2TeO6 and K2Ni2TeO6. [START_REF] Masese | Mixed Alkali-Ion Transport and Storage in Atomic-Disordered Honeycomb Layered NaKNi2TeO6[END_REF] Such observation appears not in line with the [START_REF] Yabuuchi | A Comparative Study of LiCoO2 Polymorphs: Structural and Electrochemical Characterization of O2-, O3-, and O4-Type Phases[END_REF] Na NMR in Figure 3, for which a shift of the major isotropic peak is observed (from 667 to 835 ppm). To better understand this significant difference, theoretical NMR shifts were calculated for particular configurations after DFT structural relaxations (Tables S2 andS3). For Na2Ni2TeO6 a higher NMR shift is observed when NaO6 prisms share only edges with NiO6 and TeO6 octahedra. The stacking sequence, i.e. the relative position of tellurium cations along the chex.-axis, plays a secondary role and the "zigzag" configuration also shifts the NMR peaks to higher values. Regarding NaKNi2TeO6, it appears that the introduction of potassium layers does not significantly affect the local environment of the sodium ions, as similar simulated peak shift values are obtained for both compounds. In DFT calculations, logically thicker alkali layers are observed for configurations with more face-sharing sites (Figure S8). Keeping in mind the strong ionic mobility which averages the experimental NMR shift and following our above hypothesis regarding the relative stability of NaKNi2TeO6, the higher peak shift in NaKNi2TeO6 could be related to a higher proportion of edge-sharing NaO6 prisms than in Na2Ni2TeO6.

Although the above sections clearly help to better picture the structure of NaKNi2TeO6, the analysis of the diffraction patterns is still challenging. In the aperiodic arrangement proposed from high-resolution TEM, honeycomb layers are perfectly superimposed or glided in random directions through each potassium or sodium layers, respectively. [START_REF] Masese | Mixed Alkali-Ion Transport and Storage in Atomic-Disordered Honeycomb Layered NaKNi2TeO6[END_REF] As shown in Figure S5, energy differences between Na2Ni2TeO6 with straight and "zigzag" sequences of tellurium falls within the thermal activation energy. It is the same for K2Ni2TeO6, and therefore it is reasonable to assume that both stacking sequences are actually possible for NaKNi2TeO6. It is thus necessary to evaluate the influence of the successive position of Ni/Te honeycomb layers in NaKNi2TeO6 on the diffraction patterns. To do so, we carried out the simulation of the NaKNi2TeO6 structure using the FAULTS program to evaluate how different stacking configurations could affect the XRPD patterns. Two building blocks were generated with either a sodium or a potassium layer on top of a Ni/Te honeycomb sheet (Figure S9), while different vectors and stacking probabilities were used to simulate patterns with different gliding in their stacking sequences (Figure S10). Owing to the 6-fold symmetry of the honeycomb layer, five distinct stacking models are possible, with or without gliding through sodium and/or potassium layers (Figure S11). The corresponding simulated XRPD patterns are shown in Figures S12 to S16, without or with 10 % of stacking faults. Comparing to experimental XRPD patterns, the combination of two stacking models is the only way to fully assign each peak, especially the two just around 19.5 ° (Figure 5). The first model exhibits a straight stacking, for both the potassium and the sodium layers, with a 10 % of faults for the sodium layer (Figure S12). The second is characterized by a straight sequence through the potassium layer and a "zigzag" one through the sodium layers (Figures S14). Interestingly, no model with block gliding through potassium layers is required, in agreement with the aperiodic stacking already reported. [START_REF] Masese | Mixed Alkali-Ion Transport and Storage in Atomic-Disordered Honeycomb Layered NaKNi2TeO6[END_REF] It is also in line with the different stacking sequences observed for Na2Ni2TeO6 and K2Ni2TeO6 synthesized at 800 °C (Figure S2). After the crystallographic structure, the discussion also needs to focus on the formation of NaKNi2TeO6. From the Figure 2, solid-solid reaction between Na2Ni2TeO6 and K2Ni2TeO6 precursors occurs at very moderate temperatures. This in sharp contrast with other alkali-mixed compositions. Indeed, OP4-LixNayCoO2 is formed above 800 °C from a mixture of O3-LiCoO2 and P2-NaxCoO2, [START_REF] Berthelot | Reinvestigation of the OP4-(Li/Na)CoO2-Layered System and First Evidence of the (Li/Na/Na)CoO2 Phase with OPP9 Oxygen Stacking[END_REF] and that a similarly elevated temperature range is needed to obtain O3-Li3-xNaxNi2SbO6.

56 P2-type A2M2TeO6 oxides have been recently investigated for their high ionic conductivities, around 4-11 S/m at 300 °C, [START_REF] Evstigneeva | A New Family of Fast Sodium Ion Conductors: Na2M2TeO6 (M = Ni, Co, Zn, Mg)[END_REF][START_REF] Masese | Rechargeable Potassium-Ion Batteries with Honeycomb-Layered Tellurates as High Voltage Cathodes and Fast Potassium-Ion Conductors[END_REF] thanks to the high mobility of the alkali ions within the alkali layers. This certainly plays a key role in the formation of NaKNi2TeO6 and strongly suggests a topotactic and simultaneous Na + /K + and K + /Na + ionic exchanges within the alkali layers, although such mixed-alkali occupancy is considered unstable by DFT calculations.

Conclusion

Alkali-mixed layered oxide NaKNi2TeO6 could be obtained by solid-solid reaction between analogs Na2Ni2TeO6 and K2Ni2TeO6. From the structural point of view, the alternation of sodium and potassium layers along the chex.-axis is validated through the combination of XRD, solid-state NMR and DFT calculations. Thanks to stacking faults simulations, it seems clear that multiple stacking sequences occur simultaneously in NaKNi2TeO6.

The fact that the experimental patterns result as a combination of both structural models indicates that the areas exhibiting different stacking sequences cover a significant length scale.

They could be present in different crystallites, or as intergrowth within the same crystallites.

Their relative amounts evolves with the synthesis conditions (temperature and duration).

Obtaining single crystals could be a solution to get a fine crystallographic description and to eventually validate the hypothesis of a higher fraction of edge-sharing alkali sites in NaKNi2TeO6 proposed by both DFT calculations and solid-state NMR. Contrary to the OP4-LixNayCoO2 phase, here it seems achievable as slow cooling allows the formation of NaKNi2TeO6.

The application of NaKNi2TeO6 as electrode material with liquid NaK negative electrode has been recently proposed. [START_REF] Masese | Mixed Alkali-Ion Transport and Storage in Atomic-Disordered Honeycomb Layered NaKNi2TeO6[END_REF] However, the presence of tellurium cations significantly limits the specific capacity. The family of alkali transition metal compounds is already important, but the report of NaKNi2TeO6 confirms that new compositions with alkali mixing and/or ordering could be still discovered, with eventually promising physical features and interesting applications.

Acknowledgement

3 : 8 :

 38 CIC energiGUNE, Parque Tecnológico de Álava, C/Albert Einstein 48, 01510 Miñano, Álava Spain 4 : CNRS, Bordeaux INP, ICMCB UMR5026, Université Bordeaux, F-33600 Pessac France 5 : Alistore-European Research Institute, CNRS, Amiens, 80000, France 6 : CELLS -ALBA Synchrotron 08290 Cerdanyola del Vallès, Barcelona, Spain 7 : IKERBASQUE, Basque Foundation for Science, María Díaz de Haro 3, 48013, Bilbao, Spain. Chimie du Solide et Energie, UMR8260, Collège de France, 11 place Marcelin Berthelot, 75231 Paris, France 9 : Sorbonne Université, 4 place Jussieu, F-75005 Paris, France Corresponding author:

Figure 1 :

 1 Figure 1: Crystallographic features of P2-type A2M2TeO6 layered compounds: (a) top view of the honeycomb cationic ordering between nickel and tellurium cations (in blue and yellow, respectively), (b) perspective view of the two possible stacking sequences according to the sequence of the tellurium cations in dashed red lines, with alkali ions omitted for clarity, (c) possible AO6 trigonal prismatic sites (in grey) corresponding to each stacking, sharing faces with MO6 only, TeO6 only, MO6 and TeO6, or sharing edges. In layered A2M2TeO6 compounds, the space groups corresponding to the straight and "zigzag" tellurium sequences are P63/mcm and P6322, respectively.

Figure 3 :

 3 Figure3:[START_REF] Yabuuchi | A Comparative Study of LiCoO2 Polymorphs: Structural and Electrochemical Characterization of O2-, O3-, and O4-Type Phases[END_REF] Na MAS-NMR spectra of NaKNi2TeO6 (top, in red) and Na2Ni2TeO6 (bottom, in black), with spinning sidebands (*). For Na2Ni2TeO6, weak signals are also observed in some batches (¤) and could be assigned to sodium cations in the honeycomb layer, such as in the layered compound O'3-Na3Ni1.5TeO6.[START_REF] Grundish | Structural and Electrochemical Consequences of Sodium in the Transition-Metal Layer of O′3-Na3Ni1.5TeO6[END_REF] possible sodium cations present in the honeycomb layer. Spectra are recorded at 7.05 T using a Hahn-Echo sequence synchronized with one rotor rotation period (vr = 30 kHz).

Figure 4 :

 4 Figure 4:Phase stability diagram of the Na2-xKxNi2TeO6 system with respect to the proportional mixture of Na2Ni2TeO6 and K2Ni2TeO6 (dash blue line), as obtained from DFT+U calculations and showing a significant energy destabilization for random distributions of sodium and potassium within the same alkali layer, either assuming a solid-solution behavior Na2-xKxNi2TeO6 with progressive Na/K intralayer mixing (blue circles) or focusing on a fixed NaKNi2TeO6 composition with a progressive Na/K interlayer mixing (black circles). Only configurations with edge-sharing AO6 prisms are considered for all the structure relaxation.

Figure 5 :

 5 Figure 5: Comparison between experimental SXRPD pattern and the simulated ones for stacking models #1 with 10% of stacking faults (blue line) and #3 without faults(red line). The vertical bars on the top panel highlight the peaks assigned to one model or the other. The ideal

  Matthieu Saubanère (ICGM) is acknowledged for fruitful discussion. Paula Sanz-Camacho (ICMCB) is acknowledged for technical support with the 23 Na NMR acquisitions. BL04-MSPD staff of CELLS-ALBA synchrotron is acknowledged for granting beamtime through InHouse quota (proposal 2020014011) Supporting information Crystallographic features and XRPD patterns of Na2Ni2TeO6 and K2Ni2TeO6, SXRPD pattern and TEM image of NaKNi2TeO6, additional observations from the DFT calculations, general methodology for the stacking faults simulation in NaKNi2TeO6.

TOC Synopsis

Layered oxide NaKNi2TeO6 is obtained from a stoichiometric mixture of honeycomb layered precursors Na2Ni2TeO6 and K2Ni2TeO6 and characterized by X-ray diffraction and solid-state NMR. The structure of NaKNi2TeO6 exhibits an alternation of sodium and potassium layers, and DFT calculations rationalize the existence of this cationic ordering versus phase segregation. An aperiodic stacking prevents from a detailed crystallographic description, but stacking faults simulations reveal multiple possible stacking sequences in the powder material.


Changyong Lu

Tae-Hyun Kim

Jesper Bendix

Mustapha Abdelmoula

Christian Ruby

Ulla Gro Nielsen

Hans Chr

Bruun Hansen

Ulla Gro

Hans Chr Bruun Hansen

Stability of magnetic LDH composites used for phosphate recovery

Keywords: phosphate removal, pH effect, phosphate concentration, LDH dissolution, ferrihydrite, precipitation

Introduction

Phosphorus in the form of orthophosphate is a limited resource which is mainly used as fertilizer and critical for feeding the world population [START_REF] Wyant | Phosphorus, food, and our future[END_REF][START_REF] Valsami-Jones | Phosphorus in Environmental Technology: Principles and Applications[END_REF][START_REF] Lal | Soil phosphorus[END_REF]. Phosphorus use is highly unsustainable with substantial parts of phosphorus wasted and going out of circulation [START_REF] Scholz | [END_REF], and at the same time phosphorus overuse and poor control via waste streams have caused widespread eutrophication. The presence of trace amounts of phosphate in discharged wastewater may trigger eutrophication, leading to poor water quality, toxic algal blooms, and ecosystem destabilization [START_REF] Ashley | [END_REF]. Toxic algal blooms leads to accumulation of toxic cyanopeptides produced by cyanobacteria [22]. Thus, phosphorus emission reduction and management is one of the UN sustainability goals [START_REF]Sustainable Phosphorus Use[END_REF]. The EU Water Framework defines a cut-off total P concentration for lake eutrophication at 0.1 mg P/L, while WHO set this threshold concentration at 0.05 mg P/L (here the phosphate concentration is given in terms of mass of P (in phosphate) per liter [23,[START_REF] Bartram | Toxic cyanobacteria in water: a guide to their public health consequences, monitoring and management[END_REF]. Thus, development of a phosphorus capture and recovery technology has high priority.

Phosphate removal from wastewater by adsorption is an alternative to chemical precipitation and biological treatment methods. Adsorption processes are often easier to control, faster, cheaper and scalable compared with often more complex precipitation and biological methods [START_REF] Othman | [END_REF][26][27][28][29][START_REF] Ohtake | Phosphorus recovery and recycling[END_REF]. Layered double hydroxide (LDH) with the general formula of III and A n-are di-and trivalent metal cations and n -valent anion respectively [START_REF] Mishra | [END_REF], have been intensively investigated as P i sorbents because of their anion exchange property, high sorption capacity and fast sorption kinetics [5][6][7]. In general, P i sorption to LDH mainly takes part via electrostatic interactions, interlayer anion exchange, and surface complex formation [8][9][10][11][12]. P i could also be removed via precipitation, e.g. with use of CaFe and CaAl LDHs [9]. LDH and its composites have high P i sorption capacity, ranging from 50 mg P/g to 100 mg P/g, at weak acid (pH 6) to intermediate pH (pH 7~8) [13]. On the contrary, they have low P i sorption capacity at high pH (pH 9 ~ 10), typically lower than 5 mg P/g (Table S1). Moreover, the type of interlayer ion may affect the sorption capacity. For example carbonate and sulfate decrease the adsorption capacity compared with nitrate and chloride due to competitive ion-exchange [14]. The lamellar structure and the sub-to micrometer size of LDH particles make them difficult to be used in packed filter columns and even to collect by filtration or sedimentation [9]. However, by attaching the LDHs to magnetic nanoparticles such as magnetite, the magnetic composites can be separated in a magnetic field and thus be regenerated after P i sorption [15]. Examples of such magnetic LDH P sorbents comprise magnetic nanoparticle decorated ZnFeZr LDH [15], MgAl LDH [16] and MgFeZr LDH [17] composites (Table S1).

LDHs are known to buffer pH due to dissolution in water with release of hydroxide (and metal) ions [18]: Equation 1Note, that the trivalent metal ions which are produced would form (hydr)oxide precipitates due to the negligible solubility of M b III hydroxides/oxides. The extent and rate of dissolution of the LDH is controlled by the nature of the divalent metal cations and the interlayer anions [START_REF] Jobbágy | [END_REF][32][33]. For example, the LDHs are strongly thermodynamically favored over the separate M a hydroxides in the order Mg 2+ < Mn 2+ < Co 2+ ≈ Ni 2+ < Zn 2+ , when Cl -is the interlayer anion [32].

Environmentally benign materials are required for water cleaning. This limits the suitable LDH candidates for wastewater treatment to MgAl, MgFe, CaAl, CaFe and Fe(II)Fe(III)

LDH [35]. Despite a vast number of publications on LDH P i sorbents, the stability of LDHs for practical applications has not been evaluated. Most waste waters have pH in the range from 6.5 to 8.5 [34], which clearly would challenge the use of LDH as recyclable sorbents in magnetic composites due to the solubility of the LDH phase. For example, the well studied Fe (hydr)oxides may form strong sorbents of P i . Thus, P i retention by LDHs not only comprise anion exchange and surface complexation by the LDH, but may also comprise P i sorption to dissolution residues and precipitation processes, see Fig. 1. The instability of the LDH may hamper regeneration and re-use of the LDH during P i sorption, but to which extent is still unclear. Although P i sorption to MgFe-LDH has been extensively studied due to its high sorption capacity compared with MgAl LDH [17,37], its stability and mode of P i sorption at pH and P i concentrations in wastewater needs to be critically examined. In this study, we aim to assess the stability of the potential recyclable magnetic MgFe LDH sorbent as a function of pH and P i concentration (Fig. 1). The investigation focus on quantification of LDH dissolution as a function of pH (especially at pH neutral) and P i sorption combined with solid-state analysis of alteration in the MgFe LDH structure and formation of new phases during sorption.

Materials and methods

Materials

All chemicals were purchased from Sigma and directly used without further purification.

All sample solutions were prepared with ultrapure water (18.2 MΩ•cm) (SI 1.1). with P i concentrations typical for wastewater influents [START_REF]Sustainable Phosphorus Use[END_REF]. In addition equilibration time was reduced to 1 h to have short contact time for real waste water treatment [39]. The pH of the suspensions was controlled at pH 5, 6, 7, 8, 9, 10 or 11 (± 0.1) by addition of HCl or NaOH solution (SI 1.3).

Synthesis of

The solid phases were collected by an NdFeB magnet after P i sorption and dried at 60℃ overnight, while the solutions were filtered through 0.2 μm regenerated cellulose filter (Mikrolab Aarhus A/S) and stored at 4 ℃. P i and Mg concentrations were determined in all the filtrates. P i concentrations in solution were measured by the Molybdenum-Blue method using a ultraviolet-visible spectroscopy (UV-Vis) and Mg was determined by Atomic Absorption Spectroscopy (AAS). The content of Mg in the solids after sorption was determined with ICP. For further information about solid state characterization techniques, see SI 1.5.

Solubility calculations

Visual MINTEQ equilibrium speciation software ver 3.1 was used to estimate supersaturation of solutions with respect to magnesium phosphates, such as Mg 3 (PO 4 ) 2 , MgHPO 4 , Mg(H 2 PO 4 ) 2 . The thermodynamic database supplied with the software was used (SI 1.6).

P i sorption and phase distribution versus pH and P i concentration.

All sorption data from Exp. 2 were fitted to produce a "sorption surface" with sorbed P i versus pH and equilibrium P i concentration using the nonlinear surface fit function in Origin 9.6 software using a Rational 2D Function [40]. Fitting parameters are listed in Table S8.

The extent of P i sorption in form of magnesium phosphate precipitates was estimated based on estimations of mass of LDH derived from PXRD traces of samples taken after P i sorption. First, Rietveld profile-fitting in X'pert Highscore Plus software [START_REF]X'Pert HighScore Plus[END_REF] with the Pseudo Voigt profile function was used to analyse the PXRD data (SI 1.5.3) to determine the amount of LDH and Fe 3 O 4 in crystalline phases after P i sorption (Exp. 2). Next, the theoretical concentration of Mg released was calculated (SI 5, Eq. S10), assuming that the Mg release is proportional to the amount of dissolved LDH. Last, the theoretical Mg release was compared with the observed Mg concentration after P i sorption, and the difference between these two values was then allocated to Mg in magnesium phosphate precipitates (Details in SI5). Estimates of the fraction of P i in magnesium precipitates (Eq. S11) was added to the "sorption surface" using OriginPro 2019 software.

Results and discussion

In this study, the Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH composite was first synthesized and characterized in detail (SI 2), followed by P i sorption and dissolution experiments at controlled pH, ranging from 5 to 11, with exposure to various solution P i concentrations (1~300 mg P/L). Different solid state characterization techniques, such as PXRD, TEM, 57 Fe Mössbauer spectroscopy and 1 P MAS NMR were used to characterize the composite before and after P i sorption. LDH dissolution in terms of Mg release and P i sorption was correlated with solid phase analysis to conclude on dissolution, P i phase distribution and modes of P i sorption versus pH and P i concentration. After mixing with the pH buffered P i solution (Exp. 1), the composite showed fast P i sorption, which was well fitted by pseudo second-order kinetics (Eq. S4, R 2 0.999) with a rate constant of 5.4×10 -3 g/mg P min -1 . Equilibrium was reached within 120 min (Fig. 2a), demonstrating relatively fast P i removal. Sorption isotherms (Exp. 2) showed that P i bonding to the composite was of the Freundlich type (Eq. S5, R 2 0.979~0.999 Table S2)

with no clear sorption maxima (Fig. 2b). The P i sorption capacity decreased remarkably with increasing pH from 5 to 11, with 3 to 8 times less sorption at pH 11 than pH 5 depending on the solution P i concentration (Fig. 2b and3b) which is similar with other observations in previous studies on LDHs P i sorbents [START_REF] Chitrakar | [END_REF][43][44]. In Exp. 1, when the composite was dispersed in water, pH increased from 7.0 to 9.3 within 3 h (Fig. S6). This is attributed to release of hydroxide ions during dissolution of the LDH (Eq. 1) [START_REF] Jobbágy | [END_REF][45][46][47]. However, when pH is kept constant at pH 7.2 in HEPES buffered solution, substantial parts of the Mg 3 Fe LDH in the composite dissolves as more than 60 % of the total Mg dissolves within 60 min (Fig. 3a). The kinetics of Mg release was well fitted by pseudo second-order kinetic resulting in a kinetic constant of 2.6×10 -3 L/mg min -1 (Eq. S6; R 2 0.999), demonstrating fast dissolution at pH 7.2. This is in line with our previous study of Zn doped MgFe LDH, which showed a similar fast dissolution kinetics (k = 2.53×10 -3 L/mg min -1 ) at pH 7 [46]. The PXRD data show a dramatic decrease in intensity of the Mg 3 Fe LDH reflections, compared with the LDH kept at pH 9.3 (Fig. S8). In conclusion, both the solubility and the structural degradation of the LDH confirms that the composite is unstable at pH 7.2 and dissolves rapidly.

Dissolution of

In Exp. 3, the Mg 3 Fe LDH stability was investigated in non-buffered solutions as function of constant pH ranging between 5 and 11 at a constant initial P i concentration of 10 mg P/L.

Fig 3b shows a dramatic decrease of P i sorption with increasing pH. At pH 5, almost 100 % of the P i added is removed, while at pH 11 sorption is down to 20 %. In addition, the P i sorption is followed by a concurrent release of Mg depending on pH. More than 50 % of total Mg was released at pH 5 but less than 3 % Mg released at pH 11 within 1 h. PXRD of the composite after P i sorption at pH > 8 (Fig. 3c) showed that the LDH basal (003) reflection has a relative intensity of 96 % compared with the pristine composite (Fig. S2a), demonstrating that the composite is stable at pH > 8 at low P i concentration (10 mg P/L), in agreement with our recent study [35]. However, at lower pH of 7, 6 and 5 the relative intensity of the basal reflections dropped to 37%, 20% and 0 % of the initial values, respectively (Fig. 3c). TEM images of the sorbents after P i sorption (Fig. 3c) reveal a full destruction of the LDH and formation of amorphous material at pH 5, while at pH 8 more LDH remained, but still with some amorphous material present compared with the pristine LDH (Fig. S2). At pH 11, the intact Mg 3 Fe LDH is present. In addition, the LDH platelet thickness estimated from the width of the (003) reflections was in the range 6.9-7.1 nm for pH > 8, and decreased to 5.3 nm, 4.6 nm and 0 nm when P i had sorbed at pH 7, 6 and 5, respectively (Table S3), due to dissolution. Thus, Mg 3 Fe LDH dissolves at acidic to intermediate pH, while it is stable at high pH.

The combined effect of pH and P i sorption on LDH dissolution is depicted in Fig. S10,

showing that about 90, 80 and 40 % of total Mg has dissolved at the highest P i sorption at pH 5, 7 and 9, respectively. PXRD of the residual solids after P i sorption (Fig. S11 and Fig. S12) showed a dramatic decrease in intensity of the LDH reflections in the samples from pH 5 and 7 (Fig. S11a, b; Fig. S2a), confirming the instability of the LDH structure at acid and neutral pH [35,46]. Furthermore, the (003) and ( 110) reflections of the Mg 3 Fe LDH phases weakened or disappeared when initial P i concentrations exceeded 50 mg/L at higher pH, such as pH 9 and 11, revealing that exposure to relative high P i concentrations also stimulated LDH dissolution (Fig. S11c, d and Fig. S12) [35,48]. Finally, magnetization data shows that the fraction of magnetic materials decreases from 66 to 45 wt% in the composite after P i sorption at pH 5 (Fig. S13a andb, Eq. S9). S4).

Residual phase formed at pH 5

The 57 Fe Mössbauer spectrum of the pristine composite at 140 K (Fig. 4a) shows one sharp quadrupole doublet with center shift (δ) 0.42 mm/s and quadrupole splitting (Δ) of 0.61 mm/s which is attributed to octahedrally coordinated Fe 3+ in the Mg 3 Fe LDH [START_REF] Re | hrhardt, . Bourri[END_REF]. The 125 K spectrum of the composite after P i sorption (Fig. 4b) displays two quadrupole doublets with the same δ of 0.45 mm/s and different Δ of 0.54 mm/s and 1.03 mm/s, respectively.

The Δ of 0.54 mm/s corresponds to the presence of Mg 3 Fe LDH, while the Δ of 1.03 mm/s corresponds to presence of Fe 3+ in poorly crystalized ferrihydrite [50]. The 25 K spectrum (Fig. 4c) confirms this identification with a contribution of a single-line with δ of 0.68 mm/s and a magnetic broad sextet with Δ of 2 mm/s and very low magnetic field of 132 kOe attributed to poorly crystalized ferrihydrite [50].

The high sorption capacity at low pH (79.5 mg P/g in Fig. 1b) is in agreement with the high sorption capacity for ferrihydrite at low pH [51][52][53][54]. Thus, we estimate the obtained poorly crystalized ferrihydrite has a P i sorption capacity around 344 mg P/g (details in SI 3), which is much higher than common values (34~56 mg P/L) reported for ferrihydrite [54,55]. We attribute this high sorption capacity to the possible formation of amorphous strengite (FePO 4 ) [56,57] as the Fe:P ratio is close to 1:1. Apparently, extensive Mg dissolution causes formation of ultra-small ferrihydrite particles, leading to a strengite type product [58].

Precipitation of magnesium phosphates at high pH

In Exp. 2, although the composite did not release more than about 40% of total Mg into solution at pH 9 at an initial P i concentration of 300 mg P/L (Fig S10 ), the LDH structure was almost destroyed as seen from the PXRD pattern (Fig. S11c). At pH 11, there was barely any Mg release to solution (Fig. S9), while the relative intensity of the Mg 3 Fe LDH (003) and (110) peaks decreased to 26 % and 59 % of the intensity of the pristine composite, respectively (Fig. S11d, Fig. S12), reflecting substantial disslution of the LDH phase.

According to the previous studies of MgFe-CO 3 LDH [48], precipitation of magnesium phosphates is therefore likely to take place. In order to test this, the solids after P i sorption at pH 11 and initial P i concentration of 300 mg P/L were examined by PXRD without prior washing (Fig. S14). This test confirmed the partial dissolution of the Mg 3 Fe LDH phase seen from the decrease in intensity of basal reflections. Also, the size of crystallite domains of the Mg 3 Fe LDH phase along the ab-plane and c-axis was 33 % and 66 % smaller after P i sorption than before as determined from peak broadening (Eq. S3; SI 2). Solid-state 31 P MAS NMR spectroscopy (Fig. S15) of the LDH after P i sorption at pH 9 and 11 showed the presence of narrow resonance with ( 31 P) = 2.5(2) ppm and 3.9(2) ppm at pH 9 and pH 11, respectively, demonstrating formation of an amorphous magnesium phosphate phase [35]. The Mg-P precipitate had a molar Mg/P ratio in the Mg-P precipitates of 1.4 (SI 5), which is in good agreement with formation of Mg 3 (PO 4 ) 2 . Thus, at high pH and elevated P i concentrations LDH dissolves and magnesium phosphates precipitate.

Visual MINTEQ was used to compute supersaturation with respect to magnesium phosphates using the measured solution Mg and P i concentrations at different pH (Fig. S17).

No magnesium phosphates form at pH < 6, but when pH is increased to 7, MgHPO may also be retained due to the dissolution-precipitation reaction [32].

(1)

(2) Our investigation shows that P i sorption by Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH comprises at least three reactions: P i bonding to iron hydroxides (ferrihydrite), bonding to LDH and precipitation of magnesium phosphates. By fitting the P i sorption data with the Rational 2D function (R 2 = 0.99), a sorption surface (Fig. 5) can be produced to help estimate P i sorption versus pH and equilibrium P i concertation. The colour map shows that barely no P i was removed by precipitation at acid and neutral pH (Fig. 5). When pH increased to 9, at least 40 % of sorbed P i was present in magnesium phosphates at P i concentrations > 50 mg P/L, further increasing at higher pH and P i concentrations.

We propose a set of sorption pathways for P i bonding to the composite (Fig. 6). At acidic pH, the LDH is readily and fully dissolved. Despite high Mg concentrations in solution, the solubility of magnesium phosphates does not exceeded saturation and hence P i is adsorbed to the amorphous poorly crystalized ferrihydrite. At neutral pH, the LDH only partially dissolves. Hence, P i is sorbed on the remaining LDH as well as to the poorly crystalized ferrihydrite; at higher P i concentrations, P i is also retained due to precipitation of magnesium phosphates. Finally, at alkaline pH where the LDH phase has its highest stability, Mg 3 Fe LDH is the primary sorbent of P i at low P i concentration. However, at higher P i concentration, the Mg 3 Fe LDH phase is not stable and magnesium phosphates precipitate is the main sorption pathway. Recent study of P i sorption to MgFe-NO 3 LDH showed precipitation of amorphous magnesium phosphate precipitate at a solution P i of about 100 mg P/L at pH 7.4 [35]. The solubility of LDHs decrease with the divalent cation in the order Mg 2+ > Mn 2+ > Co 2+ ≈ Ni 2+ > Zn 2+ , while for the trivalent cation the order is Fe 3+ > Al 3+ [32]. Hence, many Fe 3+ containing LDHs have higher or equal solubility as the MgFe-LDH. Most stable LDHs would be of the type with Co 2+ or Ni 2+ as divalent cation [59]. However, these compounds are not environmentally friendly. Solubility and dissolution kinetics may be decreased using larger particles and using carbonate interlayered forms rather than forms with monovalent anions [60][61][62]. On the other hand, the presence of complexing agents in solution such as humic matter will increase solubility [START_REF] Mostofa | Photobiogeochemistry of Organic Matter: Principles and Practices in Water Environments[END_REF].

Although LDHs are seen as promising material for treatment of municipal wastewater [START_REF] Zubair | [END_REF],

the significant dissolution of LDHs filter material during the sorption process at neutral pH limits its practical application as wastewater has pH 6.5~8.5 [34]. Vast amounts of Mg 2+

ions could be released to the environment in a scaled-up application, which would increase water hardness and block the filters and pipes due to formation of precipitates. In addition, the loss of relative expensive MgFe LDH materials would also increase the cost of wastewater treatment. Furthermore, high P i concentrations in wastewater would also stimulate the dissolution of LDHs at higher pH resulting in formation of precipitates. Thus, the widely reported LDH regeneration and P recovery strategy, where adsorbed P is extracted with alkali solution and the LDH re-used for P sorption, would be challenged [15,65].

Conclusions

In this study, we demonstrate the MgFe LDH has poor stability and dissolves in the pH range of wastewater (pH 6~8). In addition to the pH controlled LDH solubility, high P i concentrations causes LDH dissolution even at high pH, by formation of magnesium precipitates. 57 Fe Mössbauer spectroscopy confirms the formation of poorly crystalized ferrihydrite at acid pH while 31

  ) 2 ] x+ [A x/n •yH 2 O] x-, where M a II , M b

Figure 1 .

 1 Figure 1. Schematic illustration of the phase changes and P i sorption processes addressed in the study.

3. 1 Figure 2 .

 12 Figure 2. (a) Kinetics of P i sorption (q t ) to Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH and the

Fe 3 O 4 @Figure 3 .

 343 Figure 3. (a) Kinetics of Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH dissolution measured as the

Figure 4 .

 4 Figure 4. Mössbauer spectra of Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH before P i sorption measured

3. 5 P i sorbed to Fe 3 O 4 @Figure 5 .

 5345 Figure 5. Sorbed P i (black dots) to Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH as a function of pH and

Figure 6 .

 6 Figure 6. Pathways for P i sorption to Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH at various pH and P i concentrations.

Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH

	The Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH composite (referred to as composite in this paper) was
	synthesized by previous reported methods [35, 37, 38]. For experimental details, see SI
	(1.2).

2.3 Phosphate sorption and Fe 3 O 4 @SiO 2 -Mg 3 Fe LDH dissolution Experiment 1 (Exp. 1) The

  In this experimental series dissolution and P i sorption of the composite as function of solution P i concentration and pH was studied. The experiments were carried out as described for Exp. 1, but with initial P i concentrations at 1, 2, 5, 10, 20, 50, 100, 200 and 300 mg P/L. No HEPES buffer was used, but suspension pH was controlled at 5, 7, 9 and 11 by addition of HCl or NaOH solution (SI 1.3). All suspensions were placed on a shaking table at 250 rpm for 24 h once the pH had stabilized.

	kinetics of P i sorption and LDH dissolution from the
	composite was followed by monitoring the P

i and Mg concentrations in pH buffered suspensions. The composite (1 g/L) was dispersed in an aqueous solution of 10 mM KNO 3 electrolyte, 11.92 g/L HEPES buffer and 50 mg P/L added as KH 2 PO 4 . The pH of the solution was buffered by HEPES at pH 7.2 throughout the experiment. At various times (1, 2, 5, 10, 30 min, 1, 2, 3, 6, 12 and 24 h), 2 mL of the suspension was sampled and filtered through a 0.2 μm regenerated cellulose filter (Mikrolab Aarhus A/S) and the filtrate stored at 4 ℃ until determination of P i and Mg. In a separate experiment, the pH buffering effect of the composite was illustrated by preparing the same suspension as above but without adding HEPES buffer. The pH of the suspension was measured at 1, 2, 5, 10, 30 min, 1, 2, 3, 6, 12 and 24 h. Experiment 2 (Exp. 2) Experiment 3 (Exp. 3) This experiment was similar to Exp. 2 but only P i concentrations of 10 mg P/L was tested to avoid side reactions due to high P i concentrations and to operate

  P MAS NMR spectroscopy demonstrate the precipitation of magnesium phosphate at high pH and high P i concentration. A combined analysis of solubility and PXRD data indicate that precipitates is the main sorption pathway at high pH and high P i concentration. The main sorption pathways are summarized as sorption by ferrihydrite at acid pH, sorption by ferrihydrite and LDH at neutral pH and high pH with low P i concentration, and precipitation of magnesium phosphates at high pH and high P i concentration. This study demonstrates that LDHs are intrinsically unstable from pH 5 ~ 11 when used for P i sorption which hence prevents there use as recyclable sorbents. Future application of magnetic LDHs as recyclable sorbents is only possible if the stability of the LDH component can be substantially increased.

Acknowledgement

We acknowledge the Danish Research Council for the financial support of this project (MagS 3 , Project Number 1129331001) and the Villum Foundation for the 31 P NMR analysis in the Villum center for bioanalytical sciences (600 MHz NMR spectrometer). We greatly thank our laboratory technicians Birgitte Boje Rasmussen and Anita Schjødt Sandager for assistance with ICP and AAS.

Appendix supplementary data

The supplementary data of this paper is available on the website at DOI:


Marc-Antoine Coppo
email: coppo@unice.fr

A note on some alternating series involving zeta and multiple zeta values

Keywords: Riemann zeta function, harmonic zeta function, Stirling numbers of the first kind, Stirling numbers of the second kind, Bernoulli numbers, Bernoulli numbers of the second kind, harmonic numbers, Gregory coefficients Mathematics Subject Classification (2010) 11B73, 11B75, 11M06, 11M32, 40G99, 41A58

A note on some alternating series involving zeta and multiple zeta values

Introduction

The first part of this article is devoted to the study of the conditionally convergent alternating series ν k defined by This remarkable connection between γ and the special values at positive integers of the Riemann zeta function goes back to Euler's early works on harmonic series [START_REF] Euler | De progressionibus harmonicis observationes[END_REF]. Less famous but yet fairly well-known (cf. [8, p. 93], [START_REF] Srivastava | Sums of certain series of the Riemann zeta function[END_REF]Eq. (5.1)], [START_REF] Singh | Some series involving Riemann zeta function[END_REF]Eq. (1.5)]) is the relation

ν 1 = γ 2 - 1 2 ln 2π + 1
sometimes called Suryanarayana formula. Recently, Blagouchine [5, p. 413] gave a general expression of these series ν k in the case where k is a positive integer:

ν k = γ 2 - ln 2π k + 1 + 1 k + [ k 2 ] r=1 (-1) r k 2r -1 (2r)! r(2π) 2r ζ (2r) + [ k-1 2 ]
r=1 (-1) r k 2r (2r)! 2(2π) 2r ζ(2r + 1) . [START_REF] Agoh | Recurrence relations for Nörlund numbers and Bernoulli numbers of the second kind[END_REF] This formula seems quite cumbersome but can be much simplified using the functional equation of ζ. After some elementary transformations, we show that equation (1) can be reduced to the following equivalent (but much more pleasant) expression:

ν k = γ k + 1 - 1 2 ln 2π + k-1 j=1 (-1) j k j ζ (-j) + C k , ( 2
)
where C k is a rational number (see Proposition 1). Moreover, this expression allows to highlight a deep connection between ν 2k and the sum (in the sense of the Ramanujan summation of divergent series) of the series n 1 n 2k H n , where H n is the nth harmonic number (see Remark 2). Next, in a second part, we introduce a generalization of these series series ν k replacing the zeta values by certain multiple zeta values. A natural extension may be defined as follows: for all integers k ≥ -1 and p ≥ 0, we consider the class of series (ν k,p ) with

ν k,p := ∞ n=2 (-1) n n + k ζ(n, 1, . . . , 1 p ) , where ζ(s 1 , s 2 , • • • , s k ) = n 1 >n 2 >•••>n k ≥1 1 n s 1 1 n s 2 2 • • • n s k k ,
so that the previous series ν k become ν k,0 . Then we establish (see Proposition 2) the following identity which is the main result of this work:

ν k,p = ∞ n=1 |G (k+1) n | n p+1 , ( 3
)
where G (k) n denotes the Gregory coefficients of higher order recently introduced by Blagouchine [START_REF] Blagouchine | Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π -1[END_REF][START_REF] Blagouchine | Three notes on Ser's and Hasse's representations for the zeta-functions[END_REF]. They are defined by

G (k) n := 1 n! n j=1 s(n, j) j + k (k ≥ 0, n ≥ 1), (4)
where s(n, j) are the Stirling numbers of the first kind. Comprehensive informations on the Stirling numbers of the first and the second kind may be found in [START_REF] Agoh | Recurrence relations for Nörlund numbers and Bernoulli numbers of the second kind[END_REF][START_REF] Blagouchine | Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to π -1[END_REF][START_REF] Coppo | Shifted Mascheroni series and hyperharmonic numbers[END_REF][START_REF] Jordan | Calculus of Finite Differencies[END_REF][START_REF] Xu | Multiple zeta values and Euler sums[END_REF]. One can prove easily (see Lemma 4) that G (k) n = (-1) n+1 |G (k) n |, so that the rationals numbers G (k) n alternate in sign. As a special case of equation (3), we derive the following result:

ν k-1 = ∞ n=1 |G (k) n | n (k ≥ 0) . ( 5
)
In the case k = 1, we recover the classical Mascheroni's series for γ (cf. [5, p. 406

∞ n=2 (-1) n ζ H (n) n = γ 1 + 1 2 γ 2 + π 2 12 ,
where ζ H (s) denotes the Apostol-Vu harmonic zeta function [START_REF] Alkan | Approximation by special values of harmonic zeta function and log-sine integrals[END_REF][START_REF] Apostol | Zeta and Related Functions, NIST Handbook of Mathematical Functions[END_REF][START_REF] Apostol | Dirichlet series related to the Riemann zeta function[END_REF] and γ 1 is the first Stieltjes constant [START_REF] Blagouchine | A note on some constants related to the zeta function[END_REF][START_REF] Candelpergher | Ramanujan Summation of Divergent Series[END_REF]. Finally, in the last section, we highlight a relation between the series ν k , the Stirling numbers of the second kind S(n, k), and the shifted Mascheroni series σ r whose study was the main subject of [START_REF] Coppo | Shifted Mascheroni series and hyperharmonic numbers[END_REF] (see Proposition 3 and Example 4).

The case of a positive integer

In this section, we focus on the case of a positive integer k and give two independant proofs of our formula [START_REF] Alkan | Approximation by special values of harmonic zeta function and log-sine integrals[END_REF]. More precisely, we prove the following proposition: Proposition 1. For any positive integer k, we have

ν k = γ k + 1 - 1 2 ln 2π + k-1 j=1 (-1) j k j ζ (-j) + C k with C k = 1 k + k-1 j=1 k j B j+1 H j j + 1 , ( 6
)
where H n are the harmonic numbers,

H 0 = 0 , H n := 1 + 1 2 + • • • + 1 n (n ≥ 1) ,
and B n are the Bernoulli numbers defined by means of the exponential generating function

x e x -1 = ∞ n=0 B n x n n! (|x| < 2π) .
In particular,

B 0 = 1, B 1 = -1 /2, B 2 = 1 /6, B 2r+1 = 0 for r ≥ 1.
Proof. We can quite easily derive (2) from [START_REF] Agoh | Recurrence relations for Nörlund numbers and Bernoulli numbers of the second kind[END_REF]. Differentiation of the functional equation [3, Eq. (25.4.2)]

ζ(s) = 2(2π) s-1 Γ(1 -s)ζ(1 -s) sin πs 2
leads to the two relations

(-1) r (2r)! 2(2π) 2r ζ(2r + 1) = ζ (-2r) (r ≥ 1), and
(-1) r (2r)! r(2π) 2r ζ (2r) = -ζ (1 -2r) + B 2r 2r (H 2r-1 -γ -ln 2π) (r ≥ 1) .
Substituting these relations into (1) and grouping together the terms under the two symbols Σ, leads to the expression

ν k = γ k + 1 - 1 2 ln 2π + k-1 j=1 (-1) j k j ζ (-j) + C k ,
where the rational constant C k is given by equation ( 6). Another alternative proof of (2), independant from (1), may be deduced from the expansion in powers of z of the relation [8, p. 93

] ∞ k=0 (-1) k z k k! ∞ j=1 (-1) j-1 j ζ R (j -k) = (1 -e z ) ∞ k=0 (-1) k z k k! ζ (-k) + (1 -e z ) ∞ k=0 (-1) k z k k! 1 (k + 1) 2 + ˆ1 0 ln(t + 1)e -zt dt with ζ R (j -k) =    γ if j = k + 1 ζ(j -k) -1 j-k-1 otherwise.
Rewriting the series ν k as

ν k = ∞ j=k+2 (-1) j-k j ζ(j -k) ,
and using the well-known relations

ζ(0) = - 1 2 , ζ(1 -2r) = - B 2r 2r , and ζ (0) = - 1 2 ln 2π (cf. [3, p. 605]),
then a careful identification of the terms in z k in the previous development leads again, after some simplifications, to formula (2), and provides in addition another equivalent expression for the constant C k :

C k = H k k + 1 - k j=1 B j j(k + 1 -j) . ( 7
)
An unexpected consequence of this equivalence is the curious identity

H k k + 1 - 1 k = k-1 j=0 B j+1 j + 1 k j H j + 1 k -j (k ≥ 1)
whose direct proof does not seem obvious.

Example 1. For the first values of k, we have the following relations:

ν 1 = γ 2 - 1 2 ln 2π + 1 , ν 2 = γ 3 - 1 2 ln 2π -2ζ (-1) + 2 3 , ν 3 = γ 4 - 1 2 ln 2π -3ζ (-1) + 3ζ (-2) + 7 12 , ν 4 = γ 5 - 1 2 ln 2π -4ζ (-1) + 6ζ (-2) -4ζ (-3) + 47 90 , ν 5 = γ 6 - 1 2 ln 2π -5ζ (-1) + 10ζ (-2) -10ζ (-3) + 5ζ (-4) + 167 360 .
Remark 1. Starting from the Maclaurin series expansion [3, Eq. (25.8.5)]

ψ(x + 1) + γ = ∞ n=2 (-1) n ζ(n)x n-1 (|x| < 1)
where ψ(x) denotes the digamma function (i.e. the logarithmic derivative of the Γ-function), and multiplying each side by x k (with k ≥ 1), then an integration between 0 and 1 gives

ν k = γ k + 1 + ˆ1 0 x k ψ(x + 1) dx.
Thus, it follows from formula (2) that

ˆ1 0 x k ψ(x + 1) dx = k-1 j=0 (-1) j k j ζ (-j) + C k (k ≥ 1).
Remark 2 (Link with the Ramanujan summation: part I).

H n = 3 2 γ - 1 2 ln 2π + 1 2 ,
and for any positive integer p,

R n 1 n p H n = 1 -B p+1 p + 1 γ - 1 2 ln 2π + p j=1 (-1) j p j ζ (-j) + R p with R p ∈ Q,
where the symbol R denotes the sum of the series in the sense of the Ramanujan summation of divergent series [START_REF] Blagouchine | A note on some constants related to the zeta function[END_REF][START_REF] Candelpergher | Ramanujan Summation of Divergent Series[END_REF][START_REF] Candelpergher | A new class of identities involving Cauchy numbers, harmonic numbers and zeta values[END_REF][START_REF] Candelpergher | Ramanujan summation and the exponential generating function ∞ k=0 z k k! ζ (-k)[END_REF]. For p = 2k (with k ≥ 1), we have

B p+1 = 0 and R p = C p -Bp 2p + Bp 2 ,
then, in view of formula (2), these relations may be translated into the following identities: R n 1

H n = ν 1 + γ - 1 2 ,
and for k ≥ 1,

R n 1 n 2k H n = ν 2k + ζ (-2k) + 1 -2k 2 ζ(1 -2k) = ν 2k + ζ (-2k) + (2k -1) B 2k 4k . ( 8
)
In particular, we have

R n 1 n 2 H n = ν 2 + ζ (-2) + B 2 4 = ν 2 - ζ(3) 4π 2 + 1 24 .
2 The case k = -1

The case k = -1 behaves differently from the previous case and must be studied separately. We recall the identities [11, p. 142]

ν -1 = ˆ1 0 ψ(x + 1) + γ x dx = ∞ n=1 1 n ln(1 + 1 n ) = - ∞ n=2 ζ (n) = 1.2577468869 . . .
Another interesting representation (communicated by I. V. Blagouchine) is

ν -1 = 1 2 ˆ+∞ -∞ ζ( 3 /2 + ix) ( 1 /2 + ix) cosh(πx)
dx.

Moreover, we can write yet another relation that will be useful in the next section: for an integer p ≥ 0, let κ p be the constant defined by

κ p := ∞ n=1 |G n | n p+1 ,
where G n := G (1) n are the Bernoulli numbers of the second kind (also called Gregory coefficients). In particular, we have

κ 0 = ∞ n=1 |G n | n = γ and κ 1 = ∞ n=1 |G n | n 2 = 0.5290529699 . . .
κ 1 = ν -1 + γ 1 + 1 2 γ 2 - 1 2 ζ(2) , (9)
where γ 1 denotes the first Stieljes constant [START_REF] Apostol | Zeta and Related Functions, NIST Handbook of Mathematical Functions[END_REF][START_REF] Blagouchine | A note on some constants related to the zeta function[END_REF][START_REF] Candelpergher | Ramanujan Summation of Divergent Series[END_REF] .

γ 1 = lim n→∞    n j=1 ln j j - 1 2 ln 2 n    = -0.

Alternating series involving multiple zeta values

In this section, we consider a more general class of series of the previous type replacing zeta values with certain multiple zeta values. We prove our formula (3) and deduce some interesting consequences.

Proposition 2. For all integers p ≥ 0 and k ≥ -1, let

ν k,p := ∞ n=2 (-1) n n + k ζ(n, 1, . . . , 1 p ) ; then ν k,p = ∞ n=1 |G (k+1) n | n p+1 ,
where the rational numbers G (k) n are defined by equation ( 4). Corollary 1. In particular, for p = 0, we have

ν k-1,0 = ∞ n=1 |G (k) n | n = ν k-1 (k ≥ 0);
and for k = 0, we have

ν 0,p = ∞ n=1 |G (1) n | n p+1 = κ p (p ≥ 0) .
In order to prove Proposition 2, we begin by stating the following lemmas:

Lemma 1. For all integers j ≥ 1 and p ≥ 0, we have

ˆ1 0 ln j (1 -x) ln p (x) x dx = (-1) j+p j! p! ζ(j + 1, 1, . . . , 1 p ) . ( 10
)
Proof. This follows directly from [18,

ln j (1 + x) j! = ∞ n=j s(n, j) x n n! (|x| < 1) . ( 11
)
Lemma 3. For all integers n ≥ 1 and p ≥ 0, we have

(-1) p ˆ1 0 x n-1 ln p (x) dx = p! n p+1 (12)
Proof. This is nothing else than [7, Eq. ( 41)] in the case where p is an integer.

Lemma 4. For all integers n ≥ 1 and k ≥ 0, we have

G (k) n = (-1) n+1 n! ˆ1 0 x k (1 -x) n-1 dx , where (z) n = z(z+1)(z+2) • • • (z+n-1)
is the Pochhammer symbol. In particular, this implies that

G (k) n = (-1) n+1 |G (k) n | . ( 13
)
Proof. Integration between 0 and 1 of the expansion

x k-1 x(x -1) • • • (x -n + 1) = n j=1 s(n, j)x j+k-1
gives the required result.

Proof of Proposition 2. Using successively formulas (10)-( 13) above, we can write the following equalities:

ν k,p = ∞ j=1 (-1) j+1 j + k + 1 ζ(j + 1, 1, . . . , 1 p ) = (-1) p+1 p! ∞ j=1 1 j + k + 1 ˆ1 0 ln j (1 -x) j! ln p (x) x dx = (-1) p+1 p! ∞ j=1 1 j + k + 1 ˆ1 0   ∞ n=j (-1) n s(n, j) x n n!   ln p (x) x dx = (-1) p+1 p! ∞ j=1 1 j + k + 1 ∞ n=j (-1) n s(n, j) n! ˆ1 0 x n-1 ln p (x) dx = - ∞ j=1 1 j + k + 1 ∞ n=j (-1) n s(n, j) n! n p+1 = - ∞ n=1 (-1) n n! n p+1 n j=1 s(n, j) j + k + 1 = ∞ n=1 (-1) n+1   1 n! n j=1 s(n, j) j + k + 1   1 n p+1 = ∞ n=1 (-1) n+1 G (k+1) n n p+1 = ∞ n=1 |G (k+1) n | n p+1 .
This completes the proof.

Example 2.

For the first values of k ≥ -1, we have the following expansions in series containing only positive rational terms: and more generally, they may be computed by means of the following beautiful formula (first discovered by Euler [START_REF] Euler | Meditationes circa singulare serierum genus (1775), Eneström-Number E477[END_REF] and several times rediscovered afterwards):

ν -1 = ∞ n=1 |G (0) n | n =
2ζ H (n) = (n + 2)ζ(n + 1) - n-2 r=1 ζ(r + 1)ζ(n -r) (n ≥ 3).
Otherwise, by Proposition 2 above, we can write

∞ n=2 (-1) n ζ H (n) n = ν 0,1 -ν -1 + ζ(2) = κ 1 -ν -1 + ζ(2) ,
and thus, from equation ( 9), we derive the following elegant evaluation:

∞ n=2 (-1) n ζ H (n) n = γ 1 + 1 2 γ 2 + π 2 12 = 0.916240149 . . . . ( 14
)
Another expression of this constant is ζ (0) +

(s) = ζ H (s) - ˆ∞ 1 x -s (ψ(x + 1) + γ) dx for Re(s) > 1 .
We have the identities

ζ R H (1) = ν 0,1 = κ 1 , ζ R H (0) = ν 1 + γ - 1 2 ,
and formula (8) may be nicely rewritten

ζ R H (-2k) = ζ H (-2k) + ζ (-2k) + ν 2k .

Link with the shifted Mascheroni series

Let us consider now the forward shifted Mascheroni series which are defined by

σ r := ∞ n=1 |b n+r | n , for r = 0, 1, 2, • • • .
We have in particular σ 0 = ν 0 = γ. The study of these series σ r was the main subject of [START_REF] Coppo | Shifted Mascheroni series and hyperharmonic numbers[END_REF]. Among other things, we have established the following decomposition of ζ (-j) on the "basis" of σ r [12, Proposition 3]:

ζ (-j) = j+1 r=2
(-1) j-r (r-1)!S(j, r-1)σ r -B j+1 j + 1 γ-B j+1 (j + 1) 2 , for j = 1, 2, 3,

• • • ,
where S(j, r) are Stirling numbers of the second kind; moreover, for j = 0, we have also a similar relation:

1 2 ln 2π = -ζ (0) = σ 1 + γ 2 + 1 2 .
Then, substituting these relations into (2) enables us to write each series ν k with k ≥ 1 as an integral linear combination of γ, σ 1 , σ 2 , • • • , σ k plus a rational number D k which is closely linked to C k . In this combination, the coefficient of γ is zero since it is equal to 1 with

D k = 1 k + k j=1 k j B j H j k + 1 -j .
Example 4. For the first values of k, we have the following relations:

ν 1 = 1 2 -σ 1 , ν 2 = 1 4 -σ 1 + 2σ

ν k := ∞ n=2 (- 1 )

 n=21 n ζ(n) n + k , where ζ(s) is the Riemann zeta function and k denotes an integral parameter. By a classical result (cf. [8, p. 66], [15, p. 62]), it is well known that ν 0 is Euler's constant

The constants κ 1

 1 and ν -1 are linked by the relation [8, Eq. (3.23) p. 105]

BProposition 3 .

 3 j which vanishes by a well-known property of the Bernoulli numbers. Finally, equation (2) may be nicely rewritten in terms of σ r as follows: For all integers k ≥ 1, we have the relationν k = D k -σ 1 + k r=2(-1) r (r -1)!

  07281584548 . . . .

	In terms of the Ramanujan summation, γ 1 is	R n 1	ln n n (cf. [8, p. 67]), whereas κ 1 is
	R n 1	Hn n (cf. [8, Eq. (4.29) p. 133]).

  Eqs. (2.27), (2.28)].

	Lemma 2. The Stirling numbers of the first kind s(n, j) with fixed j ≥ 1 admit
	the (vertical) exponential generating function [1, Eq. (2.8)]

  + 30σ 2 -150σ 3 + 240σ 4 -120σ 5 , ν 6 = 23 180 -σ 1 + 62σ 2 -420σ 3 + 1560σ 4 -1800σ 5 + 720σ 6 .

				2 ,
	ν 3 =	5 24	-σ 1 + 6σ 2 -6σ 3 ,
	ν 4 =	13 72	-σ 1 + 14σ 2 -36σ 3 + 24σ 4 ,
	ν 5 =	109 720	-σ 1

Acknowledgments

The author gratefully acknowledges the referee for his careful reading, constructive comments and helpful suggestions.