A M Hashem

A E Abdel-Ghany

H M Abuzeid

R S El-Tawil

S Indris

H Ehrenberg

A Mauger

C M Julien

EDTA as chelating agent for sol-gel synthesis of spinel LiMn 2 O 4 cathode material for lithium batteries

Keywords: Lithium-ion batteries, cathode material, spinel structure, sol-gel synthesis, EDTA

EDTA as chelating agent for sol-gel synthesis of spinel

LiMn 2 O 4 cathode material for lithium batteries

Introduction

Nano-materials with controllable size and shape have received enormous attention due to their promising applications in various fields, such as catalysis, gas sensing and batteries [START_REF] Jin | Synthesis of single-crystalline octahedral LiMn 2 O 4 as high performance cathode for Li-ion battery[END_REF].

Rechargeable lithium ion batteries (LIBs) have drawn extensive attention as the most promising candidates to power the next generation of electric vehicles and electronic equipment due to their high-energy density, high working voltage, low self-discharge rate, long shelf life [START_REF] Whittingham | Lithium batteries and cathode materials[END_REF]. The various commercialized Li-ion batteries mainly differ by the chemistry of the active cathode material, which then plays a key role in the determination of both their energy and their cost.

Currently, lithium cobalt oxide (LiCoO 2 ), which is known for its high-energy density and good cycling stability, is the most commercialized cathode material for LIBs. However, the high price and toxicity of cobalt have prevented its use in next-generation LIBs [START_REF] Zhao | Effects of equimolar Mg (II) and Si (IV) co-doping on the electrochemical properties of spinel LiMn 2-2x Mg x Si x O 4 prepared by citric acid assisted sol-gel method[END_REF]. Spinel-type LiMn 2 O 4 oxide (LMO), which has three-dimensional lithium diffusion paths is considered as one of the most promising alternative cathode materials for LIBs due to the advantages such as good environmental compatibility, good thermal stability, its abundant raw material resources, low production cost, high potential plateau, and good safety [START_REF] Gao | Combustion-derived nanocrystalline LiMn 2 O 4 as a promising cathode material for lithium-ion batteries[END_REF]. However, its capacity fading during cycling, especially at high temperature, restricts its commercial usage for rechargeable Li-ion batteries [START_REF] Ebin | Comparison of 4 V and 3 V electrochemical properties of nanocrystalline LiMn 2 O 4 cathode particles in lithium ion batteries prepared by ultrasonic spray pyrolysis[END_REF]. Generally, current commercial LiMn 2 O 4 products can reach 100-120 mAh g -1 while the theoretical capacity is 148 mAh g -1 [START_REF] Kim | Spinel LiMn 2 O 4 nanorods as lithium ion battery cathodes[END_REF]. Many efforts are currently made in research to improve the capacity of LiMn 2 O 4 and its cycling life. In the present work, we report on different synthesis conditions in order to optimize the synthesis parameters leading to the best electrochemical properties of LiMn 2 O 4 . Commercial spinel LiMn 2 O 4 is synthesized by a high temperature solid-state reaction, which has advantages of facile synthesis and large-scale production. On another hand, it also has several disadvantages such as non-homogeneities, irregular morphology, long calcination times and slow reaction kinetics. Consequently, the formation of pure single-phase product is hindered, and the process leads to the formation of larger particles with broad particle size distribution. It also impedes the Li-ion diffusion into the cathode [START_REF] Cai | Long cycle life, high rate capability of truncated octahedral LiMn 2 O 4 cathode materials synthesized by a solid-state combustion reaction for lithium ion batteries[END_REF], which affects the rate capability [START_REF] Julien | Lithium Batteries: Science and Technology[END_REF]. Therefore, the reduction of the size of the particles is desirable in order to increase the performance at high rate of charge and discharge by reducing the length of the path for electrons and Li + ions within the LiMn 2 O 4 particles, and by increasing the effective surface area [START_REF] Guo | Nanostructured Materials for Electrochemical Energy Conversion and Storage devices[END_REF][START_REF] Wang | Developments in nanostructured cathode materials for high-performance lithium-ion batteries[END_REF].

To overcome these disadvantages and prepare particles of smaller size, many wetchemical methods are being used, which include micro-emulsion [START_REF] Lines | Principles and applications of ferroelectrics and related materials[END_REF][START_REF] Ke | Synthesis of spinel LiMn 2 O 4 with manganese carbonate prepared by micro-emulsion method[END_REF], hydrothermal synthesis [START_REF] Tsuji | Structural and thermal properties of LiMn 2 O 4 substituted for manganese by iron[END_REF], sol-gel [START_REF] Michalska | Nanocrystalline lithium-manganese oxide spinels for Li-ion batteries: sol-gel synthesis and characterization of their structure and selected physical properties[END_REF], microwave-induced combustion [START_REF] Fu | Comparison of the microwave-induced combustion and solid state reaction for the synthesis of LiMn 2 O 4 powder and their electrochemical properties[END_REF], and spray drying method [START_REF] Wan | Synthesis of spherical LiMn 2 O 4 cathode material by dynamic sintering of spray-dried precursors[END_REF].

Zhang et al. have outlined the fact that the electrochemical performance of α-MnO 2 strongly depends on the synthesis procedure, i.e. chelating agent used in the wet-chemical method [START_REF] Zhang | Nanotechnology of positive electrodes for Li-ion batteries[END_REF]. Actually, this exemplifies the role of the chelating agent that is important in any synthesis process. The choice of ethylene diamine tetra-acetic acid (EDTA), which is an aminopolycarboxylic acid and water-soluble solid, forms complex precursor for a low temperature synthesis of cathode materials; it has proved to be efficient to improve the properties of the LiCoO 2 cathode already 15 years ago [START_REF] Liu | Synthesis by an EDTA-based soft-chemistry route and characterization of nanosized LiCoO 2 cathode materials[END_REF]. Since then, it has also been used to synthesize Li 1-x K x CoO 2 [START_REF] Han | Preparation of lithium battery cathode materials Li 1-x K x CoO 2 by EDTA complexing method[END_REF]. A sol-gel synthesis of Li 2 MnO 3 and Li[Li 0.2 Mn 0.6 Ni 0.2 ]O 2 by a combined EDTA and citric acid complexation route has also been successful [START_REF] Jarvis | Understanding structural defects in lithium-rich layered oxide cathodes[END_REF]. Recently, an EDTA-solgel method has been used to synthesize LiV 3 O 8 [START_REF] Yuan | Preparation and characterization of LiV 3 O 8 cathode material for lithium secondary batteries through an EDTA sol-gel method[END_REF]. A comparative study of the electrochemical properties of LiNi 0.80 Co 0.15 Al 0.05 O 2 cathode synthesized by sol-gel method with different complexing agents (EDTA, glycine and citric acid) shows that the best structural properties, the largest capacity and the best cycle ability were obtained with EDTA [START_REF] Dong | Influence of complexing agent on the structure and electrochemical properties of LiNi 0[END_REF]. The beneficial effect of EDTA on lamellar compounds outlined in these works has also been tested on Li 3 V 2 (PO 4 ) 3 /C composite [START_REF] Li | Electrochemical performance of Li 3 V 2 (PO 4 ) 3 /C prepared with a novel carbon source, EDTA[END_REF]. This complexing agent has been rarely used for the synthesis of the LiMn 2 O 4 .

We found a report of this synthesis in [START_REF] Yuan | Preparation and property of normal spinel LiMn 2 O 4[END_REF], but the electrochemical properties were not explored.

In the present work, a wet chemistry method at moderate temperature and short time using EDTA and citric acid as chelating agents have been used to explore the best conditions to produce nanoparticles of LiMn 2 O 4 to optimize the electrochemical properties of this cathode material for lithium-ion batteries. Compared with the conventional wet-chemical method, the EDTA assisted sol-gel process offers many advantages such as a better control of material morphology and better electrochemical performance.

The recent progress of LiMn 2 O 4 has been obtained by surface modifications of the nanoparticles to prevent the manganese from dissolution in the electrolyte containing acidic species. A review of these modifications can be found in [START_REF] Lee | High performance LiMn 2 O 4 cathode materials grown with epitaxial layered nanostructure for Li-ion batteries[END_REF]. In this work, Lee et al. reported a novel heterostructure with epitaxially grown R-3m surface phase. This result shows that there are now coating layers that satisfy the requirement of chemical affinity between the host and the surface layer to solve the problem that limited the use of LiMn 2 O 4 in different applications, including electric vehicle. We thus believe that the cheap and easy mode of preparation of LiMn 2 O 4 that can benefit of such surface modifications is promising to promote this material in the industry, without the need of less scalable and expensive syntheses process involving hollow nanofibers with a porous structure by modified electrospinning techniques and subsequent thermal treatment [START_REF] Duan | Synthesis and electrochemical property of LiMn2O4 porous hollow nanofiber as cathode for lithium-ion batteries[END_REF], or by doping with rare earths [START_REF] Ram | Improved performance of rare earth doped LiMn2O4 cathodes for lithium-ion battery applications[END_REF].

Experimental

To compare the effect of the chelating agent, two LiMn 2 O 4 samples were prepared by sol-gel process using citric acid and EDTA as chelating agents. LiMn 2 O 4 powders were prepared using appropriate molar ratios of lithium and manganese acetate dissolved in distilled water.

Carboxylic acids, such as citric acid and EDTA, act as fuel during the formation process of transition-metal oxide powders, and they decompose the homogenous precipitate of metal complexes at temperature T<450 °C. The dissolved solution was added dropwise to a continuously stirred aqueous solution of chelating agent. The molar ratio between metal and chelating agent was adjusted to be 1:1. The COOH group of chelating agent forms a chemical bond with the metal ions and, after evaporation of the solvent by heating the prepared solution at 80 °C with magnetic stirring for about 6 h, a transparent xerogel was obtained. This gel precursor was decomposed at 450 °C for 4 h in air to eliminate the organic substances and cooled to room temperature. The decomposed powders were slightly ground using a mortar and calcined in air for 12 h at 600 and 800 °C for EDTA and citric acid, respectively. In the following, the corresponding products are named E-LiMn 2 O 4 and C-LiMn 2 O 4 , respectively.

The crystal structure of LiMn 2 O 4 samples was analyzed by X-ray diffraction (XRD) using a Philips X'Pert apparatus equipped with a Cu Kα X-ray source (λ = 154.06 pm). XRD data were collected in the 2θ range 10-80° at room temperature. Raman spectra have been recorded in spectral range 200-800 cm -1 at room temperature in a quasi-backscattering geometry.

A double monochromator (Jobin-Yvon model U1000) with holographic gratings and a computercontrolled photon-counting system was used. The laser light source was the 514.5nm line radiation from a Spectra-Physics 2020 argon-ion laser. To have a large signal to noise ratio, 12 successive scans recorded at a spectral resolution of 2 cm -1 are averaged. FTIR spectra were recorded using a Bruker model IFS113v interferometer equipped with a MCT mid-IR and DTGS/PE detectors. Specimens were prepared by the disk method using powders mixed with CsI (1:300) and pressing them at 5 tons cm -2 with good transparency for IR radiation. A JEOL, Transmission Electron Microscope (TEM, JEM-1230, Japan) was used to investigate the prepared sample. 7 Li MAS NMR spectra were measured with a Bruker Avance 200 MHz spectrometer with a magnetic field of 4.7 T corresponding to a Larmor frequency of 77.8 MHz.

Spinning was performed in 1.3 mm rotors at 60 kHz. Spectra were acquired with a rotorsynchronized Hahn-echo pulse sequence (π/2-τ-π-τ-acq.) and chemical shifts are given relative to that of a 1 mol L -1 LiCl aqueous solution. The π/2 pulse length was 1.4 μs and the recycle delay was 1 s. X-ray photoelectron spectroscopy (XPS, PHI 5800 apparatus with a monochromatic Al K (1486.6 eV) anode (250 W, 10 kV, 27 mA)) was used in evaluating the chemical state of Mn cations.

LiMn 2 O 4 electrode was prepared by mixing 80% (w/w) of the active material, 10% (w/w) super C65 carbon (TIMCAL) and 10% (w/w) polyvinylidenefluoride (Solef PVdF 6020 binder, Solvay) in N-methyl-2-pyrrolidone (NMP, Sigma-Aldrich) to get a slurry. The cathode loading of 2 mg cm -2 was obtained with a 90 µm thin film coated on aluminum foil. This film was dried over night at 80 °C then electrodes were punched out with the diameter of 1.2 cm. After subsequent drying at 100 °C under vacuum, this electrode film was finally pressed at 8 tons. The cell was assembled using the two electrode Swagelok®-type test cells in an argon-filled glove box with lithium foil (Alfa Aesar) as anode, LP30 (1M LiPF 6 in ethylene carbonate (EC):dimethylcarbonate (DMC)1:1) as electrolyte and glass microfiber filters (Whatmann®-GF/D 70 mm Ø) as separator. A VMP3 multi-channel potentiostat (Bio-Logic, France) was used to electrochemically characterize the electrodes at 25 °C in the voltage range of 3.0-4.5 V with galvanostatic cycling and cyclic voltammetry (CV) techniques.

Results and discussion

Structural properties

XRD patterns for LiMn 2 O 4 synthesized by sol-gel method using citric acid and EDTA as chelating agents at calcination temperatures of 800 and 600 °C, respectively, are shown in Fig. 1a. All the diffraction peaks of the two samples are indexed according to the single phase cubic spinel phase with Fd3m (O 7 h ) space group [START_REF] Cai | Long cycle life, high rate capability of truncated octahedral LiMn 2 O 4 cathode materials synthesized by a solid-state combustion reaction for lithium ion batteries[END_REF] (JSCD No. 050427 card). The lattice parameters were calculated via a least-squares refinement method using 10 well-defined diffraction lines with indexation in the cubic system and the results are summarized in Table 1.

The main diffraction peaks are (111), ( 311) and (400), as expected for the spinel phase. It means that the lithium ions occupy the tetrahedral (8a) sites (Wyckoff notation) and manganese ions are distributed at octahedral (16d) sites, while oxygen anions are located at the (32e) sites [START_REF] Gao | Combustion-derived nanocrystalline LiMn 2 O 4 as a promising cathode material for lithium-ion batteries[END_REF][START_REF] Liu | Preparation of sub microcrystal LiMn 2 O 4 used Mn 3 O 4 as precursor and its electrochemical performance for lithium ion battery[END_REF]. No impurities such as Li 2 MnO 3 , LiMnO 2 , Mn 2 O 3 or MnO x were detected, which gives evidence of the high purity of the synthesized products. The sharpness of main peaks indicates that both samples are well crystallized [START_REF] Yuan | Synthesis and electrochemical investigation of nanosized LiMn 2 O 4 as cathode material for rechargeable hybrid aqueous batteries[END_REF]. However, the peak intensity and the full-width at halfmaximum (FWHM) are different for C-and E-LiMn 2 O 4 , which indicates a difference in crystallinity, particle size and ordering of local structure between the two samples. This difference can be evidenced by the use of the Scherrer formula [START_REF] Yia | A literature review and test: Structure and physicochemical properties of spinel LiMn 2 O 4 synthesized by different temperatures for lithium ion battery[END_REF]:

where B is the FWHM, θ is the diffraction angle, <e 2 > denotes local strains (defined as ∆d/d with d the interplanar spacing), L is the crystallite size and K =0.88 for a spherical crystallite shape.

The plot of B 2 cos 2 θ as a function of sin 2 θ is reported in Fig. 1b for C-and E-LiMn 2 O 4 samples.

The slope, 16<e 2 >, and intercept K 2 λ 2 /L 2 , were used to estimate the strain <e 2 > and coherence length L respectively (Table 1). We find that the crystallite is larger and the strain field smaller in the E-LiMn 2 O 4 sample. Therefore, we find that the result obtained for lamellar and silicate compounds according to which EDTA leads to better structural properties than the citric acid also applies to the LiMn 2 O 4 spinel. This property is usually attributed to the fact that the calcination temperature is smaller when EDTA is used as chelating agent. In the present case, however, an additional improvement is a modification of the valence mixing of manganese ions (Mn 3+ and Mn 4+ ). Since the ionic radius of Mn 4+ (0.53 Å) is smaller than that of Mn 3+ (0.645 Å), the "a" lattice parameter decreases when the Mn 3+ /Mn 4+ ratio decreases, ranging from 8.2104 Å to 8.2446 Å between Li 0.76 Mn 1.65 O 4 and Li 1.055 Mn 1.993 O 4 , respectively [START_REF] Feng | Synthesis and electrochemical characterization of nonstoichiometry spinel phase (Li x Mn 1.93 Y 0.02 O 4 ) for lithium ion battery applications[END_REF]. A similar lattice shrinkage associated to a decrease of the amount of Mn 3+ has been observed for Y 3+ substitution [START_REF] Balaji | A comprehensive study on influence of Nd 3+ substitution on properties of LiMn 2 O 4[END_REF] and Nd 3+ substitution [START_REF] Julien | Lattice vibrations of materials for lithium rechargeable batteries III. Lithium manganese oxides[END_REF] particles have a size in the range of 300-400 nm. It implies that the difference in the calcination temperature used in the two synthesis processes did not affect neither the shape nor the size of the particles, so that the differences between the electrochemical properties of these two cathode materials will be solely attributable to structural properties. -octahedra [START_REF] Julien | Lattice vibrations of materials for lithium rechargeable batteries I. Lithium manganese oxides[END_REF][START_REF] Julien | Lattice vibrations of materials for lithium rechargeable batteries. VI: Ordered spinel[END_REF]. The smaller broadening of the A 1g band in the E-LiMn 2 O 4 is thus an evidence of the smaller strain field in this sample, in agreement with the data in Table 1. The shoulder at 587 and 583 cm -1 respectively of the T (1) 2g mode in this region is not well separated because of its low intensity. In the localized-vibration approach it is speculated that the intensity of the Raman shoulder is closely related to the manganese average oxidation state in the spinel phase. Because the intensity of this shoulder is very sensitive to the lithium stoichiometry, it is suggested that its character originates mainly from the vibration of Mn 4+ -O bonding. The RS peak with medium intensity located at 486cm -1 has T (2) 2g symmetry where the two weak bands observed at 445, 378 cm -1 and 435, 386 cm -1 for C-and E-LiMn 2 O 4 samples, respectively, have the E g and T (3) 2g symmetry. The T (3) 2g mode is related to the Li-O motion, i.e. connected to the tetrahedral cation movements. A broad and very weak band at 156±2 cm -1 is attributed to the translation mode of lattice vibration, i.e. T 2g (T) mode. ion. The presence of this ion also breaks the translational invariance and is responsible for the larger number of vibration modes revealed by the decomposition of the spectra illustrated in Figs 4 and5.

The 7 Li MAS NMR spectra of the two LiMn 2 O 4 samples are shown in Fig. 6. They are dominated by a strong peak at 524 ppm. This peak belongs to the Li ions located on the tetrahedral (8a) sites of the spinel structure. The large shift is caused by the unpaired electron spin density that is transferred from the Mn ions via Mn-O-Li bonds to the Li nuclei [START_REF] Lee | 6 Li and 7 Li MAS NMR studies of lithium manganate cathode materials[END_REF][START_REF] Lee | 6 Li magic angle spinning nuclear magnetic resonance study of the cathode materials Li 1+α Mn 2-α O 4-δ. The effect of local structure on the electrochemical properties[END_REF][START_REF] Grey | NMR studies of cathode materials for lithium-ion rechargeable batteries[END_REF].

Fast electron hopping between the Mn ions results in an average oxidation state of the Mn ions of +3.5, as seen by NMR spectroscopy. Additional peaks are observed at 655 ppm, 601 ppm, and 564 ppm. These are caused by an increased number of Mn 4+ ions (with an average Mn oxidation state +3.56), as they occur in excess-Li spinels (Li 1.05 Mn 1.95 O 4 ) [START_REF] Lee | 6 Li magic angle spinning nuclear magnetic resonance study of the cathode materials Li 1+α Mn 2-α O 4-δ. The effect of local structure on the electrochemical properties[END_REF], which is also the actual composition of the C-LMO, while the E-LMO is stoichiometric, in agreement with XPS results.

The amount of Mn 4+ is higher in the C-LiMn Spinning sidebands are marked with an asterisk.

X-ray photoelectron spectroscopy (XPS) was employed in evaluating the chemical state of Mn cations. 9a and9b, respectively. Two consecutive slowly increasing (for charge) or decreasing (for discharge) plateaus, one at 3.97/4.12 V and the other at 4.11/3.97 V are observed for E-LiMn 2 O 4 (Fig. 7a).

In Fig. 9b, the two plateaus for charge and discharge of C-LiMn 2 O 4 are observed at 3.97/4.12 V Figure 10a illustrates the cycle performance of the prepared samples between 3.0 -4.5 V vs.

Li + /Li 0 at various C-rate. The cycleability of the LMO samples recorded at 0.5C rate over 35 cycles is shown in Fig. 10b. After 35 cycles in the 4 V region, the discharge capacity of the prepared E-LiMn 2 O 4 and C-LiMn 2 O 4 reduced to 96 and 85 mAh g -1 , respectively, corresponding to 78% and 70% of the initial capacity. It is obvious that capacity fades for the two prepared samples is more significant in the first 10 cycles, probably due to the cell formation, dissolution of cathode material and side reactions [START_REF] Kim | Spinel LiMn 2 O 4 nanorods as lithium ion battery cathodes[END_REF][START_REF] Wakihara | Structural stability in partially substituted lithium manganese spinel oxide cathode[END_REF]. The coulombic efficient is also reported in Fig. 10b. Indeed, the surface area of the SEI is increased when the particles are smaller, which will result in a smaller coulombic efficiency in the first cycle during the formation of the SEI.

However, for the subsequent cycles, this is the opposite: the coulombic efficiency is increased because it is easier to extract all the lithium from a small particle than from a big particle.

Discussion

According to our results, despite the beneficial effect of EDTA on the structural properties of LiMn 2 O 4 , the impact on the electrochemical properties is marginal. One reason is that this chelating agent did not modify the size of the particles, while better results would be expected with nanostructured particles [START_REF] Gao | Combustion-derived nanocrystalline LiMn 2 O 4 as a promising cathode material for lithium-ion batteries[END_REF][START_REF] Lin | Porous LiMn 2 O 4 cubes architecture with single-crystalline nanoparticles and exhibiting excellent cyclic stability and rate capability as the cathode of a lithium ion battery[END_REF][START_REF] Cai | Facile synthesis of LiMn 2 O 4 octahedral nanoparticles as cathode materials for high capacity lithium ion batteries with long cycle life[END_REF][START_REF] Zhao | A simple, low-cost and eco-friendly approach to synthesize single-crystalline LiMn 2 O 4 nanorods with high electrochemical performance for lithium-ion batteries[END_REF]. Another reason is that the choice of EDTA fails to stabilize the spinel structure. For this purpose, different methods can be used. Doping, particularly by substituting oxygen with sulfur stabilizes the spinel structure by reducing the octahedral symmetry [START_REF] Jiang | Plasma-assisted sulfur doping of LiMn 2 O 4 for highperformance lithium-ion batteries[END_REF][START_REF] Molenda | Structural and electrochemical characterization of sulphur-doped lithium manganese spinel cathode materials for lithium ion batteries[END_REF]. The capacity fading comes from the Jahn-Teller distortion associated with the Mn 3+ ions. The reduction of such distortions can be obtained by substitution of Mn 3+ by another trivalent ion. We have already mentioned Y 3+ and Nd 3+ [START_REF] Feng | Synthesis and electrochemical characterization of nonstoichiometry spinel phase (Li x Mn 1.93 Y 0.02 O 4 ) for lithium ion battery applications[END_REF][START_REF] Balaji | A comprehensive study on influence of Nd 3+ substitution on properties of LiMn 2 O 4[END_REF]; Ni 3+ substitution for Mn 3+ also stabilizes the spinel structure [START_REF] Hernán | Synthesis, characterization and comparative study of the electrochemical properties of doped lithium manganese spinels as cathodes for high voltage lithium batteries[END_REF]. Synergetic effects combining sulfur and nickel doping have been obtained [START_REF] Sun | Effect of sulfur and nickel doping on morphology and electrochemical performance of LiNi 0.5 Mn 1.5 O 4-x S x spinel material in 3-V region[END_REF][START_REF] Raja | Influence of S and Ni co-doping on structure, band gap and electrochemical properties of lithium manganese oxide synthesized by soft chemical method[END_REF][START_REF] Bakierska | Enhancement of electrochemical performance of LiMn 2 O 4 spinel cathode material by synergetic substitution with Ni and[END_REF]. To overcome the problem of the dilution of the manganese into the electrolyte when LiMn 2 O 4 is delithiated, a surface modification is efficient [START_REF] Wang | Sputtering graphite coating to improve the elevated-temperature cycling ability of the LiMn 2 O 4 electrode[END_REF][START_REF] Zeng | A novel coating onto LiMn 2 O 4 cathode with increased lithium ion battery[END_REF][START_REF] Wang | Improving the electrochemical properties of high-voltage lithium nickel manganese oxide by surface coating with vanadium oxides for lithium ion batteries[END_REF][START_REF] Jiang | Study on the enhanced electrochemical performance of LiMn 2 O 4 cathode material at 55 °C by the nano Ag-coating[END_REF][START_REF] Wang | Improvement of the overall performances of LiMn 2 O 4 via surface-modification by polypyrrole[END_REF][START_REF] Shang | Nano-TiO 2 (B) coated LiMn 2 O 4 as cathode materials for lithium-ion batteries at elevated temperatures[END_REF]. A combination of these strategies can be done to improve both the energy density and cycle life [START_REF] Molenda | An attempt to improve electrical conductivity of the pyrolyzed carbon-LiMn 2 O 4-y S y (0≤y≤0.5) composites[END_REF].

The electrochemical properties of any cathode material do not only depend on the choice of the chelating agent. It is well known that they also depend importantly on the morphology (size and shape) of the particles, and on the synthesis temperature. These parameters are not independent. Therefore, it is not possible to change the chelating agent and fix both the morphology and the synthesis temperature at the same time. Considering the products with similar particle size, it has been currently demonstrated that LiMn 2 O 4 with good crystallinity must be prepared at T>600 °C. For example, Jin et al. [START_REF] Jin | Synthesis of single-crystalline octahedral LiMn 2 O 4 as high performance cathode for Li-ion battery[END_REF] obtained well-shaped octahedral LMO grains by a two-step synthesis at 700 °C; Zhao et al. [START_REF] Zhao | Effects of equimolar Mg (II) and Si (IV) co-doping on the electrochemical properties of spinel LiMn 2-2x Mg x Si x O 4 prepared by citric acid assisted sol-gel method[END_REF] prepared LMO at 750 °C using a sol-gel method assisted by citric acid; calcination at 600 °C for 5h achieved nanocrystalline LMO (≈80 nm) prepared by combustion method [START_REF] Gao | Combustion-derived nanocrystalline LiMn 2 O 4 as a promising cathode material for lithium-ion batteries[END_REF]; LMO nanoparticles (165 nm average size) were obtained by precipitation method and sintering at 800 °C for 10 h [START_REF] Molenda | Structural and electrochemical characterization of sulphur-doped lithium manganese spinel cathode materials for lithium ion batteries[END_REF]; nanocrystalline LMO particles were prepared by an ultrasonic spray pyrolysis method using nitrate salts at 800 °C in air atmosphere [START_REF] Ebin | Comparison of 4 V and 3 V electrochemical properties of nanocrystalline LiMn 2 O 4 cathode particles in lithium ion batteries prepared by ultrasonic spray pyrolysis[END_REF]; the solid-state combustion method requested a calcination at 800 °C for 10 h to obtain grain size ≈100 nm; well-crystallized LMO synthesized by cellulose-citric acid method was sintered at 750-900 °C for 10 h [START_REF] Lin | Porous LiMn 2 O 4 cubes architecture with single-crystalline nanoparticles and exhibiting excellent cyclic stability and rate capability as the cathode of a lithium ion battery[END_REF]; a combustion method using urea as a fuel was developed to grow LMO at T≈500 °C [START_REF] Chitra | Physical properties of LiMn 2 O 4 spinel prepared at moderate temperature[END_REF]; submicronic LMO powders were prepared using a methanolic solution of metal acetates and succinic acid with sintering at 800 °C [START_REF] Prabaharan | Synthesis and electrochemistry of LiMn 2 O 4 prepared using succinic acid as complexing agent[END_REF]; microwave-heated LMO powders were prepared at 800 °C by Fu et al. [START_REF] Fu | Comparison of the microwave-induced combustion and solid state reaction for the synthesis of LiMn 2 O 4 powder and their electrochemical properties[END_REF]; homogeneous particles (≈300-400 nm sized) were prepared by micro-emulsion method with sintering at 800 °C [START_REF] Ke | Synthesis of spinel LiMn 2 O 4 with manganese carbonate prepared by micro-emulsion method[END_REF]; 300-nm LMO nanoparticles were grown by hydrothermal synthesis following by sintering at 700 °C for 10 h [START_REF] Lin | Porous LiMn 2 O 4 cubes architecture with single-crystalline nanoparticles and exhibiting excellent cyclic stability and rate capability as the cathode of a lithium ion battery[END_REF]; LMO submicron-crystal LMO were prepared by ball milling and heat treated in the range 750-900 °C [START_REF] Rajesh | Sol-gel synthesis and characterization of nano crystalline spinel LiMn 2 O 4 for battery application[END_REF]; Molenda et al. [START_REF] Molenda | Structural and electrochemical characterization of sulphur-doped lithium manganese spinel cathode materials for lithium ion batteries[END_REF] reported the growth of nanosized LMO using a modified sol-gel method followed by calcination at 650 °C. To conclude, there is an experimental fact that 650 °C is the lowest temperature to prepare nanocrystalline LMO.

Here, we have chosen to fix the morphology, and thus adjust the synthesis temperature of the C-LiMn 2 O 4 to reproduce the same shape and size as the E-LiMn 2 O 4 prepared at 650 °C. We had to rise the temperature to 800 °C for the C-LiMn 2 O 4 for this purpose. The other approach would be to fix the synthesis temperature to 650 °C, but in that case, the morphology will change if the same synthesis process is kept. However, it is possible to find a synthesis process that gives a material synthesized at the same temperature 650 °C and morphology comparable to that of the E-LiMn 2 O 4 sample. This mode of preparation is the rheological-phase-assisted microwave synthesis method [START_REF] Cui | Synthesis and electrochemical property of LiMn 2 O 4 porous hollow nanofiber as cathode for lithium-ion batteries[END_REF]. Using this method, Cui et al. have investigated the effect of the temperature in the range 600 -900 °C, which is the typical range used to synthesize the spinel by different processes. As a result, they found that this synthesis is superior in terms of regular shapes with respect to other synthesis methods including solid-state reaction sol-gel synthesis, precipitation, and Pechini process. Although the particles have a more cubic shape than in the E-LiMn 2 O 4 sample, the size of the particles is comparable for a synthesis temperature. However, in this case the initial capacity is 100 mAh g -1 , and more importantly reduces to circa 45 mAh g -1 after 25 cycles. The measurements in [START_REF] Cui | Synthesis and electrochemical property of LiMn 2 O 4 porous hollow nanofiber as cathode for lithium-ion batteries[END_REF] have been made at rate C/3 rate. The rate capability of the E-LiMn 2 O 4 , reported in Fig. 8b up to 0.5C rate, shows the superiority of the E-LiMn 2 O 4 with respect to the results in [START_REF] Cui | Synthesis and electrochemical property of LiMn 2 O 4 porous hollow nanofiber as cathode for lithium-ion batteries[END_REF], which illustrates the efficiency of the sol-gel synthesis with the chelating agent chosen in the present work.

Conclusion

Single phase cubic spinel LiMn 2 O 4 phase with a space group Fd3m was obtained in this study by using EDTA and citric acid as chelating agents. Well-crystallized spinel powders were obtained after calcination at 650 °C by using EDTA as chelating agent, while a sintering at 800 °C was performed to get the same morphology and size of LiMn 2 O 4 particles prepared via citric acid. This reduction in preparation temperature makes this technique cost-effective. Due to its molecular configuration of EDTA, i.e. four carboxylic and two amine functions, its role as a hexadentate chelating agent provide material with better structural properties. Even if the use of EDTA as chelating agent improves the structural stability and increases slightly the capacity, the results in the present work show that the surface modifications and doping of nanostructured LiMn 2 O 4 are mandatory to obtain a cathode that is competitive for commercial use. Among the wet-chemical methods, the EDTA-assisted synthesis saves both reaction time and energy. It also gives superior electrochemical properties with respect to particles of the same size and shape prepared by other techniques or with respect to LiMn 2 O 4 particles synthesized at the same temperature.

Figure Captions

Fig. 1 (

 1 Fig. 1 (a) XRD diffraction patterns of C-LiMn 2 O 4 and E-LiMn 2 O 4 . (b) Analysis of the fullwidth at half maximum B of the Bragg peaks.

Fig. 2 .Fig. 3 .Fig. 4 .

 234 Fig. 2. SEM images of (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 samples.

Fig. 5 .

 5 Fig. 5. FTIR spectra of (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 prepared by sol-gel method using chelating agent citric acid and EDTA, respectively.

  2 O 4 sample in comparison with the E-LiMn 2 O 4 sample, which illustrates that the NMR measurements are actually more sensitive than XRD, since the error in the determination of the lattice parameter of C-LiMn 2 O 4 did not allow us to determine the Mn 4+ /Mn 3+ ratio in this sample.

Fig. 6 . 7

 67 Fig. 6. 7 Li MAS NMR spectra of the two LiMn 2 O 4 samples (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 .

Figure 7

 7 displays the XPS spectra of Mn 2p 3/2 and Mn2p 1/2 core levels for (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 samples. Deconvolution was performed with the binding energy positions of Mn 3+ and Mn 4+ ions taken at 642.1 and 643.6 eV, respectively. These binding energies correspond to that of MnIII 2 O 3 and of Mn IV O 2 as reported by Ramana et al.[START_REF] Ramana | XPS and Raman spectroscopic characterization of LiMn 2 O 4 spinels[END_REF] and Shin et al.[START_REF] Shin | XPS/EXAFS study of cycleability improved LiMn 2 O 4 thin films cathodes prepared by solution deposition[END_REF]. The analysis shows that the valence state of Mn in C-LiMn 2 O 4 is 3.56 against 3.505 in E-LiMn 2 O 4 . These results confirm the modification of the cubic lattice parameters observed by XRD and are in good agreement with the NMR measurements.

Fig. 7 .

 7 Fig. 7. XPS spectra of Mn 2p 3/2 and Mn2p 1/2 core levels for C-LiMn 2 O 4 and E-LiMn 2 O 4 samples.Deconvolution was performed with the binding energy positions of Mn 3+ and Mn 4+ ions taken at 642.1 and 643.6 eV, respectively.

and 4 .

 4 18/4.0 V, respectively. These voltage plateaus are commonly observed for LiMn 2 O 4 -based electrodes. They are attributed to the two-step cubic phase transformations of LiMn 2 O 4 /Li 0.5 Mn 2 O 4 and Li 0.5 Mn 2 O 4 /λ-MnO 2 during oxidation and the reverse during reduction [44].

Fig. 8 .

 8 Fig. 8. Cyclic voltammograms recorded at a scan rate of 0.05 mV s -1 in the voltage range 3.5-4.5 V vs. Li + /Li 0 for (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 .

Fig. 10 .

 10 Fig. 10. Cyclic performance of C-LiMn 2 O 4 and E-LiMn 2 O 4 samples: (a) Specific discharge capacity at various C-rates. (b) Specific discharge capacity vs. cycle number at rate.

Fig. 1 .

 1 Fig. 1. (a) X-ray diffraction patterns of C-LiMn 2 O 4 and E-LiMn 2 O 4 . (b) Analysis of the fullwidth at half-maximum B of the Bragg reflections.

Fig. 2 .

 2 Fig. 2. SEM images of (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 samples.

Fig. 3 .

 3 Fig. 3. TEM images of (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 samples.

Fig. 4 .

 4 Fig. 4. Raman spectra for LiMn 2 O 4 powders with chelating agent (a) citric acid and (b) EDTA prepared by sol-gel method.

Fig. 5 .

 5 Fig. 5. FTIR spectra of (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 prepared by sol-gel method using chelating agent citric acid and EDTA, respectively.

Fig. 6 . 7

 67 Fig. 6. 7 Li MAS NMR spectra of the two LiMn 2 O 4 samples (1) C-LiMn 2 O 4 and (2) E-LiMn 2 O 4 . Spinning sidebands are marked with an asterisk.

Fig. 7 .

 7 Fig. 7. XPS spectra of Mn 2p 3/2 and Mn2p 1/2 core levels for C-LiMn 2 O 4 and E-LiMn 2 O 4 samples.Deconvolution was performed with the binding energy positions of Mn 3+ and Mn 4+ ions taken at 642.1 and 643.6 eV, respectively.

Fig. 8 .Fig. 9 .Fig. 10 .

 8910 Fig. 8. Cyclic voltammograms recorded at a scan rate of 0.05 mV s -1 in the voltage range 3.5-4.5 V vs. Li + /Li 0 of (a) C-LiMn 2 O 4 and (b) E-LiMn 2 O 4 .

Table 1 .

 1 for Mn 3+ in LiMn 2 O 4 . The value of a=8.2311(1) Å in the E-LiMn 2 O 4 is the value expected for stoichiometric LiMn 2 O LiMn 2 O 4 makes difficult the estimation of the deviation from stoichiometry if any. Actually, this larger error is another evidence of the stronger local disorder of the lattice in this sample. XRD Parameters for LiMn 2 O 4 materials synthesized by wet chemistry assisted citric acid and EDTA.

[START_REF] Gao | Combustion-derived nanocrystalline LiMn 2 O 4 as a promising cathode material for lithium-ion batteries[END_REF]

[START_REF] Yia | A literature review and test: Structure and physicochemical properties of spinel LiMn 2 O 4 synthesized by different temperatures for lithium ion battery[END_REF]

and we can estimate that the average valence for this sample is +3.5. Note that the much larger error ±0.007 Å in the determination of the lattice parameter of sample C-

Acknowledgement

Some part of this work has benefited from financial support within the BMBF project "DESIREE", grant number 03SF0477B.


Gawtum Namah

ON THE UNIQUENESS OF TRAVELING FORCED CURVATURE FRONTS IN A FIBERED MEDIUM

Keywords: Mathematics Subject Classification. 35K55, 35B10 Traveling wave solutions, pulsating fronts, periodic fibered medium. 1

We investigate traveling fronts, including pulsating ones, of a forced curvature flow in a plane fibered medium. The main topic of this note is an uniqueness issue of such traveling fronts. In addition to line-shaped profiles, we also consider traveling fronts in the form of V-shaped parabolas.

Introduction

In this note, we will be interested in traveling fronts of a forced curvature flow equation [START_REF] Brauner | Propagation of a combustion front in a striated solid medium: a homogenization analysis[END_REF] V n = R + K in the plane containing periodic striations. V n is the normal velocity of a propagating interface Γ(t), K is its mean curvature and R is the driving force. For example if Γ is a flame front, then R corresponds to the combustion rate of the burning material. In all cases, we will suppose that the function R is smooth and verifies

(2) 0 < R m ≤ R ≤ R M .
Before going further, let us give a definition of a traveling front of equation [START_REF] Brauner | Propagation of a combustion front in a striated solid medium: a homogenization analysis[END_REF].

Definition 1. Γ(t), solution of (1) will be called a traveling front if there exists a constant vector v ∈ R 2 such that Γ(t) = Γ 0 + v t for all t ∈ R. Then Γ 0 is the (constant) profile of the traveling front and |v|, its speed, see Figure 1.

Note that if Γ(t) can be represented by the graph of a function u in the x-y plane, for example Γ(t) = {(x, y)/y = u(x, t)}, then V n is given by

V n = u t 1 + u 2
x , so that equation ( 1) becomes

(3)

u t -R 1 + u 2 x = u xx 1 + u 2 x , t ∈ R, x ∈ R.
Now if Γ(t) is a traveling front in the plane, we can suppose without loss of generality that v is parallel to the y-axis i.e. v = t (0, c). Then u(x, t) will be given by u(x, t) = c t + ϕ(x) so that equation (3) becomes [START_REF] Dirr | Pulsating wave for mean curvature flow in inhomogeneous medium[END_REF] c -R 1 + ϕ 2 x =

In the above, c is the speed and ϕ the constant profile of the wave. The pair (c, ϕ) will be called a traveling wave solution (TWS) of equation [START_REF] Chen | Periodic travelling wave solutions of a parabolic equation: a monotonicity result[END_REF]. Note that every solution ϕ of ( 4) is defined up to an additive constant.

y x → V Γ(t) Γ 0 | → V |t Figure 1.
A TWS: a constant profile moving with a constant speed in some given direction.

In [START_REF] Ninomiya | Traveling curved fronts of a mean curvature flow with constant driving force, Free Boundary Problems : Theory and Applications I[END_REF], the authors gave a complete classification of smooth traveling fronts in the plane in the case of a constant R i.e. for propagations in homogeneous media. They notably showed that a non stationary traveling front in the plane can only be a graph in the form of a line or of a V-shaped parabola. Different results have also been proved in the case of an interface propagating in a fibered medium with periodic striations (i.e. R is periodic in one direction). In this case, one of course expects the lines and V-shaped fronts to become undulated, the results depending also on the angle of inclination of the striations. Let us precise what we mean by an undulated line-shaped front. Definition 2. An undulated front will be termed line-shaped if there exists an angle α ∈ (-π/2, , π/2) such that the front is at a finite distance from the line y = tan α x for all x ∈ R.

The line-shaped fronts can therefore be inclined (when α = 0) or horizontal (α = 0). Essentially two types of solutions have been put forward in the case of a periodic fibered medium : traveling wave solutions (TWS) and pulsating traveling wave solutions (PTWS). Besides having some periodicity in space, these pulsating fronts also display a time periodicity namely there exists a period T and some constant Y such that u(x, t + T ) = u(x, t) + Y, for all t ∈ R and x ∈ R.

Let us point out that equations of type (1) may exhibit PTWS in other contexts where the periodicity is not necessarily carried by the function R. See for example [START_REF] Matano | Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit[END_REF] and the references therein where it is the boundary of the corresponding domain which has periodic or quasi-periodic undulations while R remains constant.

In [START_REF] Chen | Wave propagation under curvature effects in a heterogeneous medium[END_REF], which deals with vertical striations (R(x, y) = R(x), 1-periodic), it is shown that there exists a unique 'horizontal' line-shaped traveling front (c, ϕ), ϕ 1-periodic and c being the speed of the wave in the y direction. The latter traveling front corresponds therefore to the undulated version of the horizontal line in the case of a constant R.

In the more recent work [START_REF] Lou | Traveling waves of a curvature flow in almost periodic media[END_REF], the above result has been generalized to inclined fronts. The existence of inclined traveling fronts in the case of vertical striations is studied in an almost periodic context i.e. with R almost periodic in x. Among other results, it is shown that for any α ∈ (-π/2, π/2), α being the angle between the front line and the x-axis, there exists a unique (c α , ϕ α ), ϕ α

x almost periodic in x, solution of (4). The previous result easily leads to the existence of a unique traveling front Γ α given by the pair (c α , ϕ α ), with ϕ α

x periodic in the case of a periodic function R. Therefore we have a traveling front with Γ 0 = {y = ϕ α (x)} and v = t (0, c α ).

In the case of oblique striations treated in [START_REF] Namah | Convergence to periodic fronts in a class of semilinear parabolic equations[END_REF] and [START_REF] Chen | Periodic travelling wave solutions of a parabolic equation: a monotonicity result[END_REF], the authors put in evidence the fact that periodicity in space generates a time periodic regime. More precisely, let us consider periodically disposed oblique striations of period 1 and inclined by an angle β ∈ (0, π/2) from the y-axis. Set X β = 1/ cos β and Y β = 1/ sin β, so that R is X β -periodic in x and Y β -periodic in y. Then there exists a unique T β and a unique function ϕ β (x, t), T β -periodic in time and X β -periodic in x such that

u β (x, t) = c β t + ϕ β (x, t),
is a solution of (3),with c β = Y β /T β . We thus have (see Figure 2)

u β (x, t + T β ) = u β (x, t) + Y β .
Note that the uniqueness holds up to addition of constants in space and translations in time. The pair (c β , ϕ β ) or the moving front u β will be called a pulsating traveling wave solution (PTWS) of (3). Here c β represents the mean speed (over one time period) of the wave propagation in the y direction.

∃T β s.t. Γ(t + T β ) = Γ(t) + Y β ∀t.
Note that, as u β does not have a constant profile, it is not clear here whether this pulsating front is a traveling wave solution. That this is the case will be the object of the following result : Claim 1. Let Γ(t) = {y = u(x, t)} be a space-periodic solution of (1) in a periodic fibered medium such that

u(x, t + T ) = u(x, t) + Y, x ∈ R, t ∈ R,
for some given T > 0 and Y ∈ R. Then Γ(t) slides with a constant speed along the direction of the striations.

The above result implies that any periodic pulsating front is a traveling wave solution i.e. propagating with a constant speed in a given direction, here that of the striations. Now we want to enquire about the 'uniqueness' of these TWS. More precisely, consider a vertically striated medium in the x-y plane. Recall that for any given α ∈ (-π/2, π/2), there exists a line-shaped traveling front Γ α in the form of a graph {y = ϕ α (x)} traveling with speed c α in the y-direction. Can there exist a line-shaped traveling wave Γ (t) = (Γ 0 , v ) other than the (Γ α ) α ? For example Γ may be a graph in some frame x -y other than x-y and/or v may not be parallel to the y axis.

Our aim in this paper is to clarify this uniqueness issue. We will notably show Theorem 1. Let R = R(x), 1-periodic in the x-y frame. Let Γ (t) be a traveling wave solution of (1) in the form of a line-shaped graph in some x -y frame. Then there exists α ∈ (-π/2, π/2), such that Γ = Γ α .

Therefore the (Γ α ) α are the only line-shaped traveling fronts which can exist. But as pointed out before, there exist also traveling fronts in the form of V-shaped graphs. Let us recall that in the homogeneous case (R = constant), (cf. [START_REF] Ninomiya | Traveling curved fronts of a mean curvature flow with constant driving force, Free Boundary Problems : Theory and Applications I[END_REF]), for any α ∈ (0, π/2) there exists a unique function ψ α in the form of a V-shaped parabola which has as asymptotes ψ + (x) = tan α x and ψ -(x) = -tan α x when x -→ ±∞, such that Γ 0 = {y = ψ α (x)} and c α = R/ cos α.

Note that such a Γ 0 is symmetrical wrt the y axis and ψ α x (0) = 0. The result corresponding to a striated medium goes as follows:

Theorem 2. Let R = R(x) be 1-periodic. Let α ∈ (0, π/2) satisfy R M cos α < R m .
Then for any s ∈ R, (4) admits a unique V-shaped traveling wave solution ψ α such that ψ α (0) = s and ψ α x (0) = 0.

In section 2, we will deal with line-shaped traveling fronts. We will notably show that periodic pulsating fronts (other than straight lines) in a periodic fibered medium are just traveling fronts which slide down along the direction of the striations (Claim 1) and then prove Theorem 1. In section 3, we will prove Theorem 2 concerning V-shaped traveling fronts.

Line-shaped traveling fronts

2.1. The case of a homogeneous medium. Let us start by noting that when R is a constant, there is an infinite number of lines which are traveling fronts. Indeed for any α ∈ (-π/2, , π/2), the moving line Γ

(t) = Γ 0 + v t with Γ 0 = {y = tan α x)} and v = t (-R sin α, R cos α),
is a solution. Moreover the velocity v is not unique. In the above, we see that the line propagates perpendicular to itself but we could also have considered propagations in other directions. For example if we consider the line to be propagating in the y-direction, then v = t (0, R/ cos α). Likewise v = t (-R/ sin α, 0) for a propagation in the x-direction. Also, in this simple case, Γ 0 can also be written as a function of y i.e. Γ 0 = {x = cot α y)} for any α = 0.

2.2. The case of a periodic striated medium. Let us now suppose that R is 1-periodic in some direction in the plane, which can be taken to be the x direction without any loss of generality, and is constant along the y direction. We look for line-shaped traveling wave solutions of (4). We state the following Proposition 1. Let R = R(x), 1-periodic. For any α ∈ (-π/2, , π/2), there exists a unique line-shaped TWS Γ α in the form of a graph y = u(x, t), with u satisfying equation (3) and the shifted periodic condition

(5) u(x + 1, t) = u(x, t) + tan α for all real t.
In other words, there exists a unique pair (c, ϕ) solution of

c -R 1 + ϕ 2 x = ϕ xx 1 + ϕ 2 x , x ∈ R, such that ϕ(x + 1) = ϕ(x) + tan α.
Moreover we have the following estimates on c and ϕ :

R m cos α ≤ c ≤ R M cos α , |ϕ x (x)| < R 2 M R 2 m cos 2 α -1.
Of course, c and ϕ depend on α. We have omitted the superscript α as there is no ambiguity. As usual, the uniqueness of ϕ is to be understood up to an additive constant. The traveling front is therefore either periodic (when α = 0) or shiftedperiodic in which case ϕ is not periodic but it is ϕ x which is a 1-periodic function with

1 0 ϕ x dx = tan α.
Proof. Consider the ode satisfied by h = ϕ x and set

f (x, h, c) = c(1 + h 2 ) -R (1 + h 2 ) 3/2 .
Then showing the above proposition comes to proving that (6)

   h = f (x, h, c), x ∈ (0, 1) h(0) = h(1)
1 0 h dx = tan α admits a unique solution (c, h). The proof will be omitted as it goes almost along the same lines as in [START_REF] Chen | Wave propagation under curvature effects in a heterogeneous medium[END_REF] which was run for α = 0.

We can further estimate h and give a monotonicity result on c under some restrictions on α. We state Proposition 2. Let (c, h) be the solution of [START_REF] Lou | Traveling waves of a curvature flow in almost periodic media[END_REF].

Then if R M cos α < R m , we have h(x) ≥ R 2 m R 2 M cos 2 α -1, for all x ∈ [0, 1],
and c (α) > 0.

Proof. Here too, we will omit the proof as it is similar to the one given in [START_REF] Lou | Traveling waves of a curvature flow in almost periodic media[END_REF] in the case of almost periodic striations.

Remark 1. Let us remark that the result of Proposition 1 excludes the case of α = ±π/2. In fact, we know that for α = π/2 (see for example [START_REF] Brauner | Propagation of a combustion front in a striated solid medium: a homogenization analysis[END_REF]), there exists no TWS but a unique pulsating front which is a straight line propagating with an effective speed given by the harmonic mean R of R. We thus have a front of the form x = U (t) with U (t + T ) = U (t) + 1 with T given by T = 1/R.

2.3.

Proof of Claim 1. Consider a propagation through a 1-periodic fibered medium with the striations inclined by some angle β ∈ (-π/2, π/2) from the y axis, see Figure 2. Here equation ( 3) reads ( 7)

u t -R(x, y = u(x, t)) 1 + u 2 x = u xx 1 + u 2 x , t ∈ R, x ∈ R with (8) R(x, y) = R(y sin β -x cos β).
Then we know (see for example [START_REF] Chen | Periodic travelling wave solutions of a parabolic equation: a monotonicity result[END_REF], [START_REF] Namah | Convergence to periodic fronts in a class of semilinear parabolic equations[END_REF]) that there exists a unique c β > 0 and a unique space-time periodic function ϕ β such that

(9) u(x, t) = c β t + ϕ β (x, t)
is a solution of [START_REF] Matano | Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit[END_REF]. The front u is therefore a periodic pulsating one i.e. there exists

T β = Y β /c β such that u(x, t + T β ) = u(x, t) + Y β .
We proceed by the following Lemma 1. Let u be a periodic pulsating front solution of (7)- [START_REF] Namah | Convergence to periodic fronts in a class of semilinear parabolic equations[END_REF]. Then there exists a X β -periodic function W and a positive constant c such that u(x, t) = W (x -c t sin β) + c t cos β.

Note that Lemma 1 implies that the front slides along the direction of the striations with the constant speed c without changing its profile as in Figure 2. We are therefore in the presence of a traveling front with Γ 0 = {y = W (x)} and v = t (c sin β, c cos β), in the x-y plane.

Proof. So let u = u(x, t) be a PTWS of ( 7)- [START_REF] Namah | Convergence to periodic fronts in a class of semilinear parabolic equations[END_REF]. Note that in the present case, R is also given by [START_REF] Xin | Front propagation in heterogeneous media[END_REF] R(x, y) = R(x + s sin β, y + s cos β), for all s ∈ R.

Consider now, for s ∈ R, the function

(11) W s (x, t) = u(x -s sin β, t) + s cos β.
It is then not difficult to verify that W s satisfies equation [START_REF] Matano | Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit[END_REF]. Indeed, by taking the equation ( 7) at the point (x = x -s sin β, t = t), we get

W s t (x, t) -R(x -s sin β, y = u(x -s sin β, t)) 1 + W s2 x = W s xx 1 + W s2 x , t ∈ R, x ∈ R.
But as R is constant in the direction of the striations, we have

R(x -s sin β, y = u(x -s sin β, t)) = R(x, u(x -s sin β, t) + s cos β) = R(x, W s (x, t)),
so that finally W s satisfies

W s t (x, t) -R(x, W s (x, t)) 1 + W s2 x = W s xx 1 + W s2 x , t ∈ R, x ∈ R.
Moreover we have

W s (x, t + T β ) = W s (x, t) + Y β .
Therefore W s is PTWS of [START_REF] Matano | Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit[END_REF]. By the uniqueness result of [START_REF] Namah | Convergence to periodic fronts in a class of semilinear parabolic equations[END_REF], we deduce that for any real s, W s is just a shift in time of u, i.e. there exists (a constant) τ (s) such that W s (x, t) = u(x, t + τ (s)).

We conclude from (11) that for all s ∈ R, we have the following oblique shift for u :

u(x, t + τ (s)) = u(x -s sin β, t) + s cos β.
Now due to the continuity of τ with respect to s and the fact that τ (s) -→ ±∞ when s goes to ±∞, we can deduce that for any real τ, there exists a real s such that the above equation holds. Then for all real τ, the front at time t + τ is just an oblique shift of the front at time t along the direction of the striations. Let W = W (z) be the profile of the pulsating front at some reference time set to zero for simplicity. Then the front at any time t > 0 is obtained by a shift of W along the direction of the striations by a quantity s = s(t), i.e.

u(x, t) = W (x -s(t) sin β) + s(t) cos β.
As u satisfies [START_REF] Matano | Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit[END_REF], W is therefore a solution of

s (t){cos β -sin βW z } -R(z + s(t) sin β, W (z) + s(t) cos β) 1 + W 2 z = W zz 1 + W 2 z ,
which from [START_REF] Xin | Front propagation in heterogeneous media[END_REF] gives

s (t){cos β -sin βW z } -R(z, W (z)) 1 + W 2 z = W zz 1 + W 2 z , t > 0, z ∈ R.
But as W is independent of time, we deduce that s (t) is a constant so that there exists some real c with s(t) = c t. We finally end up with u(x, t) = W (x -c t sin β) + c t cos β, and the proof of the Lemma is done and so is that of Claim 1. Note that in terms of c β (which we recall was the mean speed in the y direction), we have c β = c cos β so that v = t (c β tan β, c β ).

2.4. Proof of Theorem 1. Let Γ (t) be a TWS of (3) in the form of a line-shaped graph in some tilted x -y frame and let β ∈ (-π/2, π/2), β = 0, be the angle between the x and the x axes, i.e. between the two frames. We will deal with the case β = ±π/2 afterwards. Recall that the striations are vertical in the original x-y frame so that they will be oblique inclined by an angle β from the y -axis.

Set Γ (t) = {y = u(x , t)}. The front u is therefore a solution of ( 7)-( 8). Now we can have a 'horizontal' front in the x -y frame in which case u will be periodic in x , or an 'inclined front'. We will first show that the traveling wave front Γ (t) is in fact a PTWS which propagates along the direction of the striations and then prove that it coincides with one of the fronts Γ α of Proposition 1. Let us start by considering the case of a horizontal front.

2.4.1. Case 1 : Γ horizontal in the tilted frame. Note that as Γ is a TWS, it propagates with a constant speed |v| but we do not know at this stage whether the direction of v is the same as that of the striations. But since u is periodic in x , we can say that there exists some T > 0 (in fact here T will be given by X β /v x where v x is the component of the velocity v in the x direction) such that u(x , t + T ) = u(x , t) + Y with Y = v y T. There is therefore a vertical shift (in the direction of y ) of the front after time T. But as u satisfies equation ( 7), we conclude that Γ (t) is a PTWS of [START_REF] Matano | Periodic traveling waves in a two-dimensional cylinder with saw-toothed boundary and their homogenization limit[END_REF]. Then we deduce by Claim 1 that v is parallel to the direction of the striations. Moreover by the uniqueness of the PTWS, we know that v = t (c β tan β, c β ) with c β as previously defined. Now in the context of the original x-y frame, Γ is a traveling line-shaped front inclined by an angle β from the x axis and propagating with a constant speed in the y direction. Now it suffices to prove that Γ is a graph in the x-y frame and conclude by the uniqueness result of Proposition 1 that Γ coincides with Γ β . Take t = 0 as a reference time and set Γ 0 = Γ (0). Define u 0 (x ) = u(x , 0) so that Γ 0 = {y = u 0 (x )}. We will prove that Γ 0 is a graph in the x-y coordinate frame. Indeed any point (x , y ) in the x -y frame has as coordinates x = cos β x -sin β y y = sin β x + cos β y in the x-y coordinate frame so that any point (x , u 0 (x )) on Γ 0 reads

x = cos β x -sin β u 0 (x ) = f (x ) (12) y = sin β x + cos β u 0 (x ) = g(x )
in the x-y frame. A sufficient condition to be able to write y as a function of x is that f x (x ) = 0 for all x . This ensures that we cannot have two values of x corresponding to the same x. Otherwise, one can easily verify that this latter situation would lead to the case of having one value of x corresponding to two different values of y.

We see from (12) that showing that f x (x ) does not vanish comes to proving that u 0x (x ) = cot β for all x . We therefore look for estimates on u 0x . As we know that Γ (t) slides in the direction of the striations with the speed c = c β sec β, we have for all t > 0 (t = 0 being the reference time)

u(x , t) = u 0 (x -c t sin β) + c t cos β, that is u(x , t) = u 0 (x -c β t tan β) + c β t.
We remark that the space-time periodic function ϕ β (x , t) given in ( 9) is constant along the line y = x -c β t tan β and is given by

ϕ β (x , t) = u 0 (x -c β t tan β).
By setting z = x -c β t tan β, we see from ( 7) that u 0 satisfies

c β -c β tan βu 0z -R (z, u 0 (z)) 1 + u 2 u0z = ν u 0zz 1 + u 2 0z
subject to the periodic conditions

u 0 (0) = u 0 (X β ) and u 0z (0) = u 0z (X β ). Setting h = u 0z , we obtain      h = (1 + h 2 )(c β -c β tan β h -R √ 1 + h 2 ), z ∈ (0, X β ) h(0) = h(X β ) X β 0 h dz = 0. Set h M = max z h(z) = h(z M
), the existence of z M being ensured by the periodicity of h. By writing the equation at z = z M , we have

0 = (1 + h 2 M )(c β (1 -tan β h M ) -R 1 + h 2 M ).
Suppose now that h M ≥ cot β. As c β and R are positive, the RHS will be negative which is not possible. Therefore we end up with h(z) < cot β for all z ∈ R.

We have thus proved that Γ (t) can be written as a line-shaped graph in the x-y coordinate frame sliding in the direction of the striations with a constant speed. From the uniqueness result of Proposition 1, we conclude that Γ (t) is a traveling front which coincides with one of the Γ α (t) and here precisely with Γ β .

Case 2:

Γ inclined in the tilted frame. Now let us consider the case where Γ (t) = {y = u(x , t)} is an 'inclined' TWS in some x -y coordinate frame traveling with velocity v = (v x , v y ). Let β be, as before, the angle between the tilted and the original frame and denote by γ, the angle of inclination of the front with respect to the x -axis, see Figure 3. Recall that u satisfies the same equation

(13) u t -R(x , y = u(x , t)) 1 + u 2 x = u x x 1 + u 2 x , t ∈ R, x ∈ R,
as for the previous case but this time with shifted periodic conditions, namely

(14) u(x + X γ , t) = u(x , t) + tan γ X γ
for some X γ . Here too, the periodicity in space (even if it is a shifted periodicity) will lead to a vertical shift of the front's profile after some time T , i.e. Γ (t + T ) is a vertical shift of Γ (t) for all t, so that

(15) u(x , t + T ) = u(x , t) + Y ,
for some Y which depends a priori on γ and on the velocity v . Here T will be given by T = X γ /v x . Let us start by remarking that, due to the periodicity of R, if T exists then Y is necessarily a multiple of Y β . Indeed, consider the equation (13) at the point (x , t + T ). We have

u t (x , t + T ) -R(u(x , t + T ) sin β -x cos β) 1 + u 2 x (x , t + T ) = u x x 1 + u 2 x (x , t + T ).
But as u t (x , t + T ) = u t (x , t) and likewise for the derivatives in x , we have

u t (x , t) -R(u(x , t) sin β -x cos β + Y sin β) 1 + u 2 x (x , t) = u x x 1 + u 2 x (x , t), so that necessarily Y sin β ∈ Z. Thus Y = m sin β = mY β for some m ∈ Z.
The next step will be to verify that the inclined pulsating front slides along the direction of the striations with a constant speed (given by Claim 1 for the 'horizontal' case). Suppose this is true. Then we can run the proof almost along the same lines as that for the horizontal case to show that Γ (t) is a graph in the original x-y frame. Then from the uniqueness result of Proposition 1, we can again conclude that Γ (t) coincides with one of the Γ α (t), namely Γ β+γ (t) here. It now remains to verify that Claim1 is also valid for an inclined front. What is important is the uniqueness result for pulsating fronts. In Claim 1, the uniqueness result called for had been proved for periodic fronts (cf. [START_REF] Namah | Convergence to periodic fronts in a class of semilinear parabolic equations[END_REF], [START_REF] Dirr | Pulsating wave for mean curvature flow in inhomogeneous medium[END_REF]). Also in [START_REF] Namah | Convergence to periodic fronts in a class of semilinear parabolic equations[END_REF], the analysis was carried out for viscous Hamilton-Jacobi equations. In fact, the result holds in the present case too. For completeness sake, let us give a proof of the uniqueness of the pulsating front in the inclined case.

Suppose there exist two inclined pulsating fronts u 1 and u 2 solutions of ( 13)-( 14). We know that we can write

u 1 (x , t) = c 1 t + ϕ 1 (x , t) u 2 (x , t) = c 2 t + ϕ 2 (x , t)
with ϕ 1 , T 1 -periodic (resp. ϕ 2 , T 2 -periodic) in time and both satisfying the shifted periodic condition (14). Let

t 1 = sup{t ∈ R / u 1 (., t) < u 2 (., 0)} and t 2 = inf{t ∈ R / u 1 (., t) > u 2 (., 0)}
The quantity t 1 appears therefore as the first time that u 1 (., t) touches u 2 (., 0) from below and t 2 is the last time that u 1 (., t) touches u 2 (., 0) from above. We therefore have u 1 (x , t 1 ) ≤ u 2 (x , 0) and u 1 (x , t 2 ) ≥ u 2 (x , 0)

The weak maximum principles then lead to

u 1 (x , t 1 + t) ≤ u 2 (x , t) ≤ u 1 (x , t 2 + t) for all t ∈ R, that is c 1 (t 1 + t) + ϕ 1 (x , t 1 + t) ≤ c 2 t + ϕ 2 (x , t) ≤ c 1 (t 2 + t) + ϕ 1 (x , t 2 + t).
Note that in the inclined case, the profiles ϕ 1 and ϕ 2 remain bounded in time from the line y = tan γ x . Substract then tan γ x from each term, divide by t and then let t go to infinity. We obtain

c 1 ≤ c 2 ≤ c 1 ,
so that c 1 = c 2 . The two fronts have therefore the same effective speeds. This in turn implies that the two time periods T 1 and T 2 are equal up to a multiplicative integer. More precisely, as we know that Y 2 = mY 1 for some m ∈ N (here we have supposed without loss of generality that Y 2 ≥ Y 1 ), then we will have T 2 = mT 1 .

Now let x 1 be such that u 1 (x 1 , t 1 ) = u 2 (x 1 , 0). We have at the point x = x 1 and t = t 1, ,

u 1 (x 1 , t 1 + T 2 ) = c 1 t + c 1 T 2 + ϕ 1 (x 1 , t 1 + T 2 ) = c 1 T 2 + u 1 (x 1 , t 1 ).
Likewise at the point x = x 1 and t = 0, we have

u 2 (x 1 , T 2 ) = c 2 T 2 + u 2 (x 1 , 0), so that u 1 (x 1 , t 1 +T 2 ) = u 2 (x 1 , T 2
). This situation is not allowed by the strong maximum principles applied to the linear parabolic equation satisfied by the function w(x , t) given by w(x , t) = u 1 (x , t + t 1 ) -u 2 (x , t), unless w is a constant.

2.4.3. The case β = π/2. Note that Proposition 1 does not concern TWS in the form of line-shaped graphs x = W (y, t). In fact, this comes to considering a tilted frame (x -y ) with y = x and x = -y, that is the angle β between the two frames equals to -π/2. So suppose there exists a TWS Γ (t) in the form of a line-shaped graph x = W (y, t) inclined by some angle α from the x-axis. We propose to show that Γ coincides with the Γ α given by the Proposition 1.

We know by Claim 1 that if it exists, Γ (t) slides along the direction of the striations (here in the y-direction) with a speed c, which without loss ogf generality can be taken to be positive. Let us continue our analysis for α ∈ (0, π/2), the case for negative values of α can be carried out in a similar way. We proceed by writing down the equation satisfied by W :

W t + R(W ) 1 + V 2 y = V yy 1 + V 2 y , t ∈ R, y ∈ R.
Note the positive sign before R in the above equation. Indeed as α > 0, a motion of the line y = tan α x in the upward direction corresponds to a propagation in the negative x-direction. This gives a negative speed of the line in the x-direction. The sign would change for α < 0. Now as Γ (t) has a constant profile moving with the speed c there exists a function ψ such that W (y, t) = ψ(y -c t), and where ψ satisfies the following system :

(16)    -cψ + R (ψ) 1 + ψ 2 = ψ 1+ψ 2 , z ∈ R, ψ(z + tan α) = ψ(z) + 1, z ∈ R, ψ (z + tan α) = ψ (z), z ∈ R.
Without loss of generality, as usual, we can normalize ψ such that ψ(0) = 0. We then state Lemma 2. Let α ∈ (0, π/2). If ψ is a solution of (16) then ψ (z) > 0 for all z ∈ R.

Proof. As ψ is tan α-periodic, there exists z * ∈ (0, tan α] such that

ψ (z * ) = max R ψ (z)
with ψ (z * ) = 0. As the first condition on ψ implies that tan α 0 ψ (z)dz = 1 (> 0), necessarily we must have ψ (z * ) > 0. By the equation of ψ, we recover the fact that c > 0. Now let z * correspond to the minimum i.e. ψ (z * ) = min R ψ (z). Again by using the equation of ψ, we deduce that ψ (z * ) > 0.

From the above lemma, we see that if a traveling front exists in the form of a graph x = V (y, t), we will have V (y, t) = ψ(y -c t) with ψ invertible. Thus Γ (t) may be written as Γ (t) = {y = c t + ψ -1 (x)}. We then conclude by the uniqueness result of Proposition 1 that Γ = Γ α and we are done with the proof of Theorem 1.

V-shaped TWS

As mentioned before, there exist also traveling wave solutions in the form of V-shaped parabola. Let us first review the homogenous case before going to the uniqueness result in the fibered medium.

3.1. The homogeneous case. Let α ∈ (0, π/2) and consider the two lines ψ + (x) = tan α x and ψ -(x) = -tan α x. Then there exists a unique V-shaped TWS Γ(t) = Γ 0 + c t, with c = R/ cos α and Γ 0 = {y = ψ(x)} a V-shaped parabola which has ψ + (x) and ψ -(x) as asymptotes when x -→ ±∞ (cf [START_REF] Ninomiya | Traveling curved fronts of a mean curvature flow with constant driving force, Free Boundary Problems : Theory and Applications I[END_REF]). Note that Γ 0 is symmetrical wrt the y axis with ψ (0) = 0.

3.2. The periodic case. Let us now consider the case where the forcing term R(x, y) = R(x) and is 1-periodic..We know from Proposition 1 that for every α ∈ (-π/2, π/2), there exists a unique line-shaped TWS (c, ϕ) solving the problem

P α :      c -R 1 + ϕ 2 x = ϕ xx 1 + ϕ 2 x , x ∈ R ϕ(0) = 0, ϕ(1) = tan α ϕ (0) = ϕ (1)
Remark 2. Note that the above boundary conditions are equivalent to the shifted periodic condition (5) of Proposition 1. Indeed if (c, ϕ) solves P α , then we have ϕ(x + 1) = ϕ(x) + tan α.

To see this, consider w(x) = ϕ(x + 1) -tan α. We then have w(0) = 0 and w (0) = ϕ (0). As R is 1-periodic, w also satisfies the first equation of P α . Then by uniqueness results for initial value problems for second order ode's, we know that w ≡ ϕ.

We now define for any function ϕ, the symmetrical function with respect to the y-axis ϕ sym i.e. ϕ sym (x) = ϕ(-x) and prove the following lemma: Lemma 3. Let (c 1 , ϕ 1 ) and (c 2 , ϕ 2 ) be the respective line-shaped TWS of P α1 and P α2 with α 1 > 0 and α 2 < 0 and both staifying the condition

(17) R M cos α < R m .
Then α 1 = -α 2 iff c 1 = c 2 .

Proof.

(i) α 1 = -α 2 =⇒ c 1 = c 2 .
In the homogeneous case, it's trivial because c = R/ cos α. When R is periodic, without loss of generality, we can consider R(x) = R(-x) for all x. Consider then ψ = ϕ sym Set β = -α 2 . We know that there exists a unique pair (c β , ϕ β ) solution of P β . By (i), we then have c β = c 2 and ϕ β = ϕ sym 2 . As c 1 = c 2 by assumption,, then c β = c 1. Now as α 1 and β satisfy assumption (17) and under the latter condition c(α) is strictly increasing in α (by Proposition 2), we have necessarily β = α 1 .

So for α ∈ (0, π/2), the above lemma shows that the profile of the line-shaped TWS Γ -α is symmetrical to that of Γ α , both having the same sped of propagation. It says a bit more notably that the only line-shaped TWS having the same speed as Γ α is Γ -α . Now we are ready to give the result for V-shaped TWS.

Theorem 3. Let α ∈ (0, π/2) satisfy assumption (17) and let (c, ϕ) be the solution of P α and ϕ sym be defined as above. Then for any s ∈ R, (4) admits a unique V-shaped TWS ψ such that ψ(0) = s and ψ (0) = 0. Moreover the solution satisfies the following estimates (i) ϕ sym (x) < ψ (x) < ϕ (x), (ii) there exist a unique s 0 > 0 and positive constants L and δ such that ψ(0) = s 0 and ϕ(|x|) < ψ(x) < ϕ(|x|) + L exp(-δ |x|).

Note that ψ has ϕ and ϕ sym as asymptotes when x goes to ±∞. Note also that under assumptions (17) we can show that ϕ sym (x) < 0 and ϕ (x) > 0.We will omit the proof of the theorem as it goes along the same lines as Theorem B of [START_REF] Lou | Traveling waves of a curvature flow in almost periodic media[END_REF] by remarking that in the periodic case, α 2 is necessarily equal to -α 1 by Lemma 3. It suffices then to do the neccessary modifications.

Figure 2 .

 2 Figure 2. A horizontal line-shaped pulsating front : ∃T β s.t. Γ(t + T β ) = Γ(t) + Y β ∀t.

Figure 3 .

 3 Figure 3. An inclined pulsating front in the tilted frame x -y : ∃T s.t. Γ(t + T ) = Γ(t) + Y.

1 .= ψ xx 1 + ψ 2 x 1 .

 121 It is not difficult to verify that ψ satisfiesc 1 -R 1 + ψ 2 x , x ∈ R, ψ(0) = 0, ψ(1) = -tan α 1 , ψ (0) = ψ (1), so (c 1 , ψ) is a solution of P α for α = -α 1 i.e. for α = α 2 . By uniqueness of line-shaped TWS, we deduce that c 1 = c 2 and ψ = ϕ 2 = ϕ sym (ii) c 1 = c 2 =⇒ α 1 = -α 2 .

Acknowledgement. The author would like to thank Nathaël Alibaud for our regular collaboration on problems related to propagation of fronts and for helpful discussions on this paper.


B W Jervis
email: barrie.jervis@cantab.net

Towards Reconciling Gravitational Theories

Keywords: emergent gravity, gravitational dipoles, spherical galaxy, dark matter

The aim was to demonstrate the possible compatibility between the gravitational theories of Hajdukovic and of emergent gravity, and to derive new physical relationships. These theories and necessary, associated concepts are described.

Assuming a spherical galaxy model they are shown to predict the same radial distance from the galactic centre for the onset of enhanced gravitational fields without the necessity for postulating dark matter. This agreement is used (1) to show that the virtual particle density of the vacuum at this distance depends only upon the Hubble scale and Planck's constant, (2) to calculate its value, and (3) to show that the particles consist of closely packed pions.

Introduction

Zwicky in 1933 [1] observed anomalous velocity dispersion in the Coma cluster of galaxies, and Rubin in 1970 [2] discovered that at large radii galactic material rotates faster than predicted by Newton's Laws. Both these observations suggested the presence of stronger than expected gravitational fields. The two most popular theories to explain these, and other galactic phenomena, are the presence of undiscovered matter (dark matter) or Bekenstein and Milgrom's MOND (Modified Newtonian Dynamics) theory [3] of the modification of the laws of gravity. Dark matter is postulated to consist of massive, weakly-interacting, cold, dark particles, which neither emit nor reflect light. The location of these particles, assuming they exist, has been determined by computer modelling.

Recently two alternative, promising theories of gravity have been proposed which remain to be fully tested, but have explained some of the gravitational effects of galaxies without the need for dark matter. The theory of emergent gravity is based upon a melange of the theories of thermodynamics, entropy, black holes, information, elasticity, and string theory [4]. Gravity is shown to emerge from the quantum bits of information which characterise space-time, and is, therefore, not a fundamental property of matter. According to the second theory [5], the additional gravitational force associated with galaxies is contributed by the partial alignment of gravitationally polarised virtual particle pairs. The pairs of particles and antiparticles spontaneously appear very briefly and then self-annihilate, according to Heisenberg's Uncertainty Principle and Einstein's equation of equivalence between mass and energy, E = mc 2 . The antiparticles are assumed to have negative mass, as well as negative charge. This will be called the Hajdukovic theory, after its inventor. It is these two theories which are considered in this paper.

It is shown here that, based on a spherical galactic model, the two theories predict an enhanced gravitational effect at the same location. Then, by combining equivalent formulaic results from the two theories, it is shown that the virtual particle density in the galactic halo depends upon the Planck wavelength and the size of the universe.

The particle density is calculated and the particles are shown to be pions. Some underlying concepts and the two new theories are first reviewed. The new contribution appears in Sections 4 and 5.

Underlying Concepts

The Evidence for Dark matter

Modelling shows that every galaxy is surrounded by a halo of dark matter. These haloes have a constant central surface density [6]. Also studies of the dwarf galaxies Fornax and Sculptor show dark matter distributed uniformly within the central region [7]. Thus a uniform density is characteristic of dark matter. Further evidence for possible dark matter is offered by the lensing effect of the Bullet Cluster of colliding galaxies [8] and from the cosmic background microwave radiation (CMBR) [9].

However, the existence of dark matter has not been proved. Any new theory must account for these phenomena.

Virtual Particle Pairs

It is fundamental to Hajdukovic's theory that the vacuum of space is full of virtual particle-pairs. Their existence has been shown by the Casimir Effect [10] and the Lamb shift [11].

It has been suggested that gravitational polarisation of these particles accounts for the dark matter explained by the MOND model [12] as well as for dark matter and dark energy [13], [14]. In Hajdukovic's theory these ideas are extended to the gravitational polarisation of the virtual dipoles.

Entropy, Information, Black Holes, and Entropic Force

These heading concepts are used in the theory of emergent gravity [4] and so relevant aspects are reviewed here. Thermodynamic entropy is given by the wellknown Boltzmann equation

𝑆 = 𝑘 𝐵 𝑙𝑛𝑊 (1)
in which 𝑘 𝐵 is Boltzmann's constant, 𝑊 is the number of possible states, and 𝑙𝑛 denotes logarithm. In information theory information is conveyed in strings of 𝑁 binary bits. The information entropy, is given by

𝑆 = 𝑙𝑜𝑔 2 2 𝑁 (2)
These equations are identical in form, showing that thermodynamic entropy and information entropy are interchangeable.

Microscopically, in the domain of quantum mechanics, the binary information bits become qubits taking any value between 0 and 1. When the quantum mechanical wave function represents two states simultaneously the two states are said to be entangled. Thus, their representative qubits are also entangled. In the theory of black holes [15] the entropy of a black hole may be regarded as the number of associated entangled qubits.

An increase in entropy, ∆𝑆, is associated with the work done, 𝐹∆𝑥, by the application of a force, 𝐹, in moving an object through a distance ∆𝑥. The energy increase in entropy terms is 𝑇∆𝑆, where 𝑇 is the temperature. Equating these gives

𝑇∆𝑆 = 𝐹∆𝑥 (3)
3 The New Gravitational Theories

Theory of Emergent Gravity

Verlinde presented a theory of emergent gravity [4], and a non-mathematical description has been given by Visser and Vonk [16]. According to Verlinde an empty space is filled with dark energy, which provides a volume contribution to the total entropy. It also acts as an expansive force, which drives the expansion of the universe. If some of this energy is converted into ordinary baryonic matter, the entropy change produces a force, according to equation (3). The universe is considered an elastic medium and this force results in elastic strain, which slowly relaxes over a very long timescale. At the microscopic quantum scale, black hole theory is applied, and the entropy is associated with qubits. The creation of matter displaces the qubits, leaving a vacuum in the vicinity of the mass. The qubits push to fill the vacuum, which creates an additional gravitational force. This phenomenon is significant whenever the mass is sufficiently large, and causes the large gravitational fields in the vicinity of galaxies. Thus, gravity is regarded as an emergent macro property of the microscopic behaviour of the qubits.

Verlinde has demonstrated how Newton's Laws may be derived from this theory [17].

Hajdukovic's Theory

Hajdukovic produced his theory to explain the phenomenon of dark matter [5]. The virtual particles and antiparticles are assumed to have opposite gravitational charges and to form gravitational dipoles in which their gravitational charges repel each other.

The concept of matter and antimatter being mutually repulsive has been investigated

theoretically [18] with the conclusion that antigravity is predicted by the theory of general relativity. See also [19]. Hajdukovic also assumes that the virtual particle pairs constitute a gas of virtual pions, which are quark-antiquark pairs [20]. Pions may have the smallest mass possible in nature [21], and Hajdukovic argues that pions may dominate the content of the quantum vacuum [20]. While the opposite gravitational charges of the virtual dipoles repel each other, their opposite electrical charges attract, and the separation between the poles ≅ 𝜆 𝑐 = ħ 𝑚𝑐 ⁄ , their Compton wavelength, at their equilibrium separation. Here 𝑐 is the velocity of light, ħ is the reduced Planck constant, and 𝑚 is the mass of the pion. Further, according to quantum field theory, a virtual particle pair should occupy the volume 𝜆 𝑐 3 [5]. Thus, the particle pair density in the vacuum is 𝑁 = 1 𝜆 𝑐 3 ⁄ .

These polarised particle pairs become aligned in a gravitational field to an extent which depends upon the field strength, just as electric dipoles align in an electric field. Hajdukovic used a simple spherical model of a galaxy and calculated the radial gravitational charge density. The evidence for a uniform matter density in regions of dark matter has already been mentioned above [6], [7], and so regions of uniform gravitational charge density were taken to represent dark matter regions. In fact the radial distance from the centre at which the dark matter effect was to be found was determined to be

𝑅 0 = 𝜆 𝑐 √ 𝑀 𝑏 𝑚 𝑐 (4)
in which 𝑀 𝑏 is the baryonic mass. Hajdukovic took 𝜆 𝑐 = 𝜆 𝜋 and 𝑚 𝑐 = 𝑚 𝜋 because he was considering pions.

Gravitational Theoriesthe Evidence

Table 1 gives a summary of the successes, √, failures, X, and unknown predictions,?, for the supposed dark matter phenomena, as known by the author at present, by the different theories in different instances.

Dark matter phenomenon

MOND

Emergent gravity Gravitational dipoles

Galactic haloes √ √ √

Dwarf galaxies √ ? ?

Gravitational lensing

X √ √

Milky Way disc X ? √

Table 1 The theoretical predictions compared to the locations of known dark matter

The MOND theory predicted that the Milky Way would exhibit a disc of dark matter, while Hajdukovic's theory did not predict one. Attempts to find it were unsuccessful [22]. The emergent gravity theory has been successfully tested for the case of gravitational lensing [23]. While much more work remains to be done, the table

shows that the emergent gravity and Hajdukovic's theories look promising, whilst the MOND theory failed in two instances.

Comparing the two Theories

As the theories of emergent gravity and Hajdukovic both explain galactic haloes and gravitational lensing, they should predict the same radial distance from the centres of spherical galaxies to the onset of dark matter effects. This distance was given in equation ( 4) for Hajdukovic's theory. Substituting ħ 𝑚𝑐 ⁄ for 𝜆 𝑐 , putting 𝑚 = 2.4 × 10 -28 kg as the mass of a pion, and using 𝑀 𝑏 = 1.6 × 10 42 kg, the mass of the Milky Way, gives 𝑅 0 = 1.2 × 10 20 km.

For emergent gravity, with 𝐴 as the surface area of a sphere of radius 𝑅 surrounding the galactic centre, beyond which dark matter is predicted, Verlinde's equation ( 4) in [4] describes the situation by,

𝐴 4𝐺ħ > 2𝜋𝑀 ħ𝑎 0 (5)
where 𝑀 is the enclosed mass, 𝐺 is the gravitational constant, and 𝑎 0 is the cosmological acceleration scale, given by

𝑎 0 = 𝑐 2 𝐿 ( 6
)
where 𝐿 is the Hubble scale, the radius of the universe. Upon substituting equation ( 6) into equation ( 5), putting 𝐴 = 4𝜋𝑅 2 , and rearranging gives

𝑅 > √2𝑀𝐿𝐺/𝑐 (7)
Taking the case of our galaxy, the Milky Way, 𝑀 = 1.6 × 10 42 kg, while 𝐿 = 4.4 × 10 25 m, 𝐺 = 6.67 × 10 -11 m 3 kg -1 , and 𝑐 = 3 × 10 8 m s -1 giving 𝑅 > 3.2 × 10 20 km.

Thus, 𝑅 ≅ 2.7𝑅 0 . Since 𝑅 depends upon the estimated size of the universe, this may be regarded as a good agreement.

Combining the two theories

The evidence in Table 1, and the agreement between 𝑅 and 𝑅 0 , suggest the emergent gravity and Hajdukovic theories are compatible. Since equations ( 4) and (7) give essentially the same result, they may be equated to give the new relationship

𝜆 𝑐 √ 𝑀 𝑚 𝑐 = √2𝑀𝐿𝐺/𝑐 (8)
Substituting 𝑚 𝑐 = ħ 𝜆 𝑐 𝑐 ⁄ and 𝑁 = 1 𝜆 𝑐 3 ⁄ , it is found that

𝑁𝐿 = 𝑐 3 2𝐺ħ (9)
showing that the particle density at 𝑅 0 only depends upon known constants and the radius of the spherical universe. The Planck wavelength, 𝜆 𝑃 , is

𝜆 𝑃 = √ ħ𝐺 𝑐 3 (10)
And so combining equations ( 9) and (10) gives

𝑁𝐿 = 1 2𝜆 𝑃 2 = 𝜋 2 ⁄ 𝜋𝜆 𝑃 2 = 𝜋 2 ⁄
𝑃𝑙𝑎𝑛𝑐𝑘 𝑎𝑟𝑒𝑎 (11) showing that 𝑁𝐿 is inversely proportional to the smallest meaningful area of space.

The particle density at 𝑅 0 can be estimated from equation (9) as

𝑁 = 𝑐 3 2𝐺ħ𝐿 (12)
Substituting values gives 𝑁 = 4.4𝑥10 43 particles m -3 .

So far the type of particle has not been defined. If we assume, like Hajdukovic [5],

that they are close neighbours with 𝑁 = 1 𝜆 𝑐 3 ⁄ then 𝜆 𝑐 = 𝑁 -1 3 ⁄ = 2.84𝑥10 -15 m. This is close in value to 𝜆 𝑐 for the electron (2.82𝑥10 -15 m) and for the pion (1.46𝑥10 -15 m). Uncertainties in the values mean that either of these is a possibility, but it is thought that the quantum vacuum is dominated by pions [5]. This agreement is further evidence that the two theories yield similar results and that equation ( 8) is valid. The physical meaningfulness of equation ( 11) also suggests the compatibility of the two theories.

In deriving the expression for 𝑅 0 , equation ( 4), the spacing between the virtual particle and its antiparticle was taken to be approximately 𝜆 𝑐 according to quantum field theory, and the virtual dipole pairs were assumed tightly packed so that the particle density is 𝑁 = 1 𝜆 𝑐 3 ⁄ . This second assumption can now be tested. Let the particle density be 𝑁 = 1 𝑥 3 ⁄ where 𝑥 ≥ 𝜆 𝑐 . Then from equation ( 8)

𝑥 = √2𝐿𝐺𝑚 𝑐 𝑐 ( 13
)
If pions are assumed, 𝑚 𝑐 = 2.4𝑥10 -28 kg, whence 𝑥 = 3.2𝑥10 -15 m, compared with 𝜆 𝜋 = 1.46𝑥10 -15 . Again, the factor of 2.7 difference is probably owing to the assumed value of L, and the pions are then indeed tightly packed. This is more evidence for the compatibility of the two theories. Note that 𝑅 𝑅 0 = 𝑥 𝜆 𝜋 = √2𝐿𝐺𝑚 𝜋 3 𝑐ħ = 2.7 ⁄ . ⁄ ⁄

Conclusion

The predicted agreements with the "observations" of dark matter, and the agreement between the predicted radial locations of the supposed dark matter for a spherical galactic model suggest the theories of Hajdukovic and of emergent gravity are equivalent. By equating the two formulae for the radial location of the dark matter effect it is shown that the virtual particle density in this region depends only upon the Hubble scale and the Planck area, 𝜋𝜆 𝑃 2 . This basic relationship also suggests compatibility between the two theories. The number density of virtual particle pairs was calculated to be 4𝑥10 43 m -3 , and their separation by approximately one

Compton wavelength corresponded to that of tightly packed pions or electrons.

Acknowledgements

The author thanks Dr A. J. Jervis for providing some references and comments on the manuscript, and John Edwards for feedback.