Cornelis Jacominus Van Der Beek

Jérôme Losco

Marcin Konczykowski

Patrick Pari

Takasada Shibauchi

Hiroaki Shishido

Yuji Matsuda

Magneto-optical imaging of exotic superconductors

la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.


Coélio Vallée

Matthieu Saubanère

Paula Sanz-Camacho

Yohan Biecher

Bernard Fraisse

Emmanuelle Suard

Gwenaëlle Rousse

Dany Carlier

Dr Romain Berthelot
email: romain.berthelot@umontpellier.fr

Alkali-glass behavior in honeycomb-type layered Li3-xNaxNi2SbO6 solid-solution

Keywords: Layered oxides, honeycomb ordering, solid-solution, alkali mixed occupancy

Layered oxides compositions Li3-xNaxNi2SbO6 have been prepared by solid-state synthesis. A complete solid-solution is evidenced and characterized by X-ray and neutron diffraction, as well as 7 Li and 23 Na solid-state nuclear magnetic resonance spectroscopy. The transition metal layer is characterized by the classic honeycomb Ni 2+ /Sb 5+ ordering, while a more uncommon randomly mixed occupancy of lithium and sodium is evidenced within the alkali interslab space.

In situ X-ray diffraction and DFT calculations show that this alkali disordered feature is entropically driven. Fast cooling appears then as a synthesis root to confine bidimensional alkali glass within crystalline layered oxides.

Introduction

Ionic radii of lithium and sodium cations are significantly different in chalcogenides or halides. [START_REF] Shannon | Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides[END_REF] As a consequence, there are in literature very few cases of mixed lithium/sodium occupancy in same crystallographic sites in transition metal oxides. Examples are vanadates Li1-xNaxVO3 [START_REF] Ng | Crystallographic Studies and Structural Systematics of the C2 c Alkali Metal Metavanadates[END_REF] or bronzes LixNayV3O8, the latter being widely investigated as insertion-type negative electrode materials for rechargeable lithium-ion batteries, [START_REF] Schindler | V3O8] Insertion Electrodes: Structures and Diffusion Pathways[END_REF] or niobiates perovskites Li1-xNaxNbO3 with piezoelectric properties. [START_REF] Yuzyuk | Synchrotron X-Ray Diffraction and Raman Scattering Investigations of (LixNa1x)NbO3 Solid Solutions: Evidence of the Rhombohedral Phase[END_REF][START_REF] Mishra | High Temperature Phase Stability in Li0.12Na0.88NbO3: A Combined Powder X-Ray and Neutron Diffraction Study[END_REF][START_REF] Peel | Unusual Phase Behavior in the Piezoelectric Perovskite System, Li <inf>x</Inf>Na<inf>1-X</Inf>NbO<inf>3[END_REF] Regarding the important family of alkali transition metal layered oxides, the crystallographic structure offers a large playground for solid-state chemists and physicists due to the impressive richness of chemical compositions that can tailor different properties. The lamellar organization is usually described by a stacking of edge-shared MO6 octahedra slabs, with intercalation of alkali ions in the interslab space. Chemical substitutions on the transition metal layers are possible thanks to similar ionic radii in octahedral symmetry for cations, enabling to tune some physical or chemical features. The best example might be the emerging layered compositions LiNi1-x-yCoxAlyO2 or LiNi1-x-yMnxCoyO2 which exhibit enhanced electrochemical performance compared to the standard electrode material LiCoO2. 7- 9 On the contrary, cases of alkali ions mixed occupancy within the interslab spaces are very scarce. To the best of our knowledge, only two examples have been reported, single crystals of rhomboedral titanate (Na0.73Li0.24)(Li0.32Ti0.68)O2 studied by Nalbandyan, [START_REF] Shilov | Crystal Growth and Crystal Structures of the Layered Ionic Conductors-Sodium Lithium Titanium Oxides[END_REF] and nickelates Li1-xNaxNiO2 investigated separately by Matsumura and Holzapfel in early 2000s. [START_REF] Matsumura | Structure and Physical Properties of LixNa1-xNiO2[END_REF][START_REF] Holzapfel | Decoupling of Orbital and Spin Degrees of Freedom in Li 1-xNaxNiO2[END_REF][START_REF] Holzapfel | Mixed Layered Oxide Phases Na XLi 1-XNiO 2: A Detailed Description of Their Preparation and Structural and Magnetic Identification[END_REF] In the frame of discovering new layered dual-alkali transition metal layered oxides with possible interesting physical features, this work investigates intermediate compositions between Li3Ni2SbO6 and Na3Ni2SbO6 (or LiNi2/3Sb1/3O2 and NaNi2/3Sb1/3O2, respectively).

These compounds are structurally related to O3-type [START_REF] Delmas | Structural Classification and Properties of the Layered Oxides[END_REF] LiCoO2, with an additional honeycombtype ordering of the Ni 2+ and Sb 5+ cations. [START_REF] Politaev | Mixed Oxides of Sodium, Antimony (5+) and Divalent Metals (Ni, Co, Zn or Mg)[END_REF][START_REF] Schmidt | Solid Solution Studies of Layered Honeycomb-Ordered Phases O<inf>3</Inf>-Na<inf>3[END_REF][START_REF] Ma | Ordered and Disordered Polymorphs of Na(Ni<inf>2/3</Inf>Sb<inf>1/3</Inf>)O<inf>2</Inf>: Honeycomb-Ordered Cathodes for Na-Ion Batteries[END_REF][START_REF] Zvereva | Monoclinic Honeycomb-Layered Compound Li 3Ni 2SbO 6: Preparation, Crystal Structure and Magnetic Properties[END_REF] A complete solid-solution could be obtained by ceramic route one-pot synthesis from carbonates or oxides precursors. The lithium/sodium mixed occupancy is confirmed by structural refinement and solid-state NMR spectroscopy. The alkali ions random distribution is not thermodynamically stable as decomposition is observed by in situ XRD during either heating or slow cooling. These results pave the way to design crystalline layered compounds with 2D confined alkali glass.

Experimental section

Li3Ni2SbO6, Na3Ni2SbO6 and intermediate compositions Li3-xNaxNi2SbO6 (0 ≤ x ≤ 3) have been prepared by conventional solid-state synthesis. Dried alkali carbonates (Li2CO3, Aldrich, 99.997 % or Na2CO3, Sigma-Aldrich, 99.5 %) are thoroughly mixed with nickel and antimony oxides (NiO, Prolabo and Sb2O3, Merck, 99 %), with a nominal alkali excess of 5-10 wt.% to compensate the high-temperature volatility. The resulting mixtures are then pelletized and heated for 5 hours at 900 °C in air on a platinum crucible. A rapid quench in oil bath ends the heat treatment, and the resulting pellets are then washed with ethanol and dried before storage. X-ray diffraction (XRD) was performed on a Bruker D8 Discovery diffractometer with Kα1 Cu radiation, from 5 to 120 ° (16 h of total scan). Sample powders were sieved below 100 µm for analysis. Moreover, the thermal stability of the samples was followed in situ by XRD on a PANalytical Empyrean with Kα Co radiation and equipped with an Anton Paar heating chamber.

Neutron powder diffraction (ND) was carried out at room temperature out at Institut Laue Langevin on D2B high resolution powder diffractometer (1.594 Å wavelength). For that, 3 g of as-prepared powder of the composition Li1.5Na1.5Ni2SbO6 have been sealed in a cylindrical vanadium tube. [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] Na MAS NMR spectra were recorded on a Bruker 300 Avance spectrometer at 79.49 MHz (7.05 T magnet) using a standard Bruker 2.5 MAS probe at a spinning frequency of 30 kHz and a recycle time D0 = 0.5 s, long enough to avoid T1 saturation effects. A one rotor period synchronized Hahn echo sequence was used with a 90° pulse length of 1.5 µs. The external reference was a 1 mol.L -1 NaCl aqueous solution. [START_REF] Myung | Nickel-Rich Layered Cathode Materials for Automotive Lithium-Ion Batteries: Achievements and Perspectives[END_REF] Li MAS NMR spectra were recorded on a Bruker Avance III spectrometer at 38.93 MHz (2.35 T magnet), a standard Bruker 2.5 MAS probe at a 30 kHz spinning frequency. A Hahn echo sequence was used with a 90° pulse of 1.6 µs with a relaxation time of 1 s. The 0 ppm external reference was a 1 M LiCl aqueous solution.

Density functional theory calculation (DFT+U) used the monoclinic description of Zvereva 18 with two alkali layers (C 2/m space group) as starting point. About hundred different crystallographic structures have been generated with a random distribution of lithium and sodium ions over the 24 available positions. All structures have been fully relaxed using the plane wave code "Vienna Ab-Initio Simulation Package" with PAW pseudo-potentials and the PBE functional. [START_REF] Kresse | Efficient Iterative Schemes for Ab Initio Total-Energy Calculations Using a Plane-Wave Basis Set[END_REF][START_REF] Kresse | From Ultrasoft Pseudopotentials to the Projector Augmented-Wave Method[END_REF][START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] A Ueff = 4 eV is considered to take into account electron-electron interactions on nickel. [START_REF] Dudarev | Electron-Energy-Loss Spectra and the Structural Stability of Nickel Oxide:  An LSDA+U Study[END_REF] The forces on the atoms were converged to 5 10 -3 eV/Å with a planewave energy cut-off of 600 eV and a well converged set of K-points.

Results and Discussion

An alkali ions mixed occupancy

XRD patterns obtained with a final quenching at the end of the thermal treatment are gathered in Figure 1a for all compositions from Li3Ni2SbO6 to Na3Ni2SbO6. No extra peak could be assigned to impurities like binary oxides or alkali hydroxides, and the patterns could be indexed using the monoclinic symmetry (space group C 2/m). In comparison with a classic O3-type stacking (α-NaFeO2 polytype), the Ni 2+ /Sb 5+ honeycomb ordering induces superstructure peaks visible in the 19 -32 ° 2θ region. They can be nicely observed in Figure 1b, with for Li-rich compositions a broadening that is reminiscent of stacking faults, as commonly observed in such honeycomb-ordered lamellar oxides. [START_REF] Politaev | Mixed Oxides of Sodium, Antimony (5+) and Divalent Metals (Ni, Co, Zn or Mg)[END_REF][START_REF] Schmidt | Solid Solution Studies of Layered Honeycomb-Ordered Phases O<inf>3</Inf>-Na<inf>3[END_REF][START_REF] Ma | Ordered and Disordered Polymorphs of Na(Ni<inf>2/3</Inf>Sb<inf>1/3</Inf>)O<inf>2</Inf>: Honeycomb-Ordered Cathodes for Na-Ion Batteries[END_REF] The refined cell parameters evolve regularly all along the chemical compositions, following a classical Vegard's law (Figure S1). The evolution of the corresponding monoclinic cell volume shown in inset of Figure 1c demonstrates a complete solid-solution behavior from Li3Ni2SbO6 to Na3Ni2SbO6. The question arising from this assumption is the localization of both lithium and sodium cations. Three structural models seem possible. They are pictured in Figure 2 for the particular compositions Li1.5Na1.5Ni2SbO6. The first one is a mixed occupancy of lithium and sodium ions and a random in-plane distribution within. Following the structural description of Zvereva [START_REF] Zvereva | Monoclinic Honeycomb-Layered Compound Li 3Ni 2SbO 6: Preparation, Crystal Structure and Magnetic Properties[END_REF] and Ma [START_REF] Ma | Ordered and Disordered Polymorphs of Na(Ni<inf>2/3</Inf>Sb<inf>1/3</Inf>)O<inf>2</Inf>: Honeycomb-Ordered Cathodes for Na-Ion Batteries[END_REF] , the slab thickness (basically including one transition metal and one alkali layers) is 4.856 Å and 5.344 Å for Li3Ni2SbO6 and Na3Ni2SbO6. From lithium-rich to sodium-rich end members, the increase is about 10 % and comes from an important enhancement of the alkali layer (from 2.60 Å to 3.21 Å, hence + 23.5 %) whereas the transition metal layer is slightly contracted. In our case, the linear evolution of the slab thickness also shows an increase of about 10 % (Figure 1d) and consequently supports this structural assumption.

In the second model, lithium and sodium ions always occupy the same interslab space but following an in-plane ordering (either chains, stripes, cluster, etc.). The last model also lies on alkali ions ordering, lithium and sodium ions occupy distinct interslab spaces. For the intermediate composition Li1.5Na1.5Ni2SbO6, such arrangement would result in alternate Li + /Na + layers, as it has been already evidenced for the unique case of OP4-LixNayCoO2. [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] It is important to point out that, in the two latter models, the mean slab thickness evolution (determined from the out-of-plane c parameter) could be also assumed.

In alkali transition metal layered oxides, alkali cations out-of-plane ordering like in OP4-LixNayCoO2 or alkali cations / vacancies in-plane ordering like in Na1/2CoO2 or K4/7CoO2 lead to significant superstructure peaks in X-ray diffraction pattern. [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][START_REF] Berthelot | Electrochemical Investigation of the P2-Na<inf>x</Inf>CoO<inf>2</Inf>phase Diagram[END_REF][START_REF] Blangero | First Experimental Evidence of Potassium Ordering in Layered K4Co7O14[END_REF] The two last abovementioned models involve cationic ordering and the corresponding simulated patterns clearly evidence superstructure peaks (Figure S2). The absence of such peak in our highresolution XRD pattern in Figure 1 is a first clue to discard these two hypotheses and consequently to support the model of random Li/Na distribution despite their different ionic radii (0.76 and 1.02 Å in octahedral symmetry, respectively). [START_REF] Shannon | Revised Effective Ionic Radii and Systematic Studies of Interatomic Distances in Halides and Chalcogenides[END_REF] In order to get local information on lithium and sodium environments, solid-state NMR has been further carried out for all the compositions, and corresponding [START_REF] Myung | Nickel-Rich Layered Cathode Materials for Automotive Lithium-Ion Batteries: Achievements and Perspectives[END_REF] Li and [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] Na spectra are shown in Figure 3. The peak shifts for all compounds results from the Fermi contact interaction between the paramagnetic low spin Ni 2+ (t2g 6 eg 2 ) ions and the probed nucleus. This interaction occurs through the chemical bonds and therefore denotes the local electronic structure.

Whatever the structural hypotheses presented above, the two alkaline A + ions in the system are surrounded by four Ni 2+ as first cation neighbors (90° A-O-Ni interaction) and four Ni 2+ as second cation neighbors (180° A-O-Ni interaction) with slightly different distances and angles.

For Li3Ni2SbO6, two distinct narrow peaks at 492 and 420 ppm and a larger set of signals around 300 ppm are observed by [START_REF] Myung | Nickel-Rich Layered Cathode Materials for Automotive Lithium-Ion Batteries: Achievements and Perspectives[END_REF] Li MAS NMR. According to the literature, based on their relative intensities, the two first are assigned to lithium ions respectively in 2d and 4h Wyckoff positions of the space group C 2/m, i.e. within the interslab space in octahedral insertion sites. The third broad contribution around 300 ppm could be assigned to stacking defects, similarly to stacking faults observation in the Li2MnO3 reported by Grey and coworkers. [START_REF] Br?ger | High-Resolution X-Ray Diffraction, DIFFaX, NMR and First Principles Study of Disorder in the Li2MnO3-Li[Ni1/2Mn1/2]O2 Solid Solution[END_REF] Indeed, the relative intensity of the contribution at 300 ppm compared to the two other sharp peaks depends on the synthesis conditions: when Li3Ni2SbO6 is prepared with a final slow cooling, the two first peaks are clearly sharper and the broad contribution almost disappear (Figure S3). On the opposite side of the Figure 3, [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] Na NMR spectrum for Na3Ni2SbO6 exhibits two contributions located at 1440 ppm and 1300 ppm with no second order quadrupolar lineshape, contrary to the spectrum reported by Ma et al. using a lower magnetic field (200 MHz spectrometer) and a higher spinning frequency (50 KHz). [START_REF] Ma | Ordered and Disordered Polymorphs of Na(Ni<inf>2/3</Inf>Sb<inf>1/3</Inf>)O<inf>2</Inf>: Honeycomb-Ordered Cathodes for Na-Ion Batteries[END_REF] Based on the relative peak intensities, we can also assign those two peaks to Na ions in the in 2d and 4h sites respectively. Interestingly, the difference between quenched and non-quenched Na3Ni2SbO6 is very weak as both XRD pattern and NMR data look very similar (Figure S4). Thanks to higher weight and larger ionic radii of sodium compare to lithium and in agreement with the observation of superstructure peaks shapes in inset of Figure 1, Na3Ni2SbO6 seems to ordered more easily than Li3Ni2SbO6 and is therefore less sensitive to quenching.

Intermediate compositions Li3-xNaxNi2SbO6 can be analyzed by both [START_REF] Myung | Nickel-Rich Layered Cathode Materials for Automotive Lithium-Ion Batteries: Achievements and Perspectives[END_REF] Li and [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] Na NMR. At a first glance, broader resonance peaks are clearly evidenced. In both cases, the broadness increases when compositions moves away from Li3Ni2SbO6 or Na3Ni2SbO6. Moreover, a weak but significant peak shift also accounts for variation of the (Li -O and Na -O) interatomic distances. Shifting toward lower ppm in [START_REF] Myung | Nickel-Rich Layered Cathode Materials for Automotive Lithium-Ion Batteries: Achievements and Perspectives[END_REF] Li NMR spectra corresponds to a decrease of the electronic spin transfer from low spin Ni 2+ to Li nuclei that can be attributed to an increase of the Li -O distances, whereas displacement to higher ppm in [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] Na NMR spectra is attributed to a decreasing of the Na -O distances. Therefore, the NMR study tends to demonstrate that the alkali layers in Li3-xNaxNi2SbO6 compositions do not correspond to an intergrowth of pure Li and Na layers but do contain both alkaline ions. Indeed, sodium ions present in a Li layer will increase the local Li-O distance compared to Li3Ni2SbO6 and lithium ions present in the Na layer will decrease the local Li-O distance compared to Na3Ni2SbO6. Moreover, the overall broadening of the NMR lines results from a larger distribution of environments as locally distances and angles will vary depending on the amount of surrounding Li/Na ions. In addition, it worth noting the decrease of third contribution at 300 ppm in the [START_REF] Myung | Nickel-Rich Layered Cathode Materials for Automotive Lithium-Ion Batteries: Achievements and Perspectives[END_REF] Li spectra with higher sodium amount. This accounts for less stacking faults for Na-rich compositions, which is logical when stated that Na3Ni2SbO6 is less sensitive to quenching than Li3Ni2SbO6.

In order to confirm the hypothesis of a random distribution of both lithium and sodium ions in the same interslab space suggested by the NMR analysis, structural refinements have been performed on high-resolution XRD and neutron powder diffraction patterns data for the particular composition Li1.5Na1.5Ni2SbO6. Indeed with four different chemical species (Li, Na, Ni, Sb) possibly distributed among four crystallographic sites (2d and 4h in the interslab space, and 2a and 4g in the metal layers), a unique solution cannot be found without i) combining Xray with neutrons, that offers a nice contrast (bNi = +10.3 fm, bSb = +5.57 fm, bLi = -1.90 fm, bNa = +3.63 fm), and ii) imposing some constraints that are detailed as follows. We used as a starting model the C 2/m monoclinic structural description of Li3Ni2SbO6 by Zvereva et al. [START_REF] Zvereva | Monoclinic Honeycomb-Layered Compound Li3Ni2SbO6: Preparation, Crystal Structure and Magnetic Properties[END_REF] The total number of Li, Na Ni and Sb was imposed to the nominal composition, with Li and Na being distributed on the 2d and 4h sites, and Ni and Sb on the 2a and 4g sites. All atomic positions and the relative occupancy of Li/Na and Ni/Sb were freely refined. An isotropic temperature factor was refined and imposed equal for a same chemical species/Wyckoff site.

The result of the combined XRD/neutron refinement is shown in Figure 4 and the corresponding table of atomic positions and resulting structure are presented in Table 1 and Figure 5, respectively. Although the overall Ni/Sb honeycomb pattern is confirmed within the transition metal slab with 4g and 2a sites mainly occupied by Ni and Sb, respectively; a certain Ni/Sb mixing is obtained in the structural refinement. Because the relative occupancies of Ni and Sb were not refined in previous structural refinement, this relative disorder was not reported in Li3Ni2SbO6 and Na3Ni2SbO6. As stacking faults characterized quenched layered samples, it is reasonable to assume that the high value of Ni/Sb mixing is over-estimated, as reported for a similar lamellar compound Li4FeSbO6. [START_REF] Mccalla | Novel Complex Stacking of Fully-Ordered Transition Metal Layers in Li4FeSbO6 Materials[END_REF] For comparison, structural refinement for quenched Li3Ni2SbO6 and Na3Ni2SbO6 from ND also ends up with some Ni/Sb mixing. Interestingly the Ni/Sb mixing regularly increases from Li3Ni2SbO6 to Na3Ni2SbO6 (Figure S5 and Table S1).

Regarding the interlayer space, the alkali layer is occupied by both lithium and sodium cations and the refinement indicates a Li/Na mixing very close to the statistical value of 50 %. Lithium and sodium cations are therefore randomly distributed and seem to not form any kind of ordering patterns that could be induced by the strong ionic radii difference and/or by the neighbor Ni/Sb honeycomb-like ordering. To the best of our knowledge, this is the first example of a complete solid-solution from Li to Na layered oxide with no structural change and no biphasic domain like for Li1-xNaxNiO2. [START_REF] Matsumura | Structure and Physical Properties of LixNa1-xNiO2[END_REF][START_REF] Holzapfel | Decoupling of Orbital and Spin Degrees of Freedom in Li 1-xNaxNiO2[END_REF][START_REF] Holzapfel | Mixed Layered Oxide Phases Na XLi 1-XNiO 2: A Detailed Description of Their Preparation and Structural and Magnetic Identification[END_REF] A glass-like behavior After demonstrating the mixing of lithium and sodium despite their very different ionic radii, it was important to investigate the formation process of this very uncommon crystallographic feature. First, it is worth pointing out the influence of the quenching process at the end the thermal treatment. Indeed, a slow cooling rate down to room temperature leads to very different XRD patterns, with coexistence of Li3Ni2SbO6 and Na3Ni2SbO6 as the main phases and with a very weak intermediate contribution from the mixed Li3-xNaxNi2SbO6 (see Figure S3 for a nominal composition "Li1.5Na1.5Ni2SbO6"). In their pioneering investigation on layered Li1-xNaxNiO2, Matsumura and coworkers also briefly noticed that quenching was mandatory for obtaining a monophasic product. [START_REF] Matsumura | Structure and Physical Properties of LixNa1-xNiO2[END_REF] In this current study, we decided to have much deeper grasp on the details of the stability of the obtained monophasic compositions by combining XRD and DFT approaches.

The thermal evolution of quenched Li1.5Na1.5Ni2SbO6 has been monitored by in situ XRD (Figure 6). At 300 °C, the phase decomposes as its main diffraction peak (001) splits into two peaks accounting for Li3Ni2SbO6 and Na3Ni2SbO6 domains. At higher temperatures, the synthesis conditions of a monophasic domain are gathered again and the two peaks merge into one contribution accounting for (001) of Li1.5Na1.5Ni2SbO6. During the slow cooling down, a second splitting occurs, showing the reformation of Li3Ni2SbO6 and Na3Ni2SbO6 domains. This result clearly highlights the role of the final quench for stabilizing at room temperature a monophasic composition with a random distribution of alkali ions in the interslab layer. Such thermal feature strongly recalls the behavior of the layered oxide OP4-LixNayCoO2, for which lithium and sodium are not randomly mixed but alternately ordered one layer over two in octahedral and prismatic sites, respectively. [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] Following the peak position evolution of the Li3Ni2SbO6 and Na3Ni2SbO6 domains can help in understanding the mechanism involved in the formation of a monophasic domain Li1.5Na1.5Ni2SbO6 at high temperature. First, it is important to keep in mind the thermal expansion that significantly increases the slab thickness and thus displaces the (00l) peaks towards low angles (i.e. from 20.1 ° at room temperature to 19.7 ° at 900 °C for the (001) peak of Li1.5Na1.5Ni2SbO6). This feature seems enhanced for Li3Ni2SbO6 domains (from 21 ° at 400 °C to 20.6 °C at 700 °C), whereas the trend is different for the peak related to Na3Ni2SbO6 domains, which slightly shifts to higher angles. The slab thickness contraction (expansion) accounts for lithium (sodium) diffusion into the Na3Ni2SbO6 (Li3Ni2SbO6) alkali layers, in agreement with the above NMR analysis (Figure 3). The underlying mechanism consequently involves a diffusive alkali exchange between domains, respect to a nucleation process for which only a shift due to thermal expansion would have been observed. The slow cooling monitoring shows a similar peak splitting and thus confirms that phase decomposition is thermodynamically favored. To briefly summarize, the Li/Na random distribution mimics an alkali-glass behavior, however confined within a crystalline matrix.

In parallel, DFT calculations have been performed on about 100 structures with Li1.5Na1.5Ni2SbO6 stoichiometry, within random ordering of Li/Na in the alkali layer. Figure 7 shows the formation energy of each ordered phase respect to the pure lithiated or sodiated compounds and the variation of the formation energy as a function of the number of Li-Na Nearest Neighbor (NN) bonds. As the relative formation energy is positive, all the random ordered phases are shown to be thermodynamically instable and should decompose following the equation: Li1.5Na1.5Ni2SbO6 → ½ Li3Ni2SbO6 + ½ Na3Ni2SbO6.

The most stable order consist in the out-of-plane ordered/segregated Li/Na phases (at ~3 meV/atoms) without NN Li-Na, the energy increasing barely linearly with the number of Li-Na NN bonds. In this situation, the stabilization of any ordered phase does not appear as feasible except if configuration entropy, thus Li/Na inter-mixing is considered. In fact, the latter can be evaluated within the Stirling approximation to be ~20 meV/atoms at 900 °C far enough compared to the relative energy between ordered phases. Consequently, the most probable hypothesis consists in an entropically stabilized Li/Na intermixing in agreement with the synthesis conditions and thermal behavior, structural refinement and NMR analysis, and characteristic to glassy-like systems.

Conclusion

A complete solid-solution from layered Li3Ni2SbO6 to Na3Ni2SbO6 has been obtained by conventional solid-state synthesis ended by a final quench. On the transition metal layer, the honeycomb Ni/Sb cation ordering is maintained all along the solid-solution; however with a partial Ni/Sb mixing. Focusing on the alkali layer, configuration entropy leads to Li/Na randomly occupied crystallographic positions.

It is believed that other crystalline transition metal oxides exhibiting 2D confined alkali glass behavior could now be prepared thanks to a final quenching step, and thus opening the door towards unexpected physical properties.

Figure 1 .

 1 Figure 1. (a) XRD patterns of Li3-xNaxNi2SbO6 compositions obtained after a final quench, (b)

Figure 2 .

 2 Figure 2. Structural hypotheses for the intermediate composition Li1.5Na1.5Ni2SbO6. Nickel and

Figure 3 .

 3 Figure3. Solid-state[START_REF] Myung | Nickel-Rich Layered Cathode Materials for Automotive Lithium-Ion Batteries: Achievements and Perspectives[END_REF] Li and[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] Na NMR spectra of as-quenched Li3-xNaxNi2SbO6 compositions,

Figure 4 :

 4 Figure 4: Combined Rietveld refinement of X-ray (top) and neutron (bottom) diffraction patterns of

Figure 5 :

 5 Figure 5: (a) Structural model for Li1.5Na1.5Ni2SbO6 resulting from the combined X-ray/neutron

Figure 6 .

 6 Figure 6. Thermal evolution of the quenched Li1.5Na1.5Ni2SbO6 monitored by in situ XRD.

Figure 7 .

 7 Figure 7. (a) Relative formation energy in meV/atom of randomly Na-Li ordered phase respect

Table 1 :

 1 Structural parameters for Li1.5Na1.5Ni2SbO6, deduced from the combined Rietveld refinement of the XRD and neutron powder diffraction patterns.

Figure 1 Figure 2 Figure 3 Figure 4

 1234 Figure 1

Figure 5 Figure 6

 56 Figure 5

Acknowledgment

The authors thank the French National Research Agency (STORE-EX Labex Project ANR-10-LABX-76-01).

Supporting Information

The Supporting Information section is available free of charge at DOI: xxx.

Notes

The authors declare no competing financial interest. XRD: Bragg R-factor = 8.61 %, Rf-factor = 6.23 % Neutrons: Bragg R-factor = 2.81 %, Rf-factor = 2.30 %

Figure and Table captions


Florent Mouysset

Célia Picard

Christophe Bortolaso

Frederic Migeon

Marie-Pierre Gleizes

Christine Maurel

Mustapha Derras

Investigations of Process Mining Methods to discover Process Models on a Large Public Administration Software

Keywords: Software logs analysis, Interweaving, Process Mining

Maintaining large and aging applications in a software house, with heterogeneous technologies, is very challenging. Whereas it is mandatory to continuously enhance user experience and maintain a good quality of service, the real business usage can be difficult to know precisely. To reach this goal, our project is to discover business models from the analysis of "logs". In this paper, we report on existing studies about applying process mining techniques and on our own experience with large datasets generated from daily end-user activities within an existing public services software. Our experiments led us to identify an interesting combination of features making our data hard to process with existing techniques. We conclude by providing perspectives to enable process discovery with such specific data.

Introduction

The complexity of software for public services increases with the constant evolut This has major impacts on software quality and maintenance processes. First, on the software editor side, this usually leads to an increasing number of bugs [START_REF] Subramanyam | Empirical analysis of CK metrics for objectoriented design complexity: implications for software defects[END_REF]. execution contexts and data make the bugs very hard to reproduce and to fix. As the project team designs the tests a priori, they may not reflect Thus, a gap appears between the theory and the reality (second Lehman s law [START_REF] Lehman | Programs, life cycles, and laws of software evolution[END_REF]), leading to failures on untested and unexpected cases. Then, from oftware becomes more difficult to use, interact with and understand [START_REF] Thompson | Feature Fatigue: When Product Capabilities Become Too Much of a Good Thing[END_REF]. A new paradigm emerges and tries to design intelligent software, able of adapting themselves to specific users (Salinesi, 2017). This requires observin Logger syst activities. We made the assumption that logger systems combined with process mining techniques [START_REF] Ailenei | Definition and Validation of Process Mining Use Cases[END_REF] would enable the discovery of the observed system model. The discovered business process model would allow us, in a first step, to elicit use cases like user goals or business scenarios. In a second step, it would lead to prioritization of the maintenance, more realistic tests, and business intelligence.

In this paper, we report on experiments and lessons learned from applying process mining techniques on logs extracted from an existing on-use large public [START_REF] Astromskis | A process mining approach to measure how users interact with software: an industrial case study[END_REF] who managed successfully to use process mining to understand user interaction on software. It led us to believe that their approach would be easily replicable on our dataset.

In the following pages, we first present our case study, the logged data and the associated software. Then, after presenting a general review of existing approaches, the best technique is selected to conduct experiments. Section 4 details these experiments. To conclude the paper, we stress that the presence of certain characteristics in our logs makes the process mining techniques ineffective. The problem is conceptual and not related to the logging system.

Case Study Description and Associated Data

We collected logs1 generated by a rich client application used by town hall re is composed of four modules. In this study, we focus on the analysis of logs coming from the Civil Status Management Module (hereafter called CSMM). This module contains numerous features related to civil registries, such as elections, births, deaths or court jurors. The software interface is composed of multiple tabs, each of them containing multiple forms. The navigation inside the CSMM is structured as follows: it starts with a home page containing a list of buttons. When clicking a button in this menu, a new tab opens and gets the focus. This new tab provides access to a second level menu containing itself user interface components to access forms and functional features.

The logging system is implemented in a software layer shared by all the different modules. This enabled developers to rely on a generic tracing system and to focus only on the development of functional features.

Despite the simplicity of the main navigation, the CSMM is extremely large and dense. Indeed, it is composed of about 600 different forms representing about 200,000 lines of code. We collected logs from 104 users over one year. The data contains 227,782 events for 60 mega-bytes in an XML structure. However, the software interface is designed to allow other additional navigation instances. From a form D another form C can be opened in a new tab, both being independent. Thus, it is possible to close the form D first, and after form C. Closing forms can also work with a FIFO (i.e. First In First Out) behavior. This indicates that a form closure does not necessarily imply a parent relationship. Furthermore, in ClosingTimestamp not have a closing tag). It means either that the form encountered an error, making it impossible to write the closing date or that the form was not closed before leaving the application. Finally, we observed that sometimes, even if two forms are dependent, the closing date might be incoherent. For instance, the form C is child of form A (and are dependent, no new navigation instance) but the closing of form C occurs after the closing of its parent (A2). These three scenarios illustrate the wide range of behaviors that can be found in the processed data in our case study. The next section describes the different existing approaches that use input logs to discover process models.

Discovery Process: Existing Approaches

Multiple scientific domains focus on the analysis of human activity logs. In fact, several methods and algorithms designed to discover usage patterns and process models on business activity logs exist. We identified two types of relevant paradigms: 1) web mining and 2) process mining.

Web Mining

The web mining or web usage mining offers several approaches to analyze a web application and its usages [START_REF] Srivastava | Web Usage Mining: Discovery and Applications of Usage Patterns from Web Data[END_REF]. Usually, the techniques are usercentric: all the events are related to a user. Existing techniques are most of the time based on sequence mining and pattern discovery methods. One of the typical purposes of web mining is to find the main way to accomplish a predefined goal. For instance, this purpose cou product in an optimize navigation paths and hence, increase sales [START_REF] Spiliopoulou | Improving the Effectiveness of a Web Site with Web Usage Mining[END_REF]. The goal of web mining differs from ours even though the data are similar. In web mining, the user is linked to a web session and each web session corresponds to a single use case [START_REF] Srivastava | Web Usage Mining: Discovery and Applications of Usage Patterns from Web Data[END_REF]. Moreover, in the analysis the user goal is usually known beforehand: a purchase, a web page access or more generally the achievement of a well-known special action. goals in the current navigation path. Worse, the user can perform several use cases (i.e. pursue several goals) concurrently and we do not know how to relate the events to use cases. Thus, these techniques appear irrelevant for our problem.

Process Mining

Compared to the specificities of web analytics and mining, process mining appears to be a more suitable approach to our problem. The process mining is a recent scientific domain defined as a mix between data mining and workflow analysis (van der Aalst, 2016). The process mining grasps three main aspects: 1) process discovery, 2) conformance model and 3) process enhancement. Since we seek to produce a process model from logs, we focused on the first aspect of the process mining: the process discovery. The process discovery is a collection of techniques that take logs as inputs and give process models as outputs. The input logs contain traces. Each trace is a sequence of events referring to the same case. Events are footprints left by the users when performing a business activity. Each event contains, at least, the name of the process activity, a timestamp and a specific process instance identifier2 . Generally, the inputs are process-oriented [START_REF] Pourmasoumi | Business process mining[END_REF]. This means that each trace corresponds to a process execution. Each process execution is identified by a unique id called the process instance id (or case id). The events belonging to the same process execution share the same process instance id: they are labelled. Logs containing labelled events are called labelled logs. The logs can contain various sorts of noises ((van der Aalst, 2016) p.148-151) and some data can be missing, incorrect or imprecise. This problem can occur in a continuous, intermittent or unpredictable manner. Business process models are generated as output. Various formalisms exist such as Petri Nets and Business Process Models and Notations (BPMN). Quality metrics can also be computed on these models. The most frequent metrics are the fitness (i.e. the ability to replay the behaviors seen in the logs) and the precision (i.e. the ability to forbid the behaviors unseen in the logs).

Approaches dealing with labelled logs

To enable process discovery, Cook and Wolf early proposed solutions based on neural-networks or Markov models, each solving a limited part of the problem. For example, the KTail approach finds a correct model but is noise-sensitive. On the contrary, the RNet methods find a less accurate model but are robust to noise [START_REF] Cook | Discovering models of software processes from event-based data[END_REF]. However, none of those methods can manage concurrency aspects: in a business process, some activities can be performed simultaneously, and it is important to detect and represent these concurrent activities [START_REF] Buijs | On the Role of Fitness, Precision, Generalization and Simplicity in Process Discovery[END_REF]. The -algorithm (van der Aalst et al., 2004) brings a formalism to discover WorkFlow nets (WF-nets) (van der Aalst, 1998). WF-nets are a subclass of Petri Nets that can offer some interesting properties like soundness and safeness. They provide very good results, but these methods remain extremely noise-sensitive (van der [START_REF] Van Der Aalst | Workflow Mining: Discovering Process Models from Event Log[END_REF].

algorithm (van der Aalst, 2016). The Heuristics Miner Algorithm [START_REF] Weijters | Process Mining with the Heuristics Miner-Algorithm[END_REF], brings a significant improvement by solving problems such as the short loop discovery or the mining of long-distance dependencies [START_REF] Wen | Detecting Implicit Dependencies Between Tasks from Event Logs[END_REF]. Günther to discover processes from noisy data by using a graph model formalism [START_REF] Günther | Fuzzy mining adaptive process simplification based on multi-perspective metrics[END_REF]. With two new defined metrics (the significance and correlation) the method captures the main patterns. But human intervention is required to tune parameter settings and identify the best model. Overall, probabilistic and statistical approaches are adapted to simple events but are limited when data is rich and complex.

The last family of approaches studied is the genetic algorithms. For instance, (Alves de [START_REF] Alves De Medeiros | Genetic Process Mining[END_REF] proposes to reuse the main concept of the genetic algorithms in process mining. With her method, an individual is considered as a process model on which selection and reproduction steps are applied. The main drawbacks of genetic algorithms are the long execution time and the local minimum risk. Moreover, genetic algorithms generate very complex models when using real-life logs (De [START_REF] Weerdt | A multi-dimensional quality assessment of state-of-the-art process discovery algorithms using real-life event logs[END_REF].

Approaches dealing with unlabeled logs

Because obtaining labelled logs can be very challenging [START_REF] Pérez-Castillo | Correlation of Business Activities Executed in Legacy Information Systems[END_REF], various methods try to deal with unlabeled data. The solution relies on labelling events artificially. We have identified two approaches: 1) making assumptions about structural, behavioral and temporal aspects of logged activities [START_REF] Pourmirza | Correlation Mining: Mining Process Orchestrations Without Case Identifiers[END_REF][START_REF] Walicki | Sequence Partitioning for Process Mining with Unlabeled Event Logs[END_REF], and 2) techniques based on expert [START_REF] Pérez-Castillo | Correlation of Business Activities Executed in Legacy Information Systems[END_REF].

Pourmirza [START_REF] Pourmirza | Correlation Mining: Mining Process Orchestrations Without Case Identifiers[END_REF] proposes an algorithm to label logs coming from orchestration services. Three conditions must be respected: 1) it does not work with data coming from other services than acyclic orchestration services, 2) metrics on duration per activity must be given to the algorithm, and 3) the idle time between two requests should not be equal to the activity duration. Similarly, Walicki and Ferreira propose a probabilistic approach [START_REF] Walicki | Sequence Partitioning for Process Mining with Unlabeled Event Logs[END_REF] to process discovery with unlabeled logs. The approach has a high level of abstraction because the problem can be reduced to sequence mining challenges. Their technique tries to cover the sequence with a minimal set of patterns. However, some conducted experiments show that this approach is not suitable to process large logs [START_REF] Walicki | Sequence Partitioning for Process Mining with Unlabeled Event Logs[END_REF].

When the assumptions about data and processes are too variable, experts can explicitly write rules to inform the tagging algorithms. [START_REF] Pérez-Castillo | Correlation of Business Activities Executed in Legacy Information Systems[END_REF] propose a semi-automatized process able to build correlation rules. An event must contain various attributes and the logging system must be designed to put the candidate correlation attributes into events. An expert can choose the attributes in the log and determine the correlation rules.

Synthesis on process discovery

Broadly speaking, the literature shows that a large variety of techniques are unclear in our software when and where use cases start and end. We chose to skip the early approaches of process mining for the more efficient recent ones. Process mining approaches dealing with unlabeled logs are not suitable as they require realistic temporal and structural assumptions. As described before, due to the activity interweaving and interruptions, it is impossible to make any realistic has many legacy features; it is impossible to find experts with enough business knowledge and time to choose the attributes for each code injection. The difficulties of improving logging system are detailed in section 6.3. The logs only represent the opening and closure of forms, tabs and windows. The missing of the case id is the only limitation that forbids us to apply approaches dealing with labelled data. Thus, we artificially labelled our events to execute existing process mining techniques on [START_REF] Astromskis | A process mining approach to measure how users interact with software: an industrial case study[END_REF].

Experiments with Process Mining

Based on our data and literature review, process mining methods appeared to be promising approaches to process our logs and generate the software business process model. To generate the business process model of CSMM, we have conducted experiments with several process mining techniques implemented in ProM. ProM is an academic process mining software that offers several process mining techniques through plugins.

Figure 3. The main steps of our case study experimentation

We tested four different methods aiming to discover valid process models and compared the results. Hereafter, we describe our protocol and data transformation process, the tools, and finally our results.

Protocol

As described in Figure 3, our protocol is divided into three main steps: 1) anonymization, 2) XES transformation and 3) execution of ProM.

1. Anonymization. Due to privacy concerns, the first step consists in removing data that can be related to individuals such as: emails, addresses, names, surnames, etc. This information is replaced by IDs to maintain coherence and integrity in the logs. This process is reproducible and non-reversible. . Labelling the events can be very difficult, thus we developed several strategies based on a successful study [START_REF] Astromskis | A process mining approach to measure how users interact with software: an industrial case study[END_REF]. Each of them provides a smaller granularity supposed to give a better trace. We tested different transformations methods in the next sub-sections. Each concern is addressed as a sub-step as depicted in Figure 4.

2.a Filtering: we removed some noise and execution errors from the raw logs. More precisely, we identified and removed the events where the closing date is missing as they indicate a software failure, according to the development team. Overall, 7939 events were removed, corresponding to 3.3% of the total dataset.

2.b

Event converter: various alternatives are available to translate CSMM events into XES events, each providing different advantages and semantics. We identified two strategies: 1) to the corresponding field of the XES event. This is the straightforward approach.

2) either a form opening or a form closing. Thus, the conversion may produce two XES events for each event, one a --

2.c Artificial Case

Labelling: XES logs are structured into traces, which is a missing concept in our logs. In the XES format, a trace reflects a case which is not delimited in our data. Thus, we tried five different strategies to provide each event with a case id and artificially recreate traces from our logs:

1. Hence, only one trace is created containing all the events.

2. trategy builds one trace per user. All events performed by the same user have the same case id.

3. user labelling; a case represents a use performed by the same user in the same day have the same case id. Note that a case cannot be performed over several days because town hall agents must respect working hours. Moreover, none of the use cases require several days to be accomplished.

4. the end of the previous business scenario and the beginning of a new one. All the events performed by the same user in the same day between two home page accesses have the same case id.

5. er Subprinciple as UHTD but considers the direct children forms of the home page. CSMM counts 11 sub-forms at a 2nd level of navigation. When an opening event of a subhome form occurs, all the encountered events form a trace. All the events performed by the same user in the same day between sub-home forms have the same case id. We chose to stop at the second level because it still represents a manageable size of sub-nodes. The number of forms at the 3rd level increases exponentially.

We crossed all the possibilities between our two event conversion strategies and our five artificial use case labelling strategies. We decided not to consider the combination of USHTD and 1EGRC2XES strategies. In fact, we observed that by applying this strategy 50% of the events were located between accesses of forms. This led us to believe that this strategy would not provide any significant results. The artificial use case labelling strategies can be considered as cutting functions. In fact, all the opening forms of a considering level are cut points. The forms path before the cut belongs to a use case; the forms path after the cut belongs to another use case. We restricted our study to these five strategies because more refined approaches would over-cut the logs and would have created too many traces.

2.d Sorting:

Finally, we sort the events by date in the resulting XES files. In the 1EGRC1XES condition, we sort the events by start date (i.e. opening of the form). In the 1EGRC2XES condition, we sort the events by the date kept in the event, thus opening date of the form for some events and closing date of the form for the others.

Execution. The last step consists in executing the four plugins, described in the apparatus below, on various data sets. Exactly, each one of the four listed plugins are tested with each generated XES file using the two different converters (2b) and the 5 different labelling strategies (2c), hence, a total of 4*2*5=40 experimentations.

Apparatus

To perform process mining, we rely on ProM v.6.7. ProM is an academic process mining software3 . Various plugins exist, especially for process model discovery. As explained above we aim at testing and comparing four different methods on our data (see Table 1). The experiment has been carried on a standard laptop machine including 16Gb with 6Gb RAM dedicated for JVM, Inte -4210M CPU@ 2.60GHz. Our evaluation is based on the replay fitness metric [START_REF] Buijs | On the Role of Fitness, Precision, Generalization and Simplicity in Process Discovery[END_REF]. The replay fitness indicates how much the discovered model can reproduce the behavior capture in logs. The replay fitness is ranged between 0 (the discovered model cannot replay any part of the logs) and 1 (a perfect replay).

Results

The results of the experiments are listed on Table 2. After the XES transformation step, we obtain traces corresponding to process runs in process mining. The trace number increased as the trace granularity became smaller. The event conversion produces a total of 227,782 events, considering only the opening of the forms (1EGRC1XES), and logically twice as much (455,564) when also considering closing events (1EGRC2XES). Overall, the results are disappointing, and no method allow us to discover a reliable process.

First, surprisingly, with the Heuristic Miner, the more we try to label use cases precisely, the lower the fitness is. We also observe that in over one-third of our tests, the Heuristic Miner could not provide positive fitness. Negative fitness indicates a very low success due to many remaining tokens in the associated Petri Net. The 0.55. For the Heuristic Miner, it appears simpler to deal with a unique high grain trace than to indicate various precise but noisy traces.

The ETM execution time was extremely long (several hours on our target computer), and the computation failed several times due to memory lack (i.e. java heap space). Even when reducing by 50% the dataset content, the best fitness provided by the ETM was only of about 0.59 (with Naïve labelling). This result is [START_REF] Weerdt | A multi-dimensional quality assessment of state-of-the-art process discovery algorithms using real-life event logs[END_REF] that the genetics approaches are not very suitable to process real life data. On the contrary, the Alpha Robust Miner provides no result when applying the Naïve strategy. It becomes more efficient when splitting the data into use cases. It provides results between 0.40 and 0.46. We attribute this to a better noise robustness.

Finally, the Fuzzy miner provides the best results (~73%) with the , the discovered model is unreadable and traces replays shows that many transitions noted as wrong in the model exist. Unlike the Heuristic Miner, the Fuzzy miner is better with precise tagging techniques.

Discussion and threats of validity

None of the tested techniques provides satisfying results with our data, even though we tested multiple conditions and provided ProM with logs of various granularity of events and several use case labelling strategies. We found that some of the methods have trouble to simply provide results. This could be an implementation problem or a lack of memory, but our tests highlight that with large datasets, some algorithms require very powerful machines to reach their objective without causing issues. In addition, we observed that getting reasonably good results does not mean that the resulting graph is readable. Our example with the Fuzzy miner is quite a good illustration of this limit. Our software experts indicated a high level of wrong transitions on the generated models. This brings us to a well-known problem: the Oracle. How confident in our results can we be? Moreover, 73% of replay indicates that we still have 27% of noise.

Furthermore, despite the application of an equivalent methodology and protocol, our results are in opposition with the results obtained by Astromskis [START_REF] Astromskis | A process mining approach to measure how users interact with software: an industrial case study[END_REF]. The main difference between their study and ours lies in the implementation of the logging system and the size of our software. We believe that the automatic and generic implementation of our logging system could be the main explanation to the differences between our conclusions and Astromskis et al fact, the logging system was manually added in their work. This means that the logging software was designed a posteriori for their purposes. On the opposite, in our case, the logging system was designed and implemented long time before our research. Nevertheless, the lack of information on the complexity of the analyzed software is not sufficient to explain reliably our failure. Some of the characteristics of our data might also be a reason. We detail them in the next section.

Learned Lessons

After obtaining the results we conducted a qualitative analysis to understand why process mining techniques deprived us from satisfying results. Our analysis led us to understand many specificities about our data which were directly induced by the next sub-section with the secretary scenario.

The Town Hall Agent Scenario

h explains the nature and complexity of our data. The scenario starts with a town hall agent who starts a business scenario (BS1) thus creating events. A phone call occurs interrupting the case; BS1 is suspended. During the phone call, the agent starts a new scenario (BS2) phone call ends but the agent still must perform actions related to BS2. At this point, someone enters the office and makes a new request. Again, this interruption suspends the processing of BS2. The agent starts a third scenario (BS3). When it ends, the secretary can resume BS2 and produce associated events. Finally, BS2 ends and BS1 is resumed. This sequence of actions is possible because: 1) software users can be interrupted while executing a use case and 2) the software allows to start one or several new use cases simultaneously. This flexibility between use cases and tasks is mandatory to provide efficient tools to support this kind of activities. The understandability of software logs consequently suffers from this type of practice and thus our ability to understand the tasks.

A combination of four features

Overall, following the previously described scenario, we have identified that our data is characterized by four features which, all combined, make the business model hard to discover: Unlabeled Use Cases. The first issue is related to the impossibility of labelling use cases on the fly. The users require flexible software that allow multiple use cases in parallel thus there is no clear indication in the log about when a use case starts and stops. This is a strong tendency in modern-web apps, such as Single Page Applications, enabling more flexibility, interactivity and overall a better userexperience to end-users [START_REF] Mesbah | An Architectural Style for Ajax[END_REF].

The Impossibility of making Temporal or Structural Assumptions. Our data provide no clear indication about the duration of tasks. In fact, depending on the situation, filling a form to update a civil status can take from a few minutes up to several hours due to the interruptions and the requests. Moreover, some of the forms and screens are used by multiple use cases. For example, declaring a newborn may require updating the civil status of a parent. This parent civil status update will be the same as for a new passport request. The nature of our logs does not enable to identify clearly which specific screen is used in which use case.

The Presence of Loops.

vities also involves many redundant activities. For instance, the agent can come back and forth between the home menu and the burial plot attribution form. This enables the users to fill forms in a row in order, for instance, to process a large quantity of records. This, again, is quite common in administrative activities and will have to be identified when applying analysis techniques on our logs.

Multiple Levels of Interweaving. As illustrated in the scenario on Figure 6, we found multi-level of interweaving between use cases. The logs represent a list of events where each one contains a starting timestamp and ending one. These timestamps might overlap with other ones in the entire content of the log. This interweaving is due to the ability of users to concurrently start, perform and close several use cases. Our logs do not contain one linear story but several interweaved situations. It appears that existing approaches have difficulties handling the secretary scenario depicted in our case.

These four features are today more or less manageable by different approaches. For example, Walicki [START_REF] Walicki | Sequence Partitioning for Process Mining with Unlabeled Event Logs[END_REF]) is known to be suitable for interweaving and Pourmirza approach works for cyclic applications [START_REF] Pourmirza | Correlation Mining: Mining Process Orchestrations Without Case Identifiers[END_REF]. However, the combination of these four features makes the problem very hard to solve.

Challenges with logging systems

We also investigated the logging system itself. Our analysis highlighted interesting aspects of the logging system and difficulties to enhance it. In our case study, the logging system is implemented in a transversal manner, ensuring a strong decoupling between the application modules. This enables developers to rely on a generic tracing system and to focus on functional features development. The logging system uses the minimal data provided, thus the event logs are inaccurate. To make the logs more detailed, the entire logging system would have to be redesigned; whereas it is financially impossible. This problem is not restricted to our application. In general, since the logging system is not a business feature, it is not a major concern for the production team. The fewer resources are involved to set up a logging system, the better. The existing logging systems can produce events with technical details used for debugging purposes but are not able to inform the case id necessary for process mining. Multi-layer applications increase the difficulty because the information needed to create the correct event is spread over several layers. Several logging systems may be mandatory, or a single central logging system, that may break with a decoupled architecture.

Finally, redesigning the logging system of a large part of legacy code may cause regression. In fact, in some specific situations, adding logging calls can have unexpected impact on performances or even on the reliability of the execution. This problem needs great attention, since most of the practical use cases are unknown, non-regression tests are potentially inefficient. This analysis of our logging system stresses the tension between being able to build a generic and transversal mechanism and collecting enough data to uncover the business models afterwards. This type of issues solely appears when working on large, distributed applications, where modifying the code manually is not a viable option. Overall, this highlights the need to work on logging systems architecture to enable genericity and low granularity of collected events.

Perspectives and Conclusion

In section 2, we have indicated the main reasons making it extremely difficult to understand our software logs. These difficulties amount to matching the events of a process with the software task model. This matching can be formalized by a multicriteria optimization problem where each event must be associated to a unique task instance according to the log history, and each task must try to be fully described by events. This leads to a huge search space due to the combinatorial explosion. For this reason, we believe that Artificial Intelligence approaches might be of interest to process software logs. These approaches can preprocess the log to obtain labelled logs and, hence, prepare the logs to the application of process mining techniques. Two domains are interesting: 1) the clustering and classification approaches like the artificial neural networks and support vector machines, 2) the Multi-Agent Systems (MAS), in particular the Adaptive Multi-Agent Systems (AMAS).

The first approach tries to manage and reduce the complexity and the nonstructuration of some processes. Rocío [START_REF] Rocío | Process mining through artificial neural networks and support vector machines: A systematic literature review[END_REF] have done a systematic literature review on Process Mining with artificial neuronal networks and vector machines approaches. Only two studies propose a contribution on process discovery Mining [START_REF] Song | A Comparative Study of Dimensionality Reduction Techniques to Enhance Trace Clustering Performances[END_REF].

On the contrary, the second approach does not try to reduce or simplify the complexity of data but tries to grasp the entire complexity of a problem. Some MAS rely on emergent techniques to find solutions. For example, Adaptive Multi-Agent Systems (AMAS) [START_REF] Di | Selffrom natural to artificial adaptation[END_REF] have already proved efficiency to solve some complex problems like big data analysis [START_REF] Belghache | Towards an Adaptive Multi-agent System for Dynamic Big Data Analytics[END_REF] even in very noisy environments. AMAS exploits the collective intelligence of agents to organize themselves and build a business process model.

To produce a suitable product while more complexity is needed, future software will have to dynamically adapt themselves to the users. To design these systems, we cannot rely on a known business process model. As the system cannot be intrusive, activities user traces must be taken as input. Regarding our needs, the process mining seems to be able to provide models of our software from execution logs. Because our events are unlabeled and our logging system unmodifiable, we used similar preprocessing steps as existing studies. Then, a set of process mining techniques was executed including the ProM plugins. The results showed an important quantity of wrong transitions in the discovered processes. This result highlights the complexity of our data: interweaving of use cases, lack of use case labels and various loops creating wrong trace labelling. Because of the diversity of users and use cases, we could not make temporal, structural, behavioral or semantic assumptions. To deal with the complexity and removing the noise, we consider exploring artificial intelligence approaches and particularly multi-agent systems as a potential solution.

Figure 1 .Figure 2 .

 12 Figure 1. Example of CSMM log

Figure 4 .

 4 Figure 4. Sub-steps of the XES transformation

Table 1 .

 1 ProM Plugins Used and Tested

	Name	Version	Output
	HeuristicsMiner	6.7.70	Fitness metrics
	Fuzzy	6.7.53	Replay percentage
	Alpha Robust Miner	6.7.70	Petri Net. PN Conformance Analysis plugin to compute conformance
	Evolutionary Tree Miner (ETM) 6.7.168	Process Tree / Display fitness per generation

Table 2 .

 2 Summary of experiment results

	Converter	Labelling	Nb Traces	Nb Events	Heuristics Miner Fitness (0-1)	Fuzzy Replay %	ETM Fitness 0-1	Alpha Robust Miner replay score
	1EGRC1XES Naive	1	227 782 0.4084	61.77	0.59	-
	1EGRC2XES Naive	1	455 564 0.5351	47.07	0.59	-
	1EGRC1XES UT	104	227 782 0.5554	57.54	-	0.44
	1EGRC2XES UT	104	455 564 0.4036	43.80	-	-
	1EGRC1XES UTD	6 617 227 782 -1.6116	56.76	-	0.40
	1EGRC2XES UTD	6 638 455 564 -0.3839	39.68	-	0.45
	1EGRC1XES UHTD	12 794 227 782 -0.2543	73.03	0.51	0.41
	1EGRC2XES UHTD	12 840 455 564 0.2346	56.16	-	0.46
	1EGRC1XES USHTD 8 882 141 828 -0.1706	68.78	-	0.52

Data are on open access on https://github.com/FM-BL/PublicCSMM_Logs

In any case, an id is a unique identifier.

http://www.promtools.org/doku.php