Emeric Bourasseau

Alexandre Mouret

Paul Fantou

Xavière Iltis

Renaud C Belin

Experimental and simulation study of grain boundaries in UO2

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Introduction

The fuel used in the current generation of pressurized water reactors (PWRs) is mostly uranium dioxide (UO2). UO2 fuel is a refractory ceramic, used in the form of pellets obtained by a powder metallurgy process, leading to equiaxed grains with a diameter of the order of 10 μm separated by interfaces called grain boundaries. Under irradiation, grain boundaries are strongly affected both by the thermomechanical stress induced by the temperature gradient and by the stress generated by the fission gases gathering at the grain boundaries. These two phenomena possibly induce the breaking of grain boundaries. Depending on the operating regime of the reactor, several phenomena can occur: the formation of macroscopic cracks in normal operation, a fine fragmentation of the fuel during a loss of coolant accident (LOCA), or even a generalized decohesion of grains at the periphery of the pellet in the case of reactivity-initiated accident (RIA). The opening at the grain boundaries can be very significant in accidental situations and needs to be taken into account in the models used in simulation codes as GERMINAL [START_REF] Lainet | Recent modelling improvements in fuel performance code GERMINAL for SFR oxide fuel pins[END_REF] developed in our institute in the PLEIADES numerical platform. This implies to improve the knowledge of grain boundaries properties, selectively depending on the type of boundary. According to the respective orientations of the two crystals forming the boundary, a large variety of grain boundaries coexist in a polycrystalline material, whether special or general.

The parameters that are necessary to characterize a grain boundary are called geometrical degrees of freedom of the grain boundary [START_REF] Sutton | Interfaces in Crystalline Materials[END_REF]. Five macroscopic degrees of freedom are sufficient to give a complete description of a grain boundary [START_REF] Priester | 1-Geometrical Order of Grain Boundaries[END_REF]. In materials science, the "tilt/twist" description is traditionally used to distinguish grain boundaries. A tilt boundary corresponds to a rotation of an angle θ between the two grains, along a rotation axis located in the grain boundary plane. Thus, the misorientation axis is perpendicular to the boundary plane normal. On the contrary, for a twist grain boundary, the rotation axis is perpendicular to the boundary plane and thus parallel to its normal. Tilt and twist grain boundaries are often referred as "special". We adopted this nomenclature, but it is worth noting that the use of the term "special" is controversial. In fact, depending on the authors, "special" might also refer to grain boundaries that have "special" properties or CSL grain boundaries (see next paragraph). Any other given orientation of the rotation axis with respect to the boundary plane leads to a "mixed tilt/twist" grain boundary referred here as "general" as opposed to "special".

A complementary way of describing grain boundaries is to consider the misorientation between the two crystalline networks that the boundary divides by using the Coincidence Site Lattice (CSL) theory. CSL grain boundaries are characterized by a multiplicity index Σ which is defined as the ratio between the crystal lattice site density of the two grains meeting at the boundary and the density of sites that coincide when superimposing both crystal lattices [START_REF] Brandon | The structure of high-angle grain boundaries[END_REF][START_REF] Vonlanthen | CSL grain boundary distribution in alumina and zirconia ceramics[END_REF]. It is generally admitted that grain boundaries with low Σ values have a tendency for low interfacial energy and special properties. However, the principal drawback of the CSL model is that only the orientation relationship between neighboring grains is specified while the grain boundary plane orientation, which is closely associated with atomic scale structures, is neglected [START_REF] Miyamoto | The role of grain boundary plane orientation on intergranular corrosion of symmetric and asymmetric [1 1 0] tilt grain boundaries in directionally solidified pure copper[END_REF]. For this reason, the utility of the CSL classification and of the parameter Σ is quite controversial [START_REF] Watanabe | Grain boundary engineering: historical perspective and future prospects[END_REF]. Nevertheless, it remains a practical way of categorizing special grain boundaries.

In uranium dioxide, according to the type of boundary, data on fracture properties such as fracture stress or toughness are needed. If experimental measurements are common on nonradioactive materials such as metals or yttria stabilized zirconia (YSZ) [START_REF] Shibata | Structure, energy and solute segregation behaviour of [110] symmetric tilt grain boundaries in yttria-stabilized cubic zirconia[END_REF][START_REF] Fisher | The influence of grain boundary misorientation on ionic conductivity in YSZ[END_REF], far fewer results are available on UO2 [START_REF] Nerikar | Grain Boundaries in Uranium Dioxide: Scanning Electron Microscopy Experiments and Atomistic Simulations: Grain Boundaries in Uranium Dioxide[END_REF]. To our knowledge, measurements of formation energies, toughness, or mechanical properties of grain boundaries in UO2 are not available in the literature.

To fill this lack of data, numerical simulations are a good alternative. To study the properties of grain boundaries of a material using simulation, the good scale is undoubtedly the atomic scale. Where two crystals meet, the crystal lattice is not perfect. At the atomic scale, the interface, i.e. the grain boundary, takes the form of a discontinuity in the stack of atoms and is in fact a complex crystalline defect. In most polycrystalline solids, grain boundaries are very thin (1 to 2 interatomic distances). The disorder, which corresponds to bond breaking and point or extended defects, is inevitable for geometric reasons and generates an excess of energy of several J.m -2 . However, the system simulated at this scale must achieve a minimum size to be representative. In the direction normal to the grain boundary, it must be large enough so that the grains modeled on each side have properties close to that of the bulk crystal. Similarly, the size of the system parallel to the grain boundary must be large enough to avoid commensurability problems between the two crystalline lattices in contact [START_REF] Vattré | Determining the Burgers vectors and elastic strain energies of interface dislocation arrays using anisotropic elasticity theory[END_REF]. Thus, the minimum required size (a few tens of Å in the 3 directions of space) requires systems containing from several thousand to tens of thousands of atoms. First-principles atomistic methods, such as DFT, are then impossible to use and a "classical" atomistic simulation method that uses empirical potentials to describe the interactions between atoms is required. Even though many empirical potentials are available to calculate the properties of UO2 (see for example [START_REF] Govers | Comparison of interatomic potentials for UO2. Part I: Static calculations[END_REF][START_REF] Govers | Comparison of interatomic potentials for UO2. Part II: Molecular Dynamic Simulations[END_REF][START_REF] Chernatynskiy | Critical assessment of UO2 classical potentials for thermal conductivity calculations[END_REF][START_REF] Potashnikov | Highprecision molecular dynamics simulation of UO2-PuO2: Pair potentials comparison in UO2[END_REF][START_REF] Cooper | Atomic scale simulation of irradiated nuclear fuel[END_REF]), only few papers reporting atomistic simulation results on UO2 grain boundaries are available in the literature. In 2008, Van Brutzel et al. [START_REF] Brutzel | Grain boundary influence on displacement cascades in UO2: A molecular dynamics study[END_REF] studied 6 grain boundaries using Morelon empirical potential [START_REF] Morelon | A new empirical potential for simulating the formation of defects and their mobility in uranium dioxide[END_REF]. They calculated the formation energies of the grain boundaries as well as atomic structures and investigated the influence of the presence of a grain boundary on displacement cascades. In 2011, Nerikar et al. [START_REF] Nerikar | Grain Boundaries in Uranium Dioxide: Scanning Electron Microscopy Experiments and Atomistic Simulations: Grain Boundaries in Uranium Dioxide[END_REF] studied 3 grain boundaries with the Basak potential [START_REF] Basak | Classical molecular dynamics simulation of UO2 to predict thermophysical properties[END_REF]. Only formation energies and atomic structures were calculated. In 2014, Zhang et al. [START_REF] Zhang | Molecular dynamics simulations of intergranular fracture in UO2 with nine empirical interatomic potentials[END_REF] compared the energies of two grain boundaries obtained using 9 different potentials, including those of Morelon [START_REF] Morelon | A new empirical potential for simulating the formation of defects and their mobility in uranium dioxide[END_REF], Basak [START_REF] Basak | Classical molecular dynamics simulation of UO2 to predict thermophysical properties[END_REF] and Yakub [START_REF] Yakub | Molecular dynamics simulation of premelting and melting phase transitions in stoichiometric uranium dioxide[END_REF]. These authors also studied the intergranular fracture behavior of UO2. Fracture stresses were calculated by applying an elongation to the simulated systems following the dimension normal to the boundary plane. Also in 2014, Chen et al. [START_REF] Chen | Molecular dynamics simulations of grain boundary thermal resistance in UO2[END_REF] studied 5 grain boundaries using the Yakub potential [START_REF] Yakub | Molecular dynamics simulation of premelting and melting phase transitions in stoichiometric uranium dioxide[END_REF]. Thermal resistances were calculated in addition to formation energies. In 2015, Williams et al. [START_REF] Williams | Atomistic investigation of the structure and transport properties of tilt grain boundaries of UO2[END_REF] studied 6 grain boundaries using their own potential. They reported formation energies and evaluated toughness through the calculation of the cleavage energy. They also investigated the oxygen diffusion near grain boundaries. Again in 2015, Bai et al. [START_REF] Bai | Testing thermal gradient driving force for grain boundary migration using molecular dynamics simulations[END_REF] used the Basak potential [START_REF] Basak | Classical molecular dynamics simulation of UO2 to predict thermophysical properties[END_REF] to study 5 grain boundaries. Formation energies were calculated and the thermal migration of grain boundaries was studied. Finally, in 2016, Galvin et al. [START_REF] Galvin | Pipe and grain boundary diffusion of He in UO 2[END_REF] published a paper where 4 grain boundaries were studied using the Cooper potential [START_REF] Cooper | A many-body potential approach to modelling the thermomechanical properties of actinide oxides[END_REF]. They calculated the formation energies and investigated the diffusivity of helium at the vicinity of the boundaries. Comparing some of these articles is possible since different authors studied identical grain boundaries. The formation energies obtained are always between 0.3 and 3 J.m -2 , which is a realistic order of magnitude compared to the energies measured in other systems [START_REF] Shibata | Structure, energy and solute segregation behaviour of [110] symmetric tilt grain boundaries in yttria-stabilized cubic zirconia[END_REF]. It is worth noting that different procedures were used to build and relax grain boundaries. Some authors have used specific codes like GBStudio [START_REF] Ogawa | GBstudio: a builder software on periodic models of CSL boundaries for molecular simulation[END_REF] or Metadise [START_REF] Tobyákelsey | Atomistic simulation of dislocations, surfaces and interfaces in MgO[END_REF]. In some cases, a simple 0 K energy minimization was performed [START_REF] Nerikar | Grain Boundaries in Uranium Dioxide: Scanning Electron Microscopy Experiments and Atomistic Simulations: Grain Boundaries in Uranium Dioxide[END_REF][START_REF] Zhang | Molecular dynamics simulations of intergranular fracture in UO2 with nine empirical interatomic potentials[END_REF][START_REF] Bai | Testing thermal gradient driving force for grain boundary migration using molecular dynamics simulations[END_REF] whereas, in other cases, a more complex procedure was applied involving molecular dynamics relaxations at various temperatures (300 K [START_REF] Brutzel | Grain boundary influence on displacement cascades in UO2: A molecular dynamics study[END_REF], 1500 K [START_REF] Galvin | Pipe and grain boundary diffusion of He in UO 2[END_REF], 2000 K [START_REF] Chen | Molecular dynamics simulations of grain boundary thermal resistance in UO2[END_REF], and even 3000 K [START_REF] Williams | Atomistic investigation of the structure and transport properties of tilt grain boundaries of UO2[END_REF]). These differences may explain why calculated energies values are scattered. As an example, Williams obtains energies significantly lower than other authors. The reason might be that the annealing step used to relax the initial configurations is carried out at 3000 K, i.e. close to the melting temperature, therefore inducing a significant modification of the initial system to reach a lower energy state. Thus, it is difficult to ensure that the simulated grain boundaries retained their initial geometric characteristics. Also, for a given grain boundary studied with two different potentials, the energy obtained can be very different. Each potential probably does not describe the interfaces in the same way and a deliberate choice has to be made. Two important conclusions arise from this literature review. First, only few grain boundaries were studied and the results from the literature are too scattered to be used with confidence, so that further calculations are needed. Second, the empirical potential must be chosen carefully based on the comparison of the results obtained with different potentials, but with the same calculation procedure.

To select the best potential, comparing calculations results with experimental data is very valuable. In the literature, the experimental determination of grain boundaries energies frequently involves the use of synthetic bicrystals. Actually, a bicrystal elaborated from two suitably orientated single-crystals is the only system allowing a perfect knowledge of all five geometrical degrees of freedom of a given grain boundary. Such an approach is quite documented on metals [START_REF] Miyamoto | The role of grain boundary plane orientation on intergranular corrosion of symmetric and asymmetric [1 1 0] tilt grain boundaries in directionally solidified pure copper[END_REF] but is rare on ceramic oxides such as yttria stabilized zirconia [START_REF] Shibata | Structure, energy and solute segregation behaviour of [110] symmetric tilt grain boundaries in yttria-stabilized cubic zirconia[END_REF][START_REF] Greenough | Grain-Boundary Energies in Silver[END_REF].

To our knowledge, experimental determination of energies performed on geometrically defined grain boundaries in UO2 bicrystals is not available. A few articles, however, report the characterization of CSL grain boundaries in UO2 polycrystals by Electron Back Scattered Diffraction (EBSD).

Nerikar et al. [START_REF] Nerikar | Grain Boundaries in Uranium Dioxide: Scanning Electron Microscopy Experiments and Atomistic Simulations: Grain Boundaries in Uranium Dioxide[END_REF] determined by EBSD grain sizes, grain boundaries misorientation and distribution on 2D orientation maps comprising 150 to 200 grains. They compared the number fraction of individual CSL grain boundaries obtained from the misorientation data collected manually with the length fraction statistics provided by the EBSD software. They observed that length fractions are more consistent than number fractions to estimate the occurrence of a given grain boundary. This discrepancy between length fraction and number fraction analysis is well known in various materials, e.g. metals such as Nickel-based alloys [START_REF] Randle | Sigma-Boundary Statistics by Length and Number[END_REF]. According to the authors, 16% of the UO2 grain boundaries were CSL boundaries. Among these special boundaries, a higher concentration of Σ9 was observed, compared to Σ3, Σ5, and Σ11 fractions (the Σ9 grain boundary was the most abundant and represented 18% of all CSL boundaries). This result was attributed to a microstructural evolution, limited by kinetic processes. However, due to the relatively small size of the EBSD maps, these results must be taken with caution. It is also worth noting that, with 2D-EBSD, CSL grain boundaries can be identified and their character determined (sigma index and misorientation angle) but determining the atomic structure of the grain boundary (plane orientation and rotation axis), which is necessary for understanding material properties, is not possible and requires 3D experimental techniques.

A follow-up to Nerikar's work was published by Rudman in 2013 [START_REF] Rudman | Three-dimensional characterization of sintered UO2+x : effects of oxygen content on microstructure and its evolution[END_REF]. Two samples (UO2 and UO2.14) were studied to better understand the effect of oxygen stoichiometry on the microstructure. In a first step, 2D-EBSD maps were analyzed, each composed of about 30 grains. The stoichiometric sample had more grains than the UO2.14 sample. According to the authors, this result shows that the hyperstoichiometric structure is closer to the equilibrium than the stoichiometric one. This assertion is based on the assumption that in an ideal 2D structure, each grain should be composed of six linear boundaries that meet at 120° junctions. In terms of CSL grain boundaries, the results are similar for both samples with ~ 20% of CSL linear fractions, the stoichiometric sample exhibiting a higher fraction of Σ3 and Σ9 grain boundaries. These results are different from those of Nerikar's work. However, due to an even more limited number of grains composing the 2D maps, we believe that these results must also be taken with great caution, as they clearly are not representative enough of the sample microstructure at a larger scale. In fact, the real interest of Rudman's work stands in the 3D characterization of grain boundaries with 3D-EBSD. Samples were serial sectioned using a SEM equipped with a focused ion beam. Slices were spaced by 0.5 µm for UO2.14 and 0.33 µm for UO2 and ~ 80 slices per sample were analyzed. The 3D reconstruction of the volume analyzed allowed gaining access to the orientations of grain boundaries planes. In both samples, no {111} planes were present in the 30 to 50° range. For the stoichiometric sample, most planes in this angular range were close to {112} and {113} while, for the hyperstoichiometric sample, they were close to {011} and {113}. Such a work helps in knowing which planes are more favorable. This approach is original and, to our knowledge, it is the only experimental 3D characterization of grain boundaries in UO2 available in the literature.

In the present work, we investigated grain boundaries in UO2 by measuring the length fraction of different CSL grain boundaries, on large EBSD maps acquired on seven different UO2 pellets batches. We represented their distribution as a function of their sigma index and of their misorientation angle. The occurrence of each type of grain boundary was then compared to the formation energies calculated with various empirical potentials. In parallel, we studied 26 different grain boundaries of UO2 and compared 4 empirical potentials among the more recent and/or the more used, i.e. that of Morelon et al. [START_REF] Morelon | A new empirical potential for simulating the formation of defects and their mobility in uranium dioxide[END_REF], Yakub et al. [START_REF] Yakub | Molecular dynamics simulation of premelting and melting phase transitions in stoichiometric uranium dioxide[END_REF], Potashnikov et al. [START_REF] Potashnikov | Highprecision molecular dynamics simulation of UO2-PuO2: Pair potentials comparison in UO2[END_REF] and Cooper et al. [START_REF] Cooper | A many-body potential approach to modelling the thermomechanical properties of actinide oxides[END_REF].

Samples and experimental measurements

Seven UO2 polycrystalline samples (labelled A to G) coming from seven different fabrication batches were characterized in the present work. All of them were manufactured in the "Laboratoire UO2" of the CEA Cadarache, except the sample E, which comes from an industrial batch. A classical powder metallurgy process was used. Depending on the batch/sample, some parameters varied, such as if a preliminary milling of the powder before pressing was performed or not, the sintering temperature (between 1700°C and 1900°C) and/or duration (between 4 hours and 60 hours). This yielded mean grain sizes between 9 and 22 µm. The mean density is between 0.96 and 0.986 of the theoretical UO2 density.

The samples were cut from a pellet of each batch using a diamond wire saw and mechanically polished. The final polishing step was performed with a 0.02 µm colloidal silica suspension to minimize the polishing superficial damage for EBSD characterizations.

EBSD maps were performed with a FEI Nova NanoSEM (accelerating voltage: 20 kV, beam current: 10 nA) by means of a Nordlys II Nano camera from Oxford Instruments driven by the AZTEC software. EBSD data were analyzed with the Channel 5 suite of programs (version: 5.12.63.0). The indexation rate of EBSD data (ratio of indexed pixels over total number of tested pixels) varied from 95% to 99% (prior to any data cleaning), depending on the acquisition step (which varied from 0.4 µm to 1 µm, depending on the maps) and the number and size of pores present in the examined area.

Table 1 shows the total area investigated, the total number of grains analyzed and their mean grain diameter, given in equivalent circular diameter (ECD), for all samples. Depending on the sample, several EBSD maps were collected (their size is reported in the table). Grains were detected considering a misorientation threshold of 5° to define a grain boundary, and a minimum grain surface of 10 pixels corresponding to a minimum ECD of 1.2 to 3 µm, depending on the step size.

Map size (µm x µm)

Total area (µm²) To study the effect of the size of the EBSD map on CSL grain boundaries population, we focused on a map originating from sample C and comprising about 1250 grains. The whole surface as well as various smaller surface subsets were analyzed in terms of CSL linear fractions. Table 2 shows the results obtained for each subset: measured CSL fractions are between 11% and 19% and sometimes far from the value obtained for the whole map (14.90%). This result clearly shows that a surface too small does not provide reliable values. The sum of all the considered subsets leads however to a CSL population close to that of the whole map and corresponds to an equivalent map of ~ 500 grains. Therefore, it seems reasonable to consider that for a sintered polycrystalline UO2 with a homogeneous microstructure (equiaxed grains with a limited size dispersion), analyzing a map with at least 500 grains is necessary to obtain reliable CSL fractions. If the microstructure shows marked heterogeneities (such as areas with significantly smaller or larger grains), more grains might be necessary. That also confirms the caveats already given regarding Nerikar's and Rudman's results concerning the limited number of grains analyzed by these authors.

In the present work, all CSL fractions were obtained from maps with more than 500 grains. Depending on the sample, the surface and the number of maps differed and, in certain cases, about 5000 grains or more were analyzed (cf. samples D, E and F). Figure 1 shows the total linear fraction of CSL grain boundaries for samples A to G, compared to the results of Nerikar et al. [START_REF] Nerikar | Grain Boundaries in Uranium Dioxide: Scanning Electron Microscopy Experiments and Atomistic Simulations: Grain Boundaries in Uranium Dioxide[END_REF]. The extent of the associated error bars varies due to the different number and size of EBSD maps taken into account for each sample. With respect to the uncertainties, all the values are very close to each other and in between 14,6% and 15,4%. They are also in agreement with Nerikar's results, obtained on 150 to 250 grains of UO2, and coherent with Vonlanthen's and Grobety's ones, namely between 17% and 17.8%, in YSZ samples sintered in various conditions [START_REF] Vonlanthen | CSL grain boundary distribution in alumina and zirconia ceramics[END_REF]. sequence and Vonlanthen and Grobety the Σ3, Σ5, Σ11, Σ15, Σ9 and Σ7 on YSZ. The important finding here is that the elaboration and sintering conditions do not noticeably affect the total fraction of CSL boundaries in the samples. The similarity with Nerikar's, Vonlanthen's and Grobety's results seems to demonstrate that CSL grain boundaries character distribution in oxides ceramics with a cubic fluorite structure is rather independent of the nature of the material and of the conditions of its elaboration by pressing and natural sintering. [START_REF] Hallberg | Influence of anisotropic grain boundary properties on the evolution of grain boundary character distribution during grain growth-a 2D level set study[END_REF][START_REF] Hallberg | Stability of grain boundary texture during isothermal grain growth in UO2 considering anisotropic grain boundary properties[END_REF]. Based on the classical Read-Shockley relation [START_REF] Read | Dislocation Models of Crystal Grain Boundaries[END_REF], the authors proposed a set of equations that results in the variation of grain boundary energy with misorientation in the range Σ3-Σ29 of CSL boundaries, i.e. γ/γmax = f(θ), with γmax the energy at the angle θmax differentiating between low-and high-angle boundaries. Figure 3 shows the inversed Hallberg's curve γmax/γ in red.

Area (µm²)

Number of grains total %CSL

For sample F, as an example, the linear fractions of CSL boundaries initially represented as a function of the Σ index (see Figure 2) were considered as a function of the misorientation angle up to Σ49c. The resulting plot is drawn in blue in Figure 3. The red and blue plots show an obvious similitude, clearly evidencing a correlation between the frequency of a given CSL grain boundary and its energy, as calculated by Hallberg. As an example, the Σ3 boundary is the most frequent and has the lowest energy. The same observation is true for Σ5, Σ7 et Σ9 (see Figure 3) and might be generalized to other CSL boundaries. Thus, the more frequent a CSL boundary, the lower its energy. Saylor et al. drew a similar conclusion mentioning that grain boundaries planes distribution (GBPD) in MgO exhibits a substantial inverse correlation with the grain boundary energy distribution (GBED) [START_REF] David | Distribution and Energies of Grain Boundaries in Magnesia as a Function of Five Degrees of Freedom[END_REF]. Rohrer also shows that, in general, for cubic MgO, when the energy is low, the population is high [START_REF] Rohrer | Grain boundary energy anisotropy: a review[END_REF]. A similar inverse correlation between GBPD and GBED also seems to be true in YSZ (see Helmick et al. [START_REF] Helmick | Crystallographic Characteristics of Grain Boundaries in Dense Yttria-Stabilized Zirconia[END_REF]). Nerikar et al. [START_REF] Vonlanthen | CSL grain boundary distribution in alumina and zirconia ceramics[END_REF] also state that in UO2 the more energetically stable a grain boundary, the more it is encountered in a polycrystal. Thus, the comparison of our results with those available in the literature tends to confirm that such a correlation, already observed in other ceramics and metals, is a very general phenomenon and that during grain growth, boundaries configure their orientations to minimize the total grain boundary energy.

Atomistic Simulations

In a first stage, we only focused on CSL grain boundaries, which have the advantage of being easily built knowing their geometric characteristics. EBSD measurements give explicitly information on CSL boundaries, so calculations can be simply compared to the results of experiments. To perform our atomistic simulations, we built grain boundaries using the GBStudio software [START_REF] Ogawa | GBstudio: a builder software on periodic models of CSL boundaries for molecular simulation[END_REF] that allows constructing a given grain boundary on the basis of its CSL classification and of the crystallographic structure of the material considered [START_REF] Grimmer | A method of determining the coincidence site lattices for cubic crystals[END_REF]. The system built with GBStudio respects the geometry of the anticipated grain boundary (i.e. the relative orientation of the two crystals), but the interface at the atomic scale must be relaxed. As already mentioned in the introduction, various relaxation procedures can be used to minimize the energy of the system and then obtain the energy of the corresponding grain boundary. In this work, we chose to apply a relaxation procedure in two steps. The first step is an energy minimization at 0 K, allowing the relaxation of the simulation box and the atomic positions of all the atoms. Anisotropic or isotropic minimization was applied depending on the situation. For 3 among the 26 grain boundaries considered here, a supplementary gap of 1 Å was added between the two crystals to minimize repulsive interactions, which can appear at the interface after the construction. The second step consisted in performing a thermal annealing using the molecular dynamics code LAMMPS [39]. Starting from the GBStudio configuration relaxed at 0 K, a temperature ramp was applied to reach 1870 K in 1 ns. This temperature was then kept constant during a 0.4 ns NPT simulation, as potential energetic barriers were assumed easier to overcome at this temperature. Then, the temperature was smoothly brought back to 300 K in another 1 ns and finally, energy and volume averages calculations were performed at 300 K during a last 4 ns long NPT simulation.

The formation energy of a grain boundary is the excess energy due to the presence of an interface compared with the energy of the corresponding single crystal. Practically, this energy is obtained through the following expression:

= (1)
Where, is the energy of a system of N atoms including explicitly the grain boundary, is the energy of a single crystal with the same number of atoms calculated in the same conditions. A is the surface of the simulated boundary, i.e. the section of the simulation box. Note that our simulations are performed with periodic boundary conditions to avoid free surface effects. Thus, two similar grain boundaries are simulated, explaining the factor 2 in the denominator of equation ( 1). This energy is not related to the strength of the grain boundary but rather expresses its thermodynamic stability as compared to the single crystal. The higher this energy is, the more unstable the grain boundary is from a thermodynamic point of view. To evaluate the strength of a grain boundary or its capacity to resist a normal stress, we need to calculate its cleavage energy [START_REF] Williams | Atomistic investigation of the structure and transport properties of tilt grain boundaries of UO2[END_REF]. It corresponds to the energy required to separate the boundary into two surfaces, and can be associated with the toughness of the material through the Griffith rule [START_REF] Zhang | Molecular dynamics simulations of intergranular fracture in UO2 with nine empirical interatomic potentials[END_REF], or with the theoretical breaking stress using the Orowan model [START_REF] Orowan | Fracture and strength of solids[END_REF]. In our work, it can be evaluated using the equation as follow:

= / (2)
Where, is the energy of the system including explicitly the grain boundary, and ⁄ is the energy of a half of this system, including two free surfaces corresponding to the grain boundary plane. The factor 2 appearing at the denominator comes from the periodic boundary conditions, and the factor 2 appearing at the numerator is linked to the fact that the system with the free surfaces counts half the number of atoms per unit area. The cleavage energy is assumed positive and corresponds to the energy needed to open the grain boundary.

In this work, we compared four different empirical potentials: Morelon et al. [START_REF] Morelon | A new empirical potential for simulating the formation of defects and their mobility in uranium dioxide[END_REF], Yakub et al. [START_REF] Yakub | Molecular dynamics simulation of premelting and melting phase transitions in stoichiometric uranium dioxide[END_REF], Potashnikov et al. [START_REF] Potashnikov | Highprecision molecular dynamics simulation of UO2-PuO2: Pair potentials comparison in UO2[END_REF] and Cooper et al. [START_REF] Cooper | A many-body potential approach to modelling the thermomechanical properties of actinide oxides[END_REF]. Those potentials have been chosen among the great number of available potentials because they offer an interesting compromise between simplicity and performance [START_REF] Potashnikov | Highprecision molecular dynamics simulation of UO2-PuO2: Pair potentials comparison in UO2[END_REF][START_REF] Cooper | Atomic scale simulation of irradiated nuclear fuel[END_REF]. All of them are based on the pair type Buckingham formalism, taking into account the ionic interactions between atoms through a rigid ions model with partial charges. Yakub and Cooper potentials also integrate a Morse type interaction to take into account the partially covalent character of the uranium oxygen bond. The Cooper potential goes beyond the pair potential through an Embedded Atom Model (EAM) contribution. Note that Yakub, Morelon and Cooper potentials were already used to study grain boundaries in UO2 [START_REF] Brutzel | Grain boundary influence on displacement cascades in UO2: A molecular dynamics study[END_REF][START_REF] Zhang | Molecular dynamics simulations of intergranular fracture in UO2 with nine empirical interatomic potentials[END_REF][START_REF] Chen | Molecular dynamics simulations of grain boundary thermal resistance in UO2[END_REF][START_REF] Galvin | Pipe and grain boundary diffusion of He in UO 2[END_REF] making comparison with the present work possible. To make sure that these potentials are suited to simulate grain boundaries, we first calculated surface energies. Grain boundaries being interfaces between two crystals, surface energies and formation energies of grain boundaries can be considered as similar properties at the atomic scale. {100}, {110} and {111} free surfaces of UO2 were calculated by performing a 1 ns NPT molecular dynamic simulation with the Lammps code [39]. Table 3 shows the results obtained. They are in good agreement with literature when the same potentials are used. We obtained results identical or very close to those of Zhang et al. [START_REF] Zhang | Molecular dynamics simulations of intergranular fracture in UO2 with nine empirical interatomic potentials[END_REF], who used the Yakub and Morelon potentials to calculate the {111} and {110} surfaces energies, and of Boyarchenkov et al. [START_REF] Boyarchenkov | Molecular dynamics simulation of UO2 nanocrystals surface[END_REF], who used the Potashnikov potential to calculate energies of surfaces {100} and {111}. Table 3 also shows that the four potentials lead to similar results: Yakub and Cooper potentials are very close and are around 5 to 10 % higher than the results given by Morelon and Potashnikov potentials. The comparison with DFT electronic structure calculation results coming from Bottin et al. [START_REF] Bottin | Thermodynamic stability of the UO 2 surfaces: Interplay between over-stoichiometry and polarity compensation[END_REF] shows that the three surface energies are ranked similarly (Esurf{100} > Esurf{110} > Esurf{111}). This is satisfactory, even if calculations using empirical potentials overestimate all the energies by about 20 to 40 %. We should keep in mind that the same overestimation should appear concerning the formation energies of grain boundaries. It is also worth noting that the results obtained by Sattonnay et al. [START_REF] Sattonnay | Bulk, surface and point defect properties in UO 2 from a tightbinding variable-charge model[END_REF] using a more complex and expensive potential with variable charges do not seem to be better. DFT [START_REF] Bottin | Thermodynamic stability of the UO 2 surfaces: Interplay between over-stoichiometry and polarity compensation[END_REF] Zhang [START_REF] Zhang | Molecular dynamics simulations of intergranular fracture in UO2 with nine empirical interatomic potentials[END_REF] Williams [START_REF] Williams | Atomistic investigation of the structure and transport properties of tilt grain boundaries of UO2[END_REF] Boyarchenkov [START_REF] Boyarchenkov | Molecular dynamics simulation of UO2 nanocrystals surface[END_REF] Sattonnay [START_REF] Sattonnay | Bulk, surface and point defect properties in UO 2 from a tightbinding variable-charge model[END_REF] 3. Surface energies (J.m -2 ) at 300 K calculated using molecular dynamic simulations. The empirical potential used in the calculation is indicated in brackets.

Figure 4 (and Table 1 in supplementary data) show the formation energies calculated using equation ( 1) for the 26 different CSL grain boundaries studied in this work, with the four chosen potentials. The order of magnitude of energies is correct as all results are between 0.8 and 2 J.m -2 . For some grain boundaries, formation energy is similar whatever the potential (e.g. the Σ9 grain boundary), whereas for others the energy strongly depends on the potential (e.g. the Σ5 grain boundary). Also, the Σ3 grain boundary has the lower energy regardless the potential used (except for the potential from Morelon, for which Σ3 has the second lower energy). This is consistent with the fact that this grain boundary is the most frequent in terms of linear fraction. Moreover, our results are in good agreement with those obtained by Zhang et al. [START_REF] Zhang | Molecular dynamics simulations of intergranular fracture in UO2 with nine empirical interatomic potentials[END_REF], with Yakub and Morelon potentials, and with those of Chen et al. [START_REF] Chen | Molecular dynamics simulations of grain boundary thermal resistance in UO2[END_REF], calculated with the Yakub potential. The same remark applies to the results obtained by Galvin et al. [START_REF] Galvin | Pipe and grain boundary diffusion of He in UO 2[END_REF] using the Cooper potential. Conversely, the results obtained by Van Brutzel et al. [START_REF] Brutzel | Grain boundary influence on displacement cascades in UO2: A molecular dynamics study[END_REF] using the Morelon potential are always higher than ours. A possible explanation is that the relaxation procedure used by these authors did not allow reaching the minimum formation energy. Figure 5 shows the formation energies calculated in this work as a function of the linear fraction of CSL boundaries observed experimentally using EBSD measurements. Let us recall here that EBSD analysis does not yield the boundary plane of observed grain boundaries. On the contrary, our calculations were done for a given boundary plane, the first one proposed by GBStudio for a given CSL grain boundary, generally the most favorable. Nevertheless, an inverse correlation between GBPD (measured experimentally) and GBED (calculated) is clearly evidenced in this figure. This correlation is observed apart for the Σ11 GB for all the potentials, and apart for the Σ5 for Morelon and Potashnikov potentials. This is an important result because, to our knowledge, it is the first time that such a correlation is established between calculated formation energies at the atomic scale and experimental observations for a significant number of grain boundaries in UO2. Finally, we obtained a correlation coefficient of -0.42 for Morelon, -0.45 for Potashnikov, -0.73 for Yakub and -0.74 for Cooper potentials. Moreover, if we focus on the energy of the Σ5 and Σ11 grain boundaries, the Cooper potential seems to be the most efficient. This potential also yields good results regarding thermodynamic properties [START_REF] Cooper | A many-body potential approach to modelling the thermomechanical properties of actinide oxides[END_REF]. These are the reasons why it was chosen to calculate cleavage energies in the following second step of this work. Then we have plotted in Figure 6 the formation energies calculated in this work as a function of the misorientation angle between the two crystals forming the boundary. No obvious correlation appears: almost all CSL grain boundaries have an energy between 1.5 and 2 J.m -2 (see the blue points in figure 5), except the Σ3 (60° -1.08 J.m -2 ), Σ5 (36.9° -1.28 J.m -2 ) and Σ11 (50.5° -1.13 J.m -2 ). Cleavage energy calculated using equation ( 2) and the Cooper potential as a function of the misorientation angles of grain boundaries is plotted in Figure 7. In this case, despite the scattering of the values, a trend seems to arise (with a correlation coefficient equal to -0.55): the higher the misorientation angle, the lower the cleavage energy. This result is very interesting and opens the way to a modeling of the cleavage energy usable in simulations at higher scale. The cleavage energy as a function of the CSL index calculated using equation ( 2) and the Cooper potential can be seen in Figure 8. Again, no clear correlation is observed. Only Σ3, Σ17b, Σ25b, Σ41c and Σ45c grain boundaries present a cleavage energy lower than 1 J.m -2 . These boundaries should thus be the easiest to open in UO2. In comparison, it is worth reminding that the energies of the three natural surfaces of UO2 {100}, {110} and {111} were calculated respectively equal to 1.90, 1.61 and 1.04 J.m -2 . This means that the cleavage of a single crystal of UO2 along these three planes would need 3.8, 3.22 and 2.08 J.m -2 respectively. From these results, it appears reasonable to conclude that, in a system without point defect, the opening of a grain boundary most likely needs less energy than the cleavage of a single crystal along a natural surface. Figure 9 shows the cleavage energy as a function of the formation energy calculated using the Cooper potential. Once again, no clear correlation appears. Nevertheless, the four grain boundaries with formation energies above 1.8 J.m -2 (Σ17b, Σ25b, Σ41c and Σ45c) are characterized by the lowest cleavage energies (below 0.65 J.m -2 ), which seems consistent. The Σ3 grain boundary however exhibits a low formation energy (1.16 J.m -2 ) but is also one of the easiest to open, since its cleavage energy is equal to 0.92 J.m -2 . In the first part of this work, we focused on CSL grain boundaries built using GBStudio. EBSD experiments however showed that CSL grain boundaries represent only 15 to 20 % of the linear fraction of grain boundaries observed in a UO2 polycrystal. Thus, the large majority of grain boundaries in a real material actually possesses infinite CSL indexes, i.e. no coincidence site can be found between the lattices of the two crystals forming the boundary, which is generally not symmetrical. Such boundaries can be constructed from two crystals randomly rotated with respect to each other. Unfortunately, by doing so, the general grain boundary is built among many possibilities. This procedure lacks a controlling parameter that would make it possible to compare different general boundaries or general boundaries to CSL ones.

We tried to take a step towards the simulation of general boundaries by building what we called "semi-general" boundaries. To do so, we stuck two halves of two different CSL grain boundaries constructed with GBStudio which had been studied in the previous part of this work. To be able to control both the axis and angle of rotation, two CSL boundaries with the same rotation axis were chosen. Table 4 shows the three semi-general grain boundaries built using this procedure. Green points in Figure 6 show their formation energies compared to the CSL grain boundaries studied in this work. It appears that the formation energies of these 3 semi-general grain boundaries are systematically higher than the formation energies of the CSL grain boundaries. This result is consistent with the assumption that formation energy is correlated with stability [START_REF] Nerikar | Grain Boundaries in Uranium Dioxide: Scanning Electron Microscopy Experiments and Atomistic Simulations: Grain Boundaries in Uranium Dioxide[END_REF]. Since semi-general grain boundaries are supposed to be less stable than CSL boundaries, so their formation energies should be higher. Green points in Figure 7 show their cleavage energies compared to the CSL grain boundaries studied in this work. These energies are in good agreement with the correlation observed between cleavage energy and misorientation angle. Therefore, the cleavage energy of general boundaries also seems to decrease with misorientation angle. Of course, these first results will have to be confirmed by further calculations.

CSL grain boundaries

Conclusions

Length fractions of CSL grain boundaries were obtained from reliable EBSD analysis performed on maps comprising a large number of grains acquired on several UO2 samples. Depending on the sample, 14.6% to 15.4% of grain boundaries total length were identified as CSL, suggesting that this value is rather independent of the conditions of elaboration and sintering. We also found that the most frequent CSL grain boundaries have the lowest energy, as calculated by Hallberg [START_REF] Hallberg | Stability of grain boundary texture during isothermal grain growth in UO2 considering anisotropic grain boundary properties[END_REF].

Then, formation energies calculated with Morelon [START_REF] Morelon | A new empirical potential for simulating the formation of defects and their mobility in uranium dioxide[END_REF], Yakub [START_REF] Yakub | Molecular dynamics simulation of premelting and melting phase transitions in stoichiometric uranium dioxide[END_REF], Potashnikov [START_REF] Potashnikov | Highprecision molecular dynamics simulation of UO2-PuO2: Pair potentials comparison in UO2[END_REF] and Cooper [START_REF] Cooper | A many-body potential approach to modelling the thermomechanical properties of actinide oxides[END_REF] empirical potentials were compared to linear fractions of CSL grain boundaries obtained from EBSD data. An inverse correlation between the formation energy and the linear fraction is clearly evidenced. This correlation is better with Cooper and Yakub potentials, except for the formation energy of the Σ11 grain boundary. Therefore, an interesting perspective to this work would be to measure the formation energy of this particular grain boundary and compare the experimental result with the calculated one.

The cleavage energy, representing the energy needed to open the grain boundary, was also calculated using the Cooper potential for the 26 CSL grain boundaries. Although no clear correlation is evidenced between cleavage energy and formation energy, we found that the cleavage energy decreases with the misorientation angle. This last result opens a way to the modelling of the cleavage energy as a function of the misorientation angle, which is an "easy to follow" parameter in simulations performed at the scale of a polycrystal, as in REV (Representative Elementary Volume) simulations [START_REF] Hill | Elastic properties of reinforced solids: Some theoretical principles[END_REF].

A step towards the simulation of general grain boundaries, i.e. grain boundaries without symmetry and presenting an infinite CSL index, was also taken. As such boundaries represent at least 80 % of the linear fraction observed in sintered UO2 samples, their properties have to be investigated in detail. We built 3 semi-general grain boundaries, sticking two halves of CSL grain boundaries with the same rotation axis. Their formation energies appear higher than that of CSL boundaries, as expected since they are most likely less stable. Their cleavage energies follow the correlation observed on CSL boundaries. Therefore, the cleavage energy of general boundaries probably also decreases with the misorientation angle. A perspective to this work will be to study more general grain boundaries and to investigate if the observed correlations is also found at higher temperature.

Figure 1 :

 1 Figure 1 : Total linear fraction of CSL grain boundaries for samples A to G compared to Nerikar et al. data[START_REF] Nerikar | Grain Boundaries in Uranium Dioxide: Scanning Electron Microscopy Experiments and Atomistic Simulations: Grain Boundaries in Uranium Dioxide[END_REF]

Figure 2 :

 2 Figure 2 : Average CSL grain boundaries distribution for Σ3 to Σ49, in all UO2 samples (A to G)

Figure 3 :

 3 Figure 3 : Misorientation dependence of the inverse of the grain boundary energy calculated by Hallberg (in red) [32] and linear fraction of CSL for sample F (in blue).

Figure 4 .

 4 Figure 4. Formation energies (J.m -2 ) calculated with the four different empirical potentials for the 26 CSL grain boundaries studied in this work.

Figure 5 .

 5 Figure 5. Formation energies (J.m -2 ) calculated as a function of the linear fraction measured (%) for the 26 CSL grain boundaries studied in this work.

Figure 6 .

 6 Figure 6. Misorientation dependence of formation energies (J.m -2 ) for the 26 CSL grain boundaries (blue points) and the 3 semi-general grain boundaries (in green) calculated in this work with the Cooper potential.

Figure 7 .

 7 Figure 7. Misorientation dependence of the cleavage energies (J.m -2 ) calculated with the Cooper potential for the 26 CSL grain boundaries (blue points) and the 3 semi-general grain boundaries (in green). The dashed line in red is the linear trend curve of the blue points.

Figure 8 .

 8 Figure 8. Cleavage energies (J.m -2 ) calculated with the Cooper potential for the 26 CSL grain boundaries.

Figure 9 .

 9 Figure 9. Cleavage energies (J.m -2 ) as a function of the formation energies calculated with the Cooper potential for the 26 CSL grain boundaries.

Table 1 .

 1 Characteristics of surface analyzed by EBSD on each sample.

	Total	grain diameter
	number	(mean ECD, in
	of grains	µm)

Table 2 .

 2 Linear fraction of CSL grain boundaries in the different subsets analyzed in sample C.

	Whole map	440 000	1243	14.90
	Subset 1	22 100	97	11.07
	Subset 2	27 900	122	13.20
	Subset 3	24 200	106	19.28
	Subset 4	29 300	128	14.85
	Subset 5	9 600	42	16.46
	Sum of subsets	113 100	495	14.97

Table 4 .

 4 Geometrical characteristics, formation energies, and cleavage energies of the semigeneral grain boundaries studied in this work with the Cooper potential.

	used to build the semi-general grain boundaries	Rotation axis	Misorientation angle	Formation energy (J.m -2 )	Cleavage energy (J.m -2 )
	Σ13b / Σ19b	(111)	37.3°	1.97	1.16
	Σ25a / Σ17a	(100)	22.2°	2.016	1.603
	Σ27a / Σ9	(110)	35.25°	2.12	2.32

Acknowledgements

This work was granted access to the HPC resources of CINES and TGCC under the allocation 2017-A0030906922 made by GENCI.

Data Availability

The raw/processed data required to reproduce these findings cannot be shared at this time as the data also forms part of an ongoing study.


Eric Severin
email: e.severin@wanadoo.fr

Marie-Christine Filareto-Deghaye

DETERMINANTS OF THE CHOICE LEASING VS BANK LOAN: EVIDENCE FROM THE FRENCH SME BY KACM

Keywords: Leasing, credit rationing, SME, Self organising maps (SOM) MSC: 91B60

The question of leasing credit as a substitute or complement of a banking loan has still not been resolved in the financial literature. As a continuation of these arguments, the objective of this article is, on the one hand, to determine the characteristics of firms using leasing credit and on the other hand, to better understand the relationship between leasing and credit rationing. Firstly, our results suggest that SME use leasing all the more the leasing so when they are young, leveraged, less solvent and that they present an small size and an important failure probability. Thus, leasing pushes back the limits of banking debt for firms that have no access to it. Secondly, our results suggest a strong and significant relationship between credit rationing and the use of leasing. In this framework the latter appears to be a last resort financing.

INTRODUCTION

As Cottrell et al. (1996) underline, Kohonen maps have generated many applications in finance. Areas in which they have been set up are numerous: the detection of firms in financial distress, the factors of firms' financial policy. Used as alternative methods to traditional approaches, they can allow us to deal with methodological biases found in empirical analyses. Thus, for example, in the case of the detection of firms in financial distress, the objective is to elaborate, from a population of firms, a function which discriminates between firms in financial distress and the others. [START_REF] Altman | Financial ratios, discriminant analysis and the prediction of corporate bankruptcy[END_REF] tried to undertake, with variables observed over a certain period before the date at which the observer decides to classify the firm in one category or the other, a function able to better discriminate the two groups of firms (healthy firms and firms in financial distress). The methods used, discriminant analysis and logistic regression, necessitate variables to check a certain number of statistical properties that are not always checked. In this line, the use of Kohonen maps can overcome the methodological biases of the classical analyses in finance.

In the area of corporate finance, the use of the KACM applied to the analysis of financing policies by leasing gave rise to a research paper by Cottrell et al. (1996). The authors conclude as to a significant relationship between the financial weakness of the firm and the use of leasing. If this argument has largely been accepted as an explanatory factor of the choice leasing versus banking loan, the study that we develop in this article introduces a complementary approach to this problem: it allows us to introduce a finer analysis of firms' profiles and seeks a direct measurement of the credit rationing in order to give more robustness to the relationship between the credit rationing and the using of bank loan. It also deals with other determinants of the choice leasing versus banking loan for the firm.

The article is organized as follows. In the second part, we will identify the explanatory factors of the financing policy by leasing, by focusing on the analysis of theoretical works. In the third part, we will discuss about the presentation of data and the methodology used. Finally, the fourth part will present the results of the empirical study. Finally, in a last section, we conclude and discuss possible perspectives of our study.

ANALYSIS OF THEORETICAL ARGUMENTS

A first approach: leasing as a substitute for bank loans

The question of substitutability or complementarity of leasing and debt has been studied in details by the financial literature but without reaching a consensus of unanimous reply. According to Myers et al. (1976), the use of leasing would be probably accompanied by a lesser utilization of a debt, reducing, in the same proportion, the firm's debt capacity (Levasseur and Quintart, 2000, p. 6-7). The validity of such an argument is based on the perfect substitution hypothesis between leasing and bank debt. Therefore, this perfect substitution hypothesis can be rejected in favour of an imperfect substitution and sometimes in favour a complementarity between banking loans and leasing.

The first argument against a perfect substitution between leasing and debt concerns the substantial costs borne by the (banking) creditor or (credit) lessor in case of corporate bankruptcy. For [START_REF] Krishnan | Bankruptcy costs and the financial leasing decision[END_REF], costs created by the bankruptcy of the tenant would be lesser for the lessor that for any other creditor. The quality of ownership allows him to recapture the goods in case of corporate bankruptcy when the contract is not stopped. In fact, it avoids opportunity costs associated with the slowness of the resolution process of the bankruptcy and more rapidly may allow to resale the asset. Thus, Lease et al. (1990) show, on the American market, on the one hand, that the resale value in the case of default by the lessee amounts, on average, to 37.8% of the price of its purchase price, and, on the other hand, that 12 months went on between the corporate bankruptcy and the time where the lessor covered its claim by the resale of the asset.

The second argument contradicting perfect substitutability between leasing and debt is based on properties of leasing. By the terms of the contract and rights given to the lessor, leasing has characteristics close both to the secured and unsecured debt. The secured debt, because lessor has a real guarantee in the ownership of asset in the contract; the non-secured debt, because unpaid rents prior before bankruptcy has the rank that a banking loan. In most of cases, the funds are lost by the creditor. The diversity of the debts contracted by the firm, introducing a distortion between creditors in case of bankruptcy, favours the thesis of the imperfect substitution leasing/ classical debt.

Although the perfect or imperfect substitutability between leasing and debt has been accredited by the financial literature, the work of [START_REF] Lewis | Are debt and leases substitues?[END_REF] has questioned it. In line with De Angelo and Masulis (1980) -according to them, the debt use by a firm is reduced, if it cannot benefit from substantial fiscal gains on amortizations-the authors take into account this fiscal dimension to compare leasing and bank debt. They show that leasing allows the firm to transfer fiscal gains on amortizations to lessor (who is the only one actor to have the possibility of amortizing the good). Furthermore, theses gains are retroceded back to the lessee by means of a contractual reduction in dues. In the case of financing by leasing, the lack of fiscal advantage on amortizations favours the additional debt issuing as a fiscal leverage. Leasing and the debt appear then as complements rather than substitutes. Hence a positive relationship between the use of leasing and the level of corporate debt results from this.

The analysis by [START_REF] Lewis | Are debt and leases substitues?[END_REF] has not generated empirical verification. The only studies corroborating the complementarity thesis made did so undirectedly by comparing the leasing rate and the level of corporate debt. The most prominent controversy was produced by [START_REF] Ang | The leasing puzzle[END_REF] on the American market. This paper was criticized by [START_REF] Smith | Determinants of corporate leasing activity[END_REF] and [START_REF] Stapleton | Leases, debt and taxable capacity[END_REF]. For these authors, the lack of control of debt capacity in the empirical studies generates a bias for the estimation of the leverage debt and of using of leasing. Indeed, according to these authors, if firms use leasing more due to a better debt capacity, then is normal to observe a greater use of external funds.

Afterwards, others complementary analyses corroborated [START_REF] Finucane | Some empirical evidence on the use of financial leases[END_REF], [START_REF] Filareto-Deghaye | Determinants of the leasing decision in United Kingdom listed companies[END_REF], [START_REF] Branson | An empirical examination of the leasing puzzle[END_REF] or not corroborated [START_REF] Stapleton | Leases, debt and taxable capacity[END_REF], [START_REF] Beattie V.; A | Operating leases and the assessment of lease-debt substitutability[END_REF] the complementarity thesis between leasing and debt. These empirical studies nevertheless agreed on firms' characteristics which to use, most often, leasing. They have: more debt (or more important gearing), smaller liquidity and more important risk.

The question of complementarity or substitutability of debt and leasing seems to be unsolved because there is no consensus between theoretical and empirical approaches. Leasing is often analyzed as a last resort solution, especially for very weak firms. Firms' financing preferences nevertheless depend on the characteristics. It allows lessor to gain a clientele of firms with atypical profiles. The reasons for the choice of leasing versus bank loan can therefore be several, which we develop in the following point.

The factors explaining the leasing use

As underlined by Smith and Wakeman (1985, p. 907): "the coexistence of both leased and purchased assets suggests that the net benefits of leasing are uniformly neither positive or negative ". In this line of thought understanding the choice of leasing versus bank loan for the firm, would depend on costs/advantages. Indeed, from an economic viewpoint, leasing can be analyzed as an operation of investment financing, which has the particularity of dissociating the legal ownership of the good from its economic ownership. Contrary to a classical credit operation, the lessor remains the owner of the asset during the whole duration of the transaction. This security due to asset ownership allows it to recall if there is a payment default by the lessee, so as to resell it or rent it again. The lessor appears, according to Capiez (1992, p. 38), to be "a creditor who has anticipated all possible changes of its wealth and has beforehand protected himself by the form of the contract in which its retains ownership". A limit in bank credit supply can lead the firm to increase its use of leasing. This argument is presented in the literature by [START_REF] Krishnan | Bankruptcy costs and the financial leasing decision[END_REF] who consider leasing as "the last financing resort". According to these authors, when the risk of bankruptcy for the firm increases, the unavailability of bank credit compells the firms in financial distress to turn to leasing, because costs due to the latter.

Although the conservation of the legal ownership is a protection for the lessor -it allows the firm to carry over the risk of obsolescence to the lender-. This advantage will be all the more profitable to the firm, in the case of a new technology is growing up when the contract is negotiated over a relatively short period. The acquisition of the legal ownership being optional in leasing contract, this formula of financing would seem to be completely adapted when the period of use of the good is lower from its economic life duration (Smith and Wakeman, 1995). The firm avoids transaction costs generated by marketing costs useful to resale the good on the secondary market, which would explain a greater resorting to leasing by SME which absorb the financing of expenses in their activity with greater difficulty.

Among the other advantages to leasing, may be added the entire financing of the asset. So we can consider leasing as a flexible means of financing.

Though advantages encourage firms to use leasing, the firm can however limit its use and prefer bank loan, because of an operational mobility loss for the firm. Indeed, a leasing operation involved firm for an irrevocable period. The firm loses then the ability of spreading set if the activity is no longer profitable or if the sector was affected by a circumstantial major economic shock slowing its activity. That is not the case, when the asset is financed by debt. Leasing contracts can not be cancelled without the agreement of the lessor and generates the payment of early termination penalties. This rigidity in the policy of redeployment of the firm may then limit its use. In a empirical paper, Lease et al. (1990), show that, of the 137 leasing operations whose the amount is on average $ 75000, only 46.6% operations come to an end. 31.4% of contracts are stopped before the end. In 19% of cases, the operation led to the firm's bankruptcy. This cost due to the operational flexibility loss will vary according to the firm. Indeed, the large firm possesses an internal redeployment capacity due to the fact of their more diversified activities, and for specific assets that can be the subject of an alternative use.

As our main objective is to understand financing policies by leasing, the literature allowed us to identify two main reasons in the decision of leasing, deriving from analysis supply and demand: its greater availability compared with classic banking credit and the advantages inherent in the nature of the financing.

The empirical study, proposed in the third section, based on a sample of French firms differs from previous ones, in two points.

The first concerns the methodology employed. The analysis of Cottrell et al. (1996) on this subject shows that through the first empirical works seem to confirm viewpoints advanced in the financial literature, they do not allow to develop a discrimant analysis able to distinguish between firms that use leasing and those that do not. Moreover, by considering the very particular distribution of some ratios, the authors make a comparative analysis of a technique of linear data analysis (multiple correspondence analysis) and a technique of non linear analysis (Kohonen maps). The interest is in the results obtained by the KACM. Indeed, observations by the authors suggest firstly, the existence of subgroups within the population of firms that resort to the leasing, and secondly, a clear association between the resorting to leasing as a financial technique and the financial health of firms. Even if the KACM method presents weak points (The first is that the visualization of the Kohonen map does not allow evaluation of the distance between the units. The second is the quality of the representation), its use can bring progress in the understanding of leasing by the use of Kohonen maps. The use of Kohonen maps favors a roundup of individuals according to their propensity to use leasing utilization, which allows a finer analysis of firms' profiles, and by thus of the reasons associated with the decision.

The second contribution of our study concerns the utilization of a direct measure of the rationing of credit that we associate with the characteristics of firms' groups, which makes our tests and conclusions on the association between credit rationing and the use of leasing by a firm more robust.

The next section will be devoted to the presentation of data and the methodology of the study.

DATA AND METHODOLOGY

Sample presentation and variables choice

Sample

Our study deals with a sample of 11436 French SME. We used the date from Dun and Bradstreet, for 1999. As we sought accounting data which represent leasing financing, we proceeded in several steps. Firstly, we retained all firms with a staff between 20 and 500 employees. By this, we wanted to select SME by excluding very small firms in order that the information would be less difficult to deal with. From this point, we choose to retain only industrial and services firms using leasing. Indeed, we excluded firms in the financial sector (the accounting treatment of profit for these firms is significantly different than that of other sectors). After applying these criteria 12669 were retained.

The validity of accounting data was validated by using a series of coherence tests. Hence our sample was composed of 11233 firms. Then, we worked from the Dun database.

In this database, we used the following information: number of workers, long term debt, leasing, equity, short term assets, short term liabilities, EBITDA, financial fees, fiscal debt and firm age.

The collection of these data allowed the construction of variables used in empirical tests.

Relevant variable choice

In this study, we seek to clarify explanatory factors in the propensity of the firm to finance by leasing. To take into account the intensity of credit lease utilization, we calculated the variable L (Leasing), by calculating the ratio of leasing divided by the long term debt for each observation.

The literature analysis presented in the first section allowed four arguments to be identified allowing to explain the choice of leasing/banking loan for the firm. It concerns the firm's risk level, information asymmetry between lender and borrower, the firm's debt capacity and the dissociation between judicial and economic ownership. We summarize in table 1 for each determinant the financing policy by leasing. The lack of some data and information explains we have limited our investigation to these main four arguments. Table 2 presents the variable measurement used and the expected effects on the propensity of the firm to be financed by leasing.

Table 1. Factors of financing policy using leasing.

Theoretical viewpoint

Argument Chosen Variables The risk of the borrower The use of leasing is positively associated with the bankruptcy risk [START_REF] Krishnan | Bankruptcy costs and the financial leasing decision[END_REF]Cottrell et al., 1996).

-Probability of bankruptcy -Firm's solvability

Informational asymmetry on quality of growth opportunities

Leasing is all the more used when the firms are young and small [START_REF] Sharpe | Capital market imperfections and the incentive to lease[END_REF].

-Firm size -Age

Limited debt capacity

Firms with real debt capacity should use leasing more frequently [START_REF] Krishnan | Bankruptcy costs and the financial leasing decision[END_REF].

-Leverage

The separation between judicial and economic ownership

The acquisition of judicial ownership is optional in leasing contract. The firm avoids transaction costs due to retrading the good on the second market. So the small-sized firm would probably used leasing more frequently (Smith and Wakeman, 1995). The loss of the legal ownership leads to loss in flexibility for the firm.

-Firm size Fiscal and corporate debt/going concern debt

In the case of substitution hypothesis, we expect a negative relationship between credit rationing and the amount of leasing

The use of Kohonen algorithm and its advantages

When we try to determine if a non-linear relation exists between some characteristics of the firms and leasing, we cannot use the techniques traditionally used in finance. Indeed, a great proportion of modern finance rests on the simplifying hypotheses; in particular, the normality of variables and the linearity of causality relations [START_REF] Quintart | Les fondements de la théorie financière classique et les limites de l'hypothèse générale de linéarité[END_REF].

Our hypothesis of non-linearity led us to focus our attention on self-organized maps (SOM) and more especially on one of the variants called the Kohonen Map [START_REF] Kohonen | Self organized formation of topologically correct feature maps[END_REF]1995). One of the major advantages in its use is its capacity to deal with, in particular, non-linear problems [START_REF] Quintart | Les fondements de la théorie financière classique et les limites de l'hypothèse générale de linéarité[END_REF].

Our objective was to determine several groups of homogenous individuals. Secondly, we used nonparametric tests (Wilcoxon and χ2) to highlight significant differences between our groups.

Many traditional methods assume strong hypotheses; in particular, the assumption of normality. To test this, we examined the distribution of the ratios. As our ratios do not have a normal distribution. Extreme values require the use of qualitative data. This non-normality and the presence of extreme values led us to cluster our individuals into 4 classes. Hence, we first transformed each character Xi into 4 categories (very strong, strong, weak, very weak-so, we divided the sample into four classes of equal size) and, secondly, transformed our variables into binary variables.

Thus, we used a specific kind of self-organized map (SOM) called the Kohonen map3 . The Kohonen algorithm, led to interesting applications in economics and finance, is a well-known unsupervised learning algorithm which produces a map composed of a fixed number of units (table 3, for instance, presents a onedimensional map, frequently called a string). Each unit has a specific position on the map and is associated with an n-dimensional vector Wi (which will define its position in the input space); n being the number of dimensions of the input space. Moreover, a physical neighborhood relation between the units is defined (fig. 1). Units 1 and 3 are neighbors of Unit 2 and for each unit i, Vr(i) represents the neighborhood with the radius r centered at i. After learning, each unit represents a group of individuals with similar features. The correspondence between the individuals and the units more or less respects the input space topology: individuals with similar features correspond to the same unit or to neighboring units. The final map is said to be a self-organized map that preserves the topology of the input space. The learning algorithm takes the following form: -at time 0, Wi(0) is randomly defined for each unit i, -at time t, we present a vector x(t) randomly chosen according to the input density f and we determine the winning unit i*, which minimizes the Euclidean distance between x(t) and Wi(t), -we then modify the Wi in order to move the weights of the winning unit i* and its physical neighbors towards x(t) using the following relations : 2) where ε(t) is a small positive adaptation parameter, r (t) is the radius of Vr(t), and, ε(t) and r(t) are progressively decreased during the learning (Note that For stochastic algorithm ε(t) must follow the requirements of [START_REF] Robins H | A stochastic approximation method[END_REF]. This is clearly a competitive kind of algorithm (each unit competes to be the closest to the presented individual) which will perform two interesting tasks for data analysis: 1) clustering: each unit will be associated with a similar kind of individual, the Wi vector associated with the unit converging toward the mean profile of the associated individuals. 2) reduction in the number of dimensions: the (at least local) proximities between the units will give us an idea of the proximities of clusters of individuals in the input space.

( ) [ ] *) ( for ) ( ) ( ) ( ) ( ) 1 ( ) ( i V i t W t x t t W t W t r i i i ∈ - × + = + ε (Equation 1) i t W t W i i other for ) ( ) 1 ( = + (Equation
A last remark concerning the neighborhood: it is reduced progressively to finish at value 0 (only the winning unit is displaced). The Kohonen algorithm then turns into a vectorial quantification. To assess the statistical significance of the results obtained with the Kohonen map, we used traditional tests (Wilcoxon, chisquare). If the χ² test is a test of independence that serves to determine if samples come from the same population, the Wilcoxon test is a test on ranks. Its justification is due to the non-normality of data. Tests on ranks are very robust. By arranging the different observations (i.e. by giving them a rank), one identifies the place of every observation in the sample. One substitutes rank for observation. Thus one neutralises problems concerning the accurate measure of the value for every observation. We can also note that the results of rank tests are not altered by the distributions of observations (symmetrical, non-symmetrical...).

The next section presents the results.

EMPIRICAL RESULTS

In this section, we present the main results obtained from the firms' sample.

Presentation of results

We show the results obtained for the different variables. We associated the dependent variable to the explanatory variable, by highlighting different levels: VS: very strong, S: strong, W: weak and VW: very weak.

Comments

Empirical test results highlight significant effects for the different factors explaining the use of leasing. Indeed, firms use leasing all the more so when they are young and are of a small size, leveraged and they have a smaller solvability and when they present a strong likelihood of bankruptcy. These profiles testify a stronger risk of failure. The weakness of bankruptcy costs for the lessor, in the case of default by the firm, justifies then leasing financing for firms in a situation of credit rationing [START_REF] Krishnan | Bankruptcy costs and the financial leasing decision[END_REF][START_REF] Sharpe | Capital market imperfections and the incentive to lease[END_REF].

Leasing pushes back the limits of bank debt, for firms that there have no access or no longer access (firms with a high leverage) and would be more often used when the firm can no longer bear the costs associated with the ownership good or can start up a new activity.

We have consequently sought to check if our results were robust, by verifying the association between the extent of these variables and the existence of credit. To do this, we evaluated the degree of credit rationing of the firm by the following ratio: fiscal and corporate debt/ going concern debts.

Delays in the payment of social and corporate debts generally highlight serious cash flow problems, due to credit rationing [START_REF] De | Décision de crédit et sélection adverse : une étude empirique des comportements bancaires[END_REF]. The Wilcoxon test shows significant results for the following variables: size, leverage, probability of bankruptcy and solvency. We verify consequently that relationships have their origin in credit rationing. Table 8 summarizes the results. On the French SME sample, our results suggest that the use of leasing is positively associated with credit rationing. This confirms that the use of leasing is a "last resort solution". Otherwise, the financial literature does not totally explain the motives of credit rationing. Is credit rationing due to too high leverage or informational asymmetry on the borrower? In this context, leasing could be preferred by young firms and start-ups.

CONCLUSION

We will end this paper with the advantages and the disadvantages of neuron systems and on the scope of the results.

The advantages are numerous compared to classical statistical analyses. On the one hand, they allow problems to be treated for which we have a priori non information. Thus in the framework of the detection of firms in financial distress, it is not necessary to know the distribution of variable probability contrary to the discriminant analysis. Secondly, the neuronal systems discover by themselves relationships between variables which allow us to study non-linear problems. This aspect does not allow us to specify the separation function. Moreover, as compared to classical approaches (such as ACP or ACM), the competitive learning scheme used presents advantages of avoiding the even present linearity assumption and of producing a unique graphical representation of the data. Thirdly, the uncompleted can be taken into account by the supplementary neuron addition. Fourthly, the stoppage of the iterative process, when the system produces the best results on the validation sample gives robust results. One can consider that the relevant information is integrated in the system. Fifthly, neuronal systems allow working both on qualitative or quantitative variables. Despite these advantages, several criticism can be addressed to neural networks. These are following. Firstly, it does not exist theory allowing to determine the optimal structure of the system. Especially the determination of the hidden layers number and the number of neurons are, the most often, dependant from the user and its capacity to experiment several architectures. Secondly, neural networks often assimilated to "black boxes" in which it is difficult from to extract relevant relationships among variables. Hence, one can consider that the explanatory system power of neuron systems remains limited, contrary to expert systems that are able to track the path to take the result.

However, the studies presented above show that neural networks give good results in the classification area. To be able to improve the approach by neuronal networks, efforts have to focus on: the construction of the system, the clustering of entry variables and the adjustment of learning parameters that depend greatly on human intervention.

From an empirical viewpoint essentially considering the "financial" dimension, research concerning the financing policy of leasing has the difficulty of taking into account the empirical financial gains obtained by leasing: it analyzes on the one hand, the basis of the apparent cost of the financing source, but also in function of advantages associated with the use of leasing such the financing flexibility, entire financing of the investment or fiscal advantages for the firm. Taking them into account would facilitate the understanding of the choices of corporate financing and would balance our conclusions, by excluding the too systematic association between credit rationing and leasing financing.

Table 2 .

 2 Explanatory variables and previous effects.

	Variable	Ratio	Expected sign
	Probability of bankruptcy	EBITDA / Financial expenses	positive
	(DEF)
	Solvability (SOL)	Cash flow / financial debt +	negative
		leasing
	Size (TA)	Log (total workers)	negative
	Age (AGE)	Number of years of life since the	negative
		firm's creation
	Leverage (LEV)	Long term debt + leasing / equity positive
	Control Variable	Ratio
	Credit rationing (CR)

Table 3 .

 3 Probability of the bankruptcy and the use of leasing.

	Theoretical hypothesis: The risk level of the borrower
	LVW	LW	LS	LVS
	DEFS	DEFVW	DEFW	DEFVS
	U 1	U 2	U 3	U 4
	Legend
	LVS, LS, LW, LW: Leasing very strong, strong, weak, very weak.
	DEFVS, DEFS, DEFW, DEFVW: Probability of bankruptcy very strong, strong, weak, very weak.
	U1, U2, U3 et U4: Unit 1, Unit 2, Unit 3, Unit 4.

Table 4 .

 4 Debt capacity and the use of leasing.

	Theoretical argument: The risk level of the borrower
	LVS	LS	LW	LVW
	SOLW	SOLVS	SOLS	SOLVW
	U 1	U 2	U 3	U 4
	Legend
	LVS, LS, LW, LW: Leasing very strong, strong, weak, very weak.
	SOLVS, SOLS, SOLW, SOLVW: Solvability very strong, strong, weak, very weak.
	U1, U2, U3 et U4: Unit 1, Unit 2, Unit 3, Unit 4.

Table 5 .

 5 Firm size and the use of leasing.

	Theoretical argument:-Informational asymmetry between lender and borrower
		-separation between judicial and economic ownership
	LS	LVW	LW	LVS
	TAW	TAVS	TAS	TAVW
	U 1	U 2	U 3	U 4
	Legend
	LVS, LS, LW, LW: Leasing very strong, strong, weak, very weak.
	TAVS, TAS, TAW, TAVW: Size (total assets) very strong, strong, weak, very weak.
	U1, U2, U3 et U4: Unit 1, Unit 2, Unit 3, Unit 4.

Table 6 .

 6 Firm age and the use of leasing.

	Theoretical argument: Informational asymmetry between borrower and lender
	LVS	LW	LS	LVW
	AGEVW	AGEW	AGES	AGEVS
	U 1	U 2	U 3	U 4
	Legend
	LVS, LS, LW, LW: Leasing very strong, strong, weak, very weak.
	AGEVS, AGES, AGEW, AGEVW: Age very strong, strong, weak, very weak.
	U1, U2, U3 et U4: Unit 1, Unit 2, Unit 3, Unit 4.

Table 7 .

 7 Leverage and the use of leasing.

	Theoretical argument: debt capacity of the firm
	LVW	LVS	LS	LW
	LEVVW	LEVVS	LEVS	LEVW
	U 1	U 2	U 3	U 4
	Legend
	LVS, LS, LW, LW: Leasing very strong, strong, weak, very weak.
	LEVVS, LEVS, LEVW, LEVVW: Leverage very strong, strong, weak, very weak.
	U1, U2, U3 et U4: Unit 1, Unit 2, Unit 3, Unit 4.

Table 8 .

 8 Credit rationing intensity and financing policy by leasing: Wilcoxon test.

	Variables	Wilcoxon	Interpretation
		statistics
	Size	-2.147**	Credit rationing is positively associated with a
			small size
	Leverage	-16.03***	Credit rationing is positively associated with
			highly leveraged firms
	Solvency	-5.31***	Credit rationing is negatively associated with
			firms' solvency
	Probability of Bankruptcy	-10.1***	Credit rationing is positively associated with the
			probability of bankruptcy
	Age	-1.11	Credit rationing is negatively associated with
			firms' age
	Note: *, ** and *** significant at the 10%, 5% and 1% threshold.

An extensive presentation can be found in Kohonen T., "Self organized formation of topologically correct feature maps", Biological Cybernetics,

43,1982, p. 59 or in Kohonen T., "Self-Organizing Maps", Springer Series in Information SciencesVol. 30, Springer, Berlin, 1995

ACKNOWLEDGEMENTS

The preparation of this paper has been benefited by the support of a SAMOS of Paris and of University of Lille 1 (IAE). The authors thank the anonymous referees for their comments.