Daniel Ayobami

Anna Daramola

Giovanni Fraczkiewicz

Akiyoshi Bonny

Gilles Nomoto

Christophe Adjanor

Ghiath Domain

Monnet

Ayobami Daramola
email: ayobami.daramola@emse.frghiath.monnet@edf.fr

Anna Fraczkiewicz

Giovanni Bonny

Akiyoshi Nomoto

Gilles Adjanor

Christophe Domain

Monnet D Ghiath

Ayobami Daramola

Ghiath Monnet

Atomistic investigation of elementary dislocation properties influencing mechanical behaviour of Cr15Fe46Mn17Ni22 alloy and Cr20Fe70Ni10 alloy

Keywords: High entropy alloys, Austenitic stainless steel, Molecular dynamics, Molecular statics, Edge dislocation, Stacking fault energy

à 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.

Introduction

Recent works have shown that some single-phase High Entropy Alloys (HEA) exhibit excellent mechanical properties, radiation-resistance properties and phase stability making them potential candidates for structural applications [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF][START_REF] Yeh | Recent progress in high-entropy alloys[END_REF][START_REF] Miracle | A critical review of high entropy alloys and related concepts[END_REF][START_REF] Yeh | Alloy design strategies and future trends in high-entropy alloys[END_REF][START_REF] Gao | High-entropy alloys: fundamentals and applications[END_REF][START_REF] Yao | A novel, single phase, non-equiatomic FeMnNiCoCr high-entropy alloy with exceptional phase stability and tensile ductility[END_REF][START_REF] Otto | Relative effects of enthalpy and entropy on the phase stability of equiatomic high-entropy alloys[END_REF][START_REF] Senkov | Mechanical properties of Nb25Mo25Ta25W25 and V20Nb20Mo20Ta20W20 refractory high entropy alloy[END_REF][START_REF] Senkov | Microstructure and elevated temperature properties of a refractory TaNbHfZrTi alloy[END_REF][START_REF] George | High-entropy alloys[END_REF]. However, many authors have traced the origin of these properties to the specific behaviour of dislocations [START_REF] De Cooman | Twinning-induced plasticity (TWIP) steels[END_REF][START_REF] Howie | Direct measurements of stacking-fault energies from observations of dislocation nodes[END_REF][START_REF] Smith | Atomic scale characterization and modeling of 60° dislocations in a high-entropy alloy[END_REF][START_REF] Zhao | Fluctuation in stacking fault energies improve irradiation tolerance of concentrated solid solution alloys[END_REF][START_REF] Zhao | Local-environment dependence of stacking fault energies in concentrated solid-solution alloys[END_REF][START_REF] Dou | Interaction mechanism of an edge dislocation with a void in Fe-Ni-Cr concentrated solid-solution alloy[END_REF][START_REF] Zhao | Atomic-scale dynamics of edge dislocations in Ni and concentrated solid solution NiFe alloys[END_REF][START_REF] Zeng | Effects of the stacking fault energy fluctuations on the strengthening of alloys[END_REF][START_REF] Traversier | Nitrogeninduced hardening in an austenitic CrFeMnNi high-entropy alloy (HEA)[END_REF][START_REF] Huang | Temperature dependent stacking fault energy of FeCrCoNiMn high entropy alloy[END_REF][START_REF] Oliveros | Orientation-related twinning and dislocation glide in a Cantor High Entropy Alloy at room cryogenic temperature studied by in situ TEM straining[END_REF][START_REF] Kim | Mechanical properties and deformation twinning behavior of ascast CoCrFeMnNi high-entropy alloy at low and high temperatures[END_REF][START_REF] Okamoto | Size effect, critical resolved shear stress, stacking fault energy, and solid solution strengthening in the CrMnFeCoNi high-entropy alloy[END_REF][START_REF] Schneider | Analysis of strengthening due to grain boundaries and annealing twin boundaries in the CrCoNi medium-entropy alloy[END_REF][START_REF] Varvenne | Solute strengthening in random alloys[END_REF][START_REF] Larosaa | Solid solution strengthening theories of high-entropy alloys[END_REF][START_REF] Rao | Atomistic simulations of dislocations in a model BCC multicomponent concentrated solid solution alloy[END_REF][START_REF] Leyson | Solute strengthening from first principles and application to aluminum alloys[END_REF]. Face-Centered Cubic (FCC) HEA, such as Cantor alloy consisting of equimolar CoCrFeMnNi composition exhibits excellent mechanical properties and improved irradiation resistance [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF]. On the other hand, the presence of Cobalt excludes Cantor alloy from structural applications in nuclear reactors, due to the production of 60 Co radioisotope in in-service conditions [START_REF] Traversier | Nitrogeninduced hardening in an austenitic CrFeMnNi high-entropy alloy (HEA)[END_REF][START_REF] Kumar | Microstructural stability and mechanical behaviour of FeNiMnCr high entropy alloy under ion irradiation[END_REF][START_REF] Wu | Microstructures and mechanical properties of compositionally complex Co-free FeNiMnCr18 FCC solid solution alloy[END_REF]. This is one of the reasons for the design of new Cofree HEA alloys, mainly from the CrFeMnNi family [START_REF] Traversier | Nitrogeninduced hardening in an austenitic CrFeMnNi high-entropy alloy (HEA)[END_REF][START_REF] Kumar | Microstructural stability and mechanical behaviour of FeNiMnCr high entropy alloy under ion irradiation[END_REF][START_REF] Wu | Microstructures and mechanical properties of compositionally complex Co-free FeNiMnCr18 FCC solid solution alloy[END_REF][START_REF] Liu | Strengthening and toughening the FeNiCrMn medium entropy alloy by novel ultrafine precipitate network[END_REF][START_REF] Bian | A novel cobalt free FeMnCrNi medium entropy alloy with exceptional yield strength and ductility at cryogenic temperature[END_REF][START_REF] Olszewska | Conception and Development of a Novel Grad of High Resistance High Entropy Alloy from the CrFeMnNi Family[END_REF][START_REF] Stepanov | Tensile properties of the CrFeMnNi nonequiatomic multicomponent alloys with Cr contents[END_REF][START_REF] Gao | Stacking fault energy fluctuation effect on radiation-induced dislocation loops in fcc CrFeMnNi high-entropy alloys[END_REF]. Currently, in the major types of nuclear reactor, the in-core components consist mostly of austenitic stainless steels (ASS) [START_REF] Zinkle | Materials challenges in nuclear energy[END_REF][START_REF] Garner | Radiation damage in austenitic steels[END_REF][START_REF] Raj | Ferritic steels and advanced ferritic-martensitic steels[END_REF] and some authors [START_REF] Kumar | Microstructural stability and mechanical behaviour of FeNiMnCr high entropy alloy under ion irradiation[END_REF][START_REF] Zinkle | Materials challenges in nuclear energy[END_REF][START_REF] Garner | Radiation damage in austenitic steels[END_REF] have shown that ASS can be improved to exhibit better radiation damage resistance. Few authors like Kumar et al., [START_REF] Kumar | Microstructural stability and mechanical behaviour of FeNiMnCr high entropy alloy under ion irradiation[END_REF], Gao et al., [START_REF] Gao | Stacking fault energy fluctuation effect on radiation-induced dislocation loops in fcc CrFeMnNi high-entropy alloys[END_REF] have shown that Co-free alloy can be used for structural application to overcome this extreme environment and implemented for the next-generation nuclear reactors.

Irradiation hardening and embrittlement are key issues for structure alloys for applications under extreme conditions [START_REF] Kumar | Microstructural stability and mechanical behaviour of FeNiMnCr high entropy alloy under ion irradiation[END_REF][START_REF] Gao | Stacking fault energy fluctuation effect on radiation-induced dislocation loops in fcc CrFeMnNi high-entropy alloys[END_REF][START_REF] Bonny | Interatomic potential for studying ageing under irradiation in stainless steels: the FeNiCr model alloy[END_REF][START_REF] Monnet | Multiscale modeling of irradiation hardening: Application to important nuclear materials[END_REF][START_REF] Hashimoto | Effect of stacking fault energy on damage microstructure in ionirradiated CoCrFeNiMnx concentrated solid solution alloys[END_REF][START_REF] Fernández-Caballero | High-dose ion irradiation damage in Fe28Ni28Mn26Cr18 characterised by TEM and depth-sensing nanoindentation[END_REF][START_REF] Baudouin | Molecular dynamics investigation of the interaction of an edge dislocation with Frank loops in Fe-Ni10-Cr20 alloy[END_REF]. The irradiation hardening is associated to an increase in the critical stress of the material while the embrittlement is detected by the change of fractures properties. These issues are well investigated in ASS materials from the perspective of both experimental and numerical studies [START_REF] Baudouin | Molecular dynamics investigation of the interaction of an edge dislocation with Frank loops in Fe-Ni10-Cr20 alloy[END_REF][START_REF] Zinkle | Dose dependence of the microstructural evolution in neutron-irradiated austenitic stainless steel[END_REF][START_REF] Allen | On the mechanism of radiation-induced segregation in austenitic Fe-Cr-Ni alloys[END_REF]. Findings from recent works like Zhao et al., [START_REF] Zhao | Fluctuation in stacking fault energies improve irradiation tolerance of concentrated solid solution alloys[END_REF][START_REF] Zhao | Local-environment dependence of stacking fault energies in concentrated solid-solution alloys[END_REF][START_REF] Zhao | Atomic-scale dynamics of edge dislocations in Ni and concentrated solid solution NiFe alloys[END_REF] have shown that reduction of irradiation hardening is possible using FCC multi-principal element alloys (MPEAs) like HEA. The authors explained that the fluctuation of SFE improves irradiation tolerance thereby limiting the number of defects and cluster growth under the cascade during neutron irradiation. Gao et al., [START_REF] Gao | Stacking fault energy fluctuation effect on radiation-induced dislocation loops in fcc CrFeMnNi high-entropy alloys[END_REF] have also investigated the role of local SFE on the formation of defects in Cr15Fe46Mn17Ni22 and Cr16Fe37Mn13Ni34 (at. %) under irradiation conditions. The reports from these works [START_REF] Zhao | Local-environment dependence of stacking fault energies in concentrated solid-solution alloys[END_REF][START_REF] Zhao | Atomic-scale dynamics of edge dislocations in Ni and concentrated solid solution NiFe alloys[END_REF] suggest that high fluctuation of SFEs in MPEA would increase the critical stress for dislocation glide and therefore, improve the mechanical behaviour of the material. To the knowledge of the authors, there is no detailed work that would investigate the elementary dislocation behaviour in FCC CrFeMnNi alloys, especially their dissociation and mobility under different loading conditions. Experimentally, dislocation mobility is difficult to characterize and therefore molecular dynamics (MD) is often adopted to overcome this problem. Over the years, MD has been used to investigate the dislocation behaviour of materials of different crystallographic structures: FCC [14-17, 27, 43, 46, 47], Body-Centered Cubic (BCC) [START_REF] Domain | Simulation of screw dislocation motion in iron by molecular dynamics simulations[END_REF][START_REF] Monnet | Structural and mobility of the ½<111>{112} edge dislocation in BCC iron studied by molecular dynamics[END_REF][START_REF] Queyreau | Edge dislocation mobilities in bcc Fe obtained by molecular dynamics[END_REF] as well as Hexagonal Close-Packed (HCP) [START_REF] Groh | Dislocation motion in magnesium: a study by molecular statics and molecular dynamics, model[END_REF][START_REF] Shin | Simulations of dislocation mobility in magnesium from first principle[END_REF]. In our previous work [START_REF] Daramola | Development of a Plasticityoriented Interatomic Potential for CrFeMnNi High Entropy Alloys[END_REF], we developed a new embedded atom method (EAM) type interatomic potential to describe FCC CrFeMnNi systems. This is one of the few available EAM potentials that have been able to describe the dislocation properties of CrFeMnNi systems and their subsystems. This potential was fitted to reproduce good agreement for elastic constants, stacking fault energy and phase stability compared to experimental measurement, density functional theory and thermodynamic modelling. Gao et al., [START_REF] Gao | Stacking fault energy fluctuation effect on radiation-induced dislocation loops in fcc CrFeMnNi high-entropy alloys[END_REF] recently used this potential to describe the effect of local stacking fault energies in Co-free alloys. To validate and confirm our results, some of the predictions of this potential are compared with those of a ternary potential [START_REF] Bonny | Interatomic potential to study plasticity in stainless steels: the FeNiCr model alloy[END_REF] dedicated to the CrFeNi system. In the following, this potential is referred to as EAM-11, while the potential in [START_REF] Daramola | Development of a Plasticityoriented Interatomic Potential for CrFeMnNi High Entropy Alloys[END_REF] is called EAM-21. The default potential to be considered when it is not specified is EAM-21. When EAM-11 is used, it will be explicitly mentioned. The objective of this work is to investigate and compare elementary properties affecting the dislocation behaviour such as the fluctuation of the local SFE, and dislocation dissociation and mobility in the ASS and the HEA model alloys. A constitutive description of the dislocation mobility in the two systems is reported, providing insight into the mechanical characteristics of these two model alloys.

Methods

Selection of model alloys

This present study is general and applicable to any austenitic stainless steel CrFeNi, especially 300 series of steel of American standard AlSl (304 and 316-type) and Co-free HEAs such as FCC CrFeMnNi [START_REF] Traversier | Nitrogeninduced hardening in an austenitic CrFeMnNi high-entropy alloy (HEA)[END_REF][START_REF] Kumar | Microstructural stability and mechanical behaviour of FeNiMnCr high entropy alloy under ion irradiation[END_REF][START_REF] Wu | Microstructures and mechanical properties of compositionally complex Co-free FeNiMnCr18 FCC solid solution alloy[END_REF][START_REF] Liu | Strengthening and toughening the FeNiCrMn medium entropy alloy by novel ultrafine precipitate network[END_REF][START_REF] Bian | A novel cobalt free FeMnCrNi medium entropy alloy with exceptional yield strength and ductility at cryogenic temperature[END_REF][START_REF] Olszewska | Conception and Development of a Novel Grad of High Resistance High Entropy Alloy from the CrFeMnNi Family[END_REF][START_REF] Stepanov | Tensile properties of the CrFeMnNi nonequiatomic multicomponent alloys with Cr contents[END_REF][START_REF] Gao | Stacking fault energy fluctuation effect on radiation-induced dislocation loops in fcc CrFeMnNi high-entropy alloys[END_REF]. The calculations are narrowed to Cr20Fe70Ni10 at. % and Cr15Fe46Mn17Ni22 at. %. The reasons for the selection of these alloys are as follows: a) Olszewska [START_REF] Olszewska | Conception and Development of a Novel Grad of High Resistance High Entropy Alloy from the CrFeMnNi Family[END_REF] work has reported that Cr15Fe46Mn17Ni22 alloy also called "Y3 grade" exhibit properties that are very similar to the Cantor alloy [START_REF] Cantor | Microstructural development in equiatomic multicomponent alloys[END_REF]. The interatomic potential we recently developed [START_REF] Daramola | Development of a Plasticityoriented Interatomic Potential for CrFeMnNi High Entropy Alloys[END_REF] fits the SFE of Cr15Fe46Mn17Ni22 to be 26 mJ/m 2 so it can follow the calculations of SFE by Zaddach et al., [START_REF] Zaddach | Mechanical Properties and Stacking Fault Energies of NiFeCrCoMn High-Entropy Alloy[END_REF] and Smith et al., [START_REF] Smith | Atomic scale characterization and modeling of 60° dislocations in a high-entropy alloy[END_REF] for Cantor alloy which give values in the range of 26-28 mJ/m 2 . The SFE is important to reproduce a reliable dislocation behaviour and hence, one of the reasons for Cr15Fe46Mn17Ni22 alloy selection.

b) In a previous work carried out by Kumar et al., [START_REF] Kumar | Microstructural stability and mechanical behaviour of FeNiMnCr high entropy alloy under ion irradiation[END_REF], Cr18(FeMnNi) wt. % alloy was shown to have a better radiation resistance than CrFeNi austenitic alloys. However, no work has given an in-depth understanding of irradiation conditions for Cr15Fe46Mn17Ni22 excepted Gao et al., [START_REF] Gao | Stacking fault energy fluctuation effect on radiation-induced dislocation loops in fcc CrFeMnNi high-entropy alloys[END_REF] that experimentally investigated the ratio of faulted Frank loop against perfect loop and discovered that the formation of the defects depends on the fluctuations of SFE. Yet, no information was reported on the plasticity of Cr15Fe46Mn17Ni22, especially on the collective behaviour of defect character either edge or screw dislocation. EAM-11 potential [START_REF] Daramola | Development of a Plasticityoriented Interatomic Potential for CrFeMnNi High Entropy Alloys[END_REF] and EAM-21 potential [START_REF] Bonny | Interatomic potential to study plasticity in stainless steels: the FeNiCr model alloy[END_REF] was used to calculate the dissociation distance and motion of dislocations for Cr20Fe70Ni10 and Cr15Fe46Mn17Ni22 alloy respectively. The calculations based on EAM-11 potential was used in a classical MD code developed in the Tokyo University of Science (MicroMD). EAM-21 potential was used in two different MD codes in this work: DYMOKA code [START_REF] Becquart | Massively parallel molecular dynamics simulation with EAM potentials[END_REF] for molecular statics (MS) and Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) software [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] for molecular dynamics (MD) simulations. When the box dimensions are too low, say few nanometers, the partial dislocations strongly interact with their images through the periodic boundary and with the fixed atomic layers. In this case, the box dimensions affect all the observed dislocation properties. As shown in figure 1, the simulation boxes dimensions are 45 × 30 × 20 𝑛𝑚 3 in [110], [11 ̅ 2] and [1 ̅ 11] directions, respectively, which correspond to about 2.59 million atoms in the cell. These dimensions are large enough to avoid spurious effects induced by the simulation box size. The model material is relaxed and then thermalized at given temperatures. The periodic boundary conditions are applied on the (110) and (11 ̅ 2) planes. For the dislocation motion calculations, two different methods were used to introduce the ½ <110>{111} edge dislocation in the simulation box. The method called d+ was initially reported by Osetsky et al [START_REF] Osetsky | An atomic-level model for studying the dynamics of edge dislocation in metals[END_REF] in which the dislocation is initially spread over the whole simulation width. The second one is called d-method was described by Rodney et al. [START_REF] Rodney | Dislocation pinning by small Interstitial loops: a molecular dynamics study[END_REF] and corresponds to the generation of a compact dislocation (Volterra dislocation). The atoms on the top and bottom surfaces are fixed and then displaced in the [110] direction at a constant displacement rate to apply shear deformation at a given strain rate. Since the temperature change during deformation is typically only several kelvins, no temperature control was performed during the deformation calculations. The shear stress is calculated from the component of the force on the fixed atoms parallel to the Burgers vector. The dislocation lines were identified by the common neighbour analysis (CNA) method [START_REF] Honeycutt | Molecular dynamics study of melting and freezing of small Lennard-Jones clusters[END_REF] and the dislocation extraction algorithm (DXA) function [START_REF] Stukowski | Structure identification methods for atomistic simulations of crystalline materials[END_REF].

Computational methods

The SFE calculations were carried out using the same fcc crystal simulation cell box and oriented directions. To determine the SFE, the simulation box sizes of about 10 × 5 × 10 × 12 containing around 6000 randomly distributed atoms and to analyze the role of local SFE, 6000 calculations with different random seeds within the calculation box were performed. The SFE is defined as:

𝛾 𝑆𝐹𝐸 = 𝐸 𝑓 -𝐸 𝑖 𝐴 (1)
where 𝐸 𝑓 and 𝐸 𝑖 are the energies of the system after and before creating stacking fault, respectively, and 𝐴 is the area of stacking fault.

Results and discussion

Elastic deformation response of the alloys

The shear modulus on the {111} planes in the <110> directions of the Cr20Fe70Ni10 and Cr15Fe46Mn17Ni22 alloy was investigated. The shear modulus is obtained by loading the simulation box (Fig. 1) and it equals the slope of the shear stressshear strain curve when the dislocation does not move. Figure 2 shows the obtained response of the box at 300 K with a strain rate of 10 6 𝑠 -1 . Every peak of the curves corresponds to a rearrangement of the dislocation cores. Between the peaks the dislocation is immobile, and the slope can be unambiguously calculated. From our MD calculation, the shear modulus is estimated to be 59 GPa and 79 GPa for the ASS Cr20Fe70Ni10 and HEA Cr15Fe46Mn17Ni22 alloys respectively. These values are compared with the values predicted by the anisotropic-elastic theory [START_REF] Martin | Elasticity[END_REF][START_REF] Mouhat | Necessary and sufficient elastic stability conditions in various crystal systems[END_REF] of the shear modulus on the {111} planes in the <110> directions.

In cubic symmetry, it is given by 𝜇 = (𝐶 11 -𝐶 12 + 𝐶 44 )/3. Table 1 shows the calculated values for both alloys. The latter is in close agreement with the values obtained from MD simulations and The difference in value is due to anisotropic-elastic theory's lack of a lattice force. Experimentally, the shear modulus decreases with temperature [START_REF] Teklu | Keppens Single-Crystal Elastic Constants of Fe15Ni15Cr Alloy[END_REF][START_REF] Haglund | Polycrystalline elastic moduli of a high entropy alloy at cryogenic temperatures[END_REF]. This agrees with the results presented in figure 3, in which we depict the evolution of the three elastic constants with temperature. All constants decrease significantly with temperature between 0 and 900 K. This qualitatively agrees with Haglund et al., [START_REF] Haglund | Polycrystalline elastic moduli of a high entropy alloy at cryogenic temperatures[END_REF] experimental explanation for HEA. (The approach used can be seen in the appendix).

Investigation of the stacking fault energy

The <110>{111} edge dislocation in Cr20Fe70Ni10 and Cr15Fe46Mn17Ni22 alloy dissociates into two Shockley partial dislocations separated by a stacking fault ribbon [START_REF] Baudouin | Effect of the applied stress and the friction stress on the dislocation dissociation in face centered cubic metals[END_REF]. This follows the Shockley schema as:

𝑎 2 [110] ↔ 𝑎 6 [121] + 𝑎 6 [211 ̅ ] (2)
The atom distributions in the stacking fault ribbon region play an important role in the fluctuations of the dissociation distance. Moreover, both alloys have random atoms in nature Here, the effect of local atom distributions on the SFE was investigated. As earlier stated in section 2.2, the role of the local SFE distributions can be carried out by varying the element arrangement using different random seeds in the simulation as implemented in [START_REF] Zhao | Fluctuation in stacking fault energies improve irradiation tolerance of concentrated solid solution alloys[END_REF][START_REF] Zhao | Local-environment dependence of stacking fault energies in concentrated solid-solution alloys[END_REF][START_REF] Dou | Interaction mechanism of an edge dislocation with a void in Fe-Ni-Cr concentrated solid-solution alloy[END_REF][START_REF] Zhao | Atomic-scale dynamics of edge dislocations in Ni and concentrated solid solution NiFe alloys[END_REF][START_REF] Zeng | Effects of the stacking fault energy fluctuations on the strengthening of alloys[END_REF]. The SFE distribution is estimated with a Gaussian distribution given as:

𝑔(𝑥) = 1 𝜎√2𝜋 𝑒𝑥𝑝 - (𝑥-𝜇) 2 2𝜎 2 (3)
where the values of 𝜇, and 𝜎 is the mean of the SFE and the standard deviation of the SFE distribution, respectively. In figure 4, the distribution of the SFE of Cr20Fe70Ni10 and Cr15Fe46Mn17Ni22 alloy is presented. The mean value of SFE of Cr20Fe70Ni10 is estimated to 19 mJ/m 2 and Cr15Fe46Mn17Ni22 alloy is evaluated to be 28 mJ/m 2 . However, the standard deviation which defines the fluctuations of the SFE distribution is approximately 12 mJ/m 2 for Cr20Fe70Ni10 and 18 mJ/m 2 for Cr15Fe46Mn17Ni22 alloy. The calculations in figure 4 suggest that the fluctuation of the SFE distribution increases as the chemical composition becomes complex.

Dislocation dissociation in the different alloys

Since two opposite methods were used to introduce the edge dislocation in the simulation box, it is important to check their possible effects on the dislocation behaviour. To illustrate the effect of the selected method on the resulting dissociation, we apply both methods on the ASS (Cr20Fe70Ni10) alloy using EAM-11 potential and analyze the obtained dislocation structure, relaxed at 300K (see Fig. 5). The resulting dissociation is completely altered by the construction method. Is this discrepancy a simulation spurious effect or the consequence of a hidden physical mechanism? To answer this question, we follow the analysis reported in [START_REF] Baudouin | Effect of the applied stress and the friction stress on the dislocation dissociation in face centered cubic metals[END_REF]. In pure FCC metals, the friction stress (solid solution) is negligible. The dislocation dissociation reaches equilibrium after relaxation in both methods. But in concentrated alloys, friction stress cannot be ignored, which affects simulation results as follows. When the edge dislocation in Fig. 1

F1 = Fint -  Ff + app b /2 (4) F2 = -Fint +   Ff + app b /2 (5)
where b is the norm of the Burgers vector. The division by 2 of the applied stress term results from the projection of the partial Burgers vectors on the x -axis. We will see that these equations can explain all simulation results on the dislocation dissociation after relaxation.

Let us first define what we call the equilibrium dissociation width do. By definition, it corresponds to the dissociation width under zero applied stress and zero friction stress. Imposing F1 = F2 = 0 leads to Fint = F for the two partials. Recalling that for edge dislocations [START_REF] Baudouin | Effect of the applied stress and the friction stress on the dislocation dissociation in face centered cubic metals[END_REF]:

𝐹 𝑖𝑛𝑡 = 𝐺𝑏 2 (2 + 𝜈) 24𝜋(1 -𝜈)𝑑 (6)
Then, we obtain:

𝑑 𝑜 = 𝐺𝑏 2 (2 + 𝜈) 24𝜋(1 -𝜈)𝛾 (7)
In eqs. 4 and 5, the "" sign accounts for the indetermination in the sign of the friction stress since it always opposes the dislocation motion. During relaxation of the initial atomic configuration in the d+ method [START_REF] Osetsky | An atomic-level model for studying the dynamics of edge dislocation in metals[END_REF], the dislocation is initially dissociated over the whole simulation box width as shown in fig. 4. d is thus larger than do and partials tend to move towards each other. While in the d-method [START_REF] Rodney | Dislocation pinning by small Interstitial loops: a molecular dynamics study[END_REF], the initial dislocation is compact and the two partials too close (d ≪ do). Under the effect of Fint partials tend to move away from each other. Consequently, during relaxation in the d-method, we can set the force balances as: F1 = Fint - -Ff and F2 = -Fint +  + Ff, while in the d+ method, we have F1 = Fint - + Ff and F2 = -Fint +  -Ff. Therefore, in the d-method, the partials stop spreading when Fint =  + Ff. The final dissociation width in the d-method is thus:

𝑑 -= 𝐺𝑏 2 (2 + 𝜈) 24𝜋(1 -𝜈)(𝛾 + 𝐹 𝑓 ) (8)
The spreading of the partials decreases with the friction stress and remains always lower than the equilibrium width do. The situation is different in the d+ method. During relaxation, the only driving force for motion is the stacking fault energy , which must overcome the elastic repulsion and the friction stress. In order to have an idea of the order of magnitude of the different components, consider a separation of d = 20 nm, with a shear modulus of 70 GPa and Poisson's ratio of 0.3, one obtains from eq. 6: Fint ≈ 9.5 mJ, while the average SFE is close to 28 mJ/m 2 . Converting forces per unit length to shear stresses (using the relation  = F /b), we obtain int ≈ 38 MPa and  ≈ 100 MPa.

Consequently, in the d+ method the partials will not be able to move towards each other if the friction stress is larger than 100 -38 ≈ 62 MPa. As it will be seen in the next section, this is almost always the case. In the d+ method, the obtained dissociation width is large, constant and temperature independent. Now let us study the dissociation width under a given positive applied stress, large enough to induce dislocation motion. When the partials start moving in the positive x-direction, the friction stress on both partials in the two configurations (d-and d+ methods) becomes negative (negative sign in equations 4 and 5). When the velocity is constant, the resultant force on both partials remains zero: F1

= F2 = 0. The applied stress balances completely the friction stress: app b /2 -Ff = 0. This is equivalent to the condition of dissociation under zero friction and applied stress. The condition Fint = F is recovered and the dissociation width matches with do give in equation 7. In summary, under an applied stress in the d-method (Figure 5), Partial_1 starts to move first in the positive x-direction and the separation between partials increases to do. In contrast, when the dislocation is introduced using the d+ method (Figure 5), it is Partial_2 which starts in the positive x-direction getting closer to Partial_1 and the final dissociation width decreases to do. The equilibrium dissociation width in concentrated alloys can be obtained only during dislocation motion. Of course, when the applied stress is removed after the dislocation motion, the dissociation width remains at do and the system "forgets" its initial configuration whether it was d+ or d-.

To confirm these theoretical findings, we investigate the dissociation width after relaxation, with the edge dislocation introduced using the two methods (d+ and d-) and compare it with that computed during dislocation motion do as a function of temperature in the ASS alloy. The results obtained using EAM-11 potential are shown in Fig. 6a. Fig. 6b, shows the evolution of the dissociation width during its motion as calculated for both alloys using the EAM-21 potential at different strain rates and temperatures. First, we notice that strain rate does not affect the dissociation width, which slightly increases with temperature owing to the decrease in the shear modulus. Second, the dissociation width in the ASS alloys (Cr20Fe70Ni10) remains larger than in the HEA alloy (Cr15Fe46Mn17Ni22). This is of course attributed to the fact that the SFE in the ASS alloys is lower than in the HEA alloy.

Dislocation mobility

As a general statement, molecular dynamics results show that the average shear stress required to move the dislocation is higher in the Cr15Fe46Mn17Ni22 HEA alloy than in the Cr20Fe70Ni10 ASS alloy at all temperatures and strain rates. The motion of the partial dislocations is always jerky. This is evidenced by the large number of peaks on the stress-strain curves, see Fig. 7. Partials are continuously pinned by many local strong atomic configurations. Every stress peak corresponds to an unpinning event.

Figure 7: Examples of the loading curves of the simulation box (using EAM-21) containing an edge dislocation at different temperatures of the ASS (Cr20Fe70Ni10) alloy and of the HEA (Cr15Fe46Mn17Ni22). The black, red, green and blue curves refer to simulations at 10, 300, 600 and 900 K, respectively.

In the following, the critical stress for the dislocation motion at a given stress and strain rate is computed as an average of the stress curve recorded beyond the elastic loading. However, the dislocation velocity is computed from the average dislocation position, taken as the average positions of the partial dislocations. Figure 8 shows the evolution of the critical stress as a function of the dislocation velocity at different simulation temperatures. The result shows that, as expected from the thermal activation theory, the average stress during dislocation motion decreases strongly with temperature and increases with strain rate in both alloys.

Results obtained with the two potentials (EAM-11 and EAM-21) on the ASS alloy (Cr20Fe70Ni10 alloy) using different MD codes and different simulation box dimensions are found to collapse on the same curves in Fig. 8, which confirms the consistency of the two potentials. It can be clearly seen in the figure that the evolution is not linear in both alloys. The tendency to saturation in the critical stress is not due to the limit of sound speed, since the latter is much larger than the velocities shown in the Figure . In the following, a constitutive equation was proposed for the description of behaviour shown in Figure 8.

In order to correctly describe the dislocation behaviour, two regimes must be distinguished. As widely known [START_REF] Hirth | Theory of Dislocations[END_REF], during dislocation motion in pure materials, dislocation interacts with phonons (friction regime). At velocities much lower than the sound velocity, this resistance is viscous and linear in FCC materials, see for example [START_REF] Olmsted | Atomistic simulations of dislocation mobility in Al, Ni and Al/Mg alloys[END_REF]. The velocity vf in this regime is well described by a simple friction model:

𝑣 𝑓 = 𝜏𝑏 𝐵 ( 9
)
where B is the drag coefficient. For the sake of simplicity, we consider a constant average value for the drag coefficient B = 4.5 10 -5 Pa.s. This value is consistent with other investigations (ex. [START_REF] Chu | Temperature and composition dependent screw dislocation mobility in austenitic stainless steel from large-scale molecular dynamics[END_REF]). However, in highly concentrated alloys, where some hard local configurations require unpinning by thermal activation, another regime prevails especially at low temperature/low stress (thermal regime). Although many sophisticated models were reported in the literature, it turns out that a simple Arrhenius type equation correctly predicts the behaviour of the dislocation in these simulations. The thermally activated velocity of the dislocation vth is given by:

𝑣 𝑡ℎ = 𝑣 𝑜 𝑒𝑥𝑝 - ∆𝐻 𝑘𝑇 ( 10
)
where vo is a constant, H is the stress-dependent activation enthalpy and k is the Boltzmann constant.

In general, dislocations are submitted simultaneously to the friction resistance (eq. 9) and to the thermally activated rate (eq. 10). Consequently, there is no way to completely isolate the friction regime from the thermally activated one. Away to decrease the thermally activated component is to strongly increase the stress, which results in a low activation energy. But in this case, the velocity becomes very high and may start saturating close to shear wave velocity. A superposition rule is thus required. For the dislocation to move, the applied stress must be large enough to overcome the phonon resistance and to provide enough shear rate for thermal activation. As discussed in [START_REF] Monnet | Multiscale modeling of crystal plasticity in Reactor Pressure Vessel steels: Prediction of irradiation hardening[END_REF], the two resistances must add in parallel which implies that the final dislocation velocity matches with the harmonic mean of both velocities:

1 𝑣 = 1 𝑣 𝑓 + 1 𝑣 𝑡ℎ (11)
The value of vo = 2000 m s -1 was found to ensure a smooth evolution of H with the shear stress  in all simulations (different temperatures and strain rates). Using data in Fig. 8, the obtained activation enthalpy as a function of stress is shown in Fig 9 for the two alloys. As expected, the activation enthalpy is a monotonously decreasing function of the stress. Again, results of the two potentials (EAM-11 and EAM-21) on the ASS alloy (Cr20Fe70Ni10 alloy) using different MD codes and different simulation conditions are found to collapses on the same curve in Fig. 9. Comparing the results of the two investigated alloys in Fig. 9, one notice that activation enthalpy at given stress is larger in the HEA alloy than in the ASS alloy. To provide a constitutive equation for the dislocation motion, one must provide an analytical expression for H as a function of shear stress . Following the framework of Kock et al [START_REF] Kocks | Thermodynamics and kinetics of slip[END_REF], fitting attempts show that the enthalpy barrier for both alloys can be described by the unique expression:

𝐻 = 𝐻 𝑜 (1 -√   𝑜 ) 2 (12)
in which Ho is the total activation enthalpy (i.e. under zero applied stress) and o is the theoretical threshold stress at absolute zero temperature. The other coefficients depend on the alloy composition: Ho = 0.8 eV and o = 400 MPa for the ASS (Cr20Fe70Ni10 alloy) alloy and Ho = 1.1 eV and o = 550 MPa for the HEA (Cr15Fe46Mn17Ni22) alloy. The higher value of o in the HEA alloy suggests that the local pinning forces are stronger than in the ASS model alloy. To check the values of o found in our identification process, the ideal solution is to carry out molecular static simulations (i.e. at 0 K). However, energy minimization is required in molecular statics, which is much more time consuming especially for our large box size. Instead of that, we carried out simulations at 10 K (see Fig. 7). The values found for the average flow stress are lower but quite close to the identified values of o, which validate our superposition rule (the harmonic mean). The comparison between the dislocation velocity computed in MD simulations and the predicted velocity using eq. 11 (together with eq. 9 and 10) are illustrated in Fig. 10a. The predicted values of the velocity agree closely with the computed ones over at least three orders of magnitudes. This confirms our approach in superposing the thermally activated regime to the friction regime using the harmonic mean of the velocities. To illustrate the consequences of this superposition rule, we show in Fig. 10b the evolution of the velocity as a function with the applied shear stress at two different temperatures 100 and 600 K. At low stress, the dislocation velocity is strongly dependent on temperature, which is characteristic of the thermally activated regime. However, at high stress, there is no effect of the temperature on the dislocation velocity which indicates that the controlling regime is the friction stress.

Conclusion

In this study, the elementary dislocation properties in a model alloy of austenitic stainless-steel ASS, (Cr20Fe70Ni10) and a model high entropy alloy HEA, (Cr15Fe46Mn17Ni22) were investigated using two potentials and different MD codes at different temperatures and strain rates. Based on the above presented and discussed results the following conclusions can be drawn:

I.
There is a strong fluctuation of the local SFE in both alloys, that can accurately be described by a Gaussian distribution function, in agreement with experimental observations. The SFE fluctuation is higher in Cr15Fe46Mn17Ni22 alloy compared to Cr20Fe70Ni10 alloy. II.

The critical stress for the dislocation motion of the ½<110>{111} edge dislocation is larger in the Cr15Fe46Mn17Ni22 in comparison with the Cr20Fe70Ni10 alloy. The critical stress decreases strongly with temperature and strain rate, which is consistent with experimental investigations of solid solution strengthening. III.

The method used to introduce the dislocation in the simulation box is found to strongly alter the dissociation width obtained after relaxation. A theoretical explanation is presented to explain this issue and to confirm that an equilibrium dissociation width is achieved during dislocation motion. Moreover, the equilibrium dissociation width of Cr20Fe70Ni10 alloy is larger than the width of Cr15Fe46Mn17Ni22 alloy. The difference can be traced to the SFE values, the SFE of ASS alloys is lower than that of HEA alloy. IV.

The dislocation velocity varies as a function of the applied stress and simulation temperature. The velocity is found to obey a harmonic mean superposition rule involving a viscous regime induced by phonon drag and a thermally activated regime induced by local dislocation pinning due to strong fluctuation in the local chemical composition. The provided constitutive equation is found to accurately predict the dislocation velocity over three orders of magnitudes in the Cr20Fe70Ni10 and Cr15Fe46Mn17Ni22 alloys. Therefore, the dislocation velocity is found to be faster in ASS when compared to HEA.

Data availability

All data included in this study are available upon request by contact with the corresponding author.

Figure 1 :

 1 Figure 1: Coordinate system and geometry of the calculation box used for the present study. The periodic boundary condition is along with x and y directions, while the free boundary condition is along the z-direction. An edge dislocation is inserted in the middle by joining two half crystals and stress is induced by deformation rate as explained in the text.

Figure 2 :

 2 Figure 2: Stress-stress response to shear loading of the simulation box at 300 K (𝛾̇= 10 6 𝑠 -1 ). The shear modulus is estimated between the peaks on each curve. P1 and P2 are the points to determine the slope of the line.

Figure 3 :

 3 Figure 3: Temperature-dependence of the single elastic constant and the shear modulus 𝜇 for the austenitic stainless steel, Cr20Fe70Ni10 and HEA, Cr15Fe46Mn17Ni22. (a-c) 𝐶 11 , 𝐶 12 , 𝐶 44 and (d) shear modulus 𝜇 for 0-900 K

Figure 4 :

 4 Figure 4: Distribution of stacking fault energy of a) Cr20Fe70Ni10 and b) Cr15Fe46Mn17Ni22 alloy with 6000 random realizations. The mean and standard deviation are represented by 𝜇, and 𝜎, respectively.

Figure 5 :

 5 Figure 5: a sketch of the edge dislocation construction methods in the xz plane and the obtained dislocations after relaxation at 300 K in the xy plane.

  is dissociated into two partials ("Partial_1" to the right and "Partial_2" to the left in the figure), every partial is submitted to the following force per unit length components Fapp, Fint, Ff and F corresponding respectively to the applied (Peach-Koehler) force induced by the applied shear stress app (Fapp= appb), the elastic repulsion between the partials, the friction (solid solution) force and the attraction force induced by the positive stacking fault energy (SFE) (F =  per unit length). When forces are taken in absolute values, the total force on Partial_1 and Partial_2 are:

Figure 6 :

 6 Figure 6: Dissociation width vs temperature. (a) effect of the dislocation introduction method (d+ and d-) on the dissociation width in the relaxed ASS alloy (Cr20Fe70Ni10) using the EAM-11 potential compared with the dissociation width measured during motion do. (b) effect of the strain rate and temperature on the dissociation under the motion of the edge dislocation in the HEA (Cr15Fe46Mn17Ni22) and ASS (Cr20Fe70Ni10) alloys using the EAM-21 potentialIn Fig.6a, we notice that in agreement with the predictions: (i) d+ remains constant as predicted since Ff > Fγ -Fint in the whole temperature range; (ii) d-increases significantly with temperature also as predicted by eq. 8 owing to the large decrease in the friction stress (see next section); (iii) the measured dissociation width during motion do is as expected between d+ and d-. It increases slightly with temperature due to the slight decrease in the shear modulus.

Figure 8 :

 8 Figure 8: Evolution of the critical resolved shear stress (CRSS) on the (111) plane as a function of the temperature at different strain rates for the (a) ASS alloy (Cr20Fe70Ni10) using EAM-11 and EAM-21 and the (b) HEA alloy (Cr15Fe46Mn17Ni22) using EAM-21.

Figure 9 :

 9 Figure 9: Evolution of the activation enthalpy as a function of the applied shear stress in ASS (Cr20Fe70Ni10) and HEA (Cr15Fe46Mn17Ni22) alloys.

Figure 10 :

 10 Figure 10: (a) Comparison between the predicted and computed dislocation velocities in all simulations and (b) illustration of the predicted dislocation velocity as a function of shear stress in the ASS alloy (Cr20Fe70Ni10) at two different temperatures.

Figure A1 :

 A1 Figure A1: Equilibrium lattice parameter with temperature for HEA (Cr15Fe46Mn17Ni22) and ASS (Cr20Fe70Ni10) alloys using the EAM-21 potential

Table 1 .

 1

The table contains the elastic constants predicted by EAM-21 potential, the estimated shear modulus

Acknowledgements

This work is funded under the French ANR-PRCE-HERIA project (ANR-19-CE08-0012-01). One of the authors (GB) acknowledges funding from the Euratom research and training program 2014-2018 under grant agreement No. 755269 (GEMMA project).

Appendix A

Temperature-dependence of the single elastic constant and the shear modulus 𝝁

There are three methods for estimating the elastic constants: (a) direct approach, which involves calculating the change in internal energy due to applied strain, (b) calculating the box fluctuation with Canonical ensemble; NPH (number of atoms, pressure, and enthalpy are constant), or NPT (number of atoms, pressure, and enthalpy are constant) (number of atoms, pressure, and temperature are constant) and (c) Using microcanonical ensembles and keeping volume constant, the elastic constant can be calculated as the average of stress fluctuations; NVE (number of atoms, volume, and energy are constant) or NVT (number of atoms, volume, and energy are constant) (number of atoms, volume, and temperature are constant). The third method was employed to maintain the volume of the box when temperature was applied. The simulation box comprises approximately 30000 atom crystals that are subjected to <111> tensile loading with periodic boundary conditions applied in all directions. When the temperature was applied, NPT was employed to maintain equilibrium. After the system has been relaxed, NVT is applied to the MD simulation to achieve system equilibration so that the elastic constant may be calculated with temperature. The isothermal stress fluctuations formula was utilized to analyze the MD results in order to derive the elastic constant for finite temperature, as follows [START_REF] Kart | Thermal and mechanical properties of Cu-Au intermetallic alloys[END_REF][START_REF] Dereli | Thermal and mechanical properties of Pt-Rh alloys[END_REF]:

where the first term is the Born term, which is an elastic constant at 0 K and is defined as [START_REF] Bonny | Interatomic potential to study plasticity in stainless steels: the FeNiCr model alloy[END_REF]:

〈𝐸〉 is the average energy per atom, while V is the atomic volume, 𝑁 is the number of atoms and 𝑘 𝐵 is the Boltzmann constant.

The stress tensor fluctuation is the second term in equation (A.1), and the kinetic energy is the last term.

It is important to note that, the lattice parameter used for the MD calculation are also temperature dependance, the figure A.1 shows the evolution of the equilibrium lattice parameter with temperature for the two study alloys. The method used follows the approach used in [START_REF] Baudouin | Modelling and simulation with molecular dynamics of the edge dislocation behaviour in the presence of Frank loop in austenitic steels[END_REF].


Imad Benkhaled
email: imad.benkhaled@mines-ales.frisabelle.marc@mines-ales.fr

Isabelle Marc

Dominique Lafon
email: dominique.lafon@mines-ales.fr

COLOUR CONTRAST DETECTION IN CHROMATICITY DIAGRAM: A NEW COMPUTERIZED COLOUR VISION TEST

Keywords:

Colour discrimination is one of the most important means for object detection and recognition. Acquired or congenital colour blindness may become a major cause of troubles in everyday life tasks. Digital technologies already offer some methods to overcome this impairment. However, to be efficient, these methods have to be based on a careful matching with each user's colour vision. Although the traditional Ishihara and Farnsworth tests are relatively easy to use, computerized tests seem to simplify administration and recording of colour vision tests. This work presents a new computerized colour vision test, aiming to give access to subject's detection thresholds for colour in the whole gamut of a given display device, and the preliminary works for its validation. Subjects with normal colour vision and subjects with congenital anomalous trichromacy underwent this test, and first results indicate a fairly good sensibility.

INTRODUCTION

In a world where digital images are becoming a substantial part of our environment, persons with colour vision deficiencies are prone to some extent to encounter difficulties in their everyday life or professional activities. Colour discrimination appears as an essential cue for object detection and recognition.

Colour vision tests such as Ishihara plates or Farnsworth D 15 Hue tests are usually performed by eye care practitioners during classical orthoptic assessment [START_REF] French | The evolution of colour vision testing[END_REF]. They may be sometimes administered in order to determine ability to carry out a specific occupational task. These tests have been widely studied for a long time and their reliability, reproducibility and accuracy have been proved. But for the majority of them, no quantitative data about the perception of colour contrasts are available.

This study describes a new test to give access to precise knowledge about these perception thresholds of the colour contrasts. This test is not designed for screening for congenital blindness, but the kind of information it can deliver is an important part in the design of electronic assistive devices for visually impaired or colour blind people, as they make it possible to adapt the images displayed on these devices with user visual functionalities. This paper is organized as follows: in Section 2 we present a brief review of color blindness mechanisms, and the main types of tests for the blindness assessment are listed. Afterwards, in Section 3, our method and the experimental setup are described, and results are discussed in Section 4.

COLOUR VISION BLINDNESS

Colour vision mechanisms

According to World Health Organization (WHO), about 10% of the whole human population present some kinds of colour vision deficiencies, leading to the inability to distinguish characteristic sets of colours. Colour perception mechanisms start within the eye, when light absorbed by the photoreceptors is transduced to a nervous signal, this signal is then processed by the inner layers of retina, and transmitted to the visual cortex for the final steps of processing. So deficiencies may arise from photoreceptor defects, optic nerve lesion or damage occurred in the brain.

In human retina, three types of photoreceptors are involved in colour vision, depending on their maximum spectral sensitivity: S cones (short wavelengths), M cones (medium wavelengths) and L cones (long wavelengths). Colour impairment levels vary from total colour blindness (mono-chromacy) to a slight degradation (but not a loss) of discrimination ability. Congenital colour vision deficiencies are linked to missing or abnormal cones. Dichromacy occurs when one type of cones is missing: protanopia, deuteranopia and tritanopia correspond respectively to the lack of L, M and S cones. Anomalous trichromacy is subdivided into protanomaly, deuteranomaly and tritanomaly, which is much less likely to occur. Recent studies [START_REF] Broacks | What do the Color-Blind see[END_REF] suggest that a protanomaly is characterized by L cone lack along with two varieties of M cone pigments with slightly different peak sensitivity wavelengths, and in the same way, deuteranomaly is characterized by M cone lack along with two varieties of L cone pigments. Colour blindness may also be a consequence of diseases, stroke or trauma: for example, glaucoma is responsible for a tritan like vision [START_REF] Bessler | Silent Substitution Stimulation of S-cone Pathway and L-and M-cone Pathway in Glaucoma[END_REF]. In this case of acquired deficiencies, colour vision deficiencies could get worse, or possibly improve over time, while it remains stable in case of inherited colour blindness.

Colour blindness assessment tests

Colour vision testing are intended to response to different objectives: initial diagnosis of color blindness, monitoring of some disease evolution, or ability to perform tasks, involved in particular professional areas. The so-called lamp methods are devoted to this later objective: they are still in use in armed forces (for instance in Canada and India) or for railway employee recruitment [START_REF] Dain | Color Vision and the Railways: Part 1. The Railway LED Lantern Test[END_REF]. The well-known Ishihara plates belong to the pseudo isochromatic plate families [START_REF] Ishihara | Tests for colour-blindness[END_REF].

These tests are intended to identify the subjects with modifications on the cones M and L, without indicating their severity, and are not adapted to the detection of problems related to the cones S. Other tests in the same family have been designed specifically for children, or for tritan type deficiencies. The third type of tests are named arrangement tests, among which the most commonly used has been developed by Farnsworth: its principle relies on the subject's ability to classify coloured pawns according to their chromatic proximity [START_REF] Farnsworth | The Farnsworth-Munsell 100-hue and dichotomous tests for color vision[END_REF]. The number and nature of errors in arranging the colour order make it possible to discriminate problems related to L, M or S cones and may give a quantitative assessment of their gravity [START_REF] Bowman | A method for quantitative scoring of the Farnsworth Panel D-15[END_REF]. Pseudo isochromatic and arrangement tests are liable to encounter the same limitations: their major issue is their dependence to lighting condition, reliable results can only be obtained with standardized light sources. Furthermore, handling with sweaty fingers may degrade the color of the plates or the pawns. And finally, result interpretation is largely based on the investigator's experience. The fourth type of tests are spectral tests, based on anomaloscopes: the subject is asked to realize metameric matches of a target coloured light spot (for instance yellow light) by mixing two monochromatic lights (for instance red and green light). Although very precise, this later type of test use is restricted to scientific research [START_REF] Walsh | A Performance Comparison of Color Vision Tests for Military Screening[END_REF], be-cause of the high cost and need for extensive training of examiner.

Computerized tests are often presented as a promising alternative to classical tests: They appear to be less expensive, allowing for automated treatment and storage and therefore less time consuming [START_REF] Campbell | iPad colour vision apps for dyschromatopsia screening[END_REF]. They can also be administered remotely, online: a variety of web sites propose digital versions of classical tests as Ishihara plates, or Farnsworth pawns. In a more rigorous approach, computerized tests have been validates by clinical trials and comparison with classical tests. For instance, Nakamura and al [START_REF] Nakamura | New color vision tests to evaluate faulty color recognition[END_REF] have designed pseudoisochromatic plates, Shin et al [START_REF] Shin | A new color vision test to differentiate congenital and acquired color vision deficiencies[END_REF] have developed a new kind of arrangement test.

However, these tests clearly presents some limitations, mainly due to equipment: in order to be reliable and to give precise and reproducible results, they must necessarily be administrated with computer monitors whose colorimetric characteristics are perfectly known, thanks to a precise colorimetric calibration process.

COLOUR CONTRAST DETECTION THRESHOLD TEST

A wide variety of electronic assistive devices, based on judiciously processed coloured images may be thought in order to help people with congenital or acquired colour vision impairment in their everyday life or professional activities. Whatever these assistive devices are, it appears interesting to be able to match displayed image characteristics with visual abilities of the user.

The notions of cones fundamentals (i.e imaginary stimuli that excite only one cone family) and dichromatic confusion lines are extensively used in colour vision research, as they are known to account for anomalous trichromacy behavior as well. All the colours that are indistinguishable for one type of dichromate people among the three are lying along lines of confusion through the chromaticity diagram. For a given dichromacy type, all the lines converge to a so-called "copunctal point", that stands for the chromaticity coordinates of the missing cone primary.

Our present work therefore concerns colour discrimination threshold measurement, for a range of colours distributed on the whole CIE31 chromaticity diagram along dichromate confusion lines.

Method

Subjects are seated at 70cm in front of a computer display. Stimuli are composed of two concentric coloured square patches, on a grey neutral background. Luminance of stimuli and background are maintained at a 20 cd/m² throughout the test. This level is beyond the detection range of rod photoreceptors, making it unlikely that rods participate in perception. Central and peripheral patches occupy respectively 3,3° and 8,2° in the subject's field of view. At the beginning, the two patches exhibit the same color. Then, the inner patch colour is modified step by step around the initial point, its chromatic coordinates always belonging to a given confusion line. Increment has been sized to be slightly lower than Mac Adams ellipse radius. Subjects are asked to say as soon as they detect a difference between the two patches. Afterwards, another initial colour is set for the two patches, and process goes on, until the whole confusion line within the monitor gamut has been ranged. Three confusion lines (corresponding to protanopia, deuteranopia and tritanopia) are successively studied. Copunctal point coordinates are derived from Wyszecki & Stiles [START_REF] Wyszecki | Color science[END_REF].

Materials

For the test purpose, processing and dedicated interfaces (see figure 1) have been developed under MATLAB.

Stimuli are displayed on an LCD Philips 200CS monitor, which has beforehand undergone colorimetric characterization. Spectral radiance measurements have been collected using a Konica Minolta CS-2000 spectroradiometer, enabling for the determination of monitor gamut and a precise conversion between the device dependent RGB colour space and the device independent CIE XYZ colour space. Although the XYZ color space and the xy plane are known to be not perceptually uniform, this method has been chosen because it leads to convenient comparison with data found in literature, as it will be seen later.

Results are summarized in figure 2.

The averaged thresholds are not significantly different for each of the three axes and for all the NCV subjects. DCV subjects perform poorer colour discrimination, with DT values two to three times higher. However, this discrimination exists, for points far away enough along the line. This result is not surprising: confusion line equations have been derived from experimental data gathered from metameric matches of 1° stimuli. For larger stimuli, when parafoveal vision is activated, it is known that confusions do no longer occur [START_REF] Broacks | What do the Color-Blind see[END_REF]. When looking at the relationship between color (e.g. chromatic coordinates in the CIE xy diagram) and threshold values, results are in agreement with color perception theory. The figure 3 shows the averaged DT for four points along the tritan axis within monitor gamut, whom coordinates are given in table1: distance is increasing when measures are performed from the bottom left to the top right, in agreement with increasing size of Macadam ellipses from blue to green. Increase is greater for DCV subjects, and this result is also in agreement with colour perception theory: in case of anomalous trichromaticity, L and M cone wavelength sensitivity present a greater overlap than in normal vision, leading to a smaller difference between the L and M cone excitations in the medium and long wavelength parts the spectrum.

CONCLUSION

A precise knowledge of the colour contrast detection threshold exhibited by a given colour vision deficient person may permit the development of efficient assistive digital interfaces. This paper presents a new method for contrast detection threshold measurement, and the very first results of its implementation. For a complete validation, further investigation on a larger population, and comparison with data gathered from tests already in use in clinical vision assessment are needed. However, the first results are in good agreement with well-established notions in colour vision theories.

Figure 1 :

 1 Figure 1: examiner interface, showing confusion lines and monitor gamut. Symbol * indicates xy coordinates around whom variations of colour is performed, according to the position of the slider at the bottom

Figure 2 :

 2 Figure 2: DT values for three confusion lines. NCV: averaged DT value for normal colour vision subjects (error bar: standard deviation). DCV1 to DCV3: DT values for the three subjects with deficient colour vision.

Table 1 :Figure 3 :

 13 Figure 3: DT variations for four measurement points xiyi along tritan axis. NCV: averaged DT value for normal colour vision subjects. DCV1 to DCV3: DT values for the three subjects with deficient colour vision


Daniel Diep
email: daniel.diep@mines-ales.fr

Isabelle Marc

Herve Nonon

Jonathan Delhom

Frederic Roure

Isabelle Marc Lgi2p

Hervé Nonon

Smartphones for automatic shad counting

Keywords: acoustic recognition, classification, smartphone application, shad, Alosa fallax rhodanensis I

à 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.

INTRODUCTION

Twaite shad (Alosa fallax) is a migratory fish living primarily in the seas which goes up the rivers to breed in spring. In Europe, this species has considerably declined in the midth of the 20th century due to overfishing, pollution and obstacles to migration, and for this reason is now given considerable legal protection [START_REF]LIFE-Projekt Maifisch -The re-introduction of allis shad (Alosa alosa) in the Rhine System[END_REF]. Monitoring the numbers of shads at their reproduction sites is an important indicator for measuring interannual changes in their population. In particular, it enables the evaluation of the effectiveness of structures such as sluices and fish passes, created to facilitate their annual upstream migration [START_REF]Les Aloses (Alosa alosa et Alosa fallax spp[END_REF], [START_REF] Lebel | Bilan des actions du plan migrateurs concernant l'alose feinte (Alosa fallax rhodanensis) sur le bassin Rhône-Méditerranée-Corse[END_REF].

Shad reproduce at night near to the surface of the water, turning quickly and noisily at the time of egg-laying and emitting a characteristic sound lasting a few seconds known as a "spawning splash". The currently used measuring method is manual counting, from the river bank, by an observer who listens and counts the splashes. This manual counting method is highly restrictive and costly in terms of human resources.

Recently, thanks to technological advances in the field of multimedia, particularly audio media, counting devices using microphones and portable recorders have been set up; however, they still require considerable intervention on the part of the operator (installation, monitoring of the equipment, deinstallation, listening to the recordings, spawning recognition and counting), and clearly it misses a device that would automatically recognize spawning splashes.

In this paper, we present the design of a new equipment achieved to automatically count the spawning acts of shads (SSAs) at a reproduction site. The central part of this equipment is a smartphone, and this configuration enables to -A smartphone (Samsung GT B3750) used as an audio recorder. A sampling rate as high as 44100 Hz was chosen for testing conditions. An additional preamp was inserted in order to adapt the signal level to the smartphone microphone input.

Acoustic signals generated by shad spawning acts (SSA) are particularly difficult to analyze and discriminate: they occur at random in a noisy environment populated by a lot of nocturnal animals; their statistics are non-stationary and the sounds may notably differ from one to the other. Physically, their modeling is complex, although some numerical simulation models [START_REF] Van Den | Physically-based models for liquid sounds[END_REF] are able to reproduce them.

Non-stationarity of the phenomenon within a time window of a few seconds is clearly visible on the spectrogram of Figure 1: the figure a) shows a SSA signal starting at time 0.8 s, and the figure b) represents the evolution of power spectrum versus time. To be valid, a SSA should last at least 2 s.

III. AUTOMATIC DETECTION

The method of automatic detection of shad spawning acts comprises an extraction of signal features, followed by an algorithm of classication.

A. Sound database

Our database was built from 272 records segmented manually, and is composed of sounds sampled at 44.1 KHz, having a duration between 3 s and 10 s, and variable amplitudes (Table I). Figure 2b summarizes the sequence of operations composing a MFCC calculation. After signal conditioning with a preemphasis filter and a Hamming window, the FFT spectral transform is filtered by a bank of 42 filters distributed along a Mel scale, filter centers ranging from 133 to 6854 Hz. Finally, 13 MFCC parameters are extracted after a log transform and a DCT (Discrete Cosine Transform). We used for MFCC the same time slices of 93 ms as previously.

Figure 3 shows the features extracted from the SSA signal presented in Figure 1. The energy per frame has been

C. Classification

Classification of signals in the acoustic domain is covered by a set of statistical methods such as HMM (hidden Markov models), NN (artificial neural networks) or SVM (support vector machines), which are supervised learning models (see e.g. [START_REF] Temko | Classification of acoustic events using SVM-based clustering schemes[END_REF] or [START_REF] Dufaux | Automatic sound detection and recognition for noisy environment[END_REF] for an overview). However, the nonstationarity of SSA signals indicates that extracted features evolve in a discontinuous way, feature vectors of a given shape seeming to occur intermittently.

Therefore, instant features corresponding to time slices cannot be all tagged as being parts of a SSA signal. This particularity led us to orientate the detection issue toward unsupervised learning, i.e. clustering methods. In this domain, the GMM is a well understood statistical model, and presents the advantage of computational efficiency.

The Gaussian Mixture Model (GMM) clustering method assumes that the data are generated by a mixture of probability distributions in which each component represents a different cluster.

Consider a set of N points (x 1 , …, x N ) in d to be clustered into K groups. The data is seen as N observations of a ddimensional random vector with density: [START_REF]LIFE-Projekt Maifisch -The re-introduction of allis shad (Alosa alosa) in the Rhine System[END_REF] where k Φ is the density of the normal distribution N(μ k ,Σ k ) and α k the weight of this component in the mixture ( )

1 α = k .
The parameter θ=(α,μ,Σ) defines the model, with α=(α 1 ,...,α K ), μ=(μ 1 ,...,μ Κ ) and Σ=(Σ 1 ,...,Σ Κ ).

The Expectation-Maximization or EM algorithm gives a means to estimate the parameters of the model. If we denote ( | )

k i p x θ ω
the posterior probability that given x i the point belongs to cluster ω k , the EM algorithm maximizes the likelihood function of θ :

(2)

Once clusters have been determined using the EM algorithm, it is simple to obtain a classifier. We associate to each class (SSA /non SSA) the combination of clusters, i.e. the mixture of gaussian densities that provides the highest score, the score being the sum of true positives and false negatives in the confusion matrix. Results of this classification with a varying number of clusters N c are reported in Table II, where the score has been expressed as a percentage. We observe that the scores do not progress when N c > 5. Spectrum coefficients give generally better results than MFCCs.

As an example, Table III shows the confusion matrix obtained with N c =5 and spectrum coefficients on the test set. For this classification, we obtained a True Positive Rate (TPR) of 78.6% and a False Positive Rate (FPR) of 10.0%.

IV. A SMARTPHONE IMPLEMENTATION

The target device of our application is a Samsung GT-B7350 Omnia Pro4, a cell phone running Windows Mobile 6.5. According to our needs, we developed the following software functions:

Handheld recorder: this application aims at recording and storing all sounds of interest, and to build a database for further analyses. The recorder makes use of the audio functionality of the cell phone and its facility of storage on a SDHC memory card. To reduce the volume of data, a suppression of blanks has also been included.

SSA detection and counting: this function implements the automatic detection presented in Section 3, based on spectrum

Φ = α Φ K k k k x x 1 ( | ) ( ) K k k i p x L θ ω θ = = ∏ coefficients, c) MFCCs 1 ( ) ( ) θ =
coefficients and GMM classification. To meet real-time constraints, a particular attention has been given to the programming code, e.g. by using an integer-FFT algorithm. Furthermore, some thresholds have been added on signal energy and detection trigger in order to adapt the method to online detection. Task scheduler: this component gives the ability to launch applications at pre-defined times. It allows the smartphone to remain on standby during day and to start recorder and detection at night.

Remote control via GSM and SMS: this communication tool has been designed to drive the smartphone and monitor it remotely, in case of difficulty to access the device. When solicited, the smartphone sends reports about its state, SSA counts, etc.

Autonomy: a 32 Gb SD card and a 9600 mAh battery enables the device to work in autonomy during one week. To further improve energetic autonomy, a photovoltaic sensor has been added to the equipment. On the other hand, data storage capability can be improved by using a sound compression technique.

V. RESULTS

A prototype version of the smartphone was installed in spring 2012 on a shad spawning ground located on river Cèze, in the south of France. The smartphone equipped with a parabolic microphone and a battery was mounted on a tripod placed on the bank, at a height of 3 meters.

Unitary tests of the prototype have allowed to validate the choices of configuration. More specifically, audio files have been recorded and used to evaluate the method of SSA detection. They cover 10 days were sounds have been continuously recorded between 11pm and 3 am, giving 40 hours of recordings.

At first, a training set has been selected with 40 files of 5 minutes, each of them containing at least a manually identified SSA. The classifier was then trained by extracting the spectrum coefficients and applying GMM; the on-line detection algorithm of the smartphone was then applied to the data, giving the following results. The results obtained seem fully consistent with the true positive and false positive rates obtained previously.

Another evaluation consisted to apply the automatic detection to the whole data set, and compare the result with those recorded by manual counting. Over the total period, 74 SSAs were counted automatically and 55 manually, giving an excess of 35%.

VI. CONCLUSION

The work presented in this paper describes a new application which aims at estimating and monitoring the populations of shads on their reproduction sites. Smartphones offer today a real opportunity to integrate signal processing and pattern recognition techniques. They help reduce the development time by providing increasingly integrated audio solutions, and provide in the same time new means of communication.

Although some progress has still to be made to improve the efficiency of shad counting, the first tests provide very promising results, and let consider the deployment of this new field device during the next season of reproduction.

B

  . Feature extraction Signals were analyzed in the time-frequency domain in order to extract pertinent features. In this work we considered spectrum coefficients calculated on successive time frames of 4096 samples, i.e. 93 ms, corresponding to the granularity observed in Fig.1, and related to the flapping frequency of caudal fins in water. For each frame, the energy of the signal within the frame has been computed using FFT (Fast Fourier Transform) along 10 spectral bands covering a frequency range from 100 Hz to 5000 Hz on a logarithmic scale. The 10 triangular filters are centered on 100, 274, 485, 742, 1055, 1435, 1899, 2426, 3149, 3984 Hz. The calculation of spectrum coefficients is straightforward, made by a multiplication in the Fourier domain. The operations are schematized in Figure2a. Physically, spectrum coefficients would represent the spectral decomposition of intermittent splashes occuring during a SSA.In comparison, MFCC (Mel Frequency Cepstrum Coefficients) need more computation steps. MFCCs are features commonly used to process acoustic signals such as speech or music. In association with HMM (Hidden Markov Models), they have proven to constitute an efficient tool in the domain of speech recognition[START_REF] Rabiner | Theory and applications of digital speech processing[END_REF].

Fig. 1 :Fig. 2 :

 12 Fig. 1: a SSA signal a) vs. time, b) time -frequency spectrogram

Fig. 3 :

 3 Fig. 3: Features extracted from a SSA signal, a) energy, b) spectrum coefficients, c) MFCCs

  Number of identified SSAs : 48Correctly detected: 39 (81%) Not detected: 9 (19%) False detections: 4 (8%) Contrary to the results presented in Section 3, the on-line version of the detector does not need a segmentation of signals, and the True Negative case of the confusion matrix is therefore irrelevant.

TABLE I

 I Non SSA signals are all types of environmental sounds encountered during the recordings, such as frog calls, dog barkings, wind, train passings or human voices. Some sounds overlapping with SSAs were also included in the test set.

			. SOUND DATABASE
	data set	SSA	non SSA	total
	training	56	80	136
	test	56	80	136

TABLE II .

 II PERCENTAGES OF CORRECT CLASSIFICATION

		spectrum coef.	MFCCs
	nb. clusters	training	test	training	test
	2	67.6	60.3	75.0	70.6
	3	86.8	77.9	71.3	69.9
	4	89.7	82.4	83.8	77.9
	5	91.2	85.3	88.2	89.7
	6	86.8	77.9	83.8	72.1
	7	88.2	84.6	89.0	79.4
	8	89.0	82.4	87.5	77.9
	9	90.4	88.2	85.3	75.0

TABLE III

 III

	. SAMPLE CONFUSION MATRIX
		detected	detected as
		as SSA	non SSA
	actual SSA	44	12
	actual non SSA	8	72