Yimi Wang
email: yimi.wang@cea.fr

Kangming Li
email: kangming.li@cea.fr

Frédéric Soisson
email: frederic.soisson@cea.fr

Charlotte S Becquart
email: charlotte.becquart@univ-lille.fr

Combining DFT and CALPHAD for the development of on-lattice interaction models: the case of Fe-Ni system

Chapter I Thermodynamic properties in fcc Fe-Ni alloys

This first chapter presents the thermodynamic part of our pair interaction model (PIM), which will be extended in the following chapters to deal with the formation and migration and vacancies, and diffusion and ordering phenomena.

Introduction

Atomistic simulations of the kinetics of diffusive phase transformations (phase separation or ordering) in metallic alloys require interaction models that realistically describe the thermodynamic and diffusion properties of these materials. When diffusion occurs via vacancies -the most common casethis means accurately describing how the concentration and the jump frequencies of vacancies depend on the local atomic environment. In recent years, much progress has been made in this area, largely due to ab initio computational methods. However, to simulate kinetics of thermally activated processes, the models must also be sufficiently light. Pair interaction models on rigid lattices, used in atomistic kinetic Monte Carlo (AKMC) simulations, are in this respect a good compromise between realism and numerical cost, and have proven to be an efficient tool to simulate coherent phase transformation kinetics.

Pairwise interactions can be fitted to ab initio calculations at 0 K of alloying properties (typically the enthalpies of formation of ordered or disordered structures) and / or of point defects (enthalpies of formation, migration barriers). It is, however, much more difficult to determine high temperature properties from first principle methods. Calculations of vibrational entropies are possible, but expensive, and are still mostly limited to pure metals or dilute alloys (in concentrated alloys, the number of atomic configurations to consider becomes a problem). In iron-based materials, magnetic effects are an additional difficulty for ab initio calculations: in paramagnetic materials, such as austenitic steels, magnetic disorder is added to chemical disorder. In spite of recent progress, mainly concerning pure metals [START_REF] Körmann | Free energy of bcc iron: Integrated ab initio derivation of vibrational, electronic, and magnetic contributions[END_REF], it is difficult to predict the effect of a magnetic transition on, for instance, the stability of a phase, the enthalpy of formation or migration of a vacancy, a diffusion coefficient. Magnetic interaction models, based only on Density Functional Theory (DFT) calculations, have been proposed recently (for example at the SRMP, with the work of Chu-Chun Fu and coworkers [START_REF] Schneider | Atomic Diffusion in α -iron across the Curie Point: An Efficient and Transferable Ab Initio -Based Modeling Approach[END_REF][START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF], or the Magnetic Cluster Expansion developed at Culham [START_REF] Lavrentiev | Magnetic cluster expansion simulations of FeCr alloys[END_REF]. However, these models are still too computationally costly to be used directly in AKMC simulations of precipitation or ordering kinetics, because one has to equilibrate the magnetic moments, in addition to the atomic configurations. Under these conditions, it is interesting and necessary to directly adjust the relevant parameters of the simulations on experimental data.

The idea is not new of course, but we propose here to apply it to Fe-Ni alloys with the face-centered cubic structure, in a more systematic way than in previous studies, in particular by establishing the link between our model and the CALPHAD method (both with the formalism used in CALPHAD, and the data available for Fe-Ni alloys). Fe-Ni alloys are the base of many industrial alloys. At the SRMP, their modeling is an essential step before dealing with the Fe-Cr-Ni ternary alloys (the model system for austenitic steels) and the simulation of irradiation effects in these alloys. Fe-Ni alloys have interesting magnetic properties: the γ solid solutions, stable at high temperature, are paramagnetic, but Ni rich alloys become ferromagnetic at low temperatures. Because of their industrial importance, there are many experimental studies concerning both their thermodynamic and diffusion properties (the latter in particular have been studied more extensively than those of many other alloys). Nevertheless, as shown in Figure 1, some questions remain open: as always, the phase diagram is not well known at low temperatures, when the diffusion becomes too slow to bring the system to its equilibrium state. A FeNi-L10 phase has been observed in meteorites or under irradiation, and ab initio calculations suggest that it is stable at low temperatures, but its domain of existence is debated. They are few atomistic modeling of its diffusion properties and of the kinetics of ordering, and they often used empirical interatomic potentials whose predictive power is imperfect. [START_REF] Vernyhora | Thermodynamics of f.c.c. Ni-Fe Alloys in a Static Applied Magnetic Field[END_REF] The manuscript is organized as follows: we model this system with a rigid lattice pair interaction model and AKMC simulations. The pair interactions depend on the local composition and temperature, which allows us to reproduce asymmetric mixing properties, and non-configurational entropic contributions (the entropy of configuration is treated by Monte Carlo simulation). The thermodynamic properties of Fe-Ni alloys are discussed in Chapter Ⅰ. The interatomic pair interactions that control them are fitted to DFT calculations at T = 0 K of formation enthalpies in ordered and disordered structures, performed by Kangming Li during his thesis (Li 2021). The high-temperature properties are fitted to a recent CALPHAD study [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]. This allows for an accurate description of the enthalpy and Gibbs free energy of the phases, in particular of their magnetic contribution -which are treated separately in this CALPHAD study. We build the Fe-Ni phase diagram, with Monte Carlo simulations performed in the semi-grand canonical ensemble. In particular, we study the effect of the magnetic contribution on the relative stability of the γ solid solution, the L10 phase and the L12 phase.

The diffusion properties are the subject of Chapter Ⅱ. The pair interaction model is completed to include the diffusion of vacancies and the corresponding parameters are fitted to experimental measurements of tracer diffusion coefficients in dilute and concentrated solid solutions. AKMC simulations of diffusion phenomena require the knowledge of the equilibrium vacancy concentration, and how it depends on the temperature, the alloy composition and the state of order. We find that previously used techniques, developed for systems with an unmixing tendency, are inefficient in the case of Fe-Ni, an alloy with an ordering tendency. A new method is thus proposed, based on the Widom insertion technique. We then use AKMC simulations to evaluate the interdiffusion coefficients in disordered solid solutions, and to test the Darken equation which relates the tracer coefficients, the interdiffusion coefficients, and the thermodynamic factor. Finally, we study the diffusion in the ordered FeNi3-L12 phase, for which no experimental data are available.

In the last chapter, we model the ordering kinetics in the FeNi3 alloy, with the same interaction model and AKMC simulations. We compare our results with three experimental studies that provide measurements of the time evolution of the long-range order parameter and of the size of the ordered domains in alloys of the same composition. We conclude by considering possible improvements to our model, and possible applications to other systems. 𝑔 𝑉𝑗 (𝑛) : interactions between the vacancy 𝑉 and atoms 𝑗 on the lattice (nth nearest neighbor)

List of acronyms

𝑔 𝐴𝑘 (𝑛) : interactions between the atom 𝐴 and other atoms 𝑘 on lattice (nth nearest neighbor)

𝐺 𝑉 𝑚𝑖𝑔 , 𝐻 𝑉 𝑚𝑖𝑔 , 𝑆 𝑉 𝑚𝑖𝑔 : Gibbs free energy, enthalpy and entropy of vacancy migration 𝐺 𝑉 𝑓𝑜𝑟 , 𝐻 𝑉 𝑓𝑜𝑟 , 𝑆 𝑉 𝑓𝑜𝑟 : Gibbs free energy, enthalpy and entropy of vacancy formation 𝐺 𝑉 𝐹𝑒 𝑓𝑜𝑟 , 𝐺 𝑉 𝑁𝑖 𝑓𝑜𝑟 : Gibbs free energy of vacancy formation in ordered phases: a vacancy replace an atom Fe or

Ni on its sub lattice 𝐺 𝑆-𝑉 𝑏𝑖𝑛 : Gibbs free binding energy between a solute and a vacancy (𝐺 𝑆-𝑉 𝑏𝑖𝑛 > 0 for attractive interactions)

𝐺 𝑉 𝑖 : Gibbs free energy of the system with a vacancy on site 𝑖 𝐺 0 : Gibbs free energy of the system without vacancy 𝐺 𝛾 , 𝐺 𝑟𝑒𝑓 𝛾 , 𝐺 𝑒𝑥 𝛾 , 𝐺 𝑚𝑎𝑔 𝛾 , 𝐺 𝑖𝑑 𝛾 : the total Gibbs free energy of a system and its different contribution part (reference part, excess part, magnetic part and ideal part)

𝐻 0 𝑚𝑖𝑔 , 𝐻

Abstract

We present a model of pair interactions on rigid lattice to study the thermodynamic properties of iron-nickel alloys. The pair interactions are fitted at 0 K on ab initio calculations of formation enthalpies of ordered and disordered (special quasi-random) structures. They are also systematically fitted on the Gibbs free energy of the γ Fe-Ni solid solution as described in a CALPHAD (CALculation of PHAse Diagrams) study by Cacciamani et al. This allows the effects of finite temperature, especially those of magnetic transitions, to be accurately described. We show that the ab initio and CALPHAD data for the γ solid solution and for the FeNi 3 -L1 2 ordered phase can be well reproduced, in a large domain of composition and temperature, using first and second neighbor pair interactions which depend on temperature and local alloy composition. The procedure makes it possible to distinguish and separately compare magnetic, chemical and configuration enthalpies and entropies. We discuss the remaining differences between the pair interaction model and CAL-PHAD, which are mainly due to the treatment of the short-range order and configurational entropy of the solid solution. The FCC phase diagram of the Fe-Ni system is determined by Monte Carlo simulations in the semi-grand canonical ensemble and is compared with experimental studies and other models. We especially discuss the stability of the FeNi-L1 0 phase at low temperature.

I. INTRODUCTION

Atomistic Monte Carlo simulations of phase transformation kinetics in metallic alloys (precipitation, ordering, etc. [1-13]) require interaction models that allow a precise description of the thermodynamic and kinetic properties of the materials, while remaining simple enough to model systems of a few million atoms over long periods of time (their evolution being controlled by thermally activated diffusion mechanisms). Models using effective interactions on rigid lattices -although limited to coherent problems -are among the most widely used and have become more reliable since they are systematically fitted to ab initio calculations of materials properties at 0 K (such as the formation energies of ordered phases or special quasi-random structures, point defect formation and migration energies, etc.) [START_REF] Soisson | Cu-precipitation kinetics in α-Fe from atomistic simulations: Vacancytrapping effects and Cu-cluster mobility[END_REF][START_REF] Ngayam-Happy | Formation and evolution of MnNi clusters in neutron irradiated dilute fe alloys modelled by a first principle-based AKMC method[END_REF][START_REF] Becquart | Monte carlo simulations of precipitation under irradiation[END_REF].

Evaluating non-zero temperature effects from ab initio methods is more difficult: calculations of vibration entropy, for example, are in principle possible but are computationally expensive and are usually limited to simple systems (e.g. pure metals, perfectly ordered phases, or dilute alloys). Modeling the effect of magnetic transitions and magnetic disorder -especially important in iron based alloys -is also very challenging. Alternatively, the temperature dependence of pair interactions can be adjusted on experimental data, for example on phase diagrams, but these adjustments are often made on a case-by-case basis. We propose here a new approach to systematically fit a pair interaction model both on ab initio calculations at 0 K and, for high temperatures, on a CALPHAD-type model. CALPHAD models provide an accurate description of the Gibbs free energies of the different phases of an alloy, based on empirical thermodynamic models fitted (mainly) on large numbers of experimental measurements. They also provide a specific description of some important contributions (e.g. magnetic contributions in iron-based alloys). The objective of this paper is to show how to establish a term-to-term correspondence between the empirical models used in CALPHAD and the effective interactions of a lattice model; to show the improvements that this brings to the description of a particular alloy; but also to discuss the limits of such a correspondence.

FIG. 1. The phase diagram of the Fe-Ni system (from [START_REF] Vernyhora | Thermodynamics of f.c.c. Ni-Fe Alloys in a Static Applied Magnetic Field[END_REF] and [START_REF] Yang | A revision of the Fe-Ni phase diagram at low temperatures (< 400 • c)[END_REF])

We apply this approach to Fe-Ni alloys with a face-centered cubic structure (FCC).

Recent ab initio [START_REF] Mishin | Phase stability in the Fe-Ni system: In-vestigation by first-principles calculations and atomistic simulations[END_REF][START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF] and CALPHAD [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF][START_REF] Ohnuma | Experimental determination and thermodynamic evaluation of low-temperature phase equilibria in the Fe-Ni binary system[END_REF] studies are available for this system. The thermodynamic properties of Fe-Ni alloys have been much studied, because of their industrial interest and because they still raise unresolved questions. The phase diagram of the Fe-Ni system (Fig. 1) is well known at temperatures above 400 • C [START_REF] Swartzendruber | The Fe-Ni (iron-nickel) system[END_REF]: the FCC solid solution γ is stable over the whole composition range and over a wide temperature range. The body centered cubic (BCC) solid solution α is stable below 912 • C, and only with nickel contents below approximately 10%. A two-phase domain α -γ is observed in iron-rich alloys, and an ordered FeNi 3 phase (with the L1 2 structure) is formed below 516 • C.

At lower temperatures, the phase diagram is -as usual -more difficult to establish, because of slow diffusion processes. However, irradiation experiments [START_REF] Paulevé | Une nouvelle transition ordre-désordre dans Fe-Ni (50-50 )[END_REF][START_REF] Chamberod | Electron irradiation effects on iron-nickel invar alloys[END_REF][START_REF] Reuter | Ordering in the Fe-Ni system under electron irradiation[END_REF], observations of meteorites [START_REF] Yang | A revision of the Fe-Ni phase diagram at low temperatures (< 400 • c)[END_REF][START_REF] Reuter | Determination of the Fe-Ni phase diagram below 400 • C[END_REF], and ab initio studies [START_REF] Mishin | Phase stability in the Fe-Ni system: In-vestigation by first-principles calculations and atomistic simulations[END_REF][START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF] suggest that other ordered FCC phases may be stable or metastable (notably the FeNi phase of L1 0 structure and the Fe 3 Ni phase of L1 2 structure). Many CALPHAD studies have sought to complement the Fe-Ni phase

diagram by extrapolating at low temperatures the empirical thermodynamic models fitted to experiments at high temperatures. This is especially difficult in the case of Fe-Ni, and as a result, the proposed phase diagrams show significant differences [START_REF] Yang | A revision of the Fe-Ni phase diagram at low temperatures (< 400 • c)[END_REF][START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF][START_REF] Swartzendruber | The Fe-Ni (iron-nickel) system[END_REF][START_REF] Xiong | Magnetic phase diagram of the Fe-Ni system[END_REF]. A particular difficulty of this system is that the experimental data are obtained essentially in paramagnetic phases, whereas the ordered phases are ferromagnetic. Magnetic contributions are indeed taken into account in CALPHAD approaches, but again by empirical models using experimental data and still under discussion [START_REF] Xiong | Magnetic phase diagram of the Fe-Ni system[END_REF].

Atomistic models combining the information from experiments and first principle methods may provide additional insight on these issues. A few rigid lattice interaction models, fitted to ab initio calculations, have been developed for Fe-Ni alloys: Mohri et al. [START_REF] Mohri | First-principles calculation of L1 0 -disorder phase equilibria for Fe-Ni system[END_REF] proposed a cluster expansion to study the stability of the ordered compound FeNi-L1 0 in the framework of a CVM approximation: they found an ordering temperature of 483 K (taking into account the vibration entropy, which lowers it by about 40 K). But they did not study the FeNi 3 phase, nor the effect of the ferro-to-paramagnetic transition. Effective interactions models including an explicit description of the magnetic moments have been proposed for Fe-Ni alloys, using Ising [START_REF] Lawrence | Chemical and magnetic interactions in FCC Fe-Ni alloys using the cluster variation method[END_REF][START_REF] Taylor | Magnetic and compositional order in nickel-rich Ni c Fe 1-c alloys[END_REF][START_REF] Dang | Simultaneous magnetic and chemical order-disorder phenomena in Fe 3 Ni, FeNi, and FeNi 3[END_REF][START_REF] Vernyhora | Monte carlo investigation of the correlation between magnetic and chemical ordering in nife alloys[END_REF] or Heisenberg [START_REF] Vernyhora | Monte carlo investigation of the correlation between magnetic and chemical ordering in nife alloys[END_REF][START_REF] Taylor | Monte carlo simulations of an fcc Ni c Fe 1-c alloy with vector magnetic freedom[END_REF] models for the magnetic interactions and parameters fitted to the experimental transition temperatures. Similar magnetic models have been also used in phase-field simulations of ordering and precipitation of the FeNi 3 phase [START_REF] Vernyhora | Atomic density function modeling of microstructure evolution in Ni 3-x Fe x alloys[END_REF].

More recently Lavrentiev, Wrobel et al. [33,[START_REF] Wróbel | Phase stability of ternary fcc and bcc Fe-Cr-Ni alloys[END_REF] The outline of this article is as follows: section II is devoted to the thermodynamic models of FCC phases in the Fe-Ni system. We briefly recall the Gibbs free energy models used in the CALPHAD approaches, focusing on the recent study of Cacciamani et al. [18] (II A);

then the available ab initio results on the properties of Fe-Ni alloys (in particular those of K. ) (II B). We then show how to reproduce these results with the PIM in section (II C). In section III, we use Monte Carlo simulations in the semi-canonical grand ensemble, to measure the Gibbs free energies of the FCC alloys, and to build the FCC phase diagram.

II. THERMODYNAMIC MODELS

Our PIM is built using both a CALPHAD study and ab initio calculations. We therefore recall the main information provided by these two approaches before to explain how it can be reproduced with effective interactions on a rigid lattice.

A. CALPHAD Models

Several CALPHAD-type studies have been proposed for the Fe-Ni system: the most recent are those of Cacciamani et al. [18] and Ohnuma et al. [START_REF] Ohnuma | Experimental determination and thermodynamic evaluation of low-temperature phase equilibria in the Fe-Ni binary system[END_REF] (for older ones, see the reviews in refs. [START_REF] Yang | A revision of the Fe-Ni phase diagram at low temperatures (< 400 • c)[END_REF][START_REF] Swartzendruber | The Fe-Ni (iron-nickel) system[END_REF]). Within the CALPHAD framework, a Gibbs free energy model can be defined for each of the phases to be considered (e.g. in the Fe-Ni system, the α and γ solid solutions and the different ordered phases). This gives great flexibility to fit the parameters to the experimental data. We will fit our PIM parameters to the study by Cacciamani et al. [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF] (which will be hereafter simply referred to as the "CALPHAD model" or even as "CALPHAD"), but the following presentation and procedure could easily be adapted to other CALPHAD studies.

The γ solid solution

The Gibbs free energy per atom of the solid solution γ (FCC) Fe 1-x Ni x is:

G γ = G γ ref + G γ ex + G γ mag + G γ id (1)
This is the total Gibbs free energy, including the entropy of configuration. (Note that in CALPHAD one rather uses molar energies, in J.mol -1 . We convert them in energies per atom, in eV).

G γ ref is the non-magnetic contribution of pure metals:

G γ ref = (1 -x)G γ F e -xG γ N i (2)
where G γ F e and G γ N i are the Gibbs free energy of pure Fe and pure Ni. G γ ex is the excess Gibbs free energy, written as a sum of Redlich-Kister polynomials:

G γ ex = x(1 -x) i L γ i (T )(1 -2x) i (3)
(from i = 0 to 2 in [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]) G γ mag is the magnetic contribution:

G γ mag = -k B T f (τ ) ln(β(x) + 1) (4)
τ = T /T c (x)
, where T c (x) is the Curie temperature, β(x) the average magnetic moment of the γ solid solution and f (τ ) is a polynomial function of the reduced temperature. T c (x) and β(x)

are also given by Redlich-Kister polynomials of the composition x, fitted to experimental measurements. Different expressions and values have been proposed [START_REF] Xiong | Magnetic phase diagram of the Fe-Ni system[END_REF], those used by Cacciamani et al. are given in [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF].

G γ id corresponds to an ideal entropy of configuration:

G γ id = -T S γ id = k B T [x ln x + (1 -x) ln(1 -x)] (5)
i.e. to the configuration entropy of a perfectly disordered solid solution with no short-range order. Note that the excess term (eq. 3) may include a non-ideal configurational part, but it is not identified as such.

Finally, the Gibbs free energy of mixing is:

G γ mix = G γ -(1 -x)G γ F e -xG γ N i (6)
where

G γ F e = G γ F e + G γ mag (x = 0) and G γ N i = G γ N i + G γ mag (x = 1
) are the total Gibbs free energies of the pure metals, including the magnetic part.

Ordered Phases

In the study by Cacciamani et al. [18], the ordered phases FeNi 3 (L1 2 ), Fe 3 Ni (L1 2 ) and FeNi (L1 0 ) are modeled within the framework of the Compound energy formalism (CEF) with 4 sublattices. An additional term is added to the Gibbs free energy of the γ phase, which depends on the distribution of species on the different sublattices and on interaction energies (limited to the first nearest neighbors) fitted to ab initio calculations of the formation enthalpy of the perfectly ordered phase.

In the present work, we will use more detailed ab initio studies, involving both ordered and disordered configurations and summarized in the following section.

B. Ab initio calculations

Density functional theory method

In this work, the 0 K formation enthalpies of Fe-Ni ordered and disodered phases are fitted to those computed in the ab initio study of Ref. [START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF]. The essential computational points are presented in the following.

The ab initio calculations were performed using density functional theory (DFT) with the projector augmented wave method [START_REF] Blöchl | Projector augmented-wave method[END_REF][START_REF] Kresse | From ultrasoft pseudopotentials to the projector augmented-wave method[END_REF] as implemented in the VASP (Vienna Ab-initio Simulation Package) [START_REF] Kresse | Ab initio molecular dynamics for liquid metals[END_REF][START_REF] Kresse | Efficiency of ab-initio total energy calculations for metals and semiconductors using a plane-wave basis set[END_REF][START_REF] Kresse | Efficient iterative schemes for ab initio total-energy calculations using a plane-wave basis set[END_REF]. The generalized gradient approximation for the exchangecorrelation functional in the Perdew-Burke-Ernzerhof parametrization [START_REF] Perdew | Generalized Gradient Approximation Made Simple[END_REF] was employed.

All the calculations are spin-polarized. 3d and 4s electrons were considered as valence electrons. The plane-wave basis cutoff was set to 400 eV. The Methfessel-Paxton broadening scheme with a smearing width of 0.1 eV was used [41]. The k-point grids were generated according to the cell size to achieve a k-sampling equivalent to a cubic unit cell with a 16 × 16 × 16 shifted grid following the Monkhorst-Pack scheme [START_REF] Monkhorst | Special points for Brillouin-zone integrations[END_REF]. The zero-point energies, which can be significant for light elements, have been calculated in the ordered phases. Their contribution to the mixing enthalpies is very small (typically 0.001 to 0.003 eV) and has been neglected in the following. FCC solid solutions were represented by Special Quasirandom Structures (SQS) [START_REF] Zunger | Special quasirandom structures[END_REF] minimizing Warren-Cowley short-range order parameters [START_REF] Cowley | An approximate theory of order in alloys[END_REF][START_REF] Martinez | Simulations of Decomposition Kinetics of Fe-Cr Solid Solutions during Thermal Aging[END_REF], with 128-atom and 108-atom supercells for anti-ferromagnetic-double-layer and ferromagnetic phases, respectively. [START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF]).

Ordered Phases

Ordered phase H f or (eV/atom)

Fe 7 Ni-cI32 0.039273 Fe 3 Ni-L1 2 -0.01636 Fe 3 Ni-Z1 -0.04414 Fe 2 Ni-C11 f -0.06991 FeNi-L1 1 -0.04040 FeNi-L1 0 -0.10797 Fe 2 Ni-C11 f -0.08064 FeNi 2 -L1 2 -0.10879 FeNi 7 -cI32 -0.04541
The formation enthalpies of nine ordered structures on the FCC lattice, calculated by Kangming et al. [START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF] using the DFT method presented in the previous section, are given in Table I and Fig. 2. The ordered structures are ferromagnetic, except for Fe 7 Ni-cI32 which is ferrimagnetic. The formation enthalpies are defined with the anti-ferromagnetic FCC iron and the ferromagnetic FCC nickel as reference states. The trends are the same as in a previous study by Mishin et al. [16]: only the FeNi-L1 0 and FeNi 3 -L1 2 phases are located on the convex hull (Fig. 2) and must therefore be stable at low temperature on the FCC lattice. However the FeNi 7 -cI32 and Fe 2 Ni-C11 f phases are close to the stability limit.

Special Quasi-Random Structures

The formation enthalpies H f or of special quasi-random structures (SQS) of different compositions have also been calculated in the same study with different magnetic states. These structures are representative of random solid solutions. They were generated by standard methods, with a minimization of Warren-Cowley short-range order parameters. They are described in Ref. [START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF], with a detailed analysis of their volume and magnetic moments. We just recall here the energetic results used for the PIM parametrization.

The most stable SQS are found to be double-layered anti-ferromagnetic for x < 0.184 and ferromagnetic for x > 0.184 (red circles in Fig. 3). One observes an asymmetrical evolution of H f or with the composition, as already predicted in the study by Cacciamani et al., with mainly negative values (i.e. a tendency to order) and a minimum in the vicinity of the composition of the FeNi 3 phase. However, the SQS values are significantly larger than the CALPHAD ones (Fig. 3), and are even slightly positive for x < 0.20 (as already obtained by Sansa et al. [46], using a tight-binding approach).

C. Pair interaction model

We propose to reproduce the properties of Fe-Ni alloys with a model of concentrationand temperature-dependent pair interactions on a perfect FCC lattice. FIG. 3. Formations enthalpies at 0 K of quasi-random FCC structures (DFT calculations [START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF]) and FCC solid solutions (CALPHAD [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF], with separate magnetic and excess contributions).

alloys [START_REF] Senninger | Atomistic simulations of the decomposition kinetics in Fe-Cr alloys: Influence of magnetism[END_REF][START_REF] Levesque | Simple concentrationdependent pair interaction model for large-scale simulations of fe-cr alloys[END_REF], which had however not been fitted systematically on a CALPHAD model, and did not explicitly distinguish a magnetic contribution.

The essential assumption of the PIM is that the Gibbs free energy (per atom) of a given configuration, i.e. a given distribution of n F e Fe atoms and

n N i Ni atoms (N = n F e + n N i )
on the FCC lattice, can be written as a sum of interactions g

(n)

ij (x, T ) between pairs of i and j atoms on n th neighboring sites:

G conf = 1 N ij g (n) ij (x, T ) (7)
The pair interactions depend on the temperature, T , and the Ni concentration, x (we will omit these dependencies in the following, to simplify the notations). The dependence on concentration is required to reproduce an asymmetric evolution of formation enthalpies, as observed in Fig. 2 and Fig. 3. The dependence on temperature describes the entropic contributions (electronic, vibrational and magnetic) other than the configuration entropy, so the g

(n)
ij are indeed "pair Gibbs free energies" and can be written as: g

(n) ij = h (n) ij -T s (n) ij [48].
To facilitate the comparison with CALPHAD models, each interaction g

(n)

ij is written as a sum of a non-magnetic (nm) and a magnetic term (mag). G conf is therefore the sum of: [START_REF] Martínez | Decomposition kinetics of Fe-Cr solid solutions during thermal aging[END_REF] which accounts for the chemical and vibrational contributions, and of:

G nm conf = 1 N ij g nm(n) ij (
G mag conf = 1 N ij g mag(n) ij (9)
The total Gibbs free energy of the alloy is: [START_REF] Liebscher | A hierarchical microstructure due to chemical ordering in the bcc lattice: Early stages of formation in a ferritic Fe-Al-Cr-Ni-Ti alloy[END_REF] where S conf is the entropy of configuration, which will be evaluated from Monte Carlo simulations.

G = G conf -T S conf
The Gibbs free energy of mixing G mix is:

G mix = G -(1 -x)G F e -xG N i (11)
G F e is the Gibbs free energy of pure iron, on the same FCC lattice:

G F e = G nm F e + G mag F e (12)
= n z n 2 (g nm(n) F eF e + g mag(n) F eF e ) (13)
= n z n 2 g (n) F eF e (14)
where z n is the coordination number for the nearest neighbors n. The same expressions apply to pure nickel.

1. High temperatures: fitting of the pair interactions to CALPHAD

In the PIM, the Gibbs free energy of mixing of a perfectly disordered solid solution is:

G mix (x, T ) = x(1 -x) n z n v n (x, T ) -T S id (15)
where the ordering parameters v n are defined as:

v n (x, T ) = g (n) F eN i - 1 2 g (n) F eF e + g (n) N iN i (16)
To reproduce the properties of the CALPHAD model, we identify the non-magnetic part of the ordering parameters (in eq. 15) to the excess Gibbs free energy of CALPHAD (eq. 3):

n v nm n (x, T ) = G γ ex x(1 -x) (17)
and their magnetic part to the magnetic Gibbs free energy of CALPHAD (eq. 4):

n v mag n (x, T ) = G γ mag x(1 -x) (18)
The fitting of the PIM parameters on CALPHAD can be summarized to the equations 16-18. It is worth to notice that it is based on an approximation: eq. 15 is exact only for a disordered solid solution, i.e. at sufficiently high temperatures. In the real solid solution, a short-range order may exist, and the configuration entropy does not reduced to an ideal term.

Low temperatures: fitting of the pair interactions to ab initio calculations

To reproduce the properties of a solid solution at 0 K, the same method can be used by fitting v n (x, T ) on the formation enthalpies of SQS calculated by DFT. The magnetic part remains fitted to the magnetic model of CALPHAD (eq. 18) and the non-magnetic part is fitted so that the sum of the magnetic and non-magnetic contributions of the PIM reproduces the DFT formation enthalpies. A good agreement is obtained with a Redlich-Kister polynomial of order 5 (instead of 2 for G γ ex in ref. [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]), as shown in Fig. 4 (with respectively the magnetic part, the non-magnetic part and the total mixing enthalpies of the PIM). The fitting coefficients L DF T j of the polynomial are given in Table II, corresponding to the orange curve in Fig. 4. As mentioned above, this gives mixing enthalpies above those predicted by CALPHAD at 0 K. This fit of pair interactions (using eq. 15) to the CALPHAD model or to the formation enthalpies of SQS, only involves the sum of v n and can be done with any range n of interac-

L DF T 0 L DF T 1 L DF T 2 L DF T 3 L DF T 4 L DF T

5

-0.03696 0.09631 -0.04722 0.21141 -0.04752 -0.18389

tions. However, it is well known that in FCC ordered structures, the phase diagram and in particular the order-disorder temperatures, depend strongly on the ratio = v 2 /v 1 [START_REF] Gahn | Ordering in face-centered cubic binary crystals confined to nearest-neighbour interactions-monte carlo calculations[END_REF]. The fit is also independent of the choice of the composition x (local or global) in eq. 18 and 17.

Therefore, it does not provide a very accurate description of the interatomic interactions in Fe-Ni alloys.

To get a better description of the thermodynamic properties, the range of interactions and the ratio between the interactions at different distances is fitted to the DFT calculations of the formation enthalpies of the ordered phases (Table I).

With only first nearest neighbors (nn) interactions (Fig. 5(a)), the PIM model underestimates the formation enthalpies of the ordered phases, especially those of the FeNi-L1 0 and FeNi 3 -L1 2 phases.

A better result (Fig. 5(b)) is obtained with first and second nn pair interactions, taking a constant ratio α = v 2 /v 1 = -0.7 (the agreement is very sensitive to the value of α, except in the range α ∈ [-0.6, -0.7]). The formation enthalpies obtained with first and second nn interactions are close to those of the DFT calculations, in particular for the two stable phases FeNi-L1 0 and FeNi 3 -L1 2 (Fig. 5). The least well reproduced is that of the iron-rich cI32 phase, which is unstable.

Note that in this fitting procedure, the local composition around a Fe-Ni pair has been defined as the average Ni atomic fraction around the first and second nearest neighbors of the pair, excluding the two atoms which compose it. An atom neighboring the two atoms of the pair is counted twice (so that with 12 first and 6 nearest neighbors, a pair is surrounded by 0 to 34 Ni atoms). Using this definition, the formation enthalpies of the different ordered phases are the functions of v 2 and v 1 given in Table III, together with the values of the local composition x 1 and x 2 around the first and second nn Fe-Ni pairs. Other definitions of the local composition are possible and we have tested some of them (taking into account the two atoms of the pairs, or counting each surrounding Ni atom only once). The definition chosen here gives a slightly better fit, although the differences are small.

We did not obtain significantly better results by introducing third and fourth nn interactions. In the following we will therefore restrict to the PIM with first and second nn interactions and α = -0.7, corresponding to the results shown in Fig. 4 and 5(b).

Transition between parameters at high and low temperatures

The final PIM will therefore use first and second neighbor pair interactions with a constant α = v 2 /v 1 = -0.7 ratio:

-At 0 K the variations of v 2 and v 1 with the composition are fitted to the formation enthalpies of SQS and ordered structures calculated by Li and Fu [START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF] using DFT methods,

8 3 v 1 + 2 3 v 2 FeNi-L1 1 1 2 1 2 3v 1 + 3v 2 FeNi-L1 0 1 2 0 4v 1 FeNi 2 -C11 f 11 17 11 17 8 3 v 1 + 2 3 v 2 FeNi 3 -L1 2 25 34 1 3v 1 FeNi 7 -cI32 33 34 29 34 3 2 v 1 + 3 4 v 2
as summarized in fig. 5 and4;

-At high temperatures the variations of v 2 and v 1 with the composition are fitted to the Gibbs free energies of the γ solid solution, from the CALPHAD study of Cacciamani et al. [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF];

These two sets of parameters differ only in the non-magnetic part of parameters v 2 and v 1 , which is described by Redlich-Kister polynomials having different order and coefficients (respectively L DF T i and L CALP HAD i

). The final parameters are obtained by using coefficients

L P IM i (T )
which evolves gradually from one to the other according to:

L P IM j (T ) = exp - T T j L DF T j + 1 -exp - T T j L CALP HAD j (T ) (19)
with T j = 400 K for j = 0, 1, 2 and T j = 80 K for j = 3, 4, 5. The example of coefficient

L P IM 2
is given in fig. 6. The temperatures T j have been chosen so as to give the Gibbs free energy of CALPHAD for T > 1000 K, at temperatures where it is derived from numerous and reliable experimental data. The influence of these transition temperatures on the phase diagram will be discussed later.

Let us note finally that the Gibbs free energy of mixing of the solid solution, the formation enthalpies of of SQS or ordered structures and the FCC phase diagram, depend only on the parameters v n . We have chosen to take g (n)

F eF e and g (n)

N iN i interactions independent of the concentration, but dependent on the temperature and adjusted to the free enthalpies of the pure metals (eq. 12). Only the g (n)

F eN i interactions are dependent on local concentration. This choice does not affect the results of the present study, but it allows a better description of the properties of point and diffusion defects [START_REF] Senninger | Atomistic simulations of the decomposition kinetics in Fe-Cr alloys: Influence of magnetism[END_REF], which we will address in future work. It also makes the Monte Carlo simulations slightly less time consuming.

III. MONTE CARLO SIMULATIONS

We will now use Monte Carlo simulations to measure the Gibbs free energies of the PIM (including the configuration entropy) at different compositions and temperatures and build the FCC phase diagram.

A. Semi-grand canonical isotherms

The equilibrium properties of the PIM are determined by Monte Carlo simulations carried out in the semi-grand canonical ensemble. In general, we use a system of N = 4 × 16 3 atoms, with periodic boundary conditions. Exchanges are tried between a randomly chosen atom of the system and an atom taken in a reservoir, with a given difference of chemical potential ∆µ = µ N i -µ F e . By changing ∆µ, one modifies the equilibrium concentration (Fig. 8 and9). A total of 500 increments for a interval of 1 eV in ∆µ are used to go from pure iron to pure nickel, and then 500 increments to go the other way. For each value of ∆µ, 5 × 10 6 attempts of atomic exchange (or Monte Carlo steps, MCS) are performed before measuring the equilibrium composition and order parameters. For building the phase diagram of Fig. 16, a larger system of N = 4 × 24 3 atoms and much smaller increments of ∆µ are used, in order to get a better precision (up to 100 increments for a interval of 0.04 in ∆µ).

To identify the different ordered phases, the FCC lattice is divided into 4 simple cubic sublattices, shifted by a distance a/2 in the x, y, z directions [START_REF] Inden | Atomic ordering[END_REF]. We measure the Ni concentration on each sublattice, and the long range order parameter defined as:

η = 1 4 4 i=1 x i x -1 (20)
where x i is the Ni concentration on the sublattice i. With this definition, η = 1 in the perfect FeNi-L1 0 structure and η = 0.5 in the perfect FeNi 3 -L1 2 structure.

The short range order is characterized by the Warren-Cowley parameters for the first and second nearest neighbors:

σ i = 1 - f (i) N i x (21)
where

f (i)
N i is the average fraction of Ni atoms among the i th nn of the Fe atoms. For a perfect L1 2 ordered phase, σ 1 = -0.33 and σ 2 = +1. in FeNi 3 as a function of T (Monte Carlo simulations and experiments by Kozlov et al. [51]).

The evolution of the long-range η (eq. 20), and of the short range order parameters σ 1 and σ 2 (eq. [START_REF] Paulevé | Une nouvelle transition ordre-désordre dans Fe-Ni (50-50 )[END_REF], in an alloy of composition FeNi 3 as a function of the temperature, is shown in Fig. 7. The L1 2 ordered phase is found to be stable up to 765 K (instead of 790 K for CALPHAD [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]). The evolution of η is in good agreement with the experiments of Kozlov et al. [START_REF] Kozlov | Order-discharge transition in ni 3 fe alloy[END_REF] (which gives a slightly higher ordering temperature: 807 K). The discontinuity at the order/disorder temperature indicates a first-order transition. A significant short range order remains well above the disordering temperature, especially between the first nearest neighbors.

Two examples of isotherms x = f (∆µ), at T = 1000 K and T = 600 K are shown in

B. Gibbs free energy of mixing

Using the definition of chemical potentials: µ i = (∂G/∂n i ) T,P,n j and integrating the ∆µ(x) curve, we obtain the Gibbs free energy of mixing G mix . The results obtained at different temperatures can be directly compared with the G mix of the CALPHAD study [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]. In each case, one can also compare separately, the enthalpy H mix and entropy S mix of mixing, as well as the magnetic, excess and configuration contributions.

∆µ = µ N i -µ F e .
worth to notice that the excess contribution is slightly lower in the PIM, with a minimum of -0.038 eV at x = 0.69 instead of -0.033 eV in CALPHAD. The difference is due to the fact that in the PIM, some short range order remains in the γ solid solution, even at this high temperature (Fig. 7).

-0. The effect is clearer if one separates the enthalpic and entropic contributions of G mix = H mix -T S mix . The enthalpy of mixing (Fig. 11) is dominated by the excess term. Due to the remaining short range order (σ 1 = -0.066 and σ 2 = 0.048), the PIM gives a minimum of H mix = -0.054 eV at x = 0.67 instead of -0.047 eV in CALPHAD. This discrepancy on H mix only disappear at very high temperature. At 3000 K (i.e. above the liquidus), the maximum difference between the PIM and CALPHAD is still ∆H mix = 0.0022 eV (for σ 1 = 0.06 and σ 2 = 0.013). It becomes negligible only above 5000 K.

The entropy of mixing of the the PIM (Fig. 12) is dominated by the entropy of configuration, which is very close to the ideal S id of CALPHAD. In the PIM as in CALPHAD, the excess and magnetic contributions to the entropy of mixing are less important, and negative (except below x 0.1). At x = 0.67, the discrepancy due to the short range order is only S conf -S id = 0.15 × 10 -5 eV/(atom•K), which corresponds to a difference of T (S conf -S id ) = -0.00225 eV/atom. The difference on S mix partly compensate the one on H mix , which explain the good agreement on G mix between the PIM and CALPHAD, even below 1500 K, when the short range order increases. The G mix of the PIM and CALPHAD are therefore in very good agreement in the whole range of composition and temperature where the γ solid solution is stable (Fig. 13).

- The difference between the Gibbs free energy of mixing of the PIM and CALPHAD [Fig. 14(a)] slightly increases at lower temperatures, when the ordered phase L1 2 stabilizes, i.e. when the long-range order parameter η is close to 0.5, between x 0.6 and x 0.82

[Fig. 14(b)]. This is not surprising since the PIM parameters at low temperatures are not fitted on CALPHAD, but on DFT calculations which give a different energetic landscape, especially for the disordered phase (section II C). In spite of this difference, the G mix (x)

curve of the PIM is still in good agreement with CALPHAD at T = 700 K.

At 650 K, the agreement between the Gibbs free energy of mixing of the PIM and CAL-PHAD is still quite good for the compositions where the γ solid solution and the L1 are stable [Fig. 15(a)]. However the evolution of the long-range parameter η as a function of the nickel concentration x now displays two bumps [Fig. 15(b)]. The second one (between x 0.63 and x 0.85) still corresponds to the L1 2 phase. The phase is almost perfectly ordered for the stoichiometric composition FeNi 3 (η 0.5 for x = 0.75)). However, the PIM predicts that the FeNi-L1 0 phase is stable between x 0.52 and 0.63, while it only appears at lower temperature (below 316 K) according to the CALPHAD model [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]. Note that it is not perfectly ordered (η 0.6 instead of 1 for the perfect order), because it is slightly non-stoichiometric (x > 0.5) and because 650 K is close to its order disorder-temperature (680 K).

C. FCC phase diagram

The FCC phase diagram of the Fe-Ni system predicted by the PIM is shown in Fig. 16 and compared with experimental data [START_REF] Josso | Equilibrium Diagram for the Order to Disorder Transformation of Iron-Nickels near Ni 3 Fe[END_REF][START_REF] Heumann | Karbonylverfahren und aufdampfverfahren zur bestimmung von phasengleichgewichten im temperaturbereich geringer beweglichkeit am beispiel der eisennickel-legierungen[END_REF][START_REF] Van Deen | Phase diagram of the order-disorder transition in ni3fe[END_REF][START_REF] Woude | Phase diagram of the order-disorder transition in ni3fe[END_REF]. It can be also compared to the FCC diagram of Cacciamani et al. (Fig. 8 in [18]).

The ordering temperatures of the FeNi 3 -L1 2 phase are slightly different: the PIM gives 765 K, a little lower than 790 K for the CALPHAD study of Cacciamani et al. [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF] (which is adjusted to the experimental value). As in CALPHAD, the limits of the two-phase domain FeNi 3 +γ for x > 0.75, are slightly shifted towards lower values (∆x -0.04 at 571 K) by comparison with the experiments by Heumann et al. [START_REF] Heumann | Karbonylverfahren und aufdampfverfahren zur bestimmung von phasengleichgewichten im temperaturbereich geringer beweglichkeit am beispiel der eisennickel-legierungen[END_REF]. And as in CALPHAD, the width of the two-phase domain γ+FeNi 3 at x < 0.75 is smaller than the one of the two-phase domain at x > 0.75 (earlier CALPHAD studies predicted a larger two-phase field, as in Fig. 1). Here it should be noted that, except in the vicinity of the ordering temperature at x = 0.75, the experimental data for these two-phase fields are going back to 1963 [START_REF] Heumann | Karbonylverfahren und aufdampfverfahren zur bestimmung von phasengleichgewichten im temperaturbereich geringer beweglichkeit am beispiel der eisennickel-legierungen[END_REF] and are only available for one temperature.

The discrepancy is more important for the FeNi-L1 0 phase: the PIM predicts an ordering temperature of 680 K instead of 316 K for CALPHAD. The CVM study of Mohri et al. [26] predicts an intermediate ordering temperature (483 K). A direct estimation by DFT calculations (taking into account the vibrational entropy, but not the configurational entropy of the FeNi-L1 0 phase) gives 640 K. There is no precise experimental measurement available for the evolution of the degree of order as a function of the temperature (as for the FeNi 3 phase, in Fig. 7), but the experimental observations under electron irradiation by Reuter et al. [START_REF] Reuter | Ordering in the Fe-Ni system under electron irradiation[END_REF] suggest an ordering temperature of approx. 593 K.

As in CALPHAD, the FCC phase diagram of the PIM also displays a two-phase field with an equilibrium between a ferromagnetic (γ f ) and a paramagnetic (γ p ) solid solution, at x = 0.4 and below T = 660 K. This phase separation has been first predicted by Chuang et al. [START_REF] Chuang | Magnetic contributions to the thermodynamic functions of alloys and the phase equilibria of Fe-Ni system below 1200 K[END_REF], but it has not been confirmed experimentally. The PIM is able to reproduce this two-phase field because it includes the magnetic contribution of the CALPHAD model. It is however more limited in temperatures than in ref. [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF], because of the higher stability of the FeNi-L1 0 phase in the PIM, which limits its extension below 680 K.

The phase diagram of the PIM without the magnetic contribution is shown in Fig. 17.

As in the study by Cacciamani et al., the non-magnetic phase diagram reduces the critical temperature of L1 2 by approx. 118 K and shows no γ f -γ p two-phase field.

Finally, let us recall that the parameters of the PIM and the results of section III have been obtained with parameters fitted to DFT calculations at 0 K, the CALPHAD data of ref. [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF] at high temperatures, and a transition between the high and low temperature regimes controlled by the exponential interpolation of Eq. 17. With the chosen T j temperatures, the Redlich-Kister coefficients of the PIM are almost identical to those of CALPHAD above approximately 1000 K. To assess the effect of this choice on the phase diagram, we have performed some simulations with a different set of paramaters: T j = 50 K for j = 0, 1, 2 and T j = 10 K for j = 3, 4, 5 (using the same notation as in II C 3). With these parameters, the Redlich-Kister coefficients of the PIM becomes almost identical to those of CALPHAD at a lower temperature (approx. 315 K). The resulting phase diagram is shown in Fig. 18.

It is not very different from the previous one (Fig. 16(a)), except from a moderate increase of the ordering temperatures of the L1 0 and L1 2 phases. The reason is that with the new parameters, the ordering tendency is a little more pronounced below 1000 K (Fig. 4). As a consequence, the γ p -γ f two-phase domain almost completely disappears.

IV. DISCUSSIONS AND CONCLUSIONS

We have presented in this study a method for developing a pair interaction model for Fe-Ni alloys, fitted at 0 K on the enthalpies of formation of ordered and disordered structures (computed by first-principle methods) and at high temperatures on the Gibbs free energy of the γ solid solution (as given by a CALPHAD study and its underlying experimental data).

Thanks to the temperature and concentration dependence of the pair interactions, the PIM is able to reproduce precisely these two types of energetic properties, and to distinguish between excess, magnetic and configurational entropic contributions. The identification between CALPHAD and the PIM is not perfect, because the configurational entropy and short-range order in the solid solution is described more approximately in CALPHAD methods than in the Monte Carlo simulations used to determine the equilibrium properties of the PIM. The CALPHAD method is more flexible than an atomistic model: the properties of each phase can be adjusted very precisely and independently on the experiments. The PIM imposes some constraints but, combined with Monte Carlo simulations, it ensures a consistent description of the short-and long-range order and of the entropy of configuration.

Despite these differences, the Gibbs free enthalpies of mixing of the γ solid solution, as

given by CALPHAD, are very well reproduced by the PIM, throughout the composition and temperature range of stability of the phase. The Gibbs free enthalpy of the FeNi 3 -L1 2 phase is also well reproduce, although the parameters of the PIM and CALPHAD for that phase are not fitted on the same DFT calculations. At high temperatures, the FCC phase diagram involves only these two phases, and both methods give similar results (especially for the order-disorder transition in the vicinity of FeNi 3 , and for the γ p -γ f phase separation).

On the other hand, both models predict that the FeNi-L1 0 phase is stable at low temperature, but with different ordering temperatures. Taking the electron irradiation experiments [START_REF] Reuter | Ordering in the Fe-Ni system under electron irradiation[END_REF] as a reference, it seems that CALPHAD underestimates the ordering temperature, while the PIM overestimates it. It should be noted here that alloys under irradiation may be not fully at equilibrium, so that no real experimental thermodynamic data are available for this phase; and that both the CALPHAD model and the PIM are only fitted to DFT calculations of enthalpies of formation at 0 K. The PIM and CALPHAD should therefore be both improved to properly describe this phase. One possibility is to use DFT methods to compute finite temperature contributions. These methods are computationally expansive, but can separate each energetic contribution and provide results at intermediate temperature (say, between 0 and 400 • C), where experimental results are rarer and perhaps less accurate. An example is given in the study of K. Li and C.-C. Fu [START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF], which shows that the vibrational entropy decreases the ordering temperatures of the L1 0 and L1 2 phases by respectively 280 and 200 K.

In spite of this limitation, the PIM model gives a satisfactory description of the γ solid solution and of the L1 2 phase. Of course, such a model is necessarily dependent on the CALPHAD data it uses. For the magnetic contribution for example, we rely on the recent study by Cacciamani et al., which is itself based on a large experimental database (with measurements of specific heats, Curie temperatures, magnetic moments, etc. of γ solid solutions with various compositions, described in ref. [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF][START_REF] Cacciamani | Critical evaluation of the fe-ni, fe-ti and fe-ni-ti alloy systems[END_REF]). For the same reason, our model takes into account, but cannot distinguish between energetic contributions that are not identified separately in CALPHAD (for example, harmonic and non-harmonic vibrational contributions -which are both gathered in the excess Gibbs free energy, together with the non-ideal part of the configurational entropy). However it could easily be updated to take into account future improvement on that points. It could also be easily extended to Fe-Ni-Cr ternary alloys or to other binary or ternary systems. Finally the PIM is simple enough to be used as a basis for Atomistic Kinetic Monte Carlo methods to simulate the kinetics of homogeneous ordering or of heterogeneous precipitation of the L1 2 phase; or to model the interdiffusion properties in the γ solid solution at high temperature. It is indeed not more numerically expensive than the PIM for BCC Fe-Cr alloys described in Ref. [START_REF] Levesque | Simple concentrationdependent pair interaction model for large-scale simulations of fe-cr alloys[END_REF],

which has been used for the simulations of precipitation kinetics [START_REF] Martínez | Decomposition kinetics of Fe-Cr solid solutions during thermal aging[END_REF][START_REF] Senninger | Atomistic simulations of the decomposition kinetics in Fe-Cr alloys: Influence of magnetism[END_REF] or irradiation effects [START_REF] Senninger | Modeling radiation induced segregation in iron-chromium alloys[END_REF][START_REF] Soisson | Atomistic modeling of precipitation in fe-cr alloys under charged particles and neutron irradiations: Effects of ballistic mixing and sink densities[END_REF]. As in these studies, the modeling of kinetics will require the extension of the PIM to describe the formation and migration properties of point defects.

Chapter II Diffusion properties in fcc Fe-Ni alloys II.1 Introduction

There are many experimental studies on diffusion in the Fe-Ni alloys, which have been reviewed

by Jönsson in 1994 for fcc alloys [START_REF] Jönsson | Mobilities in Fe-Ni alloys : assessment of the mobilities of Fe and Ni in fcc Fe-Ni alloys[END_REF] and in 1995 for bcc alloys [START_REF] Jönsson | Assessment of the Mobilities of Cr, Fe and Ni in bcc Cr-Fe-Ni Alloys[END_REF]. Most of them deal with alloys at relatively high temperatures, in paramagnetic (PM) Fe-Ni solid solutions. They include data on self-diffusion coefficients, impurity diffusion coefficients, and tracer and interdiffusion coefficients in the whole concentration range from Fe to Ni. All these results will be reviewed in II.2.2.

On the other hand, there are almost no experimental data at low temperatures, in ferromagnetic (FM) solid solutions and in ordered structures. As a result, diffusion coefficients at low temperatures are usually obtained by extrapolations from high temperatures, using Arrhenius expressions.

Available modelling studies will be presented in II. In the present study, we have chosen an alternative approach, presented in II.3, combining an effective interaction model and atomistic kinetic Monte Carlo (AKMC) simulations. A similar method has been used in previous studies of bcc Fe-Cr alloys [START_REF] Martínez | Decomposition kinetics of Fe-Cr solid solutions during thermal aging[END_REF][START_REF] Senninger | Atomistic simulations of the decomposition kinetics in Fe-Cr alloys: Influence of magnetism[END_REF]. In this approach, the magnetic moments are not explicitly considered. More complex magnetic interaction modelstaking into explicit account the magnetic moments and with parameters fitted to DFT calculationshave been developed very recently at the SRMP by Fu et al. [START_REF] Schneider | Atomic Diffusion in α -iron across the Curie Point: An Efficient and Transferable Ab Initio -Based Modeling Approach[END_REF][START_REF] Tran | Effective interaction model for coupled magnetism and phase stability in bcc Fe-Co systems[END_REF][START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF]. They have been used to model the effects of magnetism on the diffusion coefficients of bcc Fe-alloys [START_REF] Schneider | Atomic Diffusion in α -iron across the Curie Point: An Efficient and Transferable Ab Initio -Based Modeling Approach[END_REF] or the thermodynamic and point defect properties of fcc Fe-Ni alloys [START_REF] Li | Ground-state properties and lattice-vibration effects of disordered Fe-Ni systems for phase stability predictions[END_REF][START_REF] Li | Effects of magnetic excitations and transitions on vacancy formation: Cases of fcc Fe and Ni compared to bcc Fe[END_REF]. However, dealing with spins makes them too time consuming to model interdiffusion in concentrated alloys, or the kinetics of phase transformations. As in Chapter I, we will therefore use a simpler approach, which only considers the effects of magnetism (but also of the vibration entropy) on the energetic properties of Fe-Ni alloys, through the composition and temperature dependence of effective pair interactions. When possible however, we will use the information provided by the ab initio calculations and the magnetic interaction model developed by Kangming Li for Fe-Ni alloys, for example the work just published by Li [START_REF] Li | Effects of magnetic excitations and transitions on vacancy formation: Cases of fcc Fe and Ni compared to bcc Fe[END_REF].

To do that, it is necessary to extend the thermodynamic PIM of Chapter I into a kinetic pair interaction model (hereafter referred to as k-PIM). The k-PIM describes the vacancy migration barriers and jump frequencies in fcc Fe-Ni alloys, and the way they depend on the local atomic environments.

Self-diffusion and impurity diffusion can then be directly investigated using Le Claire diffusion model

II.2 Diffusion in Fe-Ni alloys: a bibliography II.2.1 Diffusion properties in alloys: general results

Diffusion is the transport of matter from one point to another by motion of atoms or molecules.

A phenomenological description of the diffusion at the macroscopic scale within a continuous medium is obtained by Fick's equations. The first law of Fick connects the flux of a species to its concentration gradient by:

𝑱 = -𝐷𝛁𝐶 (II-1)
where 𝑱 is the species flux, D is the diffusion coefficient and 𝐶(𝑥, 𝑦, 𝑧, 𝑡) is the concentration of the species at position (𝑥, 𝑦, 𝑧) and at the time 𝑡. The equation predicts that the diffusion of a species and its concentration gradient are in opposite directions, revealing that substances have often the tendency to diffuse spontaneously from places of high concentration to where the concentration is lower. As for Fick's second law, it relates the variation of the species concentration over time with the concentration itself by:

𝜕𝐶 𝜕𝑡 = 𝛁 • (𝐷𝛁𝐶) (II-2)
If the diffusion coefficient is independent of the concentration, the equation above simplifies to:

𝜕𝐶 𝜕𝑡 = 𝐷∆C (II-3)
where ∆ denotes the Laplace operator. By solving the diffusion equation, one gets the variation of the concentrations over time and space. Thus the evolution of the concentration profile will depend on the diffusion coefficient, 𝐷.

From an atomistic point of view, one can relate the macroscopic diffusion coefficient 𝐷 to the jump frequencies 𝜔 of the diffusing species, using random walk theories. Einstein's relation states that the diffusion coefficient is related to its mean-square displacement 〈𝑅 2 〉 during the time 𝑡 [START_REF] Mehrer | Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes[END_REF]:

𝐷 = 〈𝑅 2 〉 6𝑡 (II-4)
In the case of a completely random walk, 〈𝑅 2 〉 = 𝑛𝑑 2 , where n is the number of jumps and 𝑑 the jump distance. In a fcc lattice, 𝑛 = 12𝜔𝑡 and 𝑑 = 𝑎 0 /√2, and one simply gets: 𝐷 = 𝑎 0 2 𝜔. However, in real situations, the jump probabilities often depend on the direction of previous jumps. As a result, successive atom jumps are correlated. Therefore, correlation effects must be taken into account, which is one of the main difficulties of diffusion models.

Diffusion occurs through different mechanisms, which vary according to the type of point defects being involved. In Fe-Ni alloys, it is widely accepted that diffusion occurs through the vacancy mechanism. Furthermore, different diffusion coefficients correspond to different situations and will be considered in this study. From the simplest to most complicated, they are respectively the self-diffusion coefficients, the impurity diffusion coefficients, the intrinsic diffusion coefficients. We assign 𝐷 as the general diffusion coefficient, 𝐷 𝐴 𝐵 * as the diffusion coefficient of tracer 𝐵 in matrix 𝐴, 𝐷 𝐴𝐵 𝐵 * the tracer diffusion in the alloy A-B and 𝐷 ̃ the interdiffusion coefficient for a binary system. We will discuss whether these different diffusion coefficients can be analytically calculated by different models.

In general, the temperature dependence of diffusion coefficients is found to be ruled by the Arrhenius formula:

𝐷 = 𝐷 0 exp (- 𝑄 𝑘 𝐵 𝑇 ) (II-5)
where 𝐷 0 denotes the pre-exponential factor and 𝑄 is the activation energy of diffusion. These two quantities are assumed to be constant, at least within some temperature range.

II.2.1.2 Self-diffusion

Beginning with the simplest situation of pure A, the self-diffusion coefficient is related to the Gibbs free energy of the formation (𝐺 𝑉 𝑓𝑜𝑟 ) and migration (𝐺 𝑉 𝑚𝑖𝑔 ) of the vacancies as:

𝐷 𝐴 𝐴 * = 𝑓𝑎 0 2 𝜗 0 exp(- 𝐺 𝑉 𝑓𝑜𝑟 + 𝐺 𝑉 𝑚𝑖𝑔 𝑘 𝐵 𝑇 ) (II-6)
where 𝑎 0 is the lattice parameter, 𝑓 is the correlation factor, and 𝜗 0 is the attempt frequency.

This equation can be also written as:

𝐷 𝐴 𝐴 * = 𝑓𝑎 0 2 𝐶 𝑉 𝑒𝑞 𝜔 (II-7)
where 𝐶 𝑉 𝑒𝑞 = exp (-𝐺 𝑉 𝑓𝑜𝑟 /𝑘 𝐵 𝑇) is the equilibrium vacancy concentration and 𝜔 = 𝜗 0 exp (-𝐺 𝑉 𝑚𝑖𝑔 / 𝑘 𝐵 𝑇) is the jump frequency of the vacancy.

Here a correlation factor 𝑓 is considered. It decreases the diffusion coefficient with respect to its theoretical value in the case of a purely random walk. To better understand its origin, consider the simple case of a vacancy jumps in a fcc lattice. After the exchange of the vacancy with one of its nearest neighboring atom, the vacancy is in a position which permits a reverse jump. The probability of jumping backwards is 1/12. If it occurs, two consecutive jumps are cancelled. Therefore the correlation factor can be estimated to 𝑓 ≈ 1 -2/12 = 0.833 . Cancellations may occur after several jumps (the probability decreases rapidly), so the exact value for the self-diffusion in the fcc structure is 𝑓 = 0.7815 [START_REF] Mehrer | Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes[END_REF].

Since the Gibbs free energy can be separated into two parts, the enthalpy and the entropy, according to 𝐺 = 𝐻 -𝑇S, equation (II-6) can be set as the same form as equation (II-5), with:

𝐷 0 = 𝑓𝑎 0 2 𝜗 0 exp ( 𝑆 𝑉 𝑓𝑜𝑟 + 𝑆 𝑉 𝑚𝑖𝑔 𝑘 𝐵 ) and𝑄 = 𝐻 𝑉 𝑓𝑜𝑟 + 𝐻 𝑉 𝑚𝑖𝑔 (II-8)

II.2.1.3 Diffusion in dilute alloys

Another situation of interest is the diffusion of a solute B in a dilute solvent A. According to Le Claire, one solution is considered to be dilute when (a) all solute atoms are present as isolated atoms or as small grouping of atoms (pairs, triplets etc.), and (b) each isolated atom or group of atoms can diffuse independently (Le Claire 1978). For expressing such diffusion coefficients, there are two other factors to be introduced: the binding energy between solute-vacancy 𝐺 𝐵-𝑉 𝑏𝑖𝑛 (in the following we will use the convention 𝐺 𝐵-𝑉 𝑏𝑖𝑛 > 0 for an attraction between a solute B and a vacancy V) and the migration barrier for the solute-vacancy exchange 𝐺 2,𝑉 𝑚𝑖𝑔 . Considering the diffusion of a tracer B * into a pure metal A, we have:

𝐷 𝐴 𝐵 * = 𝑓 2 𝑎 0 2 𝜗 2 0 exp(- 𝐺 𝑉 𝑓𝑜𝑟 -𝐺 𝑆-𝑉 𝑏𝑖𝑛 𝑘 𝐵 𝑇 ) exp (- 𝐺 2,𝑉 𝑚𝑖𝑔 𝑘 𝐵 𝑇 ) (II-9)
where 𝑓 2 is the correlation factor, which is usually calculated via the five-frequency model by Le Claire

𝜔 𝑖 = 𝜗 𝑖 0 exp (- 𝐺 𝑖 𝑚𝑖𝑔 𝑘 𝐵 𝑇 ) = 𝜗 𝑖 0 exp ( 𝑆 𝑖 𝑚𝑖𝑔 𝑘 𝐵 ) exp (- 𝐻 𝑖 𝑚𝑖𝑔 𝑘 𝐵 𝑇 ) (II-10)
with 𝜗 𝑖 0 the attempt frequency. The impurity diffusion coefficient can be calculated as a function of these five frequencies:

𝐷 𝐴 𝐵 * = 𝑓 2 𝑎 0 2 𝜔 2 exp (- 𝐺 𝑉 𝑓𝑜𝑟 𝑘 𝐵 𝑇 ) 𝜔 4 𝜔 3 (II-11)
An exact expression for 𝑓 2 was derived by Manning:

𝑓 2 = 𝜔 1 + 7𝐹 3 𝜔 1 /2 𝜔 2 + 𝜔 1 + 7𝐹 3 𝜔 1 /2 (II-12)
with 𝐹 3 a function of the ratio 𝛼 = 𝜔 4 /𝜔 [START_REF] Manning | Diffusion Kinetics for Atoms in Crystals[END_REF]. The association and dissociation rates are related to the binding Gibbs free energy (𝐺 𝑆-𝑉 𝑏𝑖𝑛 ) of the vacancy and the solute:

𝜔 4 𝜔 3 = exp ( 𝐺 𝐵-𝑉 𝑏𝑖𝑛 𝑘 𝐵 𝑇 ) (II-13)
Therefore 𝜔 4 /𝜔 3 > 1 corresponds to 𝐺 𝑆-𝑉 𝑏𝑖𝑛 > 0 i.e. to attractive interactions between the solute and the vacancy. On the other hand, if vacancy-solute exchanges occur much faster than vacancy-solvent exchanges, i.e. for 𝜔 2 ≫ 𝜔 1 , 𝜔 3 , …, then we have 𝑓 2 ≪ 1 . The movement of the solute atom is then highly correlated.

For more concentrated, but still dilute alloys (with B atomic fraction limited to a few percent), other analytical models exit. The tracer diffusion coefficients A and B can be expressed as: -14) and:

𝐷 𝐴𝐵 𝐴 * (𝑥 𝐵 ) = 𝐷 𝐴 𝐴 * [1 + 𝑏 1 𝑥 𝐵 + ⋯ ] (II
𝐷 𝐴𝐵 𝐵 * (𝑥 𝐵 ) = 𝐷 𝐵 𝐵 * [1 + 𝐵 1 𝑥 𝐵 + ⋯ ] (II-15)
where the coefficients 𝑏 1 and 𝐵 1 are respectively the factors revealing the change in the jump rates of those atoms A and B influenced by the solute [START_REF] Mehrer | Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes[END_REF]).

The factor 𝑏 1 is sometimes called the linear enhancement factor of solvent diffusion and has attracted particular interest. Taking into account its relation of jump rates in the neighborhood of the solute, the following expression of 𝑏 1 has been proposed by Howard and Manning [START_REF] Howard | Kinetics of Solute-Enhanced Diffusion in Dilute Face-Centered-Cubic Alloys[END_REF]:

𝑏 1 = -18 + 𝜔 4 𝜔 [4 𝜒 1 𝑓 𝜔 1 𝜔 3 + 14 𝜒 2 𝑓 ] (II-16)
where 𝑓 is the self-diffusion correlation factor, all the jump frequencies 𝜔 are those defined in the fivefrequency model (Figure II.1), and the quantities 𝜒 1 and 𝜒 2 are the partial correlation factors given in the article of Howard and Manning [START_REF] Howard | Kinetics of Solute-Enhanced Diffusion in Dilute Face-Centered-Cubic Alloys[END_REF].

Let us point out that these results are valid within the framework of the five-frequency model, i.e. in fcc alloys with solute-vacancy interactions limited to the first-nearest neighbors. If vacancy-atom interactions between further sites are considered, more complicated models are proposed to calculate the correlation factor. For example, Manning calculated the correlation factor of impurity diffusion taking into account the association or the dissociation of the vacancy jump towards its second-nearest neighbor [START_REF] Manning | Correlation Factors for Impurity Diffusion. bcc, Diamond, and fcc Structures[END_REF]. Very recently, the KineClue code has been developed by Schuler et al. [START_REF] Schuler | KineCluE: A kinetic cluster expansion code to compute transport coefficients beyond the dilute limit[END_REF] to account for long-range interactions and for the cases of concentrated alloys.

In general, even in more complicated multiple frequency models, the correlation factor can be still written as the form of [START_REF] Allnatt | Atomic Transport in Solids[END_REF]:

𝑓 2 = 𝐻 2𝜔 𝑠 + 𝐻 (II-17)
where 𝜔 𝑠 is the jump frequency of the solute and the quantity H is independent of the solute jump frequency.

II.2.1.4 Diffusion in concentrated alloys

The interdiffusion of A and B atoms in a binary alloy is characterized by the interdiffusion coefficient, 𝐷 ̃. The interdiffusion coefficient is concentration-dependent and can be expressed in Fick's second law. In one dimension, in the x-direction:

𝜕𝐶 𝜕𝑡 = 𝜕 𝜕𝑥 [𝐷 ̃(𝐶) 𝜕𝐶 𝜕𝑥 ] (II-18)
where 𝐶 is the concentration of component B.

The interdiffusion coefficient can be measured experimentally by the Boltzmann-Matano method. The principle is the following: a binary diffusion couple consisting of two semi-infinite bars is joined at time 𝑡 = 0, with different concentrations at the two sides of the interface. During a diffusion annealing over a time 𝑡, a concentration profile 𝐶(𝑥) develops. The interdiffusion coefficient 𝐷 ̃ for each composition 𝐶 can be directly calculated from one profile 𝐶(𝑥) [START_REF] Matano | On the Relation between Diffusion-Coefficients and Concentrations of Solid Metals[END_REF].

The interdiffusion involves the diffusion of both atom A and atom B. In general they have different intrinsic diffusion coefficients, which are denoted as 𝐷 𝐴 𝐼𝑛𝑡𝑟 and 𝐷 𝐵 𝐼𝑛𝑡𝑟 respectively. These diffusion coefficients can be related to the concentration gradient by the Fick's law:

𝐽 𝐴 = -𝐷 𝐴 𝐼𝑛𝑡𝑟 𝜕𝐶 𝐴 𝜕𝑥 , 𝐽 𝐵 = -𝐷 𝐵 𝐼𝑛𝑡𝑟 𝜕𝐶 𝐵 𝜕𝑥 (II-19)
The intrinsic and interdiffusion coefficients are defined in different reference frames. The intrinsic diffusion coefficients are defined for a flux in the reference of the local crystalline lattice, while the interdiffusion coefficients are defined in the reference of the laboratory.

Diffusion models in concentrated alloys are more rigorously developed within the framework of the thermodynamics of irreversible phenomena (TIP), using Onsager transport equations:

𝑱 = ∑ 𝐿 𝑖𝑗 𝑿 𝑱 𝑛 𝑗=1 (II-20)
with 𝑱 the flux, 𝑳 the matrix of transport coefficients 𝐿 𝑖𝑗 and 𝑿 𝑱 the driving forces causing to the flux 𝑱.

For investigating the diffusion phenomenon, they deal with the real driving forces: the gradient of chemical potentials, namely 𝑿 𝑱 = -∇𝜇 𝑖 . If we consider the diffusion in a binary A-B system by vacancy diffusion, we have: (here we have used the symmetry of the matrix 𝑳, using, what is known as Onsager reciprocity theorem).

𝐽 𝐴 =
If we assume that the vacancies are always maintained at their equilibrium concentration, namely 𝑋 𝑉 = 0, the flux becomes 𝐽 𝐴 = 𝐿 𝐴𝐴 𝑋 𝐴 + 𝐿 𝐴𝐵 𝑋 𝐵 and 𝐽 𝐵 = 𝐿 𝐴𝐵 𝑋 𝐴 + 𝐿 𝐵𝐵 𝑋 𝐵 . For a real solid solution, the chemical potential can be written as:

𝜇 𝑖 = 𝜇 𝑖 0 + 𝑘 𝐵 𝑇𝑙𝑛(𝑥 𝑖 𝛾 𝑖 ) (II-22)
where 𝑥 𝑖 is the atomistic fraction of the element𝑖 and 𝛾 𝑖 the activity coefficient. Thus we have: , where 𝑟 𝐴 and 𝑟 𝐵 are vacancy-wind factors [START_REF] Manning | Diffusion Kinetics for Atoms in Crystals[END_REF].

𝐽 𝐴 =
Another important approximation used to get the Darken equation is that the non-diagonal terms 𝐿 𝑖𝑗 of the Onsager matrix are negligible.

In concentrated alloys, if the Darken equation is not verified, then by supposing that the vacancy concentration is at equilibrium, these non-diagonal terms 𝐿 𝑖𝑗 of the Onsager matrix can be measured.

II.2.2 Diffusion properties in FeNi alloys: experimental results

There are several experimental measurements of self-diffusion and impurity diffusion coefficients in pure Fe and Ni. Different methods have been also used to get some information on the formation and migration properties of point defects. They include the classical Differential Dilatometry (DD), rapid quenching, Positron Annihilation Spectroscopy (PAS) and electric resistivity measurements. We will first present those experimental results for self-diffusion in fcc Ni and fcc Fe systems respectively.

II.2.2.1 Self-diffusion in fcc

II.2.2.2 Impurity diffusion in fcc Fe and fcc Ni

The these values are part of a systematic study on diffusion in Fe-Ni solid solutions [START_REF] Růžičková | Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system[END_REF] which will be discussed in detail later.

In Figure II.6, we compare the impurity and self-diffusion coefficients in Fe and in Ni. In pure iron, the solute and the solvent diffuse approximately at the same rate (𝐷 𝐹𝑒 𝑁𝑖 * ~𝐷𝐹𝑒 𝐹𝑒 * ), but it is difficult to conclude due to the dispersion of the data. In pure nickel: according to most studies, 𝐷 𝑁𝑖 𝐹𝑒 * > 𝐷 𝑁𝑖 𝑁𝑖 * , the solute diffuses slightly faster than the solvent, at least at 𝑇 > 1200 K. There are no data for 𝐷 𝑁𝑖 𝐹𝑒 * at lower temperatures.

 Interdiffusion coefficients

Interdiffusion in Fe-Ni alloys has been measured in several studies [START_REF] Růžičková | Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system[END_REF][START_REF] Liu | Interdiffusion in nanometric Fe/Ni multilayer films[END_REF][START_REF] Ganesan | Interdiffusion in the nickel-iron system[END_REF]. Ganesan et al. found that between 1173 K and 1373 K, 𝐷 ̃ in Fe-Ni alloys increases with the iron concentration up to 30 wt% Fe, and thereafter decreases [START_REF] Ganesan | Interdiffusion in the nickel-iron system[END_REF], which is in agreement with the results of Million et al. [START_REF] Růžičková | Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system[END_REF]. Later, Jönsson did a review work for the diffusion in Fe-Ni alloys.

In his review, Jönsson [START_REF] Jönsson | Mobilities in Fe-Ni alloys : assessment of the mobilities of Fe and Ni in fcc Fe-Ni alloys[END_REF] According to the author however, this discrepancy could be explained by the effects of grain boundaries and lattice defects on the diffusion processes, which are difficult to avoid in experiments, rather than to the assumptions underlying the Darken equation.

II.2.2.4 Possible effects of chemical and magnetism ordering on the diffusion coefficients

All the experiments mentioned above (except when pointed out directly) were done at high temperatures, in paramagnetic solid solutions. To our knowledge, there are no experimental data for the diffusion in ordered phases and in ferromagnetic fcc FeNi alloys. However, it is well known that for some metals and alloys, the effect of magnetic and chemical ordering on diffusion cannot be neglected.

Such effect would affect the kinetics of ordering, e.g. in FeNi3 alloys, where diffusion occurs in a ferromagnetic L12 ordered structure.

Let us take the case of self-diffusion in bcc Fe to explain the effects that could be expected where ∆𝐻 𝑝𝑎𝑟𝑎 and 𝐷 𝑝𝑎𝑟𝑎 0 are the activation enthalpy and the exponential factor for the paramagnetic state, 𝑀 is the ferromagnetic order parameter and 𝛼 is a fitting coefficient [START_REF] Ruch | Analysis of diffusion in ferromagnets[END_REF]).

(
Figure II.12. Experimental self-diffusion coefficients in 𝛼-, 𝛾and 𝛿-Fe [START_REF] Mehrer | Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes[END_REF]). The dashed line shows the extrapolation of the diffusion coefficient in paramagnetic iron [START_REF] Iijima | Self-diffusion and isotope effect in α-iron[END_REF]) at low temperatures.

A similar effect is often observed in chemical order-disorder transitions, e.g. in the Cu-Zn system, which undergoes an A2/B2 transition below 468°C (Figure II.13). In that case, the deviation of the diffusion coefficient from the Arrhenius behavior can also be described by [START_REF] Ortega | Self-Diffusion in FCC Metals: Static and Dynamic Simulations in Aluminium and Nickel[END_REF]:

𝐷 * = 𝐷 𝑑𝑖𝑠 0 exp [- ∆𝐻 𝑑𝑖𝑠 (1 + 𝛼𝑆 2 ) 𝑘 𝐵 𝑇 ] (II-29)
where ∆𝐻 𝑑𝑖𝑠 and 𝐷 𝑑𝑖𝑠 0 are the activation enthalpy and the exponential factor for the disordered phases and S is the long range order parameter.

In the following, we will try to determine if similar effects occur in Fe-Ni alloys. and ferromagnetic (FM) states. In fcc Fe, there are DFT calculations in the NM but no calculation in the FM state as many FM states are unstable mechanically: other magnetic anti-ferromagnetic structures have been considered.

II.2.3.2 Self-diffusion in fcc Fe and fcc Ni

From section II. From these data we can conclude that the vacancy formation enthalpy in ferromagnetic fcc Ni is 𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 = 1.75 ± 0.10eV and the vacancy migration enthalpy is 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 = 1.1 ± 0.05eV. These values are slightly larger than in NM Ni (which must not be confused with PM Ni). They are also close to experimental ones, measured in paramagnetic Ni. This suggests that there are no strong differences between the FM and PM Ni.

For the activation energy, the results vary around 𝑄 = 2.5 -2.9eV, which are slightly smaller than the experimental value of 2.83 eV.

 Pure Iron

Similar studies have been carried out in fcc Fe. In 2012, Klaver et al. [START_REF] Klaver | Defect and solute properties in dilute Fe-Cr-Ni austenitic alloys from first principles[END_REF] found that the vacancy formation energy is between 1.76 and 1.95 eV with DFT calculations in a supercell of 256 atoms (Table II.9). The calculations have been done for different magnetic states, including single-layer antiferromagnetic (afmI), double-layer antiferromagnetic (afmD)

and also some ferromagnetic high-spin magnetic state. Indeed, many of the magnetic states of γ-Fe are mechanically unstable and for the FM state, the most stable is face centered tetragonal high spin. For austenite at 0 K, the fct afmI and afmD states have been proposed to be the most suitable, as they have the lowest energies. The values are given in Table II For later parametrization, we will assume that the vacancy formation enthalpy in fcc Fe is 1.943 eV from the DFT study who considers the magnetic effects for fcc Fe at 0 K (Klaver, Hepburn, and Ackland 2012). We will then consider that the vacancy migration enthalpy is 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 = 𝑄 -𝐻 𝐹𝑒,𝑉 𝑓𝑜𝑟 = 1.0 eV (with 𝑄 = 2.943 eV, from the experiments of Heumann [START_REF] Heumann | Self-diffusion and isotope effect in γ-iron[END_REF], which is representative of the average experimental values for the self-diffusion in fcc Fe). Due to the large differences between the different experimental values and between the different DFT calculations, one must admit that these values are more uncertain than those of pure Ni. For this value of 𝑄, slightly different choices of 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 and 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 would have been possible. Finally, as in pure nickel, we will assume that 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 and 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 are the same in FM and PM iron, since no data suggests a strong effect of the magnetic state.

II.2.3.3 Impurity diffusion in fcc Fe and fcc Ni

II.2.3.4 Diffusion in concentrated Fe-Ni alloys • Point defect properties:

There are very few studies relevant to diffusion in concentrated Fe-Ni alloys. Li and coworkers (Li, private communication) have recently estimated the vacancy formation and migration enthalpies in ordered structures of the FeNi system by DFT calculations. The values of the formation enthalpies are given in Table II.12.

Table II The other limitation is that these simulations are performed with a single vacancy. In such a case, time rescaling techniques must be used to get the absolute values of the Fe and Ni diffusion coefficients. The atomic diffusion coefficients are indeed proportional to the equilibrium vacancy concentration, which depends on the composition of the solid solutions and, at low temperatures, on possible ordering effects. This is not accounted for in these studies. We will try to deal with this important but non trivial issue with our model, as will be explained below.

II.2.4 Conclusions

The diffusion properties in the Fe-Ni system have been investigated using both experimental and theoretical approaches. Key parameters such as the formation and migration energy of vacancies have been measured, using various experimental methods. Let us summarize the essential results that we will retain for pure metals (Table II. [START_REF] Vernyhora | Thermodynamics of f.c.c. Ni-Fe Alloys in a Static Applied Magnetic Field[END_REF][START_REF] Yang | A revision of the Fe-Ni phase diagram at low temperatures (< 400 • c)[END_REF], and that will be used to determine the parameters of our model.

For pure fcc Ni, we have independent recommended values for the formation and migration energy of vacancies at high temperature (𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 = 1.79eV and 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 = 1.04eV ), consistent with the value of the diffusion enthalpy (𝑄 = 2.84eV). In the case of pure fcc Fe, fewer experimental data of vacancy formation and migration energy exist and those values are not in good agreement with one another. However, many experiments proposed an activation energy in the range of 2.8-3.0 eV in fcc Fe. Therefore, we use the value of 𝐻 𝐹𝑒,𝑉 𝑓𝑜𝑟 = 1.94eV which is adjusted to DFT calculation value by assuming that there is not much influence of the magnetic transition on the vacancy formation, and 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 = 1.0eV, which is adjusted on experimental value of activation energies.

For the entropies of vacancy migration and formation, theoretical results are used as references because they are not available by experiments. For example, the vacancy migration entropy in Ni of The vacancy migration is supposed to be 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 = 1 , and then the 𝐻 𝐹𝑒,𝑉 𝑓𝑜𝑟 is deduced from experimental value of activation energy [START_REF] Heumann | Self-diffusion and isotope effect in γ-iron[END_REF]). The vacancy migration entropy 𝑆 𝐹𝑒,𝑉 𝑓𝑜𝑟 in fcc Fe is the DFT value of Wang et al. [START_REF] Wang | Diffusion coefficients of rare earth elements in fcc Fe: A first-principles study[END_REF]. Finally, the vacancy formation entropy is adjusted to experimental results of Ni self-diffusion coefficients via the pre-exponential factor. in the dilute Ni-Fe system (Table II .11). We will use additional DFT calculations to parametrize the parameters of the dilute Fe-Ni system, as explained later.

In concentrated alloys, several experimental studies clearly showin spite of minor differences that the tracer diffusion coefficients of Fe and Ni change little with the alloy composition. The study by [START_REF] Růžičková | Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system[END_REF] is especially interesting because it presents a rare case of direct experimental comparison between of tracer and interdiffusion coefficients.

II.3 Atomistic modelling of diffusion in Fe-Ni alloys II.3.1 Introduction

As mentioned in II.2.1.4, there is no reliable analytical models to deal with diffusion properties in concentrated alloys. On the other hand, an atomistic modelling can be a promising approach for diffusion and phase transformation kinetics. Key points of the model are how to describe the vacancy jump frequencies and vacancy concentrations in the system.

Atomistic methods have been previously used in other systems and have improved over time.

Pareige et al. studied ordering and precipitation in the Ni-Cr-Al system [START_REF] Pareige | Ordering and phase separation in Ni-Cr-Al: Monte Carlo simulations vs three-dimensional atom probe[END_REF]). Soisson and Fu investigated the precipitation kinetics of Cu in 𝛼-Fe, taking into account the dependence of vacancy concentration and migration barriers on the local atomic environment [START_REF] Soisson | Cu-precipitation kinetics in α -Fe from atomistic simulations: Vacancytrapping effects and Cu-cluster mobility[END_REF].

Later Martinez et al. studied the decomposition in Fe-Cr system [START_REF] Martínez | Decomposition kinetics of Fe-Cr solid solutions during thermal aging[END_REF]. In these alloys the ferro-to-paramagnetic transition affects both the thermodynamic and diffusion properties: the effect on the precipitation kinetics has been simulated by Senninger et al., using a model where the effects of the magnetic transitions are taken into account through the concentration and temperature dependence of pair interactions [START_REF] Senninger | Atomistic simulations of the decomposition kinetics in Fe-Cr alloys: Influence of magnetism[END_REF]). We will use a similar approach, by extending the PIM of Chapter I to deal with vacancy formation and migration properties in Fe-Ni fcc alloys.

In the following, we will first give the general expression of the vacancy migration barriers of the k-PIM (II.3.2). Then, in section II. The vacancy migration barrier (i.e. the difference between the Gibbs free energy of the system in the transition state and in the initial state) can be expressed in terms of pair interactions as:

∆𝐺 𝐴𝑉 𝑚𝑖𝑔 = ∑ 𝑔 𝐴𝑖 𝑆𝑃 (𝑥, 𝑇) 𝑖∈𝑛(𝑆𝑃) -∑ 𝑔 𝑉𝑗 (𝑥, 𝑇) 𝑗∈𝑛(𝑉) -∑ 𝑔 𝐴𝑘 (𝑥, 𝑇) 𝑘∈𝑛(𝐴) (II-30)
where 𝑔 𝐴𝑖 𝑆𝑃 (𝑥, 𝑇) are the interactions between atom A on the saddle point and the atoms on the neighboring fcc sites (𝑖); 𝑔 𝑉𝑗 (𝑥, 𝑇) are the interactions between the vacancy and its neighboring atoms on the fcc lattice and 𝑔 𝐴𝑘 (𝑥, 𝑇) are the interactions between atom A on its initial position and its neighboring atoms on the fcc lattice.

Compared to previous studies [START_REF] Pareige | Ordering and phase separation in Ni-Cr-Al: Monte Carlo simulations vs three-dimensional atom probe[END_REF][START_REF] Senninger | Atomistic simulations of the decomposition kinetics in Fe-Cr alloys: Influence of magnetism[END_REF][START_REF] Soisson | Cu-precipitation kinetics in α -Fe from atomistic simulations: Vacancytrapping effects and Cu-cluster mobility[END_REF][START_REF] Martínez | Decomposition kinetics of Fe-Cr solid solutions during thermal aging[END_REF], we try to deal with finite temperature effects (the effect of the magnetic transition but also of the vibrational entropy) in a more systematic ways, as in Chapter I. The interatomic pair interactions, such as 𝑔 𝐴𝑘 (𝑥, 𝑇) in equation , are those of the PIM (between first and second nearest neighbors, with magnetic and non-magnetic contributions, adjusted on DFT calculations at 0 K and CALPHAD results at high temperatures). Similarly, the saddle-point and vacancy-atom interactions, are written as 𝑔 𝑖𝑗 = ℎ 𝑖𝑗 -𝑇𝑠 𝑖𝑗 (with ℎ 𝑖𝑗 and 𝑠 𝑖𝑗 the enthalpy and the entropy of the pair, each being the sum of magnetic and non-magnetic terms).

II.3.3 Application to the diffusion in pure and dilute alloys

The self-diffusion coefficients and the impurity diffusion coefficients only depend on a few jump frequencies (see section II.3.3.4 and II.3.3.5), so that it is easy to fit the relevant parameters of the k-PIM on the corresponding migration barriers. The diffusion coefficients can then be directly computed from the jump frequencies, using an exact analytical expression [equation (II-6) for 𝐷 𝐹𝑒 𝐹𝑒 * and 𝐷 𝑁𝑖 𝑁𝑖 * or the five frequency or similar models (equation (II-9) for 𝐷 𝑁𝑖 𝐹𝑒 * and 𝐷 𝐹𝑒 𝑁𝑖 * ]. AKMC simulations are not necessary, but we will use them to provide a simple illustration and to validate our code.

II.3.3.2 Atomistic Kinetic Monte Carlo simulations

The atomistic kinetic Monte Carlo (AKMC) simulations have been carried out using the residence time algorithm [START_REF] Bortz | A new algorithm for Monte Carlo simulation of Ising spin systems[END_REF] to perform exchanges between atoms and vacancies. With this algorithm, each Monte Carlo step takes place as follows. For a specific configuration of the system, 𝐶 𝑖 , we identify all available configurations {𝐶 𝑘 } 𝑖 after one possible atomvacancy exchange. Then for each possible transition state, the probability of change can be calculated using the associated jump frequencies, 𝜔 𝑘 , with a migration barrier given by equation (II-30). One of the jumps is chosen using a random number. The higher the jump frequency, the more likely the associated transition to be chosen. After the chosen transition is performed, the system continues as in the first step. For such a Monte Carlo step, the time increment is obtained by the inverse of the sum of the jump frequencies 𝜔 𝑘 ( 𝑡 𝑀𝐶 = 1/ ∑ 𝜔 𝑘 ). The residence time algorithm has the advantage of guaranteeing one exchange atom-vacancy for each Monte Carlo step.

For the Monte Carlo simulations, a simulation box with periodical condition has been used. A typical simulation box contains 𝑁 0 = 4 × 64 3 atoms and one vacancy, with periodic boundary conditions. The tracer diffusion coefficient of one species (𝑖 = 𝐴, 𝐵𝑜𝑟𝑉) can be determined by measuring the average square displacement 〈𝑅 𝑖 2 〉 of 𝑖tracers, according to the Einstein relation:

𝐷 𝑖 * = 〈𝑅 𝑖 2 〉 6𝑡 (II-31)
where 𝑡 denotes the accumulated physical time.

II.3.3.3 Vacancy concentration and time rescaling

In our simulations, only one vacancy is introduced into the system, leading to a vacancy concentration of 𝑥 𝑉 𝑀𝐶 = 1/𝑁 0 . Therefore, the accumulated time 𝑡 must be rescaled as :

𝑡 = 𝑡 𝑀𝐶 𝑥 𝑉 𝑀𝐶 𝑥 𝑉 𝑒𝑞 (II-32)
where 𝑥 𝑉 𝑒𝑞 is the real vacancy concentration (we assume that the vacancy concentration is at equilibrium on the timescale of the diffusion and ordering processes considered in this study). In the dilute limit,

𝑥 𝑉 𝑒𝑞 = exp (-𝐺 𝑉 𝑓𝑜𝑟 /𝑘 𝐵 𝑇)
where 𝐺 𝑉 𝑓𝑜𝑟 is the Gibbs free energy of vacancy formation in the pure metal, Fe or Ni.

II.3.3.4 Self-diffusion coefficient

In pure A (Fe or Ni) the migration barrier is unique. The pair interactions 𝑔 𝐴𝐴 (𝑛) have been fitted

of the cohesive energy of the pure metals. The atom-vacancy interactions 𝑔 𝐴𝑉 (𝑛) are then fitted on the Gibbs vacancy formation energies using:

𝐺 𝑉 𝑓𝑜𝑟 (𝐴) = - 1 2 ∑ 𝑔 𝐴𝐴 (𝑛) 𝑛 + ∑ 𝑔 𝐴𝑉 (𝑛) 𝑛 (II-33)
We take into account atom-vacancy interactions between first and second nearest neighbors and we assume a constant ratio: 𝛽 𝐴𝑉 = 𝑔 𝐴𝑉 (2) /𝑔 𝐴𝑉 (1) . As for the interatomic pair interactions, they are expressed as a sum of magnetic and non-magnetic term: 𝑔 𝐴𝑉 (𝑖) = 𝑔 𝐴𝑉 𝑚𝑎𝑔(𝑖) + 𝑔 𝐴𝑉 𝑛𝑚 (𝑖) . We have seen in section II.2 that existing experimental and theoretical results do not show significant effects of the magnetic transition on vacancy properties. In this study, we will therefore assume that 𝐺 𝑉 𝑓𝑜𝑟 (𝑇) and 𝐺 𝑉 𝑚𝑖𝑔 (𝑇)

show no discontinuity at the Curie temperature in pure Fe and Ni, i.e. that there is a perfect Arrhenius behavior of the self-diffusion coefficients. There are some discontinuities of the pair interactions 𝑔 𝐹𝑒𝐹𝑒 (𝑛) (𝑇) and 𝑔 𝑁𝑖𝑁𝑖 (𝑛) (𝑇) at the Curie temperature, but they are compensated by the variation of 𝑔 𝐹𝑒𝑉 (𝑛) (𝑇)

and 𝑔 𝐹𝑒𝑉 (𝑛) (𝑇).

The pair interaction 𝑔 𝐴𝐴 𝑆𝑃 can be fitted to the vacancy migration barrier in pure A using:

𝐺 0 𝑚𝑖𝑔 = 4𝑔 𝐴𝐴 𝑆𝑃 -∑ 𝑍 𝑛 𝑔 𝐴𝑉 (𝑛) 2 𝑛=1 -(𝑍 1 -1)𝑔 𝐴𝐴 (1) + 𝑍 2 𝑔 𝐴𝐴 (2)
(II-34)

The values of 𝐺 𝑉 𝑓𝑜𝑟 = 𝐻 𝑉 𝑓𝑜𝑟 -𝑇𝑆 𝑉 𝑓𝑜𝑟 and 𝐺 𝑉 𝑚𝑖𝑔 = 𝐻 𝑉 𝑚𝑖𝑔 -𝑇𝑆 𝑉 𝑚𝑖𝑔 , used to fit the 𝑔 𝐴𝑉 (𝑛) and 𝑔 𝐴𝐴 𝑆𝑃 parameters, can be found in Table II [START_REF] Tucker | Ab initio-based diffusion theory and tracer diffusion in Ni-Cr and Ni-Fe alloys[END_REF] and [START_REF] Toijer | Solute-point defect interactions, coupled diffusion, and radiation-induced segregation in fcc nickel[END_REF]. Finally, let us emphasize that in this section, for the sake of simplicity, we have considered the ℎ 𝐹𝑒𝑁𝑖 𝑆𝑃 and ℎ 𝑁𝑖𝐹𝑒 𝑆𝑃 parameters as constants, independent of the local composition. We will later introduce a dependence of these parameters on the local Ni concentration, in order to better describe the tracer diffusion coefficients in concentrated solid solution. The values given above are thus only the limits in pure Fe and Ni.

Barrier in eV

II.3.3.6 Conclusions

So far, the k-PIM model has been built and parametrized for pure metals and dilute Fe-Ni alloys.

The key parameters have been determined from experimental data and DFT calculations of formation enthalpies and vacancy migration barriers in fcc Fe-Ni systems. Other parameters, controlling the migration and formation entropies, have been fitted on experimental diffusion data. As a result, self and impurity diffusion coefficients are well reproduced in pure fcc Fe and fcc Ni, with uncertainties of the order of a factor of 2 or 3, similar to the differences between various experiments.

II.3.4 Determination of equilibrium vacancy concentration in alloys: Widom integration method

II.3.4.1 Introduction

In the following, we are going to consider diffusion in Fe-Ni concentrated solid solutions (with various degrees of short range order) and in the L12 phase (partially or perfectly ordered). As explained above, for rescaling the time in Monte Carlo simulations [eq. (II-32)], it is crucial to know the vacancy concentration in the system. This is simple in pure metals but in concentrated A-B alloys there is no general analytical expressions for 𝑥 𝑉 𝑒𝑞 and 𝐺 𝑉 𝑓𝑜𝑟 , except in limiting cases (such as a fully disordered solid solution, or a perfectly ordered structure). The other problem concerns the alloys undergoing a phase transformation, such as a phase separation or a phase ordering (Chapter III). These systems are not at equilibrium, and one must assume that a vacancy equilibrium concentration can still be defined, which evolves with the system (this assumption, often implicit in the modeling of phase transformation kinetics, is based on the fact that the vacancies diffuse much faster than the atoms). We will see in the next chapter that it poses some problems to estimate 𝑥 𝑉 𝑒𝑞 .

At equilibrium at least, some analytical models exist. In disordered solid solutions, the Bragg-Williams approximation gives the vacancy concentration [START_REF] Nastar | Atomistic modeling of phase transformations: Point-defect concentrations and the time-scale problem[END_REF]: The calculation of vacancy concentrations in perfectly ordered phases is more difficult. The main reason is that generally, at constant numbers of A and B atoms, one cannot create one single vacancy on a sublattice without creating other defects (a vacancy on another sublattice, or an antisite defect). Nevertheless efficient analytical models have been developed, such as the one of Hagen & Finnis, which is described in Appendix [START_REF] Hagen | Point defects and chemical potentials in ordered alloys[END_REF]. The method only works for almost perfectly ordered alloys (point defects, vacancies and antisites are assumed to be isolated).

𝑥 𝑉 𝑒𝑞 =
On the other hand, AKMC simulations have been used to determine the equilibrium vacancy concentration, in alloys at equilibrium or during a phase separation (LeBouar and Soisson 2002;[START_REF] Nastar | Atomistic modeling of phase transformations: Point-defect concentrations and the time-scale problem[END_REF]. The principle is the same as for the time rescaling of equation (II-32), but using the equilibrium vacancy concentration in a reference phase (for example pure A or pure B). One can rewrite equation (II-32) as, e.g. 𝑥 𝑉 𝑒𝑞 = 𝑥 𝑉 𝑀𝐶 × 𝑥 𝑉 𝑒𝑞 (𝐴)/𝑥 𝑉 𝑀𝐶 (𝐴) where 𝑥 𝑉 𝑒𝑞 is the equilibrium vacancy concentration, 𝑥 𝑉 𝑀𝐶 = 1/𝑁 the vacancy concentration in the simulation box, and 𝑥 𝑉 𝑒𝑞 (𝐴) = exp (-𝐺 𝑉,𝐴 𝑓𝑜𝑟 /𝑘 𝐵 𝑇) the equilibrium concentration in pure A. One can therefore get 𝑥 𝑉 𝑒𝑞 by measuring 𝑥 𝑉 𝑀𝐶 (𝐴), the vacancy concentration on the sites of the Monte Carlo simulation that are in a pure A environment. The method works well in systems with an unmixing tendency for example Fe-Cr (Le [START_REF] Bouar | Kinetic pathways from embedded-atom-method potentials: Influence of the activation barriers[END_REF] and Fe-Cu [START_REF] Nastar | Atomistic modeling of phase transformations: Point-defect concentrations and the time-scale problem[END_REF]. However, we found that it did not work well in systems with an ordering tendency, because fraction of pure A or pure B sites is too small.

Therefore, we propose a new method to calculate the vacancy concentration in concentrated alloys, for general cases. It is based on the Widom integration method, initially developed to estimate the chemical potential of atoms in off-lattice simulations. We will validate our method by comparing the results to Hagen & Finnis method in ordered structures at low temperatures, and to the modified Bragg-Williams approximations of equation (II-37) in random solid solutions, at high temperatures.

II.3.4.2 The Widom method

The Widom method is a statistical thermodynamic approach to the calculation of material and mixture properties. In general, the calculating of quantity 𝛹 𝑖 for a specie 𝑖 is of a form of 𝛹 𝑖 = ln〈exp(-∆𝜑 𝑖 /𝑘 𝐵 𝑇)〉, where 𝜑 𝑖 is the interaction energy of an inserted particle with all other particles in the system. The average is over all possible insertions. This can be understood conceptually as fixing the location of all molecules in the system and then inserting a particle of species 𝑖 at all locations through the system, averaging over a Boltzmann factor in its interaction energy over all of those locations. It has been proposed by Widom to give the thermodynamic functions in terms of the potential energy in fluids [START_REF] Widom | Some Topics in the Theory of Fluids[END_REF].

Applied to our PIM, the Widom integration method calculates the vacancy formation Gibbs free energy as:

𝐺 𝑉 𝑓𝑜𝑟 = -𝑘 𝐵 𝑇𝑙𝑛 〈exp (- ∆𝐺 𝑉 𝑖 𝑘 𝐵 𝑇 )〉 (II-38)
where ∆𝐺 𝑉 𝑖 is the Gibbs free energy needed to replace an atom 𝑖 in the system by a vacancy and the brackets 〈 〉 means the average value, computed on all the atoms of the system. ∆𝐺 𝑉 𝑖 is defined at constant number of atoms, i.e. :

∆𝐺 𝑉 𝑖 = 𝑔 𝑖 + 𝐺 𝑉 𝑖 -𝐺 0 (II-39)
where 𝐺 0 is the Gibbs free energy of the system without vacancy, 𝐺 𝑉 𝑖 the Gibbs free energy of the system with a vacancy on site 𝑖, and 𝑔 𝑖 the Gibbs free energy of atom 𝑖 (let us recall that all these terms are Gibbs free energies because they include the non-configurational entropic contributions of the system).

In practice, an equilibrium configuration with a certain global concentration 𝑥 𝑁𝑖 is first created by Monte Carlo simulations (using the semi-grand canonical algorithm described in chapter I). Then 𝐺 𝑉 𝑓𝑜𝑟 is computed according to equation . 𝐺 𝑉 𝑖 and 𝐺 0 are simply the sums of the pair interactions in system with and without a vacancy. In order to estimate 𝑔 𝑖 , we propose two different ways:

1. We consider it to be the chemical potential of atom 𝑖 : 𝑔 𝑖 = 𝜇 𝑖 . This choice will be hereafter referred to as "Widom 1".

2. 𝑔 𝑖 is computed for each atom 𝑖 as half the sum of the surrounding pair interactions (considering that a pair interaction 𝑔 𝑖𝑗 belongs one half to 𝑖 and one half to 𝑗). The method will be called "Widom 2".

The first choice may seem more natural, but it has two drawbacks. From a practical point of view, it requires the knowledge of the chemical potentials of A and B. They are indeed provided by semi-grand canonical Monte Carlo simulations (see Chapter I, section Ⅲ.A), but at low temperatures, in case of strong hysteresis, precise values can be difficult to obtain. In such cases, other models can be used: for example, the model of Hagen and Finnis also provides the values of 𝜇 𝐴 and 𝜇 𝐵 (see Appendix), but only for a well-ordered stoichiometric phase. The other issue is that the chemical potentials are equilibrium quantities. They can be defined in equilibrium situations, which is the case for the modeling of diffusion coefficients discussed in this chapter. However, it is less clear if they can be used during a phase transformation (as in Chapter III), in a non-equilibrium system, where chemical potentials are not the same everywhere.

The second choice has neither of these two disadvantages, and is very easy to implement in an equilibrium or kinetic Monte Carlo algorithm. It is based on an idea often used to compute vacancy formation energies in simple situations. For example in a pure metal, it is equivalent to the idea that the vacancy formation energy is the energy required to take an atom in the bulk and place it on the surface of the metals where it gets back half of its bonds [START_REF] Mehrer | Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes[END_REF]. However, to our knowledge, the validity of this idea has never been rigorously established in concentrated alloys [START_REF] Mayer | On the meaning of effective formation energies, entropies and volumes for atomic defects in ordered compounds[END_REF][START_REF] Cheng | Vacancy models for concentrated binary alloys -II long-range ordered alloys[END_REF][START_REF] Hagen | Point defects and chemical potentials in ordered alloys[END_REF].

II.3.4.3 Equilibrium vacancy concentrations in Fe-Ni alloys

We are going to apply the above Widom methods to our interaction model for Fe-Ni alloys, first in solid solutions at high temperatures, to get the evolution of 𝐺 𝑉 𝑓𝑜𝑟 (𝑥 𝑁𝑖 ) in the conditions of the experimental measurements of tracer and interdiffusion coefficients, by Million et al. and by other authors, presented in the bibliography part. Then we will model the evolution of 𝐺 𝑉 𝑓𝑜𝑟 (𝑥 𝑁𝑖 ) at lower temperatures, when the ordered L10 and L12 phases appear, as we have seen in Chapter I (section Ⅰ). We will use the same pair interactions (between atoms and between atoms and vacancies) as in section II.3.3, and we will see in the next section that these parameters allow a good reproduction of all the existing diffusion data.

In the following simulations, a system with 𝑁 = 4 × 16 3 atoms with periodic boundary conditions has been used. At a given temperature, the semi-grand canonical algorithm described in eV, because of the partial order (the model is valid for an almost perfect order, with isolated antisites).

The difference between Widom 1 and 2 is larger between 𝑥 𝑁𝑖 = 0.3 and 0.6, with a maximum of 0.07 eV, at 𝑥 𝑁𝑖 = 0.57. On the whole, the Widom 1 method seems more reliable than Widom 2, especially in the ordered phase, where the Hagen and Finnis model provides a safe reference. We will therefore use it in the following simulations of diffusion in concentrated alloys, and also for ordering kinetics of FeNi3 in the next chapter (even if it uses the equilibrium chemical potentials, we will come back to this point later).

It turns out that in these casesthe only ones for which we have some experimental observationsthe choice is not a critical one indeed, because Widom 1 and 2 happen to give almost the same results. It would be interesting to test the same methods in other systems, to see if the same conclusions apply.

II.3.5 Diffusion coefficients in concentrated alloys

II.3.5.1 Tracer diffusion coefficients in concentrated alloys

As discussed above, 𝐺 𝑉 𝑓𝑜𝑟 and the equilibrium vacancy concentration 𝑥 𝑉 𝑒𝑞 = exp(-𝐺 𝑉 𝑓𝑜𝑟 /𝑘 𝐵 𝑇) can be calculated by Widom 1 method in a concentrated alloy. The tracer diffusion coefficients of Fe and Ni can then be obtained with AKMC simulations performed with one vacancy and the appropriate time rescaling (eq. II-32).

We will use the same k-PIM parameters as in the dilute alloys, with an additional optimization of the saddle-points interactions, already mentioned in section II. The discrepancy is limited, except at 950°C, but here one must mention a very practical problem. There is an ambiguity for this temperature in the paper of Million et al.: the same data is given with a different temperature in the figure (950°C) and in the table (985°C). We have retained the value 950°C (this value is also retained, without any comment, in the Landolt-Bornstein database. Therefore, for the comparison with these experiments, we will rely more on the two other temperatures (1100 and 1250°C).

II.3.5.2 Interdiffusion coefficients. Test of the Darken equation.

We It is difficult to model in the simulations the conditions of a typical interdiffusion experiment, i.e. start from a pure Fe/pure Ni couple, simulate the interdiffusion of Fe and Ni, and deduce 𝐷 ̃(𝑥 𝑁𝑖 )

from the concentration profile with the Matano method. One of the difficulties is that at the atomic scale, it is difficult to keep a well-defined interface. It quickly becomes rough, with oscillations perpendicular to the interdiffusion direction.

For this reason, we will use the sinusoid method, which can also be used in real experiments. [START_REF] Růžičková | Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system[END_REF]) and to [START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]) Let us emphasize that these results have been obtained with the k-PIM parameters used to model diffusion at high temperatures, in the solid solutions. We have seen that these parameters give vacancy formation enthalpies in the L12-FeNi3 phase which are significantly larger than those calculated from first principles methods by Li (Table II .18). (Note that the DFT calculations have been performed at 0 K, whereas the values of the k-PIM model are given at T = 50 K, to avoid divergences coming from the magnetic contributions, see Chapter I, section Ⅱ.C.2). Contrary to the k-PIM, the DFT calculations predict that the vacancy formation enthalpies in the L12-FeNi3 ordered phase are lower than in the solid solution.

Table II

II.4 Conclusions

In One limitation of the approach is that existing DFT and experimental data are scarcer for point defects and diffusion than for thermodynamic properties, especially in concentrated alloys. Important assumptions have therefore been made to "fill the gaps": for example, based on partial experimental and theoretical evidences, we have assumed that the magnetic transition does not affect the point defect properties and the diffusion coefficients.

In spite of these limitations, we were able to reproduce with one set of parameters all existing experimental diffusion data: tracer and interdiffusion coefficients, in a large range of temperatures and compositions. The difference between the model and the experiments (typically a factor 2 or 3) are of the same order as the differences between the experiments. If new experimental or theoretical results

were to lead us change some of our assumptions, it should be easy to modify the relevant parameters within the same formalism. To give an example, the very recent results of Li et al. [START_REF] Li | Effects of magnetic excitations and transitions on vacancy formation: Cases of fcc Fe and Ni compared to bcc Fe[END_REF], based on Monte Carlo simulations DFT based effective interaction models, suggest that there is a small discontinuity of 𝐺 𝑉 𝑓𝑜𝑟 at the Curie point in pure fcc Fe and Ni, less important that in bcc Fe but not zero.

The approach is applied to a system that has actually a simple behavior: in Fe-Ni solid solutions, the evolution of the tracer diffusion with the composition is limited and almost linear. We think however that the same methods could be useful to model other systems, with more complex behaviours. We have found that the difference between the tracer and interdiffusion coefficients can be reasonably well predicted by the Darken equation. Finally, our first AKMC simulations predict a strong effect of the A1/L12 ordering transition in the FeNi3 alloy on the tracer diffusion coefficients, when using the k-PIM (i.e. an increase of the diffusion enthalpy in the L12 phase). The effect is significantly reduced however, when the k-PIM is modified to give at low temperatures the vacancy formation enthalpies calculated from first principles.

approximately 480°C (753 K) and decreases at lower temperature (due to a slower diffusion) and at higher temperature (due to the decrease of the ordering driving force). [START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF] As discussed above, these studies were carried out at different temperatures (Table III.1).

Fortunately, one is common to Calvaray and Fayart and to Morris et al. [470°C (743 K)] and one to [START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF] and Morris [START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF]) at T=470 °C (743 K).

One observes some significant differences between the two studies. [START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF] and [START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF].

III.3 AKMC simulations: methods

III.3.1 Vacancy concentrations and the Widom method

The k-PIM and k-PIM2 parameters are used in the following AKMC simulations, performed with one vacancy and the time rescaling method as described in Chapter II. However, unlike the previous chapter we now consider an alloy that is not at equilibrium, and where the vacancy concentration changes with time. We use a standard adiabatic approximation, by considering that for each transient state of the alloy, the vacancy concentration immediately adjusts to its equilibrium value, because vacancies diffuse much more rapidly than atoms.

To estimate the equilibrium value at each time, 𝑥 𝑣 𝑒𝑞 (𝑡), we use the Widom 1 method as presented in section II. 3.4.2 (Chapter II). In practice, we start from a random solid solution and performed vacancy jumps using the residence time algorithm. From time to time, 𝐺 𝑉 𝑓𝑜𝑟 (𝑡) and 𝑥 𝑣 𝑒𝑞 (𝑡) are evaluated with the Widom 1 method. The time between two evaluations must be short enough, so that the system has changed very little in between, but not too short, to save some computation time (in practice, 1000 measurements are typically done during an AKMC simulations of 6 × 10 9 MCS). We have chosen to use the Widom 1 method because we have seen that, in some cases (for example in the Fe-50%Ni alloy), it gives better estimations of 𝑥 𝑣 𝑒𝑞 . As already mentioned, this method uses the equilibrium potentials of Fe and Ni atoms, 𝜇 𝐹𝑒 and 𝜇 𝑁𝑖 , for the composition and temperature of interest. During the ordering process, the alloy is not at equilibrium and therefore 𝜇 𝐹𝑒 and 𝜇 𝑁𝑖 are not the same on the different sublattices, as long as they have not reached their equilibrium concentration (similarly, in a case of precipitation, they would not be the same in the precipitate and the matrix). The use of the equilibrium potentials can nevertheless be justified if we assume that a local equilibrium is very rapidly reached around the vacancybecause diffusion occurs rapidly in its vicinity. Note that in this specific case (the Fe-75%Ni alloy), we found that the Widom 2 method gave very close vacancy concentration and ordering kinetics. For example at T=434 °C (707 K), in the Fe-75%Ni alloy, the Gibbs free energies of vacancy formation given by Widom 1 and 2 only differ by 0.003 eV, which corresponds to a factor 1.05 on the timescale. But as indicated in section II.3.4.3 (Chapter II), Widom 1 gives a better estimation of the equilibrium vacancy concentrations on the Fe and Ni sublattices. In the Fe-50%Ni alloy, the difference between Widom 1 and 2 would also be more important (0.15 eV at 500 K). Therefore we will consider that in general, Widom 1 is a safer choice.

III.3.2 Measurement of long-range order parameter and domain size

As discussed in III.2, in the experiments the ordering process of a system is characterized by the evolution of the long-range order parameter, 𝜂, and the size of the ordered domains, 𝐿. These domains belong to one of the 4 variants of the L12 phase, which corresponds to the four sublattices defined in Chapter I (Ⅲ.A), and are separated by antiphase boundaries (APB). The LRO parameter of a given domain, 𝜂 can be measured from the concentration of Ni atoms upon the four different sublattices of the fcc structure. However, the method cannot be used to measure the averaged value of 𝜂 when several domains of different variants coexist. In such a case, it is better to measure the LRO parameter from the structure factor 𝑆(𝒌, 𝑡), of the system (Domb, Green, and Lebowitz 1976):

𝑆(𝒌, 𝑡) = 1 𝑁 〈|∑ 𝑒 𝑖𝒌•𝒓 (𝑥(𝒓, 𝑡) -𝑥̅ ) 𝒓 | 2 〉 (III-1)
where 𝑥(𝒓, 𝑡) = 0 or 1 is the Ni atomic fraction on each site 𝒓 and 𝑥̅ the average Ni atomic fraction.

𝑆(𝒌, 𝑡) is computed using a fast Fourier transform algorithm. In an ordered L12 structure, some superstructure peaks appear for specific vectors 𝒌 𝑺 in the reciprocal spacein addition to those of the fcc lattice. The intensity of one peak, 𝐼 𝑠 , is obtained by integrating the structure factor in a volume centered around 𝒌 𝒔 . Then the long-range order 𝜂 can be calculated as:

𝜂 = √ 𝐼 𝑠 𝐼 𝑠 0 (III-2)
where 𝐼 𝑠 0 is the peak intensity in a perfect L12 structure. In our simulations, we compute the intensity of around the 𝒌 𝒔 = (1,1,0) peak, over all the vectors with |𝒌 -𝒌 𝒔 | < 0.1𝒌 𝒔 [START_REF] Floc'h | Slow coarsening of B 2-ordered domains at low temperatures: A kinetic Monte Carlo study[END_REF]. The average size of domains (𝐿) is calculated with the second moment of the structure factor, 𝑘 2 :

𝐿 ∝ √ 1 𝑘 2 (III-3)
where

𝑘 2 = ∑ |𝒌| 2 𝑆(𝒌, 𝑡) 𝒌 𝑚𝑎𝑥 𝒌 ∑ 𝑆(𝒌, 𝑡) 𝒌 𝑚𝑎𝑥 𝒌 (III-4)
The summations in equation (III-4) must be done will all the values of 𝒌 in the reciprocal space where the structure factor 𝑆(𝒌, 𝑡) is less than the background values obtained for a completely disordered configuration. For the calibration of the relation given by equation (III-3), several test simulations were carried out in systems of specific domain size. We found that equation (III-3) gives the correct values of the domain size.

On the other hand, in a homogeneous system the measurement of the LRO parameter by equation (III-2) gives the same value as that measured from the concentrations on the four sublattices, used in Chapter I. There is little size effect of the size of system on the evolution of the LRO parameter:

we have compared the simulation results in systems of 𝑁 0 = 4 × 32 3 and 𝑁 0 = 4 × 64 3 and found practically no difference, except that in a larger system, there are less fluctuations.

III.4 Results and discussions

III.4.2 Long-range order parameter: comparisons between MC and experiments

Atomistic Kinetic Monte Carlo simulations of ordering kinetics have been carried out at different temperatures. All simulations were done in a system containing 𝑁 = 4 × 64 3 fcc sites (except specifically mentioned), with periodic boundary conditions. Therefore, the length of the system is about 225 Å. An example of microstructure evolution (using k-PIM2 parameters), in Fe-75%Ni, is shown in [START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF], Morris et al [START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF]) and the AKMC simulations with k-PIM and k-PIM 2 parameters [START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF] and [START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF].

III.5 Conclusions

In this chapter, we have investigated the ordering process in the L1 2 FeNi 3 alloy with AKMC simulations. Two different sets of pair interaction model, k-PIM and k-PIM2, have been used. Using a fast Fourier transformation, the LRO parameter and the domain size have been computed from the structure factor of the system.

In general, the simulations performed with the k-PIM2 parameters are in agreement with the experiments --with an uncertainty that is in the order of differences between experimental studies. The simulations performed with the k-PIM parameters are significantly slower, because they lead to an overestimation of the Gibbs free energies of the vacancy formation in the L12 phase, therefore to an underestimation of the vacancy concentrations and to a too slow diffusion.

This work needs to be completed: for example, we have not attempted a detailed analysis of the growth exponents of the domain size, partly because the experimental observations are not fully conclusive and partly for lack of time. The analysis would require costly simulations on large simulation boxes, and the determination of growth exponents can be tricky (on a log-log diagram, it is often easy to fit a power law of the type 𝐿 -𝐿 0 = 𝑘𝑡 𝛼 with several values of 𝛼, especially for transient kinetic regimes if they do not extend over several decades of the time scale). Moreover, it would be useful in connection with a detailed study of the diffusion processes in the L12 phase (diffusion in ordered phases usually occurs by vacancy jumps cycles that can affect the kinetic laws). This is left for future work.

The k-PIM could also be improved.

General conclusions and perspectives

We have developed in this work a pair interaction model, within the rigid lattice approximation.

It is used in atomistic equilibrium and kinetic Monte Carlo simulations, to describe the thermodynamic properties, diffusion coefficients and ordering kinetics in Fe-Ni alloys with face-centered cubic structures.

The parameters of the model have been fitted both on ab initio calculations and on experimental data. For the thermodynamic part, we tried to establish a direct link between the interatomic pair interactions of the model and the data of a CALPHAD study, by distinguishing in these pair interactions an enthalpic and an entropic (non-configurational) contribution on the one hand, and a magnetic and non-magnetic contribution on the other hand. We have thus been able to construct a phase diagram that is in good agreement with experiments, when available, i.e. at temperatures above about 400°C. We have also been able to predict the stability domain of the FeNi-L10 phase, at temperatures too low to reach thermodynamic equilibrium under ordinary conditions. This method has some advantages:

distinguishing the magnetic energy contributions, for example, makes it possible to reproduce a phase separation between two solid solutions, one paramagnetic and one ferromagnetic, which had already been predicted by some theoretical studies. We also believe that it could be applied to other alloys and contribute to a more methodical approach to the parameterisation of this type of atomic model.

The kinetic version of the interaction model and simulations, which so far only describes the vacancy diffusion mechanism, allowed us to reproduce all the experimentally known diffusion coefficients with good accuracy, taking into account the scattering between experimental results. For this purpose, we had to introduce a new method to describe the evolution of the equilibrium vacancy concentration, based on the Widom insertion technique. This method, unlike others previously developed for alloys with a tendency to phase separation, works well in alloys with an ordering tendency.

It allows the rescaling of the time in Monte Carlo simulations and to obtain a physical time scale, which is obviously essential to simulate diffusion properties. Here again, we believe that this method can be profitably used in other ordering alloys, including systems where the vacancy formation enthalpies and diffusion coefficients evolve much more with the composition than in Fe-Ni alloys. To give an example, one can think of fcc Ni-Al alloys, where the difference in vacancy formation enthalpy is approximately 1 eV and the difference in the self-diffusion coefficients is more than 6 orders of magnitude at 1000 K.

Finally, we were able to model the ordering kinetics of the FeNi3 phase, again in good agreement with the available experiments.

A direct perspective of this work is to extend the model to ternary Fe-Ni-Cr alloys, for the modelling of austenitic steels and to the self-interstitial diffusion mechanism, to simulate irradiation effects (in particular radiation induced segregation (RIS), as it has already been done in ferric Fe-Cr alloys). It is also possible to improve our approach in many ways. Some minor changes of the pair interactions should remove the small discrepancy with the experimental value of the critical temperature for the L12-FeN3 ordering, thus improving the ordering driving force and the simulation of ordering kinetics just below this temperature. We could also improve the description of the migration barriers in the L10 and L12 phases, by further comparison and adjustment on ab initio calculations. More fundamentally, one of the difficulties encountered in this study was the lack of data, either experimental or derived from first-principles calculations, to adjust some of the model parameters. There are little data on the entropies of formation and migration of vacancies in concentrated alloys; or on the effect of the ferro-paramagnetic transition on the same properties. This led us to make some fairly strong assumptions, in particular that the magnetic transition does not modify the enthalpies of formation and migration of vacancies at the Curie point. We also had to introduce, in the thermodynamic part of the model as well as in the kinetic part, an empirical rule for the transition between the properties at 0 K (calculated ab initio) and the properties at high temperature (experimentally measured). The transition between these two regimes directly affects the diffusion coefficients in the FeNi3 phase and the ordering kinetics. On this last point, we believe that the model of composition-and temperature-dependent pair interactions is sufficiently adaptable to integrate any new data that could be obtained on these properties, while remaining sufficiently numerically cost-effective, which is essential for simulating phase transformation kinetics. From this point of view, we can think that magnetic interaction models will provide very valuable information in the near future. We can mention for example the results of Kangming Li and Chu-Chun Fu [START_REF] Li | Effects of magnetic excitations and transitions on vacancy formation: Cases of fcc Fe and Ni compared to bcc Fe[END_REF] on the effect of the magnetic transition on the enthalpy of formation of vacancies in iron and nickel. It will be easy to modify our model slightly to take this variation into account, and see its impact on the migration of vacancies and more generally diffusion properties as well as phase transition kinetics. energy of the system where we replace one A atom on the 𝛼 sub-lattice in the perfect ordered system by a B atom. 𝑒 𝑉𝛼 is defined as the energy of the system where we replace one atom A on the sub-lattices 𝛼 in the perfect ordered system by a vacancy. Similar definition applies for 𝑒 𝐴𝛽 and 𝑒 𝑉𝛽 .

With the definition of these variables, we have:

𝑚𝑘

Figure 1 .

 1 Figure 1. The phase diagram of the Fe-Ni system[START_REF] Vernyhora | Thermodynamics of f.c.c. Ni-Fe Alloys in a Static Applied Magnetic Field[END_REF]

  𝐶: concentration of an element in a material 𝑐 𝐴𝛼 : concentration of atoms A on the 𝛼 site 𝑐 𝑉𝛼 : concentration of vacancies on the 𝛼 site𝐶 𝑉 𝑒𝑞 : equilibrium concentration of mono-vacancies 𝐶 𝑉 𝑀𝐶 : vacancy concentration in the Monte Carlo simulations 𝐶 𝐼 𝑒𝑞 : equilibrium concentration of interstitials 𝐷: self-diffusion coefficient 𝐷 0 : pre-factor of diffusion coefficient 𝐷 2 : diffusion coefficient of solute in dilute solution 𝐷 𝐴 𝐼𝑛𝑡𝑟 , 𝐷 𝐵 𝐼𝑛𝑡𝑟 : intrinsic diffusion coefficient of component A and B 𝐷 𝐴𝐵 𝐴 * : tracer diffusion coefficient of atom A in alloy A-B 𝐷 𝐴 𝐴 * : self-diffusion coefficient of A 𝐷 𝐴 𝐵 * : impurity diffusion coefficient of dilute B in element A 𝐷 ̃: inter-diffusion coefficient 𝑒 𝑖 : atom energy of atom 𝑖 𝑒 𝐴𝛼 : energy of atom A on the  sublattice 𝑓, 𝑓 2 : correlation factor of self-diffusion and interdiffusion coefficients 𝑔 𝑖 : Gibbs free energy of atom 𝑖 𝑔 𝐴𝑖 𝑆𝑃 : interactions between the atom 𝐴 on the saddle point position and atoms on its neighboring sites (𝑖)

  developed a Magnetic Cluster Expansion (MCE), based on a Heisenberg-Landau Hamiltonian, fitted to ab initio calculations. Its properties have been studied by Monte Carlo methods, but the combined equilibration of the chemical and magnetic configurations is very costly in computational time, and the complete phase diagram of the MCE model of Fe-Ni remains to be established.In the present paper, we propose a pair interaction model (PIM) based on a rigid lattice approximation, aiming at modelling the Fe-Ni system. The model does not describe explicitly the magnetic moments, nor the lattice relaxations, which makes it faster to process in Monte Carlo simulations. It only involves pair interactions between atoms that depends both on the local composition and temperature. Finite temperature effects of magnetic transitions or of lattice vibrations (harmonic and non-harmonic), on the energetic proprieties are taken into account through these dependencies.

FIG. 4 .

 4 FIG. 4. Formations enthalpies of quasi-random FCC Structures at 0 K: ab initio calculations (DFT) and pair interaction model (PIM, with the excess and magnetic contributions). The formations enthalpy of the γ solid solution of CALPHAD is also shown for comparison.

FIG. 5 .

 5 FIG. 5. Formations enthalpies of FCC ordered phases at 0 K: ab initio calculations (DFT, full circles) and pair interaction model (PIM, open circles) with (a) only first nn interactions, (b) first and second nn interactions, α = v 2 /v 1 = -0.7

FIG. 6 . 2 (

 62 FIG. 6. The evolution of the coefficient L P IM2

FIG. 7 .

 7 FIG.7. Evolution the long-range parameter η and of the short-range order parameters σ 1 and σ 2

Fig. 8 and 9 .FIG. 8 .

 98 Fig. 8 and 9. At 1000 K, x(∆µ) evolves continuously, with η 0: the disordered solid solution is stable in the whole composition range. At 600 K, η 0.76 to 0.53 between ∆µ = -0.188 and -0.068 eV, which corresponds to an over stoichiometric L1 0 phase; and η 0.49 between ∆µ = -0.05 and +0.3 eV, which corresponds to an almost stoichiometric L1 2 phase. Discontinuities and hysteresis on the x(∆µ) curve indicate first order transitions and the limits of two-phase domains.

Fig. 10 FIG. 9 .

 109 Fig. 10 for example, gives the Gibbs free energy of mixing of the Fe-Ni solid solution at T = 1500 K. The PIM is in very good agreement with the CALPHAD study. At this high temperature (well above T c ), the magnetic contribution is negligible. However it is

  FIG. 14. (a) Gibbs free energy of mixing of Calphad and the PIM and (b) long range order parameter of the PIM, for Fe-Ni alloys at 700 K.

FIG. 15 .

 15 FIG. 15. (a) Gibbs free energy of mixing of Calphad and the PIM and (b) long range order parameter of the PIM, for Fe-Ni alloys at 650 K.

  FIG. 16. (a) The Fe-Ni FCC phase diagram: comparison between the PIM and experiments (1950 Jos = [52], 1963 Heu = [53], 1980 Van = [54, 55]., (b) zoom in the region of the L1 2 ordering temperature.

FIG. 17 .FIG. 18 .

 1718 FIG. 17. The Fe-Ni FCC phase diagram of the PIM without magnetic contribution

2 . 3 .

 23 Recent analytical approaches, combining diffusion models and first principle calculations, are very promising (see e.g.[START_REF] Schuler | KineCluE: A kinetic cluster expansion code to compute transport coefficients beyond the dilute limit[END_REF][START_REF] Messina | Exact ab initio transport coefficients in bcc Fe -X ( X = Cr , Cu , Mn , Ni , P , Si ) dilute alloys[END_REF][START_REF] Ding | A computational study of impurity diffusivities for 5d transition metal solutes in α-Fe[END_REF][START_REF] Mantina | First-Principles Calculation of Self-Diffusion Coefficients[END_REF]. However, in concentrated metallic alloys, it is still very difficult to build analytical models that accurately predict the point defect concentrations and the correlation factors. It is also difficult to estimate finite temperature effects by ab initio methods. Therefore, most recent studies still deal with dilute alloys and low temperatures regimes (e.g. in bcc Fe based alloys, with ferromagnetic states). Assumptions must be madebased e.g. on Ruch's diffusion model[START_REF] Ruch | Analysis of diffusion in ferromagnets[END_REF])to model the effect of the FM/PM transition (such as in the study[START_REF] Ding | A computational study of impurity diffusivities for 5d transition metal solutes in α-Fe[END_REF], on solute diffusion in α-Fe). From that point of view, fcc Fe-Ni alloys are especially challenging, since they combine chemical and magnetic disorder. As a consequence, only the Fe and Ni diffusion coefficients in pure Ni have been estimated from first principles[START_REF] Tucker | Ab initio-based diffusion theory and tracer diffusion in Ni-Cr and Ni-Fe alloys[END_REF][START_REF] Toijer | Solute-point defect interactions, coupled diffusion, and radiation-induced segregation in fcc nickel[END_REF]).

(

  Le Claire 1978) (section II.3.2). However, for the study of diffusion in concentrated alloys, one must also know the evolution of the vacancy concentration with the alloy composition and temperature, including possible short-and long-range ordering effects. No general analytical model exists to estimate this concentration. Previous AKMC methods used for alloys with phase separation tendencies, such as Fe-Cu or Fe-Cr[START_REF] Soisson | Cu-precipitation kinetics in α -Fe from atomistic simulations: Vacancytrapping effects and Cu-cluster mobility[END_REF][START_REF] Nastar | Atomistic modeling of phase transformations: Point-defect concentrations and the time-scale problem[END_REF] do not work well for the ordering case.We therefore propose a new method based on the Widom integration technique (II.3.4). Using this method, the tracer diffusion and interdiffusion coefficients in concentrated Fe-Ni alloys are estimated and compared to experimental data (II.3.5.1). The Darken equation, which relates tracer and interdiffusion coefficients, is tested (II.3.5.2). Last, the effect of the L1 2 order-disorder transition on the tracer diffusion coefficients in FeNi 3 alloys, which has not yet measured experimentally, is predicted (II.3.5.3).

(

  Le Claire 1978), 𝑎 0 is the lattice constant of the solvent and 𝜗 2 0 is the attempt frequency of the solute in the solvent. The index 2 denotes the diffusion of the solute in a dilute solution. The model is valid when the interactions are limited to the first nearest interactions. In fcc alloys, five jump frequencies are then to be considered (Figure II.1): 𝜔 2 : solute-vacancy exchange rate 𝜔 1 : rotation rate of the solute-vacancy pair 𝜔 3 : dissociation rate of the solute-vacancy pair 𝜔 4 : association rate of the solute-vacancy pair 𝜔: vacancy-atom exchange rate in the solvent They are related to their corresponding migration barrier 𝐺 𝑖 𝑚𝑖𝑔 by:

Figure II. 1 .

 1 Figure II.1. Left: Five-frequency model for the diffusion of impurities in dilute fcc alloys. Right:Energy landscape for vacancy jumps in the neighborhood of a solute atom[START_REF] Mehrer | Diffusion in solids: fundamentals, methods, materials, diffusion-controlled processes[END_REF]

  Figure II.3. Self-diffusion coefficient in fcc Fe: experiments are represented by points and the dashed line gives the values of the k-PIM : 0.594 × 10 -4 𝑒𝑥𝑝(-2.94/𝑘 𝐵 𝑇)𝑚 2 𝑠 -1 (see section II.3.3.4)

  Figure II.5 respectively.

Figure II. 4 .

 4 Figure II.4. Impurity diffusion coefficients of Fe in fcc Ni, the black line corresponds to the assessment of Jönsson (Jönsson 1994), the purple line gives the value of our k-PIM: 1.44 × 10 -4 𝑒𝑥𝑝(-2.84/ 𝑘 𝐵 𝑇)𝑚 2 𝑠 -1 (see section II.3.3.5) and the points are different experimental studies

Figure II. 6 .

 6 Figure II.6. Self-diffusion and impurity diffusion coefficients in fcc Ni (top) and Fe (bottom): experiments are represented by points and the dashed lines correspond to the k-PIM.II.2.2.3 Diffusion in concentrated Fe-Ni alloys

Figure II. 7 .

 7 Figure II.7. Concentration dependence of the tracer diffusion coefficients in Fe-Ni solid solutions (Million et al. 1981): left: Fe tracer ; right: Ni tracer.

Figure II. 8 .

 8 Figure II.8. Comparison of the tracer diffusion coefficients 𝐷 𝐹𝑒𝑁𝑖 𝐹𝑒 * , 𝐷 𝐹𝑒𝑁𝑖 𝑁𝑖 * and the interdiffusioncoefficients 𝐷 ̃ at 1200 °C[START_REF] Jönsson | Mobilities in Fe-Ni alloys : assessment of the mobilities of Fe and Ni in fcc Fe-Ni alloys[END_REF]

Figure II. 9 .

 9 Figure II.9. Comparison of self and impurity diffusion coefficients of Million et al and other experimental studies (solid lines are the values of k-PIM). Top : Ni, bottom : Fe.

  proposed a fitting function for the interdiffusion coefficients dependence on temperature and concentration of based on different experiments, as shown in Figure II.10.

Figure II. 10 .

 10 Figure II.10. Interdiffusion coefficient 𝐷 ̃ of Jönsson (Jönsson 1994): different experiments collected are represented by points and the lines are the fitting functions of Jönsson.

Figure II. 11 :

 11 Figure II.11: Concentration dependence of the interdiffusion coefficients in Fe-Ni solid solutions at different temperatures. ○: experimental values; □: values calculated from self-diffusion and the Darken equation[START_REF] Růžičková | Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system[END_REF]

  Figure II.12). Bcc α-Fe is paramagnetic above and ferromagnetic below the Curie temperature, 𝑇 𝑐 = 1043K. The self-diffusion coefficient is a continuous function around T=1043 K. However, there is a significant curvature just below the Curie temperature. Extrapolating the Arrhenius behavior of the paramagnetic phase at low temperature (indicated by the red dashed line on Figure II.12) would lead to a significant overestimation of the diffusion coefficient (by e.g. two orders of magnitude at 800 K). The simplest model which describes this is a modified Arrhenius equation proposed byRuch et al.:

Figure 3

 3 Figure II.13. Tracer diffusion coefficients of 𝐶𝑢 65

Figure II. 14 .

 14 Figure II.14. Variation of the tracer diffusion coefficient with the concentration and temperature by MD: (a) tracer diffusion coefficients of vacancies; (b) tracer diffusion coefficients of Fe, Ni and all atoms; (c) effective activation energy of total (all atoms are tracers) and partial (Ni or Fe atom are tracers) tracer diffusion (Osetsky et al. 2018)

Figure II. 16 .

 16 FigureII.16. Magnetic contribution to vacancy formation properties in the three systems[START_REF] Li | Effects of magnetic excitations and transitions on vacancy formation: Cases of fcc Fe and Ni compared to bcc Fe[END_REF]

Ortega(

  [START_REF] Ortega | Self-Diffusion in FCC Metals: Static and Dynamic Simulations in Aluminium and Nickel[END_REF], obtained with an empirical potential, 𝑆 𝑁𝑖,𝑉 𝑚𝑖𝑔 = 1.58𝑘 𝐵 , are used since this is the only reference found for this value. Then the vacancy formation entropy is adjusted to experimental results of Ni self-diffusion coefficients via the preexponential factor 𝐷 0 as shown in equation. Then the 𝑆 𝑁𝑖,2 𝑚𝑖𝑔 is adjusted to the experimental impurity diffusion coefficient of Ni in fcc Fe.

  Finally, a few experimental and theoretical results suggest that the FM/PM magnetic transition has little effect on the vacancy properties in Fe-Ni fcc alloys, except for the recent results shown in Figure II.16.

3 . 3 ,II. 3 . 2

 3332 we will explain the parametrization of the model and compare its results with the experiments, in the relatively simple situation of self-and impurity-diffusion, where accounting for the equilibrium vacancy concentration is straightforward. In section II.3.4, we will present a new method based on Widom integration, to determine the vacancy equilibrium concentration in concentrated Fe-Ni alloys. Last, in section II.3.5, we will study diffusion properties in concentrated Fe-Ni alloys. Tracer diffusion coefficients and interdiffusion coefficients will be calculated and compared to experiments; the Darken equation will be discussed, and we will investigate the effect of the chemical order-disorder transition on diffusion. Kinetic pair interaction model for Fe-Ni alloys: jump frequencies and migration barriers We propose to reproduce and predict the diffusion properties in fcc FeNi alloys with a concentrated-and temperature-dependent pair interaction model and kinetic Monte Carlo simulations. This model is based on the PIM given in Chapter I and is an extension of it. To distinguish these two conceptions, we will call them PIM and k-PIM respectively for the rest of this thesis. The k-PIM describes the vacancy jump frequencies in the system. A vacancy-atom exchange in the fcc lattice is shown in Figure II.17: the saddle-point position (m) is surrounded by 4 nearest-neighbor atoms (i).

Figure II. 17 .

 17 Figure II.17. The vacancy migration in the fcc lattice. The vacancy (dotted circle) exchanges with an 𝐴 atom. The red sphere shows the saddle point position (𝑚) and the green ones its first nearest neighboring sites (𝑖)

Figure II. 19 .

 19 Figure II.19. Impurity diffusion coefficients in pure Fe (top) and Ni (bottom). The symbols in colors correspond to the experimental values, the black triangles to the AKMC simulations and the lines toLe Claire model.

Figure II. 20 .

 20 Figure II.20. Comparison of tracer diffusion coefficients in fcc Fe (top) and fcc Ni (bottom): small points are experimental values, large points are AKMC results and the dashed lines are k-PIM values

  chapter I gives the isotherm = 𝑓(𝜇 𝑁𝑖 -𝜇 𝐹𝑒 ) , as shown e.g. in Figure II.21. Then one can choose the value of 𝜇 𝑁𝑖 -𝜇 𝐹𝑒 to get an equilibrium configuration of the desired composition (once 𝜇 𝑁𝑖 -𝜇 𝐹𝑒 is known, the values of 𝜇 𝑁𝑖 and 𝜇 𝐹𝑒 can be deduced from 𝐺 = 𝑥 𝑁𝑖 𝜇 𝑁𝑖 + 𝑥 𝐹𝑒 𝜇 𝐹𝑒 ). Widom 1 and 2 integration methods are used to compute 𝐺 𝑉 𝑓𝑜𝑟 (𝑥 𝑁𝑖 ).

Figure II. 21 .

 21 Figure II.21. Evolution of the Ni concentration x and long-range order parameter 𝜂 of the PIM for Fe-Ni alloys at 600 K, as a function of ∆𝜇 = 𝜇 𝑁𝑖 -𝜇 𝐹𝑒

Figure II. 23 .

 23 Figure II.23. Gibbs free energies for the vacancy formation in Fe-Ni alloys at 700 K, computed with Widom 1 and Widom 2 methods, and compared with the model of Hagen and Finnis for the FeNi3-L12 phase

Figure II. 24 .

 24 Figure II.24. Gibbs free energies for the vacancy formation in Fe-Ni alloys at 500 K, computed with Widom 1 and Widom 2 methods and compared with the model of Hagen and Finnis for the L10 and L12 structure.

Figure II. 25 .

 25 Figure II.25. Gibbs free energies for vacancy formation on the Fe and Ni sublattices of the L10-FeNi and L12-FeNi3 phase, at 500 K. Top: Widom 1 method vs Hagen & Finnis, Bottom: Widom 2 method vs Hagen & Finnis model.

3 . 3 . 5 :

 335 we introduce here a dependence of 𝑔 𝑁𝑖𝐹𝑒 𝑆𝑃 and 𝑔 𝐹𝑒 𝑆𝑃 on the local concentration around the saddle-point position, 𝑥 𝑁𝑖 𝑆𝑃 , adjusted to the experiments to improve the results of the AKMC simulations. This concentration is defined by the number of Ni atoms among the four first-nearest neighbors -sites (𝑖) in Figure II.17. The concentration dependence is given by: of 𝑔 𝑁𝑖𝐹𝑒 𝑆𝑃 and 𝑔 𝐹𝑒𝑁𝑖 𝑆𝑃 are given in Table II.19, and shown in Figure II.29. The comparison between the experiments and the AKMC simulations, before and after optimization, is given in Figure II-28.

Figure II. 26 .

 26 Figure II.26. Tracer diffusion coefficients of Fe and Ni in fcc Fe-Ni alloys at different temperatures and concentrations: comparison between the AKMC simulations (full squares), the experiments of Million (open circles), and the estimation of Jönsson (dotted line). Top figures: before optimization (𝑔 𝑁𝑖𝐹𝑒 𝑆𝑃 and 𝑔 𝐹𝑒𝑁𝑖 𝑆𝑃 constant); bottom figures: after optimization (𝑔 𝑁𝑖𝐹𝑒 𝑆𝑃 and 𝑔 𝐹𝑒𝑁𝑖 𝑆𝑃 depend on 𝑥 𝑁𝑖 𝑆𝑃 )

Figure II. 27 :

 27 Figure II.27: function showing the concentration dependence for 𝑔 𝑁𝑖𝐹𝑒 𝑆𝑃 (𝑥 𝑁𝑖 𝑆𝑃 )

For

  the tracer diffusion coefficients of Ni, there are more differences between the AKMC simulations,Jönsson and Million et al., especially on the Fe rich side. At 𝑥 𝑁𝑖 = 0, this is again because the values of Million et al. for 𝐷 𝐹𝑒 𝑁𝑖 * are above those selected by Jönsson (as discussed in section II.3.3.5, see Figure II.5). The results of our simulations are between the two. At 𝑥 𝑁𝑖 = 1, the three sets of data agree for 𝐷 𝑁𝑖 𝑁𝑖 * , except Million et al at 950°C (with the problem mentioned above for this particular temperature). After optimization, our simulations give an evolution of 𝐷 𝐹𝑒𝑁𝑖 𝑁𝑖 * with 𝑥 𝑁𝑖 similar to that of Million et al, but shifted downwards by a factor of 2 or 3.

  will now use the k-PIM to model the interdiffusion coefficients in Fe-Ni solid solutions, compare them with the tracer diffusion coefficients, and test the Darken equations, as in the experiments ofMillion et al. discussed in section II.2.3.4.

  This method consists in measuring the attenuation rate of a sinusoidal concentration wave of small amplitude (Chang and Giessen 1985). The initial configuration in the 𝑥 direction is shown schematically in Figure II.28, with a concentration profile 𝑥 𝑁𝑖 (𝑥, 𝑡 = 0) = 𝑥 𝑁𝑖 + 𝐴 0 sin(2𝜋𝑥/ 𝜆), an initial amplitude 𝐴 0 of a few percent and a wavelength 𝜆.

Figure II. 28 .

 28 Figure II.28. Sinusoidal profile of Ni concentration on different planes in a lattice.

Figure II. 29 :

 29 Figure II.29: Attenuation of the concentration wave

Figure II. 30 .

 30 Figure II.30. AKMC simulation: evolution of a sinusoidal concentration profile in a Fe-98%Ni solid solution at 1523 K.

Figure II. 31 .

 31 Figure II.31. AKMC simulation: evolution of the amplitude A with time.

Figure II. 32 .

 32 Figure II.32. Interdiffusion coefficients measured by AKMC simulations (solid lines with full squares), by Million's experiments (open circles) and by Jönsson's fitting functions (dashed lines)

Figure II. 33 .

 33 Figure II.33. Evolution of the thermodynamic factor versus Ni content in Fe-Ni, according to Million et al[START_REF] Růžičková | Self-diffusion of the components in the F.C.C. phase of binary solid solutions of the Fe-Ni-Cr system[END_REF]) and to[START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]

Figure II. 34

 34 Figure II.34 shows a comparison between the interdiffusion coefficients measured by Million et al, to those calculated from their tracer diffusion coefficients and the Darken equation, with the two thermodynamic factors. The differences are slightly reduced when using Cacciamani's thermodynamic factor.

Figure II. 34 .

 34 Figure II.34. Interdiffusion coefficients directly measured (open symbols) compared to those obtained using Darken equation (full symbols) and two different thermodynamic factors: left 𝛷 of Million; right: 𝛷 of Cacciamani

Figure II. 35 .

 35 Figure II.35. Interdiffusion coefficients by direct measurement (empty symbols) and using Darken equation (full symbols) obtained by kinetic Monte Carlo simulations

Figure II. 36 : 2 (

 362 Figure II.36: AKMC simulations of the tracer diffusion coefficients of Fe and Ni in Fe-75%Ni. The dashed lines represent the extrapolation of the Arrhenius law of the disordered alloy below 𝑇 𝑐 𝐿1 2

  Modified k-PIMFor the diffusion in the ordered phase, we therefore propose a modified k-PIM, hereafter referred as k-PIM2, which includes a transition between high-temperature parameters, derived from experiments, and low temperatures parameters, derived from DFT calculations (as has been done for the thermodynamic properties and phase diagram, see Chapter I, section Ⅱ.C.3). This adjustment is introduced through the change of the vacancy formation energy in Fe and Ni, while the vacancy migration barriers will be kept the same. At low temperatures, k-PIM2 is adjusted on the vacancy formation energies in the ordered structure L1 2 FeNi 3 of DFT calculations. At temperatures higher than the order-disorder transition temperature, the k-PIM2 parameters remain the same as in k-PIM, so that the diffusion properties in solid solutions do not change. The resulting evolution of the Gibbs free energies for vacancy formation with the temperature, on the two sublattices of the ordered L1 2 FeNi 3 phase, is shown in Figure II.37. The new tracer diffusion coefficients are shown in Figure II.38, and compared with those obtained with the original k-PIM: one observes that the curvature is strongly reduced with k-PIM2. This is due to the change in the vacancy formation energy. For example, at T = 600K, the k-PIM gives 𝐺 𝑉 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 3 -𝐿1 2 ) = 1.95eV while the k-PIM 2 gives 𝐺 𝑉 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 3 -𝐿1 2 ) = 2.20eV . The difference of 0.25 eV gives a factor of exp(0.25/(𝑘 𝐵 𝑇) ≈ 125, which corresponds to the difference on the tracer diffusion coefficients in Figure II.38.

Figure II. 37 .

 37 Figure II.37. Gibbs free energies for vacancy formation on the Fe and Ni sublattices of the ordered 𝐿1 2 𝐹𝑒𝑁𝑖 3 phase, in k-PIM and k-PIM2

  this chapter, the pair interaction model (PIM) developed in Chapter I, to model the thermodynamic properties of Fe-Ni alloys, has been extended to a kinetic model (k-PIM) to model vacancy and atomic diffusion properties. As in Chapter I, our strategy has been to fit the k-PIM parameters on existing DFT calculations (usually at 0 K) and experimental data (usually at high temperatures). The additional k-PIM parameters (atom-vacancy and saddle-point pair interactions) are temperature and concentration-dependent pair interactions, divided in enthalpic and (nonconfigurational) entropic terms, and in magnetic and non-magnetic terms. The tracer and interdiffusion coefficients of the k-PIM are measured by AKMC simulations, using a time rescaling to take into account the evolution of the vacancy concentration with the alloy composition. For this purpose, we have used a new method, based on the Widom insertion technique. This new method is in good agreement with the Bragg-Williams approximation at high temperatures, in fully disordered solid solutions, and with the model of Hagen & Finnis at low temperatures, in perfectly ordered structures.

Figure III. 4 .

 4 Figure III.4. Time-Temperature-Transformation diagram of ordering kinetics in Fe-75%Ni duringisothermal annealing[START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF]

Figure III. 7 .

 7 Figure III.7. Evolution of the average domain size versus annealing time in Fe-75%Ni at 470°C (743K) and 480°C (753 K)[START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF]

Ferjani

  et al. and to Morris et al. [497°C (770 K)], which allows a direct comparison.

  Morris et al. observed a faster evolution of the LRO parameter than Ferjani et al., by approximately a factor 2 at the beginning and 10 at the end of the annealing. By contrast, the domain size evolution is about 3 times slower in Morris et al. than in Ferjani et al. Morris et al. observed two stages on the growth of domain, with a change in the growth rate at 𝑡~65 h, while Ferjani et al. reported only one stage at this temperature. By eliminating the time, one can show the evolution of the domain size as a function of the relative long-range order parameter, 𝐿(𝜂/𝜂 ∞ ). In this way, the results of Morris and Ferjani are compared in Figure III.9. The two experiments shown two different kinetic paths, Morris et al. predicting a slower growth of more ordered domains than Ferjani et al.

Figure III. 9 .

 9 Figure III.9. Evolution of the ordered domain size as a function of relative long-rang order parameter in Fe-75%Ni at T=470 °C (743 K): comparison between the observations of Ferjani et al.[START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF] and[START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF]).

Figure III. 12 .

 12 Figure III.12. To visualize the ordered domains, we use a different color for the Fe atoms located on the 4 sublattices, as in (Pareige et al. 1999). Starting from a random solution, one rapidly observes the formation of a high density of small ordered domains, equally distributed among the four variants. The evolution of LRO parameter, the domain size and the vacancy concentration are shown in Figure III.13. The system starts to be partially ordered very shortly after the simulation begins (about several hours), and becomes completely ordered (𝜂~0.8, or 𝜂/𝜂 ∞ ~1) after approximately 10 3 hours. We can see that the measurement of the domain size by equation (III-3) (Figure III.13) actually matches the one observed on the microstructure, in direct space (Figure III.12.). During the ordering transformation, one observes a decrease of the vacancy concentration, from 𝑥 𝑣 𝑒𝑞 ~1.6 × 10 -11 in the initial random solid solution, to 𝑥 𝑣 𝑒𝑞 ~4.5 × 10 -12 in the final ordered state.

Figure III. 12 .

 12 Figure III.12. AKMC simulation of ordering in Fe-75%Ni at 434°C (707 K), with the k-PIM2 parameters: evolution of the microstructure (with time in hours, only the Fe atoms are shown, with a different color for each variant).

Figure III. 13 .

 13 Figure III.13. AKMC simulation of ordering in Fe-75%Ni at T=434°C (707 K), with the k-PIM2 parameters. (a) evolution of the LRO parameter, (b) evolution of the domain size, (c) evolution of the equilibrium vacancy concentration.

Figure III. 14 .

 14 Figure III.14. Evolution of the relative long-range order parameter at T=442 °C (715 K), T=450 °C (723 K) ,T=460 °C (733 K) and T=470 °C (743 K) in Fe-75%Ni: comparisons between the experiments of Morris et al.[START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF]) and the AKMC simulations.

Figure III. 15 .

 15 Figure III.15. Evolution of the equilibrium vacancy concentration during the ordering of a Fe-Ni75% alloy at T=442°C (715 K). AKMC simulations with k-PIM and k-PIM 2 parameters.

Figure III. 16 .

 16 Figure III.16. Evolution of the long-range order parameter at T=470 °C (743 K) in Fe-75%Ni: comparison between the experiments of Ferjani et al[START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF],Morris et al[START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF]) and the AKMC simulations with k-PIM and k-PIM 2 parameters

Figure

  Figure III.17 shows the TTT diagram obtained with the AKMC simulations (with the k-PIM and k-PIM 2 parameters), compared with the one of Morris et al. When plotting the absolute values of 𝜂, for the k-PIM 2, we observe that the simulations are in good agreement with the experiments at low

Figure III. 18 .

 18 Figure III.18. Time-Temperature-Transformation diagram of LRO evolution in Fe-75%Ni comparison between the AKMC simulations and experiments of Morris et al.

Figure III. 19 .

 19 Figure III.19. Evolution of domain size with time at different temperatures (715 K, 723 K,733 K and 743K): comparison between experiments (points) and MC results (lines)

Figure III. 20 .

 20 Figure III.20. Evolution of domain size with time at T=743 K: comparison between experiments (points) and MC results (lines)

Figure III. 21 .

 21 Figure III.21. Evolution of the ordered domain size as function of relative long-rang order parameter in Fe-75%Ni at T=743 K (470 °C): comparison between the AKMC results and experiments of Ferjani et al.[START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF] and[START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF]).

  For example, the present models overestimates the critical ordering temperature by approximately 25 K: this could be easily corrected by an optimization of the pair interactions, and we hope it would improve the agreement between the simulations of the experiments and Morris et al. with regard to the absolute value of the LRO parameters. The use of the present transition between the parameters based on DFT and those derived from high temperature measurements (Figure II.37) is probably a more important issue. Magnetic interactions models may provide a more physical input for these kinds of simulations in the future.

  modèle d'interactions de paire sur ré seau rigide pour é tudier les proprié té s thermodynamiques, les proprié té s de diffusion et la ciné tique de mise en ordre dans les alliages Fe-Ni avec une structure cubique à faces centré es (fcc). La raison pour laquelle nous avons choisi ce systè me binaire à é tudier est que d'une part le systè me Fe-Ni est la base des aciers austé nitiques qui ont é té largement utilisé s dans de nombreux composants de ré acteurs nuclé aires de gé né ration actuelle et potentiellement future. De plus, il pré sente aussi lui-mê me un grand inté rê t industriel et a é té trè s é tudié au cours des derniè res dé cennies. Cependant, des questions non ré solues continuent de se poser concernant ce systè me car bien que simplement binaire, il reste complexe puisqu'il implique à la fois des transitions magné tiques, chimiques et ordre-dé sordre. Par consé quent, dans cette thè se, notre objectif est de comprendre et de pré dire ces proprié té s dans le systè me Fe-Ni dans le cadre d'un modèle d'interaction de paire simple (PIM). Le modè le dé veloppé dans ce travail peut ensuite ê tre é tendu à des systè mes ternaires tels que le systè me Fe-Ni-Cr ou à la modélisation de l'irradiation dans ces alliages. La thè se est constitué e de trois chapitres. Le chapitre I pré sente la partie thermodynamique du modè le. Le chapitre II pré sente la partie ciné tique du modè le. Puis dans le chapitre III, les ré sultats de mise en ordre dans le systè me Fe-Ni avec une concentration de 75% Ni sont pré senté s. Le chapitre I pré sente un modè le de paramé trisation systé matique des interactions de paire sure ré seaux rigides basé sur des donné es ab-initio et CALPHAD pour le systè me Fe-Ni. Pour mieux dé crire les proprié té s thermodynamiques du systè me Fe-Ni, à 0 K, le modèle d'interaction de paire est ajusté sur des calculs ab initio d'enthalpies de formation de structures ordonné es et dé sordonné es (spé cial quasi-alé atoire) ; à des tempé ratures plus é levé es, le modè le est systé matiquement adjusté sur l'é nergie libre de Gibbs des solutions solides gamma Fe-Ni donné e par la mé thode CALPHAD. Diffé rentes contributions dans les interactions de paire sont distingué es et comparé es sé paré ment. Les interactions comprennent une partie magné tique et une partie non magné tique et dé pendent de la tempé rature et de la composition locale. Ainsi, le changement magnétique n'est pas traité explicitement dans les simulations de Monte Carlo mais permet tout de mê me au modè le de dé crire l'influence des transitions magné tiques sur les proprié té s thermodynamiques. Nous montrons que les donné es ab initio et CALPHAD pour la solution solide et pour la phase ordonné e FeNi 3 -L1 2 peuvent ê tre bien reproduites, dans un large domaine de composition et de tempé rature, en utilisant des interactions de paire entre les premiè res et deuxiè mes voisins qui dé pendent de la tempé rature et de la composition locale de l'alliage. Le diagramme de phase du systè me fcc Fe-Ni est dé terminé par des simulations Monte Carlo dans l'ensemble semi-grand canonique et est comparé à des é tudes expérimentales et à d'autres modè les. Nous discutons é galement en particulier de la stabilité de la phase ordonné e FeNi-L1 0 à basse température. L'existence de cette phase est difficile à observer dans les conditions habituelles, du fait d'une diffusion lente, mais elle avait é té pré cé demment suggé ré e par des observations dans des mé té orites et par des expériences d'irradiation et notre modè le pré dit la stabilité d'une phase FeNi-L1 0 à basse tempé rature. Le chapitre II pré sente des discussions sur les proprié té s de diffusion dans le systè me Fe-Ni. L'objectif est d'é tudier les proprié té s de diffusion dans le systè me Fe-Ni par des simulations Monte Carlo ciné tiques avec le modè le d'interaction de paire. Par conséquent, le modèle d'interaction de paire du chapitre I est ensuite étendu en tant que modèle d'interaction de paire ciné tique (k-PIM) pour prendre en compte la diffusion par les lacunes. En plus des interactions de paire de la partie thermodynamique, nous introduisons des interactions atome-lacune, qui contrôlent les é nergies de formation des lacunes et les concentrations à l'é quilibre, et des interactions aux positions des cols, qui contrôlent les barriè res de migration des lacunes. Comme dans le chapitre I, les interactions magné tiques et non magné tiques sont distingué es. Les paramè tres correspondants sont ajusté s sur des calculs ab initio des barriè res de migration à 0 K, et sur des donné es expé rimentales de diffusion à hautes tempé ratures. Nous discutons é galement en particulier des concentrations de lacunes dans les alliages Fe-Ni concentré s, où aucune expression analytique générale n'existe sauf dans certains cas limites tels qu'une solution solide complè tement dé sordonné e ou une structure parfaitement ordonné e. Par consé quent, une nouvelle méthode basée sur l'inté gration de Widom est proposé e. Puis avec des simulations Monte Carlo cinétiques, nous montrons qu'il est possible de bien reproduire tous les coefficients de diffusion expé rimentaux connus : l'autodiffusion et la diffusion des impureté s dans les alliages dilué s, les coefficients de traceur et d'interdiffusion dans les alliages concentrés. L'effet de la transition ordre-dé sordre sur les coefficients de diffusion du traceur dans l'alliage FeNi 3 -L1 2 , qui n'a pas encore é té mesuré expé rimentalement, est é tudié . Pour simuler la diffusion dans les phases ordonné es, un ajustement supplémentaire du modèle d'interaction de paire ciné tiques est effectué basé sure des é nergies de formation de lacunes dans la phase ordonné e FeNi 3 -L1 2 donné es par des calculs DFT à 0 K. L'objectif est d'obtenir un nouveau modè le qui est nommé k-PIM2 dans le travail, pour amé liorer les ré sultats à des tempé ratures infé rieures à la tempé rature de transition ordre-dé sordre en maintenant les mê mes proprié té s de diffusion donné es par k-PIM à des tempé ratures plus é levé es. Enfin le chapitre III pré sente les discussions sur la ciné tique de mise en ordre dans la phase ordonné e FeNi 3 -L1 2 . Des simulations Monte Carlo ciné tiques avec le k-PIM et le k-PIM2 sont ré alisé es, avec le calcul de la concentration de lacunes dans le systè me par la mé thode de Widom proposé e au Chapitre II. Les é volutions de la taille des domaines et du degré d'ordre à longue distance obtenues avec des simulations Monte Carlo sont mesuré es par des transformations de Fourier rapides. Les simulations sont comparé es à diffé rentes ciné tiques expé rimentales à diffé rentes tempé ratures. En gé né ral, les simulations ré alisé es avec les paramè tres k-PIM2 sont en accord avec les expé riences avec une incertitude qui est de l'ordre des diffé rences entre les é tudes expé rimentales. Les simulations ré alisé es avec les paramè tres k-PIM sont significativement plus lentes, car elles conduisent à une surestimation des é nergies libres de Gibbs de la formation des lacunes dans la phase L1 2 , donc à une sous-estimation des concentrations de lacunes et à une diffusion trop lente. Notre travail donne des descriptions globalement satisfaisantes de la thermodynamique, de la diffusion et de la cinétique d'ordre dans le systè me fcc Fe-Ni avec des simulations Monte Carlo en utilisant un modèle relativement simple d'interaction de paire dé pendant à la fois de la composition et de la tempé rature. Ce travail peut ê tre potentiellement é tendu pour ê tre utilisé dans d'autres systè mes binaires ou les alliages ternaires Fe-Ni-Cr, pour la modé lisation des aciers austé nitiques. En outre, il peut é galement ê tre utilisé pour simuler l'effet de l'irradiation, en particulier la sé gré gation induite par le rayonnement, comme cela a dé jà é té fait dans les alliages ferriques Fe-Cr. disorder transformation Modé lisation atomique des proprié té s thermodynamiques, de la diffusion et de la mise en ordre dans les alliages Fe-Ni Ré sumé Nous présentons une modélisation à l'échelle atomique des propriétés thermodynamiques, de la diffusion et des ciné tiques de mise en ordre dans les alliages fer-nickel de structure cubique à faces centré es. Le model utilise des interactions de paires qui dé pendent de la tempé rature et de la composition locale. Ces interactions de paires sont ajusté es sur des calculs ab initio à 0 K, et sur des donné es expé rimentales aux tempé ratures non nulles. Le diagramme de phases du systè me est construit et comparé au diagramme de phases expé rimental. L'ensemble des coefficients diffusion expé rimentaux connus est bien reproduit : les coefficients d'autodiffusion et de diffusion d'impureté dans les alliages dilués, les coefficients de traceur et d'interdiffusion dans les alliages concentrés. Les cinétiques de mise ordre dans l'alliage FeNi 3 sont é tudies et comparé s aux expé riences disponibles. Un bon accord est obtenu pour l'évolution du paramètre d'ordre à longue distance et pour l'évolution de la taille des domaines ordonné s. Mots-Clefs : alliage Fe-Ni, dé fauts ponctuels, simulations Monte Carlo, diffusion, transformations ordre-dé sordre Atomistic modeling of thermodynamic properties, diffusion and ordering in Fe-Ni alloys Abstract We present an atomic-scale modeling of thermodynamic properties, diffusion and ordering kinetics in iron-nickel alloys of face-centered cubic structure. The model uses pair interactions that depend on temperature and local composition. These pair interactions are fitted to ab initio calculations at 0 K, and to experimental data at non-zero temperatures. The phase diagram of the system is constructed and compared to the experimental phase diagram. All known experimental diffusion coefficients are well reproduced: self-diffusion, impurity diffusion coefficients in dilute alloys, tracer and interdiffusion coefficients in concentrated alloys. The ordering kinetics in the FeNi 3 alloy are studied and compared to available experiments. A good agreement is obtained for the evolution of the long-range order parameter and for the evolution of the size of the ordered domains. Key words: Fe-Ni alloy, point defects, Monte Carlo simulations, diffusion, order-disorder transformation

TABLE I .

 I Formation enthalpies of FCC ordered phases in Fe-Ni (DFT calculations from

TABLE II .

 II The coefficients L DF T

j

of the Redlich-Kister polynomial for the excess enthalpy of mixing of the γ solid solution of the PIM (in eV).

TABLE III .

 III Formation enthalpies of FCC ordered phases in a pair interaction model with first and secong nearest neighbor interactions.

	Ordered phases x 1 x 2 H P IM f or
	Fe 7 Ni-cI32	1 34	5 34	3 2 v 1 + 3 4 v 2
	Fe 3 Ni-L1 2	9 34 0 3v 1
	Fe 3 Ni-Z1	11 34	9 34 2v 1 + 1 2 v 2
	Fe 2 Ni-C11 f	6 17	6 17

  𝐿 𝐴𝐴 𝑋 𝐴 + 𝐿 𝐴𝐵 𝑋 𝐵 + 𝐿 𝐴𝑉 𝑋 𝑉 𝐽 𝐵 = 𝐿 𝐴𝐵 𝑋 𝐴 + 𝐿 𝐵𝐵 𝑋 𝐵 + 𝐿 𝐵𝑉 𝑋 𝑉 𝐽 𝑉 = 𝐿 𝐴𝑉 𝑋 𝐴 + 𝐿 𝐵𝑉 𝑋 𝐵 + 𝐿 𝑉𝑉 𝑋 𝑉

  We can notice that equation (II-23) has the form of Fick's law, with:

	The Gibbs-Duhem relation imposes: 1 +	𝜕ln𝛾 𝐴 𝜕ln𝑥 𝐴	= 1 +	𝜕ln𝛾 𝐵 𝜕ln𝑥 𝐵	. This quantity is referred to as the
	thermodynamic factor 𝛷. 𝐷 𝐴 𝐼𝑛𝑡𝑟 = -		𝐿 𝐴𝐴 𝑥 𝐴	(1 -	𝐿 𝐴𝐵 𝑥 𝐴 𝐿 𝐴𝐴 𝑥 𝐵	) 𝑘 𝐵 𝑇𝛷
		𝐷 𝐵 𝐼𝑛𝑡𝑟 = -	𝐿 𝐵𝐵 𝑥 𝐵		(1 -	𝐿 𝐵𝐵 𝑥 𝐴 𝐿 𝐴𝐵 𝑥 𝐵	) 𝑘 𝐵 𝑇𝛷	(II-24)
	where 𝐷 𝐴 𝐼𝑛𝑡𝑟 and 𝐷 𝐵 𝐼𝑛𝑡𝑟 are the intrinsic diffusion coefficients of A and B, respectively.
	In general, the interdiffusion coefficient 𝐷 ̃ can be related to the intrinsic diffusion coefficients
	by:
		𝐷 ̃= (𝑥 𝐴 𝐷 𝐵 𝐼𝑛𝑡𝑟 + 𝑥 𝐵 𝐷 𝐴 𝐼𝑛𝑡𝑟 )	(II-25)
		𝐷 ̃= (𝑥 𝐴 𝐷 𝐴𝐵 𝐵 * + 𝑥 𝐵 𝐷 𝐴𝐵 𝐴 * )𝛷𝑆 = 𝐷 ̃𝑆	(II-27)
	with 𝑆 =	𝐶 𝐴 𝐷 𝐴𝐵 𝐵 * 𝑟 𝐵 +𝐶 𝐵 𝐷 𝐴𝐵 𝐴 * 𝑟 𝐴 𝐶 𝐴 𝐷 𝐴𝐵 𝐵 * +𝐶 𝐵 𝐷 𝐴𝐵 𝐴 *
		-(	𝐿 𝐴𝐴 𝑥 𝐴	-	𝐿 𝐴𝐵 𝑥 𝐵	) 𝑘 𝐵 𝑇 (1 +	𝜕ln𝛾 𝐴 𝜕ln𝑥 𝐴	)	𝑑𝑥 𝐴 𝑑𝑥
		𝐽 𝐵 = -( 𝐿 𝐵𝐵 𝑥 𝐵	-	𝐿 𝐴𝐵 𝑥 𝐴		) 𝑘 𝐵 𝑇 (1 +	𝜕ln𝛾 𝐵 𝜕ln𝑥 𝐵	)	𝑑𝑥 𝑑𝑥 𝐵	(II-23)

If the off-diagonal terms of the matrix 𝑳 are negligible, by relating the intrinsic and tracer diffusion coefficients of a component by the thermodynamic factor, 𝐷 ̃ can be expressed as: 𝐷 ̃= (𝑥 𝐴 𝐷 𝐴𝐵 𝐵 * + 𝑥 𝐵 𝐷 𝐴𝐵 𝐴 * )𝛷 (II-26) where 𝐷 𝐴𝐵 𝐵 * and 𝐷 𝐴𝐵 𝐴 * are the tracer diffusion coefficients of B and A in the alloy respectively. Equation (II-26) is called the Darken equation. We see that the interdiffusion coefficient will tend towards the self-diffusion coefficient 𝐷 𝐴 𝐴 * if 𝑥 tends towards 0, and the impurity diffusion coefficient 𝐷 𝐴 𝐵 * if 𝑥 tends towards 1.

The Darken equation is obtained under the assumption that the concentration of vacancies is in thermal equilibrium during the interdiffusion process. But vacancy sources and sinks exist so that a vacancy flux is created to maintain a local equilibrium. A correction term must be added against this socalled vacancy-wind effect. Manning and Brunner proposed to correct this effect by adding a factor 𝑆 and then we have the Darken-Manning equation as

[START_REF] Manning | Diffusion Kinetics for Atoms in Crystals[END_REF]

):

Fe and fcc Ni

  Table II.2. Experimental measurements of the vacancy formation enthalpy in fcc Ni[START_REF] Ehrhart | Ni[END_REF] Fe is stable between 1183 K and 1653 K and is always paramagnetic. Several experimental measurements of the self-diffusion coefficients are available for fcc Fe. A summary of the results is shown in TableII.4 and Figure II.3. An Arrhenius behavior is observed, over 3-4 orders of

	 Pure Ni Pure fcc nickel undergoes a transition from a ferromagnetic to a paramagnetic state above the Curie temperature 𝑇 𝐶𝑢𝑟𝑖𝑒 𝑁𝑖 = 633K (Dinsdale 1991). Several measurements of 𝐷 𝑁𝑖 𝑁𝑖 Table II.1. Experimentally measured self-diffusion coefficients in fcc Ni D 0 (10 -4 𝑚 2 𝑠 -1 ) Q(eV) T range (K) Method Reference 1.77 2.955 1253 K-1673 K radioisotope 𝑁𝑖 63 with electron analysis (Bakker 1968) 1.33 2.91 879 K-1193 K Ion-beam sputtering as micro-sectioning (Maier et al. 1976) 1.9 2.95 1315 K-1677 K lathe sectioning technique (Monma 1964) 1.9 2.90 773 K-923 K radioactive counting of the surface (Wazzan 1965) 0.4 2.77 1373 K-1448 K lathe sectioning technique (Reynolds, Averbach, and Cohen 1957) 3.44 3.0 1258 K-1578 K radioisotope 𝐹𝑒 59 and 𝑁𝑖 63 (Million et al. 1981) 𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 (eV) Method Reference* 1.60 Quenching 69M2 1.45-1.73 PAS 77C1 1.65-1.74 PAS 77N1 1.6-1.8 PAS 77D1 1.58-1.63 Quenching 76W 1.55 PAS 79M1 1.76 PAS 79M2 1.54 PAS 80L1 1.8 PAS 81S3 1.79±0.05 Recommended value Ehrhart 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 (eV) Method Reference* 1.05 Irradiation, cold work 59 S1 1.03 Irradiation 65 M 1.1±0.01 Cold work 66 M 1.3-1.9 Quenching 78 W 3 1.2 TEM 78 k1 1.1 Irradiation 78 A 1.04 irradiation 81 K 2 1.04±0.04 Recommended value Ehrhart * the full references can be found in (Ehrhart 1991)  Pure Fe Pure iron has two different crystal structures: bcc (α-Fe below 1183 K and δ-Fe above 1653 K) D 0 (10 -5 𝑚 2 𝑠 -1 ) Q(eV) T range (K) Method Reference 1.8 2.797 973 K-1711 K Fe electroplated absorption (Buffington, Hirano, and Cohen 1961) 4.9 2.943 1443 K-1634 K tracer-sectioning technique (Heumann and Imm 1968) 0.2 2.78 1429 K-1622 K tracer-sectioning technique (Graham and Tomlin 1963) 5.8 3.2 1243 K-1630 K radioisotope 59 𝐹𝑒 with electron analysis (Birchenall and Mehl 1950) 0.41 2.91 1373 K-1523 K tracer-sectioning technique (Fillion and Calais 1977) and fcc (γ-Fe). γ-Table II.4. Experimentally measured self-diffusion coefficients in fcc Fe 7.21 2.89 1258 K-1578 K radioisotope 59 𝐹𝑒 and
	with electron analysis

* have been carried out, only in paramagnetic Ni. They are summarized in Table II.1 and Figure II.2. One observes an almost perfect Arrhenius behavior over about 10 orders of magnitudes. Slightly different values of 𝐷 0 and 𝑄 have been proposed to fit the results with the Arrhenius law (

Table II.1). In his review, Ehrhart proposes 𝐷 𝑁𝑖 𝑁𝑖 * = 0.92 × 10 -4 exp(-2.88/𝑘 𝐵 𝑇) (Ehrhart 1991). Our k-PIM correspond to 𝐷 𝑁𝑖 𝑁𝑖 * = 0.696 × 10 -4 exp(-2.83/𝑘 𝐵 𝑇)m 2 s -1 (see section II.3.3.4). Both give practically the same values of 𝐷 𝑁𝑖 𝑁𝑖 * . The k-PIM estimation is shown in advance, for comparison, in Figure II.2.

Figure II.2. Self-diffusion coefficient in fcc Ni: experiments are represented by points and the dashed line gives the values of our k-PIM : 0.696 × 10 -4 exp(-2.83/𝑘 𝐵 𝑇)𝑚 2 𝑠 -1 (see section II.3.3.4)

The vacancy formation enthalpy 𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 (Table

II

.2) and migration enthalpy 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 (Table

II

.3) have been measured independently and reviewed by Ehrhart who recommends the values 𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 = 1.79 ± 0.05eV and 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 = 1.04 ± 0.04eV (Ehrhart 1991). The formation enthalpies were mostly obtained by positron annihilation spectroscopy and the migration enthalpies by electrical resistivity recovery after irradiation, quenching or cold work. These two values give 𝑄 = 𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 + 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 = 2.83eV, in good agreement with the measurements of diffusion coefficients (Table II.1), which confirms that diffusion in Ni operates by a mono-vacancy mechanism.

It is interesting to note that diffusion coefficients and the formation enthalpy have been measured in paramagnetic Ni, but that the migration enthalpy have been measured by resistivity recovery in ferromagnetic Ni (the peak of the corresponding stage III is located at 𝑇~400 K). The fact that the relation 𝑄 = 𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 + 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 is fulfilled therefore suggests that in pure Ni, the magnetic transition has little influence on the vacancy migration enthalpy.

*the full references can be found in

[START_REF] Ehrhart | Ni[END_REF]

Table

II

.3. Experimental measurements of the vacancy migration enthalpy in fcc Ni

[START_REF] Ehrhart | Ni[END_REF]

magnitude, much less than in pure Ni, due to the limited range of temperature for γ-Fe. The estimations of 𝐷 0 and 𝑄 (Table

II

.4) are therefore less precise. Our k-PIM gives 0.549 × 10 -4 exp(-2.94/ 𝑘 𝐵 𝑇)m 2 s -1 (see section

II.3.3.4)

.

Table II .

 II 5. Experimental vacancy formation and migration enthalpy of fcc Fe

	Method	𝐻 𝐹𝑒,𝑉 𝑓𝑜𝑟 (eV) 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 (eV)	Condition	Reference
	positron annihilation	1.7 ± 0.2	293 K-1593K	(Kim and Buyers 1978)
	Positron annihilation	1.4 ± 0.15 1.54 ± 0.15 * 1183 K-1663 K (Matter, Winter, and
				Triftshäuser 1979)

* deduced from the experimental activation energy 𝑄

[START_REF] Heumann | Self-diffusion and isotope effect in γ-iron[END_REF]

  2, equation (II-6) and (II-8), self-diffusion coefficients in pure metals depends on the Gibbs formation and migration energies of vacancies, respectively 𝐺 𝑉 𝑓𝑜𝑟 and 𝐺 𝑉 𝑚𝑖𝑔 . The activation DFT calculations have been carried out to determine the self-diffusion coefficient as well as the vacancy formation and migration enthalpies in pure fcc Ni. Calculations for both non-magnetic and ferromagnetic state exist. The results and the calculation conditions are summarized in Table II.6 and Table II.7. We also give the results obtained with an EAM potential by Ortega, which include an estimation of the migration entropy that we will use later.

	energy can be written as 𝑄 = 𝐻 𝑉 𝑓𝑜𝑟 + 𝐻 𝑉 𝑚𝑖𝑔 and the pre-exponential factor as 𝐷 0 = 𝑎 2 𝜐 0 𝑓exp[(𝑆 𝑉 𝑓𝑜𝑟 +
	𝑆 𝑉 𝑚𝑖𝑔 )/𝑘 𝐵 ].
	 Pure Nickel
		Table II.6. Vacancy formation/migration enthalpy/entropy in fcc Ni
	𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 (eV) 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 (eV) 𝑆 𝑁𝑖,𝑉 𝑓𝑜𝑟 (𝑘 𝐵 ) 𝑆 𝑁𝑖,𝑉 𝑚𝑖𝑔 (𝑘 𝐵 ) Method	Reference
	1.63-1.73 1.01-1.15		Ab-initio LDA NM	(Hargather et al. 2014) 1
	1.70-1.87 1.14-1.15		Ab-initio LDA FM	(Hargather et al. 2014) 1
	1.65	1.09	1.82	Ab-initio GGA FM	(Tucker et al. 2010) 2
	1.4	1.05	1.83	Ab-initio GGA FM	(Toijer et al. 2021) 3
	1.56	0.97	1.58	Classical MD	(Garcia Ortega, Ramos
					de Debiaggi, and Monti
					2002) 4
	1: Ecut=350 eV; supercell=2×2×2=32 atoms; k-point=9 3 ; convergence condition: 0.01 eV.
	2: Ecut=270 eV; supercell=3×3×3=108 atoms; k-point=3 3 ; convergence condition: 35 meV.
	3: Ecut=350 eV; supercell=4×4×4=256 atoms; k-point=3 3 ; convergence condition: 20 meV/Å in force
	convergence
	4: EAM potential by Voter and Chen (Voter and Chen 1986).

Table II .

 II 7. Activation energy and pre-exponential factor for the self-diffusion in fcc Ni

	𝐷 0 (𝑚 2 𝑠 -1 )	𝑄(eV)	Method	Reference
	1.99 × 10 -5 -2.14 × 10 -4 2.85-3.03	Ab-initio LDA NM	(Hargather et al. 2014) 1
	2.32 × 10 -6 -4.34 × 10 -5 2.64-2.88	Ab-initio LDA FM	(Hargather et al. 2014) 1
		2.74	Ab-initio GGA FM	(Tucker et al. 2010) 2
	7.3 × 10 -6	2.44	Ab-initio GGA FM	(Toijer et al. 2021) 3
	1.35 × 10 -5 (CO)	2.53	MD	(Garcia Ortega, Ramos de
				Debiaggi, and Monti 2002) 4
	0.30 × 10 -5 (IO)	2.53	MD	(Garcia Ortega, Ramos de
				Debiaggi, and Monti 2002) 4
	1: Ecut=350 eV; supercell=2×2×2, 32 atoms; k-point=9 3 ; convergence condition: 0.01 eV.
	2: Ecut=270 eV; supercell=3×3×3, 108 atoms; k-point=3 3 ; convergence condition: 35 meV.
	3: Ecut=350 eV; supercell=4×4×4, 256 atoms; k-point=3

3

; convergence condition: 20 meV/Å in force convergence 4: EAM potential by Voter and Chen

[START_REF] Voter | Accurate Interatomic Potentials for Ni, Al and Ni3Al[END_REF]

, calculation approximation: independent oscillators (IO) and coupled oscillators (CO).

  .8. Other calculations performed in NM Fe are summarized in Table II.9 and Table II.10.

	Table II.8. Vacancy formation energy (eV) for different magnetic states in Fe (Klaver, Hepburn, and
				Ackland 2012)
	fcc afmD		fct afmD	fct afmI	fct fm-HS
	1.76		1.82	1.95	1.69
	Ecut=400 eV; supercell=4×4×4, 256 atoms; k-point=2 3 ; convergence condition: 0.03 eV.
	Table II.9. Vacancy formation/migration enthalpy/entropy in fcc Fe by DFT calculations
	𝐻 𝐹𝑒,𝑉 𝑓𝑜𝑟 (𝑒𝑉) 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 (𝑒𝑉) 𝑆 𝐹𝑒,𝑉 𝑓𝑜𝑟 (𝑘 𝐵 ) Method	Reference
	1.80-1.95			VASP PAW GGA afm	(Klaver, Hepburn, and
					Ackland 2012) 1
	2.37	1.42		VASP PAW GGA-PBE NM (Tsuru and Kaji 2013) 2
	2.32	1.38		VASP PAW GGA-PBE NM (Angsten et al. 2014) 3
	2.36	1.39	2.47	VASP PAW GGA-PBE NM (Wang et al. 2018) 4
	2.42			GPAW with PBE NM	(Karimi and Auinger
					2020) 5
	1: Ecut=400 eV; supercell=4×4×4, 256 atoms; k-point=2 3 ; convergence condition: less than 0.03 eV.
	2: Ecut=500 eV; supercell=3×3×3, 108 atoms; k-point=5 3 ; convergence condition: 0.02eV/Å.
	3: Ecut: 1.5 times that of the maximum energy cutoff in the PAW potential file; supercell=3×3×3, 108 atoms; k-
	point=4 3 ; convergence condition: 1 meV.cell -1 .
	4: Ecut=350 eV; supercell=3×3×3, 108 atoms; k-point=9 3 ; residual atomic forces < 0.01 eV/Å.
	5: Ecut=900 eV; supercell=3×3×3, 108 atoms; k-point=6 3 ; convergence condition: < 0.02 eV for each supercell
	size
	Table II.10: Activation energy and pre-exponential factor of the self-diffusion coefficient in fcc Fe
	𝐷 0 (𝑚 2 𝑠 -1 )		𝑄(eV)	Method	Reference
	9.47 × 10 -3		3.75	VASP PAW GGA-PBE NM	(Wang et al. 2018)
	Ecut=350 eV; supercell=3×3×3, 108 atoms; k-point=9 3 ; residual atomic forces < 0.01 eV/Å.
	From these results we can conclude that the vacancy formation enthalpy in fcc Fe is about
	𝐻 𝐹𝑒,𝑉 𝑓𝑜𝑟 = 2.3 -2.4eV in non magnetic and 𝐻 𝐹𝑒,𝑉 𝑓𝑜𝑟 = 1.80 -1.95 in anti-ferromagnetic fcc Fe. The
	vacancy migration enthalpy is about 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 = 1.4eV in NM fcc Fe. Therefore, the activation energy 𝑄,
	is about 3.8eV in NM Fe, which is much higher than the experimental values of 2.80-2.95 eV (Table
	II.4): the calculations in the NM state are clearly not representative of the PM fcc Fe.

  Tucker et al.[START_REF] Tucker | Ab initio-based diffusion theory and tracer diffusion in Ni-Cr and Ni-Fe alloys[END_REF] computed the tracer diffusion coefficients of Ni, Cr and Fe in a Ni host based on ab initio calculations and the five frequency model (equations (II-11) and (II-12)). The modeling results were used to determine the matrix of phenomenological coefficients and the tracer diffusion coefficients for both vacancy and interstitial mediated diffusion in Ni based alloys, including Ni-Fe. For diffusion in Ni-Fe alloys, the authors gave the parameters needed to be used in the fivefrequency model by Le Claire (Le Claire 1978), including 𝐻 2

	Barrier in eV	𝐻 0 𝑚𝑖𝑔	𝐻 1 𝑚𝑖𝑔	𝐻 2 𝑚𝑖𝑔	𝐻 3 𝑚𝑖𝑔	𝐻 4 𝑚𝑖𝑔
	Tucker 1	1.09	1.13	0.97	1.07	1.07
	Toijer 2	1.05	1.09	0.93	1.05	1.03
	1: Ecut=270 eV; supercell=3×3×3, 108 atoms; k-point=3 3 ; convergence condition: 35 meV.
	2: Ecut=350 eV; supercell=4×4×4, 256 atoms; k-point=3 3 ; convergence condition: 20 meV/Å in force convergence

𝑚𝑖𝑔 = 0.97eV. The vacancy formation entropy is also calculated, which is 𝑆 𝑁𝑖,𝑉 𝑓𝑜𝑟 = 1.82𝑘 𝐵 . However, with these parameters, they found smaller 𝐷 𝑁𝑖 𝑁𝑖 * and 𝐷 𝑁𝑖 𝐹𝑒 * than in the experiments by approximately a factor 10. The difference between their calculation and experiments was explained by the pre-exponential factor since there is a good agreement for the slopes of the Arrhenius laws with experimental data.

Toijer et al. also calculated these different migration barriers and the impurity diffusion coefficient of Fe in fcc Ni. For the impurity diffusion coefficient 𝐷 𝑁𝑖 𝐹𝑒 * , they found a good agreement with the experiment of Bakker

[START_REF] Bakker | A Curvature in the InD versus 1/T Plot for Self-Diffusion in Nickel at Temperatures from 980 to 1400°C[END_REF]

) at high temperature, which is in the average range of experimental studies

(Figure II.4)

. The values of Tucker are a little higher than that proposed by Toijer

[START_REF] Toijer | Solute-point defect interactions, coupled diffusion, and radiation-induced segregation in fcc nickel[END_REF]

. A comparison for those values of barriers are given in Table

II

.11.

Table

II

.11. The migration barriers of the five frequency model in ferromagnetic Ni, according to DFT calculations

[START_REF] Toijer | Solute-point defect interactions, coupled diffusion, and radiation-induced segregation in fcc nickel[END_REF][START_REF] Tucker | Ab initio-based diffusion theory and tracer diffusion in Ni-Cr and Ni-Fe alloys[END_REF]

Regarding the impurity diffusion of Ni in fcc Fe, only one reference by Tsuru and Kaji (Tsuru and Kaji 2013) is available. They evaluated the binding energy between Ni and the vacancy, 𝐻 𝑁𝑖-𝑉 𝑏𝑖𝑛 = 0.114eV.

  .12. Vacancy formation energies in 𝐹𝑒𝑁𝑖 3 -𝐿1 2 and 𝐹𝑒𝑁𝑖 -𝐿1 0 ordered structures. DFT calculations.(Li et al. Private communication).

	Table II.13. Average migration energies for vacancy jumps in the simulations of Ferasat et al.
			(Ferasat et al. 2020)
	𝑥 𝐹𝑒		𝐸 𝑁𝑖 𝑚𝑖𝑔 (eV)		𝐸 𝐹𝑒 𝑚𝑖𝑔 (eV)
	0		1.182
	0.1		1.224		1.196
	0.2		1.261		1.168
	0.3		1.290		1.136
	0.4		1.303		1.090
	0.5		1.305		1.039
	𝐻 𝑉 𝐹𝑒 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 3 -𝐿1 2 ) 0.6	𝐻 𝑉 𝑁𝑖 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 3 -𝐿1 2 ) 1.301	𝐻 𝑉 𝐹𝑒 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 -𝐿1 0 ) 0.976	𝐻 𝑉 𝑁𝑖 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 -𝐿1 0 )
	1.388 0.7	1.594	1.298	1.897	0.916	1.847
	Ecut=400 eV; supercell=3×3×3, 108 atoms; k-point=16 3 ; convergence condition: 0.02eV/Å 0.8 1.278 0.834
	0.9		1.251		0.736
	where 𝐻 𝑉 𝐹𝑒 1.0				0.622

𝑓𝑜𝑟

is the formation enthalpy of a vacancy replacing a Fe atom on its sublattice.

Recently Osetsky et al. performed a series of atomistic modelling of diffusion in Fe-Ni alloys, using various EAM potentials developed by

[START_REF] Bonny | Interatomic potential to study plasticity in stainless steels: the FeNiCr model alloy[END_REF]

, classical or ab initio molecular dynamics and kinetic Monte Carlo simulations

[START_REF] Anento | Effect of nickel on point defects diffusion in Fe -Ni alloys[END_REF] Osetsky et al. 2018;[START_REF] Zhao | Diffusion of point defects in ordered and disordered Ni-Fe alloys[END_REF][START_REF] Ferasat | Accelerated kinetic Monte Carlo: A case study; vacancy and dumbbell interstitial diffusion traps in concentrated solid solution alloys[END_REF]

. The same potentials were used, with the k-ART, off lattice, kinetic Monte Carlo method to study diffusion mechanisms in the L10-FeNi phase

[START_REF] Mahmoud | Long-time point defect diffusion in ordered nickel-based binary alloys: How small kinetic differences can lead to completely long-time structural evolution[END_REF]

. These simulations show how the details of migration barriers affect the diffusion properties of vacancies and self-interstitials. The simulations of Zhao et al.

[START_REF] Zhao | Diffusion of point defects in ordered and disordered Ni-Fe alloys[END_REF]

, for example, show that the vacancy diffuse more slowly in the ordered phase FeNi3 than in a solid solution at the same composition

(Figure II.15)

. However, for a quantitative modelling of the Fe and Ni diffusion coefficients in concentrated alloys, these approaches present two limitations. The first one comes from the EAM potential, which predicts a strong decrease of the average migration barriers for the Fe-V jumps with the iron concentration (Table

II

.13)

. As a result, the simulations predict that the Fe diffusion coefficient varies by orders of magnitude between pure Ni and pure Fe (Figure II.14), in contrast to what is experimentally observed.

Table II

 II Regarding the alloys, other interesting data are the migration barriers computed byTucker et al.

		.14: parameters concerning the exchange between a vacancy and a Ni atom
	𝑎 0 𝑁𝑖 (nm) 𝜗 0 𝑁𝑖 (Hz)	𝐻 𝑁𝑖,𝑉 𝑓𝑜𝑟 (eV) 𝐻 𝑁𝑖,𝑉 𝑚𝑖𝑔 (eV) 𝑆 𝑁𝑖,𝑉 𝑓𝑜𝑟 (𝑘 𝐵 ) 𝑆 𝑁𝑖,𝑉 𝑚𝑖𝑔 (𝑘 𝐵 ) 𝐻 𝑁𝑖,2 𝑚𝑖𝑔 (eV) 𝑆 𝑁𝑖,2 𝑚𝑖𝑔 (𝑘 𝐵 )
	0.3524	0.59 × 10 13 1.79	1.04	3.22	1.58	0.96	2.7
		Table II.15: parameters concerning the exchange between a vacancy and a Fe atom
	𝑎 0 𝐹𝑒 (nm) 𝜗 0 𝐹𝑒 (Hz)	𝐻 𝐹𝑒,𝑉 𝑓𝑜𝑟 (eV) 𝐻 𝐹𝑒,𝑉 𝑚𝑖𝑔 (eV) 𝑆 𝐹𝑒,𝑉 𝑓𝑜𝑟 (𝑘 𝐵 ) 𝑆 𝐹𝑒,𝑉 𝑚𝑖𝑔 (𝑘 𝐵 ) 𝐻 𝐹𝑒,2 𝑚𝑖𝑔 (eV) 𝑆 𝐹𝑒,2 𝑚𝑖𝑔 (𝑘 𝐵 )
	0.3571	1.245 × 10 13 1.943	1.0	2.47	1.32	0.97	1.75

  .14 and Table II.15. Tucker et al. 2010). The values of the other barriers of the five frequency model are then imposed. We find (TableII.17) that the k-PIM model is in good agreement with the calculations of Tucker et al for these barriers, and also with the DFT calculations of Toijer et al.[START_REF] Toijer | Solute-point defect interactions, coupled diffusion, and radiation-induced segregation in fcc nickel[END_REF]). On another hand, the ℎ 𝑁𝑖𝐹𝑒 𝑆𝑃 parameter is fitted on the value of 𝐻 2 𝑚𝑖𝑔 in Fe, given by the DFT calculations of Kangming Li (K. Li, private communication).

	the k-PIM are the saddle-pair interactions 𝑔 𝐹𝑒𝑁𝑖 𝑆𝑃 = ℎ 𝐹𝑒𝑁𝑖 𝑆𝑃 -𝑇𝑠 𝐹𝑒𝑁𝑖 𝑆𝑃 and 𝑔 𝑁𝑖𝐹𝑒 𝑆𝑃 = ℎ 𝑁𝑖𝐹𝑒 𝑆𝑃 -𝑇𝑠 𝑁𝑖𝐹𝑒 𝑆𝑃 , which
	are involved in the 𝐻 2 𝑚𝑖𝑔 barriers in Ni and Fe. For example, the 𝑔 𝐴𝐵 𝑆𝑃 will be given by
	2	2
	𝐺 2 𝑚𝑖𝑔 = 4𝑔 𝐴𝐵 𝑆𝑃 -∑ 𝑍 𝑛 𝑔 𝐴𝐵 (𝑛)	+ 𝑔 𝐴𝐵 (1) -∑ 𝑍 𝑛 𝑔 𝐵𝑉 (𝑛)	+ 𝑔 𝐵𝑉 (1) -𝑔 𝐴𝑉 (1)	(II-35)
	𝑛=1	𝑛=1
	ℎ 𝐹𝑒𝑁𝑖 𝑆𝑃 is fitted to the DFT calculation of 𝐻 2 𝑚𝑖𝑔 in FM Ni, by Tucker (The entropic contributions are adjusted on the experimental pre-exponential factors of 𝐷 𝐹𝑒 𝑁𝑖 * and 𝐷 𝑁𝑖 𝐹𝑒 *
	(Figure II.19).

To test the model, AKMC simulations have been performed for temperatures between 500 K and 2000 K and 1% of impurities, i.e. 1% Ni in Fe and 1% of Fe in Ni. The comparison between the AKMC, the five frequency model and the experiments is shown in Figure II.19.

Table II.16: Migration barriers of the five frequency model. Comparison between the k-PIM and the DFT calculations of

Table II .

 II 17. The local concentration dependence of 𝑔 𝑁𝑖𝐹𝑒 𝑆𝑃 (𝑥 𝑁𝑖 𝑆𝑃 ) and 𝑔 𝐹𝑒𝑁𝑖 𝑆𝑃 (𝑥 𝑁𝑖 𝑆𝑃 )in eV [see eq. (II-40)]

	𝑥 𝑁𝑖 𝑆𝑃	0	0.25	0.5	0.75	1
	𝑔 𝐹𝑒𝑁𝑖 𝑆𝑃	-2.66934	-2.68207	-2.72025	-2.78389	-2.87298
	𝑔 𝑁𝑖𝐹𝑒 𝑆𝑃	-2.76156	-2.76689	-2.78287	-2.80951	-2.8468

  .[START_REF] Cacciamani | The Fe-Ni system: Thermodynamic modelling assisted by atomistic calculations[END_REF]. Vacancy formation energies on the Fe and Ni sublattices of the L12 and L10 ordered structures. Comparison between the DFT calculation and the k-PIM (unit: eV)

		𝐺 𝑉 𝐹𝑒 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 3 -𝐿1 2 ) 𝐺 𝑉 𝑁𝑖 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 3 -𝐿1 2 ) 𝐺 𝑉 𝐹𝑒 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 -𝐿1 0 )	𝐺 𝑉 𝑁𝑖 𝑓𝑜𝑟 (𝐹𝑒𝑁𝑖 -𝐿1 0 )
	Li	1.388	1.594	1.897	1.847
	This work 2.115	1.956	2.140	1.996

Table III .

 III 1. Temperature condition of different experiments (°C)The comparison between the results ofMorris et al. and those of Ferjani et al. at 𝑇 = 470°C (743 K) is shown in Figure III.8. We show the LRO parameters of Morris et al. in relative value (divided by the value at the maximum annealing time), so that they can be directly compared to those of Ferjani et al.Figure III.8. Evolution of relative long-range order (left) parameter and the domain size (right) in Fe-75%Ni: comparison between Ferjani

	Calvayrac							480		497
	Morris	434	442	450	460	470			485	497
	Ferjani					470	478
	T in K	(707)	(715)	(723)	(733)	(743)	(751)	(753)	(758)	(770)

  3 𝑐 𝑉𝛼 2(𝑚+𝑛)/𝑛 + 𝑘 1 (𝑚 -𝑚𝑥 + 𝑚𝑘 2 𝑘 3 𝑥)𝑐 𝑉𝛼 2+𝑚/𝑛 -𝑘 1 (𝑚 -𝑚𝑥 + 𝑚𝑘 2 𝑘 3 𝑥 -𝑛𝑥 -𝑛𝑘 2 𝑘 3 + 𝑛𝑘 2 𝑘 3 𝑥)𝑐 𝑉𝛼 Where 𝑘 1 ,𝑘 2 and 𝑘 3 denote respectively: By resolving equation (1), the concentration of vacancy on the 𝛼 sub-lattice is obtained. Then we can further calculate the concentration of point defects on all sites by: In addition, with Hagen & Finnis method, one is able to calculate the chemical potential as well by the following expressions:

									1+𝑚/𝑛	(1)
		+ 𝑛𝑘 1 2 (1 -𝑥)(1 -𝑘 2 𝑘 3 )𝑐 𝑉𝛼 -𝑘 1 2 𝑘 2 = 0
			𝑘 1 = exp (-𝑘 2 = exp (-𝑒 𝑉𝛼 -𝑒 𝑉𝛽 + 𝑒 𝐴𝛽 -𝑒 𝐴𝛼 𝑚𝑒 𝑉𝛼 + 𝑛𝑒 𝑉𝛽 ) 𝑛𝑘 𝐵 𝑇 𝑘 3 = exp (-𝑘 𝐵 𝑇 𝑒 𝑉𝛽 -𝑒 𝑉𝛼 + 𝑒 𝐵𝛼 -𝑒 𝐵𝛽 𝑘 𝐵 𝑇 ) )			(2)
			𝑐 𝑉𝛼 𝑚 𝑐 𝑉𝛽 𝑛 = exp (-𝑐 𝑉𝛼 𝑐 𝐴𝛽 𝑐 𝑉𝛽 𝑐 𝐴𝛼 = exp (-𝑐 𝑉𝛽 𝑐 𝐵𝛼 𝑐 𝑉𝛼 𝑐 𝐵𝛽 = exp (-	𝑚𝑒 𝑉𝛼 + 𝑛𝑒 𝑉𝛽 𝑘 𝐵 𝑇 𝑒 𝑉𝛼 -𝑒 𝑉𝛽 + 𝑒 𝐴𝛽 -𝑒 𝐴𝛼 ) 𝑘 𝐵 𝑇 𝑒 𝑉𝛽 -𝑒 𝑉𝛼 + 𝑒 𝐵𝛼 -𝑒 𝐵𝛽 ) 𝑘 𝐵 𝑇 )		(3)
			𝑐 𝐴𝛼 + 𝑐 𝐵𝛼 + 𝑐 𝑉𝛼 = 1
			𝑐 𝐴𝛽 + 𝑐 𝐵𝛽 + 𝑐 𝑉𝛽 = 1
	𝜇 𝐴 = 𝜇 𝐵 =	𝑚 𝑚 + 𝑛 𝑛 𝑚 + 𝑛	𝑒 𝐴𝛼 + 𝑒 𝐵𝛽 +	𝑛 𝑚 + 𝑛 𝑚 𝑚 + 𝑛	𝑒 𝐴𝛽 + 𝑒 𝐵𝛼 +	𝑚 𝑚 + 𝑛 𝑛 𝑚 + 𝑛	𝑘 𝐵 𝑇𝑙𝑛𝑐 𝐴𝛼 + 𝑘 𝐵 𝑇𝑙𝑛𝑐 𝐵𝛽 +	𝑛 𝑚 + 𝑛 𝑚 𝑚 + 𝑛	𝑘 𝐵 𝑇𝑙𝑛𝑐 𝐴𝛽 𝑘 𝐵 𝑇𝑙𝑛𝑐 𝐵𝛼

'European Congress on Advanced Materials and Processes'.

ACKNOWLEDGMENTS

We thank our colleagues Chu-Chun Fu and Maylise Nastar for many fruitful discussions. The research leading to these results has been carried out in the frame of EERA Joint Program for Nuclear Materials and is partly funded by the European Commission HORI-ZON 2020 Framework Programme under grant agreement No. 755269. Yimi Wang and Kangming Li are supported by the CEA NUMERICS program, which has received funding from the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie grant agreement No. 800945.

Chapter III Kinetics of ordering in fcc Fe-Ni alloys

III.1 Introduction

We are now going to study the kinetics of ordering in Fe-Ni alloys, using the k-PIM and AKMC tools developed in the two previous chapters. We will limit ourselves to the ordering of the FeNi3-L12 phase, for which three experimental studies are available, dating back to the 1970s [START_REF] Calvayrac | Structural state and mechanical properties of polycrystalline Ni3Fe Alloys[END_REF][START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF][START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF]. These studies provide detailed information on the evolution of the long range parameter, 𝜂(𝑡), and of the size of ordered domains, 𝐿(𝑡),

during isothermal annealings at temperatures slightly below the critical ordering temperature, 𝑇 𝑐 𝐿1 2 . The ordering of the FeNi-L10 phase occurs at lower temperatures, on (literally) astronomic times: is has been only observed in meteorites [START_REF] Scorzelli | A study of phase stability in invar Fe--Ni alloys obtained by non-conventional methods[END_REF], or under irradiation where it can be accelerated by the increase of point defects concentrations [START_REF] Reuter | Ordering in the Fe-Ni system under electron irradiation[END_REF]. It is thus beyond the scope of our study.

After a brief review of the experimental results (section III.2), we will give some details on the AKMC simulations (section III.3). The simulations will use the two sets of parameters developed in the previous chapter: k-PIM, derived from the modelling of diffusion coefficients in the paramagnetic solid solutions and k-PIM2, fitted on the enthalpies of vacancy formation in the L12 phase computed by DFT calculations. We will present the methods used in the simulations to measure the evolution of the long range order parameter 𝜂(𝑡) and the domain size 𝐿(𝑡), and discuss the use of the Widom method to take into account the evolution of the vacancy concentration during the ordering process (i.e. in a nonequilibrium situation). The comparison between the simulations and the experiments will be presented in section III.4.

III.2 Experimental observations of ordering process in L12-FeNi3

Calvayrac and Fayard studied the evolution of the long-range order parameter and average domain size during the ordering process of L12-FeNi3 by X-ray diffraction [START_REF] Calvayrac | Structural state and mechanical properties of polycrystalline Ni3Fe Alloys[END_REF].

The evolution of the long-range order parameter at 𝑇 = 480°C (753 K) and 𝑇 = 497°C [START_REF] Ferjani | Ordering and domain coarsening kinetics in substiteted premalloys[END_REF] The same comparison can be done between the results of Morris et al. and Ferjani et al. [START_REF] Calvayrac | Structural state and mechanical properties of polycrystalline Ni3Fe Alloys[END_REF] and [START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF]).

Figure III.11. Evolution of the domain size as a function of relative long-rang order parameter in Fe-75%Ni at T=497°C (770 K): comparison between the observations of [START_REF] Calvayrac | Structural state and mechanical properties of polycrystalline Ni3Fe Alloys[END_REF] and [START_REF] Morris | Ordering and domain growth in Ni3Fe[END_REF].

The two studies are in good agreement for the evolution of the domain size, while Morris et al.

observed an evolution of long-rang order parameter slightly slower than that of Calvayrac and Fayard.

The shapes of the 𝜂(𝑡) curves are nevertheless similar.

Appendix

The Hagen & Finnis Model: calculation of vacancy formation energy for ordered structures at their stoichiometry At high temperatures, it is relatively easy to obtain the equilibrium vacancy concentration by experiments. However, manipulation difficulties arise due to the slow rate of diffusion at low temperature, and consequently long time to reach equilibrium. Furthermore, the order-disordering process makes it even more complicated for experiments. Therefore, it is of great interest to obtain these values by means of theoretical analysis and calculations. [START_REF] Hagen | Point defects and chemical potentials in ordered alloys[END_REF] have proposed analytical expressions for the vacancy and antisite concentration in ordered systems at their stoichiometry. Both may be present to an extent which depends on the alloy and its stoichiometry as well as the temperature. The formulas are constructed based on the assumptions that several key energies can be calculated.

To clarify this theory, we consider an ordered system 𝐴 𝑚 𝐵 𝑛 . We suppose that in a perfect ordered state, all atoms A locate in the sub-lattices 𝛼 and all atoms B locate on the sub-lattices𝛽. For a real system, we denote that the global concentration of atom A of the system is 𝑥 and thus the system can be expressed as 𝐴 𝑥 𝐵 1-𝑥 . It should be noticed that the Hagen and Finnis formula is only valid within a very narrow change of concentration 𝑥 around the stoichiometry m/(m+n). With the evolution of the system, we assume that the sites can be occupied by its own atom, an atom of the other kind or a vacancy.

In this way, in total six different concentration are defined: Correspondingly, the key energies 𝑒 𝐴𝛼 , 𝑒 𝐴𝛽 ,𝑒 𝐵𝛼 , 𝑒 𝐵𝛽 ,𝑒 𝑉𝛼 , 𝑒 𝑉𝛽 are defined as well. For instance, 𝑒 𝐴𝛼 and 𝑒 𝐵𝛽 are defined in a perfect ordered system and denoted the energy of atom A on the sub-lattice 𝛼 and the energy of atom B on the sub-lattices 𝛽. As a result, the cohesive energy of the system can be expressed by the sum: 𝑒 𝑚𝑜𝑙 = 𝑚𝑒 𝐴𝛼 + 𝑛𝑒 𝐵𝛽 . However, Hagen pointed out that this is just a conventional allocation in for these two variables. As long as the sum 𝑒 𝑚𝑜𝑙 remains the same, the calculated concentration of defects will not affect any measurable properties. Then 𝑒 𝐵𝛼 is defined as the