Thermomechanical assessment of empirical potentials 

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Physical properties of MOX

The new generation of reactors are expected to work with higher Pu contents. The inĆuence of higher Pu contents are rarely investigated by the private nuclear sector. Furthermore, the efect of stoichiometry is expected to be very important to explain the processes inside the fuel (see Appendix A).
Experiments on MOX are diicult to carry out. Nonetheless, efort has to be made to get more information about this material since it is expected to be used by the new generation of reactors. First of all, a compilation of the knowledge found in the literature about MOX has to be done. This job was accomplished by a group of various European research organization in 1990. They built up a catalogue of properties of MOX, part of which is still used in models of calculation of oxide fuel codes for fast neutron reactors. A new efort was carried out by (ESNII+, 2015) to create a new catalogue of thermomechanical properties. It aimed to gather MOX fuel data required by the fuel calculation codes as GERMINAL, TRANSURANUS, MACROS, and TRAFIC. SpeciĄcally, input parameters as Temperature, Pu content, O/M ratio, porosity, burnup,. . . relevant for MOX fuels data needed for the reactor prototypes ALFRED (LFR), ALLEGRO (GFR), ASTRID (SFR), and MYRRHA (ADS). Table 1.1 shows the parameters and conditions that this catalogue comprises.

Crystallographic structure of MOX

The stoichiometric MOX has a Ćuorite structure. To be more speciĄc, the structure is cubic Fm¯3m space group 225. The atomic representation of the structure is shown in Figure 1.8. The ideal crystal adopts a face centred cubic lattice for the cations and inside the anions occupy the tetrahedral sites, which form a simple cubic sub-lattice. Each cation, U or Pu, is then inside a cube where the corners are occupied by O atoms (Figure 1.9) ZU,Pu = 8. In the same manner, the O atoms are in the centre of tetrahedrons with U or Pu in the corners ZO = 4.
This structure possesses several symmetries. Thus, it can be described by a unique parameter called « lattice parameter ». The principal crystallographic orientation are in Miller index: the ⟨1 0 0⟩, ⟨1 1 0⟩ and ⟨1 1 1⟩ . The MOX Ćuorite structure has large octahedral interstitial holes, in which interstitial ions can be introduced.

MOX potentials

For the mixed oxide compound (U,Pu)O2, several interatomic potentials are available in the literature. It exists two main families of potentials. One that considers U and Pu cations as one single entity A, hence they include only three set of parameters (C-C, C-O, and O-O) but depends on the relative percentage of Pu in the MOX. The second one treats explicitly the U and the Pu cations. Therefore, they include six set of parameters (U-U, U-Pu, Pu-Pu, U-O, Pu-O, and O-O) but do not depend on the percentage of Pu. Because we are interested in studying the spatial repartition of both cation sublattices, we will only consider and describe the second type of force Ąeld.
We have found Ąve interatomic potentials that we will coin later on by the name of their Ąrst author: Yamada (Yamada et al., 2000), Arima (Arima et al., 2005), Potashnikov (Potashnikov et al., 2011), Tiwary (Tiwary, Walle, & Jeon, 2011, 9), and Cooper (Cooper et al., 2014), (Cooper et al., 2015). These Ąve force Ąelds can be separated according the properties on which they have been Ątted. All potentials have been Ątted to reproduce correctly the thermal expansion up to the maximum temperature available by experiments at the time, which is about 2100 K.
Historically, Yamada was the Ąrst one followed by Arima and Potashnikov with some improvement at high temperature, up to the melting point. Tiwary potential includes also Ąt on the formation energy of point defects (Frenkel pairs), while Cooper potential focusses on experimental data for single crystal elastic constants.. However, with Tiwary potential it was impossible to run MD simulations, the Ćuorite structure of UO2 or PuO2 is not stable, nevertheless it gives good results using static calculations. This is probably due to the fact that the energy landscape of this potential is very rough including a lot of none physical minima. Therefore, we eliminate this potential from our study.
As we can see, the potentials were Ątted on diferent parameters. The results that each potential is able to reproduce depend strongly on what they were Ątted on. All potentials try to simulate the best, the reality. However, as a user, we have to arise some doubts of their capabilities. Generally, at the time of recalculating the properties of which they were Ątted on, all show good results. For instance, Cooper potential was Ątted using elastic constants. One would then, give more conĄdence that this potential will show better results at the time of calculating mechanical properties. Finally, when there is lack of experimental data to compare with, the conĄdence will rely rather on the overall results that they give. Obtaining diferent phenomena when simulating the same system, under the same conditions, for all the potentials, give us more statistics to consider. Thus, we can give an upper and lower limits where the real scenario is hoped to be between them. The way to explain the reality with these type of simulations is then highly dependent on the trends in the results.

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Canonical ensemble (Temperature control)

Now, we will assess the case where the temperature has to be Ąxed to a certain value. This type of situation is important to simulate various systems in nature. For example, phenomena where the systems are surrounded by a thermal bath so the system keeps its temperature constant by exchanging energy with the environment. With these conditions, including maintaining the number of particles (N) and the size of the box constants (V), we can apply the canonical ensemble (NVT) to our system.
In general, two types of algorithms are used to maintain our system at a Ąxed temperature. The Ąrst one consists in a correction of the dynamic variables after the integration step (Berendsen). The second one uses a modiĄed Lagrangian (Nosé-Hoover).

Table of contents :

Abstract i
Avant-propos
1 MOX Fuel overview 
1.1 Generation of Electricity with Nuclear Power
1.2 Why using MOX?
1.3 MOX inside the reactor
1.4 Physical properties of MOX
1.5 Crystallographic structure of MOX
1.6 Defects in MOX
1.6.1 Point defects
1.6.2 Extended defects
2 Numerical Method 
2.1 Molecular Dynamics
2.1.1 Simulation box
2.1.2 Atomic interactions
2.1.3 Pair potentials
2.1.4 EAM porential
2.1.5 ZBL porential
2.1.6 Electrostatic force
2.1.7 MOX potentials
2.1.8 LAMMPS code
2.1.9 Simulation techniques
2.2 Atomic structure analysis
2.2.1 Wigner-Seitz cell method
2.2.2 Voronoi cell method
2.2.3 Dislocation Extraction Algorithm
2.3 Simulation of X-ray powder Difraction
2.4 Can Pu atoms be randomly distributed?
3 Thermomechanical assessment of empirical potentials 
3.1 Thermodynamical properties
3.1.1 Lattice parameter
3.1.2 Thermal expansion coeicient
3.1.3 Enthalpy and speciĄc heat
3.1.4 Thermal conductivity
3.1.5 Melting point
3.1.6 Difusion in MOX
3.1.7 Phase stability (Energy-volume plots)
3.2 Mechanical properties
3.2.1 Elastic constants
3.2.2 Anisotropy factor (ZenerŠs)
3.2.3 Stress-strain curves
3.2.4 Brittle-to-ductile transition
3.2.5 Crack propagation
3.3 Conclusions
4 Damage evolution under irradiation. 
4.1 Defect formation energies
4.2 Frenkel pair recombination
4.3 Primary damage state
4.4 Dose efect
4.5 Elastic moduli vs dose
4.6 X-ray powder Difraction
4.7 Conclusions
5 Adaptive Kinetic Monte Carlo 
5.1 Simulation time scale
5.2 Transition state theory
5.3 Accelerated dynamics
5.4 The dimer method
5.5 Recycling saddle points and super basins
5.6 Monte Carlo
5.7 Adaptive Kinetic Monte Carlo
5.8 Long term recombination
5.9 Conclusions
6 Conclusions
A Phase diagram of MOX 
B Extended lattice parameter 
C Cluster analysis 

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