Thermomechanical assessment of empirical potentials 

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MOX inside the reactor

The nuclear fuel pellets for Pressurized Water Reactors (PWRs) are small cylinders of around 5 mm of diameter (NEA, 2014). For MOX, raw powders of UO2 and PuO2 are weighted and then mixed to meet the Ąnal fuel composition of 7.38% (for this case) of Pu with a Oxygen to Cation O/C ratio of 1.998% and an average grain size of 10 µm. The powder is then milled and granulated through pre-compaction, crushed and compacted again. Finally, the pellets are sintered. More details of this process can be found in (KAERI, 2009). The plutonium distribution of these pellets are shown in Figure 1.4. As we can observe, the plutonium distribution is quite homogeneous. The maximum diameter of the Pu rich spot is 13 µm with a Pu content of 18%. Under irradiation this inhomogeneity plays an important role.
An irradiation test is undertaken with an average burnup of 50 MWd/kgHM with an irradiation time of 1020 EFPD. The peak in the temperature in the fuel centre was estimated around 1673 K. The results of the irradiation show many Pu rich spots and metallic Ąssion product precipitates. The Pu rich spots were found mainly in middle and peripheral regions, in contrast to the centre where the density was lower. Pu in the spots difuses out to the UO2 matrix in the central region (KAERI, 2009). Pu spots are surrounded by a ring of small pores called « Halo » as shown in Figure 1.5 reactor. One example is the restructuring of the solid matrix. It is due to the large temperatures and temperature gradients. Voids migrate towards higher temperatures and gather in the centre. Conversely, this causes a movement of the solid towards the outer periphery and leads to the creation of a void in the centre. An illustration of the Ąnal pellet state is shown in Figure 1.6, which represents a micrography of a MOX pellet. This phenomena makes the peak in the temperature decrease, since the fuel is now closer to the heat sink. Moreover, it afects the heat transport characteristics of the pellet. In MOX, this results in the migration of oxygen atoms through the pellet and conditions the demixing of U and Pu (Welland, 2012).
Figure 1.6 Ű Microstructure of a fast reactor pelletized MOX fuel pellet (Welland, 2012). MOX fuel pellets are often divided into regions deĄned by their restructured state. Going outward from the centreline we Ąnd 1) Central void, 2) Columnar grain region, 3) Equiaxed grain growth and 4) As-fabricated microstructure.
There is a diference between MOX and UO2 pellets. In MOX, there is higher rates of Ąssion gas release, due to higher linear power rating and higher centreline temperature (caused by slightly lower thermal conductivity). Furthermore, the heterogeneity of the Ąssile material would play a role as well, to increase the rate of Ąssion gas release. This is shown in Figure 1.7 for MOX fuel fabricated by the MIMAS process. The Pu rich spots are submitted to very high burn-up. This causes restructuring, which leads to further division of grains, precipitation of gas bubbles that go to intra-granular division which make them easily freed when a power transient occurs (CEA, 2009). In overall, MOX fuel pellets are more complicated to work with, than the UO2. Their behaviour inside the reactor has to be carefully addressed, since more processes are involved at the time of the burn-up. Moreover, at the time of producing them, the alpha decays make it diicult to handle and decreases the number of labs capable of sensitize them. Therefore, it is important to understand well what are those micro-structural changes that happen inside the fuel under irradiation and surely the cause as well. This brings the need of new tools to assess the problem. In particular, modelling and computer simulations are expected to contribute to a better understanding of this phenomena.

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.

Molecular Dynamics

Molecular Dynamics is a simulation technique for atomic scales. This technique is described in some reference texts (Hoover, 1986), (Goldstein, Poole, & Safko, 2001). The theoretical basis concerns the work of great minds of analytical mechanics such as; Euler, Hamilton, Lagrange and of course Newton. MD allows us to build the trajectory of the particles in the phase space of a system composed of a great number of atoms. The simplest form of MD, that is related to structureless particles, involves the NewtonŠs equations solution (Rapaport, 2004). In MD, the atom inputs are the masses ma , position ra and velocities va see Section 2.1.9.

Simulation box

Particles are in interaction inside a simulation box. The simulation box is usually a paral-lelepiped form. This box can be described with three vectors x, y and z. The output of a MD simulation are the individual trajectories of each atom along the time described by ra (t) and va (t) (ra and va for simplicity). These trajectories and the dimension of the simulation box are required to get thermomechanical properties, such as; temperature, pressure, internal energy, etc. In addition, the MD code needs: 1) the force between atoms, 2) equations that rule the atom trajectories, 3) an integrator that solves numerically these equations with discretization of the temporal domain. Due to the intrinsic scale of MD, it is often necessary to expand the domain under study. This is done by creating inĄnitely exact replicas of the simulation box. Thus, each atom a in the original simulation box will have inĄnity number of replicas a′ inside the simulation box replicas. This type of boundary conditions are called periodic. Atoms inside the original simulation box can interact with the simulation box replicas. However, self interaction is prohibited by setting the size of the box twice the distance of the interatomic potential cut-of. Another situation that is allowed to happen is that an atom can cross the original simulation box boundaries. In this case, the atom that have left will be found in one of the neighbour replicas and strictly one atom from the replicas will go inside the original but in the opposite side. Figure 2.1 show a simulation box with its replicas and an atom crossing the limits.

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Atomic interactions

Along this work, the atoms are presented as points with a given charge. This representation is usually called rigid ion type. In addition, in this model the interactions between atoms can be taken in terms of potential energy pairs. Rigorously, the total energy of a system of N atoms interacting can be expanded in a many-body expansion: ∑ ∑ ∑ ∑ ∑ ∑ U(r ,r ,….,r ) = U a + U ab + U abc +….(2.1) 1 2 N 1 2 3 a a b>a a b>a c>b.
Each term is important for the next discussion, thus, they are explained below; 1. U1 is the one-body term. It is related to external force Ąelds or boundary conditions. For example, the wall of a container.

Simulation techniques

Along this work, we have carried out structure optimization simulations which can be coined static calculations, but also, trajectory calculations at Ąnite temperatures named as dynamic calculations. The static calculations are based in an optimization of an atomic coordinate function (positions and/or velocities). The molecular dynamic techniques are based on the numeric solutions of motion equations. In general, these equations are derived from the analytical mechanic theory formulations. These are the Lagrangian or Hamiltonian formulations. A brief description of these techniques will be presented next. Firstly, the description of the coordinate optimization. Consecutively, a description that simulates a system under thermodynamic ensembles described by the statistical mechanics theory.

Static simulations

These type of simulations are used to obtain information about a system in a stable state. They do not allow us to calculate dynamic quantities. However, they are important in the sense that they complement the dynamic calculations. For instance, we can calculate states with lowest energies. In these kind of states, the atoms are not moving, thus, we can propose that they have temperatures equal to 0. On the other hand, there is an option that allows us to perform the same type of calculations but with system at a certain temperature. This is by using the harmonic approximation in order to calculate the entropy of a system from the phonon spectrum. In general, this type of simulation is much faster than MD.
The static simulations need three elements. The inputs are the atomic positions r a , the simulation box and the potential. The output is an equation of state that can be, for instance, the internal energy, the enthalpy or the free energy. This function will be minimized by changing the system input. There are several algorithms dedicated to this minimization, such as; the conjugate gradient, the Broyden-Fletcher-Goldfarb-Shanno algorithm, the Rational Function Optimization method, etc . Along this work, we have used only the Ąrst, the Polak-Ribiere conjugate gradient version (Polak & Ribiere, 1969). The reason of our choice is that this method is one of the most popular ones for solving smooth unconstrained optimization problems due to its simplicity and low memory requirement (Yuan, Wei, & Li, 2014).

Table of contents :

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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