INVESTIGATION OF NUCLEAR FUEL MATERIALS UNDER ......INVESTIGATION OF NUCLEAR FUEL MATERIALS UNDER...

INVESTIGATION OF NUCLEAR FUEL MATERIALS

UNDER IRRADIATION AT THE ATOMIC SCALE

Marjorie Bertolus, Michel Freyss Julia Wiktor, Emerson Vathonne,

Gérald Jomard

CEA, DEN, DEC, Centre de Cadarache, France

PRACEDays 2016, Scientific session 2 | Prague, May 11th, 2016

INTRODUCTION - CONTEXT

2

NUCLEAR POWER & NUCLEAR FUELS

| PAGE 3 PRACEDays 2016, Prague, May 11th, 2016

Nuclear power

Source of low-carbon electricity

Considered worldwide as an option for mitigating climate change along renewable energies

Nuclear fuels Actinide compounds, mostly uranium dioxide in current power reactors Key role in resources optimization, waste minimization, safety and efficiency of nuclear energy Challenge for development of future nuclear reactors: improve significantly effectiveness of design and selection of innovative fuels

Cracking Fission gas bubble precipitation

High Burn up structure Radial migration

FP migration interaction

NUCLEAR FUEL BEHAVIOUR IN REACTOR

Chemistry

Complex behaviour: interconnected effects at various scales

Irradiation Temperature Mechanical effects


Fission

Fission products, in particular volatile elements (Kr, Xe, I, Cs…)

Alpha decay

Helium + Recoil nuclei Collision cascades Defects

CEA-EDF-AREVA Simulation Platform

FUEL PERFORMANCE CODES: SIMULATION OF NUCLEAR FUEL BEHAVIOUR AT THE MACRO SCALE

| PAGE 5

Progress in physical description to improve code predictivity

Justified simplifications to enable practical use

Internal pressure

PCMI

Mechanical behaviour under irradiation

External pressure

Pellet

Columnar grains

Central void

Time (days)

Radi

us (m

m)

PRACEDays 2016, Prague, May 11th, 2016

Physico-chemistry analysis Thermal analysis Mechanical calculations

Temperature distribution in fuel element

Mechanical fields Fission gas production and release

Migration, diffusion of species Composition / microstructure changes

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t dat

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om n

eutr

onic

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Use of basic research

13 September 2016

One way to get further insight into fuel behaviour: uncouple the various phenomena through separate effect investigations, in particular use of ion irradiation to simulate effects of neutron irradiation

Basic research investigations performed at Fuel behaviour law laboratory (CEA, DEN, Cadarache) since 1999

Joint experimental and modelling approach applied to the understanding of the behaviour of uranium dioxide under irradiation

Interpretation of results of precise experiments and characterization under irradiation not always straightforward

Atomic scale calculations invaluable tools to help in interpretation of experiments and determine elementary mechanisms

Mechanisms and data for higher scale models on phenomena linked to operational issues

UNDERSTANDING FUEL BEHAVIOUR


MAIN METHODS AND TECHNIQUES APPLIED

Experimental techniques

X-Ray absorption

X-ray diffraction

Thermal Desorption Spectroscopy

Positron annihilation spectroscopy Raman spectroscopy Transmission electron microscopy

Modelling

Electronic structure calculations Empirical potential methods

Rate theory methods, e.g. cluster dynamics

Schottky defect (Vu+2Vo)

Links - Synergy


CONTENT OF PRESENTATION

Atomic scale modelling methods / Application to nuclear fuels Validation of DFT+U results on UO2 Characterization of irradiation-induced point defects using positrons

Combination of atomic scale calculations and diffusion models to study Kr diffusion in UO2


ATOMIC SCALE MODELLING METHODS AND APPLICATION

TO NUCLEAR FUELS

9

ATOMIC SCALE MODELLING

N interacting nuclei and n electrons

N nuclei in electronic mean field

Electronic structure methods Haψ = Eψ

Empirical potentials E = f(ri )

Born-Oppenheimer Movement separation

Treatment of interatomic interactions

Approximate resolution of Schrödinger equation Predictive et precise

High cost Size of systems limited Density Functional Theory: 1000 atoms Exploration of config. space mainly through energy minimizations Calculations at 0 K even if ab initio molecular dynamics develops

Atoms interact through parameterized analytical potential Much quicker Possible to study large systems: 106-109 atoms and time evolution up to a few ns (molecular dynamics) Pre-existing data needed on system No description of electronic effects


DATA YIELDED BY ATOMIC SCALE METHODS


Fundamental results: total energy of a system

From this energy, its derivatives and the simulation of the system evolution in time one can get

Cohesion energies, formation enthalpies Relative stability of various phases or configurations Equilibrium structures Mechanical properties (bulk modulus, elastic constants) Formation energies of point and extended defects in materials Activation energies to migration of defects Incorporation energies and site of solutes in materials Activation energies to migration of solutes with or without defects Precipitation and resolution of solutes in materials Bonding and dissociation energies of complex/extended defects Vibration modes - phonons Thermodynamic data: free energies, calorific capacities… Simulation of irradiation damage: displacement cascades

Electronic structure methods: Schrödinger equation Hψ(r) = Eψ(r)

N-body problem due to electronic interactions

Impossible to solve exactly for systems with more than 1 electron!

Most common method for solid systems: Density Functional Theory

Principle: transform the problem into a single electron one while retaining description of electronic interaction: important in bonding

Hohenberg-Kohn theorem [1]: Replace wave functionψ for N electrons (3N variables) by electronic density ρ (3 variables)

E[ρ] = T0[ρ] + VH[ρ] + Vext[ρ] + Vxc[ρ]

DENSITY FUNCTIONAL THEORY METHOD: PRINCIPLE

Total kinetic electrons- electrons- exchange- energy energy electrons nuclei correlation

Mean-field: 1-body problem many-body term: unknown

[1] P. Hohenberg, W. Kohn, Phys. Rev. 136, B864 (1964) | PAGE 12 PRACEDays 2016, Prague, May 11th, 2016

Practical way of solving approximate Schrödinger equation: Kohn-Sham [2] Reduce intractable many-body problem of interacting electrons in a static external potential to tractable problem of non-interacting electrons moving in an effective potential (same density) Retain wavefunctions: Independent partial differential equations Standard resolution methods: self-consistent iterative processes to obtain eigenvalues: energies for each configuration (fixed nuclei positions) Energy minimization to obtain equilibrium geometry or molecular dynamics: calculations of forces

Approximation required for Vxc LDA: analytical expression for a uniform electron gas – remarkably good results GGA: takes into account gradient of electron density – improves energies Hybrid functionals: Contain a part of exact exchange DFT+U: Addition of a specific interaction term (Hubbard-like) between correlated orbitals (d or ƒ) Van der Waals functionals: description of dispersive bonds

DENSITY FUNCTIONAL THEORY METHOD IN PRACTICE

[2] W. Kohn, L.J. Sham, Phys. Rev. 140, A1133 (1965) | PAGE 13 PRACEDays 2016, Prague, May 11th, 2016

ELECTRONIC STRUCTURE CALCULATIONS AND ACTINIDE COMPOUNDS

Actinide compounds Complex systems: various phases, non-stoichiometry, magnetism Several possible oxidation states for actinide cations Electrons 5ƒ localized in numerous compounds

Strong electronic correlations

UO2: Mott insulator Insulating character due to strong electronic correlation Gap between 5ƒ bands of uranium unlike band insulators

DFT: Failure of standard approximations (LDA, GGA) Strong electronic correlations not described Mott insulators, in particular UO2, are found metallic Main fission products: rare gases which form dispersive bonds

Other approximations must be used Hybrid functionals, DFT+U, Self Interaction Correction, Van der Waals functionals Currently only DFT+U (VdW-DF+U) used for investigation of defect and solute behaviour

Calculations much heavier than on “standard” materials


ELECTRONIC STRUCTURE CALCULATIONS ON NUCLEAR FUELS AND HPC

Almost everybody can perform DFT calculations on actinide materials Open-source / academic / commercial codes available including data needed on actinide elements: ABINIT, VASP, CRYSTAL, CASTEP… New groups (for instance in China) start on this subject every month

What is difficult is doing calculations properly Convergence difficult in DFT+U method DFT+U lifts degeneracy of 5ƒ orbitals to create band gap In UO2 U4+ cation [Rn] 5ƒ2: several ways to fill the 7 orbitals with 2 e- Change from 1 configuration to another difficult Calculations can be trapped in metastable states energies wrong by several eV

Complicated magnetism (UO2: 3k AFM): distortion, low symmetry Actinide oxides exist in very large stoichiometry ranges, e.g. UO2 actually UO2±x (from UO1.95 to UO2.25) and properties depend on non-stoichiometry Interesting properties difficult to calculate: defects, solutes, diffusion, high temperature properties

You need to be a little smart and to do A LOT OF VERY LONG CALCULATIONS Our group has used CEA HPC since 2000, GENCI since 2008, PRACE in 2015 Largely responsible for prominent position in community along with proximity with experimentalists


Energy

U 5ƒ2 in UO2

ASSESSMENT OF DFT+U RESULTS

ON URANIUM DIOXIDE

M. Bertolus et al., Chapter 14, State-of-the-Art Report on Multi-scale Modelling of Nuclear Fuels OECD, NEA/NSC/R/(2015) E. Vathonne et al., PRB, submitted (2016)

STRUCTURAL, ELASTIC AND COHESIVE PROPERTIES

Cell parameter Experimentally at RT UO2 is cubic (fluorite phase) with a = b = c = 5.473 Å LDA+U: a between 5.41 and 5.45 Å GGA+U: a between 5.44 and 5.57 Å Despite quite large span, reasonable agreement between GGA+U results and exp data, result from Self Interaction Correction or DFT+DMFT methods Calculations done at 0 K while exp value for RT a = b ≠ c for several authors due to approximate 1k AFM order

cnn(GPa) c11 c12 c44 Devey 2011 361 115 64 Dorado 2013 AFM 401 132 94 Dorado 2013 PM 385 125 68 Exp Wachtmann 1965 396 121 64 Exp Fritz 1976 389 119 60

∆Hf (eV)) UO2 U4O9 LDA+U -12.08 -49.61 GGA+U -10.54 -42.46 VdW-DF+U -11.23 -45.50 Exp Guéneau 2011 -11.23 -46.53

Bulk modulus and elastic constants Calculated B between 180 and 222 Gpa compared to between 190 and 213 exp. Very good agreement on elastic constants

Enthalpy of formation of U oxides Good agreement, especially using Van der Waals functionals


COMPARISON OF DFT+U WITH DFT+DMFT

Method GGA+U AFM DFT+DMFT Exp a (Å) 5.57 5.48 5.47

a/c 1.003 1.000 1.000

B (GPa) 194 206 207

C11 (GPa) 346 373 389

C12 (GPa) 118 123 119

C44 (GPa) 58 77 60

O 2p U 5f

Energy (eV)

GGA+U

DFT+DMFT (U=3 eV)

DFT + U yields good structural (despite c/a < 1) and elastic properties Sharp 5ƒ peak due to paramagnetism, not to dynamical description DFT+U yields satisfactory DOS for AFM Confirms evaluation done against exp


PROPERTIES OF DEFECTS

Ea (eV) Vacancy Interstitial DFT+U calculations

Dorado 2011 0.67 0.88 Experimental

Kim 1981 0.51 ± 0.13 - Review Matzke 1987 0.5-0.6 0.8-1.0

Dorado 2011 0.75 ± 0.08

Ea (eV) Stoich UO2 UO2+x DFT+U calculations

Andersson 2011 7.2 4.2 Dorado 2012 3.1-3.9

Experimental results Review Belle 1969 2.3-4.5 3.0-4.6

Review Matzke 1987 5.6 2.6

O migration U migration

Structure of defects well described: linear decrease of the cell parameter of UO2+x when x increases reproduced using O interstitials, positron lifetimes for defects Diffusion coefficients: most commonly available experimental data on defects BUT experimental conditions, which influence results significantly, not always precisely known and models needed for comparison between exp and computational results

Assessment of calculation results against experiments challenging

For O self-diffusion, good agreement among exp results and between the calculated and exp activation energies; difference between I and V migration reproduced For U self-diffusion, very large scatter observed in exp and calculated energies Qualitative difference between stoichiometric UO2 and UO2+x reported by Matzke reproduced in Andersson’s calculations encouraging


CHARACTERIZATION OF POINT IRRADIATION-INDUCED DEFECTS

USING POSITRONS

20

J. Wiktor et al., Phys. Rev. B 90, 184101 (2014)

POSITRON ANNIHILATION SPECTROSCOPY (PAS)

Non-destructive experimental method enabling one to study vacancies in solids Allows studying evolution of defects with temperature (ionisation, clustering, annealing) Neutral and negative defects can be detected and distinguished, while positive ones cannot be detected

Case of fast positrons


POSITRON ANNIHILATION SPECTROSCOPY STUDIES OF UO2

Experimental PAS results on defects in unirradiated and irradiated UO2 available at room temperature [5]

No unequivocal identification of defect observed possible using PAS alone Accurate calculations of the positron lifetimes in UO2 needed

UO2 sample τ1 τ2 Conclusions

Unirradiated 170±5 ps -

Irradiated 1 MeV e- 170±5 ps - O displacements only created. Result consistent with assumption that VO are positive

Irradiated with 2.5 MeV e- or 45 MeV α particules 170±5 ps 310±5 ps

τ2 attributed to vacancy containing at least 1 VU Short lifetime close to the lattice lifetime negative ions detected (O interstitials?)

[5] M.-F. Barthe et al., Phys. Status Solidi C 2007


POSITRON LIFETIME CALCULATIONS

Electronic and positronic densities needed to compute τ can be calculated using electronic structure calculations Self-consistent scheme updated recently in electronic structure code ABINIT Quite long calculations since iterations need to be performed But when convergence reached, forces acting on atoms can be calculated and the relaxation effect can be taken into account

ions

electrons

positron

electrons

positron


POSITRON LIFETIMES YIELDED BY DFT+U IN UO2

τ (ps)

Lattice 167

VO0 199

VU0 304

VU4- 293

VU+VO0 306

VU+VO2- 301

VU+2VO0 313

VU+VU0 318

VU+VU8- 289

2VU+2VO0 339

2VU+2VO4- 319

2VU+4VO0 329

2VU+4VO2- 365

Code used: ABINIT. Cost: 5 Mh Curie


In bold, lifetimes for most stable charge states [7] J. Wiktor et al., MRS Proceedings 2014 [8] E. Vathonne et al., J. Phys.: Condens. Matter. 2014

Calculated lattice lifetime in good agreement with measured one in unirradiated UO2

Method appropriate for this system

Different defects can have similar lifetimes

τ2 of 310±5 ps observed in 2.5 MeV e- and 45 MeV α particule irradiated UO2 can correspond to 3 defects: VU+VO2-, VU+2VO or 2VU+2VO4-, which contain one U vacancy

NEW RESULTS ON TEMPERATURE EVOLUTION

UO2, 1700˚C/24h/ArH2 O/U=2.005±0.005

Irradiated with α 45 MeV, 3×1016cm-2

Evolution of positron trapping with temperature varies as a function of defect charge Knowledge of temperature dependence of annihilation characteristics can help in determining the charge state of the defect probed Application of positron trapping model to defects concentrations 3 traps needed to reproduce exp. evolution with T: negative interstitials (NI), neutral vacancy (VO) and negative vacancy (V-)

Neutral VU+2VO (Schottky defect) predominant Negative vacancies: probably mix of VU4-, VU+VO2-, and 2VU+2VO4-

CNI (cm-3) CV- (cm-3) CV0 (cm-3)

1x1019 2x1018 6.5x1019


COMBINATION OF ATOMIC SCALE CALCULATIONS AND

DIFFUSION MODELS TO STUDY KR DIFFUSION IN UO2

26 E. Vathonne et al., Phys. Rev. B, submitted (2016)


INVESTIGATION OF SOLUTE DIFFUSION IN SOLIDS USING FIVE-FREQUENCY MODEL

Diffusion models: link between elementary mechanisms at the atomic scale and macroscopic diffusion [9] Data needed: Theoretically all elementary mechanisms Approximation to limit the number of mechanisms: no interaction beyond 2nd neighbours 5-frequency model

Elementary mechanisms considered ω0 (E0, ν0): vacancy migration between adjacent lattice positions in absence of impurity ω1 (E1, ν1 ): vacancy migration between first neighbour sites of the impurity ω2 (E2, ν2): vacancy-impurity exchange ω3 (E3, ν3): transition of the vacancy from a 1st neighbour site to a 2nd neighbour site (dissociation) ω4 (E4, ν4): reverse transition of ω3 (association)

Diffusion coefficient 𝐷𝐷 can be obtained from atomic scale calculations of elementary mechanisms

[9] H. Mehrer, Diffusion in Solids, Springer Series in solid state sciences 155 (2007)

𝐷𝐷 = 𝐷𝐷0 exp( −𝐸𝐸𝑎𝑎𝑘𝑘𝑘𝑘

)

𝐸𝐸𝑎𝑎 = 𝐸𝐸𝑖𝑖 + 𝐸𝐸𝑓𝑓(𝑉𝑉𝑈𝑈) + 𝐸𝐸𝐵𝐵−∆𝐸𝐸𝑠𝑠𝑠𝑠𝑠𝑠

𝜔𝜔𝑖𝑖 = 𝜈𝜈𝑖𝑖 exp( −𝐸𝐸𝑖𝑖𝑘𝑘𝑘𝑘

)

Traps considered: most stable vacancies in UO2 and most favourable for Kr incorporation depending on non-stoichiometry: VU, VUO, VUO2. Assisting vacancy: VU Stable complex between Kr and VU2O or VU2O2 in these cases no ω2 mechanism: Diffusion model was adapted and expression of 𝐷𝐷 was derived

Defect formation and Kr solution energies calculated in DFT+U (energy minimizations) Elementary migration energies calculated in DFT+U using static method for pathway and saddle-point determination (NEB): energy minimization for all degrees of freedom except 1. Attempt frequencies obtained from phonon modes of defect at initial and saddle points using empirical potential methods “Simple” migration via interstitial mechanism also investigated


𝑉𝑉𝑈𝑈4− moves closer to Kr in Schottky defect

Migration of 𝑉𝑉𝑈𝑈4− around Kr in trap inducing Kr migration

Formation of 𝑉𝑉𝑈𝑈4− far from Kr

FIRST APPLICATION TO UO2: KR DIFFUSION

Mechanisms and trends with non-stoichiometry similar to previous study of Xe For UO2+x: migration assisted by two VU4- dominates and 0.73 < Ea < 4.09 eV Exp Ea, very probably obtained for hyper UO2, are in this interval Further experiments with precisely controlled stoichiometry in progress

RESULTS: DIFFUSION COEFFICIENTS OF KR IN UO2


VASP code 10 Mh Curie

CONCLUSIONS PERSPECTIVES

30

Atomic scale calculations, in particular using electronic structure methods, helpful in the interpretation of experimental results on defects and fission gas behaviour

Combination of modelling and experiments helps get further insight on elementary mechanisms necessary for higher scale models

HPC determinant in obtaining relevant results PERSPECTIVES

Simulate more complex properties, e.g. diffusion under irradiation, finite temperature and thermodynamic properties: Phonon calculations, Ab initio Molecular dynamics

Application to more complex fuel materials for next generation reactors, in particular mixed oxides (U,Pu)O2, (U,Am)O2, (U,Pu,Am)O2 Our HPC needs are likely to continue increasing regularly

CONCLUSIONS AND PERSPECTIVES


DEN DEC SESC

Commissariat à l’énergie atomique et aux énergies alternatives Centre de Cadarache | 13108 Saint-Paule-lez-Durance Cedex T. +33 (0)4 42 25 30 34

Etablissement public à caractère industriel et commercial | RCS Paris B 775 685 019

| PAGE 32

THANK YOU FOR YOUR ATTENTION

And thanks to Carole Valot, CEA, DEN Marc Torrent, CEA, DIF

Matthias Krack, PSI Ram Devanathan, PNNL

Pieter Glatzel, ESRF Olivier Proux, OSUG Grenoble

Dave Andersson, LANL

Investigation of �nuclear fuel materials �under irradiation �at the atomic scaleINTRODUCTION - CONTEXTNUCLEAR POWER & nuclear fuelsNUCLEAR FUEL BEHAVIOUR in reactorFuel performance codes: Simulation OF �NUCLEAR fuel behaviour at the macro scaleUNDERSTANDING FUEL BEHAVIOURMAIN METHODS AND TECHNIQUES APPLIEDCONTENT of presentationATOMIC SCALE MODELLING METHODS and APPLICATION TO NUCLEAR FUELSAtomic scale modellingData yielded by atomic scale methodSDiapositive numéro 12Diapositive numéro 13ELECTRONIC STRUCTURE CALCULATIONS �and actinide compounds ELECTRONIC STRUCTURE CALCULATIONS �ON nuclear fuels and HPC�ASSESsMENT OF DFT+U resulTs on uranium dioxide�Structural, elastic AND cohesive propertiesComparison OF DFT+U WITH DFT+DMFTPROPERTIES OF DEFECTSCHARacterization of point IRRADIATION-INDUCED defecTs using PositronsPositron Annihilation Spectroscopy (Pas)Positron annihilation �spectroscopy studies of UO2Positron lifetime calculationsDiapositive numéro 24Diapositive numéro 25Combination of atomic scale calculations and diffusion models to study kr diffusion in UO2INVESTIGATION OF SOLUTE DIFFUSION IN �SOLIDs USING Five-Frequency modelFIRST APPLICATION TO UO2: Kr diffusion RESULTS: DIFFUSION COEFFICIENTS OF KR IN UO2CONCLUSIONS �PERSPECTIVESConclusionS and perspectivesDEN�DEC�SESC

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