multiscale modelling of nuclear fuels under irradiation...workshop materials innovation for nuclear...

Centre of Excellence for Nuclear Materials

Workshop Materials Innovation for Nuclear Optimized Systems

December 5-7, 2012, CEA – INSTN Saclay, France

Michel FREYSS CEA (France)

Multiscale Modelling of Nuclear Fuels under Irradiation

Workshop organized by: Christophe GALLÉ, CEA/MINOS, Saclay – [email protected] Constantin MEIS, CEA/INSTN, Saclay – [email protected]

Article available at http://www.epj-conferences.org or http://dx.doi.org/10.1051/epjconf/20135102003

http://www.epj-conferences.orghttp://dx.doi.org/10.1051/epjconf/20135102003

Workshop Materials Innovation for Nuclear Optimized Systems December 5-7, 2012, CEA – INSTN Saclay, France

Multiscale Modelling of Nuclear Fuels under Irradiation

Michel FREYSS11CEA-DEN-DEC, Service d’Etudes et de Simulation du Comportement des Combustibles, SESC (Cadarache, France)

The fuel element under irradiation is submitted to a wide variety of coupled phenomena involving among others temperature, mechanical load, radiation damage, chemical interaction between the material and the fission products. The PLEIADES fuel performance software environment [1,2] developed by CEA can predict the behaviour of standard or innovative fuel elements under operating conditions. It is nevertheless still a challenge for R&D to refine the laws used in fuel performance codes by a more physically based description of the fuel materials, and improve both the understanding of the phenomena involved during irradiation and the capability to predict the fuel behaviour. This goal requires to decorrelate the complex phenomena involved in the material evolution by conducting studies towards the atomistic level. It also requires to couple post-irradiation examinations (PIE) with separate-effect experiments and various modeling approaches at the relevant scales. In particular, basic research on fuel materials focuses on the evolution under irradiation of the microstructure, the transport properties of defects, fission products, helium, as well as their thermochemistry.

Fig. 1: Multiscale scheme applied to oxide fuels, as implemented in the European F-BRIDGE project [3].

Nucleation of FG bubblesAtomic gas

diffusion

ProductionXe Bubble/pore destruction

re-solution

Bubble growthPore shrinking

Bubble/pore movement

Xe

Xe

Xe

U235

Xe

Xe

Big bubbles« precipitation »

Xe


diffusion


re-solution



Xe

Xe

Xe

U235

Xe

Xe


Xe

First Principles

Empirical potential / classical molecular dynamics

nm µm mm m distance

ps

µs

s

Days / years

time

ms

Thermo-mechanical Finite Element Modeling Fuel performance codes

atoms Grain(s) Pellet(s)

Thermodynamics modeling

Kinetic models/ Rate theory Cluster dynamics

Homogenization micro/macro

EPJ Web of Conferences 51, 02003 (2013) DOI: 10.1051/epjconf/20135102003 © Owned by the authors, published by EDP Sciences, 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License 2.0, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Workshop Materials Innovation for Nuclear Optimized Systems December 5-7, 2012, CEA – INSTN Saclay, France

The overall strategy for a multiscale modeling scheme of nuclear fuels will be presented, with some examples of applications making use of atomic scale approaches: First-principles electronic structure calculations used to get insight into the atomic transport properties of point defects [4] and classical molecular dynamics used to model the ballistic damage created by the recoil of fission products [5]. The importance of coupling the multiscale modeling approach to experimental studies will also be emphasized by some examples to illustrate the characterization of the oxygen diffusion [6] and the bubble formation of fission gases [7] in UO2. Such separate effects experiments should be seen as guides to orientate modeling, but also as essential tools to validate the approximations of the modeling methods.

References

[1] B. Michel, C. Nonon, J. Sercombe, F. Michel, V. Marelle. to appear in Nuclear Technology (2012). [2] L. Noirot. Nuclear Engineering and Design 241, 2099 (2011). [3] F-BRIDGE project (Basic Research for Innovative Fuel Design for GEN IV systems), project of the

Seventh Framework Programme of the European Commission, http://www.f-bridge.eu/ [4] B. Dorado, D. A. Andersson, C. R. Stanek, M. Bertolus, B. P. Uberuaga, G. Martin, M. Freyss, P.

Garcia. Phys. Rev. B 86, 035110 (2012). [5] G. Martin, P. Garcia, L. Van Brutzel, B. Dorado, S. Maillard, Nuclear Instruments and Methods.

Physics Research B 269, 1727 (2011). [6] B. Dorado, P. Garcia, M. Freyss, G. Carlot, M. Fraczkiewicz, B. Pasquet, G. Baldinozzi, D.

Simeone, M. Freyss, M. Bertolus. Phys. Rev. B 83, 035126 (2011). [7] P. Garcia, G. Martin, C. Sabathier, G. Carlot, A. Michel, P. Martin, B. Dorado, M. Freyss, M.

Bertolus, R. Skorek, J. Noirot, L. Noirot, O. Kaitasov, S. Maillard. Nuclear Instruments and Methods. Physics Research B 277, 98 (2012).

MINOS Workshop, Materials Innovation for Nuclear Optimized SystemsDecember 5-7, 2012, CEA – INSTN Saclay, France

MULTISCALE MODELLING OF NUCLEAR FUELS UNDER IRRADIATION

Fuel Study Department (DEC)Service d’Etude et de Simulation des Combustibles (SESC)

CEA, DEN, Cadarache

M. Freyss

M. Bertolus, P. Garcia, R. Skorek, S. Maillard, G. Martin, C. Sabathier, E. Vathonne, G. Carlot, A. Michel, V. Basini

B. Michel, L. Noirot, A. Bouloré, V. Blanc, R. Masson

C. Valot

| PAGE 1CEA | 10 AVRIL 2012

CEA – DEN MINOS Workshop - December 5-7, 2012, CEA – INSTN Saclay, France 2

Goal: behaviour of fuel elements under operating conditions

Multiscale approach : refinement of laws and parameters by modellingrelevant phenomena at a smaller scale.

Basis of the approach : experimental data and first-principles electronicstructure calculations

Much progress achieved lately at CEA in coupling methods :- atomistic (ab initio, MD) and mesoscopic (cluster dynamics) modelling techniques - seperate-effect experiments and atomistic modelling [oxygen transport : PRB 83, 035126,(2011)]

Links between mesoscopic and macroscopic scales need to be strengthen

MULTISCALE SCHEME FOR FUEL MODELLING


diffusion

ProductionXe

Bubble/pore destruction re-solution



Xe

Xe

Xe

U235

Xe

Xe


Xe


diffusion

ProductionXe

Bubble/pore destruction re-solution



Xe

Xe

Xe

U235

Xe

Xe


Xe

First Principles

Empiricalpotential / classical molecular dynamics

nm µm mm m

distance

ps

µs

s

Days/ years

time

ms

Thermo-mechanicalFinite Element ModelingFuel performance codes

atoms Grain(s) Pellet(s)

Thermodynamics modeling

Kinetic models/Rate theoryCluster dynamics

Homogenization micro/macro


ILLUSTRATIONS

MACROSCOPIC SCALE

Representative Elementary Volume scheme forfuel microstructure modelling

MESOSCOPIC SCALE

Between fuel performance code and atomistic simulationsFission gas behaviour by cluster dynamics

MICROSCOPIC SCALE

Recent improvements in electronic structure calculations for oxide fuels: the DFT+DMFT method and van der Waals interactions

MICROSTRUCTURE AND THERMO-

ELASTIC MODELLING OF THE FUEL

ELEMENT

« REPRESENTATIVE ELEMENTARY

VOLUME » SCHEME


REPRESENTATIVE ELEMENTARY VOLUME (REV) APPLICATIONS

The thermo-mechanical modelling of the heterogeneous fuel material is based on the homogenization theory with mean-field and/or full-field approaches

Mean-field approach: a simplified geometry is assumed for each phases of the heterogeneous material and spatial distributions of thermo-mechanical fields in each phases are described through mean values → constitutive equations of a macroscopic law can be easily derived

Full-field approach : a detailed description of the heterogeneous material is achieved within a Representative Elementary Volume (REV) using a numerical model (finite elements or FFT grid). The spatial distribution of thermo-mechanical fields is computed for each cell of the geometrical discretization → it can be used to build a mean-field model

Heterogeneous microstructure of nuclear fuelsgrains of various orientations, plutonium distribution in MOX fuels, ...

(U,Pu)O2Pu rich aggregates

U rich aggregates

Corresponding REV

Pu and U distribution in irradiated MOX fuelsElectron Probe Micro-Analysis (EPMA)

Oudinet et al., J. Nucl. Mat 375, 86 (2008)


REPRESENTATIVE ELEMENTARY VOLUME (REV) FUEL MICROSTRUCTURE

REV correspondingto a polycrystallineaggregate

(color = orientation)

Scanning electronmicroscopy image of an irradiatedUO2 (25 GWd/tU) I. Zacharie et al., J. Nucl. Mat. 255, 92 (1998)

Application of the REV scheme to describe the grain structure of the fuel

Anisotropic mechanical behaviour: viscoplastic behaviour of the grain aggregatesstrongly depends on the grain orientation

Anisotropic yieldstress in the UO 2

single crystalfor a compression test

(2.10-05 s-1, 1350°C)

Necessity to integrate the microstucture effects in fission gas release modelsand codes in order to improve the modelling of in-pile fuels

Van Mises stress (Pa)

Anisotropicstress in the UO2 REV

5 µµµµm

MESOSCOPIC MODELLING OF

FISSION GAS BEHAVIOUR

BETWEEN FUEL PERFORMANCE

CODES AND ATOMISTIC

CALCULATIONS


FISSION GAS BEHAVIOUR MODELLING

fuel element


diffusion


re-solution



Xe

Xe

Xe

U235

Xe

Xe


Xe


diffusion


re-solution



Xe

Xe

Xe

U235

Xe

Xe


Xe

Modelling the evolution of themicrostructure under in-pile or ion irradiation: populations of point defects, defect clusters and fission gas atoms and bubbles

– Essential since these populations control a large number of properties and the macroscopic evolution of the material

MARGARET code within the PLEIADES platform at CEA: Most observations are correctly simulated (gas concentration, gas release, swelling, porosity) but some phenomena are modelled empirically, e.g. bubble sink strength and bubble size distribution

Cluster Dynamics (rate theory model): Comprehensive framework to calculate defect and defect cluster (cavities / bubbles / dislocation loops) concentrations over time, with time scale and length scale appropriate for fuel study (years, grains)

For intraganular phenomena, MARGARET and cluster dynamics describe the samescales → Study using cluster dynamics of intragranular fission ga z behaviour


CLUSTER DYNAMICS (RATE THEORY)

System seen as gas of clusters of various sizes of vacancies, interstitials and solute atoms (fission gas)Spatially averaged : atomic positions are not consideredSet of differential equations on species concentrations: calculation of defect and defect clusters (cavities / bubbles / interstitial loops) concentrations over time

Equations solved numerically: maximum size of clusters is limited by computational resources

Source

term

Sink term

(for mobile

clusters)

Absorption of

mobile cluster (m,q)

by cluster (n,p)

Emission of

mobile cluster (m,q)

from cluster (n,p)

( )( ) ( )

( ) ( )pnKpnGJJdt

pndC

mobileqmqpmnqmpn

mobileqmpnqmqpmn ,,

,

,),(),(),(

,),(),(),( −+−= ∑∑ ++↔+↔+−−

Code CRESCENDO developed by CEA-EDF (cf. Thomas Jourdan’s presentation )


CLUSTER DYNAMICS: ANNEALING SIMULATIONOF KRYPTON IN UO2

Exp

Exp

Exp

Simu.

Simu.

Simu.

Ab initio

Adjustment

R. Skorek et al., J. Defect Diff. Forum, 323-325, 209 (2012)

Ab initio calculations can provide input data needed . It requires the relevant approximations to model UO2 (5f electron localization) and the bonding of rare gases

Confrontation with separate-effect experiments (PhD A. Michel, CEA-DEC / Caen University 2011)Thermal Desorption Spectroscopy experiments on UO2 samples implanted with 250 keV Kr ions

Fra

ctio

n of

Kr

rele

ased

Two Kr release regimes: initial burst + diffusion regime, related to the concentration of Kr in Schottky (VKr) and bi-Schottky (V2Kr) → Insight into diffusion mechanisms

Formation energies

Ef (V) 2.5 eV

Ef (VKr) 3.8 eV

Ef (I) 10 eV

Ef (Kr) 6.6 eV

Migration energiesEm(V) 3.0 eV

Em(V2Kr) 4.5 eV

Em(I) 0.7 eV

Em(Kr) 2.2 eV

(B. Dorado +E. VathonnePhD thesis)

RECENT IMPROVEMENTS IN

FIRST-PRINCIPLES CALCULATIONS

FOR NUCLEAR FUELS

STRONG CORRELATIONS AND

VAN DER WAALS INTERACTIONS


Density Functional Theory calculationsActinide compoundsLocalized 5ƒ electrons in many compounds: strong electron correlationsSeveral oxydation states for actinide cationsComplex structures and magnetic properties (Jahn-Teller distortion and non-collinear magnetic order in UO2)

UO2: Mott insulatorInsulating character due to strong 5ƒ electron correlationsGap between uranium 5ƒ bands

Failure of DFT within the standard approximations (LDA , GGA)Underestimate the strong electron correlationsMott insulators are found metallic, in particuliar UO2Failure in the description of dispersive bonds formed by rare gases

Approximations beyond standard DFT should be usedStrong correlations: Hybrid fonctionals, DFT+U, self interaction correction,DFT+DMFT. Up to now, only DFT+U used for the study of point defects in UO2Dispersive bonds: empirical description or non-local correlation functionals

ELECTRONIC STRUCTURE CALCULATIONS FOR ACTINIDE OXIDES AND RARE GASES


DESCRIPTION OF VAN DER WAALS (vdW) INTERACTIONS

DFT in local or semi-local approximations (LDA / GGA) can not account for long-range van der Waals (vdW ) interactions (non-local electron-electron correlations).

Failure of LDA and GGA in the description of rare gas bonds: dimers, clusters, ...

For bulk ionic solids : error on cohesiveproperties reduced by factor 2 when vdWenergy is coupled with DFT

TiO2: Moellmann et al. J. Phys. Cond. Matter 24, 424206 (2012)NaCl, MgO: Zhang et al., Phys.Rev. Lett. 107, 245501 (2011)

Benchmark for DFT description of bonds between rare gas atoms and smallmolecules Bertolus et al. Phys. Chem Chem. Phys. 14, 553 (2012)

=> Assessment of DFT treatment for rare gas incorporat ion in UO 2


vdW-DF : van der Waals density functional

Functional of the electron density taking into account non- local electron correlationeffects [ ] [ ] [ ]nEnEnE nlcDFTcc +=

Various formulations :

vdW-DF 04: Dion et al. Phys. Rev. Lett. 92, 246401 (2004)Roman-Pérez et al. Phys. Rev. Lett. 103, 096102 (2009)GGA= revPBE VASP

vdW-DF 10: J. Klimeš et al., J. Phys. Cond. Mat. 22, 022201 (2010)GGA=optPBE VASP

The vdW interaction enters through the fully non-local correction Ecnl [n] whichconcerns the correlation only

DESCRIPTION OF VAN DER WAALS INTERACTIONSWITHIN DFT

( ) ( ) ( )'',' rrrrrr nnddE cnlc Φ= ∫∫

Ar-Ar

Espejo et al, Comp. Phys. Comm 183, 480 (2012)


DFT+U calculations with occupation matrix control scheme (OMC)Code VASP

DFT functionals in DFT+U: GGA (PBE)Non-local correlation functional: vdW-DF 10Incorporation energies of Kr and Xe in UO2Various positions: interstitial, U substitution, Schottky defects (UO2 tri-vacancy)

Interstitial U substitution Schottky

DESCRIPTION OF VAN DER WAALS INTERACTIONSINCORPORATION OF RARE GASES IN UO 2


Incorporation energies of rare gases (Kr , Xe) in UO2

DFT(GGA)+U DFT(GGA)+U vdW-DF 10

Kr interstitial 6.47 6.30

Kr uranium site 2.68 2.04

Kr in Schottky 1.20 0.58

Xe interstitial 9.31 9.26

Xe uranium site 4.29 3.60

Xe in Schottky 1.77 1.03

Sharing of electron density between rare gas and neighbouring atoms: DFT OKCases of Kr and Xe at interstitial sites

Little sharing of electron density: contribution of dispersive interaction important Cases of Kr and Xe in large defects (Schottky) and rare gas clustersIn line with the study of rare gas incorporation in small molecules [Bertolus et al. Phys. Chem. Chem. Phys. 14, 553 (2012)]

DESCRIPTION OF VAN DER WAALS INTERACTIONSINCORPORATION OF RARE GASES IN UO 2

Kr Xe

Kr interstitial Xe in Schottky

DFT+U atomic structures used + total energy calculation with vdW functionalsStructure and electron density mostly determined by semilocal exchange and correlation as in standard DFT


STRONG CORRELATIONS: THE DFT+U METHOD

Addition of a Coulomb interaction term for the correlated orbitals:

EU expressed as a function of direct Coulomb U and exchange J parametersU and J parameters can be extracted from experimental spectroscopic data or calculated ab initio. For UO2, U=4.50 eV J=0.54 eV from XPS [Kotani et al.,Physica B 186, 16 (1993)]

DFT+U improves the description of strongly correlated materials: electronic structure(band gap), structural properties, energetics, etcSuccessful application to point defects in UO 2 and self-diffusion of O in UO 2

dcUDFTUDFT EEEE −+=+ [Anisimov et al., Phys. Rev. B 44, 943 (1991)]

[Dorado et al., PRB B 83, 035126 (2011), PRB 86, 035110 (2012)]

Activation energy (eV) Eact (DFT+U) Eact (exp.)

Direct interst. 3.170.75 ± 0.08

Indirect interst. 0.88

Vacancy 0.670.51 ± 0.13 *

Vacancy 2.47

Diffusion experiments at CEA: electrical conductivity measurements + SIMS experiments + control of parameters that affect the material (oxygen partial pressure and impurity content) Garcia et al., JNM. 400, 112 (2010)

Indirect interstitial mechanism vacancy mechanism

* Kim et al, J. Nucl. Mater. 102, 192 (1981)


LIMITATIONS OF THE DFT+U METHODTHE DFT+DMFT METHOD (HUBBARD I)

Limitations of the DFT+U methodLocal energy minima issue:

The system can get trapped in a local energy minimum dependingon the starting point of the calculation (5f orbital occupation)Solution: Occupation matrix control scheme (OMC): Preliminarysystematic study of ground-state and metastable occupations [Dorado et al., Phys. Rev. B 79, 235125 (2009)]

Paramagnetic order difficult to describe (UO2 paramagnetic above 30 K)

The DFT+DMFT methodCombination of DFT and dynamical mean–field theory (DMFT)

The local correlations are calculated accurately for an isolated atom (in red) in the effective field of the other atoms

Self-consistent impurity problem : Anderson impurity model solved using the Hubbard I solver to provide the lattice Green’s function (fast and accurate enough)

metastableconfiguration

ground-stateconfiguration

[Georges et al., Rev. Mod. Phys. 68, 13 (1996)]

ABINIT code , Amadon J. Phys Cond. Matter 24, 075604 (2012)


DFT+DMFT RESULTS FOR UO2BULK PROPERTIES

DFT+DMFT improves the description of the bulk and electronic properties of UO 2

Improved description of the strongcorrelations and magnetism(paramagnetism) of UO2

DFT+U (OMC) DFT+DMFT Exp.

Lattice parameter (Å) 5.57 (a=b) 5.49 (c) 5.48 5.47

Bulk modulus 194 206 207

C11 (GPa) 346 373 389

C12 (GPa) 118 123 119

C44 (GPa) 58 77 60

c/a ratio

DFT+DMFT preserves the cubic lattice of UO 2

Experimental data

Y. Baer and J. Schoenes, Solid State Com. 33, 885 (1980)

O 2p

U 5f


CONCLUSION

Toward a more physics-based description of the microstructure and the behaviour of fission gases in fuel elements

- Direct and effective links between atomistic and mesoscale simulations - Filling the gap at the mesoscopic scale : links between cluster dynamics

simulations and the MARGARET code in the fuel performance platformPLEIADES

- Coupling between mechanics and physico-chemistry

Improvements of the electronic structure description of strongly correlated UO2 and of fission gas behaviour by DFT-based methods. First application in progress of DFT+DMFT for point defects in UO2 (collaboration CEA/DAM)

Quality and accuracy of the modelling predictions can only be ensured provided some experimental data are available for validation:

- Post-irradiation examinations (macro and meso-scales)- Seperate-effect experiments (meso and micro-scales) TEM, XRD, XAS,...

(cf. Christophe Valot’s presentation)

DEN

DEC

SESC

LLCC

Commissariat à l’énergie atomique et aux énergies alternatives

Centre de Cadarache | 13108 Saint-Paul-Lez-DuranceT. +33 (0)4 42 25 70 00

Etablissement public à caractère industriel et commercial | RCS Paris B 775 685 0194 décembre 2012

| PAGE 21

Thank you for your attention

ACKNOWLEDGEMENTS

Computational resources : TGCC and CINES

PLEIADES (CEA / EDF / AREVA)

F-BRIDGE European Project (FP-7)

B. Dorado, B. Amadon CEA / DAM DIF, Bruyères-le-Châtel

J.-P. Crocombette, T Jourdan, A. BarbuCEA / DEN / DMN / SRMP, Saclay

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