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fission products (FPs). • Different interactions with the fuel matrix. 3. Olander, Fundamental Aspects of Nuclear Reactor Fuel. Elements (1985) ...

Computational Investigation of Point Defect Formation and Migration in Nuclear Fuels Pankaj Nerikar1, Minki Hong2, Taku Watanabe3, James S. Tulenko2, Blas P. Uberuaga4, Chris Stanek4, Simon R. Phillpot2 and Susan B. Sinnott2 1IBM,

India 2Department of Materials Science and Engineering, University of Florida 3Department of Chemical Engineering, Georgia Tech 4Los Alamos National Laboratory This work was funded by the DOE

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Summary of Sinnott Research Activities Electronic structure of heterogeneous interfaces

Chemical modification of polymers

Electronic properties of nanostructures

Friction at the atomic scale

Defect segregation to grain boundaries

Background

• Regions of a restructured fuel rod • Microstructure changes with temperature

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• Optical micrograph showing fission products (FPs) • Different interactions with the fuel matrix

Olander, Fundamental Aspects of Nuclear Reactor Fuel Elements (1985)

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• TEM of irradiated UO2 showing tilt boundaries. • Boundaries decorated with fission gas bubbles.

Objectives

Sonoda, Nuclear Instruments and Methods in Physics Research B 191 (2002)

• Determine the role of temperature, partial pressure and charge on point defect stability • Examine the stability of FPs and their interactions with the fuel matrix • Investigate FP segregation to grain boundaries

Time and Length Scale of Radiation Modeling

Zinkle, Workshop on Advanced Simulations (2005)

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Density Functional Theory (DFT) +

Special SchemeDFT+U: forces on site repulsion

• Limitation: Fails to describe strongly correlated systems like UO2 Mattsson, et al., Simulation in Materials Science and Engineering, 13 (2005)

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Empirical Potential Approach

• Buckingham potential that describes the energy • • • •

of the system as a function of interatomic position and ionic charge Larger supercells (10x10x10) can be treated as compared to DFT (3x3x3) Small CPU time (hours as compared to days with DFT) Guides the more expensive DFT calculations General Utility Lattice Program1 (GULP)

1Gale,

Journal of the Chemical Society: Faraday Transactions 93 (1997)

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Motivation • UO2 is used as the standard fuel in pressurized water reactors • Point defects are created by selfirradiation and influence fuel performance • Understanding the relative stabilities of defects will allow us to ultimately predict fuel performance computationally Properties • Antiferromagnetic insulator • Space group: F m 3 m • a = 5.463 Å • Melting point = 2852 C • Chemically stable

U4+ O2-

Ion

Ionic Radii (Å)

U4+

1.00

O2-

1.38

Shannon, Acta. Cryst. A32, (1975)

Bulk UO2 Lattice Parameter

The SP-GGA+U approach gives the best comparison with experiment 1Villars

and Calvert, ASM (1985)

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Electronic Structure of Bulk UO2 DOS (a.u.)

120 110 100 90 80 70 60 50 40 30 20 10 0

-7

-6

-5

-4

-3

-2

-1

UO2: SP-GGA UO2-SP-GGA Fermi energy = 0 eV Band Gap = 0 eV

0 1 Energy (eV)

DOS (a.u.)

60 55 50 45 40 35 30 25 20 15 10 5 0 -7

-6

-5

-4

-3

-2

-1

2

3

4

5

6

7

8

1Experimental

Band Gap = 2 eV

UO2:SPGGA+U UO2-SP-GGA+U Fermi energy = 0 eV Band Gap = 1.96 eV

0

1

2

3

4

5

6

7

Energy (eV)

This approach allows us to consider the charge of the point defects 1Baer

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et al., Solid State Communications, 33 (1980)

Predicted and Observed Phase Order for Bulk U Phase Temperature Range (K)

Energy/atom (eV)

α-U

298-941

-8.08

β-U

941-1049

-7.93

γ-U

1049-1408

-7.37

Nerikar et al., Journal of Nuclear Materials, 384 (2009)

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Combined DFT and Thermodynamic Calculations ∆G f

VO

= E total (VO ) − E total ( perfect ) + µO + q[ E F + EV + ∆V ]

µ O = µ O0 +

µO0 =

‘ ‘ ‘

1 k T ln ( PO 2 / P 0 ) 2

1 0 0 ( µUO2 − µU0 − ∆GUO ) + ∆GO (T ) 2 2

’ are obtained from DFT calculations ’ are obtained from thermodynamic data ’ are variables

Empirically Predicted Defect Formation Energy 1.8 Frenkel Anti-Frenkel Schottky

1.6 1.4 1.2 1 0.8 0.6 0.4

Defect formation energy converges with system size 0.2 0

1 1x1x1

2x2x2 2 2x2x2

3x3x3 3 3x3x3 System Size

4x4x4 4 4x4x4

5x5x5 5 5x5x5

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DFT+U Approach: Defect Formation Energy 16 14

1

EXPERIMENTAL

2

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Gupta:SP-GGA+U1 Freyss:GGA 3

2

10 THIS WORK-SP-GGA+U 3

8 6 4 2 0 Anti Frenkel 1Matzke,

Frenkel Defect Type

Schottky

Journal of the Chemical Society, Faraday Transactions, 83 (1987) 2 Gupta et.al., Philosophical Magazine, 87 (2007) 3Freyss et al. Journal of Nuclear Materials 347 (2005)

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Effect of Temperature 10 pO2=1 atm 8

6

4

2

0 300

O Vacancy O Interstitial U Vacancy U Interstitial

400

500

600

700

800

900 1000 1100 1200 1300 1400

-2 Temperature (K)

Phase:α-U

Phase:β-U

Nerikar et al., Journal of Nuclear Materials, 384 (2009)

Phase:γ-U

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Effect of Partial Pressure of O2 11 800 K U vacancy

9

7

U interstitial

5 O vacancy 3

1 O interstitial -23

-21

-19

-17

-15

-13

-11

-9

-7

-5

Oxygen Partial Pressure, ln (P/Po)

Nerikar et al., Journal of Nuclear Materials, 384 (2009)

-3

-1 -1

Effect of Variable Charge States • It is usual to think of point defects in metal oxides as being • •



charged. Most calculations do not take the effect of charge into account. The effect of charge is considered in two places: • DFT calculation of charged, defect supercell • Electron chemical potential The most stable defect has the lowest formation energy for a given Fermi level.

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Effect of Charge: Point Defects

•Stability of defects changes dramatically with charge • For only vacancies, +2 charged oxygen vacancy is stable when the Fermi level is near the valence band • The -4 charged uranium vacancy becomes stable when the Fermi level moves towards the conduction band

Nerikar et al., Journal of Nuclear Materials, 384 (2009)

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Effect of Charge: Defect Complex

• Charged complexes more stable than neutral ones Nerikar et al., Journal of Nuclear Materials, 384 (2009)

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Detrimental effects of fission products (FPs) • FPs detrimental under normal operating conditions. • Form bubbles, accumulate at boundaries and interfaces, affect mechanical properties. • Influence long term storage and mechanical/electrical properties of the fuel. • Gaseous FP (Xe) escapes to the surface causing damage to the cladding while solid FPs form oxides and/or metallic inclusions. • Interactions depend on stoichiometry and prevailing conditions.

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Methodology • GGA,SP-GGA+U and empirical potentials • Fission products considered: Xe, Cs, Sr, and Ru • Positions considered:

Octahedral

Oxygen Vacancy

Uranium Vacancy

Divacancy (1U+1O)

Bound Schottky (1U + 2O)

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Incorporation Energy Characteristics • Defined as the energy for taking a fission product and placing it at preexisting trap site. • Useful quantity for determining the most stable site. • A positive value corresponds to energy being required to place a fission product at a particular trap site. • Formal Definition:

Einc. = E total (α ) − E total (defect ) − Ei Limitations • Assumes excess of trap sites. • Assumption valid at only low burn-ups. Grimes and Catlow, Philosophical Transactions: Physical Sciences and Engineering, 335 (1991)

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Result: Incorporation Energy Xe Cs Sr Ru

• Higher charged fission products more stable than neutral ones • Stable phase of fission products matters Nerikar et al., Journal of Physics Condensed Matter 21 (2009) Hong et al. (in preparation)

Solution Energy

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Characteristics • Defined as the energy for placing a fission product at a particular trap site. • It depends on stoichiometry and hence burn-up. • Can be correlated with experimentally observed trends. • Formal definition:

Esolution = Einc. + Etrap • Trap site formation energy

E trap = −kT ln([ X])

• Based on the point defect model(PDM) by Matzke and Lidiard.1 Limitations • Assumes no interactions between fission products.

1Lidiard,

Journal of Nuclear Materials, 19 (1966)

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Result: Stoichiometry effect on Solution Energy

• Interstitial site independent of stoichiometry • Competition between incorporation energy and trap formation energy • Trap site stability changes with stoichiometry Nerikar et al., Journal of Physics Condensed Matter 21 (2009)

Result: Solution Energies of FPs (UO2+x) Xe Cs Sr Ru

• Stability: Sr > Cs > Ru > Xe Nerikar et al., Journal of Physics Condensed Matter 21 (2009) Hong et al. (in preparation)

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Oxide Solution Energy

Characteristics • Cs and Sr are known to react with oxygen to form secondary phases. • Can be correlated with experimentally observed trends. • Formal Definition:

E

solution SrO

=E

solution Sr

+E

solution O

−E

formation SrO

• Trap site for oxygen depends on stoichiometry • A negative energy corresponds to the fission product oxide being soluble in the UO2 matrix: does not form a stable secondary phase.

Result: Oxide Solution Energy

• SrO soluble for all the three stoichiometries

• Cs2O will form a stable secondary phase Nerikar et al., Journal of Physics Condensed Matter 21 (2009)

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Metallic Inclusion from FP Clustering

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• Ru is known to form a so-called “white inclusion” alloy with other FPs such as Mo, Tc, Rh, and Te. • Ru atoms are placed in a Schottky defect of UO2 to assess their stability: Ru clusters are insoluble and their sizes are limited by the available free volume. Configuration Ru atom Ru dimer Ru trimer

Solution Energy (eV) 5.05 6.86 11.05

1Gulden,

Binding Energy (eV) -0.51 1.34

OM image of white inclusion1

Journal of Nuclear Materials 23 (1967)

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Metallic inclusion: Partial DOS of Ru clusters

• Partial DOS of Ru trimer is similar to that of Ru bulk metal.

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Experimental Verification Experimental Fact

Prediction

Match

1.

Xe is known to be insoluble in the Predicted solution energy UO2 matrix.1 is positive and large.

Yes

2.

Although Cs is known to be insoluble, some of it is retained in high oxygen to metal ratio compounds on lattice sites.2

Uranium Vacancy is the most stable solution site at UO2+x.

Yes

3.

Solubility of Sr in UO2 was found to be 12 mol % and increases with hyperstoichiometric conditions.3

Negative solution energies for all three stoichiometries.

Yes

4.

Ru is known to be insoluble and forms metallic inclusion3

Positive solution energies for most cases and even a small Ru cluster becomes metallic

Yes

1Olander,

Fundamental Aspects of Nuclear Reactor Fuel Elements (1985) Journal of Nuclear Materials, 166 (1989) 3Kleykamp, Journal of Nuclear Materials, 206 (1993) 2Matzke,

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Interaction of FPs with Grain Boundaries • Segregation can be understood as the interaction between extrinsic impurity and structural defects like grain boundaries (GBs). • Xe migrates to grain boundaries, forms bubbles leading to swelling. • Ru segregates to GBs and forms clusters in UO2. • Determine how sensitive this segregation behavior is to the details of the GB.

Sonoda, Nuclear Instruments and Methods in Physics Research B 191 (2002)

Methodology

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• Empirical potentials: 2 different potentials. • Three different GBs considered: Σ 5 (310)/[001] Tilt, Σ 5 Twist, Amorphous. • Sigma (Σ): Reciprocal density of coincident sites at the boundary according to the coincident site lattice (CSL) model. • GBs translated to find the minimum energy structure. • Calculate segregation energy: replace each uranium site with a FP atom.

y

z

Grain Boundary Energy EGB =

(E energy / GB − nEbulk ) 2A

• Indication that the system size might be large enough

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Effect of System Size: Xe

Nerikar et al., Physical Review B 84 (2011)

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Effect of type of Boundary: Xe

• Segregation trend similar for the three boundaries • Unfavorable sites exist in the boundary

Nerikar et al., Physical Review B 84 (2011)

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-2

-1

0

Tilt 5

-3

Relative Energy (eV)

1

Effect of charge state FPs: Ru

Distance along z-axis (Å)

• Segregation trend of Ru1+ is similar to Xe case • Ru4+ ion (substituted for U4+) is always favorable at the GB Hong et al. (in preparation)

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Conclusions

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• DFT and empirical potentials were used to study point defect and fission product behavior in UO2. • Predicted the correct electronic structure of UO2. • Point defect stability was found to depend on charge, temperature and partial pressure of oxygen. • For the fission products, the most stable solution site was found to depend on stoichiometry. • In general, higher charged defects were observed to be more soluble for all the three stoichiometries. • Segregation characteristics of Xe agreed qualitatively for the three grain boundaries. • In general, there were some favorable and some unfavorable sites at the boundary compared to bulk.