Effect of Microstructure on Thermal-TransportProperties of UO2

Simon PhillpotDepartment of Materials Science and Engineering

University of FloridaGainesville FL 32611

sphil@mse.ufl.edu

1

Taku WatanabeAleksandr ChernatynskiySusan SinnottDepartment of Materials Science and Engineering, University of Florida

Daniel VegaJames TulenkoDepartment of Nuclear and Radiological Engineering, University of Florida

Robin GrimesDepartment of Materials, Imperial College London

Patrick SchellingDepartment of Physics and AMPAC, University of Central Florida

Srinivasan SrivilliputhurDepartment of Materials, U. North Texas

2

http://www.nrc.gov/reading-rm/basic-ref/students/animated-pwr.html

Pressurized-Water Reactor

4

http://coto2.files.wordpress.com/2011/03/2-fuel-pellet-assembly.jpghttp://www.kntc.re.kr/openlec/nuc/NPRT/module2/module2_2/module2_2_2/2_2_2.htm

5

Radial Fuel Temperature Profile: BOL, axial node 4/12

0

200

400

600

800

1000

1200

1400

1600

0.0 0.1 0.2 0.3 0.4 0.5 0.6

Radius [cm]

Tem

per

atu

re [

K]

FRAPCON Model

FRAPCON: Unirradiated Fuel Pellet

http://www.peakoil.org.au/news/does_nuclear_energy_produce_no_co2.htm

6

• Maximize thermodynamic efficiency

= 1 – Tcold/Thot

Highest possible fuel temperature

• There is a maximum temperature at which the fuel can be used in normal performance

Carefully control heat flow

Must understand heat transport in UO2 fuel

Motivation

7

Outline

• Phenomenology of Thermal Transport in Solids

• Phonon Mediated Thermal Transport

• Effects of Microstructure on Thermal Transport in UO2

• Phonon-phonon interactions• Phonon-point defect interactions• Phonon-dislocation interactions• Phonon-grain boundary interactions

• Bringing It All Together

8

Part 1Phenomenology of Thermal

Transport in Solids

9

Heat Transfer Mechanisms

• Convection

• Conduction

• Radiation

Convection is a mass movement of fluids (liquid or gas) rather than a real heat transfer mechanism (heat transfer is with convection rather than by convection)

Radiative heat transfer is important at high temperatures

Conduction is heat transfer by molecular or atomic motionHeat conduction dominates in solids

Three fundamental mechanisms of heat transfer:

10

Thermal Conduction

Transfer of heat through a material not involving mass transfer or emission of

electromagnetic radiation

11

Thermal Conduction

12

Thermal Conduction

Why does his tongue stick to a metal pole?

Would it stick to a wooden pole?

Dumb and Dumber

13

Phenomenology of Thermal Conductivity

heatsource

T

x

J = - dT/dx

Fourier’s Law

Heat current Thermal conductivity

14

Units

J = Heat Flux (Density)

= Heat per unit time per unit area

[J] = J s-1 m-2 = Wm-2

[dT/dx] = K m-1

Fourier’s Law

J = - dT/dx

[] = Wm-2 / K m-1 = Wm-1K-1

Also:

[] = BTU-inch/hour-square foot-°F

1 BTU-inch/hour-square foot-°F = 0.14Wm-1K-1

15

Thermal Conductivity of Solids

•Log – log plot•Only 6 order of magnitude range•Some increase with power-law dependence and then decay•Amorphous materials increase slowly

16

Water 0.6Ethylene Glycol 0.25PTFE 0.2Wood 0.2 – 0.4Engine Oil 0.15Fiberglass 0.04Air 0.03Snow 0.05 – 0.25 (T < 0C)Silica Aerogel 0.003

Solids vs. Liquids

Low materials W/mK

Liquid Na - 72 W/mK

17

Heat Carriers

•Electrons – metals only

•Lattice vibrations / phonons – all systems

18

Part 2: Phonon-mediated Thermal Transport

19

1

YSZ

Isotropic polymersAmorphous materials

Thermal conductivity(W/mK)

Phonons/vibrations

10 100

AluminaOriented polymers

phonons

Diamond

phonons

1000

Copper

electrons

Mechanisms of Thermal Conductivity

Electrical conductivity

(Cu ) ~ 5 105 (W cm)-1

(diamond) ~ 10-16 (W cm)-1

20

1

YSZ

Isotropic polymersAmorphous materials

Thermal conductivity(W/mK)

Phonons/vibrations

10 100

AluminaOriented polymers

phonons

Diamond

phonons

1000

Copper

electrons

Mechanisms of Thermal Conductivity

21

Thermal Conductivity of Oxides

Courtesy of D. R. Clarke

22

UO2 for Nuclear Fuel

• Advantages – high melting point

(~3000K)– radiation stability– chemical compatibility

• Disadvantages– difficulty of fabrication– low thermal conductivity – low fuel density

Figure from “Lecture notes on crystal structure”, ASU Intro to materials

23

Crystalline Materials: From Solids to Springs

Heat transport from atomic vibrations

Vibration of spring system similar to vibrations in solids

24

Long Wavelength Longitudinal Acoustic Phonon

25

Short Wavelength Longitudinal Acoustic Phonon

26

Longitudinal Optical Phonon

27

Acoustic vs. Optical

Which has lower energy?Why?

Lower EnergyLess Compression of Springs

28

Transverse Phonons

29

Longitudinal vs. Transverse Phonons

Which has lower energy?Why?

Lower EnergyLess Compression of Springs

30

Schematic dispersion curves for diamond

http://physics.ucsc.edu/groups/condensed/moseley/simulations

Phonons

Eigenmodes of harmonic potential

31

Phonon-defectPhonon-phonon Phonon-electron

Macroscale ***** *** *

Phonon-boundary

Phonon Scattering Mechanisms

32

Water Waves

http://learn.uci.edu/media/OC08/11004/OC0811004_Difraction.jpg

Water scattering from island defect Water waves scattering from each other

33

~ 1/3 Cv v

Thermal Conductivity

Thermal conductivity

Specific heat Velocity of sound

Phonon mean free path

34

Temperature Dependence

~ 1/3 cv v

Low T

Quantum Solid

Cv ~ T3

~ T3

High T:

Phonon-phonon

scattering

~ T-

~ TSurface Scattering

L=1mm

L=7mm

LiF

35

How Large is the Mean Free Path?

= 1/3 Cv v

~ 30 W/m.K v ~5000 m/sCv ~ 3kB = 1.9 106 J/m3K

~ 10nm

36

Part 3Effect of Microstructure on Thermal

Transport in UO2

37

Phonon-defectPhonon-phonon Phonon-electronPhonon-boundary

Phonon Scattering Mechanisms

38

Thermal conductivity from BTE

Thermal transport in UO2

Triple axis spectrometer HB-3 at HFIRPhonon dispersions

And line widths

Simulations Experiment (ORNL)

Fundamentaltheory test

Line width is affected by the microstructure

Different levels of theory

40

Phonon dispersion:

Arima et at., J. Alloys Compounds, 400 43 (2005)

Simulations: Experiment

(ORNL)

4

32

1

LA

TA

UO2: Phonon Dispersion and Lifetimes

Phonons lifetimes

Acoustic modes

Optical modes

41

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

T [K]

κ [

W/m

K]

ExperimentBuskerYamada

Thermal Conductivity of UO2

q

q

q

qqqP

qq

q

qqqP

42

Temperature Scaling

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.4 2.6 2.8 3.0 3.2 3.4

log(T )

log(

κ)

ExperimentBuskerYamada k ~ T-

Expt = 0.79Busker = 1.30Yamada = 1.14

43

Atomistic Simulations of Thermal Conductivity

40% is coming from optical modes (at 1000K)!

Detailed information about contribution to thermal conductivity from different phonons

44

Application to UO2

Force constants: Classical potentials - more than 20 is available (Govers, et al. 2007).

Experimental Data: R.L. Gibby, J. Nucl. Mater. 166, 223 (1989).Potentials: Arima1, Busker, Grimes, Morelon and Walker (Nomenclature is from Govers, et al (2007)).

45

Thermal conductivity from different potentials: Good thermal conductivity <> good phonons and vice-versa: Very sensitive

UO2: Potentials Sensitivity

46

Phonon/Point Defect Scattering

Four steps:• structure creation• initial phonon wave

packet generation– well-defined

longitudinal acoustic phonon

• MD simulation• energy analysis

doped region

48

Point Defect Scattering

• Incident phonon frequency: 2.96THz

• 1.56% dopants in doped region

•Δz = 200 unit cell

doped region

49

3000

t=0

t=26.3 ps

t=60.1 ps

t=201.3 ps

-3000 -100 100

z [a]

Snapshots

Energy trapped in the defect region becomes negligible by ~200 ps

Defects decrease efficiency of heat transport

50

0.994

0.995

0.996

0.997

0.998

0.999

1.000

1.001

1.002

1.003

-0.05 0.00 0.05 0.10 0.15 0.20 0.25

x

a(x

)/a

(0)

0

1

2

3

4

5

6

7

-0.03 -0.02 -0.01 0.00 0.01 0.02 0.03

x

κ (

W/m

K)

800 K

1600 K

Effects of Off-stoichiometry

UO2+x -0.05 < x < 0.25

Lattice Expansion Thermal Conductivity

Prototype for point defects of various types

51

Thermal Conductivity of UO2+x

• κ falls rapidly with increasing defect concentration

• Reaches plateau by x=0.10

• 800K and 1600K the same for x>0.10

0

1

2

3

4

5

6

7

0.00 0.05 0.10 0.15 0.20 0.25 0.30

x

κ (

W/m

K)

MD (800 K)

MD (1600 K)

Lucuta (773 K)

Lucuta (1673K)

Very similar to yttria-stabilized zirconia

52

0.00

0.01

0.02

0.03

0.04

0.05

0 5 10 15 20 25 30

f (THz)

DO

S (

Arb

. U

nit)

0.00

0.01

0.02

0.03

0.04

0.05

0 5 10 15 20 25 30

f (THz)

DO

S (

Arb

. Uni

t)

What do Vibrational Modes Look Like?

x=0 (x< 0.1)

• Debye DOS at low f• Highly structured DOS• wavevectors, define polarization phononscrystalline

• Non-Debye DOS at low f• Weakly structured DOS• no wave vectors, no polarization diffuse vibrational modes• similar to amorphous phase

x=0.125 (x > 0.1)

53

Thermal Conductivity of Irradiated UO2

• 10x10x20 (2 of 10x10x10)

• 250 eV PKA at 800 K

• Defect concentration: 0.75 defects/nm3 (analogous to x=0.035 in UO2+x, 6% UI and VU)

• J=3.66x10-5 eV/nm2·fs

Heat source

Heat sink

740

760

780

800

820

840

860

0 2 4 6 8 10

z (nm)

T (

K)

55

Thermal Conductivity of Irradiated UO2

• For irradiated material

κ=2.15 W/mK

• For x=0.035, k~ 3W/mK

• Radiation damage greater effect than off stoichiometry– U defects– Clustered structure

0

1

2

3

4

5

6

7

0.00 0.05 0.10 0.15 0.20 0.25 0.30

x

κ (

W/m

K)

MD (800 K)

MD (1600 K)

Lucuta (773 K)

Lucuta (1673K)

56

{110}<110> {100}<110>

Dislocations in UO2

Sawbridge and Sykes, JNM, 35 122(1970)

Nogita and Une, JNM  226 302 (1995) 

58

Thermal transport theory of dislocations

59

• T=1600K, edge dislocation

60

Colored by coordination number of U atoms (FCC)green=10Violet=11

Structure Evolution

(110)

(110)

(001)

60

• T=800K, screw dislocation

61Structure Evolution

(110)

(110)

(001)

61

• T=1600K, screw dislocation

62Structure Evolution

(110)

(110)

(001)

62

•Decrease in conductivity temperature independent, as predicted by Klemens-Callaway

•Magnitude of reduction less than predicted

MD Results 63

perfect edge

800K 5.75 5.09(-11.5%)

1000K 4.82 4.26(-11.5%)

1600K 3.39 2.99(-11.8%)

63

•Thermal conductivity for screw dislocations appears to decrease with increasing temperature.

MD Results 64

perfect edge screw

800K 5.75 5.09(-11.5%)

4.98(-13.4%)

1000K 4.82 4.26(-11.5%)

4.35(-9.8%)

1600K 3.39 2.99(-11.8%)

3.15(-6.8%)

64

65

Edge Dislocations

700 800 900 1000 1100 1200 1300 1400 1500 1600 17004

4.5

5

5.5

6

6.5

7

7.5

8

8.5

9

Temperature K

The

rmal

con

duct

ivity

Wm

-1K

-1

no dislocation

1X dislocation

2X dislocation

• Effect of dislocations independent of temperature• Effect proportional to dislocation density

65

MD vs. Classical Theory

1012

1013

1014

1015

1016

1017

0.75

0.8

0.85

0.9

0.95

1

Dislocation density (m-2)

Rel

ativ

e co

nduc

tivity

dislocation density in our model

6~44 GWd/t burnup regionin the previous model

Burnup region

Dislocation density in MD

MD results Klemens model

66

Interfacial (Kapitza) Thermal Resistance in Si

distance

dTgb

To

J = GK T

Interfacial (Kapitza) resistance temperature discontinuities at interfaces

Gk – Interfacial conductance (Wm-2K-1)Rk = 1/Gk - Interfacial resistance (m2KW-1)

68

Interface Conductance

D. G. Cahill et al., JAP 2003

69

Interface Scattering

Acoustic mismatch model

Diffuse mismatch model

c

B

cBZ = c

tAB = 4ZAZB/ (ZA + ZB)2

D DtAB() = DB()/ (DA()+DB())

-densityc -speed of sound

D() – phonon density of states

70

Long TA phonon – Si Grain Boundary

71

Short LA phonon – Si Grain Boundary

72

(100) =43° 29

LA

kz =0.35

High-Frequency LA mode

Diffuse scatteringAcoustic scattering

73

74

Effects of Interfaces: Polycrystalline UO2

* Experimental data from J.K. Fink J. Nucl. Mater. 279 (2000) 1

• d = 3.8 – 6.5 nm• (001) texture pure tilt GBs (misorientations > 15°)

75

Polycrystal

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 500 1000 1500 2000

T [K]

κ [

W/m

K]

Busker

Yamada

0

5

10

15

20

25

30

35

40

45

0 500 1000 1500 2000

T [K]

κ [

W/m

K]

ExperimentBuskerYamada

Single Crystal d=3.8nm

76

Grain Size Dependence

0.00

0.05

0.10

0.15

0.20

0.25

0.30

0.35

0.40

0.45

0.50

0 500 1000 1500 2000

T [K]

GK

[G

W/m

2K]

Busker

Yamada

• Interfacial conductance increases with temperature

• Increased anaharmonicity couples vibrational modes across the grain boundary → enhanced heat transfer

Gd1 0

0

77

Simple Model for Grain-Size Dependence

0

2

4

6

8

10

12

14

16

0 200 400 600 800 1000

d [nm]

κ [W

/mK

]

0.0

0.2

0.4

0.6

0.8

1.0

1.2

3.0 4.0 5.0 6.0 7.0

d [nm]0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 20 40 60 80 100 120

Grain size [nm]

k/k 0

Experiment: YSZ:480K

MD: MgO:300K

MD: MgO:573K

MD: NDZ:300K

MD: NDZ:573K

Gd1 0

0

78

Kapitza Length in UO2

• lκ >> d→ Grain boundaries dominate the thermal transport property

• Kapitza length approaches the nano grain size only above ~1500 K

econductanc linterfacia

tyconductivi thermal bulk

length Kapitza

G

G

l

0

10

20

30

40

50

60

70

80

90

0 500 1000 1500 2000

T [K]

lκ [

nm]

Busker

Yamada

79

Kapitza length lk = kSC/G thickness of perfect crystal offering same resistance to heat transport at the interface

Quantification of Interface Effect

1.50

1.70

1.90

2.10

2.30

2.50

0 500 1000 1500 2000

T [K]

G [

GW

/m2 K

]

0

5

10

15

20

25

30

35

0 500 1000 1500 2000

T [K]

l k (

nm

)

lk > grain size

80

FRAPCON

81

Comparison of MD model in FRAPCON

MDFRAPCON

Input model for k FRAPCON prediction at BOL

0.000

0.002

0.004

0.006

0.008

0.010

0.012

0.014

0 400 800 1200 1600Temperature (K)

Stra

in (

cm/c

m)

FRAPCON

AM/MDMDFRAPCON

82

Surface Displacement – worst case

0.0E+00

5.0E-01

1.0E+00

1.5E+00

2.0E+00

2.5E+00

3.0E+00

3.5E+00

4.0E+00

4.5E+00

0 500 1000 1500Time (days)

Su

rfac

e D

isp

lace

men

t (m

ils)

Base

Atomistic

83

MD vs. FRAPCON Model

84

Temperature in Unirradiated Pellet

85

Bringing It All Together

86

Microstructure and Thermal Properties of Nuclear Fuel Under Irradiation

Molecular Dynamics and Lattice Dynamics Simulations of Thermal

Transport, Phonon Dispersion, and Phonon Lifetimes at Temperature;

Effects of Microstructure

Materials Synthesis and Ion/Neutron Irradiation and Materials Microstructure

Characterization Using Atom-probe Microscopy, Electron Microscopy, 3D X-

ray Microscopy, and Photon Spectroscopies

87