effect of microstructure on thermal-transport properties of uo 2 simon phillpot department of...
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Effect of Microstructure on Thermal-TransportProperties of UO2
Simon PhillpotDepartment of Materials Science and Engineering
University of FloridaGainesville FL 32611
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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
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http://www.nrc.gov/reading-rm/basic-ref/students/animated-pwr.html
Pressurized-Water Reactor
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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
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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
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• 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
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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
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Part 1Phenomenology of Thermal
Transport in Solids
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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:
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Thermal Conduction
Transfer of heat through a material not involving mass transfer or emission of
electromagnetic radiation
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Thermal Conduction
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Thermal Conduction
Why does his tongue stick to a metal pole?
Would it stick to a wooden pole?
Dumb and Dumber
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Phenomenology of Thermal Conductivity
heatsource
T
x
J = - dT/dx
Fourier’s Law
Heat current Thermal conductivity
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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
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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
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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
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Heat Carriers
•Electrons – metals only
•Lattice vibrations / phonons – all systems
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Part 2: Phonon-mediated Thermal Transport
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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
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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
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Thermal Conductivity of Oxides
Courtesy of D. R. Clarke
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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
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Crystalline Materials: From Solids to Springs
Heat transport from atomic vibrations
Vibration of spring system similar to vibrations in solids
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Long Wavelength Longitudinal Acoustic Phonon
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Short Wavelength Longitudinal Acoustic Phonon
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Longitudinal Optical Phonon
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Acoustic vs. Optical
Which has lower energy?Why?
Lower EnergyLess Compression of Springs
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Transverse Phonons
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Longitudinal vs. Transverse Phonons
Which has lower energy?Why?
Lower EnergyLess Compression of Springs
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Schematic dispersion curves for diamond
http://physics.ucsc.edu/groups/condensed/moseley/simulations
Phonons
Eigenmodes of harmonic potential
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Phonon-defectPhonon-phonon Phonon-electron
Macroscale ***** *** *
Phonon-boundary
Phonon Scattering Mechanisms
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Water Waves
http://learn.uci.edu/media/OC08/11004/OC0811004_Difraction.jpg
Water scattering from island defect Water waves scattering from each other
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~ 1/3 Cv v
Thermal Conductivity
Thermal conductivity
Specific heat Velocity of sound
Phonon mean free path
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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
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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
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Part 3Effect of Microstructure on Thermal
Transport in UO2
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Phonon-defectPhonon-phonon Phonon-electronPhonon-boundary
Phonon Scattering Mechanisms
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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
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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
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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
q
qqqP
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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
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Atomistic Simulations of Thermal Conductivity
40% is coming from optical modes (at 1000K)!
Detailed information about contribution to thermal conductivity from different phonons
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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)).
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Thermal conductivity from different potentials: Good thermal conductivity <> good phonons and vice-versa: Very sensitive
UO2: Potentials Sensitivity
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Phonon/Point Defect Scattering
Four steps:• structure creation• initial phonon wave
packet generation– well-defined
longitudinal acoustic phonon
• MD simulation• energy analysis
doped region
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Point Defect Scattering
• Incident phonon frequency: 2.96THz
• 1.56% dopants in doped region
•Δz = 200 unit cell
doped region
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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
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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
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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
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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)
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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)
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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)
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{110}<110> {100}<110>
Dislocations in UO2
Sawbridge and Sykes, JNM, 35 122(1970)
Nogita and Une, JNM 226 302 (1995)
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Thermal transport theory of dislocations
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• T=1600K, edge dislocation
60
Colored by coordination number of U atoms (FCC)green=10Violet=11
Structure Evolution
(110)
(110)
(001)
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• T=800K, screw dislocation
61Structure Evolution
(110)
(110)
(001)
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• T=1600K, screw dislocation
62Structure Evolution
(110)
(110)
(001)
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•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%)
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•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%)
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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
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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
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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)
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Interface Conductance
D. G. Cahill et al., JAP 2003
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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
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Long TA phonon – Si Grain Boundary
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Short LA phonon – Si Grain Boundary
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(100) =43° 29
LA
kz =0.35
High-Frequency LA mode
Diffuse scatteringAcoustic scattering
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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°)
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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
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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
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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
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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
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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
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FRAPCON
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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
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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
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MD vs. FRAPCON Model
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Temperature in Unirradiated Pellet
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Bringing It All Together
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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