continuum-scale modeling of hydrogen and helium bubble ... · 10 18 10 19 10 20 10 21 10 22 d/m 2]...
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Continuum-scale modeling of hydrogen and helium bubble
growth in metals
R. Kolasinski(a), M. Shimada(b), D. Cowgill(a), D. Buchenauer(a), and D. Donovan(c)
(a)Sandia National Laboratories, Hydrogen & Metallurgy Science Dept., Livermore, CA
(b)Idaho National Laboratory, Fusion Safety Program, Idaho Falls, ID (c)University of Tennessee, Dept. Of Nuclear Engineering, Knoxville, TN
Y. Oya(d), T. Chikada(d), and K. Michibayashi(f)
(d)Shizuoka University, Department of Chemistry, Shizuoka, Japan (f)Shizuoka University, Department of Geosciences, Shizuoka, Japan
Tritium Focus Group Meeting September 24, 2014
SAND2014-17969 PE
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Motivation: Analysis of bubble growth in ITER-grade W samples exposed in TPE
2
exposure
type
ion energy
[eV]
duration
[min]
flux (Γi) [m-2 s-1]
fluence (Φ)
[m-2]
LF 100 60 4.9×1021 1.8×1025
HF 100 120 1.5×1022 1.1×1026
• Precipitation affects
migration through material
• Bubble growth depends on
microstructure
• Growth mechanisms critical
to developing realistic
models
• TPE plasma
exposures at INL
• Microscopy at
Shizuoka
TPE target during plasma exposure
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Retention measurements correspond closely with those obtained in other laboratories
3
Previous work by Alimov et al:
• ITER-grade W
• E = 38 eV
• Φ = 1022 D m-2 s-1
Comparable exposure conditions
1018
1019
1020
1021
1022
D r
eta
ined
[D
/m2]
6005004003002001000
exposure temperature [°C]
Alimov et al.
F=1027
F=1026
F=1026
R
V. Kh. Alimov, et al. J. Nucl. Mater. 420 (2012) 519.
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Retention measurements correspond closely with those obtained in other laboratories
4
1018
1019
1020
1021
1022
D r
eta
ined
[D
/m2]
6005004003002001000
exposure temperature [°C]
Alimov et al.
F=1027
F=1026
F=1026
R
TPE LF
HF
Previous work by Alimov et al:
• ITER-grade W
• E = 38 eV
• Φ = 1022 D m-2 s-1
Comparable exposure conditions
V. Kh. Alimov, et al. J. Nucl. Mater. 420 (2012) 519.
TPE retention measurements:
• Correspond closely with
Toyama/IPP meas.
• Confirm accepted retention
temp. dependence.
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Surface morphology variation with temperature
5
5 μm20 μm
5 μm
20 μm
20 μm
5 μm
A B
C
103 C 131 C
231 C
5 μm20 μm
D 278 C
5 μm20 μm 20 μm 5 μm
E F355 C 554 C
Key features:
• Non-uniform
coverage
• Bubbles are
small (<10 μm
dia.)
compared with
warm-rolled W
material.
• Absent at
temperature
extrema.
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EBSD measurements reveal dependence on grain orientation
6
x
y
0 100 200 300 400 500 600
0
50
100
150
200
250
300
350
400
• Grain orientation indicated by inverse pole plot.
• Bubbles visible on grains with <111> and <110> directions
aligned normal to surface
• Considerable distortion within individual grains
• Un-annealed sample showed increased distortion
SEM image of the same area
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Atomic force microscopy reveals details of surface structure
7
100
80
60
40
20
0
[m
]
100806040200 [m]
1.0
0.8
0.6
0.4
0.2
0.0
height
[m]
(a)
100
80
60
40
20
0
[m
]
100806040200 [m]
2.0
1.5
1.0
0.5
0.0
height
[m]
(b)
• Atomic force microscopy provides
information on the shape of the
deformed surface.
• Individual bubbles identified and
analyzed automatically.
corresponding
bubble size
distributions 1
2
4
6
10
2
4
6
100
co
un
ts
543210
req [m]
131 °C 231 °C
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What bubble growth mechanisms are active in W during plasma exposure?
8
near-surface plastic
deformation dislocation loop punching vacancy clustering
Figures from: J. B. Condon & T. Schober, J. Nucl. Mater. 207 (1993) 1.
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Far from the free surface, dislocation loop punching is favored
9
Three bulk precipitate growth
mechanisms considered:
– Dislocation loop punching
– Griffith nano-crack extension
– Dislocation dipole expansion
0 20 40 60 80 100
1
2
5
10
20
50
rb nmpre
ssure
GP
A
nano-crack
dipole expansion
loop punching
𝑝𝐿𝑃 ≥2𝛾
𝑟+𝜇𝑏
𝑟 ~ 1
𝑟
𝑝𝐷𝐸 ≥2𝛾
𝑠+𝜇𝑑
2𝑟 ~1
𝑟+ 𝑐
𝑝𝑁𝐶 ≥ 𝜋𝜇𝛾
1 − 𝜗 𝑟 ~1
𝑟
Based on methods developed in:
D. F. Cowgill, “Physics of He Platelets in
Metal Tritides,” in Effects of Hydrogen on
Materials (2009).
rb [nm] pre
ssure
[G
Pa]
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Near the free surface, bubbles may grow by crack extension
10
Stress calculations based on
calculations by K. Wan & Y. Mai, Acta
metall. mater. 43 (1995) 4109.
Crack extension competitive
with loop punching near
surface:
Limitations:
– Correction for thick blisters
– Effect of plasticity (blunting of
crack tip)
– Hydrogen effects
𝑝𝐵 ≥1
𝑟
4γ(𝐸ℎ)1/3
5𝐶1𝐶2
3/4
~ 1
𝑟
0.1 μm
0.5 μm
1.0 μm
rb [nm] pre
ssure
[G
Pa]
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Bubble volumes measured with AFM correlate well with blister model
11
0.1
1
10
bu
bb
le v
olu
me
(V
0)
[m
3]
543210
req [m]
131 °C
231 °C
𝑉 = 𝑦 𝑟 2𝜋𝑟𝑑𝑟 = 𝐶1𝜋𝑎2𝑦𝑐
K. Wan & Y. Mai, Acta metall. mater. 43 (1995) 4109.
Volume modeled using blister test
for thin film adhesion:
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Bubble volumes measured with AFM correlate well with deflection model
12
0.1
1
10
bu
bb
le v
olu
me
(V
0)
[m
3]
543210
req [m]
131 °C
231 °C
theory
0.1 μm
0.5 μm
1.0 μm
𝑉 = 𝑦 𝑟 2𝜋𝑟𝑑𝑟 = 𝐶1𝜋𝑎2𝑦𝑐
K. Wan & Y. Mai, Acta metall. mater. 43 (1995) 4109.
Volume modeled using blister test
for thin film adhesion:
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Diffusion and trapping modeled with a continuum-scale approach
13
Diffusion: 1-D, uniform temperature:
𝜕𝑢(𝑥, 𝑡) 𝜕𝑡 = 𝐷 𝑡 𝜕2𝑢 𝑥, 𝑡 𝜕𝑥2 − 𝑞𝑇 𝑥, 𝑡 − 𝑞𝐵(𝑥, 𝑡)
Point defects:
• 1.4 eV saturable traps, no nucleation.
• Used approach of Ogorodnikova [J. Nucl.
Mater. (2009)] to address trapping and release.
Bubbles:
Modeled using a approach of Mills [J. Appl. Phys.
(1959)].
𝑞𝐵(𝑥, 𝑡) = 𝜕𝑢𝐵(𝑥, 𝑡) 𝜕𝑡 = 4𝜋𝐷 𝑡 𝑟𝐵 𝑥, 𝑡 𝑁𝐵 𝑥 [𝑢 𝑥, 𝑡 − 𝑢𝑒𝑞 𝑥, 𝑡 ]
½ H2
void
Hm=0.39 eV
vacancy
Enthalpies for H migrating through W.
Dissolution of H in W is highly
endothermic.
Hads=0.5 eV
Hs=
1.04 eV
Hs=1.43 eV
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H equation of state takes into account non-ideal gas effects
14
0.001
0.01
0.1
1
10
pre
ssu
re [
GP
a]
1 10 100 1000
specific volume [cm3/mol]
EQUATION OF STATEDATA FOR HYDROGEN Experimental:
Michels (1959)
Mills (1977)
Mao (1988)
Curve Fits:
ideal gas Shimizu (1981) Loubeyre (1996) Tkacz (2002) San Marchi (2007)
H2 solidifiesH2 equation of state (EOS):
• P > 1 GPa expected within
small bubbles.
• At 300 K, H2 solidifies at p=5.7
GPa.
• Tkacz’s [J. Alloys &
Compounds (2002)] EOS to
provide the best fit:
𝑣 = 𝐴𝑝−1/3 + 𝐵𝑝−2/3 + 𝐶𝑝−4/3
+ 𝐷 + 𝐸𝑇 𝑝−1
• San Marchi’s simplified EOS
better at low pressure:
𝑣 =𝑅𝑇
𝑝+ b
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When is bubble growth favorable?
15
Calculation of equilibrium press.
When is precipitate in equilibrium with mobile
conc.?
• Equate chemical potentials of gas and
solution phase.
• Calculate fugacity to account for non-ideal
behavior:
ln 𝑓 𝑝 = 𝑣 𝑝, 𝑇
𝑅𝑇−1
𝑝𝑑𝑝
𝑝
0
• Equilibrium conc. given by:
𝑢𝑒𝑞 = 𝑓𝑆0exp (−𝐻𝑆 𝑅𝑇)
So and Hs from Frauenfelder [JVST, 1969].
100
102
104
f [M
Pa
]
12 4
102 4
1002 4
1000
p [MPa]
IG
25 °C
200 °C
500 °C
103
2
4
68
104
2
pe
q [
MP
a]
10-11 10
-8 10-5 10
-2
Ceq [H/W]
25 °C
200 °C
500 °C
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Summary of surface morphology findings
16
• ITER-grade W sample exposed in TPE show similar
retention to Toyama/IPP studies.
• Analysis of surface morphology:
– XPS shows implanted C reduced considerably
– SEM/EBSD illustrate non-uniform bubble growth over surface
– Bubble grow on (110) and (111) crystal planes
– AFM analysis provide bubble volumes
• Modeling of bubbles:
– Thin film adhesion model adapted to model blister grown on
tungsten.
– Model reproduces bubble sizes observed with AFM
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Acknowledgements
17
We would like to express our appreciation to:
• Our collaborators at INL and Sandia/CA:
– Brad Merrill, Robert Pawelko, Lee Cadwallader
– Richard Nygren, Josh Whaley, Jon Watkins, Thomas Felter
Sandia National Laboratories is a multi-program laboratory managed and operated by Sandia Corporation, a wholly owned subsidiary of Lockheed Martin
Corporation, for the U.S. Department of Energy's National Nuclear Security Administration under contract DE-AC04-94AL85000.
This work was prepared for the U.S. Department of Energy, Office of Fusion Energy Sciences, under the DOE Idaho Field Office contract number DE-AC07-
05ID14517.