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

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

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.

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.

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.

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

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

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.

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]

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]

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:

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:

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

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

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

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

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.