1

S. Sharafat1*, A. Takahashi2, and N. Ghoniem1

1University of California Los Angeles2Tokyo University of Science

1

18th High Average Power Laser WorkshopHosted by Los Alamos National Laboratory

Hilton Santa Fe Historical Plaza100 Sandoval Street, Santa Fe, New Mexico, USA

April 8-9, 2008

*shahrams@ucla.edu This work was supported by the US Navy/Naval Research Laboratories through a grant with UCLA.

Stress Driven Bubble Growth:Influence of Stress Gradient

on Bubble Migration

OUTLINE

1. Formulation of He-Bubble Growth in stress gradientusing Event Kinetic Monte Carlo (EKMC)

2. Single Bubble in a stress gradient

3. Collection of Bubbles in a stress gradient

Helium bubbles are treated as particles Position vector Number of helium atoms Radius (Diameter)

An event can take place at each time step Diffusion Helium Implantation Coalescence Surface pore formation

Event Kinetic Monte Carlo

:He bubble:He

x

y

z

P2r

Diffusion Implantation Coalescence Pore form.

Event (Diffusion Random Walk) Diffusion of helium bubbles

The diffusion is in 3-dimensions: random walk model

Diffusion rate of a helium bubble

BB

kT

EDDD

rD ssb exp

2

304

3/4

: Atomic volumer 　 　 : Radius of helium bubbleD0 : Diffusion Pre exponentialE : Surface migration energy

Surface diffusion

J.H. Evans, JNM 334(2004), 40-46

Stress gradient acts as a driving force on bubble migration:Change in total strain energy Bubble moves up a stress gradient

ox( )f z

Surface

Bubble

vp: Bubble volume: constant (3.02): StressE: Young’s modulusDs: Surface Diff.Dp: Pore Diff.z: Position after bubble migrationz0: Starting bubble position

23

2tot pE vE

xy

z

BmE

BmE

BmE

BmE

BmE E

BmE E

0tot totE E z E z

“Delta-Energy Rule”

5

3totP p

EF v E zz

p p pV B Fp o

p s4

D 3aB D

kT 2 kTr

*Leiden & Nichols (1971)

Event (Diffusion in Stress Gradient)

Vp 1/r

Vp 1/r4

With Stress

No Stress

Event (Diffusion in Stress Gradient)

• Helium bubble migration in matrix

• Strain Energy difference

• Diffusion of bubble

4/3

04

*0

3exp

2

log

b s s

bbm

ED D D D

r kT

DE kT

D

2 230

0

43.01

3 2i

i

rE

E

0

2

1

43

6

5(i=1,6)

* 0exp bm ii o

E ED D

kT

(Leiden, Nichols, 1971)

BmE

BmE

BmE

BmE

BmE E

BmE E

Events (Coalescence)

Clustering of two helium bubbles

Calculation of helium bubble radius

rP

nkTPV

2

NHeMHe

(N+M)He

2r

8

3nkTr

Equation of state ：

Pressure on the helium bubble surface ：

J.H. Evans, JNM 334(2004), 40-46

Influence of Stress Gradienton Bubble Migration and

Coalescence

8

9

W-Surface Stress State for HAPL

Following the heating/implantation transient the surface remains in a stressed state of about ~0.7 GPa (UMARCO):

Single Bubble Migration

in a Stress Gradient

10

Stress Gradient Effect on Single 5-nm Bubble

t=2.5 x 109 s t=2.5 x 109 s

t=1.2 x 109 s

t=4.1 x 107 s

Tungsten T = 1300 K; Initial: blueFinal : red

Collection of Bubbles:

Migration PLUS Coalescencein a Stress Gradient

12

Stress Gradient Effect on Collection of Bubbles

Initial Conditions:Depth Profile: 0.02 m – 0.5 m Number of Bubbles: 1000Ave. Bubble Radius: 1.84 nm

Bubble Distribution at 3107 s at Given Stress Gradients

d/dz = 0 MPa/m 200 MPa/m 600 MPa/m 1000 MPa/m

0

0.5

1

1.5

2

2.5

3

3.5

4

1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08

Time[s]

Ave

. por

e ra

dius

[nm

]

0.0[Mpa]100.0[Mpa]300.0[Mpa]500.0[Mpa]

Ave. Surface Pore Radius

0

1

2

3

4

5

6

7

1.0E+04 1.0E+05 1.0E+06 1.0E+07 1.0E+08Time[s]

Ave

. hel

ium

bub

ble

radi

us[n

m]

0.0[Mpa]100.0[Mpa]300.0[Mpa]500.0[Mpa]

Ave. He-Bubble Radius

15

Summary & Conclusions Influence of stress gradients on He-bubble migration has been

incorporated into the McHEROS Code

Bubble moves up the stress gradient (compressive or tensile)

Single bubble migration is significantly impacted for stressgradients > 100 MPa/m

Collection of Bubble in a stress gradient:— Bubbles move up towards the surface as a group— Stress gradient does not significantly increase bubble growth— Coalescence is not very sensitive to stress gradient— Surface pores are slightly larger for large stress gradients

With a stress gradient bubble velocity is 1/r Without a stress gradient velocity is 1/r 4 (surf. Diff.):

Stress gradient reduces the relative velocities between small and large bubbles

16

M. Narula, S. Sharafat, and N. Ghoniem

Mechanical and Aerospace Engineering DepartmentUniversity of California Los Angeles

16

18th High Average Power Laser WorkshopHosted by Los Alamos National Laboratory

Hilton Santa Fe Historical Plaza100 Sandoval Street, Santa Fe, New Mexico, USA

April 8-9, 2008

*shahrams@ucla.edu This work was supported by the US Navy/Naval Research Laboratories through a grant with UCLA.

Modeling Carbon Diffusion

Carbon Diffusion in Tungsten

• Carbon is implanted between ~0.2 to ~1 m in W• Carbon has been shown to diffuse deep after short time

anneals (~8 m at 2000 oC)• In HAPL W-surface undergoes rapid temperature transients

Goal:

– Effect of rapid temperature transient on diffusion of C.

– Investigate impact of various (realistic) grain structures.

1.Carbon Diffusion Data

2.Grain-Structure Model

Carbon Diffusion in Grain Boundary region is higher than in Grain Matrix

D(GB) >> D(Matrix)

*Adam Shepela, J Less Common Met. 26(1972) 33-43

214 3 37,800 140/ (3.45 0.12)10 cmD C W exp sRT

A. Shepela* HAPL carbon- implantation ~11016 cm-3 per shot to a depth of ~0.5 m

~31020C/cm3 at 2500 oC

~51019C/cm3 at 1810 oC

Carbon Diffusion and Solubility in Single–X W(Grain Matrix)

Gmelin Handbook of Inorganic and Organometallic Chemistry, Tungsten Suppl. vol. A 5b, 8th Ed., Springer, Berlin, 1993 (Chapters 7–9)

• Self-diffusion in POLYCRYSTALLINE tungsten is much higher (T< 2500 oC)• Implies that GB self-diffusion is faster by 103 – 104 (T < 2000 oC)

and 102 (T= 2500 oC)

• C-diffusion is taken to be a factor of 100 higher in GB region

21

Diffusion in Grain Boundary

GB Diffusion

Matrix Diff.

• Model (1): – Constant temperature anneal at 2000 oC with realistic grain

structure

• Model (2): – Impact of HAPL temperature transient 600 to 2500 oC in <10-4 s

• Model (3): – Grain size effect: Fine grain W compared with Single-X W

Carbon Diffusion Models

Grain Structure Model

* Y. Ueda et al., 2006 US-J Workshop on Fusin High Power Density components and system and Heat Removal and Plasma- Materials Interaction for Fusion, Nov. 15-17, 2006, Santa Fe, USA

Ultra-fine Grain (UFG) Tungsten*

Extracted Grain Structure

G.S. = 83 nmR.D. = 99.4 %

Tungsten Grain (UFG) Model

C-I

mp

l. R

ang

e

Tungsten Grain Structure Model used for Carbon Diffusion Simulation

1.3μm

0.5

μm

UFG -W with Grain Boundary RegionTungsten Grain Structure Model used for Carbon Diffusion Simulation

1.3 μm0.

5μm

m

Grain Boundary Regions are highlighted in red (~20 nm wide)

Model (1): C-Diffusion for Anneal at 2000 oC

• Model 1 (UFG: fine grain)

– Fine grains (with enhanced diffusion in 20-nm depth at GB)– Model dimensions (1.3 μm 0.5 μm)– Constant diffusion coefficient D = 5.24X1010 m2/s in the grains– 100 enhanced diffusion in 20-nm GB region– Initial implantation of C (in a region 0.2 μm < X < 0.4μm)– C implantation concentration is constant: 3.5 1016 cm-3

• In HAPL maximum Carbon implantation ~11017 cm-3 per shot to a depth of ~0.8 m

Carbon concentration profile as function of time

t = 5e-6 s

Initial C implantation in W (0.2 μm – 0.4 μm ): 3.51016 atoms/cm3

t = 0 sImpl. region T = 2000 oC

1.3μm

Grain boundaries are highlighted (20 nm wide)0

.5μ

m

Concentration profile along the centerline as a function of time

Initial C implantation in W (0.2μm thick), 0.0579moles/m3

~6x1015 C/cm3

~4x1016 C/cm3

Model (2): Temperature Transient Effect

Temperature Transient in HAPL

• UMARCO code provides detailed temperature transients as a function of position:

• Couple the transients with the diffusion model

Concentration profile at t = 0 s

Carbon-Diffusion coefficient variation with time (grain matrix)

Concentration profile at time 6e-5 s

Concentration profile at time 5e-5 s

C-Diffusion for HAPL

• Animation shows the C-concentration as a function of time for HAPL conditions:

~4x1016 C/cm3

~6x1015 C/cm3

~1.2x1016 C/cm3

Comparison of Concentration Profiles

(at the centerline)

Annealing “pushes” the Carbon deeper

Annealing results in a more even distribution

Rapid Temperature transient results in “pushing” carbon towards the surface

Anneal (2000 oC)

t = 5e-5 s

With Temperature Transient

t = 5e-5 s

Model (3): Comparison between UFG – and

Single –Crystal Tungsten

(annealing 2000 oC)

• UFG-Tungsten with 20 nm grain boundary regions: D = 5.24X1010 m2/s in the grains D = 5.24X1012 m2/s in GB region

• The plot shows initial C implantation with the red region (3.5x1016 C/cm3)

5.16 μm

5.16 μm

• 5.16 μm long single crystal (D = 5.24X1010 m2/s)

• The plot shows initial C implantation with the red region (3.5x1016 C/cm3)

Multiple fine grains spanning 5 microns

Initial state of fine grains model

Single Crystal spanning 5 microns

Final result: t =0.001 s

5e-7 s

1e-4 s

0.001 s

~3.5μm >5.2μm

UFG Tungsten

T = 2000 oC

C diffuses through the entire UFG-W sample (5 m) as there is rapid

diffusion along grain boundaries

5e-7 s

1e-4 s

0.001 s

~1.4μm ~3.2μm

Single-XC-concentration along the Centerline

(x direction) T = 2000 oC

Summary• Grain Boundary diffusion is significantly faster than in the matrix

• Realistic Grain structure was modeled based on UFG-W

• Modeled C-diffusion for annealing (T=2000oC) in UFG-W

• Modeled C-diffusion with HAPL temperature transient

• Compared UFG-W with Single-X W

Findings:

• C-diffuses rapidly (<10-4 s) at 2000 oC through the UFG sample ( 1.3 m)

• Near surface high temperature transients move Carbon towards surface and reduces diffusion into the W

• C concentration remains peaked for Single-X, while in UFG-W C diffuses throughout the thickness (5 m) for a 2000 oC anneal in < 0.001 s.

TOFE 2008: Submitted Abstracts

• A Unified Model for Ion Deposition and Thermomechanical Response in Dry Wall Laser IFE Chambers, J. Blanchard, Q. Hu, and N. Ghoniem

• Roughening of Surfaces under Intense and Rapid Heating,” M. Andersen, A. Takahashi, N. Ghoniem

• Thermo-mechanical Analysis of the Hibachi Foil for the Electra Laser System,” A. Aoyama, J. Blanchard, J. Sethian, N. Ghoniem, and S. Sharafat

• A Simulation of Carbon Transport in Implanted Tungsten,” M. Narula, S. Sharafat, and N. Ghoniem

• A KMC Simulation of Grain Size Effects on Bubble Growth and Gas Release of Implanted Tungsten, A. Takahashi, K. Nagasawa, S. Sharafat, and N. Ghoniem

• Simulation of Pressure Pulses in SiC due to Isochoric Heating of PbLi Using a Laser Spallation Technique, J. El-Awady, H. Kim, K. Mistry, V. Gupta, N.Ghoniem, S. Sharafat

Backup Slides for C-diffusion

modeling

42

Development of UFG Tungsten

• What is ultra fine grained tungsten?What is ultra fine grained tungsten?– Tungsten materials with very small grains (<100 nm) with some

TiC dispersoids.– Development by Dr. Kurishita (Tohoku University)

• FabricationFabrication– Mixing of powder of tungsten and TiC in Ar or H2 atmosphere

without oxygen.– Mechanical alloying – Degassing in vacuum– HIP process

• Advantages for plasma facing materialAdvantages for plasma facing material– Little or no radiation (neutron, He) hardening

– No significant blistering (H2, He)

– Superplasticity ~160 % (T>1670 K)– Higher re-crystalliation temperature (claim)

W-0.5TiC-H2

10

100

1000

1.0E-05 1.0E-04 1.0E-03 1.0E-02

Strain rate (s-1)

Flo

w s

tres

s (M

Pa)

1870K1770K1670K

m = 0.54

m = 0.53

m = 0.57

Strain rate dependence of flow stressW-0.5TiC-H2

W-0.5TiC-H2 exhibits superplasticity at and above 1670K, with a large strain rate sensitivity, m, of 0.5~0.6, that is characteristic of superplastic materials.

* Y. Ueda et al., 2006 US-J Workshop on Fusin High Power Density components and system and Heat Removal and Plasma- Materials Interaction for Fusion, Nov. 15-17, 2006, Santa Fe, USA

* Y. Ueda et al., 2006 US-J Workshop on Fusin High Power Density components and system and Heat Removal and Plasma- Materials Interaction for Fusion, Nov. 15-17, 2006, Santa Fe, USA

Superplastic deformation of UFG W-0.5TiC-H2

1970K

an initial strain rate of 5 x 10-4 s-1

I.G.L.5 mm

= 160%

Crosshead is arrested at = 160% to examine the specimen surface.I.G.L. stands for the initial gauge length of the tensile specimen.

G.S.: 0.5 mm x 1.2 mm x 5 mm

* Y. Ueda et al., 2006 US-J Workshop on Fusin High Power Density components and system and Heat Removal and Plasma- Materials Interaction for Fusion, Nov. 15-17, 2006, Santa Fe, USA

600

800

1000

1200

Vickers microhardness before and after irradiation

Before irr. After irr.

521

1039

11641108

1034

619

Vic

ker

s H

ard

nes

s (H

v)

Radiation hardening occurred in pure W, but not in W-0.5TiC-H2 and W-0.5TiC-Ar.

Radiation hardening in pure W and UFG W-0.5TiC

MA in Ar

MA in H2

pure W

HV

- 56 - 5

Tirr = 873K and fluence of 2×1024n/m2 (En > 1 MeV). (JMTR)

Backup Slides for He-bubble

Migration and Coalescence

in a Stress Gradient

47

48

McHEROS Code Simulation of IEC Surface Pores

Temperature (oC)

Implantation Rate (He/cm2-s)

Lx

(m)Ly

(m)Lz

(m)

Model-1 730 2.2x1015 0.2 1.0 1.0

Model-2 990 8.8x1015 0.2 2.5 2.5

Model-3 1160 2.6x1016 0.2 5.0 5.0

McHEROS Results:

• Good Agreement between McHEROS Simulation and Experiment

• McHEROS provides an EXPLANATION for the oversized Surface PoresKulcinski et al., 2005 IEC Facitlies, UW-Madison

40-1

60 s

30 min

1.6x1018 He/m2-s; 730 oC; t:30min (IEC)

Exp.: IEC (UW-Madison)

0-40 s

Exp*

Exp*

T:730oCt:30min

T:990oCt:7.5min

T:1160oCt:2.5min

Exp*

McHEROS Stress Gradient: Methodology

49

kT

EDD m

s exp0 sp Dr

D4

3/4

2

3

kT

DB p

p ppp FBV

effpeff p m

p p p p p p o

D EFD V B F F D D exp -

kT kT kT

Diffusion coefficient of a bubble (Dp) based on the surface diffusion (Ds) mechanism:

Velocity and mobility of a bubble in a stress gradient field

Effective diffusion coefficient of a bubble in a stress gradient field

The pre-exponential diffusion coefficient of bubble is estimated using:2

00 6

pp V

D

Vp: Volume of bubble0: Debye frequency

I. The net migration energy (Eeffm) of the bubble due to a stress-field can be

calculated using the bubble diffusion coefficient (Deffp).

II. Then we apply the “Delta-Energy Rule” to calculate the migration energy of the bubble in 6 different directions.

I. The net migration energy (Eeffm) of the bubble due to a stress-field can be

calculated using the bubble diffusion coefficient (Deffp).

II. Then we apply the “Delta-Energy Rule” to calculate the migration energy of the bubble in 6 different directions.

50

Bubble Size Near Surface vs Bulk *

*CHERNIKOV, JNM 1989

Near Surface

Bulk

• 1000 appm He Implanted in Ni at RT.

• Uniform He implantation using degrader Al-foil (28 MeV He)

• Annealing time: 0.5 – 1.5 hr

Abundance of Near Surface Vacancies

promotes rapid and large bubble growth

51

Sub-Surface Break Away Swelling Contribution

BREAK-AWAY Swelling (very rapid growth of bubbles) occurs at the subsurface

However, because the bubbles bisect the surface the swelling is stopped by venting He.

Time to BREAK-AWAY swelling DECREASES with higher Temps.

Sub-Surface Break-Away Swelling

Surface Pore Formation

He

HeAvg. Bubble Radius

52

Explaining Low-E He-Implantation Results

Abundance of near surface vacancies allow bubbles to grow rapidly to equilibrium size:

Large bubbles & low He-pressure

Near the surface, Migration & Coalescence (M&C) plus rapid growth results in super-size bubbles.

Super-large bubbles bisect the surface, thus providing a probable explanation for surface deformation and large subsurface bubbles.

A network of deep interconnecting surface pores is rapidly set up which results in drastic topographical changes of the surface

MCHEROS with Stress Gradient

53

Numerical Example:

Diffusion of single bubble— Radius: 10nm

Stress gradient in depth direction

500 MPa (at surface)z

0

10m

5m

MCHEROS with Stress Gradient

Tracking a single bubble in a stress gradient at various temperatures

SURFACE

Starting Position

54

MCHEROS with Stress Gradient

Tracking a single bubble in a stress gradient at various temperatures

• 7169507

• 11965231

• 16710670

• 23402482

Jumps to reach Surface:

SURFACE

Starting Position

55

Calculated Stress/Strain Transients in IFE FW

57

imp

i

diffi PP

Rt

log

InitializeModel

Calculate diffusion probability of He-bubble

Sum probabilities: Diffusion, Coalescence & Implantation for each He

Examine one event (Diffusion of a bubble or Implantation)

Jump with constant distance

Check Coalescence

tn+1 = tn + t

Grow He-bubbles by implantation

KMC – Calculation ProcedureKMC – Calculation Procedure Diffusion Migration:

Growth by Coalescence:

Growth by Implantation:

kT

EDD s

s exp0

4/3

b s4

3ΩD = D

2π rEs: Activation energy, 2.5eV*

D0: Pre-expon 1.25x10-2cm2/s

Surface diffusion rate

Bubble diffusion rate

Instantaneous Equilibrium Size:2

pr

R: Uniform random number (0:1) *A. Takahashi, TOFE 2006

57

Event Kinetic Monte Carlo How to pick an event

Using the event rate i and uniform random number

Physical time calculation

ii

Rt

log

0 1

Random number R （ 0 ～ 1 ）Normalized sequence

of event rates

j

j

i

59

Summary of US and Japanese Experiments of He-Implantation in W

60

UCLA He-Transport Code Development

Code Method Phenomena Comments

HEROS Rate Theory Nucleation, Growth, Transport1-D; Unified Field Parameters in Bulk Material

McHEROS Kinetic MC Growth,Transport,Coalescence3-D; Discrete bubbles; Material Geometric Features; Surfaces

Hybrid Helium Transport Code

Time

Bub

ble

Con

c.Rate-Theory KMC

Bubble Growth,Migration & Coalescence

Clustering & Nucleation

HEROS MCHEROS

Events (Surface pore formation)

B P

PB

PP

Surface pore is formed without coalescence

Events (Helium implantation) Helium implantation rate (= event rate)

Helium bubbles capture implanted helium atomsLinear relationship with the cross sectional area

B B B B

AHeI Cross sectional area: A

： Implantation rate

B B