s. sharafat 1* , a. takahashi 2 , and n. ghoniem 1 1 university of california los angeles
DESCRIPTION
Stress Driven Bubble Growth: Influence of Stress Gradient on Bubble Migration. S. Sharafat 1* , A. Takahashi 2 , and N. Ghoniem 1 1 University of California Los Angeles 2 Tokyo University of Science. 18 th High Average Power Laser Workshop Hosted by Los Alamos National Laboratory - PowerPoint PPT PresentationTRANSCRIPT
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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
*[email protected] 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
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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
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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.
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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
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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
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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
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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
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Influence of Stress Gradienton Bubble Migration and
Coalescence
8
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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):
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Single Bubble Migration
in a Stress Gradient
10
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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
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Collection of Bubbles:
Migration PLUS Coalescencein a Stress Gradient
12
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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
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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
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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
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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
*[email protected] This work was supported by the US Navy/Naval Research Laboratories through a grant with UCLA.
Modeling Carbon Diffusion
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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.
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1.Carbon Diffusion Data
2.Grain-Structure Model
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Carbon Diffusion in Grain Boundary region is higher than in Grain Matrix
D(GB) >> D(Matrix)
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*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)
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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.
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• 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
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Grain Structure Model
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* 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 %
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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
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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)
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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
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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
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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
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Model (2): Temperature Transient Effect
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Temperature Transient in HAPL
• UMARCO code provides detailed temperature transients as a function of position:
• Couple the transients with the diffusion model
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Concentration profile at t = 0 s
Carbon-Diffusion coefficient variation with time (grain matrix)
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Concentration profile at time 6e-5 s
Concentration profile at time 5e-5 s
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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
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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
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Model (3): Comparison between UFG – and
Single –Crystal Tungsten
(annealing 2000 oC)
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• 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)
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Multiple fine grains spanning 5 microns
Initial state of fine grains model
Single Crystal spanning 5 microns
Final result: t =0.001 s
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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
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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.
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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
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Backup Slides for C-diffusion
modeling
42
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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)
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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
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* 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
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* 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)
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Backup Slides for He-bubble
Migration and Coalescence
in a Stress Gradient
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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*
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McHEROS Stress Gradient: Methodology
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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.
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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
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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
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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
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MCHEROS with Stress Gradient
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Numerical Example:
Diffusion of single bubble— Radius: 10nm
Stress gradient in depth direction
500 MPa (at surface)z
0
10m
5m
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MCHEROS with Stress Gradient
Tracking a single bubble in a stress gradient at various temperatures
SURFACE
Starting Position
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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
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Calculated Stress/Strain Transients in IFE FW
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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
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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
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Summary of US and Japanese Experiments of He-Implantation in W
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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
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Events (Surface pore formation)
B P
PB
PP
Surface pore is formed without coalescence
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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