Modeling of Surface Roughening
M. Andersen, S. Sharafat, N. Ghoniem
HAPL Surface-Thermomechanics in W and Sic ArmorUCLA WorkshopMay 16th 2006
Outline
Basic mechanism for roughening—stress representation needed.
Discuss SRIM Calculation from Perkins Energy Spectra. Ion Deposition Energy Deposition
Thermal Stresses from Laser Pulse (Hector, Hetnarski).
Timeline for future work.
Additional slides on phase-field methods if interested.
Grinfeld Instability
Pioneered 1972. Considers the
movement of material. Heteroepitaxial
(thin films), atoms move along the surface.
Chemical etching, atoms move in and out of the surface.
As shown, can be cumbersome.
2
22
1
)(1
2
1
Dv
v
xgE
n
n
ssnntt
s
solid-melt
solid-vacuum
SRIM Code Ion Concentration
Goal: create a fast process for finding updated concentration of ions/vacancies and develop heat generation plots.
Any interest in this material?
Concentration 4He in W
0
2
4
6
8
0 0.25 0.5 0.75 1 1.25 1.5
Range (um)
Co
nce
ntr
atio
n (
ap
pm
/se
c)
0.0E+00
2.0E-07
4.0E-07
6.0E-07
8.0E-07
Co
nce
ntr
atio
n (
atm
/atm
)
Concentration 4He in SiC
0
1
2
3
4
0 0.5 1 1.5 2 2.5
Range (um)
Co
nce
ntr
atio
n (
ap
pm
/se
c)
0.0E+00
1.0E-07
2.0E-07
3.0E-07
4.0E-07
Co
nce
ntr
atio
n (
atm
/atm
)
.*)(~
)(
*)(*)()(~
)()(
)(
*)()(
)(
HzxCxG
xFxCxC
xFdV
dxxFxC
dx
dEEFxF
dxxFdEEF
Energy Deposition in W&SiC
Heat Deposition in W
0.00E+00
5.00E+02
1.00E+03
1.50E+03
2.00E+03
2.50E+03
0.0E+00 4.0E-05 8.0E-05 1.2E-04 1.6E-04 2.0E-04
Range (cm)
Q (
J/cm
^3)
0
2E+14
4E+14
6E+14
8E+14
1E+15
1.2E+15
Hea
t G
ener
atio
n (
W/m
^3)
1H
2H
3H
3He
4He
12C
13C
Au
Pd
Heat Deposition in SiC
0.0E+00
2.0E+02
4.0E+02
6.0E+02
8.0E+02
1.0E+03
1.2E+03
1.4E+03
0.E+00 5.E-05 1.E-04 2.E-04 2.E-04 3.E-04
Range (cm)
Q (
J/cm
^3)
0
1E+14
2E+14
3E+14
4E+14
5E+14
6E+14
7E+14
Hea
t G
ener
atio
n (
W/m
^3)
1H
2H
3H
3He
4He
12C
13C
Au
Pd
SRIM work provides volumetric heating…need this for thermal stress.
SiC experiences smaller Q over greater distance compared with W.
Detailed Temperature Profile
Single Shot Temperature Profile in 300-um Thick W-Armor
0
500
1000
1500
2000
2500
3000
1.E-10 1.E-09 1.E-08 1.E-07 1.E-06 1.E-05 1.E-04 1.E-03 1.E-02
Time (sec)
Tem
pera
ture
(C
)
W_sruf0.5um1.0um1.5um2.0um2.5um3.0um3.5um4.0um4.5um5.0um7.0um9.0um10um15um20um30um40um50um60um70um80um90um100.0um150.0um200.0um300.0um
5um
10um
50um
100 um
300 um
X-rays Ions
Discretize for Roughening Model
Formulation: Stresses due to a Laser Pulse
Thermal Field
Stress Field
Dimensionless formulation
Stresses
Temporal Pulse
zrasT
tratr
Tt
T
z
T
r
T
rr
T
,0
0,0,0
112
2
2
2
*
0
******
*
),,()(
dtzrYTt
1
1,
2
12
2
12
2
2
2
2
2
2
2
2
2
2
mmT
zr
rrr
zr
zrr
rz
zz
rr Tq
KKT
qm
KK
KK
tKtKzzKrr
cij
cij
cc
ccc
0
*
0
*
**
***
;4
;4
4;;
*****
**
**1****
0***
0 0
**0*
*2*
*******
2
)()21()()1(
)(2
1
*
)(
2)(;
**
*
*
*
2
**2**
ddt
erfcer
rJzrJz
rJz
Ge
r
zJe
GYfh
z
tzz
rr
Temporal Pulse
The rise and fall time of each pulse is accounted for.
Can be adjusted (a=0.4,b=7.0,c=3.0).
Consider a Gaussian Surface Source.
Temporal Profile
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2
t*
Y(t
*)
1
22
*
***
** )(exp)(
c
r
rr
bc
at
ttbt
ttY
z
r
Steady-State Variation of Radial Stress
Surface experiences the largest compressive radial stresses.
Explained by surface elements expanding against “cooler” sub-surface material.
Variation of Radial Stress with Radius
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 1 2 3 4 5
r*
rr*
t*=5.0, z*=0
t*=5.0, z*=1.0
Material Surface
Evolution of Surface Stresses at Selected R.
Maximum stresses are located at the center of the beam. Similar profiles away from center.
Occurs shortly after maximum energy is reached (rise time)—time needed to develop stress from absorbed energy.
Variation of Radial Stress at the Surface
-1.4
-1.2
-1
-0.8
-0.6
-0.4
-0.2
0
0 1 2 3 4 5
t*
rr*
r*=0.0, z*=0.0
r*=1.0, z*=0.0
Beam Center
Evolution of Stress Field under Surface
It is shown that small tensile stresses are developed for radial, hoop, and shear stresses.
Normal stress is still compressive.
Cold regions deep in material and around edges of beam.
Evolution of Radial Stress below the Surface
-0.04
-0.02
0
0.02
0.04
0.06
0 1 2 3 4 5
t*
rr*
r*=0.0, z*=1.5
r*=2.0, z*=1.5
Evolution of Axial Stress below Surface
-0.1
-0.08
-0.06
-0.04
-0.02
0
0.02
0 1 2 3 4 5t*
zz*
(z*=1.5, r*=0.0)
(z*=1.5, r*=2.0)
Axial Variation in Radial Stress
Notice that the radial stress reaches maximum tensile stress as the beam approaches its rise time.
Compressive stresses occur while beam deactivates.
Axial Variation of Radial Stress at Beam Center
-0.4
-0.3
-0.2
-0.1
0
0.1
0 0.25 0.5 0.75 1 1.25 1.5
t*
rr*
r*=0.0, t*=t-rise
r*=0.0, t*=1.0
t > t_rise
t = t_rise
Conclusions on Stress
Compressive radial stresses developed on surface.
Subsurface compressive axial stress develops tensile radial stress. (before deactivation)
Elastic solution—addition of plasticity and possibly wave effects (tension -> compression)
Future Research Plans
Finished: Formulation of the problem (roughening, stress field). Energy deposition calculations (SRIM). Where’s the problem? Method to fix it.
Computational Tools: Efficient elastic model – varying biaxial stress, temperature
dependence, MG ( June ’06) Elasto-plastic model using laser pulse model (August ’06) Validate with comparisons to RHEPP, XAPPER, Dragonfire (Sept.
’06)
Fatigue Analysis: Criteria to establish the transition to cracks/cusps (December
‘06) Experimental validation (January-March ’07) Extension to other materials??? (April ’07)
ATG Phase Field
Follow the total free energy of the system and account for the phase change, Kassner 2001.
Provides smoothing of sharp-interface method. Consider only the most severe location.
3
,2
1 22
dVufF ij
Energy Density
Length parameter
ATG Continued
Invariant form of free energy allows summation of elastic (fe), gravity (fgravity), double well—phase change (fdw), and equilibrium control (fc) potentials.
200
2
0
2
1
)(1)()(
)(1)(
)(2
Ehf
ghhxxf
fhfhf
gf
c
vsgrav
vaporsole
dw
ijeqkkkk
eqijij
eqijij
ij
u
u
eqijij
uuuu
udfij
eqij
)()()(
)(
2
)(
ATG Continued
H is the solid fraction function, 1 for solid and 0 for vapor as relative maxima and minima.
G accounts for the possibility for a phase transition where the two minima 0,1 correspond to the phases vapor and solid respectively
)23(2 h
22 )1( g
ATG Continued
must then solve the relaxation equation:
Which leads to:sk
R
FR
t
3
1
Essentially a time scale