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