NEEP 541 – Swelling

Fall 2002Jake Blanchard

Outline Swelling

Swelling Swelling=volume increase in a

material caused by void formation (graphite densifies first)

Process Radiation produces point defects Interstitials migrate preferentially to

sinks (dislocations, mostly) while vacancies are left to form voids

Voids grow as they absorb more vacancies

Requirements Point defects must be mobile Need preferential sink for interstitials Need sufficient defect production

rate for nucleation and growth Need trace quantities of insoluble

gases to stabilize voids (usually He from transmutation)

Observations Most metals show incubation dose

for swelling (0.005 to 50 dpa) Most metals swell in temperature

range of 0.3 Tm<T<0.55 Tm Austenitic steels typically show 1%

swelling per dpa Ferritics are usually 0.1%/dpa

Plots

V/V

dpa

incubation

V/V

T

Low diffusion

Thermal emission

Why Swelling? Excess vacancies can cause

swelling or form dislocation loops Compare formation energies

3/1

3

323

2

2

4334

108

#/1500

4

mr

rm

cmvolumeatomic

voidpervacanciesmcmergs

rE fvoid

Why Swelling?

2

2

3/21

3/2

/001.0

2

434

cmerg

latticedistorttoenergy

energyfaultstackingtensionlineT

rrTE

Loops

mKmE

sf

sf

sf

d

sfdfloop

fvoid

Stacking Fault Energy Think of crystal as a stack of layers

in a particular sequence Defects are a defect in the

stacking sequence This distorts the lattice and

introduces stored energy into the lattice

Schematic

Why Swelling? Consider FCC metal

mKmKE

mamaTE

rrTE

mararealoop

aatomarea

floop

sfdfloop

sfdfloop

32

20

20

2

20

2

20

43

432

243

43/

Why Swelling?

JmE

JmEfvoid

floop

3/220

19

109

107

0

/5.0;2

44;2

3;4

0

2

0

30

3/21

32

sf

d

fvoid

floop

mNab

GPaGGbT

Aaa

mKE

mKmKE

Why Swelling

Ef

m

void

loop

•Non-zero stacking fault energy stabilizes void•In gold: low stacking fault energy so no voids at all•In Ni: large stacking fault energy so lots of voids

• As voids grow they eventually collapse to a loop

• Gas pressure can stabilize void

Swelling Rate Theory Determine steady state defect

concentrations Find growth rate of voids, assuming

they’ve already been nucleated Keys:

Biased sinks are necessary Voids grow by vacancy absorption

Rate Theory Represent sinks by equivalent

distributions Assume initial values for

Sink density Dose rate Impurity concentrations

Fundamental Equation

Rate of change of defect concentration

= Production rate - Sink

removal

- recombination

• Thermal production

• Emission from defects

• Voids• Loops• Precipitates• Grain

boundaries• dislocations

Unknowns Xv=vacancy concentration Xi=interstitial concentration P=Pi=Pv=defect production rate d=dislocation density

Modeling Assume defect sinks are dislocations

and voids Recombination rate=XvXi Vacancy loss to dislocations=zvDvdXv

Bias factor for loss of vacancies to dislocations Diffusion

coefficient for vacancies

Modeling Vacancy loss to voids=4RNDvXv

Bubble Radius

Void Density

Resulting Equations

iii

vvv

dii

dvv

diiivii

dvvvviv

DLDL

RNzLRNzL

RNzDxxxPdtdx

RNzDxxxPdtdx

1;144

4

4

Sink strengths

Mean lifetimes

Typical values T=500 C d=5x1010 /cm2

N=1015 voids/cm3

R=100 A

kTEDD mexp0

Typical ValuesInterstitials Vacancies

z 1.1 1Do (cm2/s) 0.001 0.5

Em (eV) 0.2 1.4D (cm2/s) 5e-5 4e-10

(s) 3e-7 0.043L (/cm2) 6.7e10 6.3e10

Steady State

i

v

iiv

v

vv

v

vi

i

i

v

v

i

ivi

i

v

vvi

v

Px

xxxP

xx

xxxPdtdx

xxxPdtdx

4121

21

0

0

0

22

Steady State

14121

14121

viv

i

vii

v

Px

Px

Sink Dominant Case Assume mean lifetimes are small

ii

vi

viv

vivi

Px

PPx

PP

22

2141

Recombination Dominant Case Assume mean lifetimes are large

v

ii

i

v

i

viv

vivi

Px

PPx

PP

22

241

Swelling Rate

Assume we have determined steady state defect concentrations

dtdRNS

NRdtdS

rateswellingSVV

dtd

34

34 3

Swelling

Swelling rate = Volume

change due to vacancy absorption

- Volume change due to interstitial absorption

Volume change due to thermal vacancy emission

-

Swelling

kTEx

xRxDDxDxdtdRR

xRxRND

DRNxDRNxdtdRNR

dtdRNRS

fve

v

ev

evviivv

ev

evv

iivv

exp)(

)()(

)()(4

444

4

2

2

Thermal Emission

12exp)()()(

12exp)()()(

2exp)()(

pRkT

xxRx

kTp

RkTxxRx

pressuregasptensionsurfacevolumeatomic

kTp

RkTxRx

ev

ev

ev

ev

ev

ev

ev

ev

Thermal Emission Thermal emission rate can be

positive or negative, depending on pressure and radius

Pressure stabilizes bubble by decreasing thermal emission rate of vacancies

Sink-Dominant Swelling

iviivv

iivviivv

iivv

ii

vv

LLPDDP

dtRd

DDPDxDxdtdRR

DRNxDRNxdtdRNR

PxPx

1122

444

2

2

Ignore thermal

emission

Sink-Dominant Swelling

22

2

2

12

12

4;1

1;1

22112

xzP

xzP

dtRd

RNxxLL

zzz

LLzzP

LLLLP

LLP

dtRd

dd

d

ddvi

iv

vi

dvi

vi

vi

iv

Sink-Dominant Swelling

2

220

2

220

2

2

2

12

12

1212

xzR

xzRR

dpadosePtxztPRR

xzP

dtRd

df

df

df

d

For small

x

Sink-Dominant Swelling

2/3

2/3

2

3

12

34

34

VV

xzN

VV

RNVV

d

f

Critical Radius for Growth Small bubbles will not grow (at low pressure)

1ln2

2exp1

012exp

evv

iivv

evv

iivv

evviivv

xDDxDxp

RkT

pRkTxD

DxDx

pRkT

pxDDxDxdtdRR

Critical Radius for Growth Sink

Dominant

11

ln

2

11

ln2

1ln2

1ln2

2

2

evvd

evvd

evv

iivv

evv

iivv

xDxzPkTp

R

xDxzPkTp

R

xDDxDxkTp

R

xDDxDxp

RkT

Pressure reduces

critical radius

Effect of Swelling on Stresses Consider Beam with heating on

one surface (temperature varies through thickness

Constrain beam on both ends

Modeling

ASEdd

dd

VV

dd

dd

E

TE

TE

VVT

E

c

ct

3

3110

0

31

0

0 Initial stress

Modeling

EAASEATE

SEEAdd

exp13

exp

3