Post on 18-Jan-2018
description
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