NEEP 541 – Swelling Fall 2002 Jake Blanchard

Outline 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 T m <T<0.55 T m 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

Why Swelling?

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

Why Swelling?

Why Swelling EfEf 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 X v =vacancy concentration X i =interstitial concentration P=P i =P v =defect production rate  d =dislocation density

Modeling Assume defect sinks are dislocations and voids Recombination rate=  X v X i Vacancy loss to dislocations=z v D v  d X v Bias factor for loss of vacancies to dislocations Diffusion coefficient for vacancies

Modeling Vacancy loss to voids=4  RND v X v Bubble Radius Void Density

Resulting Equations Sink strengths Mean lifetimes

Typical values T=500 C  d =5x10 10 /cm 2 N=10 15 voids/cm 3 R=100 A

Typical Values InterstitialsVacancies z1.11 D o (cm 2 /s) E m (eV) D (cm 2 /s)5e-54e-10  (s) 3e L (/cm 2 )6.7e106.3e10

Steady State

Sink Dominant Case Assume mean lifetimes are small

Recombination Dominant Case Assume mean lifetimes are large

Swelling Rate Assume we have determined steady state defect concentrations

Swelling Swelling rate = Volume change due to vacancy absorption - Volume change due to interstitial absorption Volume change due to thermal vacancy emission -

Swelling

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 Ignore thermal emission

Sink-Dominant Swelling

For small x

Sink-Dominant Swelling

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

Critical Radius for Growth Sink Dominant 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 Initial stress

Modeling