1. NEEP 541 – Swelling Fall 2002 Jake Blanchard

2. Outline Swelling

3. 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

4. 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)

5. 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

6. Plots  V/V dpa incubation  V/V T Low diffusion Thermal emission

7. Why Swelling? Excess vacancies can cause swelling or form dislocation loops Compare formation energies

8. Why Swelling?

9. 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

10. Schematic

11. Why Swelling? Consider FCC metal

12. Why Swelling?

13. 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

14. 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

15. Rate Theory Represent sinks by equivalent distributions Assume initial values for Sink density Dose rate Impurity concentrations

16. Fundamental Equation Rate of change of defect concentration = Production rate - Sink removal - recombination Thermal production Emission from defects Voids Loops Precipitates Grain boundaries dislocations

17. Unknowns X v =vacancy concentration X i =interstitial concentration P=P i =P v =defect production rate  d =dislocation density

18. 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

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

20. Resulting Equations Sink strengths Mean lifetimes

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

22. Typical Values InterstitialsVacancies z1.11 D o (cm 2 /s)0.0010.5 E m (eV)0.21.4 D (cm 2 /s)5e-54e-10  (s) 3e-70.043 L (/cm 2 )6.7e106.3e10

23. Steady State

25. Sink Dominant Case Assume mean lifetimes are small

26. Recombination Dominant Case Assume mean lifetimes are large

27. Swelling Rate Assume we have determined steady state defect concentrations

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

29. Swelling

30. Thermal Emission

31. Thermal emission rate can be positive or negative, depending on pressure and radius Pressure stabilizes bubble by decreasing thermal emission rate of vacancies

32. Sink-Dominant Swelling Ignore thermal emission

33. Sink-Dominant Swelling

34. For small x

35. Sink-Dominant Swelling

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

37. Critical Radius for Growth Sink Dominant Pressure reduces critical radius

38. Effect of Swelling on Stresses Consider Beam with heating on one surface (temperature varies through thickness Constrain beam on both ends

39. Modeling Initial stress

40. Modeling