Grouping Method for an Approximate Numerical Calculations of

Grouping Method for an Approximate Numerical Calculations of Helium-vacancy Cluster Evolution in Solids during Irradiation or Annealing S. I. Golubov, S. J. Zinkle and R. E. Stoller Materials Science & Technology Division Oak Ridge National Laboratory, HAPL Ion Transport and Surface-Thermomechanics in W and Si. C Armor Workshop UCLA, Los Angeles May 15 -16, 2006

OUTLINE • • • Modeling of He-vacancy clustering Master Equation New grouping method Applications of the method He retention in W (UNC experiment) Conclusions

He-vacancy cluster evolution under irradiation or ageing • Mechanisms: • • • Absorption of newly created, injected or re-dissolved He atoms Absorption of point defects Thermal evaporation of He atoms and vacancies Radiation-induced He resolution Cluster coalescence • He transport: • • • He in interstitial configuration He in substitution configuration (He-2 V clusters) Brownian motion and coalescence

Methods used to describe He-vacancy cluster evolution Homogeneous approach • • Analytical models (nodal line method, di-atomic nucleation, . . ) Master Equation (discrete rate equations) Fokker-Plank equation (continuous variables) Momentum method Hybrid methods (ME for small size clusters and F-P/MM equations for larger ones) Non Homogeneous approach • Monte-Carlo Master Equation is basis for homogeneous approach

Two-dimensional Master Equation x- number of vacancies m- number of He atoms in a cluster

Conservation Laws. I. Total number of clusters - boundary fluxes

Conservation Laws. II. Total number of point defects in clusters S(t) – swelling, M(t) – total number of He atoms an clusters. , - volume fluxes , ,

Rate Equations for mobile defects Equation of State (Trinkaus et al. 1982)

New grouping method Approximation for SDF within a group - group widths in x and m spaces

Grouping Scheme + mean x, m fluxes 10 fluxes required only

Coalescence via Brownian motion

Dose dependence of defect accumulation

SDFs calculated using the grouping method

Size distribution functions after 1 h annealing at 700, 800 and 900 o. C 1/10

Comparison with experiment TEM measurements Calculations

Impact of Equation of State on bubble evolution during annealing

Experimental observations of He retention in tungsten foil (Hashimoto et al. *) Proton spectra for single crystal tungsten implanted at 850 o. C and flash annealed at 2000 o. C in 1, 100, and 10000 cycles to a total dose of 10 19 He/m 2 *) Fusion Science and Technology 47 (2005) 881

Defects and Helium generation profiles in UNC experiments

Model Description • It is assumed that the problem may be simulated by considering a foil with thickness of about 0. 7µm uniformly irradiated with He ions • Sink in the form of foil boundaries is treated by using continuum approach • Point defect generation and He generation rates in the foil are equal to Gpd =2. 5*10 -7 dpa/s and GHe =2. 5*10 -9 1/s, respectively • Total flux of He ions of 1019 ions/m 2 corresponds to accumulation of about 250 appm He • Duration of 1 step irradiation is equal 102 s in the case of 103 cycles, 103 s in the case of 102 cycles and so on; total time of irradiations is equal to 105 s • Duration of one step of flash annealing is equal 10 s • He atoms diffuse interstitially and are captured by vacancies, He-vacancy clusters and foil boundaries

Flash annealing regime used in the calculations

Sink strength of foil boundaries where is sink strength of foil interior defects

Parameters used in the calculations He dissociation energies, Edis, for m. He+1 V clusters m Edis (e. V) 1 4. 5 2 3. 75 3 3. 46 4 3. 11 5 2. 89 Evf =4. 13 e. V, Evm =1. 95 e. V, EHem=0. 24 e. V , Eim= 0. 15 e. V Ω=1. 585*10 -29 m-3 , γ=2. 8 J/m 2, Zv=1. 00, Zi =1. 25 Edis=4. 5 e. V at x>1 (Van Veen)

Evolution of cluster size distribution function during 1 step irradiation Red curve - equilibrium trajectory

He accumulation during the implantation

He accumulation during irradiation/flash annealing in 10 steps cycle He practically does not release crystal during an annealing step

He accumulation during irradiation/flash annealing in 100 steps cycle He releases crystal during each annealing step however the total amount of He accumulated in the clusters continuously increases- during 5 cycles 30% of He implanted are stored in the clusters.

He accumulation during irradiation/flash annealing in 1000 steps cycle He fully release crystal during each annealing step

Conclusions • He releases crystal already during irradiation. However the effect is significant at small implantation doses only • He fully release crystal during each annealing step in the case of 1000 cycle • He practically retained in crystal in the cases of 100, 10 and 1 cycles