The results of atomistic modeling of He-defect inter­action energies are presented in this section for both ab initio and MD and molecular statics (MS) meth­ods. The ab initio results provide interaction energies 0 K that can be used in Monte Carlo simulations that are generally restricted to very small, nonequilib­rium, high-pressure (solid) He-vacancy (V) clusters. This contrasts with the models described in Section 1.06.3 that treat He as a continuum phase within the capillary approximation. In this latter case at suffi­ciently large bubble sizes and high temperatures, and in the absence of irradiation, the pressure in a relaxed strain-free bubble is given by the capillary approximation as P = 2g/ту, where the gas pressure is balanced by the average surface tension. Note that this approximation ignores the faceted shape of small bub­bles.260 This is a lower energy cluster configuration than those at higher He pressures that are balanced by a contribution from an elastically strained matrix, P = (2g + mb)/rb, where m is the shear modulus and b is the Burgers vector. MD can simulate gas bubbles at finite temperatures.260-263 A recent MD study at tem­peratures up to 700 K shows that the He pressure is very high in 2 nm clusters but saturates at «25GPa for a He/vacancy ratio of 3, due to spontaneous SIA emission at the theoretical strength of the Fe lattice.260 The MD simulations show that the equilibrium bubble He/vacancy ratio (m/n) is <1 and decreases with
increasing temperature and bubble size as expected. However, the MD studies suggest that the pressure in small bubbles predicted by MD is a factor of ~2 times lower than that predicted by the simple capillary approximation for Van der Waals and hard sphere equations of state.263 The reasons for this discrepancy are not yet understood.

Although quantitative details differ, the most important results of the atomistic models are that (a) both substitutional and interstitial He have very high formation energies; (b) there are large positive binding energies for He and vacancies in HemVn clusters even at m/n > 1; (c) at high m/n, the clusters relax and increase n by emitting SIA, or punching SIA loops; clusters with m/n < 1 are very stable.

Ab initio calculations in the framework of density functional theory (DFT) have been used to obtain formation and binding energies (Eb) of small clusters typically containing up to combinations of four He atoms and vacancies.82,134 The results are sum­marized in Table 3. Here, the Eb of the He in the cluster is referenced to tetrahedral Hei, with an energy that is slightly lower than that for other interstitial configurations. The ab initio calculations show that Hei clusters are bound to even absent vacancies. However, the Eb for He-V clusters is much larger. The Eb for He monotonically decreases, while the vacancy Eb monotonically increases, with larger helium to vacancy ratios, m/n, except for the case HemV, where the He Eb do not change much for m < 4. These trends reflect a high level of overpres­surization in the small clusters relative to a relaxed bubble with n > m. The He Eb also increases with cluster size, from 2.3 eV at m = n = 1 to 3.05 eV at m = n = 4. The most important implications of

these results are that, except at very high tem­peratures, m > «2 and n > «2 clusters are likely the stable nucleation sites for He-vacancy cluster formation and subsequent bubble evolution, as assumed in the models described in Section 1.06.3. Once formed and equilibrated, even small He-vacancy clusters are extremely stable up to very high temperatures.

MD simulations of larger HemVn (m < 20, n < 20) clusters predict vacancy Eb trends that are very similar to those found in the ab initio calculations for small clusters.265 However, the Eb for He derived from these MD simulations are consistently «1e V higher than those from ab initio calculations. The MD simulations used an embedded atom method (EAM) potential for Fe-Fe266 and pair potentials for Fe-He267 and He-He.268 While this set of potentials has been widely used, they predict significantly larger differ­ences between Hes and Hei formation energies than the ab initio results.269 Improved EAM potentials have been developed and a three-body potential for Fe-He-Fe264,269 predicts cluster formation and binding energies that are generally closer to the ab initio results.

Note that, while we are not aware of specific simulation results, overpressurized bubbles would be expected to be biased sinks for vacancies and against SIA. Taken together, these results further reinforce the assumption that, in vacancy-rich envir­onments produced by displacement damage, stable He-V clusters nucleate at very small sizes (m — n = 2) and grow along a near-equilibrium bubble path. Thus, a critical issue is how He is transported and interacts with other microstructural features.