The previous section included examples of void swelling. Voids result from the clustering of vacancies produced by displacement damage, as characterized by the number of dpa. Atomic displacements produce equal numbers of vacancy and SIA defects. As noted previously, descriptions of swelling mechanisms, including the role of He, can be found in excellent reviews.113-116 Early RT models showed that swelling is due to an excess flux of vacancies to voids, which is a consequence of a corresponding excess flux of SIA to biased dislocation sinks.106,107 Typical dis­placement rates (Gdpa) in high-flux reactors (HFR) are «10-6-10-7dpas—1 Hence, an irradiation time of 108s («3 years) produces up to «100 dpa. Only about 30% of the primary defects survive short­time cascade recombination.137 The residual defects undergo long-range migration and almost all either recombine with each other or annihilate at sinks. However, a small fraction of SIA and vacancies cluster to form dislocation loops and cavities, respectively. Ultimate survival of only 0.1% of the dpa in the form of clustered vacancies leads to 10% swelling at 100 dpa.

Classical models138,139 demonstrated that for the low Gdpa in neutron irradiations, homogeneous void nucleation rates are very low at temperatures in the peak swelling regime for AuSS between about 500 and 600 °C. However, heterogeneous void nucleation on He bubbles is much more rapid than homogeneous nucleation.109 Indeed, nucleation is not required when the He bubbles reach a critical size (r*) and He content (m*). The CBM concept has provided a great deal of insight into the effects of He on swelling.15,109- 112,114,118,130-133,140-151 In particular, the CBM rationa­lized the extended incubation dpa in fast reactor irradiations prior to the onset of rapid swelling. As previously shown in Figure 2(d) and 2(e), here we clearly distinguish between bubbles, which shrink or grow only by the addition of He, from larger voids, which grow unstably by the continuous accumulation of vacancies. In the case of bubbles, the gas pressure (p) plus a chemical stress due to irradiation (see Section 1.06.3.4) just balances the negative capillary stress 2g/ rb, where g is the surface energy and r> the bubble radius. By definition dr,/dt = 0 for bubbles, while the growth rate is positive and negative for cavities that are slightly smaller and larger than rb, respectively. In the case of voids (v), drv/dt is positive at all rv greater than the critical radius. Voids are typically underpres­surized with p<< 2g/rv More generally, cavities include both bubbles and voids and can contain an arbitrary number of vacancies (n) and He atoms (m).

The evolution of the number of discrete vacancy (n)-He (m) cavities, N(n, m), in a two-dimensional n—m space can be numerically modeled using cluster dynamics (CD) master equations. In the simplest case of growth or shrinkage by the absorption or emission of the monomer diffusing species (He, vacancies, and SIA), an ordinary differential equation (ODE) for each n, m cluster, dN(n, m)/dt, tracks the transitions from and to all adjacent cluster classes (n ± 1 and m ± 1), as characterized by He, vacancy, and SIA rates of being absorbed (bHe, v,0 and the vacancy emission (av) rate, as dN (n, m)/dt = bne (n, m — 1)N (n, m — 1)

+ bv(n — 1, m)N (n — 1, m) — av(n + 1, m)N (n + 1, m)

+ bi(n + 1, m)N (n + 1, m) — [bHe(n, m) — bv(m, n)

— bi(n, m)]N (n, m) [1

Note that thermal SIA and He emission rates are low and need not be included in eqn [1]. However,

He may be dynamically resolutioned by displace­ment cascades.152’153 There are a total of nmax x mmax such coupled ODEs. The rate coefficients, a and b, are typically computed from solutions to the diffu­sion equation, to obtain cavity sink strengths,107’113-116 along with the concentrations of the various spe­cies in the matrix and vacancies in local thermody­namic equilibrium with the cavity surface. The local vacancy concentrations are controlled by the surface energy of the void, g, via the Gibbs Thomson effect, and the He gas pressure.109,139,141 Conservation equa­tions are used to track the matrix concentrations of the mobile He, vacancies, and SIA based on their rates of generation, clustering, loss to all the sinks present, and, for the point defects, vacancy-SIA

recombination.144

Similar RT CD methods can also be used to simultaneously model SIA clustering to form dislo­cation loops, as well as climb driven by the excess flux of SIA to network dislocations.111,144 In AuSS, loop unfaulting produces network dislocations, and net­work climb results in both production and annihila­tion of the network segments with opposite signs. Thus, dislocation structures evolve along with the cavities.

However, the a and b rate coefficients depend on a number of defect and material parameters that were not well known during the period of intense research on swelling in the 1970s and 1980s, and integrating a very large number of nmax x mmax coupled ODEs was computationally prohibitive at the time these models were first proposed. One simplified approach, based on analytically calculating the rate of void nucleation on an evolving distribution of He bubbles, coupled to a void growth model provided consider­able insight into the role of He in void swelling.109,111 These early models, which also included parametric treatments of void and bubble densities,1 — 2 led to the correct, albeit seemingly counterintuitive, predictions that higher He may decrease, or even totally suppress, swelling in some cases, while in other cases swelling is enhanced, or remains unaf­fected. These early models also predicted the forma­tion of bimodal cavity size distributions, as confirmed by subsequent modeling studies and many experi­mental observations.111’112’114’118’131’133’134’148’151

Most aspects of void formation and swelling incubation can be approximately modeled based on the CBM concept. A critical bubble is one that has grown to a radius (r*) and He content (m*), such that, upon the addition of a single He atom or vacancy, it immediately transforms into an unstably growing void (see Figure 2(d) and 2(e)) without the need for statistical nucleation. Note that while a range of n and m clusters are energeti­cally highly favorable compared with equal numbers of He atoms and vacancies in solution, bubbles rep­resent the lowest free energy configuration in the vacancy-rich environments, characteristic of mate­rials experiencing displacement damage. That is, in systems that can swell due to the presence of sink bias mechanisms that segregate excess fractions SIA and vacancies to different sinks and at low reactor relevant damage rates, cavities primarily evolve along a bubble path that can ultimately end in a conversion to voids.