Cu is of primary importance in the embrittlement of the neutron-irradiated RPV steels. It has been observed to separate into copper-rich precipitates within the ferrite matrix under irradiation. As its role was discovered more than 40 years ago,74-76 Cu precipitation in a-Fe has been studied extensively under irradiation as well as under thermal aging using atom probe tomography, small angle neutron scattering, and high resolution transmission electron microscopy. Numerical simulation techniques such as rate theory or Monte Carlo methods have also been used to investigate this problem, and we next describe one possible approach to modeling microstructure evolution in these materials.

The approach combines an MD database of pri­mary damage production with two separate KMC simulation techniques that follow the isolated and clustered SIA diffusion away from a cascade, and the subsequent vacancy and solute atom evolution, as discussed in more detail in Odette and Wirth,2 Monasterio eta/.,34 and Wirth eta/.77 Separation of the vacancy and SIA cluster diffusional time scales natu­rally leads to the nearly independent evolution of these two populations, at least for the relatively low dose rates that characterize RPV embrittlement.1,78-81

The relatively short time (~100 ns at 290 °C) evolu­tion of the cascade is modeled using OKMC with the BIGMAC code.77 This model uses the positions of vacancy and SIA defects produced in cascades obtained from an MD database provided by Stoller and coworkers82,83 and allows for additional SIA/ vacancy recombination within the cascade volume and the migration of SIA and SIA clusters away from the cascade to annihilate at system sinks. The duration of this OKMC is too short for significant vacancy migra­tion and hence the SIA/SIA clusters are the only diffus­ing defects. These OKMC simulations, which are described in detail elsewhere,77 thus provide a database of initially ‘aged’ cascades for longer time AKMC cas­cade aging and damage accumulation simulations.

The AKMC model simulates cascade aging and damage evolution in dilute Fe-Cu alloys by following vacancy — nearest neighbor atom exchanges on a bcc lattice, beginning from the spatial vacancy popula­tion produced in ‘aged’ high-energy displacement cascades and obtained from the OKMC to the ulti­mate annihilation of vacancies at the simulation cell boundary, and including the introduction of new cas­cade damage and fluxes of mobile point defects. The potential energy of the local vacancy, Cu-Fe, envi­ronment determines the relative vacancy jump prob­ability to each of the eight possible nearest neighbors in the bcc lattice, following the approach described in eqn [4]. The unrelaxed Fe-Cu vacancy lattice ener­getics are described using Finnis-Sinclair N-body type potentials. The iron and copper potentials are from Finnis and Sinclair84 and Ackland eta/.,85 respec­tively; and the iron-copper potential was developed by fitting the dilute heat of the solution of copper in iron, the copper vacancy binding energy, and the iron-copper [110] interface energy, as described else­where.55 Within a vacancy cluster, each vacancy main­tains its identity as mentioned above, and while vacancy-vacancy exchanges are not allowed, the clus­ter can migrate through the collective motion of its constituent vacancies. The saddle point energy, which is Ea0 in eqn [4], is set to 0.9 eV, which is the activation energy for vacancy exchange in pure iron calculated with the Finnis-Sinclair Fe potential.84

The time (AtAKMC) of each AKMC sweep (or step) is determined by AtAKMC = (nPmax)~ , where Pmax is the highest total probability of the vacancy population and n is an effective attempt frequency. This is slightly different than the RTA, in which an event chosen at random sets the timescale as opposed to always using the largest probability as done here. In this work, n = 1014s_1 to account for the intrinsic vibrational frequency and entropic effects associated with vacancy formation and migration, as used in the previous AKMC model by Odette and Wirth.2 As mentioned, the possible exchange of every vacancy (i) to a nearest neighbor is determined by a Metropolis random number test15 of the relative vacancy jump probability (P/Pmax) during each Monte Carlo sweep. Thus at least one, and often multiple, vacancy jumps occur during each Monte Carlo sweep, which is dif­ferent from the RTA. Finally, as mentioned above, as the total probability associated with a vacancy jump depends on the local environment, the intrinsic time­scale (AtAKMc) changes as a function of the number and spatial distribution of the vacancy population, as well as the spatial arrangement of the Cu atoms in relation to the vacancies.

The AKMC boundary conditions remove (annihi­late) a vacancy upon contact, but incorporate the ability to introduce point defect fluxes through the simulation volume that result from displacement cas­cades in neighboring regions as well as additional displacement cascades within the simulated volume. The algorithms employed in the AKMC model are described in detail in Monasterio et al3 and the remainder of this section will provide highlights of select results.

The AKMC simulations are performed in a ran­domly distributed Fe-0.3% Cu alloy at an irradiation temperature of 290 °C and are started from the spa­tial distribution ofvacancies from an 20 keV displace­ment cascade. The rate of introducing new cascade damage is 1.13 x 10~5 cascades per second, with a cascade vacancy escape probability of 0.60 and a vacancy introduction rate of 1 x 10~4 vacancies per second, which corresponds to a damage rate of this simulation at ^10~ apas~ . Thus a new cascade (with recoil energy from 100 eV to 40 keV) occurs within the simulated volume (a cube of ^86 nm edge length) every 8.8 x 104s (~1 day), while an individ­ual vacancy diffuses into the simulation volume every 1 x 104s (~3h). AKMC simulations have also been performed to study the effect of varying the cascade introduction rate from 1.13 x 10~3 to 1.13 x 10~7 cascades per second (dpa rates from 1 x 10~9 to 1 x 10_ 3dpas~ ). The simulated conditions should be compared to those experienced by RPVs in light water reactors, namely from 8 x 10~12 to 8 x 10~n dpas-1, and to model alloys irradiated in test reac­tors, which are in the range of 10~9-10~1°dpas~1.

Figures 2 and 3 show representative snapshots of the vacancy and Cu solute atom distributions as a function of time and dose at 290 °C. Note, only the Cu atoms that are part of vacancy or Cu atom clusters are presented in the figure. The main simulation volume consists of 2 x 106 atoms (100a0 x 100a0 x 100a0) of which 6000 atoms are Cu (0.3 at.%). Figure 2 demonstrates the aging evolution of a single cascade (increasing time at fixed dose prior to introducing additional diffusing vacancies or new cascade), while Figure 3 demonstrates the overall evolution with increasing time and dose. The aging of the single cascade is representative of the average behavior observed, although the number and size distribution of vacancy-Cu clusters do vary consider­ably from cascade to cascade. Further, Figure 2 is representative of the results obtained with the previ­ous AKMC models of Odette and Wirth21 and Becquart and coworkers,24,26 which demonstrated the formation and subsequent dissolution of vacancy-Cu clusters. Figure 3 represents a significant extension of that previous work21,24,26 and demonstrates the for­mation of much larger Cu atom precipitate clusters that results from the longer term evolution due to multiple cascade damage in addition to radiation enhanced diffusion.

Figure 2(a) shows the initial vacancy configura­tion from an aged 20-keV cascade. Within 200 ps at 290 °C, the vacancies begin to diffuse and cluster, although no vacancies have yet reached the cell boundary to annihilate. Eleven of the initial vacancies remain isolated, while thirteen small vacancy clusters rapidly form within the initial cascade volume. The vacancy clusters range in size from two to six vacancies. At this stage, only two of the vacancy clus­ters are associated with copper atoms, a divacancy

Figure 3 Representative vacancy (red) and clustered Cu atom (blue) evolution in an Fe-0.3% Cu alloy with increasing dose at (a) 0.2years (97 udpa), (b) 0.6years (0.32 mdpa), (c) 2.1 years (1.1 mdpa), (d) 4.0years (2.0mdpa), (e) 10.7years (4.4mdpa), and (f) 13.7years (5.3 mdpa).

Figure 2 Representative vacancy (red circles) and clustered Cu atom (blue circles) evolution in an Fe-0.3% Cu alloy during the aging of a single 20keV displacement cascade, at (a) initial (200 ns), (b) 2 ms, (c) 48 ms, (d) 55 ms, (e) 83 ms, and (f) 24.5 h.

cluster with one Cu atom and a tetravacancy cluster with two Cu atoms. From 200 ps to 2 ms, the vacancy cluster population evolves by the diffusion of isolated vacancies through and away from the cascade region, and the emission and absorption of isolated vacancies in vacancy clusters, in addition to the diffusion of the small di-, tri-, and tetravacancy clusters. Figure 2(b) shows the configuration about 2 ms after the cascade. By this time, 14 of the original vacancies have dif­fused to the cell boundary and annihilated, while 38 vacancies remain. The vacancy distribution includes six isolated vacancies and seven vacancy clusters, ranging in size from two divacancy clusters to a ten vacancy cluster. The number of nonisolated copper atoms has increased from 223 in the initial random distribution to 286 following the initial 2 ms of cas­cade aging.

The evolution from 2 to 48.8 ms involves the dif­fusion of isolated vacancies and di — and trivacancy clusters, along with the thermal emission of vacancies from the di — and trivacancy clusters. Over this time, 7 additional vacancies have diffused to the cell boundary and annihilated, and 20 additional Cu atoms have been incorporated into Cu or vacancy clusters. Figure 2(c) shows the vacancy and Cu clus­ter population at 48.8 ms, which now consists of three isolated vacancies and four vacancy clusters, includ­ing a 4V-1Cu cluster, a 6V-4Cu cluster, a 7V cluster, and an 11V—1Cu cluster. Figure 2(d) and 2(e) shows the vacancy and Cu cluster population at 54.8 and

82.8 ms, respectively. During this time, the total num­ber of vacancies has been further reduced from 31 to 21 of the original 52 vacancies, the vacancy cluster
population has been reduced to three vacancy clus­ters (a 4V—1Cu, 7V, and 9V—1Cu), and 30 additional Cu atoms have incorporated into clusters because of vacancy exchanges.

Over times longer than 100 ms, the 4V-1Cu atom cluster migrates a short distance on the order of 1 nm before shrinking by emitting vacancies, while the 7 V and 9V-1Cu cluster slowly evolve by local shape rearrangements which produces only limited local diffusion. Both the 7V and 9V-1Cu cluster are ther­modynamically unstable in dilute Fe alloys at 290 °C and ultimately will shrink over longer times. The vacancy and Cu atom evolution in the AKMC model is now governed by the relative rate ofvacancy cluster dissolution, as determined from the ‘pulsing’ algo­rithm, and the rate of new displacement damage and the diffusing supersaturated vacancy flux under irra­diation. Figure 2(f) shows the configuration about

8.8 x 104s (^24 h) after the initial 20keV cascade. Only 17 vacancies now exist in the cell, an isolated vacancy which entered the cell following escape from a 500 eV recoil introduced into a neighboring cell plus two vacancy clusters, consisting of 7V-1Cu and 9V—1Cu. Three hundred and forty-five Cu atoms (of the initial 6000) have been removed from the super­saturated solution following the initial 24 h of evolu­tion, mostly in the form ofdi — and tri-Cu atom clusters.

Figure 3(a) shows the configuration at about 0.1mdpa (0.097mdpa) and a time of 7.1 x 106s (^82 days). Ten vacancies exist in the simulation cell, consisting of eight isolated vacancies and one 2V
cluster, while 807 Cu atoms have been removed from solution in clusters, although the Cu cluster size distri­bution is clearly very fine. The majority of Cu clusters contain only two Cu atoms, while the largest cluster consists of only five Cu atoms. Figure 3(b) shows the configuration at a dose of 0.33 mdpa and time of

2.1 x 107s (245 days). Only one vacancy exists in the simulation volume, while 1210 Cu atoms are now part of clusters, including 12 clusters containing 5 or more Cu atoms. Figure 3(c) shows the evolution at 1 mdpa and 7.2 x 107s (^2.3 years). Again, only one vacancy exists in the simulation cell, while 1767 Cu atoms have been removed from supersaturated solution. A handful of well-formed spherical Cu clusters are visible, with the largest containing 13 Cu atoms. With increasing dose, the free Cu concentration in solution continues to decrease as Cu atoms join clusters and the average Cu cluster size grows. Figure 3(d) and 3(e) shows the clustered Cu atom population at about 2 and

4.4 mdpa, respectively. The growth of the Cu clusters is clearly evident when Figure 3(d) and 3(e) is com­pared. At a dose of 4.4 mdpa, 48 clusters contain more than 10 Cu atoms, and the largest cluster has 28 Cu atoms. The accumulated dose of 5.34 mdpa is shown in Figure 3(f). At this dose, more than one-third of the available Cu atoms have precipitated into clusters, the largest of which contains 42 Cu atoms, corresponding to a precipitate radius of ^0.5 nm.

Figure 4 shows the size distribution of Cu atom clusters at 5.34 mdpa, corresponding to the configu­ration shown in Figure 3(f). The vast majority of the

Cu clusters consist of di-, tri-, tetra-, and penta-Cu atom clusters. However, as shown in the inset of Figure 4 and as visible in Figure 3(f), a significant number of the Cu atom clusters contain more than five Cu atoms. Indeed, 29 clusters contain 15 or more Cu atoms (a number density of

1.2 x 1024m~3), which corresponds to a cluster con­taining a single atom with all first and second near­est neighbor Cu atoms and a radius of 0.29 nm. An additional 45 clusters contain at least nine Cu atoms (atom + all first nearest neighbors), while 9 clusters contain 23 or more atoms (number density of

3.8 x 1023 m~3). This AKMC simulation is currently continuing to reach higher doses. However, the ini­tial results are consistent with experimental obser­vations and show the formation of a high number density of Cu atom clusters, along with the contin­ual formation and dissolution of 3D vacancy-Cu clusters.

Figure 5 shows a comparison of varying the dose rate from 10~9 to 10~13 dpas-1. Each simulation was performed at a temperature of 290 °C and introduced additional vacancies into the simulation volume at the rate of 10~4s-1. The effect of increasing dose

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Figure 5 Comparison of the representative vacancy (red) and clustered Cu atom (blue) population at a dose of ~1.9 mdpa and 290 "Casa function of dose rate, at (a) 10~11 dpas~1, (b) 10~9dpas~ (c) 10^13dpas^1.

rate at an accumulated dose of 1.9 mdpa is especially pronounced when comparing Figure 5(c) (10~ 3dpa s-1) with Figure 5(a) (10~n dpas-1) and Figure 5(b) (10~9dpas-1). At the highest dose rate, a substan­tially higher number density of small 3D vacancy clusters is observed, which are often complexed with one or more Cu atoms. Vacancy cluster nucle — ation occurs during cascade aging (as described in Figure 2) and is largely independent of dose rate, but cluster growth is dictated by the cluster(s) thermal lifetime at 290 °C versus the arrival rate of additional vacancies, which is a strong function of the damage rate and vacancy supersaturation under irradiation. Thus, the higher dose rates produce a larger number of vacancies arriving at the vacancy cluster sinks, resulting in the noticeably larger number of growing vacancy clusters. Also, there is a corresponding decrease in the amount of Cu removed from the solution by vacancy diffusion. In contrast, the effect of decreasing dose rate is greatly accelerated Cu precipitation. Already at 1.9 mdpa, a number of large Cu atom clusters exist at a dose rate of 10- dpas~ , with the largest containing 35 Cu atoms, as shown in Figure 5(c). The increased Cu clustering caused by a decrease in dose rate results from a reduction in the number ofcascade vacancy clusters, which serve as vacancy sinks. Thus, a higher number of free or isolated vacancies are available to enhance Cu diffusion required for the clustering and precipi­tation of copper. While these flux effects are antici­pated and have been predicted in rate theory calculations performed by Odette and coworkers,78,79 the spatial dependences of cascade production and microstructural evolution, in addition to correlated diffusion and clustering processes involving multiple vacancies and atoms are more naturally modeled and visualized using the AKMC approach.

While the results just presented in Figures 2-5 have shown the formation of subnanometer Cu-vacancy clusters and larger growing Cu precipitate clusters that result from AKMC simulations, which only consider vacancy-mediated diffusion, Becquart and coworkers have shown that Cu atoms in tensile posi­tions can trap SIAs and therefore the Cu clustering behavior may also be influenced by interstitial — mediated transport. Ngayam Happy and coworkers63,86 have developed another AKMC model to model the behavior of FeCu under irradiation. In this model, diffusion takes place via both vacancy and self­interstitial atoms jumps on nearest neighbor sites. The migration energy of the moving species is also deter­mined using eqn [4], where the reference activation
energy Ea0 depends only on the type of the migrating species. Ea0 has been set equal to:

• the ab initio vacancy migration energy in pure Fe when a vacancy jumps towards an Fe atom (0.62 eV);

• the ab initio solute migration energy in pure Fe when a vacancy jumps towards a solute atom (0.54 eV for Cu); and

• the ab initio migration-600 rotation energy of the migrating atom in pure Fe when a dumbbell migrates (0.31 eV).

Ei and Ef are determined using pair interactions, according to the following equation:

E = e(<)(Sj — — Sk) + Edumb [5]

i=1,2 j<k

where i equals 1 or 2 and corresponds to first or second nearest neighbor interactions, respectively, and where j and k refer to the lattice sites and Sj (respectively Sk) is the species occupying site j (respectively k): Sj in {X, V} where X = Fe or Cu. A more detailed description of the model can be found in Ngayam Happy et al63

In this study, various Cu contents were simulated (0, 0.18, 0.8, and 1.34 at.%) at three different tem­peratures (300, 400, and 500 0C). Without going into too much detail here, one can state that these AKMC simulation results are qualitatively similar to those presented in Figures 2-5, which showed the formation of small, vacancy-solute clusters and copper enriched cluster/precipitate formation at 300 0C. Similarly, the effect of decreasing dose rate in high Cu content alloys was also found to accelerate Cu precipitation.

This model does show that the formation of the Cu clusters/precipitates during neutron irradiation takes place via two different mechanisms depending upon the Cu concentration. In a highly Cu supersaturated matrix, precipitation is accelerated by irradiation, whereas in the case of low Cu contents, Cu precipi­tates form by induced segregation on vacancy clusters.

The influence of temperature was investigated for an Fe-0.18wt% Cu alloy irradiated at a flux of

2.3 x 10-5dpas-1. At 400 and 500 0C, neither Cu precipitates nor Cu-vacancy clusters were formed, in agreement with the results of Xu et a/.87 At these temperatures, the model indicates that the vacancy clusters are not stable and induced segregation is thus hindered. Another interesting result obtained with this model is that the presence of Cu atoms in the matrix was found to decrease the point defect cluster sizes because of the strong interactions of Cu with both vacancies and SIAs.