In order to circumvent the problems raised in the previous paragraph and in Section 1.15.4.1 for the inclusion of defect clusters in a PFM, Badillo et al.146 have recently proposed a mixed approach that com­bines discrete and continuum treatments of the defect clusters, so that each cluster is treated as a separate entity. Point-defect cluster size is treated as a discrete quantity for cluster production, whereas the long-term fate of clusters is controlled by a continuum-based flux of free point defects. New field variables are thus introduced to describe the size of these clusters:

Np Np Np

Cp _ Nc, V; Cp _ Nc, A; Cp _ Nc, B

c, V _ Nd ’ c, A _ Nd ’ c, B _ Nd

where Nf v is the number of vacancies in the vacancy cluster in the cell p, and Nd is the number ofsubstitu- tional lattice sites per cell; N a and Np B are the numbers of A and B interstitial atoms, respectively, forming the interstitial cluster in the cell p. Each cell contains at most one cluster.

The production of point defects by irradiation takes place at a rate dictated by the irradiation flux

f in dpas-1 and by the simulated volume. In the case of irradiation conditions leading to the intracascade clustering of point defects, the total number of point defects created in a displacement cascade, the frac­tion of those defects that are clustered, and the size and spatial distribution of these clusters are used as input data. The production of Frenkel pairs is treated in the same way as defect clusters, except that the variables affected are the free vacancy and interstitial concentrations.

This treatment of defect and defect cluster pro­duction makes it possible to compare irradiation con­ditions with identical total defect production rates, that is, identical dpas-1 values, but with varied frac­tions of intracascade defect clustering and varied spatial distribution of these clusters. Furthermore, it is also very well suited for system-specific modeling, since all the above information can be directly and accurately obtained from molecular dynamic simula­tions sampling the primary recoil spectrum.1 1 In particular, one can build a library of such displace­ment cascades, so that the PFM will inherit the stochastic character in space and time of the produc­tion of defect clusters by displacement cascades.

The continuous flux of free point defects to the clusters results in the growth or shrinkage of these clusters, which translates into the continuous evolution of the cluster field variables CjP v, Cp a, and Cpp B. When any cluster field variable drops below 1 /Nd, this cluster is assumed to have dissolved, and the remaining one point defect is transferred to the corresponding free point-defect variable of that cell. For the sake ofsimplicity, defect clusters are treated as immobile, but the approach can be extended to include mobility, in particular for small interstitial clusters. Further details are available in Badillo eta/.146

The potential of the above approach is illustrated by considering a 2D A8B92 alloy with a zero heat of
mixing, so that at equilibrium, it always forms a random solid solution. The production of interstitials is, however, biased so that only A interstitial atoms are created. This could, for instance, simulate an alloy where there is a rapid conversion ofB interstitial atoms into A interstitial atoms via an interstitialcy mecha­nism. The preferential transport of A interstitial atoms to defect clusters should lead to an enrichment of A species around defect clusters, since these clusters act as defect sinks. The effect of the primary recoil spectrum on this irradiation-induced segregation is studied by comparing two cases: the first one where a small fraction of cascades, 1/Ncas = 5 x 10~4, produces defect clusters, and the second one where that fraction is 100 times higher, 1/Ncas = 5 x 10~2. In both cases, however, the displacement rate per atom per second is the same, here 10~7 dpa s_1. Figure 14 shows instanta­neous concentration maps of the A solute atoms for 1/Ncas = 5 x 10~4. In this case, the PFM uses 64 x 64 cells, each containing 7 x 7 lattice sites. Segregation of A species is clearly observed at a few locations, typi­cally 2-5. This number is close to the average number of defect clusters. The sharp peaks with high levels of segregation correspond to segregation of existing defect clusters, either interstitial or vacancy ones. This is confirmed by visualizing the defect and defect cluster fields, see Figure 15. The broader segregation profiles in Figure 14 are the remnants of sharp segre­gation profiles after their corresponding clusters shrank

A8B92 alloy with zero heat of mixing where all interstitials are created as A atoms. The two-dimensional model system contains 448 x 448 lattice sites, decomposed into 64 x 64 cells for defining the phase field variables, each cell containing 7 x 7 lattice sites. Irradiation displacement rate is 10~7 dpa s~1; the cascade frequency rate is1/Ncas = 5 x 10~4, and the irradiation dose is 6dpa. Reproduced from Badillo, A.; Bellon, P.; Averback, R. S. to be submitted.

and disappeared. As a result, the nonequilibrium segre­gation that build up on those clusters is being washed out by vacancy diffusion, as expected for an A-B alloy system with zero heat of mixing. In the case where defect sinks have a finite lifetime, as in the present case, one should thus expect a dynamical formation and elimination of segregated regions.

In the case of much higher defect cluster produc­tion rate, 1/Ncas = 5 x 10~2, a very different micro­structure is stabilized by irradiation, as illustrated in Figure 16. Now a high density of clusters is present, typically 40 interstitial clusters and 20 vacancy clus­ters, as seen in Figure 17(a) and 17(b), and the segregation measured on these clusters is reduced by about one order of magnitude compared to the previous case. These results are reminiscent of the experimental findings reported by Barbu and Ardell147 and Barbu and Martin,148 showing that, with irra­diation conditions producing displacement cascades (e. g., 500-keV Ni ion irradiation), the domain of irradiation-induced segregation and precipitation in undersaturated Ni-Si solid solutions is significantly reduced compared to the case where irradiation pro­duces only individual point defects (e. g., 1-MeV elec­tron irradiation).