As shown in the previous section, several striking observations of the damage accumulation observed in metals under cascade damage conditions can be rationalized in the framework of the PBM. This became possible because of the recognition of the importance of 1D diffusion of SIA clusters, which are continuously produced in cascades. The reaction kinetics in this case are a mixture of those for 1D and 3D migrating defects. Here, we emphasize that 1D transport is the origin of some phenomena, which are not observed in solids under FP irradiation.

One such phenomenon is the enhanced swelling observed near GBs. It is well known that GBs may have significant effect on void swelling. For example, zones denuded of voids are commonly observed adja­cent to GBs in electron-, ion-, and neutron-irradiated materials.134-137 Experimental observations on the effect of grain size on void concentration and swelling in pure austenitic stainless steels irradiated with 1 MeV electrons were also reported.138,139 In these experiments both void concentration and swelling were found to decrease with decreasing grain size. Theoretical calculations are in good agreement with the grain-size dependence of void concentration and swelling measured experimentally in austenitic stain­less steel irradiated with 1 MeV electrons.139,140

However, there is a qualitative difference between grain-size dependences of void swelling for electron irradiation and that for higher recoil energies. In particular, in the latter case, in the region adjacent to the void-denuded zone, void swelling is found to be substantially enhanced.134,136,141-147 Furthermore, in neutron-irradiation experiments on high-purity aluminum, the swelling in the grain interior increases strongly with decreasing grain size.144 This is oppo­site to the observations under 1 MeV electron irradi — ation139 and to the predictions of a model based on the dislocation bias.1 0

An important feature of the enhanced swelling near GBs under cascade irradiation is its large length scale. The width of this enhanced-swelling zone is of the order of several micrometers, whereas the mean distance between voids is of the order of 100 nm. Thus, the length scale is more than an order of magnitude longer than the mean distance between voids. The MFP of 3D diffusing vacancies and single SIAs is given by

L3D = = p2(ZdPd + 4nrcNc) —1/2 [140]

and is of the order of the mean distance between defects. Hence, 3D diffusing defects cannot explain the length scale observed. In contrast, the MFP of 1D diffusing SIA clusters is given by

L’D=yf = (pr2Pi+’"А’Г [141]

and is of the order of several micrometers, hence, exactly as required for explanation of the GB effect (see Figure 6). A possible explanation for the obser­vations would be as follows. The SIA clusters pro­duced in the vicinity of a GB, in the region of the size ~ L1D, are absorbed by it, while 3D migrating vacancies give rise to swelling rates higher than that in the grain interior. The impact of the GB on the concentration of 1D diffusing SIA clusters can be understood by using local sink strength, that is, the sink strength that depends on the distance of a local

area to the GB, /. It has been shown22 that the local sink strength in a grain of radius RClB is given by

(/ (2rGB — 1 ))1/2

As can be seen from eqn [142], the sink strength has a minimum at the center of the grain, that is, at / = RgB, and increases to infinity near the GB, when 1! 0.

The so-called grain-size effect, an increase of the swelling rate in the grain interior in grains of relatively small sizes (less than about 5 pm) with decreasing grain size, has the same origin as the GB effect discussed above. The swelling rate at the center of a grain may increase with decreasing grain size, when the grain size becomes comparable with the MFP of 1D diffusing SIA clusters and the zones of enhanced swelling of the opposite sides of GBs overlap. The swelling in the center of a grain as a function of grain size is presented in Figure 7 26 For comparison purposes, the values of the local void swelling (see Table 3 in Singh eta/.26) determined in the grain interiors by TEM are also shown. The PBM predicts a decrease of swelling with increasing grain size for grain radii bigger than 5 pm, which is in accordance with the experimental results. Note that the swelling values calculated by the FP3DM (broken curve in Figure 7) are magnified by a factor of 10.

Another striking phenomenon observed in metals under cascade damage conditions is the formation
of void lattices. It was first reported in 1971 by Evans148 in molybdenum under nitrogen ion irradi­ation, by Kulchinski et a/.149 in nickel under sele­nium ion bombardment, and by Wiffen150 in molybdenum, niobium, and tantalum under neutron irradiation. Since then it has been observed in bcc tungsten, fcc Al, hcp Mg, and some alloys.1 1-1 Jager and Trinkaus156 reviewed the characteristics of defect ordering and analyzed the theories pro­posed at that time, including those based on the elastic interaction between voids and phase instabil­ity theory. They concluded that in cubic metals, the void ordering is due to the 1D diffusion of SIA clusters along close-packed crystallographic direc­tions (first proposed by Foreman157). Two features of void ordering support this conclusion. First, the symmetry and crystallographic orientation of a void lattice are always the same as those of the host lattice. Second, the void lattices are formed under neutron and heavy-ion but not electron irradiation. This conclusion is also supported by theoretical analysis performed in Hahner and Frank15 and Barashev and Golubov.159