B to D. (B) Top view of the fractured free surface at the GB. The distance between two GB0 sites is 3.52 A, which is equal to the lattice constant of fcc Ni.

purity occupation (concentration) in the inner bulk, Cbulk, should be far less than 1.0 in this equation. In Fig. 2A, we show the occupation curves calculated from the above equation for some conditions of aging temperatures and bulk S concentrations. Here we focus on the middle line for 918 K and 25 atomic parts per million (ppm), because this condition was chosen by Heuer et al. (1, 2) for their experimental determination of the critical S concentration. From this figure, we can see that the occupations of both the GB0 and GB2 sites are very close to 1.0, whereas those of the others (GB1 and GB3 to GB6) are under 0.1 (SOM).

Next, we assume that more than one S atom segregates in the GB unit cell. When we insert S atoms one by one at four GB0 sites, the average binding energy per one S atom does not change appreciably, as in Table 1. This indicates that the GB0-GB0 distance (3.52 A) is so long that any appreciable interaction does not begin. Similar results are also obtained for the GB±2 site. On the other hand, the distance between the GB0 and GB±2 sites is 2.2 A in the clean GB case, which is even shorter than Ni-Ni nearest-

neighbor distance (2.49 A) in fcc Ni. Although there are many choices for how to insert or substitute S atoms at GB0, GB2, and GB-2 sites, we show here only one example for simplicity, which gives the largest average binding energies. We substitute S atoms for Ni atoms one by one at four GB2 sites, in addition to four GB0 sites occupied by S. The average binding energy reduces from 4.75 to 4.23 eV/S, as in Table 1. This clearly shows that there is a strong repulsive interaction between S atoms (SOM). Although in McLean's model the binding energy of impurity is assumed not to depend on the site and the occupation of impurity in the GB region, we estimate the occupation as a first approximation by substituting the average binding energy for the fixed one in McLean's equation. Despite the large energy loss, the average binding energy, 4.23 eV/S, is still sufficiently large for the eight S atoms to segregate fully at all GB0 and GB2 sites, as in Fig. 2A. If we insert or substitute 12 S atoms fully at GB0, GB2, and GB-2 sites, however, the average binding energy reduces to 3.84 eV/S. In the conditions for segregation (918 K, 25 atomic ppm) chosen by Heuer et al, it is thought to be difficult for all 12 sites to be fully occupied by S atoms as in Fig. 2A. Experimentally, it is reported that elemental S is removed from GBs by the formation of Ni3S2 precipitates after 120 hours of annealing at 918 K and 20 weight ppm of S (12). This fact is consistent with our results, in which the binding energy of S for Ni3S2 (4.06 eV/S) is larger than the above binding energy (3.84 eV/S) and is the largest among those for some Ni-S ordered alloys, as in Table 2 (SOM).

Along with the increase of occupation at GB0 and GB2 sites, the distance between adjacent GB0 and GB2 sites increases greatly. Figure 2, B to D, shows valence electron density maps when S atoms are gradually substituted for Ni atoms at all four GB2 sites in addition to fully occupied four GB0 sites by S. These figures clearly show that S atoms do not bond with each other, whereas they bond strongly with Ni atoms. The valence electron density at the middle point between two neighboring S atoms is close to zero. The distance between adjacent GB0 and GB2 sites when they are fully occupied by S is 3.3 A, which is large when compared with its original distance (2.2 A) for the clean GB case and the a i u C O

Fig. 2. (A) Occupation versus binding energy, -Eb (eV/S), at a segregation site calculated from McLean's equation with respect to the aging temperature (K) and bulk S concentration (atomic ppm, 10-4 atomic % S). The blue line with circles indicates the occupation under the conditions chosen by Heuer et al. (1, 2) to determine a critical intergranular concentration of S. Both binding energies and segregation energies of S are roughly indicated by arrows for each single site (GB0 to GB5) and for cases [(B) to (D)] in which S atoms interact with each other. (B to D) Valence electron density maps (electron/A3) on the cross-sectional plane indicated in Fig. 1A for three cases that have different S occupations. From the top layer to the bottom, gB-3, GB0, GB2, GB4, and GB6 layers are shown. (B) GB0 4/4. (C) GB0 4/4, GB2 2/4. (D) GB0 4/4, GB2 4/4.

Segregation energy (eV/atom)

Segregation energy (eV/atom)

B, GB0.2

1073 IK). 25 (al.pprn) 1073 (K). 50 [at.ppm) 918 (K>, 25 (at.ppm)

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