Technical Comments

Response to Comment on “Grain Boundary Decohesion by Impurity Segregation in a Nickel-Sulfur System”

Science  09 Sep 2005:
Vol. 309, Issue 5741, pp. 1677d
DOI: 10.1126/science.1112218

We recently concluded that the strong decohesion of a nickel (Ni) grain boundary (GB) is caused by the aggregation of sulfur (S) atoms on the GB, which repel each other (1). We further estimated the segregation concentration using the average binding energy of S atoms. However, Geng et al. (2) claim that the binding energy should be calculated not on average but incrementally (or sequentially). As they point out, the incremental binding energy (Math) when the GB2 1/4 monolayer is added to the GB0 4/4 monolayer is 3.45 eV/S, as shown in Table 1(I). The occupation possibility of this arrangement is less than 1%, according to the McLean's curve, at 918 K and 25 atomic parts per million, as shown in figure 2A in (1).

Table 1.

The calculated binding energies per one S atom. Two kinds of binding energy are shown: the average binding energies (Embedded Image) for all the segregated S atoms, and the incremental binding energies (Embedded Image) when S atoms are added one by one. The four equivalent sites are distinguished by the coordinate in the <100> direction. (a, 0.0; b, 0.25; c, 0.50; d, 0.75). See figure 1B in (1).

Average This work Embedded Image Geng et al. Embedded Image
GB0 2/4 (a,c) 4.74
GB0 4/4 4.75
GB1 4/4 3.33 3.25
GB2 4/4 4.66
GB3 4/4 3.63
GB0 4/4 GB2 4/4 4.23
GB1 4/4 GB2 4/4 4.24
GB1 4/4 GB3 4/4 4.10
Increment Embedded Image Embedded Image
(I) (GB0 4/4) → (GB0 4/4, GB2 4/4)
+ GB2a 3.45
+ GB2b 3.53
+ GB2c 3.70
+ GB2d 4.16
(II) (GB0a, GB0c) → (GB0 4/4, GB2 4/4)
+ GB2a 4.17
+ GB2c 4.07
+ GB2b 3.86
+ GB0d 3.79
+ GB0b 4.35
+ GB2d 4.15
(III) (GB2 4/4) → (GB1 4/4, GB2 4/4)
+ GB1a 3.49 4.15View inline
+ GB1b 3.42 4.23View inline
+ GB1c 4.09 4.26View inline
+ GB1d 4.17 4.55View inline
Ave.(a—d) 3.79 4.29
(IV) (GB3 4/4) → (GB1 4/4, GB3 4/4)
+ GB1a 4.33
+ GB1c 4.23
+ GB1b 4.87
+ GB1d 4.83
Ave.(a—d) 4.57
  • View inline* Geng et al. do not show the site positions in their Comment.

However, if we take another path of increasing occupation up to the (GB0 4/4, GB2 4/4) configuration as shown in Table 1(II), the minimum binding energy becomes 3.79 eV/S, the occupation of which is about 50%. Geng et al. miss the point that there are many paths to the (GB0 4/4, GB2 4/4) configuration. Only a few of all possible paths have been investigated at this stage.

Some experiments indicate that S atoms can segregate up to about two monolayers (14 atomic % S within 5Å from the GB plane, 14.4 atoms/nm2) (36). The two-monolayer (GB0 4/4, GB2 4/4) segregation can occur in our consideration using the average binding energy. However, neither the average binding energy nor the incremental binding energy has a theoretical basis, because the binding energy in McLean's equation does not take into account any atomic interactions. For quantitative estimation, the interactions among S atoms for all possible configurations should be taken into account in a statistical method like Monte Carlo simulation (e.g., Metropolis method).

Geng et al. (2) performed calculations using the same code and parameters as in (1). They claim that another two-monolayer segregation (GB1 4/4, GB2 4/4) that has a small GB expansion (0.6Å) is much more stable than the (GB0 4/4, GB2 4/4) segregation, which has a large expansion (1.2Å). They suggest that the directional change of chemical bonding in the (GB1 4/4, GB2 4/4) case, rather than the GB expansion in the (GB0 4/4, GB2 4/4) case, causes the strong decohesion. However, we have reexamined such calculations and found that the two configurations have the same structure and energy.

Table 1(III) shows our calculations as well as those of Geng et al. and reveals a large discrepancy in the incremental binding energy (Math). For example, the energies for the first added GB1 1/4 monolayer to the GB2 4/4 monolayer are 4.15 eV/S in their calculation and 3.49 eV/S in our calculation. Our results indicate that the stability of the (GB1 4/4, GB2 4/4) case is almost the same as the (GB0 4/4, GB2 4/4) case, as shown in Table 1 (I and III). The average binding energies (Math) for both cases are also almost the same. This seems reasonable, because the relaxed atomic geometries near the GB are almost the same between the two configurations (Fig. 1). In contrast, Geng et al. show neither the relaxed atomic structure nor the electron density distribution.

Fig. 1.

The relaxed atomic geometries by force minimization for the left (GB0 4/4, GB2 4/4) and the right (GB1 4/4, GB2 4/4) cases. The gray and yellow circles represent Ni and S atoms, respectively. We can see that the two geometries near the GB are almost the same. The left case includes 21 layers (Ni, 19; S, 2); the right includes 20 layers (Ni, 18; S, 2). For the left and right cases, the correspondences are as follows: GB0(S)↔GB2(S), GB2(S)↔GB1(S), GB1(Ni)↔ GB3(Ni), GB3(Ni)↔GB-2(Ni), and so on.

Geng et al. claim that the GB expansion of the (GB0 4/4, GB2 4/4) case (1.2Å) is twice as large as that of the (GB1 4/4, GB2 4/4) case (0.6Å). Here, they may miscalculate the GB expansion as well as the binding energies. The S atoms enter the GB0 interstitial vacancies in the (GB0 4/4, GB2 4/4) case, whereas S substitutes for Ni at GB1 sites in the (GB1 4/4, GB2 4/4) case. The formula used to calculate the GB expansion [Supporting Online Material in (1)] cannot be used to compare the two cases that have different numbers of Ni layers; the discrepancy of the GB expansion (0.6Å) in the Geng et al. calculations may come from the difference in the number of Ni layers, as shown in Fig. 1. The Ni interlayer distance in the <012> direction is 0.78Å, which agrees well with the difference.

As pointed out by Geng et al., the GB3 monolayer is greatly stabilized by adding the GB1 monolayer. As shown in Table 1(IV), the incremental binding energy of adding GB1 1/4 monolayer to the GB3 4/4 monolayer is about 4.2 to 4.9 eV/S, which is much larger than the average for the GB3 4/4 monolayer (3.63 eV/S) and the GB1 4/4 monolayer (3.33 eV/S). Although the occupation possibility of the GB1 4/4 or the GB3 4/4 monolayer is only about 1%, all possible paths to the (GB1 4/4, GB3 4/4) segregation should be investigated in detail. Even if this segregation is realized, the final tensile strength is about 12 GPa, which is much larger than the (GB0 4/4, GB2 4/4) and (GB1 4/4, GB2 4/4) cases (<5 GPa).

Contrary to the claim of Geng et al., we cannot find any reason why the (GB1 4/4, GB2 4/4) configuration is much more stable than the (GB0 4/4, GB2 4/4) configuration. Further details are required to make a convincing argument.

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