*3.4. Work of Separation*

Work of separation *W*oS (also called cleavage energy or cohesive energy) is another quantity which can be (and often is) used to characterize the GB cohesion. It is calculated as an energy necessary to break the crystal along a specified cleavage plane, as expressed by the following formula

$$\mathcal{W}\_{\rm cS} = \frac{E\_{\rm FS(+H)} - E\_{\rm GB(+H)}}{A} \,\mathrm{},\tag{3}$$

where *E*GB(+H) is the total energy of the fully optimized supercell containing the GB (with or without hydrogen atoms), *E*FS is the total energy of a supercell with two free surfaces (i.e., the fractured supercell), and *A* is the GB cross-section area. As has been formerly discussed [19,29], selection of the weakest cleavage (or fracture) plane is of key importance. There is no doubt about its position in the case of a clean GB but, in the case of H-decorated GB, one must consider possible redistribution of hydrogen on the created surfaces since it affects the *E*FS(+H) value. Our choice was based on the result of the tensile tests, namely, the uniaxial deformation which keeps the transverse dimensions constant and, therefore, is consistent with typical *W*oS calculations. Values computed for GB configurations in Figure 4 are listed in Table 3. To illustrate the effect of surface relaxation, we list the values for both the unrelaxed (as created) and the relaxed surfaces (relaxation reduces the value of *E*FS(+H) in Equation (3)).

**Table 3.** Work of separation (in J/m2) calculated for Σ3, Σ5, and Σ11 GBs with and without hydrogen.


Despite the fact that the Σ3 GB exhibits the greatest strength values (see Figure 6), its *W*oS values for unrelaxed surfaces are lower than values computed for the other GBs. The greatest *W*oS values were obtained for the Σ11 GB. These values fall within the range of results of fracture energies computed by Tehranchi and Curtin [12] for seven other GBs.

Let us note that values of the work of separation for bulk differ from the *W*oS values in Table 3 only by the GB energy (*γ*GB in Table 2), and one can therefore easily calculate the surface energy *γ*FS = (*W*oS + *γ*GB)/2. Values of 2.43 J/m<sup>2</sup> for (210) and 1.91 J/m<sup>2</sup> for (111) surfaces obtained this way (using the relaxed *W*oS values) agree well with the values of 2.40 J/m2 and 1.92 J/m<sup>2</sup> reported by Tran et al. [30].

Alvaro et al. [10] and Chen et al. [11] calculated the work of separation for clean and H-charged GBs in Ni using a relationship differing from Equation (3) by a factor of 2 (the definition corresponded rather to the surface energy), therefore their values correspond to one half of our *W*oS in Table 3. Values of 1.88 J/m2 [10] and 1.86 J/m2 [11] determined for the clean Σ3 GB therefore agree very well with our results of relaxed calculations. In addition, the values of 1.75 J/m<sup>2</sup> [11] and 1.8 J/m<sup>2</sup> [10] for the clean Σ5 GB and 2.04 J/m2 [11] for the clean Σ11 GB are in agreement with data in Table 3.

For H-charged GBs in fcc Ni, Chen et al. [11] predicted a complete decohesion of the Σ3 GB fully covered with hydrogen atoms (forming a monolayer), since their relevant *W*oS value was almost zero. However, our tensile tests predict only a reduction of the tensile strength (for loading of any kind). Our *W*oS values in Table 3 also show that the presence of one hydrogen monolayer at the Σ3 GB reduces its *W*oS by one third. Although such a relative reduction of *W*oS is greater than that of *σ*max, it does not predict the catastrophic crystal decohesion reported in Reference [11]. Instead, it agrees much better with the value of 1.1 J/m2 published by Alvaro et al. [10].
