3.1.1. Clean Grain Boundaries

All the constructed supercells were fully optimized in order to get their ground-state structures with equilibrium ionic positions and relaxed cell shape and dimensions. The structural parameters received from the optimization process as well as the excess volume *v*exc and the grain boundary energy *γ*GB are summarized in Table 2 and compared with available literature data.

**Table 2.** The excess volume *v*exc, GB energy *γ*GB, and the supercell parameters *a*0, *b*0, *c*<sup>0</sup> of clean Σ3, Σ5, and Σ11 GBs in fcc nickel along with the *k*-points grid and number *N* in the simulation cell.


All the GB parameters agree well with most of the ab initio results formerly published in literature [7,18,26]. Since the Σ3 GB is a coherent GB, its volume excess *v*exc and also the GB energy *γ*GB are very small (almost zero) and, therefore, these values might be strongly influenced by computational parameters (choice of pseudopotentials, size of computational supercell, etc.) as well as convergence settings during the relaxation process. Note, for example, that *v*exc computed by Chen et al. [11] is negative but their value of *γ*GB is in a good agreement with other ab initio results and also with the result *γ*GB = 0.05 J/m2 obtained by Shiga et al. [27] using the embedded atom method (EAM). Values of *γ*GB computed for the Σ5 GB are significantly greater and in a very good mutual agreement. They also reasonably agree with the EAM result [27] of 1.34 eV. The excess volume of the Σ5 GB is an order of magnitude greater than that of the Σ3. The Σ11 GB is also very compact, with very small *v*exc and relatively low *γ*GB. Former ab initio [11] as well as EAM calculations [27], giving the *γ*GB value of 0.40 eV, are in agreement with the present result.
