**2. Computational Details**

The calculations were performed using the program VASP [22] (Vienna ab initio simulation program) developed at the Fakultät für Physik, Universität Wien. In our study, the electron interactions were described by the projector-augmented waves (PAW) potentials [23] supplied with the VASP code, and the exchange correlation energy was evaluated by means of the generalized gradient approximation (GGA) with parametrization of Perdew–Burke–Ernzerhof [24]. The Methfessel–Paxton method of the first order was adopted with a smearing width of 0.1 eV.

Sampling of the Brillouin zone was done using a Monkhorst-Pack [25] scheme with 6 × 1 × 10, 3 × 1 × 7, and 3 × 1 × 10 *k*−point grids for Σ3, Σ5, and Σ11 GBs, respectively. The solution was

considered self-consistent when the difference between energies of two subsequent steps was below 10−<sup>7</sup> eV and the plane-wave basis set was expanded with the cutoff energy of 350 eV. Optimization of atomic positions in computational supercells was performed using the internal VASP procedure until all forces between atoms were lower than 10 meV/Å. For the optimization of the cell shape, we employed our own external program that cooperated with VASP via reading its output and writing new input files. This program allowed us to relax the stress tensor components to their targeted values within a tolerance of ±0.1 GPa. In all presented calculations, ferromagnetic ordering of Ni was included via spin polarization.

Before introducing the studied GBs, let us define two quantities commonly used for their characterization [7,11,26]. The first one is the excess volume *v*exc, which can be computed as

$$v\_{\rm exc} = \frac{V\_{\rm GB} - N\_{\rm GB}V\_{\rm Ni}}{2A} \,, \tag{1}$$

where *V*GB represents the volume, *N*GB is the number of atoms in a fully optimized supercell containing GB, *V*Ni is the volume per atom in a perfect bulk crystal of Ni, and *A* is the GB area. In general, larger values of the excess volume *v*exc represent more void space at GB. Another important characteristic quantity is the GB energy *γ*GB determined as

$$
\gamma\_{\rm GB} = \frac{E\_{\rm GB} - N\_{\rm GB} E\_{\rm Ni}}{2A},
\tag{2}
$$

where the *E*GB is the total energy of the optimized supercell with GB and *E*Ni is the total energy per atom of the bulk fcc nickel crystal. Thus, *γ*GB represents the energy necessary to create such a planar defect of a unit area in a perfect crystal structure. Since all simulation cells contain two identical GBs (as described in the next paragraph), right-hand sides of both equations must be divided by the factor of 2.

In our systematic study, we considered three types of symmetrical GBs in fcc nickel, namely, the Σ3(111) GB, Σ5(210) GB, and the Σ11(311) GB. Corresponding computational supercells that were constructed for the present study are illustrated in Figure 1. These supercells have orthorhombic symmetry and, in order to keep periodic boundary conditions also in the direction perpendicular to the GB plane, they contain two identical GBs. One is located in the center of the supercell and the other one at its edge (displayed as the dashed vertical lines in Figure 1).

All the computational cells were subjected to several types of tensile loading or deformation. Figure 2 illustrates the geometry of our tensile tests. The loading axis is parallel with the *x*-axis, which was set perpendicular to the GB plane in all our simulations (i.e., *x* [111], *x* [210], and *x* [311] for the Σ3, Σ5, and Σ11 GBs, respectively). Stresses *σ*<sup>2</sup> and *σ*<sup>3</sup> are thus the transverse stresses that were controlled by our computational procedure at each strain increment (of 0.01), optimizing both the cell shape and the ionic positions. Since the general loading with *σ*<sup>2</sup> = *σ*<sup>3</sup> would lead to an enormous number of triaxial stress states (and corresponding values of cohesive strength), we considered these stresses mutually dependent, keeping their ratio *k* = *σ*2/*σ*<sup>3</sup> constant as discussed hereafter. For *σ*<sup>2</sup> = *σ*<sup>3</sup> = 0, the tensile test corresponds to the so-called uniaxial loading. Another special type of loading is the isotropic (or hydrostatic) one with *σ*<sup>2</sup> = *σ*<sup>1</sup> = *σ*3. In this only case, we also controlled the axial stress *σ*1. In all the other cases, we computed *σ*<sup>1</sup> as a function of the axial strain 1. For comparative purposes, we also simulated uniaxial deformation with <sup>2</sup> = <sup>3</sup> = 0 (which implies *σ*<sup>2</sup> = *σ*<sup>3</sup> due to the crystal anisotropy) and isotropic deformation with <sup>2</sup> = <sup>1</sup> = 3. In all applied loading cases, the cohesive strength value was identified with the maximum of *σ*1, hereafter denoted *σ*max. A brief overview of all loading types is given in Table 1.

**Figure 1.** The supercells containing Σ3(111), Σ5(210), and Σ11(311) grain boundaries (GBs) used in the present ab initio calculations. The planes perpendicular to the rotation axis related to GBs are highlighted. Orientation of the supercells were *<sup>x</sup>* [111], *<sup>y</sup>* [112¯], and *<sup>z</sup>* [1¯10] for the <sup>Σ</sup>3 GB; *<sup>x</sup>* [210], *<sup>y</sup>* [<sup>120</sup> ¯ ], and *<sup>z</sup>* [001] for the <sup>Σ</sup>5 GB; and *<sup>x</sup>* [311], *<sup>y</sup>* [<sup>233</sup> ¯ ], and *<sup>z</sup>* [011¯ ] for the <sup>Σ</sup>11 GB.

**Figure 2.** Illustration of the computational supercell under triaxial tensile loading. Stress tensor components are denoted using the Voigt notation.

**Table 1.** Types of the applied loading.

