**2.2. Fundamental grain boundaries parameters**

#### *2.2.1. Grain boundary and free surface energies*

We introduce the GB energy (γGB) and free surface (FS) energy (γFS) to characterize GB cohesive properties, which are defined as the energies needed to create a GB and FS in the bulk [10]. They are given by the following relations:

The Effect of Impurities in Nickel Grain Boundary: Density Functional Theory Study http://dx.doi.org/10.5772/66427 5

$$\gamma^{\text{GB}} = \frac{E\_{\text{tot}}^{\text{GB}} - E\_{\text{tot}}^{\text{Bulk}}}{2 \, S^{\text{GB}}} \text{ and } \gamma^{\text{FS}} = \frac{E\_{\text{tot}}^{\text{FS}} - E\_{\text{tot}}^{\text{Bulk}}}{2 \, S^{\text{FS}}} \tag{1}$$

where *Etot GB*, *Etot FS* and *Etot Bulk* are total energies of the GB, FS and bulk system, respectively, and SGB and SFS refer to GB and FS areas in the GB model. In order to obtain all energies on equal footing and to make suitable comparisons, the calculations of *Etot Bulk* and *Etot FS* for the unperturbed Ni FCC ferromagnetic are performed with equal number of atoms per model.

#### *2.2.2. Segregation and binding energies*

**Table 1** presents the calculated lattice parameters for Nickel single crystal for different pseudopotentials and methods. As expected, for both approaches "ultrasoft and norm-conserving," LDA underestimates the lattice parameter 3.524 Å and the GGA overestimates it. The error for all methods is acceptable and has a maximum value of 2.66%. Moreover, the nonspin treatment results shrinking of lattice parameter. The method that most predict the lattice

**Table 1.** Nickel lattice parameters in Å with different computational parameters for the magnetic and nonmagnetic case,

**µ**

µ

LDA CA-PZ 3.516 3.451 0.740 LDA 2077.780 GGA-PBE 3.553 3.547 0.660 LSDA 7875.180 GGA-PW91 3.550 3.544 0.640 X-time 3.790

LDA CA-PZ 3.518 3.506 0.760 LSDA 7851.480 GGA-PBE 3.451 3.440 0.720 PAW-LSDA 2404.180

**Norm-conserving 50 SCF step (s)**

LDA 2057.780

From **Table 1**, we can also see the CPU time cost for 50 SCF steps using four cores 3.4 GHz and 32 GB RAM. The spin treatment costs about four times that the non-spin calculations, which explains why many calculations in the past were expensive to perform with spin treatment, particularly when the lattice parameters have close values. In order to compare the CPU time used in this work, we perform calculation with the "PAW method" implemented in VASP package, which is the most used method for GBs calculation in Ni. We use the same GB model with the closest parameters. The result shows that PAW is about two to three times much

We have to mention that the choice of LSDA using norm-conserving was not based mainly on lattice parameters nor on CPU time, but on convergence of total energy for transition metal element, especially during tensile test. The majority of other conducted methods fail to converge when the separation distance between the two surfaces is larger than 3 Å "during tensile test." Norm-conserving with LSDA was the best combination that converges well to the

We introduce the GB energy (γGB) and free surface (FS) energy (γFS) to characterize GB cohesive properties, which are defined as the energies needed to create a GB and FS in the bulk

parameter is LSDA using norm-conserving pseudopotentials with error of 0.17%.

and the time in seconds for the first 50 SCF cycle for different LDA combinations using 4 cores 3.4 GHz.

**Ultrasoft** *a* **(Å) Spin** *a* **(Å) Non-spin Magnetic moment** 

4 Study of Grain Boundary Character

NC-PP Spin Non-spin Magnetic moment

Exp 3.524 Ultra-soft

GGA-PW91 3.430 3.443 0.680 28kpt-600

faster than norm-conserving.

required tolerance in all cases.

**2.2. Fundamental grain boundaries parameters**

[10]. They are given by the following relations:

*2.2.1. Grain boundary and free surface energies*

The impurity segregation energy is defined as the difference between the total energy of a system *EI-FS/GB* with the impurity in the surface layer (or GB), and the energy *EBulk* with the impurity is in the bulk:

$$E\_{\rm t/GB} = -E\_{\rm t-FS/GB} - E\_{\rm Ball}.\tag{2}$$

Therefore, the sign convention is that the negative segregation energy corresponds to impurities that want to segregate [5].

According to the Rice-Wang model [11], the binding energies of the impurities at the grain boundary Δ *Eb GB* are defined as follows:

$$
\Delta E\_b^{\text{CB}} = E\_{\text{tCG}} - E\_{\text{cB}} - E\_t \tag{3}
$$

where *EI*/*GB* is the total energies of the GB system with segre gated impurity atoms, *EGB* is the energy of clean GB system, and *EI* is the energy for one isolated impurity atom. The larger negative value of the binding energy of the impurity at the GB means stronger bonding between the impurity and nickel atoms.

So in order to calculate these two properties (segregation and binding energies), we need to use six models which are:


#### *2.2.3. Theoretical tensile strength*

The theoretical (ideal) strength of a crystal as defined by reference [12] is determined by "*the maximum stress at elastic instability (yield or break) when applying an increasing stress to an infinite perfect crystal. It forms an upper limit to the strength of a real crystal (***Figure 2***), which is of both scientific and engineering value. The theoretical strength is an intrinsic material property, which is* 

**Figure 2.** Typical stress vs. strain diagram indicating the various stages of deformation, open source image (modified).

**Figure 3.** Representation of the GB fracture, the GB is separated in the weakest region, which corresponds to the (210) surface that attach the two grains, dz presents the separation distance.

*determined by the behavior of valence electrons and ions. Similarly, the theoretical strength of an ideal defective system containing only one defect such as a point defect, an interface, a grain boundary, etc., can be determined as the maximum stress required to reach elastic instability under increasing load without introducing extrinsic dislocations or cracks*" (**Figure 2**) [12].

Here, we recall how to calculate the cohesive energy 2γ and the tensile strength σMax (equal to the maximum tensile stress), as presented in [13]. We set a fracture plane that gives the minimum cohesive energy; then, the upper and lower half crystal blocks are rigidly separated by five equal increments (**Figure 3**). Each time in the separation process, we perform structure relaxation of the GB region, while fixing atomic layers close to the free surface in order to mimic the bulk structure. Then, the cohesive energy 2γ of the GB is the difference between the two total energies:

$$\mathcal{D}\mathbf{y} = E\_{\text{c3a}} - E\_{\text{s-c3a}} \tag{4}$$

EGB is the energy of the GB without separation (point 0), and ES-GB is the total energy for which the separating distance is so large that it does not change any more, typically after 0.5 nm separation.

The maximum tensile stress is calculated as follow. A simple function *f*(*x*) is fitted to the calculated total energy versus separation distance *x*

$$f(\mathbf{x}) = 2\gamma - 2\gamma \left( 1 + \frac{\mathbf{x}}{\lambda} \right) \exp\left(-\frac{\mathbf{x}}{\lambda}\right). \tag{5}$$

Here 2*γ* and *λ* are fitting parameters. *λ* is defined as the Thomas-Fermi screening length. This function is known as the universal binding curve proposed by Rose et al. [14]. It describes well the bonding nature between atoms and constitutes the best fit of binding energies versus atomic distances for the metallic systems. The tensile stress is the derivative of *f*(*x*):

**Figure 2.** Typical stress vs. strain diagram indicating the various stages of deformation, open source image (modified).

**Figure 3.** Representation of the GB fracture, the GB is separated in the weakest region, which corresponds to the (210)

surface that attach the two grains, dz presents the separation distance.

6 Study of Grain Boundary Character

**Figure 4.** The variation of cohesive energy and tensile stress for FeΣ3(111) GB with boron impurity segregate in site 0.

The maximum of *f*′(*x*) is at *x* = *λ* and corresponds to the maximum theoretical tensile stress or tensile strength *σMax*; therefore,

$$
\sigma\_{\lambda\alpha\alpha} = f(\lambda \ ) = e^{-1} \frac{2\gamma}{\lambda} \tag{6}
$$

The **Figure 4** represents a calculated example that shows the variation of total energy in Fe Σ3(111) GB with boron impurity segregate in site 0, and the value of the 2*γ* is 5.41 J m−2; thus, the cohesive energy *γ* is 2.70 J m−2. The tensile strength is plotted by evaluating the derivative of Rose function *f*′(*x*) with function of separation distance x.
