**2.1. Computational details**

can be quite effective in developing this understanding. We can assess whether an impurity is a cohesion enhancer and causes the strengthening of the GB or is an embrittler and induces the weakening of the GB, while experimental techniques are difficult to investigate one impu-

First-principle calculations are widely used for metals and alloys. Grain boundaries of iron make a large part of these investigations. Impurity segregation to the GB in iron is described and understood relatively well, both theoretically and experimentally. In contrast, this is not the case of GB in nickel. The experimental information is quite rare [2]. In recent years, many efforts have been devoted to theoretically determine the effect of impurities on nickel grain boundaries, motivated by the industrial challenges facing fabrication and aging of nickel-base

It is already well known, both experimentally and theoretically, that sulfur acts as an embrittling element in nickel grain boundary and boron acts as enhancing element [3]. Some other elements like phosphorus are not that clear from first-principle calculations and became a subject of debate between strengthen and embrittling [4]. In contrast to light elements, influence over Ni grain boundaries of alloying elements took less attention [5]. The main elements for this purpose are transition metals, and the most important of them are elements of col-

Even though some of these element impurities effects have been studied theoretically, it is remarkable that there are still no comparisons between the effects of theses element in the GB. This is because it is hard to make comparison between results of different works due to the differences in impurity concentrations, GB models, number of atoms per unit cell, methods of calculations and type approximations used. We think that it is important now to have a systematic study to find the most enhancing and the most embrittling elements on Ni GBs. This knowledge can be used also to optimize the material properties as we are trying to under-

This chapter is divided in two parts. Part I is about the methodology of our calculations. Part II describes and discusses the obtained results, and it is organized as follows. First, details of calculation of the effect of Vanadium and Niobium in Nickel GB are exhibited. The effect of light elements "B, P, O, N, Al, Si, S, C" in NiΣ5 GB and surface and also of transition metal elements "Ti, V, Cr, Mn. Zr, Nb, Mo, Hf, Ta, W, Re" in NiΣ5 GB and surface are presented and

This chapter describes the Nickel Σ5(210) grain boundary models used in these work. Since we are dealing with 2D defect, the constructed structure is not evident as crystals. Beside this, computational parameters used in different models are presented with the precision required and explanation the choice of methods of calculations and the adequate approximations. We briefly present how to calculate free surface energy, grain boundary energy and the tensile

rity alone without the influence of other impurities.

umns IV–VII and periods 4–6 of the periodic table.

discussed. Finally, a main conclusion is drawn.

strength for a clean and impurity decorated GB.

**2. Model, methods and computational details**

stand the mechanisms and effects of different impurity atoms.

superalloys [3].

2 Study of Grain Boundary Character

Total energy calculations and geometry optimizations have been carried out using pseudopotentials plane-wave method as implemented in Cambridge serial total energy package (CASTEP) [6].

Local density approximation LDA CA-PZ is used for the exchange-correlation potentials (CA-PZ: Ceperley-Alder [7], data as parameterized by Perdew and Zunger [8]). Norm-conserving pseudopotentials (NCP) [6] are used with 720 eV plane-wave cutoff energy for all calculations. The valence electronic configuration of Ni is 3d84s2, and all other transition metal elements are treated including the d shell in valence bonds. The k-points sampling of Ni unit cell was carried out using 8 × 8 × 8 Monkhorst Pack mesh grid, which corresponds to different sampling on different model such as 4 × 4 × 1 k-points sampling for the 22 and 80 atom model and 3 × 5 × 1 for the 44 atom model of our grain boundaries. **Figure 1** presents a sample of 44 atom model used in segregation energies; substitutional segregation sites are sited from 1 to 11. This optimized values lead to an acceptable accuracy for the calculation of the total energy since we are calculating only the difference in total energy between two calculations; thus, a precision of 1 meV is sufficient. In the calculation of density of state and Mullikan population analyses which depend on the derivative of the total energy, we used more strict tolerance such as 6 × 6 × 2 in k-point sampling. The calculations assure a high level convergence of the total energy difference with respect to the number of atoms within 10−6 eV for total energy and maximum Hellmann-Feynman force within 0.1 meV/Å for pure Ni Σ5(210) GB and energy difference of 2 × 10−5 eV per atom for segregation and tensile test calculations.

For reasonable and fast convergence of the total energy, electronic occupancies were determined according to a Gaussian scheme with an energy smearing of 0.2 eV. The Pulay scheme of density mixing was used for self-consistent field (SCF) calculations [9].

In this work, we are dealing with Nickel GBs and transition metal impurity mainly, and they provide unique challenges to compute the electronic structure and for some pseudopotentials cause a lot of trouble to converge to an acceptable criteria. The choice of LSDA, norm-conserving pseudopotentials, came after a long study; we have done based mainly of the convergence of the total energy.

**Figure 1.** Unit cell model of Ni Σ5(210) symmetrical tilt grain boundary, model used in segregation study. Unit cell shapes are shown by solid lines. Axes directions and orientations are also presented. Atomic sites used in segregation are indicated by numbers (0–11).


**Table 1.** Nickel lattice parameters in Å with different computational parameters for the magnetic and nonmagnetic case, and the time in seconds for the first 50 SCF cycle for different LDA combinations using 4 cores 3.4 GHz.

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

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 faster than norm-conserving.

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 required tolerance in all cases.
