*3.1. Grain Boundary Energy and Structure after Different Relaxation Processes*

Grain boundary energies of seven GBs calculated by using the method mentioned above are listed in Table 2, which includes the results following CG relaxation and CG-MD-CG relaxation. For comparison, the results from DFT calculations [35,36] are also listed. From this table, it is clear that CG-MD-CG relaxation results in lower energy state of GB according to Equation (1). The difference with and without MD relaxation is smaller for ∑3(111), ∑3(112) and ∑5(013) GBs while it is larger for the other cases investigated in this work. The reason for this relates to the local atomic structure change induced by MD relaxation, as indicated by CNA and displacement results, shown in Figure 2. For example, ∑3(112) and ∑5(013) grain boundaries almost retain the symmetrical character with short atomic displacement distances after MD relaxation. However, for ∑3(111) GB, after 600 K MD relaxation, some atoms located in the grain boundary area have moved a long distance, up to 1.3 Å, resulting in the loss of symmetry locally. In contrast, ∑5(012), ∑9(221), ∑11(113) and ∑17(410) GBs have almost lost the symmetrical character at the GB plane region as shown by displacement of the atoms in the GB region. The maximum atomic displacement of these cases is up to 4.81 Å. Therefore, the high temperature MD relaxations induced losses in symmetrical features in certain symmetrical GBs, through which these systems have overcome the energy barriers to lower energy states. In fact, a similar conclusion was made in previous studies [23,33], while in this work, more GBs confirm that the MD relaxation to a lower state is necessary for further analysis. The relationship between the GB formation energy and GB angle has also been investigated, as shown in Figure 2, which can be fitted by a Gaussian function (Equation (3)). Further, we have defined a new variable called Δ*σ*, which is a fit for stress peak value by GB energy, as shown in Equation (4). Δ*σ* could help explain how the free volume had coupled with stress in the GB failure process.

$$E\_{GB} = 1.228 - 0.958 \exp\left[ -0.5 \left( \frac{\theta - 109.49}{5.1696} \right)^2 \right] \tag{3}$$

**Table 2.** Grain boundary energy calculated by CG and CG-MD-CG relaxation processes. For comparison, the results from DFT calculations are also listed.


**Figure 2.** Gaussian Fit of the relationship between GB energy and GB misorientation.
