Predicting grain boundary energies of complex alloys from ab initio calculations

Investigating the grain boundary energies of pure fcc metals and their surface energies obtained from ab initio modeling, we introduce a robust method to estimate the grain boundary energies for complex multicomponent alloys. The input parameter is the surface energy of the alloy, which can easily be accessed by modern ab initio calculations based on density functional theory. The method is demonstrated in the case of paramagnetic Fe-Cr-Ni alloys for which reliable grain boundary data is available.

Starting from the present DFT results for the GBEs, we examine the correlation between the GBEs in different metals. Here, we choose Cu as the reference and compare the GBEs of the same GB structure in different materials.
We notice however that choosing another metals as reference leads practically to the same conclusions. Results for four elements, Al, Ni, Pd, and Pt, taken as examples of sp, 3d, 4d, and 5d metals, respectively, are shown in Fig. 2.
In order to strengthen our point, some previous DFT results for tilt and twist GBs [34] are also included in the figure.
We emphasize that the following observations and discussions apply to all metals considered here and the detailed results can be found in SM (Fig. S2 and Table S2). First, we confirm that there is a clear correlation between the GBEs in different metals, as demonstrated by the previous EAM results [67]. All the boundary energies in a specific metal locate approximately on a straight line passing through the origin (dashed line in Fig. 2), indicating that a single material dependent factor (δ) may be used to correlate the GBEs in a pair of metals, i.e., γ A GB (DOF) ≈ δγ Cu GB (DOF) (A stands for an fcc metal). The slopes (δ A/Cu GB(fit) ) of the linear fitting for all 9 metals are listed in Table 1. The nearly perfect scaling relation between the GBEs in different metals highlights the critical roles played by the GB structures in deciding the GBEs. The anisotropy of GBEs is, to the first order approximation, decided by the boundary structure, i.e., the five DOFs. In other words, despite of the existence of the difference in local atomic structure configuration (or even in magnetic environment), the GBEs in different materials may be described by an universal functional of the five geometric parameters in different materials. This observation follows closely the concept developed by Bulatov et al. [7].  In literature, the mean GBEs for general GBs were proposed to scale with physical parameters like shear modulus (a 0 c 44 ) or Voight average shear modulus (a 0 µ), cohesive energy (E 0 /a 2 0 ), stacking fault energy (SFE, γ SF ) or their combinations [32,34,67]. The shear moduli and cohesive energy were rescaled by the lattice parameter (a 0 ) to give the same units as the GBE. The rationale behind the relation between the GBE and shear modulus µ, e.g., γ GB ≈ ka 0 µ (k, a coefficient) is from the Read-Shockley type of dislocation model [75]. There, the GBs with low Table 1. Surface energies for the (100) and (111) surface facets (γ S(100) and γ S(111) , respectively) and their ratios relative to that of Cu for the selected fcc metals. δ A/Cu GB(fit) is the gradient of the linear fitting of the DFT GBEs in Fig. 2 and Fig. S2. γ GB (δ A/Cu S(100) ) and γ GB (δ A/Cu S(111) ) are the predicted GBEs using the (100) and (111) surface energies, respectively. γ expt.
GB are the experimental GBEs (references indicated). All the DFT calculations correspond to the static state (0 K), whereas the experimental GBEs for the general GBs are obtained by linear extrapolation to 0 K (see Fig. S3 in SM).
misorientation angles are considered to be composed of arrays of dislocations whose energies are proportional to the shear modulus [32]. Therefore, the material dependent scaling factor δ that connects the GBEs in two materials is thought to be related to the ratio of a 0 c 44 or a 0 µ (denoted as δ A/B a 0 c 44 and δ A/B a 0 µ in the following). Holm et al. [32] showed that the ratios of both a 0 c 44 and a 0 µ are very close to the actual slope of the linear fit of the EAM GBEs for metals with low SFEs; while for metals with high SFEs like Al, the ratio of a 0 c 44 acts as a better scaling factor than that of a 0 µ.
However, Ratanaphan et al. [67] reported that the ratio of the cohesive energies (δ A/B E 0 /a 2 0 ) is much better indicator than the ratios of a 0 c 44 or a 0 µ in bcc metals. It is argued that the GBEs scale with the cohesive energy based on the broken bond model of GBE [76]. But in fcc metals, EAM results [29,32] indicate that the broken bond model does not give satisfactory prediction of GBEs, i.e., GBEs do not scale with E 0 /a 2 0 , nor one can use the ratio of E 0 /a 2 0 to correlate the GBEs in different fcc metals. DFT calculations also confirm that despite a general positive correlation between the GBE and the cohesive energy may exist, the overall correlation is weak [34]. As for the SFE, it is strongly related to the coherent twin boundary energy, γ SF ≈ 2γ tw , but correlates weakly with the general GBEs [32]. The above results indicate that the ratios of a 0 c 44 , a 0 µ, E 0 /a 2 0 and SFE for a pair of materials are not likely to give a good prediction of δ that can be unambiguously used to correlate the GBEs in the randomly chosen fcc materials. In Fig. 3, we compare δ A/Cu a 0 c 44 and δ A/Cu a 0 µ with δ A/Cu GB(fit) in the studied metals with available DFT, EAM and experimental data (Table S3 in SM). Indeed, it shows that their agreement is strongly material dependent. For example, in Rh and Ir, δ A/Cu a 0 c 44 and δ A/Cu a 0 µ are about two times larger than δ A/Cu GB(fit) .  [32,77]. The surface energies are from Refs. [37,38]. Numerical values are listed in Table S5.
In high-index GBs, the geometry near the boundary is less close packed and many bulk-like bonds are missing which resembles locally a surface-like packing. Because of that, we consider the surface energy of close packed surface facets as an alternative indicator of the GBEs. Indeed, our analysis indicates that the ratio of the low-index surface energies (δ A/B S ) gives a highly accurate prediction of δ. In Fig. 2, the (111) surface energies of pure metals locate approximatively on the same lines as the GBEs. For Al, Au, Ag, Ni, Pd, Pt, Co, and Rh, the ratios of the (111) surface energies (δ A/Cu S(111) ) are very close the actually slopes of the GBEs (δ A/Cu GB(fit) ) with mean deviation of ∼9%. The largest overestimation is found for Ir, ∼25%, which is still much better than the prediction based on shear moduli or cohensive energy (Fig. 3). Similar observations apply when using the (001) surface energies, see Table 1.
We emphasize here that in the above analysis, both tilt and twist types of GBs are included. Therefore, we may anticipate that for the formation energies of the so called general GBs should also correlate with the surface energies.
In Fig. 4, we compare the ratio of the experimental GBEs with the DFT calculated gradients (δ A/B GB(fit) ) and the ratio of the surface energies (δ A/Cu S(111) and δ A/Cu S(100) , respectively). Indeed, a good agreement is reached for both (111) and (001) surface energies. The above result suggests an efficient approach for predicting the GBEs, especially the general GB for which the DOFs are not properly defined. We illustrate the approach by taking Cu as the reference system with measured general GBE of γ expt.Cu GB . Then the general GBE of an fcc metal A can be predicted as γ S is the ratio of the surface energies of metal A and Cu. This ratio changes very weakly with temperature up to the room temperature (see SM) and thus the formula is expected to apply at both low and room temperature.
Following this approach, we predicted the 0 K GBEs of all metals considered here. Results are listed in Table 1. It turns out that the proposed surface energy-based scheme gives highly reliable predictions.
The above linear expression in terms of weight percent (wt.%) becomes where x and y are the weight percent of Cr and Ni contents, respectively. The variables are within the limits (11 x 30, 4 y 34, wt.%).
In summary, we explored the correlation between the GBEs in fcc metals with ab initio calculations. Our results demonstrated that the GBEs in fcc metals are strongly correlated, with a primary origin coming from the GB structure.
A material dependent parameter δ is expected to scale the GBEs of the same GB structure in a pair of fcc metals.
Here, we found that the ratio of the low-index surfaces can give a satisfactory estimation of δ. Using ab initio surface energies and the reference data in Cu, we successfully predicted the general GBEs in other pure fcc metals and in a complex solid solution alloy. The present work introduces a feasible method for the prediction of the GBEs using ab initio calculations. We envision that with more ab initio studies for the GBs with structures varying in the space of the five DOFs in a reference metal, using the surface energy-based scaling parameters as proposed in the present work, the GBEs and anisotropy in complex alloys can be readily predicted.  Table S1. All GBs are initiated from the tilt plane with certain misorientation angles. The schematics of the selected 10 GB structures are presented in Fig. S1.  S1. Schematics of the atomic GB structures before relaxation.

The calculated GBEs
The calculated GBEs for all studied metals are tabulated in Table S2. For comparison, the available DFT results in literature are also presented .  Table S3 shows the calculated lattice parameters, bulk modulus, elastic constants, and Voight average shear modulus for the studied fcc metals in the present work. Available theoretical (DFT [25] and EAM [26]) and experimental [27] data are also included for comparison. In general, the present results show a good agreement with other theoretical and experimental data. Table S3. Comparison of lattice parameters (a 0 ), bulk modulus (B 0 ), elastic constants (c 11 , c 12 , c 44 ), and Voight average shear modulus (µ) for fcc pure metals between the present and the previous works. All theoretical and experimental data correspond to the static conditions (0 K), except those experimental results at the room temperature marked by * .

Al
Au

Surface energy
For surface energy, extensively theoretical works have been employed for fcc metals [28][29][30][31][32][33][34][35]. In Table S4, we collect several works using different exchange-correlation functionals. For using the same exchange-correlation functional, it can be seen that the calculated surface energies show a good agreement with each other. To be consistent, we adopt the surface energies from the work with results for most fcc metals [28] and the other one for Co is taken from Ref. [32]. The corresponding values are in bold. 4. The correlation between GBEs in different metals Fig. S2 shows the pairwise comparison of the calculated GBEs and the (111) surface energies for the remaining elements, Ag, Au, Co, Rh, and Ir with Cu. Previous DFT GBEs (solid symbols) for tilt and twist GBs from Ref. [1] are included in the linear fitting. DFT surface energies are taken from Refs. [28,32].  Refs. [28,32]. Table S5. The pairwise comparison of the ratios of a 0 c 44 , a 0 µ, surface energies (γ S ), and δ A/Cu GB(fit) in fcc metals. The EAM, experimental, and VASP results are from Refs. [26], [27], and [1,28,32]
6. Estimation of δ A/Cu S at room temperature.
Following the work by Tyson et al. [47], we can estimate the change of δ A/Cu S as we go from 0 K to room temperature. Tyson et al. computed the surface energy at low temperature using the surface tension data in the liquid phase and the linear temperature dependence γ S (0) − γ S (T m ) ≈ RT m /A, connecting the surface energy at melting temperature T m and at 0 K. Here R is the gas constant and A is the surface area per mole of surface atoms. Writing this expression for temperature T and taking the difference, we arrive at γ S (T ) − γ S (0) ≈ −α · T/T m , where α = RT m /A. Thus the temperature dependence of δ A/Cu S (T ) in leading order in T can be written as where the γ Cu S (0) and γ A S (0) are the 0 K surface energies for Cu and metal A, respectively. Using the α and γ S (0) values for pure metals by Tyson et al. [47], we obtain the change of δ A/Cu S when going from 0 K to room temperature. The results for the present metals are listed in Table S6. The average change of ∆δ A/Cu S is ∼0.7%, which implies that the ratio of surface energies changes very weakly when increasing the temperature to 298 K.