Statistical perspective on embrittling potency for intergranular fracture

Embrittling potency is a thermodynamic metric that assesses the influence of solute segregation to a grain boundary (GB) on intergranular fracture. Historically, authors of studies have reported embrittling potency as a single scalar value, assuming a single segregation site of importance at a GB and a particular cleavage plane. However, the topography of intergranular fracture surfaces is not generally known a priori. Accordingly, in this paper, we present a statistical ensemble approach to compute embrittling potency, where many free surface (FS) permutations are systematically considered to model fracture of a GB. The result is a statistical description of the thermodynamics of GB embrittlement. As a specific example, embrittling potency distributions are presented for Cr segregation to sites at two Ni $\langle 111\rangle$ symmetric tilt GBs using atomistic simulations. We show that the average embrittling potency for a particular GB site, considering an ensemble of FS permutations, is not equal to the embrittling potency computed using the lowest energy pair of FSs. A mean GB embrittlement is proposed, considering both the likelihood of formation of a particular FS and the probability of solute occupancy at each GB site, to compare the relative embrittling behavior of two distinct GBs.

Figure 1

(a) Lattice orientations for both top (red) and bottom (green) regions illustrating the grain boundary (GB) construction. Cr is substituted into a GB site, and energy minimization is performed to obtain $E_i^{\mathrm{GB}}$. Then Cr is swapped from the GB into the bulk, and energy minimization is performed to obtain $E^{\mathrm{Bulk}}$. (b) Example of the free surface topography permutation process with a fictitious GB with three atoms at the GB plane in one repeating unit cell. Decisions for an atom to move up or down during fracture are made within a single GB unit cell, and then those decisions are repeated in all other repeating units.

Figure 2

(a) Minimum energy structure of the Σ7 {213} and Σ21 {415} symmetric tilt grain boundaries (STGBs) in Ni. Grain boundary (GB) sites are numbered for reference and colored by {111} atomic layer to identify structural units. The structural unit nomenclature was developed by Wang et al. [39]. (b) Map of Cr solute segregation energies for Σ7 {213} and Σ21 {415} STGBs. Units are in electronvolts. Negative values indicate favorable solute segregation, while positive values indicate antisegregation. Gray atom sites are not evaluated.

Figure 3

Segregation energy vs distance from the grain boundary (GB) plane along $\langle 110\rangle$ directions for the Σ7 {213} symmetric tilt GB (STGB). The cutoff criterion is indicated above the dashed black line. Atoms in the figure inlay are colored by different {111} planes, as shown in Fig. 2.

Figure 4

Distributions of embrittling potencies for (a) Σ7 {213} symmetric tilt grain boundary (STGB) sites with a detailed histogram for site 9 and (b) Σ21 {415} STGB sites with a detailed histogram for site 5. Distributions are represented compactly as box-and-whisker plots, with plus symbols denoting statistical outliers. The dots indicate embrittling potencies associated with the relaxed minimum energy free surfaces.

Figure 5

Histograms showing relaxed free surface (FS) energies for FS topographies of the (a) Σ7 {213} and (b) Σ21 {415} symmetric tilt grain boundaries (STGBs). The distributions generally skew left toward lower FS energies. (c) Overlayed histograms of FS energies for fracture of the Σ7 {213} STGB. The periodic histogram includes the periodic grain boundary (GB) ensemble shown in Fig. 3, while the expanded histogram shows the periodic expanded ensemble that includes the GB plane atomic sites, as well as three atomic sites from the adjacent atomic plane above it. Note that the ranges of the FS energies are very similar.

Figure 6

The process of creating the mean embrittlement. Each site at the grain boundary (GB) begins with a distribution of embrittling potencies (left). Following an ensemble averaging procedure over many free surfaces, each site contains a single ensemble averaged embrittling potency (middle). Then an average is performed over all sites considering solute occupancy to obtain a mean embrittlement.

Figure 7

Examples of Weibull survival functions with different $k$ values for the Σ7 {213} symmetric tilt grain boundary (STGB). The red arrow indicates increasing $k$. Higher $k$ values result in lower weights for higher energy free surfaces.

Figure 8

The ensemble average embrittling potency $\langle \mathrm{\Delta }{E}_{\mathrm{b},\mathrm{i}}\rangle$ in electronvolts for each site $\mathrm{i}$ in the (a) Σ7 {213} symmetric tilt grain boundary (STGB) and the (b) Σ21 {415} STGB for the periodic grain boundary (GB) ensemble. Sites at the GB plane have a negative $\langle \mathrm{\Delta }{E}_{\mathrm{b},\mathrm{i}}\rangle$ (strengthening), while all other sites surrounding the GB layer have a positive $\langle \mathrm{\Delta }{E}_{\mathrm{b},\mathrm{i}}\rangle$ (embrittling). In these examples, $k=250$ for the purpose of weighting site embrittling potencies. Flat neighboring ensemble for (c) $\mathrm{\Sigma }7\left\{213\right\}$ and (d) $\mathrm{\Sigma }21\left\{415\right\}$ STGBs. (e) and (f) Difference between the flat neighboring and periodic GB ensembles for each site.

Figure 9

Ensemble average embrittling potencies for (a) sites 1–9 in the Σ7 {213} symmetric tilt grain boundary (STGB) and (b) sites 1–5 in the Σ21 {415} STGB calculated via Eq. (6) plotted against effective number of free surface topographies. Note that, for $k=250$, the Σ7 {213} STGB has 73 effective surfaces, while the Σ21 {415} STGB has 6. Gray lines mark the trajectories of sites off the grain boundary (GB), which are generally embrittling, as corroborated by Fig. 8.