Cohesive and magnetic properties of grain boundaries in bcc Fe with Cr additions

Structural, cohesive, and magnetic properties of two symmetric $\Sigma 3\left(111\right)$ and $\Sigma 5\left(210\right)$ tilt grain boundaries (GBs) in pure bcc Fe and in dilute FeCr alloys are studied from first principles. Different concentrations and positions of Cr solute atoms are considered. We found that Cr atoms placed in the GB interstice enhance the cohesion by $0.5–1.2\text{J}/{\text{m}}^2$. Substitutional Cr in the layers adjacent to the boundary shows anisotropic effect on the GB cohesion: it is neutral when placed in the (111) oriented Fe grains and enhances cohesion by $0.5\text{J}/{\text{m}}^2$ when substituted in the boundary layer of the (210) grains. The strengthening effect of the Cr solute is dominated by the chemical component of the adhesive binding energy. Our calculations show that unlike the free iron surfaces, Cr impurities segregate to the boundaries of the Fe grains. The magnetic moments on GB atoms are substantially changed and their variation correlates with the corresponding relaxation pattern of the GB planes. The moments on Cr additions are two to four times enhanced in comparison with that in a Cr crystal and are antiparallel to the moments on the Fe atoms.

Figure 1

(Color online) Side view of the supercells representing $\Sigma 3\left(111\right)$ and $\Sigma 5\left(210\right)$ boundaries between grains in bcc Fe consisting of 30 and 40 Fe atoms, respectively. The lighter and darker balls mark the atoms belonging to two different planes. The right-hand side panel shows the layers stacking in the clean, relaxed Fe $\Sigma 5\left(210\right)$ slab. The open circles indicate the positions of the Cr addition at the GB interstice. The substitutional positions in different grain layers are labeled by the numbers. Lower panels show top view of the unrelaxed $1\times 1$ supercells taken in the cross-section plane passing through the GB (broken line). The “a” and “b” on the $\Sigma 5$ top-view panel label two different sites in the GB interstice. They can be related to the octahedral sites in a bcc unit cell with site “a” placed in the middle of the edge formed by two adjacent {001} faces and site “b” at the center of the {001} face.

Figure 2

(Color online) Schematic diagram illustrating the slab configurations used to analyze different contributions to the GB adhesive energy for a substitutional impurity atom represented by dark blue balls. The white balls mark an empty space remaining after either the impurity or host atom is removed. Upper panels show, respectively (from left to right), the GB slabs: (a) the GB relaxed with an impurity atom; (b) the GB with host atoms frozen in the relaxed configuration while the impurity is removed; (c) the GB with atoms frozen in pure GB configuration and a hole for impurity insertion created by the removed host atom; and (d) the relaxed clean GB. Note that the frozen grain configurations in (b) and (c) are different. The lower panels show the corresponding slabs representing the free surfaces. The total energy differences of the corresponding GB and FS slabs give ${\gamma }_f$ quantities [cf. Eq. (1)], standing by the vertical arrows, which are used to calculate different components of the strengthening energy, $\Delta E_{\text{SE}}$ [cf. Eq. (2)]. The chemical, $\Delta E_C$, mechanical, $\Delta E_M$, and host removal, $\Delta E_R$, energy components are defined by the difference between the neighboring pairs of ${\gamma }_f$, as indicated by the labels over the vertical arrows. For an interstitial impurity there is no removal of host atom and thus the mechanical contribution is given by the difference of ${\gamma }_f^{\text{frz}}$ and ${\gamma }_f^{\text{cln}}$.

Figure 3

(Color online) Relaxations of the interplanar spacing in the Fe grains near the boundary. The (111) and (210) interplanar distances in the bulk truncated Fe are 0.822 and $0.636\text{Å}$, respectively. Also shown is the effect of high (areal) concentration of the solute Cr atoms (cf. Sec. 3B1) placed in the GB interstice (Int) or in the substitutional sites of different layers (L) across the GB slab. The BL denotes the bulk Fe layer.

Figure 4

(Color online) Magnetic moment on Fe atoms at various layers in the vicinity of the GB (cf. Fig. 1). Horizontal chain line marks the magnetic moment of the bulk Fe. The effect of a monolayer of the solute Cr atoms (cf. Sec. 3B3) placed in the interstitial (Int) or substitutional sites of different layers (L) is shown as well.

Figure 5

(Color online) Change in the excess volume per unit area of the Fe grains caused by Cr impurity placed in the GB interstice (Int) or substituted in different Fe layers (L) across the boundaries (cf. Fig. 1). The BL labels bulk (central) layer of the grain with Cr atom [L7 for the $\Sigma 3\left(111\right)$, and L9 for the $\Sigma 5\left(210\right)$ oriented slabs]. The horizontal chain line marks the calculated excess volume for the clean GB.

Figure 6

(Color online) Strengthening energy (top panels) and its chemical, mechanical, and host removal energy components for two GBs in iron doped with the Cr impurities placed accordingly at two different concentrations in the interstitial or substitutional sites across the GB slab. For explanation of the labeling of the impurity position, cf. Fig. 5. All energies are in $\text{J}/{\text{m}}^2$.

Figure 7

(Color online) Energy of segregation of the Cr solute in different layers (L) at the $\Sigma 3\left(111\right)$ and $\Sigma 5\left(210\right)$ boundaries in iron for small Cr concentration.

Figure 8

(Color online) Magnetic moment on the Cr solute atom placed at the GB interstice (Int) or substitutionally in different layers (L) at the two different boundaries in iron. Results for two different concentrations are shown.

Figure 9

(Color online) Correlation between the magnetic moment and the exchange splitting for the Fe atoms at the $\Sigma 3\left(111\right)$ and $\Sigma 5\left(210\right)$ GBs. The data points represent all examined cases of the clean and doped GBs calculated in the $1\times 1$ supercell. The magnetic moment on the bulk Fe atom is $2.24{\mu }_B$.