Core energies of dislocations in bcc metals

Accurate methods and an efficient workflow for computing and documenting dislocation core energies are developed and applied to $\frac{1}{2}\langle 111\rangle$ and $\langle 100\rangle$ dislocations in five body-centered cubic (bcc) metals W, Ta, V, Mo, and $\alpha$-Fe represented by 13 model interatomic potentials. For each dislocation type, dislocation core energies are extracted for a large number of dislocation characters thoroughly sampling the entire 2-space of crystallographic line orientations of the bcc lattice. Of particular interest, core energies of the $\frac{1}{2}\langle 111\rangle \left\{110\right\}$ dislocations are found to be distinctly asymmetric with respect to the sign of the character angle, whereas core energies of $\langle 100\rangle \left\{110\right\}$ junction dislocations exhibit marked cusps for line orientations vicinal to the closed-packed $\langle 111\rangle$ directions. Our findings furnish substantial insights for developing accurate models of dislocation core energies employed in mesoscale dislocation dynamics simulations of crystal plasticity.

Figure 1

Core energy of a $\frac{1}{2}\langle 111\rangle$ dislocation in bcc tungsten. (a) Core energies as a function of zonal plane and character angle $\theta$ for $r_c=1b$. Core energy values are averages over values computed in supercells of four different sizes (dipole separation distances). Each error bar is the standard error from the corresponding average. The dashed line is the least-squares fit of the edge-screw model [Eq. (8)] to the core energy data computed for all character angles and all three zonal planes. Inversion symmetry of the bcc lattice limits the range of distinct dislocation character angles to $-{90}^{\circ }\le \theta <+{90}^{\circ }$. (b) 3D polar plot of core energy with the pole aligned along the Burgers vector. A slightly different value of the reference radius $r_c=0.7b$ was used to sharpen the features of the core energy function plotted on the unit sphere $(\varphi ,\theta )$ of dislocation orientations, where $0\le \varphi <{360}^{\circ }$ is the zonal angle, and $\theta$ is the character angle folded into $0\le \theta <{180}^{\circ }$. The cutout slice is a symmetry-irreducible zone $0\le \varphi <{30}^{\circ }$ containing exactly one plane for each of the $\left\{110\right\}$, $\left\{112\right\}$, and $\left\{123\right\}$ families of zonal planes.

Figure 2

(a) Asymmetry of the core energy of a $\frac{1}{2}\langle 111\rangle \left\{\overline{1}10\right\}$ dislocation as a function of the character angle $\theta$ for a core radius $r_c=1b$. The inset shows variations of the difference $E^{\mathrm{core}}\left(\theta \right)-E^{\mathrm{core}}(-\theta )$ in core energies between positive and negative character angles. (b) Schematic of the (110) plane in a bcc crystal with lattice parameter $a_0$. The asymmetry of dislocation lines $\mathit{\xi }$ with positive ($+\theta$) and negative ($-\theta$) character angles with respect to the Burgers vector, e.g., $\mathit{b}=\frac{1}{2}\left[1\overline{1}1\right]$, directly results from the crystallography of the (110) plane.

Figure 3

Core energy of $\langle 100\rangle$ binary junctions as functions of the habit plane and character angle $\theta$ in (a) bcc W using the potential of Zhou et al. [20] and (b) bcc Ta using the potential of Li et al. [23]. Core energies are seen to increase monotonically from edge to screw characters in all habit planes, expect for the $\left\{110\right\}$ plane that shows a cusp at two dislocation line orientations aligned with two $\langle 111\rangle$ directions, i.e. for character angles of $\theta =\pm 54.{74}^{\circ }$.

Figure 4

Examples of core energies extracted using different treatments of the elastic energy. (a) Comparison of the core energy in bcc W for the interatomic potential of Zhou et al. [20], extracted using classical isotropic elasticity [Eq. (6)] for $r_c=b$ and nonsingular isotropic elasticity [Eq. (11)] with $a=1b$ ($\mu =159.00$ GPa, $\nu =0.28$). (b) Comparison of the core energy in bcc Ta using the interatomic potential of Li et al. [23], extracted using classical anisotropic elasticity for $r_c=b$, anisotropic elasticity with the isotropic nonsingular kernel in Eq. (10) for $a=b$, and anisotropic elasticity with the cubic nonsingular kernel in Eq. (16) for $l=b$ ($C_{11}=248.92$ GPa, $C_{12}=144.27$ GPa, $C_{44}=86.53$ GPa). (c) Same as in (b) but for the $\langle 100\rangle \left\{110\right\}$ dislocation.

Figure 5

Core energies for the $\frac{1}{2}\langle 111\rangle$ dislocation extracted for all interatomic potentials considered in this work (see Table 1). Values of the core energies are plotted for the classical anisotropic elasticity model with cutoff core treatment at $r_c=1b$. (a) bcc W, Zhou et al. [20]; (b) bcc W, Olsson [21]; (c) bcc W, Juslin et al. [22]; (d) bcc Ta, Li et al. [23]; (e) bcc Ta, Zhou et al. [24]; (f) bcc V, Olsson et al. [21]; (g) bcc V, Han et al. [25]; (h) bcc Mo, Ackland and Thetford [26]; (i) bcc Mo, Zhou et al. [24]; (j) bcc $\alpha$-Fe, Ackland et al. [27]; (k) bcc $\alpha$-Fe, Mendelev et al. [28]; (l) bcc $\alpha$-Fe, Zhou et al. [24]; and (m) bcc $\alpha$-Fe, Chamati et al. [29].

Figure 6

Core energies for the $\langle 100\rangle$ dislocation extracted for all interatomic potentials considered in this work (see Table 1). Values of the core energies are plotted for the classical anisotropic elasticity model with cutoff core treatment at $r_c=1b$. (a) bcc W, Zhou et al. [20]; (b) bcc W, Olsson [21]; (c) bcc W, Juslin et al. [22]; (d) bcc Ta, Li et al. [23]; (e) bcc Ta, Zhou et al. [24]; (f) bcc V, Olsson et al. [21]; (g) bcc V, Han et al. [25]; (h) bcc Mo, Ackland and Thetford [26]; (i) bcc Mo, Zhou et al. [24]; (j) bcc $\alpha$-Fe Ackland et al. [27]; (k) bcc $\alpha$-Fe, Mendelev et al. [28]; (l) bcc $\alpha$-Fe, Zhou et al. [24]; and (m) bcc $\alpha$-Fe, Chamati et al. [29].