Dislocations in the dominant slip system in bcc metals have (111)/2 Burgers vectors and {110} slip planes. Here, we consider an edge dislocation with Burgers vector b = 1/2 [111] (along x-axis), slip plane normal [110] (along j-axis), and line direction [T 12] (along z-axis). To prepare the atomic configuration, we first create a perfect crystal with dimensions 30[111], 40[H0], 2[ 112] along the x-, y-, z-axes. We then remove one-fourth of the atomic layers normal to the y-axis to create two free surfaces, as shown in Figure 8(a).

We introduce an edge dislocation dipole into the simulation cell by displacing the positions of all atoms according to the linear elasticity solution of the displacement field of a dislocation dipole. To satisfy PBC, the displacement field is the sum of the contributions from not only the dislocation dipole

<№

(a)

Figure 8 (a) Schematics showing the edge dislocation dipole in the simulation cell. b is the dislocation Burgers vector of the upper dislocation. Atoms in shaded regions are removed. (b) Core structure of edge dislocation (at the center) and surface atoms in FS Ta after relaxation visualized by Atomeye.41 Atoms are colored according to their central-symmetry deviation parameter. Adapted from Li, J. In Handbook of Materials Modeling;Yip, S., Ed.; Springer: Dordrecht, 2005; pp 1051-1068; Mistake-free version at http://alum. mit. edu/www/liju99/Papers/05/Li05-2.31.pdf; Kelchner, C. L.; Plimpton, S. J.; Hamilton, J. C. Phys. Rev. B 1998, 58, 11085.

inside the cell, but also its periodic images. Care must be taken to remove the spurious term caused by the conditional convergence of the sum.26,40-42 Because the Burgers vector b is perpendicular to the cut — plane connecting the two dislocations in the dipole, atoms separated from the cut-plane by <|b|/2 in the x-direction need to be removed. The resulting struc­ture contains 21414 atoms. The structure is subse­quently relaxed to a local energy minimum with zero average stress. Because one of the two dislocations in the dipole is intentionally introduced into the vacuum region, only one dislocation remains after the relaxa­tion, as shown in Figure 8(b).

The dislocation core is identified by central symmetry analysis,13 which characterizes the degree of inversion-symmetry breaking. In Figure 8(b), only atoms with a central symmetry deviation (CSD) parameter larger than 1.5 A2 are plotted. Atoms with CSD parameter between 0.6 and 6 A2 appear at the center of the cell and are identified with the disloca­tion core. Atoms with a CSD parameter between 10 and 20 A2 appear at the top and bottom of the cell and are identified with the free surfaces.

The edge dislocation thus created will move along the x-direction when the shear stress axy exceeds a critical value. To compute the Peierls stress, we apply shear stress axy by adding external forces on surface atoms. The total force on the top surface
atoms points in the x-direction and has magnitude of Fx = axyLxLz. The total force on the bottom sur­face atoms has the same magnitude but points in the opposite direction. These forces are equally distributed on the top (and bottom) surface atoms. Because we have removed some atoms when creating the edge dislocation, the bottom surface layer has fewer atoms than the top surface layer. As a result, the external force on each atom on the top surface is slightly lower than that on each atom on the bottom surface.

We apply shear stress axy in increments of 1 MPa and relax the structure using the conjugate gradient algorithm at each stress. The dislocation (as identified by the core atoms) does not move for axy < 27 MPa but moves in the x-direction during the relaxation at axy = 28 MPa. Therefore, this simulation predicts that the Peierls stress of edge dislocation in Ta (FS potential) is 28 ± 1 MPa. The Peierls stress computed in this way can depend on the simulation cell size. Therefore, we will need to repeat this calculation for several cell sizes to obtain a more reliable prediction of the Peierls stress. There are other boundary conditions that can be applied to simulate disloca­tions and compute the Peierls stress, such as PBCs in both x — and j-directions,42 and the Green’s function boundary condition.44 Different boundary conditions have different size dependence on the numerical error of the Peierls stress.

The simulation cell in this study contains two free surfaces and one dislocation. This is designed to minimize the effect of image forces from the bound­ary conditions on the computed Peierls stress. If the surfaces were not created, the simulation cell would have to contain at least two dislocations so that the total Burgers vector content was zero. On application of the stress, the two dislocations in the dipole would move in opposite directions, and the total energy would vary as a function of their relative position. This would create forces on the dislocations, in addi­tion to the Peach-Koehler force from the applied stress, and would lead to either overestimation or underestimation of the Peierls stress. On the contrary, the simulation cell described above has only one dislocation, and as it moves to an equivalent lattice site in the x-direction, the energy does not change due to the translational symmetry of the lattice. This means that, by symmetry, the image force on the dislocation from the boundary conditions is iden­tically zero, which leads to more accurate Peierls stress predictions. However, when the simulation cell is too small, the free surfaces in the y-direction
and the periodic images in the x-direction can still introduce (second-order) effects on the critical stress for dislocation motion, even though they do not produce any net force on the dislocation.