More About Periodic Boundary Conditions

Abstract

We have already used periodic boundary conditions (PBC) for the static and dynamic simulations described in Chapters 2 and 3. There, PBC were applied along one or two directions of the simulation cell. Application of PBC in all three directions holds an important advantage when one’s goal is to examine the behavior in the bulk: under fully three-dimensional (3D) PBC, the simulated solid can be free of any surfaces. By comparison, the simulations discussed in the previous chapters all contained free surfaces or artificial interfaces in the directions where PBC were not applied. Full 3D PBC are easy to implement in an atomistic simulation through the use of scaled coordinates. However, there are important technical issues specific to simulations of lattice dislocations. First, a fully periodic simulation cell can accommodate only such dislocation arrangements whose net Burgers vector is zero. Thus, the minimal number of dislocations that can be introduced in a periodic supercell is two, i.e. a dislocation dipole. Two dislocations forming a dipole are bound to interact with each other, as well as with their periodic images. Associated with these interactions are additional strain, energy, and forces whose effects can “pollute” the calculated results. The good news is that, in most cases, the artifacts of PBC can be quantified through the use of linear elasticity theory so that physical properties of dislocations can be accurately extracted. Given the simplicity and robustness of PBC, the extrawork required to extract physical results is well worth it. This chapter describes how to evaluate and eliminate the artifacts that inevitably appear when 3D PBC are used for atomistic simulations of dislocations. In the following three sections, we show how to take full advantage of PBC when one wants to calculate the displacement field induced by a dislocation (Section 5.1), the dislocation’s core energy (Section 5.2) and Peierls stress (Section 5.3). The common theme for all three case studies is an attempt to construct a solution of the elasticity equations in a periodic domain by superimposing a periodic array of solutions of an infinite domain.