dislocation_periodic_array calculation style

Lucas M. Hale, lucas.hale@nist.gov, Materials Science and Engineering Division, NIST.

Introduction

The dislocation_periodic_array calculation constructs an atomic system with a periodic array of dislocations configuration. A single dislocation is inserted into an otherwise perfect crystal, and the system is kept periodic in the two system box directions that are within the dislocation’s slip plane. The system is then statically relaxed with the atoms at the boundary perpendicular to the slip plane held fixed.

Version notes

• 2020-05-22: Notebook updated for iprPy version 0.10 and tested for hcp

• 2020-09-22: Notebook updated to reflect that calculation method has changed to now use atomman.defect.Dislocation. Setup and parameter definition cleaned up and streamlined.

• 2022-03-11: Notebook updated to reflect version 0.11.

Disclaimers

• NIST disclaimers

• This calculation was designed to be general enough to properly generate a dislocation for any crystal system but has not been fully tested yet for extreme cases.

Method and Theory

System orientation considerations

Properly constructing a periodic array of dislocations atomic configuration requires careful consideration of dislocation solutions and atomic system boundaries. Solutions for straight dislocations based on elasticity often follow the convention of using a Cartesian system ($$x', y', z'$$) in which the dislocation line is oriented along the $$z'$$-axis, and the slip plane taken to be the $$y'=0$$ plane. The dislocation’s Burgers vector, $$\vec{b}$$, is then in the $$x'z'$$-plane, with edge component in the $$x'$$-direction and screw component in the $$z'$$-direction. When the dislocation slips, the dislocation line will move in the $$x'$$-direction.

For any such dislocation solution, there will be a shearing along the slip plane resulting in disregistry, i.e. a relative displacement between the top and bottom halves. This disregistry has limits such that it is $$0$$ for $$x' \to -\infty$$ and $$\vec{b}$$ for $$x' \to +\infty$$.

Within an atomic system, the dislocation line should be aligned with one of the system’s box vectors making the dislocation infinitely long and initially perfectly straight. The slip plane can then be defined as containing that box vector and another one. This results in the third box vector being the only one with a component parallel to the slip plane’s normal.

For LAMMPS-based simulations, the most convenient orientation to use is to align the dislocation with the $$\vec{a}$$ box vector, and to define the slip plane as containing both $$\vec{a}$$ and $$\vec{b}$$. Given the limits that LAMMPS places on how system boxes can be defined, this results in favorable alignment of the system to the LAMMPS Cartesian system ($$x, y, z$$). The dislocation line will be along the $$x$$-axis, the slip plane normal parallel to the $$z$$-axis, and dislocation motion will be in the $$y$$ direction. Thus, the LAMMPS coordinates corresponds to a rotation of the theory coordinates such that $$x'=y, y'=z, z'=x$$.

Linear displacements solution

The simplest way to insert a dislocation is to cut the system in half along the slip plane and apply equal but opposite linear displacements, $$\vec{u}$$, to the two halves with end conditions

• $$\vec{u}(y=-\frac{Ly}{2}) = 0$$

• $$\vec{u}(y=\frac{Ly}{2}) = \pm \frac{\vec{b}}{2}$$

Applying these displacements results in a disregistry along the slip plane that ranges from $$0$$ to $$\vec{b}$$. While the two $$y$$ boundaries of the system both correspond to a perfect crystal, they are misaligned from each other by $$\frac{\vec{b}}{2}$$. A coherent periodic boundary along the $$\vec{b}$$ box vector can be established by adding or subtracting $$\frac{\vec{b}}{2}$$ from $$\vec{b}$$.

Note that with dislocations containing an edge component, a half-plane of atoms either needs to be inserted or removed to ensure boundary compatibility. Here, this is accomplished by always shifting $$\vec{b}$$ to be shorter in the $$y$$ direction, and removing excess atoms by identifying (near) duplicates.

Using dislocation solutions

A slightly more complicated, but ultimately more efficient, way of creating a periodic array of dislocations system is to combine the linear displacements solultion above with a more accurate linear elastic dislocation solution. The linear solution is used for the atoms at the free surfaces in the $$z$$ direction, and for ensuring periodicity across the $$\vec{b}$$ box vector direction. The linear elastic dislocation solution is then used for atoms in the middle of the system to construct an initial dislocation.