dislocation_monopole calculation style

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

Introduction

The dislocation_monopole calculation style calculation inserts a dislocation monopole into an atomic system using the anisotropic elasticity solution for a perfectly straight dislocation. The system is divided into two regions: a boundary region at the system’s edges perpendicular to the dislocation line, and an active region around the dislocation. After inserting the dislocation, the atoms within the active region of the system are relaxed while the positions of the atoms in the boundary region are held fixed at the elasticity solution. The relaxed dislocation system and corresponding dislocation-free base systems are retained in the calculation’s archived record. Various properties associated with the dislocation’s elasticity solution are recorded in the calculation’s results record.

Version notes

  • 2018-09-25: Notebook added

  • 2019-07-30: Method and Notebook updated for iprPy version 0.9.

  • 2020-09-22: Notebook updated to reflect that calculation method has changed to now use atomman.defect.Dislocation.

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

Additional dependencies

Disclaimers

  • NIST disclaimers

  • This calculation method holds the boundary atoms fixed during the relaxation process which results in a mismatch stress at the border between the active and fixed regions that interacts with the dislocation. Increasing the distance between the dislocation and the boundary region, i.e. increasing the system size, will reduce the influence of the mismatch stresses.

  • The boundary atoms are fixed at the elasticity solution, which assumes the dislocation to be compact (not spread out, dissociated into partials) and to remain at the center of the system. An alternate simulation design or boundary region method should be used if you want accurate simulations of dislocations with wide cores and/or in which the dislocation moves long distances.

  • The calculation allows for the system to be relaxed either using only static energy/force minimizations or with molecular dynamic steps followed by a minimization. Only performing a static relaxation is considerably faster than performing a dynamic relaxation, but the dynamic relaxation is more likely to result in a better final dislocation structure.

Method and Theory

Stroh theory

A detailed description of the Stroh method to solve the Eshelby solution for an anisotropic straight dislocation can be found in the atomman documentation.

Orientation

One of the most important considerations in defining an atomistic system containing a dislocation monopole system is the system’s orientation. In particular, care is needed to properly align the system’s Cartesian axes, \(x, y, z\), the system’s box vectors, \(a, b, c\), and the Stroh solution’s axes, \(u, m, n\).

  • In order for the dislocation to remain perfectly straight in the atomic system, the dislocation line, \(u\), must correspond to one of the system’s box vectors. The resulting dislocation monopole system will be periodic along the box direction corresponding to \(u\), and non-periodic in the other two box directions.

  • The Stroh solution is invariant along the dislocation line direction, \(u\), thereby the solution is 2 dimensional. \(m\) and \(n\) are the unit vectors for the 2D axis system used by the Stroh solution. As such, \(u, m\) and \(n\) are all normal to each other. The plane defined by the \(um\) vectors is the dislocation’s slip plane, i.e. \(n\) is normal to the slip plane.

  • For LAMMPS simulations, the system’s box vectors are limited such that \(a\) is parallel to the \(x\)-axis, and \(b\) is in the \(xy\)-plane (i.e. has no \(z\) component). Based on this and the previous two points, the most convenient, and therefore the default, orientation for a generic dislocation is to

    • Make \(u\) and \(a\) parallel, which also places \(u\) parallel to the \(x\)-axis.

    • Select \(b\) such that it is within the slip plane. As \(u\) and \(a\) must also be in the slip plane, the plane itself is defined by \(a \times b\).

    • Given that neither \(a\) nor \(b\) have \(z\) components, the normal to the slip plane will be perpendicular to \(z\). From this, it naturally follows that \(m\) can be taken as parallel to the \(y\)-axis, and \(n\) parallel to the \(z\)-axis.

Calculation methodology

  1. An initial system is generated based on the loaded system and uvw, atomshift, and sizemults input parameters. This initial system is saved as base.dump.

  2. The atomman.defect.Stroh class is used to obtain the dislocation solution based on the defined \(m\) and \(n\) vectors, \(C_{ij}\), and the dislocation’s Burgers vector, \(b_i\).

  3. The dislocation is inserted into the system by displacing all atoms according to the Stroh solution’s displacements. The dislocation line is placed parallel to the specified dislocation_lineboxvector and includes the Cartesian point (0, 0, 0).

  4. The box dimension parallel to the dislocation line is kept periodic, and the other two box directions are made non-periodic. A boundary region is defined that is at least bwidth thick at the edges of the two non-periodic directions, such that the shape of the active region corresponds to the bshape parameter. Atoms in the boundary region are identified by altering their integer atomic types.

  5. The dislocation is relaxed by performing a LAMMPS simulation. The atoms in the active region are allowed to relax either with nvt integration followed by an energy/force minimization, or with just an energy/force minimization. The atoms in the boundary region are held fixed at the elastic solution. The resulting relaxed system is saved as disl.dump.

  6. Parameters associated with the Stroh solution are saved to the results record.