Continuum model for the core of a straight mixed dislocation

Max Boleininger and Sergei L. Dudarev

Phys. Rev. Materials 3, 093801 – Published 3 September 2019

Abstract

Linear elasticity theory predicts a divergent strain field at the dislocation core, resulting from the continuum approximation breaking down at the atomic scale. We introduce a minimum model that includes elastic interactions and discrete lattice periodicity, and derive a set of equations that treat the core of an edge dislocation from a solely geometric perspective. We find an analytical formula for the displacement field of a straight dislocation of arbitrary mixed character, and we predict that the dislocation core widens as the screw character becomes more dominant. This finding is in qualitative and quantitative agreement with atomistic simulations of mixed dislocations in tungsten. The theory is based on a continuum form of the multistring Frenkel-Kontorova model, which is a nearest-neighbor model for atomic bonding that also takes into account the discreteness of the crystal lattice. Thus, we circumvent the need to use adjustable parameters in the treatment of a dislocation core.

Received 3 May 2019

DOI:https://doi.org/10.1103/PhysRevMaterials.3.093801

Published by the American Physical Society

Physics Subject Headings (PhySH)

Crystal defects Disclinations & dislocations Elasticity Line defects Plasticity

Body-centered cubic Metals

Molecular dynamics

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Max Boleininger^* and Sergei L. Dudarev^†

CCFE, Culham Science Centre, UK Atomic Energy Authority, Abingdon, Oxfordshire OX14 3DB, United Kingdom

^*max.boleininger@ukaea.uk
^†sergei.dudarev@ukaea.uk

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Vol. 3, Iss. 9 — September 2019

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Images

Figure 1
Local bonding environment of an atom at $r_{j, n}$ in the multistring Frenkel-Kontorova model. The Burgers vector is collinear with $\hat{z}$ . The atom interacts quadratically with neighboring atoms in the same string at $r_{j, n \pm 1}$ (the interaction is shown by springs). Neighboring atoms in the surrounding strings at $r_{j + h, n}$ contribute a periodic sinusoidal interaction, representing the periodicity of the lattice described in a mean-field picture. One out of six nearest-neighbor string vectors $h$ is shown here.
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Figure 2
bcc lattice viewed in a direction collinear with an $a / 2 [111]$ -type Burgers vector. Solid circles represent MSFK strings in a local bulklike environment, and bordered circles represent the strings adjacent to the glide surface. The glide surface of the continuum model $\partial Ω$ , the cross section of which is shown by the dashed line, is situated at midpoints of the set of neighbor vectors ${h}$ connecting pairs of strings at opposite sides of the glide surface (arrows).
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Figure 3
Triclinic simulation cell (dashed line) containing a dipole of mixed dislocations inclined at a character angle of $θ$ , as seen looking along the $- y$ direction. The coordinate system is defined in Eq. (32). The Burgers vector is not drawn to scale.
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Figure 4
Dislocation core width plotted as a function of the dislocation character angle, measured as the full width at half-maximum (FWHM) along the Burgers vector direction. The molecular-dynamics (MD) core widths are extracted from the first atomic rows immediately above or below the glide plane, which are under compression or tension, respectively.
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Figure 5
Comparison between the strain fields derived from molecular dynamics (MD) and predicted by the MSFK and Volterra continuum models for the $a / 2 [111] (10 \bar{1})$ mixed dislocation in tungsten at a few selected dislocation angles. The strain of the atomic row under tension is plotted with a negative sign. The atomistic structure (inset), colored according to the potential energy of atoms and seen in the normal direction to the glide plane, reveals the formation of kinks at large dislocation character angles. The atomistic structure is visualized with ovito [36]. Note that the core width is measured in the direction of the Burgers vector of the dislocation.
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Figure 6
MSFK prediction of the strain field assuming a mixed dislocation with staircase line shape (dotted), as an approximation to the kinked line found in atomistic simulations. Pictured here are the strain fields of two kinked lines centered at differing points; see the main text. The MSFK prediction exhibits a similar substructure to the atomistic strain field at an equivalent inclination, providing clear evidence of kink formation.
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