Volume of a dislocation network

Max Boleininger, Sergei L. Dudarev, Daniel R. Mason, and Enrique Martínez

Phys. Rev. Materials 6, 063601 – Published 8 June 2022

Abstract

We derive a simple analytical line integral expression for the relaxation volume tensor of an arbitrary interconnected dislocation network. This quantity determines the magnitude of dislocation contribution to the dimensional changes and volumetric swelling of a material, and highlights the fundamental dual role of dislocations as sources of internal strain as well as carriers of plastic deformation. To illustrate applications of the method, we compute the relaxation volume of a stacking fault tetrahedron, a defect commonly occurring in fcc metals; the volume of an unusual tetrahedral configuration formed by the $(a / 2) 〈 111 〉$ and $a 〈 001 〉$ dislocations in a bcc metal; and estimate the relative contribution of extended dislocations to the volume relaxation of heavily irradiated tungsten.

Received 15 March 2022
Accepted 13 May 2022

DOI:https://doi.org/10.1103/PhysRevMaterials.6.063601

Physics Subject Headings (PhySH)

Crystal defects Defects Disclinations & dislocations Elastic deformation Elasticity Plastic deformation Radiation damage Stacking faults Stress Vacancies

Crystalline systems Metals

Materials modeling Molecular dynamics Multiscale modeling

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Max Boleininger ^*, Sergei L. Dudarev ^†, and Daniel R. Mason ^‡

UK Atomic Energy Authority, Culham Centre for Fusion Energy, Oxfordshire OX14 3DB, United Kingdom

Enrique Martínez ^§

Department of Materials Science and Engineering, Clemson University, Sirrine Hall, Clemson, South Carolina 29634, USA

^*max.boleininger@ukaea.uk
^†sergei.dudarev@ukaea.uk
^‡daniel.mason@ukaea.uk
^§enrique@clemson.edu

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Vol. 6, Iss. 6 — June 2022

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Images

Figure 1
(a) Example of a closed dislocation network consisting of line segments $G_{i}$ with Burgers vectors $b (G_{i})$ (arrows). Triangles indicate line sense. (b) The same network represented as a superposition of closed loops $Γ_{i}$ . Also given is the relation between area vectors $A (Γ_{i})$ and $A (G_{i})$ . (c) Closed loops are split into their constitutive line segments, which are then grouped according to whether they appear uniquely or not, into sets $S_{U}$ and $S_{N}$ , respectively. The decomposition is used to prove that all three representations yield the same dipole tensor.
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Figure 2
Three example closures of an extended dislocation network consisting of two dislocation dipoles in a box with periodic boundary conditions. The network is closed by augmenting the dislocation lines with extra segments, showed by dotted blue lines, that connect the dislocation line ends, with line sense and Burgers vector chosen in accordance with the Burgers vector conservation law at junctions. As the closure lines cancel with themselves upon the repetition of the motif along the periodic cell vector $c$ , all the examples of closure are equivalent.
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Figure 3
(a) Arrangement of the faulted Frank loop and stacking fault tetrahedron corner points in the one-eighth of an fcc unit cell with the edge length of $a / 2$ . Points labeled by Greek letters are at midpoints of tetrahedral faces opposite to points labeled by the respective Roman letters. (b)–(d) Transformation of the faulted Frank loop to a stacking fault tetrahedron parametrized by the reaction coordinate $λ$ . Burgers vectors are described using the Thompson vector notation.
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Figure 4
(a) Sketch of a tetrahedral defect structure formed by dislocations with $(a / 2) 〈 111 〉$ (blue) and $a 〈 100 〉$ (red) Burgers vectors in a bcc metal. The configuration resembles a stacking fault tetrahedron, but involves no stacking faults. The topology of this dislocation configuration is not equivalent to that of a dislocation loop. Coordinates of points A, B, C, and D are given in Table 3. (b) Dislocation configuration shown in (a) represented as an equivalent superposition of two $(a / 2) 〈 111 〉$ and one $a 〈 100 〉$ loop.
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Figure 5
Dislocation networks extracted from simulated microstructure of irradiated tungsten, showed in an unwrapped representation. In this example, the dislocation microstructure is found to evolve from dispersed loops at low dose, to extended dislocation networks at intermediate dose, to a large closed interstitial loop at high dose. Networks with net interstitial and vacancy content are colored red and blue, respectively. Structures (d) and (e) contain extended networks, shown periodically repeated along $x$ direction (gray lines), with segment end points marked by black dots. Dislocations of $(a / 2) 〈 111 〉$ and $a 〈 100 〉$ type are drawn with solid and dotted lines, respectively.
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Figure 6
(a) The relaxation volume of the dislocation network and the total change of volume of the simulation cell, expressed in the units of atomic volume $Ω_{0}$ , shown as a function of dose. (b) Plot showing if the dislocation network is extended or not as a function of dose. The discontinuity in the computed relaxation volume occurs shortly after the dislocation network becomes extended, suggesting that the change of the relaxation volume is associated with the change in the topology of the network. (c) Vacancy concentration, in the units of vacancies per lattice site, shown as a function of dose.
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