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Research Papers

Topological Defects in Two-Dimensional Crystals: The Stress Buildup and Accumulation

[+] Author and Article Information
Zhigong Song

Applied Mechanics Laboratory,
Department of Engineering Mechanics,
Center for Nano and Micro Mechanics,
Tsinghua University,
Beijing 100084, China

Zhiping Xu

Applied Mechanics Laboratory,
Department of Engineering Mechanics,
Center for Nano and Micro Mechanics,
Tsinghua University,
Beijing 100084, China;
State Key Laboratory of Mechanics and
Control of Mechanical Structures,
Nanjing University of
Aeronautics and Astronautics,
Nanjing 210016, China
e-mail: xuzp@tsinghua.edu.cn

1Corresponding author.

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received May 12, 2014; final manuscript received June 4, 2014; accepted manuscript posted June 9, 2014; published online June 19, 2014. Editor: Yonggang Huang.

J. Appl. Mech 81(9), 091004 (Jun 19, 2014) (5 pages) Paper No: JAM-14-1211; doi: 10.1115/1.4027819 History: Received May 12, 2014; Revised June 04, 2014; Accepted June 09, 2014

Topological defects (TDs) arise in the growth process of two-dimensional (2D) materials, as well as after-growth heat treatment or irradiation. Our atomistic simulation results show that their mechanical modulation of material properties can be understood qualitatively through the theory of elasticity. We find that the in-plane lattice distortion and stress induced by experimentally characterized pentagon-heptagon (5|7) pairs or pentagon-octagon-pentagon (5|8|5) triplets can be captured by 2D models of dislocations or disclinations, although the out-of-plane distortion of the lattice reduces stress localization. Lineups of these TDs create nonlocal stress accumulation within a range of ∼10 nm. Interestingly, pileups of 5|7 and 5|8|5 defects show contrasting tensile and compressive buildups, which lead to opposite grain size dependence of the material strength. These findings improve our understandings of the mechanical properties of 2D materials with TDs, as well as the lattice perfection in forming large-scale continuous graphene films.

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Figures

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Fig. 1

(a) TDs in graphene and the lattice distortion they induce, including isolated pentagon and heptagon as positive and negative 60 deg disclinations, as well as the 5|7 pair and 5|8|5 triplet as an edge dislocation and a dislocation dipole in the hexagonal lattice. ((b) and (c)) Extended 5|7 and 5|8|5 TD pileups as GBs in polycrystalline graphene, illustrated as (b) atoms colored by the atomic energies and (c) edge dislocations.

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Fig. 2

Average in-plane principal stress σp = (σxx + σyy)/2 induced by single 5|7 and 5|8|5 TDs. The MD simulation results are compared with theoretical predictions for an edge dislocation and a combination of positive and negative disclinations. In the plots of theoretical predictions, the divergence within the cores of TDs is removed by truncating values higher than 20 and 0.5 GPa for 5|7 and 5|8|5 TDs, respectively.

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Fig. 3

Average in-plane principal stress distribution σp(x, y) around 5|7 and 5|8|5 defects in graphene, calculated from MD simulations without ((a) and (c)) and with ((b) and (d)) out-of-plane distortion. The values are plotted along the x ((a) and (b)) and y ((c) and (d)) axes in Fig. 2, with origins locating at the core of TDs.

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Fig. 4

The amplitudes of principal stresses plotted in the logarithmic scales to show the convergence of stress fields along both x and y direction. The results from 2D and 3D MD simulations are plotted in solid and dash lines, respectively.

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Fig. 5

(a) Illustration of stress buildup accumulation from TD pileups with length l. (b) Strength of polycrystalline graphene (Fig. 1(b)) with hexagonal grains and straight GBs consisting of 5|7 and 5|8|5 TD pileups, plotted as a function of the grain size or the GB length l. MD simulation results are plotted as symbols (circles for 5|7 pileups, squares for 5|8|5 pileups), that are fitted using the scaling relations derived from 2D elastic theory of dislocations.

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