Local curvature and stability of two-dimensional systems

Jie Guan, Zhongqi Jin, Zhen Zhu, Chern Chuang, Bih-Yaw Jin, and David Tománek

Phys. Rev. B 90, 245403 – Published 1 December 2014

Abstract

We propose a fast method to determine the local curvature in two-dimensional (2D) systems with arbitrary shape. The curvature information, combined with elastic constants obtained for a planar system, provides an accurate estimate of the local stability in the framework of continuum elasticity theory. Relative stabilities of graphitic structures including fullerenes, nanotubes, and schwarzites, as well as phosphorene nanotubes, calculated using this approach, agree closely with ab initio density functional calculations. The continuum elasticity approach can be applied to all 2D structures and is particularly attractive in complex systems with known structure, where the quality of parameterized force fields has not been established.

Received 27 March 2014
Revised 13 November 2014

DOI:https://doi.org/10.1103/PhysRevB.90.245403

Authors & Affiliations

Jie Guan¹, Zhongqi Jin², Zhen Zhu¹, Chern Chuang³, Bih-Yaw Jin⁴, and David Tománek^1,*

¹Physics and Astronomy Department, Michigan State University, East Lansing, Michigan 48824, USA
²Chemistry Department, Michigan State University, East Lansing, Michigan 48824, USA
³Department of Chemistry, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA
⁴Department of Chemistry and Center for Emerging Material and Advanced Devices, National Taiwan University, Taipei 10617, Taiwan

^*tomanek@pa.msu.edu

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Vol. 90, Iss. 24 — 15 December 2014

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Images

Figure 1
Principal radii of curvature $R_{1}, R_{2}$ and the Gaussian curvature $G$ (a) on the surface of a sphere, (b) on the surface of a cylinder, and (c) in a saddle point. (d) Determination of the local curvature at point $P$ using the atomic lattice and the dual lattice.
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Figure 2
Local Gaussian curvature $G$ (left panels) and local curvature energy $Δ E_{C} / A$ across the surface of (a) the least stable $C_{38}$ isomer, (b) the most stable $C_{38}$ isomer, (c) a (10,10) carbon nanotube, and (d) a schwarzite structure with 152 atoms per unit cell. The values of $G$ and $Δ E_{C} / A$ have been interpolated across the surface.
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Figure 3
Strain energy $Δ E$ in carbon nanostructures with respect to the graphene reference system. (a) DFT-based total strain energy $Δ E_{tot}^{DFT}$ for selected fullerenes, with the most stable isomers indicated by the larger symbols. (b) Strain energy $Δ E$ in different $C_{38}$ isomers. Total-energy differences $Δ E_{tot}^{DFT}$ based on DFT, $Δ E_{tot}^{Tersoff}$ based on the Tersoff potential, and $Δ E_{tot}^{Keating}$ based on the Keating potential are compared to curvature energies $Δ E_{C}^{Keating}$ based on Keating optimized geometries. (c) Strain energy $Δ E$ in different $C_{38}$ isomers. DFT total energies $Δ E_{tot}^{DFT}$ are compared to curvature energies $Δ E_{C}^{DFT}$ based on continuum elasticity theory for structures optimized by DFT, and $Δ E_{C}^{Keating}$ values based on continuum elasticity theory for structures optimized using the Keating potential.
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Figure 4
Strain energy $Δ E$ in carbon nanostructures with respect to the graphene reference system. (a) Comparison between DFT-based total energies $Δ E_{tot}^{DFT}$ and the curvature energy $Δ E_{C}^{Keating}$ based on Keating optimized geometries for all fullerene isomers considered in Fig. 3. (b) Comparison between DFT-based strain energies $Δ E_{tot}^{DFT} / n$ and curvature energies per atom $Δ E_{C}^{Keating} / n$ for Keating optimized geometries of fullerenes, nanotubes, and schwarzites. Dashed lines represent agreement between DFT and continuum elasticity results.
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Figure 5
(a) Perspective view of the planar structure of a blue phosphorene monolayer (top), which has been rolled up to a nanotube with radius $R$ (bottom). (b) Comparison between the strain energy per atom $Δ E_{C} / n$ based on continuum elasticity theory and $Δ E_{tot}^{DFT} / n$ based on DFT in blue phosphorene nanotubes. The dashed line represents agreement between DFT and continuum elasticity results.
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Figure 6
Average curvature energy $〈 Δ E_{C} 〉$ per area of a nanocapsule tessellated by a honeycomb lattice with different numbers of vertices. The vertical dash-dotted red line indicates that the capsule represents the $C_{120}$ structure. The inset shows how the capsule surface can be tessellated by a honeycomb lattice with 120 vertices or atoms, shown by the white lines, and also with 480 vertices, shown by the red lines. The horizontal dashed black line represents an extrapolation to an infinitely dense tessellation.
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