Fracture analysis of monolayer graphene sheets with double vacancy defects via MD simulation

Carbon nanostructures such as carbon nanotubes (CNTs) and graphene sheets have attracted great attention due to their exceptionally high strength and elastic strain. These extraordinary mechanical properties, however, can be affected by the presence of defects in their structures. When a material contains multiple defects, it is expected that the stress concentration of them superposes if the separation distances of the defects are low, which causes a more reduction of the strength. On the other hand, it is believed that if the defects are far enough such that their affected areas are distinct, their behavior is similar to a material with single defect. In this article, molecular dynamics (MD) is used to explore the influence of separation distance of double vacancy defects on the mechanical properties of single-layered graphene sheets (SLGSs). To this end, critical stress and strain of SLGSs containing double vacancy with different distances are determined and the results are compared with those of perfect SLGSs and graphene sheets with single vacancy defect. The results reveal that the ultimate strength of the SLGS with double vacancy tends to the one with a single vacancy when the separation distance becomes further. In this regard, the threshold distance beyond which double defects behave like a single one is examined. Finally, Young’s modulus of perfect, single and double vacancy defected graphene sheets with different separation distances is determined. It is concluded that this property is slightly affected by the separation distance.     

Atomistic finite element model for axial buckling and vibration analysis of single-layered graphene sheets

In this article, an atomistic model is developed to study the buckling and vibration characteristics of single-layered graphene sheets (SLGSs). By treating SLGSs as space-frame structures, in which the discrete nature of graphene sheets is preserved, they are modeled using three-dimensional elastic beam elements for the bonds. The elastic moduli of the beam elements are determined via a linkage between molecular mechanics and structural mechanics. Based on this model, the critical compressive forces and fundamental natural frequencies of single-layered graphene sheets with different boundary conditions and geometries are obtained and then compared. It is indicated that the compressive buckling force decreases when the graphene sheet aspect ratio increases. At low aspect ratios, the increase of aspect ratios will result in a significant decrease in the critical buckling load. It is also indicated that increasing aspect ratio at a given side length results in the convergence of buckling envelops associated with armchair and zigzag graphene sheets. The influence of boundary conditions will be studied for different geometries. It will be shown that the influence of boundary conditions is not significant for sufficiently large SLGSs.     

Atomistic evaluation of the stress concentration factor of graphene sheets having circular holes

Stress concentration factor concept has been developed for single-layered graphene sheets (SLGSs) with circular holes through an atomistic point of view by the application of molecular structural mechanics (MSM) approach. In this approach the response of SLGSs against unidirectional tensile loading is matched to the response of a frame-like macro structure containing beam elements by making an equivalence between strain energies of beam elements in MSM and potential energies of chemical bonds of SLGSs. Both chirality and size effects are considered and the atomistic evaluation of stress concentration factor is performed for different sizes of circular holes. Also, molecular dynamics simulations are implemented to verify the existence and location of the predicted stress concentration. The results reveal that size effects and the diameters of circular holes have a significant influence on the stress concentration factor of SLGSs and armchair SLGSs show a larger value of stress concentration than zigzag ones.     

Atomistic simulations of divacancy defects in armchair graphene nanoribbons: Stability,electronic structure,and electron transport properties

Using the first principles calculations associated with nonequilibrium Green?s function, we have studied the electronic structures and quantum transport properties of defective armchair graphene nanoribbon (AGNR) in the presence of divacancy defects. The triple pentagon–triple heptagon (555–777) defect in the defective AGNR is energetically more favorable than the pentagon–octagon–pentagon (5–8–5) defect. Our calculated results reveal that both 5–8–5-like defect and 555–777-like defect in AGNR could improve the electron transport. It is anticipated that defective AGNRs can exhibit large range variations in transport behaviors, which are strongly dependent on the distributions of the divacancy defect.     

扶手椅型石墨纳米带的双空位缺陷效应研究

采用基于密度泛函理论的第一性原理电子结构和输运性质计算,研究了扶手椅型石墨纳米带（具有锯齿边缘）的双空位缺陷效应.研究发现：双空位缺陷的存在并没有改变石墨纳米带的金属特性,但改变了费米面附近的能带结构.同时,双空位缺陷的取向对石墨纳米带的输运性质有很重要的影响.对于奇数宽度的纳米带,斜向双空位缺陷使得石墨带导电性能减弱,而垂直双空位能基本保留原有的线性伏安特性,导电性能降低较少;对于偶数宽度的纳米带,斜向双空位缺陷会使石墨带导电性能明显增强,而垂直双空位缺陷则具有完整石墨带的输运性质. 关键词： 石墨纳米带 585双空位缺陷 电子结构 输运性质     

Nonlocal plate model for nonlinear bending of single-layer graphene sheets subjected to transverse loads in thermal environments

This paper investigates the nonlinear bending behavior of a single-layer rectangular graphene sheet subjected to a transverse uniform load in thermal environments. The single-layer graphene sheet (SLGS) is modeled as a nonlocal orthotropic plate which contains small scale effect. Geometric nonlinearity in the von Kármán sense is adopted. The thermal effects are included and the material properties are assumed to be size dependent and temperature dependent, and are obtained from molecular dynamics (MD) simulations. The small scale parameter e ₀ a is estimated by matching the deflections of graphene sheets observed from the MD simulation results with the numerical results obtained from the nonlocal plate model. The numerical results show that the temperature change as well as the aspect ratio has a significant effect on the nonlinear bending behavior of SLGSs. The results reveal that the small scale parameter reduces the static large deflections of SLGSs, and the small scale effect also plays an important role in the nonlinear bending of SLGSs.     

Nonlocal plate model for free vibrations of single-layered graphene sheets

Vibration analysis of single-layered graphene sheets (SLGSs) is investigated using nonlocal continuum plate model. To this end, Eringens's nonlocal elasticity equations are incorporated into the classical Mindlin plate theory for vibrations of rectangular nanoplates. In contrast to the classical model, the nonlocal model developed in this study has the capability to evaluate the natural frequencies of the graphene sheets with considering the size-effects on the vibrational characteristics of them. Solutions for frequencies of the free vibration of simply-supported and clamped SLGSs are computed using generalized differential quadrature (GDQ) method. Then, molecular dynamics (MD) simulations for the free vibration of various SLGSs with different values of side length and chirality are employed, the results of which are matched with the nonlocal model ones to derive the appropriate values of the nonlocal parameter relevant to each boundary condition. It is found that the value of the nonlocal parameter is independent of the magnitude of the geometrical variables of the system.     

单空位缺陷对石墨纳米带电子结构和输运性质的影响

基于第一性原理电子结构和输运性质计算,研究了单空位缺陷对单层石墨纳米带(包括zigzag型和armchair型带)电子性质的影响.研究发现,单空位缺陷使石墨纳米带在费米面上出现一平直的缺陷态能带;单空位缺陷的引入使zigzag型半导体性的石墨纳米带变为金属性,这在能带工程中有重要的应用价值;奇数宽度的armchair型石墨纳米带表现出金属特性,有着很好的导电性能,同时,偶数宽度的armchair型石墨带虽有金属性的能带结构,但却有类似半导体的伏安特性;单空位缺陷使得奇数宽度的armchair石墨纳米带导电 关键词： 石墨纳米带 单空位缺陷 电子结构 输运性质     

Influence of line defects on relaxation properties of graphene: A molecular dynamics study

The relaxation properties of single layer graphene sheets containing line defects were investigated using molecular dynamics simulation with AIROBE bond-order interatomic potential. The dynamic evolution of graphene sheets during relaxation condition was analyzed. The simulation results show that the single layer graphene sheets are not perfectly flat in an ideal state, and the graphene sheet shows a significant corrugations at the verge of sheet. The graphene sheet is bent with the line defects at the end of the sheet, and the extent of this bend also increases with the increase of the defect number. Furthemore, the graphene sheet transforms into a paraboloid with the line defects at the middle of the sheet.     

Vibration analysis of defective graphene sheets using nonlocal elasticity theory

Many papers have studied the free vibration of graphene sheets. However, all this papers assumed their atomic structure free of any defects. Nonetheless, they actually contain some defects including single vacancy, double vacancy and Stone-Wales defects. This paper, therefore, investigates the free vibration of defective graphene sheets, rather than pristine graphene sheets, via nonlocal elasticity theory. Governing equations are derived using nonlocal elasticity and the first-order shear deformation theory (FSDT). The influence of structural defects on the vibration of graphene sheets is considered by applying the mechanical properties of defective graphene sheets. Afterwards, these equations solved using generalized differential quadrature method (GDQ). The small-scale effect is applied in the governing equations of motion by nonlocal parameter. The effects of different defect types are inspected for graphene sheets with clamped or simply-supported boundary conditions on all sides. It is shown that the natural frequencies of graphene sheets decrease by introducing defects to the atomic structure. Furthermore, it is found that the number of missing atoms, shapes and distributions of structural defects play a significant role in the vibrational behavior of graphene. The effect of vacancy defect reconstruction is also discussed in this paper.     

Effects of topological defects and local curvature on the electronic properties of planar graphene

A formalism is proposed to study the electronic and transport properties of graphene sheets with corrugations as the one recently synthesized. The formalism is based on coupling the Dirac equation that models the low energy electronic excitations of clean flat graphene samples to a curved space. A cosmic string analogy allows to treat an arbitrary number of topological defects located at arbitrary positions on the graphene plane. The usual defects that will always be present in any graphene sample as pentagon–heptagon pairs and Stone–Wales defects are studied as an example. The local density of states around the defects acquires characteristic modulations that could be observed in scanning tunnel and transmission electron microscopy.     

Buckling analysis of variable thickness nanoplates using nonlocal continuum mechanics

This paper presents an investigation on the buckling characteristics of nanoscale rectangular plates under bi-axial compression considering non-uniformity in the thickness. Based on the nonlocal continuum mechanics, governing differential equations are derived. Numerical solutions for the buckling loads are obtained using the Galerkin method. The present study shows that the buckling behaviors of single-layered graphene sheets (SLGSs) are strongly sensitive to the nonlocal and non-uniform parameters. The influence of percentage change of thickness on the stability of SLGSs is more significant in the strip-type nonoplates (nanoribbons) than in the square-type nanoplates.     

Comparison of Goos–Hänchen shifts of the reflected beam from graphene on dielectrics and metals

The Goos–Hänchen shifts of the reflected beam from graphene-on-dielectric (or metal) in the optical wavelength are investigated by using the stationary-phase method. For the graphene-on-dielectric substrates, it is found that the pseudo-Brewster angle and Goos–Hänchen shift are influenced greatly by the introduced graphene sheets for TM polarization. By changing number of graphene sheets, the lateral shifts can be large positive or negative near the pseudo-Brewster angle. For TE polarization, the lateral shift is still small; however it can also be positive or negative by changing the number of graphene sheets. For the graphene-on-metal substrates, graphene sheets exert a great impact on the reflectance while has little effect on the lateral shifts of both polarizations. Finally, the role of the graphene sheets on the lateral shifts for the different visible wavelengths is discussed.     

A nonlocal plate model incorporating interatomic potentials for vibrations of graphene with arbitrary edge conditions

This article presents analytical explicit frequency expressions for investigating the vibrations of single-layer graphene sheets (SLGSs). The interatomic potential is incorporated into a nonlocal continuum plate model through establishing a linkage between the strain energy density induced in the continuum and nonlocal plate constitutive relations. The model which is independent of scattered value of Young's modulus is then applied and explicit frequency formulas for the SLGSs with different edge conditions are derived using static deflection function of the nanoplate under uniformly distributed load. The reliability of the present formulation is verified by the results obtained by the molecular dynamics (MD) simulations and other research workers. The formulas are of a simple short form enabling quick and accurate evaluation of the frequency of the SLGSs and also simple calibration of scale coefficient by the use of MD simulations results.     

Zigzag型石墨纳米带电子结构和输运性质的第一性原理研究

采用第一性原理电子结构和输运性质计算研究了zigzag型单层石墨纳米带(具有armchair 边缘)的电子结构和输运性质及其边缘空位缺陷效应. 研究发现，完整边缘的zigzag型石墨纳米带是具有一定能隙的半导体带，边缘空位缺陷的存在使得纳米带能隙变小，且缺陷浓度越大，能隙越小，并发生了半导体-金属转变. 利用这些研究结果，将有助于在能带工程中实现其电子结构裁剪.     

Zigzag型石墨纳米带电子结构和输运性质的第一性原理研究

采用第一性原理电子结构和输运性质计算研究了zigzag型单层石墨纳米带(具有armchair 边缘)的电子结构和输运性质及其边缘空位缺陷效应. 研究发现,完整边缘的zigzag型石墨纳米带是具有一定能隙的半导体带,边缘空位缺陷的存在使得纳米带能隙变小,且缺陷浓度越大,能隙越小,并发生了半导体-金属转变. 利用这些研究结果,将有助于在能带工程中实现其电子结构裁剪. 关键词： 石墨纳米带 空位缺陷 电子结构 输运性质     

Molecular dynamics simulation of compression of single-layer graphene

The compression of a single-layer graphene sheet in the “zigzag” and “armchair” directions has been investigated using the molecular dynamics method. The distributions of the xy and yx stress components are calculated for atomic chains forming the graphene sheet. A graphene sheet stands significant compressive stresses in the “zigzag” direction and retains its integrity even at a strain of ～0.35. At the same time, the stresses which accompany the compressive deformation of single-layer graphene in the “armchair” direction are more than an order in magnitude lower than corresponding characteristics for the “zigzag” direction. A compressive strain of ～0.35 in the “armchair” direction fractures the graphene sheet into two parts.     

Spatial variation in the electronic structures of carpetlike graphene nanoribbons and sheets

Graphene, when deposited on a supporting substrate with a step edge, may be deformed in the presence of the step edges of the substrate. In this study, we have investigated a spatial variation in the local electronic structure near the step region, by performing first-principles calculations for carpetlike armchair graphene nanoribbons (C-AGNR) and two-dimensional periodic carpetlike graphene sheets (PCGS). Our results indicate no practical difference in the local density of states (LDOS) between those of flat and step regions. Interestingly, however, the PCGS shows a remarkable variation in the LDOS with an external electric field (E-field). Furthermore, we also discuss the dependence of the direction and the magnitude of the applied E-field on the spatial variation in the LDOS.     

The importance of defects for carbon nanoribbon based electronics

The utilization of graphene nanoribbons for next generation nanoelectronics is commonly expected to depend on the controlled synthesis that yields a low density of defects. Edge roughness and vacancies have been shown to have a large impact on the performance of graphene nanoribbon transistors. In contrast, we show how certain defects can be used to enhance the electronic and magnetic properties of graphene nanoribbons. We explore the properties of hybrid graphene nanoribbons with armchair and zigzag features joined by an array of pentagon–heptagon structural defects. The graphene nanoribbons display an increased density of states at the Fermi level, and half metallicity in absence of external fields. (© 2009 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Graphene characterization: A fully non-linear spring-based finite element prediction

In the present study, a spring-based finite element model is formulated and utilized to predict the stress–strain behavior of single-layer graphene. Generalized force–generalized displacement behavior of the developed nonlinear springs follows the relation between the first derivative of the potential energy and the corresponding bond deformation, describing interatomic interactions. A number of different loading cases are examined in order to predict mechanical properties and characterize the graphene sheet. Predicted Young's and shear moduli, tensile and shear strength, tensile and shear failure strain, etc., under tension, compression and pure shear, are compared to results found in the literature, which are based on numerical, analytical or experimental methodologies. In all the above loading cases the graphene sheet is examined as a virtually orthotropic material, exhibiting different material properties in the armchair and zigzag directions. Different behaviors in tension and compression, as suggested by the modified Morse atomic bond stretching potential, are illustrated by the predicted stress–strain curves.