Compressive Edge Stresses Research Articles

An analytic investigation for the edge effect on mechanical properties of graphene nanoribbons

We derive analytical expressions for the Young's modulus and the Poisson's ratio of the graphene nanoribbon, in which free edges are warped by the compressive edge stress. Our analytical formulas explicitly illustrate the reduction of the Young's modulus by the warped free edges, leading to the obvious width dependence for the Young's modulus of the graphene nanoribbon. The Poisson's ratio is also reduced by the warped free edges, and negative Poisson's ratio can be achieved in the graphene nanoribbon with an ultra-narrow width. These results are comparable with previous theoretical works.

Strain engineering for mechanical properties in graphene nanoribbons revisited: The warping edge effect

We investigate the strain engineering and the edge effect for mechanical properties in graphene nanoribbons. The free edges of the graphene nanoribbons are warped due to compressive edge stresses. There is a structural transformation for the free edges from the three-dimensional warping configuration to the two-dimensional planar structure at the critical strain ϵc = 0.7%, at which the applied mechanical stress is equal to the intrinsic compressive edge stress. This structural transformation leads to step-like changes in several mechanical properties studied in the present work, including the Young's modulus, the Poisson's ratio, the quality factor of nanomechanical resonators, and the phonon edge mode.

Negative Poisson's Ratio in Single-Layer Graphene Ribbons.

The Poisson's ratio characterizes the resultant strain in the lateral direction for a material under longitudinal deformation. Though negative Poisson's ratios (NPR) are theoretically possible within continuum elasticity, they are most frequently observed in engineered materials and structures, as they are not intrinsic to many materials. In this work, we report NPR in single-layer graphene ribbons, which results from the compressive edge stress induced warping of the edges. The effect is robust, as the NPR is observed for graphene ribbons with widths smaller than about 10 nm, and for tensile strains smaller than about 0.5% with NPR values reaching as large as -1.51. The NPR is explained analytically using an inclined plate model, which is able to predict the Poisson's ratio for graphene sheets of arbitrary size. The inclined plate model demonstrates that the NPR is governed by the interplay between the width (a bulk property), and the warping amplitude of the edge (an edge property), which eventually yields a phase diagram determining the sign of the Poisson's ratio as a function of the graphene geometry.

Accurate buckling analysis of thin rectangular plates under locally distributed compressive edge stresses

The buckling analysis of thin rectangular plates under locally distributed compressive edge stresses is a challenging problem if the point discrete methods are to be used. To obtain accurate buckling stress, one of the important factors is that the in-plane stress distributions within the plate prior to buckling should be accurate enough. Although it is possible to get analytical solutions for the in-plane stress distributions, but the expressions are very complicated since a stress-diffusion phenomenon exists. The differential quadrature method (DQM), being a point discrete method, has been successfully used in a variety of fields including the buckling analysis of thin rectangular plates under nonlinearly distributed edge compressions. However, it is rare to employ the DQM directly to solve problems of rectangular plates under locally distributed or point loads. To solve the challenging problem by using the DQM, novel formulations are presented in this paper. The locally distributed stress is first work-equivalently to point loads at all inner grid points on the loaded edge, then the normal stress boundary condition is numerically integrated before being discretized in terms of the differential quadrature. In this way accurate in-plane stress distributions can be obtained by the DQM without any difficulties. Buckling analysis of rectangular plates under either uniaxial or biaxial locally distributed compressive stresses is successfully performed. The accuracy of the DQ data is validated by comparing them with existing analytical solutions and finite element data. It is demonstrated that the compactness and computational efficiency of the DQM are retained. Accurate buckling loads are presented for rectangular plates with nine combinations of boundary conditions, various aspect ratios and load ratios. Some new results are also provided for references.

Rippling instabilities in suspended nanoribbons

Morphology mediates the interplay between the structure and electronic transport in atomically thin nanoribbons such as graphene as the relaxation of edge stresses occurs preferentially via out-of-plane deflections. In the case of end-supported suspended nanoribbons that we study here, past experiments and computations have identified a range of equilibrium morphologies, in particular, for graphene flakes, yet a unified understanding of their relative stability remains elusive. Here, we employ atomic-scale simulations and a composite framework based on isotropic elastic plate theory to chart out the morphological stability space of suspended nanoribbons with respect to intrinsic (ribbon elasticity) and engineered (ribbon geometry) parameters, and the combination of edge and body actuation. The computations highlight a rich morphological shape space that can be naturally classified into two competing shapes, bendinglike and twistlike, depending on the distribution of ripples across the interacting edges. The linearized elastic framework yields exact solutions for these rippled shapes. For compressive edge stresses, the body strain emerges as a key variable that controls their relative stability and in extreme cases stabilizes coexisting transverse ripples. Tensile edge stresses lead to dimples within the ribbon core that decay into the edges, a feature of obvious significance for stretchable nanoelectronics. The interplay between geometry and mechanics that we report should serve as a key input for quantifying the transport along these ribbons.

Edge-stress-induced spontaneous twisting of graphene nanoribbons

We present a continuum model for spontaneous twisting of graphene nanoribbons driven by compressive edge stresses. Based on a geometrically nonlinear theory of plates, we identify scaling laws for the dependence of twist angles on ribbon width. Strikingly, we find the existence of a critical width below which a ribbon will not undergo spontaneous twisting, preferring an in-plane stretching mode instead. The model predictions are shown to be in excellent qualitative and quantitative agreement with density-functional tight-binding simulations. More generally, our model provides a unifying picture of twisting in graphene nanoribbons with different edge orientations and chemical functionalizations that have been reported recently in the literature.

Saddles, twists, and curls: shape transitions in freestanding nanoribbons

Efforts to modulate the electronic properties of atomically thin crystalline nanoribbons requires precise control over their morphology. Here, we perform atomistic simulations on freestanding graphene nanoribbons (GNRs) to first identify the minimal shapes as a function of ribbon width, and then develop a core-edge framework based on classical plate theory to explore the effect of size and ribbon elasticity in more general systems. The elastic edge-edge interactions are central to stabilization of the flat phase in ultra-narrow ribbons, and their bifurcation to twisted and bent shapes at critical widths that vary inversely with edge stress. In the case of compressive edge stress, we uncover hitherto ignored saddle shapes that are energetically indistinguishable with twisted shapes in the vicinity of the bifurcation yet dominate the morphological space with increasing width. At much larger widths with negligible edge-edge interactions, rippling instabilities set in, i.e. edge ripples and midline dimples for compressive and tensile edge stresses, respectively. Simulations of tapering GNRs reveal the dynamics of these shape transitions. Our results capture the interplay between geometry and mechanics that sets the morphology of crystalline nanoribbons and also highlight the utility of the core-edge framework in developing a unified understanding of the interplay.

Edge stability of boron nitride nanoribbons and its application in designing hybrid BNC structures

First-principles investigations of the edge energies and edge stresses of single-layer hexagonal boron nitride (BN) are presented. The armchair edges of BN nanoribbons (BNNRs) are more stable in energy than zigzag ones. Armchair BNNRs are under compressive edge stress while zigzag BNNRs are under tensile edge stress, due to the edge reconstruction effect and edge coulomb repulsion effect. The intrinsic spin-polarization and edge saturation play important roles in modulating the edge stability of BNNRs. The edge energy difference between BN and graphene can be used to guide the design of specific hybrid BNC structures as the hybrid BNC systems prefer the low-energy edge configurations: In an armchair BNC nanoribbon (BNCNR), BN domains are expected to grow outside of C domains, while the opposite occurs in a zigzag BNCNR. More importantly, armchair BNCNRs can reproduce unique electronic properties of armchair graphene nanoribbons (GNRs), which are expected to be robust against edge functionalization or disorder. Within a certain range of C/BN ratios, zigzag BNCNRs may exhibit intrinsic half-metallicity without any external constraints. These diverse electronic properties of BNCNRs may offer unique opportunities to develop nanoscale electronics and spintronics beyond individual graphene and BN. More generally, these principles for designing BNC can also be extended to other hybrid nanostructures.

Properties on the edge: graphene edge energies, edge stresses, edge warping, and the Wulff shape of graphene flakes

It has been shown that the broken bonds of an unreconstructed graphene edge generate compressive edge stresses leading to edge warping. Here, we investigate edge energies and edge stresses of graphene nanoribbons with arbitrary orientations from armchair to zigzag, considering both flat and warped edge shapes in the presence and absence of hydrogen. We use the second generation reactive empirical bond order potential to calculate the edge energies and stresses for clean and hydrogenated edges. Using these energies, we perform a Wulff construction to determine the equilibrium shapes of flat graphene flakes as a function of hydrogen chemical potential. While edge stresses for clean, flat edges are compressive, they become tensile if allowed to warp. Conversely, we find that edge energies change little (∼1%) with edge warping. Hydrogenation of the edges virtually eliminates both the edge energy and edge stresses. For warped edges an approximately linear relationship is found between amplitudes and wavelengths. The equilibrium shape of a graphene flake is determined by the value of the hydrogen chemical potential. For very small (and large) values of it the flakes have a nearly hexagonal (dodecagon) shape with zigzag oriented edges, while for intermediate values graphene flakes are found with complex shapes.

First-principles study of graphene edge properties and flake shapes

We use density functional theory to determine the equilibrium shape of graphene flakes, through the calculation of the edge orientation dependence of the edge energy and edge stress of graphene nanoribbons. The edge energy is a nearly linear function of edge orientation angle; increasing from the armchair orientation to the zigzag orientation. Reconstruction of the zigzag edge lowers its energy to less than that of the armchair edge. The edge stress for all edge orientations is compressive, however, reconstruction of the zigzag edge reduces this edge stress to near zero. Hydrogen adsorption is favorable for all edge orientations; dramatically lowering all edge energies and all edge stresses. It also removes the reconstruction of the zigzag edge. Using the new edge energy data, we determine the equilibrium shape of a graphene sheet (with unreconstructed edges) to be hexagonal with straight armchair edges in the presence and absence of hydrogen. However, zigzag edge reconstruction produces graphene flakes with a six-fold symmetry, but with rounded edges. This shape is dominated by near zigzag edges. The compressive edge stresses will lead to edge buckling (out-of-the-plane of the graphene sheet) for all edge orientations, in the absence of hydrogen. Exposing the graphene flake to hydrogen dramatically decreases the buckling amplitude.

Material Properties in Codimension > 0: Graphene Edge Properties

AbstractWhen materials are very thin in one or more dimensions, their equilibrium shapes are often curved/bent. Such shapes commonly represent a compromise between elastic strain energy and other thermodynamic forces (e.g. related to surface stresses, electrostatic interactions, or adsorption). Examples include ZnO and SnO2nanobelts, silica/carbonate helicoids, and graphene sheets and nanoribbons. Here, we demonstrate that when the equilibrium shape of a nanomaterial is not flat/straight, important fundamental material properties may be orders of magnitude different from their bulk counterparts. We focus here primarily on the graphene edges. Graphene in three dimensions is a codimensionc= 1 material; the codimension isc = D – d= 3 – 2 = 1, whereDis the dimensionality of the space in which the material is embedded anddis the dimensionality of the object. By contrast, a flat graphene sheet hasc= 2 – 2 = 0. We use the REBO-II interatomic potential to calculate the edge orientation dependence of the edge energy and edge stresses of graphene withc= 0 andc= 1. The edge stress for all edge orientations is compressive withc= 0. Both edge energy and stresses are in reasonable agreement with DFT calculations. The compressive edge stresses inc= 0 lead to edge buckling (out-of-the-plane of the graphene sheet) for all edge orientations (c= 1). The edge buckling inc= 1 reduces all edge energies and dramatically reduces all edge stresses to near zero (more than an order of magnitude drop). We also report the effect of codimension on the free energy and entropy of a graphene sheet and the elastic properties of ZnO nanohelices.

Molecular dynamics study of energetics of graphene flakes

Molecular dynamics simulations for graphene flakes of various shapes are performed. The equilibrium structures of graphene flakes are obtained. Round, hexagonal, and rectangular graphene flakes are dealt with, and their sizes are varied from a few angstroms to 200 Å. It is shown that for round and hexagonal graphene flakes of small size, the edge configuration influences their energy in equilibrium. Graphene nanoribbons (GNRs) of various aspect ratios are equilibrated at low temperature. The energies of the equilibrated graphene flakes with zigzag (ZZ) edges are lower than the energies of the equilibrated graphene flakes with armchair (AC) edges. This result corresponds to the scanning tunneling microscopy observations in the literature. The atomic bonds on the edges of graphene flakes with both edge configurations are reconstructed. The bond lengths of such reconstructed edges are smaller than the lengths of the atomic bonds inside them. Therefore, free graphene flakes undergo compressive edge stress and vibrate at finite temperature. Constrained GNRs also vibrate, and the amplitudes of the vibrations of the GNRs with ZZ edges are smaller than those of the GNRs with AC edges. On designing and fabricating GNR devices, GNRs with ZZ edges are more favorable than GNRs with AC edges, and vibrations of GNRs should be taken into account.

Edge-Stress-Induced Warping of Graphene Sheets and Nanoribbons

We show that edge stresses introduce intrinsic ripples in freestanding graphene sheets even in the absence of any thermal effects. Compressive edge stresses along zigzag and armchair edges of the sheet cause out-of-plane warping to attain several degenerate mode shapes. Based on elastic plate theory, we identify scaling laws for the amplitude and penetration depth of edge ripples as a function of wavelength. We also demonstrate that edge stresses can lead to twisting and scrolling of nanoribbons as seen in experiments. Our results underscore the importance of accounting for edge stresses in thermal theories and electronic structure calculations for freestanding graphene sheets.

Density-functional study of edge stress in graphene

The definition of edge stress is introduced in order to quantify the energy cost to deform a pre-existing edge in a two-dimensional crystal. Using density-functional ab initio calculations, edge stresses, as well as edge energies, are determined for the armchair and zigzag edges in graphene. It is found that both edges are under compression along the edge and the magnitude of compressive edge stress of armchair edge is larger than that of zigzag edge. In addition, hydrogen termination results in stress-free state for both edges. This excess edge quantity is expected to contribute to understanding various edge-related phenomena in graphene such as edge reconstructions and edge functionalizations.

On the post-buckling behaviour of plates under stress gradient

In this paper the elastic post-buckling behaviour of plates under non-uniform compressive edge stress is investigated. The compatibility differential equations is first solved analytically and then an approximate solution of the equilibrium equation is obtained using the Galerkin method. Explicit expressions are derived for the load-deflection, ultimate strength and membrane stress distributions. Analytical effective width formulations, based on the characteristics of the stress field of the buckled plate, are proposed for this general loading condition. The predicted load-deflection expression is compared with independent test results. Results are also presented detailing the load-deflection behaviour and stress distribution for various aspect ratios.