Buckling analysis of multi-layered graphene sheets based on a continuum mechanics model

In this paper,we study the quantum size effects in multilayer graphene sheets using first principles methods within the framework of density functional theory. Four different types of functionals are adopted respectively to describe the van der Waals interactions between graphene layer sheets: the DFT-GGA(PBE), the DFT-D2, the vdW-DF and the optPBE-vdW. By inspecting the binding energy as a function of increasing graphene layers, we find that the PBE functional can not well describe the van der Waals interactions between different layers of graphene sheets. In contrast, the other three methods exhibit similar results with monotonic increasing binding energy as a function of graphene layers towards the bulk limit, concluding that the layered graphene structure is stabilized by van der Waals interactions. The density of states at zero temperature indicate that the multilayer graphene sheets is a semi-metal, which is independent of sheet layers number. The finite temperature (about 200 K) density of states at Fermi surface are studied as a function of the number of stacking graphene layers. The systematic oscillating behavior of finite temperature density of states between odd and even number of layers is a demonstration of quantum size effects. The Fermi wavelength will converge to two times the inter-layer distance of graphite, which is consistent with the theory describing the motion of particles in a quantum well. Finally, we study the adsorption of single H atom on multilayer graphene sheets to test the role of quantum size effects. The adsorption energies and the vibration frequencies are calculated for comparison with experiments. Our results shed light on understanding the stacking process of multilayer graphene in vacuum both theoretically and experimentally.

Based on an explicit formula used for describing the van der Waals (vdW) interaction between any two layers of multi-layered graphene sheets (MLGSs), the vibration properties of five- and eight-layered graphene sheets are analyzed with considering the influence of vdW interaction on the natural frequences. The present study shows that for a given combination n and m the lowest natural frequency (classical natural frequency) of a MLGSs is independent of vdW interaction, but that it is very significant for the influence of vdW interaction on the other higher frequencies (resonant frequencies). The different vibration modes, including in-phase and anti-phase ones, are given for exhibiting clearly the influence of vdW interaction.

Carbon deposits consisting of vertically oriented multilayer graphene sheets on metallic foils represent an interesting alternative to activated carbon in electrical and electrochemical devices such as super-capacitors because of the superior electrical conductivity of graphene and huge surface-mass ratio. The graphene sheets were deposited on cobalt foils by plasma-enhanced chemical vapor deposition using propane as the carbon precursor. Plasma was sustained by an inductively coupled radiofrequency discharge in the H mode at a power of 500 W and a propane pressure of 17 Pa. The precursor effectively dissociated in plasma conditions and enabled the growth of porous films consisting of multilayer graphene sheets. The deposition rate varied with time and peaked at 100 nm/s. The evolution of surface wettability was determined by the sessile drop method. The untreated substrates were moderately hydrophobic at a water contact angle of about 110°. The contact angle dropped to about 50° after plasma treatment for less than a second and increased monotonously thereafter. The maximal contact angle of 130° appeared at a treatment time of about 30 s. Thereafter, it slowly decreased, with a prolonged deposition time. The evolution of the wettability was explained by surface composition and morphology. A brief treatment with oxygen plasma enabled a super-hydrophilic surface finish of the films consisting of multilayer graphene sheets.

Nanoscale vibration characteristics of multi-layered graphene sheets

In this paper, a multiplate nonlocal shear model and molecular dynamic simulations are presented to investigate the effects of interlayer shear and nonlocality on the natural frequencies of multilayer graphene sheets (MLGSs). From one aspect in the optimal design of such structures, the interaction between graphene layers, which can significantly vary the static and dynamic behavior due to lack of solidity of layers stack, should be considered. On the other hand, it is requied that the nonlocality phenomenon which has an effective role in the mechanical analysis of nanostructures is taken into account. To this aim, the equation of motion along with corresponding boundary conditions is derived by using the Kirchhoff plate model and nonlocal continuum theory to capture these effects. For a case study, the free vibration problem of simply supported MLGSs is studied by presenting the closed-form Navier’s solution of natural frequencies and carrying out Molecular dynamics (MD) simulations to estimate the value of nonlocal parameter. Finally, the influences of interlayer shear modulus, number of layers, foundation stiffness, and nonlocal parameter on the natural frequencies of such MLGSs are examined in detail. Findings show that natural frequencies including nonlocality effect become smaller than ones of the classical continuum theory and also have good agreements with MD simulations. Moreover, it is seen that the sensitivity of natural frequency to the nonlocal parameter becomes more significant when the number of layers or interlayer shear modulus increases as well as the aspect ratio or width of sheets decreases.

Elastic buckling behaviour of multi-layered graphene sheets is rigorously investigated. Van der Waals forces are modelled, to a first order approximation, as linear physical springs which connect the nodes between the layers. Critical buckling loads and their associated modes are established and analyzed under different boundary conditions, aspect ratios and compressive loading ratios in the case of graphene sheets compressed in two perpendicular directions. Various practically possible loading configurations are examined and their effect on buckling characteristics is assessed. To model more accurately the buckling behaviour of multi-layered graphene sheets, a physically more representative and realistic mixed boundary support concept is proposed and applied. For the fundamental buckling mode under mixed boundary support, the layers with different boundary supports deform similarly but non-identically, leading to resultant van der Waals bonding forces between the layers which in turn affect critical buckling load. Results are compared with existing known solutions to illustrate the excellent numerical accuracy of the proposed modelling approach. The buckling characteristics of graphene sheets presented in this paper form a comprehensive and wholesome study which can be used as potential structural design guideline when graphene sheets are employed for nano-scale sensing and actuation applications such as nano-electro-mechanical systems.

Nanoparticle mass detection by single and multilayer graphene sheets: Theory and simulations

Article history: In this paper, a description is given of finite element method (FEM) simulations of the elastic bending modulus of multi-layered graphene sheets that were carried out to investigate the mechanical behavior of graphene sheets with different gap thicknesses through molecular mechanics theory. The interaction forces between layers with various gap thicknesses were considered based on the van der Waals interaction. A finite element (FE) model of a multi-layered rectangular graphene sheet was proposed with beam elements representing bonded interactions and spring elements representing non-bonded interactions between layers and between diagonally adjacent atoms. As a result, the average elastic bending modulus was predicted to be 1.13 TPa in the armchair direction and 1.18 TPa in the zigzag direction. The simulation results from this work are comparable to both experimental tests and numerical studies from the literature.

Analysis of nonlinear forced vibration of multi-layered graphene sheets

Holes and defects can greatly reduce the mechanical properties of multilayer graphene sheets under different loading conditions due to the stress concentration near the hole edge in each in-plane sheet and the lack of interlayer carbon–carbon bonds between the layers. Here, we report a novel design of multilayer open-hole graphene papers (MLGPs) formed through interlayer covalent bonding at the hole edges of multilayer open-hole graphene sheets (MLGSs) under high temperature using molecular dynamics (MD) simulations. Our MD results show that the hybrid sp2–sp3 interlayer bonds of MLGPs can significantly improve their both tensile strength and interlayer shear strength. The tensile strength and interlayer shear strength of MLGPs increase by around 20% and 3 times by comparison with those of MLGSs with the same number of layers, respectively, which mainly depends on the uniformity of their interlayer bond distribution. This study can provide an effective way to improve the mechanical performances of multilayer graphene sheets with flaws and also offer corresponding guidance for the design of MLGS-based nanocomposites.

The morphology of graphene sheets overlying a substrate plays an important role in controlling the electronic behaviour of graphene-based electronic devices. Atomistic simulations and energetic analysis are performed to study the morphology of graphene edges on a model substrate. We find that the van der Waals interactions are able to significantly reduce the amplitude, penetration depth and wavelength of edge ripples for a single-layer graphene sheet, but suppression of the ripples is not observed in multilayer graphene sheets with unsaturated edges. We explain this difference using an analytical model that accounts for the energetics of rippling and van der Waals interactions.

Simultaneous enhancement on thermal and mechanical properties of polypropylene composites filled with graphite platelets and graphene sheets

Since micro/nano-electromechanical devices made of nanocomposites are strongly influenced by external excitations, this paper intends to comprehensively investigate the effect of the primary and secondary resonances in performance of nano-electromechanical systems (NEMS). This model also helps to analyze nonlinear vibration behaviors of the recently built single-layered graphene sheet (SLGS) and multi-layered graphene sheet (MLGS) nanomechanical resonators. The aim of this work is to derive the effect of primary and secondary resonances in nonlinear forced vibration of piezoelectric/functionally graded (FG) porous nanocomposite subjected to a moving load and external electric voltage. At first, an FG porous core nanoplate glued with two piezoelectric layers is modeled. Then, the modeled nanocomposite is rested on a visco-Pasternak foundation. In the next step, Mindlin and Kirchhoff plate theories and Hamilton’s principle are separately employed to derive governing equation of motion. Galerkin technique and multiple time scales method are used, respectively, to solve the equation. Furthermore, primary and secondary resonances are analyzed and modulation equation of piezoelectric/FG porous nanocomposite for both of them is obtained. Finally, modulation equation under sub-harmonic and super-harmonic stimulations is studied and graphs are derived. Results show that there is a periodic relation between amplitude response and velocity of moving load. Also, even in the absence of electric voltage, the piezoelectric effect of the material can reduce the nonlinear vibration.

Interfacial thermal conductance between multi-layer graphene sheets and solid/liquid octadecane: A molecular dynamics study

Shear buckling of rippled graphene by molecular dynamics simulation