Vibration Properties of Multi-Layered Graphene Sheets

Nanoscale vibration characteristics of multi-layered graphene sheets

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.

In this paper, buckling analysis of biaxially compressed multi-layered graphene sheets with a continuum plate model is reported. The equations of motion are analytically solved to obtain closed-form solution for buckling loads of all edges simply supported multi-layered graphene sheets. The interaction of van der Waals (vdWs) pressure between the layers is incorporated in the formulation to determine the buckling behavior of simply supported MLGSs. Explicit formulae are derived for predicting the critical buckling loads of double- and triple-layered graphene sheets, and they clearly indicate the effect of vdW interaction on the critical buckling loads. The critical buckling loads are calculated for various numbers of layered graphene sheets, and the obtained results show that the vdW force has no effect on the first critical buckling load of an MLGS, but plays a significant role in all higher first critical buckling loads for all combinations of m and n.

The van der Waals Interaction between a Spherical Particle and a Cylinder

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.

Analysis of nonlinear forced vibration of multi-layered graphene sheets

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.

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.

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

Free vibration of viscoelastic double-bonded polymeric nanocomposite plates reinforced by FG-SWCNTs using MSGT, sinusoidal shear deformation theory and meshless method

The van der Waals (vdW) interaction in two-dimensional (2D)-layered materials affects key characteristics of electronic devices, such as the contact resistance, with a vertical heterostructure geometry. While various functionalizations to manipulate the properties of 2D materials have shown issues such as defect generation or have a limited spatial range for the methods, engineering the vdW interaction in nondestructive ways for device applications has not been tried or properly achieved yet. Here, we introduce the proximity engineering of the vdW interaction in multilayered graphene, which is observed as modified interlayer distances and deviated stacking orders by Raman spectroscopy. A 2D electride, [Ca2N]+·e-, possessing a low-work function of 2.6 eV, was used to trigger an avalanche of electrons over tens of graphene layers, exceeding the conventional spatial-range limit (∼1 nm) by screening with a carrier density of 1014 cm-2. Our proximity engineering reduces the vdW interaction in a nondestructive way and achieves a promising graphene-metal contact resistance of 500 Ω·μm without using complicated edge contacts, which demonstrates a way to use moderately decoupled graphene layers for device applications.

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.

Machine-learning interatomic potentials (MLIPs) enable large-scale atomistic simulations at moderate computational cost while retaining ab initio accuracy. In recent years, MLIPs trained on coupled-cluster data─particularly CCSD(T), which includes single, double, and perturbative triple excitations─have emerged as a promising route to achieve chemical accuracy (1 kcal/mol) beyond the limits of density functional theory (DFT) and to incorporate nonempirical van der Waals (vdW) interactions. Most existing approaches are, however, still not straightforwardly applicable for systems with extended covalent networks such as covalent organic frameworks (COFs) due to the limited availability of CCSD(T) under periodic boundary conditions. Here we present a methodology to train MLIPs with CCSD(T) accuracy for systems with extended covalent networks. The approach is based on the Δ-learning method with a dispersion-corrected tight-binding baseline and an MLIP trained on the differences of the target CCSD(T) energies from the baseline. This Δ-learning strategy enables training on compact molecular fragments while preserving transferability toward the periodic systems. Dispersion interactions are accounted for by including vdW-bound multimers in the training set, and the combination with a vdW-aware tight-binding baseline allows the formally local MLIP to attain CCSD(T)-level accuracy even for systems dominated by long-range vdW forces. The resulting potential yields root-mean-square energy errors below 0.4 meV/atom on both training and test sets and reproduces electronic total atomization energies, bond lengths, harmonic vibrational frequencies, and intermolecular interaction energies for benchmark molecular systems. We apply the method to a prototypical quasi-two-dimensional covalent organic framework (COF) composed of carbon and hydrogen. The COF structure, interlayer binding energies, and hydrogen absorption are analyzed at CCSD(T) accuracy. Overall, the developed methodology opens a practical route to large-scale atomistic simulations for systems with extended covalent networks and vdW interactions with chemical accuracy.

This study investigates the vibrational characteristics of multi-layered graphene sheets with variable thickness (VTGSs) by using molecular dynamics (MD) simulations. It is aimed to determine how the natural frequencies and vibration damping ratios of variable-thickness graphene change with respect to temperature. Atomistic models for six distinct geometries (1L, 3LT, 3LTB, 5LT, 5LTB, and 9LTB) were generated to analyze the influence of structural design and temperature on their natural frequencies. The simulations were performed using the Large-Scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) with an AIREBO potential to represent interatomic carbon interactions. Natural frequencies of all atomistic models were extracted by applying the Fast Fourier Transform (FFT) method to the Velocity Autocorrelation Function (VACF) data obtained from the simulations. In addition, the analysis was conducted at three different temperatures: 250 K, 300 K, and 350 K. Key findings reveal that an increase in the number of graphene layers results in a decrease in the fundamental natural frequency due to the increased mass of the structure. Moreover, it was noted that natural frequencies decrease with increasing temperature. It is attributed to the reduction in structural rigidity at higher thermal energies. These results provide critical insights into how geometric and thermal variations affect the dynamic behavior of complex multi-layered graphene structures.

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.