Double-layered Graphene Sheets Research Articles

Vibration characteristics of embedded multi-layered graphene sheets with different boundary conditions via nonlocal elasticity

A nonlocal elastic plate model accounting for the small scale effects is developed to investigate the vibrational behavior of multi-layered graphene sheets under various boundary conditions. Based upon the constitutive equations of nonlocal elasticity, derived are the Reissner–Mindlin-type field equations which include the interaction of van der Waals forces between adjacent and non-adjacent layers and the reaction from the surrounding media. The set of coupled governing equations of motion for the multi-layered graphene sheets are then numerically solved by the generalized differential quadrature method. The present analysis provides the possibility of considering different combinations of layerwise boundary conditions in a multi-layered graphene sheet. Based on exact solution, explicit expressions for the nonlocal frequencies of a double-layered graphene sheet with all edges simply supported are also obtained. The results from the present numerical solution, where possible, are indicated to be in excellent agreement with the existing data from the literature.

Geometrical nonlinear free vibration of multi-layered graphene sheets

A nonlinear continuum model is developed for the nonlinear vibration analysis of multi-layered graphene sheets (MLGSs), in which the nonlinear van der Waals (vdW) interaction between any two layers is formulated explicitly. The nonlinear equations of motion are studied by the harmonic-balance methods. Based on the present model, the nonlinear stiffened amplitude–frequency relations of double-layered graphene sheets (DLGSs) are investigated in the spectral neighbourhood of lower frequencies. The influence of the vdW interaction on the vibration properties of DLGSs is well illustrated by plotting the resulting modes' shapes, in which in-phase and anti-phase vibrations of DLGSs are studied. In particular, the large-amplitude vibration which associates with the anti-phase resonant frequencies, separating DLGS into single-layered GSs, is a promising application that needs to be explored further. In contrast, the vibration modes that are associated with the resonant frequencies are nonidentical and give various vibration patterns, which indicates that MLGSs are highly suited to being used as high-frequency resonators.

Vibration of Single- and Double-Layered Graphene Sheets

Free vibration of single- and double-layered graphene sheets is investigated by employing nonlocal continuum theory and molecular dynamics simulations. Results show that the classical elastic model overestimated the resonant frequencies of the sheets by a percentage as high as 62%. The dependence of small-scale effects, sizes of sheets, boundary conditions, and number of layers on vibrational characteristic of single- and double-layered graphene sheets is studied. The resonant frequencies predicted by the nonlocal elastic plate theory are verified by the molecular dynamics simulations, and the nonlocal parameter is calibrated through the verification process. The simulation results reveal that the calibrated nonlocal parameter depends on boundary conditions and vibrational modes. The nonlocal plate model is found to be indispensable in vibration analysis of grapheme sheets with a length less than 8 nm on their sides.

An amphiphilic pyrene sheet for selective functionalization of graphene

We have demonstrated that the aromatic amphiphile consisting of a hydrophilic dendron and an aromatic segment with a planar conformation can selectively exfoliate graphite powder into single- and double-layer graphene sheets in aqueous solution through hydrophilic functionalization of graphene surfaces.

Direct preparation of 1-PSA modified graphenenanosheets by supercritical fluidic exfoliation and its electrochemical properties

In this paper, we report on 1-pyrene sulfonic acid sodium salt (1-PSA) modified graphene nanosheets (imGNS) prepared by a novel one-pot in situ supercritical fluid exfoliation and modification reaction. The 1-PSA molecules on the surface of the graphene layers acted as electron withdrawing groups resulting in an electron transfer from the GNS surface to the 1-PSA molecules. This was confirmed by a red shift of 1572∼1576 cm−1, in the graphitic ‘G’ band of the Raman spectra. This novel approach led to an increase in the yield of mono to double layer graphene sheets (about 60%). The charge–discharge measurement shows improved Li-ion charge–discharge properties of the imGNS with an increase in the concentration of 1-PSA molecules in comparison with starting graphite materials. This report provides a new direction for graphene exfoliation and surface modification for technological applications.

Predicting nanovibration of multi-layered graphene sheets embedded in an elastic matrix

A continuum-based plate model is proposed to study the vibration behavior of multi-layered graphene sheets (MLGSs) that are embedded in an elastic matrix. A set of explicit formulas is derived to predict the natural frequencies and associated vibration modes of double-layered and triple-layered graphene sheets that are embedded in an elastic matrix. Numerical simulations are carried out to examine the effects of the van der Waals (vdW) interaction on the natural frequencies. The results show that for a given combination of wavenumbers ( m, n), the lowest natural frequency of a MLGS is independent of the vdW interaction, but all of the other higher natural frequencies depend significantly on the vdW interaction. The influence of the moduli of the surrounding matrix is also investigated, and the results indicate that the classical natural frequency depends significantly on these moduli. For lower-order resonant frequencies, the effect of the surrounding matrix is very small and can be neglected, but the higher-order resonant frequencies depend significantly on the moduli, and the influence of the surrounding matrix must thus be taken into consideration. In addition, the resonance modes of a MLGS system are dominated by the vdW interaction, whereas the moduli of the surrounding matrix only slightly affect the relative amplitude ratios of the sheets and do not change the vibration direction.

An analytical approach to determination of bending modulus of a multi-layered graphene sheet

In this paper, the bending modulus of a multi-layered graphene sheet is investigated using a geometrically based analytical approach. For this purpose, a bending potential energy is derived, based on the van der Waals interactions of atoms belonging to the two neighboring sheets of a double-layered graphene sheet. The inter-atomic spacing between the adjacent layers is determined along the line of action of the applied bending moments. The bending potential of the double-layered sheet is calculated by summing up the potentials at discrete hexagons over the length and width of the sheet. A multi-layered graphene sheet is considered as consisting of many stacking double-layers. It is observed that the bending modulus of a multi-layered graphene sheet does not depend on the length of the sheet and is a property for the multi-layered sheet.

Continuum model for the vibration of multilayered graphene sheets

The vibration analysis of multilayered graphene sheets (MLGSs) using a continuum model is reported in this paper. An explicit formula is derived to predict the van der Waals (vdW) interaction between any two sheets of a MLGS. Based on the derived formula, a continuum-plate model is developed for the vibration of MLGSs. Our investigation indicates that the lowest natural frequency (classical natural frequency) of a MLGS for a given combination of $m$ and $n$ is independent of the vdW interaction, but that all of the other higher natural frequencies (resonant frequencies) are significantly dependent on this interaction. The mode shapes that are associated with the natural frequencies are investigated for double-layered and ten-layered graphene sheets. We find that the vibration modes that are associated with the classical natural frequency of all the sheets are in the same direction and have the same amplitude, whereas the vibration modes of the sheets that are associated with the resonant frequencies are different due to the influence of the vdW interaction. Thus various resonance modes can be obtained by varying the number of layers of a MLGS.