Third Day Poster Session, 17 June 2010 - NanoTR-VI

P
P Institute
P
P Department
Poster Session, Thursday, June 17
Theme F686 - N1123
Investigation of Natural Vibration Frequency of Graphene Sheet
1
1
1,2
Arman FathizadehP P, Masoumeh OzmaianP Pand UReza NaghdabadiUP
P*
1
for NanoScience and Technology, Sharif University of Technology, Tehran, Iran
of Mechanical Engineering, Sharif University of Technology, Tehran, Iran
2
Abstract- In this study, the vibration analysis of SLGs using molecular dynamic (MD) simulation as well as beam theory is reported for different
dimensions. Using these results, parameters that affect the answers obtained by continuum theory can be modified for more accurate results and
lower computational cost.
In recent years, graphene sheets have attracted lots of
interests because of their unique properties. It could be one of
the prominent materials for the nanoelectronic devices in the
future. But still limited work has been done on studying the
mechanics of graphene sheets.
Recently, some numerical and analytical models have been
proposed for the study of vibrational behavior of single and
multilayer graphene sheets (MLGS) [1-3]. Behfar and
Naghdabadi investigated the vibration behavior of MLGS
embedded in an elastic medium [1]. Kitipornchai et al. carried
out an analysis based on a continuum-plate model for MLGSs
by considering the Van der Waals force between the plates [2].
Sakhaeepour et al. calculated fundamental frequencies of
single layer graphene sheet (SLGS) using molecular
mechanics method [3].
Modeling in this paper is carried out by two methods,
continuum theory and MD simulation. Consider a SLGS
doubly clamped in two ends. The sheet is of length L, width is
w, thickness t, density and the Young’s modulus E. The
fundamental frequency according to the Euler-Bernoulli beam
theory is given by [4]
(1)
L
Lwt
1/2
2
2
At E A T
0.57
2 2
where A is 1.03 for doubly clamped beam and T is the tension
in the graphene sheet. The thickness is taken to be 0.34 nm
(Van der Waals radius for carbon atoms), the density is 2250
3
kg/mP Pand the Young modulus is 1.02 TPa.
1/2
the system of atoms to vibrate in the first mode. Then with a
Fourier analysis on variation of position of atoms or potential
energy of the system with time, corresponding frequency of
the system can be obtained.
In order to investigate the fundamental frequency of SLGS,
results are obtained for different dimensions by MD
simulation as well as beam theory. As it can be seen in table 1,
by increasing the aspect ratios of the graphene sheets, the
results obtained by the beam theory get nearer to the MD's.
There are many adjustable parameters which can affect the
results of continuum model. The most important parameters
are the parameter A, thickness and Young modulus of the
equivalent beam (t, E), and mass distribution in the continuum
model. A has a significant effect on the fundamental
frequency. Effect of (t, E) is also important and their values
for graphene are still under discussion. The mass distribution
shows a little effect on frequency, especially for bigger sizes
in which atomic spacing is negligible relative to sheet size.
Present study may be used as a new method of adjusting these
parameters for graphene to achieve more accurate results with
continuum models.
Table 1. Comparison of the fundamental frequency obtained using
MD simulation and continuum beam theory.
Dimensions
(GHz)
Length(nm) L/W MD Beam theory
8.98 2.18 395.78 265.78
12.32 6.22 185.42 225.15
24.47 12.36 94.21 110.05
35.54 17.95 56.07 48.89
Figure 1. Molecular dynamics method used for calculating first
natural frequency of grapheme sheets. This figure shows a vibrating
graphene sheet with aspect ratio of 6.22 schematically.
The molecular dynamics simulation is performed for 4
different aspect ratios ranging from 1000 to 3000 atoms.
These simulations are done with LAMMPS software [5] and
using well-known REBO interaction potential which has been
shown to be the most accurate one for study of mechanical
properties of carbon nanostructures [6]. In order to model the
doubly clamped boundary condition, two rows of atoms in the
SLGS are fixed in two ends while the other sides are free. All
of the atoms in the sheet are placed in such a way that they are
initially at the position at the first mode shape of the sheet
with velocity equal to zero. Then in a NVE ensemble, we let
*Corresponding author: naghdabd@sharif.edu
[1] K. Behfar, R. Naghdabadi, Comp. Sci. Tech., 65, 1159–1164
(2005).
[2] S. Kitipornchai, X. Q. He, K. M. Liew, Phys. Rev. B, 72, 075443
(2005).
[3] A. Sakhaee-Pour, M.T. Ahmadian and R.
Naghdabadi,vNanotechnology, 19, 085702 (2008).
[4] S. Timoshenko and D. H. Young, W. Weaver, New York, 425–
427 (1974).
[5] LAMMPS, An open source code for molecular dynamics
simulation, HThttp://lammps.sandia.gov/TH.
[6] D. Brenner, et al., J. Phys.: Condens. Matter.,14, 783–802
(2002).
6th Nanoscience and Nanotechnology Conference, zmir, 2010 732