The field of nanotechnology has been quickly growing over the last few decades and many different functional Nano Electro Mechanical Systems(NEMS) can now be made.
To further increase the possibilities and functionality of NEMS, new materials have to be characterized. Of the new materials which become available at the nanoscale, graphene is one of the most promising. This is mainly due to the combination of its extraordinary strength, electric and thermal conductivity, and low weight.
In order to be able to use graphene's full potential in future applications, the elastic properties, such as the Young's modulus and the bending rigidity, have to be known.
These elastic properties have been obtained following different approaches. However, the obtained values are scattered. This scattering has a few reasons.
Firstly, experimental research into graphene is difficult, due to the small scale, big influence of the environment, and the difficulty of fabricating well defined graphene membranes.
Secondly, graphene is a purely two-dimensional material. Therefore, continuum theory is not easily applied. The bending rigidity for example is not related to thickness as it is in a continuum plate.
Finally, graphene exhibits strong mode coupling and is always vibrating due to Brownian motion, the motion which results from the stochastic excitation due to temperature. Parameters extracted from static measures may thus not match the reality or experiments. In conclusion, the elastic properties have to be obtained from the dynamic response, following a multi-modal approach.
As graphene, only one atom thick, is close to the atomic scale, Molecular Dynamics simulations are used to investigate its behavior. To extract parameters, these simulations are compared to an analytical continuum model. In this way, the advantages of continuum mechanics and Molecular Dynamics simulations are combined with a dynamic, multi-modal approach to extract the bending rigidity and Young's modulus of graphene.
In the continuum model, the equations of motion of a circular graphene plate and membrane are derived from a Lagrangian approach. The governing equations are discretized using admissible functions that satisfy boundary conditions. Furthermore, the geometric nonlinearity is included, as graphene is so thin, and is thus easily driven into the nonlinear regime.
The basic principle of Molecular Dynamics simulations is to solve Newton's equation of motion for every single atom of a system. The force acting on the atoms is described by a potential field. The equations of motion are then integrated over time.
The mode coupling in graphene is shown to be so strong that the energy in all modes is equal after some time. Therefore, the only steady state attainable is the state in which the energy in all modes is equal, which corresponds to the Brownian motion. The eigenfrequencies are obtained from the time response of the atoms in the graphene membrane, excited by Brownian motion. The obtained eigenfrequencies are compared to the values obtained from the continuum model. From this comparison the bending rigidity of graphene is extracted. This is done following an optimization approach, which minimizes the difference between the eigenfrequencies obtained from the continuum and the Molecular Dynamics model. Including multiple modes is shown to be a necessity for reaching convergence.
Furthermore, the Young's modulus is obtained. This is done by comparing the geometric nonlinear behavior of graphene obtained in Molecular Dynamics with a continuum prediction of this behavior.
The bending rigidity and the Young's modulus thus have been obtained, independently, from the dynamic response of graphene obtained in Molecular Dynamics.