Vibration mode localization in single- and multi-layered graphene nanoribbons

We study vibrations of single- and multi-layered rectangular graphene nanoribbons (GNRs) using molecular mechanics (MM) simulations by employing the MM3 potential. Two sets of boundary conditions are considered, namely, clamping atoms on either all four edges (CCCC) or on the two small edges (CFCF). Furthermore, we consider two scenarios for a single-layered GNR - one in which an interior atom is held stationary and the other in which a bucky-ball is covalently bonded to an interior atom. For multi-layered GNRs an interior atom only on the outermost layer is either held fixed or has a bucky-ball covalently bonded to it. For CCCC single- and multi-layered GNRs, both scenarios are found to divide the GNR into two differently vibrating regions such that in one region atoms have negligible while in the other region large out-of-plane displacements; we call this mode localization. For multi-layered GNRs, mode localization in the outermost layer leads to cooperative mode localization in the remaining layers. We also study vibrations of prestretched CFCF single-layered GNRs with and without a covalently bonded bucky-ball, and find that the attached bucky-ball localizes modes in a certain region of the single-layered GNRs. For an unstretched single-layered GNR a very interesting result from MM simulations is that one region undergoes bending while the other torsional vibration. The results for single-layered GNRs with CFCF boundary condition are correlated with those derived from continuum models, namely a stretched string-mass and a Kirchhoff plate. The frequency equation for the string-mass model is derived by solving the equation of motion using the Laplace transform technique. Frequencies of vibrations of the Kirchhoff plate are numerically found by using the finite element method. With increasing value of the prestretch the string-mass system is found to have bending mode frequencies that are closer to those of the CFCF single-layered GNR than those of the Kirchhoff plate. Using potential energy of deformation at each atom, for multi-layered GNRs with a fixed interior atom, and for single-layered GNRs with a covalently bonded bucky-ball, it is found that the classical parameter for quantifying vibration mode localization is not valid; hence a new parameter is defined. This work highlights the importance of modes of vibration for designing sensors to detect a mass attached to either a single- or a multi-layered GNR. (C) 2014 Elsevier B.V. All rights reserved.