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We describe the results of Brownian dynamics (BD) simulations of an atomic force microscope (AFM) tip scanned on locally suspended few-layer graphene. The effects of surface compliance and sample relaxation are directly related to the observed friction force. We demonstrate that the experimentally observed reduction of friction with an increasing number of graphene layers in case of a narrow scanning tip can be a result of decreased sample deformation energy due to increased local contact stiffness under the scanning tip. Simulations with varying scan rates indicate that surface relaxation at a given temperature can affect the frictional characteristics of atomically thin sheets in a manner not explained by conventional thermally activated models.

A parametric interatomic potential is constructed for graphene. The potential energy consists of two parts: a bond energy function and a radial interaction energy function. The bond energy function is based on the Tersoff-Brenner potential model. It includes angular terms and explicitly accounts for flexural deformation of the lattice normal to the plane of graphene. It determines the cohesive energy of graphene and its equilibrium lattice constant. The radial energy function has been chosen such that it does not contribute to the binding energy or the equilibrium lattice constant but contributes to the interatomic force constants. The range of interaction of each atom extends up to its fourth-neighbor atoms in contrast to the Tersoff-Brenner potential, which extends only up to second neighbors. The parameters of the potential are obtained by fitting the calculated values to the cohesive energy, lattice constant, elastic constants, and phonon frequencies of graphene. The values of the force constants between an atom and other atoms that are within its fourthneighbor distance are calculated. Analytical expressions are given for the elastic constants and the flexural rigidity of graphene. The flexural rigidity of the graphene lattice is found to be 2.13 eV, which is much higher than 0.797 eV calculated earlier using the Tersoff-Brenner potential.

This paper explores the use of lattice Green's functions for calculating the static structure of defects in lattices, in that the atoms of the lattice interact with their neighbors with an arbitrary nonlinear (short-range) potential. The method is hierarchical, in which Green s functions are calculated for the perfect lattice, for increasingly complicated defect lattices, and Bnally the nonlinear structure problem is iterated until a converged solution is found. For the case where the defect must be embedded within a very large linear system, and the slip plane, cleavage plane, nonlinear zone, etc. , can be made small compared to the system size, Green's functions are a very powerful method for studying the physics of defects and their interactions. As an illustration of the method, we report numerical calculations for an interfacial crack emitting dislocations from an interface between two joined two-dimensional hexagonal lattices. The supercell size was 4 x 10e, and the crack length was 101 lattice spacings. After the Green's functions were obtained for the defective lattice, the dislocation and crack structures were obtained in a minute or less, making possible detailed studies of the defects with various external loads, force laws, defect relative positions, etc. , with negligible computer time. With practical supercomputer times, supercell and defect sizes one or two orders larger are feasible, thus making possible a realistic calculation of three-dimensional nucleation events on cracks, etc.

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