Geometry and Self-stress of Single-Wall Carbon Nanotubes and Graphene via a Discrete Model Based on a 2nd-Generation REBO Potential

Abstract

The main purpose of this paper is to evaluate the self-stress state of single-walled Carbon NanoTubes (CNTs) and Flat Graphene Strips (FGSs) in their natural equilibrium state, that is, the state prior to the application of external loads. We model CNTs as discrete elastic structures, whose shape and volume changes are governed by a Reactive Empirical Bond-Order (REBO) interatomic potential of second generation. The kinematical variables we consider are bond lengths, bond angles, and dihedral angles; to changes of each of these variables we associate a work-conjugate nanostress. To determine the self-stress state in a given CNT, we formulate the load-free equilibrium problem as a minimum problem for the interatomic potential, whose solution yields the equilibrium nanostresses; next, by exploiting the nonlinear constitutive dependence we derive for nanostresses in terms of a list of kinematical variables, we determine the equilibrium values of the latter; finally, from the equilibrium values of the kinematical variables we deduce the natural geometry and, in particular, the natural radius. Our theoretical framework accommodates CNTs of whatever chirality. In the achiral case, the stationarity conditions implied by energy minimization are relatively easy to derive and solve numerically, because we can count on maximal intrinsic symmetries and hence the number of independent unknowns is reduced to a minimum; for chiral CNTs, we prefer to solve the minimum problem directly. The natural-radius predictions we achieve within our discrete-mechanics framework are in good agreement with the results of calculations based on Density Functional Theory (DFT) and Tight Binding (TB) theory; the same is true for our predictions of the self-energy, that is, the energy associated with self-stress (called cohesive energy in the literature); we surmise that our discrete mechanical model may serve as a source of benchmarks for Molecular-Dynamics (MD) simulation algorithms. We find that self-stress depends on changes in both bond and dihedral angles in achiral CNTs and, in addition, on changes in bond length in chiral CNTs. Our analysis applies also to FGSs, whose self-stress and self-energy we evaluate; we find that in FGSs self-stress is associated exclusively with changes in bond angle.