Investigation of Buckling Behaviors in Carbon Nanorings Using the Chebyshev-Ritz Method

Carbon nanorings (CNRs) serve as an ideal quantum system for novel electronic and magnetic properties. Although extensive theoretical studies utilizing molecular dynamics (MD) simulations have investigated the formation and structural characteristics of CNRs, systematically analyzing their properties across various toric sizes remains challenging due to the inherent complexity of this system. In this study, we introduce a novel finite element method, the Chebyshev-Ritz method, as an alternative approach to investigating the structural properties of CNRs. Previous MD simulations demonstrated that stable CNRs adopt a regular buckled shape at specific toric sizes. By meticulously selecting mechanical parameters, we observe that the critical deformation of a CNR with 50 repeated units, as determined by the Chebyshev-Ritz method, aligns with an MD simulation presenting a buckling number of 14. Additionally, the implementation of the Chebyshev-Ritz method with a constant mechanical parameter for 50 repeated units reveals a structural transition at varying toric sizes, leading to the stabilization of buckling numbers 13, 14, and 15. This structural transition across different buckling modes has also been corroborated by MD simulations. Our approach offers a reliable and accurate means of examining the structural properties of large-scale nanomaterials and paves the way for further exploration in nanoscale mechanics.