Joining curves between nano-torus and nanotube: Mathematical approaches based on energy minimization

Numerous types of carbon nanostructure have been found experimentally, including nanotubes, fullerenes, graphene and nano-torus. Each of these structures possesses unique physical, chemical and electronic properties with potential application in various nanoscale devices. Carbon nanotube is one form of the nano-scaled structures formed by rolling up a flat graphene sheet where carbon nano-torus is formed by bending carbon nanotube and connecting its two ends. To improve properties of nano-scaled structures, the combination of two or more structures is needed.
Variational calculus is utilized to determine the curve adopted by a line smoothly connecting a nano-torus and a vertical carbon nanotube, such that the arc length of the curve is specified. However, the distance in the y-direction of the join is not prescribed and determined to be part of the solution. Moreover, Willmore energy functional is utilized to predict the shape of two joined nanostructures in three-dimension. The Willmore energy functional that depends on the axial curvature and the rotational curvature.
             
Fig. 1: 3D joining between torus and tube.
The 3D joining structure between torus and tubes are shown in Fig. 1. We compare our results from both approaches by mapping 3D structure from the Willmore energy to 2D structure in order to compare with the curve obtained from the calculus of variation. The percentage difference is computed by fixing values of y and comparing the different values of x. We find approximately less than 10% difference in the positions on the joining curves. To the best of our knowledge, this combined structure has not been confirmed experimentally, and there is no real data to measure. All in all, the structures obtained by these two approaches are comparable and these mathematical derivations might be thought of as a first step to design new hybrid structures.
Ref: Panyada Sripaturad and Duangkamon Baowan 2021 Z. Angew. Math. Phys. 72:20