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

Due to a variety of applications of nanoscaled materials, several researchers further investigate a joining between two nanostructures as a candidate for new potential applications. Here, the vertically joining between the nanotube with the nano-torus is investigated. Variational calculus is used to predict the joining curve between two nanostructures based on minimizing the elastic energy. Moreover, Willmore energy is also utilized to determine the join region especially for a three-dimensional structure. Since the surface of a catenoid is a minimizer obtained by the Willmore energy function, it is used to join two symmetric nanostructures. We find that these two approaches have less than 10% difference in the positions of the joining curves. These two methods might be used to design other hybrid nano-scaled structures.