The design of many novel electronic devices will hinge on our understanding of the joining of certain nanostructures. In particular, the perpendicular joining of a carbon nanotube to a flat graphene sheet applies to the situation of connecting to an electronic platform. Connecting carbon nanostructures essentially involves a discrete geometric procedure, and the present authors have attempted to solve such problems by invoking the principle that the bond lengths and bond angles at the join are determined in such a manner that their total squared deviation from some ideal configuration is a minimum. Other authors suggest that carbon nanotubes might be deformed in such a way that their total curvature squared is minimized. From a theoretical standpoint, any continuous approach to such essentially discrete problems could be a valuable tool in obtaining the main qualitative features at the join. Here we propose a continuous variational approach to the determination of the join geometry assuming that the curvature is minimized for prescribed join lengths and defect geometries. We find that the variational model provides good overall agreement with the least-squares method in terms of the nanotube attachment height. Although the agreement in participating atomic positions is not quite as good, the absolute error in the positioning of participating atoms is less than 0.18 angstrom. Current experimental data does not exist to determine which procedure gives the more realistic results.
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