# Geometry of carbon nanoscale structures

## Geometric construction of self-similar carbon nanotube-based structures based on non-linear transformations of scalable base elements

*C. Schröppel, J. Wackerfuß*

The construction of nanotubes and other nanotube-based structures of higher order can be conceptualized as an iterative process of replacing the bonds between carbon atoms by nanotube elements. However, this intuitive approach does not provide a systematic method to construct the Y-shaped junctions that are needed to connect the nanotube elements.

It is, however, possible to construct such configurations using nonlinear transformations of scalable basic building blocks. Using this approach, a nanotube of order 0, for example, can be created by successive linear and non-linear transformations of a carbon-carbon bond, while a nanotube of order 1 can be constructed with graphenes-based unit elements, i.e. structures of order 0, as building blocks. In particular, unit cells of a given order can always be used to construct the unit element belonging to the next order, so that configurations of arbitrary order can be contructed.

The video clips presented in the following sections illustrate this method.

## Construction of a CNT junction

**Video clip 1** illustrates the construction of a Y-shaped nanotube junction, branching out into three full-fledged nano-tubes. Initially, a planar graphene sheet is constructed from a virtual half of a carbon-carbon bond (Step 1). After the construction of the graphene sheet, most virtual points (shown in red) are being eliminated in the process of combining two halfs of a carbon-carbon bond to a full bond. A few of these virtual points remain – these points can serve as docking positions where junctions can be concatenated in order to construct configurations of higher order, or they can be discarded along with the partial bonds at the end of the construction of the desired configuration.

Subsequently, the graphene sheet undergoes a sequence of non-linear transformations. The purpose of the non-linear stretching (Step 3), bending (Step 4), and shrinking (Step 5) is to map all points of the sheet that are located on the grey plane at the start of Step 3 onto points that are also located on the same plane after the completion of Step 5. This allows to construct a monolithic junction of higher order in the following steps of the procedure, which simply consist of orthogonal transformations, i.e. reflections (Step 6) and rotations (Steps 7 and 8).

## Construction of a CNT junction of order 1

**Video clip 2 **shows how a nanotube junction of order 1 can be constructed from the unit element of order 0, which results from Step 5 of Part 1 (see video clip 1). Using the same transformations as in Part 1 (with some adjustments due to the different overall size of the element), a unit element of order 1 is being constructed in Steps 1 to 5. This unit element, in turn, serves to build a nanotube junction of order 1. As with the nanotube junction of order 0, the remaining virtual points (shown in red) can be used to monolithically connect two (or more) junctions.

Publication

Schröppel, C & Wackerfuß, J 2012, 'Algebraic graph theory and itsapplications for mesh generation', PAMM Proceedings in Applied Mathematics and Mechanics, Volume 12, S. 663-664.