Geometry of nanocarbon structures

On the geometric construction of self-similar carbon nanotube-based structures using nonlinear transformations of scalable building blocks

C. Schröppel, J. Wackerfuß

The construction of nanocarbon fuel tubes (CNTs) and higher-order structures based on nanocarbon tubes can be conceptually understood as an iterative process in which the bonds between carbon atoms are replaced by nanocarbon tubes. However, this intuitive approach does not provide a systematic method to construct the Y-shaped connectors necessary to connect the nanocarbon tubes.

It is, however, possible to construct such configurations using nonlinear transformations of scalable basic building blocks. Using this approach, for example, a nanocarbon tube of order 0 can be constructed by successively applying linear and nonlinear transformations of a carbon-carbon bond. In turn, a nanocarbon tube of order 1 can be constructed from graphene-based basic elements, i.e. structures of order 0. In particular, basic elements of a given order can always be used to construct basic elements belonging to the next higher order, so that configurations of any order can be constructed using this method.

The video sequences included in the following sections illustrate the method presented here.

Video sequence 1 describes the construction of a Y-shaped nanotube interconnect that continues at the ends to form fully formed nanotubes. In the first step of the process, a planar graphene layer is constructed from an imaginary half of a carbon-carbon bond (step 1). After the graphene layer is constructed, most of the virtual dots (shown in red) are removed in the process of joining two halves of each carbon-carbon bond. However, some of the virtual dots remain - these dots can be used as junctions to which further pieces of bonds can be added to construct higher order configurations, or they can be removed along with the pieces of bonds at the end of the construction of the desired configuration.

The graphene layer is then subjected to a sequence of nonlinear transformations. The goal of the nonlinear stretching (step 3), bending (step 4) and compression (step 5) is to map all points of the layer that are on the grayed plane at the beginning of step 3 so that they are back on the same plane at the end of step 5. This makes it possible to construct a monolithic connecting piece of a higher order with the help of the following steps of the procedure, which only consist of orthogonal transformations, i.e. mirroring (step 6) and rotation (steps 7 and 8).

Video sequence 2 shows how a nanocarbon tube-based connector of order 1 can be constructed from the unit element of order 0. This unit element results from step 5 of part 1 of the construction procedure (see video sequence 1). Using the same transformations as in part 1 (with individual corrections due to the difference in size of the elements), a unit element of order 1 is constructed in steps 1 to 5. This unit element in turn serves as the starting point for the construction of the nanotube connector of order 1. As with the nanotube connector of order 0, the remaining virtual points (shown in red) can be used to join two (or more) connectors monolithically.

Publication

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