Topology optimization of branched carbon nanotubes

V. Brückmann, F. Niederhöfer, J. Wackerfuß

Carbon nanotubes (CNT) have excellent mechanical behavior and are favored for the construction of bottom-up structures. In this construction a junction of three CNTs, under angles of 120°, is an important component. A naive transfer of the hexagonal lattice of CNTs to the junction is not possible. The optimal topology of these junctions is unknown.

The aim of this project is to determine the optimal atomistic structure for junctions connecting three CNTs. The optimal solution is defined by a minimum of the internal energy of the atomic structure. The internal energy consists of a sum of interatomic potentials. As the interatomic potential depends on the topology of the atomistic structure and the atom positions, atom positions have to be considered as part of the optimization as well. Varying the atom positions (continuous variables) and the topology (binary/integer variables) of the atomistic structure leads to a mixed integer nonlinear problem (MINLP). The nonlinearity of the objective function is caused by the nonlinear interatomic potential definition. The formulated MINLP is solved with the General Algebraic Modeling System (GAMS). An approximation with a convex problem is required to ensure a global optimum.

Illustration of a branched carbon nanotubes (Y-junction): hexagonal lattice in the region of the regular tubes and the unknown lattice in the transition region.
Definition of the region, where the integer variables (representing the carbon atoms) can take place (red box).

Definition variants defining the start configuration of the MINLP.

Type 1: (8,2,1,1,0)
Type 2: (8,2,2,0,0)
Type 3: (8,2,3,1,0)
Type 4: (8,2,4,0,0)
Type 5: (8,2,5,1,0)
Type 6: (8,2,6,0,0)
Type 7: (8,2,7,1,0)
Type 8: (8,2,7,1,1)
Type 9: (8,2,10,0,1)
Type 10: (8,2,12,0,2)
Type 11: (8,2,15,1,2)

  Results of the relaxation process performed by GAMS (left: top view, right: side view)


Brückmann V.: MINLP for the Topology Optimization of Branched Carbon Nanotubes – Modelling, Implementation with GAMS and Calculation. Master Thesis 2011 (supervisor Prof. Wackerfuß & Prof. Ulbrich), Technische Universität Darmstadt.  

This project has been realized in cooperation with S. Ulbrich (Technische Universität Darmstadt, Nonlinear Optimization).