Hope College Department of Physics and Engineering  Research Experiences for Undergraduates  Summer 2002  Project Summary

Project Title: Truss Modeling of Carbon Nanotubes
Student Name: John Siehling
Student’s home institution: Hope College
Research Advisor(s): Dr. Pawloski
Source of Support (NSF-REU, or other): NASA
 

Summary
Our goals this summer were to study the behavior of several different nanotubes, including zigzag, armchair and chiral tubes using a truss model and to determine if any generalizations could be made about the basic mechanical properties of these nanotubes.
 

Background
Carbon nanotubes were discovered in Japan in 1991 and ever since have fascinated physicists, chemists and material scientists because of their extraordinary electronic properties and amazing stiffness and strength.  Their basic structure consists of carbon atoms bonded in hexagons shaped into a tube as can be seen in figure 1.
 

Figure 1: The basic structure of a nanotube	Figure 2: Interactions between the atoms

There are three different types of carbon nanotubes, and they are distinguished using pairs of integers of the form (n,m).  The zigzag tubes are all the tubes where m=0, the armchair tubes are all the tubes where m=n, and all other combinations of m and n are chiral tubes, where the hexagons spiral up the cylinder.

 There are two main interactions that the carbon atoms are subject to: type A is a short strong covalent bond between the carbon atoms, and type B is a longer and weaker interaction based on the carbon bond orientations, as can be seen in figure 2.  This is based on work by Odegard, et. all.  These two types of interactions can then be modeled as truss elements.
 

Work Accomplished
Our research this summer focused on using a truss model to analyze the nanotubes.  To do this, we first used a Maple 7 procedure to determine the coordinates of the bottom set of nodes in a specified tube.  The rest of the nodes were then propagated from this base set of nodes and the two types of elements were used to connect the nodes.  We used the finite element modeling software ANSYS to build the truss model. Using ANSYS, we tested each tube at various lengths by applying axial displacements at the ends and radial displacements along the length of the tubes.

The tubes subjected to axial displacements yield a Force/Radial Strain ratio, which is related to the modulus of elasticity, and an Axial Strain/Radial Strain ratio, which is related to Poisson’s ratio.  The force measured was the axial force reactions at the end constraints.  These were then plotted vs. radius and could be analyzed.  The Force/Radial Strain vs. Radius plot showed that there was a near linear fit of the data.  Using Maple 7, two possible fits of these points were acquired; one was a least square linear fit, and the other was a least square polynomial fit.  A common problem when trying to find the modulus of elasticity for carbon nanotubes is that the wall thickness is not very well defined, and so the stress is difficult to determine.

The tubes subjected to radial displacements yield a Force/Axial Strain ratio and a Force/(Axial Strain*Length) ratio.  The force measured was the radial force reactions at all of the nodes.  Due to time limits, these ratios were not analyzed as much as the axial displacement ratios.
 

Future Goals
From this data, we can now attempt to find an appropriate thickness to use for the walls of the nanotube, and from there we can find more mechanical properties.  The radial displacement data will also be analyzed more.  We also plan to get a larger version of ANSYS to analyze tubes with larger radii, to determine which fit is most appropriate.