CARBON NANOTUBE MECHANICS: Molecular Simulations & Continuum Models for Carbon Nanotubes

To realize the incredible structural applications potential of carbon nanotubes, it is important to characterize their material response. Molecular simulations offer advantages over physical testing due to their cost effectiveness, versatility and precision. Two continuum models for single wall nanotubes (SWNT) were previously developed based on the results from molecular simulations using two different potentials. The continuum models have been found to predict both global and local responses for buckling well. Radial expansion and contraction simulations of double wall nanotubes confirmed that the assumption of isotropy of a nanotube wall response is accurate. The SWNT continuum models were used as the basis to model multi-wall nanotubes (MWNT). In continuum mechanics, the equivalence to the van der Waals forces is pressure. Using the results from the expansion/contraction simulations the pressure between two walls was defined as a function of the wall separation. A continuum model for MWNTs was developed using finite element (FE) analysis with shell elements for the walls and truss elements to substitute for pressure. The predictions from the FE models are compared to molecular simulations. Preliminary work on nanotube/polymer composite material is also presented. Introduction Since their discovery in 1991 by Iijima1, both single wall and multi-wall carbon nanotubes (SWNTs & MWNTs) have become an active area of research. This is partly due to their having an extremely high specific strength and stiffness. These properties and their cylindrical shape allow for their potential applications in such diverse fields as fibrous reinforcement, atomic level piping and nanostructures. These structural applications of carbon nanotubes require that we ascertain their macroscopic properties. In the simplest terms, carbon nanotubes can be thought of as rolled up, closed graphite sheets. This sheet can be rolled up at different discrete angles to create the SWNT which can be described by the hexagonal base vectors (a,b)2. Thus, an SWNT is a single, closed molecule with few to no atomic imperfections, and a hexagonal ring bonding structure similar to graphite. The molecular structure of an example SWNT is shown in Figure 1, next to a fullerene which Iijima was attempting to create when he discovered nanotubes. The ring bonding structure, specifically the hybridized sigma bonds (sp), imparts the impressive mechanical properties. Multi-walled nanotubes also exist, consisting of nested SWNTs 3.4 Å apart, which widen the range of tube properties and application possibilities. Experimental studies have been performed to characterize nanotube materials. Due to the difficulty in performing these experiments, the scatter in the results is considerable and the stiffness of carbon nanotubes was found to lie between 0.5 4 TPa3-6. Stating these values underscores two common continuum assumptions, first that they are linear elastic elastic materials and have a thickness corresponding to density of graphite (t = 0.34 nm). Naturally, with such poor resolution, the validity of these assumptions can not be checked with experiment. Atomic simulations have proven to be a good vehicle for studying nanotube mechanical responses. Molecular mechanics has been used to obtain Young’s moduli of 0.75 1.25 TPa7-10, using the same assumptions as detailed above. Molecular mechanics is a method of modeling the interatomic forces, including bonding and non-bonded forces, with simple polynomial and trigonemetric expansions. This method was developed for the modeling of large molecules such as proteins and lends itself well to the study of nanotubes. The advantage of molecular mechanics is the precision with which tests can be performed. Not only can virtual tension tests be performed, but other tests such as torsion can be performed which are not yet feasible experimentally. The accuracy of these methods vary, but have been shown to predict moduli with the same precision as quantum mechanical methods11-13. A listing of the experimental and molecular simulation modulus results from various studies can be found in last years review.