Investigation of Mechanical Properties of Diverse Nanotubes via Molecular Mechanics

Moritz Armbruster

The Cooper Union for the Advancement of Science and Art

Research Experience for Undergraduates Sound and Vibration Research Summer 2004

Introduction

There are many different atomic compositions of nanotubes being discovered; additionally, there are also several different conformations.  This project concerned itself with the conformation defined by carbon, a repeating structure of hexagonal cells.  The goal is to model the nanotubes as a continuous object for which standard mechanical properties can be found.  To accomplish this, several different nanotubes were chosen and then modeled on the molecular scale to find the necessary information to model them as a continuum.  From the continuum model, the mechanical properties such as the young’s modulus and the effective thickness can be found.

Diverse Nanotubes

Carbon

Carbon nanotubes are the standard bearers for nanotubes due in part to their high strength.  Additionally, they are the most analyzed and researched of the nanotubes considered here. Despite this research, there seems to be little consensus on how to translate a molecular model into a continuum model. Which, can then be used to find standard material properties such as young’s modulus, poisson ratio, and an effective thickness of the continuum sheet. 

Boron-Nitride

Boron-Nitride has the closest structure to carbon due to the atomic similarities.  There has been some work done on boron-nitride; however, on the applied side it has been limited, in part due to boron-nitride being pyrophillic.

Aluminum-Nitride

Aluminum and boron are in the same atomic group and hence have similar properties; as a result, aluminum-nitride also shares the same conformation as boron-nitride and carbon.  Some research has been done on aluminum-nitride, but it has not been as extensive as either boron-nitride or carbon.

Aluminum-Phosphide

This atomic arrangement is speculated to share a conformation with the rest of the chosen nanotubes due to phosphorus and nitrogen sharing an atomic group.  To our knowledge there has been no work done on aluminum-phosphide as a nanotube; however, our simulations showed promising results in terms of its stability as a flat sheet.

 Zinc-Oxide

Molecular Mechanics Modeling

Nanotubes themselves are large and difficult to model due to the computational time involved.  Instead planar sheets were used as a representative element of a nanotube with the same conformation as would be used in a nanotube.  The repeating hexagonal cell structure lends itself to smaller representative sheets.  The sheets ranged in size from 20 atoms in addition to hydrogen atoms surrounding the sheet to cap off the extra bond sites, to 272 atoms in addition to the capping hydrogen atoms.  Two computational methods were used Ab-Initio method and a Force Field Method.  The sheets were strained axially, and then strained laterally in increments to find the minimum energy and hence the poisson ratio.  When the minimum is found the strain energy can be found by comparing the total energy to that of the unstrained sheet.  Similarly, the sheet can also be bent and as a result the strain energy can be found.

Ab-Initio

The Ab-Initio method estimates the wave functions for the electrons of the atoms being modeled.  This results in a very computationally intense model, but also very accurate.  The computational time grows exponentially with the number of atoms considered.  As a result, only small sheets are able to be modeled.  To remain consistent with earlier studies the size 24 sheet was used for all the Ab-Initio simulations.

Force Field Model

The program APT was used to model the entire range of sheet sizes.  APT is based upon the Universal Force Field(UFF) and its main advantage over Ab-Initio is speed, for APT is far less computationally intense than the Ab-Initio calculations.  The UFF utilizes a variety of factors ranging from bond radii, bond angles, atomic arrangement, Van der Waal’s forces...  It evaluates these factors by using empirically determined constants which are set for each arrangement of each atom.   These values were occasionally adjusted to better match the results from the Ab-Initio calculations.  As a result, the force field model presents more results and could also be used to model an entire nanotube; however, it was not possible in the time frame of this study.

Summary

The effective thicknesses were very consistent through the entire range of sheet sizes, with exceptions only for sheets which would not orient correctly.  Additionally, the effective thickness of carbon corresponded with results from other studies using similar methods.  The thickness for aluminum-phosphide is greater than that of carbon, this can be explained through the thickness being related strongly to the bending stiffness, rather than the axial stiffness.  For carbon and boron-nitride, the young’s modulus had a steady and significant increase as the sheet size increased.  This is not correlated with results from other studies and deserves further consideration to attempt an explanation.  The values for the young’s modulus for carbon are in a reasonable range based upon earlier investigations.  Aluminum-nitride, aluminum-phosphide, and zinc-oxide all have consistent young’s moduli.  Additionally, the trend between boron-nitride, aluminum-nitride, and aluminum-phosphide is of a decreasing young's modulus when going from boron to aluminum and from nitrogen to phosphorus which was predicted based upon the atomic positions and strengths.  An interesting phenomenon with the APT results is that boron-nitride overtakes carbon for the largest young’s modulus as the sheet sizes increase.  This was unexpected; but, due to lack of empirical comparisons of the two atomic configurations no real conclusion can be drawn immediately.

Zinc oxide results had some stable sheets while others preferred to be slightly bent, indicating the lack of an energy minimum.  As a result, this would indicate that zinc oxide has marginal stability at best, but would probably not take this confirmation naturally.  Zinc oxide also had significant discrepancy in the poisson ratio between the Ab-Initio and APT models, this is due to how the APT model was adjusted to match bond lengths and ignored bond orders.  Aluminum phosphide did not demonstrate any stability problems hindering it from becoming a nanotube.

Further studies can expand upon this to include the piezoelectric properties of the nanotubes; as, preliminary trials indicated a significant change in the dipole of boron-nitride which would imply piezoelectric tendencies of the sheet.  Additionally, trials on entire nanotubes would be a nice benchmark see how representative these smaller sheets are of an entire nanotube.

Solid Mechanics

With the strain energies and poisson ratio that were found using the  molecular mechanics systems, it is possible to model the sheets as continuous elements using the following equations:

Acknowledgments

I would like thank the National Science Foundation and the Army Research Office for providing the funding to make this program possible.  In addition, to my mentors I would also like to thanks Ryan Hoekstra for his help.