Energetics, Structure, Mechanical and Vibrational Properties of Single Walled Carbon Nano Tubes (SWNT)

Abstract

In this paper we present extensive molecular mechanics and molecular dynamics studies on the structure, energetic, mechanical and vibrational properties of carbon nanotubes. In our study we employed an interaction potential derived from quantum mechanics accuracy of which was shown earlier for graphite and fullerenes. The simulation studies were primarily carried out using the MPSim Program -a Massively Parallel Simulation Program developed at the Materials and Process Simulation Center.

We first explore the stable structures of armchair (n,n), zigzag (n,0), and chiral (2n,n) single-walled carbon nanotubes (SWNTs) with different n leading to different radius. Based on the variation of energy as a function of curvature, we used a classical thin plane approximation, and calculated the bending modulus of the nanotubes.

With an enthalpy optimization method, we have determined the packing, structure and lattice parameters of (10,10) armchair, similar radius (17,0) zig-zag and and (12,6) chiral forms in the bulk. Using the second derivatives of potential energy surface of these optimized structures we calculated the moduli and vibrational frequencies and modes for the armchair, zigzag and chiral forms.

We performed computational experiments on the fibers undergoing tensile, compressive and bending loadings. To address the mechanical stability of these isolated SWNT structures. Then using a least squares fit to energy-strain relationship, we determined the moduli for each studied system.

Introduction

Since its discovery in 1991 by Iijima of NEC Corporation[1] efforts in synthesis, characterization and theoretical investigation on carbon nanotubes has grown exponentially. This is mostly due to their novel mechanical and electronic properties and their tremendous potential for future technological applications. In 1993, Iijima's group[2] and an IBM team headed by Bethune[3] independently discovered that they could make the simplest kind of carbon nanotubes, namely single walled carbon nanotubes, SWNT. These SWNTs can be regarded as a rolled-up graphite sheet in the cylindrical form.

This discovery was especially important, since some specific defect-free forms of these SWNT show remarkable mechanical properties and metalic behavior. Therefore, they present tremendous potential as components for use in nano electronic and nano-mechanical device applications and or as structural elements in various devices. Thess and coworcers[4] later produced crystalline ropes of metallic carbon nanotubes which consist of 100 to 500 SWNTs as a two-dimensional triangular lattice. These highly oriented linear "ropes" are expected to have remarkable mechanical properties, as well as superior electronic and magnetic properties. There are various levels of studies performed on the properties of the SWNTs including use of classical molecular mechanics, molecular dynamcis, tight binding level Quantum Mechanical methods[5-7].

In this paper, we present a detaile study of the energetics, structures, and mechanical properties of the tubes with different radius and different chirality: (armchair (n,n), chiral (2n,n), and zigzag (n,0)). We have used a quantum mechanically derived accurate classical force field to represent the interactions between the carbon atoms [8]. These interaction potentials were used earlier in studying structure, mechanical and vibrational properties of graphite, various fullerenes and intercated compounds of fullerenes [9] and nano tubes [10]. In our studies we employed classical molecular dynamics and molecular mechanics methods as implemented in MPSim, (a massively parallel program for materials simulations) program [11]. Molecular dynamics runs are made to anneal the structures, whereas molecular mechanics, energy and/or enthalpy minimization, is applied at the end of annealing cycle to obtain the final optimized structures. Using the analytical second derivatives of the potential energy of the tubes we also calculate the vibrational modes and frequencies of almost equal radius nanotube bundles, (10,10) arm chair, (17,0) zig-zag and (12,6) chiral.

Energetics and the Stability of Circular versus Collapsed Tubes

We used a valence force field developed for graphite, and various fullerenes and their doped forms (C₆₀, C₇₀, K_nC₆₀, etc.)[8] which were earlier shown to be highly accurate in predicting the structure, energetics and mechanical properties of fullerenes.

In order to asses the energetic stability of different cross sectional shape of SWNTs, we generated two sets of initial structures for each form (n,n) armchair, (n,0) zigzag, and (2n,n), specific chiral. First set were with a perfect circular cross section and the second set were almost fully collapsed, belt like. To mimic an infinitely long isolated single nanotube, we imposed a periodic boundary condition in c-direction,(tube direction), but we set the cell parameter: a and b as 50 times the circular tube diameter so that there were no tube-tube interactions. Energy and structural optimizition on these structures were carried out using MPSim. At low radii -circumference- values only tubes with the circular cross section resulted for all three forms. The values of radii range between 10.5 Å and 11.0 Å. As this treshold value for radia is crossed both collapsed "belt-like" and circular cross sections emerged as possible stable structures at the end of the energy-structure optimizations. Upto R ~ 30.0 Å, the circular cross section single wall nanotubes were energetically more stable for all forms. Once this second treshold is crossed the collapsed form became energetically stable form. The collapsed large circumference tubes produce almost two layer structure with a comparable layer distance as in graphite in the middle section of the tubes. The collapsed structures become energetically preferred due to increasing van der Waals attraction between opposing sides of the collapsed tubes. In Figure[1], we plotted the per atom energy (is measured relative to graphite) versus the tube radius upto 170 Å.

To further illustrate our findings in Figure[2] the projections of some stable armchair (n,n)

structures along the tube axis are presented. Zigzag (n,0) and chiral (2n,n) have the same characteristic with a similar transition radius. However, the inter-plane distances between the flattened sections and the curvatures of the two ends are similar. Figure[3] are the side views of

Figure 3a: Armchair (30,30)

Figure 3b: Chiral (50,25)

Figure 3c: Zigzag (50,0)

the two attracting layers for three cases. The inter-layer stacking patterns are different due to different chirality, thus the inter-layer distance with d_(n,n) = 3.38(Å),, d_(2n,n) = 3.39(Å), and d_(n,0) = 3.41(Å). Energetically, inter-layer attraction of armchair is the best with per atom energy E = 0.7336 (kcal/mol), and the double layer stacking is almost identical to the graphite stacking. The inter-layer attraction in the zigzag form is the worst with per atom energy, E = 0.7439 (kcal/mol), since the Carbon atoms on different layers are lined up on top of each other. The attraction energy per atom for the collapsed (2n,n) chiral nanotube is in between the two. Overall, in terms of transition radius and the cross-over radius, the size of the circular tube(radius) is the dominant factor in deciding the stable forms, although the chirality does have an impact on inter-layer spacing of collapsed region.

Based on the above optimized structures and their energies, we can model the basic energetics of circular tubes by approximating the tube as a membrane with a curvature 1/R and bending modulus k [11]. Assuming the thickness of tube wall as a, and modulus of the sheet as k, the elastic energy stored in a slab of width L, is given by

The per atom energy can be written as

where N is the number of carbon atoms per slab and E_o is energy per carbon atom for tubes with 1/R ~ 0, i.e. flat sheets. Considering the number of carbon atoms per unit area of tube wall, we have

Setting a as the spacing between two graphite sheets, 3.335(Å), R₀ = 1.410(Å) as the C-C bond distance, we obtained k_(n,n) = 963.44(GPa), k_(n,0) = 911.64(GPa), and k_(2n,n) = 935.48(GPa). The results are plotted against the theoretical estimates in Figure[4].

For the collapsed tubes, the curvature is near zero except at the two ends, which are close to be circular, the cross-over radius are dictated by the balance between the van der Waals attraction energy of the flat region and the strain energy stored in the two curved ends.

Stretching and compressing tubes at cross-over radius

In order to assess the tensile and compressive strength of the SWNTs, we also performed a series of compressive and tensile loading experiments by varying the c lattice parameter, along tube axis. With respect to the optimum c lattice parameter, c_o, the strain can be defined as

At cross-over radius, we calculated the energies of collapsed and circular tubes of (45,45), (80,0), and (70,35) under compression and tensions along tube axis. In Figure[5], the armchair, Figure[6], the zigzag, and Figure[7], the chiral forms, we observe a nonlinear dependence on the strain.

The tubes are softer under compression due to buckling effect. We present views along the tube axis and perpendicular to the tube axis in Figure[8]. Based on those energy strain curves, we also calculated "elastic constants", second the derivative of energy with respect to applied strain, e, along the tube axis as

Structure and Mechanical properties of Packed SWNT crystals

The (10,10) SWNT among others is special. We studied (10,10) armchair, and similar radius (17,0) zigzag and (12,6) chiral in the bulk phase to determine specific packing structure and the crystal lattice parameters. MD and MM studies led to a triangular packing as the most stable form for all three. The triangular lattice parameter for armchair, zigzag and chiral form: a = 16.78 Å , a = 16.52 Å, a = 15.62 Å, respectively. The optimal densities are d = 1.33 (g/cm³), d = 1.34 (g/cm³), d = 1.40 (g/cm³),

More importantly we determined the Youngs modulus along the tube axis for triangular-packed SWNTs using the second derivatives of the potential energy, Y = 640.30 GPa, Y = 648.43 GPa, Y = 673.49 GPa, respectively.

Vibrational Modes and Frequencies of SWNTs

We calculated the vibrational modes and frequencies of (10,10) tube crystals and the two other cases, zigzag (17,0) and chiral (12,6) with comparable tube radius. In the table, B denotes breathing mode as displayed in Figure[8b],S stands for shearing mode as in Figure[8c], and C stands for cyclopes as in Figure[8d]. The uniform compression mode is also shown in Figure[8a] occurs at 186 cm^-1 for (10,10), which is exactly the same as the experimental frequency[13].

We tabulated the uniform compression mode and highest graphite in-plane mode in the followingTable.

Conclusion

We presented a detailed study of structure, energetics and mechanical properties of SWNTs of varying size and chirality. The determined structure and lattice parameters for closed packed (10,10) like nanotubes are in close agreement with observations. We also determined all vibrational modes and frequencies of bulk and isolated nanotubes using a highly accurate classical force field.

Acknowledgements

This research was funded by NASA (computational nanotechnology) and by NSF-GCAG (ASC 92-100368).

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