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Guanghua Gao, Tahir Cagin*, and William A. Goddard III
Materials and Process Simulation Center, Beckman Institute (139-74)
California Institute of Technology, Pasadena, California 91125
In this paper, we present extensive molecular mechanics and molecular dynamics studies on the energy, structure, mechanical and vibrational properties of single wall carbon nanotubes. In our study we employed an accurate interaction potential derived from quantum mechanics. The simulation studies were primarily carried out using the MPSim Program.
We explored the stability domains of circular and collapsed cross-section structures of armchair (n,n), zigzag (n,0), and chiral (2n,n) isolated single-walled carbon nanotubes (SWNTs) up to a circular cross section radius of 170 Å. We have found three different stability regions based on circular cross section radius. Below 10 Å radius only the circular cross section tubules are stable. Between 10 Å and 30 Å both circular and collapsed forms are possible, however the circular cross section SWNTs are energetically favorable. But, beyond 30 Å (cross-over radius) the collapsed form becomes favorable for all three types of SWNTs. We report the behavior of the SWNTs with radii close to the cross-over radius [(45,45), (80,0), (70,35)] under uniaxial compressive and tensile loads. Using classical thin plane approximation and variation of strain energy as a function of curvature, we calculated the bending modulus of the SWNTs. The calculated bending moduli are k (n,n) = 963.44(GPa), k (n,0) = 911.64(GPa), and k(2n,n) = 935.48(GPa). We also calculated the interlayer spacing between the opposite sides of the tubes and found d(n,n) = 3.38(Å), d(2n,n) = 3.39(Å), and d(n,0) = 3.41(Å)
Using an enthalpy optimization method, we have determined the crystal structure and Young's modulus of (10,10) armchair, (17,0) zig-zag and (12,6) chiral forms [which have similar diameter as (10,10)]. They all pack in a triangular pattern in 2-D. Calculated lattice parameters are a(10,10) = 16.78 Å, a(17,0) = 16.52 Å and a(12,6) = 16.52 Å. Using the second derivatives of potential we calculated the Young modulus along the tube axis and found Y(10,10) = 640.30 GPa, Y(17,0) = 648.43 GPa, and Y(12,6) = 673.94 GPa. Using the optimized structures of (10,10), (12,6) and (17,0), we determined the vibrational modes and frequencies. Here, we report the highest in plane mode, compression mode, breathing mode, shearing mode and relevant cyclop mode frequencies.
Carbon nanotubes were discovered in 1991 by Iijima of NEC Corporation . Since then, efforts in synthesis, characterization and theoretical investigation on nanotubes has grown exponentially. This is mostly due to their perceived novel mechanical and electronic properties and their tremendous potential for future technological applications. In 1993, the simplest kind of carbon nanotubes, single walled carbon nanotubes, were discovered independently by Iijima group  and an IBM team headed by Bethune . These SWNTs can be regarded as a rolled-up graphite sheet in the cylindrical form. Some specific defect-free forms of these SWNT show remarkable mechanical properties and metalic behavior . These materials present tremendous potential as components for use in nano-electronic and nano-mechanical device applications or as structural elements in various devices.
Thess and coworkers  later produced crystalline "ropes" of metallic carbon nanotubes with 100 to 500 SWNTs bundled into a two-dimensional triangular lattice. These tightly bundled linear "ropes" are expected to have remarkable mechanical properties, as well as superior electronic and magnetic properties. Various levels of studies performed on the properties of the SWNTs, including use of classical molecular mechanics (MM), lattice dynamics (LD), molecular dynamics(MD), tight binding and ab initio level Quantum Mechanical (QM) methods [5-18]. Robertson et. al. carried first studies on the structure and energetics of isolated small diameter . More recent theoretical studies include, behavior of nanotubes under high rate tensile strain and fracture , size dependent resonance raman scattering  in nanotubes, mechanical behavior of SWNT under compression using MD , and a pioneering study on the structural flexibility of carbon nanotubes using HREM and atomistic simulations addressing the role of van der Waals radii reported by Tersoff and coworkers [17-18].
In this paper, we present a detailed study of the energetics, structures, and mechanical properties of single walled carbon nanotubes with different radius and chirality (armchair (n,n), chiral (2n,n), and zigzag (n,0)). We used an accurate force field, derived through quantum mechanical calculations, to represent the interactions between the carbon atoms . These interaction potentials were used earlier in studying structure, mechanical and vibrational properties of graphite, various fullerenes and intercalated compounds of fullerenes  and nanotubes . In our studies, we employed classical molecular dynamics and molecular mechanics methods as implemented in MPSim, (a massively parallel program for materials simulations) program . MD runs are performed to anneal the structures, whereas molecular mechanics, energy and/or enthalpy minimization, are applied at the end of annealing cycle to obtain the final optimized structures. Using the analytical second derivatives of the potential energy, we also calculate the vibrational modes and frequencies of three kinds of nanotube bundles, (10,10) arm chair, (17,0) zig-zag and (12,6) chiral. These tubes have comparable cross section diameters, and are among the easiest to make.
In order to assess mechanical stability of various SWNTs, we created three chiral forms [(n,n) armchair, (n,0) zigzag, and (2n,n)] with various diameters. For each form, we studied two sets of initial structures, perfect circular cross section and elongated or collapsed cross section. For the collapsed structures, the opposte walls in the middle section are within van der Waals distance and the shape at the two ends is close to circular diameter of D ~ 10.7 Å. To mimic long isolated nanotube, we imposed a periodic boundary condition in c-direction,(tube direction). To eliminate the inter-tube interactions, we set the cell parameter a and b as 50 times of the circular tube diameter. Energy and structural optimizations were carried out using MPSim. Figure 1. is the strain energy per carbon atom versus radius of its circular form. [See also, Suplementary Material] We put the two sets (collapsed and circular) with three chiral forms ( (n,n) armchair, (n,0) zigzag, and (2n,n) ) on the same plot. For all three forms, there are regions associated with two transition radii (R1 and R2). For tubes with "circular" radius smaller than R1, only the circular form is possible, the collapsed initial structure transformed to the circular form during the structural optimization. For tubes with 'circular' radius between R1 and R2, there are two stable structures with the circular structure energetically more stable. For tubes with "circular" radius larger than R2, collapsed form becomes energetically favored while circular form becomes metastable. The structures and radii of the first transition are:
Looking at the collapsed structures of various radii along tube axis, we found that they all have two circular (or elliptical) ends of diameter D ~ 10.5 Å and flat middle section. The inner wall distances in the flat region are close to 3.4 Å, which is very similar to the inter layer spacing of adjacent sheets in graphite crystal. The end sections are highly strained compared to the circular form, thus cost energy. In addition to zero strain energy, the flat region is further stabilized by inter layer van der Waals attractions. The relative strength of these two opposite forces dictate the two structural transformations. Shown in Figure 2 are the optimized structures of armchair (n, n), tubes with collapsed initial structures viewed along c-axis. The zigzag (n,0) and chiral (2n, n) tubes are the same.
Based on the optimized structures and their energies of the circular form, we can model the basic energetics by approximating the tube as a membrane with a curvature 1/R and bending modulus of k . Assuming a as the thickness of tube wall, 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 Eo 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). These results are plotted against the theoretical estimates in Figure 3.
The bending modulus of sheets with different chirality suggest that the transition radius depends on the chirality, with (2n, n) transition radius larger thab that of (n, 0) zigzag, but smaller than that of (n, n) armchair. That is what we expected, because the higher the bending modulus, the higher the strain energy. The (n, n) armchair has higher binding energy than that of (2n, n), while the (n, 0) zigzag has lower binding energy than that of (2n, n). However, by examining the collapsed structures closely, we also found different interlayer stacking in the collapsed region. Figure 4 is the side views of the two attracting layers for three cases. The interlayer stacking patterns are different due to different chirality.
Figure 4: Side views of interlayer stacking patterns of: a) (30,30) armchair; b) (50,25) chiral and c) (50,0) zigzag SWNTs.
The inter layer distances also differ slightly, 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 stacking of the opposite walls 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, the two factors (bending modulus and van der Waals attraction) cancels out for different chirality, so that 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.
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, co, 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 5a, the armchair, Figure 5b, the zigzag, and Figure 5c, the chiral forms, we observe a nonlinear dependence on the strain.
Strain Energy vs Strain for (45,45) Armchair SWNT
Figure 5b: Strain Energy vs Strain for (80,0) Zigzag SWNT
Figure 5c: Strain Energy vs Strain for (70,35) Chiral SWNT
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 6. 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
Among various conformations, the (10,10) armchair SWNT is the easiest to make. We studied the mechanical properties of its bulk phase (10,10) tube bundles. We also calculated the bulk properties of (17,0) zigzag and (12,6) chiral tubes, with cross section radii close to that of (10,10) armchair tube. In the bulk phase we determine the specific packing, density, lattice parameters. Molecular Dynamics and Molecular Mechanics studies led to a triangular packing as the most stable form for all three forms. The triangular lattice parameter for (10,10) armchair is a = 16.78 Å with a density d = 1.33 (g/cm3). For the (17,0) zigzag, they are a = 16.52 Å, and d = 1.34 (g/cm3). For the chiral form (12,6), they are a = 16.52 Å and d = 1.40 (g/cm3). More importantly, we determined the Young's modulus along the tube axis for triangular-packed SWNTs using the second derivatives of the potential energy. They are Y = 640.30 GPa, Y = 648.43 GPa, and Y = 673.49 GPa, respectively. Normalized to carbon sheet these values are within a few percent of the graphite bulk value.
Using the second derivatives of the potential energy surface we calculated vibrational modes of a series of (n,n) tubes. The uniform compression and highest inplane modes are listed in Table 2 for n=7 to 13.
Among (n,n) tubes we focused on (10,10) tube crystal. We also calculated the vibrational frequencies and modes for the zigzag (17,0) and chiral (12,6) tube crystals due to their comparable tube radius to (10,10). We listed the breathing, shearing and cyclop modes for these equivalent radius nanotubes in Table 3. These results may be used to differentiate chiral tubes with comparable diameters. The uniform compression mode is also shown in Figure 7a occurs at 186 cm-1 for (10,10), which is exactly the same as the experimental frequency. In the Table 3, B denotes breathing mode as displayed in Figure 7b,S stands for shearing mode as in Figure 7c, and C stands for cyclopes as in Figure 7d.
Figure 7: Selected vibrational Modes for (10,10) tube, a) Top left, uniform compression mode; b) Bottom left, shearing along tube mode; c) Top right, breathing mode d) Bottom right, Cyclop mode. Click on a Figure to Animate.
We presented a detailed study of structure, energetics and mechanical properties of SWNTs of varying size and chirality. We determined the stability domains for circular and collapsed cross section SWNTs. We reported the uniaxial strain-energy behavior of the SWNTs with cross-over radius in the range of -8% strain to 8% strain. The determined structure and lattice parameters for closed packed (10,10) like nanotubes are in close agreement with observations. We also determined vibrational modes and frequencies of bulk and isolated nanotubes using a highly accurate classical force field.
This research was funded by NASA (computational nanotechnology) and by NSF-GCAG (ASC 92-100368). The facilities of the MSC are also supported by grants from NSF (CHE 95-22179), DOE-ASCI, ARO/DURIP, BP Chemical, ARO-MURI, Exxon, Seiko-Epson, Asahi Chemical, Beckman Institute, Chevron Petroleum Technology Co., Chevron Chemical Co., NASA/JPL, ONR, Avery Dennison, and Chevron Research Technology Co.
Energies as a function of radii (R up to 170 A) for (n,n), (n,0), (2n,n) tubes
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