Energetics and Structure of Single Walled Carbon Nanotoroids

*California Institute of Technology, Materials and Process Simulation Center and Division of Chemistry and Chemical Engineering
Pasadena, CA 91125 USA*

This is an abstract for a presentation given at the

Seventh Foresight Conference on Molecular Nanotechnology.

There will be a link from here to the full article when it is available on the web.

Carbon has diverse forms of structure,^{1-2} both in nature and by lab synthesize.
Three dimensional diamond and two dimensional graphite sheet are the two
well-known forms. In the past decade, the discoveries of zero dimensional
bucky balls^{3-7} and one dimensional bucky tubes^{8-9} have generated great interests
among researchers. Studies of the structures and properties of low-dimensional
carbon molecules, theorectical^{10-13} and experimental,^{14-18} showed tremendous
potential use of nano scale carbon material as components of electro-magnetic devices,
or high yielding materials. Among them, experiments done by Dai^{19} and Wang^{20}
illustrated the potential use of carbon nano tube as scanning microscopic probe. Motivated
by these exciting development in finding new forms of carbon materials and
studies of their properties, we designed a hypothetical carbon molecules,
single walled carbon nano toroids. Carbon toroids system is an ideal model for
studying the behavior of single walled nano tubes under bending. We can accurately
correlate the behavior of the tubes to its uniform curvature. In a futuristic point
of view, pure, or doped (inside the tube by other elements) forms of carbon toroids
could be synthesized and find its use as components of electro-magnetic devices
or micro machines, e.g., as nano conducting rings.

The carbon toroid can be chracterized by three integers (n, m, l), where (n, m)
defines the single-walled nano tube that is used to construct the toroid, while *l* is
the number of the smallest repeating units along tube axis.
We investigated the mechanical property of carbon toroids to investigate the bending of
(10, 10) single-walled carbon nano tube.

**Optimum Structures of Toroids**

We generated toroids with radius from 19.43 (Å) that is (10, 10, 100)
with 2,000 atoms, to 388.69 (Å), (10, 10, 2000) with 40,000 atoms.
Based on the molecular simulation forece field^{21} (MSFF), their structures are
optimized by using molecular mechanics and molecular dynamics.
Developed for graphite and fullerenes, MSFF was proved to be very accurate in calculating
vibration frequencies and predicting experimental structures. The parameters are listed in
Table 1 and Figure 1 shows the initial structures of two
toroids, *R*=38.87 (Å) of (10, 10, 200), and *R*=388.7 (Å) of (10, 10, 2000).

Table 1. Force Field Parameters for Carbon Nano Tube

van der Waals^{a} |
R_{v} |
D_{v} |
Bond^{b} |
r_{e} |
k_{r} |
E_{bond} |

C | 3.8050 | 0.06920 | C-C | 1.4114 | 720.000 | 133.0000 |

Angle^{c} |
k_{r1r2} |
|||||

C-C-C | 120.000 | 196.130 | -72.410 | -72.410 | 68.000 | |

Torsion^{d} |
V_{0} |
V_{1} |
V_{2} |
|||

C-C-C-C | 10.6400 | 0.0000 | -10.6400 |

Toroids with small radius are highly strained. To stablize the structure, harmonic bond
interactions are used at the early stage of the minimization. The more accurate
Morse potential that allows bond breaking are used at the latter stage of minimization.
By doing so, we can avoid the bias built in when the starting structure
was created. This is important for tracking down the transition radius that separates stable
toroids (though highly strained) from the unstable toroids (under Morse bond interaction,
the structure flies apart). Figure 2. is the strain energy per atom
(relative to infinite long straight (10, 10) tube) versus 1/R^{2}.
For toroids with different radius, different final structures resulted.
In the plot, we can identify three transition radii, associated with four structural
regions.

For toroids with radii larger than *R*_{s} = 183.3 (Å) (corresponds to (10, 10, 943)
toroid with 18,860 atoms), after molecular dynamics simulation and energy minimization,
smooth toriod is the only stable structure. This corresponds to the elastic bending
of isolated (10, 10) tub.

For toroids with radii smaller than *R*_{s} = 183.3 (Å) and larger than 109.6 (Å)
((10, 10, 564) with 11280 atoms),
the optimum structures obtained through minimization are smooth toroids without buckles.
However, after 20 pico seconds of molecular dynamics equilibration at 300 (K), numerous small dents
appeared along the inner wall. Take a snapshot of dynamics trajectory as the starting point of
structural minimization, we found an interesting phenomena. During the minimization, small dents
diffused along the inner tube and nucleated into larger dents when they meet. This nucleation of
deformations continues, until the optimum structure resulted. The optimum structures usually have a
number of buckles almost uniformly spaced along the tube.

Figure 3. are the snapshots at the late stage of minimization for (10, 10, 564) (toroid with radius of 109.6 (Å) and 11280 atoms). Looking at the lower left quater of each ring, we can clearly identify the diffusion of small dents. These small dents eventually moved toward the larger dent as the minimization progresses, and combined with the large dent. The snapshots are numbered according to the minimization sequence. Snapshot with smaller numbering represents structure at earlier stage of minimization. Comparing to the smooth toroids, these structures have lower strain energy per atom. This is due to the stretching of the outer surface and compression of the inner surface. Knee like buckle relaxes compression over large region at the expense of increased local strain.

Figure 4. gives a close look at a buckle, which is cut out from a optimized toroid with one buckle. At the center of the buckle, tube wall collapsed completely. The closest distance between atoms in opposite tube walls is 3.3 (Å), comparable to the distance between adjacent layers of graphite. A short distance away from the collapsed point, the tube stays almost circular. Rooms created for the inner wall at the buckles relax the stretch and compression along the rest of the toroid.

There is a strong correlation between the number of buckles and the curvature of the toroids. The higher the strain (curvature) is, the more buckles appeared in the final structure of minimization. However, for each toroid within this region, there are many stable final structures with different number of buckles, each resulted from different starting sructure. This suggests that there exist many meta stable structures for toroids in this region. The fact that the curves towards small radius in Fig. 2. are not smooth suggests that we are not connecting the points with optimum number of buckles. Generally, if we increase the radius (thus reduce strain) we get structures with smaller number of buckles and when we approach the smooth region, we should get only structures with single buckle.

In order to track down the transition point, we created structures with different number of buckles as starting point of minimization. The buckles are uniformly distributed along the circumferences. To create a buckle, we added artificial harmonic constraint on two atoms in the opposit wall of the tube to pull together the inner wall and the outer wall. After the structures are minimized to lower RMS force, where the structures are stable under Morse bond interaction, we remove the constraints and switch harmonic bond potential back to Morse potential to further optimize the structures. We could just heat up the initial structures by using molecular dynamics and then anneal them down to zero temperature. However, given the size of the toroids in the transition, the long time that takes to anneal each structure, and the fact that there could be several stable structures associated with different number of buckles, it is impractical to do so. Figure 5. shows the transition region where smooth toroids and toroids with different number of buckles co-exist. Points with same number of buckles are connected into lines. It clearly shows the overlap and shifts of lines with different number of buckles.

Towards the transition point beyond which smooth toroids resulted, we are able
to create stable structures with four buckles, three buckles, two buckles and
one buckle. At region close to the transition radius *R*_{s}, only the one buckle
toroids have the smallest strain energy per atom. Figure 6. shows
the buckled structures in this region.

If we further increase the curvature (decrease radius), at *R*_{k}=109.6 (Å),
(corresponds to (10, 10, 564) toroid with 11,280 atoms), only toroids with
various number of buckles exist. At even higher curvature, the toroids are flatterned.
In this region, there are no smooth toroids, due to the high strain built in the compression
of inner wall and tension of the outer wall. Further decrease the radius down to
the point of *R*_{b} = 38.9 (Å) (corresponds to (10, 10, 200) toroid with 4,000 atoms),
the structure breaks and atoms fly apart in the course of minimization.
Figure 7. shows toroids with more than eight buckles to the smallest
toroid that can stand the built in strain.

**Discussion**

Consider the (10, 10) tube as thin elastic rods, then the toroids are rings of thin rods.
Assuming *k* as the Young's modulus of the (10,10) tube, *I* the moment
of inertia about the axis parallel to tube cross section, the strain energy of the
rings are given by

Taking *r*_{out} = 16.70 (Å), the inter-tube distance of (10, 10) SWNT crystals,
*r*_{in} = 10.5 (Å), which assumes 13.6 (Å) as the radius of (10, 10) tube, we get Young's modulus of the 913 (GPa) for toroids of large radius, which compares to
the experimental value of 1280+/- 0.6 (GPa)^{20} for multi-walled carbon nano tubes
(MWNT).

**Conclusion**

We have investigated energetics and structures of (10, 10, n) toroids, three transition
radii are found that define the regions with different stable structures. Based on classical
elastic theory analysis, we calculated the modulus of different regions.

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^{*}Corresponding Address:

Tahir Cagin

California Institute of Technology, Materials and Process Simulation Center and Division of Chemistry and Chemical Engineering

Pasadena, CA 91125 USA

Phone: 626 395 2728; Fax: 626 395 0918

E-mail: tahir@wag.caltech.edu; Web: http://www.wag.caltech.edu/home/tahir/

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