The energy of a straight single-wall carbon nanotube (SWNT) with different helicity is proposed by making continuum limit of the curvature elastic energy of a single-layer curved graphite carbon. We find that the zigzag is the only stable state with the lowest energy for the fixed diameter tubule. Our helicity energy formula agrees well with the energy of the zigzag configuration and armchair configuration SWNT's calculated by quantum molecular dynamics numerical simulation.

A continuum model of the configurations of multiwalled carbon nanotubes (MWNT's) is studied in this paper. The conformation energy of the MWNT is proposed, and the shape equations for a MWNT are derived. From the shape equations, the solutions corresponding to the straight carbon nanotube and coiled carbon nanotube are given respectively. The configurations of the ring MWNT and the plane curve MWNT are also determined theoretically. It is revealed that there are only a very few valid values of the ring radius of the ring MWNT in the case of the fixed parameters of its cross section. The mismatched effect between the interlayer lattices enriches the variation of ring MWNT's with different size. It is found that the conformation energy per unit arc length of the ring is double of that of straight MWNT for same tube cross section. Typical configurations of the plane curve MWNT obtained from the shape equations agree well with the TEM images of the nanotubes. The total conformation energy of the plane curve MWNT is in the same order of the ring MWNT one for enough long arc length.

We investigate ring formation of single-walled carbon nanotubes theoretically in this paper. The conformation energy of the nanotubes is identified as the curvature elastic energy of a curved graphite and the intertube van der Waals interaction. The entropy of ring nanotubes depends on the total number of carbon atoms. Assuming continuity, we find that the ring single-walled carbon nanotube is stable only when the radius of the tube is valued 6.958

In this paper, we study the morphological transition of bacterial colonies exposed to ultraviolet radiation by modifying the bacteria model proposed by Delprato et al. Our model considers four factors: the lubricant fluid generated by bacterial colonies, a chemotaxis initiated by the ultraviolet radiation, the intensity of the ultraviolet radiation, and the bacteria's two-stage destruction rate with given radiation intensities. Using this modified model, we simulate the ringlike pattern formation of the bacterial colony exposed to uniform ultraviolet radiation. The following is shown. 1 Without the UV radiation the colony forms a disklike pattern and reaches a constant front velocity. 2 After the radiation is switched on, the bacterial population migrates to the edge of the colony and forms a ringlike pattern. As the intensity of the UV radiation is increased the ring forms faster and the outer velocity of the colony decreases. 3 For higher radiation intensities the total population decreases, while for lower intensities the total population increases initially at a small rate and then decreases. 4 After the UV radiation is switched off, the bacterial population grows both outward as well as into the inner region, and the colony's outer front velocity recovers to a constant value. All these results agree well with the experimental observations Phys. Rev. Lett. 87, 158102 (2001). Along with the chemotaxis, we find that lubricant fluid and the two-stage destruction rate are critical to the dynamics of the growth of the bacterial colony when exposed to UV radiation, and these were not previously considered.

The specific heat of single-walled carbon nanotubes SWNT's is studied theoretically in this paper. Making use of the continuum model, phonon dispersion relations are derived numerically and analytically. Finally, expressions for specific heat versus temperature are derived theoretically, and we find that the lattice wave propagating along the length of the SWNT plays the principle role in deciding the value of the specific heat. Our theoretical results agree well with experimental data Science 289, 1730 2000.