N 194s

where E is the elastic modulus, a is a length on the order of the interlayer graphite spacing (-3.35 A), and sic is the area associated with a single carbon atom on the tubule cylinder [19.69]. A plot of the strain energy per carbon atom using this simple model and a more detailed local density

Fig. 19.59. Strain energy per carbon atom as a function of nanotube radius calculated for unoptimized nanotube structures (open squares) and optimized nanotube structures (solid circles). The solid line depicts the inverse square dependence of the strain energy on nanotube radius [see Eq. (19.45)] drawn through the point of smallest radius [19.102],

Fig. 19.59. Strain energy per carbon atom as a function of nanotube radius calculated for unoptimized nanotube structures (open squares) and optimized nanotube structures (solid circles). The solid line depicts the inverse square dependence of the strain energy on nanotube radius [see Eq. (19.45)] drawn through the point of smallest radius [19.102],

Nanotube radius (nm)

Nanotube radius (nm)

functional (LDF) calculation is shown in Fig. 19.59. This figure shows good agreement between the simple elastic model and the more detailed LDF calculation and further shows that for d, < 2 nm the effect of strain energy exceeds that of the room temperature thermal energy. Thus it is only at small tubule diameters that the strain energy associated with tubule curvature is important. These calculations show that the strain energy per carbon atom for fullerenes of similar diameter is much larger than for the comparable nanotube. This larger strain energy for the fullerenes reflects their two-dimensional curvature in comparison with tubules which have only one-dimensional curvature. Calculations for the energetics of the stretching and compression of tubules show good agreement with an elastic continuum model based on the elastic constant of Cn for graphite [19.69,157,158], It is also found that the tubules get softer with decreasing radius, and the tubule stiffness is found to depend on chiral angle, with the zigzag (6 — 0) tubules having lower stiffness than the armchair (0 = 30°) tubules [19.69], The elastic continuum model also allows an estimate to be made for the stiffness of nanotubes E based on that for a graphene sheet E0

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