# Interdisciplinary Applied Mathematics

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where n, m are integers (0 < m < n), and ai and a2 are the graphene lattice vectors (see Figure 13.8). An armchair nanotube corresponds to the case of n = m, that is, ch = (n, n), and a zigzag nanotube corresponds to the case of m = 0, or ch = (n, 0). All other (n, m) chiral vectors correspond to chiral nanotubes. The indices (n, m) also determine the metallic or semiconducting behavior (electronic properties) of SWNT. Carbon nanotubes for which n — m = 3i, with i an integer, are metallic; all others are semiconducting. Armchair nanotubes are metallic. The electronic properties of MWNTs are rather similar to those of perfect SWNTs, because the coupling between the cylinders is weak in MWNTs.

The grand canonical ensemble (pVT) has a thermodynamic state characterized by a fixed chemical potential, p, a fixed volume, V, and a fixed temperature, T.

Molecular-dynamics simulations can be classified into:

Equilibrium MD (EMD) simulations, and

Nonequilibrium MD (NEMD) simulations.

The properties of the fluid that are not in equilibrium can be described by nonequilibrium statistical mechanics (Sadus, 1999) and calculated from NEMD simulations. Typically, NEMD involves applying a perturbation to the usual equations of motion. The perturbation can be constant throughout the simulation, it can evolve with time, or alternatively, a sinusoidally oscillating perturbation can be used.

The motion of an ensemble of atoms in MD simulations is governed by interatomic forces arising from the interaction of electrons and nuclei. Thus, the results obtained from MD simulations are linked with the ability of the potential energy function to represent the underlying system. In a classical

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