van der Waals interaction between a microparticle and a single-walled carbon nanotube

The Lifshitz-type formulas describing the free energy and the force of the van der Waals interaction between an atom (molecule) and a single-walled carbon nanotube are obtained. The single-walled nanotube is considered as a cylindrical sheet carrying a two-dimensional free-electron gas with appropriate boundary conditions on the electromagnetic field. The obtained formulas are used to calculate the van der Waals free energy and force between a hydrogen atom (molecule) and single-walled carbon nanotubes of different radii. Comparison studies of the van der Waals interaction of hydrogen atoms with single-walled and multiwalled carbon nanotubes show that depending on atom-nanotube separation distance, the idealization of graphite dielectric permittivity is already applicable to nanotubes with only two or three walls.

Figure 1

Schematic of the cylindrical graphene sheet of radius $R$ which is concentrically placed into a cylindrical cavity of radius $R+a$ in the infinite space filled with an isotropic substance.

Figure 2

The van der Waals coefficient as a function of separation for the interaction of a hydrogen atom (lines labeled 1) or a molecule (lines labeled 2) with the single-walled carbon nanotube of $R=5\mathrm{nm}$ radius (solid lines) and with a plane graphene sheet (dashed lines).

Figure 3

The coefficient $C_F$ as a function of separation for the interaction of a hydrogen atom (lines labeled 1) or a molecule (lines labeled 2) with the single-walled carbon nanotube of $R=5\mathrm{nm}$ radius (solid lines). Dotted lines are drown under the assumption that $C_F=3C_3$, i.e., that the van der Waals coefficient $C_3$ does not depend on separation.

Figure 4

The van der Waals coefficient as a function of the number of walls for the interaction of a hydrogen atom with the multiwalled carbon nanotube of $R=5\mathrm{nm}$ external radius (solid dots connected with solid lines) and with a single-walled carbon nanotube of the same radius (solid dots 1, 2, and 3) spaced at 1, 2, and $3\mathrm{nm}$ from the atom, respectively.

Figure 5

The van der Waals coefficient as a function of the number of walls for the interaction of a hydrogen molecule with the multiwalled carbon nanotube of $R=5\mathrm{nm}$ external radius (solid dots connected with solid lines) and with a single-walled carbon nanotube of the same radius (solid dots 1, 2, and 3) spaced at 1, 2, and $3\mathrm{nm}$ from the molecule, respectively.