Van der Waals and Electrostatic Forces Acting on a Carbon Nanotubes Research Center for Applied Sciences, Academia Sinica,Taipei, Taiwan contact: Evgeny Pogorelov ( Evgeny Pogorelov, Alexander Zhbanov, Yia-Chung Chang Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan Universal curves for van der Waals interaction between single-wall carbon nanotubes The continuum Lenard-Jones (LJ) model suggested by L.A. Girifalco is usually used to evaluate potential between two graphitic structures. The LJ potential for two carbon atoms in graphene-graphene structure is Figure 5. Distribution of the emission current and total emission current isolines for the work function  =4.8 eV. (a) Distribution of the relative emission current over angle . (b) Total emission current isolines on the (  E 0,  )-plane. Figure 1. Schematic drawing of interaction between two SWNTs. t 1 and t 2 are the radii, R is the distance between axes, and d is the gap between surfaces of tubes. If angle γ = 0 then tubes are in parallel. Figure 2. (a) Universal curves for potentials and (b) forces between two SWNTs. where d  1 and d  2 are the surface elements for each tube. The basic idea is to replace the cathode by a linearly charged thread in a uniform electric field and to use a set of “image” charges to reproduce the anode. Abstract: We report very simple and accurate algebraic expressions for the van der Waals (vdW) potentials and the forces between two parallel and crossed carbon nanotubes. It is found that interaction between parallel and crossed tubes are described by different universal curves which depend only on dimensionless distance. The explicit functions for equilibrium vdW distances, well depths, and maximal attractive forces have been given. These results may be used as a guide for analysis of experimental data to investigate interaction between nanotubes of various natures. We consider field emission from carbon nanotubes and other elongated nanostructures. An exact solution for the electrostatic field between a metallic hemi-ellipsoidal needle on a plate (as a cathode) and a flat anode are presented. Exact analytical formulas of the electrical field, field enhancement factor, and electrostatic force are found. The universal curve means that a plot of The VDW force: Figure 3. Schematic drawing of the cathode. (a) Current from the prolated metallic spheroid. (b) Geometry of the hemi- ellipsoidal needle. where r is a distance, A and B are the attractive and repulsive constants. The potential between two SWNTs is approximated by integration of LJ potential gives the same curve for all radii of tubes, where  0 is the minimum energy and d 0 is the equilibrium spacing for the two interacting surfaces. against The recurrent equation for equilibrium distance d 0 : where The first approximation for equilibrium spacing: The potential energy: where The potential well: The universal curve: where The universal curve: Enhancement factor, electrostatic force and emission current in a nanoneedle emitter The total detaching force acting on nanotube: The components of the electric field on the surface of the metallic spheroid: The field enhancement factor at the apex of the ellipsoid: The eccentricity of ellipsoid: The radius of curvature: Figure 4. Distribution of force and total force isolines. (a) Distribution of the relative force over the axis of the needle. (b) Isolines of the total force on the plane. The total field emission current: