General formulae for interacting spherical nanoparticles and fullerenes

Abstract

Most nanodevices under investigation adopt a computational approach such as molecular dynamics simulations, which gives a numerical value for the potential energy as calculated from the interaction of every atom on one molecule with every atom on a second molecule. Although the simulation only involves short range atom–atom interactions and ignores those interactions at longer distances, the simulation still involves significant computational time. In this paper, we determine analytical formulae for four types of Lennard–Jones interactions: (i) a solid spherical nanoparticle with an atom, (ii) two distinct radii hollow spherical fullerenes, (iii) a solid spherical nanoparticle with a hollow spherical fullerene and (iv) two distinct radii solid spherical nanoparticles. The interaction energy using the 6–12 Lennard–Jones potential for these four situations are determined using the continuum approximation, which assumes that a discrete atomic structure can be replaced by either an average atomic surface density or an average atomic volume density. Using these formulae the computational time for a simulation might be dramatically reduced for those molecular interactions involving spherical nanoparticles or fullerenes. Such formulae might be exploited in hybrid analytical-computational numerical schemes, as well as in metallofullerenes and certain assumed spherical models of molecules such as methane and ammonia. As an illustration of the formulae presented here we determine both the most stable and the maximum radii of a solid spherical nanoparticle inside a fullerene, modelling the centre of a carbon onion or metallofullerenes. We also determine new cut-off formulae for interacting spherical nanoparticles and fullerenes which might be useful in computational schemes.