Diffusive growth of fullerenes and carbon nanotubes

Abstract

The video of a growing fullerene within a carbon nanotube, initiated by a tungsten catalyst, provides a dramatic realization of a complex nanoscale process. While there may be many detailed models which can account for this growth, we propose one of the simplest possible models which is consistent with the major observed features of the growth process. In particular, we assume that the fullerene is immersed in a carbon vapor environment, and that the growth occurs as a consequence of the diffusion of the carbon vapor into the fullerene. Moreover, we assume that the classical diffusion equation applies in the region exterior to the fullerene and that a standard Stefan condition applies at the moving fullerene surface. We assume that the gaseous medium through which the carbon atoms diffuse is represented through the value of the diffusion coefficient D appearing in the classical diffusion equation. We also assume that the influence of the catalyst is felt through the value of the constant appearing in the Stefan condition. Based on these assumptions, we derive simple similarity solutions for both spherical and ellipsoidal fullerenes which are entirely consistent with the observations. A corresponding analysis is provided for the longitudinal growth of a carbon nanotube. The video of a growing fullerene within a carbon nanotube, initiated by a tungsten catalyst, provides a dramatic realization of a complex nanoscale process. While there may be many detailed models which can account for this growth, we propose one of the simplest possible models which is consistent with the major observed features of the growth process. In particular, we assume that the fullerene is immersed in a carbon vapor environment, and that the growth occurs as a consequence of the diffusion of the carbon vapor into the fullerene. Moreover, we assume that the classical diffusion equation applies in the region exterior to the fullerene and that a standard Stefan condition applies at the moving fullerene surface. We assume that the gaseous medium through which the carbon atoms diffuse is represented through the value of the diffusion coefficient D appearing in the classical diffusion equation. We also assume that the influence of the catalyst is felt through the value of the constant ␣ appearing in the Stefan condition. Based on these assumptions, we derive simple similarity solutions for both spherical and ellipsoidal fullerenes which are entirely consistent with the observations. A corresponding analysis is provided for the longitudinal growth of a carbon nanotube.

Modelling the Diffusive Growth of Carbon Fullerenes

“…In a recent paper [2] the present authors propose one of the simplest models for fullerene growth. It is assumed that the fullerene is immersed in an inert gas through which carbon vapor diffuses and carbon atoms close to the fullerene may incorporate themselves into the fullerene surface.…”

“…Here we consider a related model to that in Ref. [2] but propose a moving boundary condition [3,4] which is based on the assumption that the fullerene grows in such a way that the fullerene surface atomic density remains constant throughout the growth process which provides a physical basis for the m = −1 case in Ref. [2].…”

“…[2] but propose a moving boundary condition [3,4] which is based on the assumption that the fullerene grows in such a way that the fullerene surface atomic density remains constant throughout the growth process which provides a physical basis for the m = −1 case in Ref. [2]. This assumption can be shown to be valid when taking into account the lattice structure of the fullerene.…”