General model for diffusive growth of nanostructures

The physical properties of carbon nanotubes are known to be strongly dependent on both their radius and length. If nanotubes are to be mass produced for a wide variety of different applications, it is important that techniques are developed to reliably grow a large number of nanotubes with specific dimensions. Although there has been less research into the possible applications of different-sized fullerenes, as well as more unusual nanostructures like nanocones, it is to be expected that the dimensions of any nanostructure will be an important factor in many applications. For this reason, we need to understand how nanostructures can be forced to grow, or shrink, to any prescribed size. Here we model the diffusive growth of a general nanostructure using the classical diffusion equation coupled with a moving boundary condition that describes the growing surface of the nanostructure. The moving boundary condition plays the role of a Stefan moving boundary condition and describes a moving 2D surface with constant atomic surface density. We illustrate the model with reference to growing fullerenes, nanotubes and nanocones.