Modelling nanocrystal growth via the precipitation method

Abstract

A mathematical model for the growth of a single nanocrystal is generalised to deal with an arbitrarily large number of crystals. The basic model is a form of Stefan problem, describing diffusion of monomer over a moving domain. Various levels of approximation (an analytical solution, an ordinary differential equation model and an N particle model) are compared and shown to agree well. The N particle model and analytical solution are then shown to have excellent agreement with experimental data for the growth of CdSe nanocrystals. The theoretical solution clearly shows the effect of problem parameters on the growth process and, significantly, that there is a single controlling group. By increasing the value of N it is shown that in the absence of Ostwald ripening the single particle model may be considered as representing the average radius of a system with a large number of particles. Consequently a system with N=2 may represent either a two particle system or a bimodel initial distribution. The solution of the N=2 model provides an understanding of Ostwald ripening. In general if Ostwald ripening is expected some form of the N particle model should be employed. Finally it is shown how the analytical solution may be employed to represent a multi-stage growth process which can then guide and optimise crystal growth.