Nanoparticle evolution via the precipitation method

A mathematical model describing the evolution of an arbitrarily large number of nanocrystals from solution is presented. Initially a model for the growth of a single particle is presented and its applicability discussed. The model only requires a single fitting parameter. It is then generalised to deal with $N$ particles. By setting $N=2$ we can investigate the process of Ostwald ripening. The $N$ particle model and the analytical solution of the single particle model are compared against experimental data, both showing excellent agreement. By allowing $N$ to increase we show that the single particle model may be considered equivalent to the average radius of the system for a large number of particles. Following a similar argument the $N=2$ model could describe an initial bimodal distribution. The mathematical solution clearly shows the effect of problem parameters on the growth process and, significantly, that there is a single controlling group. This may therefore be employed to guide and optimise the process of nanocrystal growth from solution.