General nucleation-growth type kinetic models of nanoparticle formation: possibilities of finding analytical solutions

In this work, analytical solutions for the time dependences for the concentration of each chemical species are determined in a class of nucleation-growth type kinetic models of nanoparticle formation. These models have an infinitely large number of dependent variables and describe the studied process without approximations. Symbolic solutions are found for the mass kernel (where reactivity is directly proportional to the mass of a nanoparticle) and the diffusion kernel (where reactivity is independent of the size of the nanoparticle). The results show that the average particle size is primarily determined by the type of the kernel function and the ratio of the rate constants of spontaneous nucleation and particle growth. The final distribution of nanoparticle sizes is a continuously decreasing function in each studied case. Furthermore, the time dependences of the concentrations of monomeric units show the induction behavior that has already been observed in many experimental studies.