Deterministic approximation for the nucleation-growth type model of nanoparticle formation: A validation against stochastic simulations

Abstract

A deterministic approximation method is introduced here for a complicated kinetic model that describes nanoparticle formation with nucleation and growth steps. The model starts from monomer units, some of which combine in a nucleation reaction, and the number of monomer units (n) in a kinetically effective nucleus is an unbounded integer parameter in the model. The growth steps are second-order processes between a particle and a monomer unit, the rate constant of which depends on the size of the growing nanoparticle in a way that is given in a kernel function. Four different kernels are considered: diffusion kernel (size independence), Brownian kernel (direct proportionality to the linear size), surface kernel (direct proportionality to the surface), mass kernel (direct proportionality to the volume or mass). Aggregation or coagulation are not part of the model. The approximation method works well for all four kernels and all integer values of n, and yields analytical formulas for the average and the most important moments (zeroth, first, and the one that is relevant for the kernel function) of the nanoparticle size distribution at every stage of the process. The approximation is validated against stochastic simulations using the Gillespie algorithm within the framework of the continuous time discrete state approach to stochastic kinetics. The comparison shows that the obtained analytical formulas are applicable for all practically important cases of the model and provide a way to interpret the dependence of average nanoparticle size on the ratio of the rate constants of nucleation and particle growth.