A family of nucleation-growth type kinetic model describing nanoparticle formation is analyzed in this contribution based on both the deterministic and stochastic approaches to chemical kinetics. The model starts from monomer units, n of which combine in a seed formation reaction. Second-order particle growth between a particle and a monomer unit follows, 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), Brown kernel (direct proportionality to the linear size), surface kernel (direct proportionality to the surface), mass kernel (direct proportionality to the volume or mass). The two main parameters of the model are the minimum size of the viable seed (n) and the ratio of the growth and nucleation rate constants. Exact analytical solutions are derived for the time dependence of the concentrations of all different kinds of nanoparticles for n = 1 and 2 of the mass kernel and n = 1 of the diffusion kernel. All other cases are handled through introducing plausible approximations in the deterministic approach and are validated against simulations carried out by the Gillespie algorithm in the stochastic approach. Explicit formulas are explored to give the final size distribution in all of the cases. An interesting aspect of this system is that the very high number of different species guarantees that the individual concentrations or particle numbers are extremely low, yet the deterministic approach still gives a description that seems acceptable for interpreting experimental results.

Prediction of nanoparticle size distributions based on nucleation-growth type models

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