Google Meet link: https://meet.google.com/bgt-oqys-gme
A kinetic model describing nanoparticle formation is presented here using both the deterministic and stochastic approaches. The model starts from monomer units, some 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 number of monomeric units in a viable seed is also a parameter of the model.
Exact analytical solutions are derived for the time dependence of the concentrations of all different kinds of nanoparticles, as well as the cube-root number-average size of the nanoparticles in three cases, plausible approximations are used for 17 other types of models. These are compared with the results of simulations using the Gillespie algorithm and this is used for different kernel functions, as well.
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.