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

Rebeka Szabó,Gábor Lente

doi:10.1007/s10910-021-01265-z

Abstract

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.

Highlights

Nanoparticles are increasingly considered for application in various ways in new, advanced technologies, probably the most significant of them is their use as catalysts or catalyst supports, in industrial settings and as parts of user application devices or materials, e.g. self-cleaning mirrors or coatings or small-scale water purification systems [1,2,3]
A high number of nanoparticle synthesis methods have already been developed [1,2,3,4,5,6,7,8,9,10,11,12], and it is clear that the size of a nanoparticle fundamentally influences both its catalytic activity and toxicity
In our previous article [25], we showed that the analytical solution of a non-lumped kinetic model is still possible despite the fact that it has an infinite number of concentration variables

Summary

Introduction

Nanoparticles are increasingly considered for application in various ways in new, advanced technologies, probably the most significant of them is their use as catalysts or catalyst supports, in industrial settings and as parts of user application devices or materials, e.g. self-cleaning mirrors or coatings or small-scale water purification systems [1,2,3]. Keywords Nucleation kinetics · Kernel function · Moment · Nanoparticle growth · Symbolic solution · Autocatalysis In any system of kinetic ordinary differential equations, the number of dependent variables is very high.

Results

Conclusion