Modeling Sigmoidal Transients Using Dispersive Kinetic Models To Predict Nanoparticle Size Distributions

Simple methodologies for predicting the particle size distribution (PSD) of nanoparticle preparations based on their kinetics can be useful in guiding the development of new synthetic strategies. Under isothermal conditions, conversions that start off slowly and then rapidly accelerate ahead of the complete consumption of starting material give rise to "S"-shaped kinetic transients. While sigmoidal curves are readily fit to certain analytic functions, so-called "dispersive kinetic models" (DKMs) possess a unique advantage in their ability to link the continuously evolving specific rate to an underlying distribution of activation energies, g(E-a). In the case of burst nucleation, whereby the critical nuclei predicted by classical nucleation theory (CNT) are of similar size to the synthesized nanoparticles, the PSD is readily predicted from g(E-a) through application of the Gibbs-Thomson equation. A geometrical perspective on the derivation of the two DKMs discussed in this work provides connections to both classical mechanics and fractal dynamics.