Numerical simulation of particle formation processes via a population balance model incorporating nucleation pathway variation

Nucleation, the initial process of particle formation, progresses in various pathways depending on the synthetic conditions, resulting in significantly different characteristics of the final particle. Although changes in these nucleation pathways hold potential for controlling particle characteristics, design principles that leverage nucleation pathways to engineer particle properties are poorly established. This is possibly due to the lack of mathematical models capable of capturing the time evolution of particle properties while accounting for the pathway-dependent effects. Herein, we developed a population balance model to capture the effects of nucleation pathways and investigated the time evolution of the particle size distribution in model cases where the system follows either a one-step pathway to a crystal or a two-step pathway via a droplet. The nucleation pathway determines the ratio of growth time spent in the droplet to that in the crystalline state before a stable distribution is achieved. In the two-step pathway, the particle size exhibits a catching-up behavior owing to the differing growth rates of droplets and crystals, resulting in a narrower particle size distribution than that of the one-step pathway. The observed catching-up behavior in the two-step pathway underlies the size-focusing phenomenon observed in some experimental systems. The results showed that a moderate initial monomer concentration is effective for synthesizing particles with a uniform size in systems exhibiting two-step nucleation because the size-focusing effect is in a tradeoff relationship with the timing difference in nucleation, a factor emphasized in conventional particle design strategies. • A population balance model incorporating nucleation pathway variations was developed. • Effects of pathway on particle size distribution were investigated. • Two-step pathway can induce a catching-up behavior in the particle size. • Two-step pathway prefers moderate monomer concentrations for uniform particle size.