Particle Size Distributions via Mechanism-Enabled Population Balance Modeling

Mechanism-enabled population balance modeling (ME-PBM) was recently defined as the use of experimentally established (i.e., disproof-based, deliberately minimalistic, and pseudo-elementary step-based) mechanisms as more rigorous input for population balance models. ME-PBM addresses three long-sought goals: mechanism-based prediction and control of particle size distributions (PSDs), fitting PSDs by various mechanisms to test which mechanisms are refuted versus supported, and then also extracting rate constants for the most strongly supported mechanism from the wealth of kinetics information buried within the PSD. A full ordinary differential equation (ODE) approach is developed to the PBModeling that is a resurrection of Smoluchowski's 1918 approach, a little used, but powerful approach that in turn allows fitting of the information-laden PSD including its shape. The full details are reported of the 12 different particle-formation, pseudo-elementary step mechanisms tested in a disproof-based approach to the ME-PBM analysis of the PSDs and kinetics data for a prototype Ir(0)(n) nanoparticle-formation system. The ME-PBM analysis led to the discovery of a new particle formation mechanism that is a 1-step, but key, expansion of the classic Finke-Watzky two-step mechanism of nucleation, A -> B, followed by autocatalytic surface growth, A + B -> 2B. Specifically, the ME-PBM analysis yielded the new three-step particle-formation mechanism of A -> B (rate constant k(1)), A + B -> C (rate constant k(2)), and A + C -> 1.5C (rate constant k(3)), where A represents the monomeric nanoparticle precursor, B represents small nanoparticles, and C represents larger growing nanoparticles. A list of key conclusions is provided including the paradigm shifts for particle formation that (i) nucleation needs not be instantaneous or burst to achieve narrow PSDs as postulated by the classical 1950s LaMer model; that (ii) instead nucleation can be and often is continuous, as first shown in 1997, yet still leads to narrow PSDs; and critically that (iii) narrow PSDs can and do result in spite of continuous nucleation because smaller particles grow faster than larger ones (k(2) > k(3)), thereby allowing the smaller particles to catch up to the more slowly growing larger particles. Surface-ligand capping and other possible reasons that k(2) > k(3) are presented and discussed, as are additional conclusions, implications of the present studies, caveats, and needed additional studies. The goal of the present contribution is to follow up our first report of ME-PBM by providing the full details of our work to date so that others who wish to use ME-PBM in their own particle formations across nature, while fitting their own PSDs and testing their own mechanistic hypotheses, can readily do so.