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Mechanism-Enabled Population Balance Modeling of Particle Formation en Route to Particle Average Size and Size Distribution Understanding and Control

Handwerk, Derek R.
Shipman, Patrick D.
Whitehead, Christopher B.
Özkar, Saim
Finke, Richard G.
The concept of Mechanism-Enabled Population Balance Modeling (ME-PBM) is reported, illustrated by its application to a prototype Ir(0)(n) nanoparticle formation reaction. ME-PBM is defined herein as the use of now available, experimentally established, disproof-based, deliberately minimalistic mechanisms of particle formation as the required input for more rigorous Population Balance models, critically including an experimentally established nucleation mechanism. ME-PBM achieves the long-sought goal of connecting such now available experimental minimum mechanisms to the understanding and rational control of particles size and size distributions. Twelve pseudoelementary step, particle-formation mechanisms are considered so that the approach to the ME-PBM is also extensively disproof-based. Resurrection of Smoluchowski's 1918 full Ordinary Differential Equation (ODE) approach to the PBM is another, critical aspect of our approach which, in turn, allows unbiased fitting of the information-laden particle-size distribution (PSD) including its shape. The results provide one solution to the "inverse problem" in which the PSD informs one as to the correct particle formation mechanism: A new, deliberately minimalistic 3-step particle-formation mechanism has been uncovered that is a single-additional-step expansion of the now broadly used Finke-Watzky (FW) 2-step mechanism, the new 3-step mechanism being: 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" nanoparticles. The results strongly support three paradigm shifts for nucleation and growth of particles, the most critical paradigm shift being that the "burst" nucleation assumption in LaMer's 1950s model of particle formation is not required to produce narrow, near-monodisperse PSDs. Instead, narrow PSDs can be and are achieved despite continuous nucleation because smaller particles grow faster than larger ones, k(2) > k(3), thereby allowing the smaller particles to catch up in size to the more slowly growing larger particles.