Estimating reaction parameters in mechanism-enabled population balance models of nanoparticle size distributions: A Bayesian inverse problem approach.

In order to quantitatively predict nano- as well as other particle-size distributions, one needs to have both a mathematical model and estimates of the parameters that appear in these models. Here, we show how one can use Bayesian inversion to obtain statistical estimates for the parameters that appear in recently derived mechanism-enabled population balance models (ME-PBM) of nanoparticle growth. The Bayesian approach addresses the question of "how well do we know our parameters, along with their uncertainties?." The results reveal that Bayesian inversion statistical analysis on an example, prototype Ir0n nanoparticle formation system allows one to estimate not just the most likely rate constants and other parameter values, but also their SDs, confidence intervals, and other statistical information. Moreover, knowing the reliability of the mechanistic model's parameters in turn helps inform one about the reliability of the proposed mechanism, as well as the reliability of its predictions. The paper can also be seen as a tutorial with the additional goal of achieving a "Gold Standard" Bayesian inversion ME-PBM benchmark that others can use as a control to check their own use of this methodology for other systems of interest throughout nature. Overall, the results provide strong support for the hypothesis that there is substantial value in using a Bayesian inversion methodology for parameter estimation in particle formation systems.

Dust Effects on Ir(0)n Nanoparticle Formation Nucleation and Growth Kinetics and Particle Size-Distributions: Analysis by and Insights from Mechanism-Enabled Population Balance Modeling.

The effects of microfiltration removal of filterable dust on nanoparticle formation kinetics and particle-size distribution, in a polyoxometalate polyanion (P2W15Nb3O629-)-stabilized Ir(0)n nanoparticle formation system, are analyzed by the newly developed method of Mechanism-Enabled Population Balance Modeling (ME-PBM). The [(Bu4N)5Na3(1,5-COD)Ir·P2W15Nb3O62] precatalyst system produces on average Ir(0)∼200 nanoparticles of 1.74 ± 0.33 nm and hence a particle-size distribution (PSD) of ±19% dispersion when the precatalyst is reduced under H2 in unfiltered propylene carbonate solvent. But if the precatalyst is reduced in microfiltered solvent and microfiltered reagent solutions (where the filtered solvent is then also used to rinse dust from the glassware), then larger Ir(0)∼300 1.96 ± 0.16 nm nanoparticles are produced with a remarkable, 2.4-fold lowered ±8% dispersion. The results and effects of the microfiltration reduction of dust are analyzed by the newly developed method of ME-PBM. More specifically, the studies reported herein address eight outstanding questions that are listed in the Introduction. Those questions include: how easy or difficult it is to fit PSD data? What is the ability of the recently discovered alternative termolecular nucleation and two size-dependent growth steps mechanism to account for the effects of dust on the PSD? What types and amount of PSD kinetics data are needed to deconvolute the PSD into the parameters of the ME-PBM? What is the reliability of the resulting rate constants? Additional questions addressed include: if the ME-PBM results offer insights into the remarkable 2.4-fold narrowing of the PSD post simple microfiltration lowering of the dust, and if the results are likely to be more general? The Summary and Conclusions section lists nine specific insights that include comments on needed future studies.

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

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 k1), A + B → C (rate constant k2), and A + C → 1.5C (rate constant k3), 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, k2 > k3, thereby allowing the smaller particles to catch up in size to the more slowly growing larger particles.