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

Abstract

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 Ir 0 ð Þ n 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.

Solid Barium Sulfate Formation from Aqueous Solution: Re-examination of Key Literature and Kinetics of This Classic System in Search of a Minimum Mechanism of Formation

“…More broadly, it is likely that size-dependent autocatalytic surface growth is occurring as Vafa et al . used in their model and as is supported more generally by our own ME-PBM studies of other systems. − Evidence that the 4-step mechanism for (BaSO 4 ) n formation is one mechanism meriting further scrutiny as a hypothesis going forward; Evidence that two types of aggregation are required for the statistically best fits when one considers all five data sets, namely, bimolecular aggregation between similar size species (B + B → C; rate constant k 3 ) but also size-dependent, autocatalytic aggregation between smaller (B) and larger (C) species (B + C → 1.5 C; rate constant k 4 ). Looking a bit deeper and semi-quantitatively at the important concept of size-dependent (BaSO 4 ) n aggregation, in almost every case examined aggregation between particles of similar sizes (B + B → C; rate constant k 3 ) occurs with a different rate constant than does aggregation between particles of different sizes (B + C → 1.5 C; rate constant k 4 ), Table .…”

“…Can that ME-PBM then be used to predict conditions designed to lead to a specified average particle size and PSDs as desired for some application, or as needed for controlling (BaSO 4 ) n barite formation where it is undesired? Can Bayesian inversion statistical methods be employed to support the validity of the rate constant parameters found, to determine their error bars, and as a result to either provide further support for, or refute, the underlying PEStep mechanism used to create the ME-PBM model? Lastly, a referee raised the question meriting attention in future studies of “how one gets from the ‘mature’ particles discussed in the present work to the solid polycrystalline masses that actually clog pipes?” …”

“…Once deliberately minimalistic, PEStep mechanisms for (BaSO 4 ) n formation are in hand, the hope is that they can then be used as starting mechanistic hypotheses for going forward, ones where it becomes apparent what additional experimental and computational studies are needed for a better understanding of (BaSO 4 ) n and other mineral particle formation, precipitation, and crystallization. Although not a part of the present studies, once PEStep mechanisms are in hand, they can also be used to build more detailed, close to molecularly and atomically precise, mechanism-enabled population balance models (ME-PBMs). − Such ME-PBMs can, in turn, serve as an optimization tool to guide both improved syntheses as well as industrial processes.…”

Pseudoelementary Steps: A Key Concept and Tool for Studying the Kinetics and Mechanisms of Complex Chemical Systems

“…Additionally, multiparameter curve-fitting of kinetics or PSDs is fraught with classic problems of correlated variables or issues in curve fittingthat is, the classic “with four parameters I can fit an elephant, and with five I can make him wiggle his trunk” . A Bayesian inversion statistical analysis approach has recently appeared as a way to deal with this and related issues by testing statistically a given ME-PBM via the quality, error limits, and potential correlations of its rate constant (and any other) parameters Finally, as in all of science where models are built, one has to avoid becoming overenamored with one’s model or mechanism, especially one’s initial model or mechanism that has not yet been subjected to extensive attempts at disproving it.…”

“…110 A Bayesian inversion statistical analysis approach has recently appeared as a way to deal with this and related issues by testing statistically a given ME-PBM via the quality, error limits, and potential correlations of its rate constant (and any other) parameters. 111 • Finally, as in all of science where models are built, one has to avoid becoming overenamored with one's model or mechanism, especially one's initial model or mechanism that has not yet been subjected to extensive attempts at disproving it. Mechanisms must also be constructed minimalistically to startthat is, in obedience with Ockham's razor.…”