Fitting and Interpreting Transition-Metal Nanocluster Formation and Other Sigmoidal-Appearing Kinetic Data: A More Thorough Testing of Dispersive Kinetic vs Chemical-Mechanism-Based Equations and Treatments for 4-Step Type Kinetic Data

In 2005, 18 other possible mechanisms were ruled out en route to providing what at present is the more general chemical mechanism for nanoparticle formation and agglomeration, the 4-step mechanism of A → B, A + B → 2B, B + B → C, and B + C → 1.5 C with associated respective rate constants k 1, k 2, k 3, and k 4 (Besson, C.; Finney, E. E.; Finke, R. G. J. Am. Chem. Soc. 2005, 127, 8179), where in general, A is the starting metal (e.g., Ir, Rh, Pt, or other metal precursors), B is the growing nanocluster, and C is larger, agglomerated nanoclusters. The broader importance of that work is that the first two steps of the above mechanism, A → B, A + B → 2B, bear on the important problem how to best model kinetically and thereby interpret “S”-shaped, sigmoidal nucleation plus growth particle-formation curves that are ubiquitous throughout nature (the curves that are fit by the above 4-step mechanism are unusual and different from normal S-shaped, sigmoidal curves, as is discussed in the paper). Given the importance of accounting for S-shaped, sigmoidal kinetic curves for growth processes in nature, a recent paper in this journal (Skrdla, P. J.; Robertson, R. T. Chem. Mater. 2008, 20, 3) hypothesized that it is important to use what are known as dispersive kinetics treatments to examine S-shaped phase transformation kinetic data. That paper did so by re-examining a single data set digitized from a figure in the above 2005 Besson, et al., paper and examining fits to “stretched exponential” functions from dispersive kinetics. That study reached two primary conclusions, namely that (i) “This work (i.e., the 2008 Chemistry of Materials paper) demonstrates that it may be possible to utilize much simpler (i.e., containing empirical but fewer fit parameters) dispersive/solid-state kinetic models to determine the kinetics of nanoparticle formation than the four-parameter, mechanistic equation described in the original work by Besson et al.”; and that (ii) the kinetic curves fit by the 4-step mechanism have “routinely appeared in the (prior) solid-state kinetics literature since the advent of the Johnson−Mehl−Avrami−Erove’ev (JMAE, or, more simply, “Avrami”) equations in the late 1930s/early 1940s”. The validity of these two main hypotheses/claims are re-examined herein using both additional data sets and using Akaike’s Information Criterion (AIC) statistical treatment for comparing models with different numbers of parameters. In addition to those results, the 2008 Chemistry of Materials paper and the present study are of broader and fundamental interest regarding (a) the more general question of the philosophy of model building in science; (b) the application of dispersive kinetic/semiempirical vs chemical-mechanism-based models; and (c) the most important question of the role of disproof in science regardless of which approach to model building in science one is using. Hence, these topics a−c are also briefly addressed in what follows.