Is There a Minimal Chemical Mechanism Underlying Classical Avrami-Erofe’ev Treatments of Phase-Transformation Kinetic Data?

Abstract

We report herein our efforts to see if there is not a minimal chemical mechanism that can provide statistically equivalent fits to solid-state and other phase-transformation kinetic data classically treated by the Avrami-Erofe’ev (A-E) kinetic model or its derivatives. Doing so is an important, missing piece of attempts to fit and draw chemical insight from solid-state kinetics, as made apparent by citations and quotations from the literature which detail the presently confused state of solid-state kinetics and, especially, what those kinetics mean in terms of the underlying chemical mechanism(s). Specifically, we test herein the one available, minimalistic, chemical-mechanism-based kinetic model that is able to deconvolute nucleation from growth, a model originally developed for transition-metal nanocluster nucleation and growth in solution, A → B (rate constant k1) then A + B → 2B (rate constant k2). The two-step model tested is able to fit solid-state phase transformation kinetic data equally well in comparison to fits obtained using the classic A-E equation from the 1940s. Statistically, it is found that the A-E equation is statistically significantly superior for 4 of 12 literature data sets examined, but that the two-step chemical mechanism-based model (known as the Finke-Watzky (F-W) model) is statistically significantly better for 2 of those 12 data sets, and the models give arguably indistinguishable fits within experimental error to the other 6 data sets. The results provide credence to the hypothesis that the two models can be viewed as different descriptions of the same underlying chemical and physical processes. Given the evidence that the chemical-mechanism-based two-step model provides equivalent (to sometimes better) fits within experimental error in 8 of the 12 cases examined, but that the A-E and integrated F-W equations were then mathematically equated to see if this provides any insights into the question of the physical meaning of the A-E parameters k and n. The math reveals that the A-E parameters k and n can be viewed as containing a complex convolution of the F-W chemical rate constants k1 and k2′ (where k2′ = k2[A]0 of the F-W model)—at least under the experimentally supported assumption that the A-E and F-W equations are somehow equivalent. Simulated A-E data for selected values of A-E parameters k and n (that give representative sigmoidal kinetic curves), followed by fits of those simulated curves to the F-W model and then plotting the resultant F-W k1 and k2′ rate constants as a function of the A-E k and n variables, provides additional interesting evidence that the A-E k and n and F-W k1 and k2′ can be viewed as complex convolutions of one another. Specifically, k from this treatment appears as a convolution of k1, k2′, and n (as well as time, t) and n as a convolution of k1, k2′, and k (as well as t). A discussion of the advantages and limitations of each model is presented followed by a summary of the conclusions from the present studies. The significance of the results is twofold: (i) there is now an experimentally supported way to deconvolute an average nucleation, k1, from an average autocatalytic growth, k2, rate constant from solid-state kinetic data, and importantly, (ii) there now are experimentally supported and rigorously defined words/concepts that can be used to support the discussion of those rate constants in terms of the underlying physical processes, namely, “nucleation” and “autocatalytic growth”. These are not trivial results in light of concepts such as “autocatalytic nucleation” that continue to be used without adequate experimental support and which, therefore, continue to cause confusion in the solid-state kinetics literature. A section listing important caveats to this work apparent from examining the solid-state kinetic literature, plus a look ahead to needed future work, is also briefly presented.