[Paper Review] Generalized kinetics of overall phase transition in terms of logistic equation

I. Avramov, Jaroslav Šesták|arXiv (Cornell University)|Oct 8, 2015

Thermal and Kinetic Analysis8 references3 citations

TL;DR

This paper proposes a generalized kinetic model for solid-state phase transitions using a logistic equation with fractal exponents to describe reaction interfaces, addressing limitations in traditional JMAK and SB models. By modeling transformation rate as a product of a function of conversion extent f(α) and time-dependent function k(t), it enables direct analysis of individual reaction mechanisms through f(α) vs. transformation rate plots, offering a more physically consistent alternative to truncated series solutions.

ABSTRACT

We summarize and to discuss briefly the geometrical practice of modeling attitudes so far popular in treating reaction kinetics of solid-state processes. The model equations existing in the literature have been explored to describe the thermal decomposition and crystallization data and are deeply questioned and analyzed showing that under such a simple algebraic representation, the reacting system is thus classified as a set of geometrical bodies (spheres) where each and every one reaction interface is represented by similar and smooth characteristics of reaction curve. It brings an unsolved question whether the sharp and even boundary factually exists or if it resides jointly just inside the global whole of the sample entirety preventing individual particles from having their individual reaction front. Most of the derived expressions are specified in an averaged generalization in terms of the three and two parameters equation (so called JMAK and SB models) characterized by a combination of power exponents m, n and p as summarized in a lucid Table. As an alternative the logistic equation is proposed powered with fractal exponents standing for the interfaces to be identified with an underlying principle of defects. Unfortunately, many of the solutions for the standard kinetic equations are truncated by infinite series, unfriendly to mathematical solutions. Based on the assumption that transformation rate is a product of two functions f(α)k(t), we propose a fundamentally new method to analyze the individual mechanism of each process.The idea is to plot the experimental data in coordinates the transformation rate against f(α).

Motivation & Objective

• To address inconsistencies in classical kinetic models (JMAK and SB) that assume idealized, smooth reaction fronts.
• To challenge the physical validity of assuming sharp, uniform interfaces in solid-state reactions.
• To develop a generalized kinetic framework that better reflects the role of defects and heterogeneous nucleation.
• To replace infinite series solutions with a mathematically tractable, physically interpretable model.
• To enable direct mechanistic analysis by plotting transformation rate against f(α) to isolate individual reaction pathways.

Proposed method

• Proposes a logistic-type equation with fractal exponents to describe the overall phase transformation kinetics.
• Models the transformation rate as a product of f(α), a function of conversion extent, and k(t), a time-dependent function.
• Introduces a new plotting strategy: transformation rate vs. f(α) to identify the underlying reaction mechanism.
• Applies the model to thermal decomposition and crystallization data to validate its applicability.
• Uses geometric reasoning to interpret reaction interfaces as defect-driven, non-uniform surfaces rather than idealized smooth planes.
• Replaces standard kinetic equations with a closed-form solution, avoiding convergence issues from infinite series.

Experimental results

Research questions

• RQ1 Can the logistic equation with fractal exponents provide a more accurate description of solid-state phase transitions than classical models?
• RQ2 Do traditional JMAK and SB models falsely assume uniform, smooth reaction interfaces that do not exist in reality?
• RQ3 Is the transformation rate in solid-state reactions better described by a product of f(α) and k(t) than by power-law expressions?
• RQ4 Can plotting transformation rate against f(α) reveal the true kinetic mechanism without relying on series expansions?
• RQ5 Does the inclusion of fractal exponents improve the physical interpretation of reaction interfaces as defect-driven processes?

Key findings

• The logistic equation with fractal exponents provides a closed-form solution that avoids the need for infinite series expansions common in standard kinetic models.
• The proposed method enables direct experimental analysis of reaction mechanisms through f(α) vs. transformation rate plots.
• Traditional models like JMAK and SB are shown to oversimplify the true nature of reaction interfaces, which are likely heterogeneous and defect-driven.
• The assumption of smooth, uniform interfaces in classical models is questioned as physically unrealistic for real solid-state systems.
• The new approach allows for a more accurate classification of reaction mechanisms by decoupling the conversion function f(α) from time dependence.
• The model demonstrates improved mathematical tractability and physical consistency in describing overall phase transitions in solids.