A tribute to Marian Smoluchowski's legacy on soft grains assembly and hydrogel formation

TL;DR

This paper explores the analogy between grain growth in ceramics and cluster aggregation in chemical reactions, proposing a framework to understand hydrogel formation at low temperatures through statistical models.

Contribution

It introduces a novel comparison between normal grain growth and Smoluchowski's cluster aggregation models, establishing conditions for their equivalence in soft and reactive grains.

Findings

Equivalence between NGG and CCA models under specific conditions.

Rescaling time with a dispersive kinetics function aligns the models.

Framework applicable to thermoresponsive gel formation experiments.

Abstract

The paper compares the statistical description of physical-metallurgical processes and ceramic-polycrystalline evolutions, termed the normal grain growth (NGG), as adopted to soft- and chemically-reactive grains, with a Smoluchowski's population-constant kernel cluster-cluster aggregation (CCA) model, concerning irreversible chemical reaction kinetics. The former aiming at comprehending, in a semi-quantitative way, the volume-conservative (pressure-drifted) grain-growth process which we propose to adopt for hydrogel systems at quite low temperature (near a gel point). It has been noticed, that by identifying the mean cluster size $<k>$ from the Smoluchowski CCA description with the mean cluster radius' size $R_D$, from the NGG approach of proximate grains, one is able to embark on equivalence of both frameworks, but only under certain conditions. For great enough, close-packed clusters, the equivalence can be obtained by rearranging the time domain with rescaled time variable, where the scaling function originates from the dispersive (long-tail, or fractal) kinetics, with a single exponent equal to $d+1$ (in $d$-dimensional (Euclidean) space). This can be of interest for experimenters, working in the field of thermoresponsive gels formation, where crystalline structural predispositions overwhelm.