Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model

TL;DR

This paper introduces a novel analytical method to approximate solutions and correlations in nonlinear double-diffusivity models, bridging physics, biology, and materials science, and providing insights into nanocomposite diffusion processes.

Contribution

It presents a new approach using the basic reproduction number concept to analyze correlation in nonlinear coupled diffusion models, applicable across various reaction-diffusion systems.

Findings

The R0-based method closely matches numerical solutions.

It enables analytical approximation of complex nonlinear diffusion systems.

Applicable to diverse reaction-diffusion models.

Abstract

Model analogies and exchange of ideas between physics or chemistry with biology or epidemiology have often involved inter-sectoral mapping of techniques. Material mechanics has benefitted hugely from such interpolations from mathematical physics where dislocation patterning of platstically deformed metals [1,2,3] and mass transport in nanocomposite materials with high diffusivity paths such as dislocation and grain boundaries, have been traditionally analyzed using the paradigmatic Walgraef-Aifantis (W-A) double-diffusivity (D-D) model [4,5,6,7,8,9]. A long standing challenge in these studies has been the inherent nonlinear correlation between the diffusivity paths, making it extremely difficult to analyze their interdependence. Here, we present a novel method of approximating a closed form solution of the ensemble averaged density profiles and correlation statistics of coupled dynamical systems, drawing from a technique used in mathematical biology to calculate a quantity called the {\it basic reproduction number} $R_0$, which is the average number of secondary infections generated from every infected. We show that the $R_0$ formulation can be used to calculate the correlation between diffusivity paths, agreeing closely with the exact numerical solution of the D-D model. The method can be generically implemented to analyze other reaction-diffusion models.