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

Amit K. Chattopadhyay; Bidisha Kundu; Sujit Kumar Nath; Elias C. Aifantis

doi:10.3389/fams.2022.852040

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

Amit K. Chattopadhyay^*, Bidisha Kundu, Sujit Kumar Nath, Elias C. Aifantis

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

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 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. 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 basic reproduction number R0, which is the average number of secondary infections generated from every infected. We show that the R0 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.

Original language	English
Article number	852040
Journal	Frontiers in Applied Mathematics and Statistics
Volume	8
DOIs	https://doi.org/10.3389/fams.2022.852040
Publication status	Published - 3 Jun 2022

Bibliographical note

© 2022 Chattopadhyay, Kundu, Nath and Aifantis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

Funding: he authors gratefully acknowledge partial financial support from the H2020-MSCA-RISE-2016 program, Grant No. 734485, entitled Fracture Across Scales and Materials, Processes and Disciplines (FRAMED). BK acknowledges funding by UKRI Grant No. (MR/T046619/1), part of the NSF/CIHR/DFG/FRQ/UKRI-MRC Next Generation Networks for Neuroscience Program. SKN acknowledges the Leverhulme Trust (RPG-2018-137) for supporting his time and resources for research in the UK.

Keywords

Applied Mathematics and Statistics
double diffusion
reproduction number
autocorrelation
spatiotemporal correlation
Fick's diffusion

Access to Document

10.3389/fams.2022.852040Licence: CC BY 4.0

Transmissibility in Interactive Nanocomposite Diffusion
© 2022 Chattopadhyay, Kundu, Nath and Aifantis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.
Final published version, 1.17 MBLicence: CC BY 4.0

Cite this

@article{c0b2e183e65c4ff4be923c9c9979bcb8,

title = "Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model",

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 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. 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 basic reproduction number R0, which is the average number of secondary infections generated from every infected. We show that the R0 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.",

keywords = "Applied Mathematics and Statistics, double diffusion, reproduction number, autocorrelation, spatiotemporal correlation, Fick's diffusion",

author = "Chattopadhyay, {Amit K.} and Bidisha Kundu and Nath, {Sujit Kumar} and Aifantis, {Elias C.}",

note = "{\textcopyright} 2022 Chattopadhyay, Kundu, Nath and Aifantis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Funding: he authors gratefully acknowledge partial financial support from the H2020-MSCA-RISE-2016 program, Grant No. 734485, entitled Fracture Across Scales and Materials, Processes and Disciplines (FRAMED). BK acknowledges funding by UKRI Grant No. (MR/T046619/1), part of the NSF/CIHR/DFG/FRQ/UKRI-MRC Next Generation Networks for Neuroscience Program. SKN acknowledges the Leverhulme Trust (RPG-2018-137) for supporting his time and resources for research in the UK.",

year = "2022",

month = jun,

day = "3",

doi = "10.3389/fams.2022.852040",

language = "English",

volume = "8",

}

TY - JOUR

T1 - Transmissibility in Interactive Nanocomposite Diffusion: The Nonlinear Double-Diffusion Model

AU - Chattopadhyay, Amit K.

AU - Kundu, Bidisha

AU - Nath, Sujit Kumar

AU - Aifantis, Elias C.

N1 - © 2022 Chattopadhyay, Kundu, Nath and Aifantis. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms. Funding: he authors gratefully acknowledge partial financial support from the H2020-MSCA-RISE-2016 program, Grant No. 734485, entitled Fracture Across Scales and Materials, Processes and Disciplines (FRAMED). BK acknowledges funding by UKRI Grant No. (MR/T046619/1), part of the NSF/CIHR/DFG/FRQ/UKRI-MRC Next Generation Networks for Neuroscience Program. SKN acknowledges the Leverhulme Trust (RPG-2018-137) for supporting his time and resources for research in the UK.

PY - 2022/6/3

Y1 - 2022/6/3

N2 - 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 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. 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 basic reproduction number R0, which is the average number of secondary infections generated from every infected. We show that the R0 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.

AB - 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 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. 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 basic reproduction number R0, which is the average number of secondary infections generated from every infected. We show that the R0 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.

KW - Applied Mathematics and Statistics

KW - double diffusion

KW - reproduction number

KW - autocorrelation

KW - spatiotemporal correlation

KW - Fick's diffusion

UR - https://www.frontiersin.org/articles/10.3389/fams.2022.852040/full

U2 - 10.3389/fams.2022.852040

DO - 10.3389/fams.2022.852040

M3 - Article

VL - 8

JO - Frontiers in Applied Mathematics and Statistics

JF - Frontiers in Applied Mathematics and Statistics

M1 - 852040

ER -

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

Abstract

Bibliographical note

Keywords

Access to Document

Other files and links

Fingerprint

Cite this