Nearest-neighbor connectedness theory: A general approach to continuum percolation

Coupette, Fabian; de Bruijn, René; Bult, Petrus; Finner, Shari; Miller, Mark A.; van der Schoot, Paul; Schilling, Tanja

doi:10.1103/physreve.103.042115

Nearest-neighbor connectedness theory: A general approach to continuum percolation

Coupette, Fabian; de Bruijn, René; Bult, Petrus; Finner, Shari; Miller, Mark A.; van der Schoot, Paul; Schilling, Tanja

Authors

Fabian Coupette

René de Bruijn

Petrus Bult

Shari Finner

Dr Mark Miller m.a.miller@durham.ac.uk
Associate Professor

Paul van der Schoot

Tanja Schilling

Abstract

We introduce a method to estimate continuum percolation thresholds and illustrate its usefulness by investigating geometric percolation of noninteracting line segments and disks in two spatial dimensions. These examples serve as models for electrical percolation of elongated and flat nanofillers in thin film composites. While the standard contact volume argument and extensions thereof in connectedness percolation theory yield accurate predictions for slender nanofillers in three dimensions, they fail to do so in two dimensions, making our test a stringent one. In fact, neither a systematic order-by-order correction to the standard argument nor invoking the connectedness version of the Percus-Yevick approximation yield significant improvements for either type of particle. Making use of simple geometric considerations, our new method predicts a percolation threshold of ρ c l 2 ≈ 5.83 for segments of length l , which is close to the ρ c l 2 ≈ 5.64 found in Monte Carlo simulations. For disks of area a we find ρ c a ≈ 1.00 , close to the Monte Carlo result of ρ c a ≈ 1.13 . We discuss the shortcomings of the conventional approaches and explain how usage of the nearest-neighbor distribution in our method bypasses those complications.

Citation

Coupette, F., de Bruijn, R., Bult, P., Finner, S., Miller, M. A., van der Schoot, P., & Schilling, T. (2021). Nearest-neighbor connectedness theory: A general approach to continuum percolation. Physical Review E, 103(4), Article 042115. https://doi.org/10.1103/physreve.103.042115

Journal Article Type	Article
Acceptance Date	Mar 19, 2021
Online Publication Date	Apr 8, 2021
Publication Date	2021-04
Deposit Date	Mar 24, 2021
Publicly Available Date	Sep 30, 2021
Journal	Physical Review E
Print ISSN	2470-0045
Electronic ISSN	2470-0053
Publisher	American Physical Society
Peer Reviewed	Peer Reviewed
Volume	103
Issue	4
Article Number	042115
DOI	https://doi.org/10.1103/physreve.103.042115
Related Public URLs	https://arxiv.org/abs/2103.12406

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Copyright Statement
Reprinted with permission from the American Physical Society: Coupette, Fabian, de Bruijn, René, Bult, Petrus, Finner, Shari, Miller, Mark A., van der Schoot, Paul & Schilling, Tanja (2021). Nearest-neighbor connectedness theory: A general approach to continuum percolation. Physical Review E 103: 042115. © (2021) by the American Physical Society. Readers may view, browse, and/or download material for temporary copying purposes only, provided these uses are for noncommercial personal purposes. Except as provided by law, this material may not be further reproduced, distributed, transmitted, modified, adapted, performed, displayed, published, or sold in whole or part, without prior written permission from the American Physical Society.