# Extending the excluded volume for percolation threshold estimates in polydisperse systems: The binary disk system

## Abstract

For dispersions containing a single type of particle, it has been observed that the onset of percolation coincides with a critical value of volume fraction. When the volume fraction is calculated based on excluded volume, this critical percolation threshold is nearly invariant to particle shape. The critical threshold has been calculated to high precision for simple geometries using Monte Carlo simulations, but this method is slow at best, and infeasible for complex geometries. This article explores an analytical approach to the prediction of percolation threshold in polydisperse mixtures. Specifically, this paper suggests an extension of the concept of excluded volume, and applies that extension to the 2D binary disk system. The simple analytical expression obtained is compared to Monte Carlo results from the literature. In conclusion, the result may be computed extremely rapidly and matches key parameters closely enough to be useful for composite material design.

- Authors:

- Publication Date:

- Research Org.:
- Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)

- Sponsoring Org.:
- USDOE National Nuclear Security Administration (NNSA)

- OSTI Identifier:
- 1340261

- Report Number(s):
- SAND-2016-12468J

Journal ID: ISSN 0307-904X; 649809

- Grant/Contract Number:
- AC04-94AL85000

- Resource Type:
- Accepted Manuscript

- Journal Name:
- Applied Mathematical Modelling

- Additional Journal Information:
- Journal Volume: 46; Journal Issue: C; Journal ID: ISSN 0307-904X

- Publisher:
- Elsevier

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 97 MATHEMATICS AND COMPUTING; Percolation; Binary disks; Excluded volume; Percolation threshold estimates

### Citation Formats

```
Meeks, Kelsey, Pantoya, Michelle L., Green, Micah, and Berg, Jordan. Extending the excluded volume for percolation threshold estimates in polydisperse systems: The binary disk system. United States: N. p., 2017.
Web. doi:10.1016/j.apm.2017.01.046.
```

```
Meeks, Kelsey, Pantoya, Michelle L., Green, Micah, & Berg, Jordan. Extending the excluded volume for percolation threshold estimates in polydisperse systems: The binary disk system. United States. doi:10.1016/j.apm.2017.01.046.
```

```
Meeks, Kelsey, Pantoya, Michelle L., Green, Micah, and Berg, Jordan. Thu .
"Extending the excluded volume for percolation threshold estimates in polydisperse systems: The binary disk system". United States. doi:10.1016/j.apm.2017.01.046. https://www.osti.gov/servlets/purl/1340261.
```

```
@article{osti_1340261,
```

title = {Extending the excluded volume for percolation threshold estimates in polydisperse systems: The binary disk system},

author = {Meeks, Kelsey and Pantoya, Michelle L. and Green, Micah and Berg, Jordan},

abstractNote = {For dispersions containing a single type of particle, it has been observed that the onset of percolation coincides with a critical value of volume fraction. When the volume fraction is calculated based on excluded volume, this critical percolation threshold is nearly invariant to particle shape. The critical threshold has been calculated to high precision for simple geometries using Monte Carlo simulations, but this method is slow at best, and infeasible for complex geometries. This article explores an analytical approach to the prediction of percolation threshold in polydisperse mixtures. Specifically, this paper suggests an extension of the concept of excluded volume, and applies that extension to the 2D binary disk system. The simple analytical expression obtained is compared to Monte Carlo results from the literature. In conclusion, the result may be computed extremely rapidly and matches key parameters closely enough to be useful for composite material design.},

doi = {10.1016/j.apm.2017.01.046},

journal = {Applied Mathematical Modelling},

number = C,

volume = 46,

place = {United States},

year = {2017},

month = {6}

}

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