We present a large-scale computer simulation of the prototypical three-dimensional continuum percolation model consisting of a distribution of overlapping (spatially uncorrelated) spheres. By using simulations of up to particles and studying the finite-size scaling of various effective percolation thresholds, we obtain a value of . This value is significantly smaller than the values obtained for simulations that have been carried out using smaller systems. Employing this value of and systems of size L = 160 (relative to a sphere of unit radius), we also obtain estimates of the critical exponents , and for the continuum system and show that the values are different than those obtained using previous values of .

https://iopscience.iop.org/article/10.1088/0305-4470/30/16/005/pdf

Precise determination of the critical threshold and exponents in a three-dimensional continuum percolation model

Problems involving percolation in liquids (i.e., involving connectivity of some sort) range from the metal–insulator transition in liquid metals to the properties of supercooled water. A common theme, however, is that connectivity can be distinguished from interaction and that one should not be slighted in order to describe the other. In this paper we suggest a model for percolation in liquids—the model of extended spheres—which permits connectivity to be studied in the context of, but independently from, liquid structure. This model is solved exactly in the Percus–Yevick approximation, revealing the existence of an optimum liquid structure for percolation. We analyze this behavior by first deriving an explicit diagrammatic representation of the Percus–Yevick theory for connectivity and then studying how the various diagrams contribute. The predictions are in excellent qualitative agreement with recent Monte Carlo calculations.

A theory of percolation in liquids

Thermal Conductivity of Disordered Heterogeneous Media from the Microstructure

For a multicomponent system of particles in equilibrium, an exact integral equation is derived for the pair connectedness function (which measures the probability that two particles, with centers a distance r apart, are connected). The pair connectedness function, mean cluster size, and percolation thresholds for mixtures of randomly centered (noninteracting) spheres and permeable spheres are then obtained analytically in the Percus–Yevick (PY) approximation. (The permeable‐sphere model provides a one‐parameter bridge from randomly centered sphere mixtures to hard‐sphere mixtures.) For this family of models, connectedness is defined by particle overlap. It is found that, within the PY approximation, a multicomponent mixture of randomly centered spheres percolates at ξ3=π/6∑iρiR3i =1/2, independent of the concentration and size distributions of the particles. For the permeable‐sphere case the percolation threshold depends on the relative densities, size, and interparticle permeability among the species. Th...

Clustering and percolation in multicomponent systems of randomly centered and permeable spheres

We show analytically that the [0, 1], [1, 1], and [2, 1] Pade approximants of the mean cluster number S for both overlapping hyperspheres and overlapping oriented hypercubes are upper bounds on this quantity in any Euclidean dimension d. These results lead to lower bounds on the percolation threshold density ηc, which become progressively tighter as d increases and exact asymptotically as d → ∞, i.e., ηc → 2−d. Our analysis is aided by a certain remarkable duality between the equilibrium hard-hypersphere (hypercube) fluid system and the continuum percolation model of overlapping hyperspheres (hypercubes). Analogies between these two seemingly different problems are described. We also obtain Percus-Yevick-like approximations for the mean cluster number S in any dimension d that also become asymptotically exact as d → ∞. We infer that as the space dimension increases, finite-sized clusters become more ramified or “branch-like.” These analytical estimates are used to assess simulation results for ηc up to 20...

Effect of dimensionality on the continuum percolation of overlapping hyperspheres and hypercubes

An exact integral equation for the pair-connectedness function gDouble Dagger (12) is derived on the basis of the author's earlier exact equation for the pair-distribution function g(12). The new equation gives rise to a sequence of approximations for gDouble Dagger (12), starting with the Percus-Yevick approximation. Significant aspects of the first two approximations are noted, as is the relevance of the blocking functions g-gDouble Dagger for ionic and chemically associating particles.

https://iopscience.iop.org/article/10.1088/0305-4470/17/15/007/pdf

Exact equation for the pair-connectedness function

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

G Stell

Author Tools

Chat about Author

Papers

Exact equation for the pair-connectedness function