Skip to main content
U.S. flag

An official website of the United States government

Official websites use .gov
A .gov website belongs to an official government organization in the United States.

Secure .gov websites use HTTPS
A lock ( ) or https:// means you’ve safely connected to the .gov website. Share sensitive information only on official, secure websites.

Part III Chapter 3. Percolation Theory

This section discusses percolation in 2-D, using the model of randomly overlapping holes of various shapes. The beginning point is a conductive sheet, which then has holes randomly introduced into it. Percolation occurs when the sheet finally falls apart due to the solid backbone losing connectivity. The effect of hole shape on the percolation threshold is studied ("microstructure" as it affects percolation). This section shows how the percolation threshold can be made approximately universal by expressing it in terms of the critical number per unit area at percolation, weighted by an area quantity that comes from an electrical analysis of the effect of one hole on the conductivity of the sheet. There is also a review of percolation in 2-D.

(1) Universal conductivity curve for a plane containing random holes (16 pages of text, 27K of figures)

This section discusses percolation in 3-D, using the model of randomly overlapping ellipsoidal shapes. The starting point is empty space, which is then randomly filled with overlapping ellipsoidal shapes. These shapes are ellipsoids of revolution, but of arbitrary aspect ratio, ranging from very thin disks to very narrow needles. Percolation occurs when the solid shapes become continuous. The effect of aspect ratio on this percolation threshold is studied. Various shape functions, including the intrinsic conductivity of an ellipsoid, are introduced and used to scale the percolation threshold to try to construct an approximate invariant, like was done in 2-D. This attempt is not as successful as in 2-D. There is also a review of percolation in 3-D.

(2) Geometrical percolation threshold of overlapping ellipsoids (17 pages of text, 22K of figures)

This section contains an education module that uses 2-D PC and Macintosh-based algorithms to teach basic concepts of cement hydration and percolation theory. The percolation algorithms include random percolation of ellipses, rectangles, and the hard core/soft shell system for circles.

(3) Computational Materials Science of Cement-Based Materials: An Education Module (31 pages of text, 52K of figures)

Go to Part III Chapter 4. Microstructure development and probes

Go back to Part III Chapter 2. Finite element and finite difference programs for random materials


(1) E.J. Garboczi, M.F. Thorpe, M. DeVries, and A.R. Day, Physical Review A 43, 6473-6482 (1991).


(2) E.J. Garboczi, K.A. Snyder, J.F. Douglas, and M.F. Thorpe, Physical Review E 52, 819-828 (1995).


(3) D.P. Bentz, E.J. Garboczi, and R.T. Coverdale, National Institute of Standards and Technology, NIST Technical Note 1405, 1993.



Created July 20, 2017, Updated November 15, 2019