Correlated Percolation

Spring 1999


Mattan Erez


Introduction

In this project the behavior of the percolation point   as a function of the particle size is explored.

The motivation for the project are results of experiments conducted by Tamar Tepper on Composite Materials. Composite materials are artificial mixtures of two materials. In the context of this project we refer to one material as the conductor and to the second as the insulator.

The percolation point is defined as the concentration (of the conductor in the mixture) at which the sample begins to conduct electricity from top to bottom (for simplicity we deal with two dimensional samples). For a better understanding of this and related percolation concepts see[1] or Nir Yefet's page, which also includes a few percolation links.
In Tamar's experiments the particles of the two materials were not the same. The insulating particles were smaller than the conducting ones. This fact can be regarded as a kind of correlation between conducting particles - the conducting particles can be thought of as clusters of small particles (of a size equal to that of the insulating particles). The result of this correlation was a lower percolation
point than with no correlation (or equally sized conducting and insulating particles).

In this program a computer program was developed to simulate a similar correlation effect, and test the percolation point behavior as the conducting particles size (the correlation strength) is varied. The actual physical experimental model is very complex (it is three dimensional and includes a compression phase) and cannot be replicated on a computer (providing the simulations are limited by computer resources and time). Therefore a simpler model was tested.
 

Simulation Model


The model is of a two dimensional square sample of a uniform random distribution of conducting particles within a matrix. All the conducting particles are of the same shape and area. All sites in the
matrix (grid) that were not taken up by conducting particles, are assumed to be insulator particles (thus
insulator particles take up only one site - unit area). In order to allow for some of the effects of three dimensional samples, some simulations were also carried out on a 2.5D model.

A 2.5D model is an elegant and simple way of providing some of the flexibility gained by adding the third dimension. In 2.5D several sites are accommodated at each grid point (instead of only a single site per grid point in a pure 2D model). It is as if the conducting particles can overlap to a certain degree.
To illustrate the difference between the 2D, 2.5D, and 3D models look at the following example, of
arranging three conducting particles on a grid.

In the 2D model there is not enough flexibility to arrange all four particles, but both the 2.5D and 3D models allow all four particles in the matrix. In the 3D model the sample is not percolating, since
there is no connection from top to bottom, the particle touching the top and the one reaching the bottom are not on the same level and are not connected by the other particles. However, the sample is percolating in the 2.5D model.

Another parameter that greatly affects the percolation point is the type of percolation. The two types of percolation that were investigated here were: nearest-neighbor percolation and second-neighbor percolation. In nearest-neighbour percolation two sites are considered to be connected only if they share a side. In second-neighbour percolation sites that share a corner are  also considered to be connected.

Nearest-Neigbors           Second-Neigbors



 

Shapes

Blue - The Conducting Particle, Pink - Free Site,  Green - Protected Site (for the 2D optimized program)


Circular Shapes


       area = 5       area = 9               area = 13                     area = 21

Bar Shapes

                                                                    area = 3                area = 5
 
 

The Program

In general the program implements the model above for a limited number of conducting particle shapes and sizes. A full description of the program with all its functions and algorithms can be found in the readme file. The program itself has two versions: perc2_5d.c which implements the full 2.5D model, and perc2d.c which is more optimized (but less documented) version for a pure 2D model.
 
 

Results

2D - Nearest-Neighbor Percolation

Several simulations were run using this model to test the behavior of the percolation point, and also to measure the maximum concentration possible for the various shapes.

Since in a 2D model particles cannot overlap, there are inherent "holes" (insulating particles) between conducting particles, and a concentration of 1 can't be achieved (statistically).

Below is a graph showing the maximum attainable concentration as a function of the area.
It seems that the maximum concentration follows a simple power law y~x^(-0.1685).

Also, a few pictures are included to get an idea of the distribution under these conditions.

In these pictures, white - insulating, black - conducting.

Maximum Concentration:    Circle Area=9 N=200
                                             Circle Area=9 N=40
                                            Circle Area=5 N=200
                                             Circle Area=5 N=40
                                            Bar     Area=3 N=200
                                             Bar     Area=3 N=40
 
 

Percolation Point:            Circle Area=9 N=200
                                             Circle Area=9 N=40
                                            Circle Area=5 N=200
                                             Circle Area=5 N=40
                                            Bar     Area=3 N=200
                                             Bar     Area=3 N=40
 
 

In this model the results show (in contradiction of the physical experiment) that the percolation point rises with the area of the shape. However, the maximum concentration possible grows smaller as the area of the shape is increased, since there are more empty spaces between the particles. Therefore, percolation only occurs for small sizes - in the simulations the maximum area that still achieved percolation was 9.

Most samples in physical experiments are large enough to be treated as infinite. Due to the limited speed of current computer technology, and the limited time of this project, this cannot be said of the samples used in the simulations. However, the data collected from the simulations can be analyzed to gain some information on the behavior of infinite samples.

The analysis performed here is called finite size scaling ([1]). There are two variations on the same basic principle - extrapolate from the percolation point behavior for several sample sizes to the behavior as the size groes to infinity. The first method is to plot the percolation point as a function of 1/N (N is the number of sites in a grid line), the percolation point at infinty is found by extraplating to 1/N=0. The second method is based on an analytic result for the statistical properties of the percolation point measurements as the size of the sample is increased. THe property is that the percolation should behave as the measurement error (delta), so that extrapolating to a delta of 0, should give the percolation point of an infinite sample.

Analysis example:







In general for all results reported here, due to the limited time for the projects, not many grid sizes were checked and not large enough grids. Thereofre, the results may be skewed by strong finite size effects.

The table below summarizes the different percolation points (graphs used for the analysis and simulation list).
 

Shape and Area
 Perc. Point
1 (from literature)
0.5927
1
0.594
Bar 3
0.58
Bar 5
0.563
Circle 5
0.62
Circle 9
0.633

 
 

2D -  Second-Neighbor Percolation

Same analysis method as described above, but also checking for second neighbors (analysis graphs).
 
 
Shape and Area
 Perc. Point
1 (from literature)
0.5002
Circle 5
0.526
Circle 9
0.612

2.5D - Nearest-Neighbor Percolation

The added flexibility of the 2.5D model has a dramatic effect. Now the percolation point of the correlated model is lower than that of the uncorrelated one. The percolation point of the smaller of the two particles (area 5) was lower than htat of the larger one (area 9), but this may be due to finite-size effects, which are more pronounced for larger particles (analysis graphs).
 
 
Shape and Area
 Perc. Point
0.5
Circle 5
0.38
Circle 9
0.384

 

References and Links

1.  D. Stauffer and A. Aharony, "Introduction to Percolation Theory - Second Edition",
        Taylor & Francis, 1992.

2.    Tamar Tepper's web page on "Dielectric Properties of Nano-Composite Materials".

3.    Nir Yefet's web page on "3D Visualization of Percolation Clusters".



 
 

My e-mail: merez@stanford.edu