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.
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: nearestneighbor percolation and secondneighbor percolation. In nearestneighbour percolation two sites are considered to be connected only if they share a side. In secondneighbour percolation sites that share a corner are also considered to be connected.
NearestNeigbors SecondNeigbors
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).






























2. Tamar Tepper's web page on "Dielectric Properties of NanoComposite Materials".
3. Nir
Yefet's web page on "3D Visualization of Percolation Clusters".
My email: merez@stanford.edu