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COMPUTER INVESTIGATION OF THE PERCOLATION PROCESSES IN TWO- AND THREE-
DIMENSIONAL SYSTEMS WITH HETEROGENEOUS INTERNAL STRUCTURE





Konash A.V., Bagnich S.A.
e-mail: konash@imaph.bas-net.by


Institute of Molecular and Atomic Physics, NAS,
220072, Mensk, Belarus

















Introduction


Percolation processes, first discussed by Broadbent and Hammersley,
occur in diverse physical systems. The percolation model has been found
useful to characterize many disordered systems, such as porous media,
fragmentation and fractures, gelation, random-resister insulator systems,
dispersed ionic conductors, forest fires and epidemics. This model was used
successful to the description electronic properties of high doped
semiconductors. On the base of percolation theory the cluster formalism for
description energy transport in disordered media was developed by Hoshen
and Kopelman (Hoshen J., Kopelman R. // J. Chem. Phys. 65 (1976) 2817).
This model is based on the such mathematical function of the percolation
theory as percolation probability, P( , and average finite clusters size,
Iav . These functions have the critical exponents ( and ( associated with
them (Stauffer D. // Phys. Rep. 54 (1979) 1):
P( ( (C/Cc - 1((, Iav ( (C/Cc - 1(-( ,

where C is the concentration of the occupied sites in the system, Cс is the
critical concentration. In result of the investigation of the electronic
excitation energy transport in mixed molecular crystals and solid solutions
of organic molecules the values of critical exponents were obtained
corresponded to two-dimensional and tree-dimensional space (see Table).
However, the values of critical exponents differed from the theoretical
values were found when the porous glass was used as the matrix.


Table

|Critical |Percolatio|Isotopicaly|Chemically|Solid solution|Ethanol solid|
|exponents|n theory | | |of |solution of |
| | |mixed |mixed |benzaldehyde |benzaldehyde |
| | |molecular |molecular |in ethanol 3) |in porous |
| | |crystals 1)| | |glass 4) |
| | | |crystals | | |
| | | |2) | | |
| |2 D |3 D | | | | |
|( |0.14|0.41|0.13 |0.13 |0.41 |0.25 |
|( |2.1 |1.6 |2.1 |2.09 |1.7 |1.95 |

1) Ahlgren D.C., Kopelman R. // Chem. Phys. 77 (1981) 135.
2) C. von Borczyskowski, T. Kirski // Ber. Bunsenges. Phys. Chem. 93 (1989)
1377.
3) Bagnich S.A. // Chem. Phys. 185 (1994) 229.
4) Bagnich S.A. // SPIE Proc. 3176 (1997) 212.

This effect was connected (see Bagnich S.A. // Phys. Solid State 42
(2000) 1775) with the inhomogeneous properties of porous glasses. The
microscopic inhomogeneity of porous glasses determines the effective
topology of the space in which processes of percolation of electronic
excitation energy evolve.


Goal of investigation


The aim of these investigations is the study by the method of Monte-
Carlo influence of heterogeneous properties of systems on percolation
process, namely on the critical concentration, percolation probability,
average finite cluster size, the values of critical exponents, the fractal
and spectral dimensions of percolation cluster in critical point. The site
percolation on the square lattice, which contains the obstacles distributed
randomly in the simulation cell, is considered in this paper.


Determined parameters

1) Critical concentration
2) Average finite cluster size:
[pic],
where m is the cluster size, im is the number of clusters of size m.
3) Reduced average finite cluster size:
[pic],
where mmax is the size of the largest cluster.
4) Percolation probability:
[pic] ,
where L is linear size of lattice.

5) Fractal dimension of the percolation cluster from following
relation:
[pic],
where M is the number of percolation cluster sites in the square with
linear size l.
6) Spectral dimension of the percolation cluster at the critical
concentration:
[pic][pic],
where S is the number of distinct sites visited during an N-steps random
walk.


Influence of obstacles on the critical concentration

The critical concentration was found by two methods.
1) One is the maximum of reduced average finite cluster size.

The results for lattice without obstacles size 200x200 are presented below:
2) One is the point of appearance of percolation between opposite sides.


Samples of matrices with various linear size of elementary obstacle (lo)
and their relative area (Sobs)





Influence of obstacles on critical exponent ( : Iav ( (C/Cc - 1(-(


Result for lattice without obstacles





Effect of full area of obstacles
Effect of linear size of elementary obstacle


Effect of finite size of lattice (Sobs = 38%):

Influence of obstacles on percolation probability

Effect of full area of obstacles

Effect of linear size of obstacles




Influence of obstacles on the fractal dimension of percolation cluster

The fractal dimension of percolation cluster was determined by the
nested squares method. 1. Result for lattice without obstacles





2. Results for the lattices with obstacles

Influence of obstacles on the spectral dimension of percolation cluster


Conclusion


The result presented in this paper show that introduction of the
obstacles in the lattice causes strong influence on all percolation
parameters.
The presence of obstacles in a system leads to:
- increase of the critical concentration value; this effect grows with
increase of the full area of obstacles and falls with increase of the
linear size of ones.
- increase of the critical exponent ( value; this effect increase
with growth both of linear size of obstacles and of their full area.
- increase of the growth rate of percolation probability with
increase of the concentration; this effect grows with increase of the full
area of obstacles and falls with increase of the linear size of ones.
- decrease of the fractal dimension of percolation cluster; this
effect increase with growth both of linear size of obstacles and of their
full area.
The spectral dimension of percolation cluster does not depend on the
presence of obstacles in lattice what accords with the conception about
super universality of this parameter.



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