1063-7834/01/4312- $21.00 © 2001 MAIK “Nauka/Interperiodica”
2313
Physics of the Solid State, Vol. 43, No. 12, 2001, pp. 2313–2320. Translated from Fizika Tverdogo Tela, Vol. 43, No. 12, 2001, pp. 2220–2227.
Original Russian Text Copyright © 2001 by Bagnich, Konash.
1. INTRODUCTION
Percolation processes were first considered by
Broadbent and Hammersley [1]. These processes can
occur in different physical systems. The percolation
model was successfully applied to the description of
disordered systems (for example, porous media) and
related phenomena. Among these are rock fracture,
fragmentation [2] and gelation [3, 4], conduction in a
random resistance grating [5] and strongly inhomoge-
neous media [6], and propagation of forest fires [7, 8]
and epidemics [9, 10]. This approach made it possible
to describe the electronic properties of doped semicon-
ductors [11].
Relying on the percolation theory, Kopelman
et al.
[12, 13] developed a cluster formalism for describing
the electronic excitation energy transfer in inhomoge-
neous systems. This model is based on mathematical
functions, such as the percolation probability
P
∞
and
the average number
I
AV
of sites in a cluster. The depen-
dence of these quantities on the concentration
C
of sites
through which the energy migrates is determined by the
scaling relationships [14]
(1)
(2)
I
AV
C/C
c
1–∝
γ–
,
P
∞
C/C
c
1–
β
,∝
where
C
c
is the critical concentration of sites and
β
and
γ
are the critical exponents, which depend only on the
space dimension. Investigations into the transfer of
electronic excitation energy in mixed molecular crys-
tals [15, 16] and solid solutions of organic compounds
in low-molecular vitrifying solvents [17] have demon-
strated that the critical exponents determined experi-
mentally coincide with those obtained within the perco-
lation theory for two-dimensional and three-dimen-
sional spaces, respectively (see table). However, recent
studies [18–20] of similar processes in porous matrices
revealed a discrepancy between the experimental and
theoretical critical exponents. Saha
et al.
[21] also
noted that the matrix affects the topology of the energy
transfer. In [18–20, 22], this effect was explained in
terms of the inhomogeneous properties of porous
glasses used as matrices. A microscopic inhomogeneity
of porous glass brings about a change in the effective
topology of the space in which percolation processes
occur. In turn, this can affect the formation and growth
of clusters from incorporated molecules.
In this work, we performed the Monte Carlo com-
puter simulation of the percolation process on a square
lattice with inhomogeneities differing in size and rela-
tive area in order to elucidate the possible effect of
these inhomogeneities on the critical concentration
C
c
,
the average number
I
AV
of sites in a cluster, the percola-
The Influence of Inhomogeneous Properties of a System
on the Percolation Process in Two-Dimensional Space
S. A. Bagnich and A. V. Konash
Institute of Molecular and Atomic Physics, Belarussian Academy of Sciences, Minsk, 220072 Belarus
e-mail: bagnich@imaph.bas-net.by
Received January 23, 2001; in final form, March 27, 2001
Abstract
—The percolation process in a two-dimensional inhomogeneous lattice is studied by the Monte Carlo
method. The inhomogeneous lattice is simulated by a random distribution of inhomogeneities differing in size
and number. The influence of inhomogeneities on the parameters (critical concentration, average number of
sites in finite clusters, percolation probability, critical exponents, and fractal dimension of an infinite cluster)
characterizing the percolation in the system is analyzed. It is demonstrated that all these parameters essentially
depend on the linear size of inhomogeneities and their relative area.
© 2001 MAIK “Nauka/Interperiodica”.
LOW-DIMENSIONAL SYSTEMS
AND SURFACE PHYSICS
Critical exponents for mixed molecular crystals and solid solutions of organic compounds
Critical
exponents
Percolation theory Isotopically
mixed molecular
crystals
Chemically
mixed molecular
crystals
Solid solution
of benzaldehyde
in ethanol
Ethanol solution
of benzaldehyde
in porous glass
2
D
3
D
β
0.14 0.41 0.13 0.13 0.41 0.25
γ
2.1 1.6 2.1 2.09 1.7 1.95