Effective-Medium Approach for Composite Materials Containing Conductive Sticks
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EFFECTIVE-MEDIUM APPROACH FOR COMPOSITE MATERIALS CONTAINING CONDUCTIVE STICKS
A.K. SARYCHEV* AND Y.R. SMYCHKOVICH* *Institute for High Temperature, USSR Academy of Sciences, Moscow 127412, USSR
There is a growing interest in composites in which conductive component has the elongated shape (sticks) because these systems can be used to describe various physical objects [1]. The main tool to investigate the systems is MC simulation. For example in papers [2-4] the percolation threshold in such systems has been explored and it was obtained that pc• r/L, where r is the radius of stick and L is it's length. In this work, we are suggest the calculation of dielectric constant and conductivity method based on effective medium approximation. Let us consider a system, where conductive inclusions have a shape of elongated rotation ellipsoids a o b = c (this shape is useful "for an analysis and represents a stick quite good), and the dielectric is represented by spherical particles. Further on, we shall consider conductive and dielectric particles which are immersed in the medium with effective dielectric constant Eotf, and write the equations from the effective medium theory (EMT) (see, for example, [5]). Unfortunately, common EMT can't be used in this case. Indeed,the equation due to averaging the electric field in the medium will be:
E ef+nxCESoff f
(l-p)
33
1-)
2
6er
Eeff
=
Ef
Eaf +
where EM, Ed dielectric (6m=
+lE
f+n 2l Cm-e 6, f
rr
+f
1()
d,(I
are dielectric constants of conductor 1+4ILO /W), U and p are the conductive
conductivity and concentration accordingly, ellipsoid depolarization coefficients [6), nX principal axis, n - across it. The
equation
y
of
average polarization
will
nx, -
be:
Mat. Res. Soc. Symp. Proc. Vol. 195. 01990 Materials Research Society
n along
and phase are the
290
II P
P3
1
m-
ff
+
Se6
of f +nx (Er--
is
f
6
3 (Ed--so f f def 2 Soff +d
(1-p)
It
Eof
clear,
)
(~-e~
ftf •f Ee
6
2 f+ny(Sm (em-- 8ff)efd
1 I
+
+
(2)
=
(2)
that if
n
and n x
y
does
not
equal
to
1/3,
equations (1), (2) are not equivalent. It means that the self-consistence conditions are not conformed and effective medium theory appliance in this case is in problem. In order to conform the self-consistence conditions, let us suggest, that the effective medium is locally anisotropic, with the anisotropy axis, that is directed towards the stick axis. We shall discuss the simplified system (infinite prolongated inclusions directed in one side) for the illustration of such an approach (Fig. 1). It is easy to see, that such a system is characterized by two effective dielectric constant 8] and E6l but
ft
*
f
j
t
unlikely the discussing system it is globally anisotropic. We shall assume
•-that"
•
the effective surrounding the
the dielectric
*f
ff
"
"
along
its
axis
stick
medium has
constant
and
61
6]
a
cross it. To decide in what a effective surrounding is *
,,
* *
•
.
4 .,
*
the let
particles of dielectric us consider Fig. 1.
If
the external
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