A New Mixing Rule for Predicting of Frequency-
Dependent Material Parameters of Composites
Konstantin N. Rozanov
#1
, Marina Y. Koledintseva
*2
, and James L. Drewniak
*3
#
Institute for Theoretical and Applied Electromagnetic of Russian Academy of Sciences,
13 Izhorskaya ul., 125415 Moscow, Russia
*Center for Electromagnetic Compatibility, Missouri University of Science and Technology,
4000 Enterprise Dr., HyPoint, Rolla Missouri 65401, USA
—A number of mixing rules are proposed in the
literature to predict the dependence of effective material
parameters of composites, the permittivity and permeability, on
frequency and concentration. Alternatively to the mixing rules,
properties of composites can be considered in terms of the
Bergman–Milton theory (BMT), which employs the concept of
the spectral function. All known mixing rules are particular cases
of the BMT. Particularly, the Ghosh−Fuchs theory (GFT) has
been proposed based on the BMT. The GFT is shown to agree
well with measured material parameters of composites filled with
ferromagnetic metal powders. However, the GFT is not
convenient for use because of its complicated mathematical form.
Herein, a simple analytic formulation of the GFT is proposed.
The new mixing rule is based on the shape of the spectral
function typical for the Bruggeman effective medium theory with
the averaged depolarization factor of inclusions and the
percolation thresholds introduced as fitting parameters. Since
the permittivity and permeability of a composite are governed by
the same mixing rule, these fitting parameters are found from
the concentration dependence of permittivity of the composite for
further use in the analysis of the frequency dependence of
permeability. The proposed mixing law is valid for the case of
nearly spherical shape of inclusions in the composite, e. g., stone-
like inclusions.
I. INTRODUCTION
Numerous novel composite magneto-dielectric materials have
been developed recently for applications in radio frequency
and microwave electronic devices. To engineer microwave
properties of composites, it is important to be able to predict
wideband frequency response of effective material
parameters, specifically, permittivity ε
e
and permeability µ
e
, of
a composite as functions of the concentration p, permittivity ε
m
,
and permeability µ
m
of inclusions.
The effective permittivity and permeability of a composite
are conventionally assumed to be governed by the same
mixing rule. A number of mixing rules has been proposed in
literature, see, e.g., [1]. The mixing rules in the most common
use are the Maxwell Garnet equation (MG)
( ) ( ) 1 1
1
1 1
1
0 0
− +
−
=
− +
−
α
α
β
β
n
p
n
, (1)
Bruggeman's Effective Medium Theory (EMT)
( )
0
1
) 1 (
1
0 0
=
− +
−
− +
− + +
−
β β
β
β α β
β α
n
p
n
p , (2)
and Landau−Lifshits−Looyenga mixing rule (LLL), being
often referred to as asymmetrical Bruggeman’s mixing rule
( ) ( ) ( ) 1 1 1 1
3 1 3 1
− + = − + α β p . (3)
In (1)–(3), α=ε
m
–1 or µ
m
–1 and β=ε
e
–1 or µ
e
–1 are the
dielectric or magnetic susceptibilities of inclusions and
composite, respectively, all normalized to the corresponding
parameter of the host matrix; p is the concentration, and n
0
is
the shape factor (depolarization or demagnetization factor) of
inclusions. The shape factor is the same for both the
permittivity and permeability of a particular composite.
Equations (1)–(3) are written for spherical inclusions that are
characterized by a single value of n
0
; for non-spherical
inclusions, an account for orientation of these must be
included in the equations. All the mixing rules are quasi-static,
therefore the frequency dependence of effective material
parameters appears due to frequency dependence of the
material parameters of inclusions.
An agreement of a certain mixing rule with properties of a
particular composite depends on the dielectric or magnetic
contrast in the composite, which is the difference between the
corresponding material parameter of inclusions and of the host
matrix [2]. The LLL mixing rule is an exact result for the case
of small contrast (α≈0) and describes therefore the high-
frequency behavior of the composite. The result of the LLL
mixing rule is independent on the shape factor of inclusions.
With larger permeability of inclusions, the MG mixing rule
agrees closely with the measured data. This is the most
frequent occasion for the microwave permeability of magnetic
composites, as the intrinsic permeability of magnetic materials
does not exceed several units at microwaves due to its fast
decrease with frequency according to Snoek’s and Acher’s law
[2]. With these low values of intrinsic permeability, the
dependence of the effective parameters on the shape of
inclusions appears. However, the dependence is still weak and
may therefore be characterized by an averaged demagnetization
factor n
0
, which is the case of MG mixing rule [3].
2010 URSI International Symposium on Electromagnetic Theory
978-1-4244-5153-1/10/$26.00 ©2010 IEEE 584