Computer Simulations of Dielectric and Magnetic Composite Media Including Dissipation

(1)

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Composite Media Including Dissipation

H. Leinders, A. Enders

To cite this version:

H. Leinders, A. Enders. Computer Simulations of Dielectric and Magnetic Composite Media Including Dissipation. Journal de Physique I, EDP Sciences, 1995, 5 (5), pp.555-564. �10.1051/jp1:1995151�.

�jpa-00247081�

(2)

J. Phys. I France 5

(1995)

555-564 _MAY 1995, PAGE 555

Classification Physics Abstracts

02.60 77.30 77.40

Computer Simulations of Dielectric and Magnetic Composite

Media Including Dissipation

H. Leinders and A. Enders

II. Physikalisches Institute, Universitàt _zu KôIn, Zülpicher Strafle 77, 50937 KôIn, Germany

(Received

28 November 1994, received in final form 13 January 1995, accepted 16 January

1995)

Abstract. A method of simulating dissipative effective media with bath dielectric and magnetic materai properties _is presel~ted. As fat _as particles with ramdam distribution _are _con- cerned, trie effective material properties of mixtures depend only _on trie volume fractions and trie bulk values of the materais involved. Trie numerical simulations have been performed for the tintee-dimensional _case and _an algonthm _was used, which leads to _a correct representation of Afaxwell's equations in _a discrete _vector _space without _any approximation. A formula _was

denved that describes trie behavior of trie Inixture for random distribution of trie partiales. A

physical interpretation of trie effective behavior is given, based _on percolation theory.

In this _paper, the

electromagnetic

behavior of

binary

mixtures with both dielectric and _mag- netic material

properties

is studied. A value which characterizes the mixture ratio of

composites

is the volume fraction

f

_:=

Vp~rticie/l§~mpie.

^The

partiale

sizes _a should be well below the _vac-

uum

wavelength

of the

electromagnetic

_waves used. Such materials _are known _as effective media

(a

_more exact definition will follow

below),

in which the _waves "sec" several

partiales

over each

wavelength.

There is _no well-defined

analytical theory describing

^the behavior of these mixtures,

especially

_on the

problem

of

extracting

the effective material

properties

of the mixture from the

corresponding

values of the

participating

components _over _a

satisfactory large

_range of mixture ratios.

There _are _many dilferent models

dealing

with this matter

(for

_an overview, _see

il, 2]).

All have rather _severe restrictions _on

partiale shapé, size,

volume fraction and

wavelength.

It will be shown later that most theories and mixture formulae _must fail _at volume fractions

higher

thon 0.3 due to _a violation of the base

preconditions. Nevertheless, they

_are

commonly

used

over the whole volume fraction _range.

In references

[3,4],

a novel numerical

approach

and the computer _program ^COSME was presented to simulate effective media with random distribution. This program ^has been extended to include

magnetic properties

_as well and is named COSME II. The

general problem

is _to

calculate the effective values from the

given

bulk

quantities. Conceming binary

mixtures, one

component ^is denoted _as

partiale

P

(which

vanishes for

f

₌

0),

the other _one is called matrix M. Both materials _are well defined

by

their bulk _response

D,

H to the external

exciting

fields

(3)

~

Ly

LX

Fig. l. Model cavity divided into regular cubes. Ail cubes _are randomly filled with either partiale _or

matrix material, according to _a given volume fraction for trie complete system. Measured in quantities

of the cube dimension, the whole cavity bas trie dimension Lx _x Ly x Lz.

E,

B which is described

by

the

complex permittivities

_cp, _CM and

permeabilities

_pp,pM. The

atm now is to connect these

quantities

^and the volume fraction

f

ta

get

^the ^effective

complex

values e, ji ^{of the} ^mixtures ⁱⁿ ^form ^of ^functions

f ₌

F(f,Cp,EM,ÎÎLP,

_ÎLMÎ) _~~~

ÎL "

G( f,

_jlp, _j1M, _ÎEP,_CMÎ

(~)

In the most

general

_case it is necessary to _assume relations between the bulk

permittivities

and the bulk

permeabilities

of the components, so the

complex

effective

permeability

is pos-

sibly

_a function of the bulk

permittivities

and ~ice _~ersa. It will be shown here that under certain conditions this

possibility

_can be

neglected,

_so in the formulae above the

corresponding

parameters _are included in brackets.

l. Method

A resonant

cavity

with the dimensions Lx _x

Ly

_x Lz _was divided into

regular

cubes with

edges

of unit

length.

Each cube _was

randomly

filled either with material M _or

P, according

to a

given

volume fraction

(probability) f

for the whole cavity

(see Fig. l).

Then the field distribution inside the cavity had to be determined. The method used is based _on the

description

of the field distribution inside

electromagnetic

devices

developed by

Weiland

[5-10].

To each cube

a

tripod

of electric field vectors is

assigned.

These vectors _are located _at the

edges

of the cubes

(see Fig. 2).

^Another

tripod consisting

of

magnetic

field vectors is

arranged

_so that the

vectors penetrate the surfaces of the cube.

Doing

_so for ail _N

= Lx

Ly

Lz

cubes,

_a set of 3N components of the electric field _and _3N components ^{of the}

magnetic

induction is

gained.

These vectors form two

grids;

a

grid

G

consisting

of electric field vectors, and _a

grid Ô

of the

vectors of the

magnetic induction,

shifted

by

half _a cube

diagonal

with respect to G.

The

problem

_con be solved

by transforming

Maxwell's

equations

into matrix

equations (for

further information _see Ref. [3]). ^It ^is

important

that with this kind of discretization of the fields both the

tangential

component ^of E and the normal component ^of ^B

satisfy

the

boundary

conditions between the cubes

implicitly.

For

instance,

_one of the Maxwell equations is transformed into

V _x E

=

-~~

-

CDse

=

-DAÉ

,

(4)

N°5 COMPUTER SIMULATIONS OF COMPOSITE MEDIA 557

G

r~J

G --Î--/

__~__

~

~ ^Z

~

Y ~--- x

E ^°~

Z

~X

Fig. 2. Arrangement of field _vectors inside trie cavity: every cube is assigned _a tripod of E vectors at trie edges, and another _one of B vectors through trie surfaces. Ail field components of ail cubes

together form vectors e, b of _very large dimensions which contain trie whole field distribution inside the cavity ^and constituting two lattices G

(electric

field

components)

and Ô

(magnetic

field

components)

where _e and b represent the two sets of field vectors, ^C represents ^the ^rotation operator in the discrete _space and the matrices Ds>A ^contain

length (S)

and surface

(A)

elements

[3,5].

To take the

partiales

into account, discretized material equations are aise established:

d =

coD~e

b =

poD~h.

This _way, the dielectric

displacement

D and the

magnetic

field H _can be defined in vector space,

leading

to the

following equation:

CDse

~2

=

~DAD~Di~DAD~e (3)

c

This relation _can be

simplified

to

~2

~A~D~e

₌

C~Dj~Ce

,

(4)

c

for trie

simple

cubic lattice used

here,

where A is trie

edge length

of trie cubes inside trie

cavity (this

is due to trie fact that these lattices _are Ds ₌ A .13N and DA ₌ A~ .13N _are

simple diagonal

matrices, where In is the

identity

matrix of the vector space with dimension

n).

This is _a _common, but

large

dimensioned and nonhermitian

eigenvalue equation,

which

con be solved

by

_an

algorithm developed by

Stôlzle [3]. The effective material

properties

con

be calculated from the

complex

_resonance

frequencies

of the empty and filled cavity:

m

=

())~ ₍₅₎

Attention should be

paid

to the fact that

only

_a linked

quantity

q con be

calculated,

Dot values for the effective

permittivity

and

perrneability separately.

It will be

presented

in the next section

why

this

problem

could be

neglected.

(5)

e~=18-125 %~=4-13.5 e~=%y=1

o" mmm "m

m~ ~~

]

~

Ù ~

~ à

à _~

~ g ~"

f m~

E m~

«"

a2 a4 06 DB 'o DD 02 a4 06 06 'o

Volume Fraction f Volum6 Fractian f

Fig. 3. Simulated data of partiales with both magnetic and dielectric material properties (cp ₌

18 -125, _~lp = 4 -13.5, EM = ~IM = 1). The real part _(E~I)~ passes zero at f @ 0.68. The imaginary part (e~l)" remams mol~otol~ically rismg.

2. Results

With the computer program COSME

II,

several

"samples"

_(CM, _PM, _cp,

pp)

have been simulated. In most cases, the volume fraction _was varied from

f

₌ 0

(pure matrix)

to

f

₌ 1

(pure partiale material).

TO

simplify

the

calculations,

the matrix pararneters were

always

chosen _as CM = pM = 1, which is trot _a

significant

reduction, because any matrix values _cari be repro-

duced

by

_a

scaling

_process _D~ _-

D]lem.

In the _case of

e[ # 0,

this leads _to dilferent loss tangents ^which ^has ^% ^be ^considered

by appropriate

_new values for _cp. We would like to stress the fact that there is _no

difficulty

at all in

computing (cp)

_over the whole _range of volume fraction

using

^the

COSME-program,

_a ^fact which is important ^later _on. The results _were calculated

depending

_on the volume

fraction,

_so

always

functions

(cp)( f)

will be

presented.

The simulation

yields

two dilferent types ^of curves.

Depending

_on the

given

set of bulk parameters _(CM, _PM, _cp,

pp),

the _curve of the real part

(c~l'

_can _pass _zero at _a

special

volume fraction

fo (Fig. 3),

_or the _curve is smooth and

increasing monotonically (Fig. 4).

The

plot

is _more curved at smaller volume fractions and _more linear at

higher

_ones. For all

plots,

the

imaginary

part

(cp)"

is

always increasing monotonically

and

continuously;

_a

sign change

would violate _energy conservation inside the

cavity,

of _course.

Under certain conditions it is

possible

to factorize the _common value

$

_into _the _parts

e and ji

iii,12].

This is valid for _an internai

wavelength

À~,

represented by

the _wave vector k~ = 27r/À~, ^smaller ^than ^the

particle

dimensions _r ((k~

ri

«

1),

and has been tested in the

range up to k~ r m 1: Simulations have been done _to compute

samples

with the

complete

set of bulk pararneters. This _was followed

by

two more

simulations,

the first with _cp ₌ 1, _pp

#

1 to

calculate ji, ^then a second with _cp

#

_1, _pp

= 1 to obtain e. As _a _next step,

(cp)

_was

compared

with the

product

e. ji. ^The result _was that the

correspondence

_was

nearly perfect (Fig. 5).

This lits well the

precondition

for factorization

according

to

il Ii,

_{1-e-, the}

complete penetration

of ail

partides by

electric and

magnetic

fields without any

significant

decrease inside the

partiales.

These elfects _are net included in the simulations _as

long

_as the

particles

have the minimum size

(1.e.,

the size of _an

elementary

cell in the

cavity),

_so this condition is fulfilled _a

priorf.

It

cari be concluded that the simulations _are valid if the

partiale

sizes are chosen in _a range where

complete penetration

is

guaranteed.

^Under this condition the

permeability

_can be studied

separately

from the

permittivity, although

a

partiale

with _pp

#

1, _cp ₌ 1 does net exist. This

(6)

e~=15-14 %p=7-13 e~=p~=1

~ ^°°

R é

É 60 )

Î ~o

~

,'

1 $

°C

o

~. o'

«' m.

Do o 2 0 4 DE o 6 'o DD o 2 0 4 o 6 off 0

Volume Fraction f Volume Fraction f

Fig. 4. Results from simulation of partiales with trie bulk materai values _cp

= 18 -125, _~lp

=

4 -13.5, _EM _" _~IM

" 1. Trie real part

(epl'

rises continuously with _no _zero fine crossing. Trie imagmary

part (e~l)" ^looks similar to trie _one in trie previous plot.

/z~=4-13.5 e~=18-125 ^e~=/z~=l

'~°A ""

)

""

'~°À ~

b_

(

'DD '

à

a

~Î 1~

$

~ à

)

~

f 1

É

o

ce ce

E

#

Î

2c

0 2 0 4 0 6 ù 8 a 2 0 4 off DB '0

Volume Froct>on f Volume Fraction f

(Values tram Simulation) (Values tram Multiplication)

Fig. 5. Simulated data for partiales with _ep

= 18 -125, _~IP

= 4 -13.5, _EM ₌ _~IM ₌ 1. On trie left hard side trie complete simulation (e~l) ^of ^both properties together is shown. On trie nght hard, both values _were simulated atone and multiphed to give e ji. Comparmg trie results it tums out that there

is a nearly perfect agreement. So it is allowed to banale trie dielectric and magnetic eoEects separately.

has to be

kept

in mind for the

following plots,

where

"purely magnetic partiales"

_are shown.

Looking

at the

graphs,

the

similarity

with _curves of

purely

dielectric

partiales

is obvious

[3,4]

(Fig. 6). Also, they

show the _same characteristics: _a curved _region at small volume fractions and _an almost linear

region

above _some critical volume fraction

fc. Looking

_more

closely

at the linear parts of the _curves, it tutus out that

they

_are

slightly

curved _to the

right.

In

spite

of the fact that this elfect is

relatively small,

it becomes important ^for ^the

physical interpretation

of the _curves.

The simulations have to be

compared

with the most often used classical effective medium

approximations,

which is shown _in

Figure

7.

Although

the formulae _are

normally

used for

dielectrics,

it _can be

proved

that

equations

^for serres and

parallel

circuits have the _sonne _ap- pearance for

magnetic partiales

_as for dielectric material [14]. Also, the Maxwell-Garnett ar-

gumentation

can be

applied

_to

permeabilities,

_as well _as the

Looyenga

formula. The Maxwell- Garnett relation _is

plotted

for the whole range of volume fractions,

although

it is not well defined above _some maximum volume fraction which

depends

_on the

particle shape.

It is obvi-

(7)

%p=30-110 ^%M=eM=ep=1

W

'm ~

~ à

à _~

,~o

oe g

E ,"

m~

«~

Do 02 a4 off off 'o 02 04 off off 1a

Volume Fraction f Volume Fract>on f

Fig. 6. Simulated results for '~pure magnetic partiales". This is only justified by trie factorization property of trie effective materai properties. It tums out that pure magnetic _curves show trie _same

charactenstics _as dielectric materials with comparable bulk values. Trie graph is curved for lower volume fractions and almost linear above _some critical volume fraction fc. This froids for both, trie real part

ji'

and trie imaginary part

ji".

Recognize trie slightly nght curved part for higher volume fractions, _an eoEect which is _more obvious _on trie right hard.

2s

/

~

20

o o

's ^~

o o o

'o o

o o

o s

~ o o

o

a o 2 a 4 a s o s '

Volume Fraction f

Fig. 7. Comparing trie most often used classical formulae it is revealed that there _are large dioEer-

ences between trie simulated data al~d trie usual analytic forms. Especially trie Maxwell-Garnett result,

which is often used to analyze data deviates, widely from trie simulation results _even at small volume fractions. The best approximations _can be taken from trie _more _recent Looyenga form and the simple paral1e1circuit formula. Only trie real part ^{of trie} plot is shown for _pp

= 25 i10, _pM

= SP = EM = 1.

Trie deviations of trie data from trie classica1are _as drastic for trie _imagil~ary part as for trie real part shown here.

ous that these formulae _are _not suitable _to describe the measurements,

although

_some of these formulae _are used

quite

often _to

analyze

data. So _a _new formula is

mandatory

to describe the results _more

accurately.

Their _way of

setting

_up this _new formula follows

straightforwardly

from the

interpretation

of the behavior of the effective media.

(8)

N°5 COMPUTER SIMULATIONS OF COMPOSITE MEDIA ₅₆₁

i

o >

Fig. 8. - The idea _for trie interpretation of trie almost _linear, but shghtly right

curved parts

plots at higher volume fractions is to assume a _onnection

through the hole cavity

clusters from percolation _eoEects above trie percolation ^reshold. So some kind _of parallel connectivity

atenal.

3.

It is an^bvious ^act that media, where _the

of

one component _varies

from

f = 0 to f

=

1 and there _is a perfect

connection

between

two

ondition

for

the _xistence of ejfecti~e

material

values

which was

used

_until now,

namely

that

the

_particles have

to be small

compared

to

their innerwavelength.

It

follows

ercolation theory that formerly

separated particles luster and the

cluster(s)

(at

least

one) grow to infinity very apidly at a _critical ccupation

probability

which is

characteristic

for the

used lattice structure. In the present

case,

the bond percolation

(see

appropriate tool, because only _bonds _are necessary ^for connectivity and

not

some matching

areas

between

sites.

Also

we are talking about simple on this

subject

to ^mpare

the ^esults.

So _the _infinite avelength

limit

(k~ . _ri < 1 is not valid except for

very small olume fractions,

and even the

long avelength limit kvacuum r

growing_clusters, ^lready _below the _critical volume ractionl

his has to be _taken into

account

_in two dilferent ways. _First,

at

very

small

volume

f,

ail

clusters

_consist of one partiale, so most theories are applicable.

After

that, with

rising

olume

fraction,

the _clusters become larger. This _way,

a

kind of

average partiale size has

to

_be

applied to _the _theorieswhich leads to higher _values for the effective media pararneters.

Very soon, the

clusters grow _very fast and at

least

_one cluster extends over

This

follows from

the definition of the critical

ccupation

one

cluster reaches the

size _injinity. This,

of course,

can be described as

a

_parallel

circuitbetween this

cluster and the surrounding media,

smaller clusters of partiale _material.

On the other hand, the

surrounding media are no longer pure atrix, but a mixture

matrix

material

and everal

clusters

with some size stribution. This _leads to the fact hat

the straight fine

of

arallel

connectivity has to be lifted on the _side

of

f =

matrix

material

_has hanged to

higher

aterial values. This _ehavior

is

shown in _Figure

8

_in

an exaggerated way. So _the graph is not linear at higher _volume

fractions

but

slightly

an

_eifect which

ermeability or

(9)

p~=25 ^(real) -> Fit. p~~/~~~~=(f*pp~~~~~+(l-f)py~~~~ ^~)

a=0493 b=0493

Fit Function à

é °

g

E ^°

é ?

~ =

é Î

°~ LoDyenga ^°°

02 03 04 05 08 07 08 09'ù

DD o' 02 03 04 05 08 07 08 09 'a

Volume Fraction f Volume Fraction f

Fig. 9. For _pp ₌ 25, _pM ₌ _EM ₌ SP = 1 trie simulated data (dots) _are plotted with trie developed fit function

ji( f).

It is obvious that trie prediction of trie data is _very good, although trie fit parameters

a, b _were found by _a variety of dioEerent simulation data. It could be shown that these fitting parameters froid for _a _very wide range of materai properties _as long _as trie _same lattice structure is used. Trie

parameters ^therefore are connected to trie lattice. Also trie deviation from trie data is shown in trie

plot to trie nght. Trie

errons remain smaller than 4 percent.

ered.

First, only

functions defined _on the whole _range of volume fractions _are allowed.

Next,

of all classical

theories,

the

Looyenga

form _gives the best

prediction

of the simulation results.

Further,

the formula should become _a classical formula for _very small volume fraction.

And,

as a last point, ^the

growing partiale

_size should be taken into account.

Bànhegyi

_[13] has shown for dielectrics that _a

good

_way to

apply particle

sizes and

shapes

to trie

Looyenga

formula _is to

modify

its exponent. ^In

analogy

to that

approach,

it _can be concluded

by

similar arguments that the

growing particles,

1-e-,

clusters,

have their elfect _on the exponent ^of the effective media formula.

Using

this ansatz, an

equation

of the

following

form has been tested for the effective _perme-

ability

of

binary

mixtures

(as

it _was clone for

permittivities

in

[3,4])

lÂ(f)1"~~~ =

f

_Pi~~~ +

(i f)

_P[~~~

(6)

a(f)

=

a+b.f. (7)

It tutus out

that,

with _an appropriate choice of the parameters a and b, the

computed

results

could be

reproduced

within _a few percents

(Fig.

₉₎₁

Also,

this formula _can be

applied

to dielectric media with _some _new parameters c and d

by

le(f)l~~~~

=

f

_EÎ~~~ +

(1 f)

_CÉ~~

(8)

fl(f)

=

C+d.f (9)

very

successfully.

Of _course,

joining

these _two

formulae,

the

complete

result of the simulation _can be

predicted

very well

by

W(f)

=

e(f) A(f) (i°)

It should be clear that the parameters _a, b and _c and d have to

depend

_on the used

lattice,

because the

percolation

threshold varies for dilferent

grids.

For the

simple

cubic lattice that

(10)

we have

chosen,

the pararneters _were determined _to be

a = 0.493

b = 0.493

c = 0.56

d = 0.42.

(Il)

So it is _now

possible

to describe the simulated data

by

_an

analytical

formula with suitable accuracy. It should be mentioned here that for dielectrics the simulation lits _very well the

experimental

data [4], so it is reasonable _to _assume that the function is _a

good description

of real situations

including magnetic properties.

4. Conclusion

It has been shown that in _a

physical meaning

an effective medium _cannot exist above _a critical volume fraction.

Therefore,

known classical models _con be used neither for the _exact inter-

pretation

of measured data, nor for the

design

of _new

electromagnetic

materials.

So,

a new

way of

describing

the

ejfecti~e

behavior at

higher

volume fractions had _to be found. With the tools of computer simulations, mixtures could be

computed

within the whole range of volume

fractions. With this

data,

_an

interpretation

_was

given

which leads in _a

straightforward

_mariner

to _a function that describes the mixture values with

high

_accuracy for _a

given

lattice type. SO,

once the parameters for _a

given problem

with _a

special grid

_are

determined,

the _new

analytic

function _can be used to calculate the desired values without further simulation. On the other

hand,

for well-known materials it is _now very easy ^to find the

appropriate

mixture for _any given

ejfecti~e

material values.

Acknowledgments

We would like to thank S. Stôlzle for the great amount of work _on which this

investigation

is based. This work _was

supported by

the BMFT

/

Bonn

(Grant 03M273782)

and the computer

center of the

University

of

Cologné,

where ail calculations have been

performed.

R. Pelster

and G. Nimtz have also contributed their part to this work.

References

iii

Van Beek L.H.K., Progress in Dielectrics 7

(Heywood

Books, 1967) _p. 69.

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