Numerical simulation of random composite dielectrics

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Numerical simulation of random composite dielectrics

S. Stölzle, A. Enders, G. Nimtz

To cite this version:

S. Stölzle, A. Enders, G. Nimtz. Numerical simulation of random composite dielectrics. Journal de

Physique I, EDP Sciences, 1992, 2 (4), pp.401-408. �10.1051/jp1:1992153�. �jpa-00246495�

J. Phys, I France 2

(1992)

401-408 APRIL 1992, PAGE 401

Classification Physics Abstracts

02.60 77.20 77.40

Numerical simulation of random composite dielectrics

S.

St61zle,

A. Enders and G. Nimtz

II. Physikalisches Institut der Universitit _zu K61n, 5000 K61n 41, Germany

(RecHved

18 December 1991, accepted 24 December

1991)

Abstract. A novel computer ^{simulation method is} presented for the non-dissipative dielec- tric response of binary mixtures. It is based _on the discretization of Maxwell's equations and thus takes into account aII classical

long-range

interactions between different particles implicitly.

The simulations performed yield _a relationship between the dielectric functions of the _two

com-

ponents, the volume fraction and the dielectric function of the mixture, which _can be described

by _an exponential formula. It _was found that the exponent in this formula depends _on the volume fraction but not on the dielectric properties of each component. The mixture rule is in agreement with experimental data taken from literature.

I. Introduction.

A mixture of

electrc-magnetically

diTerent materials which shows _a

macroscopically

horno- geneous

behaviour,

is called "ETective Medium". For _more than _a century ^{scientists and}

engineers

have been

trying

to describe the effective dielectric _response of such _a

composite material,

for _an overview _see

ill.

Several theoretical

approaches

have been

developed

into formulas

characterizing

the

relationship

between the

complex

dielectric functions

(DF)

of the two components _{£, Em,} the

respective

volume

filling

factor

(or:

volume

fraction) f

and the

resulting

effective DF ? of the mixture.

However,

there _{are many}

approximations

_necessary

in the

analyticaI deveIopment, concerning

^the inner

topology

of the structures _as well _as

size, shape,

DF and volume fraction of the

particles.

As _a consequence none of the formulas _can be used to describe all

possible experimental

data.

They

_are all limited to small

filling

factors

f

_<

0,I) and/or

to

only

small differences between the DF'S of the components. This

problem

even occurs in the

simple

_case of

water-in-oil-microemulsions,

_as

reported

in [2, 3]. In this

paper we

investigate non-dissipative

dielectric mixtures

by

computer simulations and _compare the results with those obtained

by

the

following

formulas:

. Maxwell-Garnett [4]

? + 2£r~

~

2£ + _Em

402 JOURNAL DE PfIYSIQUE I N°4

o B6ttcher [5]

~ ~£

_{+ 2£m}

.

Bruggeman

_[6]

'-f= ~l-~~l~i)~

o

Looyenga iii

it

=

f£I

₊

_{(I f)£I}

o Lichtenecker [8]

In?=

fln£+(I- f)In£m

with _Em

being

the DF of the matrix material in which

particles

with _a bulk DF of _{£ are}

randomly

distributed.

2. The method.

For the computer simulations _we discretized the

3D-space

inside _an ideal

rectangular cavity

resonator

using

_a cubic

grid

G and _a second

grid

G*

displaced

from G

by

half the

length

A of each

grid

cell in _{z-, y-} and z-direction. The size _was chosen _as: 20 _{mm x} 20 _{mm x} 10 _mm.

On the

edges

between the nodes of

G,

_we define the electric field vectors and

on the

edges

of G* the

magnetic

field _vectors.

Every

mesh cell of G _may be filled with

a diTerent

material,

I.e.

material with diTerent

permittivity

_{£ or Em} and

permeability

_{/t or /tm.}

By

this

setting

_{~ve can}

transform Maxwell's

equations

into _a set of matrix

equations using

_a finite diTerence method

(finite integration theory

_[9]

).

By the

help

of linear

approximation

techniques for the numerical treatment of

integrals,

rot E

= -B

rot H ₌ D

(I)

D ₌ ££oE

B =

/t/toH

are transformed to:

C .e = -iw.b

C* h

= iw.d

(2)

D~ _{e =} d

D~ ^h = b

where

D~

and D~ are

diagonal

matrices

containing permittivities

and

permeabilities

inside the diTerent cells of the

grid (as

in this work

only

materials with /t = I _are studied,

D~

^{~vill be}

N°4 NUMERICAL SIMULATION OF RANDOM COMPOSITE DIELECTRICS 403

represented by

_a factor I in the

equations).

C and C* represent the discrete rot-operators _[9]

and

e,d,h,b

_are the discrete vectors

containing

^the

respective

field

strength

of

E,

D, H and B.

Reducing

the _two sets of

equations

to _{one we} eliminate the

magnetic

^{field. The result is}

one

single eigenvalue equation

for the electric field distribution:

C* C _{e =}

A~£o

_iLow~

Dee (3)

This discretization

approach

has the

advantage

^that

continuity

conditions for the fields _are

implicitly

fulfilled. Electric fields _{are never}

computed perpendicular

to surfaces between dif- ferent materials and

magnetic

flux densities _{are never} defined

parallel

to these surfaces. This

means that

only

the continuous components ^{of the}

electromagnetic

fields _are

calculated,

_a

natural property, which

always

demands _some effort ifone _uses finite element methods instead of finite differences.

If the _resonator is _now filled

only

with loss free materials

(£ real,

^different

permittivities, however)

the

eigenvalue equation

is real and

symmetric (3

x N

equations

for N

nodes).

The effective

permittivity

of the mixture is then calculated from the lowest _resonance fre-

quencies

of the empty

(wo)

and the filled

(w)

resonator:

~2

0

~ ~2

First _we used code E of the MAFIA [9] = MAxwell's

equations

Finite

Integration Algorithm) software,

which has been

developed

for the

computation

of

eigensolutions

of such systems

by

Householder-tranformation. But the MAFIA software

package only

_accepts loss free materials up ^to now and

we found numerical _accuracy to become

inacceptably

_poor for such

heteroge-

neous systems as studied here with

filling

factor

f

> 0.3.

So _{a new} computer code COSME [10] ₌

Complex

Solution of Maxwell's

Equations)

_was

developed

which finds the exact

eigenvalue

and

eigenvector belonging

to any resonator mode

using

a

conjugate gradient

method

together

with inverse iteration.

Consider She

general eigenvalue equation:

A~ ₌

qDz

First, by

_a

spectral

shift

[11],

^the

equation

is modified to have _no

eigensolutions

with _zero

as an

eigenvalue.

(A

_no

D)z

₌ q* D _~

(4)

If no is chosen _very close to the

eigenvalue

_q which is

actually

to be

determined,

_q*

= q no

results to be the smallest

eigenvalue

of the _new

problem (Eq. 4).

Then I

/q*

is found to be the

largest eigenvalue

to the inverse matrix

(A qoD)~~D

which is

easily

found

by

_a

straight

forward Mises-iteration

[11].

In the _case that the inversion of this matrix is not

possible

_{as a cause} of the

high

order of the

problem,

inverse iteration is

404 JOURNAL DE PHYSIQUE I N°4

employed:

(A

_qo D ~n+i " Dzn

~n+1 ~ ~n

In each iteration step the above

equation

is solved

using

a

conjugate gradient algorithm

and

~n converges towards the

eigenvector belonging

to q* and at the _same time to q, the

eigenvalue

of the

original problem (3).

In this case, the matrix A is identical with C* C and the matrix D with D~.

We obtain the effective

permittivity dividing

the

eigenvalue

of the empty resonator

by

_q:

~ no

This _new computer _program COSME will also enable _us to

investigate lossy

media in the

near future. Materials

showing dissipation

make

equation (3) complex

with _a

symmetric

but non-hermitian matrix.

3. Itesults.

All the mixtures _we

investigated

_were

particles

in air

(or vacuum).

It _can be

proved

mathe-

matically

that systems ^with a matrix of

higher permittivity

show the _same

behaviour,

because

normalizing

all

permittivities

to the matrix

permittivity

does not

change

the

eigenvalue prob-

lem _at all.

We studied mixtures

containing particles

^with

permittivities

_up to _{£ =} 60. There is _a

simple

reason for

choosing

this limit: the first simulations _were carried out _{on an} IBM 3084 computer

whose main _memory allowed

only

40 _x 10 _x 40 nodes _on the

grid.

As

particle

size _cannot be smaller than _one unit

length

A of _a mesh

cell,

the smallest

particles

_we simulated had _a

length

d of 0.5 _mm.

On the other hand it is obvious that

quasi-homogeneous properties

of _a mixture _can

only

be observed if the

inhomogenities

_are much smaller than the

wavelength

I used for spectroscopy.

We chose the rule

~

$

^{~ ~~}

to reduce the unwanted influence of

scattering

and reflection of the _waves

by

the

particles.

The

corresponding problem

of the simulations is the

question:

how does the number of nodes in the mesh aiect the

quality

of the obtained

results,

I-e- _{can we} be _sure of

getting

rid of

finite size effects

using

_a

sufficiently

fine mesh? We

compared

_{a very coarse} mesh

(10

x 4 _x 10

nodes) successively

^{with meshes} of

increasing

^fineness: 20 _x 16 _x

20,

40 _x 30 _x

40,

60 x 20 _x 60, 60 _x 40 _x 60. The latter sizes have become

computable

because of the _new vector computer in the Rechenzentrum at the

University

of

Cologne,

the NEC

SX3,

which allows

larger

systems

to be calculated

using

the program COSME. Above _a mesh size of 40 _x 30 _x 40 the values did not

change considerably

_{any more.} To _save computer time in

general

all _our models _were

discretized

using

that _same mesh size.

With the MAFIA programs we were able to compute ^{the effective}

permittivities

of heterostruc-

tures with _a volume

filling

factor _{up to}

f

₌ 0.3. Above that value _{accuracy was too poor to}

allow _a reasonable

interpretation

of the data.

N°4 NUMERICAL SIMULATION OF RANDOM COMPOSITE DIELECTRICS 405

3.5 ~

£ E

7

3 o f o-I

f 0 2

6

2.5

2.0 4

3 1.5

2

1.0

o.5 o

-10 0 lo 20 30 40 50 60 70 -lo 0 lo 20 30 40 50 60 70

E E

a) b)

14

I

12

f ₌ 0.3

io

8

6

4

z

0

-lo 0 lo 20 30 40 50 60 70

E

c)

Fig-I-

Computer simulation data

(o)

in comparison to Maxwell- Garnett's

(i?),

Looyenga's

(o),

Bruggeman's

(m)

B6ttcher's

(b)

and Lichtenecker's

(D)

formulas, I is plotted _{vs, e, em =} i at _a

constant volume Itaction,

a)

volume fraction f

= 0.i,

b)

volume fraction f ₌ 0.2,

c)

volume fraction f = 0.3.

First _we

compared

the data

(I,e.: ?(

_£,

f

with several of the effective medium formulas

(Fig.

I). Obviously

the results calculated with the formulas differ from each other and _none of the formulas

reproduces

the values obtained

by

the simulation.

As

Looyenga's

formula [7]

ri/3

=

j £1/3

₊

(1 f) £)3

is

clearly

the _one that

gives

closest results _to the numerical

data,

_we chose the

generalized

mixture formula

T #

f

£~ +

(l f)

_ES

to fit the data

varying

the exponent a

(0

< a <

I). Actually

for _many different

tripletts

of values

f,

_£,

?)

the exponents were found

using

Newton's method. An

example

result of this

procedure

406 JOURNAL DE PHYSIQUE I N°4

o.9

a

Da

f=0 3

07

t=o z

06

t=o 05

0 4

20 30 40 50 60

E

Fig.2.

I°

= f. e° +

(I f)

e$ is used to fit the data tripletts f, e,

I),

in this _case from MAFIA simulations. The resulting values for the exponent _{a are} plotted _{vs. e. em}

= I + I * 0.

0.8

a ^°

~~ ^.

a

o.6

o

o.5 ^°

o

DA

o '

o.3 ^~

0.2

O-O O-1 02 03 04

f

Fig.3.

Exponent _{a vs.} volume &action f, MAFIA

(o),

COSME (o)

is shown in

figure

2. We obtained the _same exponent a out of different computer

experiments provided

that the

respective filling

factors _were identical. On the other

hand,

there _was

no

significant dependence

_on ^{the value} of the

permittivity

of the

particles.

This behavjour

alight

reflect the interaction between the

particles becoming

stronger ^with

increasing

number of

particles

_per unit

volume,

I-e- with

increasing filling

factor.

Figure

I

already

shows that the relative diTerence between the simulated results and the values obtained from

analytical

formulas increase with

increasing

^volume fraction.

Obviously

all the formulas do not take into account the

many-particle

interaction. In

figure

3 the calculated exponent a is

plotted

_us. the volume fraction. Whereas _at volume fractions below 0.25 _an almost

strictly

linear behaviour is

observed,

above 0.25 the linear correlation between _a and

f

is abandoned.

If _a

Looyenga-like

formula is

applied

to describe the

relationship

between the

permittivities

of matrix and

particles,

^{the volume fraction} and the

resulting

eTective

permittivity

of _an eTective

N°4 NUMERICAL SIMULATION OF RANDOM COMPOSITE DIELECTRICS 407

2.5 O-o

E'

. ,,

E

,

o o

o ,

2.0

.

°

i.5

lo

01 0.2

f f

a) b)

igA.

-

data

powder, from

[12]). Data calculated from

exponential formula (5): a) Real

Imaginary

part

of I.

medium, _it can_only

I,e,

with an exponent a which ^aries with

the

_filling

We _{compared the} values iven by this _formula with

the

experimental data by

Nelson

and

You [12].

From

_the data shown in _figure

3

we

calculated

the

a and f

as:

a( f)

=

_1.65

5b)

a(f)

as

upper and ower mits for the

In figure 4 the effective ielectric unctions

of

^a plastic

material

^alled Kynar _mixed ^ith _air

[12]

are

ottedagainst different volume fractions in

with

the _respective effective

DF'S predicted

by our

formula (5). The reement

shows

that our

simulation

data

resulting

mixture

_{rule are} consistent with xperimental

results.

Data _{reported in} [fl

are

also

reproduced ^by formula (5).

4.

nclusion.

We modeled

alculated resonance frequencies by

solving

a discretized

form

of _{axwell's equations.} We

compared _the

resulting

_ffective permittivities with

the data

given by

several

analytic

mixture

ormulas.Looyenga's ^formula

was

^{found to} give

the

losest values

to

the

408 JOURNAL DE PHYSIQUE I N°4

computer

simulations, although

the difference

was still _very

high.

So _we present a modified

exponential

formula of the form

ga(f)

=

f ~a(f)

_{+ j}

f) ^~a(f)

where the function

a(f)

_may be

approximated by

_a linear _{or a square} root-like function. It describes well the data _we obtained out of _our computer simulations for

filling

factors below 0.3.

Generalizations to

complex

values of _£

representing dissipation

_are in progress.

Acknowledgements.

The

investigations

_were

supported by

the Deutsche

Forschungsgemeinschaft (SFB 341)

^and

by

the Bundesministerium fur

Forschung

und

Technologie /

Bonn. Discussions with D. StauTer

are also

gratefully acknowledged.

References

[1] van Beek L-K-H-,

Progress

in Dielectrics 7

(Heywood

Books, 1967) p.69.

[2] Marquardt P. and Nimtz G., Phys. Rev. B 40

(1989)

7996.

[3] Marquardt P. and Nimtz G., Phys. Rev. Lett. 57

(1986)

io36.

[4] Maxwell-Gamett J-C-, Philos. Trans. R. Sac. A203

(1904)

385.

[5] ^B6ttcher C-J-F-, Theory of Electric Polarization,

(Elsevier, Amsterdam,1952)

p.415.

[6] Bruggeman D-A-G-, Ann. Phys. Fr. 24

(1935)

636.

[7] Looyenga H., Physica

(Utrecht)

31

(196S)

401.

[8] Lichtenecker K., Phys. Z., XXVII

(192t)

its.

[9] ^Weiland T., Part. Accel. 17

(1985)

227 and 15

(1984)

245.

[10] St6Izle S., Ph. D. Th., in progress.

[ii]

Numerical Recipes, Press W-H- et al.,

(Cambridge

Univ. Press,

1986).

[12] ^{Nelson S-O- and You} T-S-, J. Phys. D, Appl. Phys. 23

(1990)

346.