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Physica B 396 (2007) 132–137

www.elsevier.com/locate/physb

Effective permittivity of random composite media: A comparative study

Ashutosh Prasad�, K. Prasad

Department of Physics, T.M. Bhagalpur University, Bhagalpur 812 007, India

Received 28 December 2006; received in revised form 7 March 2007; accepted 20 March 2007

Abstract

In the present study, experimental data for effective permittivity of amorphous, polycrystalline thick films, and ceramic form of

samples, taken from the literature, have been chosen for their comparison with those yielded by different mixture equations. In order to

test the acceptability of dielectric mixture equations for high volume fractions of the inclusion material in the mixture, eleven such

equations have been chosen. It is found that equations given by Cuming, Maxwell–Wagner, Webmann, Skipetrov and modified

Cule–Torquato show their coherence and minimal deviation from the experimental results of permittivity for all the chosen test materials

almost over the entire measurement range of volume fractions. It is further found that Maxwell–Wagner, Webmann, and Skipetrov

equations yielded equivalent results and consequently they have been combined together and reckoned as a single equation named

MWWS. The study revealed that the Cuming equation had the highest degree of acceptability (errors o71–5%) in all the cases.

r 2007 Elsevier B.V. All rights reserved.

PACS: 77.20.+y; 77.22.�d; 77.84.Lf; 77.90.+r; 78.70.Gq.

Keywords: Random composite medium; Effective permittivity; Dielectric mixture equations

1. Introduction

The problem of determining the effective physicalproperties of heterogeneous materials has received repeatedattention in the recent past. This reflects interest infabricating composite materials for particular uses. Thestudies for calculation of the effective permittivity of arandom medium have been carried out from variousstandpoints like perturbation expansion, the variationalapproach, the effective medium approximation, etc.

This paper deals with the study of macroscopic (oreffective) permittivity in random media. A theoreticalanalysis of the RF and microwave properties of micro-scopically inhomogeneous disordered materials is pre-sented. It is intended to see the acceptability of differentdielectric mixture equations for the effective permittivity ofthe medium with the help of five test materials at lower aswell as at higher volume fractions of the inclusion material,even in tough situations like high ratios of permittivity of

front matter r 2007 Elsevier B.V. All rights reserved.

ysb.2007.03.025

ng author. Tel.: +91641 2501699; fax: +91 6412620353.

sses: apd.phy@gmal.com (A. Prasad), k.prasad65@

rasad).

its components and large volume fractions of the inclusionmaterial. The first test material is Styrofoam [1], thecellular or foamed plastic, which finds its manifoldapplications in the field of microwaves such as supportsat radar ranges and anechoic chambers. A set of thick leadzirconate titanate (PZT) films [2] fabricated using sol–gelprocess is the second test material. Moldable mixtures ofparaffin wax and two allotropic forms of TiO2 (anatase andrutile) [3] constitute the third and fourth test materials,respectively. The last test material is in the form of rutileceramic samples [4].

2. A brief introduction of equations used

1.

Plonus equation [5]:

�s ¼ �eff ¼ ðVo þ �pVpÞ=ðVo þ VpÞ

¼ ½ðVo=VpÞ þ �p�=½1þ ðVp=VoÞ�

) �eff ¼ ½ð1=f Þ þ �p�=ð1þ 1=f Þ, ð1Þ

where Vo is the volume of air in the mixture; Vp thevolume of base polymer (polystyrene) in themixture; es the relative permittivity of the Styro-

ARTICLE IN PRESSA. Prasad, K. Prasad / Physica B 396 (2007) 132–137 133

foam ¼ effective permittivity of the mixture ¼ eeff;ep the relative permittivity of polystyrene.The plastic foams have two constituent materials—the base polymer that forms the cell walls, and thegas contained within the cells. Since the cellstructure is of a random nature with somepredictable average properties such as cell size anddensity, it is modeled by an aggregate of randomlydistributed spherical shells. Assemblies of scatterershave, in general, a coherent as well as incoherentscatter. Coherent scattering comes primarily fromsudden particle density changes, such as that at theboundaries, and incoherent scattering is that fromthe interior volume of the particle system. Latter isthe result of contribution of all the particles of thesystem.In Styrofoam, Vo/VpE43, i.e., Vp/VoE0.023 andhence in order to maintain the consistency ofsymbols in all the equations which are to followin the discussion, the inverse of Vo/Vp, i.e., Vp/Vo

has been chosen. It is the volume fraction of theinclusion material in the mixture, denoted by f. Ofcourse, in all the equations to follow, except the onedue to Webmann et al. [6], the equation from theeffective medium theory (EMT), the subscript ‘2’refers to the inclusion and ‘1’ refers to the host withf1+f2 ¼ 1:

2.

Knott equation [1]:

�eff ¼ �2½1� fð�2 � �1Þð1� f Þg=f�1 þ ð�2 � �1Þð1� f Þ1=3g�,

(2)

where eeff is the relative permittivity of the mixture,e2 the relative permittivity of the particles, forexample, base polymer in Styrofoam and e1 therelative permittivity of the host.In the paper containing the above Eq. (2), theauthor has assumed a model for the foam cellstructure, inserted the model between the electrodesof a parallel plate capacitor to calculate thecapacitance, and the lattice chosen is cubic innature.

3.

Cuming equation [7]:

log �eff ¼X

f i � log �i, (3a)

where ei is the relative permittivity of the ithcomponent of the mixture and fi is the volumefraction of that component. Since the foamedplastics are usually binary mixtures, the aboveequation could be rewritten as

log �eff ¼ f 1 log �1 þ f 2 log �2 ¼ f log �2. (3b)

Since the first component is generally air or a gaswhose permittivity is of the order of unity, the firstterm of the above equation vanishes. Also, asreferred to earlier, f2�f ¼ [(density of the foam)/(density of the base polymer)]. Since most of thefoams used today have fo0.06 and if the gaseous

component of the foam be supposed to have itsrelative permittivity strictly equal to unity, Eqs. (2)and (3a) are approximated to two Eqs. (4) and (5)as given below. In the present work, they have beencalled ‘deduction from Knott equation’ and ‘deduc-tion from Cuming equation’, respectively, only forreference sake.

4.

Deduction from Knott equation [1]:

�eff ¼ 1� f ð�2 � 1Þð2�2 þ 1Þ=3�2. (4)

5.

Deduction from Cuming equation [7]:

�eff ¼ 1þ f ln �2. (5)

6.

Webmann equation [6]:

�eff ¼ �B½1þ 2f Afð�A � �BÞ=ð�A

þ 2�BÞg=½1� f Afð�A � �BÞ=ð�A þ 2�BÞg� ð6aÞ

for the binary system AB in the small concentra-tion, fA, of the component A. If the componentsA and B are exchanged, fB ¼ f2 ¼ f, like othercases, supposing eB4eA (after exchange) andeA ¼ eB ¼ e1, on the lines of Hashin and Shtrik-mann [8] one gets

�eff ¼ �1½ð1þ 2f Þ�2 þ 2�1ð1� f Þ�=½�2ð1� f Þ þ ð2þ f Þ�1�.

(6b)

7.

Landau-Lifshitz equation [9]:

�eff ¼ �1½ð1þ 3f ð�2 � �1Þ=ð2�1 þ �2Þ. (7)

8.

Maxwell–Wagner equation [10]:

�eff ¼ �1½�2 þ 2�1 þ 2f ð�2 � �1Þ�=½�2 þ 2�1 � f ð�2 � �1Þ�.

(8)

9.

Skipetrov equation [11]:

�eff ¼ �1½1þ f3f ð�2 � �1Þg=f�1ð2þ f Þ þ �2ð1� f Þg�.

(9)

Eqs. have been derived using perturbative con-siderations under the assumption that ‘f’ is muchsmaller than unity, and that either correlationlength or the particle diameter, is well below thewavelength of electromagnetic waves used. TheSkipetrov equation is perhaps the first non-pertu-bative equation to the knowledge of the author,which is original and more transparent than othersand is assumed to give more correct results undertough situations (i.e., high contrast e2/e1 and largevolume fractions, f ). He has tried to fit his equationin two types of random media: (i) with e2/e1 ¼ 2;and (ii) a dilute suspension of TiO2 particles inwater at optical frequencies of measurement, andhas compared his results with those of others, likeEq. (7), to get very encouraging results.

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2.8

3.0

0.5

1.0

1.5

on

(%

)

A. Prasad, K. Prasad / Physica B 396 (2007) 132–137134

It has also been found that Eqs. (6a), (8) and (9)become equivalent and hence their common resultshave been shown under the common name ofMWWS equation.

2.40.0

via

ti

10.

0.0 0.2 0.4 0.6 0.8 1.00.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

Experimental

0.0228 0.0230 0.0232 0.0234 0.0236-1.5

-1.0

-0.5

Volume fraction

De

Knott equation

Cuming equation

MWWS equation

MCT equation

Volume fraction

� eff

Fig. 1. Effective permittivity of Styrofoam as a function of volume

fraction of base polymer, using different dielectric mixture equations.

Inset: deviation plot (& Knott; J Cuming; n MWWS; , MCT).

Cule–Torquato equation [12] based on Hashin–Shtrikmann composite cylinder construction [13]:

�eff ¼ �1½1� 2a2b=ða2bþ b2Þ� (10a)

with b ¼ (e2�e1)/(e2+e1), a is the radius of the corehaving permittivity e1, b the radius of the surround-ing concentric shells having a permittivity equal toe2, so that f2 ¼ f ¼ (a/b)2.In an attempt at finding the results through theabove Eq. (10a), it was found that the equationsometimes gives values of eeff less than unity, whichis impossible for real, passive dielectric materials.Hence, the equation needed some modification(s) inits form to fit the results with those of others. Thenecessary modification was done in the presentwork by changing the sign in the second term of thedenominator, as given below.

11.

140015

Modified Cule and Torquato (MCT) equation

�eff ¼ �1½1þ 2a2b=ðb2� a2bÞ�. (10b)

5

10

(%

)

12.

400

600

800

1000

1200

Experimental

Knott equation

Cuming equation

MWWS equation

MCT equation

-0.1 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7-15

-10

-5

0

Devia

tio

n

Volume fraction� eff

Modified deduction from Cuming Eq. (10a) of hispaper [7]:

�eff ¼ 1þ aðln �2Þa, (11)

where a is a pure number, whose value has beenchosen according to the problem at hand. Further,a�f2 ¼ f. For example, in case of Styrofoam at X-band microwave frequencies, e2 ¼ 2.53, e1 ¼ 1, anda ¼ �4.6409. The results for effective relativepermittivity as a function of volume fraction ofthe inclusion material, using the different mixtureequations, in all the test materials, are showngraphically in Figs. 1–5, respectively.

0.0 0.2 0.4 0.6 0.8 1.0

Volume fraction

Fig. 2. Effective permittivity of sol–gel–PZT composites as a function of

volume fraction of PZT, using different dielectric mixture equations. Inset:

deviation plot (& Knott; J Cuming; n MWWS; , MCT).

3. A brief description to the shape of particles in the different

mixture equations

As regards Eq. (1), the particles were assumed to be inthe form of randomly distributed spherical shells. In Eqs.(2) and (4), the unit cell was a simple 3D cubic lattice ofidentical non-overlapping spheres with dielectric constante2 embedded in a host material having the dielectricconstant e1. Eqs. (3a), (5) and (11) are nothing but generalrepresentations of logarithmic law of mixing for a chaoticor statistical mixture. The general form of the equations,called ‘Rother–Lichtenecker equation’ [14] does not con-tain the shape-dependent parameters. Further, Eq. (6a) wasderived with the consideration of medium to be randomwith the cells embedded in a medium assumed to behomogeneous on scales larger than the correlation length,but inhomogeneous on scales smaller than it, having its

permittivity eb. The cells were assumed to be spherical inshape, with dimensions of their correlation lengths �b foreach of them, centered on the points r1, r2,y, rN having aconstant value of the dielectric constant ei in each cell i. Ofcourse, Eqs. (6a) and (7) were derived using perturbativeconsiderations under the condition of volume fractions ofthe particles much smaller than unity and their correlationlengths or particle diameters (assuming the particles to bespherical) smaller than the wavelength of the electromag-netic waves used. Eq. (8) was derived almost on the samebasis of hard sphere particles embedded in the hostmedium as solutes in dilute suspensions. Eq. (9) was

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0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.400.0

2.5

5.0

7.5

10.0

12.5

15.0

17.5

20.0Experimental

Knott equation

Cuming equation

MWWS equation

MCT equation

� eff

Volume fraction

Fig. 3. Effective permittivity of paraffin wax–anatase moldable mixture as

a function of volume fraction of TiO2 (anatase), using different dielectric

mixture equations. Inset: deviation plot (&Knott; J Cuming; n MWWS;

, MCT).

0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.400

5

10

15

20

25

30

35

40

Volume fraction

Experimental

Knott equation

Cuming equation

MWWS equation

MCT equation

� eff

Fig. 4. Effective permittivity of paraffin wax–rutile moldable mixture as

a function of volume fraction of TiO2 (rutile), using different

dielectric mixture equations. Inset: Deviation plot (& Knott; J Cuming;

n MWWS; , MCT).

0.0 0.2 0.4 0.6 0.8 1.0

0

20

40

60

80

100

Experimental

Knott equation

Cuming equation

MWWS equation

MCT equation

� eff

Volume fraction

0.0 0.2 0.4 0.6 0.8 1.0-500

-400

-300

-200

-100

0

100

Devia

tio

n(

% )

Volume fraction

Fig. 5. Effective permittivity of sintered rutile ceramic samples as a

function of volume fraction of TiO2 (rutile), using different dielectric

mixture equations. Inset: deviation plot (&Knott; J Cuming; n MWWS;

, MCT).

A. Prasad, K. Prasad / Physica B 396 (2007) 132–137 135

derived with fluctuating dielectric function and applied alsoto the case of dilute suspension. The spatial scale, b, hadthe same definitions and limitations in its values asdescribed earlier. Cule–Torquato equation, Eq. (10a), andits modified form, Eq. (11), were assumed to be based onHashin–Shtrikman two-phase model when the medium asa whole was considered to be made up of compositecylinder consisting of a core of dielectric constant e2 andradius a, surrounded by a concentric shell of dielectricconstant e1 and radius b, where the ratio (a/b)2 equals thephase-II volume fraction, f2, and the composite cylinders

filling the whole space, indicating a distribution in theirsizes ranging to the infinitesimally small. Further, in theanalytical study of electric field fluctuations, a density offield states per unit volume, similar to the density of statesof phonon and electrons in solids, was considered.

4. Results and discussion

The Plonus equation showed large deviations in thevalues of effective permittivity of the mixture at highervolume fractions of the inclusion material as comparedwith experimental results and also as compared with thoseof others. Other equations, on the contrary, showed betterresults. It was also noted that in case of other testmaterials, the Plonus equation couldn’t be used in a similarfashion, and consequently, no data points corresponding tothis equation have been shown in Figs. 1–5. Further, thedeductions from Knott equation and that from Cumingequation as well as from Landau–Lifshitz equation gavequite divergent results at higher volume fractions of theinclusion material. Worse was the situation that none ofthe aforesaid equations gave coherent results at the samevolume fraction. A slight degree of improvement in the fithas been achieved by way of modification in the deductionfrom Cuming equation indicated as Eq. (11) in the presentstudy. What has been done is the introduction of theexponent ‘a’, being different for different materials used.Further, Cule–Torquato equation has been modified in aproper way so as to get acceptable results for all the testmaterials. One of the significant conclusions arrived at byway of analysis of the equations and results, in the presentstudy, is that the equations due to Maxwell–Wagner,Webmann et al., and Skipetrov, were seen to be equi-valent, thus giving 100% coherent results over the entire

ARTICLE IN PRESSA. Prasad, K. Prasad / Physica B 396 (2007) 132–137136

measurement range of volume fractions. Further, theequations given by Cuming, MCT, Maxwell–Wagner andSkipetrov gave almost coherent values. Similar was the fateof all the aforementioned equations in case of their beingused in the other test material. Inset Figs. 1–5 illustrate theerrors with the four equations giving almost coherentresults namely, Cuming, MWWS and MCT equation ascompared with the experimental results. It is quite evidentfrom the deviation plots that the results derived fromCuming equation provided quantitatively the smallest(o71–5%) errors of prediction, while others like MWWSand MCT gave slightly larger deviations. On the otherhand, Knott equation gave deviations too larger, making itquite unacceptable for such studies.

The second as well as third order polynomial regressionequations have been used for smoothening of the experi-mental data points. Tables 1–5 show the values of differentestimated statistical parameters like coefficients of regres-sion equations, coefficient of determination (r2), standard

Table 1

Estimated parameters for second and third order polynomial regression equ

frequencies

Coefficients Quadratic fit

A 0.99997

B1 1.12847

B2 0.40156

B3 —

r2 (COD) SD N p

0.99995 0.00373 11 o0.0

Table 2

Estimated parameters for second and third order polynomial regression equat

Coefficients Quadratic fit

A 397.52473

B1 570.71932

B2 370.93024

B3 —

r2 (COD) SD N p

0.99383 27.7522 8 o0

Table 3

Estimated parameters for second and third order polynomial regression equat

X-band of microwave frequencies

Coefficients Quadratic fit

A 3.01769

B1 �9.90281

B2 54.79929

B3 —

r2 (COD) SD N p

0.99863 0.06817 8 o0.

deviation (SD), number of data points (N) and mean %error of prediction (p) for all the test materials. Theseanalyses applied to experimental data for all the five testmaterials, showed excellent fits, especially with cubicmodels (r2�1 and po0.0001) as depicted.Last significant point is that instead of using the

straightforward unity value for the permittivity of thegases comprising the foamed polystyrene, a value of 1.0034(by putting e2 ¼ 2.53 and f ¼ 0.06 in Knott’s equation)produced a change only in the third order of decimal placein effective permittivity, and hence, unity value is preferredto 1.0034 in the present study, for ease in calculations,possibly without too much impairment to the results.Although only the real values of permittivity have been

used in the present study, actually they are complex innature and hence, as has rightly been remarked bySkipetrov [11], results leave space for further investiga-tions. The effect of light absorption in one (or several)mixture (or suspension) components on effective relative

ations for effective permittivity of Styrofoam at X-band of microwave

Cubic fit

1.000

�1.34393

100.61373

�106.7398

r2 (COD) SD N p

001 0.99995 0.00396 11 o0.0001

ions for effective permittivity of sol-gel-PZT composites at rf frequencies

Cubic fit

391.95688

680.09682

75.62285

194.7762

r2 (COD) SD N p

.0001 0.99408 30.3814 8 o0.0001

ions for effective permittivity of paraffin wax-anatase moldable mixture at

Cubic fit

2.4353

8.63898

�14.73487

51.85217

r2 (COD) SD N p

0001 1.00 0.0164 8 o0.0001

ARTICLE IN PRESS

Table 4

Estimated parameters for second and third order polynomial regression equations for effective permittivity of paraffin wax-rutile moldable mixture at X-

band of microwave frequencies

Coefficients Quadratic fit Cubic fit

A 3.72121 2.03203

B1 �26.89883 14.77156

B2 123.02024 �39.98284

B3 — 123.17655

r2 (COD) SD N p r2 (COD) SD N p

0.99868 1.33824 9 o0.0001 0.99997 0.2320 9 o0.0001

Table 5

Estimated parameters for second and third order polynomial regression equations for effective permittivity of sintered rutile ceramic samples at X-band of

microwave frequencies

Coefficients Quadratic fit Cubic fit

A 6.68769 �0.20573

B1 �67.82462 41.70427

B2 140.38462 �146.84149

B3 — 191.48407

r2 (COD) SD N p r2 (COD) SD N p

0.96869 5.49819 11 o0.0001 0.99802 1.47997 11 o0.0001

A. Prasad, K. Prasad / Physica B 396 (2007) 132–137 137

permittivity of the whole medium could be studied by usingequations chosen in the present study, at least theequations proposed by Cuming, MWWS and also theMCT with complex values of dielectric functions ei of thecomponents.

5. Conclusions

The dielectric mixture equations namely, Cuming,MWWS and MCT give almost identical results, except inthe case of rutile ceramic samples where only Cumingequation give excellent fit (errorso71–5%). Further, onlythe Cuming equation has the highest degree of accept-ability in all the cases, especially in the X-band microwavefrequencies, and that too for the entire range of measure-ment of volume fractions of the inclusion materials.

Acknowledgements

The authors gratefully acknowledge the help taken fromthe works of S.E. Skipetrov, Landau–Lifshitz, Cule–

Torquato, Eugene F. Knott, Plonus and all others withoutwhich the present work could not have taken its shape.

References

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