EMG 25(1) #37084

Electromagnetics, 25:69–79, 2005Copyright © 2005 Taylor & Francis Inc.ISSN: 0272-6343 print/1532-527X onlineDOI: 10.1080/02726340590522247

The Thickness Dependence of ResonanceFrequency in Anisotropic Composites

with Long Conductive Fibers

L. LIUS. M. MATITSINEY. B. GAN

Temasek LaboratoriesNational University of SingaporeSingapore

K. N. ROZANOV

The Laboratory for Microwave CompositesInstitute for Theoretical and Applied ElectromagneticsRASMoscow, Russia

The resonant frequency of long conductive fibers embedded in an anisotropic com-posite as a function of layer thickness is studied both numerically and experimentallyand is discussed in terms of the equivalent permittivity of the layer. A simple em-pirical exponential law is suggested to fit the thickness dependence of the equivalentpermittivity. The law involves a critical thickness value, below which the layer canno longer be treated as a bulk material for a fiber of given length and thickness. Thecritical thickness is retrieved from numerical data for different permittivity values ofisotropic and anisotropic layers and for different fiber length.

Keywords planar composites, dipole resonance, microwave permittivity

Introduction

Composites with polymer host matrix and long high-conductive fibers have been ex-tensively used as electromagnetic materials in various applications, such as EMC/EMI,multilayer low-reflection coatings, FSS or radomes with frequency-selective properties,

Received 30 January 2004; accepted 18 May 2004.The authors appreciate Dr. Hock Kai Meng and Dr. Kong Ling Bing, Mrs. Fang Xiao Jia, Mr.

Tan Szu Hau, Dr. Moustafaev, and Mr. Lagoisky for helping with the preparation of sample andthe measurement. They also thank Prof. A. N. Lagarkov, Dr. Qing Anyong and Mr. Xu Xin andfor the fruitful discussion. L. Liu acknowledges the help in the simulation from Dr. Istvan Bardi(with Ansoft Corporation). K. Rozanov is grateful to the program of the leading Russian scientificschool for partial support of the study according to Agreement 1694.2003.2.

Address correspondence to Dr. Lie Liu, Temasek Laboratories, National University of Singa-pore, 5 Sports Drive 2, 117508 Singapore. E-mail: tslll@nus.edu.sg

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70 L. Liu et al.

etc. (Munk, 2000; Molyneux-Child, 1997). Due to the dipole resonance of the fibers, fiber-filled composites can exhibit strong frequency dispersion of the permittivity that is usefulto develop materials with low reflection over wide bandwidth. The resonance frequencyof the effective permittivity determines the location of the dispersion frequency bandand is therefore an important performance indicator of this class of composite materialsthat has been studied by experimental, analytical, and numerical methods (Lagarkov &Sarychev, 1996; Lagarkov et al., 1998; Makhnovskiy et al., 2001; Matitsine et al., 2003).

It is well known that the resonance frequency depends on the permittivity and per-meability of the host matrix. In addition, if a composite layer is sufficiently thin, theresonance frequency is also a function of the layer thickness. This dependence has re-cently been observed experimentally at microwave frequency (Vinogradov, Makhnovskii,& Rozanov, 1999) and has been attributed to the presence of an electrodynamic boundarylayer at the boundary of inhomogeneous materials. The dependence is important for thepractical design of multilayer composite coating, which uses permittivity and permeabil-ity of each layer as input parameters for design optimization, instead of the properties ofeach component in each layer. The optimization process would not be tractable withoutthe use of such parameters as permittivity and permeability.

The boundary layers have been intensively studied for a long time in relation to theoptical properties of materials (see Mahan & Obermair, 1969, and references therein).The model applied in these studies is a regular array of point dipole scatterers, of whichthe effective permittivity is calculated. It has been shown that the effective permittivity ofa single-layer array differs greatly from that of a bulk material (Simovski, Popov, & He,2000), though this difference is completely eliminated when the sample comprises four tofive layers (Vinogradov, Dmitriev, & Romanenko, 1997). The thickness of the boundarylayer in dipole arrays is found to be about the distance between dipoles (Simovski, Popov,& He, 2000), i.e., the inhomogeneity scale of the problem.

As compared to dipole arrays, the problem is more difficult when the composite understudy contains long conducting fibers embedded in a dielectric host layer. The thicknessof the boundary layer in such composites can depend on many parameters, namely, thelength l and thickness d of the fiber, concentration p, and permittivity ε of the host layerthat is possibly anisotropic. Not much is known about these dependences, except for somenumerical and experimental data presented in Vinogradov, Makhnovskii, and Rozanov(1999) as an illustration of the phenomena. Previously, Munk, Luebbers, and Oliver(Munk, 2000; Munk, Luebbers, & Oliver, 1972) used the dielectric image technique toanalyze the problem of a dipole in a dielectric layer. Munk (2000) employed the conceptof effective dielectric constant, εeff , which is attributed to a layer of finite thickness t toaccount for the effect of t on the resonance frequency of dipoles. Based on the resultsof this study, an approximate guideline was proposed to estimate the thickness, abovewhich the boundary effect is negligible, where εeff = ε: t ≥ 0.05 λ0, λ is the wavelengthin free space, ε ≈ 3, with dielectric substrates on both sides of the dipoles.

The characterization of resonance frequency in thin layers by the effective permittiv-ity εeff , following Munk, can be somewhat confusing in treating composites, although itis very natural when scattering structures are considered, such as microstrips on a dielec-tric substrate. The reason is that the term “effective permittivity” is conventionally usedto characterize the averaged properties of composites. In Munk’s approach, the effectivepermittivity is not directly related to the electromagnetic response of the composite butdetermines the overall energy of the electric field scattered by the fiber in the presenceof the dielectric layer. Thus defined, the effective permittivity is closely related to thecapacitance of individual fiber rather than to the averaged permittivity of an assembly of

The Thickness Dependence of Composites 71

fibers. Munk’s definition of the effective permittivity allows this value to be dependenton the layer thickness that is quite inappropriate for the permittivity of composites.

For this reason, in treating the problem of a fiber embedded in a thin layer, weprefer to refer to Munk’s effective permittivity as the “equivalent permittivity of thelayer,” εeq, to avoid any misunderstandings. The equivalent permittivity can be definedas the capacitance of a fiber located within a layer normalized to the capacitance of thisfiber when it is located in free space. From the point of view of designing the composite,εeq is useful in determining the resonance frequency of multilayer composite coating withlong conductive fibers, as well as the critical thickness of composite sheet beyond whichthe composite can be considered as bulk materials. The term “effective permittivity”is used when the averaged electromagnetic response of an inhomogeneous material isconsidered.

It is well known that the electrostatic capacitance of a dipole yields rather accuratevalue for the resonance frequency. Therefore, the thickness dependence of the resonancefrequency can provide a sensitive characterization of εeq. In this paper, the finite elementmethod (FEM) is used in the study of the dielectric resonance of anisotropic slabs withlong conductive fibers as inclusions. Numerical results on the reflection coefficient offiber-filled composites are obtained and used in the extraction of data on the thicknessdependence of the equivalent permittivity that are presented and verified by measurement.The dependence is found to obey a simple exponential law involving a single parameterknown as the critical thickness t0. The curve-fitting method is used to determine the valueof t0 for various parameters in the problem. Analyses of the results enable the criticalthickness to be estimated as a function of the parameters of both the fiber and layer. Therelationship between the thickness dependence under study and the scattering propertiesof thin inhomogeneous samples is also discussed.

The dielectric constant of the layer is assumed to be anisotropic. This is of importancein practice, as actual high-permittivity materials are typically composites loaded withtiny conductive inclusions that are nonspherical in shape to achieve high permittivityproperties. Due to technological reasons, these inclusions are frequently aligned, leadingto a uniaxial anisotropic sample with high permittivity ε‖ in the plane of sheet materialand low permittivity ε⊥ in the perpendicular direction. Numerical data on isotropic layersare also presented to illustrate the effect of anisotropy on the critical thickness.

For the sake of simplicity, we consider a sample of simple geometry where all fibersare located near the center of the sample. The volume fraction of the fibers is assumed tobe small so that the dilute limit approximation is valid for their electromagnetic response,and therefore the fibers can be treated as isolated.

Method

The commercially available FEM software, High Frequency Structure Simulator 8(HFSSTM) from Ansoft, is used in the numerical study. The model includes a box ofsize 40 mm × 20 mm × 20 mm filled with air. A layer of t × 20 mm × 20 mmin size is located at the center of the box’s base. The layer thickness t is varied from0.1 mm to 6 mm. The in-plane and out-of-plane permittivity of the layer is ε‖ and ε⊥,where ε‖ ≥ ε⊥. A metallic fiber of length l and thickness d is embedded at the centerof the layer. A TEM wave with the electric field E parallel to the fiber and the wavevector k perpendicular to the layer surface illuminates the model. The perfectly matchedlayer (PML) boundary conditions are imposed at the surfaces perpendicular to the wavevector. Periodic boundary conditions (PBCs) are applied on the other surfaces of the box.

72 L. Liu et al.

Figure 1. FEM model of conductive fibers in an anisotropic layer.

A sketch of the model is given in Figure 1. The resonance frequency of the fiber embed-ded in the layer was determined by the frequency at which the reflection coefficient ofthe layer has a peak. It is calculated by Contopanagos, Zhang, and Alexopoulos (1998)and Bardi et al. (2002):

R =

√√√√√√√

∫Sin

(( �Es × �Hs∗) · n̂)ds

∫Sin

(( �Einc × �H inc∗) · n̂)ds

, (1)

where R is the reflection coefficient; Einc, H inc, Es , Hs are the electric and magneticintensity of incident and scattered field, respectively; Sin is the front surface where theincident wave enters as indicated in Figure 1; and n is the outward normal direction ofSin. Since dipole resonance in a periodical array of fibers has a high-quality factor, thisapproach gives good accuracy for the resonance frequency.

The samples for the experimental study are produced by the paint spraying tech-nique using the same method as described in Matitsine et al. (2003). Aluminum flakes(Al-flakes) or short carbon fibers (C-fibers) are added to the host polymer to achievehigh permittivity. Both types of materials are anisotropic. Measurement are made withAl-flakes composite having in-plane permittivity ε‖ = 33 and out-of-plane permittivityε⊥ = 6, and with a C-fiber composite having ε‖ = 8 and ε⊥ = 3. The loss tangent ofthe composites is taken to be frequency independent and equal to 0.1, which is foundto be close to the measurement results. A small amount of copper fibers with diameterd = 0.1 mm and of various lengths (such as 5 mm, 10 mm, 20 mm) as needed wasimmersed randomly in the middle of the layer, as shown in Figure 2. The permittivityof the samples was measured in free space by the standard technique (Nicolson & Ross,1968), from which the resonant frequency of the fibers can be found by fitting the mea-sured data according to the Lorentzian dispersion law. The details of the experimentaltechniques are given in Matitsine et al. (2003).

The Thickness Dependence of Composites 73

Figure 2. Schematic structure of composite.

Results and Discussion

The results obtained for the resonance frequency fres of copper fibers with length 10 mmand thickness d = 0.1 mm embedded in an anisotropic layer as a function of the layerthickness are given in Figure 3. The figure shows two sets of data, one related to theAl-flake composite (triangles) and another to the C-fiber composite (boxes). Numericalresults are represented by empty squares and triangles, while the experimental results areshown by the filled symbols. Close agreement between the FEM results and measureddata is obtained.

In uniaxial materials, the resonance frequency is determined by 4√

ε‖ε⊥ instead of√ε (Matitsine et al., 2003). This is in agreement with the asymptotical behavior of the

data obtained for large t . As t → 0, the resonance frequency for both composites tendsto the resonance frequency of the fiber located in free space, f0, which can be calculated

Figure 3. Thickness dependence of the resonance frequency for Al-flakes (triangles) and carbonfibers (boxes).

74 L. Liu et al.

Table 1Resonance frequency of fibers

with different lengths

Length of fibers (mm) fres (GHz)

2.5 57.45 26.8

10 13.520 6.75

by the FEM. For the fibers used in this study, the resonance frequencies of fibers withlength from 2.5 mm to 20 mm in free space are given in Table 1.

The value of fres for a fiber embedded in a layer of a finite thickness is describedin terms of the equivalent permittivity, εeq. As discussed in the introduction, this valuediffers from the effective permittivity of an inhomogeneous layer that is conventionallyobtained from reflection or transmission measurements. The physical meaning of εeq isthat it determines the overall energy of the electric field scattered by the fiber to accountfor the finite thickness of the layer. It can be determined from the measured or calculatedresonance frequency as

εeq = (fres/f0)2 (2)

and depends on t , ε‖, ε⊥, d , and l. Figure 4 plots the thickness dependence of the effectivepermittivity calculated from the data in Figure 3, with the same notation.

For very thin layers, the resonance frequency is close to that in a vacuum, implyingthat εeq = 1. As the thickness increases, the effective permittivity increases and tends to

Figure 4. Thickness dependency of the effective permittivity retrieved from the data in Figure 3.

The Thickness Dependence of Composites 75

Figure 5. The calculated thickness dependence of equivalent permittivity for different permittivityof isotropic layer (lfiber = 10 mm).

the typical value of bulk samples, i.e., εeq = √ε‖ε⊥. Based on this, we fit the thickness

dependence of effective permittivity by a simple exponential law:

εeq = √ε‖ε⊥ − (

√ε‖ε⊥ − 1) exp(−t/t0). (3)

The fitted curves obtained from the numerical data with the only fitting parameter, t0,are shown in Figure 4 by solid lines. It is seen that law (3) provides a reasonable fittingof the numerical data. The fitting of the numerical data with law (3) is much betterthan that obtained with an exponential law applied to the thickness dependence of fres

(Vinogradov, Makhnovskii, & Rozanov, 1999).The value of t0 can be referred to as the critical thickness, since a layer filled with

fibers of a given length but with thickness less than t0 cannot be treated as a bulk medium.Therefore, it cannot be described in terms of the effective material parameters. In viewof this, the values of t0 for actual composites are of certain interest. The values of criticalthickness for the data shown in Figure 4 are 0.68 for the Al-flake composite and 1.21for the C-fiber composite. Further study is aimed at understanding the critical thicknessas a function of the layer permittivity and the fiber dimensions.

In Figures 5 and 6, layers with different permittivity are used to obtain the equivalentpermittivity and critical thickness for fibers of lengths 10 mm and 20 mm, respectively.Table 2 shows the values of t0 used in the curve fitting. From the results, contrary toexpectation, t0 does not decrease when the permittivity of the matrix increases. Furtherstudy is needed to confirm if the rule still works for fibers of different dimensions. Thevariations of t0 could be due to the uncertainty in determining the resonance frequencyfrom the calculated frequency-dependent reflection coefficient, or the periodic structureof the numerical model. Another possible reason is that numerical data for the thicknessdependence of the resonance frequency are fitted with an exponential law to obtain thecritical thickness. The exponential behavior may not be typical for quasi-static problemsand is therefore an approximation to some more complex dependence. This can introduce

76 L. Liu et al.

Figure 6. The calculated thickness dependence of equivalent permittivity for different permittivityof isotropic layer (Lfiber = 20 mm).

Table 2The values of t0 for fitting curves in Figures 5 and 6

Permittivity of the layer t0 for l = 10 mm t0 for l = 20 mm

ε = 1.5 0.94 0.81ε = 3.5 0.7 0.91ε = 6 0.78 1.15ε = 11 1.06 1.17ε = 22 1.10 1.4ε = 33 0.75 0.95

Ave. 0.89 1.07

error in the values obtained for t0. However, the average critical thickness, the values ofwhich are given in the last row of Table 2, gives a good prediction of the resonance inthe design of the composite with fiber inclusions.

The dependency of t0 on the length of the fibers is given in Figure 8. The criticalthicknesses are obtained from the calculated equivalent permittivity given in Figure 7.It is observed that the critical thickness decreases as the length of fibers l decreases.A possible reason for this observation is that for shorter fibers with small ratio of l/t , asignificant part of the electric fringing fields between the two ends of the fiber lies withinthe slab, while for larger l/t ratio, the electric fringing fields can be found more in thefree space outside the slab. Hence, the equivalent permittivity is higher for smaller l/t

ratio, which implies that the critical thickness t0 is correspondingly smaller. However, itshould be noted that several other wave mechanisms (such as surface wave, etc.) existin such a structure which could affect the value of t0. The explanation based on fringingfields is a simplified view of the phenomenon.

The Thickness Dependence of Composites 77

Figure 7. The calculated thickness dependence of equivalent permittivity for fiber with differentlength within aluminum flakes layer.

Figure 8. Dependency of t0 on the length of fibers.

A few values of εeq (black marks) calculated from the measurement are also plottedin Figure 7. Reasonable agreement with the calculated values (white marks) is obtained.Some of the measured εeq are slightly smaller than the calculated values, which can beattributed to the deviation of the fibers’ locations from the central layer in the experimentand to the air gap due to the sandwich structure of the thick sample. Though a linear fitgives a good description of the dependence in Figure 8, the relationship may not be sosimple, since the resonance is affected not only by the length of the fiber but also by its

78 L. Liu et al.

aspect ratio. Further analytical study is needed to confirm and explain the dependence.For the embedded copper fibers employed in this study, the empirical equation of (4)may describe the dependence of equivalent permittivity of various composites matrices

εeq = √ε‖ε⊥ − (

√ε‖ε⊥ − 1) exp

( −5

0.06l

), (4)

where l is the length of embedded copper fibers. The empirical model could be of useto the practical design of composites.

Concluding Remarks

The critical thickness t0 introduced in this paper can be understood as a boundary be-tween planar and bulk inhomogeneous materials. In planar samples, evanescent modespropagating beyond the layer have a significant effect on the effective permittivity. Strictlyspeaking, such materials cannot be described adequately in terms of the effective permit-tivity, as the permittivity is a parameter that is independent of thickness. The thicknessdependence of the permittivity implies also that the permittivity is a function of theproperties of the surrounding media.

In bulk materials of which the thickness is large as compared to the critical thickness,most of the contributions to the effective properties are due to the fibers located deepinside the sample, and the boundary layer has negligible effect. Therefore, the samplecan be described in terms of effective permittivity.

In this study, a particular sample geometry was considered, with all fibers locatedin the center of the sample. Equivalent permittivity (εeq) instead of effective permittivity(εeff ) was proposed to describe the shift in the resonance frequency of the compositewith conductive fibers in the presence of an anisotropic dielectric layer. According tothe simulation and experimental results obtained in the study, t0 increases linearly asthe length of the fibers increases, but it is kept almost constant when the anisotropic orisotropic permittivity of the dielectric layer increases. The variations of t0 might comefrom the numerical model or the exponential law used to fit the calculated εeq.

It is suggested that the results obtained can be a good estimate for other possiblegeometries, namely those with fibers distributed over the entire volume of the sample,located at the sample surface, etc. To validate this suggestion, more studies are necessary.

The results obtained can be useful to the design of multilayer inhomogeneous elec-tromagnetic materials in predicting the location of the dipole resonance, as well as therange of thickness of individual layers that provides the correct description in termsof εeff .

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