Journal of ELECTRONIC MATERIALS, Vol. 39, No. 8, 2010

DOI: 10.1007/s11664-010-1229-x  2010 TMS

Effective Permittivity Calculation of Composites with Interpenetrating Phases YU ZENG,1 HONGTAO ZHANG,1,3 HONGYAN ZHANG,2 and ZHIPENG HU1 1.—Institute of Materials Research and Application Technology (IMRAT), Jiangxi Blue Sky University (JXBSU), Nanchang 330098, People’s Republic of China. 2.—Department of Mechanical, Industrial and Manufacturing Engineering, University of Toledo, Toledo, OH 43606, USA. 3.—e-mail: [email protected]

The effective permittivity of composites with interpenetrating phases was determined through finite-element simulation. The spectra of effective permittivity for such composites were calculated, and relationships among the effective permittivity eeff, permittivity ratio x of the matrix to network phase, and volume fraction of the network f were developed using the numerical results. The simulation has been validated by comparing the results with previously published experiments, and excellent agreement has been achieved. The electric field distributions in the composites were also explored in this work. Key words: Interpenetrating phase composite, effective permittivity, finite-element method (FEM)

INTRODUCTION From natural objects in biology to geology to electronics, bridges, and many other artifacts, most structures involve the use of composite materials. Some of the materials found in nature may need artificial modifications for efficient use, while others are entirely synthetic, created by chemical and physical processes for particular purposes. Theoretical attempts to characterize the dielectric properties of composite materials are as old as the theory of electromagnetism itself. Such a composite or mixture generally consists of two dielectric components, of which one is considered as the matrix phase (environment) and the other as the inclusion. In the literature, many models can be found for calculating the effective dielectric permittivity of such a mixture. For example, the Maxwell–Garnett formula1 calculates it from the electric-field average taken over both components of the mixture; the

(Received November 8, 2009; accepted April 3, 2010; published online April 27, 2010)

Bottcher formula2 assumes that particles of both constituents are dispersed in an effective medium with dielectric constant eeff; and the Bruggeman formula3 was derived assuming that the Maxwell– Garnett formula holds in the dilute limit of an infinitesimal amount of inclusions added to the mixture. In addition, many other models have been reported for predicting the effective permittivity of two-phase composites.4–8 In general, the most popular effective permittivity formulas are the Maxwell–Garnett mixing rule ei  ee (1) eeff ¼ ee þ 3f ee ei þ 2ee  f ðei  ee Þ and the Bruggeman formula ee  eeff ei  eeff ð1  f Þ þf ¼ 0: ee þ 2eeff ei þ 2eeff

(2)

Here, the permittivity of the environment is ee and that of the inclusion is ei, with volume fraction f. However, for the same given mixture, different mixing models predict different permittivity values, with certain bounds limiting the range of the predictions. The loosest bounds are the so-called 1351

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MATHEMATICAL MODELING The model used in computer simulation of the effective dielectric behavior of composites was constructed by considering a capacitor with simple configuration. Consider a parallel-plate capacitor with conducting plates of area S and separation d. The capacitor is filled with a composite dielectric material (Fig. 2), consisting of a matrix and a reinforcement network phase with dielectric constants emat and enet, respectively. A constant potential difference U0 is maintained between the capacitor plates. The effective dielectric constant of the composite eeff can be determined from the charge Q stored on the capacitor plates as eeff ¼

Fig. 1. The model of interpenetrating phase composites: (a) matrix, (b) network phase, and (c) assembled composite.

Wiener bounds.9 bounds are

These

effective

permittivity

ei ee f ee þ ð1  f Þei

(3)

eeff ;max ¼ f ei þ ð1  f Þee ;

(4)

eeff ;min ¼ and

corresponding to capacitors connected in parallel and series, respectively, in a circuit. Recently, a new type of composite material that contains at least two phases, each of which is continuous, interwoven in three dimensions to construct a topologically continuous network throughout the microstructure,10–12 has attracted many researchers. Seawater, earth, tree, air, and many other examples from nature fall into this category of composites. Figure 1 shows a model of a composite with tetrakaidecahedron structure of interpenetrating phases. The aforementioned existing effective permittivity theories, derived by making significant simplifications, have proven unsuitable for computing the effective permittivity of such composites. Although a number of numerical studies13–15 have been published on analyzing the effective response of a mixture, calculation of the effective permittivity for such composites has rarely been reported. In this work, the effective permittivity of such composites at various permittivity ratios of matrix to network was computed, with the ultimate goal of obtaining a spectrum of the effective permittivity of such composites for any permittivity ratio of the matrix to network in the range of 1:2 to 1:100 and 2:1 to 100:1, with volume fraction of 8.0% to 51.0% of the network phase. Such a spectrum may serve as guidance for the design of new composite materials, avoiding blind experimentation. This simulation also makes it possible to describe the electric field in such composites, which would be very hard, if not impossible, to observe experimentally.

Qd ; e0 SU0

(5)

where e0 ¼ 8:854  1012 F=m is the permittivity of vacuum. The charge Q can be obtained, knowing the space distribution of charge density r on the capacitor plate, as Z (6) Q ¼ r  ds; S

where the integration is over the plate area of the capacitor, S. Then the effective dielectric constant can be expressed as R d s r  ds : (7) eeff ¼ e0 SU0 Edge effects can be eliminated by periodic extension of the capacitor through the condition n  rU ¼ 0

(8)

at the edge planes, where n denotes the unit vector normal to the surface considered. In this paper, the finite-element method (FEM) was used to calculate the effective permittivity of two types of mixtures: raisin pudding, in which the network permittivity is higher than that of the

Φ= U0 n

S d

Φ=0 n• ∇ Φ

U0=const

Φ=0 Fig. 2. Illustration of the calculation of the effective permittivity of the composite.

Effective Permittivity Calculation of Composites with Interpenetrating Phases 2.0 1.9

effective permittivity εeff

matrix (enet > emat), and Swiss cheese, an inverted mixture with a network permittivity lower than that of the matrix (enet < emat). The effective dielectric permittivity of the composite eeff is assumed to be a function of the dielectric ratio between the matrix and network materials, emat/enet, and the volume fraction of the network phase f. The volume fraction of the network phase was altered in the range of 8.0% to 51.0%, and the permittivity ratio of the matrix to network was varied between 1:2 and 1:100 and between 2:1 and 100:1. The volume fraction of the network phase must be confined to the aforementioned range because even maximum packing allows a network phase with a volume fraction of only about 51.0%. Exceeding this value would result in a composite foam with closed cells.

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FEM Min Max

1.8 1.7 1.6 1.5 1.4 1.3 0.1

0.2

0.3

0.4

0.5

network phase volume fraction f Fig. 4. Simulated FEM results of a Swiss cheese mixture (enet = 1, emat = 2) compared with the Wiener bounds.

RESULTS AND DISCUSSION Calculations using various combinations of volume fraction and permittivity ratios, as described in the previous section, were performed using finiteelement models developed for these cases. The models were first tested and validated by comparing the numerical simulation results with those predicted by existing effective permittivity theories for composites of the same mixtures. An advantage of numerical simulation is that the distribution of electric field can be obtained for a mixture along with other electromagnetic properties, and such results are shown for selected cases in this work. Wiener Bounds Figure 3 shows the effective permittivity obtained from simulations for a raisin pudding mixture of emat/enet = 1/2 (emat = 1 and enet = 2). In this investigation, the permittivity e used in the simulation is a value relative to the free-space permittivity e0. The figure shows that the effective permittivity of mixtures always lies between the Wiener bounds for

the composite under consideration. In Fig. 4, showing the simulation results for a mixture with inverted permittivity ratio of emat/enet = 2/1 (emat = 2 and enet = 1) from the previous case, the results again fall mainly between the Wiener bounds. Mixing Models In Figs. 5–8, FEM simulation results of raisin pudding and Swiss cheese mixtures are compared with those predicted by the theoretical models (1) and (2). The Maxwell–Garnet and Bruggeman models produce similar results to the numerical simulations when the difference in dielectric properties between matrix and network is small, such as emat/enet = 2/1 or 1/2, as shown in Figs. 5 and 6, respectively. As can be seen from Figs. 7 and 8, the theoretical model predictions and numerical results differ increasingly as the difference in permittivity values of the matrix and network grows larger.

1.5

effective permittivity εeff

effective permittivity εeff

1.5

FEM Min Max

1.4 1.3 1.2

FEM MG Bruggeman

1.4

1.3

1.2

1.1

1.1

1.0 1.0

0.1

0.2

0.3

0.4

0.5

network phase volume fraction f Fig. 3. Simulated FEM results of a raisin pudding mixture (enet = 2, emat = 1) bounded by the Wiener model.

0.1

0.2

0.3

0.4

0.5

network phase volume fraction f Fig. 5. FEM results of a raisin pudding mixture (enet = 2, emat = 1) compared with the Maxwell–Garnett mixing rule and the Bruggeman model.

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9

FEM MG Bruggeman

1.8 1.7 1.6 1.5

effective permittivity εeff

effective permittivity εeff

1.9

FEM MG Bruggeman

8 7 6 5 4

1.4 0.1

0.2

0.3

0.4

3

0.5

0.1

network phase volume fraction f

0.2

0.3

0.4

0.5

network phase volume fraction f

Fig. 6. FEM results of a Swiss cheese mixture (enet = 1, emat = 2) compared with the Maxwell–Garnett mixing rule and the Bruggeman model.

Fig. 8. Simulated FEM results of a Swiss cheese mixture (enet = 1, emat = 10) compared with the Maxwell–Garnett mixing rule and the Bruggeman model.

5

1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10

effective permittivity εeff

effective permittivity εeff

4.5

FEM MG Bruggeman

4.0 3.5 3.0 2.5 2.0

4

3

2

1.5 1.0

1 0.1

0.2

0.3

0.4

0.5

network phase volume fraction f Fig. 7. FEM results of a raisin pudding mixture (enet = 10, emat = 1) compared with the Maxwell–Garnett mixing rule and the Bruggeman model.

None of the theoretical models were in good agreement with the FEM simulations in the case of raisin pudding or Swiss cheese mixture, and therefore a new model is needed for accurately predicting the effective permittivity of mixtures. Fitting and Discussion A new model can be developed based on the results of finite-element modeling of composites with various mixtures. The values of effective permittivity were calculated systematically for interpenetrating network structural composites of various ratios of emat/enet, with network phase volume fractions from 8.0% to 51.0%. Although such modeling was conducted based on certain assumptions, the results reveal important characteristics of

0.1

0.2

0.3

0.4

0.5

network phase volume fraction f Fig. 9. FEM results of raisin pudding mixtures with emat/enet = 1/2, 1/3, 1/4, 1/5, 1/6, 1/7, 1/8, 1/9, and 1/10, as a function of the volume fraction of the network phase.

such composites, and can serve as valuable guidance in the application of such materials and help the design of new composites. The effective permittivity values are plotted in Figs. 9 and 11 as a function of volume fraction of network phase for the raisin pudding mixtures. In these mixtures, the permittivity of the matrix is fixed at emat = 1, while that of the network phase enet varies from 2 to 100, resulting in a permittivity ratio emat/enet = 1/2, 1/3,…, 1/100. The dependence of effective permittivity on volume fraction of the network phase for the Swiss cheese mixtures is shown in Figs. 10 and 12. In these simulations, the permittivity of the network phase in the mixtures is taken as enet = 1, while that of the matrix emat varies from 2 to 100, or emat/enet = 2/1, 3/1,…, 100/1.

Effective Permittivity Calculation of Composites with Interpenetrating Phases

effective permittivity εeff

8 7 6 5 4 3

20/1 30/1 40/1 50/1 60/1 70/1 80/1 90/1 100/1

80 70 60 50 40 30 20

2

10

0.1

0.2

0.3

0.4

0.5

network phase volume fraction f Fig. 10. FEM results of Swiss cheese mixtures with emat/enet = 2/1, 3/1, 4/1, 5/1, 6/1, 7/1, 8/1, 9/1, and 10/1, as a function of the volume fraction of the network phase.

40

1/20 1/30 1/40 1/50 1/60 1/70 1/80 1/90 1/100

35

effective permittivity εeff

90

effective permittivity εeff

2/1 3/1 4/1 5/1 6/1 7/1 8/1 9/1 10/1

9

1355

30 25 20 15 10 5 0

0.1

0.2

0.3

0.4

0.5

network phase volume fraction f Fig. 11. FEM results of raisin pudding mixtures with emat/enet = 1/20, 1/30, 1/40, 1/50, 1/60, 1/70, 1/80, 1/90, and 1/100, as a function of the volume fraction of the network phase.

In a raisin pudding mixture, the network has a higher permittivity than the matrix, i.e., enet > emat. As a result, the effective permittivity of the mixture increases with the volume fraction of the network phase. On the other hand, in a Swiss cheese mixture, the network permittivity is lower than that of the matrix enet < emat, so the effective permittivity decreases with the volume fraction of the network phase. Although simulations were performed for mixtures with limited discrete number of permittivity ratios, such as emat/enet = 1/2, 1/3,…, 1/100, and 2/1, 3/1,…, 100/1, the effective permittivity of a mixture with a permittivity ratio of an arbitrary value in the simulated range, e.g., 38.2/1, can be obtained by interpolating the calculation results for the mixtures with permittivity ratios nearest to 38.2/1. As seen from the figures, the curves are fairly smooth

0.1

0.2

0.3

0.4

0.5

network phase volume fraction f Fig. 12. FEM results of Swiss cheese mixtures with emat/enet = 20/1, 30/1, 40/1, 50/1, 60/1, 70/1, 80/1, 90/1, and 100/1, as a function of the volume fraction of the network phase.

and their trends are consistent. Therefore, the effective permittivity of the mixture of a permittivity ratio of 38.2/1 can be calculated by interpolating between those of the mixtures with permittivity ratios of 38/1 and 39/1. The results can also be used to deduce effective permittivity values for mixtures with the same permittivity ratios but different permittivity magnitudes from those used in the simulation. Consider a mixture which has a volume fraction of network phase of 20% with permittivity values of the matrix and network of emat = 2 and enet = 1, respectively; i.e., a permittivity ratio of the matrix to network of emat/enet = 2/1. The calculated effective permittivity of this composite is eeff = 1.76. Then the effective permittivity of any mixture with the same ratio, such as emat = 4 and enet = 2, or emat = 20 and enet = 10, the respective effective permittivity values of these composites are eeff = 1.76 9 2 = 3.52 or eeff = 1.76 9 10 = 17.6. Using the results from simulations, the dependence of effective permittivity of a Swiss cheese mixture on the volume fraction and the permittivity of the network phase was derived, as shown in the following equation: eeff ¼ ðp1 þ p2 x p3 þ p4 f p5 þ p6 xp3 f p5 Þ  enet ;

(9)

p2 = 1.090, p3 = 0.994, where p1 = 0.190, p4 = 2.089, p5 = 0.761, and p6 = 1.345. These constants were determined from the results shown in Figs. 10 and 12, using a least-squares method. A similar equation can be developed for the raisin pudding mixture using the results of Figs. 9 and 11: eeff ¼ ðp1 þ p2 x p3 þ p4 f p5 þ p6 x p3 f p5 Þ  emat ;

(10)

where p1 = 1.069, p2 = 0.012, p3 = 0.990, p4 = 0.735, p5 = 1.444, and p6 = 0.959. In these formulas, x is the permittivity ratio of the matrix to network (in formula 9) or of the network to matrix (in formula 10), and f is the volume fraction of the network.

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Effective permittivity εeff

900

Calculated results Experimental results

800 700 600 500 400 0.0

0.1

0.2

0.3

0.4

0.5

0.6

volume fraction of the network phase PZT( f )

Fig. 14. Electric flux density contours (a) and vector diagram distribution (b) of the raisin pudding mixture with emat = 1 and enet = 10, and volume fraction of the network of 0.28.

Fig. 13. Calculated and experimental effective permittivity eeff of sol– gel–PZT composites as a function of the volume fraction f of the network phase PZT.

The R2 values, which measure how successful the fit is in explaining the variation of the data, were very close to unity (0.99996) for both formulas, indicating an excellent fit to the calculated data. Although it is difficult to derive any physical meaning from the fitted formulas, the dependence of the effective permittivity on the permittivity ratio x of the matrix to network (or of the network to matrix), and the volume fraction of the network f is clearly seen. In order to validate these formulas, effective permittivity values, eeff, of sol–gel–lead zirconate titanate (PZT) composites which were previously investigated experimentally for their radiofrequency (RF) absorption capabilities16 were taken for comparison. The RF wavelength is much larger than the characteristic dimensions of a sol–gel–PZT microstructure. According to dispersion relations, one can draw an analogy between the effective permittivity of a sol–gel–PZT under the field of RF frequencies and that under a static electrical field. Therefore, such a comparison is reasonable. The sol–gel–PZT composites can be regarded as a twophase interpenetrating composite of PZT and sol– gel where PZT acts as the reinforcement phase. The permittivity of PZT is enet = ePZT = 1400 F/m, and that of sol–gel is emat = esol–gel = 400 F/m. So formula 10 with enet > emat should be used, with x = enet/emat = ePZT/esol–gel = 1400/400 = 3.5. The results are illustrated in Fig. 13. From the figure it is easy to see that, although they follow the same trend, there are deviations of the experimentally measured effective permittivity values from those calculated using formula (10). However, the differences are generally small and do not affect the ability of the formulas to predict effective properties of such composites with two solid phases. Therefore, formulas 9 and 10 can be used to predict the effective permittivity of composites with interpenetrating phases.

Fig. 15. Electric flux density contours (a) and vector diagram distribution (b) of the Swiss cheese mixture with emat = 10 and enet = 1, and volume fraction of the network of 0.28.

Field Distribution The space distribution of the complex electrostatic potential was obtained by FEM for interpenetrating composites with volume fraction of 28% of the network phase. The distribution of electric flux density in the composites was also investigated and is shown in Figs. 14 (raisin pudding mixture) and 15 (Swiss cheese mixture). It was found that the global electric flux is one dimensional, parallel to the potential gradient. However, the local distribution depends strongly on the azimuth of the struts and the intrinsic permittivity. As expected, the electric flux density has a tendency to favor highpermittivity materials. CONCLUSIONS The effective permittivities of interpenetrating phase composites were calculated by finite-element method (FEM). The interpenetrating network structural composites at different permittivity ratios of matrix to network, emat/enet, with network phase volume fractions ranging from 8.0% to 51.0% were systematically investigated. The effective permittivity spectrum of these composites was depicted,

Effective Permittivity Calculation of Composites with Interpenetrating Phases

and the relationships among the effective permittivity eeff, permittivity ratio x of the matrix to network phase (or of the network to matrix), and the volume fraction of the network f were developed by fitting the calculated data. The accuracy of the simulations was validated and the results agreed well with published experimental findings. The electric field distribution in such composites was analyzed, and it was found that the local electric flux distribution depends strongly on the azimuth of the struts and the intrinsic permittivity. The electric flux density has a tendency to favor highpermittivity materials. REFERENCES 1. 2. 3. 4. 5.

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