CALCULATING THE EFFECTIVE PERMITTIVITY AND PERMEABILITY OF COM- POSITES BASED ON THE DILUTION PROCESS MODEL

Transcription

1 CALCULATING THE EFFECTIVE PERMITTIVITY AND PERMEABILITY OF COM- POSITES BASED ON THE DILUTION PROCESS MODEL Liming Yuan 1, Bin Wang 2, Yonggang Xu 1, Qilin Wu 3 1 Science and Technology on Electromagnetic Scattering Laboratory, Shanghai , PR China 2 Shanghai Radio Equipment Research Institute, Shanghai , PR China 3 Key Laboratory of High Performance Fibres & Products, Ministry of Education, Donghua University, Shanghai , PR China Received 20 September 2017; accepted 30 November 2017 ABSTRACT In order to solve the engineering problem of calculating the effective permittivity and permeability of composites, this paper introduced a dilution process model (DPM) which sets the composite of the highest particle volume fraction as the new matrix medium and the old matrix medium as the inclusion. With DPM, complex permittivity and permeability of composites could be calculated by the Maxwell Garnett (MG) formula, the Bruggeman (BG) formula and the generalized effective medium theory (GEMT) with a fitting parameter of V which is a second-degree polynomial of new particle volume fraction. With the proposed method, the effective complex permittivity and permeability of composites consisting of isotropic spherical carbonyl iron particles (SCIPs) or anisotropic flaky carbonyl iron particles (FCIPs) embedded in paraffin are calculated. Comparisons are performed among the calculated results by different formulae and the measured results. It indicates that the calculated results by the GEMT formula with DPM are in great agreement with the measured results. The achievement of this paper has am important theoretical significance and engineering application. Keywords: Dilution process model; the effective permittivity and permeability; composites Letter 1. INTRODUCTION Accurate prediction on the effective permittivity and permeability is one of the most basic requirements for the electromagnetic composite performance design and the electromagnetic scattering characteristics analysis. Since J. C. Maxwell firstly studied the conductivity of composites consisting of conducting particles and non-conducting matrix medium, many formulae had been proposed for predicting the effective permittivity and permeability of composites [1-10].The most classic formulae should be the Maxwell Garnett (MG) formula and the Bruggeman (BG) formula [2], and most of other formulae were proposed on the basis of them. When the filling particles have spherical shape, the MG formula and the BG formula could be unified into the generalized effective medium theory (GEMT) formula, eff m i m p 2 V 2 i V eff m eff m i m eff m where, f eff denotes the effective permittivity or permeability of composites, f m is the permittivity or permeability of the matrix medium, f i is the permittivity or permeability of particles, p i is the particle volume fraction, and V is a dimensionless parameter. When V=0, the MG formula could be obtained: eff m i m pi 2 2 eff m i m When V=2, the BG formula could be obtained: (1) (2) i eff m eff pi 1 pi i eff m eff However, it is difficult to accurately predicting the effective permittivity and permeability of composites in engineering by existing formulae. One reason is that these formulae are derived under certain assumptions such as the particle concentration is low, and the other is that some necessary parameters which have great effect on the calculated results cannot be obtained in engineering such as the permittivity and permeability of particles, particle geometric distribution, morphological distribution and so on. In order to solve this problem, a novel dilution process model (DPM) is proposed in this paper, and the effective permittivity and permeability of composites could be calculated accurately by the generalized effective (GEMT) with DPM. The validity of the presented method is proved by calculating the effective permittivity and permeability of two kind composites, one of which includes isotropic spherical carbonyl iron particles (SCIPs) and the other includes anisotropic flaky carbonyl iron particles (FCIPs). 2. THEORETICAL METHODS When particles and the matrix medium are mixed very well, although particles are always anisotropic in engineering, composites could be thought to be isotropic and homogeneous in macroscopic scale. If the composite with the highest volume fraction is viewed as the new matrix medium and the old matrix medium is viewed as the inclusion, and the effective permittivity and permeability of other composites with lower particle concentration could be obtained by adding matrix medium into the composite (3) Advanced Composites Letters, Vol. 25, Iss.6,

2 with the highest particle volume fraction. The calculation model is called DPM here. Fig. 1 shows the schematic of DPM. p 1 pl ph 4. RESULTS AND DISCUSSIONS 4.1 SCIPs/paraffin composites The effective permittivity and permeability of composites consisting of SCIPs and paraffin are calculated by the GEMT formula with DPM as well as the MG formula and the BG formula with DPM. In the calculations, f m and f h are the measured permittivity or permeability of the paraffin and the composite with 35vol% SCIPs. Meanwhile, these calculated results are compared with that calculated by a conventional GEMT formula, where the intrinsic elec- ph Fig.1: The schematic of DPM. In Fig.1, p h and p l are the particle volume fractions of the composite with high and low particle concentration, respectively. p is the added volume fraction of the matrix medium to obtain the composite with low particle concentration, i.e., p=1-p l /p h. Considering both of the matrix medium and the composite with high particle concentration are isotropic, the effective permittivity and permeability of a composite with lower particle concentration could be calculated by the MG formula and the BG formula. The formulae are of the following forms. l h m h p 2 2 l h m h pl (4) The method proposed here avoids some complicated mathematical problems such as calculating the intrinsic permittivity and permeability of particles, mathematically characterizing particle geometric and morphological distributions in composites, and so on. It could be applied to macroscopic homogeneous composites including isotropic particles, anisotropic particles and multiple particles with the same proportion. Generally, the mechanical property of composites will be weakened as the particle concentration increases. So the particle concentration should not be too high in engineering. Under the constraint of the mechanical property, the method proposed in this paper could fully meet the requirement of calculating the permittivity and permeability of composites in electromagnetic material design. 3. EXPERIMENT 3.1 Material preparation The SCIPs and FCIPs were purchased from Shaanxi Xinghua Chemical Co. Ltd. and Shenyang Hangda technology Co. Ltd., respectively. Figure 2 shows the microstructures of the two particles. The SCIPs are regular spherical shape with the particle size 1~5 μm distributed uniformly. The FCIPs are significantly anisotropic and the particle diameter is mostly about 10 μm, the thickness about 0.3 μm. about 10μm, the thickness about 0.3 μm. (a) (b) m l h l p 1 p m l h l (5) Where, f h and f l is the effective permittivity or permeability of the composite with high and low particle concentration. In engineering applications, both of f h and f m can be obtained by experimental measurement. When there is a great difference between ph and pl, p would be a large value, then Equation (4) and (5) may be out of the application ranges of the MG formula and the BG formula. In order to achieve more accurate results, the GEMT formula is used with DPM, the expression could be written as following, l h m h p (6) 2 V 2 V l h l h m h l h where, V is defined as a fitting parameter. This paper aims to study an effective methodology to accurately predict the effective permittivity and permeability of a composite in engineering. Hence, the parameter V is given a great freedom to let the GEMT formula much more flexible. After some studies it turned out that a second-order polynomial is good enough to describe the variation of V in volume fraction direction [11,12]. Therefore, V is written to be of form V=ap 2 +bp+c, in which a, b and c are dimensionless parameters and obtained by fitting experimental data of composites with different particle concentrations. Fig.2: The particle SEM photographs: (a) SCIPs; (b) FCIPs. 3.2 The effective permittivity and permeability of composites measurement SCIPs/paraffin composites and FCIPs/paraffin composites could be obtained by mixing these two kinds of particles with paraffin respectively. Composites are pressed into toroidal shape in a mold with an outer diameter of 7 mm, inner diameter of 3 mm and thickness of 2 mm for microwave measurement, the effective permittivity and permeability could be obtained using the transmission/reflection method. Advanced Composites Letters, Vol. 25, Iss.6,

3 tromagnetic parameters of SCIPs are obtained by fitting the measured electromagnetic parameters of composites. The intrinsic electromagnetic parameters of SCIPs can be represented as [13] i 1 0 n1,2, M s Ha i n n 2 2 Ha i n n n i where ε 0 is the permittivity of free space, s is the electric conductivity, ω is the angular frequency, λ is the gyromagnetic ratio, M s, H a and a are the nth saturation magnetization, magnetic anisotropy field and damping coefficient, respectively. Then the effective permittivity and permeability of a composite can be calculated by [14] a i m p p eff m 3pi 3 1 a a m eff m a i c i m where a is a pending parameter, fc is the percolation threshold where a metal-dielectric composite begins to conduct dc current. Figure 3 plots the calculated results and the measured results of the effective permittivity and permeability of composites with different particle concentrations at 2 GHz. It could be seen that the calculated results by the formulae with DPM are always the same with the measured results, and the calculated results by the GEMT formula are in best agreement with the measured results than that by the MG formula, BG formula and the conventional GEMT, which is ascribed to the fitting dimensional parameter of V. Meanwhile, there are obvious deviations between the calculated results by the conventional GEMT and the measured results, which may be ascribed to the complicated electromagnetic properties of SCIPs along with irregular geometric and morphological distributions. (7) (8) (9) Fig.4: the fitted values of V for calculating the effective permittivity and permeability of composites with different SCIP concentrations at 2 GHz. Figure 4 shows the fitted values of V for calculating the effective permittivity and permeability of different composites at 2 GHz. It could be seen that the increase rate for calculating the effective permittivity become more and more large while that for calculating the effective permeability become more and more small. When V is close to 1, the calculated result by the GEMT formula is close to that by the MG formula, and when V is close to 2, the calculated result by the GEMT formula is close to that by the BG formula. Synthesizing Fig.3 and Fig.4, it could be found that the calculated result decreases as V increases. Figure 5 shows the calculated results and the measured results of the composite with 20vol% SCIPs in 2~8GHz. It could be seen that the calculated results by the GEMT formula with DPM are very consistent with the measured results in the whole frequency band, while there are some deviations calculated by the MG formula with DPM, the BG formula with DPM and the conventional GEMT, especially for the real part of permittivity. The statistical average errors of the effective permittivity calculated by the MG formula with DPM, the BG formula with DPM, the GEMT formula with DPM and the conventional GEMT are 12.3%, 9.7%, 2.3% and 9.6%, and that of the effective permeability are 9.7%, 8.9%, 2.1% and 2.7%, respectively. It could be concluded that the GEMT formula with DPM could fully meet the requirement of electromagnetic material design in engineering. Fig.3: the calculated results and the measured results of composites with different SCIP concentrations at 2 GHz: (a) the effective permittivity (b) the effective permeability. Fig.5: the calculated results and the measured results of the composite with 20 vol% SCIPs in 2~8 GHz: (a) the effective permittivity; (b) the effective permeability. Advanced Composites Letters, Vol. 25, Iss.6,

4 accurate results for composites including anisotropic particles. Fig.6: the fitted values of V for calculating the effective permittivity and permeability of the composite with 20vol% SCIPs in 2~8 GHz. Fig. 6 shows the fitted values of V for calculating the effective permittivity and permeability of the composite with 20 vol% SCIPs in 2~8 GHz. It could be seen that the values of V for calculating the effective permittivity tend to be a constant, approximately 3.25, while have some fluctuation with a decreasing trend versus frequency for calculating the effective permeability. It is ascribed to that there is an obvious frequency dispersion characteristics of the effective permeability of composites while the effective permittivity remained almost unchanged in frequency range of 2~8 GHz. 4.2 FCIPs/paraffin composites The effective permittivity and permeability of composites consisting of anisotropic FCIPs and paraffin are calculated here. It is worth mentioning that the anisotropic particles are distributed randomly and homogeneously to form an isotropic and homogeneous composite. In following calculations, f m and f h are the measured permittivity or permeability of the paraffin and the composite with 45 vol% FCIPs, respectively. Fig.7: the calculated results and the measured results of composites with different FCIP concentrations at 2 GHz: (a) the effective permittivity (b) the effective permeability. Figure 7 shows the calculated results and the measured results of the effective permittivity and permeability of different composites at the frequency of 2 GHz. It could be seen that the calculated results by the GEMT formula with DPM are in better agreement with the measured results than that by the MG formula and BG formula, which means that the GEMT formula with DPM could achieve Figure 8 shows the fitted values of V for calculating the effective permittivity and permeability of different composites at 2 GHz. It could be seen that the change trends of V here are very different from that for SCIPs/paraffin composites. The values of V for calculating the effective permittivity decreases as the FCIP concentration reduces, while that for calculating the effective permeability increases firstly and then decreases slightly as particle concentration increases and is kept in the range of 0.3~1.2. Fig.8: the fitted values of V for calculating the effective permittivity and permeability of composites with different FCIP concentrations at 2 GHz. Fig.9: the calculated results and the measured results of the composite with 20vol% FCIPs in 2~8 GHz: (a) the effective permittivity; (b) the effective permeability. Figure 9 shows the calculated results and the measured results of the effective permittivity and permeability of the composite with 20 vol% FCIPs in 2~8 GHz. It could be seen that the GEMT formula with DPM could obtain better calculated results of the complex permittivity and the imaginary part of permeability than both the MG formula and the BG formula with DPM. But there are obvious deviations between the calculated results of the real part of permeability by the three formulae with DPM and the measured results. The deviations could be ascribed to that it is difficult to form a perfectly isotropic and homogeneous composite which include FCIPs and there would be a difference in particle distribution between the calculated composite and the new matrix medium, i.e., the composite with 45 vol% FCIPs. The statistical average errors of the effective permittivity calculated by the MG formula with DPM, the BG formula with DPM and the GEMT formula with DPM are 21.5%, 3.7%, and 4.1%, and that of the ef- Advanced Composites Letters, Vol. 25, Iss.6,

5 fective permeability are 7.4%, 11.7%,and 6.5%, respectively. Figure 10 shows the fitted values of V for calculating the effective permittivity and permeability of the composite with 20 vol% FCIPs in 2~8 GHz. It could be seen that the values of V for calculating the effective permittivity tend to be a constant, approximately 2.25, while V had the first decrement and later increment trend for calculating the effective permeability, the values are in range of 0~1.5. It is also ascribed to that there is an obvious frequency dispersion characteristics of the effective permeability of composites while the effective permittivity remained almost unchanged in frequency range of 2~8 GHz. ost unchanged in frequency range of 2~8GH Fig.10: the fitted values of V for calculating the effective permittivity and permeability of the composite with 20 vol% FCIPs in 2~8 GHz. 5. CONCLUSIONS In order to solve the problem of calculating the effective permittivity and permeability of engineering composites, a novel DPM is proposed. It avoids complicated mathematical problems such as calculating the intrinsic permittivity and permeability of particles, mathematically characterizing particle geometric and morphological distributions in composites, and so on. The effective permittivity and permeability of composites consisting of isotropic SCIPs embedded in the paraffin are calculated. Comparisons are performed among the calculated results by the MG formula with DPM, the BG formula with DPM, the GEMT with DPM and a conventional GEMT. The calculated results by the GEMT formula with DPM are in best agreement with the measured results. Furthermore, the effective permittivity and permeability of composites consisting of anisotropic FCIPs embedded in the paraffin are calculated. The calculated results by the GEMT formula with DPM are still in great agreement with the measured results. It could be concluded that the GEMT formula with DPM proposed in this paper can accurately predict the effective EM parameters of a composite in engineering. This achievement has important theoretical significance and engineering application, such as in the field of perfect electromagnetic absorbing material design, scale nonmetal material design for the scale measurement of electromagnetic scattering properties and so on. ACKNOWLEDGEMENTS This work was supported by the National Natural Science Foundation of China under Grant NO , NO and NO , and Shanghai Municipal Science and Technology Commission under Grant NO , NO.14ZR and NO.15ZR References: 1. J. M. Garnett, Colours in metal glasses and metal films [J], Transactions of the Royal Society, 1904, CCIII: Bruggeman D A G. Dielectric constant and conductivity of mixtures of isotropic materials [J]. Ann. Phys.(Leipzig), 1935, 24: Elliott R J, Krumhansl J A, Leath P L. The theory and properties of randomly disordered crystals and related physical systems [J]. Reviews of modern physics, 1974, 46(3): Ari H. Sihvola, Jin Au Kong. Effective permittivity of dielectric mixtures [J]. IEEE Transactions on Geoscience and Remote Sensing, 1988, 26(4): Sareni B, Krähenbühl L, Beroual A, et al. Effective dielectric constant of random composite materials [J]. Journal of Applied Physics, 1997, 81(5): Liu X, Wu Y, Zhang Z. Theoretical study of the interface effect on the electromagnetic wave absorbing characteristics [J]. Physica B: Condensed Matter, 2010, 405(20): Yan L, Wang J, Han X, et al. Enhanced microwave absorption of Fe nanoflakes after coating with SiO 2 nanoshell [J]. Nanotechnology, 2010, 21(9): Tinga W R, Voss W A G, Blossey D F. Generalized approach to multiphase dielectric mixture theory [J]. Journal of Applied Physics, 1973, 44(9): Morse P M, Feshbach H. Methods of theoretical physics, Vol. II [J] Acher O, Adenot A L. Bounds on the dynamic properties of magnetic materials [J]. Physical Review B, 2000, 62(17): Karkkainen K K, Sihvola A H, Nikoskinen K I. Effective permittivity of mixtures: Numerical validation by the FDTD method [J]. IEEE Transactions on Geoscience and Remote Sensing, 2000, 38(3): Karkkainen K, Sihvola A, Nikoskinen K. Analysis of a three-dimensional dielectric mixture with finite difference method [J]. IEEE Transactions on Geoscience and Remote Sensing, 2001, 39(5): Chen P, Liu M, Wang L, et al. Frequency dispersive complex permittivity and permeability of ferromagnetic metallic granular composite at microwave frequencies [J]. Journal of Magnetism and Magnetic Materials, 2011, 323(23): Liu T, Zhou P H, Xie J L, et al. Extrinsic permeability of Fe-based flake composites from intrinsic parameters: A comparison between the aligned and random cases [J]. Journal of Magnetism and Magnetic Materials, 2012, 324(4): Advanced Composites Letters, Vol. 25, Iss.6,