ISSN 10628738, Bulletin of the Russian Academy of Sciences. Physics, 2014, Vol. 78, No. 11, pp. 1209–1217. © Allerton Press, Inc., 2014. Original Russian Text © A.T. Morchenko, 2014, published in Izvestiya Rossiiskoi Akademii Nauk. Seriya Fizicheskaya, 2014, Vol. 78, No. 11, pp. 1482–1490.
Modeling the Structure and Conditions of the Absorption of Electromagnetic Radiation in Magnetic Composites Using Effective Medium Approximations A. T. Morchenko Department of Electronics Materials Technology, National University of Science and Technology (MISiS), Moscow, 119049 Russia email:
[email protected] Abstract—The development of ultrawideband composite absorbers of electromagnetic waves depends largely on finding the best combination of a medium’s characteristics that determine its absorption capacity. It is possible to achieve a higher level of absorption in a wider frequency band by combining a variety of mech anisms that increase the loss of electromagnetic field energy by, e.g., combining specific constituents in a dielectric matrix. Different models of mixing are analyzed for a material’s parameters in the effective medium approximation using the example of ferrite–dielectric composite. DOI: 10.3103/S1062873814110203
INTRODUCTION
materials needed to protect personnel and measuring instruments from EMR rereflections. Anechoic chambers are also used for testing and tuning such sen sitive precision systems satellites, georadars, ships’ radiobuoys, and so on.
The interest in absorbers of electromagnetic radia tion (EMR) is due mainly to the attention devoted by industrially developed countries to solving the prob lem of cloaking different weapons systems and military equipment (i.e., stealth technology) [1]. Mathemati cal modeling of the scattering of electromagnetic waves is vital in optimizing the shape and electrody namic characteristics of an object. Calculation models are based on rigorously solving boundary problems of the physical theory of the EMR diffraction on bodies with complex shape using the edge wave approach elaborated by P.Ya. Ufimtsev in the middle of the last century [2].
In many countries, screening (i.e., the localization of electromagnetic waves within a certain area by erecting barriers to the propagation of EMRs) is used to protect the health of people near electrical and elec tronic instruments, and the personnel who service radio engineering devices. Another problem related to the above is protecting against the unauthorized inter ception of information (from, e.g., PCs) forces us to use special screened premises.
Coatings that contain special radarabsorbing materials (RAMs) play a major role in reducing the radar visibility of a camouflaged facility. RAMs can also be used to coat the internal surfaces of the anechoic chambers needed for testing electro and radio devices for electromagnetic compatibility. RAMs are also used in manufacturing environmen tally friendly defense installations. According to the RF Law On Governmental Regulations in the Field of Ensuring the Electromagnetic Compatibility of Tech nical Facilities [3], requirements on the EMR levels of electronic and electric instruments are to be tightened. The law is aimed at coordinating the operation of instruments and devices, and to protecting humans from technogenic radiation of ever increasing inten sity that upsets the nervous system and quite often leads to cancerous neoplasms [4]. Tests on compliance with these requirements are performed in anechoic chambers whose walls are covered with radar absorb ing coatings (RACs) and contain radar absorbing
An EM wave penetrating into and interacting with a medium is converted to other types of energy. The absorption, scattering (due to structural and geomet ric inhomogeneities of the material) and interference of the waves occur simultaneously. Purely interfer ence RACs are, as a rule, narrowband. At the same time, there is a tendency for the spectrum of frequen cies against which protection from EMR must be provided to expand. The main requirement for an RAM is therefore broadbandness, i.e., a high level of absorption over a broad range of frequencies. How ever, the ability of a material to absorb radiofre quency radiation depends on its composition and structure [5], and no universal RAM suitable for use in all ranges of frequencies exists in nature [6]. Achieving broadbandness while reducing the cost, mass, and dimensional factors of RAMs is thus one of the most complex problems that arise in developing new materials of this type.
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Ferrites, in which the processes associated with magnetic, dielectric losses, and losses on eddy cur rents can occur simultaneously, have rather broad absorption bands. Unlike many other RAMs, they are effective in the lowfrequency region of the microwave range (100 kHz–1 GHz). In the range of frequencies below 1 GHz, RAMs based on nickel–zinc ferrite are of the greatest interest, so materials based on them that absorb EMR due to the resonance encountered at the level of domains and atoms have recently been in great demand [7]. Unfortunately, they are rather expensive, and the nickel compounds used during their produc tion are additionally quite toxic [8. 9]. To solve the problem of raising the level of and expanding the EMR absorption band, we must ensure the frequency dispersion of a RAM’s parame ters: dielectric permittivity and/or magnetic perme ability, along with electric conductivity (ε, μ, σ). This can be achieved by, e.g., using complex systems man ufactured on the basis of magnetic particles, wires, and different binders. By combining the components of such artificially created materials, we can produce a wide variety of physical mechanisms that cause the loss of electromagnetic field energy, leading to a syn ergic effect and substantial enhancement of absorp tion capabilities [10]. Meanwhile, analyzing the electromagnetic properties of multicomponent het erogeneous systems (HSes) is a very complicated problem [11]. Composite materials (CMs) in particular can have rather complex structures that differ in composition, shape, size, and the distribution of individual compo nents. To determine levels of absorbing ability, it is desirable to know the generalized material parameters of HSes with discrete structure, i.e., those that describe its properties with respect to a particular con tinuous medium that responds the same to the impact of EM waves. An approach that uses the socalled effective medium approximation to describe very diverse objects, allowing us to derive them from known data on the parameters of the initial components, is very productive for estimating specified values [1114]. Two types of HSes are usually considered as the main model representations in the theory of an effec tive medium: matrix, in which each element of the inclusion medium is surrounded on all sides by a par ticular matrix medium and the interaction between elements of the filler can be ignored, and statistical, in which all components of the medium are considered equal. In this work, we consider models that allow us to estimate the effective characteristics of RAMs consist ing of dielectric matrices with conducting and non conducting ferromagnetic fillers.
MODELING THE STRUCTURE OF A FERRITE–DIELECTRIC RAM The efficiency of composite RAMs prepared on the basis of magnetic particles and different binders is determined not only by the properties of the initial components but by the structure of the CM as well, since the scattering of electromagnetic waves depends on the dispersion and shape of the powder particles, their concentration, the type of distribution in the scattering medium, the average distance between indi vidual particles in the matrix, and the thickness of the dielectric or conducting layers between them. For example, different mechanisms of magnetic loss take effect when the distance between particles changes, since the frequency spectrum depends on whether or not the fluctuations in the magnetization of individual particles are interconnected. Consequently, informa tion on the structural parameters of the medium is required to estimate the influence different factors have on the final characteristics. Let us consider a matrix model of a CM consisting of filler particles, e.g., ferrite phase (1) that are round or cubic and of uniform size (D = 2R), distributed in dielectric medium (2) chaotically (or regularly in par ticular), forming an ordered spatial lattice of the sim ple cubic (SC), bodycentered cubic (BCC), or dense packing (DP) type. The CM can in this case contain macro or microscopic particles of the ferromagnetic filler, or it could be a nanocomposite. At a given volume concentration of the filler (CV), there is a certain volume per each particle, and it is separated from other particles by a layer with thickness d. The highest possible value of CV at the specified type of round particle packing is determined by compact ness factor Kc, which is 0.5236 for SC, 0.6802 for BCC, and 0.7405 for DP structures. The upper esti mate of CV for an arbitrary distribution of particles lies within the above limits. When approximating the par ticles by cubes, Kc = 1 can be reached. Since the parameter expressing the volume concentration of the filler is important in the effective medium model, and it is more convenient to use the concept of component mass fractions Cm in CM technology, to transform one quantity into another we use the expression CV = ρ1Cm/[ρ2Cm + ρ1(1 – Cm)], (1) where ρ1 and ρ2 are the density of CM components. Depending on ratio k = ρ1/ρ2, the values of the mass and volume content of the magnetic phase will differ more or less. The magnetic densities that are important in prac tice lie in the limits of 3.6–5.8 for ferrite–spinels and hexaferrites, 5.2–7.2 for ferrite–garnets, and up to 9 g/cm3 for metals and alloys (Fe, 7.8; Ni and Co, 8.9; permalloys, 8.2–8.7; cementite, 7.8). The density of the polymer and other materials used in a dielectric matrix is ~1 g/cm3 (paraffin, 0.9; wax, 0.95–0.97; silicone sealant, 0.95–1.05; PVA glue, 1–1.1; PVA, 1.29). CM compositions created on the basis of
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CV, % 100 D δ
80 h
60
D δ
40 1
2.5
4.5
10
d
a
a
30
20
0
d
L
20
40
60
80
Fig. 2. Scheme of particle packing and characteristic sizes in the CM structure.
100 Cm, %
Fig. 1. Region of compositions of ferrite–dielectric com posites in the diagram CV = f(Cm) for different values of ratio k between phase densities.
the above components and thus of the greatest interest would lie in the dashed region of the diagram shown in Fig. 1. For example, we used Mn–Zn ferrites of 700NM and 2000NM grade to prepare composite RAMs [15] with ρ1 ~ 4.3–4.7 g/cm3; k ~ 4.5 (the mid dle of the dashed region in Fig. 1) is therefore a reason able choice when performing estimates. To estimate the electrophysical characteristics of a CM with a filler consisting of conducting particles, the system can be presented in the form of either a capac itor with thickness h, containing a dielectric CM layer with effective value of permittivity ε, or a set of capac itors consisting of adjacent particles of the conducting phase and interparticle layers with thickness d = 2δ and dielectric permittivity ε2. A twodimensional image of the geometry of such a structure is shown schematically in Fig. 2 for a CM with cubic and round particles. The average distance between the centers of parti cles (the period of the CM structure) and the mini mum gap between particles a = D(Kc/CV)1/3, d = D[(Kc/CV)1/3 – 1]
(2)
grow along with the size of the particles, lowering their concentration; the generalized (effective) electromag netic characteristics of a CM therefore depend strongly on these parameters because of the effect they have on the strength of the interaction between indi vidual filler particles. For cubic particles with SC packing, we estimate the minimum thickness of the layer using the expression d = D(1 – CV1 3 )/CV1 3,
L
(3)
The expression for the average thickness of the layer between particles, assuming the usual plane sur face of the particles [8], davg = D(1 – CV)/3CV, (4) differs from the one obtained using formula (2), since it is overestimated by allowing for the part of the dielectric phase outside the region of the interface between neighboring particles. It is therefore prefera ble to use (2) as the thickness of the layer when dealing with a regular distribution of the CM particles. A rigorous estimate of the averaged thickness of layers with allowance for the sphericity of the surface near round particles yields the expression davg = D{[6Kc/(πCV)]1/3 – 1}, (5) which, unlike (3), considers the distribution of the particles in the structure. The estimate of davg accord ing to formula (5) for SC structures with the packing of round particles of diameter D coincides with the min imum thickness of layers between cubic particles with D calculated according to formula (4). Figure 3 shows the results from calculating the thicknesses of the interparticle layers for different types of particle packing. The average size of filler par ticles D = 400 μm, k = 4.5 was chosen as an example. We should also bear in mind that the orientation of a regular spatial CM structure with respect to the plane of the sample can differ from the one shown in Fig. 2. It makes sense in this case to use different estimates of the average thickness of dielectric shells or interparti cle layers to determine the limits of applicability for any model of the medium in further calculations. MATERIAL PARAMETERS FOR HETEROGENEOUS SYSTEMS IN THE EFFECTIVE MEDIUM APPROXIMATION The revival of interest in the electrodynamics of heterogeneous media is mainly due to the growth of studies in the field of creating artificial materials.
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d, μm 1000 900 800 700 600 500 400 300 200 100 0
a d
R' R
r
r
Fig. 4. Geometry of a capacitor consisting of round filler particles in a CM.
10 20 30 40 50 60 70 80 90 100 Cm, %
Fig. 3. Dependence of the thickness of the interparticle layer on the mass fraction of filler in the composite (D = 400 µm, k = 4.5), calculated according to (2) for different round particle packings: –䉬– SC, – – BCC, – – DP, FCC. Calculated davg, using (4): –×– ; davg calculated using (5) for spherical particles and d using (3) for the SC packing of cubic particles coincide: .
There are many works devoted to analyzing the prop erties of mixed systems that have long been classics, and many works considering this problem have come out only recently. Unfortunately, deep analysis of the effective values of electromagnetic characteristics is usually hampered due to the lack of reliable informa tion on the material parameters of the initial compo nents and structural elements of CMs. In addition, the initial idealized prerequisites of many theoretical models’ heterogeneous systems often do not corre spond completely to the real structure of an investi gated material. Estimating the Effective Dielectric Permittivity of a CM with Conducting Filler Particles As an example of a CM that contains conducting inclusions (e.g., filler with particles of weakly con ducting Mn–Zn ferrite), let us consider a system con sisting of cubic particles distributed in accordance with the SC packing shown schematically in the left hand side of Fig. 2. On the one hand, the sample is a capacitor filled with a medium having effective dielectric permittivity ε with thickness h and plate area L2. On the other, since the phase of the ferrite is in this case is assumed to be conducting, this system in turn consists of a set of similarly bound capacitors with plate area D2 and dis tance d between them. A. At sufficiently high concentrations of the filler phase (e.g., if the average size of the particles and their concentration in the CM are such that the conditions L Ⰷ h Ⰷ D Ⰷ d are met), we can use the formula for a
flat capacitor to determine its capacity. The total num ber of capacitors combined in series or in parallel can be considered equal to number of particles N = CVL2h/D3. The capacity of such a system is CCM = CVL2(d + D)ε2/{4πdD[h/(d + D) – 1]}, (6) which with allowance for (3) yields the expression CCM = L2CVε2/[4π(1 – CV1 3 )(hCV1 3 – D)]. (7) On the other hand, such a system can be viewed as a capacitor with plates L × L filled with a dielectric that has additional effective permittivity Δε and thickness h – d. Then CCM = ΔεL2/[4π(h – d)]. (8) Comparing (7) and (8), we obtain Δε = ε2(D/d)CV1 3 = ε2CV2 3 /(1 – CV1 3 ), (9) where the total value of effective dielectric permittivity ε = ε2 + Δε. Using the highest possible filler concentrations with particles of the same size (depending on the type of the structure, CV ~ 52–74%), we can improve the effective dielectric permittivity of a CM with a filler of conducting ferrite particles by 3–10 times, relative to the initial permittivity of a dielectric CM matrix. When using particles of different size, the filler content is not limited by the geometric factor; we can therefore achieve enormous values for a CM’s effective dielec tric permittivity (up to 60ε2 at CV ~ 95%). B. Let us find the effective dielectric permittivity at low filler concentrations using the example of a CM containing round particles in the approximation of widely separated balls (a Ⰷ R), according to the scheme shown in Fig. 4. The interaction between oppositely charged balls in this case does not redistrib ute the charge over their surfaces, and we may consider the field between them to be produced by uniformly charged spheres, charges +q and –q of which are con centrated at their centers at distance a. The charges over the balls’ surfaces are in fact distributed nonuni formly in chains of particles that act as the plates of capacitors joined in series, so we may generally con sider that the equivalent sites of the positive and nega tive charges of a particle lie on a line connecting the centers of the particles at distance R' < R on both sides of its center: R' → 0 and d → a at CV → 0.
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The electric field strength as a function of distance r from the center of a particle to the left in the direction to a particle on the right can be written as 2
E ( r ) = q/ [ ε 2 ( r – R' ) ] 2
– ( – q )/ [ ε 2 ( a – R' – r ) ] = ( q/ε 2 ) 2
Estimating the Material Parameters of a CM with Nonconducting Inclusions
2
∫ E ( r ) dr = ( q/ε ) ∫ [ 1/ ( r – R' ) ∂
for the difference between possible types of packing for cubic and round particles with all other things (parti cle size D and concentration CV) being equal can raise the estimated value of effective dielectric permittivity Δε to (6/π)1/3 ≈ 1.24 times.
(10)
× [ 1/ ( r – R' ) + 1/ ( r + R' – a ) ]. The potential difference between the balls is U =
1213
2
(11)
2
+ 1/ ( r + R' – a ) ]dr, the choice of the integration limit depends on the dis tribution of the charge over the surface of the particles and the coordinates of the selected levels on the parti cles in the direction of the raxis. With a uniform charge distribution over the surface of the conducting body, its potential is the same on all level lines, and the limits of integration therefore cor respond to the points on the surfaces of the particles being at the minimum distance from one another, i.e., having coordinates r = R and a – R (the ball poles). The electric field between particles corresponds to the one created by the point charges at the centers of the balls; i.e., R' = 0. With a nonuniform charge distribu tion, we must choose as our integration limits the val ues of r associated with the level on the surface that corresponds to an equivalent distribution of charges of like sign on each of semispheres of a particle; i.e., r = R' and a – R'. Within limits, we may assume that the maximum fraction of the charge of each semi sphere of two balls is concentrated near its pole, i.e., at r = ± R and r = a ± R. In this situation as well, the inte gration limits can remain the same but only if we con sider that now R' = R. After integration in the specified limits, we obtain U = (2q/ε2)(2R – a)/[(R – R')(R + R' – a)], (12) Since the mutual electric capacity of the system of two balls is C = q/U, C = ε2(R – R')(a – R – R')/[2(a – 2R)], (13) and its highest possible value is reached at R' = 0: C = ε2D(d + D/2)/(4d). (14) At low concentrations, assuming that h Ⰷ (d, D) and R' = 0, calculations similar to those in the previ ous case yield Δε = ε2πCV1 3 (1 – CV1 3 /2)/(1 – CV1 3 ), (15) which for a CM with a filler of conducting ferrite par ticles at mass content Сm = 40% (СV = 13%) elevates the effective dielectric permittivity by 3.5 times, rela tive to the permittivity of a dielectric CM matrix: ε = 3.5ε2. Since average layer thickness davg coincides with minimum thickness d of the layers between cubic par ticles with sides D (see expressions (2) and (5)) in SC packing of round particles with diameter D, allowing
The approaches elaborated earlier for estimating the magnetic permeability and dielectric permittivity of combined materials based on the properties of their individual components led to the expressions known as, e.g., the Lorentz, Rayleigh, Kondorskii– Odelevskii, Lichtenecker, Ollendorf, Bruggemann, Maxwell, Maxwell–Garnett, and Wagner mixing for mulas [8, 10–12, 16]). Many of the relations derived, e.g., for dielectric permittivity ε can also be used to find electric conductivity σ and magnetic permeability μ. This was noted long ago in comparing the formulas obtained independently by researchers for different classes of materials. On the other hand, it has been found that some of the models initially elaborated for limited conditions of application or with insufficiently rigorous substantiation under arbitrary assumptions sometimes work well beyond the relevant limits as well. In this section, we analyze the applicability of the different models described in the literature to RAMs we have developed on the basis of granulated ferrite powders in a dielectric binder. Because a discrete medium in the phenomenological approach is pre sented in the form of a continuous continuum with more or less uniformly distributed parameters, the considered model should describe both materials con sisting of ferrite granules 10–650 μm in size distributed throughout a dielectric medium and nanocomposites consisting of much smaller particles, since the EMR wavelength is in any case much greater than the scale of inhomogeneity of the medium for the range of radio frequencies (100 kHz–10 GHz, or even 30 GHz) that interest us. If the dimensions of the RAM coatings or the thickness of the studied samples approach that wavelength, it could also be important to allow for interference and diffraction. A schematic description of the twophase medium can be provided by, e.g., the models shown in Fig. 5: either (a) as a mixture consisting (according to Licht enecker) of particles of two components in the shape of rectangular prisms, taken in equal amounts per volume and distributed in a chessboardlike structure, or (b) as particles of inclusions 1 distributed in the matrix 2, the continuous medium of a dielectric binder. Each phase is described by the corresponding material parameters ε1, μ1, σ1, ε2, μ2, σ2. The average properties of a twophase medium can in the first case be considered using the example of electric conductivity, assuming the same probability for series and parallel connections [16]. If in one case
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(a) 1
2 1
2
2
1
Fig. 5. Schematic representation of a twocomponent (twophase) medium: (a) chessboard structure; (b) inclu sion of phase 1 into the matrix of phase 2.
the cells are first considered to be connected pairwise in series and then the pairs are connected in parallel, and vice versa in the other (first parallel and then in series), we obtain two estimates of the electric conductiv σ1 σ2 ity, lower σmin = 2 and upper σmax = (σ1 + σ2)/2. σ1 + σ2 Due to the statistical nature of the phase distribution, the average (effective) parameters of the medium are assumed to be equal to the geometric mean of these values, i.e., σ=
σ min σ max =
σ1 σ2 .
(16)
If the probabilities of the parallel (u) and in series (1 – u) connections and volume concentrations of the components p1 ≡ p and p2 = 1 – p differ, the effective conductivity is determined by the expression u
σ = ( p 1 σ 1 + p 2 σ 2 ) ( p 1 /σ 1 + p 2 /σ 2 )
u–1
.
(17)
When both types of connections are equally proba ble, u = 0.5; therefore σ=
( σ 1 σ 2 ) ( p 1 σ 1 + p 2 σ 2 )/ ( p 1 σ 2 + p 2 σ 1 ).
(18)
Similar formulas can be written for the effective values of magnetic permeability and dielectric permit tivity. The formula in the logarithmic form obtained by Lichtenecker for the magnetic permeability of a com posite consisting of equal components (two sorts of filler particles) is symmetrical with respect to the con tent and permeability of both materials: ln μ eff = p ln μ 1 + ( 1 – p ) ln μ 2 , (19) p (1 – p) or μ eff = μ 1 μ 2 and is generally met well in practice. Among the different mixing formulas being com pared, we also verified two relations that provide the effective value of the CM parameter as a simple weighted average quantity derived from the character istics of equal components, ε = ε1p + ε2(1 – p) (20) and obtained for the matrix model of the system (Fig. 2b) in the form of inclusions of the phase with dielectric permittivity ε1 and volume fraction p1 = p,
distributed throughout a medium with permittivity ε2. Let us assume that at such a ratio of the volumes of the two phases, one filler particle in the shape of a cube with side D is found in the matrix’s cubic region with linear sizes D + d (as in Fig. 2). The filler content (phase 1) in the system is p1 ≡ p = [D/(D + d)]3 (corre sponding to CV). Let us consider the selected region to be a system of three capacitors connected in series with plate area D2. One is filled with material having dielectric permittiv ity ε1 and thickness D, while the two others have dielectric permittivity ε2 and total thickness d. Along side these is a capacitor connected in parallel and filled with the dielectric having permittivity ε2 and plate area (D + d)2 – D2; the distance between them is D + d. Their total capacity is the same as the system’s, which in the effective medium approximation can be written as the capacity of a capacitor filled with a dielectric having effective permittivity ε: 2
2
C CM = ε ( D + d ) / ( D + d ) = D / [ d/ε 2 + D/ε 1 ] 2
2
+ ε 2 ( D + d ) [ 1 – D / ( D + d ) ],
(21)
In terms of the volume concentration of the filler, it follows that p ( ε1 – ε2 ) ε = ε 2 1 + 1/3 1/3 ε1 ( 1 – p ) + ε2 p p ( ε1 – ε2 ) . = ε 2 1 + 1/3 ε 1 – ( ε 1 – ε 2 )p
(22)
If the filler is conducting (e.g., Mn–Zn ferrite), the effective value of the medium’s dielectric permittivity can be found using (22) via the formal transition to the limit ε1 → ∞: ε = ε2[1 + p/(1 – p1/3)].
(23)
The authors of [9] solved a similar problem in studying the permittivity of polycrystalline ferrite con sisting of weakly conducting grains and intergrain boundaries with the properties of a dielectric. In their model, the ferrite material consists of cubic particles distributed in accordance with the packing in the form of a simple cubic lattice and acts as a set of capacitor plates in a dielectric medium with permittivity ε2. However, they incorrectly used the formula for a flat capacitor by assuming there were two interacting plates and a corresponding increase in electrical capacity due to the contribution from the mutual capacity during the transition in the limit to a single particle, which has an incomparably lower value of individual capacity, so the formula derived from their model, Δε = ε2p1/3/(1 – p1/3)
(24)
differs from formula (9) obtained in this work by hav ing no coefficient p1/3 and yields a somewhat overesti mated effective permittivity.
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With conducting inclusions, the effective dielectric permittivity of a CM at low concentrations of inclu sions can be found using expression (15). At high con centrations of inclusions, formulas (9), (23), and (24) produce similar results. Analysis of different mixing formulas (including those obtained by other authors) showed that the majority of them yield almost identical results at low values of filler concentration p and/or when there were slight differences between the permittivities of the ini tial components. In this work, model parameters were compared to experimental data on the dielectric permittivity and magnetic permeability of initial components and a CM with 12.5 vol % of 2000NMgrade granulated fer rite powder distributed throughout a silicone her metic. Inserting experimental values ε and μ of the individual CM components into the formulas of dif ferent models leads to satisfactory agreement between the calculated values of effective permittivity and those obtained only in some regions of the dispersion curve in experiments with CMs consisting of these phases, and the frequency ranges for the dielectric permittivity and magnetic permeability characterizing them often do not coincide. This is likely due to features of the electric and magnetic interactions between filler parti cles, resulting in differences between their depen dences on the frequency of electromagnetic vibrations and affecting the applicability of each model elabo rated under its own limitations. Since the granulated Mn–Zn ferrite powders that we used in our experiments had weak conductivity, we also analyzed the relations derived for systems with conducting inclusions. The results from calculating effective permittivity ε were in this case closest to the corresponding results from measurements. It should be remembered that a sintered massive sample of the highdensity ferrite can have parameters different from those of granules of the same grade (700NM or 2000NM) used as filler. Granules are assumed to be round or cubic, though their shape is actually far from regular. In addition, the structure of granules has yet to be thoroughly explored. For exam ple, it is thought that their individual particles are 2– 3 μm in size, while Xray phase analysis reveals the nanometer scale of coherent scattering regions. Even granules themselves should probably be considered as loose ferrite composites in a gas medium or porous polycrystals with intergrain layers having permittivities different from those of the grains. It would be logical first to use the effective medium approximation for the material of granules by estimating the concentration of their ferrite phase from the ratio of Xrays to appar ent density, and then to use the procedure a second time for granules in the dielectric matrix.
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ABSORPTION CAPABILITY OF A COMPOSITE RAM Studying and predicting the behavior of ferrite– dielectric CMs in electromagnetic fields in the micro wave range is an important task, since it allows us to study the properties of such combined systems, estab lish the mechanisms responsible for the loss of electro magnetic radiation in an absorbing medium, and solve problems of materials science and technology associ ated with the development of promising materials that have desireable electromagnetic characteristics. Like theoretically deriving the analytical form of depen dences using different models to describe empirical data, interpreting and approximating experimental dispersion spectra remains a difficult problem of great interest [17–22]. The complex values of magnetic permeability and dielectric permittivity, conductivity, and transmission coefficients are determined by radiophysical means. One of these involves determining the logarithmic reflection coefficient of an electromagnetic wave from a sample of material placed on a metallic plate (“mirror”) in the mode of measuring decays in trans mission. The reflected signal can be presented as part of the radiation transmitted through a layer of material twice: directly and after reflecting off the metallic plate in the opposite direction. According to the law of EMR absorption, in the geometry of our experiments P1/P0 = exp (–2αh), (25) where P0 is the intensity of the wave entering a sample, P1 is the intensity of the propagated wave, α is a linear factor (coefficient) for the absorption of the medium, and 2h is the double thickness of the sample. The value of the measured reflection coefficient contains information on the drop in the energy of the wave due to absorption and is usually a decimal loga rithm of the ratio of radiation fluxes, K ref = 10lg ( P 1 /P 0 ) (26) = – 10α ( 2h ) log ( e ) = – 8.686αh. Kref ≤ 0 when Kref = 0 corresponds to no absorption. On the other hand, we can write refraction coefficient n in the complex form n = n'+in'' =|n|e iδ = |n|(cosδ + isinδ), (27) where tanδ = sinδ/cosδ = n''/n', and express it in terms of the complex values of magnetic permeability and dielectric permittivity ε = ε' – iε'' and μ = μ' – iμ'' as n = ( εμ )
1/2
= [ ( μ' – iμ'' ) ( ε' – iε'' ) ]
1/2
1/2
= [ ( ε'μ' – ε''μ'' ) – i ( ε'μ'' + ε''μ' ) ] . After transformations, from (28) we obtain
(28)
n = (|ε||μ|)1/2ei(δ0 + 2kπ)/2, (29) where |ε| = (ε'2+ε''2)1/2, |μ| = (μ'2+μ''2)1/2, tan δ0 = ⎯ (ε'μ'' + ε''μ')/(ε'μ' – ε''μ''), k = 0, ±1.
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MORCHENKO
Such values of k mean that the solutions to the argument of the CM’s complex refraction coefficient differ from its main value δ0 by angle π. The value and sign of the tangent remain the same, but the signs before the imaginary and real parts of refraction change synchronously; therefore, 1/2 iδ
n = (ε μ) e
1/2 iδ0/2
= ±( ε μ ) e
(30)
The concepts and approaches considered in this work could be useful in predicting the properties of composite materials during their development, and in interpreting the results from experimental studies of such materials and systems. ACKNOWLEDGMENTS This work was supported by the Russian Founda tion for Basic Research, project no. 130301316.
1/2
= ± ( ε μ ) ( cos δ 0 /2 + i sin δ 0 /2 ). REFERENCES
It is easy to show that tan δ 0 = – ( ε'μ'' + ε''μ' ) / ( ε'μ' – ε''μ'' ) = – ( tan δ ε + tan δ μ )/ ( 1 – tan δ ε tan δ μ )
(31)
= – tan ( δ ε + δ μ ) , where tanδε = ε''/ε' and tanδμ = μ''/μ' are the tangents of the angles of dielectric and magnetic losses. From (29), we find with accuracy up to angle π that n' = ( ε μ )
1/2
cos δ = ± ( ε μ )
1/2
cos ( δ 0 /2 )
2
1/2
2
(32)
= ± ( ε μ ) [ 1 – tan ( δ 0 /4 ) ]/ [ 1 + tan ( δ 0 /4 ) ], n'' = ( ε μ )
1/2
sin δ = ± ( ε μ )
1/2
sin ( δ 0 /2 ) 2
1/2
(33)
= ± ( ε μ ) [ 2 tan ( δ 0 /4 ) ]/ [ 1 + tan ( δ 0 /4 ) ]. According to [23–25], the relationship between the linear absorption coefficient and the imaginary part of the complex refraction coefficient (sometimes referred to as the main absorption coefficient) can be written as n'' = cα/(2ω) = cα/(4πν) = αλ/(4π),
(34)
where c is the velocity of the electromagnetic wave in a vacuum, ω and ν are the circular and linear EMR fre quencies, respectively; and λ is the length of the elec tromagnetic wave in a vacuum. It follows from (26), (32), and (33) that Kref[dB] = –109.15νh(|ε||μ|)1/2sin(δ0/2)/c.
(35)
Knowing the experimental or calculated values of complex permeability, we can thus find its absorption coefficient α and estimate the value of Kref[dB] that characterizes the absorption of an existing composite material or one under development. CONCLUSIONS The possibility of using different models with effec tive medium approximations in the study and develop ment of electromagnetic material parameters for dif ferent types of radarabsorbing composite materials with different scales of their structural components was analyzed.
1. Lagar’kov, A.N. and Pogosyan, M.A., Vestn. Ross. Akad. Nauk, 2003, vol. 73, no. 9, p. 848. 2. Ufimtsev, P.L., Metod kraevykh voln v fizicheskoi teorii difraktsii (Boundary Waves Method in Physical Theory of Diffraction), Moscow: Sov. radio, 1962. 3. Federal’nyi zakon (Federal Law) On State Control on Securing Electromagnetic Compatibility of Technical Means, Dec. 1, 1999. 4. Men’shova, S.B. and Zyabirova, A.R., Mater. Vseros. Nauchnoprakt. konf. “Aktual’nye problemy nauki v Rossii” (Proc. AllRussian Sci.Pract. Conf. “Topical Scientific Problems in Russia”), Kuznetsk, 2008, vol. 4, pp. 49–61. 5. Shneiderman, Ya.A., Zarubezh. Radioelektron., 1969, no. 6. 6. Maizel’s, E.N. and Torgovanov, V.A., Zarubezh. Radio elektron., 1972, no. 7. 7. Andreev, V.G., Kostishyn, V.G., Podgornaya, S.V., Ver gazov, R.M., Bibikov, S.B., and Morchenko, A.T., Izv. Vyssh. Uchebn. Zaved. Mater. Elektron. Tekhn., 2010, no. 4, pp. 18–21. 8. Tekhnologiya proizvodstva materialov magnitoelektroniki (The Way to Produce Materials for Magnetoelectron ics), Letyuk, L.M., Ed., Moscow: Metallurgiya, 1994. 9. Antsiferov, V.N., Letyuk, L.M., Andreev, V.G., Kos tishyn, V.G., et al., Problemy poroshkovogo materia lovedeniya (Problems of Powder Materials Science), part 4: Tekhnologiya proizvodstva poroshkovykh ferri tovykh materialov (The Way to Produce Powdered Fer rite Materials), Yekaterinburg: Ural Branch RAS, 2004. 10. Mikhailin, Yu.A., Spetsial’nye polimernye kompozitsion nye materialy (Special Polymeric Composite Materi als), St. Petersburg: Izd. Nauchn. osnovy tekhnol., 2009. 11. Vinogradov, A.P., Elektrodinamika kompozitnykh mate rialov (Electrodynamics of Composite Materials), Kat senelenbaum, B.Z., Ed., Moscow: Editorial URSS, 2001. 12. Lagar’kov, A.N., Panina, L.V., and Sarychev, A.K., Zh. Eksp. Teor. Fiz., 1987, vol. 93, no. 1(7), pp. 215– 221. 13. Zlenko, V.A., Demydenko, M.H., and Protsenko, S.I., J. Nano Electron. Phys., 2013, vol. 5, no. 3, p. 030055. 14. Panina, L.V., Morchenko, A.T., Kozhitov, L.V., and Ryapolov, P.A., J. Nano Electron. Phys, 2013, vol. 5, no. 4, p. 04003. 15. Kostishyn, V.G., Vergazov, R.M., Andreev, V.G., Bibikov, S.B., Morchenko, A.T., Terent’ev, D.S., and
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MODELING THE STRUCTURE AND CONDITIONS OF THE ABSORPTION Prokof’ev, M.V., Tr. IX Mezhdunar. Konf. “Perspek tivnye tekhnologii, oborudovanie i analiticheskie sistemy dlya materialovedeniya i nanomaterialov” (Proc. 9th Int. Conf. “Promising Technologies, Equipment and Analytical Schemes for Material Science and Nanomaterials”), Astrakhan, 2012. 16. Polivanov, K.M., Ferromagnetiki (Ferrimagnetics), MoscowLeningrad: Gosenergoizdat, 1957. 17. Nikolaeva, E.V. and Kotov, L.N., Tez. 8i Vseros. nauch. konf. studentovradiofizikov (Proc. 8th AllRus sian Students Radiophysicists Conf.), St. Petersburg: V.A. Fock Institute of Physics, 2004. 18. Kotov, L.N. and Bazhukov, K.Yu., Radiotekh. Elek tron., 1999, vol. 4, no. 7, pp. 41–46. 19. Pokusin, D.N., Chukhlebov, E.A., and Zalesskii, M.Yu., Radiotekh. Elektron., 1991, vol. 36, no. 11, pp. 2085– 2091.
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20. Pokusin, D.N. and Romanov, A.M., Radiopromyshlen nost’, 2004, no. 2, pp. 160–165. 21. Zotov, I.S., The way to research electrodynamical char acteristics of composite material with regular structure, Extended Abstract of Cand. Sci. (Phys.Math.) Disserta tion, Chelyabinsk: Chelyabinsk State Univ., 2011. 22. Yakovlev, Yu.M. and Gendelev, S.Sh., Monokristally ferritov v radioelektronike (Ferrite Monocrystals in Radioelectronics), Moscow: Sov. radio, 1975. 23. Korolev, F.A., Teoreticheskaya optika (Theoretical Optics), Moscow: Vysshaya shkola, 1966. 24. Yavorskii, B.M. and Detlaf, A.A., Spravochnik po fizike (Handbook on Physics), Moscow: Nauka, 1974. 25. Fizicheskaya entsiklopediya (Physical Encyclopedia), Moscow: Bol’shaya rossiiskaya entsiklopediya, 1992, vol. 3.
BULLETIN OF THE RUSSIAN ACADEMY OF SCIENCES. PHYSICS
Translated by L. Mosina
Vol. 78
No. 11
2014
