The effective ellipsoid: a method for calculating the permittivity of composites with multilayer ellipsoids

1 Introduction

The effective permittivity and permeability occupy an important role in designing electromagnetic absorbing functional materials [1], [2], [3]. Because of the duality in the electromagnetic theory, formulas usually make no distinction between effective permittivity and effective permeability [4]. The effective medium theory (EMT) is widely used to predict the effective permittivity and permeability of a mixture. When the inclusions in the mixture are spherical, the most famous formula can be written in the following form [5], [6], [7]:

where ε_m, ε_i, and ε_eff are the permittivities of the matrix, inclusions, and mixture, respectively; f is the inclusion concentration; and v is a constant parameter. Equation (1) evolves to the Maxwell-Garnett equation, the Bruggeman equation, or the coherent potential approximation equation when v=0, 2, or 3, respectively. Furthermore, some analysis is extended to a mixture with ellipsoidal inclusions. When ellipsoids are oriented randomly, the effective permittivity of the mixture can be expressed by [8]

where ε_v=ε_m+v(ε_eff−ε_m), v is a constant parameter greatly depending on the inclusion concentration, and N_k (k=x, y, z) denotes the depolarization factors.

Along with the development of electromagnetic absorbing materials, more and more absorbents are multiphase and multilayer [9], [10], [11], [12], [13], [14]. This kind of absorbents may improve the electromagnetic wave absorbing ability as there is a good interface effect between inclusions and the matrix. There are a number of publications studying the effective transport and elastic parameters of a mixture with multilayer ellipsoids [15], [16], [17], [18], [19], [20]. In these publications, an effective ellipsoid that substitutes the original multilayer ellipsoid is introduced into the calculated procedure. The concept of the effective ellipsoid is very successful. Although there are not many reports studying the method of calculating the effective permittivity of a mixture with multilayer inclusions, it is may be an effective way to solve the problem by using the concept of the effective ellipsoid.

In this paper, a self-consistent approximation (SCA) is proposed on the basis of the Bruggeman’s analytical model [21]. In order to calculate the effective permittivity of a mixture with multilayer ellipsoids oriented randomly, the effective ellipsoid is introduced and the method for calculating the permittivity of the effective ellipsoid is presented. To verify the effectiveness of the SCA method, results calculated by the SCA method are compared to that calculated by existing theories about the effective permittivity of a mixture with uniphase or bilayer ellipsoids.

2 Theoretical considerations

For the inclusion, diameter is always much smaller than the electromagnetic wavelength, and the electromagnetic problem can be solved under static or quasi-static condition. Here, electromagnetic correlations among inclusions are not taken into account; i.e. the inclusion concentration is not very high. The next problem is finding the analytical solution for the multilayer ellipsoid in the electrostatic field. The solving details are shown as following.

2.1 SCA for a mixture with aligned multilayer ellipsoids

First, let’s investigate a mixture with bilayer ellipsoidal inclusions. When ellipsoids are aligned, as shown in Figure 1A, the mixture is anisotropic and the effective permittivity is different from each other in a different direction. Here, Bruggeman’s analytical model is selected, electromagnetic correlations among inclusions are neglected, and it is assumed that each inclusion is surrounded by the same effective medium, as shown in Figure 1B. ε₂ represents the permittivity of the kernel, ε₁ is the shell permittivity, and ε₀ is the permittivity of the host matrix.

Figure 1:

Scheme of a mixture with aligned bilayer ellipsoids (A) and the Bruggeman’s analytical model (B).

In Figure 1B, the ellipsoidal interfaces are confocal and they can be written by a specific u in the following equation:

(3)x2a2+u+y2b2+u+z2c2+u=1,

where a, b, and c are the semiaxes of the ellipsoidal kernel. Equation (3) represents the ellipsoidal boundary S₂ when u=0. Meanwhile, it is assumed that Eq. (3) represents S₁ or S₀ when u=t₁ or t₂, respectively. Here, introduce the electric potential φ and the electric field strength E=−∇φ. Because there is no charge source, Laplace’s equation is available. It is expressed as follows:

(4)∇2φ=0.

It is given that the electric field E₀ is applied in the x direction. The key to this problem is finding out the potential for the interior and exterior of the ellipsoid in Figure 1B. After neglecting higher-order terms, the potential for the exterior of the boundary S₀ is expressed as

(5)φout=−E0x.

Because all the ellipsoidal boundaries are confocal, the ellipsoidal coordinate system (ECS) [22] can be built up and used. Appendix gives some rules about the ECS. In the ECS, the following solution to Eq. (4) is obtained for Figure 1B:

(6)φ={φ0=A0φout+B0L(u,a)φout(t1<u≤t2)φ1=A1φout+B1L(u,a)φout(0<u≤t1)φ2=A2φout(−a2<u≤0),

where L(u,a) is defined as follows:

L(m,n)=∫m∞ds(s+n2)R(s)

and

R(s)=(a2+s)(b2+s)(c2+s).

A₀, A₁, A₂, B₀ and B₁ are constant parameters. The boundary condition of the electric field on the interfaces S₀, S₁ and S₂ is expressed as follows:

(7a)S0(u=t2): {φ0=φoutε0∂φ0∂n=εx∂φout∂n,

(7b)S1(u=t1): {φ1=φ0ε1∂φ1∂n=ε0∂φ0∂n,

and

(7c)S2(u=0): {φ2=φ1ε2∂φ2∂n=ε1∂φ1∂n.

It is necessary to point out that ε_x in Eq. (7a) is the effective permittivity in this principal direction. By substituting Eq. (6) into Eqs. (7a), (7b) and (7c), a linear system of equations can be built up in the matrix form as follows:

(8)[1−ε02ε0R(t2)−ε0L(t2,a)00001L(t2,a)0000ε0ε0L(t1,a)−2ε0R(t1)−ε12ε1R(t1)−ε1L(t1,a)001L(t1,a)−1−L(t1,a)0000ε1ε1L(0,a)−2ε1R(0)−ε20001L(0,a)−1](εxA0B0A1B1A2)=(010000).

By solving Eq. (8), the effective permittivity in the x direction ε_x of the mixture can be derived, along with the electric potential. And the effective permittivity in the y or z direction can also be obtained through the same procedure.

Figure 2 shows the model of a mixture with aligned three-layer ellipsoids. Similar to Eq. (6), the potential solution is written as follows.

Figure 2:

Scheme of the Bruggeman’s analytical model for a mixture with aligned three-layer ellipsoids.

In Figure 2, a, b, and c are the semi-axes of the ellipsoidal kernel. Equation (3) represents the ellipsoidal boundary S₃ when u=0, and it is assumed that Eq. (3) represents S₂, S₁, or S₀ when u=t₁, t₂, or t₃, respectively.

(9)φ={φ0=A0φout+B0L(u,a)φout(t2<u≤t3)φ1=A1φout+B1L(u,a)φout(t1<u≤t2)φ2=A2φout+B2L(u,a)φout(0<u≤t1)φ3=A3φout(−a2<u≤0),

where A₀, A₁, A₂, A₃, B₀, B₁, and B₂ are constant parameters, and by applying the boundary conditions of the electric field on the interfaces S₀, S₁, S₂ and S₃, which are similar to Eqs. (7a), (7b) and (7c), a linear system of equations can also be built up for the model in Figure 2, shown as follows:

(10)[1−ε02ε0R(t3)−ε0L(t3,a)00001L(t3,a)0000ε0ε0L(t2,a)−2ε0R(t2)−ε12ε1R(t2)−ε1L(t2,a)001L(t2,a)−1−L(t2,a)0000ε1ε1L(t1,a)−2ε1R(t1)−ε20001L(t1,a)−100000ε2000001 000000002ε2R(t1)−ε2L(t1,a)0−L(t1,a)0ε2L(0,a)−2ε2R(0)−ε3L(0,a)−1](εxA0B0A1B1A2B2A3)=(01000000).

The effective permittivity in the x direction of a mixture with aligned three-layer ellipsoids can be obtained by solving Eq. (10). There is a similar procedure for calculating the effective permittivity in the y or z direction.

For a mixture with aligned ellipsoids of arbitrary number layers, a potential similar to Eq. (6) can be written and a linear system of equations similar to Eq. (8) can be built up. The effective permittivity in the principal direction can be derived from the equation system.

2.2 SCA for a mixture with multilayer ellipsoids oriented randomly

Actually, ellipsoidal inclusions in a mixture are always oriented randomly, as shown in Figure 3.

Figure 3:

Scheme of a mixture with bilayer ellipsoids oriented randomly.

In this case, the mixture is regarded to be homogenous and isotropic from a macro point of view, because it contains a great mass of inclusions that are distributed homogeneously and oriented randomly. In order to calculate effective permittivity, the effective ellipsoid is introduced to substitute the original multilayer ellipsoid, as shown in Figure 4.

Figure 4:

Scheme of a bilayer ellipsoid (A) and its effective ellipsoid (B).

For the effective ellipsoid in Figure 4B, the potential expression similar to Eq. (6) is written as follows:

(11)φ′={A′0φout+B′0L(u,a)φout(t1<u≤t2)A′1φout(−a2<u≤t1).

And a linear system of equations is built up by combining Eq. (11) with the boundary conditions of the electric field, shown as follows:

(12)(1−ε02ε0R(t2)−ε0L(t2,a)001L(t2,a)00ε0ε0L(t1,a)−2ε0R(t1)−εx,ie01L(t1,a)−1)(εxA0′B0′A1′)=(0100),

where the effective permittivity in the x direction ε_x is derived from Eq. (8) and ε_x,ie denotes the permittivity of the effective ellipsoid in the x direction. Using a similar method, the permittivity of the effective ellipsoid in the y or z direction can be obtained as well. There are four features for this method to calculate the permittivity of the effective ellipsoid: (I) For a mixture with bilayer ellipsoids, when t₁→0, the calculated permittivity of the effective ellipsoid approaches the permittivity of the kernel as a limit; on the contrary, if t₁ is much greater than the size of the kernel, the calculated permittivity of the effective ellipsoid approaches the permittivity of the shell as a limit. (II) For different t₂ values, i.e. different inclusion concentrations in the mixture, the calculated permittivity of the effective ellipsoid in the principal direction still remains the same. (III) If the permittivity of each layer is the same with each other, the calculated permittivity is also the same with the permittivity of any layer. (IV) The electric potential calculated by Eq. (12) for the exterior of the effective ellipsoid is the same with that for the exterior of the original bilayer ellipsoid. All of these features meet the theoretical demands of an effective ellipsoid. This method is appropriate to determine the effective ellipsoid of an ellipsoid that is constituted of arbitrary number of layers.

After obtaining the permittivity of the effective ellipsoid in the three principal directions, the expression can be obtained for calculating the effective permittivity of a mixture with multilayer ellipsoids oriented randomly by combining the Maxwell-Garnett formula [8], shown as follows:

(13)εeff=εm+εmf∑i=x,y,zεi,ie−εmεm+Ni(εi,ie−εm)3−f∑i=x,y,zNi(εi,ie−εm)ε0+Ni(εi,ie−εm),

where

Ni=abc2L(0,ai) (i=x,y,z)

are depolarization factors that are widely used. It is necessary to point out that the Maxwell-Garnett formula is valid when the mixture is dilute. Usually, the inclusion concentration is <10%. Thus, further research is needed into calculating the effective permittivity of a mixture with a high inclusion concentration.

3 Results and discussions

The effective permittivities are calculated by the above SCA method for mixtures with different bilayer ellipsoid concentrations. Here, it is assumed that the permittivity of the host matrix is 2, the bilayer ellipsoidal inclusions are composed of the kernel with ε₂=20 and the shell with ε₁=200, and the aspect ratio of ellipsoids remains a=b=c/2. In these calculations, five different configurations are selected, which correspond to different ratios between the volume of the kernel and that of the shell. Results are plotted in Figure 5. In the derivative process of the SCA method, the correlation between inclusions is neglected and the mixture is decomposed to a single ellipsoidal model. So this method should be reevaluated and improved when the inclusion concentration is very high.

Figure 5:

The effective permittivity of a mixture with aligned bilayer ellipsoids in the x or y direction (A) and z direction (B), along with that of a mixture with bilayer ellipsoids oriented randomly (C) for different inclusion concentrations.

The shape of ellipsoids has a significant influence on the effective permittivity of a mixture. It is assumed that ellipsoids are axisymmetric, and the shape of an ellipsoid is specified by the aspect ratio a/c of the kernel. The ellipsoid is penny shaped as a/c is greater than 1, while the ellipsoid is needle shaped as a/c is smaller than 1 and spherical as a/c equals to 1. It is assumed that the permittivity of the host matrix is 2, and the bilayer ellipsoidal inclusion is composed of the kernel ε₂=20 and the shell ε₁=200. In the calculating processes, the kernel volume remains a half of the whole ellipsoid. Calculations of the effective permittivity are performed by changing the aspect ratio a/c under nine different inclusion concentrations from 0.1 to 0.9. Calculated results are plotted in Figure 6. The horizontal values represent denary logarithms of a/c.

Figure 6:

The effective permittivity of a mixture with aligned bilayer ellipsoids in the x or y direction (A) and z direction (B), along with that of a mixture with bilayer ellipsoids oriented randomly (C) for different aspect ratios.

As shown in Figure 6, the effective permittivity in the principal direction increases with the susceptibility enhancing and vice versa, as shown in Figure 6A and B; but for the mixture with ellipsoids oriented randomly, the effective permittivity is smallest when the ellipsoids are spherical, as shown in Figure 6C.

Figure 7 shows the results calculated by the SCA method and by the EMT formula of Eq. (2) for a mixture with aligned uniphase ellipsoids. It reveals that the results obtained by the SCA method are very close to that by the EMT formula of Eq. (2). There is just a small difference between them when the inclusions are penny shaped or needle shaped, and they are the same completely when the inclusions are spherical. For mixtures with ellipsoidal and not spherical inclusion, the results obtained by the SCA method are a little greater than that by the EMT formula of Eq. (2) in the direction of strong susceptibility, and vice versa.

Figure 7:

The effective permittivity of mixtures with aligned uniphase ellipsoids in the x direction (A) and z direction (B) is calculated by the SCA method (solid line) and the EMT formula (dash line) for different inclusion concentrations. The permittivity of inclusions is 20 and that of the host matrix is 2.

There are not many researches that study on calculating the effective permittivity of a mixture with multilayer ellipsoids except for the method developed by Jones and Friedman (JF) [23]. Here, the calculated effective permittivity of mixtures with bilayer ellipsoids by the SCA method is compared to that by the JF method. It is still assumed that the permittivity of the host matrix is 2, the bilayer ellipsoidal inclusion is composed of the kernel ε₂=20 and the shell ε₁=200, and the kernel volume remains a half of the whole ellipsoid. The results are plotted in Figure 8. It is found that there is very little difference between them.

Figure 8:

The effective permittivity of mixtures with bilayer ellipsoids oriented randomly calculated by the SCA method (□) and the method developed by Jones and Friedman (∗).

The SCA method can be used to calculate the effective permittivity of a mixture with multilayer ellipsoids, and on some classic problems, such as calculating the effective permittivity of a mixture with uniphase or bilayer ellipsoids, the SCA results show good coincidence with that by existing theories. Further research into calculating the effective electromagnetic parameters of a mixture with a high inclusion concentration or the complex electromagnetic parameters in the dynamical field by the SCA method is still on the way.

4 Conclusions

In this paper, an SCA on the basis of Bruggeman’s analytical model is proposed to calculate the effective permittivity of a mixture with multilayer ellipsoids. When multilayer ellipsoids are aligned in the mixture, the effective permittivity in the principle direction can be derived from a linear system of equations, which are built using the boundary condition of the electric field on the confocal ellipsoidal interface. While multilayer ellipsoids are oriented randomly in the mixture, an effective ellipsoid is introduced to substitute the original multilayer ellipsoid, and the permittivity of the effective ellipsoid is derived by jointly solving the two linear systems of equations for the situation of the original multilayer ellipsoid and that of the effective ellipsoid, then the effective permittivity of the mixture can be calculated by the existing Maxwell-Garnett formula. Results of comparisons reveal that the SCA method has good agreement with existing theories.

Further research into calculating the effective electromagnetic parameters of a mixture with a high inclusion concentration or the complex electromagnetic parameters in the dynamical field by the SCA method is still on the way.