A New Numerical-Homogenization Method to Predict the Effective Permittivity of Composite Materials

Abstract

The effective permittivity of composite materials depends highly on the geometry, the arrangement, and the permittivity of each component. This research proposes a new numerical method that takes into account those dependencies through assemblies of virtual capacitors (electrical circuit). Then, the effective permittivity is calculated from the equivalent capacity of the suggested circuits. The new presented method delimits the effective permittivity of heterogeneous materials through two obtained different summation expressions. This new method was applied to some inclusions to investigate its validity by comparing its results to the existing models and the results of the finite element method. The new results are shown in good agreement with the literature. Moreover, the presented model takes less time in simulation to estimate the effective permittivity compared to the finite element method. The new model can be applied to all kinds of composite materials including the ones that involve multiple phases and complex geometries.

Introduction

Owing to the important role played by the effective permittivity in physics and other different science fields like geology [1, 2], biology [3,4,5], hydrology [6] etc., considerable research efforts have been devoted for studying and predicting the effective permittivity of heterogeneous structures as well as other physical properties. Generally, the heterogeneous biphasic structures show an effective permittivity's value that is between those of the constituents (at least for non lossy mixtures [7]). The first suggested rules known to delimit the effective permittivity using the volume fraction of the constituents were the ones established by Wiener in 1912; in this case, the effective permittivity bounds are given by the following expressions [8].

$$ \varepsilon_{{W,{\text{par}}}} = \left( {1 - \phi_{{{\text{inc}}}} } \right) \cdot \varepsilon_{{{\text{mat}}}} + \phi_{{{\text{inc}}}} \cdot \varepsilon_{{{\text{inc}}}} $$
(1)
$$ \varepsilon_{{W,{\text{ser}}}} = \left( {\frac{{1 - \phi_{{{\text{inc}}}} }}{{\varepsilon_{{{\text{mat}}}} }} + \frac{{\phi_{{{\text{inc}}}} }}{{\varepsilon_{{{\text{inc}}}} }}} \right)^{ - 1} $$
(2)

where \(\phi_{{{\text{inc}}}}\), εinc, \(\phi_{{{\text{mat}}}}\), and εmat denote, respectively, the volume fraction of the inclusion, the relative permittivity of the inclusion, the volume fraction of the host matrix, and the relative permittivity of the host matrix. The Eq. (1) corresponds to the exact permittivity of two capacitors mounted in parallel to form one capacitor as shown in the Fig. 1a, and the Eq. (2) gives the exact permittivity of two capacitors mounted in series which corresponds to the Fig. 1b.

Fig. 1
figure1

Illustration of capacitors mounted in parallel at left and series at right

In 1962, based on a variational treatment of functional energy for mixtures that involve particles distributed in 3D host matrix, Hashin and Shtrikman [9] suggested other bounds that are narrower than those suggested by Wiener. For the 3D case, the bounds read as follows:

$$ \varepsilon_{HS,1} = \varepsilon_{{{\text{mat}}}} + \frac{{\phi_{{{\text{inc}}}} }}{{\left( {\frac{1}{{\varepsilon_{{{\text{inc}}}} - \varepsilon_{{{\text{mat}}}} }}} \right) + \left( {\frac{{1 - \phi_{{{\text{inc}}}} }}{{3 \cdot \varepsilon_{{{\text{mat}}}} }}} \right)}} $$
(3)
$$ \varepsilon_{HS,2} = \varepsilon_{{{\text{inc}}}} + \frac{{1 - \phi_{{{\text{inc}}}} }}{{\left( {\frac{1}{{\varepsilon_{{{\text{mat}}}} - \varepsilon_{{{\text{inc}}}} }}} \right) + \left( {\frac{{\phi_{{{\text{inc}}}} }}{{3 \cdot \varepsilon_{{{\text{inc}}}} }}} \right)}} $$
(4)

After, Krischer and Kast suggested an expression for the heat conductivity [10] that when it is applied to the relative effective permittivity, the expression becomes [11]:

$$ \varepsilon_{{{\text{Krischer}}}} = \left( {\frac{1 - S}{{\varepsilon_{{W,{\text{par}}}} }} + \frac{S}{{\varepsilon_{{W,{\text{ser}}}} }}} \right)^{ - 1} $$
(5)

where S denotes the structure parameter and which can be defined by fitting experiment data. This parameter depends highly on the shape and the arrangement of the inclusions and reveals the behavior of a given system that can be one of the following cases:

  • A system of capacitors mounted in series (S = 1).

  • A system of capacitors mounted in parallel (S = 0).

  • A system that has a share in parallel and a remaining part in series (0 < S < 1).

Mathematically speaking, the Eq. (5) is a weighted harmonic mean of the two Wiener bounds. Other means were also mentioned in the literature [12]; more precisely, the arithmetic weighted mean and the geometric weighted mean. The simplest form of the latter (S = 1/2) for equal weights in the case of two phases is used to estimate the effective permittivity of lamellar composites [13].

Another interesting model, which is used in remote sensing and classified as power-law models, is that one suggested by Looyenga [14]. For this model, the effective permittivity of mixtures reads as follows:

$$ \varepsilon_{{{\text{Lyng}}}}^{1/3} = \left( {1 - \phi_{{{\text{inc}}}} } \right) \cdot \varepsilon_{{{\text{mat}}}}^{1/3} + \phi_{{{\text{inc}}}} \cdot \varepsilon_{{{\text{inc}}}}^{1/3} $$
(6)

The estimation of the effective permittivity using capacitors is still used in many studies, not so far, Patil et al. [15] calculated the effective permittivity based on the resulting capacitance of an appropriate circuit suggested for a spherical inclusion. As described by the authors, the model can be extended to other types of inclusions, but it seems to be a hard task even impossible to solve through analytic methods especially, for inclusions with non regular geometries.

Since the effective permittivity of mixtures plays a major role in many research areas, the validity and the efficiency of the most known related models have been investigated. For example: Araujo et al. [16] classified many models into three groups, and they found that only few models (notably, the ones that involve fitting or structure parameters) can reproduce the experimental results in the "quasi-static limit"; but none of them was able to represent the experiment data for composites with small inclusions (10 nm). So, they highly advised to establish a new model that takes into account the effect that occurs at the interface of ceramic/polymer composite. In a study related to the scattering of light by inhomogeneous particles, Kolokolova and Gustafson [17] recommended that the effective medium theories should be used carefully when estimating the color of the scattered light or when interpreting the observations linked to the cosmic dust in the visual. Also, they concluded that none of the discussed effective medium theories worked well in the back scattering domain or for some specific scattering angles. At last but not least, in a study related to the unfrozen water content, He et al. [18] evaluated five composite dielectric mixing rules and checked their applicability in the science of frozen soil. They concluded that the Sihvola concrete and confocal models [19] are the most appropriate ones when compared to the law and sphere models.

Concerning random composites, Karkkainen et al. [20] reported results of an extensive numerical analysis of electromagnetic fields in random dielectric materials. They tested how well the existing available models are valid. Throughout thousands of finite difference time domain (FDTD) simulations, they agreed that none of the checked mixing models could offer the absolute truth for a certain volume fraction because in random composite materials; all values between the bounds of Wiener are possible. In relation to the random composite too, but from the percolation theory’s point of view, Myroshnychenko and Brosseau [21] tested the McLachlan two exponent phenomenological percolation equation (TEPPE) [22, 23] by comparing its prediction of the effective permittivity to those obtained by simulating systems of overlapping disks; they found that larger discrepancies occurred when the degree of impenetrability (λ) was lower and the imaginary part of the disk permittivity (εinc″) was higher. They explained the obtained results by pointing to the various degree of disk aggregation present in the equilibrium distributions. In a nutshell, the applicability and the validity of a given permittivity model are limited to a restricted number of systems.

The second way that can be followed to study the response of heterogeneous structures is the one involving the use of numerical methods such as finite difference time domain method (FDTD) [24,25,26,27] or finite element method (FEM) [28,29,30]. The latter was used in this paper to compare the results of the present model. In fact, the FEM has been widely used and discussed in several works even if it is known for its consumption of memory and the time taken to solve the Laplace’ s equation:

$$ \nabla \left( {\varepsilon_{0} \varepsilon_{r} } \right)\nabla V = 0 $$
(7)

where εr, V and ε0 are, respectively, the local relative permittivity, the local potential, and (ε0 = 8.854187817·10−12 F/m) the vacuum permittivity constant (Fig. 2).

Fig. 2
figure2

Homogenization of a multiphase system consisting of different particles of permittivity \(\varepsilon_{{{\text{inc}}_{i} }}\) and randomly placed in a host matrix of permittivity εmat

As described in the Fig. 3, the heterogeneous structure is placed in an electric field created by a potential difference (V1 − V2) applied on the top and on the bottom faces of the heterogeneous structure; the remaining faces obey to the condition \(\left( {\frac{\partial V}{{\partial n}} = 0} \right)\). Once the system has been set and simulated, the next step is to calculate numerically (through the computed potential values) the electrostatic energy We that is given by the following expression:

$$ W_{e} = \frac{1}{2}\varepsilon_{0} \int {\int {\int {\varepsilon \left( {x,y,z} \right)\left[ {\left( {\frac{\partial V}{{\partial x}}} \right)^{2} + \left( {\frac{\partial V}{{\partial y}}} \right)^{2} + \left( {\frac{\partial V}{{\partial z}}} \right)^{2} } \right]{\text{d}}x{\text{d}}y{\text{d}}z} } } $$
(8)
Fig. 3
figure3

Random inclusion in a host matrix

Then, the effective permittivity is evaluated using the Eq. (9) that corresponds to the energy of a cuboid capacitor whereabouts a potential difference (V1 − V2) is applied in the y-axis direction.

$$ W_{e} = \tfrac{1}{2}\varepsilon_{0} \varepsilon_{{{\text{eff}}}} \left( {\tfrac{WD}{H}} \right)\left( {V_{1} - V_{2} } \right)^{2} $$
(9)

where (W), (H), and (D) denote respectively, the width, the height, and the depth of the capacitor.

In this work, the new model is introduced and the method used to obtain the two new bounds is exhibited. Then, the obtained results are compared for three different inclusions (cube, sphere, and cylinder) and for four different values of the contrast (k = εinc/εmat). In a separate section, the rapidity of this method is checked and compared to the finite element method’s rapidity. At the end, a conclusion summarizes the paper.

Methodology

The main idea of this method is to subdivide the whole system (capacitor) into small tiny virtual capacitors (Fig. 4). Accordingly, the one need to perform three subdivisions. The first step is to cut the initial structure into slices. Then, the second step is to subdivide the obtained slices again to get smaller columns. Finally, the last step is subdividing those columns to get tiny cuboids (capacitors).

Fig. 4
figure4

Subdividing the whole system into small capacitors Cijk

Depending on which side is subdivided first and which one is subdivided later, the one could obtain different expressions for the effective permittivity. The subscripts (W), (H), and (D) of a given effective permittivity expression, indicates the order of the subdivisions. For instance, the effective permittivity εDWH is obtained by subdividing the depth (D) in the first time, then the width (W), and at the end the height (H).

The present model will be helpful when estimating the effective permittivity of any system that involves few or many different inclusions including the ones with random shapes (Figs. 2, 3) and for which is difficult to estimate the effective permittivity's value using the majority of the classical models. Thus, this new model will provide, in a quick way, more guidance to engineers willing to design new systems. Besides, the new method can be extended and applied easily to others physical properties such as thermal conductivity.

First Bound of the Effective Permittivity (εDWH)

To obtain the first bound of the effective permittivity (εDWH), it is more convenient to use the Fig. 5 that illustrates an inclusion of permittivity (εinc) and volume fraction \((\phi_{{{\text{inc}}}} )\) placed in a host matrix of permittivity (εmat) and volume fraction \((\phi_{{{\text{mat}}}} )\). An electric field was created by a difference of potential (V1 − V2) applied on top and bottom faces of the heterogeneous structure. The remaining faces obey to the condition \((\partial V/\partial n = 0)\). In the Fig. 5a, the initial cuboid is substituted by \((C_{{N_{z} }} )\) cuboids (slices) obtained by subdividing the depth (D) (Nz) times. These cuboids keep the same value of the width (W) and the height (H) of the initial heterogeneous structure and have a new depth value that is equal to (D/Nz). So, they have a volume (Vα) which is equal to (VT/Nz). Where (VT) denotes the total volume of the initial cuboid (capacitor).

Fig. 5
figure5

Domain's subdivision to (NxNyNz) cuboids through DWH procedure

Assuming that these new cuboids (slices) are separable and taking into account the disposition of these cuboids and the direction of the electric field, the viewer could easily consider that these slices (capacitors) are mounted in parallel (Fig. 5a). So, the equivalent effective permittivity of the initial structure (Fig. 5a) could be written as follows:

$$ \varepsilon_{{{\text{DWH}}}} = \sum\limits_{\alpha = 1}^{{N_{z} }} {\varphi_{\alpha } \varepsilon_{\alpha } } $$
(10)

where \((\varphi_{\alpha } )\) denotes the volume fraction of the cuboid (Cα), and (εα) its permittivity.

Knowing that the cuboids (Cα) have the same volume value and that exist (Nz) cuboids, the volume fraction \((\varphi_{\alpha } )\) of the cuboid (Cα) could be written as follows:

$$ \varphi_{\alpha } = \frac{{V_{\alpha } }}{{V_{T} }} = \frac{{(V_{T} /N_{z} )}}{{V_{T} }} = \frac{1}{{N_{z} }} $$
(11)

where (VT) denotes the volume of the initial main capacitor and (Nz) the number of the subdivisions performed along the depth.

The first subdivision (Fig. 5a) led to new slices Cα for which the effective permittivity εα depends on its components. The next step (Fig. 5b) is to subdivide each cuboid Cα to other small ones by subdividing uniformly this time the width (W) (Nx) times. So, each cuboid (Cα) of volume fraction \(\varphi_{\alpha }\) and a permittivity εα are subdivided (Nx) times to Cαβ cuboids of volume fraction \(\varphi_{\alpha \beta }\) and partial effective permittivity εαβ.

Now, if a person assumes that these new cuboids (Cαβ) are separable and takes into account the disposition of these new cuboids (Cαβ) regarding the direction of the electric field (Fig. 5b), he or she will consider easily that these new cuboids (capacitors) are mounted in parallel. So, the equivalent effective permittivity (εα) of each cuboid (Cα) (Fig. 5b) could be written as follows:

$$ \varepsilon_{\alpha } = \sum\limits_{\beta = 1}^{{N_{x} }} {\varphi_{\alpha \beta } \cdot \varepsilon_{\alpha \beta } } $$
(12)

where (εαβ) is the equivalent permittivity of the cuboid (Cαβ), and \((\varphi_{\alpha \beta } )\) its volume fraction evaluated in relation to the volume of the cuboid (Cα).

Knowing that the cuboids (Cαβ) have the same volume value and that exist Nx cuboids for each cuboid (Cα), the volume fraction \((\varphi_{\alpha \beta } )\) of the cuboid (Cαβ) evaluated in relation to the volume of the cuboid (Cα) could be written as follows:

$$ \varphi_{\alpha \beta } = \frac{{V_{\alpha \beta } }}{{V_{\alpha } }} = \frac{{(V_{\alpha } /N_{x} )}}{{V_{\alpha } }} = \frac{1}{{N_{x} }} $$
(13)

Now, the next step is to subdivide each obtained cuboid (Cαβ) to other ones that are smaller by subdividing the height H in this case (Ny) times (Fig. 5c). Each tiny obtained cuboid (Cαβγ) is characterized by a volume fraction \((\varphi_{\alpha \beta \gamma } )\) and a permittivity (εαβγ) taken at its center.

Assuming this time that these new tiny cuboids (Cαβ\) are separable and keeping in mind the disposition of these new tiny cuboids (Cαβγ) and the direction of the electric field (Fig. 5c), it will be easy to consider that these new cuboids (capacitors) are mounted in series. So, the equivalent effective permittivity (εαβ) of each cuboid (Cαβ) (Fig. 5c) could be written as follows:

$$ \varepsilon_{\alpha \beta }^{ - 1} = \sum\limits_{\gamma = 1}^{{N_{y} }} {\frac{{\varphi_{\alpha \beta \gamma } }}{{\varepsilon_{\alpha \beta \gamma } }}} $$
(14)

Each cuboid (Cαβ) was uniformly subdivided (Ny) times. So, the volume fraction \((\varphi_{\alpha \beta \gamma } )\) of the cuboid (Cαβγ) evaluated in relation to the volume of the cuboid (Cαβ) can be written as follows:

$$ \varphi_{\alpha \beta \gamma } = \frac{{V_{\alpha \beta \gamma } }}{{V_{\alpha \beta } }} = \frac{{(V_{\alpha \beta } /N_{y} )}}{{V_{\alpha \beta } }} = \frac{1}{{N_{y} }} $$
(15)

Finally, using the Eqs. (1015), the effective permittivity (εDWH) can be written as follows:

$$ \varepsilon_{{{\text{DWH}}}} = \frac{{N_{y} }}{{N_{x} N_{z} }}\sum\limits_{\alpha = 1}^{{N_{z} }} {\sum\limits_{\beta = 1}^{{N_{x} }} {\left( {\sum\limits_{\gamma = 1}^{{N_{y} }} {\left( {\varepsilon_{\alpha \beta \gamma } } \right)^{ - 1} } } \right)^{ - 1} } } $$
(16)

where (εαβγ) denotes the permittivity taken at the center of the small cuboid (Cαβγ) E.g. If the permittivity at the center of the small cuboid is equal to εinc, then the permittivity of the whole small cuboid is set to εinc, else if the permittivity at the center of the small cuboid is equal to εmat, then the permittivity of the whole small cuboid is set to εmat.

For example, when (Nx = Ny = Nz = 8), the equivalent electrical circuit corresponding to the relative permittivity (εDWH) is shown in the Fig. 6. The first subdivision, which was performed along the z-axis, leads to parallel slices (capacitors mounted in parallel) as is shown in the top of the Fig. 6. The second subdivision leads to cuboids mounted in parallel, and the last slicing to small tiny cuboids mounted in series. Then, each capacitor, from (c1) to the capacitor \((c_{{N_{z} }} )\), has a different capacitance circuit. The equivalent circuit of the capacitor (cα) of the slice (Cα) is given as an example in the bottom of the Fig. 6. In that figure, (cα, β, γ) denotes the capacitance of the cuboid (Cαβγ) of permittivity (εαβγ) taken at its center.

Fig. 6
figure6

The equivalent capacitance circuit corresponding to the relative permittivity (εDHW) for the example (Nx = Ny = Nz = 8)

Second Bound of the Effective Permittivity (εDHW)

The bound (εDHW) is obtained by changing the sequence of the two last steps of the bound (εDWH). In other words, the first step to do here is similar to the first step of the previous bound. So, the expression of the effective permittivity using the parameters of the Fig. 7a can be rewritten as follows:

$$ \varepsilon_{{{\text{DHW}}}} = \sum\limits_{\delta = 1}^{{N_{z} }} {\varphi_{\delta } \varepsilon_{\delta } } $$
(17)
Fig. 7
figure7

Domain's subdivision to (NxNyNz) cuboids through DHW method

where (εδ) denotes the equivalent permittivity of the cuboid (Cδ), and \((\varphi_{\delta } )\) its volume fraction given by the following expression:

$$ \varphi_{\delta } = \frac{{V_{\delta } }}{{V_{T} }} = \frac{{(V_{T} /N_{z} )}}{{V_{T} }} = \frac{1}{{N_{z} }} $$
(18)

Now, to evaluate the partial permittivity (εδ) of the cuboid (Cδ), it is important to focus on the second step described in the Fig. 7b. The height (H) of the cuboid (Cδ) is subdivided (Ny) times. This subdivision led to obtain smaller cuboids that are assumed to be separable capacitors. So, by taking into account their disposition and the direction of the electric field, the one could easily consider that these capacitors (Cδ,ζ) are mounted in series and the equivalent permittivity of the cuboid (Cδ) Fig. 7b in this case can be written as follows:

$$ \varepsilon_{\delta }^{ - 1} = \sum\limits_{\zeta = 1}^{{N_{y} }} {\frac{{\varphi_{\delta \zeta } }}{{\varepsilon_{\delta \zeta } }}} $$
(19)

where (εδζ) denotes the equivalent permittivity of the cuboid (Cδζ), and \((\varphi_{\delta \zeta } )\) its volume fraction given by the following expression:

$$ \varphi_{\delta \zeta } = \frac{{V_{\delta \zeta } }}{{V_{\delta } }} = \frac{{(V_{\delta } /N_{y} )}}{{V_{\delta } }} = \frac{1}{{N_{y} }} $$
(20)

In order to obtain tiny cuboids (Cδζη) that have the same width (Fig. 7c), the last subdivision has to be done uniformly. So, each tiny obtained cuboid (Cδζη) has a width equals to (W/Nx).

Taking into consideration, the disposition of these new tiny cuboids and the direction of the electric field, and assuming that these cuboids (Cδζη) are separable, the one could easily consider that these tiny cuboids in this case (Fig. 7c) are mounted in parallel, and the equivalent permittivity of the cuboid (Cδζ) can be written as follows:

$$ \varepsilon_{\delta \zeta } = \sum\limits_{\eta = 1}^{{N_{x} }} {\varphi_{\delta \zeta \eta } \cdot \varepsilon_{\delta \zeta \eta } } $$
(21)

Each cuboid (Cδζ) was uniformly subdivided (Nx) times. So, the volume fraction \((\varphi_{\delta \zeta \eta } )\) of the cuboid (Cδζη) evaluated in relation to the volume of the cuboid (Cδζ) could be written as follows:

$$ \varphi_{\delta \zeta \eta } = \frac{{V_{\delta \zeta \eta } }}{{V_{\delta \zeta } }} = \frac{{(V_{\delta \zeta } /N_{x} )}}{{V_{\delta \zeta } }} = \frac{1}{{N_{x} }} $$
(22)

where (Vδζη), (Vδζ), and (εδζη) denote respectively, the volume of the cuboid (Cδζη), the volume of the cuboid (Cδζ), and the permittivity taken at the center of the tiny cuboid (Cδζη).

Finally, using the Eqs. (1722), the bound εDHW of the effective permittivity can be written as follows:

$$ \varepsilon_{{{\text{DHW}}}} = \frac{{N_{y} }}{{N_{x} N_{z} }}\sum\limits_{\delta = 1}^{{N_{z} }} {\left( {\sum\limits_{\zeta = 1}^{{N_{y} }} {\left( {\sum\limits_{\eta = 1}^{{N_{x} }} {\varepsilon_{\delta \zeta \eta } } } \right)^{ - 1} } } \right)^{ - 1} } $$
(23)

where (εδζη) denotes the permittivity taken at the center of the small cuboid (Cδζη). E.g. If the permittivity at the center of the small cuboid is equal to εinc, then the permittivity of the whole small cuboid is set to εinc, else if the permittivity at the center of the small cuboid is equal to εmat, then the permittivity of the whole small cuboid is set to εmat.

For example, when (Nx = Ny = Nz = 8), the circuit corresponding to the relative permittivity (εDHW) is shown in the top of the Fig. 8. The bottom circuit of the same figure corresponds to the equivalent capacitance of the capacitor (cδ). Where (cδ,ζ,η) denotes the capacitance of the cuboid (Cδζη) of permittivity (εδζη) taken at its center. Note that each capacitor from (c1) to the capacity \((c_{{N_{z} }} )\) of the top circuit has an equivalent capacitance circuit like is shown for the capacitor (cδ) of the cuboid (Cδ).

Fig. 8
figure8

The equivalent capacitance circuit corresponding to the relative permittivity (εDHW) for the example (Nx = Ny = Nz = 8)

This second bound (εDHW) was obtained by changing the order of the last two steps in the first bound (εDWH). The first bound (εDWH) was obtained by slicing the depth (D) in the first place, followed by the width (W), and finally the height (H). The second bound (εDHW) was obtained by slicing the depth (D), then the height (H), and finally the width (W). It is worth mentioning that other expressions of permittivity can be obtained through other different subdivisions. E.g. instead of slicing the depth (D) in the first place, the one could start by slicing the width (W) instead. Then, by following the same logic in the previous bounds, which consists of using the appropriate Wiener formula for the obtained slices, the one could easily obtain other expressions of the effective permittivity. It is also worth mentioning that when we studied the impact of the subdivision’s order of the slicing process (not shown in this paper), we found that the bound (εDWH) gave the same results of (εWDH) and the bound (εDHW) gave the same results of (εWHD) only when the values of the depth (D), the width (W) and the height (H) are equal and the conditions (Nx = Ny = Nz) are verified. These conditions, (Nx = Ny = Nz) and (W = H = D), are set to ensure that all subdivision processes lead to the same number of the small cuboids of identical dimensions. However, we recommend using the bounds (εDWH) and (εDHW) that were presented in this manuscript. We consider them as the default bounds because they are the only ones that can be used to find easily, the equations corresponding to the bounds of the 2D case [31]. In fact, on one hand, either by setting (Nz = 1) in the bound (εDWH) or setting (εαβγ = εij) for each (α), the one could easily switch to the bound εWH given by the following expression [31]:

$$ \varepsilon_{{{\text{WH}}}} = \frac{{N_{y} }}{{N_{x} }}\sum\limits_{i = 1}^{{N_{x} }} {\left( {\sum\limits_{j = 1}^{{N_{y} }} {\frac{1}{{\varepsilon_{ij} }}} } \right)^{ - 1} } $$
(24)

where (εij) denotes the permittivity taken at the center of the rectangle (Rij) and (εWH) denotes the permittivity bound in the 2D case obtained by slicing the width (W) then the height (H).

And on the other hand, either by setting (Nz = 1) in the bound (εDHW) or setting (εδζη = εpq) for each (δ), the one could easily switch to the bound εHW given by the following expression [31]:

$$ \varepsilon_{{{\text{HW}}}} = \frac{{N_{y} }}{{N_{x} }}\left( {\sum\limits_{p = 1}^{{N_{y} }} {\left( {\sum\limits_{q = 1}^{{N_{x} }} {\varepsilon_{{{\text{pq}}}} } } \right)^{ - 1} } } \right)^{ - 1} $$
(25)

where (εpq) denotes the permittivity taken at the center of the rectangle (Rpq) and (εHW) denotes the permittivity bound in the 2D case obtained by slicing the height (H) then the width (W).

Application to Arbitrary Shapes and Comparison

It is obvious to check the new model's validity. So, the new presented bounds were used to estimate the effective permittivity of three different systems involving different arbitrary inclusions as shown in the Fig. 9 (cube, sphere, and cylinder) and for different values of permittivity. The inclusions are prohibited from overlapping the lateral faces of the host matrix. Then, the maximal accepted (R) value is (A/2). So, the available range of the volume fraction of the inclusion \((\phi_{{{\text{inc}}}} )\) is:

  • From 0% to 99.99% for the cube.

  • From 0% to 52.35% for the sphere.

  • From 0% to 78.53% for the cylinder.

Fig. 9
figure9

The simulated heterogeneous structures

Note that the cylinder’s height always equals two times the radius of the cylinder base, and that all simulations were done using a personal computer (DELL INSPIRON N5050), equipped by (4 Gb of RAM + 4 Gb of SWAP), running under Ubuntu 16.04 (64bits), and powered by an Intel© Core i5-2450 M ™ CPU @ 2.50 GHz × 4.

For all simulation cases, the relative permittivity (εmat) of the host matrix was set to the value 102; and to study the impact of the contrast given by (k = εinc/εmat), four values (104, 103, 10, 1) were chosen for the inclusion's relative permittivity (εinc).

The present model was applied to the three suggested inclusions using the Eqs. (16 and 23) for an equal number of subdivisions (Nx = Ny = Nz); the four chosen contrast values lead to four graphs for each inclusion type. The Fig. 10 regroups the results of the cube, from which anyone could easily notice that the FEM results are always between the Hashin–Shtrikman limits and also between the new presented bounds. Moreover, the new limits are narrower than the Hashin–Shtrikman bounds. The Looyenga's rule gives results far from the FEM results, but always between the Hashin–Shtrikman limits. Also, the Hashin–shtrikman bound (εHS,2) is the furthest one from the FEM results for all cases of the contrast (k). Also, from the two last graphs corresponding to the two last cases of contrast (k = 10/102 and k = 1/102) the bound (εDWH) is slightly less than the FEM results for \((\phi_{{{\text{inc}}}} < 0.5)\), but this difference is still acceptable. Moreover, for all values of contrast (k), the bound (εDHW) gives approximately the same results of those obtained by applying the FEM.

Fig. 10
figure10

Comparison of cube's results

The results corresponding to the sphere are regrouped in the Fig. 11. In this figure, both the Hashin–Shtrikman bounds and the new bounds delimit the FEM results, and similarly to the cubic inclusion, the new bounds are always narrower than the Hashin–Shtrikman bounds for the whole available range. Moreover, the Looyenga rule gives results far from the FEM results especially, for the contrast value (k = 104/102), but it gives better results when compared to the (εDWH) for the contrast value (k = 10/102) and they intersect with each other approximately at the volume fraction \((\phi_{{{\text{inc}}}} = 0.2)\) for the contrast value (k = 1/102). In the graph corresponding to the contrast case (k = 103/102), on which an effective permittivity (εmean = (εDWH + εDHW)/2) was added, the one could easily notice that (εmean) reproduces perfectly the FEM results. It is noted that for the contrast case (k = 104/102) the results were presented using a logarithmic scale for the (εeff) axis.

Fig. 11
figure11

Comparison of sphere's results

Regarding the results shown in the Fig. 12, which are related to the cylinder, the viewer could easily notice that the (εDHW) bound reproduces perfectly the FEM results for all suggested contrast values, while the Hashin–Shtrikman bound (εHs,2) and the Looyenga rule (εLyng) failed together. Besides, the higher the contrast between phases, the larger the difference between the FEM results and both Hashin–Shtrikman bound (εHS,2) and Looyenga rule (εLyng).

Fig. 12
figure12

Comparison of cylinder's results

Indeed, for all studied cases, the FEM results are practically between the new narrower presented bounds. These latter ones are also delimited by the Hashin–Shtrikman limits. Moreover, as can be observed from the Figs. 10, 11, and 12 the new bounds vary accordingly to the structure of the studied system. In other words, they are depending on the distribution of the different constituents. Moreover, for almost cases, the results obtained by the FEM can be approximated by the (εDHW) bound except the two cases of contrast (k = 104/102) and (k = 103/102) for the sphere's inclusion.

It is worth mentioning that the difference between the results of the two bounds εDWH and εDHW is essentially due to the order of slicing process and is slightly impacted by the total number of the small cuboids Ntot = NxNyNz. In fact, after studying the convergence of this model (not shown in this paper), we noticed that the bounds εDWH and εDHW started to stabilize when the total number of the small cuboids Ntot reached the value 106. However, the bounds εDWH and εDHW did not converge toward an identical value of permittivity no matter how much we increased the number of the small cuboids Ntot.

Computation Time

To show the advantage of this new model regarding the computing time, the time taken by this model to estimate the effective permittivity was compared to that one taken by the finite element method (FEM). The simulated systems consist of identical spheres that were evenly spaced in the 3 directions of space. Moreover, the volume fraction of the inclusions \((\phi_{{{\text{inc}}}} )\) was fixed at 45%, and the contrast (k = εinc/εmat = 102/10).

The Fig. 13 shows the difference between the times taken by the two methods. In this figure, the reader could easily notice and conclude that the greater the number of the inclusions in the matrix, the wider is the difference between the two times. Moreover, the present model is approximately 5 times faster than the FEM.

Fig. 13
figure13

Computation time

It is noted that the difference between those two methods is too huge in the 3D case when it is compared to the 2D case [31]. This is normal because the 3D case is characterized by a higher degree of freedom due to the third dimension that generates more independent variables. Also, it is noted that the meshing parameters of the FEM were all the time chosen to produce a mesh leading to a degree of freedom 10 times less than the product (Nx·Ny·Nz). This latter product represents the total number of the small tiny cuboids.

Conclusion

Based on the Wiener's formulas for the effective permittivity, a new simulation method to predict the effective permittivity of 3D composite and nanocomposite materials was introduced. The results obtained by applying the new numerical method were compared to the ones obtained by applying the finite element method (FEM), the Hashin-Shtrickmann bounds, and Looyenga rule. Many simulations were performed to check the model's validity. The set of obtained data reveals how the results of the new model vary depending on the shape of the studied inclusions. This flexibility was expected since the expression of each bound involves the local permittivity introduced via the permittivity εijk of each small cuboid (Cijk). In other words, systems that have different dispositions of small cuboids, have consequently different assemblies of capacitors for each bound, and since each system has a unique assembly of capacitors, then there will be two unique bounds of effective permittivity for each system. The presented bounds, (εDWH) and (εDHW), can be used to find easily the 2D ones that were previously published [31]. Moreover, the investigation that was made to check the code's rapidity, allows concluding that the new model is faster than the FEM. So, for huge systems that involve a lot of parameters, for which even the most powerful computers cannot deal with using the finite element method, the present model would be a good alternative tool to predict their effective permittivity. Thus, this new model will provide, in a quick way, more guidance to engineers willing to design new sophisticated systems.

Data Availability

The data that support the findings of this study are available on request from the corresponding author. The data are not publicly available as the data also form part of an ongoing study.

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Jarmoumi, Y., Derouiche, A. & Benzouine, F. A New Numerical-Homogenization Method to Predict the Effective Permittivity of Composite Materials. Integr Mater Manuf Innov 9, 423–434 (2020). https://doi.org/10.1007/s40192-020-00194-0

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Keywords

  • Effective properties
  • Simulation techniques
  • Dielectric constant
  • Circuit-based models
  • Heterogeneous media