Mathematical Methods in the Applied Sciences
Volume 46, Issue 6 pp. 6613-6638
RESEARCH ARTICLE
Open Access

Conductivity gain predictions for multiscale fibrous composites with interfacial thermal barrier resistance

Ernesto Iglesias-Rodríguez, 

Corresponding Author

Ernesto Iglesias-Rodríguez

Unidad Académica del Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas en el Estado de Yucatán, Universidad Nacional Autónoma de México, Yucatán, Mexico

Correspondence

Ernesto Iglesias-Rodríguez, Unidad Académica del Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas en el Estado de Yucatán, Universidad Nacional Autónoma de México, Yucatán, Mexico.

Email: [email protected]

Communicated by: R. Rodriguez

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Julián Bravo-Castillero, 

Julián Bravo-Castillero

Unidad Académica del Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas en el Estado de Yucatán, Universidad Nacional Autónoma de México, Yucatán, Mexico

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Manuel E. Cruz, 

Manuel E. Cruz

Departamento de Engenharia Mecânica, Politécnica/COPPE, Universidade Federal de Rio de Janeiro, Rio do Janeiro, Brazil

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Raúl Guinovart-Díaz, 

Raúl Guinovart-Díaz

Facultad de Matemática y Computación, Universidad de La Habana, La Habana, Cuba

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First published: 21 December 2022
https://doi.org/10.1002/mma.8928
Citations: 2

Abstract

Nanocomposites are heterogeneous media with two or more micro-structural levels. For instance, a nano-level is characterized by isolated nano-inclusions, and a micro-level is represented by the clusters resulting from aggregation processes. Based on the reiterated homogenization method, we present a procedure to study the influence of this aggregation process and interfacial thermal resistance on the effective thermal conductivity for 2-D square arrays of circular cylinders. First, an effective intermediate thermal property is obtained by taking into account only the influence of the individual nano-inclusions in the matrix. Second, the final effective thermal coefficient ( k ^ R H $$ {\hat{k}}_{RH} $$ ) is calculated considering the clusters immersed in the intermediate effective medium derived in the first step. The conductivity gain ( k g a i n $$ {k}_{gain} $$ ) is defined as the quotient ( k ^ R H / k ^ C H $$ {\hat{k}}_{RH}/{\hat{k}}_{CH} $$ ) where k ^ C H $$ {\hat{k}}_{CH} $$ is the effective thermal coefficient computed considering only one microstructural level with the same volumetric fraction of inclusions. We apply the theory of complex variable functions deriving in an infinite system of equations solvable by truncation method. The analytical formulas of the effective coefficient used in the calculations generalize other well-known formulas reported in the literature. We also provide formulas for several truncation orders. The main novel contribution is in the reiterative application of the obtained formulas to illustrate the gain effect in nanocomposites, expressed as a function of the Biot number, thermal conductivities, volumetric fibers fraction, and an aggregation parameter. Furthermore, this result can be used to assess numerical computations and for nano-reinforced fibers and nanofluids applications. An appendix is included showing similarities and differences with a three-phase model.

1 INTRODUCTION

The study of the effective thermal conductivity in heterogeneous media with various micro-structural scales1, 2 has increasing interest because of its importance in heat transfer applications, especially in the areas of nanocomposites3-7 and nanofluids.8-12

There is also an increasing interest in the influence of the thermal barriers at the interfaces.13, 14 In previous studies,15, 16 an ad hoc homogenization model was developed to analyze the role of aggregation processes and interfacial thermal resistance on the effective thermal conductivity of nanofluids and nanocomposites. Based on the relationship between aggregation process and the existence of multiple spatial scales, their model showed a significant enhancement of the effective thermal conductivity with respect to fully dispersed media. A similar effect was described by Eapen et al.17 based in several models and experimental results present in the literature.

The methodology to be followed in this work is based on the reiterated homogenization method (RH). This is a mathematical method of homogenization developed in Bensoussan et al.1 The ad hoc multiscale model formulated by Evans et al.16 was an inspiration for the application of RH in some of previous works due to the proved influence of aggregation and interfacial thermal resistance. For instance, in Iglesias-Rodríguez et al.,18 the RH was applied to investigate the macroscopic behavior of the strong form Fourier heat conduction problem with periodic and rapidly oscillating coefficients depending on two fast variables representing two microstructural scales. Laminate composites were considered, and no improvement of the effective thermal conductivity was observed relative to conventional laminates. However, in Nascimento et al.,19 perfect interfacial contact was considered, and an improved effective thermal conductivity was observed for multiscale fibrous and particulate composites with ordered microstructures by combining the reiterated homogenization method with known analytical formulas. Similar gain of the effective thermal conductivity was also observed in Mattos et al.20 for two-dimensional multiscale heterogeneous media through reiterated homogenization and finite element methods. A similar approach was taken by previous studies21, 22 for square cross-section hierarchy in the cases of square fiber inclusions and porous (perforated) media, respectively.

In Álvarez-Borges et al.,14 a one-dimensional walk-through of the RH, involving numerical examples, mathematical justifications and examples of thermal conductivity gain can be seen. Recently, in Iglesias-Rodríguez et al.,23 the methodology used in Iglesias-Rodríguez et al.18 was extended by considering the effects of thermal barriers at the interfaces. In Iglesias-Rodríguez et al.,23 the formal procedure to derive the homogenized problem, local problems, and effective coefficients was described for a general three-dimensional problem although the application to nanofluid was derived only for the case of laminated composites. In this case, an enhancement of the effective conductivity was also detected.

In the current work, the analysis in Nascimento et al.19 is generalized to the case of imperfect interfacial contact. For convenience, we consider the same square periodic arrangement of circular cylinders for both microstructural levels. However, the formulas that we will use for the effective coefficients were calculated following the method described by Pobedrya,24, Chap. 6 which was extended by previous studies25-27 for imperfect contact. This method is sometimes called after Natanzon-Filshtinsky28-31 for their contribution to this approach in the field of linear elasticity. They study separately the so-called plane and antiplane problems of linear elasticity theory, being the second one equivalent to the conductive case. A detailed explanation of a historical development can be found at previous works.31-33

Unlike literature,26, 27 we derive in a conductive context five types of explicit analytical formulas for the effective thermal coefficient with different approximation orders. These formulas were obtained considering the imperfect spring-type contact condition at the interface.34 Furthermore, these new formulas are expressed in the style of those reported by previous studies35, 36 for perfect contact in a three-phase conductive problem. In this way, the equivalence between the imperfect spring-type contact model with the three-phase model is detailed.34, 37-39

The application of this method considering a thermal barrier (imperfect contact) is one of the main novelties of this work. This paper extends the earlier results, considering a square arrays of fibers with thermal barrier. The formulas obtained here are not explicitly written in other works although we compare ours with at least the most familiar ones26, 36, 40-42 and numerical results. For media with two or more micro-structural levels, the effective thermal conductivity tensor was calculated from the solutions of recurrent local problems. This allows us to analyze the gain effect on effective properties by comparing the problem of media with two microscales respect to the case where only one scale is considered. The theoretical gain result add novelty to the present work too and have several applications. For instance, we provide a mathematically supported methodology that could facilitate, based on the components geometric and conductive characteristics, the design43, 44 of composite materials with the appropriate volume fractions to achieve a maximum gain in effective thermal conductivity.

1.1 Physical motivation

Let us consider a two-dimensional conductive medium formed from a matrix and inclusions with a periodic configuration. The inclusions in the nano-scale can be grouped into aggregates periodically distributed in the micro-scale. In this manner, we have two-scale periodic media where the bigger micro-structure is formed by a larger inclusion with the same conductivity as the smaller nano-structure.

To distinguish a medium that is fully aggregated, fully disperse, or in an intermediate phase, we will define an aggregation parameter (Figure 1). Following previous studies,16, 45 for the described model, we set
α = ϕ c ϕ = 1 − ϕ n c ϕ , $$ \alpha =\frac{\phi_c}{\phi }=1-\frac{\phi_{nc}}{\phi }, $$ (1)
where ϕ $$ \phi $$ is the volume fraction of the inclusions in the medium, ϕ c $$ {\phi}_c $$ is the volume fraction of the aggregates, and ϕ n c $$ {\phi}_{nc} $$ is the volume fraction of the inclusions that are not clustered in an aggregate ( ϕ = ϕ c + ϕ n c $$ \phi ={\phi}_c+{\phi}_{nc} $$ ). In this way, as seen in Figure 1, when α = 0 $$ \alpha =0 $$ , we have a fully disperse medium with only one scale formed by nano-scale inclusions in the matrix (becomes monodisperse), and when α = 1 $$ \alpha =1 $$ , we have a fully aggregated medium with one scale formed by micro-scale aggregates in the matrix (also becomes monodisperse).
Details are in the caption following the image
FIGURE 1
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Schematic representation of the influence of aggregation in the base geometry (in black are marked the clustered inclusions to ease viewing)

For modeling purposes, we will assume a periodic bimodal heterodisperse real medium; that is, the inclusions have two different geometries and sizes (modes). The periods in each mode are assumed proportional, and then we will define the scales from the ratio between these periods. In this model, the largest inclusions would represent the clusters, for which we will consider the same conductivity as for an isolated particle, and which have managed to form a regular network. An example of this model is proposed in Figure 2.

Details are in the caption following the image
FIGURE 2
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Schematic representation of a real material (left), proposed model (center) with-ordered microstructure, and basic cells Y and Z (on the right). The d $$ d $$ and D $$ D $$ scales characterize the nano-level and micro-level, respectively. The volume fraction in each basic cells depends on aggregation [Colour figure can be viewed at wileyonlinelibrary.com]
In order to calculate the effective conductivity of a periodic medium with two local scales, we will apply the RH. For example, Figure 2 (left) illustrates a well-ordered two-phase periodic fibrous composite that can be abstracted as a periodic bimodal heterodisperse medium in the plane as seen in Figure 2 (center). The mathematical procedure of homogenization will be described in Section 3.1. Based on this methodology, the calculation of the effective conductivity of the medium is carried out in two stages. In the first stage, an “intermediate” effective conductivity is calculated over a periodic cell formed by an isolated fiber, depicted as a small circle (cell Z in Figure 2; right). Finally, the effective conductivity is calculated over a periodic cell formed by an aggregate, depicted as a large circle (cell Y in Figure 2; right) embedded in a matrix whose thermal conductivity is that obtained in the first step. Both periodic cells are considered unit squares. In the first homogenization stage, the volume fraction of the isolated inclusion (cell Z) is related with the aggregation parameter α $$ \alpha $$ by the formula
( 1 − α ) ϕ 1 − ϕ c . $$ \frac{\left(1-\alpha \right)\phi }{1-{\phi}_c}. $$
In the second stage, the aggregates' volume fraction or volume fraction of the aggregate (cell Y) is equal to
ϕ c = α ϕ . $$ {\phi}_c=\alpha \phi . $$
Aggregation relates directly with multiple spatial scales, and a significantly enhancement of the overall thermal conductivity was found in media with multiple micro-scales, see, for instance, previous studies.3, 5, 16, 19 For both scales, we will consider a thermal barrier at the interfaces between matrix and inclusions, as will be described in the next section. Our interest is to compare media with multiple micro-scales against a single micro-scale; more details on this property gain will be given in Section 4.

In the case of fibrous composite, we will only analyze fibers with circular cross-sections in the nanoscale. In the microscale, we consider clusters resulting from the aggregation process that will be represented as larger fibers with the same shape. The same situation can be found in advanced composite material46; for example, the addition of nanotubes to the matrix of fiber composites modifies their thermal and mechanical properties, even for low concentrations.47, 48

The study of random composites is also of great interest. It was studied in literature32, 49, 50 in the perfect contact case. An extension of this method is proposed in previous studies51, 52 for a wide range of random and periodic materials, based on the comprehensive study53 of boundary problems for the Laplace equation in singularly perturbed domains. This is one of the techniques applied in recent works54, 55 to obtain analytical formulas for effective properties of dispersed composites. Here and hereafter, we will only refer to periodic composites.

The paper continues as follows. In Section 2, we pose the physical and mathematical formulation of the problem. Section 3 is devoted to the solution methodology, with special emphasis on reiterated homogenization. Analytical formulas are derived for the effective coefficient of isotropic two-phase fibrous composites with an interfacial thermal barrier. More details on these formulas and a comparison with the triphasic model can be seen in Appendix A. In Section 4, results on thermal conductivity gain will be given, in order to study the role of aggregation processes and interfacial thermal resistance in the effective coefficient. Finally, some concluding remarks are presented in Section 5.

2 PROBLEM STATEMENT

We will model the heat conduction based on the Fourier heat equation in the interior of the body and Dirichlet conditions on the boundaries (perfect thermal contact with the environment). We will also consider that there is an imperfect contact between the matrix and such inclusions, whether they are individual or clusters. That is assumed as a thermal barrier that induces a spring-type contact condition; that is, the heat flow through each interface is proportional to the corresponding jump of temperature. The thermal barrier will be characterized by the non-dimensional Biot number56-58:
B i = h S ⟦ K ⟧ , $$ Bi=\frac{hS}{\left\llbracket K\right\rrbracket }, $$ (2)
that measures the thermal barrier at the interface with respect to the thermal conductivities of the components. Here, ⟦ K ⟧ $$ \left\llbracket K\right\rrbracket $$ represents the jump in thermal conductivities, h $$ h $$ is the interfacial conductance, and S $$ S $$ is the surface area of the contact. In this formulation, B i → ∞ $$ Bi\to \infty $$ indicates the presence of a perfect contact.

Here and henceforth, we used the contrast notation ⟦ ( · ) ⟧ = ( · ) ( m ) − ( · ) ( i ) $$ \left\llbracket \left(\cdotp \right)\right\rrbracket ={\left(\cdotp \right)}^{(m)}-{\left(\cdotp \right)}^{(i)} $$ where the indices indicates the expression being evaluated inside the domain representing an inclusion ( · ) ( i ) $$ {\left(\cdotp \right)}^{(i)} $$ or inside the domain representing the matrix ( · ) ( m ) $$ {\left(\cdotp \right)}^{(m)} $$ . The next section will be devoted to the mathematical formulation of the heat transfer problem; for more details, we refer to literature.23, 59

2.1 Mathematical formulation

We will consider a conductive medium formed by a matrix and an arrangement of parallel fibers, assuming a periodic configuration. There is no loss of generality for this case if we consider the two-dimensional problem, that is, considering only a plane perpendicular to the fibers. These fibers can be grouped into clusters that will be considered individual fibers with arbitrary shape, assuming then that each fiber is in a single periodic cell. We will also consider that there is an imperfect contact between the matrix and such fibers, whether they are individual or clusters. We are interested in solving local problems in the principal directions (perpendicular to the directions of the fiber) and calculating the final effective properties.

For our study, we will consider only a uniaxially reinforced composite. This can be interpreted as a two-phase medium formed by a connected phase (matrix) and the non-connected phase formed by the repetition of the same domain (inclusions). These types of media are known as matrix-inclusion systems. We will assume a periodic arrangement of parallel cylindrical inclusions distributed evenly and periodically in a plane perpendicular to the axes of the cylinders.

Let the position of a typical point on the body be denoted by three coordinates ( x 1 , x 2 , x 3 ) $$ \left({x}_1,{x}_2,{x}_3\right) $$ in a Cartesian coordinate system. We can assume, for example, that the axis x 3 $$ {x}_3 $$ corresponds to the transverse axis of symmetry in the direction of the cylindrical fiber. This means that the dimension of the problem is two and the cylinders are distributed in a periodic network of the transversal cross-section plane. In this sense, we will consider a domain Ω ⊂ ℝ 2 $$ \Omega \subset {\mathbb{R}}^2 $$ with piecewise smooth boundaries ∂ Ω $$ \mathrm{\partial \Omega } $$ . Said domain extends over the spatial coordinate x = ( x 1 , x 2 ) ∈ ℝ 2 $$ x=\left({x}_1,{x}_2\right)\in {\mathbb{R}}^2 $$ (known as a slow variable) and cylinders reduce to circles.

In this context, we will consider a complex structure that can be periodically characterized by two microstructural levels: a first level with period D $$ D $$ and a second level with period d $$ d $$ . In this way, for a medium with characteristic length L $$ L $$ , we can consider the small parameters:
ε 1 = D L , ε 2 = d L , $$ {\varepsilon}_1=\frac{D}{L},\kern2em {\varepsilon}_2=\frac{d}{L}, $$
where ε 1 ≪ 1 $$ {\varepsilon}_1\ll 1 $$ (the parameter is small) and ε 2 ≪ ε 1 $$ {\varepsilon}_2\ll {\varepsilon}_1 $$ (there is a clear scales separation).60, 61 For simplicity, we will consider ε 1 = ε $$ {\varepsilon}_1=\varepsilon $$ and ε 2 = ε 2 $$ {\varepsilon}_2={\varepsilon}^2 $$ (i.e., the case d = ε D $$ d=\varepsilon D $$ ).
The presence of two micro-structural periodic scales is related to the existence of two periodic cells ε Y $$ \varepsilon Y $$ , ε 2 Z $$ {\varepsilon}^2Z $$ whose repeated application covers the domain Ω $$ \Omega $$ .6, 23 We will consider unitary squares for these cells
− 1 2 , 1 2 × − 1 2 , 1 2 , $$ \left(-\frac{1}{2},\frac{1}{2}\right)\times \left(-\frac{1}{2},\frac{1}{2}\right), $$
and restrict the analysis to a periodic conductive composite with two components with known behavior (two-phase), although the extension to other multiphase media is direct. This means that each cell is formed by non-empty subsets,62, 63
Y = Y ( m ) ∪ Y ( i ) ( ∂ Y ( i ) ∩ ∂ Y = ∅ ) , Z = Z ( m ) ∪ Z ( i ) ( ∂ Z ( i ) ∩ ∂ Z = ∅ ) , $$ {\displaystyle \begin{array}{cc}\hfill Y=& \kern0.2em {Y}^{(m)}\cup {Y}^{(i)}\kern0.30em \left(\partial {Y}^{(i)}\cap \partial Y=\varnothing \right),\hfill \\ {}\hfill Z=& \kern0.2em {Z}^{(m)}\cup {Z}^{(i)}\kern0.30em \left(\partial {Z}^{(i)}\cap \partial Z=\varnothing \right),\hfill \end{array}} $$ (3)
where Y ( i ) $$ {Y}^{(i)} $$ and Z ( i ) $$ {Z}^{(i)} $$ represent the region occupied by the inclusions at each scale. For the model defined in Section 1.1, we denote by K ( m ) $$ {K}^{(m)} $$ and K ( i ) $$ {K}^{(i)} $$ the thermal conductivities of matrix and inclusions, respectively. Notice that the volume fractions can be expressed as
ϕ c = | Y ( i ) | | Y | = | Y ( i ) | , ϕ n c = ( 1 − ϕ c ) | Z ( i ) | | Z | = ( 1 − ϕ c ) | Z ( i ) | , $$ {\phi}_c=\frac{\mid {Y}^{(i)}\mid }{\mid Y\mid }=\mid {Y}^{(i)}\mid, \kern2em {\phi}_{nc}=\left(1-{\phi}_c\right)\frac{\mid {Z}^{(i)}\mid }{\mid Z\mid }=\left(1-{\phi}_c\right)\mid {Z}^{(i)}\mid, $$
where | ω | = ∫ ω d V $$ \mid \omega \mid ={\int}_{\omega } dV $$ .

We will consider the steady-state ε $$ \varepsilon $$ -dependent multiscale heat conduction family of boundary value problem23; that is,

Global Problem: Find u ε ∈ C 2 ( Ω \ Γ ε ) $$ {u}^{\varepsilon}\in {C}^2\left(\Omega \backslash {\Gamma}^{\varepsilon}\right) $$ , such as
L ε u ε ≡ − ∇ · ( K ε ∇ u ε ) = f in Ω \ Γ ε , $$ {L}^{\varepsilon }{u}^{\varepsilon}\equiv -\nabla \cdotp \left({K}^{\varepsilon}\nabla {u}^{\varepsilon}\right)=f\kern2em \mathrm{in}\kern0.5em \Omega \backslash {\Gamma}^{\varepsilon }, $$ (4)
K ε ∇ u ε · n = − B ε ⟦ u ε ⟧ on Γ ε = Γ Y ε ∪ Γ Z ε , $$ \kern5em {K}^{\varepsilon}\nabla {u}^{\varepsilon}\cdotp n=-{B}^{\varepsilon}\left\llbracket {u}^{\varepsilon}\right\rrbracket \kern2em \mathrm{on}\kern0.5em {\Gamma}^{\varepsilon }={\Gamma}_Y^{\varepsilon}\cup {\Gamma}_Z^{\varepsilon }, $$ (5)
⟦ K ε ∇ u ε · n ⟧ = 0 on Γ ε = Γ Y ε ∪ Γ Z ε , $$ \kern7em \left\llbracket {K}^{\varepsilon}\nabla {u}^{\varepsilon}\cdotp n\right\rrbracket =0\kern2em \mathrm{on}\kern0.5em {\Gamma}^{\varepsilon }={\Gamma}_Y^{\varepsilon}\cup {\Gamma}_Z^{\varepsilon }, $$ (6)
u ε = 0 on ∂ Ω , $$ \kern7em {u}^{\varepsilon }=0\kern2em \mathrm{on}\kern0.5em \mathrm{\partial \Omega }, $$ (7)
where K ε = K ( x / ε , x / ε 2 ) $$ {K}^{\varepsilon }=K\left(x/\varepsilon, x/{\varepsilon}^2\right) $$ and f $$ f $$ are the thermal conductivity and heat output, respectively. In these equations x $$ x $$ represents the spatial variable on a global scale and
B ε = β ε − 1 , on Γ Y ε β ε − 2 , on Γ Z ε $$ {B}^{\varepsilon }=\left\{\begin{array}{l}\beta {\varepsilon}^{-1},\kern0.5em \mathrm{on}\kern0.5em {\Gamma}_Y^{\varepsilon}\\ {}\beta {\varepsilon}^{-2},\kern0.5em \mathrm{on}\kern0.5em {\Gamma}_Z^{\varepsilon}\end{array}\right. $$
is the Biot number.56, 58 The thermal conductivity tensor K ε $$ {K}^{\varepsilon } $$ is symmetric, positive definite, and it is worth noting that for small ε $$ \varepsilon $$ its coefficients are rapidly oscillating. The perfect contact case is obtained for β → ∞ $$ \beta \to \infty $$ , which implies that ⟦ u ε ⟧ = 0 $$ \left\llbracket {u}^{\varepsilon}\right\rrbracket =0 $$ in Equation (5).

Now, we are going to present the fundamental theory, which allows us to pass from problem (4)–(7) to an equivalent (homogenized) one whose coefficients are constant when ϵ → 0 $$ \epsilon \to 0 $$ . This constant coefficients are known as the effective coefficients of the macro-scale or global effective coefficients and will be obtained in the next section (Equation (13)). The derived problem will be solved using the theory of complex variable functions and will present various estimates of the effective coefficients.

3 ASYMPTOTIC HOMOGENIZATION METHOD

Homogenization is an upscaling procedure that provides mathematical models that allow the calculations of effective properties of a composite from known properties of its components. In this manner, the macroscopic behavior of a heterogeneous medium is obtained as an equivalent homogeneous material based on the relationships at the smallest scales.

From the mathematical point of view, the homogenization process transforms problems involving systems of partial differential equations with rapidly oscillating coefficients in a homogenized problem with constant ones, called effective coefficients. In a previous work,23 the formal procedure to derive the homogenized problem and effective coefficients are described for a general three-dimensional problem. The solution of local problems in each micro-structural scale is used to calculate the effective coefficients. This is the case for the model defined in Section 1.1, when the aggregation parameter α ∈ { 0,1 } $$ \alpha \in \left\{0,1\right\} $$ .

The name homogenization was introduced in literature,64-66 who worked a similar approach to the prior work of Sanchez-Palencia.67 Since then, there have been great advances in this theory, and it has proven to be very useful in all kinds of applications. For our work, we use the asymptotic homogenization method (A.H.M), which was stated by previous works1, 68, 69 and systematically formalized to handle homogenization of contour problems with rapidly oscillating periodic coefficients.70, 71 This is based on the asymptotic expansion of the solution of the boundary problem, and the idea is to obtain a homogenized problem whose solution u 0 $$ {u}_0 $$ is the limit (when ε → 0 $$ \varepsilon \to 0 $$ ) of the solutions u ε $$ {u}^{\varepsilon } $$ of (4)–(7). Homogenization is, thus, one of those distinguished methods that keep theoretical rigor with no ad-hoc hypotheses, leading to useful practical and general applications.

3.1 Reiterated homogenization

The reiterated homogenization method (RHM) is a rigorous mathematical technique for investigating the macroscopic behavior of periodic composites with different micro-structural levels.1 In our case, and due to the presence of multiple scales, we will apply this formalism in the context of the reiterated homogenization. More details on this procedure can be found in Iglesias-Rodríguez et al.,23 but it can be stated briefly as follow.

An asymptotic expansion for the solution with the form
u ( x , y , z ) = u 0 ( x ) + ε u 1 ( x , y ) + ε 2 u 2 ( x , y , z ) + O ( ε 3 ) $$ u\left(x,y,z\right)={u}_0(x)+\varepsilon {u}_1\left(x,y\right)+{\varepsilon}^2{u}_2\left(x,y,z\right)+O\left({\varepsilon}^3\right) $$ (8)
will be proposed, where each u i $$ {u}_i $$ is considered Y $$ Y $$ -periodic respect to y = x / ε $$ y=x/\varepsilon $$ and Z $$ Z $$ -periodic respect to z = x / ε 2 $$ z=x/{\varepsilon}^2 $$ (these new variables are known as fast variables).

Formally applying the operators in (4)–(6) to the series described, and after some manipulations, we obtain a recurring chain of problems whose solutions are the coefficients of (8). In particular, u 0 $$ {u}_0 $$ is the solution to the next problem

Homogenized Problem: find u ∈ C 2 ( Ω ) $$ u\in {C}^2\left(\Omega \right) $$ , such as
− ∇ · ( K ^ ∇ u 0 ) = f in Ω , $$ -\nabla \cdotp \left(\hat{K}\nabla {u}_0\right)=f\kern2em \mathrm{in}\kern0.5em \Omega, $$ (9)
u 0 = 0 on ∂ Ω , $$ \kern5em {u}_0=0\kern2em \mathrm{on}\kern0.5em \mathrm{\partial \Omega }, $$ (10)
where the K ^ $$ \hat{K} $$ is the global effective tensor for the macro-scale. This is a symmetric, positive definite, constant tensor, in contrast with K ε $$ {K}^{\varepsilon } $$ , which is rapidly oscillating.
In terms of the proposed model (Section 1.1), we consider, in the nano-scale,
K ( y , z ) = K ( m ) , z ∈ Z ( m ) , y ∈ Y ( m ) (i.e., matrix) , K ( i ) , z ∈ Z ( i ) , y ∈ Y ( m ) (i.e., isolated inclusion) , K ( i ) , y ∈ Y ( i ) (i.e., aggregates) . $$ K\left(y,z\right)=\left\{\begin{array}{clr}{K}^{(m)},& \kern5.14pt z\in {Z}^{(m)},y\in {Y}^{(m)}& \left(\mathrm{i}.\mathrm{e}.,\mathrm{matrix}\right),\\ {}{K}^{(i)},& \kern5.14pt z\in {Z}^{(i)},y\in {Y}^{(m)}& \kern2em \left(\mathrm{i}.\mathrm{e}.,\mathrm{isolated}\ \mathrm{inclusion}\right),\\ {}{K}^{(i)},& \kern5.14pt y\in {Y}^{(i)}& \left(\mathrm{i}.\mathrm{e}.,\mathrm{aggregates}\right).\end{array}\right. $$ (11)
We will consider the same conductivity for the aggregates as for an isolated inclusion, and they are represented as larger inclusions in the micro-scale with the form
K ¯ ( y ) = K 1 , y ∈ Y ( m ) (i.e., effective matrix) , K ( i ) , y ∈ Y ( i ) (i.e., aggregates) . $$ \overline{K}(y)=\left\{\begin{array}{llr}{K}^1,& y\in {Y}^{(m)}& \left(\mathrm{i}.\mathrm{e}.,\mathrm{effective}\ \mathrm{matrix}\right),\\ {}{K}^{(i)},& y\in {Y}^{(i)}& \left(\mathrm{i}.\mathrm{e}.,\mathrm{aggregates}\right).\end{array}\right. $$ (12)
We find K ^ $$ \hat{K} $$ in the form
k ^ i j = k ¯ i j − k ¯ i k ∂ N j ∂ y k Y , $$ {\hat{k}}_{ij}={\left\langle {\overline{k}}_{ij}-{\overline{k}}_{ik}\frac{\partial {N}_j}{\partial {y}_k}\right\rangle}_Y, $$ (13)
but K ¯ ( y ) $$ \overline{K}(y) $$ depends on the micro-scale through K 1 $$ {K}^1 $$ . This intermediate effective coefficients have the form
k i j 1 = k i j − k i k ∂ N j y ∂ z k Z . $$ {k}_{ij}^1={\left\langle {k}_{ij}-{k}_{ik}\frac{\partial {N}_j^y}{\partial {z}_k}\right\rangle}_Z. $$ (14)
The given formulas for both effective tensors ( K 1 , K ^ ) $$ \left({K}^1,\hat{K}\right) $$ are depending on functions ( N j y , N j ) $$ \left({N}_j^y,{N}_j\right) $$ that will be obtained as solutions of what are known in A.H.M as local problems.
Local Problem 1: Find the family { N j y } y ∈ Y Z $$ {\left\{{N}_j^y\right\}}_{y\in Y}\kern0.1em Z $$ -periodic such as
− ∇ z · K − K ∇ z N j y = 0 , in Z \ Γ Z , $$ \kern1em -{\nabla}_z\cdotp \left(K-K{\nabla}_z{N}_j^y\right)=0,\kern2em \mathrm{in}\kern0.5em Z\backslash {\Gamma}_Z, $$ (15)
K − K ∇ z N j y . n = − β ⟦ N j y ⟧ , on Γ Z , $$ \kern0.1em \left(K-K{\nabla}_z{N}_j^y\right).n=-\beta \left\llbracket {N}_j^y\right\rrbracket, \kern2.30em \mathrm{on}\kern0.5em {\Gamma}_Z, $$ (16)
⟦ K − K ∇ z N j y ⟧ . n = 0 , on Γ Z , $$ \kern2em \left\llbracket K-K{\nabla}_z{N}_j^y\right\rrbracket .n=0,\kern2em \mathrm{on}\kern0.5em {\Gamma}_Z, $$ (17)
N j y Z = 0 . $$ {\left\langle {N}_j^y\right\rangle}_Z=0. $$ (18)
Local Problem 2: Find N j Y $$ {N}_jY $$ -periodic such as
∇ y · K ¯ − K ¯ ∇ y N j = 0 , in Y \ Γ Y , $$ \kern2em {\nabla}_y\cdotp \left(\overline{K}-\overline{K}{\nabla}_y{N}_j\right)=0,\kern2em \mathrm{in}\kern0.5em Y\backslash {\Gamma}_Y, $$ (19)
K ¯ − K ¯ ∇ y N j . n = − β ⟦ N j ⟧ , on Γ Y , $$ \kern0.1em \left(\overline{K}-\overline{K}{\nabla}_y{N}_j\right).n=-\beta \left\llbracket {N}_j\right\rrbracket, \kern2.30em \mathrm{on}\kern0.5em {\Gamma}_Y, $$ (20)
⟦ K ¯ − K ¯ ∇ y N j ⟧ . n = 0 , on Γ Y , $$ \kern2em \left\llbracket \overline{K}-\overline{K}{\nabla}_y{N}_j\right\rrbracket .n=0,\kern2em \mathrm{on}\kern0.5em {\Gamma}_Y, $$ (21)
N j Z = 0 . $$ {\left\langle {N}_j\right\rangle}_Z=0. $$ (22)

It is worth noting that a single microstructural scale (conventional homogenization)69, 72, 73 become special cases of the more general treatment presented here. We will consider both single microstructural scale, and later, we will extend these results to the case of two microstructural scales.

Our final goal is to analyze the role of aggregation processes (single scale against multiple scales) and interfacial thermal resistance in the effective thermal conductivity. The results of this comparison will be given in Section 4.

The previous development is valid for any geometry, number of dimensions and any material.23 We will solve this problem in the case of homogeneous isotropic components, inclusion with circular cross section, and assuming constant conductance at the interface.

3.2 Analytical solution of the local problems

The geometric arrangement described in Section 2.1 is equivalent, in the complex plane, to the structures known as lattices.70, 74 We will use the theory of complex variable functions to solve the local problem in the isotropic case. We will follow the methodology proposed by previous studies,24, 26 which is a continuation of the Rayleigh's method75 (often called Natanzon-Filshtinsky method29, 31). However, the notations and development follow the works of previous works.25, 59, 76

As we are solving the problem for homogeneous isotropic materials, we will denote by W = Y , Z $$ W=Y,Z $$ and by
K = ( κ ( ξ ) δ i j ) , $$ \mathcal{K}=\left(\kappa \left(\xi \right){\delta}_{ij}\right), $$
where K = K , K ¯ , ξ = y , z $$ \mathcal{K}=K,\overline{K},\xi =y,z $$ , and
κ ( ξ ) = κ ( m ) , ξ ∈ W ( m ) (i.e., in the matrix) , κ ( i ) , ξ ∈ W ( i ) (i.e., in the inclusion) . $$ \kappa \left(\xi \right)=\left\{\begin{array}{rlr}{\kappa}^{(m)},& \xi \in {W}^{(m)}& \kern0.30em \left(\mathrm{i}.\mathrm{e}.,\mathrm{in}\ \mathrm{the}\ \mathrm{matrix}\right),\\ {}{\kappa}^{(i)},& \xi \in {W}^{(i)}& \kern0.30em \left(\mathrm{i}.\mathrm{e}.,\mathrm{in}\ \mathrm{the}\ \mathrm{inclusion}\right).\end{array}\right. $$
We assumed here that κ ( i ) ≠ 0 $$ {\kappa}^{(i)}\ne 0 $$ , the case of empty fibers (perforated material) is extensible analyzed in literature.77-79

By isotropy, (15) and (19) are equivalent to Laplace's equation and N k $$ {N}_k $$ are harmonic functions. Under this conditions, the problems in question have the form:

Find N k , W − periodic $$ {N}_k,\kern3.0235pt W-\mathrm{periodic} $$ such as
∇ 2 N k = 0 , in W \ Γ $$ \kern7em {\nabla}^2{N}_k=0,\kern2em \mathrm{in}\kern0.5em W\backslash \Gamma $$ (23)
− κ ∂ N k ∂ ξ i η i = ⟦ κ ⟧ η k on Γ $$ -\left[\kern-3pt \left[\kappa \frac{\partial {N}_k}{\partial {\xi}_i}\right]\kern-3pt \right]{\eta}_i=\left\llbracket \kappa \right\rrbracket {\eta}_k\kern2em \mathrm{on}\kern0.5em \Gamma $$ (24)
− κ ∂ N k ∂ ξ i η i = κ η k − β ⟦ N k ⟧ . on Γ $$ -\kappa \frac{\partial {N}_k}{\partial {\xi}_i}{\eta}_i=\kappa {\eta}_k-\beta \left\llbracket {N}_k\right\rrbracket .\kern2em \mathrm{on}\kern0.5em \Gamma $$ (25)
Although we will only solve this problem below for k = 1 $$ k=1 $$ , the case k = 2 $$ k=2 $$ is analogous.
We will consider that each inclusion is centered at the origin of coordinates of the periodic cell, in the sense that said point belongs to the domain represented by the inclusion, as seen in Figure 3. We will denote then by
ξ 1 = R e ( z ) , ξ 2 = I m ( z ) . $$ {\xi}_1=\mathcal{R}e(z),\kern2em {\xi}_2=\mathcal{I}m(z). $$
The inclusions are located in a network of the plane generated by a pair of vectors ω 1 , ω 2 ∈ ℂ $$ {\omega}_1,{\omega}_2\in \mathbb{C} $$ such as I m ( ω 2 / ω 1 ) ≠ 0 $$ \mathcal{I}m\left({\omega}_2/{\omega}_1\right)\ne 0 $$ . Double periodic harmonic functions are sought in the regions formed by the matrix ( Y ( m ) $$ {Y}^{(m)} $$ ) and the inclusion ( Y ( i ) $$ {Y}^{(i)} $$ ) for each local problem. We also know that there must be some analytical function in the complex plane Φ ( z ) $$ \Phi (z) $$ such as
N 1 = R e Φ ( z ) , $$ {N}_1=\mathcal{R}e\left\{\Phi (z)\right\}, $$ (26)
and I m Φ ( z ) $$ \mathcal{I}m\left\{\Phi (z)\right\} $$ is its conjugate harmonic.74 We will propose solutions of the form
Φ ( z ) = Φ i n = ∑ n = 0 ∞ c 2 n + 1 z 2 n + 1 , z ∈ Y ( i ) (inclusion) , Φ e x = a 0 z + ∑ n = 0 ∞ a 2 n + 1 ( 2 n ) ! d 2 n ζ d z 2 n ( z ) , z ∈ Y ( m ) (matrix) . $$ \Phi (z)=\left\{\begin{array}{lllr}{\Phi}_{in}& =\sum \limits_{n=0}^{\infty }{c}_{2n+1}{z}^{2n+1},& z\in {Y}^{(i)}& \left(\mathrm{inclusion}\right),\\ {}{\Phi}_{ex}& ={a}_0z+\sum \limits_{n=0}^{\infty}\frac{a_{2n+1}}{(2n)!}\frac{d^{2n}\zeta }{d{z}^{2n}}(z),& z\in {Y}^{(m)}& \left(\mathrm{matrix}\right).\end{array}\right. $$ (27)
where ζ ( z ) = ζ ( z , ω 1 , ω 2 ) $$ \zeta (z)=\zeta \left(z,{\omega}_1,{\omega}_2\right) $$ is the Weierstrass Zeta76, 80 function and the coefficients a i , c i $$ {a}_i,{c}_i $$ are reals.
Details are in the caption following the image
FIGURE 3
Open in figure viewerPowerPoint
Square periodic cell with circular inclusion [Colour figure can be viewed at wileyonlinelibrary.com]
The term
a 0 = δ 1 ω 1 a 1 $$ {a}_0=\frac{\delta_1}{\omega_1}{a}_1 $$
is obtained from the periodicity condition from the Legendre relation80:
δ 1 ω 2 − δ 2 ω 1 = 2 π i . $$ {\delta}_1{\omega}_2-{\delta}_2{\omega}_1=2\pi i. $$
The remaining indeterminate coefficients a k $$ {a}_k $$ and c k $$ {c}_k $$ will be obtained from evaluating the proposal (27) formally in the contact conditions (24)–(25). For this, we will use Lauren's series development, which is of the form25, 81:
Φ e x = ∑ n = 0 ∞ a 2 n + 1 z − 2 n − 1 + σ 2 n + 1 z 2 n + 1 , $$ {\Phi}_{ex}=\sum \limits_{n=0}^{\infty}\left({a}_{2n+1}{z}^{-2n-1}+{\sigma}_{2n+1}{z}^{2n+1}\right), $$ (28)
where
σ l = ∑ k = 1 ∞ o a k η k l ≡ ∑ n = 0 ∞ a 2 n + 1 η 2 n + 1 l , $$ {\sigma}_l={\sum \limits_{k=1}^{\infty}}^o{a}_k{\eta}_{kl}\equiv \sum \limits_{n=0}^{\infty }{a}_{2n+1}{\eta}_{2n+1\kern2.56804pt l}, $$ (29)
η 11 = δ 1 ω 1 , $$ {\eta}_{11}=\frac{\delta_1}{\omega_1}, $$ (30)
η k l = − ( k + l − 1 ) ! ( k − 1 ) ! l ! S k + l = − k l S k + l , ( k + l > 2 ) , $$ {\eta}_{kl}=-\frac{\left(k+l-1\right)!}{\left(k-1\right)!l!}{S}_{k+l}=-\left(\left[\genfrac{}{}{0ex}{}{k}{l}\right]\right){S}_{k+l},\kern0.30em \left(k+l>2\right), $$ (31)
S λ = ∑ m 2 + n 2 ≠ 0 m ω 1 + n ω 2 − λ , ( λ > 2 ) . $$ {S}_{\lambda }=\sum \limits_{m^2+{n}^2\ne 0}{\left(m{\omega}_1+n{\omega}_2\right)}^{-\lambda },\kern0.30em \left(\lambda >2\right). $$ (32)
We have used here the notation ∑ o $$ {\sum}^o $$ to indicate that the sum is carried out only on the odd index. k l $$ \left(\kern0.1em \left[\genfrac{}{}{0ex}{}{k}{l}\right]\kern0.1em \right) $$ is known as the multiset number82 and is related to the generalization of combinatorial numbers.83 The sums S λ $$ {S}_{\lambda } $$ are known as lattice sum.
For the case of a circular cross section and constant conductance at the interface (Figure 3), we can parameterize the boundary in such a way that
Γ = z = R e i θ : R = constant ∈ 0 , 1 2 , − π ≤ θ ≤ π . $$ \Gamma =\left\{z=R{e}^{i\theta}:R=\mathrm{constant}\in \left(0,\frac{1}{2}\right),-\pi \le \theta \le \pi \right\}. $$
Note also that, on Γ $$ \Gamma $$ , η = ( η 1 , η 2 ) = ( cos θ , sin θ ) = 1 R d ξ 2 d θ , − d ξ 1 d θ . $$ \eta =\left({\eta}_1,{\eta}_2\right)=\left(\cos \theta, \sin \theta \right)=\frac{1}{R}\left(\frac{d{\xi}_2}{d\theta},-\frac{d{\xi}_1}{d\theta}\right). $$
Substituting the series (27), (28) in (24), (25), and by the Cauchy-Riemann relations, the following equations written in the form of expansion in cosine series are obtained:
∑ k = 1 ∞ o κ ( m ) [ − a k R − 2 k + σ k ] − κ ( i ) c k + ( κ ( m ) − κ ( i ) ) δ 1 k k R k cos ( k θ ) = 0 , $$ {\sum \limits_{k=1}^{\infty}}^o\left({\kappa}^{(m)}\left[-{a}_k{R}^{-2k}+{\sigma}_k\right]-{\kappa}^{(i)}{c}_k+\left({\kappa}^{(m)}-{\kappa}^{(i)}\right){\delta}_{1k}\right)k{R}^k\cos \left( k\theta \right)=0, $$ (33)
κ ( i ) ∑ k = 1 ∞ o ( c k + δ 1 k ) k R k cos ( k θ ) = β ∑ k = 1 ∞ o − c k + a k R − 2 k + σ k R k cos ( k θ ) . $$ {\kappa}^{(i)}{\sum \limits_{k=1}^{\infty}}^o\left({c}_k+{\delta}_{1k}\right)k{R}^k\cos \left( k\theta \right)=\beta {\sum \limits_{k=1}^{\infty}}^o\left(-{c}_k+{a}_k{R}^{-2k}+{\sigma}_k\right){R}^k\cos \left( k\theta \right). $$ (34)
Using here the orthogonality of { cos ( k θ ) } k $$ {\left\{\cos \left( k\theta \right)\right\}}_k $$ as square-integrable functions, we have
κ ( i ) κ ( m ) c k = − a k R − 2 k + σ k + ( 1 − ρ ) δ 1 k , $$ \frac{\kappa^{(i)}}{\kappa^{(m)}}{c}_k=-{a}_k{R}^{-2k}+{\sigma}_k+\left(1-\rho \right){\delta}_{1k}, $$ (35)
c k = β ( β + κ ( i ) k ) − 1 a k R − 2 k + σ k − κ ( i ) β δ 1 k . $$ {c}_k=\beta {\left(\beta +{\kappa}^{(i)}k\right)}^{-1}\left({a}_k{R}^{-2k}+{\sigma}_k-\frac{\kappa^{(i)}}{\beta }{\delta}_{1k}\right). $$ (36)
Note that to use the orthogonality as square-integrable functions, we require that the series formed by the sum of the squares of the coefficients multiplying cos ( k θ ) $$ \cos \left( k\theta \right) $$ to be convergent. For this, it is sufficient to consider the bounding of { a k } k $$ {\left\{{a}_k\right\}}_k $$ and { c k } k $$ {\left\{{c}_k\right\}}_k $$ .
Finally equating the coefficients c k $$ {c}_k $$ in (35) and (36), we obtain the following infinite linear system of equations:
β k ∑ n = 0 ∞ a 2 n + 1 η 2 n + 1 k + δ 1 k − a k R − 2 k = 0 . $$ {\beta}_k\left(\sum \limits_{n=0}^{\infty }{a}_{2n+1}{\eta}_{2n+1\kern2.56804pt k}+{\delta}_{1k}\right)-{a}_k{R}^{-2k}=0. $$ (37)
where k $$ k $$ are odd and
β k = ( 1 − ρ ) β + κ ( i ) k ( 1 + ρ ) β + κ ( i ) k ρ = κ ( i ) κ ( m ) . $$ {\beta}_k=\frac{\left(1-\rho \right)\beta +{\kappa}^{(i)}k}{\left(1+\rho \right)\beta +{\kappa}^{(i)}k}\kern.2em \left(\rho =\frac{\kappa^{(i)}}{\kappa^{(m)}}\right). $$ (38)
This result agree with those obtained in previous studies26, 81 for elastic media. When β → ∞ $$ \beta \to \infty $$ , all the β k $$ {\beta}_k $$ collapse into the Rayleigh contrast parameter31, 84:
lim β → ∞ β k = 1 − ρ 1 + ρ = κ ( m ) − κ ( i ) κ ( m ) + κ ( i ) ≡ ϱ , $$ \underset{\beta \to \infty }{\lim }{\beta}_k=\frac{1-\rho }{1+\rho }=\frac{\kappa^{(m)}-{\kappa}^{(i)}}{\kappa^{(m)}+{\kappa}^{(i)}}\equiv \varrho, $$ (39)
and this formulation coincides with those reported in literature.25, 76
For ρ → ∞ $$ \rho \to \infty $$ :
lim ρ → ∞ β k = 1 , $$ \underset{\rho \to \infty }{\lim }{\beta}_k=1, $$
and, on the other hand, for ρ = 0 $$ \rho =0 $$ :
β k = ϱ , $$ {\beta}_k=\varrho, $$
as in (39). These limit cases correspond to media whose inclusions are superconductive or thermally insulating, respectively. In both cases the system does not depend on β $$ \beta $$ .
Additionally, it is convenient, in some contexts, to write the system (37) in its matrix form. Let us denote the following infinite matrices by
I = ( δ k l ) , H = ( η k l ) , P = ( δ k l R k ) , B = ( δ k l β k ) . $$ I=\left({\delta}_{kl}\right),\kern2em H=\left({\eta}_{kl}\right),\kern2em P=\left({\delta}_{kl}{R}^k\right),\kern2em B=\left({\delta}_{kl}{\beta}_k\right). $$
With this notation, the vectors being infinite a = ( a 1 , a 3 , … ) ⊤ $$ a={\left({a}_1,{a}_3,\dots \right)}^{\top } $$ and e 1 ≡ ( 1,0 , 0 , … ) ⊤ , $$ {e}_1\equiv {\left(1,0,0,\dots \right)}^{\top }, $$ the system can be written as
M a ≡ ( I − B P 2 H ) a = v , $$ Ma\equiv \left(I-B{P}^2H\right)a=v, $$ (40)
where v = B P 2 e 1 = ( β 1 R 2 , 0,0 , … ) ⊤ $$ v=B{P}^2{e}_1={\left({\beta}_1{R}^2,0,0,\dots \right)}^{\top } $$ and B P 2 H = ( β k R 2 k η k l ) $$ B{P}^2H=\left({\beta}_k{R}^{2k}{\eta}_{kl}\right) $$ .

3.3 Effective coefficient approximations

In this section, we will obtain approximations to the solutions of the infinite system and the effective coefficients associated with these approximations through finite order truncations.

To calculate the effective properties (13) and (14), we can use
κ ^ i j = κ i j W + κ i k ∂ N j ∂ y k W = δ i j κ W + κ ∂ N j ∂ y i W . $$ {\displaystyle \begin{array}{cc}\hfill {\hat{\kappa}}_{ij}=& \kern0.2em {\left\langle {\kappa}_{ij}\right\rangle}_W+{\left\langle {\kappa}_{ik}\frac{\partial {N}_j}{\partial {y}_k}\right\rangle}_W\hfill \\ {}\hfill =& \kern0.2em {\delta}_{ij}{\left\langle \kappa \right\rangle}_W+{\left\langle \kappa \frac{\partial {N}_j}{\partial {y}_i}\right\rangle}_W.\hfill \end{array}} $$ (41)
For κ 11 $$ {\kappa}_{11} $$ , we can apply the Green theorem to get
κ ∂ N 1 ∂ y 1 W = − ∫ − π π ∑ k = 1 ∞ o κ ( m ) a 1 η 11 δ 1 k + a k R − 2 k + σ k − κ ( m ) a 1 η 11 δ 1 k − a k R − 2 k + σ k + ( 1 − ρ ) δ 1 k R k cos ( k θ ) R cos θ d θ = − ∑ k = 1 ∞ o κ ( m ) 2 a k R − 2 k − ( 1 − ρ ) δ 1 k R k + 1 ∫ − π π cos ( k θ ) cos θ d θ , $$ {\displaystyle \begin{array}{ll}{\left\langle \kappa \frac{\partial {N}_1}{\partial {y}_1}\right\rangle}_W=& -\underset{-\pi }{\overset{\pi }{\int }}{\sum \limits_{k=1}^{\infty}}^o\left({\kappa}^{(m)}\left[{a}_1{\eta}_{11}{\delta}_{1k}+{a}_k{R}^{-2k}+{\sigma}_k\right]\right.\\ {}& -{\kappa}^{(m)}\left.\left[{a}_1{\eta}_{11}{\delta}_{1k}-{a}_k{R}^{-2k}+{\sigma}_k+\left(1-\rho \right){\delta}_{1k}\right]\right){R}^k\cos \left( k\theta \right)R\cos \theta d\theta \\ {}=& -{\sum \limits_{k=1}^{\infty}}^o{\kappa}^{(m)}\left[2{a}_k{R}^{-2k}-\left(1-\rho \right){\delta}_{1k}\right]{R}^{k+1}\underset{-\pi }{\overset{\pi }{\int }}\cos \left( k\theta \right)\cos \theta d\theta, \end{array}} $$
and ∫ − π π cos ( k θ ) cos θ d θ = π δ k 1 $$ \underset{-\pi }{\overset{\pi }{\int }}\cos \left( k\theta \right)\cos \theta d\theta =\pi {\delta}_{k1} $$ . It is obtained then
κ ^ 11 = κ ( m ) − ( κ ( m ) − κ ( i ) ) π R 2 + ( κ ( m ) − κ ( i ) ) π R 2 − 2 κ ( m ) π a 1 = κ ( m ) ( 1 − 2 π a 1 ) . $$ {\displaystyle \begin{array}{cc}\hfill {\hat{\kappa}}_{11}=& \kern0.2em {\kappa}^{(m)}-\left({\kappa}^{(m)}-{\kappa}^{(i)}\right)\pi {R}^2+\left({\kappa}^{(m)}-{\kappa}^{(i)}\right)\pi {R}^2-2{\kappa}^{(m)}\pi {a}_1\hfill \\ {}\hfill =& \kern0.2em {\kappa}^{(m)}\left(1-2\pi {a}_1\right).\hfill \end{array}} $$ (42)
For the rest of coefficients
κ ^ 12 = κ ^ 21 = 0 , $$ {\hat{\kappa}}_{12}={\hat{\kappa}}_{21}=0, $$ (43)
κ ^ 22 = κ ^ 11 , $$ {\hat{\kappa}}_{22}={\hat{\kappa}}_{11}, $$ (44)
This means that it is not necessary to solve the entire infinite system but only to find the solution for the first variable a 1 $$ {a}_1 $$ . For small order truncations it is possible to find formulas where few terms are involved, however, for the rest of truncations, it is convenient to use the system in its matrix form (40) and solve it by numerical methods.
Take, for example, a first-order truncation of the system (37), that is, when we keep only a single unknown ( a 1 $$ {a}_1 $$ ) in (37). Then
β 1 a 1 η 11 + 1 − a 1 R − 2 = 0 , $$ {\beta}_1\left[{a}_1{\eta}_{11}+1\right]-{a}_1{R}^{-2}=0, $$ (45)
Thus,
a 1 = − β 1 ( β 1 η 11 − R − 2 ) − 1 = ( R − 2 β 1 − 1 − η 11 ) − 1 , $$ {a}_1=-{\beta}_1{\left({\beta}_1{\eta}_{11}-{R}^{-2}\right)}^{-1}={\left({R}^{-2}{\beta}_1^{-1}-{\eta}_{11}\right)}^{-1}, $$ (46)
where
β 1 = ( 1 − ρ ) β + κ ( i ) ( 1 + ρ ) β + κ ( i ) . $$ {\beta}_1=\frac{\left(1-\rho \right)\beta +{\kappa}^{(i)}}{\left(1+\rho \right)\beta +{\kappa}^{(i)}}. $$
For the square cell, from (42), it is obtained
κ ^ 11 κ ( m ) = 1 − π R 2 β 1 1 + π R 2 β 1 = 1 − ϕ β 1 1 + ϕ β 1 . $$ \frac{{\hat{\kappa}}_{11}}{\kappa^{(m)}}=\frac{1-\pi {R}^2{\beta}_1}{1+\pi {R}^2{\beta}_1}=\frac{1-\phi {\beta}_1}{1+\phi {\beta}_1}. $$ (47)
This result agrees with the classic results of previous studies,41, 85 who derive them independently for a spheroid of which the fiber is the limiting case when its major axis tends to infinity. Equivalent formulas were obtained by previous works25, 75, 76 in the perfect contact case ( β → ∞ $$ \beta \to \infty $$ ):
κ ^ 11 κ ( m ) = 1 − π R 2 β 1 1 + π R 2 β 1 = 1 − ϕ ϱ 1 + ϕ ϱ . $$ \frac{{\hat{\kappa}}_{11}}{\kappa^{(m)}}=\frac{1-\pi {R}^2{\beta}_1}{1+\pi {R}^2{\beta}_1}=\frac{1-\phi \varrho}{1+\phi \varrho}. $$ (48)
which is the well known Clausius-Mossotti approximation42, 86, 87 (also recognized as Van Fo Fy formula24, 40 or Maxwell-Garnet approximation35, 49).
Also of interest are the cases κ ( i ) ∈ { 0 , ∞ } $$ {\kappa}^{(i)}\in \left\{0,\infty \right\} $$ :
lim κ ( i ) → 0 κ ^ 11 κ ( m ) = 1 − ϕ 1 + ϕ lim κ ( i ) → ∞ κ ^ 11 κ ( m ) = 1 + ϕ 1 − ϕ , $$ \underset{\kappa^{(i)}\to 0}{\lim}\frac{{\hat{\kappa}}_{11}}{\kappa^{(m)}}=\frac{1-\phi }{1+\phi}\kern2em \underset{\kappa^{(i)}\to \infty }{\lim}\frac{{\hat{\kappa}}_{11}}{\kappa^{(m)}}=\frac{1+\phi }{1-\phi }, $$
and agree with classic results.77-79
In the case of a second-order truncation of the system (37), we have
β 1 a 1 η 11 + a 3 η 31 + 1 − a 1 R − 2 = 0 , $$ {\beta}_1\left[{a}_1{\eta}_{11}+{a}_3{\eta}_{31}+1\right]-{a}_1{R}^{-2}=0, $$ (49)
β 3 a 1 η 13 + a 3 η 33 − a 3 R − 6 = 0 . $$ {\beta}_3\left[{a}_1{\eta}_{13}+{a}_3{\eta}_{33}\right]-{a}_3{R}^{-6}=0. $$ (50)
We shall notice that, for a square cell η 11 = − π , η 31 = − 3 S 4 , η 13 = − S 4 $$ {\eta}_{11}=-\pi, {\eta}_{31}=-3{S}_4,{\eta}_{13}=-{S}_4 $$ and η 33 = 0 $$ {\eta}_{33}=0 $$ , where
S 4 = Γ 1 4 8 960 π 2 ≈ 3 . 1512120021538975 … $$ {S}_4=\frac{\Gamma {\left(\frac{1}{4}\right)}^8}{960{\pi}^2}\approx 3.1512120021538975\dots $$
Finally, by Cramer's rule, we get
κ ^ 11 κ ( m ) = 1 − ϕ β 1 − 3 S 4 2 π − 4 β 1 β 3 ϕ 4 1 + ϕ β 1 − 3 S 4 2 π − 4 β 1 β 3 ϕ 4 . $$ \frac{{\hat{\kappa}}_{11}}{\kappa^{(m)}}=\frac{1-\phi {\beta}_1-3{S}_4^2{\pi}^{-4}{\beta}_1{\beta}_3{\phi}^4}{1+\phi {\beta}_1-3{S}_4^2{\pi}^{-4}{\beta}_1{\beta}_3{\phi}^4}. $$ (51)
Similarly, for a third-order truncation, we obtain
κ ^ 11 κ ( m ) = 1 − ϕ β 1 − 3 S 4 2 π − 4 β 1 β 3 ϕ 4 / Δ 1 1 + ϕ β 1 − 3 S 4 2 π − 4 β 1 β 3 ϕ 4 / Δ 1 , $$ \frac{{\hat{\kappa}}_{11}}{\kappa^{(m)}}=\frac{1-\phi {\beta}_1-3{S}_4^2{\pi}^{-4}{\beta}_1{\beta}_3{\phi}^4/{\Delta}_1}{1+\phi {\beta}_1-3{S}_4^2{\pi}^{-4}{\beta}_1{\beta}_3{\phi}^4/{\Delta}_1}, $$ (52)
where
Δ 1 = 1 − 735 S 8 2 π − 8 β 3 β 5 ϕ 8 , $$ {\Delta}_1=1-735{S}_8^2{\pi}^{-8}{\beta}_3{\beta}_5{\phi}^8, $$ (53)
and
S 8 = 3 7 S 4 2 ≈ 4 . 2557730353651895 … $$ {S}_8=\frac{3}{7}{S}_4^2\approx 4.2557730353651895\dots $$
Furthermore, a fourth-order truncation leads to the following formula:
κ ^ 11 κ ( m ) = 1 − ϕ β 1 − C 4 β 1 β 3 ϕ 4 / Δ 2 − C 8 β 1 β 7 ϕ 8 Δ 1 ′ / Δ 2 1 + ϕ β 1 − C 4 β 1 β 3 ϕ 4 / Δ 2 − C 8 β 1 β 7 ϕ 8 Δ 1 ′ / Δ 2 , $$ \frac{{\hat{\kappa}}_{11}}{\kappa^{(m)}}=\frac{1-\phi {\beta}_1-{C}_4{\beta}_1{\beta}_3{\phi}^4/{\Delta}_2-{C}_8{\beta}_1{\beta}_7{\phi}^8{\Delta}_1^{\prime }/{\Delta}_2}{1+\phi {\beta}_1-{C}_4{\beta}_1{\beta}_3{\phi}^4/{\Delta}_2-{C}_8{\beta}_1{\beta}_7{\phi}^8{\Delta}_1^{\prime }/{\Delta}_2}, $$ (54)
where
C 4 = 3 S 4 2 π − 4 , C 8 = 7 S 8 2 π − 8 , Δ 1 ′ ( ϕ ) = Δ 1 ( ϕ ) − b 12 β 5 β 7 ϕ 12 , Δ 2 = 1 − 735 S 8 2 π − 8 β 3 β 5 ϕ 8 − 6 6 2 · 35 S 12 2 π − 12 β 5 β 7 ϕ 12 , $$ {\displaystyle \begin{array}{ll}{C}_4=& \kern0.2em 3{S}_4^2{\pi}^{-4},\kern0.30em {C}_8=7{S}_8^2{\pi}^{-8},\\ {}{\Delta}_1^{\prime}\left(\phi \right)=& \kern0.2em {\Delta}_1\left(\phi \right)-{b}_{12}{\beta}_5{\beta}_7{\phi}^{12},\\ {}{\Delta}_2=& \kern0.2em 1-735{S}_8^2{\pi}^{-8}{\beta}_3{\beta}_5{\phi}^8-6{6}^2\cdotp 35{S}_{12}^2{\pi}^{-12}{\beta}_5{\beta}_7{\phi}^{12},\end{array}} $$
and
S 12 = 18 143 S 4 3 ≈ 3 . 9388490128279704 … . $$ {S}_{12}=\frac{18}{143}{S}_4^3\approx 3.9388490128279704\dots . $$
The fourth-order formula (54) will be used in Section 4 to obtain the effective coefficient via conventional and reiterated homogenization. As a matter of validation, we compare next our approach with analytical formulas, variational bounds, numerical results, and limiting cases found in the literature.

Higher order formulas can be found from this procedure, but the current ones are sufficient for our goal. For detailed information of these formulas, up to fifth order, we refer to Table A1.

The similarities in the structure of (47), (51), and (52) are not casual. Let us denote by Δ 11 $$ {\Delta}_{11} $$ the ( 1,1 ) $$ \left(1,1\right) $$ -th minor from the infinite matrix M $$ M $$ in (40) (i.e., the determinant of the ( n − 1 ) × ( n − 1 ) $$ \left(n-1\right)\times \left(n-1\right) $$ matrix formed by removing the first row and column of M $$ M $$ ) by Cramer's rule, and taking into account that M 11 = ( 1 − β 1 R 2 η 11 ) $$ {M}_{11}=\left(1-{\beta}_1{R}^2{\eta}_{11}\right) $$ , we have
a 1 = β 1 R 2 Δ 11 ( 1 − β 1 R 2 η 11 ) Δ 11 − T , $$ {a}_1=\frac{\beta_1{R}^2{\Delta}_{11}}{\left(1-{\beta}_1{R}^2{\eta}_{11}\right){\Delta}_{11}-T}, $$
where T $$ T $$ is a weighted sum of terms of order not less than R 2 $$ {R}^2 $$ . Notice that this is also valid for any truncation of the system. From here:
κ ^ κ ( m ) = 1 − ϕ β 1 − τ 1 + ϕ β 1 − τ , $$ \frac{\hat{\kappa}}{\kappa^{(m)}}=\frac{1-\phi {\beta}_1-\tau }{1+\phi {\beta}_1-\tau }, $$ (55)
where τ $$ \tau $$ is a weighted sum of terms of order not less than ϕ 2 $$ {\phi}^2 $$ . This representation will be used in Table A1.
Finally, we analyze if there are values of β $$ \beta $$ such as κ ^ κ ( m ) = 1 $$ \frac{\hat{\kappa}}{\kappa^{(m)}}=1 $$ . From (55), that is possible if and only if β 1 = 0 $$ {\beta}_1=0 $$ , which, from (38) gives
( ρ − 1 ) β = κ ( i ) . $$ \left(\rho -1\right)\beta ={\kappa}^{(i)}. $$
That is, for ρ ≥ 1 $$ \rho \ge 1 $$ , there exists a critical value for the Biot number, independent of ϕ $$ \phi $$ :
β c = κ ( i ) κ ( m ) κ ( i ) − κ ( m ) $$ {\beta}_c=\frac{\kappa^{(i)}{\kappa}^{(m)}}{\kappa^{(i)}-{\kappa}^{(m)}} $$ (56)
such as the effective conductivity k ^ 11 $$ {\hat{k}}_{11} $$ is equal to the conductivity of the matrix k ( m ) $$ {k}^{(m)} $$ . A similar result have been reported in literature.26, 51

3.4 Validation

As mentioned in Section 3.2, the system (37) agree with the ones presented by previous studies.26, 81 In this section, we will evaluate the behavior of the different proposed truncations, comparing them with the results obtained there and others present in the literature.

Similar developments to ours are found in the literature for elastic materials. In Table 1, it is compared with the effective longitudinal shear modulus for a lattice of square cells and thermally insulating fibers (voids in the elastic equivalent). In this case, even second formulas are very accurate comparing them with the results present in the literature, and small improvement are obtained from higher order.

TABLE 1. Comparison of effective conductivities in the case of thermally insulating fibers ( ρ = 1 0 − 6 / 3 ) $$ \left(\rho =1{0}^{-6}/3\right) $$ for different approximations and known results
Manteufel and
ϕ $$ \phi $$ Equation (47) Equation (51) Equation (52) Equation (54) Todreas36 López-Realpozoet al.26 Jiang et al.88
0.1 0.818182 0.818177 0.818177 0.818177 0.818177 0.818177 0.818177
0.2 0.666667 0.666531 0.666531 0.666531 0.666531 0.666530 0.666530
0.3 0.538462 0.537580 0.537580 0.537580 0.537580 0.537580 0.537580
0.4 0.428572 0.425358 0.425355 0.425351 0.425351 0.425350 0.425350
0.5 0.333333 0.324729 0.324681 0.324655 0.324655 0.324654 0.324653
0.6 0.25 0.230949 0.230477 0.230329 0.230322 0.230318 0.23032
  • a Tab. 1, sixth-order formula, at Manteufel and Todreas.36
  • b Tab. 2, at López-Realpozo et al.26
  • c Tab. 7, at Jiang et al.88

In Tables 2 and 3, the effective conductivities obtained for an arrangement of square cells with the current approximations are shown, comparing them with results present in the literature.26, 36, 56 We have considered ϕ = 0 . 3 $$ \phi =0.3 $$ and ϕ = 0 . 75 $$ \phi =0.75 $$ as representative of the small and large volume fractions, respectively. For small values of the volume fraction, all approximations are very similar. Even first-order formulas result in very accurate results for small concentrations, as we will see next from Maclaurin series analysis. For high volume fraction, the difference is more notable with increasing ρ $$ \rho $$ . This difference should vanish for higher order truncations, as seen in Table 4 near percolation. In this case, the numerical solution given by Guinovart-Díaz et al.27 for N 0 = 0,1 , 2,3 $$ {N}_0=0,1,2,3 $$ are equivalent to Equations (47), (51), (52) and Equation (54), respectively.

TABLE 2. Comparison of effective conductivities in the case of low volume fraction ( ϕ = 0 . 3 ) $$ \left(\phi =0.3\right) $$ for different approximations
Manteufel and López-Realpozo Rocha and
Bi ρ $$ \rho $$ Equation (47) Equation (51) Equation (52) Equation (54) Todreas36 et al.26 Cruz56
10−7 2 0.538462 0.537580 0.537580 0.537580 0.537580 0.537580 0.537620
10−5 0.538465 0.537584 0.537584 0.537584 0.537584 0.537584 0.537620
0.001 0.538816 0.537937 0.537937 0.537936 0.537936 0.537936 0.537970
0.01 0.541993 0.541129 0.541129 0.541128 0.541128 0.541128 0.541170
0.1 0.572212 0.571480 0.571480 0.571479 0.571479 0.571479 0.571510
1 0.772152 0.771995 0.771995 0.771995 0.771995 0.771995 0.772020
10 1.111801 1.111799 1.111799 1.111799 1.111799 1.111800 1.111800
100 1.209302 1.209353 1.209353 1.209353 1.209353 1.209350 1.209300
1000 1.220908 1.220974 1.220974 1.220974 1.220974 1.220970 1.220900
105 1.222209 1.222277 1.222277 1.222277 1.222277 1.222280 1.222200
107 1.222222 1.222290 1.222290 1.222290 1.222290 1.222290 1.222300
10−7 50 0.538462 0.537580 0.537580 0.537580 0.537580 0.537580 0.537620
10−5 0.538465 0.537584 0.537584 0.537584 0.537584 0.537584 0.537620
0.001 0.538816 0.537937 0.537937 0.537936 0.537936 0.537936 0.537970
0.01 0.541993 0.541129 0.541129 0.541128 0.541128 0.541128 0.541170
0.1 0.572212 0.571480 0.571480 0.571479 0.571479 0.571479 0.571510
1 0.772152 0.771995 0.771995 0.771995 0.771995 0.771995 0.772020
10 1.111801 1.111799 1.111799 1.111799 1.111799 1.111800 1.111800
100 1.209302 1.209353 1.209353 1.209353 1.209353 1.209350 1.209300
1000 1.220908 1.220974 1.220974 1.220974 1.220974 1.220970 1.220900
105 1.222209 1.222277 1.222277 1.222277 1.222277 1.222280 1.222200
107 1.222222 1.222290 1.222290 1.222290 1.222290 1.222290 1.222300
  • a Tab. 1, sixth-order formula, at Manteufel and Todreas.36
  • b Tab. 4, at López-Realpozo et al.26
  • c Tab. 1, at Rocha and Cruz.56
TABLE 3. Comparison of effective conductivities in the case of high volume fraction ( ϕ = 0 . 75 ) $$ \left(\phi =0.75\right) $$ for different approximations
Manteufel and López-Realpozo
Bi ρ $$ \rho $$ Equation (47) Equation (51) Equation (52) Equation (54) Todreas36 et al.26 FEM56
10−7 2 0.142857 0.092688 0.083926 0.080420 0.080038 0.078424 0.078423
10−5 0.142862 0.092693 0.083932 0.080426 0.080044 0.078431 0.078438
0.001 0.143347 0.093254 0.084512 0.081013 0.080632 0.079022 0.079030
0.01 0.147739 0.098331 0.089761 0.086328 0.085955 0.084381 0.084389
0.1 0.190311 0.147216 0.140183 0.137336 0.137030 0.135768 0.135770
1 0.513514 0.502053 0.500960 0.500430 0.500383 0.500212 0.500220
10 1.305085 1.304799 1.304799 1.304772 1.304772 1.304770 1.304800
100 1.620690 1.627780 1.627851 1.627939 1.627942 1.627940 1.627900
1000 1.661941 1.671344 1.671489 1.671646 1.671652 1.671660 1.671600
105 1.666619 1.676312 1.676468 1.676635 1.676641 1.676650 1.676600
107 1.666666 1.676362 1.676518 1.676685 1.676691 1.676700 1.676700
10−7 50 0.142857 0.092688 0.083926 0.080420 0.080038 0.078424 0.078423
10−5 0.142862 0.092693 0.083932 0.080426 0.080044 0.078431 0.078438
0.001 0.143347 0.093254 0.084512 0.081013 0.080632 0.079023 0.079030
0.01 0.147751 0.098344 0.089775 0.086342 0.085968 0.084395 0.084402
0.1 0.191441 0.148481 0.141474 0.138636 0.138332 0.137074 0.137080
1 0.595773 0.588092 0.587389 0.587039 0.587009 0.586898 0.586900
10 2.842520 2.895365 2.895347 2.894793 2.894785 2.894780 2.894800
100 5.500000 6.978402 7.189395 7.262994 7.270545 7.286740 7.287200
1000 6.084922 8.384497 8.897360 9.113386 9.137249 9.224800 9.225600
105 6.157156 8.579661 9.149738 9.396639 9.424103 9.532260 9.533100
107 6.157887 8.581664 9.152350 9.399586 9.427089 9.535470 9.536300
  • a Tab. 1, sixth-order formula, at Manteufel and Todreas.36
  • b Tab. 4, at López-Realpozo et al.26
  • c Tab. 1, at Rocha and Cruz.56
TABLE 4. Comparison of effective conductivities for volume fractions near percolation ( ϕ = 0 . 78 $$ \phi =0.78 $$ ) with different approximations27
Bi ρ $$ \rho $$ Equation (47) Equation (51) Equation (52) Equation (54) Guinovart-Díaz et al.27
0 6 0.1236 0.0641 0.0487 0.0405 0.0279
400/399 0.8883 0.8878 0.8878 0.8877 0.8877
6/5 1 1 1 1 1
6 2.2787 2.3029 2.3029 2.3026 2.3026
20 3.0176 3.1835 3.1912 3.1942 3.1946
120 3.4214 3.7491 3.7843 3.8001 3.8068
400 3.4872 3.8488 3.8925 3.9129 3.9233
1012 3.5161 3.8935 3.9415 3.9643 3.9771
0 400 0.1236 0.0641 0.0487 0.0405 0.0279
400/399 1 1 1 1 1
6/5 1.1500 1.1493 1.1492 1.1492 1.1492
6 3.4872 3.6452 3.6462 3.6455 3.6454
20 5.7156 7.3012 7.5026 7.5662 7.5823
120 7.4488 12.5760 14.7625 16.0557 17.3740
400 7.7811 14.0994 17.5498 20.0453 24.2400
1012 7.9329 14.8821 19.1605 22.6171 31.0040
  • a Tab. 3, N 0 = 20 $$ {N}_0=20 $$ , at Guinovart-Díaz et al.27
To analyze the approximation orders obtained in (47), (51), (52), and (54), it is necessary to write the developments in Maclaurin series around ϕ = 0 $$ \phi =0 $$ . For example, in the case of (47):
1 − ϕ β 1 1 + ϕ β 1 = 1 + 2 ∑ j = 1 ∞ − β 1 ϕ j . $$ \frac{1-\phi {\beta}_1}{1+\phi {\beta}_1}=1+2\sum \limits_{j=1}^{\infty }{\left(-{\beta}_1\phi \right)}^j. $$ (57)
The complexity of these developments increases as the truncation is of a higher order. However, we were able to obtain an expression up to the eighth term, which is true for all higher-order truncations. Moreover, the Maclaurin series of (54) and those of the highest order formulas proposed by Manteufel and Todreas36 (see Table A1) coincide up to the 12th term, and have the form:
κ ^ 11 κ ( m ) = 1 + 2 ∑ j = 1 12 − β 1 j ϕ j − 2 ∑ j = 1 8 3 j S 4 2 π − 4 − β 1 j + 1 β 3 ϕ 4 + j + 2 ∑ j = 1 4 9 j ( j + 1 ) 2 S 4 4 π − 8 − β 1 j + 2 β 3 2 − 7 j S 8 2 π − 8 − β 1 j + 1 β 7 ϕ 8 + j + O ( ϕ 13 ) . $$ {\displaystyle \begin{array}{cc}\hfill \frac{{\hat{\kappa}}_{11}}{\kappa^{(m)}}=& \kern0.2em 1+2\sum \limits_{j=1}^{12}{\left(-{\beta}_1\right)}^j{\phi}^j-2\sum \limits_{j=1}^8\left(3j{S}_4^2{\pi}^{-4}{\left(-{\beta}_1\right)}^{j+1}{\beta}_3\right){\phi}^{4+j}\hfill \\ {}\hfill & +2\sum \limits_{j=1}^4\left(9\frac{j\left(j+1\right)}{2}{S}_4^4{\pi}^{-8}{\left(-{\beta}_1\right)}^{j+2}{\beta}_3^2-7j{S}_8^2{\pi}^{-8}{\left(-{\beta}_1\right)}^{j+1}{\beta}_7\right){\phi}^{8+j}\hfill \\ {}\hfill & +O\left({\phi}^{13}\right).\hfill \end{array}} $$ (58)

 

The development (58) is a generalization of Mityushev's89 (Equation 35) and Godin's90, (Equation 76) result and coincides exactly with them in the context of perfect contact, that is, when β → ∞ $$ \beta \to \infty $$ .

Additionally, Table 5 compares the results from Drygas and Mityushev51 (formula 81) with those derived from formulas in the present work for ρ = 19 $$ \rho =19 $$ . In fact, although (54) was used for consistency, second-order formula (51) is sufficient to reproduce this results for ϕ = 0 . 126 $$ \phi =0.126 $$ and ϕ = 0 . 212 $$ \phi =0.212 $$ and third-order formula up to ϕ = 0 . 322 $$ \phi =0.322 $$ . We shall remark that the periodic cell analyzed in Drygas and Mityushev51 is different from our model, depicted in Figure 3. In their case, inclusions are represented by four identical not overlapping circles in the unit square aligned over the axis (see fig. 2 in Drygas and Mityushev51). Thus, the volumetric fraction used in the calculations in Table 5 equals the joined areas of the four circles.

TABLE 5. Comparison of effective conductivities for different volume fractions in the 4-inclusion model from51 r = ϕ 4 π $$ \left(r=\sqrt{\frac{\phi }{4\pi }}\right) $$ and formula (54)
B i $$ Bi $$ ϕ = 0 . 126 $$ \phi =0.126 $$ ϕ = 0 . 212 $$ \phi =0.212 $$ ϕ = 0 . 322 $$ \phi =0.322 $$ ϕ = 0 . 454 $$ \phi =0.454 $$
(54) ( r = 0 . 1 ) $$ \left(r=0.1\right) $$ 51 (54) ( r = 0 . 13 ) $$ \left(r=0.13\right) $$ 51 (54) ( r = 0 . 16 ) $$ \left(r=0.16\right) $$ 51 (54) ( r = 0 . 19 ) $$ \left(r=0.19\right) $$ 51
1.357 1.03000 1.0300 1.05120 1.0512 1.07870 1.0786 1.11270 1.1126
1.267 1.02180 1.0218 1.03710 1.0371 1.05670 1.0567 1.08090 1.0808
1.188 1.01410 1.0141 1.02390 1.0239 1.03640 1.0364 1.05170 1.0516
1.118 1.00680 1.0068 1.01150 1.0115 1.01750 1.0175 1.02480 1.0248
19 / 18 1 1 1 1 1 1 1 1
1.000 0.99358 0.9935 0.98917 0.9891 0.98364 0.9836 0.97700 0.9769
0.950 0.98751 0.9875 0.97899 0.9789 0.96834 0.9683 0.95563 0.9555
0.905 0.98178 0.9817 0.96940 0.9693 0.95400 0.9539 0.93571 0.9355
0.864 0.97635 0.9763 0.96034 0.9603 0.94054 0.9404 0.91711 0.9167
  • a Tab. 1 at Drygas and Mityushev.51

The obtained values for critical β = β c $$ \beta ={\beta}_c $$ in Tables 4 and 5 are consistent with (56). For ρ = 2 $$ \rho =2 $$ and ρ = 50 $$ \rho =50 $$ , the critical values are β c = 2 $$ {\beta}_c=2 $$ and β c = 50 / 49 ≈ 1 . 02 $$ {\beta}_c=50/49\approx 1.02 $$ , respectively.

4 RESULTS AND DISCUSSION

In Evans et al.,16 an ad hoc homogenization model was developed to analyze the role of aggregation processes and interfacial thermal resistance on the effective thermal conductivity of nanofluids and nanocomposites. Gain has also been reported for multiscale fibrous and particulate composites19-21 considering perfect contact and for laminates14, 23 taking into account imperfect contact.

Given K ( m ) , K ( i ) $$ {K}^{(m)},{K}^{(i)} $$ the thermal conductivities of the matrix and inclusions, let us denote by K ^ C H , K ^ R H $$ {\hat{K}}_{CH},{\hat{K}}_{RH} $$ the effective coefficient of the whole medium obtained via conventional (one-level) and reiterated (multi-level) homogenization. For our isotropic medium case, we can write them in terms of their scalar equivalents:
K ( m ) = k ( m ) I , K ( i ) = k ( i ) I , K ^ C H = k ^ C H I , K ^ R H = k ^ R H I , $$ {\displaystyle \begin{array}{ll}{K}^{(m)}={k}^{(m)}I,& \kern0.1em {K}^{(i)}={k}^{(i)}I,\\ {}{\hat{K}}_{CH}={\hat{k}}_{CH}I,& \kern0.1em {\hat{K}}_{RH}={\hat{k}}_{RH}I,\end{array}} $$
where I $$ I $$ is the identity matrix.

These quantities will depend on the volume fraction of inclusions ( ϕ $$ \phi $$ ), the phase conductivity ratio ( ρ = k ( i ) / k ( m ) $$ \rho ={k}^{(i)}/{k}^{(m)} $$ ), the geometry, and the thermal barrier (characterized by β $$ \beta $$ ). The geometry is characterized by the shape of the inclusions and for k ^ R H $$ {\hat{k}}_{RH} $$ will also depend on the shape of the clusters and the aggregation parameter ( α $$ \alpha $$ ), previously discussed in Section 1.1.

Both k ^ C H $$ {\hat{k}}_{CH} $$ and k ^ R H $$ {\hat{k}}_{RH} $$ can be calculated by means of the formulas obtained in Section 3.2, or other means. We obtain k ^ C H $$ {\hat{k}}_{CH} $$ by direct application of the fourth order formula (54) and k ^ R H $$ {\hat{k}}_{RH} $$ applying the same formula twice: first in (11) to obtain the intermediate effective coefficient (14) and finally in (12) to obtain the global effective coefficient (13). Notice that, from (43)–(44), they have the isotropy property and only one coefficient should be calculated in each step.

For comparison, we consider the cases where the volume fraction ϕ $$ \phi $$ of the inclusions is the same and the complexity of the second media is characterized by the aggregation parameter α $$ \alpha $$ . We are particularly interested in the benefits or inconvenience of considering two micro-scales (reiterated homogenization) against only one micro-scale (conventional homogenization). For this purpose, we define a gain function as in Nascimento et al.19 in the form:
k g a i n ( α , ϕ , ρ , β ) ≡ k ^ R H ( α , ϕ , ρ , β ) k ^ C H ( ϕ , ρ , β ) . $$ {k}_{gain}\left(\alpha, \phi, \rho, \beta \right)\equiv \frac{{\hat{k}}_{RH}\left(\alpha, \phi, \rho, \beta \right)}{{\hat{k}}_{CH}\left(\phi, \rho, \beta \right)}. $$ (59)
Notice that, by definition, for α ∈ { 0,1 } $$ \alpha \in \left\{0,1\right\} $$ : k ^ R H = k ^ C H $$ {\hat{k}}_{RH}={\hat{k}}_{CH} $$ and k g a i n = 1 $$ {k}_{gain}=1 $$ .

We will analyze the effect of the thermal barrier at the interface for two-phase media, which, as typical in practical applications, have much higher conductivity than the matrix (we propose ρ = 500 $$ \rho =500 $$ ). It does appear that the set of results that are calculated and analyzed here represent a novel contribution of the present work to the field of thermal composite media.

Figures 4-6 illustrate k g a i n $$ {k}_{gain} $$ -curves (for different concentrations ϕ $$ \phi $$ ) versus the aggregation parameter α $$ \alpha $$ .

For greater Biot numbers (better contact), it is observed that the gain tends to grow with the concentration and, for the higher concentrations, oscillates and starts to decrease reporting loss near the maximum possible value of ϕ $$ \phi $$ (percolation), as shown in Figure 4. These gains have a maximum up to 9% for ϕ ∈ ( 0 . 5,0 . 6 ) $$ \phi \in \left(0.5,0.6\right) $$ and α ≈ 0 . 6 $$ \alpha \approx 0.6 $$ .

For low concentration volumes, the increasing behavior (as ϕ $$ \phi $$ increase) is in agreement with the results of the literature.32, 49 It is interesting that this behavior is also manifested for higher concentration until a certain threshold (which, under the conditions of Figure 4, seems to be close to 0.68). For even higher concentrations ( ϕ ≥ 0 . 7 $$ \phi \ge 0.7 $$ ), under the conditions of this model, loss is obtained for all value of aggregation.

Details are in the caption following the image
FIGURE 4
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Gain for ρ = 500 $$ \rho =500 $$ as a function of the aggregation for different concentrations and high Biot numbers B i ∈ { 1 0 3 ; 1 0 2 } $$ Bi\in \left\{1{0}^3;1{0}^2\right\} $$ ) [Colour figure can be viewed at wileyonlinelibrary.com]
Details are in the caption following the image
FIGURE 5
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Gain for ρ ∈ { 2,6 , 50,500 } $$ \rho \in \left\{2,6,\mathrm{50,500}\right\} $$ as a function of the aggregation for different concentrations. In this case the perfect contact ( B i → ∞ $$ Bi\to \infty $$ ) is computed using the by approximation B i = 1 0 10 $$ Bi=1{0}^{10} $$ [Colour figure can be viewed at wileyonlinelibrary.com]
Details are in the caption following the image
FIGURE 6
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Gain for ρ = 500 $$ \rho =500 $$ as a function of the aggregation for different concentrations and small Biots numbers ( B i ∈ { 1 0 − 1 ; 1 0 − 2 ; 1 0 − 5 } $$ Bi\in \left\{1{0}^{-1};1{0}^{-2};1{0}^{-5}\right\} $$ ) [Colour figure can be viewed at wileyonlinelibrary.com]

A similar behavior was observed in the perfect contact case: The gain continues its tendency to grow with ϕ $$ \phi $$ for small concentrations and then oscillates around the values obtained for ϕ = 0 . 5 $$ \phi &#x0003D;0.5 $$ and starts to decrease reporting loss near percolation. Figure 5 shows the influence of aggregation in the property gain in the case of perfect contact with high inclusions volume fraction around ϕ ∈ [ 0 . 5,0 . 7 ] $$ \phi \in \left[0.5,0.7\right] $$ . In each case, it is shown behaviors where there are alternating regions with gain/loss (i.e., k g a i n > 1 $$ {k}_{gain}&gt;1 $$ or k g a i n < 1 $$ {k}_{gain}&lt;1 $$ , respectively). It is worth to notice that these alternating regions were reported too for high volume fraction by Berlyand and Mityushev49 in the perfect contact case.

For Figure 5, we consider different contrasts ( ρ ∈ { 2,6 , 50,500 } $$ \rho \in \left\{2,6,\mathrm{50,500}\right\} $$ ) to analyze the influence of this parameter. It can be seen that an increase of the contrast ρ $$ \rho $$ contributes to the increase in the length of gain-zones. Notice, for instance, the behavior of the curves corresponding to ϕ = 0 . 68 $$ \phi &#x0003D;0.68 $$ . For ρ = 2 $$ \rho &#x0003D;2 $$ , k g a i n ≤ 1 $$ {k}_{gain}\le 1 $$ for all α $$ \alpha $$ ; for ρ = 6 $$ \rho &#x0003D;6 $$ , k g a i n ≥ 1 $$ {k}_{gain}\ge 1 $$ for α ∈ ( 0 . 8,0 . 9 ] $$ \alpha \in \left(0.8,0.9\right] $$ ; for ρ = 50 $$ \rho &#x0003D;50 $$ , k g a i n ≥ 1 $$ {k}_{gain}\ge 1 $$ for α > 0 . 3 $$ \alpha &gt;0.3 $$ ; and for ρ = 500 $$ \rho &#x0003D;500 $$ , k g a i n ≥ 1 $$ {k}_{gain}\ge 1 $$ for all α $$ \alpha $$ .

For smaller Biot numbers (worst contact), the behavior is the opposite: The gain decreases for small concentrations with certain loss and starts to grow for higher ones (Figure 6). The concentration value where this change of behavior happens decreases when the Biot number decreases, reporting gains for each fixed small Biot at higher concentrations. On the other hand, the maximum gain increases when the Biot number decreases, being higher than 25% in the case of B i → 0 $$ Bi\to 0 $$ , which represents decoupling of the phases. Independence of ρ $$ \rho $$ for no gain/loss value also occurs in this case.

It is physically expected that the gain should increase as the Biot number decreases and this behavior is reflected in our results. In fact, the three-scale geometric arrangement is better conductive than the associated two-scale counterpart. Thus, the better conductive three-scale geometric arrangement will be relatively more important as the interfacial thermal barrier increases.

5 CONCLUDING REMARKS

In the present work, the reiterated homogenization method was applied to investigate the macroscopic behavior of fibrous composites with aggregation. Although the model and procedures are very general, due to the limitation of analytical procedures employed, for application purposes, we restrict the present work to parallel isotropic long fibers with two microstructural levels. A thermal barrier was considered assuming imperfect spring-type contact condition at the interface.34

The partial differential equations derived from reiterated homogenization method (RH) were solved via an analytical method, following a Rayleigh's type methodology. For this, we construct a quasiperiodic harmonic solution using a combination of the Weierstrass Z function and its derivatives, and substituting in the conditions, we arrive to an infinite system of equations that can be solved by a truncation method. We derive five types of explicit analytical formulas for the effective thermal coefficient with different approximation orders (Equations (47), (51), (52), (54), and a fifth-order truncation, see Table A1).

The size of the clusters allows for rapid heat flow over large distances. On the other hand, long surface areas with interfacial resistance will slow the heat flow significantly. The cluster morphology and nanoparticles shape will affect this behavior although we left this question as a matter of study for future research.

These formulas are expressed in the style of those reported by Manteufel and Todreas36 for perfect contact in a three-phase conductive problem, establishing an equivalence relationship between both models. The comparison with results in the literature, including the finite element method,56 showed an improvement of the accuracy as truncation order increase. In particular, for small concentrations, the obtained Maclaurin series expansion is a generalization of the results in previous studies89, 90 (Equation (58)).

The two micro-structural levels' analysis in Nascimento et al.19 is generalized to the case of imperfect interfacial contact. The results reveal the role of aggregation processes and imperfect contact in thermal conductivity. In fact, we can identify the ranges of problem parameters in which the arrangement of the inclusions in three scales leads to gain/loss relative to the conventional two-scales arrangement. It has been found that the values of the gain are highly dependent on Biot numbers, with attained maximums that go from 9% when B i → ∞ $$ Bi\to \infty $$ (perfect contact) to beyond 25% when B i → 0 $$ Bi\to 0 $$ (decoupling). Besides, while Nascimento et al.19 considered low volume fraction, our method is also feasible in high volume fraction cases.

The present reiterated homogenization-based research is a work-in-progress, and formulas for other type of lattices can be obtained from this methodology. We are currently working on the study of thermal conductivity gain in periodic three-dimensional composites.30, 91-94 We are also interested in the study of non-uniform thermal barriers13, 57, 95 and non-spring type imperfect contact.96-98

The asymptotic homogenization method is a mathematically rigorous tool to investigate processes that occur in periodic media. Advantages and disadvantages of this method are explained, for example, in the introduction of Bakhvalov and Panasenko.69 The reiterated homogenization method (RH) that we apply here is an asymptotic homogenization method useful for investigating processes that occur in periodic structures provided with several microstructural levels. This methodology simplifies the calculations and offers mathematical models to estimate the effective conductivity of the modeled composite. However, not considering small and large circles in the same cell could be a limitation for the study of the interactions between both types of inclusions. A study of the limitations of RH implementation can be seen in Mityushev.55

ACKNOWLEDGEMENTS

E. Iglesias-Rodríguez would like to thank CONACYT for support his PhD studies at UNAM. Prof. M. E. Cruz is grateful to CNPq-Brazilian Council for Development of Science and Technology for Grant PQ-305089-2020/0. Support Program for Research and Technological innovation projects (PAPIIT), Dirección General de Asuntos del Personal Academico (DGAPA), UNAM Project Number IN101822 is also recognized. Part of this work was done when Professor R. Ginovart-Díaz visited the Unidad Académica del IIMAS del Estado de Yucatán, Mérida. Professor Ginovart-Díaz is grateful for the hospitality and support of his visit.

    CONFLICT OF INTEREST

    This work does not have any conflicts of interest.

    APPENDIX A: COMPARISON WITH THREE-PHASE METHOD

    In the literature, there are different ways of modeling imperfect contact. So far it has been considered the imperfection as a surface with certain effect that separates two phases, one representing the matrix and the other formed by the inclusions. Another model can be based on considering a three-phase medium, the ones mentioned and a third that separates both, with its own characteristics and properties.25, 35, 36, 39 In this way, when the thickness of the intermediate layer is very thin, both models should be equivalent (Figure A1).

    Details are in the caption following the image
    FIGURE A1
    Open in figure viewerPowerPoint
    Cross section of an equivalent three-phase medium [Colour figure can be viewed at wileyonlinelibrary.com]

    For this three-phase model, there are two main difficulties: From the theoretical point of view, the difficulty is to guarantee the equivalence of both in the limit of zero intermediate-material thickness (existence of said limit, that the problems are well-posed, etc.). From a practical point of view, the numerical methods some times do not converge to the correct value when the dimensions of the layer are very small (due to discretization, aliasing, and other numerical artifacts).

    In this section, we will compare the analytical results of Manteufel and Todreas36 for a three-phase model with our proposal. An equivalent result can be found in Nicorovici et al.35 in the dielectric case via a similar three-phase model. In this way, for small thicknesses, we can validate the results and look for conditions that guarantee the equivalence of both models, at least in the limiting cases, when the thickness tends to zero.

    We will denote by
    κ ( y ) = κ ( m ) , y ∈ Y ( m ) (matrix) , κ ( Γ ) , y ∈ Y ( Γ ) (interface) , κ ( i ) , y ∈ Y ( i ) (inclusion) , $$ \kappa (y)&#x0003D;\left\{\begin{array}{rlll}{\kappa}&#x0005E;{(m)},&amp; \kern6.05pt &amp; y\in {Y}&#x0005E;{(m)}&amp; \kern0.30em \left(\mathrm{matrix}\right),\\ {}{\kappa}&#x0005E;{\left(\Gamma \right)},&amp; \kern6.05pt &amp; y\in {Y}&#x0005E;{\left(\Gamma \right)}&amp; \kern0.30em \left(\mathrm{interface}\right),\\ {}{\kappa}&#x0005E;{(i)},&amp; \kern6.05pt &amp; y\in {Y}&#x0005E;{(i)}&amp; \kern0.30em \left(\mathrm{inclusion}\right),\end{array}\right. $$
    and by
    r o = r + δ 2 , r i = r − δ 2 . $$ {r}_o&#x0003D;r&#x0002B;\frac{\delta }{2},\kern2em {r}_i&#x0003D;r-\frac{\delta }{2}. $$
    Hypothesis: The following developments will be carried out under the assumption that there is a certain normalization constant C ( r ) $$ C(r) $$ such that, for the models presented, the following limit exists:
    lim δ → 0 κ Γ δ β = C ( r ) . $$ \underset{\delta \to 0}{\lim}\frac{\kappa&#x0005E;{\Gamma}}{\delta \beta}&#x0003D;C(r). $$ (A1)
    In Manteufel and Todreas,36 we obtain approximations of the effective property, pretty similar to (55), having the form
    F cond ≡ κ ^ κ ( m ) = 1 − ϕ v 1 − t e r m s 1 + ϕ v 1 − t e r m s = 1 − 2 ϕ v 1 1 + ϕ v 1 − t e r m s . $$ {F}_{\mathrm{cond}}\equiv \frac{\hat{\kappa}}{\kappa&#x0005E;{(m)}}&#x0003D;\frac{1-\phi {v}_1- terms}{1&#x0002B;\phi {v}_1- terms}&#x0003D;1-\frac{2\phi {v}_1}{1&#x0002B;\phi {v}_1- terms}. $$ (A2)
    Here, the expression t e r m s $$ terms $$ is referred as a weighted sum of terms of ϕ $$ \phi $$ whose coefficients are calculated using the following formulas:
    v k = χ o + χ i r i r o 2 k 1 + χ o χ i r i r o 2 k , $$ {v}_k&#x0003D;\frac{\chi_o&#x0002B;{\chi}_i{\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}}{1&#x0002B;{\chi}_o{\chi}_i{\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}}, $$ (A3)
    where
    χ o = κ ( m ) − κ ( Γ ) κ ( m ) + κ ( Γ ) , χ i = κ ( Γ ) − κ ( i ) κ ( Γ ) + κ ( i ) , $$ {\chi}_o&#x0003D;\frac{\kappa&#x0005E;{(m)}-{\kappa}&#x0005E;{\left(\Gamma \right)}}{\kappa&#x0005E;{(m)}&#x0002B;{\kappa}&#x0005E;{\left(\Gamma \right)}},\kern2em {\chi}_i&#x0003D;\frac{\kappa&#x0005E;{\left(\Gamma \right)}-{\kappa}&#x0005E;{(i)}}{\kappa&#x0005E;{\left(\Gamma \right)}&#x0002B;{\kappa}&#x0005E;{(i)}}, $$
    are the equivalents of the Rayleigh parameter (39). In Table A1, we present the explicit formulas for these t e r m s $$ terms $$ .

    The authors' intention is to find the relationship between v k $$ {v}_k $$ and β k $$ {\beta}_k $$ from (38) and (A3), as long as δ < < 1 $$ \delta &lt;&lt;1 $$ .

    Let us rewrite
    v k = ( κ ( m ) − κ ( Γ ) ) ( κ ( Γ ) + κ ( i ) ) + ( κ ( m ) + κ ( Γ ) ) ( κ ( Γ ) − κ ( i ) ) r i r o 2 k ( κ ( m ) + κ ( Γ ) ) ( κ ( Γ ) + κ ( i ) ) + ( κ ( m ) − κ ( Γ ) ) ( κ ( Γ ) − κ ( i ) ) r i r o 2 k = 2 ( 1 − ρ ) κ ( Γ ) + κ ( i ) 1 − r i r o 2 k − κ ( Γ ) A k − 2 ( 1 + ρ ) κ ( Γ ) + κ ( i ) 1 − r i r o 2 k − κ ( Γ ) A k + , $$ {\displaystyle \begin{array}{ll}{v}_k&#x0003D;&amp; \kern0.2em \frac{\left({\kappa}&#x0005E;{(m)}-{\kappa}&#x0005E;{\left(\Gamma \right)}\right)\left({\kappa}&#x0005E;{\left(\Gamma \right)}&#x0002B;{\kappa}&#x0005E;{(i)}\right)&#x0002B;\left({\kappa}&#x0005E;{(m)}&#x0002B;{\kappa}&#x0005E;{\left(\Gamma \right)}\right)\left({\kappa}&#x0005E;{\left(\Gamma \right)}-{\kappa}&#x0005E;{(i)}\right){\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}}{\left({\kappa}&#x0005E;{(m)}&#x0002B;{\kappa}&#x0005E;{\left(\Gamma \right)}\right)\left({\kappa}&#x0005E;{\left(\Gamma \right)}&#x0002B;{\kappa}&#x0005E;{(i)}\right)&#x0002B;\left({\kappa}&#x0005E;{(m)}-{\kappa}&#x0005E;{\left(\Gamma \right)}\right)\left({\kappa}&#x0005E;{\left(\Gamma \right)}-{\kappa}&#x0005E;{(i)}\right){\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}}\\ {}&#x0003D;&amp; \kern0.2em \frac{2\left(1-\rho \right){\kappa}&#x0005E;{\left(\Gamma \right)}&#x0002B;{\kappa}&#x0005E;{(i)}\left(1-{\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}\right)-{\kappa}&#x0005E;{\left(\Gamma \right)}{A}_k&#x0005E;{-}}{2\left(1&#x0002B;\rho \right){\kappa}&#x0005E;{\left(\Gamma \right)}&#x0002B;{\kappa}&#x0005E;{(i)}\left(1-{\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}\right)-{\kappa}&#x0005E;{\left(\Gamma \right)}{A}_k&#x0005E;{&#x0002B;}},\end{array}} $$
    where
    A k − = 1 − κ ( i ) − κ ( Γ ) κ ( m ) 1 − r i r o 2 k , and A k + = 1 + κ ( i ) − κ ( Γ ) κ ( m ) 1 − r i r o 2 k . $$ {A}_k&#x0005E;{-}&#x0003D;\left(1-\frac{\kappa&#x0005E;{(i)}-{\kappa}&#x0005E;{\left(\Gamma \right)}}{\kappa&#x0005E;{(m)}}\right)\left(1-{\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}\right),\kern1em \mathrm{and}\kern1em {A}_k&#x0005E;{&#x0002B;}&#x0003D;\left(1&#x0002B;\frac{\kappa&#x0005E;{(i)}-{\kappa}&#x0005E;{\left(\Gamma \right)}}{\kappa&#x0005E;{(m)}}\right)\left(1-{\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}\right). $$
    In this way, the limits
    lim δ → 0 v k = ( 1 − ρ ) r C ( r ) β + k κ ( i ) ( 1 + ρ ) r C ( r ) β + k κ ( i ) $$ \underset{\delta \to 0}{\lim }{v}_k&#x0003D;\frac{\left(1-\rho \right) rC(r)\beta &#x0002B;k{\kappa}&#x0005E;{(i)}}{\left(1&#x0002B;\rho \right) rC(r)\beta &#x0002B;k{\kappa}&#x0005E;{(i)}} $$ (A4)
    are obtained.
    This means that in (A1), using the normalization
    κ ( Γ ) = δ β r , $$ {\kappa}&#x0005E;{\left(\Gamma \right)}&#x0003D;\frac{\delta \beta}{r}, $$ (A5)
    it is possible to establish a parallelism between the described model and the three-phase model proposed by Manteufel and Todreas36 being
    lim δ → 0 v k = β k , $$ \underset{\delta \to 0}{\lim }{v}_k&#x0003D;{\beta}_k, $$
    with β k $$ {\beta}_k $$ as in (38). This normalization (A5) also coincides with the proposals in literature.26, 37, 38 In other words, given a sufficiently small thickness δ $$ \delta $$ and if we know β $$ \beta $$ in an imperfect contact model, it is possible to calculate a κ ( Γ ) = δ β / r $$ {\kappa}&#x0005E;{\left(\Gamma \right)}&#x0003D;\delta \beta /r $$ , for which the three-phase model is equivalent, and vice versa (find β $$ \beta $$ knowing κ ( Γ ) $$ {\kappa}&#x0005E;{\left(\Gamma \right)} $$ ).

    From the numerical point of view, the approximations obtained in Section 3.3 coincide reasonably with the results presented by Manteufel and Todreas.36 In fact, the accompanying terms in (A2) coincide for the first and second truncation of the system (Equations (47) and (51), respectively). For third-order truncation, some differences appear between the two approximations, since while Manteufel and Todreas36 propose the same approximation of the previous order, in our case, it was obtained (52). The fourth- and fifth-order differ even more, and an exact equivalence cannot be established between its results and ours. In Figure A2, some values from the literature are also shown together with the approximations in a close range.

    Details are in the caption following the image
    FIGURE A2
    Open in figure viewerPowerPoint
    Comparison of effective conductivities estimates for high volume fractions. For the computation of the dashed curves (marked with *); the formulas from Manteufel and Todreas36 were used instead [Colour figure can be viewed at wileyonlinelibrary.com]

    Table A1 presents a summary of this comparison between the formulas in Manteufel and Todreas36 and the current ones: (47), (51)–(54).

    TABLE A1. Comparison between Manteufel and Todreas36 (square cell) and the present work
    Trunc. order Equations (47), (51)–(54), and 5th Manteufel-Todreas36 Coefficients and auxiliary functions
    κ ^ 11 κ ( m ) = 1 − ϕ β 1 − τ 1 + ϕ β 1 − τ $$ \frac{{\hat{\kappa}}_{11}}{\kappa&#x0005E;{(m)}}&#x0003D;\frac{1-\phi {\beta}_1-\tau }{1&#x0002B;\phi {\beta}_1-\tau } $$ κ ^ 11 κ ( m ) = 1 − ϕ v 1 − τ 1 + ϕ v 1 − τ $$ \frac{{\hat{\kappa}}_{11}}{\kappa&#x0005E;{(m)}}&#x0003D;\frac{1-\phi {v}_1-\tau }{1&#x0002B;\phi {v}_1-\tau } $$
    1st τ = 0 $$ \tau &#x0003D;0 $$ τ = 0 $$ \tau &#x0003D;0 $$
    2nd τ = C 4 β 1 β 3 ϕ 4 $$ \tau &#x0003D;{C}_4{\beta}_1{\beta}_3{\phi}&#x0005E;4 $$ τ = C 4 v 1 v 3 ϕ 4 $$ \tau &#x0003D;{C}_4{v}_1{v}_3{\phi}&#x0005E;4 $$ C 4 = 3 S 4 2 π − 4 ≈ 0 . 305827833 $$ {C}_4&#x0003D;3{S}_4&#x0005E;2{\pi}&#x0005E;{-4}\approx 0.305827833 $$
    3rd τ = C 4 β 1 β 3 ϕ 4 / Δ 1 ( ϕ ) $$ \tau &#x0003D;{C}_4{\beta}_1{\beta}_3{\phi}&#x0005E;4/{\Delta}_1\left(\phi \right) $$ Δ 1 ( ϕ ) = 1 − b 8 β 3 β 5 ϕ 8 b 8 = 735 S 8 2 π − 8 ≈ 1 . 40295995 $$ {\displaystyle \begin{array}{cc}\hfill {\Delta}_1\left(\phi \right)&#x0003D;&amp; \kern0.2em 1-{b}_8{\beta}_3{\beta}_5{\phi}&#x0005E;8\hfill \\ {}\hfill {b}_8&#x0003D;&amp; \kern0.2em 735{S}_8&#x0005E;2{\pi}&#x0005E;{-8}\approx 1.40295995\hfill \end{array}} $$
    4th τ = C 4 β 1 β 3 ϕ 4 / Δ 2 ( ϕ ) + C 8 β 1 β 7 ϕ 8 Δ 1 ′ ( ϕ ) Δ 2 ( ϕ ) $$ {\displaystyle \begin{array}{cc}\hfill \tau &#x0003D;&amp; \kern0.2em {C}_4{\beta}_1{\beta}_3{\phi}&#x0005E;4/{\Delta}_2\left(\phi \right)\hfill \\ {}\hfill &amp; &#x0002B;{C}_8{\beta}_1{\beta}_7{\phi}&#x0005E;8\frac{\Delta_1&#x0005E;{\prime}\left(\phi \right)}{\Delta_2\left(\phi \right)}\hfill \end{array}} $$ τ = C 4 v 1 v 3 ϕ 4 / D 1 ( ϕ ) + C 8 v 1 v 7 ϕ 8 $$ {\displaystyle \begin{array}{cc}\hfill \tau &#x0003D;&amp; \kern0.2em {C}_4{v}_1{v}_3{\phi}&#x0005E;4/{D}_1\left(\phi \right)\hfill \\ {}\hfill &amp; &#x0002B;{C}_8{v}_1{v}_7{\phi}&#x0005E;8\hfill \end{array}} $$ C 8 = 7 S 8 2 π − 8 ≈ 0 . 013361523 D 1 ( ϕ ) = 1 − b 8 v 3 v 5 ϕ 8 Δ 2 ( ϕ ) = Δ 1 ( ϕ ) − b 12 β 5 β 7 ϕ 12 Δ 1 ′ ( ϕ ) = 1 − 71 13 2 b 8 β 3 β 5 ϕ 8 $$ {\displaystyle \begin{array}{cc}\hfill {C}_8&#x0003D;&amp; \kern0.2em 7{S}_8&#x0005E;2{\pi}&#x0005E;{-8}\approx 0.013361523\hfill \\ {}\hfill {D}_1\left(\phi \right)&#x0003D;&amp; \kern0.2em 1-{b}_8{v}_3{v}_5{\phi}&#x0005E;8\hfill \\ {}\hfill {\Delta}_2\left(\phi \right)&#x0003D;&amp; \kern0.2em {\Delta}_1\left(\phi \right)-{b}_{12}{\beta}_5{\beta}_7{\phi}&#x0005E;{12}\hfill \\ {}\hfill {\Delta}_1&#x0005E;{\prime}\left(\phi \right)&#x0003D;&amp; \kern0.2em 1-{\left(\frac{71}{13}\right)}&#x0005E;2\kern0.1em {b}_8{\beta}_3{\beta}_5{\phi}&#x0005E;8\hfill \end{array}} $$
    b 12 = 152460 S 12 2 π − 12 ≈ 2 . 55915216 - $$ {b}_{12}&#x0003D;\kern0.70em 152460{S}_{12}&#x0005E;2{\pi}&#x0005E;{-12}\approx 2.55915216\hbox{-} $$
    5th τ = C 4 β 1 β 3 ϕ 4 Δ 0 ( ϕ ) Δ 3 ( ϕ ) + C 8 β 1 β 7 ϕ 8 Δ 2 ′ ( ϕ ) Δ 3 ( ϕ ) $$ {\displaystyle \begin{array}{cc}\hfill \tau &#x0003D;&amp; \kern0.2em {C}_4{\beta}_1{\beta}_3{\phi}&#x0005E;4\frac{\Delta_0\left(\phi \right)}{\Delta_3\left(\phi \right)}\hfill \\ {}\hfill &amp; &#x0002B;{C}_8{\beta}_1{\beta}_7{\phi}&#x0005E;8\frac{\Delta_2&#x0005E;{\prime}\left(\phi \right)}{\Delta_3\left(\phi \right)}\hfill \\ {}\hfill &amp; \hfill \\ {}\hfill &amp; \hfill \end{array}} $$ Δ 0 ( ϕ ) = 1 − b 12 β 5 β 7 ϕ 12 − b 16 β 7 β 9 ϕ 16 Δ 3 ( ϕ ) = Δ 2 ( ϕ ) − b 12 ′ β 3 β 9 ϕ 12 − b 16 β 7 β 9 ϕ 16 − C 24 β 3 β 5 β 7 β 9 ϕ 24 Δ 2 ′ ( ϕ ) = Δ 1 ′ ( ϕ ) + 168 13 b 8 β 3 β 5 ϕ 8 + 1001 17 b 12 ′ β 3 β 9 ϕ 12 $$ {\displaystyle \begin{array}{cc}\hfill {\Delta}_0\left(\phi \right)&#x0003D;&amp; \kern0.2em 1-{b}_{12}{\beta}_5{\beta}_7{\phi}&#x0005E;{12}-{b}_{16}{\beta}_7{\beta}_9{\phi}&#x0005E;{16}\hfill \\ {}\hfill {\Delta}_3\left(\phi \right)&#x0003D;&amp; \kern0.2em {\Delta}_2\left(\phi \right)-{b}_{12}&#x0005E;{\prime }{\beta}_3{\beta}_9{\phi}&#x0005E;{12}\hfill \\ {}\hfill &amp; -{b}_{16}{\beta}_7{\beta}_9{\phi}&#x0005E;{16}-{C}_{24}{\beta}_3{\beta}_5{\beta}_7{\beta}_9{\phi}&#x0005E;{24}\hfill \\ {}\hfill {\Delta}_2&#x0005E;{\prime}\left(\phi \right)&#x0003D;&amp; \kern0.2em {\Delta}_1&#x0005E;{\prime}\left(\phi \right)&#x0002B;\frac{168}{13}{b}_8{\beta}_3{\beta}_5{\phi}&#x0005E;8\hfill \\ {}\hfill &amp; &#x0002B;\frac{1001}{17}{b}_{12}&#x0005E;{\prime }{\beta}_3{\beta}_9{\phi}&#x0005E;{12}\hfill \end{array}} $$
    b 12 ′ = 5 84 b 12 ≈ 0 . 15233049 b 16 = 32207175 S 16 2 π − 16 ≈ 5 . 76867559 C 24 = 2858625 ( 22 S 12 2 − 91 S 16 S 8 ) 2 π − 24 ≈ 4 . 93057350 $$ {\displaystyle \begin{array}{cc}\hfill {b}_{12}&#x0005E;{\prime }&#x0003D;&amp; \kern0.2em \frac{5}{84}{b}_{12}\approx 0.15233049\hfill \\ {}\hfill {b}_{16}&#x0003D;&amp; \kern0.2em 32207175{S}_{16}&#x0005E;2{\pi}&#x0005E;{-16}\approx 5.76867559\hfill \\ {}\hfill {C}_{24}&#x0003D;&amp; \kern0.2em 2858625{\left(22{S}_{12}&#x0005E;2-91{S}_{16}{S}_8\right)}&#x0005E;2{\pi}&#x0005E;{-24}\hfill \\ {}\hfill &amp; \approx 4.93057350\hfill \end{array}} $$
    6th and 7th τ = C 4 v 1 v 3 ϕ 4 / D 2 ( ϕ ) + C 8 v 1 v 7 ϕ 8 + C 12 ( v 1 ) 2 ϕ 12 + C 16 v 1 v 3 v 5 v 7 ϕ 16 / D 3 ( ϕ ) + C 20 v 1 v 5 ( v 7 ) 2 ϕ 20 / D 0 ( ϕ ) + C 28 v 1 v 3 ( v 5 ) 2 ( v 7 ) 2 ϕ 28 / D 4 ( ϕ ) $$ {\displaystyle \begin{array}{cc}\hfill \tau &#x0003D;&amp; \kern0.2em {C}_4{v}_1{v}_3{\phi}&#x0005E;4/{D}_2\left(\phi \right)\hfill \\ {}\hfill &amp; &#x0002B;{C}_8{v}_1{v}_7{\phi}&#x0005E;8\hfill \\ {}\hfill &amp; &#x0002B;{C}_{12}{\left({v}_1\right)}&#x0005E;2{\phi}&#x0005E;{12}\hfill \\ {}\hfill &amp; &#x0002B;{C}_{16}{v}_1{v}_3{v}_5{v}_7{\phi}&#x0005E;{16}/{D}_3\left(\phi \right)\hfill \\ {}\hfill &amp; &#x0002B;{C}_{20}{v}_1{v}_5{\left({v}_7\right)}&#x0005E;2{\phi}&#x0005E;{20}/{D}_0\left(\phi \right)\hfill \\ {}\hfill &amp; &#x0002B;{C}_{28}{v}_1{v}_3{\left({v}_5\right)}&#x0005E;2{\left({v}_7\right)}&#x0005E;2{\phi}&#x0005E;{28}/{D}_4\left(\phi \right)\hfill \end{array}} $$ C 12 ≈ 0 . 000184643 ; C 16 ≈ 0 . 242252 ; C 20 ≈ 0 . 0341942 ; C 28 ≈ 0 . 0479731 ; $$ {\displaystyle \begin{array}{cc}\hfill {C}_{12}\approx &amp; \kern3.0235pt 0.000184643;{C}_{16}\approx \kern3.0235pt 0.242252;\hfill \\ {}\hfill {C}_{20}\approx &amp; \kern3.0235pt 0.0341942;{C}_{28}\approx \kern3.0235pt 0.0479731;\hfill \end{array}} $$
    D 0 ( ϕ ) = 1 − b 12 v 5 v 7 ϕ 12 , D 2 ( ϕ ) = 1 − b 8 v 3 v 5 ϕ 8 / D 0 ( ϕ ) − b 12 ′ v 3 v 9 ϕ 12 D 3 ( ϕ ) = 1 − b 8 v 3 v 5 ϕ 8 / D 0 ( ϕ ) − b 12 v 5 v 7 + b 12 ′ v 3 v 9 ϕ 12 + b 20 v 3 v 5 2 v 7 ϕ 20 / D 0 ( ϕ ) + b 24 v 3 v 5 v 7 v 9 ϕ 24 D 4 ( ϕ ) = 1 − b 8 v 3 v 5 ϕ 8 / D 0 ( ϕ ) − [ 2 b 12 v 5 v 7 + b 12 ′ v 3 v 9 ] ϕ 12 + 2 b 20 v 3 ( v 5 ) 2 v 7 ϕ 20 / D 0 ( ϕ ) + [ 2 b 24 v 3 v 5 v 7 v 9 + b 24 ′ ( v 5 ) 2 ( v 7 ) 2 ] ϕ 24 − b 32 v 3 ( v 5 ) 3 ( v 7 ) 2 ϕ 32 / D 0 ( ϕ ) − b 36 v 3 ( v 5 ) 2 ( v 7 ) 2 v 9 ϕ 36 $$ {\displaystyle \begin{array}{cc}\hfill {D}_0\left(\phi \right)&#x0003D;&amp; \kern0.2em 1-{b}_{12}{v}_5{v}_7{\phi}&#x0005E;{12},\hfill \\ {}\hfill {D}_2\left(\phi \right)&#x0003D;&amp; \kern0.2em 1-{b}_8{v}_3{v}_5{\phi}&#x0005E;8/{D}_0\left(\phi \right)-{b}_{12}&#x0005E;{\prime }{v}_3{v}_9{\phi}&#x0005E;{12}\hfill \\ {}\hfill {D}_3\left(\phi \right)&#x0003D;&amp; \kern0.2em 1-{b}_8{v}_3{v}_5{\phi}&#x0005E;8/{D}_0\left(\phi \right)\hfill \\ {}\hfill &amp; -\left[{b}_{12}{v}_5{v}_7&#x0002B;{b}_{12}&#x0005E;{\prime }{v}_3{v}_9\right]{\phi}&#x0005E;{12}\hfill \\ {}\hfill &amp; &#x0002B;{b}_{20}{v}_3{\left({v}_5\right)}&#x0005E;2{v}_7{\phi}&#x0005E;{20}/{D}_0\left(\phi \right)\hfill \\ {}\hfill &amp; &#x0002B;{b}_{24}{v}_3{v}_5{v}_7{v}_9{\phi}&#x0005E;{24}\hfill \\ {}\hfill {D}_4\left(\phi \right)&#x0003D;&amp; \kern0.2em 1-{b}_8{v}_3{v}_5{\phi}&#x0005E;8/{D}_0\left(\phi \right)\hfill \\ {}\hfill &amp; -\left[2{b}_{12}{v}_5{v}_7&#x0002B;{b}_{12}&#x0005E;{\prime }{v}_3{v}_9\right]{\phi}&#x0005E;{12}\hfill \\ {}\hfill &amp; &#x0002B;2{b}_{20}{v}_3{\left({v}_5\right)}&#x0005E;2{v}_7{\phi}&#x0005E;{20}/{D}_0\left(\phi \right)\hfill \\ {}\hfill &amp; &#x0002B;\Big[2{b}_{24}{v}_3{v}_5{v}_7{v}_9\hfill \\ {}\hfill &amp; &#x0002B;{b}_{24}&#x0005E;{\prime }{\left({v}_5\right)}&#x0005E;2{\left({v}_7\right)}&#x0005E;2\Big]{\phi}&#x0005E;{24}\hfill \\ {}\hfill &amp; -{b}_{32}{v}_3{\left({v}_5\right)}&#x0005E;3{\left({v}_7\right)}&#x0005E;2{\phi}&#x0005E;{32}/{D}_0\left(\phi \right)\hfill \\ {}\hfill &amp; -{b}_{36}{v}_3{\left({v}_5\right)}&#x0005E;2{\left({v}_7\right)}&#x0005E;2{v}_9{\phi}&#x0005E;{36}\hfill \end{array}} $$
    b 20 = 3 . 59039 , b 24 = 0 . 389837 , b 24 ′ = 84 5 b 24 , b 32 = 9 . 18835 , b 36 = 0 . 997652 $$ {\displaystyle \begin{array}{cc}\hfill {b}_{20}&#x0003D;&amp; \kern0.2em 3.59039,\kern3.0235pt {b}_{24}&#x0003D;0.389837,\kern3.0235pt {b}_{24}&#x0005E;{\prime }&#x0003D;\frac{84}{5}{b}_{24},\hfill \\ {}\hfill {b}_{32}&#x0003D;&amp; \kern0.2em 9.18835,\kern3.0235pt {b}_{36}&#x0003D;0.997652\hfill \end{array}} $$
    where (Equations (38), (A3)) β k = ( 1 − ρ ) β + κ ( i ) k ( 1 + ρ ) β + κ ( i ) k , v k = χ o + χ i r i r o 2 k 1 + χ o χ i r i r o 2 k , ρ = κ ( i ) κ ( m ) χ o = κ ( m ) − κ ( Γ ) κ ( m ) + κ ( Γ ) , χ i = κ ( Γ ) − κ ( i ) κ ( Γ ) + κ ( i ) , $$ \kern2em {\beta}_k&#x0003D;\frac{\left(1-\rho \right)\beta &#x0002B;{\kappa}&#x0005E;{(i)}k}{\left(1&#x0002B;\rho \right)\beta &#x0002B;{\kappa}&#x0005E;{(i)}k},\kern2em {v}_k&#x0003D;\frac{\chi_o&#x0002B;{\chi}_i{\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}}{1&#x0002B;{\chi}_o{\chi}_i{\left(\frac{r_i}{r_o}\right)}&#x0005E;{2k}},\kern2em \left(\rho &#x0003D;\frac{\kappa&#x0005E;{(i)}}{\kappa&#x0005E;{(m)}}\kern3.0235pt {\chi}_o&#x0003D;\frac{\kappa&#x0005E;{(m)}-{\kappa}&#x0005E;{\left(\Gamma \right)}}{\kappa&#x0005E;{(m)}&#x0002B;{\kappa}&#x0005E;{\left(\Gamma \right)}},\kern3.0235pt {\chi}_i&#x0003D;\frac{\kappa&#x0005E;{\left(\Gamma \right)}-{\kappa}&#x0005E;{(i)}}{\kappa&#x0005E;{\left(\Gamma \right)}&#x0002B;{\kappa}&#x0005E;{(i)}},\right) $$

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