Effective thermal conductivity of periodic composites with highly conducting imperfect interfaces

https://doi.org/10.1016/j.ijthermalsci.2011.03.009Get rights and content

Abstract

The purpose of this work is to determine the effective conductivity of periodic composites accounting for highly conducting imperfect interfaces between the matrix and inclusions phases and to study the dependencies of the effective conductivity on the size and distribution of inhomogeneities in the matrix phase in different cases: squared, hexagonal, cubic and random inclusion distributions. The local solution of the periodic conduction problem is found in Fourier space by using the Green operators and closed-form expressions of factors depending on the size and shape of the inclusions. The numerical results of size-dependent effective thermal conductivity are finally compared with an analytical estimation obtained from the generalized self-consistent model. The method elaborated and results provided by the present work are directly applicable to other physically analogous transport phenomena, such as electric conduction, dielectrics, magnetism, diffusion and flow in porous media and to the mathematically identical phenomenon of anti-plane elasticity.

Introduction

Recently, different works have been devoted to study the size-dependent mechanical behavior in nanosystems incorporating surface/interface energies. Indeed, when the inclusion size is diminished to the nano-scale, due to the large surface-to-volume ratio, the matrix–inclusion interface energy can no longer be neglected. This fact has been emphasized and exploited in recent investigations on nanomaterials and nano-structural elements (see, e.g. [11], [24]). In this context, in order to estimate the size-dependent overall elastic properties of nanocomposites and nano-structural elements accounting for the surface/interface energies, the classical perfect interface is modified by adopting a coherent interface model in which the displacement vector field is continuous across an interface while the stress vector field is discontinuous across the same interface ([2], [29], [31]). The thermal conduction counterpart of the interface stress and energy model is the highly conducting (HC) interface model, which is the subject of the present paper. More precisely, according to this imperfect interface model, the temperature is continuous across this interface but the normal heat flux component is discontinuous across the same interface due to the possibility of having a surface heat flux along the interface whose surface energy conservation equation gives rise to the generalized Young–Laplace equation. These interface conditions for the HC interface model are completely contrary to the ones of the well-known Kapitza interface thermal resistance model which has been recognized to be of a great theoretical and practical importance in physics and materials science (see e.g. [17]). By accounting for the thermal resistance appearing at the interface between two bulk media, the Kapitza interface thermal resistance model stipulates that the temperature suffers a jump across the interface while the normal heat flux component is continuous across the same interface and usually taken to be proportional to the temperature jump. Thus, the HC interface model can be viewed as dual with respect to the Kapitza interface thermal resistance model. The physical background of both HC interface model and thermal resistance interface model can be clarified by considering the general imperfect interface model in which a very thin interphase of uniform thickness is situated between two bulk phases. By applying an asymptotic approach to this interphase to obtain appropriate temperature and normal heat flux component jump conditions for an interface of zero thickness replacing the interphase, Sanchez-Palencia [30] and Pham Huy and Sanchez-Palencia [28] showed that the general imperfect interface model reduces to the HC interface model or the thermal resistance interface model according as the interphase is highly conducting or slowly conducting with respect to the surrounding phases. In other words, the HC interface model and the thermal resistance interface model may be considered as the two limiting cases of the general imperfect interface model.

There are several possible methodologies in deriving the size-dependent overall thermal properties of composites affected by the significant size effects appearing in conducting composites containing HC imperfect interfaces. These possible interests can be classified into three categories.

The first category includes all analytical estimation methods based either on the generalized Eshelby’s formalism, such as diluted, self-consistent, generalized self-consistent, differential schemes and those of Mori-Tanaka or on the ellipsoidal harmonic solution for the Laplace equation. In ref. [18], the Eshelby’s results and formalism for an circular or spherical inhomogeneity embedded in an elastic infinite matrix are extended to the thermal conduction phenomenon accounting for the HC imperfect interface between matrix and inclusions. Quite different from the relevant results of elasticity, Le Quang et al. [18] showed that the generalized Eshelby’s conduction tensor fields and localization tensor fields inside circular and spherical inhomogeneities remain uniform even in the presence of the HC imperfect interface. Then, the analytical closed-form expressions for thermal effective conductivity has been derived as functions of interface properties and of the inhomogeneities size. Another approach employing the ellipsoidal harmonic and analytical solution for the Laplace equation to calculate the effective thermal conductivity has been presented in refs. [9], [25]. However, these last two works are limited to the case where the phases constituting the composite are isotropic.

The second category contains all methods based on variational principles to determine the bounds of size-dependent effective thermal properties. In refs. [6], [19], [20], [21], by using the generalized Hashin-Shtrikman variational principles and by constructing appropriate trial fields, the explicit upper and lower bounds are obtained for the effective thermal conductivity of composite materials consisting of two anisotropic phases. Recently, these results have been generalized by Lipton and Talbot [22] aiming at finding the upper and lower bounds for the effective thermal conductivity of composite with a nonlinear imperfect interface.

The third category associated to the present study includes all the numerical methods. Previous methods use mainly the finite element method with surface elements or the level set method to describe the HC imperfect interface (e.g. [35]), while the method used in the present paper is based on the fast Fourier transform (FFT) of the solution. The method based on FFT has been used frequently to compute the effective properties of periodic composites in the context of linear or nonlinear elasticity (e.g. [3], [23], [27]) and has been generalized recently to the context of piezoelectricity by Brenner [5]. However, the results presented in refs. [3], [5], [23], [27] were limited to the case where the interfaces between matrix and inclusion phases are perfect.

The present work is concerned with the thermal conduction phenomenon which plays an important area in the fields of materials science and solid-state physics. The elaborated method and the results obtained for heat conduction are directly applicable to other transport phenomena like electric conduction, dielectrics, magnetism, diffusion and flow in porous media and to anti-plane elasticity problem. This is due to the fact that the thermal conduction problem is physically analogous to other problems involving transport phenomena such as electric conduction, dielectrics, magnetism, diffusion and flow in porous media and that there is a correspondence between 2D conduction and anti-plane elasticity (see, e.g. [26]). The present work has two objectives. First, it aims at extending the alternate method based on Fourier series ([3], [23], [27]) in the context of linear elasticity to the thermal conduction phenomenon including the effect of the highly conducting imperfect interface. The generalized method proposed uses the FFT and an iterative process to solve the local solution of periodic conduction problem in the three following cases: with squarely, hexagonally and randomly distributed inclusions. Second, it has the purpose of employing the solution of the localization problem to determine and study the effects of the interface, size and distribution of inclusions on the effective thermal properties of periodic composites.

The paper is organized as follows. Section 2 is dedicated to specifying the constitutive laws of the constituent phases of composites under investigation, the HC interface model and the general form of the effective thermal behavior. In Section 3, the solution of the localization periodic problem is presented in the Fourier space in the context of conduction phenomenon with HC imperfect interface. Then, the size-dependent effective conductivities are calculated from the local solution. In Section 4, the HC interface and inhomogeneities size effects as well as the dependence of the distribution of inhomogeneities in the matrix phase on the effective conductivity of periodic composites are numerically discussed and illustrated; in addition, the numerically obtained results are compared with the results provided by using the estimation with the help of the generalized self-consistent model (GSCM) and matrix/interphase/inclusion configuration. In Section 5, a few conclusions are given.

Section snippets

Local constitutive laws

In this section, we specify the local constitutive laws of the two-dimensional (2D) or three-dimensional (3D) periodic composite studied in this work. Let Ω be a representative volume element (RVE) consisting of a host matrix medium in which M (≥1) inclusions are embedded. The matrix, referred to phase 2 and occupied by Ω(m), and inclusions, denoted as phase 1, are assumed to be individually homogeneous and have the linear thermal behavior described by Fourier’s lawq(x)=K(x)ɛ(x).

Here ɛ(x) being

Solution of the localization problem

First, let Ω be subjected to the following uniform intensity field boundary conditionT(x)=E·x,xΩ,where E is a constant intensity vector. Under the boundary condition Eq. (18), as for the classical case with perfect interfaces, owing to the fact that the temperature field T(x) is continuous across the coherent imperfect interfaces Γ(i), it is immediate from Eq.(15) that the macroscopic intensity field is equal to E.

In order to calculate the local intensity and heat flux fields in Ω, we

Fiber–matrix composites

To numerically illustrate the features of the results obtained above, we consider the first example of a fiber composite submitted to a heat flow normal to the direction of the fibers. The fibers are assumed cylindrical with circular sections having the same radius, i.e. Ri = R, and introduced periodically into the host matrix phase. For this 2D problem, three typical microstructures with squared, hexagonal and random distributions of the inhomogeneities are studied in detail. In addition, the

Conclusions

The effective conductivity of periodic composites with highly conducting imperfect interfaces has been studied by using a solution in Fourier space and the known Fourier transform of the Green’s tensor for conductivity. The plane problem of periodic fiber–matrix composites with squared lattice, hexagonal lattice and periods containing random distributions so as the 3D problem of particle–matrix composites with cubic SC or FCC lattices and random distributions were studied. The results show that

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