Theory for Zero-Index Conductive Cloaks: Constant-Temperature Scheme

Abstract

In this chapter, we propose an exact approach to an effectively infinite thermal conductivity with a constant-temperature boundary condition, which an external thermostatic sink can easily realize. Since (effectively) infinite thermal conductivity corresponds to zero refractive indexes in photonics, it has direct applications in designing zero-index thermal metamaterials. Therefore, we experimentally demonstrate zero-index thermal cloaks, which can work in highly conductive backgrounds with simple structures. These results provide insights into thermal management with effectively infinite thermal conductivities.

1 Opening Remarks

Thermal conductivity plays a crucial role in heat transfer, and extreme (zero and infinite) thermal conductivities are always a research focus due to their excellent properties. For low thermal conductivities, a recent study reported that the thermal conductivity of ceramic aerogel could be as low as 0.0024 W m$^{-1}$ K$^{-1}$ [1]. For high thermal conductivities, there is still a long way ahead. Although many materials have high thermal conductivity, such as boron nitride with 600 W m$^{-1}$ K$^{-1}$ [2], carbon nanotube with 2300 W m$^{-1}$ K$^{-1}$ [3], and graphene with 5300 W m$^{-1}$ K$^{-1}$ [4], they are still far from infinite thermal conductivities.

A recent study reported that the effective thermal conductivity of moving fluids could approximately tend to infinity [5]. Such an effectively infinite thermal conductivity requires the velocity of moving fluids to be also infinite, which cannot be exactly realized. To go further, we propose an exact approach to effectively infinite thermal conductivities with simple structures. By applying a constant-temperature boundary condition to an object with a finite thermal conductivity, the object can effectively have infinite thermal conductivity. Meanwhile, an external thermostatic sink can easily realize the constant-temperature boundary condition, which is beneficial for practical applications.

Since (effectively) infinite thermal conductivities are in analogue of zero refractive indexes in photonics [6,7,8,9,10,11], they can be used to design zero-index thermal metamaterials. We take thermal cloaking [12,13,14,15,16,17,18,19,20,21] as an example, which can be realized by transformation thermotics [12, 13] or scattering cancellation [15,16,17]. Here, we use infinite thermal conductivity to realize zero-index thermal cloaks, which can work in highly conductive backgrounds with simple structures. Specifically, if the previous bilayer scheme [15,16,17] is applied to a highly conductive background (such as copper, 400 W m$^{-1}$ K$^{-1}$), the thermal conductivity of the inner shell is zero, and that of the outer shell should be larger than 400 W m$^{-1}$ K$^{-1}$. However, few common materials have thermal conductivities higher than 400 W m$^{-1}$ K$^{-1}$ [5]. Although some rare materials like diamond have high thermal conductivities, the cost and difficulty of practical applications also increase. In contrast, if the zero-index scheme is applied, the core with a constant-temperature boundary condition can effectively have infinite thermal conductivity. Therefore, the thermal conductivity of the outer shell can be smaller than 400 W m$^{-1}$ K$^{-1}$, and many common materials such as aluminum can be applied. Therefore, the zero-index scheme is free from the thermal conductivities of backgrounds.

2 Thermal Zero Index Connotation

The Fourier law describes thermal conduction, namely $\boldsymbol{J}=-\kappa \boldsymbol{\nabla } T$, where $\boldsymbol{J}$ is the heat flux, $\kappa$ is the thermal conductivity, and T denotes temperature. To understand the temperature field effect of infinite thermal conductivity (i.e., zero-index thermal conductivity), we put a two-dimensional elliptical particle (with thermal conductivity $\kappa _p=\infty$, actually set as $10^{10}$ W m$^{-1}$ K$^{-1}$) in the background (with thermal conductivity $\kappa _b$) and apply a horizontal thermal field $\boldsymbol{K}_0$. Consequently, the isotherms are all repelled, and the black arrows (denoting the directions of heat fluxes) are always perpendicular to the exterior boundary of the particle (Fig. 5.1a). The particle is isothermal, and a brief proof is as follows. We denote the temperature distribution of the particle as $T_p$. By solving the Laplace equation $\boldsymbol{\nabla }\cdot \left( -\kappa \boldsymbol{\nabla } T\right) =0$, we can derive $T_p$ as

$$\begin{aligned} T_p=\frac{-\kappa _b}{L_{p1}\kappa _p+\left( 1-L_{p1}\right) \kappa _b}K_0x_1+T_0, \end{aligned}$$

(5.1)

where $K_0=|\boldsymbol{K}_0|$, $T_0$ is the reference temperature, and $\left( x_1,\,x_2,\,x_3\right)$ denote the Cartesian coordinates. $L_{p1}$ is the shape factor of the particle along $x_1$ axis, which will be discussed later. Equation (5.1) indicates that whatever value $L_{p1}$ takes on, if $\kappa _p=\infty$, $T_p$ is always a constant $T_0$. Physically, since heat fluxes $\left( \boldsymbol{J}=-\kappa \boldsymbol{\nabla } T\right)$ do not diverge, a direct conclusion from $\kappa =\infty$ is $\boldsymbol{\nabla } T=0$. In other words, a finite thermal conductivity with a constant-temperature boundary condition is equivalent to an infinite thermal conductivity. For comparison, we reset $\kappa _p$ to a finite value ($\kappa _p<\infty$, actually set as 0.026 W m$^{-1}$ K$^{-1}$) and apply a constant-temperature boundary condition on the boundary of the particle (Fig. 5.1b). As a result, the temperature profile and directions of heat fluxes are the same as those in Fig. 5.1(a), thus achieving an effectively infinite thermal conductivity with a constant-temperature boundary condition. Note that such an equivalence is only exact for temperature distributions.

Fig. 5.1

Adapted from Ref. [22]

a Temperature profile with an elliptical particle ($\kappa _p=\infty$, actually set as $10^{10}$ W m$^{-1}$ K$^{-1}$) embedded in the background ($\kappa _b$ $=$ 400 W m$^{-1}$ K$^{-1}$). b Temperature profile with a common particle ($\kappa _p<\infty$, actually set as 0.026 W m$^{-1}$ K$^{-1}$) and a constant-temperature boundary condition with temperature (Max+Min)/2 embedded in the same background. Here, Max and Min denote the temperatures of the left and right boundaries, respectively. Rainbow surfaces denote temperature distributions, and white lines represent isotherms. c Schematic diagram of the zero-index thermal cloak. A constant-temperature boundary condition is applied to the core boundary, so the core has an effectively infinite thermal conductivity.

3 Zero-Index Thermal Cloak

Zero-index metamaterials have been widely explored to manipulate electromagnetic waves due to their excellent properties [6,7,8,9,10,11]. We know that the directions of heat fluxes are always perpendicular to the exterior boundary of the particle with an (effectively) infinite thermal conductivity (Fig. 5.1a and b). This phenomenon follows zero refractive indexes in photonics, where electromagnetic waves travel outward vertically from materials with zero refractive indexes. Therefore, (effectively) infinite thermal conductivities can be directly used to design zero-index thermal metamaterials.

Zero-index thermal cloaks are a typical example of zero-index thermal metamaterials, which can be realized by introducing thermal convection [5]. Such a scheme requires the velocity of moving fluids to be infinite, which cannot be exactly realized, thus called near-zero-index thermal cloaks. In contrast, the present approach can realize exact-zero-index thermal cloaks with simple structures because only an external thermostatic sink is required to realize a constant-temperature boundary condition. In a word, thermal zero-index parameters indicate that thermal conductivities are (effectively) infinite. We apply a constant-temperature boundary condition on the core to realize an effectively infinite thermal conductivity, so the present thermal cloaks are also called zero-index thermal cloaks.

Zero-index thermal cloaks are essentially a core-shell structure (Fig. 5.1c). We denote the thermal conductivities of the core and shell as $\kappa _c$ and $\kappa _s$, respectively. The subscript c (or s) represents the core (or shell) throughout this chapter. For generality, we consider an ellipsoidal case in three dimensions. The semi axes of the core and shell along $x_i$ axis $\left( i=1,\,2,\,3\right)$ are denoted as $r_{ci}$ and $r_{si}$, respectively. The effective thermal conductivity of such a core-shell structure (denoted as $\boldsymbol{\kappa }_e$) is anisotropic, and the component along $x_i$ axis (denoted as $\kappa _{ei}$) can be calculated by

$$\begin{aligned} \kappa _{ei}=\kappa _s\frac{L_{ci}\kappa _c+\left( 1-L_{ci}\right) \kappa _s+f\left( 1-L_{si}\right) \left( \kappa _c-\kappa _s\right) }{L_{ci}\kappa _c+\left( 1-L_{ci}\right) \kappa _s-fL_{si}\left( \kappa _c-\kappa _s\right) }, \end{aligned}$$

(5.2)

where $f=r_{c1}r_{c2}r_{c3}/\left( r_{s1}r_{s2}r_{s3}\right)$ is core fraction. $L_{ci}$ and $L_{si}$ are, respectively, the shape factors of the core and shell along $x_i$ axis, which can be calculated by

$$\begin{aligned} L_{wi}=\dfrac{r_{w1}r_{w2}r_{w3}}{2} \displaystyle \int \limits _0^\infty \frac{du}{\left( u+r_{wi}^2\right) \sqrt{\left( u+r_{w1}^2\right) \left( u+r_{w2}^2\right) \left( u+r_{w3}^2\right) }}, \end{aligned}$$

(5.3)

where the subscript w can take c or s, representing the shape factor of the core or shell. Note that only when the core-shell structure is concentric or confocal, can Eq. (5.2) predict the effective thermal conductivities exactly.

When a constant-temperature boundary condition is applied, the thermal conductivity of the core turns to infinity, namely $\kappa _c=\infty$. Then, Eq. (5.2) becomes

$$\begin{aligned} \kappa _{ei}=\kappa _s\frac{L_{ci}+f\left( 1-L_{si}\right) }{L_{ci}-fL_{si}}. \end{aligned}$$

(5.4)

Equation (5.4) can also be applied to two dimensions as long as we take $r_{w3}=\infty$ and $f=r_{c1}r_{c2}/\left( r_{s1}r_{s2}\right)$. Then, Eq. (5.3) can be reduced to $L_{w1}=r_{w2}/\left( r_{w1}+r_{w2}\right)$, $L_{w2}=r_{w1}/\left( r_{w1}+r_{w2}\right)$, and $L_{w3}=0$. As an intrinsic property, $L_{w1}+L_{w2}+L_{w3}=1$ is always valid no matter in two or three dimensions.

Fig. 5.2

Adapted from Ref. [22]

Simulations of zero-index thermal cloak. The system size is $20\times 20$ cm$^2$. The thermal conductivities of the core and shell are $\kappa _c=0.026$ and $\kappa _s=203$ W m$^{-1}$ K$^{-1}$, respectively. The thermal conductivities of the background in a–b and c–f are $\boldsymbol{\kappa }_b=400$ and $\boldsymbol{\kappa }_b=\textrm{diag}\left( 358,\,270\right)$ W m$^{-1}$ K$^{-1}$, respectively. The inner and outer radii of the shell in a and b are $r_{c1}=r_{c2}=4$ and $r_{s1}=r_{s2}=7$ cm, respectively. The inner and outer semiaxes of the elliptical shell in c–f are $r_{c1}=4$, $r_{c2}=2$, $r_{s1}=7$, and $r_{s2}=6$ cm, respectively. The left and right columns show the temperature profiles without and with a constant-temperature boundary condition, respectively. The constant-temperature boundary condition is set at 298 K. The high and low temperatures are set at 313 and 283 K, respectively. The other boundaries are insulated.

4 Finite-Element Simulation

We perform simulations with COMSOL Multiphysics to confirm these theoretical analyses. Without loss of generality, we discuss two two-dimensional cases, including a circular one and an elliptical one. Figure 5.2a and b show the circular case where the thermal conductivities of the core and shell are $\kappa _c=0.026$ and $\kappa _s=203$ W m$^{-1}$ K$^{-1}$, respectively. The thermal conductivity of the background is set as $\boldsymbol{\kappa }_b=\boldsymbol{\kappa }_e=400$ W m$^{-1}$ K$^{-1}$ which is derived from Eq. (5.4). When a constant-temperature boundary condition is not applied, the isotherms are contracted due to the smaller effective thermal conductivity of the core-shell structure (Fig. 5.2a). However, if we apply a constant temperature boundary condition to the boundary of the core, thermal cloaking can be achieved because the core has an effectively infinite thermal conductivity (Fig. 5.2b).

We further discuss the elliptical case with the same thermal conductivities of the core-shell structure, namely $\kappa _c=0.026$ and $\kappa _s=203$ W m$^{-1}$ K$^{-1}$. Unlike the circular case, the effective thermal conductivity of the elliptical core-shell structure is anisotropic. Therefore, we set the thermal conductivity of the background as (expressed in the Cartesian coordinates) $\boldsymbol{\kappa }_b=\boldsymbol{\kappa }_e=\textrm{diag}\left( 358,\,270\right)$ W m$^{-1}$ K$^{-1}$, which is also derived from Eq. (5.4). In the presence of a horizontal thermal field, the smaller effective thermal conductivity of the core-shell structure makes the isotherms contracted (Fig. 5.2c), whereas a constant-temperature boundary condition helps us achieve thermal cloaking (Fig. 5.2d). The results are similar if the system is in the presence of a vertical thermal field (Fig. 5.2e and f).

Fig. 5.3

Adapted from Ref. [22]

Schematic diagrams of six samples and experimental setup. The thermal conductivities of air and copper are 0.026 and 400 W m$^{-1}$ K$^{-1}$, respectively. The size of each sample is $20\times 20\times 4$ cm$^3$ with a copper thickness of 2 mm. The central air hole in a and b has a radius of 4 cm, and that in c–f has semiaxes of $r_{c1}=4$ and $r_{c2}=2$ cm. The effective shell radius in a and b is 7 cm, and the effective shell semiaxes in c–f are $r_{s1}=7$ and $r_{s2}=6$ cm. The air holes in the shell regions in a–f have the same radius of 1.6 mm, thus making the effective thermal conductivity of the shells to be 203 W m$^{-1}$ K$^{-1}$. The air holes in the background regions in c–f have a major semi axis of 2.9 mm and a minor one of 0.8 mm, thus making the effective thermal conductivity of the backgrounds to be $\textrm{diag}\left( 358,\,270\right)$ W m$^{-1}$ K$^{-1}$. The distance between air holes in the shell region is 5 mm, and that in the background region is 10 mm. The temperatures of the hot, medium and cold sources are set at 313, 298, and 283 K, respectively. g–i Sample photos of b, d, and f, respectively.

5 Laboratory Experiment

For experimental demonstration, we fabricate six samples to confirm the six simulations in Fig. 5.2. We use integrated fabrication technology, indicating that the samples have no weld joints. The three samples without a constant-temperature boundary condition are presented in Fig. 5.3a, c, and e. Air holes are drilled on the copper plate to realize the designed thermal conductivities of the shell and background. Another three samples with a constant-temperature boundary condition are presented in Fig. 5.3b, d, and f. By immersing the central hollow cylinders in an external thermostatic sink with medium temperature, a constant-temperature boundary condition can be obtained, and effectively infinite thermal conductivity is achieved. Compared with other active schemes [23,24,25], our scheme does not require complicated temperature settings. These six samples’ upper and lower surfaces are covered with transparent and foamed plastic to reduce environmental interferences. The sample photos of Fig. 5.3b, d, and f with top view are presented in Fig. 5.3g–i, respectively.

Fig. 5.4

Adapted from Ref. [22]

Measured results (left two columns) and simulated results (the third and fourth columns) of the six samples in Fig. 5.3. Dashed lines are plotted for the convenience of comparison. (m) and (n) show the temperature distributions at $x_1=-8$ cm (the origin is in the center of each simulation), and (o) shows the temperature distributions at $x_2=-8$ cm. Each line corresponds to a figure shown in the legend.

Then, we use the Flir E60 infrared camera to detect temperature distributions. The measured results corresponding to the six samples in Fig. 5.3 are presented in the left two columns of Fig. 5.4. We also perform finite-element simulations according to these six samples, and corresponding results are shown in the third and fourth columns of Fig. 5.4. For quantitative analyses, we plot the temperature distributions at $x_1=-8$ cm for the first two rows and $x_2=-8$ cm for the last row (the origin is in the center of each simulation). The experiments and simulations agree well with each other (Fig. 5.4m–o), thus confirming the feasibility of realizing zero-index thermal cloaks with effectively infinite thermal conductivities.

The cloaking effect is also robust under more complicated conditions such as different directions of external fields, point heat sources, and three dimensions. Furthermore, thermal cloaking can be extended to other functions such as thermal camouflaging. Nevertheless, the scheme is applicable for only stable states because the temperature of a constant-temperature boundary condition is fixed.

6 Conclusion

We have shown that an effectively infinite thermal conductivity can be precisely achieved by applying a constant-temperature boundary condition to a common material. Meanwhile, an external thermostatic sink can easily realize the constant-temperature boundary condition. The current approach has direct applications in designing zero-index thermal cloaks, which can work in highly conductive backgrounds with simple structures. These features, such as accuracy and simplicity, benefit practical applications. This work applies a constant-temperature boundary condition to realize effectively infinite thermal conductivity, which is expected to design more zero-index thermal metamaterials.

7 Exercise and Solution

Exercise

1. Derive Eq. (5.2).

Solution

1. Suppose the semiaxes of the core and shell along $x_i$ to be $r_{ci}$ and $r_{si}$, respectively. The conversion between the Cartesian coordinates and ellipsoidal (or elliptical) coordinates can be expressed as

$$\begin{aligned} \sum _i\dfrac{x_i^2}{\rho _j+r_{ci}^2}=1, \end{aligned}$$

(5.5)

with $j=1,\,2$ for two dimensions and $j=1,\,2,\,3$ for three dimensions. $\rho _1\left( >-r_{ci}^2\right)$ represents the boundary of an ellipse or an ellipsoid. For example, $\rho _1=\rho _{c}\left( =0\right)$ and $\rho _1=\rho _{s}$ can represent the inner and outer boundaries of the shell, respectively. In the presence of a thermal field along $x_i$, the heat conduction equation can be expressed as

$$\begin{aligned} \dfrac{\partial }{\partial \rho _1}\left( g\left( \rho _1\right) \dfrac{\partial T}{\partial \rho _1}\right) +\dfrac{g\left( \rho _1\right) }{\rho _1+r_{ci}^2}\dfrac{\partial T}{\partial \rho _1}=0, \end{aligned}$$

(5.6)

with $g\left( \rho _1\right) =\prod \limits _i\left( \rho _1+r_{ci}^2\right) ^{1/2}$. Accordingly, the temperatures of the core $T_{ci}$, shell $T_{si}$, and matrix $T_{si}$ can be expressed as

$$\begin{aligned} T_{ci}&=A_{ci}x_i, \end{aligned}$$

(5.7a)

$$\begin{aligned} T_{si}&=\left( A_{si}+B_{si}\phi _i\left( \rho _1\right) \right) x_i, \end{aligned}$$

(5.7b)

$$\begin{aligned} T_{mi}&=\left( A_{mi}+B_{mi}\phi _i\left( \rho _1\right) \right) x_i, \end{aligned}$$

(5.7c)

with $\phi _i\left( \rho _1\right) =\displaystyle \int _{\rho _{c}}^{\rho _1}\left( \left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) \right) ^{-1}{d}\rho _1$. $A_{ci}$, $A_{si}$, $B_{si}$, and $B_{mi}$ are determined by the following boundary conditions,

$$\begin{aligned} T_{ci}\left( \rho _1=\rho _{c}\right)&=T_{si}\left( \rho _1=\rho _{c}\right) ,\end{aligned}$$

(5.8a)

$$\begin{aligned} T_{mi}\left( \rho _1=\rho _{s}\right)&=T_{si}\left( \rho _1=\rho _{s}\right) , \end{aligned}$$

(5.8b)

$$\begin{aligned} \kappa _{c}\dfrac{\partial T_{ci}}{\partial \rho _1}\left( \rho _1=\rho _{c}\right)&=\kappa _{s}\dfrac{\partial T_{si}}{\partial \rho _1}\left( \rho _1=\rho _{c}\right) , \end{aligned}$$

(5.8c)

$$\begin{aligned} \kappa _{m}\dfrac{\partial T_{mi}}{\partial \rho _1}\left( \rho _1=\rho _{s}\right)&=\kappa _{s}\dfrac{\partial T_{si}}{\partial \rho _1}\left( \rho _1=\rho _{s}\right) . \end{aligned}$$

(5.8d)

We also need the following two mathematical skills,

$$\begin{aligned} \dfrac{\partial x_i}{\partial \rho _1}&=\dfrac{x_i}{2\left( \rho _1+r_{ci}^2\right) }, \end{aligned}$$

(5.9a)

$$\begin{aligned} \dfrac{\partial }{\partial \rho _1}\left( \phi _i\left( \rho _1\right) x_i\right)&=\dfrac{x_i}{2\left( \rho _1+r_{ci}^2\right) }\phi _i\left( \rho _1\right) +\dfrac{x_i}{\left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) } \nonumber \\&=\dfrac{x_i}{2\left( \rho _1+r_{ci}^2\right) }\left( \phi _i\left( \rho _1\right) +\dfrac{2}{g\left( \rho _1\right) }\right) . \end{aligned}$$

(5.9b)

Based on Eqs. (5.7) and (5.9), Eq. (5.8) can be written as

$$\begin{aligned} A_{ci}&=A_{si}+B_{si}\phi _i\left( \rho _{c}\right) , \end{aligned}$$

(5.10a)

$$\begin{aligned} A_{mi}+B_{mi}\phi _i\left( \rho _{s}\right)&=A_{si}+B_{si}\phi _i\left( \rho _{s}\right) , \end{aligned}$$

(5.10b)

$$\begin{aligned} \kappa _{c}A_{ci}&=\kappa _{s}\left( A_{si}+B_{si}\phi _i\left( \rho _{c}\right) +\dfrac{2B_{si}}{g\left( \rho _{c}\right) }\right) , \end{aligned}$$

(5.10c)

$$\begin{aligned} \kappa _{m}\left( A_{mi}+B_{mi}\phi _i\left( \rho _{s}\right) +\dfrac{2B_{mi}}{g\left( \rho _{s}\right) }\right)&=\kappa _{s}\left( A_{si}+B_{si}\phi _i\left( \rho _{s}\right) +\dfrac{2B_{si}}{g\left( \rho _{s}\right) }\right) . \end{aligned}$$

(5.10d)

To further simplify $\phi _i\left( \rho _{c}\right)$ and $\phi _i\left( \rho _{s}\right)$, we define the shape factors of the core and shell along $x_i$ as $L_{ci}$ and $L_{si}$, respectively. They can be written as

$$\begin{aligned} L_{ci}&=\dfrac{g\left( \rho _{c}\right) }{2}\displaystyle \int \limits _{\rho _{c}}^{\infty }\dfrac{{d}\rho _1}{\left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) }, \end{aligned}$$

(5.11a)

$$\begin{aligned} L_{si}&=\dfrac{g\left( \rho _{s}\right) }{2}\displaystyle \int \limits _{\rho _{s}}^{\infty }\dfrac{{d}\rho _1}{\left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) }, \end{aligned}$$

(5.11b)

with $g\left( \rho _{c}\right) =\prod \limits _ir_{ci}$, $g\left( \rho _{s}\right) =\prod \limits _ir_{si}$, and $\sum \limits _iL_{ci}=\sum \limits _iL_{si}=1$. For two dimensions, the shape factors can be further reduced to

$$\begin{aligned} L_{c1}&=\dfrac{r_{c2}}{r_{c1}+r_{c2}}, \end{aligned}$$

(5.12a)

$$\begin{aligned} L_{c2}&=\dfrac{r_{c1}}{r_{c1}+r_{c2}}, \end{aligned}$$

(5.12b)

$$\begin{aligned} L_{s1}&=\dfrac{r_{s2}}{r_{s1}+r_{s2}}, \end{aligned}$$

(5.12c)

$$\begin{aligned} L_{s2}&=\dfrac{r_{s1}}{r_{s1}+r_{s2}}. \end{aligned}$$

(5.12d)

Based on Eq. (5.11), we can derive

$$\begin{aligned} \phi _i\left( \rho _{c}\right)&=\int \limits _{\rho _{c}}^{\rho _{c}}\dfrac{{d}\rho _1}{\left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) }=0, \end{aligned}$$

(5.13a)

$$\begin{aligned} \phi _i\left( \rho _{s}\right)&=\left( \int \limits _{\rho _{c}}^{\infty }-\int \limits _{\rho _{s}}^{\infty }\right) \dfrac{{d}\rho _1}{\left( \rho _1+r_{ci}^2\right) g\left( \rho _1\right) } =\dfrac{2L_{ci}}{g\left( \rho _{c}\right) }-\dfrac{2L_{si}}{g\left( \rho _{s}\right) }. \end{aligned}$$

(5.13b)

With Eq. (5.10), we can derive $A_{ci}$, $A_{si}$, $B_{si}$, and $B_{mi}$. By setting $B_{mi}=0$, we can further derive the effective thermal conductivity of the core-shell structure,

$$\begin{aligned} \kappa _{e}=\kappa _{m}=\kappa _{s}\dfrac{L_{ci}\kappa _{c}+\left( 1-L_{ci}\right) \kappa _{s}+\left( 1-L_{si}\right) \left( \kappa _{c}-\kappa _{s}\right) f}{L_{ci}\kappa _{c}+\left( 1-L_{ci}\right) \kappa _{s}-L_{si}\left( \kappa _{c}-\kappa _{s}\right) f}, \end{aligned}$$

(5.14)

with $f=g\left( \rho _{c}\right) /g\left( \rho _{s}\right) =\prod \limits _ir_{ci}/r_{si}$, indicating area fraction for two dimensions and volume fraction for three dimensions.