Enhanced Thermal Invisibility Effect in an Isotropic Thermal Cloak\\ with Bulk Materials

Qingru Shan(单庆茹), Chunrui Shao(邵春瑞)$^{\hyperlink{s}{*}}$, Jun Wang(王军)$^{\hyperlink{s}{*}}$, and Guodong Xia(夏国栋)

doi:10.1088/0256-307X/40/10/104401

⇦Chinese Physics Letters, 2023, Vol. 40, No. 10, Article code 104401⇨ Enhanced Thermal Invisibility Effect in an Isotropic Thermal Cloak with Bulk Materials Qingru Shan (单庆茹), Chunrui Shao (邵春瑞)*, Jun Wang (王军)*, and Guodong Xia (夏国栋) Affiliations MOE Key Laboratory of Enhanced Heat Transfer and Energy Conservation, Beijing Key Laboratory of Heat Transfer and Energy Conversion, Beijing University of Technology, Beijing 100124, China Received 15 August 2023; accepted manuscript online 18 September 2023; published online 3 October 2023 ^*Corresponding authors. Email: chunruishao@emails.bjut.edu.cn; jwang@bjut.edu.cn Citation Text: Shan Q R, Shao C R, Wang J et al. 2023 Chin. Phys. Lett. 40 104401 Abstract A thermal cloak is well known for hiding objects from thermal signature. A bilayer thermal cloak made from inner insulation layer and outer isotropic homogeneous layer could realize such thermal protection. However, its thermal protection performance can be suppressed for low-thermal-conductivity surrounding media. We propose a tri-layer thermal cloak model by adding a transition layer between the insulation layer and the outer layer. Numerical simulations and theoretical analysis show that, under the same geometry size and surrounding thermal conductivity, the performance of the thermal cloak can be significantly enhanced by introducing a transition layer with higher thermal conductivity and an outer-layer with lower thermal conductivity. The tri-layer cloak proposed provides a design guidance to realize better thermal protection using isotropic bulk materials.

DOI:10.1088/0256-307X/40/10/104401 © 2023 Chinese Physics Society Article Text In recent years, several significant devices of thermal management, such as thermal cloaks,[1-25] thermal expanders,[25,26] thermal camouflage,[27-31] thermal rotators,[8] thermal concentrators,[8-12,16] thermal diodes,[32-35] and thermoregulatory textiles, have been designed and attracted intensive attention.[36] Zhang et al.[37] constructed a nanoscale thermal cloak based on silicon carbide film, and the functional area was constructed by transforming crystalline silicon carbide into amorphous through the “melt and quench” technique. A thermal cloak refers to the phenomenon that any object concealed inside the cloak will not alter the temperature distribution and heat flux outside the cloak. By introducing the concept of transformation into thermal transport, a thermal cloak can be designed, which can guide the heat flow around the internal object according to form invariability of the steady-state thermal conductive equation. Xu et al.[7] provided active and controllable components to thermal wave cloaks, which can be further used to design more active thermal wave metamaterials. Thermal cloaks can find potential applications in thermal protection, misleading infrared detection, and heat preservation or dissipation.[6,27-31] Based on transformation thermotics, many extensions of thermal cloaks have been proposed, such as anisotropic background,[2] ellipsoidal thermal concentrators, and cloaks,[3,4] non-steady states,[15-18] arbitrary shapes,[11,13,15] temperature-dependent cloaks,[6] and bifunctional cloaks.[14,15] However, the theoretical solution of transformation has strict limitations on non-uniformity and extreme anisotropy of thermal parameters of materials, which hardly exist in nature. Guenneau et al.[17] proposed a simplified construction which could be designed only with multilayers of homogeneous materials, so that extreme constitutive parameters are no longer required. Such a thermal cloak was experimentally studied[8] by alternately folding 40 layers of silicone elastomers and natural latex rubber, which bring the theoretical thermal cloak to practical applications. Considering that the thermal conduction under steady state condition obeys the Laplace equation, Han et al.[19] and Xu et al.[20] experimentally demonstrated a bilayer thermal cloak, which is made of two layers of homogeneous bulk materials. This thermal cloak consists of an inner layer with very low thermal conductivity and an outer layer with higher thermal conductivity. The central cloaked region can be protected by the inner thermal insulation layer. By directly solving the governing Laplace equation, the effective outer-layer thermal conductivity could be determined, which can eliminate the external-field distortion. It is very important for a bilayer thermal cloak that the thermal conductivity of the inner layer must be approximately zero, otherwise the thermal cloak phenomenon could not be achieved. Note that this bilayer thermal cloak model does not rely on transformation thermotics. On the basis of bilayer thermal cloaks, a functional camouflage device is proposed, in which the origin thermal signature could be transformed as illusion objects and its transient extension has also been theoretically studied.[26,27] Although the bilayer thermal cloak could realize thermal protection under certain conditions, its thermal protection performance is suppressed for higher inner-layer thermal conductivity or lower surrounding thermal conductivity. In this Letter, we propose a tri-layer thermal cloak model by adding a transition layer between the insulation layer and the outer layer. Based on theoretical analysis and numerical simulations, it is found that the thermal protection performance can be significantly enhanced by introducing the transition layer into a thermal cloak. The bilayer thermal cloak model proposed in Refs. [19,20] is schematically shown in Fig. 1. The central gray area ($r < a$) represents the cloaked region with thermal conductivity $k_{0}$, the yellow shell ($a < r < b$) represents the inner layer with thermal conductivity $k_{1}$, and the orange shell ($b < r < d$) indicates the outer layer with thermal conductivity $k_{2}'$. Figure 1(b) illustrates the cross-sectional view of the bilayer thermal cloak model, in which the blue rectangle ($r>d$) represents the surrounding area with thermal conductivity $k_{4}$. High source temperature $T_{\rm H}$ and low source temperature $T_{\rm L}$ are respectively applied to the upper and lower ends of the surrounding area.

Fig. 1. (a) Three-dimensional schematic diagram and (b) cross-sectional view of a bilayer thermal cloak model.

For a perfect thermal cloak, the inner layer should be a perfect thermal insulation layer ($k_{1}=0$ W$\cdot$m$^{-1}\cdot$K$^{-1}$), by which the central region can be protected and its influence on the surrounding area can be neglected. However, if there is no outer layer between the inner layer and the surrounding media, then the insulation layer will repel the heat flow and the external isotherms can be distorted. Thus, another outer layer with higher thermal conductivity is necessary, which can attract the heat flow that is repelled by the insulation layer. As a result, the thermal profiles of the surrounding media can restore isothermal lines as if nothing were there. The theoretical framework of the bilayer thermal cloak is given in Refs. [19,20]. It is assumed that $k_{1}=0$ W$\cdot$m$^{-1}\cdot$K$^{-1}$ ensures that the external temperature field cannot be affected by the central area. Then, the thermal conductivity of the outer layer of the thermal cloak is given by \begin{align} k_{2}' =\frac{2d^{3}+b^{3}}{2(d^{3}-b^{3})}k_{4}, \tag {1} \end{align} which can eliminate the distortion in the external temperature field. However, there is no perfect thermal insulator for the existing bulk materials. As a result, the thermal protection effect of the bilayer thermal cloak depends on $k_{1}$ and $k_{4}$. The numerical simulation of the heat transfer through the present model is conducted based on the finite element method (software COMSOL Multiphysics). The model sizes and temperature sources are set the same as those in Ref. [19], namely, $a=6$ mm, $b=9.5$ mm, $d=12$ mm, $T_{\rm H}=333$ K, and $T_{\rm L}=273$ K. The thermal conductivity of the central part is set as $k_{0}=10$ W$\cdot$m$^{-1}\cdot$K$^{-1}$. The external area is $15\,{\rm mm} \times 30$ mm, dash line at $r=\delta$ ($\delta =13$ mm) and $r=\delta '$ ($\delta '=5$ mm) are used for data collection. Figure 2 shows the magnitude of the local temperature gradient at $r=\delta$ versus $\varphi$ with varying $k_{1}$ and $k_{4}$, wherein $\varphi$ is illustrated in Fig. 1(b).

Fig. 2. Temperature gradient at $r=\delta$ versus $\varphi$ in a bilayer thermal cloak (a) for varying inner thermal conductivity $k_{1}$ ($k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$) and (b) for varying surrounding thermal conductivity $k_{4}$ ($k_{1}=0.03$ W$\cdot$m$^{-1}\cdot$K$^{-1}$). The thermal conductivity of the outer layer $k_{2}'$ is determined with Eq. (1).

For a perfect thermal cloak, the temperature gradient at $r=\delta$ should have a magnitude of 2000 K/m, which is equal to the temperature difference divided by the spatial distance between the hot and cold thermal reservoirs. In Fig. 2(a), the surrounding thermal conductivity is set as $k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$ (ACCSILICONES–AS1802), and $k_{2}'$ is determined with Eq. (1). To ensure that the external temperature field would not be affected by the central area, $k_{1}=0$ W$\cdot$m$^{-1}\cdot$K$^{-1}$ is required. However, owing to the nonzero thermal conductivity of the inner insulation layer, the performance of the thermal cloak is weakened since the inner layer cannot repel the heat flow quite well. The temperature gradient in the environment differs greatly from the reference temperature gradient for relatively high $k_{1}$. With decreasing $k_{1}$, the cloak performance could be improved, and for an extremely small $k_{1}$, the temperature gradient is almost the same as the reference gradient. Assume that the inner layer is a good thermal insulator with $k_{1}=0.03$ W$\cdot$m$^{-1}\cdot$K$^{-1}$ (expanded polystyrene). The dependence of the temperature gradient at $r=\delta$ on $k_{4}$ is presented in Fig. 2(b). For higher $k_{4}$, the magnitude of the temperature gradient is very close to 2000 K/m, and a good thermal cloak can be realized. However, for lower $k_{4}$, the performance of the bilayer thermal cloak gets worse. For $k_{4}=0.5$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, there is a significant difference between the temperature gradient magnitude at $r=\delta$ and the reference temperature gradient. Here, the poor thermal protection of the bilayer thermal cloak at higher $k_{1}$ or lower $k_{4}$ is due to the breakdown of Eq. (1). In other words, the thermal insulation layer could not act as a perfect insulator at higher $k_{1}$ or lower $k_{4}$. In the present study, the ratio of the difference between the maximum and minimum temperature gradient magnitude at $r=\delta$ to the reference temperature gradient magnitude (i.e. 2000 K/m) is defined to evaluate the performance of a thermal cloak, \begin{align} \varepsilon =\frac{\nabla T_{\max}|{_{r=\delta} -\nabla T_{\min}|{_{r=\delta}}}}{\nabla T_{\rm ref}}. \tag {2} \end{align} Apparently, $\varepsilon =0$ indicates that temperature gradient in the external environment is exactly equal to the reference temperature gradient (a perfect thermal cloak). Figure 3 shows the performance of the bilayer cloak versus contact thermal resistance under varying surroundings conductivity. For simplicity, we assume that the contact thermal resistances between different layers are equal to each other. The thermal conductivity of the outer layer $k_{2}'$ is determined with Eq. (1). It is found that $\varepsilon$ decreases and then increases along with contact resistance $R$. This issue can be understood heuristically as follows. For a perfect thermal cloak, the heat flux at $r=\delta$ should have a magnitude of 1000 W/m$^{2}$, 4600 W/m$^{2}$, 20000 W/m$^{2}$, corresponding to the surroundings conductivity $k_{4}=0.5$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, $k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, and $k_{4}=10$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, respectively. However, the averaged heat flux at $r=\delta$ with various conductivities of surroundings are 1022.35 W/m$^{2}$, 4622.76 W/m$^{2}$, and 20022.85 W/m$^{2}$, which are higher than those for perfect thermal cloaks with $k_{1}=0$ W$\cdot$m$^{-1}\cdot$K$^{-1}$. Therefore, lower interfacial thermal resistance can prevent heat from flowing into the cloaked area, which could enhance the performance of the bilayer cloak to a certain extent. For higher interfacial thermal resistance, which may dominate the thermal transport through the system, poor thermal cloak performance should be expected, as shown in Fig. 3.

Fig. 3. Characteristics of $\varepsilon$ versus the contact thermal resistance $R$ in a bilayer thermal cloak for varying $k_{4}$.

Additionally, the size of bilayer thermal cloaks could also affect the thermal protecting performance. Figure 4 plots $\varepsilon$ versus $b$ in a bilayer thermal cloak for relatively lower $k_{4}$, where $a=6$ mm, $d=12$ mm, and $k_{1}=0.03$ W$\cdot$m$^{-1}\cdot$K$^{-1}$. The thermal conductivity of the outer layer $k_{2}'$ under different geometric sizes is determined with Eq. (1). It can be seen that the performance of the bilayer cloak can be enhanced by increasing $b$, and further enhancement of the cloaking performance cannot be achieved by adjusting the geometric size for $b > 9.5$ mm.

Fig. 4. Behavior of $\varepsilon$ versus $b$ in a bilayer thermal cloak with varying $k_{4}$, for $a=6$ mm, $d=12$ mm, and $k_{1}=0.03$ W$\cdot$m$^{-1}\cdot$K$^{-1}$.

Fig. 5. Exhibition of $\varepsilon$ versus $k_{4}/k_{1}$ in a bilayer thermal cloak for $k_{4}=0.5$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, $k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, and $k_{4}=10$ W$\cdot$m$^{-1}\cdot$K$^{-1}$. Here $k_{1}$ is varied from 0.003 W$\cdot$m$^{-1}\cdot$K$^{-1}$ to 0.3 W$\cdot$m$^{-1}\cdot$K$^{-1}$, $b=9.5$ mm. The dashed line denotes the power-law dependence with $\alpha \approx -0.88$.

In Fig. 5, the parameter $\varepsilon$ is plotted versus $k_{4}/k_{1}$ by gradually varying $k_{1}$ or $k_{4}$, where $b=9.5$ mm. A power-law dependence of $\varepsilon$ on $k_{4}/k_{1}$ is found, i.e., $\varepsilon \sim (k_{4}/k_{1})^{\alpha}$ and $\alpha \approx -0.88$ in Fig. 3, as shown by the dashed line. As mentioned above, for higher inner-layer thermal conductivity $k_{1}$ or lower surrounding thermal conductivity $k_{4}$ in the bilayer thermal cloak model, the performance of the cloak effect gets worse because the thermal insulation layer could not act as a perfect insulator. In this study, we propose a tri-layer thermal cloak model, as shown in Fig. 6, in which the original outer layer of the bilayer thermal cloak (with thermal conductivity $k_{2}'$) is replaced by a transition layer ($b < r < c$) with thermal conductivity $k_{2}$ and an outer layer ($c < r < d$) with thermal conductivity $k_{3}$, as shown in Fig. 6(b).

Fig. 6. (a) Three-dimensional schematic diagram of a tri-layer thermal cloak, (b) the schematic design of the tri-layer thermal cloak. Here, the original outer layer of the bilayer thermal cloak is split into a transition layer and an outer layer.

For steady-state heat conduction without internal heat source, we can take $\nabla\cdot(k\nabla T)=0$ as the governing equation. In a polar coordinate system, it can be expressed as \begin{align} T_{i} =\sum\limits_{m=1}^\infty {\big[A_{m}^{i} } r^{m}+B_{m}^{i} r^{-m-1}\big]P_{m} (\cos \theta), \tag {3} \end{align} where $A^i_m$ and $B^i_m$ are Laplace solution constants. $P_{m}$ is the $m$-th order Legendre polynomials. $T_{i}$ represents the temperature in different regions, where the subscript $i=0$, 1, 2, 3, 4 represent the central cloak region, inner layer, transition layer, outer layer, and surrounding area, respectively. In the surrounding area, $A^4_1=-\nabla T_{4}$ and $B^4_1$ represents the temperature gradient distortion. $B^4_1=0$ indicates that the temperature gradient of surrounding area is uniform (i.e., perfect thermal protection). The closer the $B^4_1$ is to zero, the better the thermal protection is achieved. The boundary conditions of Eq. (3) is given by \begin{align} &T_{1} |_{r\to 0} = {\rm finity},\nonumber\\ &T_{4} |_{r\to \infty} =-|{\nabla T_{4}}|r\cos \theta. \tag {4} \end{align} Then, it could be obtained that $m=1$, $B^4_1$=0. At the interface between two adjacent layers, the temperature and heat flow should be continuous, so \begin{align} &T_{i} |_{r={a,b,c,d}}=T_{i+1}|_{r={a,b,c,d}},\nonumber\\ &k_{i} \frac{\partial T_{i} }{\partial r}|_{r={a,b,c,d}}=k_{i+1} \frac{\partial T_{i} }{\partial r}|_{r={a,b,c,d}}. \tag {5} \end{align} Assuming that the inner layer of the thermal cloak is completely insulated ($k_{1}=0$), then using Eqs. (3)-(5) and $B^4_1=0$, we have \begin{align} k_{3}=\,&\big[c^{3}d^{3}(4+2\beta)+b^{3}d^{3}(2-2\beta)+c^{6}(2-2\beta)\nonumber\\ &+b^{3}c^{3}(1+2\beta)\big]\big[c^{3}d^{3}(4+2\beta)+b^{3}d^{3}(2-2\beta)\nonumber\\ &+c^{6}(4\beta -4)-b^{3}c^{3}(2+4\beta)\big]^{-1}k_{4}, \tag {6} \end{align} where $\beta =k_{2}/k_{3}$. For $\beta =1$ ($k_{2}=k_{3}$), the tri-layer thermal cloak could be simplified to a bilayer cloak, and Eq. (6) can reduce to \begin{align} k_{2} =k_{3} =\frac{2d^{3}+b^{3}}{2(d^{3}-b^{3})}k_{4}, \tag {7} \end{align} which is consistent with Eq. (1).

Fig. 7. (a)–(d) Temperature distributions, wherein black lines represent the isotherm; (e)–(h) temperature gradient distributions, (i)–(l) local heat flux distributions, wherein the arrows denotes the direction of the heat flux. $T_{\rm H}=333$ K and $T_{\rm L}=273$ K.

Assuming that the transition layer and the outer layer have equal thickness, i.e., $c=(b+d)/2=10.75$ mm, and the thermal conductivity of the insulation layer $k_{1}=0.03$ W$\cdot$m$^{-1}\cdot$K$^{-1}$. For a given surrounding thermal conductivity $k_{4}$, Eq. (6) presents the relationship between $k_{2}$ and $k_{3}$, which is required for a perfect thermal cloak. Generally, the thermal conductivity of a common bulk material is less than that of copper and the commonly used thermal insulation materials cannot be completely insulated. Thus, only the calculation results with the thermal conductivity in the range of 0.03–400 W$\cdot$m$^{-1}\cdot$K$^{-1}$ are considered in the present study. For $k_{1}=0.03$ W$\cdot$m$^{-1}\cdot$K$^{-1}$ and $k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, Fig. 7 plots the temperature distributions, the temperature gradient distributions, and heat flux distributions for four combinations of $k_{2}$ and $k_{3}$ given by Eq. (6), corresponding to $\beta =0.003$ ($k_{2}=0.03328$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, $k_{3}=10.14$ W$\cdot$m$^{-1}\cdot$K$^{-1}$), $\beta =1$ ($k_{2}=k_{3}=5.698$ W$\cdot$m$^{-1}\cdot$K$^{-1}$), $\beta =100$ ($k_{2}= 37.28$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, $k_{3}= 0.3728$ W$\cdot$m$^{-1}\cdot$K$^{-1}$), and $\beta =1000$ ($k_{2}=276.04$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, $k_{3}=0.27604$ W$\cdot$m$^{-1}\cdot$K$^{-1}$), respectively. It can be seen that heat flux through the transition layer increases and the temperature gradient in the cloaked area decreases with the increasing $\beta$. The heat flux repelled by the thermal insulation layer could be transferred to the outer layer with lower $\beta$, while for higher $\beta$, the repelled flux could flow in the transition layer, which improves the cloaking performance significantly.

Fig. 8. Temperature gradient with $\varphi$ at $r=\delta$. The dotted line represents the reference temperature gradient.

Figure 8 plots the corresponding local temperature gradient magnitude at $r=\delta$ for $\beta =0.003$, $\beta =1$, $\beta =100$, and $\beta =1000$, respectively. It can be seen that for lower or higher $\beta$, the external temperature gradient is closer to $\nabla T_{\rm ref}$ compared with $\beta =1$. Especially for $\beta =1000$, the local temperature gradient basically coincides with the reference temperature gradient, which indicates a perfect thermal cloak. For higher $\beta$, the transition layer has a higher thermal conductivity, while the outer layer has lower thermal conductivity. In this case, the transition layer plays the role of a heat conduction layer with high thermal conductivity. The heat flow that is repelled by the thermal insulation layer can be transferred through the transition layer, as shown in Fig. 7. The outer insulation layer protects the temperature file of the surrounding area from the heat flow through the transition layer. Therefore, the thermal cloak effect can be improved by the tri-layer structure with higher $\beta$. In the case of lower $\beta$, the transition layer is close to a thermal insulation layer ($k_{2}=0.03328$ W$\cdot$m$^{-1}\cdot$K$^{-1}$ for $\beta =0.003$), which enhances the insulation effect and in turn enhances the thermal cloak effect. To further analyzing the performance of the tri-layer cloak, the temperature distribution of the cloaked region is checked, and a line at $r=\delta '$ ($\delta '=5$ mm) is used for data collection (see the red line in Fig. 1). Another parameter $\omega$, i.e., the ratio of the difference between the maximum and minimum temperature magnitudes at $r=\delta '$ to the reference temperature magnitude (i.e., 303 K), is defined to evaluate the performance of a thermal cloak of the cloaked region, \begin{align} \omega =\frac{T_{\max}|{_{r=\delta'}-T_{\min}|{_{r=\delta'}}}}{T_{\rm ref} }. \tag {8} \end{align} Apparently, $\omega =0$ indicates that temperature in the cloaked region is exactly uniform (a perfect thermal cloak). Due to the nonzero thermal conductivity of $k_{1}$, a perfect thermal protection could not be achieved. Figure 9 illustrates $B^4_1$, $\varepsilon$, and $\omega$ versus $\beta$ under different $k_{4}$. $B^4_1$ and $\varepsilon$ are determined to evaluate the ambient cloaking performance, while $\omega$ is defined to evaluate the performance in the cloaked region. It is found that the constant $B^4_1$, $\varepsilon$ and $\omega$ have the similar trends, and they increase with $\beta$ at first, then decrease after maximal values at about $\beta =1$. Therefore, the thermal cloak effect can be enhanced by introducing a transition layer, and the thermal cloak effect can be significantly improved by increasing $\beta$ in the case of $\beta >1$. This is due to the fact that a larger $\beta$ is equivalent to introducing another insulation layer (outer layer with $k_{3}$), which could effectively reduce the impact of the nonzero thermal conductivity of the insulation layer. For higher $k_{4}$, the thermal conductivity of the insulation layer is quite smaller than those of other layers. Then, the influence of the parameter $\beta$ is weakened on the thermal protection. For lower $k_{4}$, as the thermal conductivity of the surrounding area decreases, the thermal conductivity of the transition layer and the outer layer also gradually decreases. Then, the influence of the insulation layer is gradually strengthened and it is necessary to introduce a transition layer to improve the thermal cloak effect.

Fig. 9. (a) $B^4_1$ versus $\beta$ under different $k_{4}$, (b) $\varepsilon$ versus $\beta$ under different $k_{4}$, (c) $\omega$ versus $\beta$ under different $k_{4}$ in the tri-layer thermal cloak.

Figure 10 shows the performance of the tri-layer cloak versus the contact thermal resistance for $k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$. It is shown that $\varepsilon$ decreases with $R$, then increases along with contact resistance $R$. The thermal cloak performance can be improved by a lower contact thermal resistance, while it becomes worse for very high contact thermal resistance.

Fig. 10. Variation of $\varepsilon$ versus the contact thermal resistance $R$ in a tri-layer thermal cloak for $k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$.

The performance of cloaks could also be adjusted by changing the geometric size. Assuming that the transition layer and the outer layer have equal thickness, i.e., $a=6$ mm, $d=12$ mm, $c=(b+d)/2$, and the thermal conductivity of the insulation layer is $k_{1}=0.03$ W$\cdot$m$^{-1}\cdot$K$^{-1}$. For a given ambient thermal conductivity $k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$. Equation (6) presents the relationships between $k_{2}$ and $k_{3}$. Figure 11 illustrates $\varepsilon$ and $\omega$ versus $\beta$ under various $b$. It can be found that the thermal cloak effect can be enhanced by introducing a transition layer under different geometric size, and the thermal cloak effect can be significantly improved by increasing $\beta$ while $\beta >1$. Here $\beta =1$ refers to the corresponding bilayer cloaks. Obviously the tri-layer cloak has better performance compared to the bilayer cloak when $b>9.5$ mm.

Fig. 11. Plots of (a) $\varepsilon$ and (b) $\omega$ versus $\beta$ under different geometric sizes for $k_{4}=2.3$ W$\cdot$m$^{-1}\cdot$K$^{-1}$, $a=6$ mm, $d=12$ mm, $c=(b+d)/2$, and $k_{1}=0.03$ W$\cdot$m$^{-1}\cdot$K$^{-1}$.

Figure 12 presents the parameter $\varepsilon$ versus $k_{4}/k_{1}$ with various $\beta$, where $b=9.5$ mm. It could be seen that power-law relationship between $\varepsilon$ and $k_{4}/k_{1}$ could also be found in the tri-layer thermal cloak, which means that better thermal protect performance could be expected for high $k_{4}/k_{1}$. For low $k_{4}$ or high $k_{1}$, the thermal cloak performance can be enhanced by increasing $\beta$ (by choosing a higher $k_{2}$ or lower $k_{3}$).

Fig. 12. Plots of $\varepsilon$ versus $k_{4}/k_{1}$ in the tri-layer thermal cloak

In summary, we propose a tri-layer thermal cloak design by introducing a transition layer into the bi-layer thermal cloak model. The effect of the tri-layer thermal cloak is analyzed based on the numerical simulations and theoretical analysis. It is found that the thermal protection performance can be greatly enhanced with higher transition-layer thermal conductivity and lower outer-layer thermal conductivity. The transition layer plays the role of a heat conduction layer, through which the heat flow that is repelled by the thermal insulation layer can be transferred from the high to the low temperature source. The outer layer with lower thermal conductivity could effectively reduce the negative effect of the nonzero thermal conductivity of the inner insulation layer. References Shaped graded materials with an apparent negative thermal conductivityCloak for curvilinearly anisotropic media in conductionEllipsoidal Thermal Concentrator and Cloak with Transformation MediaAnalysis of elliptical thermal cloak based on entropy generation and entransy dissipation approach*Geometric phase, effective conductivity enhancement, and invisibility cloak in thermal convection-conductionTemperature-Dependent Transformation Thermotics: From Switchable Thermal Cloaks to Macroscopic Thermal DiodesActive Thermal Wave CloakHeat Flux Manipulation with Engineered Thermal MaterialsIsotropic Thermal Cloaks with Thermal Manipulation FunctionTheory of transformation thermal convection for creeping flow in porous media: Cloaking, concentrating, and camouflageDesign of square-shaped heat flux cloaks and concentrators using method of coordinate transformationThermal cloak-concentratorDesign of two-dimensional open cloaks with finite material parameters for thermodynamicsA bifunctional cloak using transformation mediaArbitrarily polygonal transient thermal cloaks with natural bulk materials in bilayer configurationsTransformation thermodynamics: cloaking and concentrating heat fluxExperiments on Transformation Thermodynamics: Molding the Flow of HeatTransient heat flux shielding using thermal metamaterialsExperimental Demonstration of a Bilayer Thermal CloakUltrathin Three-Dimensional Thermal CloakEffect of Interfacial Thermal Resistance in a Thermal CloakAnalysis of a trilayer thermal cloakExperimental Demonstration of a Multiphysics Cloak: Manipulating Heat Flux and Electric Current SimultaneouslyHomogeneous Thermal Cloak with Constant Conductivity and Tunable Heat LocalizationMetamaterials for Manipulating Thermal Radiation: Transparency, Cloak, and ExpanderThermal expanderFull Control and Manipulation of Heat Signatures: Cloaking, Camouflage and Thermal MetamaterialsTransient thermal camouflage and heat signature controlCamouflage thermotics: A cavity without disturbing heat signatures outsideStructured thermal surface for radiative camouflageIllusion ThermoticsEnhancement of thermal rectification effect based on zigzag interfaces in bi-segment thermal rectifier using bulk materialsThermal diodes, regulators, and switches: Physical mechanisms and potential applicationsColloquium : Phononics: Manipulating heat flow with electronic analogs and beyondFive thermal energy grand challenges for decarbonizationDesigning heat transfer pathways for advanced thermoregulatory textilesConstruction and mechanism analysis on nanoscale thermal cloak by in-situ annealing silicon carbide film

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