⇦Chinese Physics Letters, 2023, Vol. 40, No. 9, Article code 094401 Autonomously Tuning Multilayer Thermal Cloak with Variable Thermal Conductivity Based on Thermal Triggered Dual Phase-Transition Metamaterial Qi Lou (娄琦)1 and Ming-Gang Xia (夏明岗)1,2,3* Affiliations 1Department of Applied Physics, School of Physics, Xi'an Jiaotong University, Xi'an 710049, China 2MOE Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter, School of Physics, Xi'an Jiaotong University, Xi'an 710049, China 3Shaanxi Province Key Laboratory of Quantum Information and Optoelectronic Quantum Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, China Received 30 June 2023; accepted manuscript online 23 August 2023; published online 7 September 2023 *Corresponding author. Email: xiamg@mail.xjtu.edu.cn Citation Text: Lou Q and Xia M G 2023 Chin. Phys. Lett. 40 094401    Abstract Thermal cloaks offer the potential to conceal internal objects from detection or to prevent thermal shock by controlling external heat flow. However, most conventional natural materials lack the desired flexibility and versatility required for on-demand thermal manipulation. We propose a solution in the form of homogeneous multilayer thermodynamic cloaks. Through an ingenious design, these cloaks achieve exceptional and extreme parameters, enabling the distribution of multiple materials in space. We first investigate the effects of important design parameters on the thermal shielding effectiveness of conventional thermal cloaks. Subsequently, we introduce an autonomous tuning function for the thermodynamic cloak, accomplished by leveraging two phase transition materials as thermal conductive layers. Remarkably, this tuning function does not require any energy input. Finite element analysis results demonstrate a significant reduction in the temperature gradient inside the thermal cloak compared to the surrounding background. This reduction indicates the cloak's remarkable ability to manipulate the spatial thermal field. Furthermore, the utilization of materials undergoing phase transition leads to an increase in thermal conductivity, enabling the cloak to achieve the opposite variation of the temperature field between the object region and the background. This means that, while the temperature gradient within the cloak decreases, the temperature gradient in the background increases. This work addresses a compelling and crucial challenge in the realm of thermal metamaterials, i.e., autonomous tuning of the thermal field without energy input. Such an achievement is currently unattainable with existing natural materials. This study establishes the groundwork for the application of thermal metamaterials in thermodynamic cloaks, with potential extensions into thermal energy harvesting, thermal camouflage, and thermoelectric conversion devices. By harnessing phonons, our findings provide an unprecedented and practical approach to flexibly implementing thermal cloaks and manipulating heat flow.
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 DOI:10.1088/0256-307X/40/9/094401 © 2023 Chinese Physics Society Article Text The ability to control and manipulate heat distribution in thermal fields in a flexible and diverse way plays a pivotal role in contemporary energy technologies. Macroscopic heat transfer processes underpin numerous thermal management issues encountered in daily life and industrial production, including military applications, energy conversion, and thermal management of electronic components.[1-3] The drive within materials science and physics for heat transfer research has positioned thermal metamaterials and transformation thermodynamics as distinctive avenues for macroscopic heat field control, particularly in the manipulation of heat transfer situations.[4] These two concepts have evolved from metamaterials and transformation optics within thermal fields. To enhance comprehension of the uniqueness of thermal metamaterials and transformational thermodynamics in macroscopic thermal field manipulation, a brief outline of metamaterials and transformation optics is provided in the following. Progress in energy management technology is often facilitated by advancements in the realm of functional materials. Apart from natural materials, which rely on the concoction of new chemical elements, new composite materials can be precisely formed through the combination of natural materials. In contrast to natural materials, which undergo variations at atomic or molecular scales, new composite materials are directly designed and arranged into unique geometric configurations at the macroscopic scale. These purposefully designed composites boast distinctive properties, surpassing those of natural materials, and are termed metamaterials. The metamaterials concept was conceived in the field of electromagnetism.[5-8] Researchers have demonstrated that artificial metamaterials with spatially varying optical properties can modulate light fields almost arbitrarily, which has the most quintessential application of electromagnetic cloak.[9] The mapping between different coordinate systems in transformation optics often results in inhomogeneous, anisotropic, or even singular parameter distributions in the transformation space due to spatial distortions.[10] However, the distinctive attributes of metamaterials can largely compensate for this disadvantage. Metamaterials can be reverse-engineered, wherein transformation optical technology is employed to obtain the spatial distribution of energy control parameters on demand. Subsequently, metamaterials matching the required parameter distribution are designed and constructed to realize arbitrary manipulation of the electromagnetic field. Metamaterials and transformation optics rapidly expanded into various wave fields, including sound and elastic fields, in their early stages and continue to flourish.[11-19] As per the definitional concept of metamaterials, the geometric size of a metamaterial protocell should be smaller than the characteristic length which pertains to the incident wavelength of the wave in the wave's physical field. For a thermal field, the characteristic length refers to the length of heat diffusion related to time during heat conduction and the length of geometric heat flow in the medium, proportional to time, in the heat conduction process.[20] Drawing inspiration from the electromagnetic cloak, some researchers in 2008 applied transformation technology to the heat conduction process, proposed the concepts of transformation thermodynamics and thermal metamaterials, and demonstrated the abnormal thermal phenomenon of a thermal cloak.[21] Artificial thermal metamaterials, constructed with special structures, introduce new thermal properties and functions absent in natural materials or compounds, thereby offering a fresh perspective on thermal field manipulation.[22,23] Research in this domain has rapidly increased, with studies on nanoscale thermal management,[24] thermal rotators,[25,26] thermal encoding,[27] thermal illusions,[28-30] thermal camouflage,[31,32] and other thermal metamaterials and devices.[33,34] The outcomes have innovatively spurred and promoted the development of thermal metamaterial devices. Today, thermal metamaterials have been widely explored by research teams globally, and their substantial value in the flexible and efficient control and management of thermal energy has been widely acknowledged by the academic community. Huang et al. designed an anisotropic stealthy thermal cloak on the basis of the principle of transformation thermodynamics. This definition method bestowed ideal thermodynamic properties on thermal metamaterials. However, special thermodynamic parameters pose significant challenges to the creation of metamaterials, as the spatially continuous change in thermal conductivity necessitates a proportional continuous change in each material component. Therefore, searching suitable natural materials and achieving the target thermodynamic parameters with a specific macroscopic spatial geometry is of challenge.[35] Han et al. employed multilayer homogeneous metamaterials to attain the difference in radial and tangential thermal conductivity of the ideal thermal cloak by stacking thermal conductivity and thermal resistance layers alternately. On a macro level, heat flow tends to be conducted in the high thermal conductivity layer rather than between the heat conduction layer and the heat resistance layer. That work showed the thermal cloak function of metamaterials that could be realized by using only two common natural materials. Nonetheless, that thermal cloak had a single function and limited application scenarios, with neither principles of material selection nor effects of metamaterial-related parameters on thermodynamic properties provided. Based on this, the present work investigates the effects of thermal conductivity, the number of layers, and the thickness of thermal barrier layers on macroscopic heat transfer for a homogeneous multilayer thermodynamic cloak. A thermodynamic cloak with dual phase-transition properties is designed, offering potential applications in realizing autonomous thermal management.
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Fig. 1. (a) Schematic of the bare sensor and cloaked sensor with topology-optimized freeform thermal metamaterials. (b)–(d) Temperature distributions of the observed area without cloak for $\kappa_{\rm o}=\kappa_{\rm b}$, $\kappa_{\rm o} < \kappa_{\rm b}$, and $\kappa_{\rm o} > \kappa_{\rm b}$. (e)–(g) Temperature distributions of the observed area with cloak for $\kappa_{\rm o}=\kappa_{\rm b}$, $\kappa_{\rm o} < \kappa_{\rm b}$, and $\kappa_{\rm o} > \kappa_{\rm b}$.
In this study, we scrutinize the geometric parameters of structures from the referenced literature. Furthermore, we elucidate the effects of altering the geometric parameters of a three-dimensional thermal cloak on the behavior of an approximate two-dimensional condition. Previous work established the feasibility of an analytical approach that reduces a three-dimensional thermal diffusion issue to a two-dimensional context.[36-38] In our current work, we define an object region, and give the diverse working scenarios of the thermal cloak, we vary the thermal conductivity of the object arbitrarily in contrast with the background thermal conductivity. Figures 1(b)–1(d) present the temperature distributions in the background area upon placing objects with different thermal conductivities and reaching steady state. The temperature field is distorted because of the difference between the thermal conductivity of the object ($\kappa_{\rm o}$) and that of the background ($\kappa_{\rm b}$). The disturbance of the temperature field worsens as this difference increases. A conventional thermal cloak is then set up around the object region, as shown in Figs. 1(e)–1(g), where the thermal conductivity of the thermal conductive layer ($\kappa_{\rm c}$) is 7.87 W/(m$\cdot$K), and the thermal conductivity of the thermal resistive layer ($\kappa_{\rm r}$) is 0.13 W/(m$\cdot$K). The thermal conductive layer is thermal epoxy, and the thermal resistive layer is natural rubber, which are common commercially available materials. The results show that the temperature gradient in the object region is remarkably reduced, which proves that the incorporation of the thermal cloak is equivalent to providing a channel for the heat flow in the space, so that heat flux is inclined to flow along the thermal cloak from the hot end to the cold end, thus reducing the flows of heat through the central region. In most cases, $\kappa_{\rm o}$ differs from $\kappa_{\rm b}$, and different thermal conductivities lead to variation in the temperature gradient around the object; such variation is the reason why most objects are not thermally stealthy in the background environment.[39-41] The above-mentioned studies demonstrated the ability of thermal cloaks to manipulate the thermal field in space. In the present work, the effect of thermal cloak geometrical parameters is studied to achieve enhanced performance in a limited space.
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Fig. 2. Simulated temperature difference in the object area for the background thermal conductivity: (a) $\kappa_{\rm b} = 0.023$ W/(m$\cdot$K), (b) $\kappa_{\rm b} = 1$ W/(m$\cdot$K), and (c) $\kappa_{\rm b} =10$ W/(m$\cdot$K). (d) Simulated variation in the temperature difference in the object region versus $C$. (e)–(h) Temperature distribution in space at different values of $C$. (i) Simulated variation in the temperature difference in the object region with N$_{\rm total}$. (j)–(m) Temperature distributions in space at different values of $N_{\rm total}$.
It is known that multilayer thermal cloaks have been designed before by the symmetric method with one thermal conductive layer and one thermal resistive layer as the minimum unit. In the present work, we employ an asymmetric design method, i.e., the ratio of the numbers of thermal conductive layers and thermal resistive layers is not 1. The total number of layers of the thermal cloak, $N_{\rm total}$, is set to 10, where the number of thermal conductive layers is $N$, and the number of thermal resistive layers is $10-N$. According to the definition, there is the following inequality: $\kappa_{\rm c} >\kappa_{\rm r}$, so there are three kinds of relationships for $\kappa_{\rm b}$, $\kappa_{\rm c}$, and $\kappa_{\rm r}$, which are $\kappa_{\rm b} < \kappa_{\rm r} < \kappa_{\rm c}$, $\kappa_{\rm r} < \kappa_{\rm b} < \kappa_{\rm c}$, and $\kappa_{\rm r} < \kappa_{\rm c} < \kappa_{\rm b}$, corresponding to Figs. 2(a)–2(c). The simulation results in Fig. 2(a) show that under the previous condition, the larger the $N$ is, the smaller the temperature difference in the object region is. Such a phenomenon occurs because air with a thermal conductivity of 0.023 W/(m$\cdot$K) is chosen as the background material in the simulation, and the thermal conductivity of the thermal barrier layer is 0.13 W/(m$\cdot$K), which is still larger than the background thermal conductivity. This phenomenon can be attributed to the thermal resistive layer playing a certain role in thermal conduction. Meanwhile, the thermal conductive layer has high thermal conductivity, so when $N$ increases, the thermal cloak works out better. The $\kappa_{\rm b}$ is set to 1 W/(m$\cdot$K) to obtain generalizable findings, but a different result is obtained. The simulation results in Fig. 2(b) show that the thermal cloak achieves the best heat shielding function at $N = 5$. Then in Fig. 2(c), the $\kappa_{\rm b}$ is set to 10 W/(m$\cdot$K) and the result shows that the smaller the $N$ is, the better the thermal cloak performance is. Therefore, the optimal $N$ should be determined by $\kappa_{\rm c}$, $\kappa_{\rm r}$, and $\kappa_{\rm b}$ collectively. Considering the practical application scenarios of the thermal cloak, $\kappa_{\rm b}$ is consistent with that shown in Fig. 2(a), i.e., $\kappa_{\rm b}=\kappa_{\rm air} = 0.023$ W/(m$\cdot$K), unless stated otherwise. Predicting the optimal $N$ is an extremely complicated task that requires consideration of multiple factors. The above findings indicate that this symmetric design approach with a fixed minimum unit does not yield the best working performance of the thermal cloak in all scenarios. As mentioned above, the geometric design of the thermal cloak should be based on the properties of the material, especially thermal conductivity. Therefore, the effect of thermal conductivity of the materials on temperature of the object region is evaluated. The conductive layer provides a pathway for heat flow to transfer in the direction of low thermal resistance. During this process, a portion of the heat flows to the central region. Meanwhile, the thermal resistive layer provides a barrier in the radial direction to prevent this portion of heat from flowing to the central region. Hence, depending on the desired behavior of the heat flow in different layers of the cloak, this artificial manipulation of heat flow requires the contrast of thermal conductivity of the two functional layer materials. In this work, $\kappa_{\rm c}/\kappa_{\rm r}$ is further defined as $C$. The alteration of the heat shielding function of the thermal cloak when $C$ varies is explored. Figures 2(d)–2(h) present the curve of the temperature difference in the object region with $C$ and the temperature distribution at several specific values of $C$. When $C = 1$, i.e., both the thermal conductivities are equal, and a large temperature gradient is observed in the object region, almost the same as the external gradient. When $C$ gradually increases, the temperature gradient in the central region decreases, and the heat shielding function gradually manifests. This phenomenon is particularly obvious when the value of $C$ is relatively small. Therefore, two materials with contrasting high thermal conductivity correspond to superior thermal performance when designing multilayer thermal cloaks. Considering that the payoff is not obvious at large $C$, pursuing extremely large thermal conductivity contrast is not needed, and trade-offs can be performed appropriately in accordance with the actual demands. As mentioned above, the single-layer thickness of the cloak has a negligible effect on performance. Furthermore, the effect of $N_{\rm total}$ on the temperature distribution is determined. The number of thermally conductive and resistive layers varies in equal amounts and synchronously. The simulation results in Figs. 2(i)–2(m) show that, as is expected, the temperature differences inside the cloak gradually decrease when $N_{\rm total}$ increases. Increasing $N_{\rm total}$ can significantly enhance the heat shielding effect with a small $N_{\rm total}$. However, the shielding enhancement of the thermal cloak gradually decreases as $N_{\rm total}$ increases, very similar to the case of $C$ variation. The properties that a superb thermal cloak should possess and the parameters that influence the realization have been described above. However, a monofunctional thermal cloak is obviously insufficient to meet the application requirements of practical scenarios. For the purpose of maintaining and even achieving improved thermal manipulation in complex and extreme temperature fields, nonlinear materials, whose conductivity relies on temperature, must be adopted. Faced to the requirement of materials with thermally responsive conductivity, researchers gave a unified solution where the deformation of shape memory alloys drove the materials to connect or disconnect near a critical temperature.[42-44] Connecting and disconnecting can be equivalently viewed as localized thermal conductivity shifts. However, faced with the same requirement, we have designed a tuneable thermal cloak by utilizing the change in thermal conductivity that accompanies the phase transition of materials, which is a novel method. Since no shape memory alloys are used, this thermal cloak operates without moving parts. This is a significant difference between our work and aforementioned works.
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Fig. 3. Temperature distribution in space at (a) $T_{\rm h}= 280$ K and (b) $T_{\rm h}=320$ K. The inset illustrates the construction of thermal cloak, where blue and grey represent Ni$_{1-x}$Fe$_{x}$S and natural rubber, respectively. Simulations of the temperature and the temperature difference in the object region with the temperature of the heat source under the condition of dual thermal conductive materials for (c) $\Delta T_{\rm source}$ fixed at 50 K and (d) $T_{\rm c}$ fixed at 270 K.
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Fig. 4. Temperature distribution in space at (a) $T_{\rm h}= 280$ K (before phase transition) and (b) $T_{\rm h}=320$ K (after phase transition). The insert illustrates the construction of thermal cloak, where blue, orange, and grey represent Ni$_{1-x}$Fe$_{x}$S, W$_{x}$V$_{1-x}$O$_{2}$, and natural rubber, respectively. Simulations of temperature and the temperature difference in the object region with the temperature of the heat source under the condition of mono thermal conductive material for (c) $\Delta T_{\rm source}$ fixed at 50 K and (d) $T_{\rm c}$ fixed at 270 K. Simulations of the temperature and the temperature difference in the object region with the temperature of the heat source under the condition of dual thermal conductive materials for (e) $\Delta T_{\rm source}$ fixed at 50 K and (f) $T_{\rm c}$ fixed at 180 K. Yellow and green shades represent the partial phase transitions of Ni$_{1-x}$Fe$_{x}$S and W$_{x}$V$_{1-x}$O$_{2}$, respectively.
Table 1. Thermal conductivity of phase-transition materials.
Materials Critical temperature (K) $\kappa$ [W/(m$\cdot$K)]
Before phase transition After phase transition
Ni$_{1-x}$Fe$_{x}$S 281 4.2 15.0
W$_{x}$V$_{1-x}$O$_{2}$ 240 1.9 4.0
In this work, 0.5 wt% Ag/Ni$_{0.9}$Fe$_{0.1}$S and W$_{0.045}$V$_{0.955}$O$_{2}$ are selected as the phase-transition thermal conductive layer materials that exhibit structural transitions at 281 and 240 K, respectively, as listed in Table 1, accompanied by an abrupt change in thermal conductivity, which can be utilized to achieve autonomous tuning of the thermal cloak.[45,46] Since the thermal conductivities of these two materials are much higher than $r$, it is possible to obtain ideally large values of $C$. The phase transition accompanied by a sudden change in thermal conductivity is importantly reversible, which is a valuable property for thermal management functional materials. Thermal cloaks designed on the basis of this property can be applied in thermal fields with recurrent temperature changes.[47] Moreover, the first-order phase transition process is accompanied by a large amount of heat absorption or discharge, allowing for self-regulating thermal functions in response to sudden shocks of heat. However, transient conditions are beyond the scope of the present study, which focuses on the thermal cloak in a temperature field up to steady-state conditions. The latent heat of phase change, as a nonpermanent heat source, can be neglected in steady-state cases. Based on the previous conclusions, we determine the design parameters of the smart thermal cloak, where $N_{\rm total}=10$, $N= 8$, $\kappa_{\rm r} = 0.13$ W/(m$\cdot$K), $\kappa_{\rm b}= 0.1$ W/(m$\cdot$K). It should be specially noted that $N$ is taken as 8 instead of 10 in order to keep the multilayer structure, and the difference between the two values is very small as can be seen in Fig. 2(a). Here, $\kappa_{\rm b}$ is taken as 0.1 W/(m$\cdot$K) instead of 0.023 W/(m$\cdot$K) to avoid the huge difference in thermal conductivity that leads to divergent results. Since the relationship of $\kappa_{\rm b} < \kappa_{\rm r} < \kappa_{\rm c}$ is still satisfied, the trend of the results is not affected. Figures 4(a) and 4(b) demonstrate the distribution of the temperature in the object region before and after the phase transition. The trend of increasing temperature gradient inside the thermal cloak is suppressed after the phase transition. In order to demonstrate that this tuning function is caused by the phase transition of the material, assuming that the material does not undergo phase transition at the critical temperature under the same conditions. Figure 3 illustrates the results of the simulation. Figures 3(a) and 3(b) show that at the same temperature, the autonomous tuning function cannot be realized for the conventional material thermal cloak. By comparing Figs. 3(c) and 3(d) with Figs. 4(c) and 4(d), the autonomous tuning dependence on the material phase transition manifests itself more clearly. The temperature difference between the two ends of the thermal field $\Delta T_{\rm source}$ is fixed at 50 K. The temperature of the cold end $T_{\rm c}$ increases, and the temperature of the hot end $T_{\rm h}$ changes synchronously. In this process, the phase transition of the thermal conductive layer occurs because of a temperature gradient in space. The phase transition starts from the hot side first, then the phase-transition interface gradually moves to the cold side. Finally, at $T_{\rm c} = 270$ K, the thermally conductive layer fully transforms into a high thermal conductivity phase. Given that $\Delta T_{\rm source}$ is fixed, the temperature difference in the object region $\Delta T_{\rm object}$ hardly changes before the phase transition occurs and after the phase transition is completed. However, due to the increase in $\kappa_{\rm c}$, as shown in Fig. 4(c), $\Delta T_{\rm object}$ decreases significantly after the phase transition. Figure 4(d) shows the temperature change inside the cloak when $T_{\rm c}$ is fixed at 270 K and $T_{\rm c}$ is gradually increased. During the phase transition, $\Delta T_{\rm object}$ paradoxically decreases as $\Delta T_{\rm source}$ increases, and this phenomenon occurs precisely because of the sudden change in thermal conductivity brought about by the phase transition. However, in conventional material thermal cloaks, the internal temperature difference and temperature gradient vary exactly with the trend of the external thermal field and no tuning function is observed (Fig. 3). Then, the same investigation is conducted for the thermal cloak with Ni$_{0.9}$Fe$_{0.1}$S/W$_{0.045}$V$_{0.955}$O$_{2}$ dual thermal conductivity material, where two thermal conductivity materials are placed alternately. The results in Figs. 4(e) and 4(f) show that the phase transition of the two materials proceeds sequentially during the heat-up process. In addition, the thermal cloak achieves the repression of the heat-up tendency in the object region twice, which is a novel phenomenon reflecting the autonomous regulation of the thermal cloak over a wide temperature range given by the addition of the phase transition layer. Notably, this autonomous regulation function is achieved without additional energy input but by the material itself. The results imply that the proposed dual phase-transition thermodynamic cloak enables internal temperature control without artificial intervention when the external temperature changes, which is a function of smart metamaterials. The accomplishment of this work enables effective overheating protection of electronic components at high temperatures and the object region to maintain stealth in high-gradient temperature fields. In summary, we have investigated the effects of geometrical parameters and material properties on performance of the homogeneous multilayer thermodynamic cloak, providing design and preparation guidance for high-performance thermal cloaks. Innovatively, we introduce a phase-transition dual-material thermal conductive layer and validates it through steady-state temperature field simulations. Finite element simulation results demonstrate that our proposed diffusion field cloak achieves an autonomous tuning function across a large temperature range without any additional energy input. This cloak has the potential for widespread, cost-effective applications in solving practical problems of thermodynamic cloaking, heat transfer blocking, and component protection in many related industries, including the infrared cloaking of military units, mechanical power systems, and semiconductor devices. Acknowledgements. This work was supported by the National Natural Science Foundation of China (Grant No. 11774278), and the Fundamental Research Funds for Central Universities (Grant No. 2012jdgz04). 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