Keywords

1 Opening Remarks

The concept of invisibility, historically shrouded in tales and legend, found its scientific grounding with the publication of two seminal papers in 2006 [1, 2]. At the heart of this scientific renaissance was Professor Pendry’s pioneering use of the coordinate transformation theory. This theoretical leap sparked an explosion of research in metamaterials, diversifying into myriad sectors such as optics [3], acoustics [4], thermodynamics [5], and mass transport [6]. Particularly in thermodynamics, the revolutionary idea of thermal invisibility surfaced in 2008, trailblazed by Fan and colleagues [7]. Their work underscored that the heat conduction equation also retains transformation invariance. This insight extended the applicability of the coordinate transformation theory to heat transfer, marking the dawn of thermal metamaterials.

In traditional terms, thermal cloak, when related to heat conduction, can be characterized as such: an object, endowed with distinctive thermal characteristics, influences the background’s temperature distribution due to its varied thermal conductivities. The quintessential role of a thermal invisibility apparatus is to counteract this influence, ensuring that the ambient temperature persists unaltered, as if the intervening entity were absent. The overarching aim is to shield the presence of the intruding object from an observer who relies solely on the temperature gradient, thereby realizing thermal invisibility. Broadly, the assessment of thermal invisibility pivots on two foundational tenets: firstly, the ambient temperature remains undisturbed by the intrusive object; and secondly, the object’s temperature stands autonomously from the surrounding temperature landscape.

The advent of thermal metamaterials [8,9,10,11,12] has laid a theoretical groundwork for crafting thermal invisibility devices. As previously highlighted, the transformation theory [13,14,15,16] served as an initial blueprint for devising thermal cloaks. Nevertheless, intrinsic attributes of this theory lead to material anisotropy and inhomogeneity, posing hurdles for practical implementation. In response to this challenge, scholars integrated the scattering cancellation theory [17, 18] into heat conduction, yielding a thermal cloak design that leans on bulk materials alone. Furthermore, to circumvent the constraints of standardized shapes in invisibility apparatus conceptualized via analytical approaches, the scientific community has embraced topology optimization [19,20,21]. This inclusion paves the way for a richer variety of invisibility device designs.

In addressing engineering challenges, various factors come into play. For instance, the enclosed design of thermal cloaks based on the discussed theories can lead to difficulties in manufacturing. To counteract this, the “carpet cloak” concept emerged [22,23,24,25], tailored to conceal objects on insulating surfaces. Moreover, the implications of interface thermal resistance (ITR) [26, 27] in tangible settings are critical. This resistance is prevalent not only between the thermal invisibility device and its surroundings but also notably inside the device, particularly in layered configurations. To navigate this, the notion of an ITR-free cloak was introduced [28], bypassing ITR challenges. In recent developments, the concept of thermal domes [29] has been proposed for engineering applications, addressing the challenge that traditional thermal cloaks do not have the capability to protect objects with self-generated heat. The field has also witnessed the deployment of sophisticated methods to boost the versatility and intelligence of thermal invisibility. This encompasses leveraging adjustable thermal convection to achieve optimal thermal conductivity [30, 31] and integrating machine learning for smarter operations [32, 33]. In essence, the evolution of thermal invisibility has shifted from two-dimensional designs to three-dimensional constructs, and from complex, hard-to-fabricate materials to more accessible and simpler alternatives, positioning it aptly for real-world integration.

This chapter delves into thermal invisibility technology from both theoretical methods and advanced techniques perspectives. Section 5.2 discusses coordinate transformation techniques, scattering cancellation, and topology optimization. In Sect. 5.3, we introduce three typical devices in the development history of thermal invisibility: carpet cloaks, ITR-free thermal cloaks, and thermal domes. These technologies have played a significant role in bringing thermal invisibility into practical applications. Lastly, we explore potential applications in this field and provide an outlook on its future developments.

2 Foundations of Theory: The Pillars of Thermal Invisibility

2.1 Transformation Theory: The Key to Controlling Heat Flow

Designing thermal metamaterials using transformation theory is essentially an inverse problem, much like optical transformation theory. The heat conduction equation informs us about how heat flows under a specific thermal conductivity coefficient \(\kappa \). However, with transformation theory, we obtain a specific thermal conductivity coefficient based on a predetermined heat flow direction. For this process to work, there is a prerequisite: the heat conduction equation must remain unchanged under coordinate transformations so that solving the inverse problem can proceed smoothly. In general, the application of this theory involves the following steps:

Step 1: Checking if heat conduction’s governing equations remain consistent under coordinate transformations.

Step 2: Crafting transformations to steer the heat flow’s desired direction in the transformed virtual space.

Step 3: Applying the transformation matrix to the thermal conductivity, substituting thermal conductivity’s anisotropy for spatial non-uniformity, hence reproducing the same heat flow direction in the actual and virtual spaces.

This process is depicted in Fig. 5.1. Researchers have already demonstrated that the heat conduction equation remains invariance under coordinate transformations, whether in steady state or transient conditions [7, 34, 35]. Generally, the transient heat conduction is described by the Fourier equation:

$$\begin{aligned} \rho C\frac{\partial T}{\partial t} - \nabla \cdot (\kappa \cdot \nabla T) = 0, \end{aligned}$$
(5.1)

where t is time, \(\rho \), C, T and \(\kappa \) are the density, heat capacity, temperature and thermal conductivity. Post-coordinate transformation, this equation evolves into:

$$\begin{aligned} \rho ^{'} C^{'}\frac{\partial T}{\partial t} - \nabla ^{'} \cdot (\kappa ^{'} \cdot \nabla ^{'} T) = 0. \end{aligned}$$
(5.2)

\(\rho ^{'} C^{'} = \rho C/\mathrm{{det}}J\), \(\kappa ^{'}\)=\(J \kappa J^\textrm{T}\)/jetJ,  [38]. J is the Jacobian transformation matrix. Therefore, achieving the desired functionalities can be accomplished by manipulating either the thermal conductivity \(\kappa \) or the \(\rho C\). For example, when designing a spherical thermal cloak, one can utilize the following transformation: \(r = R_{1} + (R_{2}-R_{1})r^{'}/R_2\). In this case, a point in the original space is stretched into a circular region with a radius of \(R_1\). Within this region, heat flow will not penetrate, and its interior won’t affect the external background, thereby achieving the invisibility effect. Finally, we need to apply the transformation matrix to the material coefficients within the shell region (\(R_1 < r < R_2\)) to replace the spatial non-uniformity: \(\kappa _{rr}/\kappa _0 = \kappa _0/\kappa _{\theta \theta } = (r-R_1)/r\), \(\kappa _{r\theta } = \kappa _{\theta r} = 0\) and \(\rho c_{p} = \rho _0c_{p0}[R_2/(R_2 - R_1)]^2(r - R_1)/r\) [35]. \(\kappa _0\), \(\rho _0\), \(c_{p0}\) are the thermal conductivity, density and specific heat capacity of the background, respectively.

Fig. 5.1
A schematic of the transformation theory in heat transfer. It demonstrates heat flow transitioning from a homogeneous medium in physical space, to an inhomogeneous medium in virtual space, and back to a homogeneous medium in physical space.

Transformation theory schematic

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Fan et al. were at the forefront of utilizing this approach to design thermal cloaks [7]. Chen et al. subsequently broadened the scope of transformation thermotics to accommodate anisotropic backgrounds, moving beyond merely isotropic ones, signifying preliminary advancements in thermal invisibility [34]. However, the central challenge lies in achieving anisotropic thermal conductivity in practical applications.

Employing the effective medium theory, Narayana and his team, in 2012, showcased an experimental model of thermal invisibility using a stratified material composition [36]. This technique entailed alternating between two isotropic substances with distinct thermal conductivities to realize anisotropic thermal conductivity.

Guenneau et al. explored the realm of transient thermal invisibility, unveiling the transformation principles associated with density and heat capacity during such operations [35]. Schittny et al. engineered a microstructured thermal cloak that redirects heat flow around an encased object within a metal plate, providing transient thermal shielding [37]. Their contributions cemented the foundational principles of the transformation theory for heat conduction, furthering the development of thermal metamaterials.

The subsequent research further enriched the field. For instance, Li and his team introduced nonlinear elements into the fundamental equations, allowing the thermal conductivity to vary with temperature [39, 40]. This innovation established the foundation for adaptive thermal invisibility solutions and deepened our understanding of nonlinear thermal phenomena and intelligent thermal cloaks.

2.2 Scattering Cancellation: A Streamlined Approach for Implementation

Thermal cloaks, developed using transformation theory, typically require materials characterized by anisotropic, inhomogeneous, and singular thermal conductivities, complicating their production. A significant breakthrough occurred when two research teams [17, 18], inspired by prior studies on scattering cancellation methods for static magnetic field shielding [41], incorporated this idea into heat conduction. Consequently, they crafted bilayer thermal cloaks using common materials.

The core principle of scattering cancellation warrants elucidation. Scattering phenomena are prevalent across many domains, particularly in electromagnetics and acoustics. In these contexts, waves change direction when meeting varied media or obstacles. Regarding heat conduction, materials with distinct thermal conductivities from their environment can cause deviations in heat flow. Observing through isotherm lines, one can discern these perturbations: isotherms are drawn towards areas with higher thermal conductivity and repelled by those with lower. As illustrated in Fig. 5.2, strategic placement of these materials facilitates a state of thermal invisibility, mimicking an environment undisturbed by thermal fluctuations.

Fig. 5.2
A schematic of the scattering cancellation theory. It has a low conductivity medium and a high conductivity medium, each with an isotherm. The final panel demonstrates the combined effect on the isotherms due to scattering cancellation.

A schematic diagram of the scattering cancellation theory

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To craft a thermal cloak using the scattering cancellation approach, one must:

Step 1: Establish the governing equation for heat conduction.

Step 2: Derive the general solution for temperature distribution.

Step 3: Identify the coefficients of this solution, based on boundary conditions, ensuring the external thermal field remains unaffected. This also helps deduce cloak parameters like its shape and thermal conductivities.

Using a 3D bilayer cloak as an illustration, in the absence of heat sources, the heat conduction equation transforms into Laplace’s equation:

$$\begin{aligned} \nabla (-\kappa \nabla T) = 0. \end{aligned}$$
(5.3)

The temperature within various regions can be generalized as:

$$\begin{aligned} T_{i} = \sum ^{\infty }_{m=1}[\mathrm{{A}}^{i}_{m}r^{m} + \mathrm{{B}}^{i}_{m}r^{-m-1}]P_m(\cos \theta ), \end{aligned}$$
(5.4)

where \(\mathrm{{A}}^{i}_{m}\) and \(\mathrm{{B}}^{i}_{m}\) ( i=1, 2, 3, 4) are constants to be determined by the boundary conditions and \(T_i\) denotes the temperature in different regions. The boundary conditions primarily encompass four aspects: Finite temperature at r\(\rightarrow \)0, uniform external temperature field, temperature equality at interfaces between different regions and equal heat flux at interfaces. From the established boundary conditions, the design blueprint for a bilayer cloak is inferred: its inner layer functions as an adiabatic barrier. The relationship between the outer layer’s thermal conductivity and that of the backdrop is described by [18]:

$$\begin{aligned} \kappa _s = \frac{2R^3_2 + R^3_1}{2(R^3_2 - R^3_1)} \kappa _b. \end{aligned}$$
(5.5)

\(\kappa _s\) represents the thermal conductivity of the outer layer of the thermal cloak, \(\kappa _b\) represents the thermal conductivity of the background, and \(R_1\) and \(R_2\) represent the inner and outer radii of the outer layer.

In thermal cloaking, the 2D bilayer cloak is a notable milestone. Han and his team spearheaded its theoretical and empirical validation [18]. The thermal cloak they designed is cylindrical, with the direction of heat flow following a radial path. This means that when viewed in cross-section, it appears as a circular cloak within a two-dimensional thermal field. The entire cloak is made from bulk materials, greatly simplifying the manufacturing process. Both simulation and experimental results have demonstrated its exceptional thermal invisibility performance.

Shifting to a three-dimensional perspective, Xu’s team was the inaugural group to experimentally verify a 3D bilayer cloak [17]. Adopting an innovative approach, they used a compression technique to produce half of the cloak. These halves were then merged to complete a full 3D bilayer cloak.

A shared feature between the 2D and 3D bilayer cloaks is their architecture: both utilize an adiabatic inner layer paired with a high thermal conductive outer layer, facilitating scattering cancellation. There’s merit in designating the inner layer as adiabatic: it offers the cloak adaptability for various objectives. Certainly, we can also opt for a combination of an outer layer made of low thermal conductivity material and an inner layer made of high thermal conductivity material. Likewise, to make this cloak applicable to any target, the ideal scenario would require the thermal conductivity of the inner material to be infinitely high. A cloak designed with this approach is termed a “zero-index cloak”, which was proposed and realized by Li and his colleagues [42]. To approximate an infinitely high thermal conductivity, they ingeniously incorporated thermal convection as an equivalent.

It’s worth noting that although the scattering cancellation theory avoids the use of singular materials, it still has limitations. Firstly, cloaks designed based on it are only suitable for steady-state situations, and their performance is not optimal in transient scenarios. Secondly, only some specific shapes of thermal cloaks can obtain exact analytical solutions from this method; designing cloaks of more complex shapes is extremely challenging.

2.3 Topology Optimization: Crafting Thermal Cloaks for Every Shape

Up to now, the design of thermal cloaks has mainly relied on materials with anisotropic thermal conductivity or specially shaped thermal cloaks made from natural materials. With the advancement of computational technology and the increasing power of computers, numerical optimization has emerged as an innovative method for designing thermal stealth devices. In 2014, Dede and his colleagues utilized numerical optimization techniques to develop composite structures that can manipulate heat flux to achieve functionalities like shielding, concentrating, and rotating [19]. These anisotropic thermal composite materials consist of elliptical inclusions that are uniformly integrated into the primary matrix.

Advancing this methodology, Fujii and his group employed the Covariance Matrix Adaptation Evolution Strategy (CMA-ES)–a sophisticated optimization algorithm–to refine the material design structure [20]. Their primary objective was to produce thermal cloaks using naturally existing materials. The typical steps of this optimization method include:

Step 1: Obtain a reference temperature distribution, which represents the undisturbed temperature field.

Step 2: Formulate an objective function that measures the discrepancy between the reference and optimized temperature fields. This function is crucial as it gauges the effectiveness of the thermal cloak.

Step 3: Apply advanced optimization techniques, like CMA-ES, to enhance the material structure. The primary aim is to ensure the optimized temperature profile closely matches the reference.

Step 4: Continue optimization until the required accuracy in the objective function is attained.

Using this optimization procedure, the thermal cloak’s structure can be perfected, enabling it to achieve thermal invisibility using natural materials. This approach offers a pathway to develop versatile thermal cloak designs, eliminating the need for complex material structures. Researchers can customize thermal cloaks to various shapes based on precision requirements, utilizing common materials such as aluminum and iron. This approach significantly enhances the diversity of thermal cloaks available.

As previously mentioned, the optimization approach heavily depends on the prior knowledge of the background temperature field. Achieving the desired accuracy might not always be possible for all cloak shapes. Addressing this challenge, Sha and his colleagues proposed an innovative method that uses topology optimization to directly design foundational functional units [21]. These units are then assembled to achieve thermal invisibility. Their detailed process includes: First, determining the necessary thermal conductivity tensor for the cloak based on transformation theory. Next, they segment the entire area into foundational units and use topology optimization to shape topological functional cells (TFCs). These TFCs have distinct microstructures specific to each unit. Finally, these units are combined to realize thermal invisibility. Remarkably, the TFCs combine two different materials: Die steel (H13) and polydimethylsiloxane. This groundbreaking strategy offers unparalleled flexibility, facilitating the creation of thermal solutions previously deemed challenging. This pioneering approach sets the stage for a new chapter in the adaptive fabrication of thermal cloaking devices.

Furthermore, many recent studies are merging deep learning with topology optimization [33], paving the way for a more sophisticated application of the optimization technique in thermal invisibility technology.

3 From Blueprint to Reality: Advancements in Thermal Cloaking Technology

3.1 The Revolutionary Thermal Carpet Cloak: Concealment on Surfaces

The concept of the “carpet cloak” originated from the study of optical invisibility cloaks [43]. In terms of heat conduction, a carpet cloak is a device designed to conceal objects placed on an insulating surface. It emerged as a solution to the challenges faced in crafting traditional thermal cloaks and to enhance their applicability in specific scenarios. The foundational principles of designing carpet cloaks are derived from the methodologies we discussed in the previous section, which we will further elaborate on in the following sections.

Transformation theory laid the groundwork for one of the initial iterations of the thermal carpet cloak [23, 25]. The cloak’s distinct shape derives from a transformation applied to the y-coordinate, described by the equation [23]:

$$\begin{aligned} y^{'} = \frac{c-a}{c}y + \frac{b-x\mathrm{{sgn}}(x )}{b}a, \end{aligned}$$
(5.6)

where a, b, and c denote geometric parameters. It’s important to highlight that devices tailored for thermal invisibility via the outlined methods exhibit strong directionality. As demonstrated in Ref. [23], they are more effective in a vertical temperature gradient than a horizontal one.

Drawing inspiration from the bilayer thermal cloak, the bilayer thermal carpet cloak emerged. Analyzing the heat flow within a bilayer thermal cloak uncovers a remarkable symmetry. For example, when a temperature gradient aligns with the major axis of the elliptical bilayer cloak, the heat flow symmetrically aligns as well. Modifying the major axis to an adiabatic state does not hinder the heat flow direction. Building on this observation, Han and his team developed a carpet cloak based on the principle of scattering cancellation [24]. This model features a bilayer composition: an inner adiabatic layer complemented by an outer layer possessing high thermal conductivity. Set atop an adiabatic foundation, it ensures perfect invisibility against a horizontal temperature gradient. Made entirely from natural materials, this invention accentuates the practicality of carpet cloaks.

Topology optimization further facilitates the crafting of carpet cloaks using natural materials. In a similar vein, Fujii and his team employed the covariance matrix adaptation evolution strategy to refine carpet cloak designs [44]. These cloaks showcase outstanding performance under diverse heat flow angles.

Of course, some entirely new methods have also been employed in the design of carpet cloaks, such as the law of thermal refraction [22]. The law of thermal refraction states that heat flow will propagate in the direction of the least thermal resistance. Based on this, one can directly control the direction of heat flow to achieve the purpose of invisibility.

3.2 ITR-Free Thermal Cloak: Overcoming Interface Thermal Resistance

In practical applications, heat flow can be influenced by the interface thermal resistance (ITR) between different materials. Unfortunately, most existing thermal cloaks require the use of two or more materials to manufacture [26]. For example, using scattering cancellation methods or topology optimization typically necessitates at least two materials–one with high thermal conductivity and one with low thermal conductivity. Similarly, cloaks designed based on coordinate transformation theory often require multiple layers of materials, exacerbating the interface thermal resistance effect. Consequently, thermal cloaking devices designed under ideal conditions (neglecting interface thermal resistance) may face functionality issues when applied in real-world scenarios.

At the macro level, ITR primarily arises from incomplete contact between material surfaces and their intervening voids. This resistance is intrinsically tied to the involved materials’ mechanical and thermophysical attributes. Furthermore, factors like contact surface shape, interfacial pressure, temperature, and the intervening medium can modulate ITR. In scenarios where ultra-precision isn’t paramount, various techniques, such as stress application during installation, inter-material gap filling using thermally conductive grease, precision enhancement in manufacturing, and surface roughness reduction, can mitigate ITR.

For applications that require high precision or when the above methods are not effective, it becomes necessary to consider the impact of interface thermal resistance during the design of thermal invisibility devices. Zheng and Li carefully examined the boundary conditions that need to be satisfied for temperature and heat flux at different interfaces when interface ITR is present [26]. In the presence of ITR, temperature continuity is disrupted while heat flux continuity is maintained, as expressed by the following equations:

$$\begin{aligned} \begin{aligned} T_{i+1}\vert _{\partial \Omega } = T_{i}\vert _{\partial \Omega } + R_{i,i+1}{} {\textbf {n}}\cdot (\kappa _i \nabla T_i)\vert _{\partial \Omega },\\ {\textbf {n}}\cdot (\kappa _i \nabla T_i)\vert _{\partial \Omega } = {\textbf {n}}\cdot (\kappa _{i+1} \nabla T_{i+1})\vert _{\partial \Omega }, \end{aligned} \end{aligned}$$
(5.7)

where \(R_{i,i+1}\) is the ITR between the ith region and the \((i+1)\)th region. After considering the above boundary conditions, modifications can be made to the design formula for the thermal cloak. For example, the modification of a bilayer cloak designed using the scattering cancellation method can be expressed in the following form:

$$\begin{aligned} \frac{\kappa _1}{\kappa _2} = \frac{b^2 + a^2}{(b^2 - a^2)(1-R_i\kappa _2/b)}, \end{aligned}$$
(5.8)

where a is the inner radius and b is the outer radius of the outer shell. \(\kappa _1\) is the thermal conductivity of the outer shell, and \(\kappa _2\) is the thermal conductivity of the background region. \(R_i\) denotes the ITR which includes thermal contact resistance and thermal boundary resistance. The introduction of \(\mathrm{{Q}}\) \( = 1-R_i\kappa _2/b\) is used to represent the impact of ITR on the thermal cloak. When Q = 1, the above equation transforms into the design formula for the ideal case.

However, even when ITR has been considered in the design, determining its specific impact can still be challenging. To address this, Han and colleagues proposed a method to circumvent ITR [28]. Their design inspiration draws from the concept of equivalent thermal conductivity, which is achieved by varying the cross section of heat conduction. In simple terms, for two-dimensional heat conduction, we can achieve an equivalent high thermal conductivity by adding a third dimension, namely increasing the thickness h. According to the law of energy conservation, we can derive the following relationship to achieve the equivalence of a material with thermal conductivity \(\kappa _2\) to a material with thermal conductivity \(\kappa _1\):

$$\begin{aligned} \frac{\kappa _1}{\kappa _2} = \frac{t + h}{t} = 1+\frac{h}{t}, \end{aligned}$$
(5.9)

where t represents the thickness of the background material, h represents the added interface thickness. In this process, the same material is used throughout, thereby eliminating the presence of interfacial thermal resistance. Consequently, a method for achieving any thermal conductivity without ITR is obtained. When applied to a bilayer thermal cloak, this approach results in a thermal invisibility device without interfacial thermal resistance.

3.3 The Thermal Dome: A New Horizon in Thermal Shielding

Despite the innovative designs of various thermal cloaks with some undergoing successful experimental validation, their foundational design principle hinders widespread engineering application. This principle, involving the complete encasement of the target object with an insulating material followed by heat flow redirection around it, inevitably leads to a fully enclosed hidden area. This aspect presents significant manufacturing, installation, and reusability challenges, especially in three-dimensional structures. Further, these cloaks are unsuitable for providing thermal invisibility to objects with heat sources, an overlooked common reality in prior research. Additionally, these cloaks’ custom design for specific environments renders them ineffective when environmental conditions or background thermal conductivity changes. Therefore, traditional thermal invisibility devices still have a long way to go before they can be engineered.

To address these issues, Zhou et al. proposed a novel thermal invisibility device tailored for engineering applications called the “thermal dome” [29]. This device features an open hidden region, facilitating easy installation and reuse. Importantly, it represents a significant advancement in thermal invisibility technology as it achieves thermal invisibility for objects containing heat sources, marking a major breakthrough in this field. Furthermore, due to the open structure of the thermal dome, concepts involving multiple layers can be applied to make it reconfigurable, similar to LEGO blocks, enabling it to adapt to dynamic environmental changes.

Similar to the bilayer cloak, the thermal dome is also constructed using natural materials only, and its structure is depicted in Fig. 5.3. The semi-axis of the core (dome) is specified as \(l_{ci}\) (\(l_{di}\)) along the \(x_i\) axis, where \(i=1,2,3\) represents the three dimensions. After solving the heat conduction equation under special boundary conditions, we obtain the design requirement for the thermal dome:

$$\begin{aligned} \kappa _{b}=\frac{L_{ci}\kappa _{c}+\left( 1-L_{ci}\right) \kappa _{d}+\left( 1-L_{di}\right) \left( \kappa _{c}-\kappa _{d}\right) f}{L_{ci}\kappa _{c}+\left( 1-L_{ci}\right) \kappa _{d}-L_{di}\left( \kappa _{c}-\kappa _{d}\right) f}\kappa _{d}, \end{aligned}$$
(5.10)

where \(f=g\left( \rho _{c}\right) /g\left( \rho _{d}\right) =\prod \limits _il_{ci}/l_{di}\) denotes the volume fraction, \(L_{ci}\) and \(L_{di}\) are the shape factors. \(\kappa _b\) denotes the thermal conductivity of the background, \(\kappa _d\) denotes the thermal conductivity of the thermal dome.

Fig. 5.3
A schematic and a cross-sectional view of a thermal dome with dimensions and layers. A highlights the dimensions, and b highlights the hot plane, cold plane, background and cross-section of the dome, and core.

(from Ref. [29], licensed under CC-BY 4.0)

a Presents a schematic representation of the thermal dome, while b exhibits its cross-sectional view.

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The method described above can be easily extended to accommodate a core-shell structure with n layers. Computational software can be used to determine the parameters of the thermal domes in each layer. Another rapid method for designing multi-layer thermal domes is to use effective medium theory, where the design process can be done layer by layer in an iterative manner.

As mentioned earlier, a prominent feature of the thermal dome is its ability to achieve thermal invisibility for objects containing heat sources. As shown in Fig. 5.4a and c, the open design of the thermal dome allows it to directly contact the cold source, enabling the heat flux generated by internal heat sources to be absorbed through the bottom. In contrast, traditional thermal cloaks lose their functionality when there are internal heat sources, and the temperature inside their enclosed space continues to rise, leading to catastrophic consequences as shown in Fig. 5.4b and d.

Fig. 5.4
A schematic of the performance of the thermal dome and a thermal cloak with a heat source. Panels A and B present heat dispersion at 5 and 10 minutes. These present a hemispherical structure transitioning to a spherical structure. Panels c and d plot temperature variations over time, indicating the thermal cloak maintains lower temperatures more effectively than the dome.

(from Ref. [29], licensed under CC-BY 4.0)

a and b Compare the performance of the thermal dome and thermal cloak with a heat source in the hidden area at 5 and 10 min respectively. The heat source in the thermal dome and thermal cloak emitting heat outward at a rate of 500 kW per square meter. In c and d, we quantitatively show the variation of temperature at different positions with time.

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4 Conclusion and Outlook

In this chapter, we present three mainstream theories for the design of thermal cloaks. Subsequently, by introducing three specific thermal invisible devices, we illustrate the primary challenges and solutions encountered when adapting the traditional thermal cloak to practical engineering applications. It’s noteworthy that some novel theories, such as topological phenomena [45,46,47] and conformal mapping theory [51], have emerged and are being employed in the design and understanding of thermal cloaks, injecting new vitality into the field of thermal invisibility. Furthermore, the content of statistical physics can also be utilized to manufacture the required materials [48,49,50].

It should be noted that the research on thermal invisibility holds significant practical importance. Firstly, it offers a novel method for precise control of heat flow, which is crucial for thermal management. This can be used to regulate the distribution of heat flow in chips to protect vital components, or combined with solar panels to enhance power generation efficiency. Secondly, the concept of “thermal invisibility” itself plays a significant role, especially in the defense sector, where it can shield crucial targets from being detected by infrared cameras.

Lastly, as previously mentioned, the advancement of thermal invisibility has been in tandem with the development of thermal metamaterials [52, 53]. Theories utilized for designing thermal cloaks, such as transformation theory or scattering cancellation theory, among others, have also been employed to design other functional devices. These include thermal concentrators [54, 55], thermal sensors [56,57,58], thermal transparency [59], thermal chameleonlike devices [60, 61] and thermal illusions [62, 63], greatly enriching the content of thermal metamaterials. Similar theories can even be extended to economics and music research [64, 65]. It is hoped that this chapter on thermal invisibility technology will serve as a stepping stone, offering a fresh perspective on the development of thermal metamaterials.