Transformation thermotics: Thermal metamaterials

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1 From: Rachel Seah Mei Hui Sent: Monday, 6 November, :46 PM To: Nurul Affiah Binte Sahrin <affiah@wspc.com> Subject: FW: RE: FW: Re: Fw: 巜物理学报 软物质专题英文版 Dear Affiah, International Journal of Modern Could Physics we try implementing B the new layout that you suggested, but add to the back of the citation: Pl Vol. 32 (2018) (36 pages) c World Scientific Publishing Let s Company do a sample layout before we typeset all the papers. Thanks DOI: /S Best regards, Rachel 发件人 : Rachel Seah Mei Hui 发送时间 : :39:05 收件人 : Steven Shi Hong Bing - EXT Transformation thermotics: Thermal metamaterials 抄送 : 主题 : and RE: FW: their Re: Fw: applications 巜物理学报 软物质专题英文版 Dear Steven, I do agree with their recommendation to add the Chinese citation, although their case seems to be sl QinThanks! Ji, Xiang-Ying Shen and Ji-Ping Huang Department of Physics and State Key Laboratory Best regards, of Surface Physics, Fudan University, Shanghai , P. R. China Rachel jphuang@fudan.edu.cn From: Accepted Steven Shi Hongbing 26 September [mailto:hbshi@worldscientific.com.cn] 2017 Sent: Monday, Published November 5 December 6, : AM To: Rachel Seah Mei Hui <mhseah@wspc.com> Translated fromsubject: Acta Physica Re: FW: Re: Sinica Fw: 巜物理学报 软物质专题英文版 ( ), 2016, 65(17): Please reference the original doi: /aps ( ). Dear Rachel, In this paper, we reviewthank someyou recent for your achievements and in thank thermal you for metamaterials, the advice. I including have also asked the editor of AP Below is their suggestion and I think it will be easy to do since very litter change needed. novel thermal devices, simplified experimental method, macroscopic thermal diode based on temperature-dependent 关于 transformation DOI 号, 我咨询了同时刊登中英文的一本期刊 thermotics, and the, important 他们采用的办法也是两个 role that soft DOI 号, 但是统计引用的时候 matters play in the experimental 下 confirmations of thermal metamaterials. These works pave the developments in transformation mapping theory and can surely inspire more designs of thermal metamaterials. And the pointer WhatURL is more, link on some the original approaches DOI number provide is good, more please flexibility in controlling heat flow, and they may also be useful in other fields that are closely keep it. related to temperature Thank gradient, you and suchbest as regards the Seebeck effect and many other domains where transformation theory is valid. Steven Keywords: Transformation thermotics; thermal metamaterial; soft matter. PACS numbers: Xj, fc, c Steven Shi Hongbing 1. Introduction 发件人 : Rachel Seah Mei Hui 发送时间 : :36:10 收件人 : Steven Shi Hong Bing - EXT 抄送 : 主题 : FW: Re: Fw: 巜物理学报 软物质专题英文版 Heat energy is one of the most common forms of energy in the world, and it is also the most neglected energy. Since most forms of energy dissipate heat, it is often seen as a waste energy and is difficult to use. However, with the development file:////aihua/aihua/fwrefw~1.htm[11/7/2017 1:16:01 PM] of science technology and industrial civilization, nonrenewable resources are being consumed at an alarming rate. Therefore, it is considered as an important goal and task of scientific research that how to make use of waste energy. By means of arranging a certain structure of the natural material, metamaterial 1 can realize the Corresponding author

2 Q. Ji, X.-Y. Shen & J.-P. Huang peculiar unnatural material function, which may be the key method to solve some traditional heat transfer problems. There are three ways of heat transfer: heat conduction, heat convection and heat radiation. This paper mainly discusses the heat conduction. Heat transfer is a diffusion process, which makes it difficult to collect and utilize heat energy. This property also greatly limits the efficiency of thermoelectric materials and solar cells. On the other hand, it is a significant research direction that how to make the heat transfer more quickly than to accumulate and cause the damage to the industrial devices, such as the cooling method of the chip. Then, the ability to control the flow of heat will become very urgent and important. As with the optical metamaterial can control the propagation of light, we will show in this paper how to design a variety of thermal metamaterials by means of the transformation thermotics. Transformation thermotics is based on coordinate transformation. This paper will detail the use of different transformation design of different functional thermal metamaterial: thermal cloak (can guide the heat flow around the cloaked object); thermal concentrator (heat flux can be gathered in a particular region); thermal rotator (heat flux can be rotated at a specific angle); thermal lens (heat flux can be converged to the smaller region). The experimental methods of these metamaterials will also be given in this paper. Different materials with different thermal properties are widely accepted as common sense. Low thermal conductivity of the material is applied to heat preservation, to achieve the purpose of cool in summer and warm in winter. However, the traditional materials that have been used for thousands of years seem to be inadequate in the face of growing thirst for heat flow control. In the cooling of the chip, we need heat to dissipate quickly. In the solar cells and thermoelectric materials, we hope that the heat energy can be gathered quickly to improve the efficiency of energy conversion. Therefore, it is necessary to find new materials beyond the traditional materials to meet the requirements of heat flow control, which makes the metamaterials emerge as the times require. Metamaterials are made up of natural materials combined in a series of special ways. They can realize the special properties of materials, and the essence of these properties comes from the special structure of the material, rather than its physical properties. Of course, due to the complexity of the heat transfer and diffusion (such as heat conduction), preparation of the materials becomes more difficult. Fortunately, in 2006, Leonhardt and Pendry et al. independently proposed a new type of optical metamaterial called invisibility cloak. 2,3 This device was known for bending light around an object to make it invisible. Such a significant progress soon enlightened a lot of scientists in different aspects since it offers a powerful approach to design metamaterials. The design of the cloak implied new design concept of metamaterials. 4 8 The principle of invisibility cloak, which is concluded as transformation optics 9 15 has been applied to light waves, 16 acoustic, seismic waves, hydrodynamics, 29 and even matter waves, as they all satisfy with wave equation. Mathematically, the wave equation is a hyperbolic differential

3 Transformation thermotics: thermal metamaterials and their applications equation, and the diffusion equation is a parabolic differential equation. This natural difference leads to the difficulty of the transformation theory in the related fields. In 2008, based on the previous work, Fan et al. established the theory of steadystate transformation thermotics and put forward the theory of thermal cloak for the first time In 2012, Guenneau et al. established the theory of unsteady transformation thermotics, which has been widely concerned in the field of science and has been reported by some public media such as BBC. 38 Since then, a large number of new thermal materials have been proposed In this paper, we will introduce the development of transformation thermotics in the past years, including a large number of theoretical design and experimental verification. Soft materials are widely used in the experiment of thermal metamaterials. They can be used as fillers to adjust the thermal conductivity of materials to reduce the contact thermal resistance, but also as a protective film to prevent the oxidation of metal samples, as well as against thermal convection dissipation. More importantly, the metal samples covered with soft materials can be imaged by infrared thermal imager, which can be used to verify the rationality of the theoretical design through experiments. Some of the latest developments have injected new blood and vitality into the traditional heat transport. 2. Transformation Thermotics and Thermal Metamaterials 2.1. Transformation thermotics theory Heat transfer has three forms: convection, conduction and radiation. Among them, the radiation is microwave which is naturally in line with the requirements of the transformation optics theory. The convection equation is more complex, and the role in the solid material is relatively weak. Therefore, the theory of transformation thermotics only considers the heat conduction process. For the general consideration, according to Fourier s law, a heat conduction equation in an anisotropic medium has the following form: ρc T t = (κ T ). (1) In the formula, ρ is the mass density of the material, c is the specific heat, T is the temperature, t is the time, and κ is the thermal conductivity of the material. It is proved that the equation of heat conduction can be changed in the form of coordinate transformation. The matrix form of inverter base in Cartesian coordinate system is the Jacobian matrix in the curve system. Under the coordinate transformation, if the heat conduction equation is kept constant, the heat conductivity should be changed, [ κ ij ] = JκJ T det(j). (2)

4 Q. Ji, X.-Y. Shen & J.-P. Huang The thermal conductivity tensor is expressed by the matrix form. From the above discussion, it is found that even the heat conduction and diffusion equation are parabolic partial differential equation. They still can remain unchanged in the coordinate transformation. This is the most important prerequisite for transformation thermotics. It is noteworthy that the transformation changes the thermal conductivity tensor. This change is reflected in mathematics in introducing the Jacobian matrix J, and J is precisely mathematical tool to characterize the coordinate transform. This theory allows for the design of thermal metamaterials with specific thermal properties by adjusting the anisotropic thermal conductivity of the heterogeneous materials. In the following, we will give the method of designing the thermal metamaterials based on the transformation thermotics. The whole process can be divided into three steps. First of all, according to different needs, design a suitable coordinate transformation. In the second step, a new thermal conductivity tensor is calculated according to Eq. (2). Finally, the simulation or experimental method is used to verify whether the thermal metamaterial with new thermal conductivity can be designed to achieve the desired function. These three steps can be used to guide the design of various new materials, including invisibility cloak, concentrator, reverser, rotator and thermal lens Thermal metamaterial The method of using the gradient material to realize the electromagnetic cloak has attracted much attention. 54 Some studies of the scattering cross-section 55 and its two-dimensional case were even extended to other fields such as the field of electrical conductivity followed by this method In spite of the difference from the wave equation, Fan et al. and Guenneau et al. successfully extended Pendry et al. s electromagnetic cloak and transformation theory to the thermal field. 34,38 Heat conduction is the phenomenon of heat transfer in the medium without macroscopic motion, which can occur in solid, liquid and gas. Strictly speaking, only the solid is pure heat conduction. When there is a thermal gradient, the thermal energy spontaneously flows from the high-temperature to the low-temperature. For the sake of simplicity, the static thermal field distribution in the natural state is often more important, so we often consider the static heat conduction process in the design of the material. In this framework, Eq. (2) can be reduced to the form of the Laplace equation, 0 = (κ T ). (3) Under the influence of the invisibility cloak, most of the thermal metamaterials are designed in the two-dimensional polar coordinates. It has two main reasons. On the one hand, the two-dimensional systems are relatively simple in theoretical calculation, simulation and experimental verification. On the other hand, it is very easy to extend the two-dimensional polar coordinates to the three-dimensional cylindrical or spherical coordinates system. Therefore, in the process of discussing several

5 Transformation thermotics: thermal metamaterials and their applications kinds of main metamaterials, we only discuss the two-dimensional polar coordinates of the static heat conduction, which can be simply extended to three-dimensional condition and unsteady state. In fact, there are a lot of problems in the actual operation of optical cloak. The transformation theory based on the wave equation needs to consider frequency in the realization. At present, the experimental materials are often corresponding to a single or narrow frequency band. Therefore, optical cloak is still a long way to go. On the contrary, based on the heat conduction equation, there is no frequency problem in the thermal metamaterials. This particularity makes the material designed based on the theory of transformation thermotics more applicable, and the thermal metamaterial can perform well in the wide enough boundary conditions. This makes that the material can be applied in production more quickly and has a wider range of adaptation and value Thermal cloak Thermal cloak is a thermodynamic extension of optical cloak. In 2008 and 2012, Fan et al. and Guenneau et al. set up the steady-state and unsteady-state transformation thermotics theory, respectively. The direct application of these theories is the design and manufacture of thermal cloak. Compared with the traditional cloak device, thermal cloak has two characteristics. First, the temperature gradient in the cloak region r < R 1 is none. Second, no matter what kind of object placed in the cloak region, it will not have any effect on the temperature distribution of the field. In other words, an observer is not able to detect the object in the invisibility cloak through the temperature distribution outside the cloak. According to the transformation of optical cloak, we can also write the transformation equation r = r(r 2 R 1 ) + R 1, R 2 (4) θ = θ. The above formula compresses a circular region into the annular region. According to the above discussion, we can calculate the Jacobian matrix of complex coordinate transformation. We can solve the transformation matrix T to give the material parameters we need T 1 = R(θ )diag(f r/r, f 1 r /r)r(θ ) 1, (5) where f = R2 R1 R 2. For the purpose of simplifying, assuming that the thermal conductivity of the background is 1 W/mK, we can finally get the eigenvalues of the matrix, κ r = r R 1 r, κ θ = r r R 1. (6) These are the parameters we need to build a thermal cloak to protect the flow of heat. As shown in Fig. 1, the thermal cloak designed on this basis can be simulated

6 Q. Ji, X.-Y. Shen & J.-P. Huang (a) (b) Fig. 1. (Color online) (a) Schematic diagram of thermal cloak, where R 1 = 1 cm, R 2 = 2 cm. (b) The temperature profile of the thermal cloak when the heat diffuses from a temperature of 400 K to 300 K. and verified (by simulation using a commercial multiphysical field coupled finite element analysis software COMSOL). It is obvious that the existence of thermal cloak does not change the temperature distribution of the external field. Because of infinitely high thermal conductivity in tangential direction and zero thermal conductivity in radial direction, the heat flow avoids the protected object. So that the temperature gradient inside the clock region is zero

7 Transformation thermotics: thermal metamaterials and their applications Thermal concentrator The special properties and the transformation theory of the thermal cloak have stimulated the research of other thermal metamaterials with special properties. In the following, we will introduce a series of thermal metamaterials derived from the theory of transformation thermotics. For instance, a thermal concentrator can aggregate heat and increase the thermal gradient sharply in a certain region without affecting the temperature distribution in the external field, while the thermal rotator can reverse a heat flow to a specific angle. All of these interesting properties are derived from the theory of the design of nonlinear inhomogeneous materials with large degrees of freedom. All methodologies are also summed up in the success of the thermal cloak. Looking back on the thermal cloak, it allows the flow of heat to avoid the middle of the area to ensure that there is no temperature gradient inside the cloak. A heat concentrator device makes the temperature gradient in the middle region increases instead of dropping to zero. Affected by this idea, in 2012, Guenneau et al. first proposed the concept of thermal concentrator. 38 This can be used as a nonintrusive thermal device to serve the thermoelectric effect, so as to improve the energy conversion efficiency. As shown in Fig. 2, we consider a circular model with three rings, and there is the coordinate transformation, r = rr 1 (0 r R 2 ), R 2 r = r(r 3 R 1 ) + (R (7) 1 R 2 )R 3 (0 r R 2 ). R 3 R 2 R 3 R 2 In the whole transformation space, the tension and compression compensate each other, which leads to the consistency of the original space and the transformation space. Therefore, the thermal concentrator consists of two parts. The transformation matrix T can be calculated by using the previous method. Then we can calculate the parameters of the thermal concentrator, κ r = 1, κ θ = 1 (0 r R 2 ), κ r = r + (R2 R1)R3 R 3 R 2 r, κ θ = r (R 2 r R 3 ). r + (R2 R1)R3 R 3 R 2 The thermal conductivity given above will not be eliminated or infinite singularities. Similar to the treatment of thermal cloak, we use finite element simulation to show the performance of the concentrator in the thermal field. As can be seen in Fig. 2, the temperature gradient of the middle region is enhanced because the space in the middle region is squeezed. The degree of gradient enhancement is determined by the geometric parameters of the radius of three concentric circles. In addition, because of the compression and stretch cancellation, the determinant of (8)

8 Q. Ji, X.-Y. Shen & J.-P. Huang (a) (b) Fig. 2. (Color online) (a) A schematic diagram of thermal concentrator where R 1 = 1 cm, R 2 = 1.5 cm, R 2 = 2 cm. (b) The temperature profile of the thermal concentrator when the heat diffuses from a temperature of 400 K to 300 K. the thermal conductivity matrix is the same as that of the background, so it will not affect the temperature distribution of the external field as the thermal cloak Thermal rotator In 2013, a new type of thermal metamaterial with special function was proposed by Guenneau et al. through the finite element method to solve the problem of cylindrical diffusion. 41 It allows the flow of heat to rotate at a specific angle, so it is

9 Transformation thermotics: thermal metamaterials and their applications called a thermal rotator. The implementation method is still started by the geometric transformation, and then the thermal conductivity tensor is obtained. In a series of simulations, there is an apparent negative thermal conductivity phenomenon. In other words, heat may flow from low-temperature to high-temperature. If an object is placed in the center area of the material, outside observers view the internal space as turning a specific angle, which leads to a kind of mirage phenomenon. (About the thermal illusion, a more detailed discussion will be given in later chapters.) According to the experience of designing a thermal cloak and a thermal concentrator, it is necessary to make the transformation area conservative (the product of mass density and specific heat constant). This will greatly simplify the control of heat flow in the transient state, and will not have an impact on the external field, just like a thermal cloak and a thermal concentrator. The device also has a ring structure as the first two metamaterials. Now look at the transformation of thermal rotator. Fig. 3. (Color online) Temperature distribution for a heat source located on the top, which diffuses heat in a medium containing a rotator for different time: (a) t = s, (b) t = 0.01 s and (c) t = 0.1 s. The rotation angle is θ 0 = π. (d) The mesh formed by heat flux streamlines and 2 isothermal lines in the long-time regime, t 0.1 s, illustrates the deformation of the transformed thermal space. Parameters: R 1 = m and R 2 = m

10 Q. Ji, X.-Y. Shen & J.-P. Huang For the inner ring region r < R 1, r = r, θ = θ + θ 0. (9) There is no coordinate transformation for the outer ring region r > R 2. For the region R 1 < r < R 2, r = r, θ = θ 0(f(R 2 ) f(r)) (10) + θ. f(r 2 ) f(r 1 ) With the coordinate transformation, the next step is to calculate the transformation matrix, and obtain the thermal conductivity tensor. As a result of the Fig. 4. (Color online) Temperature distribution for a heat source located on the top, which diffuses heat in a medium containing a rotator for different time: (a) and (b) t = s; (c) and (d) t = 0.1 s. The rotation angle is θ 0 = π. The rectangular object has a diffusivity one hundred 2 times smaller than that of the surrounding medium, and it is rotated by an angle of π in (b) and 2 (d) when compared to (a) and (c). The temperature distribution outside the rotator is the same in both (a) and (b), or in both (c) and (d). Parameters: R 1 = m and R 2 = m

11 Transformation thermotics: thermal metamaterials and their applications needs of the production, we had better write it in the Cartesian coordinate system. Finally, based on the material parameters, we can create this kind of thermal rotator device by means of experiment and simulation. Through the prior empirical equation, we can get the transformation matrix. The finite element method is used to simulate the anisotropy thermal conductivity by using the software COMSOL. The performance of the thermal rotator in the thermal field is obtained. Figure 3 shows the case of rotation angle and the temperature range is normalized. It is very clear that the direction of heat flow is rotated in the device as compared to the outside by observing the white isotherm. This phenomenon also causes the apparent negative thermal conductivity, as the heat flows from a lower temperature to a higher temperature. This apparent negative thermal conductivity is significantly enhanced during the unsteady state. What is more important is that the heat flow in the annular region is very smooth. And because of not affecting to an external field, the thermal rotator is invisible to the outside observer. As shown in Fig. 4, when an object is placed in the inner region of the thermal device, the outside observers view the heat flow as the object is rotated 90 degrees. This phenomenon is actually a kind of thermal illusion. The temperature distribution of an object that has been placed horizontally seems as it is rotated an angle. Based on the equivalence between the geometric transformation and the material properties, we can calculate the thermal response of thermal rotator to the thermal field. Guenneau et al. also further improved the rotator so that it can work very well near the heat source, and greatly enhance the apparent negative thermal conductivity effect in a short time. So the thermal rotator has the ability of changing the direction of heat flow. 3. Theoretical Design and Experimental Preparation of Thermal Metamaterials It is clear that to design these materials in theory is not enough, since it is always the most important part in the application of materials science. The material parameters mentioned in the previous section are inhomogeneous and anisotropic, which is not possible in nature. Therefore, it is necessary to realize the material properties of these materials by other means. Fortunately, the effective medium theory provides us with a powerful tool for the realization of the metamaterials based on the theory of transformation thermotics. This method can be used to simulate the anisotropic complex thermal conductivity by using a variety of homogeneous and isotropic materials. From our previous discussion, we can see that the parameters of the thermal cloak are very difficult to realize in the experiment. Fortunately, in the study of soft materials, we have developed a set of effective medium approximation theory which is very effective and easy to operate. The theory is based on the idea of approximation of material properties. For example, when a set of resistors in a

12 Q. Ji, X.-Y. Shen & J.-P. Huang circuit are connected in series, the electrical conductivity can be regarded as a result of the equivalent resistance of a resistor. The same way, we can use the same method to achieve some material which does not exist naturally. That is to say, the distribution of the properties of a material, such as the thermal cloak, is regarded as the equivalent property of several secondary structural components. In this way, we can obtain the anisotropic distribution, which is related to the structure, the geometry of the material, the different physical properties of different materials, the composition of different materials and so on Effective medium approximation theory: Two media mixing Symmetry microstructure Let us now consider the material of two kinds of media. First of all, it is assumed that a mixture contains two sizes of nano- or micro-sized particles of 1 and 2, and the two particles are randomly doped as shown in Fig. 5. The thermal conductivities of particles 1 and 2 are κ 1 and κ 2. Then the effective thermal conductivity of the mixture of the two components can be expressed as P 1 (κ 1 κ e ) 2κ e + κ 1 + P 2(κ 2 κ e ) 2κ e + κ 2 = 0. (11) Here P 1 and P 2 represent the volume fraction of particles 1 and 2, P 1 + P 2 = 1. For the above equation, if the index of 1 and 2 are exchanged, the equation is still set up. Therefore, the equation is symmetric microscopically, which is actually a copy of dielectric Bruggeman effective medium approximation theory in the system of dielectric system. 59 It is very easy to extend the two-component mixture Fig. 5. (Color online) Schematic graph showing a model symmetrical microstructure. A twocomponent composite where particles 1 (in red) and 2 (in blue) are randomly distributed in the whole system. The size of particles are different, so that they could fill in the whole composite

13 Transformation thermotics: thermal metamaterials and their applications to multiple components. Suppose that each component has a corresponding thermal conductivity. Then, the effective thermal conductivity of the mixture with N particles is satisfied, N i=1 P i (κ i κ e ) 2κ e + κ i = 0. (12) Here P i is the volume fraction of the ith component, and N i=1 P i = Asymmetric microstructure We continue to consider a class of components consisting of two components. The mixture is composed of nanoscale or microscale spherical particles randomly distributed in the substrate. The thermal conductivities of particle and substrate are κ 1 and κ 0. The mixture of the two components has the effective thermal conductivity κ e and satisfies the relationship κ e κ 0 κ e + 2κ 0 = (κ 1 κ 0 )P 1 2κ 0 + κ 1. (13) The volume fraction of particle 1 is P 1. In the equation, if we exchange 1 and 2, the equation will change. Therefore, this equation is also known as an asymmetric microscopic effective medium equation. Equation (13) is actually the electrical Maxwell Garnett effective medium approximation theory to the thermal field. This approximate equation cannot be used to predict the percolation threshold of the structure. Similarly, Eq. (13) can be easily extended from the two components to the mixture of N + 1 components, which contains N kinds of filler particles and a substrate. The thermal conductivity of each particle is κ i, and the substrate remains κ 0. Then the effective thermal conductivity of the mixture of the N + 1 component is the following: κ e κ 0 κ e + 2κ 0 = N i=1 Here P i is the volume fraction of the ith particles. (κ i κ 0 )P i κ i + 2κ 0. (14) 3.2. Dual cloak based on the effective medium theory Li et al. proposed the use of nanomaterials based on the effective medium theory to design invisibility cloak. 36 They hope to use the combination of the two materials to perfectly fit the anisotropic parameters generated by the coordinate transformation. In this design, the thermal conductivity of the nonspherical nanoparticles is spread over a homogeneous medium of thermal conductivity. The effective thermal conductivity of the mixture are derived from the Bruggeman effective medium approximation theory

14 Q. Ji, X.-Y. Shen & J.-P. Huang Considering the thermal resistance of the nanoparticles at the interface of the mixture, the thermal conductivity of a layer of nanoparticles with a layer thickness and thermal conductivity can be calculated. 63,64 The thermal resistance at the interface is determined by the aspect ratio of the particles. Of course, if the aspect ratio is large enough, the equations can also be explained by approximate theoretical calculations. Because of the thermal resistance of the nanoparticles, the effective thermal conductivity of the whole hybrid system can be given by the theory, (1 P )(κ i κ r,t ) P (κ m κ r,t ) + = 0. (15) Γ r,t (κ i κ r,t ) + κ r,t Γ r,t (κ m κ r,t ) + κ r,t In the formula, P is the volume fraction of nonspherical nanoparticles. Γ r and Γ t are the radial and tangential particle shape factor, respectively. In this work, the radial shape factor is always larger than 1/3, which means that the nanoparticle in the mixture is an oblate ellipsoid, and its three principal axes satisfy the relation c < b = a. On the other hand, the shape factor are the equation of aspect ratio γ, and γ = a/c. 65 Then, the shape of the nanoparticles doped in the mixture can be determined to make the cloak capsid. Next, by adjusting the shape and volume fraction of the nanoparticles, which are determined by the shape factor, we can embed them into the substrate to achieve anisotropic inhomogeneous material properties. In order to meet the theoretical requirements of thermal conductivity, by using the nonspherical nanoparticles with different shapes and volume fractions, any given anisotropic thermal conductivity can be fitted as close as possible Experimental realization of steady-state thermal metamaterials Since the thermal metamaterial based on transformation theory has been put forward, it has been a great challenge to the realization of the extreme anisotropic thermal conductivity. In this section, we continue to explore how to achieve various kinds of stable metamaterials under steady-state heat transfer. Due to the influence of above-mentioned simplified methods, Han et al. developed the method of thermal cloak based on the theory of steady-state transformation thermotics. 42 This method only uses a uniform, nonsingular and widespread thermal conductive materials. The thermal cloak made of this material has the properties of anisotropy and can be any geometric size, which is not the same as the previous thermal cloak theory. The design is based on the invariance of the heat conduction equation in the coordinate transformation. A perfect thermal cloak parameter is shown in Eq. (6). It is obvious that the thermal conductivity is a variable related to the spatial parameters, and there is a singularity when r is equal to the inner diameter. All these make it difficult to design such a thermal device in engineering. We will illustrate how to use a finite thermal conductivity as a constant material to approximate the composition of a thermal material close to the thermal cloak. There is a very interesting relationship between the radial and tangential

15 Transformation thermotics: thermal metamaterials and their applications Fig. 6. (Color online) Temperature profile for a thermal cloak: (a) ideal conductivity; (b) κ r = 0.1, κ θ = 10; (c) κ r = 0.2, κ θ = 5; (d) κ r = 0.3, κ θ = 3.3. Isothermal lines are represented in green. Parameters: R 1 = 1 cm, R 2 = 2 cm. 42 parts mentioned in the perfect thermal conductivity tensor. Thus, a method for realizing the thermal cloak can be developed. In Fig. 6, we show the temperature distribution of four kinds of thermal cloak. Diagram (a) shows a perfect thermal cloak, and (b) (d) show the uniform thermal cloaks with various parameters. As shown in Fig. 6, even a uniform thermal cloak can deflect heat flow from the protected area and prevent heat from entering the inclusion. Obviously, this design greatly simplifies the parameters. In the picture above, we can see that when C = κr κ θ is very small, the heat flow is limited to a narrow range within the thermal cloak. Of course, in the case of the large C, heat flow may enter into the cloak region. Therefore, the parameter C must not exceed a specific value. Based on the above discussion, the most important problem in designing a uniform thermal cloak is to find the threshold of C. A rigorous theoretical analysis and finite element simulation have shown that the thermal cloak performed well when C < We also need more steady-state thermal cloak to achieve the proposed and experimental verification, whether the thermal cloak can achieve clock function

16 Q. Ji, X.-Y. Shen & J.-P. Huang Fig. 7. (Color online) (a) A sketch to show how to fabricate an anisotropic thermal cloak; (b) cross-sectional schematic graph of a multilayered thermal concentrator. 39 also need to create a real thermal device to test. To this end, in 2012, Narayana and Sato created the first true sense of the thermal clock device 39 using two different thermal conductivity of materials. As previously pointed out, the layered structure with alternating layers of concentric circles can be used to realize the compression transformation. Imagine a mixture consisting of two alternating layers of A and B. In the vertical direction, the two kinds of alternate thermal conductivity will produce the compression effect, while in the tangential direction is tensile. Thus, even though we have been using isotropic homogeneous materials, the effective thermal conductivity of the whole mixture becomes anisotropic, and the flow of heat is also bypass the invisible object as expected. When this effect is achieved, it may be possible to inspire people to design some special heat diffusion paths. As shown in Fig. 7, they designed a concentric layered structure. The combination of two kinds of thermal conductivity, κ A and κ B, produced an anisotropic thermal cloak. 39 For background material, they used 5% of the agar water, whose thermal conductivity is 0.56 W/mK. In order to ensure that the device does not affect the distribution of the external field, 66 the thermal resistance of the substrate material needs to be matched with the values calculated by the effective medium theory of the two materials. As shown in Fig. 8, their column metamaterials were designed as 40-layer, 0.36 mm thick natural rubber film with thermal conductivity of 0.13 W/mK, and 0.38 mm thick nickel containing silicone membrane with thermal conductivity of 2.5 W/mK, which arranged alternately. The inner diameter of the cylinder is 0.8 cm, the outer diameter is the size of 2.7 cm. The whole thermal cloak material is 5 cm in length and placed on an agar plate. Two other materials were used as controls

17 Transformation thermotics: thermal metamaterials and their applications Fig. 8. (Color online) Experimentally measured temperature map for the reported thermal cloak and concentrator. 39 The first is a hollow cylinder contrast material. The other is polyurethane. The two materials were also placed on the agar plate as the sample. In the experiment, the temperature of both sides was controlled at 41 C and 0 C. The actual temperature distribution measured by the experimental results is shown in Fig. 8. It is obvious that the thermal cloak does not distort the temperature distribution of the external field, but the copper and polyurethane does. The high thermal conductivity of the copper ring can successfully allow the heat flow to bypass the central region. According to the measurement, in the r < a region, the temperature gradient is only 0.25 K/cm. As a contrast, the thermal gradient of the polyurethane is about 3.5 K/cm. According to the simulation, there is no temperature gradient inside the thermal cloak. Based on the experimental results, the internal temperature gradient is approximately equal to 0.38 K/cm. So, the thermal cloak on heat flow shielding effect is comparable to copper ring. In addition, the group also uses the same two materials through a special arrangement [see Fig. 7(b)] to create a thermal concentrator. In the design, the radial stack stretch transformation is achieved by alternately stacking 4-layer A material and 2-layer B material. They provide an external temperature gradient and measure the temperature distribution of the thermal concentrator. In Fig. 8, the heat exchanger compresses the heat flow in the whole ring to a very small area. And when the internal energy density is greatly improved, the field can still be kept uniform. It is quite clear that the heat flow line of the outer ring is bent in the opposite direction with the thermal cloak. This can improve the internal temperature gradient without affecting the outside. The experimental results show that the thermal gradient in the r < a region can be increased by 44%

18 Q. Ji, X.-Y. Shen & J.-P. Huang 3.4. Experimental performance of nonsteady thermal cloak Guenneau et al. tried to give an effective method to manufacture the unsteady thermal cloak. 38 The design of his thermal cloak is similar to Narayana et al. s static thermal cloak, 38 but in detail is more complex. They want to use a large number of uniform thermal diffusivity coefficient of the layered material to form a wide area of thermal cloak. In order to achieve this goal, the whole design is divided into two steps. The first step is normalization and homogenization of the thermal conductivity, and getting the anisotropic but uniform thermal diffusivity. This method reduces the numerical calculation. When we use the multilayered cloak to approximate the ideal cloak, we will use the M-layer uniform concentric ring. The second step is using homogeneous thin N-layers to approximate each layer of M. This means that there will be a total of more than N M of the same kind of thin layers. Of course, this figure may be very large, we use the parameters of N = 2 and M = 10. A corresponding mathematical argument can be found in acoustic and Schrödinger s equations, 31 which describe the homogenization process and method in detail. Such a design requires the use of the diffusion coefficient of the material. Of course, a radial change or a determinant associated with R is a major problem in the homogenization process. This approximation is obtained by allowing and multiplying the determinant of radial variation. After doing so, we can only use the Eq. (12) as the parameters of the thermal cloak, so as to remove the determinant of mass and heat capacity. Of course, such a step simplification of the obtained parameters is based on the following approximation: (κ T ) = det(j) (κ T ) + κ det(j) T. (16) This approximation is valid only if the perturbation is small enough. This condition is not available in the outer boundary of the thermal cloak, which results in the interface thermal resistance between the thermal cloak and the background. Of course, we will see in the rear, however, the thermal protection of the cloak is still working. The next problem that needs to be addressed and solved is the thermal conductivity of the thermal cloak. The equivalent material parameters of this mixture is shown as follows: 1 κ r = 1 κ a + 1 κ b d b d a + 1, κ θ = κ a + κ b d b d a + 1. (17) The cloak effect of the approximate design was detected by the finite element simulation of COMSOL (Fig. 9). According to the simulation results, by using two different types of homogeneous isotropic concentric arrangement of the thin layer of material, we can achieve good performance of anisotropic nonuniform invisibility cloak. In order to be able to verify the above theory, Schittny et al. in 2013 released the results of their unsteady-state heat conduction cloak. 40 By theoretical work and corresponding experimental methods, they chose 10-layer ring structure with different thermal conductivity of each thin layer. The thermal conductivity of each

19 Transformation thermotics: thermal metamaterials and their applications ring is defined by Eq. (16), which can best fit the material parameters. In order to make the approximation accurate enough, the distance between the ring and the ring should be smaller than that of the temperature gradient. For the different thermal conductivity obtained by using two kinds of materials, they design a hybrid structure by several holes in the copper and injected into the soft material polydimethylsiloxane to the hole. They adjusted the volume fraction of two kinds of materials to meet the requirements of the thermal conductivity. The thermal conductivity of copper plate is 394 W/mK, and the thermal conductivity of filler polydimethylsiloxane is 0.15 W/mK. According to the previous introduction, the scale has no effect on the thermal metamaterial. With this in mind, in Figs. 10(a) 10(c), the experimental setup of the unsteady state heat transfer cloak at room-temperature is given. In the experiment, the scale of the whole device was limited, as R 1 = 2.5 cm, R 2 = 5 cm. The thickness of each ring is 2.5 mm. In order to measure the unsteady temperature data, they used a conventional thermal infrared camera (FLIR A320). According to Kirchhoff s law, it is very important to obtain an approximate 100% absorption equivalent Fig. 9. (Color online) Diffusion of heat from the left toward a multilayered cloak, which consists of 20 homogeneous layers of equal thickness, at time (a) t = s, (b) t = s, (c) t = 0.02 s and (d) t = 0.05 s. The mesh is formed by heat flux streamlines and isothermal lines. Parameters: R 1 = m and R 2 = m

20 Q. Ji, X.-Y. Shen & J.-P. Huang to radiation. The existence of polydimethylsiloxane can effectively eliminate the reflection on the surface of copper plate, which is favorable for the imaging of the camera. In addition, a layer of 100 µm polydimethylsiloxane film can also resist the heat dissipation effect. In the process of heat conduction due to the presence of the outside air, there will inevitably be a thermal convection and radiation effects, which will make the heat transfer in fact more complex than the theory. In order to overcome these problems, the thickness of the device is set to 2 mm, which greatly improves the heat capacity of the whole system, thus greatly reducing the influence of convection and radiation dissipation on the experimental results. Figure 10(d) gives the results of the experimental results of the unsteady-state thermal cloak and the results of the copper plate as a control group at different time points. By observing the experimental results, it is obvious that the middle region of the thermal cloak is lower than the ambient temperature in the whole experiment. The presence of a thermal cloak protects the hidden area in the middle from the influence Fig. 10. (Color online) (a) (c) The experiment installation of Schittny et al. They fill polydimethylsiloxane into the hole drilled in a copper plate and an infrared camera is used to measure the heat distribution of the device; (d) the experiment results of unsteady-state heat flow cloak and a copper case as reference are presented

21 Transformation thermotics: thermal metamaterials and their applications of external heat flow. A region with no temperature gradient can be generated by this hybrid structure. All in all, the experimental results are in good agreement with the theoretical calculations Bilayer thermal cloak Bilayer thermal cloak only contains two layers of the same material composition, which is undoubtedly to simplify the design of the ultimate thermal cloak. 67 The bilayer thermal cloak is composed of an inner layer (a < r < b) and an outer layer (b < r < c). This design uses only the most basic materials with a simple arrangement, which is undoubtedly a very great progress. Based on the first principle, the parameter of bilayer thermal cloak is deduced by solving the steady-state heat conduction equation. This method is very robust and can greatly reduce the difficulty of the experimental design. After that, many experiments are based on the design of bilayer thermal cloak. This method promotes the design of simplified metamaterials and makes it possible to realize the free control of heat flux. For the steady-state thermal diffusion without heat source, the main equation mentioned in the previous section can satisfy the Laplace equation. The effect of thermal clock can be achieved by adjusting the geometry of the double layer structure directly instead of changing the material used. Two-dimensional thermal cloak can be verified by finite element simulation and experiment. As shown in Fig. 11, we can find that there is no temperature gradient in the covered area as the traditional cloak, and the external field will not be distorted by the influence of the object in the cloak area. The actual performance of samples is also very good; this kind of structure is of great significance in the realization of heat conduction cloak function, because it is deduced from first principle rather than thermal transformation. Only with two kinds of natural thermal material can achieve a thermal cloak Soft material in the experiment of thermal metamaterials Through the introduction of the theoretical part, we can know that the value of thermal conductivity in the transform space depends on the matrix J. The thermal conductivity can be written as an anisotropic two-order tensor. But in almost all materials in nature, the thermal conductivity of the material is uniform and homogeneous. According to the effective medium theory, two or more different homogeneous materials are used to construct an equivalent anisotropic thermal conductivity material. In the experiment, the extensive use of polydimethylsiloxane soft material as the basic material to prepare the experimental device is mainly due to the following reasons. For two different solid materials, even when the surface is smooth, the contact thermal resistance will come to appear. There are many voids in the solid-solid contact interface. As is known to all, the air is a poor heat conductor, whose thermal conductivity is close to zero. So that it will cause the temperature difference of the

22 Q. Ji, X.-Y. Shen & J.-P. Huang Fig. 11. (Color online) Simulated and measured temperature distributions for the steady state. (a) and (b) The reference structure of the object inside. (c) and (d) The reference structure of inner layer. (e) and (f) The proposed bilayer cloak. (g) and (h) The proposed bilayer thermal cloak in the presence of a point heat source, emitting cylindrical heat fronts. Isothermal lines are also represented with white color in the panel

23 Transformation thermotics: thermal metamaterials and their applications two solid contact interface, resulting in a large number of energy losses and unnecessary experimental errors. However, using the soft materials can be a good solution to this problem, because the soft material and solid interface can make contact with no air resistance. Temperature is continuous at the interface between them. Therefore, people will choose soft materials to prepare the experimental samples. Soft materials usually have relatively low thermal conductivity. According to the effective medium approximation theory, the experimental parameters are often filled with high thermal conductivity materials. Using the soft materials can reduce the contact resistance. On the other hand, the natural heat insulating materials can be used as filling material to reduce the thermal conductivity of the whole system, so as to obtain the expected parameters. Soft materials can also be used as a protective film covering the entire sample. They can protect the metal from the sample surface by air oxidation. Furthermore, this layer of soft material protective film resists the heat dissipation caused by air convection in the experiment. It is helpful to improve the accuracy of experimental measurement and help to control the experimental variables. The infrared thermal imager is needed to observe the experimental results. If the sample is made of metal, the metal surface will cause reflection, reducing the precision of the imager. However, when the soft material is covered with the sample, the reflection will be stopped, so that the experimental measurement becomes feasible. In the experiment, in order to be able to capture the results of temperature distribution, the surface of all the samples need to cover a soft material. 4. Nonlinear Transformation Thermodynamics Li and Shen 68,69 recently considered a more general heat conduction process, i.e., the thermal conductivity will change with the temperature. Thus, a very interesting new transformation thermal theory, temperature-dependent nonlinear transformation thermotics, has been developed. Using this new theory tool, we can design a new type of thermal metamaterial which cannot be realized by the former transformation thermotics. This process enriches the design of the thermal metamaterials, and the new self-coupling method can also be extended to other fields, such as acoustics, electromagnetics, optics and so on. Below we will detail the nonlinear transformation thermodynamics Nonlinear transformation thermodynamics theory Let us consider the generalized Fourier law in steady-state heat conduction. One of the most notable points in the equation is that the thermal conductivity is a function of temperature. In Ref. 68, we theoretically proved that the equation still satisfies form-invariant under transformation. Only in this way could we continue to obtain the thermal conductivity tensor in the new transform space. Next, we achieved the temperature-dependent thermal conductivity tensor in a given physical space,

24 Q. Ji, X.-Y. Shen & J.-P. Huang which has the same form of traditional transformation thermodynamics, κ(t ) = Jκ(T )J T det(j). (18) To this end, we have basically completed the nonlinear transformation thermodynamics. This formula shows a correlation between thermal conductivity and temperature through the temperature-dependent transform is introduced. This inference will directly lead to the introduction of the next section of the switching thermal cloak and the macroscopic thermal diode Switch thermal cloak and macroscopic thermal diode The traditional thermal cloak can protect the heat flux of middle region from outside, and can guarantee the stability in the region at a constant temperature. Meanwhile, the internal object has no external temperature disturbance. In order to achieve the cloak effect, we need to carry out a simple compression transformation in the radial direction. The transformation formula of the thermal cloak has been discussed in previous sections. In order to achieve different response to different direction through the heat flux, thus making macroscopic thermal diode, we need two different thermal cloaks. The first one can show cloak function at high-temperature (type A thermal cloak), and it becomes ordinary at low-temperature. The second one shows cloak function in low-temperature (type B thermal cloak), and it becomes the common material at high-temperature. The switching function of two kinds of thermal cloak is automatically activated by the change of temperature. One way to do this is to change Eq. (4) according to the contents mentioned in the previous section, r = R 2 R 1 (T ) R 2 + R 1 (T ). (19) It is shown that the upper form and Eq. (4) have the same form. Here the A type switch thermal cloak has R 1 1 (T ) = R 1 (1 ), while for the B type e β(t Tc) +1 R R 1 (T ) = 1 e β(t Tc) +1. Here T c is the phase change point of the switch invisibility cloak, which determines the opening and closing of the thermal cloak. β is a scaling factor, which is used to adjust the approaching speed of phase transition. So far, in order to realize the switching function in the stealth device, we need to obtain the thermal conductivity tensor, r κ r (T ) = κ R 1 (T ) 0 r, (20) r κ θ (T ) = κ 0 r R 1 (T ), where κ 0 denotes thermal conductivity of the temperature-independent background material

25 Transformation thermotics: thermal metamaterials and their applications Fig. 12. (Color online) Switchable thermal cloaks obtained by two-dimensional finite element simulations: (a) and (c) Switch on for the temperature above 340 K; (b) and (d) switch off for the temperature below 320 K. The color surface denotes the distribution of temperature, where isothermal lines are indicated; heat diffuses from left to right. The upper and lower boundaries are the thermal insulation. In (a) (d), an object with thermal conductivity 0.01 W/mK is set in the central region with radius R 1. Parameters: κ 0 = 1 W/mK, κ a = 0.1 W/mK, κ b = 10 W/mK, R 1 = 1 cm, R 2 = 2 cm and T c = 330 K. 68 Next, we use COMSOL to perform the finite element simulations. As shown in Fig. 12, heat flow from the left side of the high-temperature zone to the right side of the low-temperature region. At the same time, the upper and lower boundary conditions of the simulation environment are thermal insulation. The results of the A thermal cloak are shown in Figs. 12(a) and 12(b). In the picture, in a hightemperature environment (T = K), we can observe that the thermal cloak can hide the inner area and achieve cloak function. However, when the environment temperature is low enough (T = K), the cloak function is closed, which means that temperature field in the thermal cloak is distorted by the external fields,

26 Q. Ji, X.-Y. Shen & J.-P. Huang like the thermal cloak did not even exist. Like Eq. (20), the thermal conductivity is neither uniform nor anisotropic, but also temperature dependent. This makes the experimental implementation very difficult. In fact, in order to create such a material, we can simply repeat the alternating arrangement of two kinds of homogeneous material layer, which must be related to the temperature. The thickness of the two layers are respectively d 1 and d 2, and the thermal conductivity κ a and κ b. In order to simplify the calculation, we choose d 1 = d 2. In order to be able to give the traditional cloak with a temperature switch function, some of the corresponding similar to our previous mathematical operations must be applied on κ a and κ b. Thus we can obtain the temperature-dependent thermal conductivity, κ 1 (T ) = κ 0 κ a e β(t Tc) κ a, κ 2 (T ) = κ b κ 0 e β(t Tc) κ b. (21) In the case of high-temperature, the thermal cloak effect is activated, and the cloak becomes the same material as the background at low-temperature. Equation (21) provides a very convenient and practical guidance for the experimental realization. Next, we show the simulation results of a multilayered structure to achieve an A-type switching thermal cloak, in Figs. 12(c) and 12(d). It is shown that the effective medium theory is very close to the theoretical prediction, which proves the feasibility of this method. By the same method, we can also achieve a B-type switching thermal cloak. The discussion on the switching thermal cloak can help us to design a kind of macroscopic thermal diode. As shown in Fig. 13, the diode device contains three regions: I, II and III. The areas I and II are part of the A-type switching thermal cloak and the B-type switching thermal cloak, while the III region is an ordinary thermal conductor. Compared with the whole system, the thermal cloak effect still exists in the diode, but not perfect. There will still be a small amount of heat flow through the middle of the insulation. However, the thermal cloak is truncated to the two part of the A type and B type. The asymmetry of the two part and different temperature responses cause the diode effect. The transformation theory plays a very important role in this process. It is just because of the introduction of anisotropic structure, this structure will guide the heat flux to the boundary so as to greatly increase the heat shielding effect. As shown in Fig. 13, macroscopic thermal diode allows heat flow from the left to the right. And when the heat flows reversely from the right to left, this material will limit the flux, showing the property of thermal insulation. In addition, for the different temperature difference, we calculated the heat flux density J, which is obtained by the integral of all x components along the x = 0 direction, as shown in Fig. 13(b). The device shows a very clear rectification effect, according to the current parameters, the rectifier ratio can reach to about 30. In order to realize this kind of macroscopic thermal diode, we also use the effective medium theory. As discussed previously, the switching thermal cloak can be obtained by alternately arranging two thin layers. The thermal conductivity of

27 Transformation thermotics: thermal metamaterials and their applications Fig. 13. (Color online) Sketch of a thermal diode, which is the rectangular area enclosed by the solid black lines. The blurred area outside is a reference and actually does not exist in the design. I, II and III represent three regions, respectively. Here, the arrows indicate the direction of heat flow; the length of the arrows represents the amount of heat flux: the heat flux transferred from right to left (upper panel, the insulating case) is much smaller than that from left to right (lower panel, the conducting case); (b) heat current J versus temperature bias T ; (c) and (d) thermal diode obtained by two-dimensional finite element simulations: (c) the insulating case and (d) the conducting case. The color surface denotes the distribution of temperature; the white arrows represent the direction of heat flow; the length of the white arrows indicates the amount of heat flux; the upper and lower boundaries are the thermal insulation. Thermal conductivities are calculated according to Eq. (3); an object with thermal conductivity 10 W/mK is set in the central region with radius R 1. Parameters: κ 0 = 1 W/mK, R 1 = 3.6 cm, R 2 = 4 cm and T c = 330 K. 68 the thin layer needs to be very sensitive to the temperature and to undergo drastic numerical changes at the phase transition point. It is true that this phenomenon can be found in certain phase change materials. However, the phase change material may encounter a lot of constraints and inconvenience in operation. Therefore, under the guidance of the spirit of the metamaterial (by means of the specific structure of the traditional materials to achieve a new function), we try to use a constant thermal conductivity of the material to achieve the macroscopic thermal conductivity. Our approach is to use the phase transition in the structure instead of the phase

28 Q. Ji, X.-Y. Shen & J.-P. Huang transition to the physical properties of the material. According to the design requirements, the geometric structure of the device needs to change very rapidly when the temperature changes. Fortunately, we found that shape memory alloy can provide the shape change we want. As shown in Fig. 14, in the experiment, the first section of the thermal cloak is made of copper and foamed plastic foam. In the vicinity of the phase transition temperature, the shape memory alloy sheet will be deformed, the deformation can be driven to paste on the copper on the shape memory alloy to connect or disconnect the connection with the copper. This connection and disconnection state can be equivalent to the phase transition of local thermal conductivity. Therefore, the temperature-dependent thermal material can be achieved by a homogeneous isotropic material. At the same time, because the macroscopic thermal diode is a part of the thermal cloak, a whole piece of material can be realized in the same way. The experimental results are shown in Fig. 14. In the case of heat insulation, heat flow is almost impossible to enter the inner region. On the contrary, there is a very obvious thermal gradient in the middle region in Fig. 14(d). In summary, a macroscopic thermal diode with a rectifier function is devised. One such diode has many potential applications in thermal protection, heat dissipation, and even thermal imaging, such as efficient refrigerators, solar components, energy efficient buildings or military camouflage. Therefore, the temperaturedependent thermal transformation theory can be used to construct a material with nonlinear effect, which can be used to produce the macroscopic thermal structure materials. Due to the wide application of the transformation theory, the switching cloak and rectifier diode mentioned in this paper may also be extended to other fields such as seismic wave, acoustic wave, electromagnetic wave and even matter wave Nonlinear transformation thermodynamics The steady-state temperature-dependent transformation theory only considers the steady-state heat conduction. We will extend the theory to the field of unsteady heat conduction. The instantaneous heat conduction equation can be obtained by taking into account the n-dimensional generalized Fourier law. By representing the Fourier function through a corresponding transformation in a curvilinear coordinate system, we can obtain the desired thermal conductivity under the unsteady state. Based on the above expression, we can create a completely different thermal structure material at different temperatures. We will use the switching thermal concentrator as an example to demonstrate the thermal theory of unsteady nonlinear transformation. The principle of the switching thermal concentrator is shown in Fig. 15. All discussions are in polar coordinates. The device is a gray ring with a diameter of R 1 and a diameter of R 3. The collector can improve the temperature gradient of the inner region and has no influence on the temperature distribution of the external field. The specific thermal concentrator transformation can refer to the previous chapters. Based on the previous idea, the instantaneous nonlinear transformation

29 Transformation thermotics: thermal metamaterials and their applications (a) (c) Fig. 14. (Color online) (a) and (b) Scheme of experimental demonstration of the macroscopic thermal diode: (a) insulating case and (b) conducting case. Both the copper-made concentric layered structure and the central copper plate (both displayed in orange) are placed on an EPS plate which, for the sake of clarity, is not shown. The left and right sides of the diode are stuck in water to promote constant temperature boundary conditions. (a) When cold water is filled in the left container (light blue) and hot water in the right container (pale red), the bimetallic strips of shape memory alloy and copper (white) warp up and the device blocks heat from right to left. (b) When the two containers swap their locations, the bimetallic strips (white) flatten and the device conducts heat from left to right. (c) and (d) Experimentally measured temperature distribution of the device: (c) insulating case and (d) conducting case. 68 (b) (d) heat endows the function of traditional thermal concentrator at one temperature and can turn off to be ordinary materials at another temperature. Specifically, the material can be designed to be a conventional thermal concentrator when the temperature is lower than a specific value, and when the temperature is higher than this value, it becomes the same material as the background thermal conductivity. By replacing the thermal conductivity of traditional thermal concentrator, we can construct a new function R (T ), whose value can be switched remarkably when the temperature changes. This one has the following form: R (T ) = R 2 R 1 e β(t Tc) + 1, (22)

30 Q. Ji, X.-Y. Shen & J.-P. Huang (a) (b) Fig. 15. (Color online) Schematic graphs of a switchable thermal concentrator when the concentrating effect is switched (a) on or (b) off. The red lines with arrows represent the flow of heat. R 1 and R 3 denote the interior radius and exterior radius, respectively. R 2 and r are also indicated. 69 where β is the scaling factor to control the sensitivity of temperature. Then we can achieve a new transformation for switching thermal concentrator RR 1 R (r < R 1 ), T F (r, T ) = r(r 3 R 1 ) R 3 R T + R 3(R 1 R T ) R 3 R T (R 1 < r < R 3 ). (23) Next, the thermal conductivity distribution of the device in the polar coordinates is obtained. For r < R 1, { κr (T ) = κ 0, κ θ (T ) = κ 0. (24)

31 Transformation thermotics: thermal metamaterials and their applications Fig. 16. (Color online) Results of the finite element simulations for the switchable concentrator. All the physical quantities adopted are nondimensionalized, as listed in the text. Heat conducts from left to right throughout the time and the temperatures on two lateral boundaries are fixed. Upper and lower boundaries are thermally insulated. The critical temperature T c is set to be 1. (a) (d) The device functions as a concentrator at temperature below 0.9 and (e) (h) the device is turned to be the same as the background when the temperature is higher than 1.1. The temperature distribution is denoted by the rainbow color surfaces, while the isothermal lines are indicated by the white lines. We take the snapshots of temperature distributions at time (a), (e) t = 2; (b), (f) t = 6; (c), (g) t = 10; and (d), (h) t =