A Thermal-Electric Cloak via Nonlinear Transformation

A thermal-electric invisibility cloaking is designed by a nonlinear transformation, which can guide both electric current and heat flux around the concealed region and restore their original diffusion directions. Compared with previous thermal-electric cloaks designed by linear transformation, the proposed thermal-electric cloak by nonlinear transformation can introduce some parameters during the coordinate transformations, thus providing more degrees of freedom for subsequent material design and selection. As an example, we further design a 2N-layered structure consisting of four isotropic homogeneous materials for implementing the proposed cloak, which shows expected good cloaking effect for both electric current and heat flux by numerical simulations. The simulated results notably shows that the proposed thermal-electric cloak can manipulate the waves without any agitation either the highly conductive material placed in a vacuum or waves can generate from any other direction, which verify the expected thermal-electric invisibility cloaking simultaneously in the steady-state situation.

A Thermal-Electric Cloak via Nonlinear Transformation

Muhammad Ahsan and Fei Sun
Abstract-A thermal-electric invisibility cloaking is designed by a nonlinear transformation, which can guide both electric current and heat flux around the concealed region and restore their original diffusion directions.Compared with previous thermalelectric cloaks designed by linear transformation, the proposed thermal-electric cloak by nonlinear transformation can introduce some parameters during the coordinate transformations, thus providing more degrees of freedom for subsequent material design and selection.As an example, we further design a 2N-layered structure consisting of four isotropic homogeneous materials for implementing the proposed cloak, which shows expected good cloaking effect for both electric current and heat flux by numerical simulations.The simulated results notably shows that the proposed thermal-electric cloak can manipulate the waves without any agitation either the highly conductive material placed in a vacuum or waves can generate from any other direction, which verify the expected thermal-electric invisibility cloaking simultaneously in the steady-state situation.

I. INTRODUCTION
T HE ability to design an artificial material with specific properties has led to the development of novel devices, such as invisibility cloaking [1], [2], which are mainly designed by transformation optics.In addition, other methods can also be utilized to design invisibility cloaking, such as solving Laplace equation [3], numerical methods [4], and optimizations [5], [6].Among these methods, linear transformations (LT) are commonly used, which is mathematically simpler than the non-linear transformation (NLT) [7] and have been extended to design invisibility cloaking for other physics (e.g., electric, thermal, acoustic, etc.) [8], [9], [10].Most of the early studies on invisibility cloaking are based LT, such as the experimental demonstration of the first microwave invisibility cloaking [3], conformal electromagnetic cloaking for arbitrarily shaped objects [11], open cloaking [12], elliptical acoustic/ cloaking [13], cylindrical/spherical cloaking for static fields [14], [15], thermal cloaking [16], and a light-controlled tunable DC cloak [17].Then, the nonlinear transformation (NLT) has been proposed to weaken the effect of removing singularities on the performance of the cloak, and to eliminate the scattering introduced by the impedance mismatch at the outer boundary of the simplified cloak based on LT and eikonal approximation [18], [19].Compared to LTs, NLTs can introduce more tunable parameters during the coordinate transformations, thus providing more degrees of freedom for subsequent material design and selection [18], [19].In addition, it has also been shown that NLT-based devices can provide some novel effects, e.g., cloaking an optical soliton, modeling nonlinear solution to Einstein's field equation, and control of transport in Debye solid [7].
In this study, the electric-thermal invisibility cloaking is designed with the aid of NLT, in which both fields have different distorted mapping.A spherical cloak is designed that executes the electric and thermal fields simultaneously in a steady-state situation.The designing process of the nonlinear thermal-electric cloak can be divided into two steps: the first step is to derive the required media by using NLT and the second step is to realize the required media by using a 2N-layered structure by effective-medium theory.

II. THEORY AND SIMULATED RESULTS
The structure of the proposed cloak is composed of two concentric spherical shells, i.e., the Region I (a<r<b, colored yellow) and the Region II (b<r<c, colored blue) in Fig. 1, where a, b and c are the radii of the spheres.The concealed region (r<a) is filled with vacuum or any material, and the outside region (r>c, colored green) is the background.In Region I (Region II), the electric and thermal conductivities are denoted as σ 1 (σ 2 ) and k 1 (k 2 ) respectively.
In the steady-state situation, both potential voltage for electric field and temperature for thermal field satisfy the conduction equation for both fields, which is given as: Here λ represents the electric or thermal conductivity of the virtual space, and Φ indicates the potential voltage or temperature.∇ = êx ∂ x + êy ∂ y is the 2D vector operator.( 1) is forminvariance under the coordinate transformation [22].Here, we adapt to the following NLT in the spherical coordinate system; where Here w 1 and w 2 are two modulation factors in NLT, which can provide more degrees of freedom for subsequent material selection.Herein, the r maps the circle of radius b (virtual space) to Region I (physical space), and r maps the circle of radius c (virtual space) to Region II (physical space).To keep the form invariant under transformation the constitutive conductivities are determined by [1]: where λ is the electric or thermal conductivity of Region I or Region II in the physical space, J is the Jacobian matrix, J T is the transpose of J and det(J) is the determinant of the Jacobian matrix.
Although, we are to execute two fields for both models simultaneously.If the field waves of each physics flow through the different regions, then it would be more convenient to analyze the behavior of the electric and thermal fields simultaneously.Therefore, we have adopted a strategy according to which the manipulation of the thermal waves is restricted through Region I and Region II restricts the electric waves.To obtain such results, two physical conditions are required: where k b is the thermal conductivity of the background.The condition given in ( 5) is required to get the uniform flow of thermal waves in the background and Region II.Whereas, the condition given in ( 6) is implemented to avoid the flow of electric waves from Region I. Next, the thermal conductivity of Region I and the electric conductivity of Region II will be calculated by using ( 2)-( 4).The thermal conductivity of Region I is determined by using (2) in (4).Then Jacobian matrix J is given as: Thus, by using ( 4) and ( 7), we have Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
The electric and thermal conductivities of Region I are given in ( 6) and ( 8), respectively, which can be summarized as Similarly, to find the electric conductivity of Region II we use (3) in (4): Hence together with thermal conductivity: for Region II.(11) Next, we use numerical simulations to verify the performance of the thermal-electric cloak whose electric and thermal conductivities of Region I and Region II are given in ( 9) and ( 11), respectively.All numerical simulations in this study are 2D cases and conducted by using the Multiphysics software COMSOL with license number 9406999.As an example, the radii of each region are chosen as a = 3 mm, b = 4 mm and c = 5 mm.The background of the device is incorporated with the material appropriate to the conductivities k b = 0.6 W/(mK) and σ b = 0.12 S/m.The inner region (r<a) of the model is a vacuum.and a point of copper material is taken to the left top of the rectangle that has sources voltage (1V) and temperature (373K) to generate the detecting thermal/electric field.
The simulated cases for the thermal fields are shown in Fig. 2. As shown in Fig. 2(a) and (b), the thermal fluxes (i.e., white lines) generated by a point high temperature source are smoothly guided around the concealed region (i.e., the vacuum r<a), and then guided back to their original pathway without creating any distortion when the designed cloak is applied.If the designed cloak is removed in Fig. 2(c), the thermal fluxes (i.e., the white lines) will be distorted obviously.The simulated cases for the electric fields are shown in Fig. 3.We have noticed from Fig. 3(a), the lines are moving uniformly and gradually around the vacuum through Region II.To verify this, the point source test is also performed by embedding an elliptical copper in the vacuum as represented in Fig. 3(b).It can be seen that the electric waves are not influenced by the embedded copper.Whereas, the electric current lines are distorting without the cloak in Fig. 3(c).The simulated results in Figs. 2 and 3 have verified the performance of the thermal-electric cloak in ( 9) and (11).
In the second step, a 2N-layered structure with four isotropic materials A, B, C, D is proposed to realize the required media in ( 9) and (11), where the second and third components are the same, then with the help of Sten's formula [25]: Suppose each region (Region I and Region II) is realized by two materials: the materials A and B with equal thicknesses are staggered in Region I to achieve the required material parameters in (9); the materials C and D with equal thicknesses are inserted in an alternative way in Region II to achieve the required material parameters in (11).Assuming four materials have the same thickness, then the conductivities of materials A, B, C, D can be obtained with the help of ( 12) and ( 13): , for material A, (14a) , for material B, (14b) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
, for material C, (14c) , for material D, (14d) where two modulation factors introduced in NLT are chosen as w 2 1 = −1 and w 2 2 = −1 to to ensure that the material parameters in ( 14) are all positive real numbers.
The performance of the a 2N-layered structure with four isotropic materials in (14) is verified by numerical simulations in Fig. 4, which show the 2N-layered structure can work as a cloak for both electric and thermal fields.As shown in Fig. 4(a) and (b), the thermal fluxes generated by a high temperature point can be guided around the concealed region, and then restored to their original directions with the help of the designed 2N-layered structure.Similarly, the 2N-layered structure can also guide the electric currents around the concealed region, and redirect them to their original pathway without creating any distortion in Fig. 4(c) and (d).
The main limitation of the proposed nonlinear thermalelectric cloak is that it has specific requirements for both the electrical and thermal conductivity, as specified in (14), which poses challenges for experimental validation.As both thermal conductivity and electrical conductivity of the cloak vary with spatial location, it becomes challenging to find corresponding materials in nature.
The question of how to realize the designed cloak is really a question of how to realize the materials A, B, C, D in (14).As we use a nonlinear transformation, the thermal and electrical conductivities in (14) are non-negative.To achieve materials with extremely low electrical conductivity in regions A and B, while maintaining positive and gradient thermal conductivity, a challenging feat to find naturally, one can employ active thermal metasurfaces for their realization (see "Thermal tunneling effect by thermal complementary media" in Supporting Information [38]).For regions C and D whose thermal conductivity is the same as the background material but electrical conductivity is positive and gradient, this material can be equivalently realized by overlaying the background thermally conductive material with a varying mixture of electrically conductive but thermally insulating materials.The electrical conducting but thermally insulating materials can be realized by metallic vanadium dioxide [39], and, nanoscale granular nickel [40].

III. CONCLUSION
A non-linear transformation is applied to design a thermalelectric cloak, which can guide both electric and thermal fields around the concealed region without any distortion.Then, a 2N-layered structure consisting of four isotropic homogeneous materials is proposed to realize the designed thermal-electric cloak, which numerically shows good cloaking effect for both electric and thermal fields.As a preliminary attempt, the nonlinear transformation can introduce some modulation factors during the coordinate transformations, thus providing more degrees of freedom for subsequent material selection.The method used in this study, i.e., non-linear transformation with modulation factors, can be extended to other dual-physical controlling structures in addition to thermal-electric cloak here.

Fig. 1 .
Fig. 1.Schematic diagram of spherical biphysics cloak for (a) the virtual space and (b) the physical space.

Fig. 2 .
Fig. 2. Numerical simulations of the thermal field for the ideal state (a) with the designed cloak (b) in the presence of elliptical copper (major axis a/3 and minor axis 2a/3) at the center of the vacuum and (c) without the cloak.The white lines indicate the flow pattern of heat fluxes.The point with the fixed high temperature 373K is used to generate detecting thermal field.

Fig. 3 .
Fig. 3. Simulations of the electric field; (a) with the cloak, (b) in the presence of an elliptical copper material and (c) without the cloak.The black lines represent the flow pattern of current density.The detecting source is generated by a point with the fixed voltage 1V.

Fig. 4 .
Fig. 4. Simulated results for the distributions of the temperature and voltage when the designed 2N-layered cloak in (14) is applied with N = 20 in this case.(a), (b) are the temperature field when the concealed region is vacuum and an elliptical copper, respectively.(c), (d) are the normalized voltage when the concealed region is vacuum and an elliptical copper, respectively.The elliptical copper is at the center of the vacuum with major axis a/3 and minor axis 2a/3.The white lines are showing the trajectory of heat flux and the gray lines are indicating the trajectory of current density.