1. Introduction

Usually, the realization of physical functionality means to manipulate and control the relevant physical fields in a desired way. To achieve such a goal, various structures and geometries are designed based on available materials. However, the capacity and functional diversities are usually limited by the material it is made of. As a result, some desirable properties seem impossible to achieve with materials available at present. For example, a material with independently controllable thermal and electric conductivity is difficult to come by. Once such material becomes available, numerous intriguing possibilities might be opened up.

Fortunately, such problem might be solved due to the rapid development of metamaterials, which are composites consisting of well-arranged inclusions that can be considered as “effective” material. Over the past years, metamaterials have been widely used in manipulation of physical fields ranging from electromagnetic wave [1–3], acoustic wave [4], elastic wave [5] to matter wave [6]. Recently, metamaterial was introduced to control fields such as static magnetic field [7–10], dc electric field [11–13], thermal field [14–19], electrostatic field [20] and diffusive field [21–23]. However, bifunctional or multifunctional metamaterials are rarely reported and studied, except for several reports in recently [24–28]. It is interesting to explore the possibility of creating novel metamaterials to achieve independently controllable thermal and electric conductivity. Recently, Moccia introduced “transformation multiphysics” framework and concept of bifunctional metamaterial. Furthermore, a shell consisting of thousands of thermal and electric elements was theoretically designed to act as thermal concentrator and electrical invisibility cloak [24]. Such study provides an encouraging example of metamaterials transcending their natural limitations, which may open up novel possibilities in the largely unexplored phase space of multifunctional/ multi-physical devices, and realize considerable potential applications. Up to date, however, the corresponding experimental demonstration is still unexplored. This might be attributed to the fact that the transformation-based devices require anisotropic, gradient and extreme parameters, which greatly complicates the fabrication. Consequently, it is highly desirable to explore the possibility of a simpler and general scheme and perform corresponding experiment.

Here, we proposed and experimentally demonstrated a bifunctional metamaterial to manipulate thermal and electric fields simultaneously and independently. A composite with simple structure was designed with independently controllable thermal and electric conductivity. Based on this composite, we designed, fabricated and characterized a simple bifunctional device capable of shielding thermal flux and concentrating electric current simultaneously, which confirmed the feasibility of our scheme.

2. Bifunctional metamaterials

The key to manipulating the thermal and electric fields independently is constructing a medium with independently controllable thermal and electric conductivity. To achieve such goal, we introduce a bifunctional metamaterial composed of fan-like inclusions in the associated (ρ, ϕ, z) cylindrical coordinate system, as shown in Fig. 1(a). A fan-like unit cell with periodicity of Δρ × Δϕ is illustrated in the magnified details, which consists of several fan-like inclusions made of four types of materials (A, B, C, and D). The thermal and electric conductivity for materials A, B, C, and D are (κ_A, σ_A), (κ_B, σ_B), (κ_C, σ_C) and (κ_D, σ_D) respectively. Here, a thermal and electrical insulation with σ_A = 0 S/m and κ_A = 0W/mK is used as material A. A fan-like inclusion (namely material D with a high thermal conductivity and electric conductivity) with a size of a × α is placed in the center. To manipulate thermal and electric fields independently, one can employ two special materials (material B and material C). Material-B inclusion has good electric conductivity but good thermal insulation conductivity (κ_B = 0W/mK). As is shown in this picture, material-B-based fan-like inclusion (with a size of α in ϕ direction) is placed along ρ direction and touches the material D. Material C has a high thermal conductivity and good electrical insulation performance (σ_C = 0 S/m). Such inclusion (with length of a in ρ direction) is placed along ϕ direction and touches the material-D inclusion. Clearly, this composite with such cell would show good thermal conductivity in theϕ direction and poor thermal conductivity in ρ direction. Meanwhile, it has good electric conductivity in ρ direction and good electrical insulation performance in ϕ direction. According to effective medium theory (EMT) [29], one can get the thermal and electric conductivity tensors:

 

Fig. 1 (a) The proposed bifunctional metamaterial with independently controllable thermal and electric conductivity in the associated (ρ, ϕ, z) cylindrical coordinate system.The unit cell is composed of severial fanlike inclusions made ofmaterial A, B, C, and D.The thermal and electric conductivity for materials A, B, C, and D are (κ_A, σ_A), (κ_B, σ_B), (κ_C, σ_C) and (κ_D, σ_D) respectively. The corresponding geometrical parameters can be seen in the insert. The principle for bifunctional device behaving as thermal cloak and electric concentrator: (b) The corresponding physical model. the space is divided into three parts: interior region (ρ<R1), shell (R1<ρ<R2), and exterior region (ρ>R2) (see Fig. 1(b)). The thermal and electric conductivity for background medium (interior and external regions) is κ₀, σ₀, while the one for the bifunctional shell is κ₁, σ₁, respectively.(c) The bifunctional device applied with temperature gradient and electric potential gradient. (d) The thermal flux distribution. (e) The current distribution.

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Here, $m$ is a small value compared to ${\kappa }_{\varphi }$ ($m\ll {\kappa }_{\varphi }$). Apparently, ${\kappa }_{\varphi }$ is dependent on $\Delta \rho$, $\Delta \varphi$, ${\kappa }_C$, ${\kappa }_D$, $a$and $\alpha$, but independent of ${\kappa }_B$. Meanwhile, ${\sigma }_{\varphi }$ is dependent on $\Delta \rho$, $\Delta \varphi$, ${\sigma }_B$, ${\sigma }_D$, $a$ and $\alpha$, but independent of ${\sigma }_C$. Additionally, because of ${\sigma }_{\varphi }=0$ and ${\kappa }_{\rho }\text{=0}$, therefore, one can design the effective thermal and electric conductivity tensors independently by adjusting ${\kappa }_C$, ${\sigma }_B$, $a$, $\alpha$, $\Delta \rho$, $\Delta \varphi$. Consequently, by using such composite, one can manipulate thermal and electric fields independently. In practice, we should find proper materials to construct such bifunctional metamaterial. In our study, silicon carbide (α-SiC) with thermal conductivity of κ₀ = 20W/mK and electric conductivity of σ₀ = 1.0 S/m is chosen as the background material. Clearly, it is easy to find a material with high thermal conductivity and good electrical insulation performance (material C). We can choose aluminum nitride (AIN), which has high thermal conductivity (κ_C = 190W/mK) and good electrical insulation performance (σ_C = 0 S/m). Other materials with these properties can also be used here, for instance, ceramics (beryllium oxide, boron nitride, aluminum oxide). However, it is hard to find a material with good thermal insulation conductivity (κ_B = 0W/mK). Inspired by the previous work [24], we choose conductive silver adhesive which has good electric conductivity (σ_B = 5.5e5 S/m) but poor thermal conductivity (κ_B = 1W/mK). Since κ_B<<κ₀, the conductive silver adhesive can be considered as a thermal insulation approximatively. Electroinsulating material–Acrylonitrile Butadiene Styrene (ABS) with σ_A = 0 S/m is used as material A. Similarly, due to its poor thermal conductivity (κ_A = 0.25W/mK<<κ₀), ABS is treated as a thermal insulation (κ_A = 0W/mK) approximatively. Aluminum (κ_D = 207 W/mK, σ_D = 2.1e7 S/m) is chosen as material D due to its good thermal and electric conductivity. Therefore, metamaterial with independently controllable thermal and electric conductivity is obtained approximatively, which is made of several specific materials: A: ABS (κ_A = 0.25 W/mK, σ_A = 0 S/m), B: silver paste (κ_B = 1 W/mK, σ_B = 5.5e5 S/m), C: aluminum nitride (AIN) (κ_C = 190W/mK, σ_C = 0 S/m) and D: aluminum (κ_D = 207 W/mK, σ_D = 2.1e7 S/m).

3. Bifunctional device

As a typical example, a bifunctional device capable of shielding thermal flux and concentrating current simultaneously is designed in this work. Figure 1b-e shows the schematic diagram of the proposed bifunctional device. Suppose that the space is divided into three parts: interior region (ρ<R1), shell (R1<ρ<R2), and exterior region (ρ>R2) (Fig. 1(b)). The thermal and electric conductivity for background medium (interior and external regions) are κ₀ and σ_0, while those for the shell are κ₁ and σ_1, respectively. In Fig. 1(c), the thermal flux flows from left to right, and the electric current is produced from left to right. As shown in Fig. 1d, the thermal flux flows circumvent the interior region, keeping their original paths without any distortions. The distortion of thermal fluxes only occurs in the shell, indicating that a thermal cloaking effect is achieved. As for electric domain (see Fig. 1(e)), the current touching the shell is concentrated into the interior region, resulting in increased current density, while the exterior current keeps its original path without distortion. In other words, the shell functions as electric concentrator. To implement such a device, one can employ the so-called “transformation multiphysics” [24]. In such a scheme, transformations are applied to thermal and electric fields simultaneously, and the required parameters can be achieved with a bifunctional metamaterial. However, such a scheme suffers from gradient and extreme parameters. In addition, the requirement of independently controllable thermal and electric conductivity makes it even more difficult. Therefore, here we employ anisotropic but homogeneous medium. It is assumed that the shell material is homogeneous but with anisotropic thermal conductivities κ_ρ, κ_ϕ. Here, makeκ_ρ⋅κ_ϕ = κ₀², to ensure that the external thermal flux is kept undistorted. Then, the only remaining task is to make thermal flux flow around the interior region. We introduce a variable c, where c = κ_ρ /κ_ϕ. According to the work [27], one can find that when c<9/100, a nearly perfect cloak can be obtained. Moreover, the smaller c corresponds to the better performance. Similarly, as for electric current, we assume that the shell is made of homogeneous but anisotropic electric material with σ_ρ, σ_ϕ, where σ_ρ⋅σ_ϕ = σ₀². According to the previous work [18], when σ_ρ >σ_ϕ, the current density in the interior region is higher than the exterior one. The smaller value of c (c = κ_ϕ /κ_ρ) leads to better concentrator efficiency. Based on the above analysis, using anisotropic but homogeneous metamaterial will greatly simplify the corresponding realization of the function.

To achieve the bifunctional device above, one can use the aforementioned metamaterial. First, to obtain electric concentrator, it is easy to satisfy conditions: 1) σ_ρ⋅σ_ϕ = σ₀²; 2) σ_ρ >σ_ϕ, since σ_ϕtends to be zero. According to Eq. (4), one can tailor σ_r independently by adjusting the geometrical parameters of B inclusion to make σ_ρ⋅σ_ϕ = σ₀².Because material A and material B are treated as thermal insulation, κ_r tends to be zero, therefore one can tailor κ_ϕ by adjusting the geometrical parameters of C inclusion. According to Eqs (1)-(4) and simulation below, the corresponding optimized geometry parameters are:$\Delta \rho \text{=4mm}$, $\Delta \varphi =\text{4}0°$, $a=3mm$, and $\alpha =\text{2}0°$.Here, one can obtain: σ_ϕ = 0S/m, σ_ρ = 3.63e5 S/m, which satisfy two conditions above. It is worth mentioning that, it is not very easy to obtain the thermal cloak by designing the thermal conductivity tensor precisely according to Eq. (1) because material A and material B are treated as thermal insulation approximatively. However, we find that, with the help of simulation, one can obtain a thermal cloak easily by adjusting the geometry parameter of C inclusion. The schematic illustration for practical realization and fabricated sample are shown in Figs. 2(a) and 2(b), respectively. First, simulations are carried out to obtain the temperature profile distribution. For comparison, the case for homogeneous background material is also simulated and shown in Fig. 3(a). As expected, a uniform temperature gradient is generated from left to right. The corresponding thermal flux distribution is provided in Fig. 3(c), where a uniform value can be obtained. The simulation results for our bifunctional device are shown in Figs. 3(b),(d). From Fig. 3(b), it can be found that the exterior temperature profile distribution has not been altered with the presence of bifunctional device, and the distortion for the isothermal lines only occurs inside the bifunctional device. As seen in Fig. 3(d), the thermal flux in the inner region is reduced by a factor of 0.03, while that in the exterior region remains unchanged.The results given above show that this device can function as a thermal cloak. To demonstrate its capacity of concentrating the electric current in the inner region, 1V potential is applied between the two sides, as schematically illustrated in Fig. 1(c). For comparison, we first simulate the potential distribution and current density distribution in homogeneous background material, as shown in Figs. 3(e) and (g). As expected, the isopotential lines are straight and paralleled to each other, and the current densities are uniform. The simulation results of potential distribution and current density for our bifunctional device are provided in Figs. 3(f) and 3(h). First, the exterior isopotential lines remain straight and paralleled to each otherwith no distortions (Fig. 3(f)). Second, the isopotential lines touching the device are compressed in the inner region. This leads to the increased current densities in the inner region, which can be verified by the simulated current density distribution as shown in Fig. 3(f). And, the current density in the inner region is enhanced by a factor of 2. Consequently, this device can function as an electric current concentrator. It is worth mentioning that the smaller cells would lead to better performance, one can optimize the performance by reducing the size of cells. In our study, the size we used is for the sake of easy fabrication.

 

Fig. 2 (a) The schematic illustration for practical realization of bifunctional device. the corresponding geometry parameters are optimized as follow: $\Delta \rho \text{=4mm}$, $\Delta \varphi =\text{4}0°$, $a=3mm$, $\alpha =\text{2}0°$. (b) The photogragh for the fabricated sample.

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Fig. 3 Thermal simulation results for background material: a) temperature profile. c) thermal flux distribution. Thermal simulation results for bifunctional device: b) temperature profile. d) thermal flux distribution. Electric simulation results for background material: e) electric potential distribution. g) current density. Electric simulation results for bifunctional device: f) electric potential distribution. h) current density.

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In the experiment, the left side of sample is put into contact with hot water (333 K), and the right side is with ice water (273 K). To characterize the thermal property, an infrared heat camera (Fluke Ti300) is employed to measure the temperature profile distribution. To reduce the reflection of sample for the operating wavelengths of the thermal heat camera, thin electrical insulation tape with emissivity higher than 95% is attached to the surface of the sample. The measured results are presented in Figs. 4(a)−-4(b) and the corresponding simulated ones are presented in Figs. 4(c)−-4(d). Clearly, the temperature gradient in the core region is decreased greatly and the external field is nearly undistorted. Good consistency can be obtained between the simulation and measured results. To experimentally confirm the electric property, a Multimeter (Agilent 34410A, 6, 1/2Digit Multimeter) is used to obtain the potential distribution. The performance of electric concentrator can be evaluated by the potential distribution along the lines x = −20mm, x = 20mm and y = 0, as given in the insets of Fig. (5). Clearly, the isopotential lines (both for line x = −20mm and x = 20mm) in homogeneous background material are straight and those for our device also remain straight due to the fact that there is no distortion in the external field. As for y = 0mm, the potential gradient is considerably enhanced in our device compared to that in homogeneous background material, indicating a good electric current concetration effect for our device. The simulation and experimental results show good consistency, confirming the feasibility of our design. It is noted that the deviation of simulation results and experimental results can be attributed to the fabrication.

 

Fig. 4 (a) Measured temperature profile for background material. (b) Measured temperature profile for bifunctional device. (c) The corresponding simulated temperature profile for background material. (d) The simulated corresponding temperature profile for bifunctional device.

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Fig. 5 The simulation and experiment results for background material and the one with electric concentrator: simulated electric potential values for the different cases at corresponding positions: (a) x = −20mm, (b) x = 20mm and (c) y = 0mm. d, e, f) Corresponding experimental potential values, respectively. The white lines in inserts represent observed lines.

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At this point, it has been demonstrated that by employing bifunctional metamaterial, the thermal and electric fields can be manipulated simultaneously and independently. In practice, applications may range from thermoelectric materials to multifunctional electronic components [24]. As for the designed bifunctional device, anisotropic but homogeneous thermal and electric parameters were designed to concentrate or cloak physical field. Compared with TO method, our method can avoid many problems, such as gradient and extreme parameters, thus greatly simplifying the fabrication process. It is pointed out that the independently controllable thermal and electric conductivity tensors are the keys to manipulating thermal and electric fields independently. One can obtain various independently controllable thermal and electric conductivity tensors by adjusting the geometrical parameters, structures and materials. As a result, the design idea used here can been extended to the design of bifunctional devices like thermal concentrator/ electric cloak, thermal rotator/ electric cloak, thermal illusion/ electric cloak and so on.

4. Conclusion

In conclusion, we have provided the theoretical and experimental realization of simultaneous and independent manipulation of thermal and electric fields with bifunctional metamaterials. We introduced a bifunctional metamaterial with independently controllable thermal and electric conductivity to achieve independent manipulation of thermal and electric fields. Based on this concept, we demonstrated a bifunctional device capable of shielding thermal flux and concentrating electric current simultaneously. This work provides a novel approach towards independently tailoring material properties, providing a broad platform for manipulation of multi-physics field.