Experimental demonstration of an ultra-thin. three-dimensional thermal cloak

Transcription

1 Experimental demonstration of an ultra-thin three-dimensional thermal cloak Hongyi Xu 1, Xihang Shi 1, Fei Gao 1, Handong Sun 1,2, *, Baile Zhang 1,2, * 1. Division of Physics and Applied Physics, School of Physical and Mathematical Sciences, Nanyang Technological University, Singapore , Singapore. 2. Centre for Disruptive Photonic Technologies, Nanyang Technological University, Singapore , Singapore *Electronic mail of corresponding author: hdsun@ntu.edu.sg; blzhang@ntu.edu.sg Abstract We report the first experimental realization of a three-dimensional thermal cloak shielding an air bubble in a bulk metal without disturbing external thermal flux. The cloak is made of a thin layer of homogeneous and isotropic material with simple mechanical manufacturing. The cloak s thickness is 200 μm while the cloaked air bubble has a diameter of 1 cm, achieving the ratio between dimensions of the cloak and the cloaked object 2 orders smaller than previous thermal cloaks which were mainly realized at a two-dimensional plane. This work can find applications in novel thermal devices in the three-dimensional physical space. 1

2 Invisibility cloaks 1,2 have the capability of routing electromagnetic energy around an object without disturbing the external electromagnetic environment. Similar concepts have been extended to the control of acoustic waves 3, elastic waves 4, and matter waves 5. In the past few years, remarkable effort has been made in developing cloaks based on various wave equations Recently the possibility of manipulating energy flux based on diffusion equations such as direct current 13, static magnetic field 14,15 and heat flux has attracted a lot of attention, because of their potential applications in electric/magnetic shielding and thermal management. While various electric shielding techniques have been widely used in electrical engineering, not until recently has thermal cloaking been proposed based on the method of transformation thermodynamics 18 that manipulates heat flux by constructing an effectively curved space similar to previous transformation optics. Subsequent two-dimensional (2D) implementations of inhomogeneous and anisotropic thermal cloaks have been successfully demonstrated in experiment 19,22,23. Nevertheless, a three-dimensional (3D) thermal cloak that can hide a 3D object in a thermal environment still remains unrealized. Here we report the first experimental realization of a 3D thermal cloak that can hide an air bubble in a bulk metal without disturbing the external thermal flux. We choose an air bubble as the object to be hidden, since it is well known that stationary air is a poor conductor of heat, and air gaps or air deflects can degrade the efficiency of a heat exchange system and cause local over-heating. Moreover, similar to most previous demonstrations of electromagnetic cloaks 3,7-12 where a perfect electric conductor (PEC) that does not allow any field penetration was shielded, any object put inside the cloaked air bubble will also be cloaked from external thermal flux because of the poor thermal conductivity of stationary air. As it has been pointed out 14, invisibility cloaks based on diffusion equations have their distinct properties compared 2

3 to previous cloaks based on wave equations. Therefore it is possible to implement a 3D cloak for current like flux using widely available materials with simple manufacturing. Results We start with the physical model under the assumption that the thermal cloak works to shield spherical objects in homogenous heat flux in steady state. This corresponds to the scenario of cloaking a small-size round-shape 3D component such as a transistor foot or a soldering point in a circuit board under small temporal fluctuation of temperature. Illustration of the thermal cloak is shown in the inset of Fig. 1. The air bubble, in approximation to a thermal insulator with thermal conductivity 1 0, locates at the center with radius R 1, surrounded by a single-layer cloak with thermal conductivity 2 and outer radius R 2. Homogeneous heat flux diffuses from top to bottom through the background material with 0. The thermal diffusion equation in the homogeneous background takes the form as: 2 T=0. (1) where T is the temperature distribution in the homogeneous background. With the presence of the thermal cloak, the heat flux is expected to bend around the air bubble and return to its original path after passing the cloak. By adopting an analytic method similar to that used in designing a static magnetic cloak 14, we can derive the thermal conductivity 2 of the cloak as follows: (2 R R ). (2) ( R2 R1) To fix the geometry, we first plot the dependence of the relative conductivity of the cloak / on the thickness ratio R 2 /R 1 in Fig. 1. We assume the background material is stainless 2 0 3

4 steel with thermal conductivity 0 15 W/mK. The thermal conductivity of the cloak can be chosen from a variety of natural materials marked in red color. Here we choose copper with thermal conductivity W/mK in both simulation and experiment. Radius R 2 and R 1 are 0.5cm and 0.51cm, respectively, meaning a thin layer of copper with 200 μm thickness is sufficient to cloak an object with 1 cm diameter. We perform 3D simulations with the commercial FEM solver COMSOL Multiphysics to verify effectiveness of the designed thermal cloak in Fig. 2. A heat source with constant temperature T 1 at 100 Celsius degree contacts the left side of the background material, and a heat sink with constant temperature T 2 at 0 Celsius degree contacts the right side of the background material. The rest of boundaries are defined as insulating boundaries. In Fig. 2ac, when there is only an idealized air bubble ( 1 0 ) without the cloak, distortion of temperature distribution can be observed around the air bubble which can cause local temperature inhomogeneity. In contrast, Fig. 2d-f show that when the cloak is in operation to shield the air bubble, the distortion becomes negligible and all heat flux trajectories are recovered after passing the air bubble as if it does not exist. We notice that at the beginning of transient state, there is still some distortion around the air bubble (Fig. 2d) since the thermal cloak is designed to work in thermal equilibrium. However, during most time in establishing the thermal equilibrium, the function of the cloak to suppress temperature distortion is still valid. Now we proceed to implement the 3D thermal cloak. Practically, stationary air has nonzero thermal conductivity 0.02 W/mK, which is close to zero in experiment. Our simulation has verified that this approximation is valid. Pure copper with conductivity 385 W/mK is chosen to implement the single-layer cloak and stainless steel ASTM 301 with conductivity 15 W/mK is chosen for the background material. The molding process of the 4

5 cloak is shown in Fig. 3a, where a thin layer of copper is hit by a molding rod to fit into the stainless steel mold. Two identical half blocks (Fig. 3b) are then combined together as shown in Fig. 3c to form a complete 3D cloak. The dimension is 0.5 / 0.51cm for the inner/ outer radius of the copper spherical layer, and 2 2 2cm for the stainless steel block. In the experimental characterization the bottom and top surfaces of the stainless steel block were attached with a hot plate and an ice tank, respectively (Fig. 3c). An electrical hot plate (Barnstead RT-elite) provided constant temperature T 1 at 100 Celsius degree, and a steel ice tank provided temperature T 2 around 0 Celsius degree that was maintained by the continuous supply of ice. Top and bottom surfaces of the stainless steel block were coated with conductive thermal compound to ensure good thermal contact with heat sources. After the system was turned on for about 1 min, the temperature distribution did not change much. Thereafter the sample was separated into half blocks where the cross-section surface temperature was captured by the infrared camera FLIR T640 that was calibrated before measurement with a laboratory black body. For the correction of low emissivity of stainless steel, surfaces to be measured were coated with 3M black tape with near-unitary emissivity around Distribution of normalized temperature and its contours at the cross-section plane are shown in Fig. 4. Distortion can be observed around the air bubble without the cloak, as shown in the experimental result in Fig. 4(a) and the simulated result in (b). On the contrary, when the cloak shields the air bubble, the temperature distribution is much less disturbed, as shown in the experimental result in Fig. 4(c) which matches well with the simulation in Fig. 4(d). 5

6 Discussion Because an air bubble has nonzero thermal conductivity, its temperature will raise slowly rather than staying constant. This is a common observation in most of previous thermal cloak experiments up to date. However, the raise of temperature of the air bubble has little influence on the temperature distribution in the external region. Moreover, because of the slow rising speed of temperature inside the air bubble compared to the surrounding background, this cloak can still provide effective thermal protection for a relatively long time. In conclusion we present the design, fabrication and experimental characterization of an ultra-thin 3D thermal cloak that can shield an air bubble in a bulk metal. To the best of our knowledge, it is the first realization of a thermal cloak in a 3D sphere. Because of its ultrathin thickness and simple manufacturing, this cloak can find wide cost-effective applications in addressing practical issues of distorted temperature distributions caused by air defects in mechanical engine systems and semiconductor devices. 6

7 Figure 1 Material candidates to realize a 3D thermal cloak with the background material of stainless steel. The black curve shows relative thermal conductivity required to implement a 3D thermal cloak with different thickness ratio of the cloak. The inset figure illustrates the cross section of the cloak. Red/ blue region denotes high/ low temperature, and dashed arrows represent the heat flux. 7

8 Figure 2 Simulated temperature distribution and trajectories of heat flux (red curves) for cases without the cloak (a-c) and cases with the cloak (d-f) at different temporal frames 2s, 10s and 30s. Heat flux diffuses from left to right. 8

9 Figure 3 (a) Molding process of half of the 3D thermal cloak: a thin copper layer is hit into the stainless steel base. (b) Illustration and snapshot of half of the thermal cloak after molding. (c) Illustration and snapshot of the full cloak by combining two half blocks. The red/ blue plate represents high/ low temperature at the top/ bottom surface. 9

10 Figure 4 Distribution and contours (white curves) of temperature for cases without the 3D thermal cloak (a: experiment and b: simulation) and cases with the 3D thermal cloak (c: experiment and d: simulation). 10

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