Introduction

Researchers have been pursuing effective methodologies to control thermal flux for multifarious applications1,2,3,4,5,6,7,8,9,10,11,12,13. The manipulation of heat flow is essential in technology development in many areas such as thermoelectricty1, fuel cells2, thermal barrier coatings3, solar cells4, electronics and optoelectronics5 and low thermal conductivity materials6. In addition, the ability to precisely control heat flow can potentially lead to the development of thermal analogues of electrical circuit components7, such as thermal diodes8,9,10, thermal transistors11 and thermal memory12. More recently, thermo-crystals were theoretically proposed for thermal management, guiding thermal wave just as photonic crystals guide light13.

While the conduction of heat has been known for a long time, the advance in the control of heat flow has been very slow7,8,9,10,11,12,13,14,15,16. Most recently, by using a multilayered composite approach, cloaking, concentration and reversal of heat flux were experimentally demonstrated in thick composites17 and planar structures18. Considering that the heat conduction equation is form invariant under coordinate transformations, thermal cloaks were realized with inhomogeneous anisotropic thermal conductivities19,20,21. On the basis of exact methodology (directly solving heat conduction equation), bilayer thermal cloaks made of bulk isotropic materials have been reported22,23. Though significant progress has been made toward the manipulation of heat flow, different functionalities have to rely on different structures14,15,16,17,18,19,20,21,22,23,24. This motivates us to explore a general class of thermal metamaterial units, with which a set of interesting functionalities in advanced control of heat conduction can be experimentally demonstrated by positioning and combining identical unit.

In this paper, we introduce a new class of thermal metamaterials composed of two regular materials in a simplified planar geometry. The thermal metamaterial unit, designed with transformation optics25,26, is capable of manipulating thermal energy and heat flux in a versatile variety of fashions by positioning and combining identical unit sensu-elements (which is a traditional folding hand-fan in Japan). We experimentally demonstrate that the combination of thermal metamaterial units exhibits novel properties such as the creation of a uniform heating region, heat flux focusing and heat flux concentration. On one hand, these proposed metamaterials are fabricated using regular materials and can hence be easily accessed and followed. On the other hand, these novel properties are remarkably robust to the geometrical sizing of the proposed metamaterials without having to change the material compositions.

Results

Figure 1(a) shows the schematic of a random cluster of sensu-shaped thermal metamaterial (SSTM) units made of two regular bulk materials. The design of a SSTM unit is based on transformation optics25,26, which is schematically illustrated in the inset of Fig. 1(a). The semi-annular region () in virtual space is extruded to a larger region (a ≤ r ≤ b) in real space, where the virtual space and real space are denoted as and , respectively. The transformation equations can be expressed as , and . Then the thermal conductivity in the region (a ≤ r ≤ b) is obtained as . When δ → 0, we obtain . It is rational to set κr=κb2n and κθ=κb2−n so long as n is large enough, where κb is the thermal conductivity of the background. On the basis of effective medium theory (EMT), an anisotropic material may be practically realized by alternately stacking two materials A and B in the azimuthal direction. For practical realization, when the background is stainless steel, we choose copper and polydimethylsiloxane (PDMS) as materials A and B, respectively. Thus we can obtain the SSTM unit composed of 18 copper wedges and 18 PDMS wedges, shown in Fig. 1(a). The conductivities of copper, PDMS and stainless steel are κCu=394 W/Km, κPDMS=0.15 W/Km and κb=16 W/Km, respectively. We choose a = 1.2 cm and b = 6 cm for both simulations and experiments throughout.

Figure 1
figure 1

(a) Schematic of a random cluster of sensu-shaped thermal metamaterial units made of two regular bulk materials. The inset schematically shows the transformation principle of thermal metamaterial unit. (b) Two heat-sources placed near the inner boundary of a thermal metamaterial unit, analogous to the performance of an optical hyperlens. (c) A heat-source placed near the outer boundary of a thermal metamaterial unit, demonstrating harvesting property.

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To understand the functionalities of the SSTM unit, numerical simulations based on finite element modelling are demonstrated in Fig. 1(b,c). In the simulation setup, the four sides of the stainless steel (background) are set as fixed temperature at 0 °C. The top and bottom surfaces are set as adiabatic boundary. This ensures that the heat is mainly conducted rather than convected. When two point heat-sources are placed near the inner boundary of a SSTM unit, the simulated thermal profile is shown in Fig. 1(b). It is clearly seen that the thermal fields of the two heat-sources are almost perfectly transferred from inner-boundary to outer-boundary, analogous to the performance of an optical or acoustic hyperlens27,28,29,30. In contrast to the conceptual thermocrystal superlens for near-field imaging13, our scheme may provide a more effective strategy for far-field thermal imaging. Inversely, when a point heat-source is placed near the outer boundary of the SSTM unit, the thermal energy is transferred into the center with almost no attenuation, as shown in Fig. 1 (c). This example only represents a small part of the potential of an SSTM unit and more interesting functionalities that could be achieved through the combination of SSTM units will be demonstrated in the following.

We first experimentally demonstrate how to form a uniform heating region between four distant heat-sources by separating and enclosing them with four of our SSTM units. The fabricated structure is schematically illustrated in Fig. 2(a). It is noted that we have thermally isolated all of the fabricated samples by using an approximately 100 μm thick layer of PDMS. The benefits of the PDMS layer are twofold -- heat convection by air is significantly reduced and the PDMS layer on top of the sample is nearly “black” (i.e., 100% emissivity) for the wavelengths detected by the infra-red thermal imaging camera. The temperature distributions of four distant heat-sources with and without the SSTM units are quantitatively calculated in Fig. 2(b), in which the distance between two adjacent heat-sources (from the center of the left heat-source to the center of the right heat-source) is 13 cm. We can see that the valley-shaped temperature distribution for the case without the SSTM units become uniform after incorporating the four SSTM units. In the experimental setup, we use four brass cylinders with radius of 1 cm as heat-sources and the four boundaries are connected to a tank filled with ice water (0 °C). The four brass cylinders are connected to a hotplate fixed at 60°. The cross-sectional temperature profile is captured with a Flir i60 infrared camera. The measured results with and without SSTM units are shown in Fig. 2(c,d), respectively, which is as expected and agrees very well with the theoretical results in Fig. 2(b). From the coordinate transformation perspective, this is because the geometrical size of heat-sources have been enlarged by b/a = 5 times by using the SSTM. This property may find potential application in medical techniques like thermal therapy, where uniform heating is required over a body region.

Figure 2
figure 2

Experimental demonstration of forming a uniform heating region between four distant heat-sources enclosed by SSTM units. (a) Schematic of the fabricated sample. (b) Quantitative contrast of four distant heat-sources with and without SSTM. (c) Measured thermal profile of four distant heat-sources separately enclosed by SSTM units. (d) Measured thermal profile of four distant heat-sources without SSTM.

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A natural question is whether a uniform heating region can be formed by only using copper of the same size. To answer this question, we simulated the case in which the SSTM units are replaced by four copper quadrants of the same size as that of Fig.2(a); see Fig. S1 in Supplementary Materials. It is clearly demonstrated that sensu-patterned copper exhibits higher temperature and better heating uniformity performance than the unpatterned bare copper counterparts of the same size overall. It is also important to examine the performance of SSTM cluster in Fig. 2(a) with larger distance between two adjacent units, which is demonstrated in Fig. S2 in Supplementary Materials. For quantitative comparison, the temperature of the center is marked out. In general, the performance is naturally degraded with an increase in d. In all individual cases of different separation distances, the temperature distribution for the case with SSTM is always much higher than the case without SSTM.

We next experimentally demonstrate heat flux focusing by using SSTM units. Consider a heat-source (brass cylinder) being enclosed by a SSTM unit and a second SSTM is placed beside the first SSTM unit. We want to see that the enhanced thermal field of the heat-source is harvested by the second SSTM unit and focused into its center. The fabricated structure is schematically illustrated in Fig. 3(a). A brass cylinder with radius of 1 cm is placed at the center of the left SSTM unit and the four boundaries are connected to ice water (0 °C). The image captured by the infrared camera is shown in Fig. 3(c). For contrast, we measured the case where the two SSTM units are absent, as shown in Fig. 3(d). The temperature distribution along the semicircular red line in Fig. 3(c,d) is also calculated in Fig. 3(b), which agrees well with the experiment. Assuming the heat-source (placed in the first dotted circle) acts as a transmitter and the second dotted circle as a receiver, it is clear that the heat has been directed from the transmitter to the receiver by using the SSTM. When a thermal sensor is placed in the receiver position, the temperature is greatly enhanced, thus increasing the sensitivity of the temperature measurement. We also examine the cases where the SSTM units are replaced by copper wedges or copper quadrants (see Fig. S3 in Supplementary Materials). Obviously, there is no focusing effect with bare copper.

Figure 3
figure 3

Experimental demonstration of thermal focusing. (a) Schematic of the fabricated sample. (b) Quantitative contrast of temperature distribution along semicircular red line in (c) and (d). (c) Measurement result of thermal focusing by enclosing a heat-source with SSTM unit and placing a second SSTM unit beside the first SSTM. (d) Measurement result of (c) without SSTM.

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Moreover, we demonstrate an efficient heat flux concentrator by positioning two identical SSTM units. A concentrator is achieved by a combination of two back-to-back SSTM units, as shown in Fig. 4(a). For the measurement of the concentration effect, local heating on the left side is achieved by a hot plate fixed at 60 °C and the right side is connected to a tank filled with ice water (0 °C). The experimental result on heat flux concentration is presented in Fig. 4(b), which clearly shows the significant role of the SSTM’s presence. Nearly all heat flux in the region (0 ≤ r ≤ b) is concentrated into the inner core (0 ≤ r ≤ a) without any reflection and distortion, indicating nearly perfect concentration. For quantitative comparison, the temperature distributions along x-axis are presented in Fig. 4(c) for both cases (i.e., with and without SSTM), which clearly reveals that the presence of SSTM (solid line) confines and builds up the temperature within the central area more efficiently than the case without the SSTM (dashed line). We also examine the concentration behavior in the presence of a point heat-source, as shown in Fig. 4(d). In the experimental setup of Fig. 4(d), the left and right boundaries are connected to ice water (0 °C). Fig. 4(d) demonstrates the effectiveness of the proposed scheme in a non-uniform thermal field. In parallel, we study the cases in which the SSTM units are replaced by copper or PDMS, as shown in Fig. S4 in Supplementary Materials. Obviously, there is no concentration effect with bare copper or PDMS. Beyond the thermal regime, it is noted that dc electric field concentrator based on resistor networks31 and dc magnetic field concentrator based on superconductor- ferromagnetic metamaterials32,33 have been demonstrated recently.

Figure 4
figure 4

Experimental demonstration of an efficient thermal concentrator. (a) Schematic of the fabricated concentrator by the combination of two SSTM units. (b) Experimental verification of concentrating property in uniform thermal field. (c) Calculated temperature distribution along x-axis. The inset shows the heat flux lines. (d) Experimental verification of concentrating property in the presence of a point heat-source, emitting cylindrical heat fronts.

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Discussion

The SSTM concept can be extended -- we find that the proposed SSTM approach can also be applied to manipulate dc currents, which has been numerically verified in Fig. S5 in Supplementary Materials. In the simulation setup, we use the same geometrical parameters and material components as those in the thermal regime. This means that the proposed SSTM is able to manipulate both thermal fields and dc currents simultaneously, demonstrating bifunctional property34,35.

In summary, we have proposed a new class of thermal metamaterials by using two regular bulk materials (copper and PDMS) and experimentally demonstrated its unique properties in terms of forming a uniform heating region, heat flux focusing and concentration. These novel properties are robust to geometrical sizing of the SSTM unit whether scaled up or down. More interestingly the proposed SSTM is capable of manipulating both thermal field and dc currents simultaneously, exhibiting multi-functional property. Our scheme may find straightforward applications in technological devices such as thermoelectric devices, solar cells, thermal sensors, thermal imagers, as well as in thermal therapy applications. Our results can also provide novel ways to control other fields, e.g., dc magnetic fields, spin waves in spintronic devices, electrons in semiconductors.

Methods

The numerical simulations are performed by a commercial finite element method solver COMSOL Multiphysics. The SSTM unit, embedded in the stainless steel, is fabricated with 18 copper wedges and 18 PDMS wedges. The conductivities of copper, PDMS and stainless steel are κCu=394 W/Km, κPDMS=0.15 W/Km and κb=16 W/Km, respectively. We choose a = 1.2 cm and b = 6 cm for both simulations and experiments throughout. For measurements, the cross-sectional temperature profile is captured with a Flir i60 infrared camera.

Additional Information

How to cite this article: Han, T. et al. Manipulating Steady Heat Conduction by Sensu-shaped Thermal Metamaterials. Sci. Rep. 5, 10242; doi: 10.1038/srep10242 (2015).