Blackhole-Inspired Thermal Trapping with Graded Heat-Conduction Metadevices

Black holes are one of the most intriguing predictions of general relativity. So far, metadevices have enabled analogous black holes to trap light or sound in laboratory spacetime. However, trapping heat in a conductive ambient is still challenging because diffusive behaviors are directionless. Inspired by black holes, we construct graded heat-conduction metadevices to achieve thermal trapping, resorting to the imitated advection produced by graded thermal conductivities rather than the trivial solution of using insulation materials to confine thermal diffusion. We experimentally demonstrate thermal trapping for guiding hot spots to diffuse towards the center. Graded heat-conduction metadevices have advantages in energy-efficient thermal regulation because the imitated advection has a similar temperature field effect to the realistic advection that is usually driven by external energy sources. These results also provide insights into correlating transformation thermotics with other disciplines such as cosmology for emerging heat control schemes.


Introduction
Efficient heat utilization is crucial for industrial production and daily life [1][2][3]. However, heat control is still facing many challenges because thermal diffusion, a primary heat transfer scheme, is directionless. Thermal advection, another fundamental heat transfer mode, can break the intrinsic space-reversal symmetry of thermal diffusion and realize asymmetric heat transfer.
However, both the actual advection induced by mass transfer [4][5][6] and the effective advection generated by spatiotemporal modulation [7][8][9][10] require the implementation of inconvenient and energy-consuming external drives. Therefore, it is not only a different subject but also a big challenge to realize asymmetric diffusion in a pure and passive heat-conduction environment.
On the other side, black holes are well-known for their ability to trap light in their horizons.
Due to the development of metadevices, analogous black holes have been explored in electromagnetics [11][12][13][14][15][16][17] and acoustics [18][19][20][21], thus trapping light and sound in laboratory spacetime. However, because of the vast difference between wave and diffusion, these analogous black holes have no natural correspondence in heat conduction. Therefore, it is still intrinsically elusive to realize thermal trapping.
Inspired by black holes, we propose purely heat-conduction-based metadevices with graded thermal parameters [22][23][24] to realize thermal trapping. Though the designed structures are stationary solids without external drives, the graded thermal conductivities can still generate the imitated advection that brings about a similar temperature field effect to the realistic advection induced by mass transfer. As an alternative understanding, graded parameters can also be mathematically linked to curvilinear spacetime by the transformation theory [25][26][27][28].
Then we design the imitated advection to point towards the center, and surrounding heat can 4 also be trapped towards the center, thereby achieving thermal trapping. Inspired by rotational black holes, we further perform a rotation transformation to realize rotational thermal trapping.
These schemes are validated numerically and experimentally.

The imitated advection
The space-reversal symmetry makes thermal diffusion directionless. Specifically, hot spots in a homogeneous structure only dissipate but do not move ( Fig. 1a and b). An insulatorconductor-insulator structure can introduce anisotropy [29] and make thermal diffusion distinctly different in two perpendicular directions: one direction very conductive and the other very insulating. Nevertheless, if a hot spot is in the conductive region bounded by insulation walls, heat still goes forward and backward along the conductive pathway (the right-top and left-bottom hot spots in Fig. 1c and d). If a hot spot is in the region with a very low thermal conductivity, heat diffuses slightly in an isotropic way (the left-top and right-bottom hot spots in Fig. 1c and d).
In contrast to these conventional structures, a graded structure can break the space-reversal symmetry of thermal diffusion and trap hot spots towards the center even with rotation ( Fig. 1e and f). The underlying mechanism lies in the imitated advection induced by graded thermal conductivities. We compare wave propagation and heat diffusion to understand the imitated advection. In photonics, a graded refractive index can generate an effective momentum [30][31][32].
Hence, the bending effect can be achieved despite the vertical incidence of waves (Fig. 2a). The imitated advection (Fig. 2b) shares a similar conception with the effective momentum.
Heat conduction in one dimension is described by 0 0 + (− 0 ) = 0, where 0 , 0 , and 0 are the mass density, heat capacity, and thermal conductivity of a homogeneous medium. , , and denote temperature, time, and coordinate. To achieve the imitated advection, we consider a graded thermal conductivity of ( ), a graded mass density of ( ), and a graded heat capacity of ( ) . Hence, the heat conduction equation becomes We consider the diffusion-advection equation of + 0 − 0 2 = 0, where 0 is the advection velocity, and 0 = 0 /( 0 0 ) is the thermal diffusivity of a homogeneous medium.
Hence, the second term on the left side of Eq. (1) denotes the imitated advection. The imitated advection velocity of is ( The graded thermal conductivity with ( ) ≠ 0 is the key to the imitated advection. For a constant velocity, we assume the parameters have exponential forms, where is a constant. The substitution of Eqs. (3a) and (3b) into Eq. (2) yields a constant velocity of 0 , On the one hand, Eqs. (3a) and (3b) lead to a global constant thermal diffusivity of 0 = ( )/( ( ) ( )) = 0 /( 0 0 ) . Hence, the designed structure is still homogeneous on a thermal-diffusivity level. In general, thermal diffusivity reflects the ability of heat to diffuse, and a constant value means the same diffusive properties, thereby making the imitated advection unexpected. On the other hand, Eqs. (3a) and (3b) ensure a constant imitated 6 advection velocity. Since the imitated advection is attributed to graded thermal conductivities, the exponential variation is not mandatory as long as ( ) ≠ 0.
The imitated advection has nearly the same temperature field effect as the realistic advection (Supplementary Part I), so it can also break the space-reversal symmetry of thermal diffusion and generate asymmetric temperature profiles. This equivalence is only on a temperature-field level rather than a heat-flux level because graded thermal conductivities cannot replace fluid flows in fluid dynamics.
To demonstrate the imitated advection, we fabricate a graded heat-conduction metadevice, i.e., a copper plate drilled with air holes (Fig. 2c). The effective graded thermal conductivity of the fabricated sample is determined by the Maxwell-Garnett formula , where 1 and 2 are the thermal conductivities of air and copper, and is the area fraction of the air holes [33]. Hence, we can use only two natural materials to achieve a graded thermal conductivity with graded hole size. The theoretical distribution of the graded thermal conductivity is / 0 = −1.4 / , where is the structure length ( Fig. 2d).
We define the normalized temperature as * = /( ℎ − ) − /( ℎ − ), where is the realistic temperature, and ℎ (or ) is the temperature of the hot (or cold) source. For thermal diffusion in a homogeneous medium, the forward and backward cases are symmetric, with temperature distributions of * = − / + 1 and * = / , respectively. Therefore, * = 0.5 should appear at / = 0.5 . Intriguingly, the graded heat-conduction metadevice can produce the imitated advection and break the space-reversal symmetry of heat conduction.
Therefore, we can draw two conclusions from Fig. 2. One is that we experimentally demonstrate the imitated advection in a pure and passive heat-conduction environment, which is similar to the realistic advection in terms of the temperature field effect. The other is that 8 energy loss in graded heat-conduction metadevices can further generate asymmetric heat fluxes.

Blackhole-inspired thermal trapping
Inspired by black holes, we guide the imitated advection to point towards the center to We first perform two-dimensional simulations to show directionless thermal diffusion in a homogeneous medium by COMSOL Multiphysics (Fig. 3a-c). The four hot spots are almost stationary, indicating no trapping effect in a homogeneous medium. Then we design the imitated advection (represented by the dashed arrows in Fig. 3d) to be centrally pointing, and the parameters are radially distributed, e.g., the thermal conductivity of ( )/ 0 = (1− / ) , where denotes the radial coordinate, and is the radius of the circular plate. Hence, the four hot spots can be trapped towards the center ( Fig. 3e and f).
Inspired by rotational black holes, we perform a rotation transformation on the scheme shown in Fig. 3d to demonstrate rotational thermal trapping (Fig. 3g). The rotation transformation is = ′ and = ′ + ( − ), where ( , ) and ( ′ , ′ ) are cylindrical coordinates in physical and virtual spaces, respectively. We define = 0 ( 0 − ), where 0 is the rotation angle, and 0 is a constant radius. With the transformation thermotics theory [36][37][38], the transformed parameters are ⃡ = ′ † / det and = ′ ′ / det , where the superscript of ′ denotes those parameters in virtual space, and † and det are the transpose 9 and determinant of , respectively. Since is the Jacobian transformation matrix expressed as = [( / ′ , /( ′ ′ )), ( / ′ , /( ′ ′ ))] , the transformed thermal conductivity for rotational thermal trapping is a tensor, not only graded but also anisotropic. The effect of rotational thermal trapping can be observed in Fig. 3h and i.
The simulated temperature evolution of these two types of thermal trapping is also exported as animations (Supplementary Gifs I for normal thermal trapping and II for rotational thermal trapping). Besides, thermal trapping can still be achieved with periodic temperature excitation (Supplementary Gifs III for normal thermal trapping and IV for rotational thermal trapping) because the major mechanism is the imitated advection generated by graded parameters, which is robust against different excitations.
We fabricate two samples ( Fig. 4a and d) to demonstrate thermal trapping. Three common materials (i.e., copper, iron, and steel) are used to increase the thermal conductivity gradient to ~− 4.8(1− / ) . We use four bigger initial hot spots to reduce the effect of heat dissipation ( Fig.   4a and d). For normal thermal trapping, the hot spots are trapped towards the center directly ( Fig. 4b and c). For rotational thermal trapping, the hot spots are both trapped and rotated ( Fig.   4e and f). To measure these two samples, we put them into an ice water tank (acting as a cold source, Fig. 4g). Heat guns generate hot spots, which can output a constant temperature for the initial setting. Two types of thermal trapping can be observed in Fig. 4h and i and Fig. 4j-l. Two factors mainly affect the experimental results. The first factor is the heat dissipation by natural convection and thermal radiation, making these hot spots decay too fast to observe. The second factor is the approximate parameters restricted by materials. Since our proof-of-concept experiments realize a thermal conductivity gradient much lower than the simulations in Fig. 3, the trapping effect is not as ideal as the simulations in Fig. 3. Nevertheless, these two factors do not affect our main conclusions.

Discussion and conclusion
Thermal advection is crucial for non-Hermitian physics and nonreciprocal transport, helping reveal various intriguing thermal phenomena such as exceptional points [39], topological transitions [40], and diffusive Fizeau drag [41]. Since the imitated advection has almost the same temperature field effect as the realistic advection, it is promising to uncover similar phenomena [39][40][41] with graded heat-conduction metadevices. The major advantage is zero energy consumption because external drives are not required. Moreover, due to the asymmetric thermal diffusion induced by the imitated advection, graded heat-conduction metadevices have potential applications in thermal funneling [42]. The imitated advection may also provide insights into controlling near-field thermal radiation [43][44][45] and fluid flows [46].
To summarize, we have revealed blackhole-inspired thermal trapping with graded heat- When we compare the imitated advection with the realistic advection in Fig. 2 and Fig. S1, the template of heat transfer in fluids is also applied.

Fabrication
The samples are prepared with a milling machine. Three materials are finely joined together, reducing the effect of thermal interfacial resistance as much as possible. The samples are also covered with plastic films to reduce thermal radiation reflection. We use an infrared camera to detect the temperature profiles. The hot and cold sources in Fig. 2 are water baths. Heat guns generate the initial hot spots in Fig. 4.