Temperature-Dependent Transformation Thermotics: From Switchable Thermal Cloaks to Macroscopic Thermal Diodes

The macroscopic control of ubiquitous heat flow remains poorly explored due to the lack of a fundamental theoretical method. Here, by establishing temperature-dependent transformation thermotics for treating materials whose conductivity depends on temperature, we show analytical and simulation evidence for switchable thermal cloaking and a macroscopic thermal diode based on the cloaking. The latter allows heat flow in one direction but prohibits the flow in the opposite direction, which is also confirmed by our experiments. Our results suggest that the temperature-dependent transformation thermotics could be a fundamental theoretical method for achieving macroscopic heat rectification, and it could provide guidance both for the macroscopic control of heat flow and for the design of the counterparts of switchable thermal cloaks or macroscopic thermal diodes in other fields like seismology, acoustics, electromagnetics, and matter waves.

Figure 1

Schematic graph depicting a thermal cloak between radius $R_1$ and $R_2$. The red lines with arrows denote the flow of heat: the cloak does not disturb the heat flow at a region with a radius larger than $R_2$; the heat flux cannot enter a central region with a radius smaller than $R_1$.

Figure 2

Switchable thermal cloaks obtained by two-dimensional finite-element simulations: (a),(c) switch on for the temperature above 340 K and (b),(d) switch off for the temperature below 320 K. The color surface denotes the distribution of temperature, where isothermal lines are indicated; heat diffuses from left to right; the upper and lower boundaries are the thermal insulation. (a) and (b) show the results of thermal conductivities calculated according to Eq. (3); (c) and (d) show the results of ten alternating layers of two sublayers, with ${\kappa }_1(T)$ and ${\kappa }_2(T)$ given by Eq. (4) (EMT). In (a)–(d), an object with thermal conductivity $0.01\mathrm{W}/\mathrm{mK}$ is set in the central region with radius $R_1$. Parameters: ${\kappa }_0=1\mathrm{W}/\mathrm{mK}$, ${\kappa }_a=0.1\mathrm{W}/\mathrm{mK}$, ${\kappa }_b=10\mathrm{W}/\mathrm{mK}$, $R_1=1\mathrm{cm}$, $R_2=2\mathrm{cm}$, and $T_c=330\mathrm{K}$.

Figure 3

(a) Sketch of a thermal diode, which is the rectangular area enclosed by the solid black lines. The blurred area outside is a reference and actually does not exist in the design. I, II, and III represent three regions, respectively. Here, the arrows indicate the direction of heat flow; the length of the arrows represents the amount of heat flux: the heat flux transferred from right to left (upper panel, the insulating case) is much smaller than that from left to right (lower panel, the conducting case). (b) Heat current $J$ versus temperature bias $\mathrm{\Delta }T$. (c),(d) Thermal diode obtained by two-dimensional finite-element simulations: (c) the insulating case and (d) the conducting case. The color surface denotes the distribution of temperature; the white arrows represent the direction of heat flow; the length of the white arrows indicates the amount of heat flux; the upper and lower boundaries are the thermal insulation. Thermal conductivities are calculated according to Eq. (3); an object with thermal conductivity $10\mathrm{W}/\mathrm{mK}$ is set in the central region with radius $R_1$. Parameters: ${\kappa }_0=1\mathrm{W}/\mathrm{mK}$, $R_1=3.6\mathrm{cm}$, $R_2=4\mathrm{cm}$, and $T_c=330\mathrm{K}$.

Figure 4

(a),(b) Scheme of experimental demonstration of the macroscopic thermal diode: (a) insulating case and (b) conducting case. Both the copper-made concentric layered structure and the central copper plate (both displayed in orange) are placed on an EPS plate which, for the sake of clarity, is not shown. The left and right sides of the diode are stuck in water to promote constant temperature boundary conditions. (a) When cold water is filled in the left container (light blue) and hot water in the right container (pale red), the bimetallic strips of SMA and copper (white) warp up and the device blocks heat from right to left. (b) When the two containers swap their locations, the bimetallic strips (white) flatten and the device conducts heat from left to right. (c),(d) Experimentally measured temperature distribution of the device: (c) insulating case and (d) conducting case.