Diffusive nonreciprocity and thermal diode

Wave propagation and diffusion in linear materials preserve local reciprocity in terms of a symmetric Green's function. For wave propagations, the relation between the fields entering and leaving a system is more relevant than the detailed information about the fields inside it. In such cases, the global reciprocity of the scattering off a system through several ports is more important, which is defined as the symmetric transmission between the scattering channels. When a two-port system supports nonreciprocal (electromagnetic, acoustic) wave propagation, it is a (optical, phonon) diode directly following the definition. However, to date no concrete definition or discussion has been made on the global reciprocity of diffusive processes through a multiple-port system. It thus remains unclear what are the differences and relations between the three concepts, namely, local nonreciprocity, global nonreciprocity, and diode effect in diffusion. Here, we provide theoretical analysis on the frequency-domain Green's function and define the global reciprocity of heat diffusion through a two-port system, which has a different setup from that of a thermal diode. We further prove the equivalence between a heat transfer system with broken steady-state global reciprocity and a thermal diode, assuming no temperature-dependent heat generation. The validities of some typical mechanisms in breaking the diffusive reciprocity and making a thermal diode have been discussed. Our results set a general background for future studies on symmetric and asymmetric diffusive processes.

Figure 1

Local reciprocity in heat transfer.

Figure 2

Global reciprocity of heat transfer through a two-port system.

Figure 3

Thermal diode.

Figure 4

Effects of nonlinearity. (a) Nonlinear materials 1 and 2 with temperature dependent thermal conductivities ${\kappa }_1\left(T\right)$ and ${\kappa }_2\left(T\right)$. (b) Steady-state temperature profiles for a system built of material 1 with asymmetric geometry. The forward $(\mathrm{\Delta }T=25\mathrm{K})$ and backward $(\mathrm{\Delta }T=-25\mathrm{K})$ cases give slightly different amounts of heat. (c) Steady-state temperature profiles for a system built of materials 1 and 2. The forward $(\mathrm{\Delta }T=25\mathrm{K})$ and backward $(\mathrm{\Delta }T=-25\mathrm{K})$ cases give different amounts of heat. (d) Heat (per unit thickness) across the system.

Figure 5

Effects of asymmetric thermal conductivity tensor: local reciprocity. (a) Steady-state temperature profiles generated by a point heat source with all boundaries at fixed temperature. The reciprocity is preserved. (b) Steady-state temperature profiles generated by a point heat source with the left side at fixed temperature and the other three boundaries thermally insulated. The reciprocity is broken. (c) Temperature profiles at $t=9120\mathrm{s}$ generated by a time-harmonic point heat source with all boundaries thermally insulated. The reciprocity is broken. Boundaries with fixed temperatures are indicated with purple lines. The target positions where temperatures are measured are indicated by orange dots.

Figure 6

Effects of asymmetric thermal conductivity tensor: global reciprocity and diode effect. (a) Temperature profiles at $t=9110\mathrm{s}$ generated by a time-harmonic point heat source with all boundaries thermally insulated. The global reciprocity is broken. (b) As the oscillating frequency ω of the heat source approaches zero, the ratio between the two amplitudes approaches one, indicating a preserved global reciprocity at steady state. (c) Steady-state temperature profiles with the left and lower boundaries at fixed temperatures. The forward $(\mathrm{\Delta }T=25\mathrm{K})$ and backward $(\mathrm{\Delta }T=-25\mathrm{K})$ cases are symmetric, so the system is not a thermal diode. (d) Heat (per unit thickness) across the system. The target positions where temperatures are measured are indicated by orange dots.

Figure 7

Effects of heat convection. (a) Temperature profiles on a rotating ring (with rotation speed Ω) at $t=9110\mathrm{s}$ generated by a time-harmonic point heat source (at angular frequency ω) with all boundaries thermally insulated. The local reciprocity is broken. (b) Temperature profiles on a rotating ring attached with two channels at 9080 s generated by a time-harmonic point heat source with all boundaries thermally insulated. The global reciprocity is broken. (c) Steady-state temperature profiles with the left and lower boundaries at fixed temperatures. The forward $(\mathrm{\Delta }T=25\mathrm{K})$ and backward $(\mathrm{\Delta }T=-25\mathrm{K})$ cases are symmetric, so the system is not a thermal diode. The target positions where temperatures are measured are indicated by orange dots. (d) Heat (per unit thickness) across the system at different rotation speeds Ω.

Figure 8

Effects of temperature-dependent heat source. (a) Temperature profiles on a two-port system with an interior hole at fixed temperature generated by a point heat source with all boundaries thermally insulated. The global reciprocity is preserved. (b) Steady-state temperature profiles with the left and right boundaries at fixed temperatures. The forward $(\mathrm{\Delta }T=25\mathrm{K})$ and backward $(\mathrm{\Delta }T=-25\mathrm{K})$ cases are asymmetric. (c) Heat (per unit thickness) across the system. Boundaries with fixed temperatures are indicated with purple lines. The target positions where temperatures are measured are indicated by orange dots.