Thermal-rectification coefficients in solid-state thermal rectifiers

The thermal rectifier is an analog of the electrical rectifier, in which heat flux in a forward direction is larger than that in the reverse direction. Owing to the controllability of the heat flux, the solid-state thermal rectifier is promising from both theoretical and applicational points of view. In this paper, we examine analytical expressions of thermal-rectification coefficients $R$ for thermal rectifiers with typical linear and nonlinear model functions as nonuniform thermal conductivities against temperature $T$. For the thermal rectifier with linear (quadratic) temperature-dependent thermal conductivity, a maximum value of $R$ is calculated to be 3 ($\simeq 14$). With use of a structural-phase-transition material, a maximum value of $R$ is found to ideally reach to ${\kappa }_2/{\kappa }_1$, where ${\kappa }_1$ (${\kappa }_2$) is the minimum (maximum) value of its $\kappa \left(T\right)$. Values of $R$ for the thermal rectifiers with an inverse $T$-dependent function and an exponential function of $\kappa$ are also analytically examined.

Figure 1

(a) Schematic figure of a solid-state thermal rectifier consisting of juxtaposing material A and material B with nonuniform thermal conductivities $\kappa$ against position $x$ and temperature $T$. For the forward (f) direction, the edge (at $x=0$) of material A is contacted with a heat bath with high temperature $T_{\mathrm{H}}$, and the heat flux $J_{\mathrm{q}}^{\mathrm{f}}$ flows to a heat bath with low temperature $T_{\mathrm{L}}$ at $x=l$, where an edge of the material B is contacted. The temperature at the junction of materials A and B is defined as $T_{\mathrm{M}}^{\mathrm{f}}$ for the forward direction. For the reverse (r) direction, positions of materials A and B are exchanged. The heat flux $J_{\mathrm{q}}^{\mathrm{r}}$ flows to an edge of material A with $T_{\mathrm{L}}$ through the junction at temperature $T_{\mathrm{M}}^{\mathrm{r}}$. (b) Linear $T$-dependent $\kappa$ of material A (${\kappa }_{\mathrm{A}}$) and material B (${\kappa }_{\mathrm{B}}$). $T_0$ is the temperature at the intersection point of ${\kappa }_{\mathrm{A}}\left(T\right)$ and ${\kappa }_{\mathrm{B}}\left(T\right)$.

Figure 2

(a) Linear $T$-dependent ${\kappa }_{\mathrm{A}}$ and ${\kappa }_{\mathrm{B}}$. $T_0$ is the temperature at the intersection point of ${\kappa }_{\mathrm{A}}$ and ${\kappa }_{\mathrm{B}}$. (b) Thermal rectification coefficient $R$ as a function of $\frac{\mathrm{\Delta }T}{T_0}$.

Figure 3

(a) Quadratic $T$-dependent ${\kappa }_{\mathrm{A}}$ and ${\kappa }_{\mathrm{B}}$. (b) $R$ as a function of $\frac{\mathrm{\Delta }T}{T_0}$.

Figure 4

(a) Quadratic $T$-dependent ${\kappa }_{\mathrm{A}}$ and ${\kappa }_{\mathrm{B}}$. (b) $R$ as a function of $\frac{T_{\mathrm{A}}}{2\mathrm{\Delta }T}$.

Figure 5

(a) Inverse $T$-dependent ${\kappa }_{\mathrm{A}}$ and ${\kappa }_{\mathrm{B}}$. (b) $R$ as a function of $\frac{\mathrm{\Delta }T}{T_0}$.

Figure 6

(a) Exponentially $T$-dependent ${\kappa }_{\mathrm{A}}$ and ${\kappa }_{\mathrm{B}}$. (b) $R$ as a function of $\mathrm{\Delta }T$.

Figure 7

(a) Step-function-like $T$-dependent ${\kappa }_{\mathrm{A}}$ and ${\kappa }_{\mathrm{B}}$. (b) $R$ as a function of $\frac{\mathrm{\Delta }T}{T_0}$.