Theory for Negative Thermal Transport: Complex Thermal Conductivity

Abstract

In this chapter, we coin a complex thermal conductivity whose imaginary part corresponds to the real part of a complex refractive index. Therefore, the thermal counterpart of a negative refractive index is just a negative imaginary thermal conductivity, featuring the opposite directions of energy flow and wave vector in thermal conduction and advection, thus called negative thermal transport herein. We design an open system with energy exchange and explore three different cases to reveal negative thermal transport to avoid violating causality. We further provide experimental suggestions with a solid ring structure. All finite-element simulations agree with the theoretical analyses, indicating that negative thermal transport is physically feasible. These results have potential applications such as designing the inverse Doppler effect in thermal conduction and advection.

1 Opening Remarks

Negative refraction is one of the most attractive phenomena in wave systems, which was first revealed with negative permeability and permittivity [1]. With the proposal of metal wire arrays [2] and split rings [3], negative refractive index was soon designed and fabricated [4,5,6], which gave birth to broad applications like breaking diffraction limit [7,8,9]. One representative property of a negative refractive index is the opposite directions of energy flow (or Poynting vector) and wave vector [Fig. 19.1(a)]. Based on this property, related phenomena were revealed intensively, such as the inverse Doppler effect [10], the inverse Cerenkov radiation [11], and the abnormal Goos-Hanschen shift [12].

Refractive phenomena were also studied with a thermal wave with a time-periodic heat source [13]. Moreover, multilayered structures were also proposed to guide heat flow [14,15,16,17], yielding practical applications such as thermal lens [18] and thermal cloaks [19]. These studies attempted to connect thermal phenomena and electromagnetic phenomena. However, some basic concepts are still ambiguous, especially the correspondence between thermal conductivity and refractive index.

Fig. 19.1

Adapted from Ref. [20]

Comparison between a wave system and b diffusion system. n and $\kappa$ denote complex refractive index and complex thermal conductivity, respectively. $\boldsymbol{J}$ and $\boldsymbol{\beta }$ denote energy flow and wave vector, respectively.

To promote the related physics in thermotics with a clear physical picture, we manage to coin a complex thermal conductivity $\kappa$ as the thermal counterpart of a complex refractive index n (Fig. 19.1). The imaginary part of a complex thermal conductivity is analogous to the real part of a complex refractive index. Therefore, the thermal counterpart of a negative refractive index is just a negative imaginary thermal conductivity, which is characterized by the opposite directions of energy flux $\boldsymbol{J}$ and wave vector $\boldsymbol{\beta }$, thus called negative thermal transport. We design an open system with energy exchange to observe negative thermal transport and provide experimental suggestions with a three-dimensional solid ring structure. All theoretical analyses and finite-element simulations indicate that negative thermal transport is physical.

2 Complex Thermal Conductivity

Thermal conduction-advection process is dominated by the famous equation

$$\begin{aligned} \rho C \partial T/\partial t+\boldsymbol{\nabla }\cdot \left( -\sigma \boldsymbol{\nabla }T+\rho C \boldsymbol{v}T \right) =0, \end{aligned}$$

(19.1)

where $\rho$, C, $\sigma$, $\boldsymbol{v}$, T, and t are density, heat capacity, thermal conductivity, convective velocity, temperature, and time, respectively. Equation (19.1) indicates the energy conservation of thermal conduction and advection. We assume that the convective term ($\rho C \boldsymbol{v}T$) results from the motion of solid, so density and heat capacity can be seen as two constants which do not depend on time or temperature [21, 22]. Therefore, the mass and momentum conservations of thermal advection are naturally satisfied.

To proceed, we apply a plane-wave solution for temperature [21,22,23],

$$\begin{aligned} T=A_0\textrm{e}^{\textrm{i}\left( \boldsymbol{\beta }\cdot \boldsymbol{r}-\omega t\right) }+T_0, \end{aligned}$$

(19.2)

where $A_0$, $\boldsymbol{\beta }$, $\boldsymbol{r}$, $\omega$, and $T_0$ are the amplitude, wave vector, position vector, frequency, and reference temperature of wave-like temperature profile, respectively. $\textrm{i}=\sqrt{-1}$ is imaginary unit. Only the real part of Eq. (19.2) makes sense. We substitute Eq. (19.2) into Eq. (19.1), and derive a dispersion relation,

$$\begin{aligned} \omega =\boldsymbol{v}\cdot \boldsymbol{\beta }-\textrm{i}\frac{\sigma \beta ^2}{\rho C}. \end{aligned}$$

(19.3)

With the wave-like temperature profile described by Eq. (19.2), we can derive $\boldsymbol{\nabla }T=\textrm{i}\boldsymbol{\beta }T$ ($T_0$ is neglected for brevity). Then, Eq. (19.1) can be rewritten as

$$\begin{aligned} \rho C \partial T/\partial t+\boldsymbol{\nabla }\cdot \left( -\textrm{i}\sigma \boldsymbol{\beta }T+\rho C \boldsymbol{v}T \right) =0. \end{aligned}$$

(19.4)

With the mass conservation of thermal advection, we can obtain $\boldsymbol{\nabla }\cdot \left( \rho \boldsymbol{v}\right) =0$ or $\boldsymbol{\nabla }\cdot \boldsymbol{v}=0$ (for $\rho$ is a constant). Meanwhile, $\boldsymbol{\beta }$ is a constant vector, so Eq. (19.4) can be reduced to

$$\begin{aligned} \rho C \partial T/\partial t-\textrm{i}\sigma \boldsymbol{\beta }\cdot \boldsymbol{\nabla }T+\rho C \boldsymbol{v}\cdot \boldsymbol{\nabla }T=0, \end{aligned}$$

(19.5)

which can be further reduced with $\boldsymbol{\nabla }T=\textrm{i}\boldsymbol{\beta }T$ to

$$\begin{aligned} \rho C \partial T/\partial t+\sigma \boldsymbol{\beta }^2T+\textrm{i}\rho C \boldsymbol{v}\cdot \boldsymbol{\beta }T=0. \end{aligned}$$

(19.6)

Now, it is natural to coin a complex thermal conductivity $\kappa$ as

$$\begin{aligned} \kappa =\sigma +\textrm{i}\frac{\rho C \boldsymbol{v}\cdot \boldsymbol{\beta }}{\beta ^2}, \end{aligned}$$

(19.7)

with which Eq. (19.6) can be simplified as

$$\begin{aligned} \rho C \partial T/\partial t+\kappa \boldsymbol{\beta }^2T=0. \end{aligned}$$

(19.8)

With $\boldsymbol{\nabla }T=\textrm{i}\boldsymbol{\beta }T$, Eq. (19.8) is equivalent to the familiar equation of thermal conduction,

$$\begin{aligned} \rho C \partial T/\partial t+\boldsymbol{\nabla }\cdot \left( -\kappa \boldsymbol{\nabla }T\right) =0. \end{aligned}$$

(19.9)

Clearly, the thermal conduction-advection equation (Eq. (19.1)) is converted to the complex thermal conduction equation (Eq. (19.9)) with a complex thermal conductivity (Eq. (19.7)). Although thermal conductivity is generally defined by fixing moving parts (advection) as zero, advection can be mathematically regarded as a complex form of conduction. Conduction and advection are mathematically unified within a conductive framework (despite different physical mechanisms). Therefore, coining complex thermal conductivity makes sense by treating advection as a complex form of conduction.

With the substitution of Eq. (19.2) into Eq. (19.9), we can derive a dispersion relation,

$$\begin{aligned} \omega ={-\mathrm i}\frac{\kappa \beta ^2}{\rho C}=\boldsymbol{v}\cdot \boldsymbol{\beta }-\textrm{i}\frac{\sigma \beta ^2}{\rho C}, \end{aligned}$$

(19.10)

which is in accordance with the result of Eq. (19.3), indicating that complex thermal conductivity is physical.

To understand the complex frequency $\omega$, we substitute Eq. (19.10) into Eq. (19.2), and the wave-like temperature profile becomes

$$\begin{aligned} T=A_0\textrm{e}^{\textrm{Im}\left( \omega \right) t}\textrm{e}^{\textrm{i}\left[ \boldsymbol{\beta }\cdot \boldsymbol{r}-\textrm{Re}\left( \omega \right) t\right] }+T_0. \end{aligned}$$

(19.11)

Obviously, $\textrm{Re}(\omega )$ and $\textrm{Im}(\omega )$ determine propagation and dissipation, respectively. Meanwhile, $\textrm{Re}(\omega )$ and $\textrm{Im}(\omega )$ are related to $\textrm{Im}\left( \kappa \right)$ and $\textrm{Re}\left( \kappa \right)$, respectively. In other words, $\textrm{Re}\left( \kappa \right)$ and $\textrm{Im}\left( \kappa \right)$ are related to dissipation and propagation, respectively. The physical connotation can be clearly understood with Fig. 19.1b. Positive (or negative) $\textrm{Re}\left( \kappa \right)$ means loss (or gain), indicating the amplitude decrement (or increment) of wave-like temperature profile. $\textrm{Im}\left( \kappa \right)$ is of our interest, which is discussed later.

We further confirm complex thermal conductivity in a thermal conduction-advection system with COMSOL Multiphysics. The system is shown in Fig. 19.2a, which has width L and height H. The left and right ends are set with a periodic boundary condition. Then, the wave vector can take on only discrete values, say, $\beta =2\pi m/L$ with m being any positive integers. We take on $m=5$, and initial temperature is set at $T=40\cos \left( \beta x\right) +323$ K (Fig. 19.2b, f).

Fig. 19.2

Adapted from Ref. [20]

a Schematic diagram. Temperature evolutions with b–e $\boldsymbol{v}//\boldsymbol{\beta }$ and f–i $\boldsymbol{v}\perp \boldsymbol{\beta }$. Parameters: $L=0.5$ m, $H=0.25$ m, $\rho C=10^6$ J m$^{-3}$ K$^{-1}$, $\sigma =1$ W m$^{-1}$ K$^{-1}$, and $v=1$ mm/s. PBC in (a) means periodic boundary condition.

We discuss two cases with $\boldsymbol{v}//\boldsymbol{\beta }$ (Fig. 19.2b–e) and $\boldsymbol{v}\perp \boldsymbol{\beta }$ (Fig. 19.2f–i). When $\boldsymbol{v}//\boldsymbol{\beta }$, $\textrm{Im}\left( \kappa \right)$ appears due to $\boldsymbol{v}\cdot \boldsymbol{\beta }\ne 0$, as predicted by Eq. (19.7). Therefore, propagation occurs and the period of wave-like temperature profile is $t_0=2\pi /\textrm{Re}\left( \omega \right) =2\pi /\left( \boldsymbol{v}\cdot \boldsymbol{\beta }\right) =100$ s, as predicted by Eq. (19.10). The wave-like temperature profiles at $t=0.5t_0=50$ s and $t=t_0=100$ s are shown in Fig. 19.2c, d, respectively. The temperature distributions along x axis in Fig. 19.2b–d are plotted in Fig. 19.2e. Clearly, the wave-like temperature profile has amplitude decrement because of the positive $\textrm{Re}\left( \kappa \right)$. Meanwhile, the wave-like temperature profile propagates along x axis due to the positive $\textrm{Im}\left( \kappa \right)$. After propagating for a period (100 s), the wave-like temperature profile approximately gains a phase difference of $2\pi$, thus going back to the initial position (Fig. 19.2b, d).

When $\boldsymbol{v}\perp \boldsymbol{\beta }$, $\textrm{Im}\left( \kappa \right)$ vanishes due to $\boldsymbol{v}\cdot \boldsymbol{\beta }=0$. Therefore, propagation does not occur, and the period is $t_0=2\pi /\left( \boldsymbol{v}\cdot \boldsymbol{\beta }\right) =\infty$ s, as predicted by Eq. (19.10). The wave-like temperature profiles at $t=50$ s and $t=100$ s are presented in Fig. 19.2g, h, respectively. The temperature distributions along x axis in Fig. 19.2f–h are plotted in Fig. 19.2i. The wave-like temperature profile has also amplitude decrement due to the positive $\textrm{Re}\left( \kappa \right)$. However, the wave-like temperature profile does not propagate because of the zero $\textrm{Im}\left( \kappa \right)$. Therefore, the behaviors of thermal conduction and advection can be well described by using complex thermal conductivity. When wave vector and convective velocity are with an arbitrary angle $\alpha$, the velocity component $v\cos \alpha$ contributes to propagation.

We further discuss the energy flow in Fig. 19.2b–d. Relative energy flow $\boldsymbol{J}'$ can be calculated with periodicity average,

$$\begin{aligned} \boldsymbol{J}'=\frac{1}{\lambda }\int \limits _0^\lambda \left( -\kappa \boldsymbol{\nabla }T\right) dx=0, \end{aligned}$$

(19.12)

where $\lambda =2\pi /\beta$ is wavelength. Here, we only take on the real part of $\boldsymbol{J}'$ because the imaginary part does not make sense. Although conductive flow is irrelevant to reference temperature, convective flow ($\rho C \boldsymbol{v}T$) is closely associated with reference temperature [24,25,26,27,28]. Therefore, absolute energy flow $\boldsymbol{J}$ is

$$\begin{aligned} \boldsymbol{J}=\rho C \boldsymbol{v} T_0. \end{aligned}$$

(19.13)

In what follows, we discuss absolute energy flow and neglect the expression of absolute for brevity. Clearly, $\boldsymbol{J}$ and $\boldsymbol{v}$ have the same direction. In other words, only thermal advection contributes to energy flow. As we can imagine from Fig. 19.2b–d, thermal advection results in the motion of wave-like temperature profile, so the direction of wave vector $\boldsymbol{\beta }$ follows that of convective velocity $\boldsymbol{v}$, yielding positive thermal transport ($\textrm{Im}\left( \kappa \right) >0$; see Fig. 19.1b). To some extent, positive thermal transport is the result of causality, so negative thermal transport ($\textrm{Im}\left( \kappa \right) <0$; see Fig. 19.1b) is unique.

Fig. 19.3

Adapted from Ref. [20]

Two-dimensional negative thermal transport. a Schematic diagram with $a_u=a_d=2$ mm, $a_i=1$ mm, $L=0.5$ m, $\sigma _u=\sigma _d=10$ W m$^{-1}$ K$^{-1}$, $\sigma _i=0.1$ W m$^{-1}$ K$^{-1}$, and $\rho _u C_u=\rho _d C_d=\rho _i C_i=10^6$ J m$^{-3}$ K$^{-1}$. These parameters lead to $g/\beta =4$ mm/s. b $v_u=-v_d=5$ mm/s. c $v_u=0.5$ mm/s and $v_d=-1.5$ mm/s. d $v_u=-v_d=1$ mm/s. Circles and stars denote the trajectories of Max($T_u$) and Max($T_d$), respectively.

3 Negative Thermal Transport

It might be very difficult to reveal negative thermal transport in an isolated system (Fig. 19.2a), so we consider an open system (Fig. 19.3a) where an intermediate layer allows heat exchange between up and down layers. The complex thermal conductivities of up layer $\kappa _u$ and down layer $\kappa _d$ are denoted as

$$\begin{aligned} \kappa _u&=\sigma _u+\textrm{i}\frac{\rho _u C_u \boldsymbol{v}_u\cdot \boldsymbol{\beta }_u}{\beta _u^2},\end{aligned}$$

(19.14a)

$$\begin{aligned} \kappa _d&=\sigma _d+\textrm{i}\frac{\rho _d C_d \boldsymbol{v}_d\cdot \boldsymbol{\beta }_d}{\beta _d^2}, \end{aligned}$$

(19.14b)

where the subscripts u and d denote the parameters in up and down layers, respectively. We set the wave-like temperature profiles in up layer $T_u$ and down layer $T_d$ as

$$\begin{aligned} T_u&=A_u\textrm{e}^{\textrm{i}\left( \boldsymbol{\beta }_u\cdot \boldsymbol{x}-\omega t\right) }+T_0,\end{aligned}$$

(19.15a)

$$\begin{aligned} T_d&=A_d\textrm{e}^{\textrm{i}\left( \boldsymbol{\beta }_d\cdot \boldsymbol{x}-\omega t\right) }+T_0. \end{aligned}$$

(19.15b)

The heat exchange between up and down layers is along z axis, which is not of our concern. Then, the energy flows along x axis in up layer $\boldsymbol{J}_u$ and down layer $\boldsymbol{J}_d$ can be calculated as

$$\begin{aligned} \boldsymbol{J}_u&=\rho _u C_u \boldsymbol{v}_u T_0,\end{aligned}$$

(19.16a)

$$\begin{aligned} \boldsymbol{J}_d&=\rho _d C_d \boldsymbol{v}_d T_0. \end{aligned}$$

(19.16b)

Clearly, the directions of energy flow in up and down layers are opposite due to $\boldsymbol{v}_u=-\boldsymbol{v}_d$.

The thermal conduction-advection processes in up and down layers can be described by the complex thermal conduction equation,

$$\begin{aligned} \rho _u C_u\partial T_u/\partial t+\boldsymbol{\nabla }\cdot \left( -\kappa _u\boldsymbol{\nabla }T_u\right)&=s_u,\end{aligned}$$

(19.17a)

$$\begin{aligned} \rho _d C_d\partial T_d/\partial t+\boldsymbol{\nabla }\cdot \left( -\kappa _d\boldsymbol{\nabla }T_d\right)&=s_d, \end{aligned}$$

(19.17b)

where $s_u$ and $s_d$ are two heat sources, reflecting the heat exchange between up and down layers [21, 22]. Since the three layers in Fig. 19.3a are thin enough ($L\gg a_{u,\,i,\,d}$), the temperature variance along z axis can be neglected, yielding $\partial ^2 T/\partial z^2=0$. The energy flow from down layer to up layer $j_{u}$ can be calculated as $j_u=-\sigma _i\left( T_u-T_d\right) /a_i$, where $\sigma _i$ and $a_i$ are the thermal conductivity and width of stationary intermediate layer, respectively. It is also reasonable to suppose that energy flow is uniformly distributed in up layer due to thin thickness, so the heat source in up layer $s_u$ can be expressed as $s_u=j_u/a_u=-\sigma _i\left( T_u-T_d\right) /\left( a_ia_u\right)$, where $a_u$ is the width of up layer. Similarly, the heat source in down layer $s_d$ can be derived as $s_d=j_d/a_d=-\sigma _i\left( T_d-T_u\right) /\left( a_ia_d\right)$, where $a_d$ is the width of down layer. With these analyses, Eq. (19.17) can be reduced to

$$\begin{aligned} \rho _u C_u\partial T_u/\partial t-\kappa _u\partial ^2T_u/\partial x^2&=h_u\left( T_d-T_u\right) ,\end{aligned}$$

(19.18a)

$$\begin{aligned} \rho _d C_d\partial T_d/\partial t-\kappa _d\partial ^2T_d/\partial x^2&=h_d\left( T_u-T_d\right) , \end{aligned}$$

(19.18b)

where $h_u=\sigma _i/\left( a_ia_u\right)$ and $h_d=\sigma _i/\left( a_ia_d\right)$, reflecting the exchange rate of heat energy. We take on the same material parameters of up and down layers, say, $\sigma _u=\sigma _d\left( =\sigma \right)$, $\rho _u C_u=\rho _d C_d\left( =\rho C\right)$, $a_u=a_d\left( =a\right)$, and $h_u=h_d\left( =h\right)$. We also suppose $\boldsymbol{v}_u=-\boldsymbol{v}_d\left( =\boldsymbol{v}\right)$ and $\boldsymbol{\beta }_u=\boldsymbol{\beta }_d\left( =\boldsymbol{\beta }\right)$, thus yielding $\kappa _u=\overline{\kappa }_d\left( =\kappa \right)$ where $\overline{\kappa }_d$ is the conjugate of $\kappa _d$. The substitution of Eq. (19.15) into Eq. (19.18) yields an eigenequation $\boldsymbol{\hat{H}}\boldsymbol{\psi }=\omega \boldsymbol{\psi }$, where the Hamiltonian $\boldsymbol{\hat{H}}$ reads

$$\begin{aligned} \boldsymbol{\hat{H}}= \left[ \begin{matrix} -\textrm{i}\left( g+\eta \beta ^2\right) &{} \textrm{i}g\\ \textrm{i}g &{} -\textrm{i}\left( g+\overline{\eta }\beta ^2\right) \end{matrix} \right] , \end{aligned}$$

(19.19)

where $\eta =\kappa /\left( \rho C\right)$ and $g=h/\left( \rho C\right)$. The eigenvalue of Eq. (19.19) is

$$\begin{aligned} \omega _\pm =-\textrm{i}\left[ g+\textrm{Re}\left( \eta \right) \beta ^2\pm \sqrt{g^2-\mathrm{Im^2}\left( \eta \right) \beta ^4}\right] , \end{aligned}$$

(19.20)

where $\textrm{Re}\left( \eta \right) =\sigma /\left( \rho C\right)$ and $\textrm{Im}\left( \eta \right) \beta ^2=v\beta$.

With Eq. (19.20), we can obtain three different cases of negative thermal transport. We discuss the first case with $g^2-\mathrm{Im^2}\left( \eta \right) \beta ^4<0$, say, $v>g/\beta$. The eigenvector is

$$\begin{aligned} \boldsymbol{\psi }_+&=\left[ 1,\,\textrm{e}^{\textrm{i} \pi /2-\delta }\right] ^{\varsigma },\end{aligned}$$

(19.21a)

$$\begin{aligned} \boldsymbol{\psi }_-&=\left[ 1,\,\textrm{e}^{\textrm{i} \pi /2+\delta }\right] ^{\varsigma }, \end{aligned}$$

(19.21b)

where $\delta =\cosh ^{-1}\left[ \textrm{Im}\left( \eta \right) \beta ^2/g\right]$, and $\varsigma$ denotes transpose. The eigenvectors in Eq. (19.21) indicate that the wave-like temperature profiles in up and down layers move with a constant phase difference of $\pi /2$ but with different amplitudes. We take on $\beta =2\pi m/L$ with $m=1$ in what follows. The initial wave-like temperature profiles in up and down layers are set as the eigenvector described by Eq. (19.21b), say, $T_u=40\cos \left( \beta x\right) +323$ K and $T_d=\textrm{e}^{\delta }40\cos \left( \beta x+\pi /2\right) +323$ K. We track the motion of maximum temperature in up layer Max($T_u$) and down layer Max($T_d$) to observe the directions of wave vector. Since the amplitude of wave-like temperature profile in down layer (with $\textrm{e}^{\delta }>1$) is larger than that in up layer, the directions of wave vector in up and down layers are both leftward. Therefore, negative thermal transport occurs in up layer, and the transport in down layer is still positive (Fig. 19.3b).

We discuss the second case with $g^2-\mathrm{Im^2}\left( \eta \right) \beta ^4>0$, say, $0<v<g/\beta$. The corresponding eigenvector is

$$\begin{aligned} \boldsymbol{\psi }_+&=\left[ 1,\,\textrm{e}^{\textrm{i} \left( \pi -\alpha \right) }\right] ^{\varsigma },\end{aligned}$$

(19.22a)

$$\begin{aligned} \boldsymbol{\psi }_-&=\left[ 1,\,\textrm{e}^{\textrm{i} \alpha }\right] ^{\varsigma }, \end{aligned}$$

(19.22b)

where $\alpha =\sin ^{-1}\left[ \textrm{Im}\left( \eta \right) \beta ^2/g\right]$. The eigenvectors in Eq. (19.22) indicate that the wave-like temperature profiles in up and down layers are motionless with a constant phase difference of $\pi -\alpha$ or $\alpha$. To make the wave-like temperature profiles move, we give the system a reference velocity $\boldsymbol{v}_0$, resulting in $\boldsymbol{v}_u=\boldsymbol{v}_u'+\boldsymbol{v}_0$ and $\boldsymbol{v}_d=\boldsymbol{v}_d'+\boldsymbol{v}_0$, where $\boldsymbol{v}_u'$ and $\boldsymbol{v}_d'$ are original convective velocities. This operation does not affect the essence of eigenvectors in Eq. (19.22), and only gives a reference velocity $\boldsymbol{v}_0$ to wave-like temperature profiles. We set the initial wave-like temperature profiles in up and down layers to be the eigenvector described by Eq. (19.22b), say, $T_u=40\cos \left( \beta x\right) +323$ K and $T_d=40\cos \left( \beta x+\alpha \right) +323$ K. We also take on $\boldsymbol{v}_0=-0.5\boldsymbol{v}_u'$, so the wave-like temperature profiles in up and down layers still maintain a constant phase difference of $\alpha$ but with leftward motion. The trajectories of Max($T_u$) and Max($T_d$) are presented in Fig. 19.3c. Clearly, negative thermal transport occurs in up layer.

These two cases are related to eigenvectors, indicating that negative thermal transport occurs in one layer (say, up layer). However, thermal transport is still positive if we regard up and down layers as a whole. We further explore the third case, related to non-eigenvectors and their dynamics, to reveal negative thermal transport in up and down layers. For this purpose, we set the initial wave-like temperature profiles by adding the eigenvector described by Eq. (19.22b) with an extra phase $\gamma$, say, $\boldsymbol{\psi }_-'=\left[ 1,\,\textrm{e}^{\textrm{i} \left( \alpha +\gamma \right) }\right] ^{\varsigma }$, yielding $T_u=40\cos \left( \beta x\right) +323$ K and $T_d=40\cos \left( \beta x+\alpha +\gamma \right) +323$ K. In this way, even if we do not give a reference velocity to the system, the wave-like temperature profile still moves to reach the eigenvector. One principle of the evolution route is to make the temperature profile decay as slowly as possible. Therefore, the eigenvector $\boldsymbol{\psi }_+$ with a phase difference of $\pi -\alpha$ described by Eq. (19.22a) becomes a key due to its large decay rate (say, the $\omega _+$ of Eq. (19.20)). The evolution route should try to avoid $\boldsymbol{\psi }_+$ to survive longer. When $\gamma \in \left( 0,\,\pi -2\alpha \right)$, negative thermal transport will not make the temperature profile go through $\boldsymbol{\psi }_+$, but positive thermal transport will make the temperature profile go through $\boldsymbol{\psi }_+$ twice. Therefore, the evolution route naturally chooses negative thermal transport in both up and down layers to decay more slowly (Fig. 19.3d). Nevertheless, the wave-like temperature profile remains motionless after reaching the eigenvector, so negative thermal transport is no longer present. In other words, negative thermal transport in both up and down layers is transient.

4 Experimental Suggestion

We also suggest experimental demonstration with a three-dimensional solid ring structure (Fig. 19.4a), which can naturally meet the requirement of periodic boundary conditions. Up ring (with width $a_u$) and down ring (with width $a_d$) rotate with opposite angular velocities ($\Omega _u$ and $\Omega _d$), which are connected by a stationary intermediate layer (with width $a_i$). The inner and outer radii of the ring structure are $r_1$ and $r_2$, respectively. Like two dimensions, we track Max($T_u$) and Max($T_d$) on the interior edge of the solid ring structure. The parametric settings for Fig. 19.4b–d are basically the same as those for Fig. 19.3b–d, respectively. Therefore, the results are also similar. Negative thermal transport occurs in the up ring of Fig. 19.4b, c and occurs in both up and down rings of Fig. 19.4d. Three-dimensional and two-dimensional results agree well with theoretical analyses, confirming the feasibility of negative thermal transport in thermal conduction and advection.

Fig. 19.4

Adapted from Ref. [20]

Experimental suggestions with a three-dimensional solid ring structure. a Schematic diagram with $r_1=80$ mm, $r_2=82$ mm. Other parameters are kept the same as those for Fig. 19.3a. b $\Omega _u=-\Omega _d=0.063$ rad/s. c $\Omega _u=0.006$ rad/s and $\Omega _d=-0.019$ rad/s. d $\Omega _u=-\Omega _d=0.013$ rad/s.

Negative thermal transport may enlighten the inverse Doppler effect in thermal conduction and advection. Since energy flow is generated from the energy source, a detector with the opposite direction of energy flow is getting close to the energy source. Positive thermal transport makes wave vector (regarded as a thermal signal) follow the direction of energy flow. Therefore, the detector and wave vector directions are opposite, yielding frequency increment (the Doppler effect). However, negative thermal transport leads to the same detector and wave vector directions. As a result, the frequency decreases even though the detector gets close to the energy source (the inverse Doppler effect). These results may also provide guidance to extend transformation thermotics [29,30,31] to complex regime [32] and regulate thermal imaging [33,34,35,36,37,38,39]. Other thermal systems, such as those with periodic structures [40,41,42,43], are also good candidates to explore negative thermal transport. Nevertheless, here we reveal negative thermal transport in an open system with energy exchange, so there is a difference from wave systems where no energy exchange is required to realize negative refraction. Whether negative thermal transport can exist in an isolated system remains studied.

5 Conclusion

In summary, we have established the thermal counterpart of a complex refractive index by coining a complex thermal conductivity. As a result, a negative imaginary thermal conductivity is just the thermal counterpart of a negative refractive index, featuring the opposite directions of energy flow and wave vector in thermal conduction and advection. Negative thermal transport seems to violate causality, but it can occur in an open system with heat exchange. We further reveal negative thermal transport in three cases and provide three-dimensional experimental suggestions, confirming its physical feasibility. These results provide a different perspective to cognize conduction and advection. They may enlighten outspread explorations of negative thermal transport, such as the inverse Doppler effect in thermal conduction and advection.

6 Exercise and Solution

Exercise

• 1. Explain the left half in Fig. 19.1b.

Solution

• 1. In the left half, the real part of a complex thermal conductivity is negative, so the temperature field amplitude increases. This effect does not occur naturally because the second law of thermodynamics is violated. However, it may happen with external energy sources.