Theory for Chameleonlike Thermal Rotators: Extremely Anisotropic Conductivity

Abstract

In this chapter, we propose a mechanism for intelligent thermal regulation based on transformation-invariant metamaterials, which possess highly anisotropic thermal conductivities. As an application, we design intelligent thermal rotators that can guide heat flux direction with different environmental parameters. Since the adaptive behavior is similar to chameleons, the present rotators are called chameleonlike rotators. We further perform finite-element simulations and laboratory experiments to validate the scheme and demonstrate the chameleonlike behavior. These results have potential applications for implementing adaptive and adjustable thermal metamaterials. Similar behaviors can also be expected in other fields, such as hydrodynamics.

1 Opening Remarks

Transformation thermotics [1, 2] provides a fundamental and powerful method to control heat flux at will. Initial explorations mainly focused on thermal conduction, and many functions were proposed, such as cloaking, concentrating, and rotating [3]. For the sake of practical applications, convection [4,5,6] and radiation [7, 8] have also been considered to develop corresponding transformation theories.

Although transformation-thermotics-based metamaterials have achieved great success, the lack of intelligence remains a problem. Specifically, the key equation of transformation thermotics is $\boldsymbol{\kappa }'={\textbf{J}}\boldsymbol{\kappa }{\textbf{J}}^\tau /\det {\textbf{J}}$, where $\boldsymbol{\kappa }'$ is transformed thermal conductivity, $\boldsymbol{\kappa }$ is environmental thermal conductivity, ${\textbf{J}}$ is the Jacobian matrix, and $\tau$ denotes transpose [1, 2]. The transformed parameter $\left( \boldsymbol{\kappa }'\right)$ is crucially dependent on the environmental parameter $\left( \boldsymbol{\kappa }\right)$. In other words, once the environmental parameter changes, the transformed parameter should change accordingly, making the original design fail in the new environment. This limitation is fatal because one device applies to only one environment. Similar problems also exist in the fields of electromagnetism and thermal radiation, and some inspiring studies [9, 10] gave insights.

To improve intelligence, we propose a mechanism based on thermal transformation-invariant metamaterials [11, 12], whose thermal conductivities are highly anisotropic [13,14,15], i.e., 0 W m$^{-1}$ K$^{-1}$ in one direction and $\infty$ W m$^{-1}$ K$^{-1}$ in the other. Transformation-invariant (i.e., highly anisotropic) metamaterials have aroused broad interest in various fields, such as electromagnetism [16, 17] and acoustics [18, 19]. For a two-dimensional case, highly anisotropic thermal conductivities have adaptive responses to environmental changes [20, 21], just like chameleons. We perform coordinate transformations based on transformation-invariant metamaterials, which can keep the chameleonlike behavior. Therefore, the designed devices have adaptive responses to environmental changes. We take thermal rotators [22,23,24,25,26] as an example, which can guide heat flux direction. Existing designs only apply to a particular environment, which cannot cope with environmental changes. Here, we propose the concept of thermal chameleonlike rotators. Going beyond a normal isotropic shell with near-zero thermal conductivity (Fig. 9.1a), we start the rotation transformation from a transformation-invariant shell (Fig. 9.1b). In this way, the designed rotator can work in different environments (Fig. 9.1c and d), thus called a chameleonlike rotator. The environment denotes the regions except for the rotator, and the environmental parameter refers to its thermal conductivity.

Fig. 9.1

Adapted from Ref. [27]

Schematic diagram of the thermal chameleonlike rotator. a Isotropic shell with near-zero thermal conductivity. b Transformation-invariant shell with near-zero thermal conductivity in the tangential direction and near-infinite thermal conductivity in the radial direction. c and d Thermal chameleonlike rotator working in different environments. Lines with arrows indicate heat flow. The environmental thermal conductivities of c and d are $\kappa _1$ and $\kappa _2$, respectively.

2 Chameleonlike Behavior Origin

We consider a passive and stable conduction process in two dimensions, which is governed by the Fourier law,

$$\begin{aligned} \nabla \cdot \left( -\boldsymbol{\kappa }\cdot \nabla T\right) =0. \end{aligned}$$

(9.1)

The whole system is divided into three regions, i.e., core, shell, and background, with tensorial thermal conductivities of $\boldsymbol{\kappa }_{1}=\kappa _{1}\boldsymbol{I}$, $\boldsymbol{\kappa }_{2}=\textrm{diag}\left( \kappa _{ rr},\kappa _{\theta \theta }\right)$, and $\boldsymbol{\kappa }_{3}=\kappa _{3}\boldsymbol{I}$, respectively. We treat the core and background as the environment, and suppose their thermal conductivities to be the same, i.e., $\kappa _1=\kappa _3$. $\boldsymbol{\kappa }_{2}$ is expressed in cylindrical coordinates $\left( r,\,\theta \right)$. By solving the Laplace equation, the effective thermal conductivity of the core and shell $\kappa _e$ can be expressed as

$$\begin{aligned} \kappa _{e} = \kappa _{rr}\frac{n_1\left( \kappa _{1}-n_{2}\kappa _{rr}\right) -n_2\left( \kappa _{1}-n_{1}\kappa _{rr}\right) p^{\left( n_{1}-n_{2}\right) /2}}{\kappa _{1}-n_{2}\kappa _{rr}-\left( \kappa _{1}-n_{1}\kappa _{rr}\right) p^{\left( n_1-n_2\right) /2}}, \end{aligned}$$

(9.2)

where $n_{1,\,2} = \pm \sqrt{\kappa _{\theta \theta }/\kappa _{rr}}$ and $p = \left( R_{1}/R_{2}\right) ^{2}$. $R_1$ and $R_2$ are the inner and outer radii of the shell, respectively. The thermal conductivity of a transformation-invariant metamaterial is

$$\begin{aligned} \boldsymbol{\kappa }_{2} = \begin{pmatrix} \infty &{} 0\\ 0 &{} 0 \end{pmatrix}. \end{aligned}$$

(9.3)

The substitution of Eq. (9.3) into Eq. (9.2) yields

$$\begin{aligned} \kappa _{e} \approx \kappa _{1}, \end{aligned}$$

(9.4)

which means that the effective thermal conductivity of the core-shell structure can adaptively change with the environment. In other words, two-dimensional transformation-invariant metamaterials (Eq. (9.3)) have a chameleonlike behavior (Eq. (9.4)). We then consider an arbitrary two-dimensional coordinate transformation,

$$\begin{aligned} r'=R\left( r,\,\theta \right) ,\end{aligned}$$

(9.5a)

$$\begin{aligned} \theta '=\Theta \left( r,\theta \right) , \end{aligned}$$

(9.5b)

where $\left( r',\,\theta '\right)$ are physical coordinates and $\left( r,\,\theta \right)$ are virtual coordinates. We can express the Jacobian matrix ${\textbf{J}}$ as

$$\begin{aligned} {\textbf{J}}= \begin{pmatrix} \dfrac{\partial r'}{\partial r} &{} \dfrac{\partial r'}{r\partial \theta }\\ \dfrac{r' \partial \theta '}{\partial r} &{} \dfrac{r' \partial \theta '}{r\partial \theta } \end{pmatrix}. \end{aligned}$$

(9.6)

The transformed thermal conductivity is

$$\begin{aligned} \boldsymbol{\kappa }_{2}' = \frac{{\textbf{J}}\boldsymbol{\kappa }_{2}{\textbf{J}}^\tau }{\det {\textbf{J}}}, \end{aligned}$$

(9.7)

which can be expressed in detail as

$$\begin{aligned} \boldsymbol{\kappa }_{2}' = \dfrac{1}{\det {\textbf{J}}} \begin{pmatrix} \kappa _{rr}\left( \dfrac{\partial r'}{\partial r}\right) ^{2} + \kappa _{\theta \theta }\left( \dfrac{\partial r'}{r\partial \theta }\right) ^{2} &{} \kappa _{rr}\left( \dfrac{\partial r'}{\partial r}\right) \left( \dfrac{r'\partial \theta ' }{\partial r}\right) + \kappa _{\theta \theta }\left( \dfrac{\partial r'}{r\partial \theta }\right) \left( \dfrac{r'\partial \theta ' }{r\partial \theta }\right) \\ \kappa _{rr}\left( \dfrac{\partial r'}{\partial r}\right) \left( \dfrac{r'\partial \theta ' }{\partial r}\right) + \kappa _{\theta \theta }\left( \dfrac{\partial r'}{r\partial \theta }\right) \left( \dfrac{r'\partial \theta ' }{r\partial \theta }\right) &{} \kappa _{rr}\left( \dfrac{r'\partial \theta ' }{\partial r}\right) ^{2} + \kappa _{\theta \theta }\left( \dfrac{r'\partial \theta ' }{r\partial \theta }\right) ^{2} \end{pmatrix}. \end{aligned}$$

(9.8)

With Eq. (9.3), the eigenvalues of Eq. (9.8) are

$$\begin{aligned} \lambda _{1}&= \frac{\kappa _{rr}}{\det {\textbf{J}}} \left[ \left( \frac{\partial r'}{\partial r}\right) ^{2}+\left( \frac{r' \partial \theta '}{\partial r}\right) ^{2}\right] ,\end{aligned}$$

(9.9a)

$$\begin{aligned} \lambda _{2}&\approx \frac{\kappa _{\theta \theta }}{\det {\textbf{J}}}. \end{aligned}$$

(9.9b)

Due to $\kappa _{rr} = \infty$ and $\kappa _{\theta \theta } = 0$, Eq. (9.9) can be further reduced to

$$\begin{aligned} \lambda _{1}&= \infty ,\end{aligned}$$

(9.10a)

$$\begin{aligned} \lambda _{2}&= 0. \end{aligned}$$

(9.10b)

An arbitrary coordinate transformation does not change the eigenvalues.

We then design a thermal chameleonlike rotator with transformation-invariant metamaterials. The coordinate transformation of rotating can be expressed as

$$\begin{aligned} r'&= r,\end{aligned}$$

(9.11a)

$$\begin{aligned} \theta '&= \theta + \theta _{0}~~~~\left( r < R_{1}\right) ,\end{aligned}$$

(9.11b)

$$\begin{aligned} \theta '&= \theta + \theta _{0}\left( R_{2} - r\right) /\left( R_{2}-R_{1}\right) ~~~~\left( R_{1}< r < R_{2}\right) , \end{aligned}$$

(9.11c)

where $\theta _{0}$ is rotation angle. With Eqs. (9.6) and (9.7), we can derive the thermal conductivity of the rotator as

$$\begin{aligned} \boldsymbol{\kappa }_{2}' = \begin{pmatrix} \kappa _{rr} &{} \kappa _{rr}\dfrac{r'\theta _{0}}{R_{2}-R_{1}}\\[1em] \kappa _{rr}\dfrac{r'\theta _{0}}{R_{2}-R_{1}} &{} \kappa _{rr}\left( \dfrac{r'\theta _{0}}{R_{2}-R_{1}}\right) ^{2} + \kappa _{\theta \theta } \end{pmatrix}, \end{aligned}$$

(9.12)

which is the key parameter for a thermal chameleonlike rotator as long as $\boldsymbol{\kappa }_{2}$ satisfies Eq. (9.3).

Fig. 9.2

Adapted from Ref. [27]

Simulation results of a–c chameleonlike rotator and d–f normal rotator. White lines represent isotherms, and the values in each simulation are the corresponding thermal conductivities. The system size is $1\times 1$ m$^2$. The outer and inner diameters of the shell are 0.3 and 0.6 m, respectively.

3 Finite-Element Simulation

To verify the scheme, we first perform simulations with COMSOL Multiphysics. The system is the same as Fig. 9.1c. We compare the difference between a chameleonlike rotator and a normal rotator (Fig. 9.2). Before performing the rotation transformation, the thermal conductivities for the chameleonlike rotator and normal rotator are $\textrm{diag}\left( 10^6,\,10^{-3}\right)$ and 100 W m$^{-1}$ K$^{-1}$, respectively. The radial thermal conductivity of the transformation-invariant metamaterial should be much larger than the environmental thermal conductivity (at least two orders of magnitude), or the chameleonlike rotator may fail. We then change the environmental thermal conductivity from 10 to 1000 W m$^{-1}$ K$^{-1}$, and the chameleonlike rotator can always work, i.e., rotating heat flux without distorting the environmental temperature profile (Fig. 9.2a–c). Therefore, the simulation results confirm the chameleonlike property. However, the normal rotator fails. When the environmental thermal conductivity is 100 W m$^{-1}$ K$^{-1}$, it behaves like a traditional rotator (Fig. 9.2e). When the environment changes, the temperature profile is distorted (Fig. 9.2d and f). Therefore, the normal rotator has no response to environmental changes.

4 Laboratory Experiment

For experimental verification, it is not easy to find a material in nature that satisfies Eq. (9.12). Therefore, we use the effective medium theory to realize the corresponding parameter. Drawing on the multilayered structure [22], we design the chameleonlike rotator as shown in Fig. 9.3a. As required by Eqs. (9.3) and (9.12), we choose two materials with extremely large ($\kappa _l \approx 10^6$ W m$^{-1}$ K$^{-1}$) and extremely small ($\kappa _s \approx 10^{-3}$ W m$^{-1}$ K$^{-1}$) thermal conductivities to approximately satisfy Eq. (9.3), and then use the helical structure to approximately satisfy Eq. (9.12). The simulation results are shown in Fig. 9.3b–g. Among them, Fig. 9.3b–d show the results of chameleonlike rotator-1, rotating heat flux 90 $^\circ$C. Figure 9.3e, f presents the results of chameleonlike rotator-2, rotating heat flux 180 $^\circ$C. Therefore, it is feasible to fabricate chameleonlike rotators with multilayered composite structures.

Fig. 9.3

Adapted from Ref. [27]

Simulation results of chameleonlike rotators with multilayered structures. a Schematic diagram. The structure is composed of two kinds of material with thermal conductivities of $10^6$ and $10^{-3}$ W m$^{-1}$ K$^{-1}$, respectively. Simulations results of b–d chameleonlike rotator-1 and e–g chameleonlike rotator-2 in different environments. The composite materials in b–g are the same as those in a.

Restricted by experimental conditions, we choose copper ($\kappa _{cu}\approx$ 400 W m$^{-1}$ K$^{-1}$) and air ($\kappa _{air}\approx$ 0.026 W m$^{-1}$ K$^{-1}$) to fabricate a multilayered composite structure to realize a small-angle rotator. According to the series/parallel connection formula [28], the effective thermal conductivity of the composite structure is about $\textrm{diag}\left( 200,\,0.052\right)$ W m$^{-1}$ K$^{-1}$ before transformation, putting a limit on the variation range of the environmental thermal conductivity. We calculate $\kappa _{e}$ with $\kappa _{1}$ changing from 0.1 to 50 W m$^{-1}$ K$^{-1}$, and confirm that the chameleonlike rotator works well from 0.1 to 5 W m$^{-1}$ K$^{-1}$, as shown in Fig. 9.4b. The difference $|\kappa _{e} - \kappa _{1}|$ is smaller than 0.05 W m$^{-1}$ K$^{-1}$ (denoted by the star, with a deviation smaller than 1%). Therefore, we conduct experiments with environmental thermal conductivities of 1 and 5 W m$^{-1}$ K$^{-1}$. The system is designed as shown in Fig. 9.4a. The chameleonlike rotator is composed of air and copper, fabricated by laser cutting. The environment is colloidal materials obtained by mixing silica gel ($\kappa _{gel} = 0.15$ W m$^{-1}$ K$^{-1}$ and density $\rho _{gel} = 1.14\times 10^{3}$ kg m$^{-3}$) and white copper powder ($\kappa _{wcu} =33$ W m$^{-1}$ K$^{-1}$ and $\rho _{wcu} = 8.65\times 10^{3}$ kg m$^{-3}$). The thermal conductivity of the mixture is determined by the Bruggeman formula [29],

$$\begin{aligned} p_{gel}\frac{\kappa _{gel}-\kappa _{mix}}{\kappa _{gel}+2\kappa _{mix}} + (1 - p_{gel})\frac{\kappa _{wcu}-\kappa _{mix}}{\kappa _{wcu}+2\kappa _{mix}} = 0, \end{aligned}$$

(9.13)

where $p_{gel}$ is the volume fraction of silica gel in the mixture. By setting $\kappa _{mix} = 1$ or 5 W m$^{-1}$ K$^{-1}$, we can derive the composition ratio of silica gel, which helps us fabricate the colloidal materials. Although interface thermal conductance [30, 31] exists, the mixture in regions I and III has a little fluidity, ensuring good contact between the object and copper. We then fill two hot and ice water tanks as hot and cold sources. The FLIR E60 infrared camera measures the temperature profile of the sample. The experimental results are shown in Fig. 9.4d ($\kappa _{mix} = 1$ W m$^{-1}$ K$^{-1}$) and 9.4f ($\kappa _{mix} = 5$ W m$^{-1}$ K$^{-1}$). The corresponding simulation results are presented in Fig. 9.4c and e. Heat dissipation exists because the sample is connected to the hot and cold tanks with two copper plates. Moreover, the natural convection between the sample and air also results in heat dissipation. Therefore, there is a small difference between the computational and experimental values, but this does not affect the expected results. The isotherms still keep straight even though the environmental thermal conductivity changes. Meanwhile, heat flux is rotated as expected. Therefore, the experimental results are consistent with the simulation results, verifying the feasibility of chameleonlike rotators.

Fig. 9.4

Adapted from Ref. [27]

Laboratory experiments of chameleonlike rotator. a Experimental setup. The structure is composed of copper ($\kappa _{cu}\approx 400$ W m$^{-1}$ K$^{-1}$) and air ($\kappa _{air}\approx 0.026$ W m$^{-1}$ K$^{-1}$). b $\kappa$ as a function of $\kappa _{1}$. The blue (top) and red (middle) lines correspond to $\kappa _1$ and $\kappa _e$, respectively. The black (bottom) line refers to $|\kappa _{e} - \kappa _{1}|$. The coordinate of $*$ is $\left( 5,\,0.047\right)$. c and e Simulation results and d and f experimental results of the samples. The arrows indicate the direction of heat flux. The inner and outer diameters of the shell are 0.075 and 0.15 m, respectively.

5 Discussion

The major difference of our scheme is to start the coordinate transformation from a highly anisotropic parameter, which is proved to have a chameleonlike behavior. Therefore, the designed rotator based on this parameter can also have a chameleonlike behavior. Meanwhile, no matter how the shape of the rotator changes, the chameleonlike behavior still exists. Therefore, the present scheme can also design chameleonlike rotators with arbitrary shapes. Nevertheless, a perfect transformation-invariant (i.e., highly anisotropic) shell is described by Eq. (9.3), indicating that the higher anisotropy yields a better chameleonlike behavior and a wider working range. Due to the lack of highly conductive materials, the working range of the fabricated rotator is from 0.1 to 5 W m$^{-1}$ K$^{-1}$. More methods [32,33,34,35,36] can be applied to enhance thermal conductivities.

Moreover, the scheme can be extended to transient regimes by taking density and heat capacity into account [23, 37,38,39,40,41,42]. The scheme is also not limited to conductive systems. Recent studies explored convective-diffusive systems [43,44,45], hydrodynamic systems [46, 47], and acoustic systems [18, 19] to design functional devices. Therefore, it is also promising to design chameleonlike rotators in these fields. Although these results are obtained at the macroscopic scale described by the Fourier law, intelligence may also be helpful for heat manipulations with nanostructures [48, 49].

6 Conclusion

We have designed thermal chameleonlike rotators based on transformation-invariant metamaterials. With a highly anisotropic thermal conductivity, the designed rotator can work in different environments, saving time and labor. Both simulations and experiments verify the feasibility of the scheme. These results improve the intelligence of traditional thermal metamaterials and have potential applications in designing intelligent metamaterials. The proposed scheme can also be extended to other fields, such as hydrodynamics, where the critical parameter (permeability or viscosity) plays a similar role as thermal conductivity in thermotics.

7 Exercise and Solution

Exercise

1. Calculate the Jacobian transformation matrix of Eq. (9.11).

Solution

1. For the region $r<R_1$, ${\textbf{J}}={\textbf{I}}$. For the region $R_1<r<R_2$,

$$\begin{aligned} {\textbf{J}}= \begin{pmatrix} 1 &{} 0\\ -\theta _0/\left( R_2-R_1\right) &{} 1 \end{pmatrix}. \end{aligned}$$

(9.14)