Theory for Invisible Thermal Sensors: Monolayer Scheme

Abstract

In this chapter, we propose an anisotropic monolayer scheme to prevent thermal sensors from distorting local and background temperature profiles, making them accurate and thermally invisible. We design metashells with anisotropic thermal conductivity and perform finite-element simulations in two or three dimensions for arbitrarily given thermal conductivity of sensors and backgrounds. We further experimentally fabricate a metashell with an anisotropic thermal conductivity based on the effective medium theory, which confirms the feasibility of our scheme. Our results are beneficial to improving the performance of thermal detection and may also guide other diffusive physical fields.

1 Opening Remarks

Temperature measurement has broad applications, requiring high sensitivity for thermal sensors. However, the distortion of temperature profiles resulting from thermal sensors cannot be avoided by only improving sensitivity. A severe problem lies in the thermal-conductivity mismatch between sensors and backgrounds. Similar problem (parametric mismatch) also occurs in some other fields, and promotes relevant researches in electromagnetism [1, 2], magnetics [3], and acoustics [4,5,6].

Many schemes were proposed based on neutral inclusion [7] or transformation thermotics [8]. Furthermore, a multiphysical scheme was proposed by coating a sensor with an isotropic shell [9]. Though these schemes improve the performance of thermal detection, they are also faced with problems of complex parameters and technological difficulty. For the scheme based on transformation thermotics [8], anisotropic, inhomogeneous, and even negative thermal conductivity is required, which makes its experimental realization extremely difficult. For the scheme based on an isotropic shell [7, 9,10,11], local temperature profiles are still different from the original ones, making thermal detection inaccurate. Here, “local” indicates the region occupied by a sensor. To improve accuracy, one should minimize the thickness of the isotropic shell, which, however, still cannot completely remove inaccuracy. In this sense, to date, thermal sensors with both accuracy and invisibility are still experimentally lacking. Here, accuracy and invisibility respectively indicate that local and background temperature profiles are not distorted.

Different from existing methods [7,8,9,10,11], we propose an anisotropic monolayer scheme that can accurately measure local temperature profiles without disturbing background thermal fields. It is worth noting that a similar scheme has been successfully applied to design other thermal functions, such as cloaking [12,13,14], concentrating [13,14,15], and chameleon [16, 17]. The present scheme is applicable for arbitrarily given thermal conductivities of backgrounds and sensors, which is confirmed by finite-element simulations in two or three dimensions. Furthermore, we experimentally fabricate a metashell with anisotropic thermal conductivity based on the effective medium theory [18,19,20,21,22], and the experimental results agree well with the theory and finite-element simulations.

2 Theoretical Foundation

We start by discussing a two-dimensional system shown in Fig. 11.1. The system is divided by a metashell (Area II) into three areas, whose thermal conductivities are $\kappa _1$ for sensor (Area I), $\boldsymbol{\kappa }_2=\textrm{diag}\left( \kappa _{rr},\,\kappa _{\theta \theta }\right)$ for metashell (Area II), and $\kappa _3$ for background (Area III). $\boldsymbol{\kappa }_2$ is expressed in cylindrical coordinates $\left( r,\,\theta \right)$, where $\kappa _{rr}$ and $\kappa _{\theta \theta }$ are radial and tangential thermal conductivities, respectively. We consider the known equation describing passive heat conduction at steady states,

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

(11.1)

where $\boldsymbol{\kappa }$ and T are thermal conductivity and temperature, respectively.

Equation (11.1) can be expanded in cylindrical coordinates as

$$\begin{aligned} \dfrac{1}{r}\dfrac{\partial }{\partial r}\left( r\kappa _{rr}\dfrac{\partial T}{\partial r}\right) +\dfrac{1}{r}\dfrac{\partial }{\partial \theta }\left( \kappa _{\theta \theta }\dfrac{\partial T}{r\partial \theta }\right) =0. \end{aligned}$$

(11.2)

The general solution to Eq. (11.2) is

$$\begin{aligned} T&= A_0+B_0\ln r+\sum \limits _{i=1}^\infty \left[ A_i\sin \left( i\theta \right) +B_i\cos \left( i\theta \right) \right] r^{im_1} \\ \nonumber&\quad +\sum \limits _{j=1}^\infty \left[ C_j\sin \left( j\theta \right) +D_j\cos \left( j\theta \right) \right] r^{jm_2}, \end{aligned}$$

(11.3)

where $m_{1}=\sqrt{\kappa _{\theta \theta }/\kappa _{rr}}$ and $m_{2}=-\sqrt{\kappa _{\theta \theta }/\kappa _{rr}}$, representing anisotropy degree.

Fig. 11.1

Adapted from Ref. [23]

Schematic diagram of the anisotropic monolayer scheme.

The temperature profiles of sensor (Area I), metashell (Area II), and background (Area III) are respectively denoted as $T_1$, $T_2$, and $T_3$, which satisfy the general solution in Eq. (11.3). Especially, $T_1$ and $T_3$ can be given by the right side of Eq. (11.3) with $m_{1}=1$ and $m_{2}=-1$. Boundary conditions are determined by the continuities of temperature and normal heat flux,

$$\begin{aligned} {\left\{ \begin{array}{ll}T_1\left( R_1\right) =T_2\left( R_1\right) ,\\ T_2\left( R_2\right) =T_3\left( R_2\right) ,\\ \left( -\kappa _1\dfrac{\partial T_1}{\partial r}\right) _{R_1}=\left( -\kappa _{rr}\dfrac{\partial T_2}{\partial r}\right) _{R_1},\\ \left( -\kappa _{rr}\dfrac{\partial T_2}{\partial r}\right) _{R_2}=\left( -\kappa _3\dfrac{\partial T_3}{\partial r}\right) _{R_2}.\end{array}\right. } \end{aligned}$$

(11.4)

Considering the symmetry of boundary conditions, we only keep certain terms in Eq. (11.3) as the temperature profiles of three areas,

$$\begin{aligned} T_1&= A_0+Ar\cos \theta ,\end{aligned}$$

(11.5a)

$$\begin{aligned} T_2&= A_0+Br^{m_1}\cos \theta +Cr^{m_2}\cos \theta ,\end{aligned}$$

(11.5b)

$$\begin{aligned} T_3&= A_0+Dr\cos \theta +Er^{-1}\cos \theta , \end{aligned}$$

(11.5c)

where the temperature at $\theta =\pm \pi /2$ is defined as $A_0$, and $D=|\boldsymbol{\nabla }T_0|$ is the modulus of an external linear thermal field $\boldsymbol{\nabla }T_0$.

We have six undetermined coefficients (i.e., A, B, C, E, $\kappa _{rr}$, and $\kappa _{\theta \theta }$) and only four equations (Eq. (11.4)). The other two equations are to make thermal sensors accurate and thermally invisible,

$$\begin{aligned} {\left\{ \begin{array}{ll}A=D,\\ E=0,\end{array}\right. } \end{aligned}$$

(11.6)

where $A=D$ indicates that the local temperature profile is the same as the background temperature profile, making a thermal sensor accurate; and $E=0$ indicates that the background temperature profile is undistorted, making a thermal sensor invisible.

Then, the six unknown coefficients can be uniquely determined by six equations (Eqs. (11.4) and (11.6)), including the anisotropic thermal conductivity of the metashell $\boldsymbol{\kappa }_2=\textrm{diag}\left( \kappa _{rr},\,\kappa _{\theta \theta }\right)$. Therefore, thermal sensors with accuracy and thermal invisibility in two dimensions are designed.

On the same footing, we extend the two-dimensional theory to three dimensions. Accordingly, the thermal conductivity of the metashell is denoted as $\boldsymbol{\kappa }_2=\textrm{diag}\left( \kappa _{rr},\,\kappa _{\theta \theta },\,\kappa _{\varphi \varphi }\right)$ with $\kappa _{\theta \theta }=\kappa _{\varphi \varphi }$. Then, Eq. (11.1) can be expanded in spherical coordinates $\left( r,\,\theta ,\,\varphi \right)$ as

$$\begin{aligned} \dfrac{1}{r^{2}}\dfrac{\partial }{\partial r}\left( r^{2}\kappa _{rr}\dfrac{\partial T}{\partial r}\right) +\dfrac{1}{r}\dfrac{1}{\sin \theta }\dfrac{\partial }{\partial \theta }\left( \sin \theta \kappa _{\theta \theta }\dfrac{\partial T}{r\partial \theta }\right) =0, \end{aligned}$$

(11.7)

where $\varphi$ is neglected because we consider a rotational symmetric case.

The general solution to Eq. (11.7) is

$$\begin{aligned} T=\sum \limits _{i=0}^\infty \left( A_ir^{s_1}+B_ir^{s_2}\right) P_i\left( \cos \theta \right) , \end{aligned}$$

(11.8)

where $s_{1}=\left[ -1+\sqrt{1+4i\left( i+1\right) \kappa _{\theta \theta }/\kappa _{rr}}\right] /2$, $s_{2}=\left[ -1-\sqrt{1+4i\left( i+1\right) \kappa _{\theta \theta }/\kappa _{rr}}\right] /2$, i is summation index, and $P_i$ is the Legendre polynomials. Since three-dimensional boundary conditions are the same as two-dimensional ones, the temperature profiles for three areas can be expressed as

$$\begin{aligned} T_1&= A_0+Ar\cos \theta ,\end{aligned}$$

(11.9a)

$$\begin{aligned} T_2&= A_0+Br^{s_1}\cos \theta +Cr^{s_2}\cos \theta ,\end{aligned}$$

(11.9b)

$$\begin{aligned} T_3&= A_0+Dr\cos \theta +Er^{-2}\cos \theta , \end{aligned}$$

(11.9c)

where $s_{1}=\left[ -1+\sqrt{1+8\kappa _{\theta \theta }/\kappa _{rr}}\right] /2$, $s_{2}=\left[ -1-\sqrt{1+8\kappa _{\theta \theta }/\kappa _{rr}}\right] /2$, and the six unknown coefficients (i.e., A, B, C, E, $\kappa _{rr}$, and $\kappa _{\theta \theta }$) can also be solved by Eqs. (11.4) and (11.6). Therefore, we also make three-dimensional thermal sensors accurate and invisible. We use Mathematica to solve the six equations (Eqs. (11.4) and (11.6)) and obtain the required thermal conductivity of the metashell.

3 Finite-Element Simulation

To confirm our theory, we further perform finite-element simulations with COMSOL Multiphysics. We can derive the anisotropic thermal conductivity of metashells and perform finite-element simulations for arbitrarily given thermal conductivity of backgrounds and sensors.

Fig. 11.2

Adapted from Ref. [23]

Case with $\kappa _1<\kappa _3$. The simulation box is 18$\times$18 cm$^{2}$, $R_1=3$ cm, and $R_2=6$ cm. Black lines represent isotherms. The temperatures of cold source (left boundary) and hot source (right boundary) are set at 283 and 313 K, respectively. The thermal conductivities of a reference (all areas) and b reference shell (Area II) are set to be the same, i.e., 222.5 W m$^{-1}$ K$^{-1}$. The thermal conductivities of sensor (Area I) and background (Area III) in b–d are set to be 108.2 and 222.5 W m$^{-1}$ K$^{-1}$, respectively. The thermal conductivities of c isotropic shell and d anisotropic metashell are 277.3 and $\textrm{diag}\left( 178.0,\,349.0\right)$ W m$^{-1}$ K$^{-1}$, respectively.

Fig. 11.3

Adapted from Ref. [23]

Quantitative comparison of Fig. 11.2. Temperature-difference distributions with a the temperature in Fig. 11.2c minus that in Fig. 11.2a and b the temperature in Fig. 11.2d minus that in Fig. 11.2a. c Temperature-gradient distributions on Line 1 in Fig. 11.2. d Temperature distributions on Line 2 in Fig. 11.2.

For comparison, we show a reference with uniform thermal conductivity in Fig. 11.2a. The finite-element simulations with a reference shell (a bare sensor embedded in the background, i.e., bareness), an isotropic shell (calculated with the theory in Ref. [9]), and an anisotropic metashell (calculated by our theory) are presented in Fig. 11.2b–d, respectively. Figure 11.2b indicates that a bare thermal sensor indeed distorts local (Area I) and background (Area III) temperature profiles. Comparing Fig. 11.2a and c, although the isotropic shell keeps the background temperature profile undistorted, the local temperature profile is still changed. Fortunately, our scheme (Fig. 11.2d) makes local and background temperature profiles the same as those in Fig. 11.2a, thus making the sensor accurate and thermally invisible.

To compare different schemes quantitatively, we also plot temperature-difference profiles with the temperature in Fig. 11.2c minus that in Fig. 11.2a (Fig. 11.3a) and the temperature in Fig. 11.2d minus that in Fig. 11.2a (Fig. 11.3b). The temperature difference in the local region of Fig. 11.3a is nonzero, indicating that the detected temperature is not the original one. The temperature difference in the local region of Fig. 11.3b is zero, making a thermal sensor accurate. Certainly, the temperature profile of the metashell has a small difference. We export the temperature-gradient distributions along x axis on Line 1 in Fig. 11.2, as shown in Fig. 11.3c. The results indicate that the isotropic and anisotropic shells (Fig. 11.2c and d) show the same advantage of thermal invisibility (i.e., the constant temperature-gradient distribution as that in reference). However, due to the thermal-conductivity mismatch between the sensor and background, a bare sensor distorts the heat flow of the original background thermal field, so the temperature gradient at the position of Line 1 in Fig. 11.2b is not a constant. We also export the temperature distributions on Line 2 in Fig. 11.2, as shown in Fig. 11.3d. The results show that the anisotropic scheme is better than the isotropic one because what the thermal sensor in Fig. 11.2d detects is completely consistent with the reference. However, the detection in Fig. 11.2c deviates from the reference.

Fig. 11.4

Adapted from Ref. [23]

Case with $\kappa _1>\kappa _3$. The thermal conductivities of a reference (all areas) and b reference shell (Area II) are set to be the same, i.e., 251.8 W m$^{-1}$ K$^{-1}$. The thermal conductivities of sensor (Area I) and background (Area III) in b–d are set to be 397.0 and 251.8 W m$^{-1}$ K$^{-1}$, respectively. The thermal conductivities of c isotropic shell and d anisotropic metashell are 217.5 and $\textrm{diag}\left( 308.0,\,104.0\right)$ W m$^{-1}$ K$^{-1}$, respectively. Other parameters are the same as those for Fig. 11.2.

Fig. 11.5

Adapted from Ref. [23]

Quantitative comparison of Fig. 11.4. Temperature-difference distributions with a the temperature in Fig. 11.4c minus that in Fig. 11.4a and b the temperature in Fig. 11.4d minus that in Fig. 11.4a. c Temperature-gradient distributions on Line 1 in Fig. 11.4. d Temperature distribution on Line 2 in Fig. 11.4.

Moreover, we also discuss the case that the thermal conductivity of the sensor (Area I) is larger than that of the background (Area III) to ensure completeness; see Figs. 11.4 and 11.5. The results are similar to Figs. 11.2 and 11.3. That is, a bare thermal sensor will distort temperature profiles of all areas (Fig. 11.4b). The existing isotropic scheme can keep the background temperature profile undistorted, but the local one is changed (Fig. 11.4c). The present scheme can ensure both local and background temperature profiles undistorted (Fig. 11.4d). Factual data can be found in Fig. 11.5.

Fig. 11.6

Adapted from Ref. [23]

Three-dimensional simulations. The simulation box is 18$\times$18$\times$18 cm$^{3}$, $R_1=3$ cm, and $R_2=6$ cm. The thermal conductivities of a reference (all areas) and b reference shell (Area II) are set to be the same, i.e., 222.5 W m$^{-1}$ K$^{-1}$. The thermal conductivities of sensor (Area I) and background (Area III) in b-d are set to be 108.2 and 222.5 W m$^{-1}$ K$^{-1}$, respectively. The thermal conductivities of c isotropic shell and d anisotropic metashell are 242.5 and $\textrm{diag}\left( 183.7,\,269.0,\,269.0\right)$ W m$^{-1}$ K$^{-1}$, respectively. Temperature-difference distributions with e the temperature in c minus that in a (i.e., $\Delta T_1=T_1-T_0$) and f the temperature in d minus that in a (i.e., $\Delta T_2=T_2-T_0$). g Temperature-gradient distributions on Line 1 in a–d. h Temperature distributions on Line 2 in a-d.

To go further, we also perform three-dimensional finite-element simulations. The temperature profiles in the sensor (Area I) in Fig. 11.6b and c are different from that in Fig. 11.6a, which means that the sensor cannot accurately measure local temperature distributions. Fortunately, the temperature profiles in the sensor (Area I) and background (Area III) in Fig. 11.6d are identical to those in Fig. 11.6a. We also perform quantitative analyses on “accurate” and “invisible” properties of the sensor (Fig. 11.6e–h), and the results are what we expect, just like the two-dimensional case. To sum up, the accuracy and invisibility of three-dimensional thermal sensors are also confirmed by simulations.

Fig. 11.7

Adapted from Ref. [23]

Laboratory experiments. a Experimental setup. b and c Real photos of two samples. d and e (or f and g) are measured results (or finite-element simulations) corresponding to the two samples shown in b and c, respectively. White lines represent isotherms. The sensor (or background) in b and c is carved with air circles with radius 0.21 cm (or 0.15 cm). The anisotropic metashell is carved with air ellipses with major (or minor) semiaxis of 0.21 cm (or 0.044 cm). The thermal conductivities of copper and air are 397 and 0.026 W m$^{-1}$ K$^{-1}$, respectively. These parameters cause the tensorial thermal conductivity of the anisotropic metashell in c to be $\textrm{diag}\left( 178.0,\,349.0\right)$ W m$^{-1}$ K$^{-1}$, and the thermal conductivities of sensor (or background) in b and c to be 108.2 W m$^{-1}$ K$^{-1}$ (or 222.5 W m$^{-1}$ K$^{-1}$). The sample size in b and c is the same as that for Fig. 11.2b and d.

4 Laboratory Experiment

To experimentally validate the finite-element simulations in Fig. 11.2b and d, we set up a device shown in Fig. 11.7a. By utilizing laser cutting, we fabricate two samples (Fig. 11.7b and c) based on ellipse-embedded structures [18]. The holes in Areas I and III are uniformly distributed with circular shapes, ensuring that the effective thermal conductivity is isotropic. The holes in Area II have anisotropic (elliptical) geometry, so the effective thermal conductivity is also anisotropic. Therefore, the perforated structure indeed follows the theory. To eliminate infrared reflection and thermal convection as much as possible, we also apply transparent and foamed plastic films (insulating materials) on the upper and lower surfaces of the two samples, respectively. Then, we measure the temperature profiles of these two samples with the infrared camera Flir E60. The measured results with a reference shell and with an anisotropic metashell are shown in Fig. 11.7d and e, respectively. We also perform finite-element simulations based on these two samples, as shown in Fig. 11.7f and g. Both finite-element simulations (Fig. 11.2b and d, and Fig. 11.7f and g) and experiments (Fig. 11.7d and e) prove that with the present scheme, a thermal sensor does accurately detect the local temperature profile without disturbing the background thermal field.

We have discussed the scheme in steady heat conduction, and extending it to transient states is promising, which should consider density and heat capacity. Furthermore, topology optimization is also a powerful method to design metamaterials [24,25,26,27] beyond transformation method [28,29,30,31], which could be applied to design accurate and invisible sensors.

5 Conclusion

We have proposed an anisotropic monolayer scheme to make thermal sensors accurate and thermally invisible. By coating a thermal sensor with a metashell with anisotropic thermal conductivity, the thermal sensor can accurately measure local temperature profiles without disturbing surrounding thermal fields. The present scheme is validated by two-dimensional simulations and experiments, which also apply to three dimensions. These results may advance the performance of thermal detection and provide guidance to thermal camouflage [32,33,34,35,36,37,38,39]. On the same basis, this work also offers hints to obtaining counterparts in other diffusive fields.

6 Exercise and Solution

Exercise

1. Solve the unknown numbers in Eqs. (11.5) and (11.9).

Solution

1. Since $\kappa _{rr}$ and $\kappa _{\theta \theta }$ appear in the exponent, it is difficult to analytically express them. To solve the problem, we treat $\kappa _1$ and $\kappa _3$ as two undetermined coefficients, which together with other coefficients can be respectively expressed in two and three dimensions as

$$\begin{aligned} {\left\{ \begin{array}{ll} A=|\boldsymbol{\nabla }T_0|,\\ \\ B=\dfrac{R_1^{m_2}R_2-R_1R_2^{m_2}}{R_1^{m_2}R_2^{m_1}-R_1^{m_1}R_2^{m_2}}|\boldsymbol{\nabla }T_0|,\\ \\ C=-\dfrac{R_1^{m_1}R_2-R_1R_2^{m_1}}{R_1^{m_2}R_2^{m_1}-R_1^{m_1}R_2^{m_2}}|\boldsymbol{\nabla }T_0|,\\ \\ E=0,\\ \\ \kappa _1=\kappa _{rr}\dfrac{m_1\left( -R_1^{m_1}R_2^{m_2}+R_1^{m_1+m_2-1}R_2\right) +m_2\left( R_1^{m_2}R_2^{m_1}-R_1^{m_1+m_2-1}R_2\right) }{R_1^{m_2}R_2^{m_1}-R_1^{m_1}R_2^{m_2}},\\ \\ \kappa _3=\kappa _{rr}\dfrac{m_1\left( R_1^{m_2}R_2^{m_1}-R_1R_2^{m_1+m_2-1}\right) +m_2\left( -R_1^{m_1}R_2^{m_2}+R_1R_2^{m_1+m_2-1}\right) }{R_1^{m_2}R_2^{m_1}-R_1^{m_1}R_2^{m_2}},\\ \end{array}\right. } \end{aligned}$$

(11.10)

$$\begin{aligned} {\left\{ \begin{array}{ll} A=|\boldsymbol{\nabla }T_0|,\\ \\ B=\dfrac{R_1^{s_2}R_2-R_1R_2^{s_2}}{R_1^{s_2}R_2^{s_1}-R_1^{s_1}R_2^{s_2}}|\boldsymbol{\nabla }T_0|,\\ \\ C=-\dfrac{R_1^{s_1}R_2-R_1R_2^{s_1}}{R_1^{s_2}R_2^{s_1}-R_1^{s_1}R_2^{s_2}}|\boldsymbol{\nabla }T_0|,\\ \\ E=0,\\ \\ \kappa _1=\kappa _{rr}\dfrac{s_1\left( -R_1^{s_1}R_2^{s_2}+R_1^{s_1+s_2-1}R_2\right) +s_2\left( R_1^{s_2}R_2^{s_1}-R_1^{s_1+s_2-1}R_2\right) }{R_1^{s_2}R_2^{s_1}-R_1^{s_1}R_2^{s_2}},\\ \\ \kappa _3=\kappa _{rr}\dfrac{s_1\left( R_1^{s_2}R_2^{s_1}-R_1R_2^{s_1+s_2-1}\right) +s_2\left( -R_1^{s_1}R_2^{s_2}+R_1R_2^{s_1+s_2-1}\right) }{R_1^{s_2}R_2^{s_1}-R_1^{s_1}R_2^{s_2}},\\ \end{array}\right. } \end{aligned}$$

(11.11)

Equations (11.10) and (11.11) have similar forms with only different $m_{1,\,2}$ and $s_{1,\,2}$.