Keywords

1 Opening Remarks

Precision measurement is indispensable in many fields, so high-performance sensors are crucial. Generally, when a sensor is put in a physical field, it will distort the physical field. Therefore, the measured value is not the original one, thus making the sensor inaccurate. In addition to inaccuracy, the perturbation induced by the sensor also makes the sensor “visible”, which is adverse in many practical applications. The methods of scattering cancellation [1] and transformation optics [2] were proposed to design invisible electromagnetic sensors. Invisible acoustic sensors [3,4,5] and invisible magnetic sensors [6] were also presented successively.

Invisible thermal sensors also attracted research interest. The methods of scattering cancellation [7,8,9], neutral inclusion [10], and transformation thermotics [11] were put forward to design invisible thermal sensors. Furthermore, an invisible multiphysical sensor was also fabricated for both thermal and electric detection [12]. These studies focused on thermal invisibility because it is particularly important to fight against infrared detection. For example, invisibility can protect the sensor from being discovered when a thermal sensor detects temperature. However, accuracy is almost neglected in these schemes, so the detected temperature has deviations from the original one, thus making thermal sensors inaccurate. Meanwhile, invisible thermal sensors for nonlinear cases are still lacking, limiting practical applications. Here, “nonlinear” means that thermal conductivities are temperature-dependent.

To solve the problem, we propose a bilayer scheme to design invisible thermal sensors, even considering geometrically anisotropic and thermally nonlinear cases. These two points benefit practical applications because thermal sensors do not have to be geometrically isotropic, and nonlinear thermal conductivity is common. In fact, bilayer scheme has achieved great success in designing thermal cloaks [13,14,15,16,17,18], thermal concentrators [19], and chameleonlike metashells [20, 21]. Cloaks make the temperature gradient in the center zero; concentrators make the temperature gradient in the center steeper than that in the matrix; invisible sensors keep the same temperature gradient in the sensor and matrix. We derive two groups of thermal conductivities by solving the linear Laplace equation, making thermal sensors both accurate and invisible. Moreover, we prove that the bilayer scheme can be directly extended to thermally nonlinear cases as long as the ratio between the nonlinear thermal conductivities of the sensor and matrix is a temperature-independent constant.

Fig. 10.1
figure 1

Adapted from Ref. [22]

Schematic diagrams of a geometrically isotropic case and b geometrically anisotropic case.

Full size image

2 Linear and Geometrically Isotropic Case

We discuss the case shown in Fig. 10.1a. The Cartesian coordinates are denoted as \(x_i\) (\(i=1,\,2\) for two dimensions and \(i=1,\,2,\,3\) for three dimensions). The radii of the core, inner shell, and outer shell are denoted as \(\lambda _a\), \(\lambda _b\), and \(\lambda _c\), respectively. The thermal conductivities of the core, inner shell, outer shell, and matrix are denoted as \(\kappa _a\), \(\kappa _b\), \(\kappa _c\), and \(\kappa _d\), respectively. Since the geometry is isotropic, we discuss the case in cylindrical coordinates \(\left( r,\,\theta \right) \) or spherical coordinates \(\left( r,\,\theta ,\,\varphi \right) \). Here, two dimensions and three dimensions are similar because \(\varphi \) essentially does not matter. In the presence of an external linear thermal field \(\boldsymbol{G}_0\), the temperature profiles in different regions can be expressed as

$$\begin{aligned} T_a&=u_a r\cos \theta , \end{aligned}$$
(10.1a)
$$\begin{aligned} T_b&=\left( u_b r+v_b r^{-\tau }\right) \cos \theta , \end{aligned}$$
(10.1b)
$$\begin{aligned} T_c&=\left( u_c r+v_c r^{-\tau }\right) \cos \theta , \end{aligned}$$
(10.1c)
$$\begin{aligned} T_d&=\left( u_d r+v_d r^{-\tau }\right) \cos \theta , \end{aligned}$$
(10.1d)

where \(T_a\), \(T_b\), \(T_c\), and \(T_d\) are the temperatures in the core, inner shell, outer shell, and matrix, respectively. \(\tau =1\) for two dimensions and \(\tau =2\) for three dimensions. \(u_a\), \(u_b\), \(v_b\), \(u_c\), \(v_c\), \(u_d\), and \(v_d\) are seven coefficients to be determined by the following boundary conditions,

$$\begin{aligned} u_a \lambda _a&=u_b \lambda _a+v_b\lambda _a^{-\tau }, \end{aligned}$$
(10.2a)
$$\begin{aligned} u_b \lambda _b+v_b\lambda _b^{-\tau }&=u_c \lambda _b+v_c\lambda _b^{-\tau }, \end{aligned}$$
(10.2b)
$$\begin{aligned} u_c \lambda _c+v_c\lambda _c^{-\tau }&=u_d \lambda _c+v_d\lambda _c^{-\tau }, \end{aligned}$$
(10.2c)
$$\begin{aligned} \kappa _a u_a&=\kappa _b\left( u_b-\tau v_b\lambda _a^{-\tau -1}\right) , \end{aligned}$$
(10.2d)
$$\begin{aligned} \kappa _b\left( u_b-\tau v_b\lambda _b^{-\tau -1}\right)&=\kappa _c\left( u_c-\tau v_c\lambda _b^{-\tau -1}\right) , \end{aligned}$$
(10.2e)
$$\begin{aligned} \kappa _c\left( u_c-\tau v_c\lambda _c^{-\tau -1}\right)&=\kappa _d\left( u_d-\tau v_d\lambda _c^{-\tau -1}\right) , \end{aligned}$$
(10.2f)
$$\begin{aligned} u_d&=G_0, \end{aligned}$$
(10.2g)
$$\begin{aligned} v_d&=0, \end{aligned}$$
(10.2h)
$$\begin{aligned} u_a&=u_d. \end{aligned}$$
(10.2i)

Equations (10.2a)–(10.2c) and (10.2d)–(10.2f) indicate the continuities of temperature and heat flux, respectively. Equations (10.2g) and (10.2h) ensure a linear thermal field in the matrix, thus making the sensor thermally invisible. Equation (10.2i) makes the temperature in the sensor the same as that in the matrix, thus ensuring accurate detection. We take \(\kappa _b\) and \(\kappa _c\) as other two coefficients which (together with the seven coefficients in Eqs. (10.1a)–(10.1d)) can be determined by the nine equations in Eq. (10.2). Therefore, \(\kappa _b\) and \(\kappa _c\) can be solved as

$$\begin{aligned} \kappa _b^{\left( 1\right) }=\frac{\kappa _a \alpha _3-\kappa _d \alpha _1+\sqrt{\left( \kappa _a-\kappa _d\right) \left( \kappa _a \alpha _2^2-\kappa _d \alpha _1^2\right) }}{\alpha _5}, \end{aligned}$$
(10.3a)
$$\begin{aligned} \kappa _c^{\left( 1\right) }=\frac{\kappa _a \alpha _2-\kappa _d \alpha _4-\sqrt{\left( \kappa _a-\kappa _d\right) \left( \kappa _a \alpha _2^2-\kappa _d \alpha _1^2\right) }}{\alpha _6}, \end{aligned}$$
(10.3b)

or

$$\begin{aligned} \kappa _b^{\left( 2\right) }=\frac{\kappa _a \alpha _3-\kappa _d \alpha _1-\sqrt{\left( \kappa _a-\kappa _d\right) \left( \kappa _a \alpha _2^2-\kappa _d \alpha _1^2\right) }}{\alpha _5}, \end{aligned}$$
(10.4a)
$$\begin{aligned} \kappa _c^{\left( 2\right) }=\frac{\kappa _a \alpha _2-\kappa _d \alpha _4+\sqrt{\left( \kappa _a-\kappa _d\right) \left( \kappa _a \alpha _2^2-\kappa _d \alpha _1^2\right) }}{\alpha _6}, \end{aligned}$$
(10.4b)

where

$$\begin{aligned} \alpha _1&=\lambda _a^{1+\tau }\left( \lambda _b^{1+\tau }+\tau \lambda _c^{1+\tau }\right) +\lambda _b^{1+\tau }\left[ \tau \lambda _b^{1+\tau }-\left( 2\tau +1\right) \lambda _c^{1+\tau }\right] ,\end{aligned}$$
(10.5a)
$$\begin{aligned} \alpha _2&=\lambda _a^{1+\tau }\left[ \left( 2\tau +1\right) \lambda _b^{1+\tau }-\tau \lambda _c^{1+\tau }\right] -\lambda _b^{1+\tau }\left( \tau \lambda _b^{1+\tau }+\lambda _c^{1+\tau }\right) ,\end{aligned}$$
(10.5b)
$$\begin{aligned} \alpha _3&=\lambda _a^{1+\tau }\left[ 2\tau \lambda _a^{1+\tau }-\left( 2\tau -1\right) \lambda _b^{1+\tau }-\tau \lambda _c^{1+\tau }\right] +\lambda _b^{1+\tau }\left( \tau \lambda _b^{1+\tau }-\lambda _c^{1+\tau }\right) ,\end{aligned}$$
(10.5c)
$$\begin{aligned} \alpha _4&=\lambda _b^{1+\tau }\left( \lambda _a^{1+\tau }-\tau \lambda _b^{1+\tau }\right) +\lambda _c^{1+\tau }\left[ \tau \lambda _a^{1+\tau }+\left( 2\tau -1\right) \lambda _b^{1+\tau }-2\tau \lambda _c^{1+\tau }\right] ,\end{aligned}$$
(10.5d)
$$\begin{aligned} \alpha _5&=2\tau \left( \lambda _a^{1+\tau }-\lambda _b^{1+\tau }\right) \left( \lambda _a^{1+\tau }-\lambda _c^{1+\tau }\right) ,\end{aligned}$$
(10.5e)
$$\begin{aligned} \alpha _6&=2\tau \left( \lambda _a^{1+\tau }-\lambda _c^{1+\tau }\right) \left( \lambda _b^{1+\tau }-\lambda _c^{1+\tau }\right) . \end{aligned}$$
(10.5f)

When \(\kappa _a<\kappa _d\), Eqs. (10.3) and (10.4) are always positive. When \(\kappa _a=\kappa _d\), the sensor has the same thermal conductivity as the matrix, resulting in \(\kappa _b=\kappa _c=\kappa _a=\kappa _d\), so the bilayer scheme is not necessary. When \(\kappa _a>\kappa _d\), Eqs. (10.3b) and (10.4a) are negative. Negative thermal conductivity means that the direction of heat flux is from low temperature to high temperature, which can be effectively realized by introducing extra energy [23]. Also, we do not need to worry about complex values as long as the value of \(\lambda _b\) is appropriately chosen. Physically, when \(\kappa _a<\kappa _d\), the temperature gradient in the sensor is larger than that in the matrix, and the bilayer scheme can reduce the temperature gradient to make the temperature gradients in the sensor and matrix the same. When \(\kappa _a>\kappa _d\), the temperature gradient in the sensor is smaller than that in the matrix, but the bilayer scheme cannot enhance the temperature gradient with only positive thermal conductivities.

3 Linear and Geometrically Anisotropic Case

We discuss the case shown in Fig. 10.1b. The semi axes of the core, inner shell, and outer shell are denoted as \(\lambda _{ai}\), \(\lambda _{bi}\), and \(\lambda _{ci}\), respectively (\(i=1,\,2\) for two dimensions and \(i=1,\,2,\,3\) for three dimensions). Since the geometry is anisotropic, we discuss the case in elliptical coordinates \(\left( \rho ,\,\xi \right) \) or ellipsoidal coordinates \(\left( \rho ,\,\xi ,\,\eta \right) \). Here, although two dimensions and three dimensions are different, we can remove the terms associated with \(\eta \) and \(x_3\) to reduce three dimensions to two dimensions. The ellipsoidal coordinates \(\left( \rho ,\,\xi ,\,\eta \right) \) can be expressed as

$$\begin{aligned} \frac{x_1^2}{\rho +\lambda _{a1}^2}+\frac{x_2^2}{\rho +\lambda _{a2}^2}+\frac{x_3^2}{\rho +\lambda _{a3}^2}&=1,\end{aligned}$$
(10.6a)
$$\begin{aligned} \frac{x_1^2}{\xi +\lambda _{a1}^2}+\frac{x_2^2}{\xi +\lambda _{a2}^2}+\frac{x_3^2}{\xi +\lambda _{a3}^2}&=1,\end{aligned}$$
(10.6b)
$$\begin{aligned} \frac{x_1^2}{\eta +\lambda _{a1}^2}+\frac{x_2^2}{\eta +\lambda _{a2}^2}+\frac{x_3^2}{\eta +\lambda _{a3}^2}&=1, \end{aligned}$$
(10.6c)

where \(\rho =\textrm{constant}\) denotes an ellipsoidal surface, and \(\lambda _i\) is the semi axis of the ellipsoid (\(\rho =\textrm{constant}\)) along \(x_i\) axis. Accordingly, the Cartesian coordinates can be expressed as

$$\begin{aligned} x_1^2&=\frac{\left( \rho +\lambda _{a1}^2\right) \left( \xi +\lambda _{a1}^2\right) \left( \eta +\lambda _{a1}^2\right) }{\left( \lambda _{a1}^2-\lambda _{a2}^2\right) \left( \lambda _{a1}^2-\lambda _{a3}^2\right) },\end{aligned}$$
(10.7a)
$$\begin{aligned} x_2^2&=\frac{\left( \rho +\lambda _{a2}^2\right) \left( \xi +\lambda _{a2}^2\right) \left( \eta +\lambda _{a2}^2\right) }{\left( \lambda _{a2}^2-\lambda _{a1}^2\right) \left( \lambda _{a2}^2-\lambda _{a3}^2\right) },\end{aligned}$$
(10.7b)
$$\begin{aligned} x_3^2&=\frac{\left( \rho +\lambda _{a3}^2\right) \left( \xi +\lambda _{a3}^2\right) \left( \eta +\lambda _{a3}^2\right) }{\left( \lambda _{a3}^2-\lambda _{a1}^2\right) \left( \lambda _{a3}^2-\lambda _{a2}^2\right) }. \end{aligned}$$
(10.7c)

In the presence of an external linear thermal field \(\boldsymbol{G}_0\) along \(x_i\) axis, the temperature profiles in different regions can be expressed as [24]

$$\begin{aligned} T_a&=u_a x_i, \end{aligned}$$
(10.8a)
$$\begin{aligned} T_b&=\left[ u_b+v_b\int _{\rho _a}^\rho \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\right] x_i, \end{aligned}$$
(10.8b)
$$\begin{aligned} T_c&=\left[ u_c+v_c\int _{\rho _a}^\rho \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\right] x_i, \end{aligned}$$
(10.8c)
$$\begin{aligned} T_d&=\left[ u_d+v_d\int _{\rho _a}^\rho \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\right] x_i, \end{aligned}$$
(10.8d)

where \(g\left( \rho \right) =\sqrt{\left( \rho +\lambda _{a1}^2\right) \left( \rho +\lambda _{a2}^2\right) \left( \rho +\lambda _{a3}^2\right) }=\lambda _1\lambda _2\lambda _3\), and \(\rho _a\) \(\left( =0\right) \) denotes the ellipsoidal core surface with semiaxes \(\lambda _{ai}\). As explained above, \(g\left( \rho \right) =\sqrt{\left( \rho +\lambda _{a1}^2\right) \left( \rho +\lambda _{a2}^2\right) }=\lambda _1\lambda _2\) for two dimensions.

We use two mathematical skills to proceed. The first one is associated with the temperature derivations in Eq. (10.8),

$$\begin{aligned} \frac{\partial x_i}{\partial \rho }&=\frac{x_i}{2\left( \rho +\lambda _{ai}^2\right) },\end{aligned}$$
(10.9a)
$$\begin{aligned} \frac{\partial }{\partial \rho }\left[ x_i\int \limits _{\rho _a}^\rho \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\right]&=\frac{x_i}{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) } \\ \nonumber&\quad +\frac{x_i}{2\left( \rho +\lambda _{ai}^2\right) }\int \limits _{\rho _a}^\rho \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }. \end{aligned}$$
(10.9b)

The second one is related to the integrations in Eqs. (10.8b)–(10.8d) which can be rewritten as

$$\begin{aligned} \int \limits _{\rho _a}^\rho \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }&=\int \limits _{\rho _a}^\infty \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }-\int \limits _{\rho }^\infty \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\\ \nonumber&=\frac{2L_{ai}}{g\left( \rho _a\right) }-\frac{2L_{i}}{g\left( \rho \right) }, \end{aligned}$$
(10.10)

where \(L_{ai}\) and \(L_i\) are shape factors along \(x_i\) axis,

$$\begin{aligned} L_{ai}&=\frac{g\left( \rho _a\right) }{2}\int \limits _{\rho _a}^\infty \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) },\end{aligned}$$
(10.11a)
$$\begin{aligned} L_i&=\frac{g\left( \rho \right) }{2}\int \limits _{\rho }^\infty \frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }. \end{aligned}$$
(10.11b)

Then, the boundary conditions can be expressed as

$$\begin{aligned} u_a&=u_b, \end{aligned}$$
(10.12a)
$$\begin{aligned} u_b+v_b \int \limits _{\rho _a}^{\rho _b}\frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }&=u_c+v_c \int \limits _{\rho _a}^{\rho _b}\frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }, \end{aligned}$$
(10.12b)
$$\begin{aligned} u_c+v_c\int \limits _{\rho _a}^{\rho _c}\frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }&=u_d+v_d \int \limits _{\rho _a}^{\rho _c}\frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }, \end{aligned}$$
(10.12c)
$$\begin{aligned} \kappa _a u_a&=\kappa _b\left[ u_b+\frac{2 v_b}{g\left( \rho _a\right) }\right] , \end{aligned}$$
(10.12d)
$$\begin{aligned} \kappa _b\left[ u_b+\frac{2 v_b}{g\left( \rho _b\right) }+v_b\int \limits _{\rho _a}^{\rho _b}\frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\right]&=\kappa _c\left[ u_c+\frac{2 v_c}{g\left( \rho _b\right) }+v_c\int \limits _{\rho _a}^{\rho _b}\frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\right] , \end{aligned}$$
(10.12e)
$$\begin{aligned} \kappa _c\left[ u_c+\frac{2 v_c}{g\left( \rho _c\right) }+v_c\int \limits _{\rho _a}^{\rho _c}\frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\right]&=\kappa _d\left[ u_d+\frac{2 v_d}{g\left( \rho _c\right) }+v_d\int \limits _{\rho _a}^{\rho _c}\frac{d\rho }{\left( \rho +\lambda _{ai}^2\right) g\left( \rho \right) }\right] , \end{aligned}$$
(10.12f)
$$\begin{aligned} u_d&=G_0, \end{aligned}$$
(10.12g)
$$\begin{aligned} v_d&=0, \end{aligned}$$
(10.12h)
$$\begin{aligned} u_a&=u_d. \end{aligned}$$
(10.12i)

The physical understanding of Eq. (10.12) is similar to Eq. (10.2). Similarly, we can derive two groups of thermal conductivities as

$$\begin{aligned} \kappa _b^{\left( 1\right) }=\beta ^{\left( 1\right) }\left( \kappa _a,\,\kappa _d,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right) , \end{aligned}$$
(10.13a)
$$\begin{aligned} \kappa _c^{\left( 1\right) }=\gamma ^{\left( 1\right) }\left( \kappa _a,\,\kappa _d,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right) , \end{aligned}$$
(10.13b)

or

$$\begin{aligned} \kappa _b^{\left( 2\right) }=\beta ^{\left( 2\right) }\left( \kappa _a,\,\kappa _d,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right) , \end{aligned}$$
(10.14a)
$$\begin{aligned} \kappa _c^{\left( 2\right) }=\gamma ^{\left( 2\right) }\left( \kappa _a,\,\kappa _d,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right) , \end{aligned}$$
(10.14b)

where \(\beta ^{\left( 1\right) }\), \(\gamma ^{\left( 1\right) }\,\left[ <\beta ^{\left( 1\right) }\right] \), \(\beta ^{\left( 2\right) }\), and \(\gamma ^{\left( 2\right) }\,\left[ >\beta ^{\left( 2\right) }\right] \) are four functions determined by Eq. (10.12). The physical understanding of Eqs. (10.13) and (10.14) are consistent with that of Eqs. (10.3) and (10.4). The isotropic case with Eqs. (10.3) and (10.4) is very complicated, let alone the anisotropic case with Eqs. (10.13) and (10.14). Therefore, we use Mathematica to calculate thermal conductivities with determined \(\left( \kappa _a,\,\kappa _d,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right) \) when performing simulations. Certainly, the anisotropic case with Eqs. (10.13) and (10.14) can be reduced to the isotropic case with Eqs. (10.3) and (10.4). We do not start from Eqs. (10.13) and (10.14) to derive Eqs. (10.3) and (10.4) because Eqs. (10.13) and (10.14) are too complicated to simplify.

4 Nonlinear Case

We discuss the thermally nonlinear case where thermal conductivities are dependent on temperature. This consideration is necessary because many common materials, such as silicon and germanium, are nonlinear. We suppose the thermal conductivity of the matrix to be \(\kappa _d\left( T\right) =\kappa _d f\left( T\right) \), where \(f\left( T\right) \) can be any temperature-dependent functions. Then, we prove that the bilayer scheme can also be applied for thermally nonlinear cases as long as the ratio between the nonlinear thermal conductivities of core and matrix is a temperature-independent constant, namely \(\kappa _d\left( T\right) /\kappa _a\left( T\right) =\kappa _d/\kappa _a\). Therefore, the thermal conductivity of the core should be \(\kappa _a\left( T\right) =\kappa _a f\left( T\right) \).

We directly substitute \(\kappa _d\left( T\right) \) and \(\kappa _a\left( T\right) \) into Eqs. (10.3) and (10.4). Then, we can also derive two groups of \(\kappa _b\left( T\right) \) and \(\kappa _c\left( T\right) \) which satisfy

$$\begin{aligned} \kappa _b\left( T\right) =\kappa _bf\left( T\right) , \end{aligned}$$
(10.15a)
$$\begin{aligned} \kappa _c\left( T\right) =\kappa _cf\left( T\right) . \end{aligned}$$
(10.15b)

Here, superscripts are omitted because both two groups of thermal conductivities satisfy this property. More generally, we substitute \(\kappa _d\left( T\right) \) and \(\kappa _a\left( T\right) \) into Eqs. (10.13) and (10.14). \(\kappa _b\left( T\right) \) and \(\kappa _c\left( T\right) \) also satisfy

$$\begin{aligned} \nonumber \kappa _b\left( T\right)&=\beta \left[ \kappa _af\left( T\right) ,\,\kappa _df\left( T\right) ,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right] \\&=\beta \left[ \kappa _a,\,\kappa _d,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right] f\left( T\right) =\kappa _bf\left( T\right) ,\end{aligned}$$
(10.16a)
$$\begin{aligned} \kappa _c\left( T\right)&=\gamma \left[ \kappa _af\left( T\right) ,\,\kappa _df\left( T\right) ,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right] \nonumber \\&=\gamma \left[ \kappa _a,\,\kappa _d,\,\lambda _{ai},\,\lambda _{bi},\,\lambda _{ci}\right] f\left( T\right) =\kappa _cf\left( T\right) . \end{aligned}$$
(10.16b)

Such a property allows us to transform the nonlinear Laplace equation into the linear Laplace equation. Meanwhile, general solutions are consistent in different regions. The nonlinear Laplace equation in different regions can be expressed as

$$\begin{aligned} \nonumber \boldsymbol{\nabla }\cdot \left[ -\kappa _{a,\,b,\,c,\,d}\left( T\right) \boldsymbol{\nabla }T\right]&=\boldsymbol{\nabla }\cdot \left[ -\kappa _{a,\,b,\,c,\,d}f\left( T\right) \boldsymbol{\nabla }T\right] \\&=\boldsymbol{\nabla }\cdot \left[ -\kappa _{a,\,b,\,c,\,d}\boldsymbol{\nabla }h\left( T\right) \right] =0, \end{aligned}$$
(10.17a)

where \(\partial h\left( T\right) /\partial T=f\left( T\right) \). In other words, as long as we replace T with \(h\left( T\right) \), the nonlinear Laplace equation can be transformed into the linear Laplace equation. Therefore, the above theories can be applied without any correction. The only assumption is that the ratio between the nonlinear thermal conductivities of sensor and matrix is a temperature-independent constant.

5 Finite-Element Simulation

We use the template of solid heat transfer in COMSOL Multiphysics to confirm these theoretical analyses. Without loss of generality, we perform simulations in two dimensions. Although interfacial thermal resistance may exist in practice [25, 26], its macroscopic effect is usually small. Therefore, we neglect the interfacial thermal resistance in simulations.

Fig. 10.2
figure 2

Adapted from Ref. [22]

Simulations of geometrically isotropic case. a Sensor embedded in the matrix. b Sensor coated by the monolayer scheme proposed in Ref. [12] with inner and outer radii of \(\lambda _a\) and \(\lambda _c\), respectively. The thermal conductivity of the single layer is 161.1 W m\(^{-1}\) K\(^{-1}\). c Sensor coated by two layers designed with Eq. (10.3). d Sensor coated by two layers designed with Eq. (10.4). e Temperature gradients on the dashed lines in a–d as a function of \(x_1\). The simulation size is \(10\times 10\) cm\(^2\). The temperatures of the left and right boundaries are set at 313 and 283 K. Other boundaries are insulated. \(\lambda _a=2\), \(\lambda _b=2.5\), \(\lambda _c=3\) cm, and \(\kappa _a=50\), \(\kappa _d=100\) W m\(^{-1}\) K\(^{-1}\). \(\kappa _b^{\left( 1\right) }=378.5\), \(\kappa _c^{\left( 1\right) }=58.5\), and \(\kappa _b^{\left( 2\right) }=26.7\), \(\kappa _c^{\left( 2\right) }=346.3\) W m\(^{-1}\) K\(^{-1}\).

Full size image

Firstly, we discuss the geometrically isotropic case in Fig. 10.2. A thermal sensor is embedded in the matrix for temperature detection. Since the thermal conductivity of the sensor is different from that of the matrix, the whole temperature profile is distorted (Fig. 10.2a). Therefore, the sensor is not only thermally visible but also inaccurate. When a pioneering monolayer scheme [12] is applied, it can ensure thermal invisibility. However, it does not perform well in detecting accuracy because the temperature in the sensor is still different from the original one (Fig. 10.2b). Then, we resort to the bilayer scheme. We coat the sensor with the bilayer scheme whose thermal conductivities are designed according to Eq. (10.3), and the simulation result is shown in Fig. 10.2c. The temperature in the matrix becomes linear again, thus making the sensor thermally invisible. Meanwhile, the temperature in the sensor is the same as the original one, thus ensuring accurate detection. We also design the thermal conductivities of the two layers according to Eq. (10.4), and the same effect can be obtained (Fig. 10.2d). For quantitative comparison, we export the data on the dashed lines in Fig. 10.2a–d. Since the temperature difference is not significant enough to observe, we export temperature gradient \(\partial T/\partial x_1\) for comparison. The result is presented in Fig. 10.2e, indicating that the bilayer scheme can indeed simultaneously ensure thermal invisibility and accurate detection. The inset of Fig. 10.2e shows the temperature profile of a pure matrix with a linear thermal field of \(-300\) K/m.

Fig. 10.3
figure 3

Adapted from Ref. [22]

Simulations of geometrically anisotropic case. a Pure matrix. b Sensor embedded in the matrix. c Sensor coated by the monolayer scheme proposed in Ref. [16] whose thermal conductivity is 149.5 W m\(^{-1}\) K\(^{-1}\). d Sensor coated by two layers designed with Eq. (10.13). e Sensor coated by two layers designed with Eq. (10.14). f Temperature difference with the temperature in c minus that in a. g Temperature difference with the temperature in d minus that in a. h Temperature difference with the temperature in e minus that in a. \(\lambda _{a1}=2\), \(\lambda _{a2}=1\), \(\lambda _{b1}=2.5\), \(\lambda _{b2}=1.8\), \(\lambda _{c1}=3\), \(\lambda _{c2}=2.45\) cm, and \(\kappa _a=5\), \(\kappa _d=100\) W m\(^{-1}\) K\(^{-1}\). \(\kappa _b^{\left( 1\right) }=274.5\), \(\kappa _c^{\left( 1\right) }=61.8\), and \(\kappa _b^{\left( 2\right) }=2.4\), \(\kappa _c^{\left( 2\right) }=342.1\) W m\(^{-1}\) K\(^{-1}\).

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Then, we discuss the geometrically anisotropic case in Fig. 10.3, which is more practical. The results are similar to the geometrically isotropic case. Figure 10.3a and b demonstrate the temperature profiles without and with a sensor embedded in the matrix, respectively. The sensor distorts the whole temperature profile, resulting in thermal visibility and inaccurate sensor detection. When the monolayer scheme [16] is applied, it can ensure thermal invisibility, but the temperature in the sensor is still changed (Fig. 10.3c). Figure 10.3d and e shows the results coated by two layers designed with Eqs. (10.13) and (10.14), respectively. Again, the temperatures in the matrix and sensor become the same. Therefore, the sensor is thermally invisible and accurate. For clarity, we plot the temperature difference \(\Delta \) with the temperature in Fig. 10.3c (Fig. 10.3d or e) minus that in Fig. 10.3a, which is shown in Fig. 10.3f (Fig. 10.3g or h). Our scheme ensures that the temperature difference \(\Delta \) in the matrix and sensor is always zero, confirming an accurate and thermally invisible sensor.

Fig. 10.4
figure 4

Adapted from Ref. [22]

Simulations of thermally nonlinear case. The temperatures of the left and right boundaries are set at 2283 and 283 K, respectively. \(f\left( T\right) =1+10^{-9}T^3\). The other parameters are the same as those for Fig. 10.3.

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Finally, we discuss the thermally nonlinear case in Fig. 10.4. Nonlinear (temperature-dependent) thermal conductivities, whether weak or strong, are common in nature. Here, “strong” (or “weak”) means that the nonlinear (or linear) term of thermal conductivity is dominant. Therefore, it is necessary to extend our scheme to thermally nonlinear cases. To make nonlinear properties clear, we discuss strong nonlinearity directly. A typical case of strong nonlinearity is the thermal radiation described by the Rosseland diffusion approximation, which is proportional to \(T^3\) [27,28,29]. Therefore, we take on \(f\left( T\right) =\mu +\nu T^3\) with \(\mu \) and \(\nu \) being two constants. We set a high temperature at 2283 K K, and aerogel (or ceramic), with excellent tolerance to high temperatures, can be applied to observe thermal nonlinearity. As proved in Eq. (10.16), we can directly multiply the original thermal conductivities with \(f\left( T\right) \) to proceed.

Since the thermal conductivity of the matrix is nonlinear, the temperature gradient is no longer a constant (Fig. 10.4a). When an elliptical sensor is embedded in the matrix, the straight isotherms are distorted (Fig. 10.4b). Then, we coat the sensor with two layers designed with Eq. (10.16). The simulation results are presented in Fig. 10.4c and d, respectively. The distorted isotherms in the matrix and sensor restore. Similarly, we also plot the temperature difference \(\Delta \) with the temperature in Fig. 10.4c (or Fig. 10.4d) minus that in Fig. 10.4a, and the results are shown in Fig. 10.4e (or Fig. 10.4f). We can observe zero temperature difference \(\Delta \) in the matrix and sensor, so the bilayer scheme performs satisfactorily.

The bilayer scheme can be extended to transient regimes by considering heat capacity and density [30,31,32]. Since invisibility is a special case of camouflage, these results may guide thermal camouflage [33,34,35,36,37,38,39,40]. The present scheme is dependent on elliptical/ellipsoid shapes because the Laplace equation can be analytically handled. Therefore, other methods remain to be explored for complex shapes [41, 42], such as combining neutral inclusion [10] and transformation thermotics [43, 44].

6 Conclusion

We have proposed a bilayer scheme to design invisible thermal sensors. Compared with existing schemes, the present one is accurate and applicable for geometrically anisotropic and thermally nonlinear cases. Thermal invisibility can protect sensors from being detected, and accurate detection benefits practical applications. The extensions to geometric anisotropy and thermal nonlinearity make thermal sensors more widely applicable. Moreover, we unify two/three-dimensional cases, isotropic/anisotropic cases, and linear/nonlinear cases with a single theoretical framework, laying a solid foundation for designing thermal metamaterials under different conditions.

7 Exercise and Solution

Exercise

1. Take the interfacial thermal resistance discussed in Ref. [26] into account and derive the corresponding parameters for invisible thermal sensors.

Solution

1. When interfacial thermal resistance is taken into consideration, temperature jumps will occur at the interfaces of the system. Therefore, the boundary conditions associated with the continuity of temperatures (Eqs. (10.2a)–(10.2c)) should be rewritten as [26]

$$\begin{aligned} u_a \lambda _a+R_{ab}\kappa _a u_a&=u_b \lambda _a+v_b\lambda _a^{-\tau }, \end{aligned}$$
(10.18a)
$$\begin{aligned} u_b \lambda _b+v_b\lambda _b^{-\tau }+R_{bc}\kappa _b\left( u_b-\tau v_b \lambda _b^{-\tau -1}\right)&=u_c \lambda _b+v_c\lambda _b^{-\tau }, \end{aligned}$$
(10.18b)
$$\begin{aligned} u_c \lambda _c+v_c\lambda _c^{-\tau }+R_{cd}\kappa _c\left( u_c-\tau v_c \lambda _c^{-\tau -1}\right)&=u_d \lambda _c+v_d\lambda _c^{-\tau }, \end{aligned}$$
(10.18c)

where \(R_{ab}\), \(R_{bc}\), and \(R_{cd}\) are the interfacial thermal resistances of the sensor and first layer, the first layer and second layer, and the second layer and matrix, respectively. The other boundary conditions are unchanged.