Theory for Active Thermal Control: Thermal Dipole Effect

Abstract

In this chapter, we establish a theory for thermal-dipole-based thermotics. Tailoring the thermal dipole moment allows thermal invisibility without the requirements of singular and uncommon thermal conductivities. Furthermore, finite-element simulations and laboratory experiments both validate the theoretical analyses. Thermal-dipole-based thermotics offers a distinct mechanism to achieve thermal invisibility and provides guidance to other physical fields, such as electrostatics, magnetostatics, and particle diffusion. These results also pave the way for heat regulation with thermal dipoles, and potential applications can be expected in thermal protection, infrared detection, etc.

1 Opening Remarks

With growing concerns about energy issues, many researchers have turned their research focus to heat management. The emerging field of thermal metamaterials mainly drove this trend in the last decade. The most representative example is thermal invisibility [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16], which has almost run through the development of thermal metamaterials. Thermal invisibility is characterized by the uniform thermal field of the matrix. Many schemes have been proposed for this realization, but they have shortcomings. The initial exploration is based on transformation thermotics [1,2,3,4,5,6], which is the thermal counterpart of transformation optics [17]. However, transformation thermotics leads to four severe problems, thus limiting practical applications. The first is anisotropy which requires different radial and tangential components of the tensorial thermal conductivity. The second is inhomogeneity which means a spatially-distributed thermal conductivity. The third is singularity which requires zero and/or infinite thermal conductivities. The fourth is extremely large thermal conductivities which are uncommon. Thermal conductivities of natural materials range only from 0.026 W m$^{-1}$ K$^{-1}$ (air) to 430 W m$^{-1}$ K$^{-1}$ (silver). Thermal conductivities out of this range are uncommon [16].

Fig. 17.1

Adapted from Ref. [19]

Approaches to thermal invisibility with a a bilayer cloak, b the concept of neutral inclusion, c a near-zero-index cloak, and d a thermal dipole. None of these approaches can simultaneously remove the problem of singular and uncommon thermal conductivities except for our thermal-dipole-based scheme.

Although these problems restrict practical applications, they also promote the development of thermal metamaterials by solving them. Fortunately, the issues of anisotropy and inhomogeneity were solved soon [7,8,9,10,11,12,13,14,15,16]. However, the issue of singular and uncommon thermal conductivity still cannot be solved simultaneously. For example, we discuss a matrix with a very high thermal conductivity, such as copper (400 W m$^{-1}$ K$^{-1}$) because high thermal conductivities correspond to the high efficiency of heat transfer. When a bilayer cloak [7,8,9,10,11] is designed, the thermal conductivities of the inner and outer shells are, respectively, 0 and 2615 W m$^{-1}$ K$^{-1}$, which are singular and inexistent; see Fig. 17.1a. When the concept of neutral inclusion [12,13,14,15] is used, the thermal conductivity of the shell should be 727 W m$^{-1}$ K$^{-1}$, which is also uncommon; see Fig. 17.1b. When a near-zero-index cloak [16] is designed, the thermal conductivity of the inner shell should tend to infinity, which is singular; see Fig. 17.1c. These two problems (singular and uncommon) largely restrict the further development of thermal metamaterials because uncommon thermal conductivity is difficult to achieve, and the realization of singularity (mainly the infinite thermal conductivity) depends on very complex devices, such as thermal convection [16].

To completely solve these two problems, we propose a theory for thermal-dipole-based thermotics, which can simultaneously remove the requirements of singular and uncommon thermal conductivities. We even do not require to design any shell (or metamaterial), and a thermal dipole is enough; see Fig. 17.1d. This advantage originates from the particularity of the thermal field of a thermal dipole, which can just offset the influence of a particle by designing the thermal dipole moment (M). In what follows, we establish the theory for thermal-dipole-based thermotics in two dimensions. Finite-element simulations and laboratory experiments further validate the approach. The thermal dipole effect provides a distinct mechanism for controlling heat with heat, inspiring the thermal counterpart of coherent perfect absorbtion [18].

2 Thermal-Dipole-Based Thermotics

Thermal invisibility aims to keep the thermal field of the matrix undistorted. Therefore, we focus on the thermal field of the matrix in what follows. In the presence of an external uniform thermal field $\boldsymbol{G}_0$, when there is a particle (with thermal conductivity $\kappa _p$ and radius $r_p$) embedded in the matrix (with thermal conductivity $\kappa _m$), it will distort the uniform thermal field of the matrix. The thermal field of the matrix (generated by the external uniform thermal field), $\boldsymbol{G}_{me}$, can be expressed as

$$\begin{aligned} \boldsymbol{G}_{me}=-\boldsymbol{\nabla }T_{me}. \end{aligned}$$

(17.1)

$T_{me}$ is the temperature distribution given by [20]

$$\begin{aligned} T_{me}=-G_0 r \cos \theta -\frac{\kappa _m-\kappa _p}{\kappa _m+\kappa _p}r_p^2G_0r^{-1}\cos \theta +T_0, \end{aligned}$$

(17.2)

where $\left( r,\,\theta \right)$ denotes cylindrical coordinates whose origin is at the center of the particle. $G_0=|\boldsymbol{G}_0|$, and $T_0$ is the temperature at $\theta =\pm \pi /2$.

When there is a thermal dipole (with hot source power Q and distance l) at the center of the particle, it will generate a thermal field in the matrix. The thermal field of the matrix (generated by the thermal dipole), $\boldsymbol{G}_{md}$, can be expressed as

$$\begin{aligned} \boldsymbol{G}_{md}=-\boldsymbol{\nabla }T_{md}. \end{aligned}$$

(17.3)

$T_{md}$ is the temperature distribution given by

$$\begin{aligned} T_{md}=\frac{M}{\pi \left( \kappa _m+\kappa _p\right) } r^{-1}\cos \theta +T_0, \end{aligned}$$

(17.4)

where M is the thermal dipole moment given by $M=Ql$. Equation (17.4) is valid only when $r\gg l$, and details will be shown in the discussion part.

Because of the superposition principle of vector fields, the thermal field of the matrix (generated by the external uniform thermal field and the thermal dipole), $\boldsymbol{G}_s$, can be expressed as

$$\begin{aligned} \boldsymbol{G}_s=\boldsymbol{G}_{me}+\boldsymbol{G}_{md}=-\boldsymbol{\nabla }T_s. \end{aligned}$$

(17.5)

$T_s$ is the temperature distribution given by

$$\begin{aligned} T_s=-G_0r\cos \theta -\left[ \frac{\kappa _m-\kappa _p}{\kappa _m+\kappa _p}r_p^2G_0-\frac{M}{\pi \left( \kappa _m+\kappa _p\right) }\right] r^{-1}\cos \theta +T_0. \end{aligned}$$

(17.6)

As mentioned at the very beginning, thermal invisibility is characterized by the uniform thermal field of the matrix, and thus the second term on the right side of Eq. (17.6) should be zero,

$$\begin{aligned} \frac{\kappa _m-\kappa _p}{\kappa _m+\kappa _p}r_p^2G_0-\frac{M}{\pi \left( \kappa _m+\kappa _p\right) }=0. \end{aligned}$$

(17.7)

Solving Eq. (17.7), we can derive the thermal dipole moment,

$$\begin{aligned} M=\left( \kappa _m-\kappa _p\right) f G_0, \end{aligned}$$

(17.8)

where $f=\pi r_p^2$ is the acreage of the particle. When the thermal dipole moment is set as required by Eq. (17.8), thermal invisibility can be achieved.

Fig. 17.2

Adapted from Ref. [19]

Finite-element simulations in the presence of a and d an external uniform thermal field, b and e a thermal dipole, and c and f both an external uniform thermal field and a thermal dipole. The simulation box is $20\times 20$ cm$^2$, $r_p=6$ cm, and $l=2$ cm. The thermal conductivities of the particle and the matrix are 200 and 400 W m$^{-1}$ K$^{-1}$, respectively. The thermal dipole moment should be 452.4 W m as required by Eq. (17.8), which leads to $Q=22620$ W. Each source of the thermal dipole has a radius of 0.5 cm. White lines represent isotherms. Temperatures higher than 323 K are shown as 323 K, and temperatures lower than 283 K are shown as 283 K.

3 Finite-Element Simulation

We further perform finite-element simulations with COMSOL Multiphysics to validate the theoretical analyses. In Fig. 17.2a, d, the temperatures of the left and right boundaries are set at 323 and 283 K, and the top and bottom boundaries are insulated. If there is a particle with different thermal conductivity from the matrix in the center, isotherms are contracted due to the smaller thermal conductivity of the particle; see Fig. 17.2d. The distorted temperature profile makes the particle visible with infrared detection. Then, we explore the thermal profile of a thermal dipole; see Fig. 17.2b, e. All boundaries are insulated, and we set the temperature at $\theta =\pm \pi /2$ to 303 K as the reference temperature. The temperature profile is presented in Fig. 17.2e. Finally, we combine the structures shown in Fig. 17.2a, b and obtain the structure presented in Fig. 17.2c. As predicted by Eq. (17.8), the distorted temperature profile is restored; see Fig. 17.2f. Therefore, the particle becomes invisible with infrared detection, and thermal invisibility is achieved.

Fig. 17.3

Adapted from Ref. [19]

Laboratory experiments. a Schematic diagrams of the sample and experimental devices. b and c are the measured results without and with a thermal dipole, respectively. d and e are the corresponding finite-element simulations based on the structure in (a). Copper: thermal conductivity 400 W m$^{-1}$ K$^{-1}$, density 8960 kg m$^{-3}$, and heat capacity 385 J kg$^{-1}$ K$^{-1}$; air: thermal conductivity 0.026 W m$^{-1}$ K$^{-1}$, density $1.29\times 10^{-3}$ kg m$^{-3}$, and heat capacity 1005 J kg$^{-1}$K$^{-1}$. The radius of the 256 air holes is 0.22 cm, and the distance between air holes is 2/3 cm.

4 Laboratory Experiment

We also perform laboratory experiments to validate the theoretical analyses and finite-element simulations. We fabricate the sample based on a copper plate (400 W m$^{-1}$ K$^{-1}$); see Fig. 17.3a. Air holes (0.026 W m$^{-1}$ K$^{-1}$) are engraved on the copper plate by laser cut, which makes the effective thermal conductivity of the corresponding region to be 200 W m$^{-1}$ K$^{-1}$. The upper and lower surfaces are covered with transparent and foamed plastic (insulated) to reduce infrared reflection and thermal convection.

The thermal dipole is realized by a ceramic heater and a semiconductor cooler. The designed power of a heater (or cooler) is 22620 (or −22620) W, which is an extremely large (or small) value. On the one hand, it maintains the uniform field of the matrix. On the other hand, it generates a higher (or lower) temperature inside the heater (or cooler) than the hot (or cold) source. However, the higher (or lower) temperature inside the heater (or cooler) does not contribute to the effect of thermal invisibility because only the edge temperature of the heater (or cooler) makes sense. Therefore, we only need to keep the temperature of the heater (or cooler) at 325 (or 281) K, as ensured by the uniqueness theorem in thermotics. The two temperatures can be directly obtained from finite-element simulations, depending on the heater’s size (or cooler).

We measure the temperature profile with an infrared camera (FLIR E60) between the hot source (323 K) and the cold source (283 K). The measured results without and with a thermal dipole are presented in Fig. 17.3b, c. We also perform finite-element simulations based on the structure presented in Fig. 17.3a; see Fig. 17.3d, e. The experimental results (Fig. 17.3b, c) and finite-element simulations (Fig. 17.2d, f, d, e) both validate that the thermal dipole is reliable and flexible to achieve thermal invisibility.

Fig. 17.4

Adapted from Ref. [19]

Effects of the thermal dipole on thermal invisibility. a shows the thermal-dipole-based temperature distribution. b presents the temperature distribution when the thermal conductivities of the matrix and particle are the same (say, 400 W m$^{-1}$ K$^{-1}$). c exhibits the temperature-difference distribution of the matrix. In d and e, we explore the effects of two parameters (l and $r_d$) on thermal invisibility. The upper panel in (e) is with $r_d=0$ cm, say two point sources of the dipole. The lower panel in (e) is with $l=2$ cm.

5 Discussion

There is only one approximation (say, $r\gg l$) in the whole process to ensure the validity of Eq. (17.4). Therefore, we discuss the effect of this approximation on thermal invisibility. First, we compare our thermal-dipole-based result (Fig. 17.4a) with a reference (Fig. 17.4b). The temperature distributions of the matrix are the same. We also plot the temperature-difference distribution ($\Delta T=T_1-T_2$) of the matrix (Fig. 17.4c) to perform quantitative analyses. The maximum value of the temperature difference ($\Delta T_{max}$) is 0.04 K. Compared with the temperature difference between the hot and cold source (40 K), the relative error is only 0.1%, which shows the excellent performance of the thermal-dipole-based scheme.

We also find that the maximum value of the temperature difference ($\Delta T_{max}$) can reflect the effect of the thermal dipole on thermal invisibility. Therefore, we calculate $\Delta T_{max}$ with different $r_d$ (the radius of the source) and l (the distance between sources); see Fig. 17.4d, and plot two curves, showing $\Delta T_{max}$ changing with $l/2r_p$ and $2r_d/l$; see Fig. 17.4e. The top curve in Fig. 17.4e shows that the performance of the thermal dipole decreases with the increment of $l/2r_p$. When $l/2r_p=0$ (say, $l=0$), $\Delta T_{max}=0$, which indicates the perfect performance. However, the bottom curve in Fig. 17.4e shows that the performance of the thermal dipole keeps unchanged with the increment of $2r_d/l$. Therefore, only one parameter (say, the distance l) mainly influences the effect of the thermal dipole on thermal invisibility, and the shorter, the better.

Further explorations on thermal dipoles could be surely expected. For example, thermal dipoles might be used to realize other thermal phenomena beyond thermal invisibility, such as thermal camouflage. Thermal dipoles may also exhibit novel properties in different systems, such as thermal Janus structures [21] and many-particle systems [22]. The properties of thermal quadrupoles may contain other interesting points.

Although the concept of a dipole originates from electromagnetism, its development in thermotics may, in turn, promote the further development of electrostatics [23] and magnetostatics [24, 25]. Indeed, the concept of a dipole may also be extended to other physical fields such as heat and mass diffusive fields [26, 27].

6 Conclusion

We have established a theory for thermal-dipole-based thermotics, which helps realize thermal invisibility by tailoring thermal dipole moments. The thermal-dipole-based scheme removes the requirements of singular and uncommon thermal conductivities, contributing to practical applications and further developments in thermal management. Both finite-element simulations and laboratory experiments validate the theoretical analyses. The potential applications of thermal dipoles are to mislead infrared detections, simplify the fabrication of thermal metamaterials, enhance the efficiency of heat management, etc.

7 Exercise and Solution

Exercise

• 1. Derive the three-dimensional dipole moment for thermal invisibility.

Solution

• 1. The thermal field of the matrix (generated by the external uniform thermal field), $\boldsymbol{G}_{me}'$, can be expressed as

  $$\begin{aligned} \boldsymbol{G}_{me}'=-\boldsymbol{\nabla }T_{me}'. \end{aligned}$$

  (17.9)

  $T_{me}'$ is the temperature distribution given by

  $$\begin{aligned} T_{me}'=-G_0' r \cos \theta -\frac{\kappa _m'-\kappa _p'}{2\kappa _m'+\kappa _p'}r_p'^3G_0'r^{-2}\cos \theta +T_0'. \end{aligned}$$

  (17.10)

  The thermal field of the matrix (generated by the thermal dipole), $\boldsymbol{G}_{md}'$, can be expressed as

  $$\begin{aligned} \boldsymbol{G}_{md}'=-\boldsymbol{\nabla }T_{md}'. \end{aligned}$$

  (17.11)

  $T_{md}'$ is the temperature distribution given by

  $$\begin{aligned} T_{md}'=\frac{3M'}{4\pi \left( 2\kappa _m'+\kappa _p'\right) } r^{-2}\cos \theta +T_0'. \end{aligned}$$

  (17.12)

  Detailed derivations of Eq. (17.12) are as follows. The general solution to the heat conduction equation in three dimensions is

  $$\begin{aligned} T=\sum _{i=0}^\infty \left( A_i r^{-1/2+\sqrt{1/4+i\left( i+1\right) }}+B_i r^{-1/2-\sqrt{1/4+i\left( i+1\right) }}\right) P_i\left( \cos \theta \right) , \end{aligned}$$

  (17.13)

  where $P_i$ is Legendre polynomial. Then, we perform similar limit analyses to determine the forms of $T_{pd}'$ and $T_{md}'$. We suppose $r_p'\rightarrow \infty$, and then the temperature distribution of the particle generated by the thermal dipole in three dimensions can be expressed as

  $$\begin{aligned} T_{pd}'\left( r_p'\rightarrow \infty \right) =\frac{Q'}{4\pi \kappa _p'}r_+'^{-1} +\frac{-Q'}{4\pi \kappa _p'}r_-'^{-1}=\frac{Q'l'}{4\pi \kappa _p'}r^{-2}\cos \theta =\frac{M'}{4\pi \kappa _p'}r^{-2}\cos \theta . \end{aligned}$$

  (17.14)

  Equation (17.14) is valid only when $r\gg l'$ (or $l'\rightarrow 0$). The temperature distribution of a thermal dipole in three dimensions is characterized by $r^{-2}\cos \theta$. Further, we consider a finite $r_p$. Similar to the analyses in two dimensions, $T_{pd}'$ and $T_{md}'$ can be concluded as

  $$\begin{aligned} T_{pd}'=\frac{M'}{4\pi \kappa _p'}r^{-2}\cos \theta +\alpha ' r\cos \theta +T_0', \end{aligned}$$

  (17.15)
  $$\begin{aligned} T_{md}'=\beta ' r^{-2}\cos \theta +T_0'. \end{aligned}$$

  (17.16)

  The boundary conditions are given by the continuous temperatures and heat fluxes,

  $$\begin{aligned} T_{pd}'\left( r_p\right) =T_{md}'\left( r_p\right) , \end{aligned}$$

  (17.17)
  $$\begin{aligned} \left( -\kappa _p\partial T_{pd}'/\partial r\right) _{r_p}=\left( -\kappa _m\partial T_{md}'/\partial r\right) _{r_p}, \end{aligned}$$

  (17.18)

  Therefore, the undetermined coefficients can be calculated,

  $$\begin{aligned} \alpha '=\frac{-M'\left( \kappa _m'-\kappa _p'\right) }{2\pi r_p'^3\kappa _p'\left( 2\kappa _m'+\kappa _p'\right) }, \end{aligned}$$

  (17.19)
  $$\begin{aligned} \beta '=\frac{3M'}{4\pi \left( 2\kappa _m'+\kappa _p'\right) }. \end{aligned}$$

  (17.20)

  Then, Eq. (17.16) turns to

  $$\begin{aligned} T_{md}'=\frac{3M'}{4\pi \left( 2\kappa _m'+\kappa _p'\right) } r^{-2}\cos \theta +T_0', \end{aligned}$$

  (17.21)

  which is just Eq. (17.12).

  Because of the superposition principle, the thermal field of the matrix (generated by the external uniform thermal field and the thermal dipole), $\boldsymbol{G}_s'$, can be expressed as

  $$\begin{aligned} \boldsymbol{G}_s'=\boldsymbol{G}_{me}'+\boldsymbol{G}_{md}'=-\boldsymbol{\nabla }T_s'. \end{aligned}$$

  (17.22)

  $T_s'$ is the temperature distribution given by

  $$\begin{aligned} T_s'=-G_0'r\cos \theta -\left[ \frac{\kappa _m'-\kappa _p'}{2\kappa _m'+\kappa _p'}r_p'^3G_0'-\frac{3M'}{4\pi \left( 2\kappa _m'+\kappa _p'\right) }\right] r^{-2}\cos \theta +T_0'. \end{aligned}$$

  (17.23)

  Thermal invisibility requires the second term on the right side of Eq. (17.23) to be zero,

  $$\begin{aligned} \frac{\kappa _m'-\kappa _p'}{2\kappa _m'+\kappa _p'}r_p'^3G_0'-\frac{3M'}{4\pi \left( 2\kappa _m'+\kappa _p'\right) }=0, \end{aligned}$$

  (17.24)

  Solving Eq. (17.24), we can derive the thermal dipole moment,

  $$\begin{aligned} M'=\left( \kappa _m'-\kappa _p'\right) f' G_0', \end{aligned}$$

  (17.25)

  where $f'=4\pi r_p'^3/3$ is the volume of the particle.