Keywords

1 Opening Remarks

Metamaterials have shown superior control ability beyond naturally occurring materials in both wave [1,2,3,4,5,6,7,8,9] and diffusion [10,11,12,13,14,15,16,17,18] systems. The transformation theory [1,2,3,4, 10, 11] and scattering-cancellation method [8, 9, 12,13,14], as two common approaches for manipulating physical fields, have achieved great success in artificial structure design. In particular, the latter is based on solving steady-state governing equations directly under given boundary conditions, leading to isotropic and homogeneous design parameters. However, if multiple fields act on an individual system, for example, there exist heat and electric fluxes simultaneously [19,20,21], the governing equations are hard to handle because of the newly-introduced coupling terms induced by thermoelectric (TE) effects. Appropriate theoretical methods need to be developed for designing such multiphysical metamaterials.

Early research on tailoring TE fields focused on the decoupled cases, which means that heat and current flows transfer independently without interaction [22,23,24,25,26]. This simplified hypothesis facilitates the generalization of transformation theory or scattering-cancellation method from extensively-studied single physics to multiphysics. Nevertheless, it usually deviates from actual situations because the coupling terms are omitted. Recently, transformed TE metamaterials were reported [27, 28], which extended the transformation theory from controlling a single field to coupled TE field. The form invariance of TE governing equations under coordinate transformation remains valid, and corresponding transformation rules on physical parameters are deduced. However, inhomogeneous and anisotropic TE materials are still required, just as their counterparts in single physics. Although some laminar-structure schemes with natural TE materials are proposed for mimicking the predicated TE parameters [27,28,29,30], experimental realization remains lacking due to the complexity of manufacture and availability of materials. Considering the challenges mentioned above, the scattering-cancellation method, which facilitates manufacture with simplified structures and homogeneous isotropic materials, could be a feasible route to practical implementation in TE control.

We propose a bilayer scheme based on the scattering-cancellation method for manipulating TE fields with naturally occurring TE materials. By introducing a generalized auxiliary potential, we construct Laplacian-form governing equations. We then derive the required thermal conductivity, electrical conductivity, and the Seeback coefficient for achieving cloaking, concentrating, and sensing functionalities. Finite-element simulations confirm our theoretical design and show the robustness of the proposed bilayer design under various conditions. Compared with the transformation TE theory, anisotropy and inhomogeneity are no longer necessities, making the manufacturing more convenient. The theoretical results and device behaviors can be naturally extended to other coupled multiphysics.

2 Theoretical Foundation

Let us consider a steady TE transport process where physical parameters are scalar at each local position. That is, the isotropy of TE materials is stipulated. In such an isotropic system, the governing equations can be described by [21]

$$\begin{aligned} \boldsymbol{j}&=-\sigma \boldsymbol{\nabla }\mu -{\sigma }S\boldsymbol{\nabla }{T}, \end{aligned}$$
(7.1)
$$\begin{aligned} \boldsymbol{\nabla }\cdot \boldsymbol{j}&=0, \end{aligned}$$
(7.2)
$$\begin{aligned} \boldsymbol{q}&= -\kappa \boldsymbol{\nabla }{T} + TS\boldsymbol{j}, \end{aligned}$$
(7.3)
$$\begin{aligned} \boldsymbol{\nabla }\cdot \boldsymbol{q}&=-\boldsymbol{\nabla }\mu \cdot \boldsymbol{j}, \end{aligned}$$
(7.4)

where \(\boldsymbol{q}\) and \(\boldsymbol{j}\) are thermal and electric flows respectively, T and \(\mu \) refer to temperature and electric potentials, and \(\kappa \) and \(\sigma \) denote to scalar thermal and electrical conductivities. S is Seebeck coefficient for coupling heat and current flows. We define U as an auxiliary generalized potential, which is expressed as

$$\begin{aligned} U ={\mu } + TS. \end{aligned}$$
(7.5)

Combining Eqs. (7.1)–(7.5), two identical relations about U can be obtained as

$$\begin{aligned} \sigma {\boldsymbol{\nabla }}^2{U}=0 \end{aligned}$$
(7.6)

and

$$\begin{aligned} \kappa {\boldsymbol{\nabla }}^2{T} = \sigma \boldsymbol{\nabla }{U}\cdot \boldsymbol{\nabla }{U}. \end{aligned}$$
(7.7)

Note that Eq. (7.6) has a Laplacian form, so it is possible to map the field distribution of U by tailoring \(\sigma \) in a bilayer structure with a similar method employed in single-physics cases [12, 13]. Then we resort to remolding Eq. (7.7) for detecting the direction relation between \(\boldsymbol{\nabla } T\) and \(\boldsymbol{\nabla } U\). The Poisson equation Eq. (7.7) has the solution consisting of two parts. One is the general solution of its corresponding Laplace equation

$$\begin{aligned} \kappa {\boldsymbol{\nabla }}^2{T} =0. \end{aligned}$$
(7.8)

The other is the particular solution. We can see the identical relation

$$\begin{aligned} \kappa \boldsymbol{\nabla }{T} = \sigma {U}\boldsymbol{\nabla }{U} \end{aligned}$$
(7.9)

should always be valid to make Eq. (7.7) be satisfied. This can be deduced by taking the divergence of Eq. (7.9) in both sides as

$$\begin{aligned} \kappa {\boldsymbol{\nabla }}^2{T} =\sigma \boldsymbol{\nabla }(U\boldsymbol{\nabla }{U}) =\sigma (\boldsymbol{\nabla }{U}\cdot \boldsymbol{\nabla }{U}+U{\boldsymbol{\nabla }}^2{U}) =\sigma \boldsymbol{\nabla }{U}\cdot \boldsymbol{\nabla }{U}. \end{aligned}$$
(7.10)

Then we can conclude that \(\boldsymbol{\nabla }{T}\) is always parallel to \(\boldsymbol{\nabla }{U}\) in its particular solution. Now we are in the position to discuss the relation between \(\boldsymbol{\nabla }{T}\) and \(\boldsymbol{\nabla }{U}\) in the general solution. T will thus be manipulated like U. Combining Eqs. (7.6) and (7.8), which are both Laplace equations, we can get the following conditions to make \(\boldsymbol{\nabla }{T}\) parallel to \(\boldsymbol{\nabla }{U}\). Condition I is

$$\begin{aligned} S = S_0, \end{aligned}$$
(7.11)

indicating that S keeps invariant in the whole space. Condition II is

$$\begin{aligned} \sigma = C\kappa , \end{aligned}$$
(7.12)

where C is a constant for keeping \(\sigma \) and \(\kappa \) proportional in the whole space. Condition III relies on boundary condition settings. It means that external thermal and electrical fields should be parallel for ensuring homodromous \(\boldsymbol{\nabla }{U}\) and \(\boldsymbol{\nabla }{T}\) at each point. These three conditions enable us to map T distribution by tailoring U, which is described by a Laplacian-form governing equation. Then, we can define \(f(\boldsymbol{r})\), a coordinate-dependent scalar function, to denote the relationship between \(\boldsymbol{\nabla }{U}\) and \(\boldsymbol{\nabla }{T}\) as

$$\begin{aligned} \boldsymbol{\nabla }{U} = f(\boldsymbol{r})\boldsymbol{\nabla }{T}. \end{aligned}$$
(7.13)

Next, we will handle the electrical potential \(\mu \). Note that S is constant, by combining Eqs. (7.5) and (7.13) together, we can obtain

$$\begin{aligned} \boldsymbol{\nabla }\mu = (f(\boldsymbol{r})-S)\boldsymbol{\nabla }{T}. \end{aligned}$$
(7.14)

Evidently, \(\boldsymbol{\nabla }\mu \) is also parallel to \(\boldsymbol{\nabla }{T}\) and \(\boldsymbol{\nabla }{U}\). So once Eqs. (7.11) and (7.12) are satisfied simultaneously, and the boundary temperature and potential fields are parallel, we can manipulate TE flows. Since bilayer is the most simplified structure for realizing specific functionalities such as cloaking, concentrating, and sensing with isotropic materials in a single field [12, 32, 33], we design TE cloaking, invisible sensing, and concentrating devices with bilayer configurations for verification. More layers will achieve the same effects but cannot improve the behaviors, which has been discussed sufficiently in many single-field metamaterial research.

We design three different functionalities in a size-fixed bilayer structure with background thermal conductivity \(\kappa _b\) and electrical conductivity \(\sigma _b\), as shown in Fig. 7.1. For simplification without loss of generality, we only consider two-dimensional cases, which can readily be transferred to three-dimensional systems. According to the deductions above, the parameter requirements, i.e., Eqs. (7.11) and (7.12), should be satisfied simultaneously. And some additional conditions for realizing different functions are required. We set \(\sigma _0\), \(\sigma _1\), \(\sigma _2\) as respective electrical conductivities from the center to the outer layer. Same definitions are employed for \(\kappa _0\), \(\kappa _1\), \(\kappa _2\). Detailed parameter settings are as follows.

For cloaking [12], which prevents TE flows from running into the center without distorting the ambient temperature and potential distributions outside, as shown in Fig. 7.1b, the additional conditions for the inner layer should be

$$\begin{aligned} \sigma _{1} \approx 0, \end{aligned}$$
(7.15)

which make the inner layer a nearly-perfect thermal/electrical insulation material. And the outer layer should be

$$\begin{aligned} \sigma _2 = \sigma _b({r_2}^2+{r_1}^2)/({r_2}^2-{r_1}^2), \end{aligned}$$
(7.16a)

guarantying no distortion of ambient temperature and potential outside.

Fig. 7.1
figure 1

Adapted from Ref. [31]

a Schematic diagram of bilayer TE metamaterials. The core is marked as region III with electrical conductivity \({\sigma }_0\) and thermal conductivity \(\kappa _0\). The inner layer is marked as region II with \({\sigma }_1\) and \(\kappa _1\). The outer layer is marked as region I with \({\sigma }_2\) and \(\kappa _2\). The background in gray has \({\sigma }_b\) and \(\kappa _b\). The electrical conductivity, thermal conductivity, and Seeback coefficient S are homogeneous isotropic scalars in each region. The Cartesian coordinate (x-y) is built on designed metamaterials with the overlapping origin and center point. b Illustration of a TE cloak. c Illustration of a TE invisible sensor. d Illustration of a TE concentrator. Red and blue lines represent heat and electrical fluxes, respectively, in b–d.

Full size image
Fig. 7.2
figure 2

Adapted from Ref. [31]

Simulation results of the TE cloak under parallel external thermal and electrical fields. Isothermal or isopotential lines are marked in white. a Temperature distribution of the matrix plus cloak. b Temperature distribution of the pure matrix. c Temperature distribution of the bare perturbation. d Temperature difference distribution between a and b. e Potential distribution of the matrix with a cloak. f Potential distribution of the pure matrix as a reference. g Potential distribution of the bare perturbation. h Potential difference distribution between d and e. i Quantitative temperature comparison between a and b at the chosen line, which crosses the origin along the x axis. j Quantitative potential comparison between d and e at the chosen line, which crosses the origin along the x axis. Different regions (I, II, and III) are indicated in i and j, corresponding to the model in Fig. 7.1a. Backgrounds are outside region I.

Full size image

For invisible sensing [32], which maintains the original temperature and potential in both center and background regions for obtaining accurate sensor effects, as shown in Fig. 7.1c, the additional conditions are found as

$$\begin{aligned} \sigma _1=\left[ \sigma _0 A_3-\sigma _b A_1+\sqrt{\left( \sigma _0-\sigma _b\right) \left( \sigma _0 A_2^2-\sigma _b A_1^2\right) }\right] /A_5, \end{aligned}$$
(7.17a)
$$\begin{aligned} \sigma _2=\left[ \sigma _0 A_2-\sigma _b A_4-\sqrt{\left( \sigma _0-\sigma _b\right) \left( \sigma _0 A_2^2-\sigma _b A_1^2\right) }\right] /A_6, \end{aligned}$$
(7.17b)

where

$$\begin{aligned}&A_1=r_0^2(r_1^2+r_2^2) +r_1^2(r_1^2-3 r_2^2),\end{aligned}$$
(7.18a)
$$\begin{aligned}&A_2=r_0^2(3r_1^2-r_2^2) -r_1^2(r_1^2+r_2^2), \end{aligned}$$
(7.18b)
$$\begin{aligned}&A_3=r_0^2(2r_0^2-r_1^2-r_2^2) +r_1^2(r_1^2-r_2^2), \end{aligned}$$
(7.18c)
$$\begin{aligned}&A_4=r_1^2(r_0^2-r_1^2) +r_2^2(r_0^2+r_1^2-2r_2^2), \end{aligned}$$
(7.18d)
$$\begin{aligned}&A_5=2(r_0^2-r_1^2)(r_0^2-r_2^2),\end{aligned}$$
(7.18e)
$$\begin{aligned}&A_6=2(r_0^2-r_2^2)(r_1^2-r_2^2). \end{aligned}$$
(7.18f)

\(\kappa _1\) and \(\kappa _2\) follow the formally-similar parameter requirements as \(\sigma _1\) and \(\sigma _2\). It is noted that two sets of parameters are available in sensing design within a fixed geometry structure. We arbitrarily adopt one of them here.

Fig. 7.3
figure 3

Adapted from Ref. [31]

Simulation results of the TE invisible sensor under parallel thermal and electrical boundary conditions. All figure arrangements are the same as Fig. 7.2 except the functionality of the central device.

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For concentrating [33], which can enhance the gradients of temperature and potential in the center without distorting the ambient ones, as shown in Fig. 7.1d, the additional condition for \(\sigma _0\), \(\sigma _1\), and \(\sigma _2\) can be written as

$$\begin{aligned} \begin{aligned} \sigma _0&=[ r_2^2 r_0^2\left( \sigma _2-\sigma _1\right) \left( \sigma _2-\sigma _b\right) /r_1^2-r_0^2 \left( \sigma _1+\sigma _2\right) \left( \sigma _b+\sigma _2\right) \\&+r_1^2\left( \sigma _1-\sigma _2\right) \left( \sigma _b+\sigma _2\right) +r_2^2\left( \sigma _1+\sigma _2\right) \left( \sigma _2-\sigma _b\right) ] \sigma _1 /\\&[r_2^2 r_0^2\left( \sigma _2-\sigma _1\right) \left( \sigma _2-\sigma _b\right) /r_1^2-r_0^2 \left( \sigma _1+\sigma _2\right) \left( \sigma _b+\sigma _2\right) \\&+r_1^2\left( \sigma _2-\sigma _1\right) \left( \sigma _b+\sigma _2\right) -r_2^2 \left( \sigma _1+\sigma _2\right) \left( \sigma _2-\sigma _b\right) ], \end{aligned} \end{aligned}$$
(7.19)

which is obtained by solving the Laplacian equation and then set the coefficient of the nonlinear term of the ambient potential zero. Similar forms of the relation between \(\kappa _0\), \(\kappa _1\), and \(\kappa _2\) are also requested. Given that all the required conditions are met, the ratio of the temperature/potential gradient in the center to the temperature/potential gradient in the background, which can describe the efficiency of concentrating, can be tailored by changing the dimensions and conductivities of the layers. So far, we have listed three sets of parameters for achieving three functionalities in TE transport. It is noted that the rationality of generalization from single physics to coupled multiphysics is established on the basis that Eqs. (7.11) and (7.12) should be satisfied simultaneously.

3 Finite-Element Simulation

Table 7.1 Simulation parameter settings of TE cloaking, invisible sensing and concentrating. For simplicity, Seebeck coefficient is set as 1 in all regions. (This value is much larger than common materials, which will induce stong coupling effects between heat and electricity.) Adapted from Ref. [31]
Full size table

We perform finite-element simulations with the commercial software COMSOL Multiphysics to confirm the proposed theoretical models. A two-dimensional bilayer structure of \(r_0=0.02\) m, \(r_1=0.025\) m, and \(r_2=0.03\) m is employed. The bilayer structure is embedded at the center of a matrix, whose length is 0.11 m, as shown in Fig. 7.1a. To demonstrate the functionalities of the cloak, invisible sensor, and concentrator, we obtain three sets of thermal conductivity, electrical conductivity, and Seebeck coefficient for each case, as listed in Table 7.1. For boundary conditions, temperature and potential gradients should be parallel. So we set boundary conditions as follows. The temperatures of the left and right boundaries are 273.15 K and 333.15 K. The potentials of the left and right boundaries are 0 V and 50 V. Upper and lower boundaries are thermally and electrically insulated. To show the effectiveness and accuracy of these three metamaterials, we also compare them with bare-perturbation and pure-background results. We perform simulations of these references under the same boundary conditions and plot the temperature and potential distribution of metamaterials and references. Differences in temperature and potential distribution illustrate the changes in temperature and potential between the metamaterials and pure backgrounds. These simulation results of cloak, concentrator and invisible sensor are demonstrated in Figs. 7.2, 7.3 and 7.4.

Fig. 7.4
figure 4

Adapted from Ref. [31]

Simulation results of the TE concentrator under parallel thermal and electrical boundary conditions. All figure arrangements are the same as Fig. 7.2 except the functionality of the central device.

Full size image

As shown in Figs. 7.2d and h, 7.3d and h, 7.4d and h, both the temperature and potential differences in backgrounds are nearly zero, which means none of these three metamaterials have distorted the ambient temperatures or potentials. This is also confirmed by the overlapping parts of the curves in Figs. 7.2i and j, 7.3i and j, 7.4i and j. As contrast, in Fig. 7.2c and g, 7.3c and g, 7.4c and g, the ambient temperatures and potentials are manifestly distorted by the bare perturbations. For the cloak, we can see in Fig. 7.2a and e or i and j, the temperature and potential gradients at the center are nearly zero, which means that thermal and electric flows are prevented from running into the center. For the sensor, which refers to the core region coated by the bilayer structure in Fig. 7.3a and e, it can be intuitively seen that the core temperature and potential are consistent before and after the sensor is embedded. In Fig. 7.3i and j, the curves of metamaterials and references fit well at the core and ambient regions. Therefore, we may safely say that the sensor can measure the ambient temperature and potential without introducing any distortion. For the concentrator, Fig. 7.4a and e show that both the temperature and potential gradients in the core are greater than the ambient. From Fig. 7.4i and j, we can see more clearly that along the x-axis, the temperature and potential gradients are enhanced at the center.

Fig. 7.5
figure 5

Adapted from Ref. [31]

a–f Simulation results of the TE cloak under the perpendicular boundary temperature and potential fields. Isothermal or isopotential lines are marked in white. a Temperature distribution of the matrix plus cloak. b Temperature distribution of the pure matrix. c Quantitative temperature comparison between a (cloak) and b (reference) at the chosen line, which crosses the origin along the x axis. d Potential distribution of the matrix with a cloak. e Potential distribution of the pure matrix. f Quantitative potential comparison between d (cloak) and e (reference) at the chosen line, which crosses the origin along the x axis. Different regions (I, II, and III) are indicated in g and h, corresponding to the model in Fig. 7.1a. g–l Simulation results of the TE cloak under the y-direction external potential fields and point heat sources at the left-bottom corner.

Full size image

To verify that only under the condition \(\boldsymbol{\nabla }{T}\) is parallel to \(\boldsymbol{\nabla }{\mu }\) can our design be exactly effective, we perform two simulations for the cloak when \(\boldsymbol{\nabla }{T}\) is not parallel to \(\boldsymbol{\nabla }{\mu }\), see Fig. 7.5. We set the upper and lower boundary temperatures in the upper two panels as 273.15 K and 333.15 K and potentials as 0 V and 50 V, respectively. In the lower two panels, a linear point heat source with the power of \(6\times 10^6\) W m\(^{-3}\) K\(^{-1}\) is applied at the left-bottom corner of the matrix, whose position is \((-0.049,\,-0.049)\) cm. The neighbor sides of the source are insulated, and the temperature of the remaining two sides is kept at 300 K. The results are shown in Fig. 7.5. Along the x axis, the difference between the ambient temperature (potential) of the pure matrix and the matrix with a cloak has some minor gaps. The designed schemes are not strictly accurate under nonparallel external fields. But it can still be regarded as a well approximated result based on the curves in Fig. 7.5c, f, i, and l, showing great accordance at background regions. The robustness of our design makes it adaptive under multiple complicated conditions.

4 Discussion

Although actual materials may not perfectly meet the requirements put forward in our theory, we further verify that it is possible that practical realization to an approximate extent can be achieved. Many TE materials, such as ionic-conducting materials, can yield a large variety of TE characteristics due to various mechanisms and tuning methods such as changing the doping ratio [34] or humidity [35]. Therefore, this provides the physical possibility for searching for available materials. Compared with transformation optics requiring extremely anisotropic and inhomogeneous properties, though the proposed scattering cancellation methodology cannot achieve some effects such as rotating, our scheme will yield isotropic and homogeneous parameters to achieve the same effects of cloaking, concentrating, and sensing. Once we have suitable TE materials, the bilayer design will make it easier to manufacture corresponding metamaterials. Another issue is that the role of contact resistance, especially the thermal contact resistance (TCR), may affect the practical results [36]. TCR arises due to limited contact areas at the interface and lattice mismatch at the boundaries of different materials. According to the acoustic mismatch or diffusive mismatch model, the latter is usually too slight to be considered at the macroscale. In most reported macroscale experiments, the former is usually eliminated by “solid plus soft matter” structures. Even without such structures, the experimental results of a decoupled TE sensor, based on common metals, are in accord with the theory, ignoring the contact resistance [25].

5 Conclusion

In conclusion, we have built a scattering-cancellation method for manipulating coupled TE transport and designed three representative devices with bilayer schemes. Considering that TE governing equations are no longer Laplacian forms, additional constraint conditions are required beyond single-field cases. Our deduced requirements of constant Seebeck coefficient and proportional thermal/electrical conductivities echo with the results of the transformation TE method [27, 28] under homogeneous isotropic background conditions. And we further point out that the external TE distribution will not be affected only by applying parallel external thermal and electrical fields on the devices. However, simulation results also verify the robustness of our design under other boundary conditions, which can broaden the practical application range. Our work may provide hints for manipulating coupled multiphysical fields beyond single-physics Laplacian transport, which doubtlessly simplifies the requirements on materials and structures of existing transformation metamaterials. Moreover, since TE effects are widely utilized in practical applications, ranging from generating electric power from waste heat to solid-state-based cooling down, our work may help facilitate device preparation and raise energy conversion efficiency.

6 Exercise and Solution

Exercise

1. Derive the clear relations about U, including boundary conditions and parameter requirements.

Solution

1. The introduction of auxiliary generalized potential U and the analyses on corresponding boundary condition settings are provided. First, we consider U in a certain domain. Combing Eqs. (7.1) and (7.2), we can obtain

$$\begin{aligned} {\boldsymbol{\nabla }}\cdot \sigma {\boldsymbol{\nabla }}(\mu +ST)=0. \end{aligned}$$
(7.20)

Considering Eq. (7.5), we can write

$$\begin{aligned} {\boldsymbol{\nabla }}\cdot \sigma {\boldsymbol{\nabla }}U=0. \end{aligned}$$
(7.21)

Substituting Eq. (7.4) into Eq. (7.3), we have

$$\begin{aligned} -\kappa {\boldsymbol{\nabla }}^2{T}+S\boldsymbol{\nabla }{T}\cdot \boldsymbol{j}+ST\boldsymbol{\nabla }\cdot \boldsymbol{j} = -\boldsymbol{\nabla }\mu \cdot \boldsymbol{j}. \end{aligned}$$
(7.22)

According to Eq. (7.1), that is \(\boldsymbol{\nabla }\cdot \boldsymbol{j}=0\), Eq. (7.22) can be simplified as

$$\begin{aligned} \kappa {\boldsymbol{\nabla }}^2{T} = (\boldsymbol{\nabla }\mu +S\boldsymbol{\nabla }{T})\cdot \boldsymbol{j}. \end{aligned}$$
(7.23)

Substituting Eqs. (7.2) and (7.5) into Eq. (7.23), we can thus obtain another equation about U as

$$\begin{aligned} \kappa {\boldsymbol{\nabla }}^2{T} = \sigma \boldsymbol{\nabla }{U}\cdot \boldsymbol{\nabla }{U}. \end{aligned}$$
(7.24)

Now let us discuss the boundary condition settings of U. Apparently, U is a combination of T and \(\mu \). For T and \(\mu \), the boundary behaviors are already known as

$$\begin{aligned} T_i&= T_{i+1}, \end{aligned}$$
(7.25a)
$$\begin{aligned} {\kappa }_{i}\frac{\partial {T_i}}{\partial {r}}&= {\kappa }_{i+1}\frac{\partial {T_{i+1}}}{\partial {r}}, \end{aligned}$$
(7.25b)
$$\begin{aligned} {\mu }_i&= {\mu }_{i+1},\end{aligned}$$
(7.25c)
$$\begin{aligned} {\sigma }_{i}\frac{\partial {{\mu }_i}}{\partial {r}}&= {\sigma }_{i+1}\frac{\partial {{\mu }_{i+1}}}{\partial {r}}, \end{aligned}$$
(7.25d)

where i and \(i+1\) denote two adjacent domains. Because U satisfies Laplace equation Eq. (7.6), to make U be manipulated by tailoring \(\sigma \) in a way similar to that proposed by Ref. [12], similar boundary behaviors will also be required

$$\begin{aligned} {U}_i&= {U}_{i+1},\end{aligned}$$
(7.26a)
$$\begin{aligned} {\sigma }_{i}\frac{\partial {{U}_i}}{\partial {r}}&= {\sigma }_{i+1}\frac{\partial {{U}_{i+1}}}{\partial {r}}. \end{aligned}$$
(7.26b)

According to Eq. (7.5), we can rewrite Eq. (7.26) as

$$\begin{aligned} {\mu }_i + {S_i}T_i&= {\mu }_{i+1}+S_{i+1}T_{i+1}, \end{aligned}$$
(7.27a)
$$\begin{aligned} {\sigma }_{i}\frac{\partial {{\mu }_i}}{\partial {r}}+{\sigma }_{i}S_{i}\frac{\partial {T_i}}{\partial {r}}&= {\sigma }_{i+1}\frac{\partial {{\mu }_{i+1}}}{\partial {r}}+{\sigma }_{i+1}S_{i+1}\frac{\partial {T_{i+1}}}{\partial {r}}. \end{aligned}$$
(7.27b)

Substituting Eqs. (7.25a), (7.25c) into (7.27a), we have

$$\begin{aligned} S_i = S_{i+1}, \end{aligned}$$
(7.28)

from which the conclusion that S should keep invariant in all domains can be easily deduced. Meanwhile, Eq. (7.27b) can be rewritten as

$$\begin{aligned} {\sigma }_{i}\frac{\partial {{\mu }_i}}{\partial {r}}+\frac{{\sigma }_{i}S_{i}}{\kappa _{i}}\frac{\kappa _{i}\partial {T_{i}}}{\partial {r}}= {\sigma }_{i+1}\frac{\partial {{\mu }_{i+1}}}{\partial {r}}+\frac{{\sigma }_{i+1}S_{i+1}}{\kappa _{i+1}}\frac{\kappa _{i+1}\partial {T_{i+1}}}{\partial {r}}. \end{aligned}$$
(7.29)

Hence substituting Eqs. (7.25b), (7.25d) into (7.27b), we have

$$\begin{aligned} \frac{{\sigma }_{i}S_{i}}{\kappa _{i}}=\frac{{\sigma }_{i+1}S_{i+1}}{\kappa _{i+1}}. \end{aligned}$$
(7.30)

Making use of the Eq. (7.28), we can eventually obtain

$$\begin{aligned} \frac{{\sigma }_{i}}{\kappa _{i}}=\frac{{\sigma }_{i+1}}{\kappa _{i+1}}, \end{aligned}$$
(7.31)

from which a generalized conclusion that \(\sigma \) and \(\kappa \) are proportional between different domains, i.e., condition II or Eq. (7.13), can be easily deduced. For condition III, since \(\boldsymbol{\nabla }{T}\) and \(\boldsymbol{\nabla }\mu \) should be parallel where there are sources or boundary temperatures/potentials, it is obvious that the sources or boundary temperatures/potentials should appear in pairs.