Abstract
In this chapter, we theoretically design bilayer thermoelectric metamaterials based on the generalized scattering-cancellation method. By solving the governing equations directly, we formulate the specific parameter requirements for desired functionalities beyond existing single-field or decoupled multi-field Laplacian metamaterials. Unlike the recently reported transformation thermoelectric flows, bilayer schemes do not require inhomogeneity and anisotropy in constitutive materials. Finite-element simulations confirm the analytical results and show robustness under various exterior conditions. Feasible experimental design with naturally occurring materials is also proposed for further proof-of-principle verification. Our theoretical method may be extended to other coupled multiphysical systems such as thermo-optics, thermomagnetics, and optomechanics.
You have full access to this open access chapter, Download chapter PDF
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]
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
Combining Eqs. (7.1)–(7.5), two identical relations about U can be obtained as
and
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
The other is the particular solution. We can see the identical relation
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
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
indicating that S keeps invariant in the whole space. Condition II is
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
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
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
which make the inner layer a nearly-perfect thermal/electrical insulation material. And the outer layer should be
guarantying no distortion of ambient temperature and potential outside.
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
where
\(\kappa _1\) 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.
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
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
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.
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.
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
Considering Eq. (7.5), we can write
Substituting Eq. (7.4) into Eq. (7.3), we have
According to Eq. (7.1), that is \(\boldsymbol{\nabla }\cdot \boldsymbol{j}=0\), Eq. (7.22) can be simplified as
Substituting Eqs. (7.2) and (7.5) into Eq. (7.23), we can thus obtain another equation about U as
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
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
According to Eq. (7.5), we can rewrite Eq. (7.26) as
Substituting Eqs. (7.25a), (7.25c) into (7.27a), we have
from which the conclusion that S should keep invariant in all domains can be easily deduced. Meanwhile, Eq. (7.27b) can be rewritten as
Hence substituting Eqs. (7.25b), (7.25d) into (7.27b), we have
Making use of the Eq. (7.28), we can eventually obtain
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.
References
Pendry, J.B., Schurig, D., Smith, D.R.: Controlling electromagnetic fields. Science 312, 1780 (2006)
Chen, H.Y., Chan, C.T., Sheng, P.: Transformation optics and metamaterials. Nat. Mater. 9, 387 (2010)
Pendry, J.B., Aubry, A., Smith, D.R., Maier, S.A.: Transformation optics and subwavelength control of light. Science 337, 549 (2012)
Xu, L., Chen, H.Y.: Conformal transformation optics. Nat. Photonics 9, 15 (2015)
Yang, Z., Mei, J., Yang, M., Chan, N.H., Sheng, P.: Membrane-type acoustic metamaterial with negative dynamic mass. Phys. Rev. Lett. 101, 204301 (2008)
Zigoneanu, L., Popa, B.I., Cummer, S.A.: Three-dimensional broadband omnidirectional acoustic ground cloak. Nat. Mater. 13, 352 (2014)
Cummer, S.A., Christensen, J., Alù, A.: Controlling sound with acoustic metamaterials. Nat. Rev. Mater. 1, 16001 (2016)
Alù, A., Engheta, N.: Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623 (2005)
Gomory, F., Solovyov, M., Souc, J., Navau, C., Prat-Camps, J., Sanchez, A.: Experimental realization of a magnetic cloak. Science 335, 1466 (2012)
Fan, C.Z., Gao, Y., Huang, J.P.: Shaped graded materials with an apparent negative thermal conductivity. Appl. Phys. Lett. 92, 251907 (2008)
Chen, T.Y., Weng, C.-N., Chen, J.-S.: Cloak for curvilinearly anisotropic media in conduction. Appl. Phys. Lett. 93, 114103 (2008)
Han, T.C., Bai, X., Gao, D.L., Thong, J.T.L., Li, B.W., Qiu, C.-W.: Experimental demonstration of a bilayer thermal cloak. Phys. Rev. Lett. 112, 054302 (2014)
Xu, H.Y., Shi, X.H., Gao, F., Sun, H.D., Zhang, B.L.: Ultrathin three-dimensional thermal cloak. Phys. Rev. Lett. 112, 054301 (2014)
Su, C., Xu, L.J., Huang, J.P.: Nonlinear thermal conductivities of core-shell metamaterials: rigorous theory and intelligent application. EPL 130, 34001 (2020)
Maldovan, M.: Sound and heat revolutions in phononics. Nature 503, 209 (2013)
Li, Y., Li, W., Han, T.C., Zheng, X., Li, J.X., Li, B.W., Fan, S.H., Qiu, C.-W.: Transforming heat transfer with thermal metamaterials and devices. Nat. Rev. Mater. 6, 488 (2021)
J. Wang, G. L. Dai, and J. P. Huang, Thermal metamaterial: fundamental, application, and outlook. Isience 23, 101637 (2020)
Yang, S., Wang, J., Dai, G.L., Yang, F.B., Huang, J.P.: Controlling macroscopic heat transfer with thermal metamaterials: theory, experiment and application. Phys. Rep. 908, 1 (2021)
Bell, L.E.: Cooling, heating, generating power, and recovering waste heat with thermoelectric systems. Science 321, 1457 (2008)
Domenicali, C.A.: Irreversible thermodynamics of thermoelectricity. Rev. Mod. Phys. 26, 1103 (1954)
Biswas, K., He, J.Q., Blum, I.D., Wu, C.-I., Hogan, T.P., Seidman, D.N., Dravid, V.P., Kanatzidis, M.G.: High-performance bulk thermoelectrics with all-scale hierarchical architectures. Nature 489, 11439 (2012)
Li, J.Y., Gao, Y., Huang, J.P.: A bifunctional cloak using transformation media. J. Appl. Phys. 108, 074504 (2010)
Moccia, M., Castaldi, G., Savo, S., Sato, Y., Galdi, V.: Independent manipulation of heat and electrical current via bifunctional metamaterials. Phys. Rev. X 4, 021025 (2014)
Ma, Y.G., Liu, Y.C., Raza, M., Wang, Y.D., He, S.L.: Experimental demonstration of a multiphysics cloak: manipulating heat flux and electric current simultaneously. Phys. Rev. Lett. 113, 205501 (2014)
Yang, T.Z., Bai, X., Gao, D.L., Wu, L.Z., Li, B.W., Thong, J.T.L., Qiu, C.-W.: Invisible sensors: simultaneous sensing and camouflaging in multiphysical fields. Adv. Mater. 27, 7752 (2015)
Lan, C.W., Bi, K., Fu, X.J., Li, B., Zhou, J.: Bifunctional metamaterials with simultaneous and independent manipulation of thermal and electric fields. Opt. Express 24, 23072 (2016)
Stedman, T., Woods, L.M.: Cloaking of thermoelectric transport. Sci. Rep. 7, 6988 (2017)
Shi, W., Stedman, T., Woods, L.M.: Transformation optics for thermoelectric flow. J. Phys-Energy 1, 025002 (2019)
Wang, J., Shang, J., Huang, J.P.: Negative energy consumption of thermostats at ambient temperature: electricity generation with zero energy maintenance. Phys. Rev. Appl. 11, 024053 (2019)
Shi, W., Stedman, T., Woods, L.M.: Thermoelectric transport control with metamaterial composites. J. Appl. Phys. 128, 025104 (2020)
Qu, T., Wang, J., Huang, J.P.: Manipulating thermoelectric fields with bilayer schemes beyond Laplacian metamaterials. EPL 135, 54004 (2021)
Xu, L.J., Yang, S., Huang, J.P.: Effectively infinite thermal conductivity and zero-index thermal cloak. EPL 131, 24002 (2020)
Xu, G.Q., Zhou, X., Zhang, J.Y.: Bilayer thermal harvesters for concentrating temperature distribution. Int. J. Heat Mass Transf. 142, 118434 (2019)
He, X., Cheng, H., Yue, S., Ouyang, J.: Quasi-solid state nanoparticle/(ionic liquid) gels with significantly high ionic thermoelectric properties. J. Mater. Chem. A 8, 10813 (2020)
Kim, S.L., Lin, H.T., Yu, C.: Thermally chargeable solid-state supercapacitor. Adv. Energy Mater. 6, 1600546 (2016)
Zheng, X., Li, B.W.: Effect of interfacial thermal resistance in a thermal cloak. Phys. Rev. Appl. 13, 024071 (2020)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.
The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
Copyright information
© 2023 The Author(s)
About this chapter
Cite this chapter
Xu, LJ., Huang, JP. (2023). Theory for Coupled Thermoelectric Metamaterials: Bilayer Scheme. In: Transformation Thermotics and Extended Theories. Springer, Singapore. https://doi.org/10.1007/978-981-19-5908-0_7
Download citation
DOI: https://doi.org/10.1007/978-981-19-5908-0_7
Published:
Publisher Name: Springer, Singapore
Print ISBN: 978-981-19-5907-3
Online ISBN: 978-981-19-5908-0
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)