Keywords

2.1 Theoretical Thermotics

Theoretical thermotics originates from the theory of transformation thermotics [1, 2]. With the artificial heat regulation development, the connotation of theoretical thermotics has been greatly extended, not limited to those theories for designing thermal cloaks, concentrators, and rotators. Therefore, theoretical thermotics is the summarization of “transformation thermotics and extended theories”. For clarity, we mainly divide theoretical thermotics into three levels according to the historical development.

The first level (LV1) is those transformation-related theories for designing cloaking, concentrating, rotating, etc. Since the theory of transformation thermotics was proposed for controlling steady and passive heat conduction in 2008 [1, 2], extended transformation theories have been developed successively from steady and passive to transient and active heat conduction [3]. Then, temperature-dependent (nonlinear) thermal conductivities were considered for developing nonlinear transformation thermotics [4]. These coordinate transformations were all time-independent, making it challenging to deal with time-dependent coordinate transformations. Thus, spatiotemporal coordinate transformations were discussed [5]. Beyond conduction, convection is also a primary heat transfer mode, so researchers developed a transformation theory for convection control [6]. Nevertheless, it was still challenging to guide convective velocities directly. Therefore, the Darcy law in porous media was introduced to transform convection and ensure feasibility [7, 8]. Another convective model with creeping flows was also explored [9]. The last basic heat transfer scheme is radiation, and researchers also proposed a transformation theory to regulate the radiation described by the Rosseland diffusion approximation [10]. With these efforts, conduction, convection, and radiation can be unified in the transformation framework [11]. Besides, heat transfer may also be accompanied by other physical processes, such as electric transport. Therefore, a transformation theory was put forward to regulate thermal and electric fields simultaneously [12]. Researchers further studied the coupling between thermal and electric fields, i.e., the thermoelectric effect, and proposed a transformation theory [13]. Therefore, most thermal phenomena can be manipulated by transformation theories.

The second level (LV2) is other theories for designing functions predicted by the transformation theory. Although the transformation theory is powerful, it still has some limitations. For example, the parameters for thermal cloaking should be anisotropic, inhomogeneous, and even singular. Thus, other theories beyond the transformation theory were proposed. We take thermal cloaking as an example. A bilayer scheme was proposed [14,15,16] to remove anisotropic and inhomogeneous parameters. Then, an active scheme was developed to remove all parametric requirements because only active temperature control was required [17]. Furthermore, a dipole-based scheme was considered to simplify the active temperature control [18]. Besides these analytical theories, topology optimization is an indispensable method that largely simplifies the design [19, 20]. These theories and schemes are distinctly different from the transformation theory, but they are still applied to design functions predicted by the transformation theory.

The third level (LV3) is other theories for designing functions not predicted by the transformation theory. With the development of theoretical thermotics, many phenomena and functions beyond the predictions of transformation thermotics were revealed, such as the anti-parity-time symmetry in diffusive systems [21, 22], diffusive geometric phases [23, 24], thermal wave nonreciprocity [25,26,27,28,29], thermal edge states [30,31,32,33,34], and thermal skin effects [35, 36]. These emerging theories may guide the future development of theoretical thermotics.

2.2 Characteristic Length

Compared with traditional thermodynamics (A in Fig. 2.1), theoretical thermotics focuses on the active control of heat based on transformation thermotics and extended theories (B in Fig. 2.1). Since theoretical thermotics also designs artificial structures for heat regulation, what is the relation between theoretical thermotics and the emerging field of metamaterials? The answer is the characteristic length.

Fig. 2.1
figure 1

Theoretical thermotics meets metamaterials. A: Focus on the passive description of heat based on the four laws of thermodynamics. B: Focus on the active control of heat based on transformation thermotics and extended theories. C: Artificial structures with characteristic lengths have novel properties. C1: Diffusive systems. Right area: thermotics (thermal metamaterials or metamaterial-based devices, which are interdisciplinary with physics, thermophysics, and materials science); left area: other diffusions (particle/plasma diffusion, etc.). C2: Wave systems (electromagnetic/optical waves). C3: Wave systems (acoustic/elastodynamic waves, etc.). C4: Systems other than diffusion and wave systems

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Metamaterials generally refer to those artificial structures with a structural unit size (much) smaller than the characteristic length (C in Fig. 2.1). In this way, an artificial structure has novel parameters that do not exist in nature or chemical compounds according to effective media, such as negative permittivities. Electromagnetic metamaterials (C2 in Fig. 2.1) originate from the research on negative refractive index [37,38,39]. Then, the metamaterial research was extended to other wave systems (C3 in Fig. 2.1), such as acoustics [40, 41] and elastodynamics [42, 43]. In 2008, transformation thermotics and thermal cloaking were proposed [1, 2], extending the metamaterial physics from wave to diffusion systems (right part of C1 in Fig. 2.1). Therefore, thermal metamaterials are an interdisciplinary product of metamaterials and theoretical thermotics [44, 45]. Certainly, characteristic lengths should be available to distinguish thermal metamaterials from other thermal materials. For heat conduction, the characteristic length is the thermal diffusion length \(L=\sqrt{\kappa t/\rho C}\), where \(\kappa \), t, \(\rho \), and C are thermal conductivity, time, density, and heat capacity, respectively. The characteristic length for thermal convection is the geometric length of fluid migration. For thermal radiation, the characteristic length is the wavelength of electromagnetic waves. Beyond heat transfer, metamaterials were also studied in other diffusive systems, such as mass diffusion [46] and light diffusion [47] (left part of C1 in Fig. 2.1). Besides, there are also metamaterials beyond wave and diffusion (C4 in Fig. 2.1), such as origami metamaterials [48] and robotic metamaterials [49].

2.3 Book Organization

We divide this book into two parts according to characteristic lengths, i.e., inside and outside metamaterials. Those theories with a characteristic length (much) larger than the structural unit size belong to Part I (inside metamaterials). The others belong to Part II (outside metamaterials). See Fig. 2.2.

Fig. 2.2
figure 2

Book framework

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In Part I (inside metamaterials), we introduce fourteen theories, classified into three levels for logical clarity. We start from the transformation theory, the foundation of theoretical thermotics (LV1), for dealing with complex thermal materials (Chap. 3) and thermoelectric materials (Chap. 4). Although the transformation theory is powerful, the required complicated parameters make practical fabrications challenging. Therefore, we introduce other theories beyond the transformation theory (i.e., effective medium theory) but still design functions predicted by the transformation theory (LV2), such as cloaks (Chap. 5–7), concentrators (Chap. 8), rotators (Chap. 9), sensors (Chaps. 10–12), and metasurfaces (Chap. 13). Based on LV1 and LV2, we develop the wavelike diffusion theory for designing functions not predicted by the transformation theory (LV3), such as advectionlike behavior (Chap. 14), diffusive Fizeau drag (Chap. 15), and thermal refraction effect (Chap. 16).

In Part II (outside metamaterials), we propose six theories, starting from the invisibility function. Thermal invisibility can be realized with metamaterials designed by the transformation theory or the effective medium theory, see Part I. A natural question is whether it can achieve thermal invisibility without metamaterials. The first theory (active dipole theory, Chap. 17) in Part II answers this question by demonstrating that thermal invisibility can be realized by an active dipole (without metamaterials, LV2). Based on nonlinear thermal conductivity (Chap. 18) and complex thermal conductivity (Chap. 19), we then develop theories for achieving functions not predicted by the transformation theory (LV3). With these foundations, we explore the topology-related approach for uncovering three-port thermal nonreciprocity (Chap. 20), thermal geometric phases (Chap. 21), and thermal edge states (Chap. 22).

Finally, we summarize this book and make an outlook in Chap. 23.

The research paradigms of theoretical thermotics can be extended to other diffusive systems. To provide more insights, we add an appendix to introduce other diffusion systems, including particle diffusion (Appendix A) and plasma diffusion (Appendix B).